LMFDB - Embedded newform 177.4.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,4,Mod(1,177)]

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, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("177.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	177.a (trivial)

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:	yes
Analytic conductor:	$10.4433380710$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 49x^{6} + 89x^{5} + 648x^{4} - 1023x^{3} - 1476x^{2} + 1940x - 384 $ x^8 - 2x^7 - 49x^6 + 89x^5 + 648x^4 - 1023x^3 - 1476x^2 + 1940*x - 384
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.21744$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.21744 q^{2} -3.00000 q^{3} +9.78684 q^{4} +17.5635 q^{5} +12.6523 q^{6} +14.8996 q^{7} -7.53588 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-4.21744 q^{2} -3.00000 q^{3} +9.78684 q^{4} +17.5635 q^{5} +12.6523 q^{6} +14.8996 q^{7} -7.53588 q^{8} +9.00000 q^{9} -74.0731 q^{10} -18.9001 q^{11} -29.3605 q^{12} +31.7591 q^{13} -62.8381 q^{14} -52.6905 q^{15} -46.5125 q^{16} -84.1910 q^{17} -37.9570 q^{18} +126.537 q^{19} +171.891 q^{20} -44.6987 q^{21} +79.7102 q^{22} -21.1777 q^{23} +22.6076 q^{24} +183.477 q^{25} -133.942 q^{26} -27.0000 q^{27} +145.820 q^{28} -12.6033 q^{29} +222.219 q^{30} +215.817 q^{31} +256.451 q^{32} +56.7003 q^{33} +355.071 q^{34} +261.689 q^{35} +88.0815 q^{36} -3.62282 q^{37} -533.661 q^{38} -95.2774 q^{39} -132.357 q^{40} -381.728 q^{41} +188.514 q^{42} +168.813 q^{43} -184.972 q^{44} +158.072 q^{45} +89.3156 q^{46} +613.793 q^{47} +139.538 q^{48} -121.003 q^{49} -773.804 q^{50} +252.573 q^{51} +310.821 q^{52} +270.545 q^{53} +113.871 q^{54} -331.952 q^{55} -112.281 q^{56} -379.610 q^{57} +53.1537 q^{58} -59.0000 q^{59} -515.674 q^{60} +311.206 q^{61} -910.195 q^{62} +134.096 q^{63} -709.468 q^{64} +557.802 q^{65} -239.130 q^{66} +375.157 q^{67} -823.963 q^{68} +63.5330 q^{69} -1103.66 q^{70} -987.919 q^{71} -67.8229 q^{72} -359.475 q^{73} +15.2790 q^{74} -550.431 q^{75} +1238.39 q^{76} -281.604 q^{77} +401.827 q^{78} +933.998 q^{79} -816.923 q^{80} +81.0000 q^{81} +1609.92 q^{82} +1262.64 q^{83} -437.459 q^{84} -1478.69 q^{85} -711.960 q^{86} +37.8099 q^{87} +142.429 q^{88} +273.294 q^{89} -666.658 q^{90} +473.197 q^{91} -207.262 q^{92} -647.450 q^{93} -2588.64 q^{94} +2222.43 q^{95} -769.353 q^{96} -897.200 q^{97} +510.323 q^{98} -170.101 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9}+O(q^{10}) $ 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9	\( 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9} + 29 q^{10} - 27 q^{11} - 114 q^{12} + 89 q^{13} - 37 q^{14} + 36 q^{15} + 362 q^{16} + 79 q^{17} + 18 q^{18} + 288 q^{19} + 457 q^{20} - 159 q^{21} + 596 q^{22} + 202 q^{23} - 9 q^{24} + 264 q^{25} + 270 q^{26} - 216 q^{27} + 702 q^{28} - 114 q^{29} - 87 q^{30} + 538 q^{31} + 316 q^{32} + 81 q^{33} + 498 q^{34} - 196 q^{35} + 342 q^{36} + 395 q^{37} + 397 q^{38} - 267 q^{39} + 918 q^{40} - 39 q^{41} + 111 q^{42} + 527 q^{43} + 64 q^{44} - 108 q^{45} - 539 q^{46} + 860 q^{47} - 1086 q^{48} + 347 q^{49} - 591 q^{50} - 237 q^{51} - 644 q^{52} - 812 q^{53} - 54 q^{54} + 536 q^{55} - 2218 q^{56} - 864 q^{57} - 1154 q^{58} - 472 q^{59} - 1371 q^{60} - 460 q^{61} - 2014 q^{62} + 477 q^{63} - 451 q^{64} - 986 q^{65} - 1788 q^{66} + 1934 q^{67} - 69 q^{68} - 606 q^{69} - 1028 q^{70} - 1687 q^{71} + 27 q^{72} + 1980 q^{73} - 2400 q^{74} - 792 q^{75} - 940 q^{76} - 821 q^{77} - 810 q^{78} + 3319 q^{79} - 2119 q^{80} + 648 q^{81} + 429 q^{82} + 2057 q^{83} - 2106 q^{84} + 566 q^{85} - 6690 q^{86} + 342 q^{87} + 1189 q^{88} + 1668 q^{89} + 261 q^{90} + 2427 q^{91} - 980 q^{92} - 1614 q^{93} + 332 q^{94} + 2146 q^{95} - 948 q^{96} + 1956 q^{97} - 2026 q^{98} - 243 q^{99}+O(q^{100}) \) 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9 + 29 * q^10 - 27 * q^11 - 114 * q^12 + 89 * q^13 - 37 * q^14 + 36 * q^15 + 362 * q^16 + 79 * q^17 + 18 * q^18 + 288 * q^19 + 457 * q^20 - 159 * q^21 + 596 * q^22 + 202 * q^23 - 9 * q^24 + 264 * q^25 + 270 * q^26 - 216 * q^27 + 702 * q^28 - 114 * q^29 - 87 * q^30 + 538 * q^31 + 316 * q^32 + 81 * q^33 + 498 * q^34 - 196 * q^35 + 342 * q^36 + 395 * q^37 + 397 * q^38 - 267 * q^39 + 918 * q^40 - 39 * q^41 + 111 * q^42 + 527 * q^43 + 64 * q^44 - 108 * q^45 - 539 * q^46 + 860 * q^47 - 1086 * q^48 + 347 * q^49 - 591 * q^50 - 237 * q^51 - 644 * q^52 - 812 * q^53 - 54 * q^54 + 536 * q^55 - 2218 * q^56 - 864 * q^57 - 1154 * q^58 - 472 * q^59 - 1371 * q^60 - 460 * q^61 - 2014 * q^62 + 477 * q^63 - 451 * q^64 - 986 * q^65 - 1788 * q^66 + 1934 * q^67 - 69 * q^68 - 606 * q^69 - 1028 * q^70 - 1687 * q^71 + 27 * q^72 + 1980 * q^73 - 2400 * q^74 - 792 * q^75 - 940 * q^76 - 821 * q^77 - 810 * q^78 + 3319 * q^79 - 2119 * q^80 + 648 * q^81 + 429 * q^82 + 2057 * q^83 - 2106 * q^84 + 566 * q^85 - 6690 * q^86 + 342 * q^87 + 1189 * q^88 + 1668 * q^89 + 261 * q^90 + 2427 * q^91 - 980 * q^92 - 1614 * q^93 + 332 * q^94 + 2146 * q^95 - 948 * q^96 + 1956 * q^97 - 2026 * q^98 - 243 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−4.21744	−1.49109	−0.745546	−	0.666454i	$-0.767811\pi$
$2$	−4.21744	−1.49109	−0.745546	+	0.666454i	$0.767811\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	9.78684	1.22335
$5$	17.5635	1.57093	0.785464	−	0.618907i	$-0.212424\pi$
$5$	17.5635	1.57093	0.785464	+	0.618907i	$0.212424\pi$
$6$	12.6523	0.860882
$7$	14.8996	0.804501	0.402251	−	0.915530i	$-0.368228\pi$
$7$	14.8996	0.804501	0.402251	+	0.915530i	$0.368228\pi$
$8$	−7.53588	−0.333042
$9$	9.00000	0.333333
$10$	−74.0731	−2.34240
$11$	−18.9001	−0.518054	−0.259027	−	0.965870i	$-0.583402\pi$
$11$	−18.9001	−0.518054	−0.259027	+	0.965870i	$0.583402\pi$
$12$	−29.3605	−0.706304
$13$	31.7591	0.677569	0.338784	−	0.940864i	$-0.389984\pi$
$13$	31.7591	0.677569	0.338784	+	0.940864i	$0.389984\pi$
$14$	−62.8381	−1.19958
$15$	−52.6905	−0.906976
$16$	−46.5125	−0.726758
$17$	−84.1910	−1.20114	−0.600568	−	0.799574i	$-0.705059\pi$
$17$	−84.1910	−1.20114	−0.600568	+	0.799574i	$0.705059\pi$
$18$	−37.9570	−0.497031
$19$	126.537	1.52787	0.763934	−	0.645294i	$-0.223265\pi$
$19$	126.537	1.52787	0.763934	+	0.645294i	$0.223265\pi$
$20$	171.891	1.92180
$21$	−44.6987	−0.464479
$22$	79.7102	0.772467
$23$	−21.1777	−0.191993	−0.0959967	−	0.995382i	$-0.530604\pi$
$23$	−21.1777	−0.191993	−0.0959967	+	0.995382i	$0.530604\pi$
$24$	22.6076	0.192282
$25$	183.477	1.46782
$26$	−133.942	−1.01032
$27$	−27.0000	−0.192450
$28$	145.820	0.984190
$29$	−12.6033	−0.0807025	−0.0403512	−	0.999186i	$-0.512848\pi$
$29$	−12.6033	−0.0807025	−0.0403512	+	0.999186i	$0.512848\pi$
$30$	222.219	1.35238
$31$	215.817	1.25038	0.625191	−	0.780472i	$-0.285021\pi$
$31$	215.817	1.25038	0.625191	+	0.780472i	$0.285021\pi$
$32$	256.451	1.41671
$33$	56.7003	0.299099
$34$	355.071	1.79100
$35$	261.689	1.26381
$36$	88.0815	0.407785
$37$	−3.62282	−0.0160970	−0.00804848	−	0.999968i	$-0.502562\pi$
$37$	−3.62282	−0.0160970	−0.00804848	+	0.999968i	$0.502562\pi$
$38$	−533.661	−2.27819
$39$	−95.2774	−0.391195
$40$	−132.357	−0.523185
$41$	−381.728	−1.45405	−0.727023	−	0.686613i	$-0.759097\pi$
$41$	−381.728	−1.45405	−0.727023	+	0.686613i	$0.759097\pi$
$42$	188.514	0.692581
$43$	168.813	0.598692	0.299346	−	0.954145i	$-0.403231\pi$
$43$	168.813	0.598692	0.299346	+	0.954145i	$0.403231\pi$
$44$	−184.972	−0.633764
$45$	158.072	0.523643
$46$	89.3156	0.286280
$47$	613.793	1.90491	0.952457	−	0.304674i	$-0.0985475\pi$
$47$	613.793	1.90491	0.952457	+	0.304674i	$0.0985475\pi$
$48$	139.538	0.419594
$49$	−121.003	−0.352778
$50$	−773.804	−2.18865
$51$	252.573	0.693476
$52$	310.821	0.828907
$53$	270.545	0.701173	0.350587	−	0.936530i	$-0.385982\pi$
$53$	270.545	0.701173	0.350587	+	0.936530i	$0.385982\pi$
$54$	113.871	0.286961
$55$	−331.952	−0.813826
$56$	−112.281	−0.267933
$57$	−379.610	−0.882115
$58$	53.1537	0.120335
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	−515.674	−1.10955
$61$	311.206	0.653211	0.326605	−	0.945161i	$-0.394095\pi$
$61$	311.206	0.653211	0.326605	+	0.945161i	$0.394095\pi$
$62$	−910.195	−1.86443
$63$	134.096	0.268167
$64$	−709.468	−1.38568
$65$	557.802	1.06441
$66$	−239.130	−0.445984
$67$	375.157	0.684071	0.342035	−	0.939687i	$-0.388884\pi$
$67$	375.157	0.684071	0.342035	+	0.939687i	$0.388884\pi$
$68$	−823.963	−1.46942
$69$	63.5330	0.110847
$70$	−1103.66	−1.88446
$71$	−987.919	−1.65133	−0.825665	−	0.564160i	$-0.809200\pi$
$71$	−987.919	−1.65133	−0.825665	+	0.564160i	$0.809200\pi$
$72$	−67.8229	−0.111014
$73$	−359.475	−0.576348	−0.288174	−	0.957578i	$-0.593048\pi$
$73$	−359.475	−0.576348	−0.288174	+	0.957578i	$0.593048\pi$
$74$	15.2790	0.0240020
$75$	−550.431	−0.847444
$76$	1238.39	1.86912
$77$	−281.604	−0.416775
$78$	401.827	0.583307
$79$	933.998	1.33016	0.665082	−	0.746770i	$-0.268397\pi$
$79$	933.998	1.33016	0.665082	+	0.746770i	$0.268397\pi$
$80$	−816.923	−1.14169
$81$	81.0000	0.111111
$82$	1609.92	2.16811
$83$	1262.64	1.66979	0.834893	−	0.550412i	$-0.185529\pi$
$83$	1262.64	1.66979	0.834893	+	0.550412i	$0.185529\pi$
$84$	−437.459	−0.568222
$85$	−1478.69	−1.88690
$86$	−711.960	−0.892705
$87$	37.8099	0.0465936
$88$	142.429	0.172534
$89$	273.294	0.325496	0.162748	−	0.986668i	$-0.447964\pi$
$89$	273.294	0.325496	0.162748	+	0.986668i	$0.447964\pi$
$90$	−666.658	−0.780799
$91$	473.197	0.545105
$92$	−207.262	−0.234876
$93$	−647.450	−0.721908
$94$	−2588.64	−2.84040
$95$	2222.43	2.40017
$96$	−769.353	−0.817935
$97$	−897.200	−0.939143	−0.469571	−	0.882895i	$-0.655592\pi$
$97$	−897.200	−0.939143	−0.469571	+	0.882895i	$0.655592\pi$
$98$	510.323	0.526024
$99$	−170.101	−0.172685
$100$	1795.66	1.79566
$101$	296.784	0.292387	0.146194	−	0.989256i	$-0.453298\pi$
$101$	296.784	0.292387	0.146194	+	0.989256i	$0.453298\pi$
$102$	−1065.21	−1.03404
$103$	−36.8226	−0.0352257	−0.0176128	−	0.999845i	$-0.505607\pi$
$103$	−36.8226	−0.0352257	−0.0176128	+	0.999845i	$0.505607\pi$
$104$	−239.333	−0.225659
$105$	−785.066	−0.729663
$106$	−1141.01	−1.04551
$107$	−479.349	−0.433088	−0.216544	−	0.976273i	$-0.569479\pi$
$107$	−479.349	−0.433088	−0.216544	+	0.976273i	$0.569479\pi$
$108$	−264.245	−0.235435
$109$	946.780	0.831973	0.415987	−	0.909371i	$-0.363436\pi$
$109$	946.780	0.831973	0.415987	+	0.909371i	$0.363436\pi$
$110$	1399.99	1.21349
$111$	10.8684	0.00929358
$112$	−693.017	−0.584678
$113$	2131.61	1.77456	0.887279	−	0.461234i	$-0.152593\pi$
$113$	2131.61	1.77456	0.887279	+	0.461234i	$0.152593\pi$
$114$	1600.98	1.31531
$115$	−371.954	−0.301608
$116$	−123.346	−0.0987277
$117$	285.832	0.225856
$118$	248.829	0.194124
$119$	−1254.41	−0.966315
$120$	397.070	0.302061
$121$	−973.786	−0.731620
$122$	−1312.49	−0.973997
$123$	1145.18	0.839493
$124$	2112.16	1.52966
$125$	1027.06	0.734904
$126$	−565.543	−0.399862
$127$	2054.51	1.43550	0.717751	−	0.696300i	$-0.245172\pi$
$127$	2054.51	1.43550	0.717751	+	0.696300i	$0.245172\pi$
$128$	940.532	0.649469
$129$	−506.440	−0.345655
$130$	−2352.50	−1.58714
$131$	−602.309	−0.401710	−0.200855	−	0.979621i	$-0.564372\pi$
$131$	−602.309	−0.401710	−0.200855	+	0.979621i	$0.564372\pi$
$132$	554.917	0.365904
$133$	1885.34	1.22917
$134$	−1582.20	−1.02001
$135$	−474.215	−0.302325
$136$	634.453	0.400029
$137$	1919.26	1.19689	0.598444	−	0.801164i	$-0.295786\pi$
$137$	1919.26	1.19689	0.598444	+	0.801164i	$0.295786\pi$
$138$	−267.947	−0.165284
$139$	−448.518	−0.273689	−0.136845	−	0.990593i	$-0.543696\pi$
$139$	−448.518	−0.273689	−0.136845	+	0.990593i	$0.543696\pi$
$140$	2561.10	1.54609
$141$	−1841.38	−1.09980
$142$	4166.49	2.46228
$143$	−600.251	−0.351017
$144$	−418.613	−0.242253
$145$	−221.358	−0.126778
$146$	1516.07	0.859388
$147$	363.009	0.203676
$148$	−35.4559	−0.0196923
$149$	−1613.45	−0.887105	−0.443553	−	0.896248i	$-0.646282\pi$
$149$	−1613.45	−0.887105	−0.443553	+	0.896248i	$0.646282\pi$
$150$	2321.41	1.26362
$151$	−1522.22	−0.820374	−0.410187	−	0.912002i	$-0.634537\pi$
$151$	−1522.22	−0.820374	−0.410187	+	0.912002i	$0.634537\pi$
$152$	−953.565	−0.508844
$153$	−757.719	−0.400379
$154$	1187.65	0.621450
$155$	3790.50	1.96426
$156$	−932.464	−0.478570
$157$	−79.4233	−0.0403737	−0.0201868	−	0.999796i	$-0.506426\pi$
$157$	−79.4233	−0.0403737	−0.0201868	+	0.999796i	$0.506426\pi$
$158$	−3939.08	−1.98340
$159$	−811.634	−0.404823
$160$	4504.18	2.22554
$161$	−315.538	−0.154459
$162$	−341.613	−0.165677
$163$	2254.35	1.08328	0.541639	−	0.840611i	$-0.317804\pi$
$163$	2254.35	1.08328	0.541639	+	0.840611i	$0.317804\pi$
$164$	−3735.91	−1.77881
$165$	995.857	0.469863
$166$	−5325.10	−2.48981
$167$	−382.224	−0.177110	−0.0885550	−	0.996071i	$-0.528225\pi$
$167$	−382.224	−0.177110	−0.0885550	+	0.996071i	$0.528225\pi$
$168$	336.844	0.154691
$169$	−1188.36	−0.540901
$170$	6236.29	2.81354
$171$	1138.83	0.509289
$172$	1652.15	0.732413
$173$	−929.824	−0.408631	−0.204316	−	0.978905i	$-0.565497\pi$
$173$	−929.824	−0.408631	−0.204316	+	0.978905i	$0.565497\pi$
$174$	−159.461	−0.0694753
$175$	2733.73	1.18086
$176$	879.092	0.376500
$177$	177.000	0.0751646
$178$	−1152.60	−0.485344
$179$	−509.106	−0.212583	−0.106291	−	0.994335i	$-0.533898\pi$
$179$	−509.106	−0.212583	−0.106291	+	0.994335i	$0.533898\pi$
$180$	1547.02	0.640601
$181$	824.865	0.338739	0.169369	−	0.985553i	$-0.445827\pi$
$181$	824.865	0.338739	0.169369	+	0.985553i	$0.445827\pi$
$182$	−1995.68	−0.812801
$183$	−933.618	−0.377131
$184$	159.592	0.0639419
$185$	−63.6294	−0.0252872
$186$	2730.58	1.07643
$187$	1591.22	0.622254
$188$	6007.09	2.33038
$189$	−402.288	−0.154826
$190$	−9372.97	−3.57888
$191$	−3681.09	−1.39453	−0.697263	−	0.716815i	$-0.745599\pi$
$191$	−3681.09	−1.39453	−0.697263	+	0.716815i	$0.745599\pi$
$192$	2128.40	0.800022
$193$	−3051.17	−1.13797	−0.568985	−	0.822348i	$-0.692664\pi$
$193$	−3051.17	−1.13797	−0.568985	+	0.822348i	$0.692664\pi$
$194$	3783.89	1.40035
$195$	−1673.40	−0.614538
$196$	−1184.23	−0.431573
$197$	−5332.57	−1.92858	−0.964290	−	0.264849i	$-0.914678\pi$
$197$	−5332.57	−1.92858	−0.964290	+	0.264849i	$0.914678\pi$
$198$	717.391	0.257489
$199$	−5371.72	−1.91352	−0.956761	−	0.290874i	$-0.906054\pi$
$199$	−5371.72	−1.91352	−0.956761	+	0.290874i	$0.906054\pi$
$200$	−1382.66	−0.488844
$201$	−1125.47	−0.394949
$202$	−1251.67	−0.435976
$203$	−187.784	−0.0649252
$204$	2471.89	0.848367
$205$	−6704.48	−2.28420
$206$	155.297	0.0525247
$207$	−190.599	−0.0639978
$208$	−1477.20	−0.492429
$209$	−2391.56	−0.791519
$210$	3310.97	1.08799
$211$	1896.92	0.618907	0.309454	−	0.950915i	$-0.399854\pi$
$211$	1896.92	0.618907	0.309454	+	0.950915i	$0.399854\pi$
$212$	2647.78	0.857784
$213$	2963.76	0.953396
$214$	2021.63	0.645775
$215$	2964.95	0.940503
$216$	203.469	0.0640940
$217$	3215.57	1.00593
$218$	−3992.99	−1.24055
$219$	1078.43	0.332755
$220$	−3248.76	−0.995598
$221$	−2673.83	−0.813852
$222$	−45.8371	−0.0138576
$223$	−2808.85	−0.843472	−0.421736	−	0.906719i	$-0.638579\pi$
$223$	−2808.85	−0.843472	−0.421736	+	0.906719i	$0.638579\pi$
$224$	3821.01	1.13974
$225$	1651.29	0.489272
$226$	−8989.95	−2.64603
$227$	3302.83	0.965711	0.482856	−	0.875700i	$-0.339600\pi$
$227$	3302.83	0.965711	0.482856	+	0.875700i	$0.339600\pi$
$228$	−3715.18	−1.07914
$229$	−5780.19	−1.66797	−0.833987	−	0.551785i	$-0.813947\pi$
$229$	−5780.19	−1.66797	−0.833987	+	0.551785i	$0.813947\pi$
$230$	1568.70	0.449725
$231$	844.811	0.240625
$232$	94.9769	0.0268773
$233$	4892.84	1.37571	0.687856	−	0.725847i	$-0.258552\pi$
$233$	4892.84	1.37571	0.687856	+	0.725847i	$0.258552\pi$
$234$	−1205.48	−0.336772
$235$	10780.4	2.99248
$236$	−577.423	−0.159267
$237$	−2801.99	−0.767970
$238$	5290.40	1.44086
$239$	−6461.39	−1.74876	−0.874378	−	0.485246i	$-0.838730\pi$
$239$	−6461.39	−1.74876	−0.874378	+	0.485246i	$0.838730\pi$
$240$	2450.77	0.659152
$241$	−60.7729	−0.0162437	−0.00812184	−	0.999967i	$-0.502585\pi$
$241$	−60.7729	−0.0162437	−0.00812184	+	0.999967i	$0.502585\pi$
$242$	4106.89	1.09091
$243$	−243.000	−0.0641500
$244$	3045.72	0.799108
$245$	−2125.23	−0.554189
$246$	−4829.75	−1.25176
$247$	4018.69	1.03524
$248$	−1626.37	−0.416429
$249$	−3787.91	−0.964052
$250$	−4331.57	−1.09581
$251$	−3782.13	−0.951100	−0.475550	−	0.879689i	$-0.657751\pi$
$251$	−3782.13	−0.951100	−0.475550	+	0.879689i	$0.657751\pi$
$252$	1312.38	0.328063
$253$	400.260	0.0994630
$254$	−8664.80	−2.14046
$255$	4436.07	1.08940
$256$	1709.10	0.417261
$257$	−72.7427	−0.0176559	−0.00882795	−	0.999961i	$-0.502810\pi$
$257$	−72.7427	−0.0176559	−0.00882795	+	0.999961i	$0.502810\pi$
$258$	2135.88	0.515404
$259$	−53.9784	−0.0129500
$260$	5459.11	1.30215
$261$	−113.430	−0.0269008
$262$	2540.20	0.598986
$263$	−4787.49	−1.12247	−0.561235	−	0.827657i	$-0.689673\pi$
$263$	−4787.49	−1.12247	−0.561235	+	0.827657i	$0.689673\pi$
$264$	−427.287	−0.0996125
$265$	4751.72	1.10149
$266$	−7951.32	−1.83281
$267$	−819.883	−0.187925
$268$	3671.60	0.836861
$269$	3320.18	0.752546	0.376273	−	0.926509i	$-0.377205\pi$
$269$	3320.18	0.752546	0.376273	+	0.926509i	$0.377205\pi$
$270$	1999.97	0.450795
$271$	7147.95	1.60224	0.801120	−	0.598504i	$-0.204238\pi$
$271$	7147.95	1.60224	0.801120	+	0.598504i	$0.204238\pi$
$272$	3915.94	0.872936
$273$	−1419.59	−0.314716
$274$	−8094.39	−1.78467
$275$	−3467.73	−0.760408
$276$	621.787	0.135606
$277$	4611.19	1.00021	0.500107	−	0.865963i	$-0.333294\pi$
$277$	4611.19	1.00021	0.500107	+	0.865963i	$0.333294\pi$
$278$	1891.60	0.408096
$279$	1942.35	0.416794
$280$	−1972.05	−0.420903
$281$	8307.67	1.76368	0.881840	−	0.471548i	$-0.156305\pi$
$281$	8307.67	1.76368	0.881840	+	0.471548i	$0.156305\pi$
$282$	7765.91	1.63991
$283$	−8181.70	−1.71856	−0.859279	−	0.511507i	$-0.829087\pi$
$283$	−8181.70	−1.71856	−0.859279	+	0.511507i	$0.829087\pi$
$284$	−9668.60	−2.02016
$285$	−6667.28	−1.38574
$286$	2531.52	0.523399
$287$	−5687.58	−1.16978
$288$	2308.06	0.472235
$289$	2175.12	0.442728
$290$	933.565	0.189037
$291$	2691.60	0.542214
$292$	−3518.13	−0.705078
$293$	−264.862	−0.0528102	−0.0264051	−	0.999651i	$-0.508406\pi$
$293$	−264.862	−0.0528102	−0.0264051	+	0.999651i	$0.508406\pi$
$294$	−1530.97	−0.303700
$295$	−1036.25	−0.204517
$296$	27.3011	0.00536096
$297$	510.303	0.0996996
$298$	6804.62	1.32276
$299$	−672.584	−0.130089
$300$	−5386.98	−1.03672
$301$	2515.24	0.481649
$302$	6419.87	1.22325
$303$	−890.351	−0.168810
$304$	−5885.54	−1.11039
$305$	5465.87	1.02615
$306$	3195.64	0.597001
$307$	4379.69	0.814209	0.407104	−	0.913382i	$-0.366539\pi$
$307$	4379.69	0.814209	0.407104	+	0.913382i	$0.366539\pi$
$308$	−2756.01	−0.509864
$309$	110.468	0.0203375
$310$	−15986.2	−2.92889
$311$	10544.6	1.92261	0.961303	−	0.275494i	$-0.0888413\pi$
$311$	10544.6	1.92261	0.961303	+	0.275494i	$0.0888413\pi$
$312$	717.999	0.130284
$313$	−6496.71	−1.17321	−0.586607	−	0.809872i	$-0.699537\pi$
$313$	−6496.71	−1.17321	−0.586607	+	0.809872i	$0.699537\pi$
$314$	334.963	0.0602009
$315$	2355.20	0.421271
$316$	9140.88	1.62726
$317$	2628.81	0.465769	0.232884	−	0.972504i	$-0.425184\pi$
$317$	2628.81	0.465769	0.232884	+	0.972504i	$0.425184\pi$
$318$	3423.02	0.603628
$319$	238.204	0.0418083
$320$	−12460.7	−2.17680
$321$	1438.05	0.250044
$322$	1330.76	0.230312
$323$	−10653.2	−1.83518
$324$	792.734	0.135928
$325$	5827.06	0.994546
$326$	−9507.59	−1.61527
$327$	−2840.34	−0.480340
$328$	2876.65	0.484258
$329$	9145.25	1.53250
$330$	−4199.97	−0.700608
$331$	−3679.05	−0.610933	−0.305466	−	0.952203i	$-0.598812\pi$
$331$	−3679.05	−0.610933	−0.305466	+	0.952203i	$0.598812\pi$
$332$	12357.2	2.04274
$333$	−32.6053	−0.00536565
$334$	1612.01	0.264087
$335$	6589.08	1.07463
$336$	2079.05	0.337564
$337$	1323.74	0.213972	0.106986	−	0.994261i	$-0.465880\pi$
$337$	1323.74	0.213972	0.106986	+	0.994261i	$0.465880\pi$
$338$	5011.84	0.806532
$339$	−6394.83	−1.02454
$340$	−14471.7	−2.30835
$341$	−4078.96	−0.647765
$342$	−4802.95	−0.759397
$343$	−6913.44	−1.08831
$344$	−1272.16	−0.199390
$345$	1115.86	0.174133
$346$	3921.48	0.609307
$347$	5541.30	0.857270	0.428635	−	0.903478i	$-0.358995\pi$
$347$	5541.30	0.857270	0.428635	+	0.903478i	$0.358995\pi$
$348$	370.039	0.0570005
$349$	−6590.45	−1.01083	−0.505414	−	0.862877i	$-0.668660\pi$
$349$	−6590.45	−1.01083	−0.505414	+	0.862877i	$0.668660\pi$
$350$	−11529.3	−1.76077
$351$	−857.496	−0.130398
$352$	−4846.95	−0.733930
$353$	−5633.11	−0.849349	−0.424674	−	0.905346i	$-0.639611\pi$
$353$	−5633.11	−0.849349	−0.424674	+	0.905346i	$0.639611\pi$
$354$	−746.488	−0.112077
$355$	−17351.3	−2.59412
$356$	2674.69	0.398197
$357$	3763.23	0.557902
$358$	2147.13	0.316981
$359$	2672.65	0.392916	0.196458	−	0.980512i	$-0.437056\pi$
$359$	2672.65	0.392916	0.196458	+	0.980512i	$0.437056\pi$
$360$	−1191.21	−0.174395
$361$	9152.52	1.33438
$362$	−3478.82	−0.505091
$363$	2921.36	0.422401
$364$	4631.10	0.666856
$365$	−6313.65	−0.905402
$366$	3937.48	0.562337
$367$	−2881.92	−0.409904	−0.204952	−	0.978772i	$-0.565704\pi$
$367$	−2881.92	−0.409904	−0.204952	+	0.978772i	$0.565704\pi$
$368$	985.027	0.139533
$369$	−3435.55	−0.484682
$370$	268.353	0.0377055
$371$	4031.00	0.564095
$372$	−6336.49	−0.883149
$373$	7349.59	1.02023	0.510117	−	0.860105i	$-0.329602\pi$
$373$	7349.59	1.02023	0.510117	+	0.860105i	$0.329602\pi$
$374$	−6710.88	−0.927837
$375$	−3081.18	−0.424297
$376$	−4625.47	−0.634416
$377$	−400.269	−0.0546815
$378$	1696.63	0.230860
$379$	6001.29	0.813366	0.406683	−	0.913569i	$-0.366685\pi$
$379$	6001.29	0.813366	0.406683	+	0.913569i	$0.366685\pi$
$380$	21750.5	2.93626
$381$	−6163.54	−0.828787
$382$	15524.8	2.07937
$383$	−7844.21	−1.04653	−0.523264	−	0.852171i	$-0.675286\pi$
$383$	−7844.21	−1.04653	−0.523264	+	0.852171i	$0.675286\pi$
$384$	−2821.60	−0.374971
$385$	−4945.95	−0.654724
$386$	12868.2	1.69682
$387$	1519.32	0.199564
$388$	−8780.75	−1.14890
$389$	−574.603	−0.0748934	−0.0374467	−	0.999299i	$-0.511922\pi$
$389$	−574.603	−0.0748934	−0.0374467	+	0.999299i	$0.511922\pi$
$390$	7057.49	0.916333
$391$	1782.97	0.230610
$392$	911.863	0.117490
$393$	1806.93	0.231927
$394$	22489.8	2.87569
$395$	16404.3	2.08959
$396$	−1664.75	−0.211255
$397$	−14329.0	−1.81147	−0.905735	−	0.423844i	$-0.860680\pi$
$397$	−14329.0	−1.81147	−0.905735	+	0.423844i	$0.860680\pi$
$398$	22654.9	2.85324
$399$	−5656.02	−0.709663
$400$	−8533.98	−1.06675
$401$	8440.51	1.05112	0.525560	−	0.850757i	$-0.323856\pi$
$401$	8440.51	1.05112	0.525560	+	0.850757i	$0.323856\pi$
$402$	4746.61	0.588904
$403$	6854.15	0.847219
$404$	2904.57	0.357693
$405$	1422.64	0.174548
$406$	791.967	0.0968095
$407$	68.4716	0.00833910
$408$	−1903.36	−0.230957
$409$	−3995.02	−0.482985	−0.241492	−	0.970403i	$-0.577637\pi$
$409$	−3995.02	−0.482985	−0.241492	+	0.970403i	$0.577637\pi$
$410$	28275.8	3.40595
$411$	−5757.79	−0.691024
$412$	−360.377	−0.0430935
$413$	−879.075	−0.104737
$414$	803.840	0.0954266
$415$	22176.3	2.62312
$416$	8144.66	0.959915
$417$	1345.56	0.158015
$418$	10086.3	1.18023
$419$	−1885.71	−0.219864	−0.109932	−	0.993939i	$-0.535063\pi$
$419$	−1885.71	−0.219864	−0.109932	+	0.993939i	$0.535063\pi$
$420$	−7683.31	−0.892637
$421$	−9268.38	−1.07295	−0.536477	−	0.843915i	$-0.680245\pi$
$421$	−9268.38	−1.07295	−0.536477	+	0.843915i	$0.680245\pi$
$422$	−8000.16	−0.922848
$423$	5524.14	0.634971
$424$	−2038.79	−0.233520
$425$	−15447.1	−1.76305
$426$	−12499.5	−1.42160
$427$	4636.84	0.525509
$428$	−4691.31	−0.529821
$429$	1800.75	0.202660
$430$	−12504.5	−1.40238
$431$	−3346.11	−0.373959	−0.186980	−	0.982364i	$-0.559870\pi$
$431$	−3346.11	−0.373959	−0.186980	+	0.982364i	$0.559870\pi$
$432$	1255.84	0.139865
$433$	−927.848	−0.102978	−0.0514891	−	0.998674i	$-0.516397\pi$
$433$	−927.848	−0.102978	−0.0514891	+	0.998674i	$0.516397\pi$
$434$	−13561.5	−1.49994
$435$	664.074	0.0731952
$436$	9265.98	1.01780
$437$	−2679.75	−0.293341
$438$	−4548.20	−0.496168
$439$	10124.5	1.10072	0.550361	−	0.834927i	$-0.314490\pi$
$439$	10124.5	1.10072	0.550361	+	0.834927i	$0.314490\pi$
$440$	2501.55	0.271038
$441$	−1089.03	−0.117593
$442$	11276.7	1.21353
$443$	−6492.44	−0.696310	−0.348155	−	0.937437i	$-0.613192\pi$
$443$	−6492.44	−0.696310	−0.348155	+	0.937437i	$0.613192\pi$
$444$	106.368	0.0113693
$445$	4800.01	0.511331
$446$	11846.2	1.25769
$447$	4840.34	0.512170
$448$	−10570.8	−1.11478
$449$	−17708.0	−1.86123	−0.930614	−	0.366003i	$-0.880726\pi$
$449$	−17708.0	−1.86123	−0.930614	+	0.366003i	$0.880726\pi$
$450$	−6964.23	−0.729549
$451$	7214.70	0.753275
$452$	20861.7	2.17091
$453$	4566.66	0.473643
$454$	−13929.5	−1.43996
$455$	8311.00	0.856321
$456$	2860.70	0.293781
$457$	10785.3	1.10397	0.551984	−	0.833855i	$-0.313871\pi$
$457$	10785.3	1.10397	0.551984	+	0.833855i	$0.313871\pi$
$458$	24377.6	2.48710
$459$	2273.16	0.231159
$460$	−3640.25	−0.368973
$461$	−14512.5	−1.46619	−0.733097	−	0.680124i	$-0.761926\pi$
$461$	−14512.5	−1.46619	−0.733097	+	0.680124i	$0.761926\pi$
$462$	−3562.94	−0.358794
$463$	−9637.81	−0.967402	−0.483701	−	0.875233i	$-0.660708\pi$
$463$	−9637.81	−0.967402	−0.483701	+	0.875233i	$0.660708\pi$
$464$	586.211	0.0586512
$465$	−11371.5	−1.13407
$466$	−20635.3	−2.05131
$467$	−7260.33	−0.719417	−0.359709	−	0.933065i	$-0.617124\pi$
$467$	−7260.33	−0.719417	−0.359709	+	0.933065i	$0.617124\pi$
$468$	2797.39	0.276302
$469$	5589.68	0.550336
$470$	−45465.6	−4.46207
$471$	238.270	0.0233098
$472$	444.617	0.0433584
$473$	−3190.59	−0.310155
$474$	11817.2	1.14511
$475$	23216.6	2.24263
$476$	−12276.7	−1.18215
$477$	2434.90	0.233724
$478$	27250.6	2.60756
$479$	12069.4	1.15128	0.575641	−	0.817703i	$-0.304753\pi$
$479$	12069.4	1.15128	0.575641	+	0.817703i	$0.304753\pi$
$480$	−13512.5	−1.28492
$481$	−115.057	−0.0109068
$482$	256.306	0.0242208
$483$	946.614	0.0891769
$484$	−9530.28	−0.895030
$485$	−15758.0	−1.47533
$486$	1024.84	0.0956536
$487$	−6015.96	−0.559773	−0.279886	−	0.960033i	$-0.590297\pi$
$487$	−6015.96	−0.559773	−0.279886	+	0.960033i	$0.590297\pi$
$488$	−2345.21	−0.217547
$489$	−6763.05	−0.625430
$490$	8963.06	0.826346
$491$	4022.81	0.369749	0.184875	−	0.982762i	$-0.440812\pi$
$491$	4022.81	0.369749	0.184875	+	0.982762i	$0.440812\pi$
$492$	11207.7	1.02700
$493$	1061.08	0.0969347
$494$	−16948.6	−1.54363
$495$	−2987.57	−0.271275
$496$	−10038.2	−0.908725
$497$	−14719.6	−1.32850
$498$	15975.3	1.43749
$499$	−14112.8	−1.26608	−0.633042	−	0.774117i	$-0.718194\pi$
$499$	−14112.8	−1.26608	−0.633042	+	0.774117i	$0.718194\pi$
$500$	10051.7	0.899048
$501$	1146.67	0.102255
$502$	15950.9	1.41818
$503$	689.493	0.0611192	0.0305596	−	0.999533i	$-0.490271\pi$
$503$	689.493	0.0611192	0.0305596	+	0.999533i	$0.490271\pi$
$504$	−1010.53	−0.0893109
$505$	5212.56	0.459319
$506$	−1688.07	−0.148308
$507$	3565.08	0.312289
$508$	20107.2	1.75613
$509$	−12620.3	−1.09899	−0.549495	−	0.835497i	$-0.685180\pi$
$509$	−12620.3	−1.09899	−0.549495	+	0.835497i	$0.685180\pi$
$510$	−18708.9	−1.62440
$511$	−5356.03	−0.463673
$512$	−14732.3	−1.27164
$513$	−3416.49	−0.294038
$514$	306.788	0.0263266
$515$	−646.735	−0.0553370
$516$	−4956.44	−0.422859
$517$	−11600.8	−0.986849
$518$	227.651	0.0193097
$519$	2789.47	0.235923
$520$	−4203.53	−0.354494
$521$	2186.17	0.183835	0.0919174	−	0.995767i	$-0.470700\pi$
$521$	2186.17	0.183835	0.0919174	+	0.995767i	$0.470700\pi$
$522$	478.383	0.0401116
$523$	−18639.4	−1.55840	−0.779202	−	0.626773i	$-0.784375\pi$
$523$	−18639.4	−1.55840	−0.779202	+	0.626773i	$0.784375\pi$
$524$	−5894.70	−0.491433
$525$	−8201.18	−0.681769
$526$	20191.0	1.67370
$527$	−18169.8	−1.50188
$528$	−2637.28	−0.217373
$529$	−11718.5	−0.963139
$530$	−20040.1	−1.64243
$531$	−531.000	−0.0433963
$532$	18451.5	1.50371
$533$	−12123.3	−0.985216
$534$	3457.81	0.280214
$535$	−8419.06	−0.680351
$536$	−2827.14	−0.227824
$537$	1527.32	0.122735
$538$	−14002.7	−1.12211
$539$	2286.97	0.182758
$540$	−4641.06	−0.369851
$541$	−6609.92	−0.525291	−0.262646	−	0.964892i	$-0.584595\pi$
$541$	−6609.92	−0.525291	−0.262646	+	0.964892i	$0.584595\pi$
$542$	−30146.1	−2.38909
$543$	−2474.59	−0.195571
$544$	−21590.9	−1.70166
$545$	16628.8	1.30697
$546$	5987.05	0.469271
$547$	8883.37	0.694379	0.347190	−	0.937795i	$-0.387136\pi$
$547$	8883.37	0.694379	0.347190	+	0.937795i	$0.387136\pi$
$548$	18783.5	1.46422
$549$	2800.85	0.217737
$550$	14625.0	1.13384
$551$	−1594.78	−0.123303
$552$	−478.777	−0.0369168
$553$	13916.2	1.07012
$554$	−19447.4	−1.49141
$555$	190.888	0.0145995
$556$	−4389.58	−0.334819
$557$	13797.8	1.04961	0.524805	−	0.851222i	$-0.324138\pi$
$557$	13797.8	1.04961	0.524805	+	0.851222i	$0.324138\pi$
$558$	−8191.75	−0.621478
$559$	5361.36	0.405655
$560$	−12171.8	−0.918487
$561$	−4773.66	−0.359258
$562$	−35037.1	−2.62981
$563$	−666.036	−0.0498581	−0.0249290	−	0.999689i	$-0.507936\pi$
$563$	−666.036	−0.0498581	−0.0249290	+	0.999689i	$0.507936\pi$
$564$	−18021.3	−1.34545
$565$	37438.6	2.78770
$566$	34505.9	2.56253
$567$	1206.87	0.0893890
$568$	7444.84	0.549962
$569$	−18945.7	−1.39586	−0.697932	−	0.716164i	$-0.745896\pi$
$569$	−18945.7	−1.39586	−0.697932	+	0.716164i	$0.745896\pi$
$570$	28118.9	2.06626
$571$	23781.4	1.74295	0.871473	−	0.490443i	$-0.163165\pi$
$571$	23781.4	1.74295	0.871473	+	0.490443i	$0.163165\pi$
$572$	−5874.56	−0.429419
$573$	11043.3	0.805130
$574$	23987.0	1.74425
$575$	−3885.61	−0.281811
$576$	−6385.21	−0.461893
$577$	12773.4	0.921601	0.460800	−	0.887504i	$-0.347562\pi$
$577$	12773.4	0.921601	0.460800	+	0.887504i	$0.347562\pi$
$578$	−9173.45	−0.660148
$579$	9153.52	0.657007
$580$	−2166.39	−0.155094
$581$	18812.7	1.34335
$582$	−11351.7	−0.808491
$583$	−5113.33	−0.363246
$584$	2708.96	0.191948
$585$	5020.21	0.354804
$586$	1117.04	0.0787448
$587$	11287.3	0.793658	0.396829	−	0.917893i	$-0.370111\pi$
$587$	11287.3	0.793658	0.396829	+	0.917893i	$0.370111\pi$
$588$	3552.70	0.249169
$589$	27308.7	1.91042
$590$	4370.31	0.304954
$591$	15997.7	1.11347
$592$	168.506	0.0116986
$593$	−23122.5	−1.60123	−0.800614	−	0.599181i	$-0.795493\pi$
$593$	−23122.5	−1.60123	−0.800614	+	0.599181i	$0.795493\pi$
$594$	−2152.17	−0.148661
$595$	−22031.8	−1.51801
$596$	−15790.5	−1.08524
$597$	16115.2	1.10477
$598$	2836.58	0.193974
$599$	−28372.3	−1.93533	−0.967665	−	0.252239i	$-0.918833\pi$
$599$	−28372.3	−1.93533	−0.967665	+	0.252239i	$0.918833\pi$
$600$	4147.98	0.282234
$601$	−15958.6	−1.08314	−0.541569	−	0.840656i	$-0.682170\pi$
$601$	−15958.6	−1.08314	−0.541569	+	0.840656i	$0.682170\pi$
$602$	−10607.9	−0.718182
$603$	3376.42	0.228024
$604$	−14897.7	−1.00361
$605$	−17103.1	−1.14932
$606$	3755.01	0.251711
$607$	12001.9	0.802540	0.401270	−	0.915960i	$-0.368569\pi$
$607$	12001.9	0.802540	0.401270	+	0.915960i	$0.368569\pi$
$608$	32450.5	2.16454
$609$	563.351	0.0374846
$610$	−23052.0	−1.53008
$611$	19493.5	1.29071
$612$	−7415.67	−0.489805
$613$	−17040.5	−1.12277	−0.561384	−	0.827555i	$-0.689731\pi$
$613$	−17040.5	−1.12277	−0.561384	+	0.827555i	$0.689731\pi$
$614$	−18471.1	−1.21406
$615$	20113.4	1.31878
$616$	2122.13	0.138804
$617$	−23366.1	−1.52461	−0.762304	−	0.647219i	$-0.775932\pi$
$617$	−23366.1	−1.52461	−0.762304	+	0.647219i	$0.775932\pi$
$618$	−465.892	−0.0303251
$619$	3475.30	0.225661	0.112831	−	0.993614i	$-0.464008\pi$
$619$	3475.30	0.225661	0.112831	+	0.993614i	$0.464008\pi$
$620$	37097.0	2.40298
$621$	571.797	0.0369491
$622$	−44471.3	−2.86678
$623$	4071.97	0.261862
$624$	4431.59	0.284304
$625$	−4895.84	−0.313334
$626$	27399.5	1.74937
$627$	7174.67	0.456984
$628$	−777.303	−0.0493913
$629$	305.008	0.0193346
$630$	−9932.92	−0.628154
$631$	16882.7	1.06512	0.532561	−	0.846392i	$-0.321230\pi$
$631$	16882.7	1.06512	0.532561	+	0.846392i	$0.321230\pi$
$632$	−7038.49	−0.443000
$633$	−5690.77	−0.357326
$634$	−11086.9	−0.694504
$635$	36084.5	2.25507
$636$	−7943.33	−0.495242
$637$	−3842.94	−0.239031
$638$	−1004.61	−0.0623400
$639$	−8891.27	−0.550443
$640$	16519.0	1.02027
$641$	19928.1	1.22794	0.613971	−	0.789328i	$-0.289571\pi$
$641$	19928.1	1.22794	0.613971	+	0.789328i	$0.289571\pi$
$642$	−6064.89	−0.372838
$643$	−27843.0	−1.70765	−0.853827	−	0.520556i	$-0.825725\pi$
$643$	−27843.0	−1.70765	−0.853827	+	0.520556i	$0.825725\pi$
$644$	−3088.12	−0.188958
$645$	−8894.86	−0.543000
$646$	44929.5	2.73642
$647$	28838.3	1.75232	0.876160	−	0.482021i	$-0.160097\pi$
$647$	28838.3	1.75232	0.876160	+	0.482021i	$0.160097\pi$
$648$	−610.406	−0.0370047
$649$	1115.11	0.0674449
$650$	−24575.3	−1.48296
$651$	−9646.72	−0.580776
$652$	22062.9	1.32523
$653$	17697.2	1.06056	0.530280	−	0.847823i	$-0.322087\pi$
$653$	17697.2	1.06056	0.530280	+	0.847823i	$0.322087\pi$
$654$	11979.0	0.716231
$655$	−10578.7	−0.631057
$656$	17755.1	1.05674
$657$	−3235.28	−0.192116
$658$	−38569.6	−2.28511
$659$	21288.4	1.25839	0.629196	−	0.777247i	$-0.283384\pi$
$659$	21288.4	1.25839	0.629196	+	0.777247i	$0.283384\pi$
$660$	9746.29	0.574809
$661$	−3564.16	−0.209727	−0.104864	−	0.994487i	$-0.533441\pi$
$661$	−3564.16	−0.209727	−0.104864	+	0.994487i	$0.533441\pi$
$662$	15516.2	0.910957
$663$	8021.49	0.469878
$664$	−9515.07	−0.556109
$665$	33113.2	1.93094
$666$	137.511	0.00800068
$667$	266.908	0.0154943
$668$	−3740.76	−0.216668
$669$	8426.54	0.486979
$670$	−27789.1	−1.60237
$671$	−5881.83	−0.338399
$672$	−11463.0	−0.658030
$673$	803.878	0.0460434	0.0230217	−	0.999735i	$-0.492671\pi$
$673$	803.878	0.0460434	0.0230217	+	0.999735i	$0.492671\pi$
$674$	−5582.80	−0.319053
$675$	−4953.88	−0.282481
$676$	−11630.3	−0.661713
$677$	−14587.1	−0.828106	−0.414053	−	0.910253i	$-0.635887\pi$
$677$	−14587.1	−0.828106	−0.414053	+	0.910253i	$0.635887\pi$
$678$	26969.8	1.52768
$679$	−13367.9	−0.755541
$680$	11143.2	0.628416
$681$	−9908.48	−0.557554
$682$	17202.8	0.965878
$683$	19187.3	1.07494	0.537469	−	0.843284i	$-0.319381\pi$
$683$	19187.3	1.07494	0.537469	+	0.843284i	$0.319381\pi$
$684$	11145.5	0.623042
$685$	33709.0	1.88023
$686$	29157.1	1.62277
$687$	17340.6	0.963005
$688$	−7851.93	−0.435105
$689$	8592.26	0.475093
$690$	−4706.09	−0.259649
$691$	19785.3	1.08924	0.544622	−	0.838682i	$-0.316673\pi$
$691$	19785.3	1.08924	0.544622	+	0.838682i	$0.316673\pi$
$692$	−9100.03	−0.499901
$693$	−2534.43	−0.138925
$694$	−23370.1	−1.27827
$695$	−7877.56	−0.429946
$696$	−284.931	−0.0155176
$697$	32138.0	1.74651
$698$	27794.9	1.50724
$699$	−14678.5	−0.794267
$700$	26754.5	1.44461
$701$	−14311.7	−0.771104	−0.385552	−	0.922686i	$-0.625989\pi$
$701$	−14311.7	−0.771104	−0.385552	+	0.922686i	$0.625989\pi$
$702$	3616.44	0.194436
$703$	−458.419	−0.0245940
$704$	13409.0	0.717857
$705$	−32341.1	−1.72771
$706$	23757.3	1.26646
$707$	4421.95	0.235226
$708$	1732.27	0.0919530
$709$	−11747.7	−0.622277	−0.311139	−	0.950365i	$-0.600710\pi$
$709$	−11747.7	−0.622277	−0.311139	+	0.950365i	$0.600710\pi$
$710$	73178.3	3.86807
$711$	8405.98	0.443388
$712$	−2059.51	−0.108404
$713$	−4570.49	−0.240065
$714$	−15871.2	−0.831884
$715$	−10542.5	−0.551423
$716$	−4982.53	−0.260064
$717$	19384.2	1.00964
$718$	−11271.7	−0.585874
$719$	1263.52	0.0655374	0.0327687	−	0.999463i	$-0.489568\pi$
$719$	1263.52	0.0655374	0.0327687	+	0.999463i	$0.489568\pi$
$720$	−7352.31	−0.380562
$721$	−548.642	−0.0283391
$722$	−38600.3	−1.98969
$723$	182.319	0.00937829
$724$	8072.82	0.414398
$725$	−2312.41	−0.118456
$726$	−12320.7	−0.629838
$727$	−15608.6	−0.796275	−0.398138	−	0.917326i	$-0.630343\pi$
$727$	−15608.6	−0.796275	−0.398138	+	0.917326i	$0.630343\pi$
$728$	−3565.96	−0.181543
$729$	729.000	0.0370370
$730$	26627.5	1.35004
$731$	−14212.6	−0.719111
$732$	−9137.17	−0.461365
$733$	5555.30	0.279931	0.139966	−	0.990156i	$-0.455301\pi$
$733$	5555.30	0.279931	0.139966	+	0.990156i	$0.455301\pi$
$734$	12154.3	0.611205
$735$	6375.70	0.319961
$736$	−5431.03	−0.271998
$737$	−7090.51	−0.354386
$738$	14489.2	0.722705
$739$	−6833.98	−0.340179	−0.170089	−	0.985429i	$-0.554406\pi$
$739$	−6833.98	−0.340179	−0.170089	+	0.985429i	$0.554406\pi$
$740$	−622.730	−0.0309352
$741$	−12056.1	−0.597694
$742$	−17000.5	−0.841117
$743$	35475.6	1.75165	0.875823	−	0.482633i	$-0.160319\pi$
$743$	35475.6	1.75165	0.875823	+	0.482633i	$0.160319\pi$
$744$	4879.11	0.240426
$745$	−28337.8	−1.39358
$746$	−30996.5	−1.52126
$747$	11363.7	0.556596
$748$	15573.0	0.761237
$749$	−7142.10	−0.348420
$750$	12994.7	0.632666
$751$	−33087.6	−1.60770	−0.803850	−	0.594832i	$-0.797218\pi$
$751$	−33087.6	−1.60770	−0.803850	+	0.594832i	$0.797218\pi$
$752$	−28549.1	−1.38441
$753$	11346.4	0.549118
$754$	1688.11	0.0815351
$755$	−26735.5	−1.28875
$756$	−3937.13	−0.189407
$757$	−11497.2	−0.552013	−0.276007	−	0.961156i	$-0.589011\pi$
$757$	−11497.2	−0.552013	−0.276007	+	0.961156i	$0.589011\pi$
$758$	−25310.1	−1.21280
$759$	−1200.78	−0.0574250
$760$	−16747.9	−0.799358
$761$	7865.73	0.374681	0.187341	−	0.982295i	$-0.440013\pi$
$761$	7865.73	0.374681	0.187341	+	0.982295i	$0.440013\pi$
$762$	25994.4	1.23580
$763$	14106.6	0.669323
$764$	−36026.3	−1.70600
$765$	−13308.2	−0.628966
$766$	33082.5	1.56047
$767$	−1873.79	−0.0882119
$768$	−5127.30	−0.240906
$769$	8020.94	0.376128	0.188064	−	0.982157i	$-0.439779\pi$
$769$	8020.94	0.376128	0.188064	+	0.982157i	$0.439779\pi$
$770$	20859.3	0.976254
$771$	218.228	0.0101936
$772$	−29861.3	−1.39214
$773$	−17731.7	−0.825053	−0.412526	−	0.910946i	$-0.635354\pi$
$773$	−17731.7	−0.825053	−0.412526	+	0.910946i	$0.635354\pi$
$774$	−6407.64	−0.297568
$775$	39597.4	1.83533
$776$	6761.19	0.312774
$777$	161.935	0.00747670
$778$	2423.36	0.111673
$779$	−48302.5	−2.22159
$780$	−16377.3	−0.751798
$781$	18671.8	0.855479
$782$	−7519.57	−0.343861
$783$	340.289	0.0155312
$784$	5628.15	0.256384
$785$	−1394.95	−0.0634242
$786$	−7620.61	−0.345825
$787$	−10866.2	−0.492170	−0.246085	−	0.969248i	$-0.579144\pi$
$787$	−10866.2	−0.492170	−0.246085	+	0.969248i	$0.579144\pi$
$788$	−52189.0	−2.35934
$789$	14362.5	0.648058
$790$	−69184.1	−3.11577
$791$	31760.1	1.42763
$792$	1281.86	0.0575113
$793$	9883.63	0.442595
$794$	60431.9	2.70107
$795$	−14255.2	−0.635947
$796$	−52572.1	−2.34092
$797$	−36334.0	−1.61483	−0.807414	−	0.589985i	$-0.799134\pi$
$797$	−36334.0	−1.61483	−0.807414	+	0.589985i	$0.799134\pi$
$798$	23854.0	1.05817
$799$	−51675.8	−2.28806
$800$	47052.8	2.07946
$801$	2459.65	0.108499
$802$	−35597.4	−1.56731
$803$	6794.13	0.298580
$804$	−11014.8	−0.483162
$805$	−5541.96	−0.242644
$806$	−28907.0	−1.26328
$807$	−9960.53	−0.434483
$808$	−2236.53	−0.0973772
$809$	29433.4	1.27914	0.639569	−	0.768734i	$-0.279113\pi$
$809$	29433.4	1.27914	0.639569	+	0.768734i	$0.279113\pi$
$810$	−5999.92	−0.260266
$811$	−6295.17	−0.272569	−0.136284	−	0.990670i	$-0.543516\pi$
$811$	−6295.17	−0.272569	−0.136284	+	0.990670i	$0.543516\pi$
$812$	−1837.81	−0.0794266
$813$	−21443.8	−0.925053
$814$	−288.775	−0.0124344
$815$	39594.3	1.70175
$816$	−11747.8	−0.503990
$817$	21361.1	0.914723
$818$	16848.8	0.720175
$819$	4258.77	0.181702
$820$	−65615.6	−2.79439
$821$	39899.7	1.69611	0.848057	−	0.529905i	$-0.177772\pi$
$821$	39899.7	1.69611	0.848057	+	0.529905i	$0.177772\pi$
$822$	24283.2	1.03038
$823$	−1122.81	−0.0475560	−0.0237780	−	0.999717i	$-0.507569\pi$
$823$	−1122.81	−0.0475560	−0.0237780	+	0.999717i	$0.507569\pi$
$824$	277.491	0.0117316
$825$	10403.2	0.439022
$826$	3707.45	0.156173
$827$	−11050.5	−0.464649	−0.232325	−	0.972638i	$-0.574633\pi$
$827$	−11050.5	−0.464649	−0.232325	+	0.972638i	$0.574633\pi$
$828$	−1865.36	−0.0782920
$829$	28472.9	1.19289	0.596444	−	0.802654i	$-0.296580\pi$
$829$	28472.9	1.19289	0.596444	+	0.802654i	$0.296580\pi$
$830$	−93527.4	−3.91131
$831$	−13833.6	−0.577474
$832$	−22532.1	−0.938893
$833$	10187.3	0.423734
$834$	−5674.80	−0.235614
$835$	−6713.19	−0.278227
$836$	−23405.8	−0.968308
$837$	−5827.05	−0.240636
$838$	7952.88	0.327837
$839$	6906.38	0.284189	0.142095	−	0.989853i	$-0.454616\pi$
$839$	6906.38	0.284189	0.142095	+	0.989853i	$0.454616\pi$
$840$	5916.16	0.243008
$841$	−24230.2	−0.993487
$842$	39088.9	1.59987
$843$	−24923.0	−1.01826
$844$	18564.9	0.757143
$845$	−20871.7	−0.849716
$846$	−23297.7	−0.946800
$847$	−14509.0	−0.588589
$848$	−12583.7	−0.509584
$849$	24545.1	0.992210
$850$	65147.3	2.62886
$851$	76.7228	0.00309051
$852$	29005.8	1.16634
$853$	−22035.5	−0.884504	−0.442252	−	0.896891i	$-0.645820\pi$
$853$	−22035.5	−0.884504	−0.442252	+	0.896891i	$0.645820\pi$
$854$	−19555.6	−0.783582
$855$	20001.9	0.800057
$856$	3612.32	0.144237
$857$	39983.9	1.59373	0.796864	−	0.604159i	$-0.206491\pi$
$857$	39983.9	1.59373	0.796864	+	0.604159i	$0.206491\pi$
$858$	−7594.57	−0.302185
$859$	−17967.4	−0.713668	−0.356834	−	0.934168i	$-0.616144\pi$
$859$	−17967.4	−0.713668	−0.356834	+	0.934168i	$0.616144\pi$
$860$	29017.5	1.15057
$861$	17062.7	0.675373
$862$	14112.0	0.557608
$863$	42533.4	1.67770	0.838850	−	0.544363i	$-0.183229\pi$
$863$	42533.4	1.67770	0.838850	+	0.544363i	$0.183229\pi$
$864$	−6924.18	−0.272645
$865$	−16331.0	−0.641930
$866$	3913.15	0.153550
$867$	−6525.36	−0.255609
$868$	31470.3	1.23061
$869$	−17652.7	−0.689097
$870$	−2800.69	−0.109141
$871$	11914.7	0.463505
$872$	−7134.82	−0.277082
$873$	−8074.80	−0.313048
$874$	11301.7	0.437398
$875$	15302.8	0.591231
$876$	10554.4	0.407077
$877$	48368.0	1.86234	0.931169	−	0.364589i	$-0.118791\pi$
$877$	48368.0	1.86234	0.931169	+	0.364589i	$0.118791\pi$
$878$	−42699.6	−1.64128
$879$	794.585	0.0304900
$880$	15439.9	0.591455
$881$	29172.9	1.11562	0.557810	−	0.829969i	$-0.311642\pi$
$881$	29172.9	1.11562	0.557810	+	0.829969i	$0.311642\pi$
$882$	4592.90	0.175341
$883$	−17067.5	−0.650471	−0.325235	−	0.945633i	$-0.605444\pi$
$883$	−17067.5	−0.650471	−0.325235	+	0.945633i	$0.605444\pi$
$884$	−26168.3	−0.995630
$885$	3108.74	0.118078
$886$	27381.5	1.03826
$887$	19672.0	0.744671	0.372335	−	0.928098i	$-0.378557\pi$
$887$	19672.0	0.744671	0.372335	+	0.928098i	$0.378557\pi$
$888$	−81.9033	−0.00309515
$889$	30611.4	1.15486
$890$	−20243.8	−0.762441
$891$	−1530.91	−0.0575616
$892$	−27489.7	−1.03187
$893$	77667.3	2.91046
$894$	−20413.9	−0.763693
$895$	−8941.68	−0.333953
$896$	14013.5	0.522499
$897$	2017.75	0.0751067
$898$	74682.4	2.77526
$899$	−2720.00	−0.100909
$900$	16160.9	0.598553
$901$	−22777.4	−0.842205
$902$	−30427.6	−1.12320
$903$	−7545.73	−0.278080
$904$	−16063.6	−0.591002
$905$	14487.5	0.532134
$906$	−19259.6	−0.706245
$907$	−6730.37	−0.246393	−0.123196	−	0.992382i	$-0.539315\pi$
$907$	−6730.37	−0.246393	−0.123196	+	0.992382i	$0.539315\pi$
$908$	32324.2	1.18141
$909$	2671.05	0.0974623
$910$	−35051.2	−1.27685
$911$	5094.05	0.185262	0.0926308	−	0.995701i	$-0.470472\pi$
$911$	5094.05	0.185262	0.0926308	+	0.995701i	$0.470472\pi$
$912$	17656.6	0.641085
$913$	−23864.0	−0.865040
$914$	−45486.2	−1.64612
$915$	−16397.6	−0.592446
$916$	−56569.8	−2.04052
$917$	−8974.14	−0.323176
$918$	−9586.91	−0.344679
$919$	−18873.8	−0.677464	−0.338732	−	0.940883i	$-0.609998\pi$
$919$	−18873.8	−0.677464	−0.338732	+	0.940883i	$0.609998\pi$
$920$	2803.00	0.100448
$921$	−13139.1	−0.470084
$922$	61205.7	2.18623
$923$	−31375.4	−1.11889
$924$	8268.02	0.294370
$925$	−664.703	−0.0236274
$926$	40646.9	1.44248
$927$	−331.404	−0.0117419
$928$	−3232.13	−0.114332
$929$	42077.8	1.48603	0.743017	−	0.669272i	$-0.233394\pi$
$929$	42077.8	1.48603	0.743017	+	0.669272i	$0.233394\pi$
$930$	47958.6	1.69100
$931$	−15311.3	−0.538998
$932$	47885.4	1.68298
$933$	−31633.9	−1.11002
$934$	30620.0	1.07272
$935$	27947.4	0.977516
$936$	−2154.00	−0.0752196
$937$	−16791.9	−0.585452	−0.292726	−	0.956196i	$-0.594562\pi$
$937$	−16791.9	−0.585452	−0.292726	+	0.956196i	$0.594562\pi$
$938$	−23574.2	−0.820601
$939$	19490.1	0.677355
$940$	105506.	3.66087
$941$	−21099.3	−0.730943	−0.365472	−	0.930822i	$-0.619092\pi$
$941$	−21099.3	−0.730943	−0.365472	+	0.930822i	$0.619092\pi$
$942$	−1004.89	−0.0347570
$943$	8084.10	0.279167
$944$	2744.24	0.0946159
$945$	−7065.60	−0.243221
$946$	13456.1	0.462470
$947$	−13159.7	−0.451567	−0.225783	−	0.974178i	$-0.572494\pi$
$947$	−13159.7	−0.451567	−0.225783	+	0.974178i	$0.572494\pi$
$948$	−27422.6	−0.939500
$949$	−11416.6	−0.390516
$950$	−97914.5	−3.34396
$951$	−7886.43	−0.268912
$952$	9453.08	0.321824
$953$	−14229.9	−0.483685	−0.241843	−	0.970315i	$-0.577752\pi$
$953$	−14229.9	−0.483685	−0.241843	+	0.970315i	$0.577752\pi$
$954$	−10269.1	−0.348505
$955$	−64652.9	−2.19070
$956$	−63236.6	−2.13935
$957$	−714.611	−0.0241380
$958$	−50901.9	−1.71667
$959$	28596.2	0.962898
$960$	37382.2	1.25678
$961$	16785.8	0.563453
$962$	485.248	0.0162630
$963$	−4314.15	−0.144363
$964$	−594.774	−0.0198718
$965$	−53589.3	−1.78767
$966$	−3992.29	−0.132971
$967$	−43103.2	−1.43341	−0.716704	−	0.697377i	$-0.754350\pi$
$967$	−43103.2	−1.43341	−0.716704	+	0.697377i	$0.754350\pi$
$968$	7338.33	0.243660
$969$	31959.7	1.05954
$970$	66458.4	2.19985
$971$	−24953.4	−0.824710	−0.412355	−	0.911023i	$-0.635294\pi$
$971$	−24953.4	−0.824710	−0.412355	+	0.911023i	$0.635294\pi$
$972$	−2378.20	−0.0784782
$973$	−6682.73	−0.220183
$974$	25372.0	0.834673
$975$	−17481.2	−0.574201
$976$	−14475.0	−0.474726
$977$	−7739.34	−0.253432	−0.126716	−	0.991939i	$-0.540444\pi$
$977$	−7739.34	−0.253432	−0.126716	+	0.991939i	$0.540444\pi$
$978$	28522.8	0.932574
$979$	−5165.29	−0.168625
$980$	−20799.3	−0.677969
$981$	8521.02	0.277324
$982$	−16966.0	−0.551330
$983$	19799.0	0.642410	0.321205	−	0.947010i	$-0.395912\pi$
$983$	19799.0	0.642410	0.321205	+	0.947010i	$0.395912\pi$
$984$	−8629.96	−0.279587
$985$	−93658.7	−3.02966
$986$	−4475.06	−0.144538
$987$	−27435.8	−0.884792
$988$	39330.3	1.26646
$989$	−3575.07	−0.114945
$990$	12599.9	0.404497
$991$	10262.8	0.328968	0.164484	−	0.986380i	$-0.447404\pi$
$991$	10262.8	0.328968	0.164484	+	0.986380i	$0.447404\pi$
$992$	55346.4	1.77142
$993$	11037.1	0.352722
$994$	62079.0	1.98091
$995$	−94346.2	−3.00601
$996$	−37071.6	−1.17938
$997$	−18748.2	−0.595549	−0.297774	−	0.954636i	$-0.596244\pi$
$997$	−18748.2	−0.595549	−0.297774	+	0.954636i	$0.596244\pi$
$998$	59520.0	1.88785
$999$	97.8160	0.00309786

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.a.c.1.2	✓	8
3.2	odd	2		531.4.a.f.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.c.1.2	✓	8	1.1	even	1	trivial
531.4.a.f.1.7		8	3.2	odd	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q-4.21744 q^{2} -3.00000 q^{3} +9.78684 q^{4} +17.5635 q^{5} +12.6523 q^{6} +14.8996 q^{7} -7.53588 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-4.21744 q^{2} -3.00000 q^{3} +9.78684 q^{4} +17.5635 q^{5} +12.6523 q^{6} +14.8996 q^{7} -7.53588 q^{8} +9.00000 q^{9} -74.0731 q^{10} -18.9001 q^{11} -29.3605 q^{12} +31.7591 q^{13} -62.8381 q^{14} -52.6905 q^{15} -46.5125 q^{16} -84.1910 q^{17} -37.9570 q^{18} +126.537 q^{19} +171.891 q^{20} -44.6987 q^{21} +79.7102 q^{22} -21.1777 q^{23} +22.6076 q^{24} +183.477 q^{25} -133.942 q^{26} -27.0000 q^{27} +145.820 q^{28} -12.6033 q^{29} +222.219 q^{30} +215.817 q^{31} +256.451 q^{32} +56.7003 q^{33} +355.071 q^{34} +261.689 q^{35} +88.0815 q^{36} -3.62282 q^{37} -533.661 q^{38} -95.2774 q^{39} -132.357 q^{40} -381.728 q^{41} +188.514 q^{42} +168.813 q^{43} -184.972 q^{44} +158.072 q^{45} +89.3156 q^{46} +613.793 q^{47} +139.538 q^{48} -121.003 q^{49} -773.804 q^{50} +252.573 q^{51} +310.821 q^{52} +270.545 q^{53} +113.871 q^{54} -331.952 q^{55} -112.281 q^{56} -379.610 q^{57} +53.1537 q^{58} -59.0000 q^{59} -515.674 q^{60} +311.206 q^{61} -910.195 q^{62} +134.096 q^{63} -709.468 q^{64} +557.802 q^{65} -239.130 q^{66} +375.157 q^{67} -823.963 q^{68} +63.5330 q^{69} -1103.66 q^{70} -987.919 q^{71} -67.8229 q^{72} -359.475 q^{73} +15.2790 q^{74} -550.431 q^{75} +1238.39 q^{76} -281.604 q^{77} +401.827 q^{78} +933.998 q^{79} -816.923 q^{80} +81.0000 q^{81} +1609.92 q^{82} +1262.64 q^{83} -437.459 q^{84} -1478.69 q^{85} -711.960 q^{86} +37.8099 q^{87} +142.429 q^{88} +273.294 q^{89} -666.658 q^{90} +473.197 q^{91} -207.262 q^{92} -647.450 q^{93} -2588.64 q^{94} +2222.43 q^{95} -769.353 q^{96} -897.200 q^{97} +510.323 q^{98} -170.101 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9}+O(q^{10}) \) 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9	\( 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9} + 29 q^{10} - 27 q^{11} - 114 q^{12} + 89 q^{13} - 37 q^{14} + 36 q^{15} + 362 q^{16} + 79 q^{17} + 18 q^{18} + 288 q^{19} + 457 q^{20} - 159 q^{21} + 596 q^{22} + 202 q^{23} - 9 q^{24} + 264 q^{25} + 270 q^{26} - 216 q^{27} + 702 q^{28} - 114 q^{29} - 87 q^{30} + 538 q^{31} + 316 q^{32} + 81 q^{33} + 498 q^{34} - 196 q^{35} + 342 q^{36} + 395 q^{37} + 397 q^{38} - 267 q^{39} + 918 q^{40} - 39 q^{41} + 111 q^{42} + 527 q^{43} + 64 q^{44} - 108 q^{45} - 539 q^{46} + 860 q^{47} - 1086 q^{48} + 347 q^{49} - 591 q^{50} - 237 q^{51} - 644 q^{52} - 812 q^{53} - 54 q^{54} + 536 q^{55} - 2218 q^{56} - 864 q^{57} - 1154 q^{58} - 472 q^{59} - 1371 q^{60} - 460 q^{61} - 2014 q^{62} + 477 q^{63} - 451 q^{64} - 986 q^{65} - 1788 q^{66} + 1934 q^{67} - 69 q^{68} - 606 q^{69} - 1028 q^{70} - 1687 q^{71} + 27 q^{72} + 1980 q^{73} - 2400 q^{74} - 792 q^{75} - 940 q^{76} - 821 q^{77} - 810 q^{78} + 3319 q^{79} - 2119 q^{80} + 648 q^{81} + 429 q^{82} + 2057 q^{83} - 2106 q^{84} + 566 q^{85} - 6690 q^{86} + 342 q^{87} + 1189 q^{88} + 1668 q^{89} + 261 q^{90} + 2427 q^{91} - 980 q^{92} - 1614 q^{93} + 332 q^{94} + 2146 q^{95} - 948 q^{96} + 1956 q^{97} - 2026 q^{98} - 243 q^{99}+O(q^{100}) \) 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9 + 29 * q^10 - 27 * q^11 - 114 * q^12 + 89 * q^13 - 37 * q^14 + 36 * q^15 + 362 * q^16 + 79 * q^17 + 18 * q^18 + 288 * q^19 + 457 * q^20 - 159 * q^21 + 596 * q^22 + 202 * q^23 - 9 * q^24 + 264 * q^25 + 270 * q^26 - 216 * q^27 + 702 * q^28 - 114 * q^29 - 87 * q^30 + 538 * q^31 + 316 * q^32 + 81 * q^33 + 498 * q^34 - 196 * q^35 + 342 * q^36 + 395 * q^37 + 397 * q^38 - 267 * q^39 + 918 * q^40 - 39 * q^41 + 111 * q^42 + 527 * q^43 + 64 * q^44 - 108 * q^45 - 539 * q^46 + 860 * q^47 - 1086 * q^48 + 347 * q^49 - 591 * q^50 - 237 * q^51 - 644 * q^52 - 812 * q^53 - 54 * q^54 + 536 * q^55 - 2218 * q^56 - 864 * q^57 - 1154 * q^58 - 472 * q^59 - 1371 * q^60 - 460 * q^61 - 2014 * q^62 + 477 * q^63 - 451 * q^64 - 986 * q^65 - 1788 * q^66 + 1934 * q^67 - 69 * q^68 - 606 * q^69 - 1028 * q^70 - 1687 * q^71 + 27 * q^72 + 1980 * q^73 - 2400 * q^74 - 792 * q^75 - 940 * q^76 - 821 * q^77 - 810 * q^78 + 3319 * q^79 - 2119 * q^80 + 648 * q^81 + 429 * q^82 + 2057 * q^83 - 2106 * q^84 + 566 * q^85 - 6690 * q^86 + 342 * q^87 + 1189 * q^88 + 1668 * q^89 + 261 * q^90 + 2427 * q^91 - 980 * q^92 - 1614 * q^93 + 332 * q^94 + 2146 * q^95 - 948 * q^96 + 1956 * q^97 - 2026 * q^98 - 243 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more