LMFDB - Embedded newform 51.4.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [51,4,Mod(1,51)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(51, 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("51.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 51 = 3 \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	51.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:	$3.00909741029$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.5912.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 14x - 10 $ x^3 - x^2 - 14*x - 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.75985$ of defining polynomial
Character	$\chi$	$=$	51.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.75985 q^{2} +3.00000 q^{3} +14.6562 q^{4} -7.65616 q^{5} +14.2795 q^{6} -31.5852 q^{7} +31.6823 q^{8} +9.00000 q^{9} -36.4422 q^{10} -7.18910 q^{11} +43.9685 q^{12} +84.3331 q^{13} -150.341 q^{14} -22.9685 q^{15} +33.5537 q^{16} +17.0000 q^{17} +42.8386 q^{18} -37.0838 q^{19} -112.210 q^{20} -94.7557 q^{21} -34.2190 q^{22} +150.218 q^{23} +95.0469 q^{24} -66.3832 q^{25} +401.413 q^{26} +27.0000 q^{27} -462.918 q^{28} -11.5846 q^{29} -109.326 q^{30} -53.2865 q^{31} -93.7478 q^{32} -21.5673 q^{33} +80.9174 q^{34} +241.822 q^{35} +131.905 q^{36} -99.2134 q^{37} -176.513 q^{38} +252.999 q^{39} -242.565 q^{40} +118.249 q^{41} -451.023 q^{42} -456.016 q^{43} -105.365 q^{44} -68.9054 q^{45} +715.014 q^{46} +571.014 q^{47} +100.661 q^{48} +654.627 q^{49} -315.974 q^{50} +51.0000 q^{51} +1236.00 q^{52} +462.867 q^{53} +128.516 q^{54} +55.0409 q^{55} -1000.69 q^{56} -111.251 q^{57} -55.1407 q^{58} +48.0674 q^{59} -336.630 q^{60} +59.5236 q^{61} -253.636 q^{62} -284.267 q^{63} -714.655 q^{64} -645.668 q^{65} -102.657 q^{66} -740.787 q^{67} +249.155 q^{68} +450.653 q^{69} +1151.03 q^{70} -930.437 q^{71} +285.141 q^{72} -697.419 q^{73} -472.241 q^{74} -199.150 q^{75} -543.506 q^{76} +227.070 q^{77} +1204.24 q^{78} +1036.04 q^{79} -256.893 q^{80} +81.0000 q^{81} +562.849 q^{82} -22.2043 q^{83} -1388.75 q^{84} -130.155 q^{85} -2170.57 q^{86} -34.7537 q^{87} -227.767 q^{88} -369.726 q^{89} -327.979 q^{90} -2663.68 q^{91} +2201.62 q^{92} -159.860 q^{93} +2717.94 q^{94} +283.920 q^{95} -281.244 q^{96} +1139.56 q^{97} +3115.93 q^{98} -64.7019 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 5 q^{2} + 9 q^{3} + 13 q^{4} + 8 q^{5} + 15 q^{6} - 8 q^{7} + 33 q^{8} + 27 q^{9} - 38 q^{10} + 34 q^{11} + 39 q^{12} + 36 q^{13} - 104 q^{14} + 24 q^{15} - 79 q^{16} + 51 q^{17} + 45 q^{18} - 142 q^{19}+ \cdots + 306 q^{99}+O(q^{100}) $ 3 * q + 5 * q^2 + 9 * q^3 + 13 * q^4 + 8 * q^5 + 15 * q^6 - 8 * q^7 + 33 * q^8 + 27 * q^9 - 38 * q^10 + 34 * q^11 + 39 * q^12 + 36 * q^13 - 104 * q^14 + 24 * q^15 - 79 * q^16 + 51 * q^17 + 45 * q^18 - 142 * q^19 - 126 * q^20 - 24 * q^21 - 248 * q^22 + 110 * q^23 + 99 * q^24 - 193 * q^25 + 154 * q^26 + 81 * q^27 - 472 * q^28 + 90 * q^29 - 114 * q^30 - 148 * q^31 - 151 * q^32 + 102 * q^33 + 85 * q^34 + 416 * q^35 + 117 * q^36 + 110 * q^37 + 80 * q^38 + 108 * q^39 - 202 * q^40 + 720 * q^41 - 312 * q^42 - 146 * q^43 - 192 * q^44 + 72 * q^45 + 748 * q^46 + 500 * q^47 - 237 * q^48 + 379 * q^49 - 385 * q^50 + 153 * q^51 + 1218 * q^52 + 610 * q^53 + 135 * q^54 + 430 * q^55 - 1368 * q^56 - 426 * q^57 + 1006 * q^58 - 216 * q^59 - 378 * q^60 - 18 * q^61 - 904 * q^62 - 72 * q^63 + 377 * q^64 - 966 * q^65 - 744 * q^66 - 1404 * q^67 + 221 * q^68 + 330 * q^69 + 1472 * q^70 - 960 * q^71 + 297 * q^72 - 794 * q^73 - 1874 * q^74 - 579 * q^75 - 392 * q^76 + 48 * q^77 + 462 * q^78 - 276 * q^79 - 1130 * q^80 + 243 * q^81 + 382 * q^82 - 1552 * q^83 - 1416 * q^84 + 136 * q^85 + 16 * q^86 + 270 * q^87 + 1724 * q^88 + 1394 * q^89 - 342 * q^90 - 3968 * q^91 + 2244 * q^92 - 444 * q^93 + 3960 * q^94 - 602 * q^95 - 453 * q^96 + 402 * q^97 + 4109 * q^98 + 306 * q^99

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 $

$2$	4.75985	1.68286	0.841430	−	0.540366i	$-0.181714\pi$
$2$	4.75985	1.68286	0.841430	+	0.540366i	$0.181714\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	14.6562	1.83202
$5$	−7.65616	−0.684788	−0.342394	−	0.939557i	$-0.611238\pi$
$5$	−7.65616	−0.684788	−0.342394	+	0.939557i	$0.611238\pi$
$6$	14.2795	0.971600
$7$	−31.5852	−1.70544	−0.852721	−	0.522366i	$-0.825049\pi$
$7$	−31.5852	−1.70544	−0.852721	+	0.522366i	$0.825049\pi$
$8$	31.6823	1.40017
$9$	9.00000	0.333333
$10$	−36.4422	−1.15240
$11$	−7.18910	−0.197054	−0.0985271	−	0.995134i	$-0.531413\pi$
$11$	−7.18910	−0.197054	−0.0985271	+	0.995134i	$0.531413\pi$
$12$	43.9685	1.05772
$13$	84.3331	1.79921	0.899607	−	0.436700i	$-0.143853\pi$
$13$	84.3331	1.79921	0.899607	+	0.436700i	$0.143853\pi$
$14$	−150.341	−2.87002
$15$	−22.9685	−0.395362
$16$	33.5537	0.524277
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	42.8386	0.560954
$19$	−37.0838	−0.447769	−0.223885	−	0.974616i	$-0.571874\pi$
$19$	−37.0838	−0.447769	−0.223885	+	0.974616i	$0.571874\pi$
$20$	−112.210	−1.25454
$21$	−94.7557	−0.984638
$22$	−34.2190	−0.331615
$23$	150.218	1.36185	0.680925	−	0.732353i	$-0.261578\pi$
$23$	150.218	1.36185	0.680925	+	0.732353i	$0.261578\pi$
$24$	95.0469	0.808390
$25$	−66.3832	−0.531066
$26$	401.413	3.02783
$27$	27.0000	0.192450
$28$	−462.918	−3.12440
$29$	−11.5846	−0.0741792	−0.0370896	−	0.999312i	$-0.511809\pi$
$29$	−11.5846	−0.0741792	−0.0370896	+	0.999312i	$0.511809\pi$
$30$	−109.326	−0.665340
$31$	−53.2865	−0.308727	−0.154364	−	0.988014i	$-0.549333\pi$
$31$	−53.2865	−0.308727	−0.154364	+	0.988014i	$0.549333\pi$
$32$	−93.7478	−0.517889
$33$	−21.5673	−0.113769
$34$	80.9174	0.408154
$35$	241.822	1.16787
$36$	131.905	0.610673
$37$	−99.2134	−0.440827	−0.220413	−	0.975407i	$-0.570741\pi$
$37$	−99.2134	−0.440827	−0.220413	+	0.975407i	$0.570741\pi$
$38$	−176.513	−0.753533
$39$	252.999	1.03878
$40$	−242.565	−0.958821
$41$	118.249	0.450425	0.225213	−	0.974310i	$-0.427692\pi$
$41$	118.249	0.450425	0.225213	+	0.974310i	$0.427692\pi$
$42$	−451.023	−1.65701
$43$	−456.016	−1.61725	−0.808626	−	0.588323i	$-0.799789\pi$
$43$	−456.016	−1.61725	−0.808626	+	0.588323i	$0.799789\pi$
$44$	−105.365	−0.361007
$45$	−68.9054	−0.228263
$46$	715.014	2.29180
$47$	571.014	1.77215	0.886073	−	0.463545i	$-0.153423\pi$
$47$	571.014	1.77215	0.886073	+	0.463545i	$0.153423\pi$
$48$	100.661	0.302691
$49$	654.627	1.90853
$50$	−315.974	−0.893710
$51$	51.0000	0.140028
$52$	1236.00	3.29620
$53$	462.867	1.19962	0.599809	−	0.800143i	$-0.295243\pi$
$53$	462.867	1.19962	0.599809	+	0.800143i	$0.295243\pi$
$54$	128.516	0.323867
$55$	55.0409	0.134940
$56$	−1000.69	−2.38792
$57$	−111.251	−0.258520
$58$	−55.1407	−0.124833
$59$	48.0674	0.106065	0.0530325	−	0.998593i	$-0.483111\pi$
$59$	48.0674	0.106065	0.0530325	+	0.998593i	$0.483111\pi$
$60$	−336.630	−0.724312
$61$	59.5236	0.124938	0.0624689	−	0.998047i	$-0.480103\pi$
$61$	59.5236	0.124938	0.0624689	+	0.998047i	$0.480103\pi$
$62$	−253.636	−0.519545
$63$	−284.267	−0.568481
$64$	−714.655	−1.39581
$65$	−645.668	−1.23208
$66$	−102.657	−0.191458
$67$	−740.787	−1.35077	−0.675384	−	0.737466i	$-0.736022\pi$
$67$	−740.787	−1.35077	−0.675384	+	0.737466i	$0.736022\pi$
$68$	249.155	0.444330
$69$	450.653	0.786265
$70$	1151.03	1.96536
$71$	−930.437	−1.55525	−0.777623	−	0.628730i	$-0.783575\pi$
$71$	−930.437	−1.55525	−0.777623	+	0.628730i	$0.783575\pi$
$72$	285.141	0.466724
$73$	−697.419	−1.11817	−0.559087	−	0.829109i	$-0.688848\pi$
$73$	−697.419	−1.11817	−0.559087	+	0.829109i	$0.688848\pi$
$74$	−472.241	−0.741850
$75$	−199.150	−0.306611
$76$	−543.506	−0.820322
$77$	227.070	0.336065
$78$	1204.24	1.74812
$79$	1036.04	1.47549	0.737747	−	0.675077i	$-0.235890\pi$
$79$	1036.04	1.47549	0.737747	+	0.675077i	$0.235890\pi$
$80$	−256.893	−0.359018
$81$	81.0000	0.111111
$82$	562.849	0.758003
$83$	−22.2043	−0.0293643	−0.0146822	−	0.999892i	$-0.504674\pi$
$83$	−22.2043	−0.0293643	−0.0146822	+	0.999892i	$0.504674\pi$
$84$	−1388.75	−1.80388
$85$	−130.155	−0.166085
$86$	−2170.57	−2.72161
$87$	−34.7537	−0.0428274
$88$	−227.767	−0.275910
$89$	−369.726	−0.440346	−0.220173	−	0.975461i	$-0.570662\pi$
$89$	−369.726	−0.440346	−0.220173	+	0.975461i	$0.570662\pi$
$90$	−327.979	−0.384134
$91$	−2663.68	−3.06846
$92$	2201.62	2.49494
$93$	−159.860	−0.178244
$94$	2717.94	2.98228
$95$	283.920	0.306627
$96$	−281.244	−0.299003
$97$	1139.56	1.19283	0.596415	−	0.802676i	$-0.296591\pi$
$97$	1139.56	1.19283	0.596415	+	0.802676i	$0.296591\pi$
$98$	3115.93	3.21180
$99$	−64.7019	−0.0656847
$100$	−972.923	−0.972923
$101$	703.083	0.692667	0.346334	−	0.938111i	$-0.387427\pi$
$101$	703.083	0.692667	0.346334	+	0.938111i	$0.387427\pi$
$102$	242.752	0.235648
$103$	−897.160	−0.858250	−0.429125	−	0.903245i	$-0.641178\pi$
$103$	−897.160	−0.858250	−0.429125	+	0.903245i	$0.641178\pi$
$104$	2671.87	2.51921
$105$	725.465	0.674268
$106$	2203.18	2.01879
$107$	−1901.21	−1.71773	−0.858864	−	0.512203i	$-0.828829\pi$
$107$	−1901.21	−1.71773	−0.858864	+	0.512203i	$0.828829\pi$
$108$	395.716	0.352572
$109$	584.555	0.513671	0.256836	−	0.966455i	$-0.417320\pi$
$109$	584.555	0.513671	0.256836	+	0.966455i	$0.417320\pi$
$110$	261.986	0.227086
$111$	−297.640	−0.254511
$112$	−1059.80	−0.894124
$113$	−63.4225	−0.0527990	−0.0263995	−	0.999651i	$-0.508404\pi$
$113$	−63.4225	−0.0527990	−0.0263995	+	0.999651i	$0.508404\pi$
$114$	−529.540	−0.435052
$115$	−1150.09	−0.932578
$116$	−169.785	−0.135898
$117$	758.998	0.599738
$118$	228.793	0.178493
$119$	−536.949	−0.413631
$120$	−727.694	−0.553576
$121$	−1279.32	−0.961170
$122$	283.323	0.210253
$123$	354.748	0.260053
$124$	−780.976	−0.565594
$125$	1465.26	1.04845
$126$	−1353.07	−0.956674
$127$	175.543	0.122653	0.0613266	−	0.998118i	$-0.480467\pi$
$127$	175.543	0.122653	0.0613266	+	0.998118i	$0.480467\pi$
$128$	−2651.67	−1.83107
$129$	−1368.05	−0.933721
$130$	−3073.28	−2.07342
$131$	1865.42	1.24414	0.622070	−	0.782961i	$-0.286292\pi$
$131$	1865.42	1.24414	0.622070	+	0.782961i	$0.286292\pi$
$132$	−316.094	−0.208428
$133$	1171.30	0.763644
$134$	−3526.03	−2.27315
$135$	−206.716	−0.131787
$136$	538.599	0.339592
$137$	1057.57	0.659519	0.329760	−	0.944065i	$-0.393032\pi$
$137$	1057.57	0.659519	0.329760	+	0.944065i	$0.393032\pi$
$138$	2145.04	1.32317
$139$	904.833	0.552136	0.276068	−	0.961138i	$-0.410968\pi$
$139$	904.833	0.552136	0.276068	+	0.961138i	$0.410968\pi$
$140$	3544.18	2.13955
$141$	1713.04	1.02315
$142$	−4428.74	−2.61726
$143$	−606.279	−0.354543
$144$	301.983	0.174759
$145$	88.6932	0.0507970
$146$	−3319.61	−1.88173
$147$	1963.88	1.10189
$148$	−1454.09	−0.807603
$149$	809.001	0.444805	0.222402	−	0.974955i	$-0.428610\pi$
$149$	809.001	0.444805	0.222402	+	0.974955i	$0.428610\pi$
$150$	−947.922	−0.515984
$151$	−352.121	−0.189769	−0.0948847	−	0.995488i	$-0.530248\pi$
$151$	−352.121	−0.189769	−0.0948847	+	0.995488i	$0.530248\pi$
$152$	−1174.90	−0.626954
$153$	153.000	0.0808452
$154$	1080.82	0.565550
$155$	407.970	0.211413
$156$	3708.00	1.90306
$157$	537.882	0.273424	0.136712	−	0.990611i	$-0.456346\pi$
$157$	537.882	0.273424	0.136712	+	0.990611i	$0.456346\pi$
$158$	4931.41	2.48305
$159$	1388.60	0.692599
$160$	717.748	0.354644
$161$	−4744.66	−2.32256
$162$	385.548	0.186985
$163$	1922.74	0.923933	0.461966	−	0.886897i	$-0.347144\pi$
$163$	1922.74	0.923933	0.461966	+	0.886897i	$0.347144\pi$
$164$	1733.08	0.825188
$165$	165.123	0.0779078
$166$	−105.689	−0.0494160
$167$	−2971.76	−1.37702	−0.688509	−	0.725228i	$-0.741734\pi$
$167$	−2971.76	−1.37702	−0.688509	+	0.725228i	$0.741734\pi$
$168$	−3002.08	−1.37866
$169$	4915.07	2.23717
$170$	−619.517	−0.279499
$171$	−333.754	−0.149256
$172$	−6683.45	−2.96284
$173$	988.564	0.434446	0.217223	−	0.976122i	$-0.430300\pi$
$173$	988.564	0.434446	0.217223	+	0.976122i	$0.430300\pi$
$174$	−165.422	−0.0720725
$175$	2096.73	0.905702
$176$	−241.221	−0.103311
$177$	144.202	0.0612367
$178$	−1759.84	−0.741042
$179$	−1937.65	−0.809089	−0.404545	−	0.914518i	$-0.632570\pi$
$179$	−1937.65	−0.809089	−0.404545	+	0.914518i	$0.632570\pi$
$180$	−1009.89	−0.418182
$181$	−2180.07	−0.895267	−0.447634	−	0.894217i	$-0.647733\pi$
$181$	−2180.07	−0.895267	−0.447634	+	0.894217i	$0.647733\pi$
$182$	−12678.7	−5.16379
$183$	178.571	0.0721329
$184$	4759.24	1.90683
$185$	759.594	0.301873
$186$	−760.907	−0.299959
$187$	−122.215	−0.0477927
$188$	8368.86	3.24661
$189$	−852.801	−0.328213
$190$	1351.41	0.516010
$191$	1675.78	0.634845	0.317423	−	0.948284i	$-0.397183\pi$
$191$	1675.78	0.634845	0.317423	+	0.948284i	$0.397183\pi$
$192$	−2143.97	−0.805872
$193$	−257.961	−0.0962094	−0.0481047	−	0.998842i	$-0.515318\pi$
$193$	−257.961	−0.0962094	−0.0481047	+	0.998842i	$0.515318\pi$
$194$	5424.12	2.00737
$195$	−1937.00	−0.711342
$196$	9594.32	3.49647
$197$	−693.466	−0.250799	−0.125399	−	0.992106i	$-0.540021\pi$
$197$	−693.466	−0.250799	−0.125399	+	0.992106i	$0.540021\pi$
$198$	−307.971	−0.110538
$199$	−240.295	−0.0855984	−0.0427992	−	0.999084i	$-0.513628\pi$
$199$	−240.295	−0.0855984	−0.0427992	+	0.999084i	$0.513628\pi$
$200$	−2103.17	−0.743584
$201$	−2222.36	−0.779866
$202$	3346.57	1.16566
$203$	365.901	0.126508
$204$	747.464	0.256534
$205$	−905.335	−0.308446
$206$	−4270.34	−1.44432
$207$	1351.96	0.453950
$208$	2829.69	0.943287
$209$	266.599	0.0882348
$210$	3453.10	1.13470
$211$	268.114	0.0874774	0.0437387	−	0.999043i	$-0.486073\pi$
$211$	268.114	0.0874774	0.0437387	+	0.999043i	$0.486073\pi$
$212$	6783.86	2.19772
$213$	−2791.31	−0.897922
$214$	−9049.47	−2.89070
$215$	3491.33	1.10747
$216$	855.422	0.269463
$217$	1683.07	0.526516
$218$	2782.39	0.864437
$219$	−2092.26	−0.645578
$220$	806.688	0.247213
$221$	1433.66	0.436374
$222$	−1416.72	−0.428307
$223$	−5524.43	−1.65894	−0.829468	−	0.558554i	$-0.811357\pi$
$223$	−5524.43	−1.65894	−0.829468	+	0.558554i	$0.811357\pi$
$224$	2961.05	0.883229
$225$	−597.449	−0.177022
$226$	−301.882	−0.0888534
$227$	−384.400	−0.112394	−0.0561972	−	0.998420i	$-0.517898\pi$
$227$	−384.400	−0.112394	−0.0561972	+	0.998420i	$0.517898\pi$
$228$	−1630.52	−0.473613
$229$	1395.48	0.402690	0.201345	−	0.979520i	$-0.435469\pi$
$229$	1395.48	0.402690	0.201345	+	0.979520i	$0.435469\pi$
$230$	−5474.26	−1.56940
$231$	681.209	0.194027
$232$	−367.025	−0.103864
$233$	3409.39	0.958613	0.479307	−	0.877648i	$-0.340888\pi$
$233$	3409.39	0.958613	0.479307	+	0.877648i	$0.340888\pi$
$234$	3612.71	1.00928
$235$	−4371.77	−1.21354
$236$	704.483	0.194313
$237$	3108.13	0.851877
$238$	−2555.80	−0.696083
$239$	−1509.18	−0.408456	−0.204228	−	0.978923i	$-0.565468\pi$
$239$	−1509.18	−0.408456	−0.204228	+	0.978923i	$0.565468\pi$
$240$	−770.678	−0.207279
$241$	3406.91	0.910615	0.455307	−	0.890334i	$-0.349529\pi$
$241$	3406.91	0.910615	0.455307	+	0.890334i	$0.349529\pi$
$242$	−6089.35	−1.61751
$243$	243.000	0.0641500
$244$	872.387	0.228889
$245$	−5011.93	−1.30694
$246$	1688.55	0.437633
$247$	−3127.39	−0.805633
$248$	−1688.24	−0.432271
$249$	−66.6129	−0.0169535
$250$	6974.42	1.76440
$251$	3394.43	0.853605	0.426802	−	0.904345i	$-0.359640\pi$
$251$	3394.43	0.853605	0.426802	+	0.904345i	$0.359640\pi$
$252$	−4166.26	−1.04147
$253$	−1079.93	−0.268358
$254$	835.560	0.206408
$255$	−390.464	−0.0958894
$256$	−6904.30	−1.68562
$257$	1778.91	0.431772	0.215886	−	0.976419i	$-0.430736\pi$
$257$	1778.91	0.431772	0.215886	+	0.976419i	$0.430736\pi$
$258$	−6511.71	−1.57132
$259$	3133.68	0.751804
$260$	−9463.01	−2.25719
$261$	−104.261	−0.0247264
$262$	8879.11	2.09372
$263$	−4316.88	−1.01213	−0.506065	−	0.862495i	$-0.668900\pi$
$263$	−4316.88	−1.01213	−0.506065	+	0.862495i	$0.668900\pi$
$264$	−683.302	−0.159297
$265$	−3543.79	−0.821483
$266$	5575.22	1.28511
$267$	−1109.18	−0.254234
$268$	−10857.1	−2.47463
$269$	6546.31	1.48378	0.741888	−	0.670524i	$-0.233931\pi$
$269$	6546.31	1.48378	0.741888	+	0.670524i	$0.233931\pi$
$270$	−983.938	−0.221780
$271$	3785.20	0.848466	0.424233	−	0.905553i	$-0.360544\pi$
$271$	3785.20	0.848466	0.424233	+	0.905553i	$0.360544\pi$
$272$	570.413	0.127156
$273$	−7991.04	−1.77157
$274$	5033.86	1.10988
$275$	477.236	0.104649
$276$	6604.85	1.44045
$277$	−3521.06	−0.763755	−0.381878	−	0.924213i	$-0.624722\pi$
$277$	−3521.06	−0.763755	−0.381878	+	0.924213i	$0.624722\pi$
$278$	4306.87	0.929168
$279$	−479.579	−0.102909
$280$	7661.47	1.63521
$281$	−2922.30	−0.620391	−0.310195	−	0.950673i	$-0.600394\pi$
$281$	−2922.30	−0.620391	−0.310195	+	0.950673i	$0.600394\pi$
$282$	8153.81	1.72182
$283$	735.075	0.154402	0.0772008	−	0.997016i	$-0.475402\pi$
$283$	735.075	0.154402	0.0772008	+	0.997016i	$0.475402\pi$
$284$	−13636.6	−2.84924
$285$	851.759	0.177031
$286$	−2885.80	−0.596646
$287$	−3734.93	−0.768175
$288$	−843.731	−0.172630
$289$	289.000	0.0588235
$290$	422.166	0.0854843
$291$	3418.67	0.688681
$292$	−10221.5	−2.04852
$293$	−8702.92	−1.73526	−0.867628	−	0.497213i	$-0.834357\pi$
$293$	−8702.92	−1.73526	−0.867628	+	0.497213i	$0.834357\pi$
$294$	9347.78	1.85433
$295$	−368.011	−0.0726320
$296$	−3143.31	−0.617234
$297$	−194.106	−0.0379231
$298$	3850.72	0.748545
$299$	12668.3	2.45026
$300$	−2918.77	−0.561717
$301$	14403.4	2.75813
$302$	−1676.04	−0.319355
$303$	2109.25	0.399912
$304$	−1244.30	−0.234755
$305$	−455.722	−0.0855559
$306$	728.257	0.136051
$307$	2516.95	0.467916	0.233958	−	0.972247i	$-0.424832\pi$
$307$	2516.95	0.467916	0.233958	+	0.972247i	$0.424832\pi$
$308$	3327.97	0.615677
$309$	−2691.48	−0.495511
$310$	1941.88	0.355778
$311$	6593.31	1.20216	0.601081	−	0.799188i	$-0.294737\pi$
$311$	6593.31	1.20216	0.601081	+	0.799188i	$0.294737\pi$
$312$	8015.60	1.45447
$313$	4392.99	0.793312	0.396656	−	0.917967i	$-0.370171\pi$
$313$	4392.99	0.793312	0.396656	+	0.917967i	$0.370171\pi$
$314$	2560.24	0.460135
$315$	2176.39	0.389289
$316$	15184.4	2.70314
$317$	2601.23	0.460882	0.230441	−	0.973086i	$-0.425983\pi$
$317$	2601.23	0.460882	0.230441	+	0.973086i	$0.425983\pi$
$318$	6609.54	1.16555
$319$	83.2825	0.0146173
$320$	5471.51	0.955834
$321$	−5703.63	−0.991731
$322$	−22583.9	−3.90854
$323$	−630.425	−0.108600
$324$	1187.15	0.203558
$325$	−5598.30	−0.955502
$326$	9151.97	1.55485
$327$	1753.66	0.296568
$328$	3746.41	0.630674
$329$	−18035.6	−3.02229
$330$	785.959	0.131108
$331$	−4670.49	−0.775568	−0.387784	−	0.921750i	$-0.626759\pi$
$331$	−4670.49	−0.775568	−0.387784	+	0.921750i	$0.626759\pi$
$332$	−325.430	−0.0537960
$333$	−892.921	−0.146942
$334$	−14145.1	−2.31733
$335$	5671.58	0.924989
$336$	−3179.41	−0.516223
$337$	1801.67	0.291226	0.145613	−	0.989342i	$-0.453485\pi$
$337$	1801.67	0.291226	0.145613	+	0.989342i	$0.453485\pi$
$338$	23395.0	3.76485
$339$	−190.268	−0.0304835
$340$	−1907.57	−0.304272
$341$	383.082	0.0608360
$342$	−1588.62	−0.251178
$343$	−9842.83	−1.54945
$344$	−14447.7	−2.26443
$345$	−3450.27	−0.538424
$346$	4705.41	0.731112
$347$	168.340	0.0260431	0.0130216	−	0.999915i	$-0.495855\pi$
$347$	168.340	0.0260431	0.0130216	+	0.999915i	$0.495855\pi$
$348$	−509.355	−0.0784606
$349$	−4447.85	−0.682200	−0.341100	−	0.940027i	$-0.610799\pi$
$349$	−4447.85	−0.682200	−0.341100	+	0.940027i	$0.610799\pi$
$350$	9980.12	1.52417
$351$	2276.99	0.346259
$352$	673.963	0.102052
$353$	−10509.5	−1.58460	−0.792298	−	0.610135i	$-0.791115\pi$
$353$	−10509.5	−1.58460	−0.792298	+	0.610135i	$0.791115\pi$
$354$	686.380	0.103053
$355$	7123.57	1.06501
$356$	−5418.76	−0.806723
$357$	−1610.85	−0.238810
$358$	−9222.93	−1.36158
$359$	8342.99	1.22654	0.613268	−	0.789875i	$-0.289855\pi$
$359$	8342.99	1.22654	0.613268	+	0.789875i	$0.289855\pi$
$360$	−2183.08	−0.319607
$361$	−5483.79	−0.799503
$362$	−10376.8	−1.50661
$363$	−3837.95	−0.554932
$364$	−39039.3	−5.62147
$365$	5339.55	0.765712
$366$	849.969	0.121390
$367$	352.402	0.0501232	0.0250616	−	0.999686i	$-0.492022\pi$
$367$	352.402	0.0501232	0.0250616	+	0.999686i	$0.492022\pi$
$368$	5040.36	0.713987
$369$	1064.24	0.150142
$370$	3615.55	0.508009
$371$	−14619.8	−2.04588
$372$	−2342.93	−0.326546
$373$	12563.2	1.74397	0.871983	−	0.489537i	$-0.162834\pi$
$373$	12563.2	1.74397	0.871983	+	0.489537i	$0.162834\pi$
$374$	−581.724	−0.0804284
$375$	4395.78	0.605326
$376$	18091.0	2.48131
$377$	−976.961	−0.133464
$378$	−4059.21	−0.552336
$379$	−1770.57	−0.239969	−0.119984	−	0.992776i	$-0.538284\pi$
$379$	−1770.57	−0.239969	−0.119984	+	0.992776i	$0.538284\pi$
$380$	4161.17	0.561746
$381$	526.630	0.0708139
$382$	7976.47	1.06836
$383$	4330.57	0.577759	0.288880	−	0.957365i	$-0.406717\pi$
$383$	4330.57	0.577759	0.288880	+	0.957365i	$0.406717\pi$
$384$	−7955.00	−1.05717
$385$	−1738.48	−0.230133
$386$	−1227.85	−0.161907
$387$	−4104.15	−0.539084
$388$	16701.5	2.18529
$389$	10295.5	1.34191	0.670957	−	0.741496i	$-0.265883\pi$
$389$	10295.5	1.34191	0.670957	+	0.741496i	$0.265883\pi$
$390$	−9219.84	−1.19709
$391$	2553.70	0.330297
$392$	20740.1	2.67228
$393$	5596.26	0.718305
$394$	−3300.79	−0.422060
$395$	−7932.12	−1.01040
$396$	−948.282	−0.120336
$397$	93.1792	0.0117797	0.00588983	−	0.999983i	$-0.498125\pi$
$397$	93.1792	0.0117797	0.00588983	+	0.999983i	$0.498125\pi$
$398$	−1143.77	−0.144050
$399$	3513.90	0.440890
$400$	−2227.40	−0.278426
$401$	−13320.9	−1.65889	−0.829443	−	0.558591i	$-0.811342\pi$
$401$	−13320.9	−1.65889	−0.829443	+	0.558591i	$0.811342\pi$
$402$	−10578.1	−1.31241
$403$	−4493.82	−0.555466
$404$	10304.5	1.26898
$405$	−620.149	−0.0760875
$406$	1741.63	0.212896
$407$	713.255	0.0868667
$408$	1615.80	0.196063
$409$	9272.21	1.12098	0.560491	−	0.828161i	$-0.310613\pi$
$409$	9272.21	1.12098	0.560491	+	0.828161i	$0.310613\pi$
$410$	−4309.26	−0.519071
$411$	3172.70	0.380774
$412$	−13148.9	−1.57233
$413$	−1518.22	−0.180888
$414$	6435.12	0.763935
$415$	170.000	0.0201083
$416$	−7906.05	−0.931793
$417$	2714.50	0.318776
$418$	1268.97	0.148487
$419$	11325.4	1.32049	0.660244	−	0.751051i	$-0.270453\pi$
$419$	11325.4	1.32049	0.660244	+	0.751051i	$0.270453\pi$
$420$	10632.5	1.23527
$421$	6934.54	0.802776	0.401388	−	0.915908i	$-0.368528\pi$
$421$	6934.54	0.802776	0.401388	+	0.915908i	$0.368528\pi$
$422$	1276.18	0.147212
$423$	5139.12	0.590715
$424$	14664.7	1.67967
$425$	−1128.52	−0.128802
$426$	−13286.2	−1.51108
$427$	−1880.07	−0.213074
$428$	−27864.4	−3.14691
$429$	−1818.84	−0.204695
$430$	16618.2	1.86373
$431$	−2776.55	−0.310306	−0.155153	−	0.987890i	$-0.549587\pi$
$431$	−2776.55	−0.310306	−0.155153	+	0.987890i	$0.549587\pi$
$432$	905.950	0.100897
$433$	−3252.22	−0.360951	−0.180476	−	0.983579i	$-0.557764\pi$
$433$	−3252.22	−0.360951	−0.180476	+	0.983579i	$0.557764\pi$
$434$	8011.15	0.886054
$435$	266.080	0.0293277
$436$	8567.32	0.941056
$437$	−5570.65	−0.609795
$438$	−9958.82	−1.08642
$439$	−13345.7	−1.45093	−0.725464	−	0.688260i	$-0.758375\pi$
$439$	−13345.7	−1.45093	−0.725464	+	0.688260i	$0.758375\pi$
$440$	1743.82	0.188940
$441$	5891.65	0.636178
$442$	6824.02	0.734356
$443$	11639.6	1.24834	0.624169	−	0.781290i	$-0.285438\pi$
$443$	11639.6	1.24834	0.624169	+	0.781290i	$0.285438\pi$
$444$	−4362.26	−0.466270
$445$	2830.68	0.301544
$446$	−26295.4	−2.79176
$447$	2427.00	0.256808
$448$	22572.6	2.38048
$449$	−6937.72	−0.729201	−0.364601	−	0.931164i	$-0.618794\pi$
$449$	−6937.72	−0.729201	−0.364601	+	0.931164i	$0.618794\pi$
$450$	−2843.77	−0.297903
$451$	−850.106	−0.0887582
$452$	−929.531	−0.0967289
$453$	−1056.36	−0.109563
$454$	−1829.69	−0.189144
$455$	20393.6	2.10124
$456$	−3524.70	−0.361972
$457$	−10285.8	−1.05284	−0.526422	−	0.850224i	$-0.676467\pi$
$457$	−10285.8	−1.05284	−0.526422	+	0.850224i	$0.676467\pi$
$458$	6642.28	0.677671
$459$	459.000	0.0466760
$460$	−16855.9	−1.70850
$461$	−625.833	−0.0632277	−0.0316138	−	0.999500i	$-0.510065\pi$
$461$	−625.833	−0.0632277	−0.0316138	+	0.999500i	$0.510065\pi$
$462$	3242.45	0.326520
$463$	−6055.97	−0.607872	−0.303936	−	0.952692i	$-0.598301\pi$
$463$	−6055.97	−0.607872	−0.303936	+	0.952692i	$0.598301\pi$
$464$	−388.705	−0.0388904
$465$	1223.91	0.122059
$466$	16228.2	1.61321
$467$	−815.966	−0.0808531	−0.0404265	−	0.999183i	$-0.512872\pi$
$467$	−815.966	−0.0808531	−0.0404265	+	0.999183i	$0.512872\pi$
$468$	11124.0	1.09873
$469$	23397.9	2.30366
$470$	−20809.0	−2.04223
$471$	1613.65	0.157862
$472$	1522.88	0.148509
$473$	3278.35	0.318686
$474$	14794.2	1.43359
$475$	2461.74	0.237795
$476$	−7869.61	−0.757779
$477$	4165.81	0.399872
$478$	−7183.49	−0.687375
$479$	−16219.1	−1.54712	−0.773559	−	0.633724i	$-0.781525\pi$
$479$	−16219.1	−1.54712	−0.773559	+	0.633724i	$0.781525\pi$
$480$	2153.25	0.204754
$481$	−8366.97	−0.793142
$482$	16216.4	1.53244
$483$	−14234.0	−1.34093
$484$	−18749.9	−1.76088
$485$	−8724.63	−0.816835
$486$	1156.64	0.107956
$487$	2725.13	0.253568	0.126784	−	0.991930i	$-0.459535\pi$
$487$	2725.13	0.253568	0.126784	+	0.991930i	$0.459535\pi$
$488$	1885.84	0.174935
$489$	5768.23	0.533433
$490$	−23856.0	−2.19940
$491$	−8344.13	−0.766935	−0.383468	−	0.923554i	$-0.625270\pi$
$491$	−8344.13	−0.766935	−0.383468	+	0.923554i	$0.625270\pi$
$492$	5199.24	0.476423
$493$	−196.937	−0.0179911
$494$	−14885.9	−1.35577
$495$	495.368	0.0449801
$496$	−1787.96	−0.161858
$497$	29388.1	2.65238
$498$	−317.067	−0.0285304
$499$	13762.0	1.23461	0.617306	−	0.786723i	$-0.288224\pi$
$499$	13762.0	1.23461	0.617306	+	0.786723i	$0.288224\pi$
$500$	21475.1	1.92079
$501$	−8915.29	−0.795022
$502$	16157.0	1.43650
$503$	−11909.5	−1.05570	−0.527852	−	0.849336i	$-0.677003\pi$
$503$	−11909.5	−1.05570	−0.527852	+	0.849336i	$0.677003\pi$
$504$	−9006.24	−0.795972
$505$	−5382.92	−0.474330
$506$	−5140.31	−0.451610
$507$	14745.2	1.29163
$508$	2572.79	0.224703
$509$	742.666	0.0646721	0.0323360	−	0.999477i	$-0.489705\pi$
$509$	742.666	0.0646721	0.0323360	+	0.999477i	$0.489705\pi$
$510$	−1858.55	−0.161369
$511$	22028.1	1.90698
$512$	−11650.1	−1.00560
$513$	−1001.26	−0.0861732
$514$	8467.35	0.726612
$515$	6868.80	0.587719
$516$	−20050.3	−1.71060
$517$	−4105.07	−0.349209
$518$	14915.8	1.26518
$519$	2965.69	0.250827
$520$	−20456.2	−1.72513
$521$	−4815.73	−0.404953	−0.202477	−	0.979287i	$-0.564899\pi$
$521$	−4815.73	−0.404953	−0.202477	+	0.979287i	$0.564899\pi$
$522$	−496.266	−0.0416111
$523$	−16249.5	−1.35858	−0.679291	−	0.733869i	$-0.737713\pi$
$523$	−16249.5	−1.35858	−0.679291	+	0.733869i	$0.737713\pi$
$524$	27339.9	2.27929
$525$	6290.19	0.522908
$526$	−20547.7	−1.70327
$527$	−905.871	−0.0748773
$528$	−723.663	−0.0596466
$529$	10398.4	0.854637
$530$	−16867.9	−1.38244
$531$	432.606	0.0353550
$532$	17166.8	1.39901
$533$	9972.33	0.810412
$534$	−5279.51	−0.427841
$535$	14556.0	1.17628
$536$	−23469.8	−1.89131
$537$	−5812.96	−0.467128
$538$	31159.4	2.49699
$539$	−4706.18	−0.376085
$540$	−3029.67	−0.241437
$541$	−5458.04	−0.433751	−0.216876	−	0.976199i	$-0.569587\pi$
$541$	−5458.04	−0.433751	−0.216876	+	0.976199i	$0.569587\pi$
$542$	18017.0	1.42785
$543$	−6540.21	−0.516883
$544$	−1593.71	−0.125606
$545$	−4475.44	−0.351756
$546$	−38036.2	−2.98131
$547$	17237.0	1.34735	0.673677	−	0.739026i	$-0.264714\pi$
$547$	17237.0	1.34735	0.673677	+	0.739026i	$0.264714\pi$
$548$	15499.9	1.20825
$549$	535.712	0.0416460
$550$	2271.57	0.176109
$551$	429.600	0.0332152
$552$	14277.7	1.10091
$553$	−32723.7	−2.51637
$554$	−16759.7	−1.28529
$555$	2278.78	0.174286
$556$	13261.4	1.01152
$557$	−8452.71	−0.643003	−0.321502	−	0.946909i	$-0.604188\pi$
$557$	−8452.71	−0.643003	−0.321502	+	0.946909i	$0.604188\pi$
$558$	−2282.72	−0.173182
$559$	−38457.3	−2.90978
$560$	8114.01	0.612285
$561$	−366.644	−0.0275931
$562$	−13909.7	−1.04403
$563$	−8547.32	−0.639834	−0.319917	−	0.947446i	$-0.603655\pi$
$563$	−8547.32	−0.639834	−0.319917	+	0.947446i	$0.603655\pi$
$564$	25106.6	1.87443
$565$	485.573	0.0361561
$566$	3498.85	0.259836
$567$	−2558.40	−0.189494
$568$	−29478.4	−2.17762
$569$	19464.8	1.43411	0.717055	−	0.697017i	$-0.245490\pi$
$569$	19464.8	1.43411	0.717055	+	0.697017i	$0.245490\pi$
$570$	4054.24	0.297919
$571$	−3839.06	−0.281366	−0.140683	−	0.990055i	$-0.544930\pi$
$571$	−3839.06	−0.281366	−0.140683	+	0.990055i	$0.544930\pi$
$572$	−8885.72	−0.649529
$573$	5027.35	0.366528
$574$	−17777.7	−1.29273
$575$	−9971.94	−0.723232
$576$	−6431.90	−0.465270
$577$	−18797.7	−1.35625	−0.678127	−	0.734945i	$-0.737208\pi$
$577$	−18797.7	−1.35625	−0.678127	+	0.734945i	$0.737208\pi$
$578$	1375.60	0.0989918
$579$	−773.882	−0.0555465
$580$	1299.90	0.0930611
$581$	701.328	0.0500791
$582$	16272.4	1.15895
$583$	−3327.60	−0.236390
$584$	−22095.8	−1.56564
$585$	−5811.01	−0.410693
$586$	−41424.6	−2.92020
$587$	4217.92	0.296580	0.148290	−	0.988944i	$-0.452623\pi$
$587$	4217.92	0.296580	0.148290	+	0.988944i	$0.452623\pi$
$588$	28783.0	2.01869
$589$	1976.07	0.138238
$590$	−1751.68	−0.122230
$591$	−2080.40	−0.144799
$592$	−3328.98	−0.231115
$593$	−3011.92	−0.208575	−0.104287	−	0.994547i	$-0.533256\pi$
$593$	−3011.92	−0.208575	−0.104287	+	0.994547i	$0.533256\pi$
$594$	−923.914	−0.0638193
$595$	4110.97	0.283249
$596$	11856.8	0.814891
$597$	−720.886	−0.0494203
$598$	60299.3	4.12345
$599$	15137.1	1.03253	0.516266	−	0.856428i	$-0.327322\pi$
$599$	15137.1	1.03253	0.516266	+	0.856428i	$0.327322\pi$
$600$	−6309.52	−0.429309
$601$	−18980.1	−1.28821	−0.644104	−	0.764938i	$-0.722770\pi$
$601$	−18980.1	−1.28821	−0.644104	+	0.764938i	$0.722770\pi$
$602$	68558.0	4.64155
$603$	−6667.08	−0.450256
$604$	−5160.74	−0.347661
$605$	9794.65	0.658197
$606$	10039.7	0.672996
$607$	−4593.63	−0.307166	−0.153583	−	0.988136i	$-0.549081\pi$
$607$	−4593.63	−0.307166	−0.153583	+	0.988136i	$0.549081\pi$
$608$	3476.53	0.231894
$609$	1097.70	0.0730397
$610$	−2169.17	−0.143979
$611$	48155.3	3.18847
$612$	2242.39	0.148110
$613$	−25654.3	−1.69032	−0.845160	−	0.534513i	$-0.820495\pi$
$613$	−25654.3	−1.69032	−0.845160	+	0.534513i	$0.820495\pi$
$614$	11980.3	0.787437
$615$	−2716.01	−0.178081
$616$	7194.09	0.470549
$617$	−7170.97	−0.467897	−0.233948	−	0.972249i	$-0.575165\pi$
$617$	−7170.97	−0.467897	−0.233948	+	0.972249i	$0.575165\pi$
$618$	−12811.0	−0.833876
$619$	−13560.6	−0.880525	−0.440263	−	0.897869i	$-0.645115\pi$
$619$	−13560.6	−0.880525	−0.440263	+	0.897869i	$0.645115\pi$
$620$	5979.27	0.387312
$621$	4055.88	0.262088
$622$	31383.2	2.02307
$623$	11677.9	0.750985
$624$	8489.07	0.544607
$625$	−2920.36	−0.186903
$626$	20910.0	1.33503
$627$	799.798	0.0509424
$628$	7883.28	0.500919
$629$	−1686.63	−0.106916
$630$	10359.3	0.655119
$631$	−1414.98	−0.0892700	−0.0446350	−	0.999003i	$-0.514212\pi$
$631$	−1414.98	−0.0892700	−0.0446350	+	0.999003i	$0.514212\pi$
$632$	32824.3	2.06595
$633$	804.342	0.0505051
$634$	12381.4	0.775600
$635$	−1343.99	−0.0839914
$636$	20351.6	1.26886
$637$	55206.7	3.43386
$638$	396.412	0.0245989
$639$	−8373.93	−0.518416
$640$	20301.6	1.25389
$641$	−21708.4	−1.33764	−0.668822	−	0.743422i	$-0.733201\pi$
$641$	−21708.4	−1.33764	−0.668822	+	0.743422i	$0.733201\pi$
$642$	−27148.4	−1.66895
$643$	7537.23	0.462270	0.231135	−	0.972922i	$-0.425756\pi$
$643$	7537.23	0.462270	0.231135	+	0.972922i	$0.425756\pi$
$644$	−69538.5	−4.25497
$645$	10474.0	0.639401
$646$	−3000.73	−0.182759
$647$	−32667.3	−1.98498	−0.992491	−	0.122316i	$-0.960968\pi$
$647$	−32667.3	−1.98498	−0.992491	+	0.122316i	$0.960968\pi$
$648$	2566.27	0.155575
$649$	−345.561	−0.0209006
$650$	−26647.1	−1.60798
$651$	5049.20	0.303984
$652$	28180.1	1.69266
$653$	−25845.8	−1.54889	−0.774443	−	0.632644i	$-0.781970\pi$
$653$	−25845.8	−1.54889	−0.774443	+	0.632644i	$0.781970\pi$
$654$	8347.17	0.499083
$655$	−14281.9	−0.851972
$656$	3967.70	0.236148
$657$	−6276.77	−0.372725
$658$	−85846.7	−5.08610
$659$	−15741.2	−0.930485	−0.465243	−	0.885183i	$-0.654033\pi$
$659$	−15741.2	−0.930485	−0.465243	+	0.885183i	$0.654033\pi$
$660$	2420.06	0.142729
$661$	23495.2	1.38254	0.691269	−	0.722598i	$-0.257052\pi$
$661$	23495.2	1.38254	0.691269	+	0.722598i	$0.257052\pi$
$662$	−22230.8	−1.30517
$663$	4300.99	0.251940
$664$	−703.483	−0.0411151
$665$	−8967.67	−0.522934
$666$	−4250.17	−0.247283
$667$	−1740.21	−0.101021
$668$	−43554.7	−2.52272
$669$	−16573.3	−0.957788
$670$	26995.9	1.55663
$671$	−427.921	−0.0246195
$672$	8883.15	0.509933
$673$	−7057.34	−0.404221	−0.202110	−	0.979363i	$-0.564780\pi$
$673$	−7057.34	−0.404221	−0.202110	+	0.979363i	$0.564780\pi$
$674$	8575.67	0.490092
$675$	−1792.35	−0.102204
$676$	72036.0	4.09855
$677$	20756.4	1.17833	0.589167	−	0.808011i	$-0.299456\pi$
$677$	20756.4	1.17833	0.589167	+	0.808011i	$0.299456\pi$
$678$	−905.645	−0.0512995
$679$	−35993.2	−2.03430
$680$	−4123.60	−0.232548
$681$	−1153.20	−0.0648909
$682$	1823.41	0.102378
$683$	−7013.65	−0.392928	−0.196464	−	0.980511i	$-0.562946\pi$
$683$	−7013.65	−0.392928	−0.196464	+	0.980511i	$0.562946\pi$
$684$	−4891.56	−0.273441
$685$	−8096.91	−0.451631
$686$	−46850.4	−2.60751
$687$	4186.44	0.232493
$688$	−15301.0	−0.847888
$689$	39035.0	2.15837
$690$	−16422.8	−0.906093
$691$	−4897.47	−0.269622	−0.134811	−	0.990871i	$-0.543043\pi$
$691$	−4897.47	−0.269622	−0.134811	+	0.990871i	$0.543043\pi$
$692$	14488.5	0.795913
$693$	2043.63	0.112022
$694$	801.273	0.0438269
$695$	−6927.54	−0.378096
$696$	−1101.08	−0.0599658
$697$	2010.24	0.109244
$698$	−21171.1	−1.14805
$699$	10228.2	0.553456
$700$	30730.0	1.65926
$701$	17525.5	0.944264	0.472132	−	0.881528i	$-0.343485\pi$
$701$	17525.5	0.944264	0.472132	+	0.881528i	$0.343485\pi$
$702$	10838.1	0.582706
$703$	3679.21	0.197388
$704$	5137.73	0.275050
$705$	−13115.3	−0.700640
$706$	−50023.4	−2.66665
$707$	−22207.1	−1.18130
$708$	2113.45	0.112187
$709$	32564.3	1.72493	0.862466	−	0.506115i	$-0.168919\pi$
$709$	32564.3	1.72493	0.862466	+	0.506115i	$0.168919\pi$
$710$	33907.1	1.79227
$711$	9324.40	0.491832
$712$	−11713.8	−0.616561
$713$	−8004.58	−0.420440
$714$	−7667.39	−0.401884
$715$	4641.77	0.242786
$716$	−28398.5	−1.48227
$717$	−4527.55	−0.235822
$718$	39711.4	2.06409
$719$	14498.2	0.752007	0.376004	−	0.926618i	$-0.377298\pi$
$719$	14498.2	0.752007	0.376004	+	0.926618i	$0.377298\pi$
$720$	−2312.03	−0.119673
$721$	28337.0	1.46370
$722$	−26102.0	−1.34545
$723$	10220.7	0.525744
$724$	−31951.5	−1.64015
$725$	769.020	0.0393941
$726$	−18268.1	−0.933872
$727$	−25787.6	−1.31556	−0.657778	−	0.753212i	$-0.728503\pi$
$727$	−25787.6	−1.31556	−0.657778	+	0.753212i	$0.728503\pi$
$728$	−84391.6	−4.29637
$729$	729.000	0.0370370
$730$	25415.4	1.28859
$731$	−7752.28	−0.392241
$732$	2617.16	0.132149
$733$	−4177.45	−0.210502	−0.105251	−	0.994446i	$-0.533565\pi$
$733$	−4177.45	−0.210502	−0.105251	+	0.994446i	$0.533565\pi$
$734$	1677.38	0.0843504
$735$	−15035.8	−0.754563
$736$	−14082.6	−0.705287
$737$	5325.59	0.266175
$738$	5065.64	0.252668
$739$	14115.5	0.702636	0.351318	−	0.936256i	$-0.385734\pi$
$739$	14115.5	0.702636	0.351318	+	0.936256i	$0.385734\pi$
$740$	11132.7	0.553037
$741$	−9382.18	−0.465132
$742$	−69587.9	−3.44293
$743$	17992.0	0.888376	0.444188	−	0.895934i	$-0.353492\pi$
$743$	17992.0	0.888376	0.444188	+	0.895934i	$0.353492\pi$
$744$	−5064.72	−0.249572
$745$	−6193.84	−0.304597
$746$	59799.0	2.93485
$747$	−199.839	−0.00978810
$748$	−1791.20	−0.0875571
$749$	60050.2	2.92949
$750$	20923.3	1.01868
$751$	2055.99	0.0998989	0.0499495	−	0.998752i	$-0.484094\pi$
$751$	2055.99	0.0998989	0.0499495	+	0.998752i	$0.484094\pi$
$752$	19159.6	0.929095
$753$	10183.3	0.492829
$754$	−4650.19	−0.224602
$755$	2695.89	0.129952
$756$	−12498.8	−0.601292
$757$	12132.4	0.582508	0.291254	−	0.956646i	$-0.405927\pi$
$757$	12132.4	0.582508	0.291254	+	0.956646i	$0.405927\pi$
$758$	−8427.65	−0.403834
$759$	−3239.79	−0.154937
$760$	8995.23	0.429331
$761$	9319.01	0.443908	0.221954	−	0.975057i	$-0.428757\pi$
$761$	9319.01	0.443908	0.221954	+	0.975057i	$0.428757\pi$
$762$	2506.68	0.119170
$763$	−18463.3	−0.876037
$764$	24560.5	1.16305
$765$	−1171.39	−0.0553618
$766$	20612.9	0.972288
$767$	4053.67	0.190834
$768$	−20712.9	−0.973193
$769$	38790.8	1.81903	0.909514	−	0.415672i	$-0.136454\pi$
$769$	38790.8	1.81903	0.909514	+	0.415672i	$0.136454\pi$
$770$	−8274.90	−0.387282
$771$	5336.73	0.249284
$772$	−3780.71	−0.176257
$773$	12857.4	0.598252	0.299126	−	0.954214i	$-0.403305\pi$
$773$	12857.4	0.598252	0.299126	+	0.954214i	$0.403305\pi$
$774$	−19535.1	−0.907204
$775$	3537.33	0.163954
$776$	36103.8	1.67017
$777$	9401.04	0.434054
$778$	49005.2	2.25825
$779$	−4385.14	−0.201687
$780$	−28389.0	−1.30319
$781$	6689.00	0.306468
$782$	12155.2	0.555844
$783$	−312.783	−0.0142758
$784$	21965.2	1.00060
$785$	−4118.11	−0.187238
$786$	26637.3	1.20881
$787$	−26862.0	−1.21668	−0.608339	−	0.793677i	$-0.708164\pi$
$787$	−26862.0	−1.21668	−0.608339	+	0.793677i	$0.708164\pi$
$788$	−10163.5	−0.459468
$789$	−12950.6	−0.584353
$790$	−37755.7	−1.70036
$791$	2003.22	0.0900457
$792$	−2049.91	−0.0919700
$793$	5019.81	0.224790
$794$	443.519	0.0198235
$795$	−10631.4	−0.474284
$796$	−3521.81	−0.156818
$797$	−12471.5	−0.554285	−0.277142	−	0.960829i	$-0.589387\pi$
$797$	−12471.5	−0.554285	−0.277142	+	0.960829i	$0.589387\pi$
$798$	16725.7	0.741957
$799$	9707.23	0.429809
$800$	6223.29	0.275033
$801$	−3327.53	−0.146782
$802$	−63405.4	−2.79168
$803$	5013.82	0.220341
$804$	−32571.3	−1.42873
$805$	36325.9	1.59046
$806$	−21389.9	−0.934772
$807$	19638.9	0.856658
$808$	22275.3	0.969854
$809$	−24760.8	−1.07607	−0.538037	−	0.842921i	$-0.680834\pi$
$809$	−24760.8	−1.07607	−0.538037	+	0.842921i	$0.680834\pi$
$810$	−2951.81	−0.128045
$811$	11237.3	0.486556	0.243278	−	0.969957i	$-0.421777\pi$
$811$	11237.3	0.486556	0.243278	+	0.969957i	$0.421777\pi$
$812$	5362.70	0.231766
$813$	11355.6	0.489862
$814$	3394.99	0.146185
$815$	−14720.8	−0.632698
$816$	1711.24	0.0734134
$817$	16910.8	0.724156
$818$	44134.3	1.88645
$819$	−23973.1	−1.02282
$820$	−13268.7	−0.565079
$821$	39976.4	1.69937	0.849687	−	0.527287i	$-0.176791\pi$
$821$	39976.4	1.69937	0.849687	+	0.527287i	$0.176791\pi$
$822$	15101.6	0.640789
$823$	36877.0	1.56191	0.780955	−	0.624588i	$-0.214733\pi$
$823$	36877.0	1.56191	0.780955	+	0.624588i	$0.214733\pi$
$824$	−28424.1	−1.20170
$825$	1431.71	0.0604190
$826$	−7226.49	−0.304409
$827$	−14311.9	−0.601781	−0.300890	−	0.953659i	$-0.597284\pi$
$827$	−14311.9	−0.601781	−0.300890	+	0.953659i	$0.597284\pi$
$828$	19814.5	0.831646
$829$	−12629.7	−0.529130	−0.264565	−	0.964368i	$-0.585228\pi$
$829$	−12629.7	−0.529130	−0.264565	+	0.964368i	$0.585228\pi$
$830$	809.172	0.0338395
$831$	−10563.2	−0.440954
$832$	−60269.1	−2.51136
$833$	11128.7	0.462888
$834$	12920.6	0.536455
$835$	22752.3	0.942965
$836$	3907.32	0.161648
$837$	−1438.74	−0.0594146
$838$	53907.4	2.22220
$839$	−18683.1	−0.768786	−0.384393	−	0.923170i	$-0.625589\pi$
$839$	−18683.1	−0.768786	−0.384393	+	0.923170i	$0.625589\pi$
$840$	22984.4	0.944092
$841$	−24254.8	−0.994497
$842$	33007.3	1.35096
$843$	−8766.90	−0.358183
$844$	3929.52	0.160260
$845$	−37630.6	−1.53199
$846$	24461.4	0.994092
$847$	40407.5	1.63922
$848$	15530.9	0.628932
$849$	2205.22	0.0891438
$850$	−5371.56	−0.216756
$851$	−14903.6	−0.600340
$852$	−40909.9	−1.64501
$853$	−35648.7	−1.43094	−0.715468	−	0.698646i	$-0.753786\pi$
$853$	−35648.7	−1.43094	−0.715468	+	0.698646i	$0.753786\pi$
$854$	−8948.83	−0.358575
$855$	2555.28	0.102209
$856$	−60234.7	−2.40512
$857$	−49860.8	−1.98741	−0.993707	−	0.112010i	$-0.964271\pi$
$857$	−49860.8	−1.98741	−0.993707	+	0.112010i	$0.964271\pi$
$858$	−8657.39	−0.344474
$859$	21487.0	0.853466	0.426733	−	0.904378i	$-0.359664\pi$
$859$	21487.0	0.853466	0.426733	+	0.904378i	$0.359664\pi$
$860$	51169.6	2.02892
$861$	−11204.8	−0.443506
$862$	−13216.0	−0.522201
$863$	−15067.6	−0.594330	−0.297165	−	0.954826i	$-0.596041\pi$
$863$	−15067.6	−0.594330	−0.297165	+	0.954826i	$0.596041\pi$
$864$	−2531.19	−0.0996677
$865$	−7568.60	−0.297503
$866$	−15480.1	−0.607431
$867$	867.000	0.0339618
$868$	24667.3	0.964588
$869$	−7448.23	−0.290752
$870$	1266.50	0.0493544
$871$	−62472.8	−2.43032
$872$	18520.0	0.719229
$873$	10256.0	0.397610
$874$	−26515.4	−1.02620
$875$	−46280.6	−1.78808
$876$	−30664.4	−1.18271
$877$	−24852.6	−0.956912	−0.478456	−	0.878111i	$-0.658803\pi$
$877$	−24852.6	−0.956912	−0.478456	+	0.878111i	$0.658803\pi$
$878$	−63523.7	−2.44171
$879$	−26108.8	−1.00185
$880$	1846.83	0.0707460
$881$	1840.51	0.0703842	0.0351921	−	0.999381i	$-0.488796\pi$
$881$	1840.51	0.0703842	0.0351921	+	0.999381i	$0.488796\pi$
$882$	28043.3	1.07060
$883$	49803.7	1.89811	0.949054	−	0.315115i	$-0.102043\pi$
$883$	49803.7	1.89811	0.949054	+	0.315115i	$0.102043\pi$
$884$	21012.0	0.799445
$885$	−1104.03	−0.0419341
$886$	55402.7	2.10078
$887$	36314.7	1.37467	0.687333	−	0.726342i	$-0.258781\pi$
$887$	36314.7	1.37467	0.687333	+	0.726342i	$0.258781\pi$
$888$	−9429.93	−0.356360
$889$	−5544.58	−0.209178
$890$	13473.6	0.507456
$891$	−582.317	−0.0218949
$892$	−80966.9	−3.03921
$893$	−21175.4	−0.793512
$894$	11552.2	0.432172
$895$	14835.0	0.554054
$896$	83753.6	3.12278
$897$	38005.0	1.41466
$898$	−33022.5	−1.22714
$899$	617.300	0.0229011
$900$	−8756.31	−0.324308
$901$	7868.75	0.290950
$902$	−4046.38	−0.149368
$903$	43210.2	1.59241
$904$	−2009.37	−0.0739278
$905$	16691.0	0.613068
$906$	−5028.12	−0.184380
$907$	33679.5	1.23297	0.616487	−	0.787365i	$-0.288555\pi$
$907$	33679.5	1.23297	0.616487	+	0.787365i	$0.288555\pi$
$908$	−5633.83	−0.205909
$909$	6327.75	0.230889
$910$	97070.3	3.53610
$911$	32437.2	1.17968	0.589841	−	0.807519i	$-0.299190\pi$
$911$	32437.2	1.17968	0.589841	+	0.807519i	$0.299190\pi$
$912$	−3732.90	−0.135536
$913$	159.629	0.00578636
$914$	−48958.8	−1.77179
$915$	−1367.17	−0.0493957
$916$	20452.4	0.737736
$917$	−58919.7	−2.12181
$918$	2184.77	0.0785492
$919$	−3511.45	−0.126042	−0.0630208	−	0.998012i	$-0.520073\pi$
$919$	−3511.45	−0.126042	−0.0630208	+	0.998012i	$0.520073\pi$
$920$	−36437.5	−1.30577
$921$	7550.86	0.270151
$922$	−2978.87	−0.106403
$923$	−78466.6	−2.79822
$924$	9983.90	0.355461
$925$	6586.11	0.234108
$926$	−28825.5	−1.02296
$927$	−8074.44	−0.286083
$928$	1086.03	0.0384166
$929$	13527.2	0.477733	0.238866	−	0.971052i	$-0.423224\pi$
$929$	13527.2	0.477733	0.238866	+	0.971052i	$0.423224\pi$
$930$	5825.63	0.205408
$931$	−24276.1	−0.854583
$932$	49968.6	1.75620
$933$	19779.9	0.694069
$934$	−3883.87	−0.136064
$935$	935.695	0.0327278
$936$	24046.8	0.839737
$937$	−8862.80	−0.309002	−0.154501	−	0.987993i	$-0.549377\pi$
$937$	−8862.80	−0.309002	−0.154501	+	0.987993i	$0.549377\pi$
$938$	111371.	3.87674
$939$	13179.0	0.458019
$940$	−64073.4	−2.22324
$941$	22824.0	0.790693	0.395346	−	0.918532i	$-0.370625\pi$
$941$	22824.0	0.790693	0.395346	+	0.918532i	$0.370625\pi$
$942$	7680.71	0.265659
$943$	17763.1	0.613412
$944$	1612.84	0.0556074
$945$	6529.18	0.224756
$946$	15604.4	0.536305
$947$	−26308.4	−0.902754	−0.451377	−	0.892333i	$-0.649067\pi$
$947$	−26308.4	−0.902754	−0.451377	+	0.892333i	$0.649067\pi$
$948$	45553.3	1.56066
$949$	−58815.5	−2.01184
$950$	11717.5	0.400176
$951$	7803.68	0.266090
$952$	−17011.8	−0.579155
$953$	11947.5	0.406106	0.203053	−	0.979168i	$-0.434914\pi$
$953$	11947.5	0.406106	0.203053	+	0.979168i	$0.434914\pi$
$954$	19828.6	0.672930
$955$	−12830.1	−0.434734
$956$	−22118.9	−0.748300
$957$	249.848	0.00843932
$958$	−77200.5	−2.60358
$959$	−33403.5	−1.12477
$960$	16414.5	0.551851
$961$	−26951.5	−0.904688
$962$	−39825.5	−1.33475
$963$	−17110.9	−0.572576
$964$	49932.2	1.66826
$965$	1974.99	0.0658830
$966$	−67751.6	−2.25660
$967$	21505.3	0.715163	0.357581	−	0.933882i	$-0.383601\pi$
$967$	21505.3	0.715163	0.357581	+	0.933882i	$0.383601\pi$
$968$	−40531.7	−1.34580
$969$	−1891.27	−0.0627002
$970$	−41527.9	−1.37462
$971$	45685.3	1.50990	0.754950	−	0.655783i	$-0.227661\pi$
$971$	45685.3	1.50990	0.754950	+	0.655783i	$0.227661\pi$
$972$	3561.45	0.117524
$973$	−28579.4	−0.941636
$974$	12971.2	0.426719
$975$	−16794.9	−0.551659
$976$	1997.24	0.0655020
$977$	31710.3	1.03839	0.519193	−	0.854657i	$-0.326233\pi$
$977$	31710.3	1.03839	0.519193	+	0.854657i	$0.326233\pi$
$978$	27455.9	0.897693
$979$	2657.99	0.0867721
$980$	−73455.7	−2.39434
$981$	5260.99	0.171224
$982$	−39716.8	−1.29065
$983$	46038.9	1.49381	0.746904	−	0.664932i	$-0.231540\pi$
$983$	46038.9	1.49381	0.746904	+	0.664932i	$0.231540\pi$
$984$	11239.2	0.364120
$985$	5309.28	0.171744
$986$	−937.392	−0.0302765
$987$	−54106.8	−1.74492
$988$	−45835.6	−1.47593
$989$	−68501.8	−2.20246
$990$	2357.88	0.0756952
$991$	14394.9	0.461420	0.230710	−	0.973023i	$-0.425895\pi$
$991$	14394.9	0.461420	0.230710	+	0.973023i	$0.425895\pi$
$992$	4995.50	0.159886
$993$	−14011.5	−0.447775
$994$	139883.	4.46359
$995$	1839.74	0.0586167
$996$	−976.289	−0.0310591
$997$	−33473.4	−1.06330	−0.531652	−	0.846963i	$-0.678428\pi$
$997$	−33473.4	−1.06330	−0.531652	+	0.846963i	$0.678428\pi$
$998$	65505.0	2.07768
$999$	−2678.76	−0.0848371

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	51.4.a.e.1.3	✓	3
3.2	odd	2		153.4.a.f.1.1		3
4.3	odd	2		816.4.a.s.1.1		3
5.4	even	2		1275.4.a.q.1.1		3
7.6	odd	2		2499.4.a.n.1.3		3
12.11	even	2		2448.4.a.bd.1.3		3
17.16	even	2		867.4.a.k.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.a.e.1.3	✓	3	1.1	even	1	trivial
153.4.a.f.1.1		3	3.2	odd	2
816.4.a.s.1.1		3	4.3	odd	2
867.4.a.k.1.3		3	17.16	even	2
1275.4.a.q.1.1		3	5.4	even	2
2448.4.a.bd.1.3		3	12.11	even	2
2499.4.a.n.1.3		3	7.6	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:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	51.a (trivial)

\(f(q)\)	\(=\)	\(q+4.75985 q^{2} +3.00000 q^{3} +14.6562 q^{4} -7.65616 q^{5} +14.2795 q^{6} -31.5852 q^{7} +31.6823 q^{8} +9.00000 q^{9} -36.4422 q^{10} -7.18910 q^{11} +43.9685 q^{12} +84.3331 q^{13} -150.341 q^{14} -22.9685 q^{15} +33.5537 q^{16} +17.0000 q^{17} +42.8386 q^{18} -37.0838 q^{19} -112.210 q^{20} -94.7557 q^{21} -34.2190 q^{22} +150.218 q^{23} +95.0469 q^{24} -66.3832 q^{25} +401.413 q^{26} +27.0000 q^{27} -462.918 q^{28} -11.5846 q^{29} -109.326 q^{30} -53.2865 q^{31} -93.7478 q^{32} -21.5673 q^{33} +80.9174 q^{34} +241.822 q^{35} +131.905 q^{36} -99.2134 q^{37} -176.513 q^{38} +252.999 q^{39} -242.565 q^{40} +118.249 q^{41} -451.023 q^{42} -456.016 q^{43} -105.365 q^{44} -68.9054 q^{45} +715.014 q^{46} +571.014 q^{47} +100.661 q^{48} +654.627 q^{49} -315.974 q^{50} +51.0000 q^{51} +1236.00 q^{52} +462.867 q^{53} +128.516 q^{54} +55.0409 q^{55} -1000.69 q^{56} -111.251 q^{57} -55.1407 q^{58} +48.0674 q^{59} -336.630 q^{60} +59.5236 q^{61} -253.636 q^{62} -284.267 q^{63} -714.655 q^{64} -645.668 q^{65} -102.657 q^{66} -740.787 q^{67} +249.155 q^{68} +450.653 q^{69} +1151.03 q^{70} -930.437 q^{71} +285.141 q^{72} -697.419 q^{73} -472.241 q^{74} -199.150 q^{75} -543.506 q^{76} +227.070 q^{77} +1204.24 q^{78} +1036.04 q^{79} -256.893 q^{80} +81.0000 q^{81} +562.849 q^{82} -22.2043 q^{83} -1388.75 q^{84} -130.155 q^{85} -2170.57 q^{86} -34.7537 q^{87} -227.767 q^{88} -369.726 q^{89} -327.979 q^{90} -2663.68 q^{91} +2201.62 q^{92} -159.860 q^{93} +2717.94 q^{94} +283.920 q^{95} -281.244 q^{96} +1139.56 q^{97} +3115.93 q^{98} -64.7019 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 5 q^{2} + 9 q^{3} + 13 q^{4} + 8 q^{5} + 15 q^{6} - 8 q^{7} + 33 q^{8} + 27 q^{9} - 38 q^{10} + 34 q^{11} + 39 q^{12} + 36 q^{13} - 104 q^{14} + 24 q^{15} - 79 q^{16} + 51 q^{17} + 45 q^{18} - 142 q^{19}+ \cdots + 306 q^{99}+O(q^{100}) \) 3 * q + 5 * q^2 + 9 * q^3 + 13 * q^4 + 8 * q^5 + 15 * q^6 - 8 * q^7 + 33 * q^8 + 27 * q^9 - 38 * q^10 + 34 * q^11 + 39 * q^12 + 36 * q^13 - 104 * q^14 + 24 * q^15 - 79 * q^16 + 51 * q^17 + 45 * q^18 - 142 * q^19 - 126 * q^20 - 24 * q^21 - 248 * q^22 + 110 * q^23 + 99 * q^24 - 193 * q^25 + 154 * q^26 + 81 * q^27 - 472 * q^28 + 90 * q^29 - 114 * q^30 - 148 * q^31 - 151 * q^32 + 102 * q^33 + 85 * q^34 + 416 * q^35 + 117 * q^36 + 110 * q^37 + 80 * q^38 + 108 * q^39 - 202 * q^40 + 720 * q^41 - 312 * q^42 - 146 * q^43 - 192 * q^44 + 72 * q^45 + 748 * q^46 + 500 * q^47 - 237 * q^48 + 379 * q^49 - 385 * q^50 + 153 * q^51 + 1218 * q^52 + 610 * q^53 + 135 * q^54 + 430 * q^55 - 1368 * q^56 - 426 * q^57 + 1006 * q^58 - 216 * q^59 - 378 * q^60 - 18 * q^61 - 904 * q^62 - 72 * q^63 + 377 * q^64 - 966 * q^65 - 744 * q^66 - 1404 * q^67 + 221 * q^68 + 330 * q^69 + 1472 * q^70 - 960 * q^71 + 297 * q^72 - 794 * q^73 - 1874 * q^74 - 579 * q^75 - 392 * q^76 + 48 * q^77 + 462 * q^78 - 276 * q^79 - 1130 * q^80 + 243 * q^81 + 382 * q^82 - 1552 * q^83 - 1416 * q^84 + 136 * q^85 + 16 * q^86 + 270 * q^87 + 1724 * q^88 + 1394 * q^89 - 342 * q^90 - 3968 * q^91 + 2244 * q^92 - 444 * q^93 + 3960 * q^94 - 602 * q^95 - 453 * q^96 + 402 * q^97 + 4109 * q^98 + 306 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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