LMFDB - Embedded newform 354.5.d.a.235.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [354,5,Mod(235,354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(354, 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("354.235");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 354 = 2 \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	354.d (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:	$36.5929669317$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			235.20
Character	$\chi$	$=$	354.235
Dual form			354.5.d.a.235.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} +5.19615 q^{3} -8.00000 q^{4} +26.1412 q^{5} +14.6969i q^{6} -51.7193 q^{7} -22.6274i q^{8} +27.0000 q^{9} +O(q^{10})$	\(q+2.82843i q^{2} +5.19615 q^{3} -8.00000 q^{4} +26.1412 q^{5} +14.6969i q^{6} -51.7193 q^{7} -22.6274i q^{8} +27.0000 q^{9} +73.9384i q^{10} +34.2542i q^{11} -41.5692 q^{12} -269.411i q^{13} -146.284i q^{14} +135.834 q^{15} +64.0000 q^{16} +472.055 q^{17} +76.3675i q^{18} +44.0001 q^{19} -209.129 q^{20} -268.741 q^{21} -96.8854 q^{22} +650.891i q^{23} -117.576i q^{24} +58.3616 q^{25} +762.010 q^{26} +140.296 q^{27} +413.754 q^{28} +1255.43 q^{29} +384.195i q^{30} -441.475i q^{31} +181.019i q^{32} +177.990i q^{33} +1335.17i q^{34} -1352.00 q^{35} -216.000 q^{36} -801.627i q^{37} +124.451i q^{38} -1399.90i q^{39} -591.508i q^{40} +2404.73 q^{41} -760.115i q^{42} -3073.44i q^{43} -274.033i q^{44} +705.812 q^{45} -1841.00 q^{46} +785.833i q^{47} +332.554 q^{48} +273.884 q^{49} +165.072i q^{50} +2452.87 q^{51} +2155.29i q^{52} +4187.19 q^{53} +396.817i q^{54} +895.445i q^{55} +1170.27i q^{56} +228.631 q^{57} +3550.90i q^{58} +(1707.94 + 3033.20i) q^{59} -1086.67 q^{60} +322.102i q^{61} +1248.68 q^{62} -1396.42 q^{63} -512.000 q^{64} -7042.72i q^{65} -503.431 q^{66} +392.975i q^{67} -3776.44 q^{68} +3382.13i q^{69} -3824.04i q^{70} -1960.97 q^{71} -610.940i q^{72} +445.202i q^{73} +2267.34 q^{74} +303.256 q^{75} -352.001 q^{76} -1771.60i q^{77} +3959.52 q^{78} +2734.81 q^{79} +1673.04 q^{80} +729.000 q^{81} +6801.60i q^{82} +9316.32i q^{83} +2149.93 q^{84} +12340.1 q^{85} +8693.00 q^{86} +6523.42 q^{87} +775.083 q^{88} -2930.73i q^{89} +1996.34i q^{90} +13933.7i q^{91} -5207.13i q^{92} -2293.97i q^{93} -2222.67 q^{94} +1150.21 q^{95} +940.604i q^{96} -2290.43i q^{97} +774.662i q^{98} +924.862i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 144 q^{15} + 2560 q^{16} + 480 q^{17} - 792 q^{19} - 1024 q^{22} + 3400 q^{25} + 768 q^{26} + 640 q^{28} + 1608 q^{29} - 5760 q^{35} - 8640 q^{36} + 6264 q^{41} + 7040 q^{46} + 17912 q^{49} + 1296 q^{51} - 1104 q^{53} + 5040 q^{57} + 13584 q^{59} + 1152 q^{60} - 12288 q^{62} - 2160 q^{63} - 20480 q^{64} + 1152 q^{66} - 3840 q^{68} + 35352 q^{71} + 4608 q^{74} + 3168 q^{75} + 6336 q^{76} - 12672 q^{78} - 15720 q^{79} + 29160 q^{81} - 26872 q^{85} + 18432 q^{86} + 7776 q^{87} + 8192 q^{88} - 18432 q^{94} - 19128 q^{95}+O(q^{100}) $ 40 * q - 320 * q^4 - 80 * q^7 + 1080 * q^9 - 144 * q^15 + 2560 * q^16 + 480 * q^17 - 792 * q^19 - 1024 * q^22 + 3400 * q^25 + 768 * q^26 + 640 * q^28 + 1608 * q^29 - 5760 * q^35 - 8640 * q^36 + 6264 * q^41 + 7040 * q^46 + 17912 * q^49 + 1296 * q^51 - 1104 * q^53 + 5040 * q^57 + 13584 * q^59 + 1152 * q^60 - 12288 * q^62 - 2160 * q^63 - 20480 * q^64 + 1152 * q^66 - 3840 * q^68 + 35352 * q^71 + 4608 * q^74 + 3168 * q^75 + 6336 * q^76 - 12672 * q^78 - 15720 * q^79 + 29160 * q^81 - 26872 * q^85 + 18432 * q^86 + 7776 * q^87 + 8192 * q^88 - 18432 * q^94 - 19128 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/354\mathbb{Z}\right)^\times$.

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

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 $

$2$			2.82843i			0.707107i
$2$			2.82843i			0.707107i
$3$	5.19615			0.577350
$3$	5.19615			0.577350
$4$	−8.00000			−0.500000
$5$	26.1412			1.04565			0.522824	−	0.852441i	$-0.324879\pi$
$5$	26.1412			1.04565			0.522824	+	0.852441i	$0.324879\pi$
$6$			14.6969i			0.408248i
$7$	−51.7193			−1.05550			−0.527748	−	0.849401i	$-0.676963\pi$
$7$	−51.7193			−1.05550			−0.527748	+	0.849401i	$0.676963\pi$
$8$		−	22.6274i		−	0.353553i
$9$	27.0000			0.333333
$10$			73.9384i			0.739384i
$11$			34.2542i			0.283092i	0.989932	+	0.141546i	$0.0452074\pi$
$11$			34.2542i			0.283092i	−0.989932	+	0.141546i	$0.954793\pi$
$12$	−41.5692			−0.288675
$13$		−	269.411i		−	1.59415i	−0.603881	−	0.797074i	$-0.706380\pi$
$13$		−	269.411i		−	1.59415i	0.603881	−	0.797074i	$-0.293620\pi$
$14$		−	146.284i		−	0.746348i
$15$	135.834			0.603705
$16$	64.0000			0.250000
$17$	472.055			1.63341			0.816705	−	0.577056i	$-0.195799\pi$
$17$	472.055			1.63341			0.816705	+	0.577056i	$0.195799\pi$
$18$			76.3675i			0.235702i
$19$	44.0001			0.121884			0.0609420	−	0.998141i	$-0.480590\pi$
$19$	44.0001			0.121884			0.0609420	+	0.998141i	$0.480590\pi$
$20$	−209.129			−0.522824
$21$	−268.741			−0.609391
$22$	−96.8854			−0.200176
$23$			650.891i			1.23042i	0.788364	+	0.615209i	$0.210928\pi$
$23$			650.891i			1.23042i	−0.788364	+	0.615209i	$0.789072\pi$
$24$		−	117.576i		−	0.204124i
$25$	58.3616			0.0933786
$26$	762.010			1.12723
$27$	140.296			0.192450
$28$	413.754			0.527748
$29$	1255.43			1.49279			0.746393	−	0.665505i	$-0.231784\pi$
$29$	1255.43			1.49279			0.746393	+	0.665505i	$0.231784\pi$
$30$			384.195i			0.426884i
$31$		−	441.475i		−	0.459392i	−0.973262	−	0.229696i	$-0.926227\pi$
$31$		−	441.475i		−	0.459392i	0.973262	−	0.229696i	$-0.0737731\pi$
$32$			181.019i			0.176777i
$33$			177.990i			0.163443i
$34$			1335.17i			1.15499i
$35$	−1352.00			−1.10368
$36$	−216.000			−0.166667
$37$		−	801.627i		−	0.585557i	−0.956180	−	0.292778i	$-0.905420\pi$
$37$		−	801.627i		−	0.585557i	0.956180	−	0.292778i	$-0.0945798\pi$
$38$			124.451i			0.0861849i
$39$		−	1399.90i		−	0.920382i
$40$		−	591.508i		−	0.369692i
$41$	2404.73			1.43053			0.715267	−	0.698851i	$-0.246305\pi$
$41$	2404.73			1.43053			0.715267	+	0.698851i	$0.246305\pi$
$42$		−	760.115i		−	0.430904i
$43$		−	3073.44i		−	1.66222i	−0.556111	−	0.831108i	$-0.687707\pi$
$43$		−	3073.44i		−	1.66222i	0.556111	−	0.831108i	$-0.312293\pi$
$44$		−	274.033i		−	0.141546i
$45$	705.812			0.348549
$46$	−1841.00			−0.870037
$47$			785.833i			0.355741i	0.984054	+	0.177871i	$0.0569209\pi$
$47$			785.833i			0.355741i	−0.984054	+	0.177871i	$0.943079\pi$
$48$	332.554			0.144338
$49$	273.884			0.114071
$50$			165.072i			0.0660286i
$51$	2452.87			0.943049
$52$			2155.29i			0.797074i
$53$	4187.19			1.49063			0.745317	−	0.666710i	$-0.232298\pi$
$53$	4187.19			1.49063			0.745317	+	0.666710i	$0.232298\pi$
$54$			396.817i			0.136083i
$55$			895.445i			0.296015i
$56$			1170.27i			0.373174i
$57$	228.631			0.0703697
$58$			3550.90i			1.05556i
$59$	1707.94	+	3033.20i	0.490646	+	0.871359i
$59$	1707.94	+	3033.20i	0.490646	+	0.871359i
$60$	−1086.67			−0.301852
$61$			322.102i			0.0865634i	0.999063	+	0.0432817i	$0.0137813\pi$
$61$			322.102i			0.0865634i	−0.999063	+	0.0432817i	$0.986219\pi$
$62$	1248.68			0.324839
$63$	−1396.42			−0.351832
$64$	−512.000			−0.125000
$65$		−	7042.72i		−	1.66692i
$66$	−503.431			−0.115572
$67$			392.975i			0.0875417i	0.999042	+	0.0437709i	$0.0139372\pi$
$67$			392.975i			0.0875417i	−0.999042	+	0.0437709i	$0.986063\pi$
$68$	−3776.44			−0.816705
$69$			3382.13i			0.710382i
$70$		−	3824.04i		−	0.780417i
$71$	−1960.97			−0.389004			−0.194502	−	0.980902i	$-0.562309\pi$
$71$	−1960.97			−0.389004			−0.194502	+	0.980902i	$0.562309\pi$
$72$		−	610.940i		−	0.117851i
$73$			445.202i			0.0835432i	0.999127	+	0.0417716i	$0.0133002\pi$
$73$			445.202i			0.0835432i	−0.999127	+	0.0417716i	$0.986700\pi$
$74$	2267.34			0.414051
$75$	303.256			0.0539121
$76$	−352.001			−0.0609420
$77$		−	1771.60i		−	0.298803i
$78$	3959.52			0.650808
$79$	2734.81			0.438201			0.219100	−	0.975702i	$-0.429688\pi$
$79$	2734.81			0.438201			0.219100	+	0.975702i	$0.429688\pi$
$80$	1673.04			0.261412
$81$	729.000			0.111111
$82$			6801.60i			1.01154i
$83$			9316.32i			1.35235i	0.736742	+	0.676174i	$0.236363\pi$
$83$			9316.32i			1.35235i	−0.736742	+	0.676174i	$0.763637\pi$
$84$	2149.93			0.304695
$85$	12340.1			1.70797
$86$	8693.00			1.17536
$87$	6523.42			0.861861
$88$	775.083			0.100088
$89$		−	2930.73i		−	0.369995i	−0.982739	−	0.184998i	$-0.940772\pi$
$89$		−	2930.73i		−	0.369995i	0.982739	−	0.184998i	$-0.0592278\pi$
$90$			1996.34i			0.246461i
$91$			13933.7i			1.68262i
$92$		−	5207.13i		−	0.615209i
$93$		−	2293.97i		−	0.265230i
$94$	−2222.67			−0.251547
$95$	1150.21			0.127448
$96$			940.604i			0.102062i
$97$		−	2290.43i		−	0.243429i	−0.992565	−	0.121715i	$-0.961161\pi$
$97$		−	2290.43i		−	0.243429i	0.992565	−	0.121715i	$-0.0388393\pi$
$98$			774.662i			0.0806603i
$99$			924.862i			0.0943641i
$100$	−466.893			−0.0466893
$101$		−	11137.8i		−	1.09183i	−0.837840	−	0.545916i	$-0.816182\pi$
$101$		−	11137.8i		−	1.09183i	0.837840	−	0.545916i	$-0.183818\pi$
$102$			6937.77i			0.666837i
$103$		−	3145.00i		−	0.296446i	−0.988954	−	0.148223i	$-0.952645\pi$
$103$		−	3145.00i		−	0.296446i	0.988954	−	0.148223i	$-0.0473554\pi$
$104$	−6096.08			−0.563617
$105$	−7025.22			−0.637208
$106$			11843.2i			1.05404i
$107$	−2727.31			−0.238214			−0.119107	−	0.992881i	$-0.538003\pi$
$107$	−2727.31			−0.238214			−0.119107	+	0.992881i	$0.538003\pi$
$108$	−1122.37			−0.0962250
$109$			16847.7i			1.41804i	0.705188	+	0.709020i	$0.250863\pi$
$109$			16847.7i			1.41804i	−0.705188	+	0.709020i	$0.749137\pi$
$110$	−2532.70			−0.209314
$111$		−	4165.38i		−	0.338071i
$112$	−3310.03			−0.263874
$113$			12321.1i			0.964926i	0.875916	+	0.482463i	$0.160258\pi$
$113$			12321.1i			0.964926i	−0.875916	+	0.482463i	$0.839742\pi$
$114$			646.667i			0.0497589i
$115$			17015.1i			1.28658i
$116$	−10043.5			−0.746393
$117$		−	7274.10i		−	0.531383i
$118$	−8579.19	+	4830.78i	−0.616144	+	0.346939i
$119$	−24414.4			−1.72406
$120$		−	3073.56i		−	0.213442i
$121$	13467.7			0.919859
$122$	−911.043			−0.0612095
$123$	12495.3			0.825920
$124$			3531.80i			0.229696i
$125$	−14812.6			−0.948006
$126$		−	3949.67i		−	0.248783i
$127$	−10959.8			−0.679509			−0.339755	−	0.940514i	$-0.610344\pi$
$127$	−10959.8			−0.679509			−0.339755	+	0.940514i	$0.610344\pi$
$128$		−	1448.15i		−	0.0883883i
$129$		−	15970.1i		−	0.959681i
$130$	19919.8			1.17869
$131$		−	9051.19i		−	0.527428i	−0.964601	−	0.263714i	$-0.915053\pi$
$131$		−	9051.19i		−	0.527428i	0.964601	−	0.263714i	$-0.0849475\pi$
$132$		−	1423.92i		−	0.0817217i
$133$	−2275.65			−0.128648
$134$	−1111.50			−0.0619013
$135$	3667.51			0.201235
$136$		−	10681.4i		−	0.577497i
$137$	−3053.93			−0.162712			−0.0813558	−	0.996685i	$-0.525925\pi$
$137$	−3053.93			−0.162712			−0.0813558	+	0.996685i	$0.525925\pi$
$138$	−9566.11			−0.502316
$139$	22981.1			1.18944			0.594719	−	0.803934i	$-0.297263\pi$
$139$	22981.1			1.18944			0.594719	+	0.803934i	$0.297263\pi$
$140$	10816.0			0.551838
$141$			4083.31i			0.205387i
$142$		−	5546.45i		−	0.275067i
$143$	9228.45			0.451291
$144$	1728.00			0.0833333
$145$	32818.5			1.56093
$146$	−1259.22			−0.0590740
$147$	1423.14			0.0658589
$148$			6413.02i			0.292778i
$149$			2041.48i			0.0919543i	0.998942	+	0.0459772i	$0.0146401\pi$
$149$			2041.48i			0.0919543i	−0.998942	+	0.0459772i	$0.985360\pi$
$150$			857.737i			0.0381216i
$151$		−	11395.6i		−	0.499784i	−0.968274	−	0.249892i	$-0.919605\pi$
$151$		−	11395.6i		−	0.499784i	0.968274	−	0.249892i	$-0.0803951\pi$
$152$		−	995.608i		−	0.0430925i
$153$	12745.5			0.544470
$154$	5010.84			0.211285
$155$		−	11540.7i		−	0.480362i
$156$			11199.2i			0.460191i
$157$		−	24766.1i		−	1.00475i	−0.864649	−	0.502376i	$-0.832459\pi$
$157$		−	24766.1i		−	1.00475i	0.864649	−	0.502376i	$-0.167541\pi$
$158$			7735.22i			0.309855i
$159$	21757.3			0.860618
$160$			4732.06i			0.184846i
$161$		−	33663.6i		−	1.29870i
$162$			2061.92i			0.0785674i
$163$	−12373.1			−0.465698			−0.232849	−	0.972513i	$-0.574805\pi$
$163$	−12373.1			−0.465698			−0.232849	+	0.972513i	$0.574805\pi$
$164$	−19237.8			−0.715267
$165$			4652.87i			0.170904i
$166$	−26350.5			−0.956254
$167$	32300.2			1.15817			0.579086	−	0.815267i	$-0.303410\pi$
$167$	32300.2			1.15817			0.579086	+	0.815267i	$0.303410\pi$
$168$			6080.92i			0.215452i
$169$	−44021.3			−1.54131
$170$			34903.0i			1.20772i
$171$	1188.00			0.0406280
$172$			24587.5i			0.831108i
$173$		−	35261.1i		−	1.17816i	−0.808076	−	0.589078i	$-0.799491\pi$
$173$		−	35261.1i		−	1.17816i	0.808076	−	0.589078i	$-0.200509\pi$
$174$			18451.0i			0.609427i
$175$	−3018.42			−0.0985607
$176$			2192.27i			0.0707731i
$177$	8874.72	+	15761.0i	0.283275	+	0.503079i
$178$	8289.37			0.261626
$179$			23386.3i			0.729886i	0.931030	+	0.364943i	$0.118912\pi$
$179$			23386.3i			0.729886i	−0.931030	+	0.364943i	$0.881088\pi$
$180$	−5646.50			−0.174275
$181$	23628.6			0.721241			0.360621	−	0.932713i	$-0.382565\pi$
$181$	23628.6			0.721241			0.360621	+	0.932713i	$0.382565\pi$
$182$	−39410.6			−1.18979
$183$			1673.69i			0.0499774i
$184$	14728.0			0.435018
$185$		−	20955.5i		−	0.612286i
$186$	6488.33			0.187546
$187$			16169.9i			0.462406i
$188$		−	6286.66i		−	0.177871i
$189$	−7256.01			−0.203130
$190$			3253.30i			0.0901191i
$191$		−	69259.3i		−	1.89850i	−0.314517	−	0.949252i	$-0.601843\pi$
$191$		−	69259.3i		−	1.89850i	0.314517	−	0.949252i	$-0.398157\pi$
$192$	−2660.43			−0.0721688
$193$	−47890.8			−1.28569			−0.642847	−	0.765994i	$-0.722247\pi$
$193$	−47890.8			−1.28569			−0.642847	+	0.765994i	$0.722247\pi$
$194$	6478.30			0.172131
$195$		−	36595.1i		−	0.962395i
$196$	−2191.07			−0.0570355
$197$	−42609.8			−1.09793			−0.548967	−	0.835844i	$-0.684979\pi$
$197$	−42609.8			−1.09793			−0.548967	+	0.835844i	$0.684979\pi$
$198$	−2615.91			−0.0667255
$199$	−41679.0			−1.05247			−0.526237	−	0.850338i	$-0.676398\pi$
$199$	−41679.0			−1.05247			−0.526237	+	0.850338i	$0.676398\pi$
$200$		−	1320.57i		−	0.0330143i
$201$			2041.96i			0.0505422i
$202$	31502.4			0.772042
$203$	−64930.1			−1.57563
$204$	−19623.0			−0.471525
$205$	62862.5			1.49584
$206$	8895.40			0.209619
$207$			17574.1i			0.410139i
$208$		−	17242.3i		−	0.398537i
$209$			1507.19i			0.0345044i
$210$		−	19870.3i		−	0.450574i
$211$		−	42185.6i		−	0.947543i	−0.880648	−	0.473772i	$-0.842892\pi$
$211$		−	42185.6i		−	0.947543i	0.880648	−	0.473772i	$-0.157108\pi$
$212$	−33497.5			−0.745317
$213$	−10189.5			−0.224591
$214$		−	7713.99i		−	0.168442i
$215$		−	80343.3i		−	1.73809i
$216$		−	3174.54i		−	0.0680414i
$217$			22832.8i			0.484886i
$218$	−47652.6			−1.00271
$219$			2313.34i			0.0482337i
$220$		−	7163.56i		−	0.148007i
$221$		−	127177.i		−	2.60390i
$222$	11781.5			0.239053
$223$	−63139.8			−1.26968			−0.634839	−	0.772645i	$-0.718933\pi$
$223$	−63139.8			−1.26968			−0.634839	+	0.772645i	$0.718933\pi$
$224$		−	9362.19i		−	0.186587i
$225$	1575.76			0.0311262
$226$	−34849.5			−0.682306
$227$		−	2721.09i		−	0.0528069i	−0.999651	−	0.0264034i	$-0.991595\pi$
$227$		−	2721.09i		−	0.0528069i	0.999651	−	0.0264034i	$-0.00840545\pi$
$228$	−1829.05			−0.0351849
$229$			6934.39i			0.132232i	0.997812	+	0.0661161i	$0.0210608\pi$
$229$			6934.39i			0.132232i	−0.997812	+	0.0661161i	$0.978939\pi$
$230$	−48125.9			−0.909752
$231$		−	9205.51i		−	0.172514i
$232$		−	28407.2i		−	0.527780i
$233$			63544.9i			1.17049i	0.810855	+	0.585247i	$0.199002\pi$
$233$			63544.9i			1.17049i	−0.810855	+	0.585247i	$0.800998\pi$
$234$	20574.3			0.375744
$235$			20542.6i			0.371980i
$236$	−13663.5	−	24265.6i	−0.245323	−	0.435679i
$237$	14210.5			0.252995
$238$		−	69054.2i		−	1.21909i
$239$	−26421.5			−0.462554			−0.231277	−	0.972888i	$-0.574290\pi$
$239$	−26421.5			−0.462554			−0.231277	+	0.972888i	$0.574290\pi$
$240$	8693.35			0.150926
$241$	−101486.			−1.74732			−0.873659	−	0.486539i	$-0.838259\pi$
$241$	−101486.			−1.74732			−0.873659	+	0.486539i	$0.838259\pi$
$242$			38092.3i			0.650438i
$243$	3788.00			0.0641500
$244$		−	2576.82i		−	0.0432817i
$245$	7159.66			0.119278
$246$			35342.2i			0.584013i
$247$		−	11854.1i		−	0.194301i
$248$	−9989.44			−0.162419
$249$			48409.0i			0.780778i
$250$		−	41896.4i		−	0.670342i
$251$	39507.0			0.627085			0.313542	−	0.949574i	$-0.398484\pi$
$251$	39507.0			0.627085			0.313542	+	0.949574i	$0.398484\pi$
$252$	11171.4			0.175916
$253$	−22295.7			−0.348322
$254$		−	30999.0i		−	0.480486i
$255$	64121.0			0.986097
$256$	4096.00			0.0625000
$257$	−112548.			−1.70400			−0.852002	−	0.523538i	$-0.824612\pi$
$257$	−112548.			−1.70400			−0.852002	+	0.523538i	$0.824612\pi$
$258$	45170.1			0.678597
$259$			41459.6i			0.618053i
$260$			56341.8i			0.833459i
$261$	33896.7			0.497595
$262$	25600.6			0.372948
$263$	−24961.1			−0.360872			−0.180436	−	0.983587i	$-0.557751\pi$
$263$	−24961.1			−0.360872			−0.180436	+	0.983587i	$0.557751\pi$
$264$	4027.45			0.0577860
$265$	109458.			1.55868
$266$		−	6436.52i		−	0.0909678i
$267$		−	15228.5i		−	0.213617i
$268$		−	3143.80i		−	0.0437709i
$269$			100859.i			1.39383i	0.717152	+	0.696917i	$0.245446\pi$
$269$			100859.i			1.39383i	−0.717152	+	0.696917i	$0.754554\pi$
$270$			10373.3i			0.142295i
$271$	44438.0			0.605085			0.302542	−	0.953136i	$-0.402165\pi$
$271$	44438.0			0.605085			0.302542	+	0.953136i	$0.402165\pi$
$272$	30211.5			0.408352
$273$			72401.9i			0.971459i
$274$		−	8637.83i		−	0.115054i
$275$			1999.13i			0.0264348i
$276$		−	27057.0i		−	0.355191i
$277$	−33851.6			−0.441184			−0.220592	−	0.975366i	$-0.570799\pi$
$277$	−33851.6			−0.441184			−0.220592	+	0.975366i	$0.570799\pi$
$278$			65000.4i			0.841059i
$279$		−	11919.8i		−	0.153131i
$280$			30592.3i			0.390208i
$281$	68982.5			0.873628			0.436814	−	0.899552i	$-0.356107\pi$
$281$	68982.5			0.873628			0.436814	+	0.899552i	$0.356107\pi$
$282$	−11549.3			−0.145231
$283$		−	84196.7i		−	1.05129i	−0.850704	−	0.525644i	$-0.823824\pi$
$283$		−	84196.7i		−	1.05129i	0.850704	−	0.525644i	$-0.176176\pi$
$284$	15687.7			0.194502
$285$	5976.69			0.0735819
$286$			26102.0i			0.319111i
$287$	−124371.			−1.50992
$288$			4887.52i			0.0589256i
$289$	139315.			1.66803
$290$			92824.8i			1.10374i
$291$		−	11901.4i		−	0.140544i
$292$		−	3561.61i		−	0.0417716i
$293$	−131401.			−1.53061			−0.765303	−	0.643670i	$-0.777411\pi$
$293$	−131401.			−1.53061			−0.765303	+	0.643670i	$0.777411\pi$
$294$			4025.26i			0.0465693i
$295$	44647.6	+	79291.4i	0.513043	+	0.911134i
$296$	−18138.8			−0.207026
$297$			4805.73i			0.0544811i
$298$	−5774.17			−0.0650215
$299$	175357.			1.96147
$300$	−2426.05			−0.0269561
$301$			158956.i			1.75446i
$302$	32231.5			0.353400
$303$		−	57873.6i		−	0.630370i
$304$	2816.01			0.0304710
$305$			8420.14i			0.0905148i
$306$			36049.7i			0.384998i
$307$	40942.6			0.434409			0.217204	−	0.976126i	$-0.430306\pi$
$307$	40942.6			0.434409			0.217204	+	0.976126i	$0.430306\pi$
$308$			14172.8i			0.149401i
$309$		−	16341.9i		−	0.171153i
$310$	32642.0			0.339667
$311$	−188654.			−1.95050			−0.975250	−	0.221107i	$-0.929033\pi$
$311$	−188654.			−1.95050			−0.975250	+	0.221107i	$0.929033\pi$
$312$	−31676.1			−0.325404
$313$		−	143775.i		−	1.46755i	−0.679391	−	0.733777i	$-0.737756\pi$
$313$		−	143775.i		−	1.46755i	0.679391	−	0.733777i	$-0.262244\pi$
$314$	70049.2			0.710467
$315$	−36504.1			−0.367892
$316$	−21878.5			−0.219100
$317$	119650.			1.19068			0.595339	−	0.803474i	$-0.297018\pi$
$317$	119650.			1.19068			0.595339	+	0.803474i	$0.297018\pi$
$318$			61538.9i			0.608549i
$319$			43003.8i			0.422596i
$320$	−13384.3			−0.130706
$321$	−14171.5			−0.137533
$322$	95215.1			0.918320
$323$	20770.5			0.199086
$324$	−5832.00			−0.0555556
$325$		−	15723.3i		−	0.148859i
$326$		−	34996.5i		−	0.329299i
$327$			87543.4i			0.818706i
$328$		−	54412.8i		−	0.505770i
$329$		−	40642.7i		−	0.375483i
$330$	−13160.3			−0.120848
$331$	87826.1			0.801618			0.400809	−	0.916162i	$-0.368729\pi$
$331$	87826.1			0.801618			0.400809	+	0.916162i	$0.368729\pi$
$332$		−	74530.6i		−	0.676174i
$333$		−	21643.9i		−	0.195186i
$334$			91358.9i			0.818951i
$335$			10272.8i			0.0915378i
$336$	−17199.4			−0.152348
$337$			178405.i			1.57090i	0.618926	+	0.785450i	$0.287568\pi$
$337$			178405.i			1.57090i	−0.618926	+	0.785450i	$0.712432\pi$
$338$		−	124511.i		−	1.08987i
$339$			64022.5i			0.557100i
$340$	−98720.7			−0.853985
$341$	15122.4			0.130050
$342$			3360.18i			0.0287283i
$343$	110013.			0.935094
$344$	−69544.0			−0.587682
$345$			88412.9i			0.742809i
$346$	99733.3			0.833083
$347$			30133.6i			0.250260i	0.992140	+	0.125130i	$0.0399348\pi$
$347$			30133.6i			0.250260i	−0.992140	+	0.125130i	$0.960065\pi$
$348$	−52187.4			−0.430930
$349$		−	190810.i		−	1.56657i	−0.621660	−	0.783287i	$-0.713541\pi$
$349$		−	190810.i		−	1.56657i	0.621660	−	0.783287i	$-0.286459\pi$
$350$		−	8537.38i		−	0.0696929i
$351$		−	37797.3i		−	0.306794i
$352$	−6200.67			−0.0500441
$353$			188128.i			1.50974i	0.655872	+	0.754872i	$0.272301\pi$
$353$			188128.i			1.50974i	−0.655872	+	0.754872i	$0.727699\pi$
$354$	−44578.8	+	25101.5i	−0.355731	+	0.200306i
$355$	−51262.0			−0.406761
$356$			23445.9i			0.184998i
$357$	−126861.			−0.995384
$358$	−66146.4			−0.516108
$359$	122754.			0.952462			0.476231	−	0.879320i	$-0.342003\pi$
$359$	122754.			0.952462			0.476231	+	0.879320i	$0.342003\pi$
$360$		−	15970.7i		−	0.123231i
$361$	−128385.			−0.985144
$362$			66831.7i			0.509995i
$363$	69980.0			0.531081
$364$		−	111470.i		−	0.841308i
$365$			11638.1i			0.0873568i
$366$	−4733.92			−0.0353393
$367$			127882.i			0.949459i	0.880132	+	0.474730i	$0.157454\pi$
$367$			127882.i			0.949459i	−0.880132	+	0.474730i	$0.842546\pi$
$368$			41657.0i			0.307604i
$369$	64927.7			0.476845
$370$	59271.1			0.432952
$371$	−216559.			−1.57336
$372$			18351.8i			0.132615i
$373$	104165.			0.748692			0.374346	−	0.927289i	$-0.377867\pi$
$373$	104165.			0.748692			0.374346	+	0.927289i	$0.377867\pi$
$374$	−45735.3			−0.326970
$375$	−76968.5			−0.547332
$376$	17781.4			0.125774
$377$		−	338228.i		−	2.37972i
$378$		−	20523.1i		−	0.143635i
$379$	43450.2			0.302492			0.151246	−	0.988496i	$-0.451671\pi$
$379$	43450.2			0.302492			0.151246	+	0.988496i	$0.451671\pi$
$380$	−9201.72			−0.0637238
$381$	−56948.8			−0.392315
$382$	195895.			1.34244
$383$	67869.2			0.462674			0.231337	−	0.972874i	$-0.425690\pi$
$383$	67869.2			0.462674			0.231337	+	0.972874i	$0.425690\pi$
$384$		−	7524.83i		−	0.0510310i
$385$		−	46311.7i		−	0.312442i
$386$		−	135456.i		−	0.909123i
$387$		−	82982.8i		−	0.554072i
$388$			18323.4i			0.121715i
$389$	171549.			1.13368			0.566838	−	0.823829i	$-0.308167\pi$
$389$	171549.			1.13368			0.566838	+	0.823829i	$0.308167\pi$
$390$	103506.			0.680516
$391$			307257.i			2.00978i
$392$		−	6197.29i		−	0.0403302i
$393$		−	47031.4i		−	0.304511i
$394$		−	120519.i		−	0.776357i
$395$	71491.2			0.458204
$396$		−	7398.90i		−	0.0471820i
$397$		−	276684.i		−	1.75551i	−0.479109	−	0.877755i	$-0.659040\pi$
$397$		−	276684.i		−	1.75551i	0.479109	−	0.877755i	$-0.340960\pi$
$398$		−	117886.i		−	0.744212i
$399$	−11824.6			−0.0742749
$400$	3735.14			0.0233446
$401$			90247.1i			0.561235i	0.959820	+	0.280617i	$0.0905392\pi$
$401$			90247.1i			0.561235i	−0.959820	+	0.280617i	$0.909461\pi$
$402$	−5775.53			−0.0357388
$403$	−118938.			−0.732338
$404$			89102.3i			0.545916i
$405$	19056.9			0.116183
$406$		−	183650.i		−	1.11414i
$407$	27459.1			0.165767
$408$		−	55502.1i		−	0.333418i
$409$			189869.i			1.13503i	0.823364	+	0.567514i	$0.192095\pi$
$409$			189869.i			1.13503i	−0.823364	+	0.567514i	$0.807905\pi$
$410$			177802.i			1.05772i
$411$	−15868.7			−0.0939415
$412$			25160.0i			0.148223i
$413$	−88333.4	−	156875.i	−0.517875	−	0.919715i
$414$	−49706.9			−0.290012
$415$			243540.i			1.41408i
$416$	48768.6			0.281808
$417$	119413.			0.686722
$418$	−4262.97			−0.0243983
$419$			163874.i			0.933431i	0.884407	+	0.466716i	$0.154563\pi$
$419$			163874.i			0.933431i	−0.884407	+	0.466716i	$0.845437\pi$
$420$	56201.7			0.318604
$421$		−	90195.4i		−	0.508885i	−0.967088	−	0.254443i	$-0.918108\pi$
$421$		−	90195.4i		−	0.508885i	0.967088	−	0.254443i	$-0.0818920\pi$
$422$	119319.			0.670014
$423$			21217.5i			0.118580i
$424$		−	94745.4i		−	0.527019i
$425$	27549.9			0.152525
$426$		−	28820.2i		−	0.158810i
$427$		−	16658.9i		−	0.0913673i
$428$	21818.5			0.119107
$429$	47952.4			0.260553
$430$	227245.			1.22902
$431$			329019.i			1.77119i	0.464454	+	0.885597i	$0.346251\pi$
$431$			329019.i			1.77119i	−0.464454	+	0.885597i	$0.653749\pi$
$432$	8978.95			0.0481125
$433$	−261055.			−1.39238			−0.696188	−	0.717860i	$-0.745122\pi$
$433$	−261055.			−1.39238			−0.696188	+	0.717860i	$0.745122\pi$
$434$	−64580.9			−0.342866
$435$	170530.			0.901202
$436$		−	134782.i		−	0.709020i
$437$			28639.3i			0.149968i
$438$	−6543.10			−0.0341064
$439$	−270625.			−1.40423			−0.702116	−	0.712063i	$-0.747761\pi$
$439$	−270625.			−1.40423			−0.702116	+	0.712063i	$0.747761\pi$
$440$	20261.6			0.104657
$441$	7394.88			0.0380236
$442$	359711.			1.84123
$443$		−	193450.i		−	0.985737i	−0.870104	−	0.492868i	$-0.835948\pi$
$443$		−	193450.i		−	0.985737i	0.870104	−	0.492868i	$-0.164052\pi$
$444$			33323.0i			0.169036i
$445$		−	76612.8i		−	0.386885i
$446$		−	178586.i		−	0.897798i
$447$			10607.8i			0.0530898i
$448$	26480.3			0.131937
$449$	165610.			0.821473			0.410737	−	0.911754i	$-0.365272\pi$
$449$	165610.			0.821473			0.410737	+	0.911754i	$0.365272\pi$
$450$			4456.93i			0.0220095i
$451$			82372.0i			0.404973i
$452$		−	98569.1i		−	0.482463i
$453$		−	59213.1i		−	0.288550i
$454$	7696.39			0.0373401
$455$			364245.i			1.75942i
$456$		−	5173.33i		−	0.0248794i
$457$			286984.i			1.37412i	0.726599	+	0.687061i	$0.241100\pi$
$457$			286984.i			1.37412i	−0.726599	+	0.687061i	$0.758900\pi$
$458$	−19613.4			−0.0935024
$459$	66227.5			0.314350
$460$		−	136121.i		−	0.643292i
$461$	8531.29			0.0401433			0.0200717	−	0.999799i	$-0.493611\pi$
$461$	8531.29			0.0401433			0.0200717	+	0.999799i	$0.493611\pi$
$462$	26037.1			0.121986
$463$			333548.i			1.55595i	0.628292	+	0.777977i	$0.283754\pi$
$463$			333548.i			1.55595i	−0.628292	+	0.777977i	$0.716246\pi$
$464$	80347.7			0.373197
$465$		−	59967.2i		−	0.277337i
$466$	−179732.			−0.827664
$467$			37826.1i			0.173443i	0.996233	+	0.0867217i	$0.0276391\pi$
$467$			37826.1i			0.173443i	−0.996233	+	0.0867217i	$0.972361\pi$
$468$			58192.8i			0.265691i
$469$		−	20324.4i		−	0.0923999i
$470$	−58103.2			−0.263030
$471$		−	128689.i		−	0.580094i
$472$	68633.5	−	38646.3i	0.308072	−	0.173470i
$473$	105278.			0.470561
$474$			40193.4i			0.178895i
$475$	2567.92			0.0113813
$476$	195315.			0.862028
$477$	113054.			0.496878
$478$		−	74731.4i		−	0.327075i
$479$	−394038.			−1.71738			−0.858692	−	0.512492i	$-0.828723\pi$
$479$	−394038.			−1.71738			−0.858692	+	0.512492i	$0.828723\pi$
$480$			24588.5i			0.106721i
$481$	−215967.			−0.933464
$482$		−	287046.i		−	1.23554i
$483$		−	174921.i		−	0.749805i
$484$	−107741.			−0.459929
$485$		−	59874.5i		−	0.254541i
$486$			10714.1i			0.0453609i
$487$	395509.			1.66762			0.833812	−	0.552048i	$-0.186154\pi$
$487$	395509.			1.66762			0.833812	+	0.552048i	$0.186154\pi$
$488$	7288.34			0.0306048
$489$	−64292.7			−0.268871
$490$			20250.6i			0.0843423i
$491$	−204960.			−0.850172			−0.425086	−	0.905153i	$-0.639756\pi$
$491$	−204960.			−0.850172			−0.425086	+	0.905153i	$0.639756\pi$
$492$	−99962.7			−0.412960
$493$	592634.			2.43833
$494$	33528.5			0.137392
$495$			24177.0i			0.0986716i
$496$		−	28254.4i		−	0.114848i
$497$	101420.			0.410592
$498$	−136921.			−0.552093
$499$	−161651.			−0.649199			−0.324599	−	0.945852i	$-0.605229\pi$
$499$	−161651.			−0.649199			−0.324599	+	0.945852i	$0.605229\pi$
$500$	118501.			0.474003
$501$	167837.			0.668671
$502$			111743.i			0.443416i
$503$			443586.i			1.75324i	0.481183	+	0.876620i	$0.340207\pi$
$503$			443586.i			1.75324i	−0.481183	+	0.876620i	$0.659793\pi$
$504$			31597.4i			0.124391i
$505$		−	291155.i		−	1.14167i
$506$		−	63061.9i		−	0.246301i
$507$	−228742.			−0.889875
$508$	87678.4			0.339755
$509$			47300.9i			0.182572i	0.995825	+	0.0912860i	$0.0290977\pi$
$509$			47300.9i			0.182572i	−0.995825	+	0.0912860i	$0.970902\pi$
$510$			181361.i			0.697276i
$511$		−	23025.5i		−	0.0881795i
$512$			11585.2i			0.0441942i
$513$	6173.04			0.0234566
$514$		−	318333.i		−	1.20491i
$515$		−	82214.0i		−	0.309978i
$516$			127760.i			0.479841i
$517$	−26918.0			−0.100708
$518$	−117265.			−0.437029
$519$		−	183222.i		−	0.680209i
$520$	−159359.			−0.589344
$521$	209712.			0.772587			0.386293	−	0.922376i	$-0.373755\pi$
$521$	209712.			0.772587			0.386293	+	0.922376i	$0.373755\pi$
$522$			95874.3i			0.351853i
$523$	−52941.7			−0.193550			−0.0967752	−	0.995306i	$-0.530853\pi$
$523$	−52941.7			−0.193550			−0.0967752	+	0.995306i	$0.530853\pi$
$524$			72409.5i			0.263714i
$525$	−15684.2			−0.0569040
$526$		−	70600.7i		−	0.255175i
$527$		−	208401.i		−	0.750374i
$528$			11391.4i			0.0408609i
$529$	−143818.			−0.513928
$530$			309595.i			1.10215i
$531$	46114.4	+	81896.4i	0.163549	+	0.290453i
$532$	18205.2			0.0643240
$533$		−	647861.i		−	2.28048i
$534$	43072.8			0.151050
$535$	−71295.1			−0.249087
$536$	8892.00			0.0309507
$537$			121519.i			0.421400i
$538$	−285273.			−0.985590
$539$			9381.68i			0.0322926i
$540$	−29340.1			−0.100617
$541$			497053.i			1.69828i	0.528171	+	0.849138i	$0.322878\pi$
$541$			497053.i			1.69828i	−0.528171	+	0.849138i	$0.677122\pi$
$542$			125690.i			0.427859i
$543$	122778.			0.416409
$544$			85451.1i			0.288749i
$545$			440420.i			1.48277i
$546$	−204783.			−0.686925
$547$	−50557.5			−0.168971			−0.0844853	−	0.996425i	$-0.526925\pi$
$547$	−50557.5			−0.168971			−0.0844853	+	0.996425i	$0.526925\pi$
$548$	24431.5			0.0813558
$549$			8696.76i			0.0288545i
$550$	−5654.39			−0.0186922
$551$	55239.2			0.181947
$552$	76528.9			0.251158
$553$	−141443.			−0.462519
$554$		−	95746.7i		−	0.311964i
$555$		−	108888.i		−	0.353503i
$556$	−183849.			−0.594719
$557$	306221.			0.987018			0.493509	−	0.869741i	$-0.335714\pi$
$557$	306221.			0.987018			0.493509	+	0.869741i	$0.335714\pi$
$558$	33714.4			0.108280
$559$	−828018.			−2.64982
$560$	−86528.2			−0.275919
$561$			84021.1i			0.266970i
$562$			195112.i			0.617748i
$563$		−	203946.i		−	0.643426i	−0.946837	−	0.321713i	$-0.895741\pi$
$563$		−	203946.i		−	0.643426i	0.946837	−	0.321713i	$-0.104259\pi$
$564$		−	32666.5i		−	0.102694i
$565$			322089.i			1.00897i
$566$	238144.			0.743374
$567$	−37703.4			−0.117277
$568$			44371.6i			0.137534i
$569$		−	490032.i		−	1.51356i	−0.653669	−	0.756781i	$-0.726771\pi$
$569$		−	490032.i		−	1.51356i	0.653669	−	0.756781i	$-0.273229\pi$
$570$			16904.6i			0.0520303i
$571$		−	231627.i		−	0.710422i	−0.934786	−	0.355211i	$-0.884409\pi$
$571$		−	231627.i		−	0.710422i	0.934786	−	0.355211i	$-0.115591\pi$
$572$	−73827.6			−0.225646
$573$		−	359882.i		−	1.09610i
$574$		−	351774.i		−	1.06768i
$575$			37987.0i			0.114895i
$576$	−13824.0			−0.0416667
$577$	−361495.			−1.08580			−0.542901	−	0.839797i	$-0.682674\pi$
$577$	−361495.			−1.08580			−0.542901	+	0.839797i	$0.682674\pi$
$578$			394043.i			1.17947i
$579$	−248848.			−0.742296
$580$	−262548.			−0.780464
$581$		−	481833.i		−	1.42740i
$582$	33662.3			0.0993796
$583$			143429.i			0.421987i
$584$	10073.8			0.0295370
$585$		−	190154.i		−	0.555639i
$586$		−	371658.i		−	1.08230i
$587$			197018.i			0.571780i	0.958262	+	0.285890i	$0.0922892\pi$
$587$			197018.i			0.571780i	−0.958262	+	0.285890i	$0.907711\pi$
$588$	−11385.2			−0.0329294
$589$		−	19425.0i		−	0.0559924i
$590$	−224270.	+	126282.i	−0.644269	+	0.362776i
$591$	−221407.			−0.633893
$592$		−	51304.1i		−	0.146389i
$593$	44722.4			0.127179			0.0635895	−	0.997976i	$-0.479745\pi$
$593$	44722.4			0.127179			0.0635895	+	0.997976i	$0.479745\pi$
$594$	−13592.6			−0.0385240
$595$	−638220.			−1.80276
$596$		−	16331.8i		−	0.0459772i
$597$	−216571.			−0.607646
$598$			495985.i			1.38697i
$599$	217424.			0.605973			0.302987	−	0.952995i	$-0.402016\pi$
$599$	217424.			0.605973			0.302987	+	0.952995i	$0.402016\pi$
$600$		−	6861.90i		−	0.0190608i
$601$		−	200834.i		−	0.556018i	−0.960578	−	0.278009i	$-0.910325\pi$
$601$		−	200834.i		−	0.556018i	0.960578	−	0.278009i	$-0.0896745\pi$
$602$	−449595.			−1.24059
$603$			10610.3i			0.0291806i
$604$			91164.5i			0.249892i
$605$	352060.			0.961848
$606$	163691.			0.445739
$607$	−51921.8			−0.140920			−0.0704600	−	0.997515i	$-0.522447\pi$
$607$	−51921.8			−0.140920			−0.0704600	+	0.997515i	$0.522447\pi$
$608$			7964.87i			0.0215462i
$609$	−337387.			−0.909690
$610$	−23815.7			−0.0640036
$611$	211712.			0.567105
$612$	−101964.			−0.272235
$613$		−	358505.i		−	0.954056i	−0.878888	−	0.477028i	$-0.841714\pi$
$613$		−	358505.i		−	0.954056i	0.878888	−	0.477028i	$-0.158286\pi$
$614$			115803.i			0.307173i
$615$	326643.			0.863621
$616$	−40086.8			−0.105643
$617$	242674.			0.637459			0.318730	−	0.947846i	$-0.396744\pi$
$617$	242674.			0.637459			0.318730	+	0.947846i	$0.396744\pi$
$618$	46221.9			0.121024
$619$	−259624.			−0.677584			−0.338792	−	0.940861i	$-0.610018\pi$
$619$	−259624.			−0.677584			−0.338792	+	0.940861i	$0.610018\pi$
$620$			92325.5i			0.240181i
$621$			91317.5i			0.236794i
$622$		−	533595.i		−	1.37921i
$623$			151575.i			0.390528i
$624$		−	89593.7i		−	0.230095i
$625$	−423695.			−1.08466
$626$	406656.			1.03772
$627$			7831.57i			0.0199211i
$628$			198129.i			0.502376i
$629$		−	378412.i		−	0.956454i
$630$		−	103249.i		−	0.260139i
$631$	−421676.			−1.05906			−0.529529	−	0.848292i	$-0.677631\pi$
$631$	−421676.			−1.05906			−0.529529	+	0.848292i	$0.677631\pi$
$632$		−	61881.7i		−	0.154927i
$633$		−	219203.i		−	0.547064i
$634$			338422.i			0.841937i
$635$	−286502.			−0.710527
$636$	−174058.			−0.430309
$637$		−	73787.5i		−	0.181846i
$638$	−121633.			−0.298821
$639$	−52946.1			−0.129668
$640$		−	37856.5i		−	0.0924230i
$641$	−486873.			−1.18495			−0.592475	−	0.805589i	$-0.701849\pi$
$641$	−486873.			−1.18495			−0.592475	+	0.805589i	$0.701849\pi$
$642$		−	40083.1i		−	0.0972503i
$643$	69165.6			0.167289			0.0836446	−	0.996496i	$-0.473344\pi$
$643$	69165.6			0.167289			0.0836446	+	0.996496i	$0.473344\pi$
$644$			269309.i			0.649350i
$645$		−	417476.i		−	1.00349i
$646$			58747.8i			0.140775i
$647$	377718.			0.902318			0.451159	−	0.892444i	$-0.351011\pi$
$647$	377718.			0.902318			0.451159	+	0.892444i	$0.351011\pi$
$648$		−	16495.4i		−	0.0392837i
$649$	−103900.	+	58504.1i	−0.246675	+	0.138898i
$650$	44472.1			0.105259
$651$			118643.i			0.279949i
$652$	98985.1			0.232849
$653$	−74227.6			−0.174076			−0.0870380	−	0.996205i	$-0.527740\pi$
$653$	−74227.6			−0.174076			−0.0870380	+	0.996205i	$0.527740\pi$
$654$	−247610.			−0.578913
$655$		−	236609.i		−	0.551504i
$656$	153903.			0.357634
$657$			12020.4i			0.0278477i
$658$	114955.			0.265507
$659$			764602.i			1.76062i	0.474403	+	0.880308i	$0.342664\pi$
$659$			764602.i			1.76062i	−0.474403	+	0.880308i	$0.657336\pi$
$660$		−	37222.9i		−	0.0854521i
$661$	279344.			0.639347			0.319674	−	0.947528i	$-0.396427\pi$
$661$	279344.			0.639347			0.319674	+	0.947528i	$0.396427\pi$
$662$			248410.i			0.566830i
$663$		−	660831.i		−	1.50336i
$664$	210804.			0.478127
$665$	−59488.3			−0.134520
$666$	61218.3			0.138017
$667$			817150.i			1.83675i
$668$	−258402.			−0.579086
$669$	−328084.			−0.733049
$670$	−29055.9			−0.0647270
$671$	−11033.3			−0.0245054
$672$		−	48647.4i		−	0.107726i
$673$		−	820958.i		−	1.81255i	−0.422684	−	0.906277i	$-0.638912\pi$
$673$		−	820958.i		−	1.81255i	0.422684	−	0.906277i	$-0.361088\pi$
$674$	−504607.			−1.11079
$675$	8187.91			0.0179707
$676$	352171.			0.770654
$677$	205394.			0.448136			0.224068	−	0.974574i	$-0.428066\pi$
$677$	205394.			0.448136			0.224068	+	0.974574i	$0.428066\pi$
$678$	−181083.			−0.393929
$679$			118459.i			0.256939i
$680$		−	279224.i		−	0.603859i
$681$		−	14139.2i		−	0.0304881i
$682$			42772.5i			0.0919594i
$683$			807368.i			1.73073i	0.501140	+	0.865366i	$0.332914\pi$
$683$			807368.i			1.73073i	−0.501140	+	0.865366i	$0.667086\pi$
$684$	−9504.02			−0.0203140
$685$	−79833.4			−0.170139
$686$			311163.i			0.661211i
$687$			36032.2i			0.0763444i
$688$		−	196700.i		−	0.415554i
$689$		−	1.12808e6i		−	2.37629i
$690$	−250069.			−0.525245
$691$		−	108519.i		−	0.227273i	−0.993522	−	0.113637i	$-0.963750\pi$
$691$		−	108519.i		−	0.227273i	0.993522	−	0.113637i	$-0.0362499\pi$
$692$			282088.i			0.589078i
$693$		−	47833.2i		−	0.0996009i
$694$	−85230.6			−0.176961
$695$	600754.			1.24373
$696$		−	147608.i		−	0.304714i
$697$	1.13517e6			2.33665
$698$	539693.			1.10774
$699$			330189.i			0.675785i
$700$	24147.4			0.0492803
$701$			607738.i			1.23674i	0.785885	+	0.618372i	$0.212208\pi$
$701$			607738.i			1.23674i	−0.785885	+	0.618372i	$0.787792\pi$
$702$	106907.			0.216936
$703$		−	35271.7i		−	0.0713700i
$704$		−	17538.1i		−	0.0353865i
$705$			106742.i			0.214763i
$706$	−532105.			−1.06755
$707$			576038.i			1.15242i
$708$	−70997.7	−	126088.i	−0.141637	−	0.251540i
$709$	428441.			0.852313			0.426156	−	0.904650i	$-0.359867\pi$
$709$	428441.			0.852313			0.426156	+	0.904650i	$0.359867\pi$
$710$		−	144991.i		−	0.287623i
$711$	73839.9			0.146067
$712$	−66314.9			−0.130813
$713$	287352.			0.565244
$714$		−	358816.i		−	0.703843i
$715$	241243.			0.471891
$716$		−	187090.i		−	0.364943i
$717$	−137290.			−0.267056
$718$			347201.i			0.673492i
$719$			620288.i			1.19987i	0.800047	+	0.599937i	$0.204808\pi$
$719$			620288.i			1.19987i	−0.800047	+	0.599937i	$0.795192\pi$
$720$	45172.0			0.0871373
$721$			162657.i			0.312898i
$722$		−	363128.i		−	0.696602i
$723$	−527337.			−1.00881
$724$	−189029.			−0.360621
$725$	73269.1			0.139394
$726$			197933.i			0.375531i
$727$	−32479.1			−0.0614519			−0.0307259	−	0.999528i	$-0.509782\pi$
$727$	−32479.1			−0.0614519			−0.0307259	+	0.999528i	$0.509782\pi$
$728$	315285.			0.594895
$729$	19683.0			0.0370370
$730$	−32917.5			−0.0617706
$731$		−	1.45083e6i		−	2.71508i
$732$		−	13389.5i		−	0.0249887i
$733$	83129.9			0.154721			0.0773605	−	0.997003i	$-0.475351\pi$
$733$	83129.9			0.154721			0.0773605	+	0.997003i	$0.475351\pi$
$734$	−361704.			−0.671369
$735$	37202.7			0.0688652
$736$	−117824.			−0.217509
$737$	−13461.0			−0.0247824
$738$			183643.i			0.337180i
$739$			781734.i			1.43143i	0.698393	+	0.715715i	$0.253899\pi$
$739$			781734.i			1.43143i	−0.698393	+	0.715715i	$0.746101\pi$
$740$			167644.i			0.306143i
$741$		−	61595.8i		−	0.112180i
$742$		−	612520.i		−	1.11253i
$743$	60485.0			0.109565			0.0547823	−	0.998498i	$-0.482554\pi$
$743$	60485.0			0.109565			0.0547823	+	0.998498i	$0.482554\pi$
$744$	−51906.7			−0.0937729
$745$			53366.6i			0.0961518i
$746$			294622.i			0.529405i
$747$			251541.i			0.450782i
$748$		−	129359.i		−	0.231203i
$749$	141054.			0.251433
$750$		−	217700.i		−	0.387022i
$751$		−	336022.i		−	0.595783i	−0.954600	−	0.297891i	$-0.903717\pi$
$751$		−	336022.i		−	0.595783i	0.954600	−	0.297891i	$-0.0962833\pi$
$752$			50293.3i			0.0889353i
$753$	205284.			0.362048
$754$	956652.			1.68272
$755$		−	297894.i		−	0.522597i
$756$	58048.1			0.101565
$757$	−486133.			−0.848327			−0.424163	−	0.905586i	$-0.639432\pi$
$757$	−486133.			−0.848327			−0.424163	+	0.905586i	$0.639432\pi$
$758$			122896.i			0.213894i
$759$	−115852.			−0.201104
$760$		−	26026.4i		−	0.0450595i
$761$	729172.			1.25910			0.629551	−	0.776959i	$-0.283239\pi$
$761$	729172.			1.25910			0.629551	+	0.776959i	$0.283239\pi$
$762$		−	161076.i		−	0.277408i
$763$		−	871353.i		−	1.49674i
$764$			554074.i			0.949252i
$765$	333182.			0.569323
$766$			191963.i			0.327160i
$767$	817178.	−	460138.i	1.38908	−	0.782163i
$768$	21283.4			0.0360844
$769$			75567.2i			0.127785i	0.997957	+	0.0638926i	$0.0203515\pi$
$769$			75567.2i			0.127785i	−0.997957	+	0.0638926i	$0.979648\pi$
$770$	130989.			0.220930
$771$	−584815.			−0.983807
$772$	383127.			0.642847
$773$		−	1.08985e6i		−	1.82393i	−0.410268	−	0.911965i	$-0.634565\pi$
$773$		−	1.08985e6i		−	1.82393i	0.410268	−	0.911965i	$-0.365435\pi$
$774$	234711.			0.391788
$775$		−	25765.2i		−	0.0428973i
$776$	−51826.4			−0.0860653
$777$			215430.i			0.356833i
$778$			485214.i			0.801630i
$779$	105808.			0.174359
$780$			292761.i			0.481198i
$781$		−	67171.3i		−	0.110124i
$782$	−869053.			−1.42113
$783$	176132.			0.287287
$784$	17528.6			0.0285177
$785$		−	647416.i		−	1.05062i
$786$	133025.			0.215322
$787$	338449.			0.546442			0.273221	−	0.961951i	$-0.411911\pi$
$787$	338449.			0.546442			0.273221	+	0.961951i	$0.411911\pi$
$788$	340878.			0.548967
$789$	−129702.			−0.208349
$790$			202208.i			0.323999i
$791$		−	637241.i		−	1.01848i
$792$	20927.2			0.0333627
$793$	86777.9			0.137995
$794$	782581.			1.24133
$795$	568761.			0.899903
$796$	333432.			0.526237
$797$			965801.i			1.52045i	0.649662	+	0.760223i	$0.274910\pi$
$797$			965801.i			1.52045i	−0.649662	+	0.760223i	$0.725090\pi$
$798$		−	33445.1i		−	0.0525203i
$799$			370956.i			0.581071i
$800$			10564.6i			0.0165072i
$801$		−	79129.8i		−	0.123332i
$802$	−255257.			−0.396853
$803$	−15250.0			−0.0236504
$804$		−	16335.7i		−	0.0252711i
$805$		−	880007.i		−	1.35798i
$806$		−	336408.i		−	0.517841i
$807$			524080.i			0.804731i
$808$	−252019.			−0.386021
$809$			1.10763e6i			1.69238i	0.532880	+	0.846191i	$0.321110\pi$
$809$			1.10763e6i			1.69238i	−0.532880	+	0.846191i	$0.678890\pi$
$810$			53901.1i			0.0821538i
$811$			351312.i			0.534135i	0.963678	+	0.267068i	$0.0860547\pi$
$811$			351312.i			0.534135i	−0.963678	+	0.267068i	$0.913945\pi$
$812$	519441.			0.787815
$813$	230907.			0.349346
$814$			77666.0i			0.117215i
$815$	−323449.			−0.486956
$816$	156984.			0.235762
$817$		−	135232.i		−	0.202597i
$818$	−537029.			−0.802586
$819$			376211.i			0.560872i
$820$	−502900.			−0.747918
$821$			244956.i			0.363414i	0.983353	+	0.181707i	$0.0581622\pi$
$821$			244956.i			0.363414i	−0.983353	+	0.181707i	$0.941838\pi$
$822$		−	44883.5i		−	0.0664267i
$823$		−	425374.i		−	0.628017i	−0.949420	−	0.314008i	$-0.898328\pi$
$823$		−	425374.i		−	0.628017i	0.949420	−	0.314008i	$-0.101672\pi$
$824$	−71163.2			−0.104810
$825$			10387.8i			0.0152621i
$826$	443709.	−	249845.i	0.650337	−	0.366193i
$827$	−1.04760e6			−1.53174			−0.765868	−	0.642997i	$-0.777691\pi$
$827$	−1.04760e6			−1.53174			−0.765868	+	0.642997i	$0.777691\pi$
$828$		−	140592.i		−	0.205070i
$829$	−601444.			−0.875158			−0.437579	−	0.899180i	$-0.644164\pi$
$829$	−601444.			−0.875158			−0.437579	+	0.899180i	$0.644164\pi$
$830$	−688834.			−0.999904
$831$	−175898.			−0.254718
$832$			137938.i			0.199269i
$833$	129289.			0.186325
$834$			337752.i			0.485586i
$835$	844367.			1.21104
$836$		−	12057.5i		−	0.0172522i
$837$		−	61937.3i		−	0.0884099i
$838$	−463506.			−0.660035
$839$		−	166605.i		−	0.236681i	−0.992973	−	0.118341i	$-0.962243\pi$
$839$		−	166605.i		−	0.236681i	0.992973	−	0.118341i	$-0.0377575\pi$
$840$			158962.i			0.225287i
$841$	868832.			1.22841
$842$	255111.			0.359836
$843$	358444.			0.504389
$844$			337485.i			0.473772i
$845$	−1.15077e6			−1.61167
$846$	−60012.1			−0.0838490
$847$	−696537.			−0.970907
$848$	267980.			0.372659
$849$		−	437499.i		−	0.606962i
$850$			77922.9i			0.107852i
$851$	521772.			0.720480
$852$	81515.9			0.112296
$853$	−1.10373e6			−1.51692			−0.758462	−	0.651717i	$-0.774049\pi$
$853$	−1.10373e6			−1.51692			−0.758462	+	0.651717i	$0.774049\pi$
$854$	47118.5			0.0646064
$855$	31055.8			0.0424825
$856$			61711.9i			0.0842212i
$857$			797165.i			1.08539i	0.839929	+	0.542696i	$0.182596\pi$
$857$			797165.i			1.08539i	−0.839929	+	0.542696i	$0.817404\pi$
$858$			135630.i			0.184239i
$859$			174411.i			0.236368i	0.992992	+	0.118184i	$0.0377072\pi$
$859$			174411.i			0.236368i	−0.992992	+	0.118184i	$0.962293\pi$
$860$			642747.i			0.869046i
$861$	−646250.			−0.871755
$862$	−930606.			−1.25242
$863$		−	129046.i		−	0.173270i	−0.996240	−	0.0866351i	$-0.972389\pi$
$863$		−	129046.i		−	0.173270i	0.996240	−	0.0866351i	$-0.0276114\pi$
$864$			25396.3i			0.0340207i
$865$		−	921766.i		−	1.23194i
$866$		−	738375.i		−	0.984558i
$867$	723903.			0.963035
$868$		−	182662.i		−	0.242443i
$869$			93678.7i			0.124051i
$870$			482332.i			0.637246i
$871$	105872.			0.139554
$872$	381221.			0.501353
$873$		−	61841.5i		−	0.0811431i
$874$	−81004.1			−0.106043
$875$	766097.			1.00062
$876$		−	18506.7i		−	0.0241169i
$877$	1.41859e6			1.84441			0.922203	−	0.386707i	$-0.126388\pi$
$877$	1.41859e6			1.84441			0.922203	+	0.386707i	$0.126388\pi$
$878$		−	765443.i		−	0.992941i
$879$	−682780.			−0.883696
$880$			57308.4i			0.0740037i
$881$			652835.i			0.841108i	0.907267	+	0.420554i	$0.138164\pi$
$881$			652835.i			0.841108i	−0.907267	+	0.420554i	$0.861836\pi$
$882$			20915.9i			0.0268868i
$883$	1.04498e6			1.34026			0.670129	−	0.742245i	$-0.266239\pi$
$883$	1.04498e6			1.34026			0.670129	+	0.742245i	$0.266239\pi$
$884$			1.01742e6i			1.30195i
$885$	231996.	+	412010.i	0.296206	+	0.526044i
$886$	547159.			0.697021
$887$		−	636188.i		−	0.808608i	−0.914625	−	0.404304i	$-0.867514\pi$
$887$		−	636188.i		−	0.808608i	0.914625	−	0.404304i	$-0.132486\pi$
$888$	−94251.7			−0.119526
$889$	566833.			0.717219
$890$	216694.			0.273569
$891$			24971.3i			0.0314547i
$892$	505118.			0.634839
$893$			34576.7i			0.0433591i
$894$	−30003.5			−0.0375402
$895$			611345.i			0.763204i
$896$			74897.5i			0.0932935i
$897$	911183.			1.13245
$898$			468415.i			0.580869i
$899$		−	554243.i		−	0.685773i
$900$	−12606.1			−0.0155631
$901$	1.97659e6			2.43482
$902$	−232983.			−0.286359
$903$			825960.i			1.01294i
$904$	278796.			0.341153
$905$	617679.			0.754164
$906$	167480.			0.204036
$907$	−365131.			−0.443847			−0.221924	−	0.975064i	$-0.571234\pi$
$907$	−365131.			−0.443847			−0.221924	+	0.975064i	$0.571234\pi$
$908$			21768.7i			0.0264034i
$909$		−	300720.i		−	0.363944i
$910$	−1.03024e6			−1.24410
$911$	−494618.			−0.595982			−0.297991	−	0.954569i	$-0.596317\pi$
$911$	−494618.			−0.595982			−0.297991	+	0.954569i	$0.596317\pi$
$912$	14632.4			0.0175924
$913$	−319123.			−0.382839
$914$	−811714.			−0.971651
$915$			43752.3i			0.0522587i
$916$		−	55475.2i		−	0.0661161i
$917$			468121.i			0.556698i
$918$			187320.i			0.222279i
$919$			17914.2i			0.0212113i	0.999944	+	0.0106056i	$0.00337594\pi$
$919$			17914.2i			0.0212113i	−0.999944	+	0.0106056i	$0.996624\pi$
$920$	385007.			0.454876
$921$	212744.			0.250806
$922$			24130.1i			0.0283856i
$923$			528306.i			0.620129i
$924$			73644.1i			0.0862569i
$925$		−	46784.3i		−	0.0546785i
$926$	−943418.			−1.10023
$927$		−	84915.0i		−	0.0988155i
$928$			227258.i			0.263890i
$929$			578380.i			0.670165i	0.942189	+	0.335083i	$0.108764\pi$
$929$			578380.i			0.670165i	−0.942189	+	0.335083i	$0.891236\pi$
$930$	169613.			0.196107
$931$	12050.9			0.0139034
$932$		−	508359.i		−	0.585247i
$933$	−980276.			−1.12612
$934$	−106988.			−0.122643
$935$			422699.i			0.483513i
$936$	−164594.			−0.187872
$937$			46478.7i			0.0529389i	0.999650	+	0.0264694i	$0.00842647\pi$
$937$			46478.7i			0.0529389i	−0.999650	+	0.0264694i	$0.991574\pi$
$938$	57486.0			0.0653366
$939$		−	747075.i		−	0.847292i
$940$		−	164341.i		−	0.185990i
$941$			713989.i			0.806329i	0.915128	+	0.403164i	$0.132090\pi$
$941$			713989.i			0.806329i	−0.915128	+	0.403164i	$0.867910\pi$
$942$	363986.			0.410188
$943$			1.56522e6i			1.76016i
$944$	109308.	+	194125.i	0.122662	+	0.217840i
$945$	−189681.			−0.212403
$946$			297771.i			0.332737i
$947$	499095.			0.556523			0.278262	−	0.960505i	$-0.410242\pi$
$947$	499095.			0.556523			0.278262	+	0.960505i	$0.410242\pi$
$948$	−113684.			−0.126498
$949$	119942.			0.133180
$950$			7263.16i			0.00804783i
$951$	621720.			0.687439
$952$			552434.i			0.609546i
$953$	−823253.			−0.906458			−0.453229	−	0.891394i	$-0.649728\pi$
$953$	−823253.			−0.906458			−0.453229	+	0.891394i	$0.649728\pi$
$954$			319766.i			0.351346i
$955$		−	1.81052e6i		−	1.98517i
$956$	211372.			0.231277
$957$			223454.i			0.243986i
$958$		−	1.11451e6i		−	1.21437i
$959$	157947.			0.171741
$960$	−69546.8			−0.0754631
$961$	728621.			0.788959
$962$		−	610848.i		−	0.660059i
$963$	−73637.3			−0.0794045
$964$	811888.			0.873659
$965$	−1.25192e6			−1.34438
$966$	494752.			0.530192
$967$		−	1.36674e6i		−	1.46161i	−0.682584	−	0.730807i	$-0.739144\pi$
$967$		−	1.36674e6i		−	1.46161i	0.682584	−	0.730807i	$-0.260856\pi$
$968$		−	304738.i		−	0.325219i
$969$	107927.			0.114943
$970$	169351.			0.179988
$971$	−62894.8			−0.0667077			−0.0333539	−	0.999444i	$-0.510619\pi$
$971$	−62894.8			−0.0667077			−0.0333539	+	0.999444i	$0.510619\pi$
$972$	−30304.0			−0.0320750
$973$	−1.18857e6			−1.25545
$974$			1.11867e6i			1.17919i
$975$		−	81700.5i		−	0.0859439i
$976$			20614.5i			0.0216408i
$977$			1.36683e6i			1.43194i	0.698129	+	0.715972i	$0.254016\pi$
$977$			1.36683e6i			1.43194i	−0.698129	+	0.715972i	$0.745984\pi$
$978$		−	181847.i		−	0.190121i
$979$	100390.			0.104743
$980$	−57277.3			−0.0596390
$981$			454889.i			0.472680i
$982$		−	579715.i		−	0.601162i
$983$		−	1.47801e6i		−	1.52957i	−0.644285	−	0.764786i	$-0.722845\pi$
$983$		−	1.47801e6i		−	1.52957i	0.644285	−	0.764786i	$-0.277155\pi$
$984$		−	282737.i		−	0.292007i
$985$	−1.11387e6			−1.14805
$986$			1.67622e6i			1.72416i
$987$		−	211186.i		−	0.216785i
$988$			94832.9i			0.0971505i
$989$	2.00047e6			2.04522
$990$	−68382.9			−0.0697713
$991$			1.23654e6i			1.25911i	0.776958	+	0.629553i	$0.216762\pi$
$991$			1.23654e6i			1.25911i	−0.776958	+	0.629553i	$0.783238\pi$
$992$	79915.6			0.0812097
$993$	456358.			0.462815
$994$			286859.i			0.290332i
$995$	−1.08954e6			−1.10052
$996$		−	387272.i		−	0.390389i
$997$	564455.			0.567857			0.283928	−	0.958845i	$-0.408362\pi$
$997$	564455.			0.567857			0.283928	+	0.958845i	$0.408362\pi$
$998$		−	457218.i		−	0.459053i
$999$		−	112465.i		−	0.112690i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.5.d.a.235.20	yes	40
3.2	odd	2		1062.5.d.b.235.5		40
59.58	odd	2	inner	354.5.d.a.235.19	✓	40
177.176	even	2		1062.5.d.b.235.6		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.5.d.a.235.19	✓	40	59.58	odd	2	inner
354.5.d.a.235.20	yes	40	1.1	even	1	trivial
1062.5.d.b.235.5		40	3.2	odd	2
1062.5.d.b.235.6		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:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	354.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} +5.19615 q^{3} -8.00000 q^{4} +26.1412 q^{5} +14.6969i q^{6} -51.7193 q^{7} -22.6274i q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q+2.82843i q^{2} +5.19615 q^{3} -8.00000 q^{4} +26.1412 q^{5} +14.6969i q^{6} -51.7193 q^{7} -22.6274i q^{8} +27.0000 q^{9} +73.9384i q^{10} +34.2542i q^{11} -41.5692 q^{12} -269.411i q^{13} -146.284i q^{14} +135.834 q^{15} +64.0000 q^{16} +472.055 q^{17} +76.3675i q^{18} +44.0001 q^{19} -209.129 q^{20} -268.741 q^{21} -96.8854 q^{22} +650.891i q^{23} -117.576i q^{24} +58.3616 q^{25} +762.010 q^{26} +140.296 q^{27} +413.754 q^{28} +1255.43 q^{29} +384.195i q^{30} -441.475i q^{31} +181.019i q^{32} +177.990i q^{33} +1335.17i q^{34} -1352.00 q^{35} -216.000 q^{36} -801.627i q^{37} +124.451i q^{38} -1399.90i q^{39} -591.508i q^{40} +2404.73 q^{41} -760.115i q^{42} -3073.44i q^{43} -274.033i q^{44} +705.812 q^{45} -1841.00 q^{46} +785.833i q^{47} +332.554 q^{48} +273.884 q^{49} +165.072i q^{50} +2452.87 q^{51} +2155.29i q^{52} +4187.19 q^{53} +396.817i q^{54} +895.445i q^{55} +1170.27i q^{56} +228.631 q^{57} +3550.90i q^{58} +(1707.94 + 3033.20i) q^{59} -1086.67 q^{60} +322.102i q^{61} +1248.68 q^{62} -1396.42 q^{63} -512.000 q^{64} -7042.72i q^{65} -503.431 q^{66} +392.975i q^{67} -3776.44 q^{68} +3382.13i q^{69} -3824.04i q^{70} -1960.97 q^{71} -610.940i q^{72} +445.202i q^{73} +2267.34 q^{74} +303.256 q^{75} -352.001 q^{76} -1771.60i q^{77} +3959.52 q^{78} +2734.81 q^{79} +1673.04 q^{80} +729.000 q^{81} +6801.60i q^{82} +9316.32i q^{83} +2149.93 q^{84} +12340.1 q^{85} +8693.00 q^{86} +6523.42 q^{87} +775.083 q^{88} -2930.73i q^{89} +1996.34i q^{90} +13933.7i q^{91} -5207.13i q^{92} -2293.97i q^{93} -2222.67 q^{94} +1150.21 q^{95} +940.604i q^{96} -2290.43i q^{97} +774.662i q^{98} +924.862i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} - 80 q^{7} + 1080 q^{9}+O(q^{10}) \) 40 * q - 320 * q^4 - 80 * q^7 + 1080 * q^9	\( 40 q - 320 q^{4} - 80 q^{7} + 1080 q^{9} - 144 q^{15} + 2560 q^{16} + 480 q^{17} - 792 q^{19} - 1024 q^{22} + 3400 q^{25} + 768 q^{26} + 640 q^{28} + 1608 q^{29} - 5760 q^{35} - 8640 q^{36} + 6264 q^{41} + 7040 q^{46} + 17912 q^{49} + 1296 q^{51} - 1104 q^{53} + 5040 q^{57} + 13584 q^{59} + 1152 q^{60} - 12288 q^{62} - 2160 q^{63} - 20480 q^{64} + 1152 q^{66} - 3840 q^{68} + 35352 q^{71} + 4608 q^{74} + 3168 q^{75} + 6336 q^{76} - 12672 q^{78} - 15720 q^{79} + 29160 q^{81} - 26872 q^{85} + 18432 q^{86} + 7776 q^{87} + 8192 q^{88} - 18432 q^{94} - 19128 q^{95}+O(q^{100}) \) 40 * q - 320 * q^4 - 80 * q^7 + 1080 * q^9 - 144 * q^15 + 2560 * q^16 + 480 * q^17 - 792 * q^19 - 1024 * q^22 + 3400 * q^25 + 768 * q^26 + 640 * q^28 + 1608 * q^29 - 5760 * q^35 - 8640 * q^36 + 6264 * q^41 + 7040 * q^46 + 17912 * q^49 + 1296 * q^51 - 1104 * q^53 + 5040 * q^57 + 13584 * q^59 + 1152 * q^60 - 12288 * q^62 - 2160 * q^63 - 20480 * q^64 + 1152 * q^66 - 3840 * q^68 + 35352 * q^71 + 4608 * q^74 + 3168 * q^75 + 6336 * q^76 - 12672 * q^78 - 15720 * q^79 + 29160 * q^81 - 26872 * q^85 + 18432 * q^86 + 7776 * q^87 + 8192 * q^88 - 18432 * q^94 - 19128 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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