LMFDB - Embedded newform 177.5.c.a.58.33

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,5,Mod(58,177)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("177.58");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	177.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.2964834658$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			58.33
Character	$\chi$	$=$	177.58
Dual form			177.5.c.a.58.8

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.96663i q^{2} +5.19615 q^{3} -8.66741 q^{4} +41.3974 q^{5} +25.8074i q^{6} +1.08721 q^{7} +36.4182i q^{8} +27.0000 q^{9} +O(q^{10})$	\(q+4.96663i q^{2} +5.19615 q^{3} -8.66741 q^{4} +41.3974 q^{5} +25.8074i q^{6} +1.08721 q^{7} +36.4182i q^{8} +27.0000 q^{9} +205.606i q^{10} -142.708i q^{11} -45.0372 q^{12} +35.2627i q^{13} +5.39978i q^{14} +215.107 q^{15} -319.555 q^{16} +527.661 q^{17} +134.099i q^{18} +76.0375 q^{19} -358.808 q^{20} +5.64932 q^{21} +708.780 q^{22} +138.991i q^{23} +189.235i q^{24} +1088.75 q^{25} -175.137 q^{26} +140.296 q^{27} -9.42332 q^{28} -1044.28 q^{29} +1068.36i q^{30} +1364.14i q^{31} -1004.42i q^{32} -741.534i q^{33} +2620.69i q^{34} +45.0078 q^{35} -234.020 q^{36} -2076.33i q^{37} +377.650i q^{38} +183.230i q^{39} +1507.62i q^{40} -1604.49 q^{41} +28.0581i q^{42} +746.077i q^{43} +1236.91i q^{44} +1117.73 q^{45} -690.318 q^{46} -613.862i q^{47} -1660.45 q^{48} -2399.82 q^{49} +5407.39i q^{50} +2741.80 q^{51} -305.636i q^{52} -3306.58 q^{53} +696.799i q^{54} -5907.75i q^{55} +39.5944i q^{56} +395.102 q^{57} -5186.56i q^{58} +(330.884 + 3465.24i) q^{59} -1864.42 q^{60} -3021.56i q^{61} -6775.17 q^{62} +29.3547 q^{63} -124.304 q^{64} +1459.78i q^{65} +3682.93 q^{66} +3679.74i q^{67} -4573.45 q^{68} +722.220i q^{69} +223.537i q^{70} +1827.67 q^{71} +983.293i q^{72} +2692.51i q^{73} +10312.4 q^{74} +5657.29 q^{75} -659.048 q^{76} -155.154i q^{77} -910.037 q^{78} +3087.31 q^{79} -13228.7 q^{80} +729.000 q^{81} -7968.90i q^{82} +7955.09i q^{83} -48.9650 q^{84} +21843.8 q^{85} -3705.49 q^{86} -5426.25 q^{87} +5197.19 q^{88} -5141.10i q^{89} +5551.35i q^{90} +38.3380i q^{91} -1204.70i q^{92} +7088.27i q^{93} +3048.82 q^{94} +3147.75 q^{95} -5219.11i q^{96} -13063.6i q^{97} -11919.0i q^{98} -3853.13i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9}+O(q^{10}) $ 40 * q - 320 * q^4 + 80 * q^7 + 1080 * q^9	$ 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9} + 360 q^{12} + 144 q^{15} + 3944 q^{16} - 528 q^{17} + 444 q^{19} + 444 q^{20} + 1304 q^{22} + 4880 q^{25} - 1452 q^{26} - 1160 q^{28} - 996 q^{29} + 10320 q^{35} - 8640 q^{36} - 5196 q^{41} - 10476 q^{46} + 576 q^{48} + 5104 q^{49} + 936 q^{51} - 2184 q^{53} - 2520 q^{57} - 11736 q^{59} - 11448 q^{60} + 15240 q^{62} + 2160 q^{63} - 81012 q^{64} + 17352 q^{66} + 29568 q^{68} - 5964 q^{71} + 14376 q^{74} - 2736 q^{75} + 3480 q^{76} + 37692 q^{78} + 19020 q^{79} + 33096 q^{80} + 29160 q^{81} + 25128 q^{84} + 20220 q^{85} - 65880 q^{86} + 1512 q^{87} - 14932 q^{88} - 17864 q^{94} + 11004 q^{95}+O(q^{100}) $ 40 * q - 320 * q^4 + 80 * q^7 + 1080 * q^9 + 360 * q^12 + 144 * q^15 + 3944 * q^16 - 528 * q^17 + 444 * q^19 + 444 * q^20 + 1304 * q^22 + 4880 * q^25 - 1452 * q^26 - 1160 * q^28 - 996 * q^29 + 10320 * q^35 - 8640 * q^36 - 5196 * q^41 - 10476 * q^46 + 576 * q^48 + 5104 * q^49 + 936 * q^51 - 2184 * q^53 - 2520 * q^57 - 11736 * q^59 - 11448 * q^60 + 15240 * q^62 + 2160 * q^63 - 81012 * q^64 + 17352 * q^66 + 29568 * q^68 - 5964 * q^71 + 14376 * q^74 - 2736 * q^75 + 3480 * q^76 + 37692 * q^78 + 19020 * q^79 + 33096 * q^80 + 29160 * q^81 + 25128 * q^84 + 20220 * q^85 - 65880 * q^86 + 1512 * q^87 - 14932 * q^88 - 17864 * q^94 + 11004 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			4.96663i			1.24166i	0.783946	+	0.620829i	$0.213204\pi$
$2$			4.96663i			1.24166i	−0.783946	+	0.620829i	$0.786796\pi$
$3$	5.19615			0.577350
$3$	5.19615			0.577350
$4$	−8.66741			−0.541713
$5$	41.3974			1.65590			0.827948	−	0.560805i	$-0.189508\pi$
$5$	41.3974			1.65590			0.827948	+	0.560805i	$0.189508\pi$
$6$			25.8074i			0.716871i
$7$	1.08721			0.0221880			0.0110940	−	0.999938i	$-0.496469\pi$
$7$	1.08721			0.0221880			0.0110940	+	0.999938i	$0.496469\pi$
$8$			36.4182i			0.569035i
$9$	27.0000			0.333333
$10$			205.606i			2.05606i
$11$		−	142.708i		−	1.17941i	−0.807620	−	0.589704i	$-0.799244\pi$
$11$		−	142.708i		−	1.17941i	0.807620	−	0.589704i	$-0.200756\pi$
$12$	−45.0372			−0.312758
$13$			35.2627i			0.208655i	0.994543	+	0.104327i	$0.0332690\pi$
$13$			35.2627i			0.208655i	−0.994543	+	0.104327i	$0.966731\pi$
$14$			5.39978i			0.0275499i
$15$	215.107			0.956032
$16$	−319.555			−1.24826
$17$	527.661			1.82581			0.912907	−	0.408167i	$-0.133832\pi$
$17$	527.661			1.82581			0.912907	+	0.408167i	$0.133832\pi$
$18$			134.099i			0.413886i
$19$	76.0375			0.210630			0.105315	−	0.994439i	$-0.466415\pi$
$19$	76.0375			0.210630			0.105315	+	0.994439i	$0.466415\pi$
$20$	−358.808			−0.897021
$21$	5.64932			0.0128103
$22$	708.780			1.46442
$23$			138.991i			0.262744i	0.991333	+	0.131372i	$0.0419382\pi$
$23$			138.991i			0.262744i	−0.991333	+	0.131372i	$0.958062\pi$
$24$			189.235i			0.328533i
$25$	1088.75			1.74199
$26$	−175.137			−0.259078
$27$	140.296			0.192450
$28$	−9.42332			−0.0120195
$29$	−1044.28			−1.24171			−0.620857	−	0.783924i	$-0.713215\pi$
$29$	−1044.28			−1.24171			−0.620857	+	0.783924i	$0.713215\pi$
$30$			1068.36i			1.18706i
$31$			1364.14i			1.41950i	0.704454	+	0.709749i	$0.251192\pi$
$31$			1364.14i			1.41950i	−0.704454	+	0.709749i	$0.748808\pi$
$32$		−	1004.42i		−	0.980876i
$33$		−	741.534i		−	0.680931i
$34$			2620.69i			2.26704i
$35$	45.0078			0.0367410
$36$	−234.020			−0.180571
$37$		−	2076.33i		−	1.51668i	−0.651860	−	0.758339i	$-0.726011\pi$
$37$		−	2076.33i		−	1.51668i	0.651860	−	0.758339i	$-0.273989\pi$
$38$			377.650i			0.261530i
$39$			183.230i			0.120467i
$40$			1507.62i			0.942263i
$41$	−1604.49			−0.954485			−0.477242	−	0.878772i	$-0.658364\pi$
$41$	−1604.49			−0.954485			−0.477242	+	0.878772i	$0.658364\pi$
$42$			28.0581i			0.0159059i
$43$			746.077i			0.403503i	0.979437	+	0.201751i	$0.0646633\pi$
$43$			746.077i			0.403503i	−0.979437	+	0.201751i	$0.935337\pi$
$44$			1236.91i			0.638901i
$45$	1117.73			0.551965
$46$	−690.318			−0.326237
$47$		−	613.862i		−	0.277891i	−0.990300	−	0.138946i	$-0.955629\pi$
$47$		−	613.862i		−	0.277891i	0.990300	−	0.138946i	$-0.0443713\pi$
$48$	−1660.45			−0.720683
$49$	−2399.82			−0.999508
$50$			5407.39i			2.16296i
$51$	2741.80			1.05413
$52$		−	305.636i		−	0.113031i
$53$	−3306.58			−1.17714			−0.588569	−	0.808447i	$-0.700308\pi$
$53$	−3306.58			−1.17714			−0.588569	+	0.808447i	$0.700308\pi$
$54$			696.799i			0.238957i
$55$		−	5907.75i		−	1.95298i
$56$			39.5944i			0.0126258i
$57$	395.102			0.121607
$58$		−	5186.56i		−	1.54178i
$59$	330.884	+	3465.24i	0.0950543	+	0.995472i
$59$	330.884	+	3465.24i	0.0950543	+	0.995472i
$60$	−1864.42			−0.517895
$61$		−	3021.56i		−	0.812028i	−0.913867	−	0.406014i	$-0.866918\pi$
$61$		−	3021.56i		−	0.812028i	0.913867	−	0.406014i	$-0.133082\pi$
$62$	−6775.17			−1.76253
$63$	29.3547			0.00739600
$64$	−124.304			−0.0303476
$65$			1459.78i			0.345511i
$66$	3682.93			0.845484
$67$			3679.74i			0.819724i	0.912148	+	0.409862i	$0.134423\pi$
$67$			3679.74i			0.819724i	−0.912148	+	0.409862i	$0.865577\pi$
$68$	−4573.45			−0.989068
$69$			722.220i			0.151695i
$70$			223.537i			0.0456198i
$71$	1827.67			0.362560			0.181280	−	0.983431i	$-0.441976\pi$
$71$	1827.67			0.362560			0.181280	+	0.983431i	$0.441976\pi$
$72$			983.293i			0.189678i
$73$			2692.51i			0.505256i	0.967563	+	0.252628i	$0.0812949\pi$
$73$			2692.51i			0.505256i	−0.967563	+	0.252628i	$0.918705\pi$
$74$	10312.4			1.88320
$75$	5657.29			1.00574
$76$	−659.048			−0.114101
$77$		−	155.154i		−	0.0261687i
$78$	−910.037			−0.149579
$79$	3087.31			0.494682			0.247341	−	0.968928i	$-0.420443\pi$
$79$	3087.31			0.494682			0.247341	+	0.968928i	$0.420443\pi$
$80$	−13228.7			−2.06699
$81$	729.000			0.111111
$82$		−	7968.90i		−	1.18514i
$83$			7955.09i			1.15475i	0.816478	+	0.577377i	$0.195924\pi$
$83$			7955.09i			1.15475i	−0.816478	+	0.577377i	$0.804076\pi$
$84$	−48.9650			−0.00693949
$85$	21843.8			3.02336
$86$	−3705.49			−0.501012
$87$	−5426.25			−0.716904
$88$	5197.19			0.671124
$89$		−	5141.10i		−	0.649047i	−0.945878	−	0.324524i	$-0.894796\pi$
$89$		−	5141.10i		−	0.649047i	0.945878	−	0.324524i	$-0.105204\pi$
$90$			5551.35i			0.685352i
$91$			38.3380i			0.00462964i
$92$		−	1204.70i		−	0.142332i
$93$			7088.27i			0.819548i
$94$	3048.82			0.345046
$95$	3147.75			0.348782
$96$		−	5219.11i		−	0.566309i
$97$		−	13063.6i		−	1.38842i	−0.719772	−	0.694210i	$-0.755754\pi$
$97$		−	13063.6i		−	1.38842i	0.719772	−	0.694210i	$-0.244246\pi$
$98$		−	11919.0i		−	1.24105i
$99$		−	3853.13i		−	0.393136i
$100$	−9436.60			−0.943660
$101$			6212.97i			0.609055i	0.952504	+	0.304527i	$0.0984985\pi$
$101$			6212.97i			0.609055i	−0.952504	+	0.304527i	$0.901502\pi$
$102$			13617.5i			1.30887i
$103$		−	12027.0i		−	1.13366i	−0.823834	−	0.566831i	$-0.808169\pi$
$103$		−	12027.0i		−	1.13366i	0.823834	−	0.566831i	$-0.191831\pi$
$104$	−1284.21			−0.118732
$105$	233.867			0.0212125
$106$		−	16422.6i		−	1.46160i
$107$	−5915.89			−0.516717			−0.258359	−	0.966049i	$-0.583182\pi$
$107$	−5915.89			−0.516717			−0.258359	+	0.966049i	$0.583182\pi$
$108$	−1216.00			−0.104253
$109$		−	17887.0i		−	1.50552i	−0.658298	−	0.752758i	$-0.728723\pi$
$109$		−	17887.0i		−	1.50552i	0.658298	−	0.752758i	$-0.271277\pi$
$110$	29341.6			2.42493
$111$		−	10788.9i		−	0.875655i
$112$	−347.424			−0.0276964
$113$		−	24933.1i		−	1.95262i	−0.216368	−	0.976312i	$-0.569421\pi$
$113$		−	24933.1i		−	1.95262i	0.216368	−	0.976312i	$-0.430579\pi$
$114$			1962.33i			0.150995i
$115$			5753.88i			0.435076i
$116$	9051.23			0.672653
$117$			952.093i			0.0695517i
$118$	−17210.6	+	1643.38i	−1.23604	+	0.118025i
$119$	573.679			0.0405112
$120$			7833.83i			0.544016i
$121$	−5724.67			−0.391003
$122$	15007.0			1.00826
$123$	−8337.17			−0.551072
$124$		−	11823.6i		−	0.768962i
$125$	19197.8			1.22866
$126$			145.794i			0.00918330i
$127$	22413.1			1.38961			0.694806	−	0.719197i	$-0.255490\pi$
$127$	22413.1			1.38961			0.694806	+	0.719197i	$0.255490\pi$
$128$		−	16688.0i		−	1.01856i
$129$			3876.73i			0.232963i
$130$	−7250.21			−0.429006
$131$			17802.3i			1.03737i	0.854965	+	0.518685i	$0.173578\pi$
$131$			17802.3i			1.03737i	−0.854965	+	0.518685i	$0.826422\pi$
$132$			6427.18i			0.368870i
$133$	82.6689			0.00467346
$134$	−18275.9			−1.01782
$135$	5807.89			0.318677
$136$			19216.5i			1.03895i
$137$	−11891.9			−0.633595			−0.316797	−	0.948493i	$-0.602607\pi$
$137$	−11891.9			−0.633595			−0.316797	+	0.948493i	$0.602607\pi$
$138$	−3587.00			−0.188353
$139$	−24841.5			−1.28572			−0.642862	−	0.765982i	$-0.722253\pi$
$139$	−24841.5			−1.28572			−0.642862	+	0.765982i	$0.722253\pi$
$140$	−390.101			−0.0199031
$141$		−	3189.72i		−	0.160441i
$142$			9077.34i			0.450176i
$143$	5032.28			0.246089
$144$	−8627.97			−0.416087
$145$	−43230.6			−2.05615
$146$	−13372.7			−0.627355
$147$	−12469.8			−0.577066
$148$			17996.4i			0.821605i
$149$		−	26032.6i		−	1.17259i	−0.810099	−	0.586293i	$-0.800587\pi$
$149$		−	26032.6i		−	1.17259i	0.810099	−	0.586293i	$-0.199413\pi$
$150$			28097.6i			1.24878i
$151$		−	12278.3i		−	0.538497i	−0.963071	−	0.269249i	$-0.913225\pi$
$151$		−	12278.3i		−	0.538497i	0.963071	−	0.269249i	$-0.0867753\pi$
$152$			2769.15i			0.119856i
$153$	14246.8			0.608605
$154$	770.594			0.0324926
$155$			56471.8i			2.35054i
$156$		−	1588.13i		−	0.0652586i
$157$			21189.6i			0.859655i	0.902911	+	0.429827i	$0.141426\pi$
$157$			21189.6i			0.859655i	−0.902911	+	0.429827i	$0.858574\pi$
$158$			15333.5i			0.614226i
$159$	−17181.5			−0.679621
$160$		−	41580.3i		−	1.62423i
$161$			151.113i			0.00582976i
$162$			3620.67i			0.137962i
$163$	22822.0			0.858970			0.429485	−	0.903074i	$-0.358695\pi$
$163$	22822.0			0.858970			0.429485	+	0.903074i	$0.358695\pi$
$164$	13906.8			0.517057
$165$		−	30697.6i		−	1.12755i
$166$	−39510.0			−1.43381
$167$	−28368.2			−1.01718			−0.508592	−	0.861008i	$-0.669834\pi$
$167$	−28368.2			−1.01718			−0.508592	+	0.861008i	$0.669834\pi$
$168$			205.738i			0.00728948i
$169$	27317.5			0.956463
$170$			108490.i			3.75398i
$171$	2053.01			0.0702100
$172$		−	6466.56i		−	0.218583i
$173$			7580.94i			0.253297i	0.991948	+	0.126649i	$0.0404221\pi$
$173$			7580.94i			0.253297i	−0.991948	+	0.126649i	$0.959578\pi$
$174$		−	26950.2i		−	0.890150i
$175$	1183.70			0.0386513
$176$			45603.1i			1.47221i
$177$	1719.32	+	18005.9i	0.0548796	+	0.574736i
$178$	25533.9			0.805894
$179$		−	34939.2i		−	1.09045i	−0.838289	−	0.545226i	$-0.816444\pi$
$179$		−	34939.2i		−	1.09045i	0.838289	−	0.545226i	$-0.183556\pi$
$180$	−9687.83			−0.299007
$181$	37840.7			1.15505			0.577527	−	0.816372i	$-0.304018\pi$
$181$	37840.7			1.15505			0.577527	+	0.816372i	$0.304018\pi$
$182$	−190.411			−0.00574843
$183$		−	15700.5i		−	0.468825i
$184$	−5061.82			−0.149510
$185$		−	85954.8i		−	2.51146i
$186$	−35204.8			−1.01760
$187$		−	75301.6i		−	2.15338i
$188$			5320.59i			0.150537i
$189$	152.532			0.00427008
$190$			15633.7i			0.433067i
$191$		−	3569.74i		−	0.0978521i	−0.998802	−	0.0489261i	$-0.984420\pi$
$191$		−	3569.74i		−	0.0978521i	0.998802	−	0.0489261i	$-0.0155799\pi$
$192$	−645.901			−0.0175212
$193$	24451.4			0.656430			0.328215	−	0.944603i	$-0.393553\pi$
$193$	24451.4			0.656430			0.328215	+	0.944603i	$0.393553\pi$
$194$	64882.3			1.72394
$195$			7585.26i			0.199481i
$196$	20800.2			0.541447
$197$	−29706.3			−0.765449			−0.382724	−	0.923863i	$-0.625014\pi$
$197$	−29706.3			−0.765449			−0.382724	+	0.923863i	$0.625014\pi$
$198$	19137.0			0.488140
$199$	−22181.5			−0.560125			−0.280062	−	0.959982i	$-0.590355\pi$
$199$	−22181.5			−0.560125			−0.280062	+	0.959982i	$0.590355\pi$
$200$			39650.2i			0.991255i
$201$			19120.5i			0.473268i
$202$	−30857.5			−0.756237
$203$	−1135.36			−0.0275512
$204$	−23764.4			−0.571039
$205$	−66421.7			−1.58053
$206$	59733.8			1.40762
$207$			3752.77i			0.0875812i
$208$		−	11268.4i		−	0.260456i
$209$		−	10851.2i		−	0.248419i
$210$			1161.53i			0.0263386i
$211$			81724.8i			1.83565i	0.396989	+	0.917824i	$0.370055\pi$
$211$			81724.8i			1.83565i	−0.396989	+	0.917824i	$0.629945\pi$
$212$	28659.5			0.637671
$213$	9496.83			0.209324
$214$		−	29382.1i		−	0.641586i
$215$			30885.6i			0.668159i
$216$			5109.34i			0.109511i
$217$			1483.11i			0.0314959i
$218$	88838.3			1.86933
$219$			13990.7i			0.291710i
$220$			51205.0i			1.05795i
$221$			18606.7i			0.380965i
$222$	53584.7			1.08726
$223$	−54373.6			−1.09340			−0.546699	−	0.837329i	$-0.684116\pi$
$223$	−54373.6			−1.09340			−0.546699	+	0.837329i	$0.684116\pi$
$224$		−	1092.02i		−	0.0217637i
$225$	29396.1			0.580664
$226$	123833.			2.42449
$227$		−	55801.8i		−	1.08292i	−0.840726	−	0.541460i	$-0.817872\pi$
$227$		−	55801.8i		−	1.08292i	0.840726	−	0.541460i	$-0.182128\pi$
$228$	−3424.51			−0.0658763
$229$			102771.i			1.95975i	0.199624	+	0.979873i	$0.436028\pi$
$229$			102771.i			1.95975i	−0.199624	+	0.979873i	$0.563972\pi$
$230$	−28577.4			−0.540215
$231$		−	806.205i		−	0.0151085i
$232$		−	38030.9i		−	0.706579i
$233$			60636.1i			1.11691i	0.829534	+	0.558457i	$0.188606\pi$
$233$			60636.1i			1.11691i	−0.829534	+	0.558457i	$0.811394\pi$
$234$	−4728.69			−0.0863593
$235$		−	25412.3i		−	0.460159i
$236$	−2867.91	−	30034.7i	−0.0514922	−	0.539261i
$237$	16042.1			0.285605
$238$			2849.25i			0.0503010i
$239$	12138.8			0.212510			0.106255	−	0.994339i	$-0.466114\pi$
$239$	12138.8			0.212510			0.106255	+	0.994339i	$0.466114\pi$
$240$	−68738.5			−1.19338
$241$	−13471.2			−0.231938			−0.115969	−	0.993253i	$-0.536997\pi$
$241$	−13471.2			−0.231938			−0.115969	+	0.993253i	$0.536997\pi$
$242$		−	28432.3i		−	0.485491i
$243$	3788.00			0.0641500
$244$			26189.1i			0.439886i
$245$	−99346.2			−1.65508
$246$		−	41407.6i		−	0.684243i
$247$			2681.29i			0.0439490i
$248$	−49679.5			−0.807745
$249$			41335.9i			0.666697i
$250$			95348.6i			1.52558i
$251$	−20999.2			−0.333315			−0.166657	−	0.986015i	$-0.553297\pi$
$251$	−20999.2			−0.333315			−0.166657	+	0.986015i	$0.553297\pi$
$252$	−254.430			−0.00400651
$253$	19835.2			0.309882
$254$			111317.i			1.72542i
$255$	113504.			1.74554
$256$	80894.5			1.23435
$257$	−39315.0			−0.595240			−0.297620	−	0.954684i	$-0.596193\pi$
$257$	−39315.0			−0.595240			−0.297620	+	0.954684i	$0.596193\pi$
$258$	−19254.3			−0.289260
$259$		−	2257.42i		−	0.0336521i
$260$		−	12652.5i		−	0.187168i
$261$	−28195.6			−0.413905
$262$	−88417.5			−1.28806
$263$	37861.6			0.547378			0.273689	−	0.961818i	$-0.411756\pi$
$263$	37861.6			0.547378			0.273689	+	0.961818i	$0.411756\pi$
$264$	27005.4			0.387474
$265$	−136884.			−1.94922
$266$			410.586i			0.00580284i
$267$		−	26713.9i		−	0.374727i
$268$		−	31893.8i		−	0.444055i
$269$		−	123756.i		−	1.71026i	−0.518413	−	0.855130i	$-0.673477\pi$
$269$		−	123756.i		−	1.71026i	0.518413	−	0.855130i	$-0.326523\pi$
$270$			28845.7i			0.395688i
$271$	4855.26			0.0661110			0.0330555	−	0.999454i	$-0.489476\pi$
$271$	4855.26			0.0661110			0.0330555	+	0.999454i	$0.489476\pi$
$272$	−168616.			−2.27909
$273$			199.210i			0.00267292i
$274$		−	59062.9i		−	0.786707i
$275$		−	155373.i		−	2.05452i
$276$		−	6259.78i		−	0.0821752i
$277$	97089.7			1.26536			0.632680	−	0.774414i	$-0.281955\pi$
$277$	97089.7			1.26536			0.632680	+	0.774414i	$0.281955\pi$
$278$		−	123378.i		−	1.59643i
$279$			36831.7i			0.473166i
$280$			1639.10i			0.0209069i
$281$	−108342.			−1.37209			−0.686045	−	0.727559i	$-0.740655\pi$
$281$	−108342.			−1.37209			−0.686045	+	0.727559i	$0.740655\pi$
$282$	15842.2			0.199212
$283$		−	67893.8i		−	0.847729i	−0.905726	−	0.423864i	$-0.860673\pi$
$283$		−	67893.8i		−	0.847729i	0.905726	−	0.423864i	$-0.139327\pi$
$284$	−15841.1			−0.196404
$285$	16356.2			0.201369
$286$			24993.5i			0.305559i
$287$	−1744.42			−0.0211781
$288$		−	27119.3i		−	0.326959i
$289$	194905.			2.33360
$290$		−	214710.i		−	2.55304i
$291$		−	67880.7i		−	0.801605i
$292$		−	23337.1i		−	0.273704i
$293$	51965.3			0.605311			0.302655	−	0.953100i	$-0.402127\pi$
$293$	51965.3			0.605311			0.302655	+	0.953100i	$0.402127\pi$
$294$		−	61933.0i		−	0.716518i
$295$	13697.7	+	143452.i	0.157400	+	1.64840i
$296$	75616.4			0.863043
$297$		−	20021.4i		−	0.226977i
$298$	129294.			1.45595
$299$	−4901.21			−0.0548227
$300$	−49034.0			−0.544823
$301$			811.144i			0.00895293i
$302$	60981.7			0.668629
$303$			32283.5i			0.351638i
$304$	−24298.1			−0.262921
$305$		−	125085.i		−	1.34463i
$306$			70758.8i			0.755679i
$307$	122608.			1.30089			0.650445	−	0.759553i	$-0.274582\pi$
$307$	122608.			1.30089			0.650445	+	0.759553i	$0.274582\pi$
$308$			1344.79i			0.0141759i
$309$		−	62494.2i		−	0.654520i
$310$	−280475.			−2.91857
$311$	−159419.			−1.64823			−0.824117	−	0.566420i	$-0.808328\pi$
$311$	−159419.			−1.64823			−0.824117	+	0.566420i	$0.808328\pi$
$312$	−6672.93			−0.0685499
$313$		−	17723.3i		−	0.180907i	−0.995901	−	0.0904534i	$-0.971168\pi$
$313$		−	17723.3i		−	0.180907i	0.995901	−	0.0904534i	$-0.0288316\pi$
$314$	−105241.			−1.06740
$315$	1215.21			0.0122470
$316$	−26759.0			−0.267976
$317$	−112577.			−1.12029			−0.560145	−	0.828394i	$-0.689255\pi$
$317$	−112577.			−1.12029			−0.560145	+	0.828394i	$0.689255\pi$
$318$		−	85334.1i		−	0.843856i
$319$			149028.i			1.46449i
$320$	−5145.85			−0.0502524
$321$	−30739.9			−0.298327
$322$	−750.523			−0.00723856
$323$	40122.0			0.384572
$324$	−6318.54			−0.0601904
$325$			38392.1i			0.363475i
$326$			113348.i			1.06655i
$327$		−	92943.7i		−	0.869210i
$328$		−	58432.7i		−	0.543135i
$329$		−	667.398i		−	0.00616585i
$330$	152464.			1.40003
$331$	−126890.			−1.15817			−0.579083	−	0.815268i	$-0.696589\pi$
$331$	−126890.			−1.15817			−0.579083	+	0.815268i	$0.696589\pi$
$332$		−	68950.1i		−	0.625545i
$333$		−	56061.0i		−	0.505560i
$334$		−	140895.i		−	1.26299i
$335$			152332.i			1.35738i
$336$	−1805.27			−0.0159905
$337$			145883.i			1.28454i	0.766480	+	0.642268i	$0.222006\pi$
$337$			145883.i			1.28454i	−0.766480	+	0.642268i	$0.777994\pi$
$338$			135676.i			1.18760i
$339$		−	129556.i		−	1.12735i
$340$	−189329.			−1.63779
$341$	194674.			1.67417
$342$			10196.5i			0.0871768i
$343$	−5219.51			−0.0443651
$344$	−27170.8			−0.229607
$345$			29898.0i			0.251191i
$346$	−37651.7			−0.314509
$347$		−	118230.i		−	0.981905i	−0.871186	−	0.490953i	$-0.836649\pi$
$347$		−	118230.i		−	0.981905i	0.871186	−	0.490953i	$-0.163351\pi$
$348$	47031.5			0.388357
$349$			24509.0i			0.201222i	0.994926	+	0.100611i	$0.0320797\pi$
$349$			24509.0i			0.201222i	−0.994926	+	0.100611i	$0.967920\pi$
$350$			5878.99i			0.0479917i
$351$			4947.22i			0.0401557i
$352$	−143339.			−1.15685
$353$			38007.7i			0.305015i	0.988302	+	0.152508i	$0.0487349\pi$
$353$			38007.7i			0.305015i	−0.988302	+	0.152508i	$0.951265\pi$
$354$	−89428.7	+	8539.24i	−0.713625	+	0.0681417i
$355$	75660.7			0.600362
$356$			44560.1i			0.351597i
$357$	2980.92			0.0233892
$358$	173530.			1.35397
$359$	192503.			1.49365			0.746826	−	0.665020i	$-0.231577\pi$
$359$	192503.			1.49365			0.746826	+	0.665020i	$0.231577\pi$
$360$			40705.8i			0.314088i
$361$	−124539.			−0.955635
$362$			187941.i			1.43418i
$363$	−29746.3			−0.225745
$364$		−	332.292i		−	0.00250794i
$365$			111463.i			0.836652i
$366$	77978.4			0.582120
$367$			18295.0i			0.135831i	0.997691	+	0.0679157i	$0.0216349\pi$
$367$			18295.0i			0.135831i	−0.997691	+	0.0679157i	$0.978365\pi$
$368$		−	44415.3i		−	0.327972i
$369$	−43321.2			−0.318162
$370$	426906.			3.11838
$371$	−3594.96			−0.0261183
$372$		−	61437.0i		−	0.443960i
$373$	57630.7			0.414225			0.207112	−	0.978317i	$-0.433593\pi$
$373$	57630.7			0.414225			0.207112	+	0.978317i	$0.433593\pi$
$374$	373995.			2.67376
$375$	99754.9			0.709368
$376$	22355.8			0.158130
$377$		−	36824.2i		−	0.259090i
$378$			757.569i			0.00530198i
$379$	170818.			1.18920			0.594599	−	0.804023i	$-0.297311\pi$
$379$	170818.			1.18920			0.594599	+	0.804023i	$0.297311\pi$
$380$	−27282.9			−0.188940
$381$	116462.			0.802293
$382$	17729.6			0.121499
$383$	−221122.			−1.50742			−0.753709	−	0.657208i	$-0.771737\pi$
$383$	−221122.			−1.50742			−0.753709	+	0.657208i	$0.771737\pi$
$384$		−	86713.6i		−	0.588065i
$385$		−	6422.99i		−	0.0433327i
$386$			121441.i			0.815061i
$387$			20144.1i			0.134501i
$388$			113228.i			0.752126i
$389$	87151.4			0.575937			0.287969	−	0.957640i	$-0.407020\pi$
$389$	87151.4			0.575937			0.287969	+	0.957640i	$0.407020\pi$
$390$	−37673.2			−0.247687
$391$			73340.2i			0.479721i
$392$		−	87397.2i		−	0.568755i
$393$			92503.6i			0.598926i
$394$		−	147540.i		−	0.950425i
$395$	127807.			0.819143
$396$			33396.6i			0.212967i
$397$			126155.i			0.800431i	0.916421	+	0.400216i	$0.131065\pi$
$397$			126155.i			0.800431i	−0.916421	+	0.400216i	$0.868935\pi$
$398$		−	110167.i		−	0.695483i
$399$	429.560			0.00269823
$400$	−347913.			−2.17446
$401$		−	27343.9i		−	0.170048i	−0.996379	−	0.0850242i	$-0.972903\pi$
$401$		−	27343.9i		−	0.170048i	0.996379	−	0.0850242i	$-0.0270967\pi$
$402$	−94964.4			−0.587637
$403$	−48103.2			−0.296185
$404$		−	53850.4i		−	0.329933i
$405$	30178.7			0.183988
$406$		−	5638.90i		−	0.0342091i
$407$	−296310.			−1.78878
$408$			99851.7i			0.599840i
$409$		−	104932.i		−	0.627282i	−0.949542	−	0.313641i	$-0.898451\pi$
$409$		−	104932.i		−	0.627282i	0.949542	−	0.313641i	$-0.101549\pi$
$410$		−	329892.i		−	1.96247i
$411$	−61792.3			−0.365806
$412$			104243.i			0.614120i
$413$	359.741	+	3767.45i	0.00210907	+	0.0220875i
$414$	−18638.6			−0.108746
$415$			329320.i			1.91215i
$416$	35418.5			0.204665
$417$	−129080.			−0.742313
$418$	53893.8			0.308451
$419$			203320.i			1.15812i	0.815286	+	0.579058i	$0.196580\pi$
$419$			203320.i			1.15812i	−0.815286	+	0.579058i	$0.803420\pi$
$420$	−2027.02			−0.0114911
$421$		−	7915.89i		−	0.0446617i	−0.999751	−	0.0223309i	$-0.992891\pi$
$421$		−	7915.89i		−	0.0446617i	0.999751	−	0.0223309i	$-0.00710872\pi$
$422$	−405897.			−2.27924
$423$		−	16574.3i		−	0.0926304i
$424$		−	120420.i		−	0.669833i
$425$	574488.			3.18056
$426$			47167.3i			0.259909i
$427$		−	3285.07i		−	0.0180173i
$428$	51275.5			0.279913
$429$	26148.5			0.142080
$430$	−153398.			−0.829625
$431$			166950.i			0.898734i	0.893347	+	0.449367i	$0.148350\pi$
$431$			166950.i			0.898734i	−0.893347	+	0.449367i	$0.851650\pi$
$432$	−44832.3			−0.240228
$433$	−47184.7			−0.251666			−0.125833	−	0.992051i	$-0.540160\pi$
$433$	−47184.7			−0.251666			−0.125833	+	0.992051i	$0.540160\pi$
$434$	−7366.05			−0.0391071
$435$	−224633.			−1.18712
$436$			155034.i			0.815558i
$437$			10568.5i			0.0553417i
$438$	−69486.6			−0.362204
$439$	168825.			0.876010			0.438005	−	0.898973i	$-0.355685\pi$
$439$	168825.			0.876010			0.438005	+	0.898973i	$0.355685\pi$
$440$	215150.			1.11131
$441$	−64795.1			−0.333169
$442$	−92412.7			−0.473028
$443$			307500.i			1.56689i	0.621462	+	0.783444i	$0.286539\pi$
$443$			307500.i			1.56689i	−0.621462	+	0.783444i	$0.713461\pi$
$444$			93512.2i			0.474354i
$445$		−	212828.i		−	1.07475i
$446$		−	270053.i		−	1.35763i
$447$		−	135269.i		−	0.676993i
$448$	−135.145			−0.000673352
$449$	−243616.			−1.20841			−0.604203	−	0.796830i	$-0.706509\pi$
$449$	−243616.			−1.20841			−0.604203	+	0.796830i	$0.706509\pi$
$450$			146000.i			0.720986i
$451$			228974.i			1.12573i
$452$			216105.i			1.05776i
$453$		−	63799.8i		−	0.310902i
$454$	277147.			1.34462
$455$			1587.10i			0.00766620i
$456$			14388.9i			0.0691988i
$457$			240817.i			1.15307i	0.817073	+	0.576535i	$0.195595\pi$
$457$			240817.i			1.15307i	−0.817073	+	0.576535i	$0.804405\pi$
$458$	−510426.			−2.43333
$459$	74028.7			0.351378
$460$		−	49871.3i		−	0.235686i
$461$	105419.			0.496043			0.248021	−	0.968755i	$-0.420220\pi$
$461$	105419.			0.496043			0.248021	+	0.968755i	$0.420220\pi$
$462$	4004.12			0.0187596
$463$		−	177261.i		−	0.826895i	−0.910528	−	0.413447i	$-0.864325\pi$
$463$		−	177261.i		−	0.826895i	0.910528	−	0.413447i	$-0.135675\pi$
$464$	333705.			1.54998
$465$			293436.i			1.35709i
$466$	−301157.			−1.38682
$467$			387563.i			1.77709i	0.458792	+	0.888544i	$0.348282\pi$
$467$			387563.i			1.77709i	−0.458792	+	0.888544i	$0.651718\pi$
$468$		−	8252.18i		−	0.0376771i
$469$			4000.66i			0.0181880i
$470$	126213.			0.571360
$471$			110105.i			0.496322i
$472$	−126198.	+	12050.2i	−0.566459	+	0.0540892i
$473$	106471.			0.475894
$474$			79675.4i			0.354624i
$475$	82785.4			0.366916
$476$	−4972.31			−0.0219455
$477$	−89277.7			−0.392379
$478$			60288.8i			0.263865i
$479$	346182.			1.50881			0.754404	−	0.656411i	$-0.227926\pi$
$479$	346182.			1.50881			0.754404	+	0.656411i	$0.227926\pi$
$480$		−	216057.i		−	0.937749i
$481$	73217.1			0.316463
$482$		−	66906.3i		−	0.287987i
$483$			785.207i			0.00336581i
$484$	49618.1			0.211811
$485$		−	540801.i		−	2.29908i
$486$			18813.6i			0.0796524i
$487$	−204643.			−0.862855			−0.431428	−	0.902148i	$-0.641990\pi$
$487$	−204643.			−0.862855			−0.431428	+	0.902148i	$0.641990\pi$
$488$	110040.			0.462072
$489$	118586.			0.495927
$490$		−	493416.i		−	2.05504i
$491$	−124429.			−0.516128			−0.258064	−	0.966128i	$-0.583085\pi$
$491$	−124429.			−0.516128			−0.258064	+	0.966128i	$0.583085\pi$
$492$	72261.7			0.298523
$493$	−551026.			−2.26714
$494$	−13317.0			−0.0545696
$495$		−	159509.i		−	0.650992i
$496$		−	435917.i		−	1.77190i
$497$	1987.06			0.00804449
$498$	−205300.			−0.827809
$499$	−410778.			−1.64970			−0.824851	−	0.565350i	$-0.808741\pi$
$499$	−410778.			−1.64970			−0.824851	+	0.565350i	$0.808741\pi$
$500$	−166396.			−0.665583
$501$	−147406.			−0.587271
$502$		−	104295.i		−	0.413863i
$503$			245979.i			0.972213i	0.873899	+	0.486107i	$0.161583\pi$
$503$			245979.i			0.972213i	−0.873899	+	0.486107i	$0.838417\pi$
$504$			1069.05i			0.00420859i
$505$			257201.i			1.00853i
$506$			98514.2i			0.384767i
$507$	141946.			0.552214
$508$	−194263.			−0.752771
$509$			32679.0i			0.126134i	0.998009	+	0.0630672i	$0.0200882\pi$
$509$			32679.0i			0.126134i	−0.998009	+	0.0630672i	$0.979912\pi$
$510$			563730.i			2.16736i
$511$			2927.33i			0.0112106i
$512$			134764.i			0.514085i
$513$	10667.8			0.0405358
$514$		−	195263.i		−	0.739084i
$515$		−	497887.i		−	1.87723i
$516$		−	33601.2i		−	0.126199i
$517$	−87603.2			−0.327747
$518$	11211.7			0.0417844
$519$			39391.7i			0.146241i
$520$	−53162.8			−0.196608
$521$	−113050.			−0.416479			−0.208240	−	0.978078i	$-0.566773\pi$
$521$	−113050.			−0.416479			−0.208240	+	0.978078i	$0.566773\pi$
$522$		−	140037.i		−	0.513928i
$523$	237542.			0.868435			0.434218	−	0.900808i	$-0.357025\pi$
$523$	237542.			0.868435			0.434218	+	0.900808i	$0.357025\pi$
$524$		−	154300.i		−	0.561957i
$525$	6150.67			0.0223154
$526$			188044.i			0.679656i
$527$			719802.i			2.59174i
$528$			236961.i			0.849979i
$529$	260522.			0.930966
$530$		−	679851.i		−	2.42026i
$531$	8933.87	+	93561.4i	0.0316848	+	0.331824i
$532$	−716.525			−0.00253168
$533$		−	56578.6i		−	0.199158i
$534$	132678.			0.465283
$535$	−244903.			−0.855630
$536$	−134010.			−0.466452
$537$		−	181549.i		−	0.629573i
$538$	614651.			2.12356
$539$			342474.i			1.17883i
$540$	−50339.4			−0.172632
$541$			203519.i			0.695360i	0.937613	+	0.347680i	$0.113030\pi$
$541$			203519.i			0.695360i	−0.937613	+	0.347680i	$0.886970\pi$
$542$			24114.3i			0.0820873i
$543$	196626.			0.666871
$544$		−	529991.i		−	1.79090i
$545$		−	740477.i		−	2.49298i
$546$	−989.404			−0.00331886
$547$	−388812.			−1.29947			−0.649733	−	0.760163i	$-0.725119\pi$
$547$	−388812.			−1.29947			−0.649733	+	0.760163i	$0.725119\pi$
$548$	103072.			0.343227
$549$		−	81582.0i		−	0.270676i
$550$	771680.			2.55101
$551$	−79404.6			−0.261543
$552$	−26302.0			−0.0863198
$553$	3356.57			0.0109760
$554$			482209.i			1.57114i
$555$		−	446634.i		−	1.44999i
$556$	215311.			0.696493
$557$	307150.			0.990011			0.495006	−	0.868890i	$-0.335166\pi$
$557$	307150.			0.990011			0.495006	+	0.868890i	$0.335166\pi$
$558$	−182930.			−0.587511
$559$	−26308.7			−0.0841929
$560$	−14382.4			−0.0458624
$561$		−	391278.i		−	1.24325i
$562$		−	538093.i		−	1.70367i
$563$			224936.i			0.709647i	0.934933	+	0.354824i	$0.115459\pi$
$563$			224936.i			0.709647i	−0.934933	+	0.354824i	$0.884541\pi$
$564$			27646.6i			0.0869128i
$565$		−	1.03216e6i		−	3.23334i
$566$	337203.			1.05259
$567$	792.578			0.00246533
$568$			66560.4i			0.206310i
$569$		−	48390.1i		−	0.149463i	−0.997204	−	0.0747313i	$-0.976190\pi$
$569$		−	48390.1i		−	0.149463i	0.997204	−	0.0747313i	$-0.0238099\pi$
$570$			81235.2i			0.250032i
$571$		−	344185.i		−	1.05565i	−0.849353	−	0.527825i	$-0.823008\pi$
$571$		−	344185.i		−	1.05565i	0.849353	−	0.527825i	$-0.176992\pi$
$572$	−43616.8			−0.133310
$573$		−	18548.9i		−	0.0564949i
$574$		−	8663.89i		−	0.0262960i
$575$			151326.i			0.457697i
$576$	−3356.20			−0.0101159
$577$	68280.6			0.205091			0.102545	−	0.994728i	$-0.467301\pi$
$577$	68280.6			0.205091			0.102545	+	0.994728i	$0.467301\pi$
$578$			968019.i			2.89753i
$579$	127053.			0.378990
$580$	374697.			1.11384
$581$			8648.88i			0.0256217i
$582$	337138.			0.995319
$583$			471876.i			1.38833i
$584$	−98056.5			−0.287509
$585$			39414.2i			0.115170i
$586$			258093.i			0.751589i
$587$		−	275118.i		−	0.798441i	−0.916855	−	0.399220i	$-0.869281\pi$
$587$		−	275118.i		−	0.798441i	0.916855	−	0.399220i	$-0.130719\pi$
$588$	108081.			0.312604
$589$			103726.i			0.298989i
$590$	−712472.	+	68031.6i	−2.04675	+	0.195437i
$591$	−154359.			−0.441932
$592$			663502.i			1.89321i
$593$	84870.1			0.241349			0.120674	−	0.992692i	$-0.461494\pi$
$593$	84870.1			0.241349			0.120674	+	0.992692i	$0.461494\pi$
$594$	99439.0			0.281828
$595$	23748.8			0.0670823
$596$			225635.i			0.635206i
$597$	−115258.			−0.323388
$598$		−	24342.5i		−	0.0680711i
$599$	−75105.1			−0.209322			−0.104661	−	0.994508i	$-0.533376\pi$
$599$	−75105.1			−0.209322			−0.104661	+	0.994508i	$0.533376\pi$
$600$			206028.i			0.572301i
$601$		−	94230.3i		−	0.260880i	−0.991456	−	0.130440i	$-0.958361\pi$
$601$		−	94230.3i		−	0.260880i	0.991456	−	0.130440i	$-0.0416391\pi$
$602$	−4028.65			−0.0111165
$603$			99353.0i			0.273241i
$604$			106421.i			0.291711i
$605$	−236986.			−0.647460
$606$	−160340.			−0.436614
$607$	247136.			0.670747			0.335374	−	0.942085i	$-0.391137\pi$
$607$	247136.			0.670747			0.335374	+	0.942085i	$0.391137\pi$
$608$		−	76373.4i		−	0.206602i
$609$	−5899.49			−0.0159067
$610$	621249.			1.66957
$611$	21646.4			0.0579834
$612$	−123483.			−0.329689
$613$		−	349727.i		−	0.930696i	−0.885128	−	0.465348i	$-0.845929\pi$
$613$		−	349727.i		−	0.930696i	0.885128	−	0.465348i	$-0.154071\pi$
$614$			608947.i			1.61526i
$615$	−345137.			−0.912518
$616$	5650.45			0.0148909
$617$	−645133.			−1.69465			−0.847323	−	0.531078i	$-0.821787\pi$
$617$	−645133.			−1.69465			−0.847323	+	0.531078i	$0.821787\pi$
$618$	310386.			0.812690
$619$	−473269.			−1.23517			−0.617585	−	0.786504i	$-0.711889\pi$
$619$	−473269.			−1.23517			−0.617585	+	0.786504i	$0.711889\pi$
$620$		−	489464.i		−	1.27332i
$621$			19499.9i			0.0505650i
$622$		−	791774.i		−	2.04654i
$623$		−	5589.47i		−	0.0144011i
$624$		−	58552.1i		−	0.150374i
$625$	114275.			0.292544
$626$	88024.9			0.224624
$627$		−	56384.4i		−	0.143425i
$628$		−	183659.i		−	0.465687i
$629$		−	1.09560e6i		−	2.76917i
$630$			6035.50i			0.0152066i
$631$	775739.			1.94831			0.974153	−	0.225891i	$-0.0725294\pi$
$631$	775739.			1.94831			0.974153	+	0.225891i	$0.0725294\pi$
$632$			112435.i			0.281492i
$633$			424655.i			1.05981i
$634$		−	559128.i		−	1.39102i
$635$	927842.			2.30105
$636$	148919.			0.368160
$637$		−	84624.0i		−	0.208552i
$638$	−740166.			−1.81839
$639$	49347.0			0.120853
$640$		−	690842.i		−	1.68663i
$641$	368716.			0.897380			0.448690	−	0.893688i	$-0.351891\pi$
$641$	368716.			0.897380			0.448690	+	0.893688i	$0.351891\pi$
$642$		−	152674.i		−	0.370420i
$643$	−81254.3			−0.196528			−0.0982640	−	0.995160i	$-0.531329\pi$
$643$	−81254.3			−0.196528			−0.0982640	+	0.995160i	$0.531329\pi$
$644$		−	1309.76i		−	0.00315806i
$645$			160487.i			0.385762i
$646$			199271.i			0.477506i
$647$	−176909.			−0.422612			−0.211306	−	0.977420i	$-0.567772\pi$
$647$	−176909.			−0.422612			−0.211306	+	0.977420i	$0.567772\pi$
$648$			26548.9i			0.0632261i
$649$	494518.	−	47219.9i	1.17407	−	0.112108i
$650$	−190679.			−0.451312
$651$			7706.46i			0.0181841i
$652$	−197807.			−0.465316
$653$	467222.			1.09571			0.547857	−	0.836572i	$-0.315444\pi$
$653$	467222.			1.09571			0.547857	+	0.836572i	$0.315444\pi$
$654$	461617.			1.07926
$655$			736970.i			1.71778i
$656$	512722.			1.19145
$657$			72697.8i			0.168419i
$658$	3314.72			0.00765588
$659$		−	41300.9i		−	0.0951018i	−0.998869	−	0.0475509i	$-0.984858\pi$
$659$		−	41300.9i		−	0.0951018i	0.998869	−	0.0475509i	$-0.0151416\pi$
$660$			266069.i			0.610810i
$661$	−298921.			−0.684153			−0.342077	−	0.939672i	$-0.611130\pi$
$661$	−298921.			−0.684153			−0.342077	+	0.939672i	$0.611130\pi$
$662$		−	630215.i		−	1.43805i
$663$			96683.4i			0.219950i
$664$	−289711.			−0.657095
$665$	3422.28			0.00773877
$666$	278434.			0.627732
$667$		−	145146.i		−	0.326253i
$668$	245879.			0.551022
$669$	−282533.			−0.631274
$670$	−756575.			−1.68540
$671$	−431201.			−0.957712
$672$		−	5674.28i		−	0.0125653i
$673$		−	814423.i		−	1.79812i	−0.437821	−	0.899062i	$-0.644249\pi$
$673$		−	814423.i		−	1.79812i	0.437821	−	0.899062i	$-0.355751\pi$
$674$	−724549.			−1.59495
$675$	152747.			0.335247
$676$	−236772.			−0.518129
$677$	−345729.			−0.754324			−0.377162	−	0.926147i	$-0.623100\pi$
$677$	−345729.			−0.754324			−0.377162	+	0.926147i	$0.623100\pi$
$678$	643456.			1.39978
$679$		−	14203.0i		−	0.0308063i
$680$			795512.i			1.72040i
$681$		−	289955.i		−	0.625225i
$682$			966873.i			2.07874i
$683$		−	239945.i		−	0.514363i	−0.966363	−	0.257181i	$-0.917206\pi$
$683$		−	239945.i		−	0.514363i	0.966363	−	0.257181i	$-0.0827938\pi$
$684$	−17794.3			−0.0380337
$685$	−492295.			−1.04917
$686$		−	25923.4i		−	0.0550863i
$687$			534014.i			1.13146i
$688$		−	238412.i		−	0.503677i
$689$		−	116599.i		−	0.245616i
$690$	−148492.			−0.311893
$691$			695558.i			1.45672i	0.685193	+	0.728362i	$0.259718\pi$
$691$			695558.i			1.45672i	−0.685193	+	0.728362i	$0.740282\pi$
$692$		−	65707.1i		−	0.137215i
$693$		−	4189.17i		−	0.00872290i
$694$	587206.			1.21919
$695$	−1.02837e6			−2.12902
$696$		−	197614.i		−	0.407944i
$697$	−846626.			−1.74271
$698$	−121727.			−0.249848
$699$			315075.i			0.644850i
$700$	−10259.6			−0.0209379
$701$		−	839756.i		−	1.70890i	−0.519533	−	0.854451i	$-0.673894\pi$
$701$		−	839756.i		−	1.70890i	0.519533	−	0.854451i	$-0.326106\pi$
$702$	−24571.0			−0.0498596
$703$		−	157879.i		−	0.319458i
$704$			17739.2i			0.0357922i
$705$		−	132046.i		−	0.265673i
$706$	−188770.			−0.378725
$707$			6754.82i			0.0135137i
$708$	−14902.1	−	156065.i	−0.0297290	−	0.311342i
$709$	340569.			0.677505			0.338753	−	0.940875i	$-0.389995\pi$
$709$	340569.			0.677505			0.338753	+	0.940875i	$0.389995\pi$
$710$			375778.i			0.745444i
$711$	83357.5			0.164894
$712$	187230.			0.369331
$713$	−189603.			−0.372964
$714$			14805.1i			0.0290413i
$715$	208323.			0.407498
$716$			302832.i			0.590712i
$717$	63075.0			0.122693
$718$			956093.i			1.85460i
$719$		−	155487.i		−	0.300770i	−0.988627	−	0.150385i	$-0.951949\pi$
$719$		−	155487.i		−	0.300770i	0.988627	−	0.150385i	$-0.0480514\pi$
$720$	−357176.			−0.688996
$721$		−	13075.9i		−	0.0251537i
$722$		−	618541.i		−	1.18657i
$723$	−69998.2			−0.133909
$724$	−327981.			−0.625708
$725$	−1.13696e6			−2.16306
$726$		−	147739.i		−	0.280299i
$727$	−61349.1			−0.116075			−0.0580376	−	0.998314i	$-0.518484\pi$
$727$	−61349.1			−0.116075			−0.0580376	+	0.998314i	$0.518484\pi$
$728$	−1396.20			−0.00263443
$729$	19683.0			0.0370370
$730$	−553595.			−1.03884
$731$			393675.i			0.736722i
$732$			136082.i			0.253969i
$733$	39246.3			0.0730451			0.0365225	−	0.999333i	$-0.488372\pi$
$733$	39246.3			0.0730451			0.0365225	+	0.999333i	$0.488372\pi$
$734$	−90864.4			−0.168656
$735$	−516218.			−0.955561
$736$	139605.			0.257719
$737$	525130.			0.966789
$738$		−	215160.i		−	0.395048i
$739$			856850.i			1.56898i	0.620144	+	0.784488i	$0.287074\pi$
$739$			856850.i			1.56898i	−0.620144	+	0.784488i	$0.712926\pi$
$740$			745006.i			1.36049i
$741$			13932.4i			0.0253740i
$742$		−	17854.8i		−	0.0324300i
$743$	598825.			1.08473			0.542366	−	0.840143i	$-0.317529\pi$
$743$	598825.			1.08473			0.542366	+	0.840143i	$0.317529\pi$
$744$	−258142.			−0.466352
$745$		−	1.07768e6i		−	1.94168i
$746$			286230.i			0.514325i
$747$			214788.i			0.384918i
$748$			652670.i			1.16651i
$749$	−6431.83			−0.0114649
$750$			495446.i			0.880792i
$751$			332338.i			0.589250i	0.955613	+	0.294625i	$0.0951947\pi$
$751$			332338.i			0.589250i	−0.955613	+	0.294625i	$0.904805\pi$
$752$			196162.i			0.346880i
$753$	−109115.			−0.192439
$754$	182892.			0.321701
$755$		−	508289.i		−	0.891696i
$756$	−1322.06			−0.00231316
$757$	824719.			1.43918			0.719588	−	0.694401i	$-0.244331\pi$
$757$	824719.			1.43918			0.719588	+	0.694401i	$0.244331\pi$
$758$			848387.i			1.47658i
$759$	103067.			0.178910
$760$			114636.i			0.198469i
$761$	1.09768e6			1.89542			0.947711	−	0.319130i	$-0.103391\pi$
$761$	1.09768e6			1.89542			0.947711	+	0.319130i	$0.103391\pi$
$762$			578422.i			0.996173i
$763$		−	19447.0i		−	0.0334044i
$764$			30940.4i			0.0530078i
$765$	589782.			1.00779
$766$		−	1.09823e6i		−	1.87170i
$767$	−122194.	+	11667.9i	−0.207710	+	0.0198336i
$768$	420340.			0.712654
$769$			237713.i			0.401976i	0.979594	+	0.200988i	$0.0644153\pi$
$769$			237713.i			0.401976i	−0.979594	+	0.200988i	$0.935585\pi$
$770$	31900.6			0.0538043
$771$	−204287.			−0.343662
$772$	−211930.			−0.355597
$773$			998180.i			1.67051i	0.549860	+	0.835257i	$0.314681\pi$
$773$			998180.i			1.67051i	−0.549860	+	0.835257i	$0.685319\pi$
$774$	−100048.			−0.167004
$775$			1.48520e6i			2.47276i
$776$	475755.			0.790060
$777$		−	11729.9i		−	0.0194290i
$778$			432849.i			0.715117i
$779$	−122001.			−0.201043
$780$		−	65744.6i		−	0.108061i
$781$		−	260823.i		−	0.427606i
$782$	−364254.			−0.595649
$783$	−146509.			−0.238968
$784$	766873.			1.24765
$785$			877196.i			1.42350i
$786$	−459431.			−0.743661
$787$	−292759.			−0.472673			−0.236337	−	0.971671i	$-0.575947\pi$
$787$	−292759.			−0.472673			−0.236337	+	0.971671i	$0.575947\pi$
$788$	257477.			0.414654
$789$	196734.			0.316029
$790$			634769.i			1.01709i
$791$		−	27107.5i		−	0.0433248i
$792$	140324.			0.223708
$793$	106548.			0.169434
$794$	−626566.			−0.993861
$795$	−711269.			−1.12538
$796$	192256.			0.303427
$797$			105074.i			0.165416i	0.996574	+	0.0827079i	$0.0263569\pi$
$797$			105074.i			0.165416i	−0.996574	+	0.0827079i	$0.973643\pi$
$798$			2133.47i			0.00335027i
$799$		−	323911.i		−	0.507378i
$800$		−	1.09355e6i		−	1.70868i
$801$		−	138810.i		−	0.216349i
$802$	135807.			0.211142
$803$	384244.			0.595903
$804$		−	165725.i		−	0.256375i
$805$			6255.69i			0.00965347i
$806$		−	238911.i		−	0.367761i
$807$		−	643056.i		−	0.987419i
$808$	−226265.			−0.346573
$809$		−	526375.i		−	0.804263i	−0.915582	−	0.402131i	$-0.868269\pi$
$809$		−	526375.i		−	0.804263i	0.915582	−	0.402131i	$-0.131731\pi$
$810$			149886.i			0.228451i
$811$			725298.i			1.10274i	0.834260	+	0.551372i	$0.185895\pi$
$811$			725298.i			1.10274i	−0.834260	+	0.551372i	$0.814105\pi$
$812$	9840.61			0.0149248
$813$	25228.7			0.0381692
$814$		−	1.47166e6i		−	2.22106i
$815$	944770.			1.42237
$816$	−876156.			−1.31583
$817$			56729.8i			0.0849899i
$818$	521160.			0.778869
$819$			1035.13i			0.00154321i
$820$	575704.			0.856193
$821$		−	299805.i		−	0.444787i	−0.974957	−	0.222393i	$-0.928613\pi$
$821$		−	299805.i		−	0.444787i	0.974957	−	0.222393i	$-0.0713870\pi$
$822$		−	306900.i		−	0.454206i
$823$			416111.i			0.614341i	0.951655	+	0.307170i	$0.0993821\pi$
$823$			416111.i			0.614341i	−0.951655	+	0.307170i	$0.900618\pi$
$824$	438003.			0.645094
$825$		−	807342.i		−	1.18618i
$826$	−18711.5	+	1786.70i	−0.0274252	+	0.00261874i
$827$	1.32953e6			1.94396			0.971982	−	0.235057i	$-0.0755277\pi$
$827$	1.32953e6			1.94396			0.971982	+	0.235057i	$0.0755277\pi$
$828$		−	32526.8i		−	0.0474439i
$829$	212633.			0.309401			0.154700	−	0.987961i	$-0.450559\pi$
$829$	212633.			0.309401			0.154700	+	0.987961i	$0.450559\pi$
$830$	−1.63561e6			−2.37424
$831$	504493.			0.730555
$832$		−	4383.28i		−	0.00633217i
$833$	−1.26629e6			−1.82492
$834$		−	641093.i		−	0.921698i
$835$	−1.17437e6			−1.68435
$836$			94051.7i			0.134572i
$837$			191383.i			0.273183i
$838$	−1.00982e6			−1.43798
$839$		−	916109.i		−	1.30144i	−0.759319	−	0.650719i	$-0.774468\pi$
$839$		−	916109.i		−	1.30144i	0.759319	−	0.650719i	$-0.225532\pi$
$840$			8517.04i			0.0120706i
$841$	383244.			0.541856
$842$	39315.3			0.0554546
$843$	−562960.			−0.792177
$844$		−	708343.i		−	0.994394i
$845$	1.13088e6			1.58380
$846$	82318.2			0.115015
$847$	−6223.93			−0.00867557
$848$	1.05663e6			1.46937
$849$		−	352786.i		−	0.489437i
$850$			2.85327e6i			3.94916i
$851$	288592.			0.398497
$852$	−82313.0			−0.113394
$853$	1.10136e6			1.51367			0.756833	−	0.653609i	$-0.226746\pi$
$853$	1.10136e6			1.51367			0.756833	+	0.653609i	$0.226746\pi$
$854$	16315.7			0.0223713
$855$	84989.4			0.116261
$856$		−	215446.i		−	0.294030i
$857$			1.13182e6i			1.54105i	0.637410	+	0.770525i	$0.280006\pi$
$857$			1.13182e6i			1.54105i	−0.637410	+	0.770525i	$0.719994\pi$
$858$			129870.i			0.176414i
$859$		−	574395.i		−	0.778439i	−0.921145	−	0.389220i	$-0.872745\pi$
$859$		−	574395.i		−	0.778439i	0.921145	−	0.389220i	$-0.127255\pi$
$860$		−	267699.i		−	0.361951i
$861$	−9064.28			−0.0122272
$862$	−829178.			−1.11592
$863$			792718.i			1.06438i	0.846625	+	0.532190i	$0.178631\pi$
$863$			792718.i			1.06438i	−0.846625	+	0.532190i	$0.821369\pi$
$864$		−	140916.i		−	0.188770i
$865$			313831.i			0.419434i
$866$		−	234349.i		−	0.312483i
$867$	1.01275e6			1.34730
$868$		−	12854.7i		−	0.0170617i
$869$		−	440585.i		−	0.583432i
$870$		−	1.11567e6i		−	1.47400i
$871$	−129758.			−0.171039
$872$	651414.			0.856691
$873$		−	352719.i		−	0.462807i
$874$	−52490.1			−0.0687154
$875$	20872.1			0.0272616
$876$		−	121263.i		−	0.158023i
$877$	−17892.2			−0.0232629			−0.0116314	−	0.999932i	$-0.503702\pi$
$877$	−17892.2			−0.0232629			−0.0116314	+	0.999932i	$0.503702\pi$
$878$			838494.i			1.08770i
$879$	270020.			0.349476
$880$			1.88785e6i			2.43782i
$881$			1.36036e6i			1.75268i	0.481697	+	0.876338i	$0.340021\pi$
$881$			1.36036e6i			1.75268i	−0.481697	+	0.876338i	$0.659979\pi$
$882$		−	321813.i		−	0.413682i
$883$	125594.			0.161082			0.0805408	−	0.996751i	$-0.474335\pi$
$883$	125594.			0.161082			0.0805408	+	0.996751i	$0.474335\pi$
$884$		−	161272.i		−	0.206374i
$885$	71175.5	+	745398.i	0.0908750	+	0.951703i
$886$	−1.52724e6			−1.94554
$887$		−	1.43325e6i		−	1.82169i	−0.412750	−	0.910844i	$-0.635432\pi$
$887$		−	1.43325e6i		−	1.82169i	0.412750	−	0.910844i	$-0.364568\pi$
$888$	392914.			0.498278
$889$	24367.8			0.0308327
$890$	1.05704e6			1.33448
$891$		−	104034.i		−	0.131045i
$892$	471278.			0.592308
$893$		−	46676.5i		−	0.0585323i
$894$	671833.			0.840594
$895$		−	1.44639e6i		−	1.80567i
$896$		−	18143.5i		−	0.0225998i
$897$	−25467.4			−0.0316519
$898$		−	1.20995e6i		−	1.50043i
$899$		−	1.42455e6i		−	1.76261i
$900$	−254788.			−0.314553
$901$	−1.74475e6			−2.14924
$902$	−1.13723e6			−1.39777
$903$			4214.83i			0.00516898i
$904$	908018.			1.11111
$905$	1.56651e6			1.91265
$906$	316870.			0.386033
$907$	780561.			0.948838			0.474419	−	0.880299i	$-0.342658\pi$
$907$	780561.			0.948838			0.474419	+	0.880299i	$0.342658\pi$
$908$			483657.i			0.586633i
$909$			167750.i			0.203018i
$910$	−7882.51			−0.00951880
$911$	414587.			0.499550			0.249775	−	0.968304i	$-0.419643\pi$
$911$	414587.			0.499550			0.249775	+	0.968304i	$0.419643\pi$
$912$	−126257.			−0.151798
$913$	1.13526e6			1.36192
$914$	−1.19605e6			−1.43172
$915$		−	649959.i		−	0.776325i
$916$		−	890759.i		−	1.06162i
$917$			19354.9i			0.0230172i
$918$			367673.i			0.436291i
$919$			56347.3i			0.0667179i	0.999443	+	0.0333590i	$0.0106205\pi$
$919$			56347.3i			0.0667179i	−0.999443	+	0.0333590i	$0.989380\pi$
$920$	−209546.			−0.247573
$921$	637088.			0.751069
$922$			523580.i			0.615915i
$923$			64448.4i			0.0756500i
$924$			6987.72i			0.00818448i
$925$		−	2.26060e6i		−	2.64204i
$926$	880388.			1.02672
$927$		−	324730.i		−	0.377887i
$928$			1.04890e6i			1.21797i
$929$			1.10326e6i			1.27834i	0.769067	+	0.639168i	$0.220721\pi$
$929$			1.10326e6i			1.27834i	−0.769067	+	0.639168i	$0.779279\pi$
$930$	−1.45739e6			−1.68504
$931$	−182476.			−0.210526
$932$		−	525558.i		−	0.605047i
$933$	−828364.			−0.951608
$934$	−1.92488e6			−2.20653
$935$		−	3.11729e6i		−	3.56577i
$936$	−34673.5			−0.0395773
$937$			912935.i			1.03983i	0.854219	+	0.519913i	$0.174036\pi$
$937$			912935.i			1.03983i	−0.854219	+	0.519913i	$0.825964\pi$
$938$	−19869.8			−0.0225833
$939$		−	92092.8i		−	0.104447i
$940$			220259.i			0.249274i
$941$			1.63562e6i			1.84716i	0.383409	+	0.923579i	$0.374750\pi$
$941$			1.63562e6i			1.84716i	−0.383409	+	0.923579i	$0.625250\pi$
$942$	−546849.			−0.616262
$943$		−	223010.i		−	0.250785i
$944$	−105735.	−	1.10733e6i	−0.118652	−	1.24261i
$945$	6314.42			0.00707082
$946$			528804.i			0.590898i
$947$	−927647.			−1.03439			−0.517193	−	0.855869i	$-0.673023\pi$
$947$	−927647.			−1.03439			−0.517193	+	0.855869i	$0.673023\pi$
$948$	−139044.			−0.154716
$949$	−94945.2			−0.105424
$950$			411165.i			0.455584i
$951$	−584967.			−0.646800
$952$			20892.4i			0.0230523i
$953$	−1.38318e6			−1.52297			−0.761487	−	0.648180i	$-0.775530\pi$
$953$	−1.38318e6			−1.52297			−0.761487	+	0.648180i	$0.775530\pi$
$954$		−	443409.i		−	0.487201i
$955$		−	147778.i		−	0.162033i
$956$	−105212.			−0.115119
$957$			774371.i			0.845523i
$958$			1.71936e6i			1.87342i
$959$	−12929.1			−0.0140582
$960$	−26738.6			−0.0290133
$961$	−937353.			−1.01498
$962$			363642.i			0.392938i
$963$	−159729.			−0.172239
$964$	116760.			0.125644
$965$	1.01222e6			1.08698
$966$	−3899.83			−0.00417918
$967$			671415.i			0.718023i	0.933333	+	0.359011i	$0.116886\pi$
$967$			671415.i			0.718023i	−0.933333	+	0.359011i	$0.883114\pi$
$968$		−	208482.i		−	0.222494i
$969$	208480.			0.222033
$970$	2.68596e6			2.85467
$971$	−819673.			−0.869365			−0.434683	−	0.900584i	$-0.643139\pi$
$971$	−819673.			−0.869365			−0.434683	+	0.900584i	$0.643139\pi$
$972$	−32832.1			−0.0347509
$973$	−27007.9			−0.0285276
$974$		−	1.01638e6i		−	1.07137i
$975$			199491.i			0.209853i
$976$			965552.i			1.01362i
$977$			209664.i			0.219651i	0.993951	+	0.109826i	$0.0350293\pi$
$977$			209664.i			0.219651i	−0.993951	+	0.109826i	$0.964971\pi$
$978$			588975.i			0.615771i
$979$	−733678.			−0.765491
$980$	861075.			0.896579
$981$		−	482950.i		−	0.501839i
$982$		−	617991.i		−	0.640854i
$983$		−	777728.i		−	0.804861i	−0.915451	−	0.402431i	$-0.868165\pi$
$983$		−	777728.i		−	0.804861i	0.915451	−	0.402431i	$-0.131835\pi$
$984$		−	303625.i		−	0.313579i
$985$	−1.22976e6			−1.26750
$986$		−	2.73674e6i		−	2.81501i
$987$		−	3467.90i		−	0.00355986i
$988$		−	23239.8i		−	0.0238078i
$989$	−103698.			−0.106018
$990$	792224.			0.808309
$991$		−	1.67117e6i		−	1.70166i	−0.525442	−	0.850829i	$-0.676100\pi$
$991$		−	1.67117e6i		−	1.70166i	0.525442	−	0.850829i	$-0.323900\pi$
$992$	1.37016e6			1.39235
$993$	−659339.			−0.668668
$994$			9869.00i			0.00998850i
$995$	−918257.			−0.927509
$996$		−	358275.i		−	0.361159i
$997$	43262.3			0.0435230			0.0217615	−	0.999763i	$-0.493073\pi$
$997$	43262.3			0.0435230			0.0217615	+	0.999763i	$0.493073\pi$
$998$		−	2.04018e6i		−	2.04837i
$999$		−	291301.i		−	0.291885i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.5.c.a.58.33	yes	40
3.2	odd	2		531.5.c.d.235.8		40
59.58	odd	2	inner	177.5.c.a.58.8	✓	40
177.176	even	2		531.5.c.d.235.33		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.8	✓	40	59.58	odd	2	inner
177.5.c.a.58.33	yes	40	1.1	even	1	trivial
531.5.c.d.235.8		40	3.2	odd	2
531.5.c.d.235.33		40	177.176	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	177.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.96663i q^{2} +5.19615 q^{3} -8.66741 q^{4} +41.3974 q^{5} +25.8074i q^{6} +1.08721 q^{7} +36.4182i q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q+4.96663i q^{2} +5.19615 q^{3} -8.66741 q^{4} +41.3974 q^{5} +25.8074i q^{6} +1.08721 q^{7} +36.4182i q^{8} +27.0000 q^{9} +205.606i q^{10} -142.708i q^{11} -45.0372 q^{12} +35.2627i q^{13} +5.39978i q^{14} +215.107 q^{15} -319.555 q^{16} +527.661 q^{17} +134.099i q^{18} +76.0375 q^{19} -358.808 q^{20} +5.64932 q^{21} +708.780 q^{22} +138.991i q^{23} +189.235i q^{24} +1088.75 q^{25} -175.137 q^{26} +140.296 q^{27} -9.42332 q^{28} -1044.28 q^{29} +1068.36i q^{30} +1364.14i q^{31} -1004.42i q^{32} -741.534i q^{33} +2620.69i q^{34} +45.0078 q^{35} -234.020 q^{36} -2076.33i q^{37} +377.650i q^{38} +183.230i q^{39} +1507.62i q^{40} -1604.49 q^{41} +28.0581i q^{42} +746.077i q^{43} +1236.91i q^{44} +1117.73 q^{45} -690.318 q^{46} -613.862i q^{47} -1660.45 q^{48} -2399.82 q^{49} +5407.39i q^{50} +2741.80 q^{51} -305.636i q^{52} -3306.58 q^{53} +696.799i q^{54} -5907.75i q^{55} +39.5944i q^{56} +395.102 q^{57} -5186.56i q^{58} +(330.884 + 3465.24i) q^{59} -1864.42 q^{60} -3021.56i q^{61} -6775.17 q^{62} +29.3547 q^{63} -124.304 q^{64} +1459.78i q^{65} +3682.93 q^{66} +3679.74i q^{67} -4573.45 q^{68} +722.220i q^{69} +223.537i q^{70} +1827.67 q^{71} +983.293i q^{72} +2692.51i q^{73} +10312.4 q^{74} +5657.29 q^{75} -659.048 q^{76} -155.154i q^{77} -910.037 q^{78} +3087.31 q^{79} -13228.7 q^{80} +729.000 q^{81} -7968.90i q^{82} +7955.09i q^{83} -48.9650 q^{84} +21843.8 q^{85} -3705.49 q^{86} -5426.25 q^{87} +5197.19 q^{88} -5141.10i q^{89} +5551.35i q^{90} +38.3380i q^{91} -1204.70i q^{92} +7088.27i q^{93} +3048.82 q^{94} +3147.75 q^{95} -5219.11i q^{96} -13063.6i q^{97} -11919.0i q^{98} -3853.13i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9}+O(q^{10}) \) 40 * q - 320 * q^4 + 80 * q^7 + 1080 * q^9	\( 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9} + 360 q^{12} + 144 q^{15} + 3944 q^{16} - 528 q^{17} + 444 q^{19} + 444 q^{20} + 1304 q^{22} + 4880 q^{25} - 1452 q^{26} - 1160 q^{28} - 996 q^{29} + 10320 q^{35} - 8640 q^{36} - 5196 q^{41} - 10476 q^{46} + 576 q^{48} + 5104 q^{49} + 936 q^{51} - 2184 q^{53} - 2520 q^{57} - 11736 q^{59} - 11448 q^{60} + 15240 q^{62} + 2160 q^{63} - 81012 q^{64} + 17352 q^{66} + 29568 q^{68} - 5964 q^{71} + 14376 q^{74} - 2736 q^{75} + 3480 q^{76} + 37692 q^{78} + 19020 q^{79} + 33096 q^{80} + 29160 q^{81} + 25128 q^{84} + 20220 q^{85} - 65880 q^{86} + 1512 q^{87} - 14932 q^{88} - 17864 q^{94} + 11004 q^{95}+O(q^{100}) \) 40 * q - 320 * q^4 + 80 * q^7 + 1080 * q^9 + 360 * q^12 + 144 * q^15 + 3944 * q^16 - 528 * q^17 + 444 * q^19 + 444 * q^20 + 1304 * q^22 + 4880 * q^25 - 1452 * q^26 - 1160 * q^28 - 996 * q^29 + 10320 * q^35 - 8640 * q^36 - 5196 * q^41 - 10476 * q^46 + 576 * q^48 + 5104 * q^49 + 936 * q^51 - 2184 * q^53 - 2520 * q^57 - 11736 * q^59 - 11448 * q^60 + 15240 * q^62 + 2160 * q^63 - 81012 * q^64 + 17352 * q^66 + 29568 * q^68 - 5964 * q^71 + 14376 * q^74 - 2736 * q^75 + 3480 * q^76 + 37692 * q^78 + 19020 * q^79 + 33096 * q^80 + 29160 * q^81 + 25128 * q^84 + 20220 * q^85 - 65880 * q^86 + 1512 * q^87 - 14932 * q^88 - 17864 * q^94 + 11004 * q^95

Introduction

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