LMFDB - Embedded newform 399.2.a.e.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [399,2,Mod(1,399)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("399.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(399, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 399 = 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	399.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,1,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.18603104065$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
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			$2.86620$ of defining polynomial
Character	$\chi$	$=$	399.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.34889 q^{2} +1.00000 q^{3} +3.51730 q^{4} -0.517304 q^{5} +2.34889 q^{6} -1.00000 q^{7} +3.56399 q^{8} +1.00000 q^{9} -1.21509 q^{10} -3.73240 q^{11} +3.51730 q^{12} +5.73240 q^{13} -2.34889 q^{14} -0.517304 q^{15} +1.33682 q^{16} +4.51730 q^{17} +2.34889 q^{18} -1.00000 q^{19} -1.81952 q^{20} -1.00000 q^{21} -8.76700 q^{22} -7.73240 q^{23} +3.56399 q^{24} -4.73240 q^{25} +13.4648 q^{26} +1.00000 q^{27} -3.51730 q^{28} -6.18048 q^{29} -1.21509 q^{30} -6.69779 q^{31} -3.98793 q^{32} -3.73240 q^{33} +10.6107 q^{34} +0.517304 q^{35} +3.51730 q^{36} +9.46479 q^{37} -2.34889 q^{38} +5.73240 q^{39} -1.84366 q^{40} +2.26760 q^{41} -2.34889 q^{42} +1.30221 q^{43} -13.1280 q^{44} -0.517304 q^{45} -18.1626 q^{46} -4.18048 q^{47} +1.33682 q^{48} +1.00000 q^{49} -11.1159 q^{50} +4.51730 q^{51} +20.1626 q^{52} +13.6453 q^{53} +2.34889 q^{54} +1.93078 q^{55} -3.56399 q^{56} -1.00000 q^{57} -14.5173 q^{58} +6.06922 q^{59} -1.81952 q^{60} -5.46479 q^{61} -15.7324 q^{62} -1.00000 q^{63} -12.0409 q^{64} -2.96539 q^{65} -8.76700 q^{66} +6.43018 q^{67} +15.8887 q^{68} -7.73240 q^{69} +1.21509 q^{70} +0.947489 q^{71} +3.56399 q^{72} +3.03461 q^{73} +22.2318 q^{74} -4.73240 q^{75} -3.51730 q^{76} +3.73240 q^{77} +13.4648 q^{78} +6.06922 q^{79} -0.691542 q^{80} +1.00000 q^{81} +5.32636 q^{82} +6.24970 q^{83} -3.51730 q^{84} -2.33682 q^{85} +3.05876 q^{86} -6.18048 q^{87} -13.3022 q^{88} -7.80161 q^{89} -1.21509 q^{90} -5.73240 q^{91} -27.1972 q^{92} -6.69779 q^{93} -9.81952 q^{94} +0.517304 q^{95} -3.98793 q^{96} +15.3956 q^{97} +2.34889 q^{98} -3.73240 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{3} + 9 q^{4} + q^{6} - 3 q^{7} - 9 q^{8} + 3 q^{9} + 10 q^{10} + 4 q^{11} + 9 q^{12} + 2 q^{13} - q^{14} + 13 q^{16} + 12 q^{17} + q^{18} - 3 q^{19} - 16 q^{20} - 3 q^{21} - 8 q^{22}+ \cdots + 4 q^{99}+O(q^{100}) $ 3 * q + q^2 + 3 * q^3 + 9 * q^4 + q^6 - 3 * q^7 - 9 * q^8 + 3 * q^9 + 10 * q^10 + 4 * q^11 + 9 * q^12 + 2 * q^13 - q^14 + 13 * q^16 + 12 * q^17 + q^18 - 3 * q^19 - 16 * q^20 - 3 * q^21 - 8 * q^22 - 8 * q^23 - 9 * q^24 + q^25 + 10 * q^26 + 3 * q^27 - 9 * q^28 - 8 * q^29 + 10 * q^30 - 8 * q^31 - 27 * q^32 + 4 * q^33 - 6 * q^34 + 9 * q^36 - 2 * q^37 - q^38 + 2 * q^39 + 14 * q^40 + 22 * q^41 - q^42 + 16 * q^43 - 12 * q^46 - 2 * q^47 + 13 * q^48 + 3 * q^49 - 9 * q^50 + 12 * q^51 + 18 * q^52 + q^54 + 12 * q^55 + 9 * q^56 - 3 * q^57 - 42 * q^58 + 12 * q^59 - 16 * q^60 + 14 * q^61 - 32 * q^62 - 3 * q^63 + 29 * q^64 - 12 * q^65 - 8 * q^66 - 8 * q^67 + 52 * q^68 - 8 * q^69 - 10 * q^70 - 26 * q^71 - 9 * q^72 + 6 * q^73 + 18 * q^74 + q^75 - 9 * q^76 - 4 * q^77 + 10 * q^78 + 12 * q^79 - 52 * q^80 + 3 * q^81 - 2 * q^82 + 2 * q^83 - 9 * q^84 - 16 * q^85 - 24 * q^86 - 8 * q^87 - 52 * q^88 - 2 * q^89 + 10 * q^90 - 2 * q^91 - 36 * q^92 - 8 * q^93 - 40 * q^94 - 27 * q^96 + 22 * q^97 + q^98 + 4 * 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 $

$2$	2.34889	1.66092	0.830460	−	0.557079i	$-0.188078\pi$
$2$	2.34889	1.66092	0.830460	+	0.557079i	$0.188078\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.51730	1.75865
$5$	−0.517304	−0.231345	−0.115673	−	0.993287i	$-0.536902\pi$
$5$	−0.517304	−0.231345	−0.115673	+	0.993287i	$0.536902\pi$
$6$	2.34889	0.958932
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	3.56399	1.26006
$9$	1.00000	0.333333
$10$	−1.21509	−0.384246
$11$	−3.73240	−1.12536	−0.562680	−	0.826675i	$-0.690230\pi$
$11$	−3.73240	−1.12536	−0.562680	+	0.826675i	$0.690230\pi$
$12$	3.51730	1.01536
$13$	5.73240	1.58988	0.794940	−	0.606688i	$-0.207502\pi$
$13$	5.73240	1.58988	0.794940	+	0.606688i	$0.207502\pi$
$14$	−2.34889	−0.627768
$15$	−0.517304	−0.133567
$16$	1.33682	0.334205
$17$	4.51730	1.09561	0.547804	−	0.836607i	$-0.315464\pi$
$17$	4.51730	1.09561	0.547804	+	0.836607i	$0.315464\pi$
$18$	2.34889	0.553640
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.81952	−0.406856
$21$	−1.00000	−0.218218
$22$	−8.76700	−1.86913
$23$	−7.73240	−1.61232	−0.806158	−	0.591700i	$-0.798457\pi$
$23$	−7.73240	−1.61232	−0.806158	+	0.591700i	$0.798457\pi$
$24$	3.56399	0.727496
$25$	−4.73240	−0.946479
$26$	13.4648	2.64066
$27$	1.00000	0.192450
$28$	−3.51730	−0.664708
$29$	−6.18048	−1.14769	−0.573844	−	0.818965i	$-0.694548\pi$
$29$	−6.18048	−1.14769	−0.573844	+	0.818965i	$0.694548\pi$
$30$	−1.21509	−0.221845
$31$	−6.69779	−1.20296	−0.601479	−	0.798888i	$-0.705422\pi$
$31$	−6.69779	−1.20296	−0.601479	+	0.798888i	$0.705422\pi$
$32$	−3.98793	−0.704972
$33$	−3.73240	−0.649727
$34$	10.6107	1.81971
$35$	0.517304	0.0874403
$36$	3.51730	0.586217
$37$	9.46479	1.55600	0.778001	−	0.628263i	$-0.216234\pi$
$37$	9.46479	1.55600	0.778001	+	0.628263i	$0.216234\pi$
$38$	−2.34889	−0.381041
$39$	5.73240	0.917918
$40$	−1.84366	−0.291509
$41$	2.26760	0.354140	0.177070	−	0.984198i	$-0.443338\pi$
$41$	2.26760	0.354140	0.177070	+	0.984198i	$0.443338\pi$
$42$	−2.34889	−0.362442
$43$	1.30221	0.198585	0.0992927	−	0.995058i	$-0.468342\pi$
$43$	1.30221	0.198585	0.0992927	+	0.995058i	$0.468342\pi$
$44$	−13.1280	−1.97912
$45$	−0.517304	−0.0771151
$46$	−18.1626	−2.67793
$47$	−4.18048	−0.609786	−0.304893	−	0.952387i	$-0.598621\pi$
$47$	−4.18048	−0.609786	−0.304893	+	0.952387i	$0.598621\pi$
$48$	1.33682	0.192953
$49$	1.00000	0.142857
$50$	−11.1159	−1.57203
$51$	4.51730	0.632549
$52$	20.1626	2.79605
$53$	13.6453	1.87432	0.937162	−	0.348896i	$-0.113443\pi$
$53$	13.6453	1.87432	0.937162	+	0.348896i	$0.113443\pi$
$54$	2.34889	0.319644
$55$	1.93078	0.260347
$56$	−3.56399	−0.476258
$57$	−1.00000	−0.132453
$58$	−14.5173	−1.90622
$59$	6.06922	0.790145	0.395072	−	0.918650i	$-0.370720\pi$
$59$	6.06922	0.790145	0.395072	+	0.918650i	$0.370720\pi$
$60$	−1.81952	−0.234898
$61$	−5.46479	−0.699695	−0.349848	−	0.936807i	$-0.613767\pi$
$61$	−5.46479	−0.699695	−0.349848	+	0.936807i	$0.613767\pi$
$62$	−15.7324	−1.99802
$63$	−1.00000	−0.125988
$64$	−12.0409	−1.50511
$65$	−2.96539	−0.367812
$66$	−8.76700	−1.07914
$67$	6.43018	0.785572	0.392786	−	0.919630i	$-0.371511\pi$
$67$	6.43018	0.785572	0.392786	+	0.919630i	$0.371511\pi$
$68$	15.8887	1.92679
$69$	−7.73240	−0.930871
$70$	1.21509	0.145231
$71$	0.947489	0.112446	0.0562231	−	0.998418i	$-0.482094\pi$
$71$	0.947489	0.112446	0.0562231	+	0.998418i	$0.482094\pi$
$72$	3.56399	0.420020
$73$	3.03461	0.355174	0.177587	−	0.984105i	$-0.443171\pi$
$73$	3.03461	0.355174	0.177587	+	0.984105i	$0.443171\pi$
$74$	22.2318	2.58439
$75$	−4.73240	−0.546450
$76$	−3.51730	−0.403462
$77$	3.73240	0.425346
$78$	13.4648	1.52459
$79$	6.06922	0.682840	0.341420	−	0.939911i	$-0.389092\pi$
$79$	6.06922	0.682840	0.341420	+	0.939911i	$0.389092\pi$
$80$	−0.691542	−0.0773168
$81$	1.00000	0.111111
$82$	5.32636	0.588198
$83$	6.24970	0.685994	0.342997	−	0.939337i	$-0.388558\pi$
$83$	6.24970	0.685994	0.342997	+	0.939337i	$0.388558\pi$
$84$	−3.51730	−0.383769
$85$	−2.33682	−0.253464
$86$	3.05876	0.329834
$87$	−6.18048	−0.662617
$88$	−13.3022	−1.41802
$89$	−7.80161	−0.826969	−0.413485	−	0.910511i	$-0.635688\pi$
$89$	−7.80161	−0.826969	−0.413485	+	0.910511i	$0.635688\pi$
$90$	−1.21509	−0.128082
$91$	−5.73240	−0.600918
$92$	−27.1972	−2.83550
$93$	−6.69779	−0.694528
$94$	−9.81952	−1.01281
$95$	0.517304	0.0530743
$96$	−3.98793	−0.407016
$97$	15.3956	1.56318	0.781592	−	0.623790i	$-0.214408\pi$
$97$	15.3956	1.56318	0.781592	+	0.623790i	$0.214408\pi$
$98$	2.34889	0.237274
$99$	−3.73240	−0.375120
$100$	−16.6453	−1.66453
$101$	−11.9821	−1.19226	−0.596132	−	0.802887i	$-0.703296\pi$
$101$	−11.9821	−1.19226	−0.596132	+	0.802887i	$0.703296\pi$
$102$	10.6107	1.05061
$103$	15.4648	1.52379	0.761896	−	0.647700i	$-0.224269\pi$
$103$	15.4648	1.52379	0.761896	+	0.647700i	$0.224269\pi$
$104$	20.4302	2.00334
$105$	0.517304	0.0504837
$106$	32.0513	3.11310
$107$	−14.5173	−1.40344	−0.701720	−	0.712452i	$-0.747584\pi$
$107$	−14.5173	−1.40344	−0.701720	+	0.712452i	$0.747584\pi$
$108$	3.51730	0.338453
$109$	−6.49940	−0.622530	−0.311265	−	0.950323i	$-0.600753\pi$
$109$	−6.49940	−0.622530	−0.311265	+	0.950323i	$0.600753\pi$
$110$	4.53521	0.432415
$111$	9.46479	0.898358
$112$	−1.33682	−0.126318
$113$	−2.18048	−0.205123	−0.102561	−	0.994727i	$-0.532704\pi$
$113$	−2.18048	−0.205123	−0.102561	+	0.994727i	$0.532704\pi$
$114$	−2.34889	−0.219994
$115$	4.00000	0.373002
$116$	−21.7386	−2.01838
$117$	5.73240	0.529960
$118$	14.2559	1.31237
$119$	−4.51730	−0.414101
$120$	−1.84366	−0.168303
$121$	2.93078	0.266435
$122$	−12.8362	−1.16214
$123$	2.26760	0.204463
$124$	−23.5582	−2.11559
$125$	5.03461	0.450309
$126$	−2.34889	−0.209256
$127$	18.4302	1.63541	0.817707	−	0.575634i	$-0.195245\pi$
$127$	18.4302	1.63541	0.817707	+	0.575634i	$0.195245\pi$
$128$	−20.3068	−1.79489
$129$	1.30221	0.114653
$130$	−6.96539	−0.610905
$131$	14.6107	1.27654	0.638270	−	0.769813i	$-0.279650\pi$
$131$	14.6107	1.27654	0.638270	+	0.769813i	$0.279650\pi$
$132$	−13.1280	−1.14264
$133$	1.00000	0.0867110
$134$	15.1038	1.30477
$135$	−0.517304	−0.0445224
$136$	16.0996	1.38053
$137$	−3.39558	−0.290104	−0.145052	−	0.989424i	$-0.546335\pi$
$137$	−3.39558	−0.290104	−0.145052	+	0.989424i	$0.546335\pi$
$138$	−18.1626	−1.54610
$139$	2.96539	0.251521	0.125761	−	0.992061i	$-0.459863\pi$
$139$	2.96539	0.251521	0.125761	+	0.992061i	$0.459863\pi$
$140$	1.81952	0.153777
$141$	−4.18048	−0.352060
$142$	2.22555	0.186764
$143$	−21.3956	−1.78919
$144$	1.33682	0.111402
$145$	3.19719	0.265512
$146$	7.12797	0.589915
$147$	1.00000	0.0824786
$148$	33.2906	2.73647
$149$	−10.3610	−0.848804	−0.424402	−	0.905474i	$-0.639516\pi$
$149$	−10.3610	−0.848804	−0.424402	+	0.905474i	$0.639516\pi$
$150$	−11.1159	−0.907609
$151$	13.8950	1.13076	0.565379	−	0.824831i	$-0.308730\pi$
$151$	13.8950	1.13076	0.565379	+	0.824831i	$0.308730\pi$
$152$	−3.56399	−0.289077
$153$	4.51730	0.365202
$154$	8.76700	0.706465
$155$	3.46479	0.278299
$156$	20.1626	1.61430
$157$	−8.06922	−0.643994	−0.321997	−	0.946741i	$-0.604354\pi$
$157$	−8.06922	−0.643994	−0.321997	+	0.946741i	$0.604354\pi$
$158$	14.2559	1.13414
$159$	13.6453	1.08214
$160$	2.06297	0.163092
$161$	7.73240	0.609398
$162$	2.34889	0.184547
$163$	−8.76700	−0.686685	−0.343342	−	0.939210i	$-0.611559\pi$
$163$	−8.76700	−0.686685	−0.343342	+	0.939210i	$0.611559\pi$
$164$	7.97585	0.622809
$165$	1.93078	0.150311
$166$	14.6799	1.13938
$167$	−14.5686	−1.12735	−0.563677	−	0.825995i	$-0.690614\pi$
$167$	−14.5686	−1.12735	−0.563677	+	0.825995i	$0.690614\pi$
$168$	−3.56399	−0.274968
$169$	19.8604	1.52772
$170$	−5.48894	−0.420983
$171$	−1.00000	−0.0764719
$172$	4.58027	0.349243
$173$	−3.80161	−0.289031	−0.144516	−	0.989503i	$-0.546162\pi$
$173$	−3.80161	−0.289031	−0.144516	+	0.989503i	$0.546162\pi$
$174$	−14.5173	−1.10055
$175$	4.73240	0.357736
$176$	−4.98954	−0.376101
$177$	6.06922	0.456190
$178$	−18.3252	−1.37353
$179$	−3.55191	−0.265482	−0.132741	−	0.991151i	$-0.542378\pi$
$179$	−3.55191	−0.265482	−0.132741	+	0.991151i	$0.542378\pi$
$180$	−1.81952	−0.135619
$181$	19.3956	1.44166	0.720831	−	0.693111i	$-0.243760\pi$
$181$	19.3956	1.44166	0.720831	+	0.693111i	$0.243760\pi$
$182$	−13.4648	−0.998077
$183$	−5.46479	−0.403969
$184$	−27.5582	−2.03161
$185$	−4.89618	−0.359974
$186$	−15.7324	−1.15356
$187$	−16.8604	−1.23295
$188$	−14.7040	−1.07240
$189$	−1.00000	−0.0727393
$190$	1.21509	0.0881521
$191$	−9.80161	−0.709220	−0.354610	−	0.935014i	$-0.615386\pi$
$191$	−9.80161	−0.709220	−0.354610	+	0.935014i	$0.615386\pi$
$192$	−12.0409	−0.868974
$193$	−14.8604	−1.06967	−0.534836	−	0.844956i	$-0.679627\pi$
$193$	−14.8604	−1.06967	−0.534836	+	0.844956i	$0.679627\pi$
$194$	36.1626	2.59632
$195$	−2.96539	−0.212356
$196$	3.51730	0.251236
$197$	9.32636	0.664476	0.332238	−	0.943196i	$-0.392196\pi$
$197$	9.32636	0.664476	0.332238	+	0.943196i	$0.392196\pi$
$198$	−8.76700	−0.623044
$199$	−8.36097	−0.592693	−0.296347	−	0.955080i	$-0.595768\pi$
$199$	−8.36097	−0.592693	−0.296347	+	0.955080i	$0.595768\pi$
$200$	−16.8662	−1.19262
$201$	6.43018	0.453550
$202$	−28.1447	−1.98025
$203$	6.18048	0.433785
$204$	15.8887	1.11243
$205$	−1.17304	−0.0819287
$206$	36.3252	2.53089
$207$	−7.73240	−0.537439
$208$	7.66318	0.531346
$209$	3.73240	0.258175
$210$	1.21509	0.0838493
$211$	−13.5340	−0.931720	−0.465860	−	0.884859i	$-0.654255\pi$
$211$	−13.5340	−0.931720	−0.465860	+	0.884859i	$0.654255\pi$
$212$	47.9946	3.29628
$213$	0.947489	0.0649209
$214$	−34.0996	−2.33100
$215$	−0.673639	−0.0459418
$216$	3.56399	0.242499
$217$	6.69779	0.454676
$218$	−15.2664	−1.03397
$219$	3.03461	0.205060
$220$	6.79115	0.457859
$221$	25.8950	1.74188
$222$	22.2318	1.49210
$223$	14.1626	0.948397	0.474198	−	0.880418i	$-0.342738\pi$
$223$	14.1626	0.948397	0.474198	+	0.880418i	$0.342738\pi$
$224$	3.98793	0.266454
$225$	−4.73240	−0.315493
$226$	−5.12173	−0.340692
$227$	−19.8258	−1.31588	−0.657941	−	0.753069i	$-0.728572\pi$
$227$	−19.8258	−1.31588	−0.657941	+	0.753069i	$0.728572\pi$
$228$	−3.51730	−0.232939
$229$	−18.3610	−1.21333	−0.606663	−	0.794959i	$-0.707492\pi$
$229$	−18.3610	−1.21333	−0.606663	+	0.794959i	$0.707492\pi$
$230$	9.39558	0.619526
$231$	3.73240	0.245574
$232$	−22.0272	−1.44615
$233$	9.10382	0.596411	0.298206	−	0.954502i	$-0.403612\pi$
$233$	9.10382	0.596411	0.298206	+	0.954502i	$0.403612\pi$
$234$	13.4648	0.880221
$235$	2.16258	0.141071
$236$	21.3473	1.38959
$237$	6.06922	0.394238
$238$	−10.6107	−0.687788
$239$	−20.2318	−1.30869	−0.654343	−	0.756198i	$-0.727055\pi$
$239$	−20.2318	−1.30869	−0.654343	+	0.756198i	$0.727055\pi$
$240$	−0.691542	−0.0446389
$241$	7.39558	0.476391	0.238195	−	0.971217i	$-0.423444\pi$
$241$	7.39558	0.476391	0.238195	+	0.971217i	$0.423444\pi$
$242$	6.88410	0.442527
$243$	1.00000	0.0641500
$244$	−19.2213	−1.23052
$245$	−0.517304	−0.0330493
$246$	5.32636	0.339596
$247$	−5.73240	−0.364744
$248$	−23.8708	−1.51580
$249$	6.24970	0.396059
$250$	11.8258	0.747927
$251$	22.7491	1.43591	0.717955	−	0.696089i	$-0.245078\pi$
$251$	22.7491	1.43591	0.717955	+	0.696089i	$0.245078\pi$
$252$	−3.51730	−0.221569
$253$	28.8604	1.81444
$254$	43.2906	2.71629
$255$	−2.33682	−0.146337
$256$	−23.6169	−1.47606
$257$	19.6274	1.22432	0.612161	−	0.790733i	$-0.290300\pi$
$257$	19.6274	1.22432	0.612161	+	0.790733i	$0.290300\pi$
$258$	3.05876	0.190430
$259$	−9.46479	−0.588114
$260$	−10.4302	−0.646853
$261$	−6.18048	−0.382562
$262$	34.3189	2.12023
$263$	−18.6978	−1.15296	−0.576478	−	0.817113i	$-0.695573\pi$
$263$	−18.6978	−1.15296	−0.576478	+	0.817113i	$0.695573\pi$
$264$	−13.3022	−0.818695
$265$	−7.05876	−0.433616
$266$	2.34889	0.144020
$267$	−7.80161	−0.477451
$268$	22.6169	1.38155
$269$	−19.6274	−1.19670	−0.598351	−	0.801234i	$-0.704177\pi$
$269$	−19.6274	−1.19670	−0.598351	+	0.801234i	$0.704177\pi$
$270$	−1.21509	−0.0739482
$271$	−11.4648	−0.696437	−0.348218	−	0.937413i	$-0.613213\pi$
$271$	−11.4648	−0.696437	−0.348218	+	0.937413i	$0.613213\pi$
$272$	6.03882	0.366157
$273$	−5.73240	−0.346940
$274$	−7.97585	−0.481839
$275$	17.6632	1.06513
$276$	−27.1972	−1.63708
$277$	−3.66318	−0.220099	−0.110050	−	0.993926i	$-0.535101\pi$
$277$	−3.66318	−0.220099	−0.110050	+	0.993926i	$0.535101\pi$
$278$	6.96539	0.417756
$279$	−6.69779	−0.400986
$280$	1.84366	0.110180
$281$	23.2151	1.38490	0.692448	−	0.721468i	$-0.256532\pi$
$281$	23.2151	1.38490	0.692448	+	0.721468i	$0.256532\pi$
$282$	−9.81952	−0.584744
$283$	29.8950	1.77707	0.888536	−	0.458807i	$-0.151723\pi$
$283$	29.8950	1.77707	0.888536	+	0.458807i	$0.151723\pi$
$284$	3.33261	0.197754
$285$	0.517304	0.0306424
$286$	−50.2559	−2.97170
$287$	−2.26760	−0.133852
$288$	−3.98793	−0.234991
$289$	3.40604	0.200355
$290$	7.50986	0.440994
$291$	15.3956	0.902505
$292$	10.6736	0.624627
$293$	14.2676	0.833522	0.416761	−	0.909016i	$-0.363165\pi$
$293$	14.2676	0.833522	0.416761	+	0.909016i	$0.363165\pi$
$294$	2.34889	0.136990
$295$	−3.13963	−0.182796
$296$	33.7324	1.96066
$297$	−3.73240	−0.216576
$298$	−24.3368	−1.40979
$299$	−44.3252	−2.56339
$300$	−16.6453	−0.961016
$301$	−1.30221	−0.0750582
$302$	32.6378	1.87810
$303$	−11.9821	−0.688353
$304$	−1.33682	−0.0766719
$305$	2.82696	0.161871
$306$	10.6107	0.606572
$307$	−4.76700	−0.272067	−0.136034	−	0.990704i	$-0.543436\pi$
$307$	−4.76700	−0.272067	−0.136034	+	0.990704i	$0.543436\pi$
$308$	13.1280	0.748036
$309$	15.4648	0.879761
$310$	8.13843	0.462232
$311$	11.1459	0.632025	0.316012	−	0.948755i	$-0.397656\pi$
$311$	11.1459	0.632025	0.316012	+	0.948755i	$0.397656\pi$
$312$	20.4302	1.15663
$313$	−20.7912	−1.17519	−0.587593	−	0.809157i	$-0.699924\pi$
$313$	−20.7912	−1.17519	−0.587593	+	0.809157i	$0.699924\pi$
$314$	−18.9537	−1.06962
$315$	0.517304	0.0291468
$316$	21.3473	1.20088
$317$	−6.71569	−0.377191	−0.188595	−	0.982055i	$-0.560393\pi$
$317$	−6.71569	−0.377191	−0.188595	+	0.982055i	$0.560393\pi$
$318$	32.0513	1.79735
$319$	23.0680	1.29156
$320$	6.22878	0.348200
$321$	−14.5173	−0.810277
$322$	18.1626	1.01216
$323$	−4.51730	−0.251350
$324$	3.51730	0.195406
$325$	−27.1280	−1.50479
$326$	−20.5928	−1.14053
$327$	−6.49940	−0.359418
$328$	8.08171	0.446238
$329$	4.18048	0.230478
$330$	4.53521	0.249655
$331$	10.4302	0.573295	0.286647	−	0.958036i	$-0.407459\pi$
$331$	10.4302	0.573295	0.286647	+	0.958036i	$0.407459\pi$
$332$	21.9821	1.20642
$333$	9.46479	0.518667
$334$	−34.2201	−1.87244
$335$	−3.32636	−0.181738
$336$	−1.33682	−0.0729295
$337$	20.4302	1.11290	0.556452	−	0.830880i	$-0.312162\pi$
$337$	20.4302	1.11290	0.556452	+	0.830880i	$0.312162\pi$
$338$	46.6499	2.53742
$339$	−2.18048	−0.118428
$340$	−8.21931	−0.445754
$341$	24.9988	1.35376
$342$	−2.34889	−0.127014
$343$	−1.00000	−0.0539949
$344$	4.64106	0.250229
$345$	4.00000	0.215353
$346$	−8.92959	−0.480058
$347$	−15.2330	−0.817750	−0.408875	−	0.912590i	$-0.634079\pi$
$347$	−15.2330	−0.817750	−0.408875	+	0.912590i	$0.634079\pi$
$348$	−21.7386	−1.16531
$349$	−0.465991	−0.0249439	−0.0124720	−	0.999922i	$-0.503970\pi$
$349$	−0.465991	−0.0249439	−0.0124720	+	0.999922i	$0.503970\pi$
$350$	11.1159	0.594170
$351$	5.73240	0.305973
$352$	14.8845	0.793348
$353$	10.5865	0.563464	0.281732	−	0.959493i	$-0.409091\pi$
$353$	10.5865	0.563464	0.281732	+	0.959493i	$0.409091\pi$
$354$	14.2559	0.757695
$355$	−0.490140	−0.0260139
$356$	−27.4406	−1.45435
$357$	−4.51730	−0.239081
$358$	−8.34307	−0.440945
$359$	−22.6620	−1.19605	−0.598027	−	0.801476i	$-0.704048\pi$
$359$	−22.6620	−1.19605	−0.598027	+	0.801476i	$0.704048\pi$
$360$	−1.84366	−0.0971697
$361$	1.00000	0.0526316
$362$	45.5582	2.39448
$363$	2.93078	0.153826
$364$	−20.1626	−1.05681
$365$	−1.56982	−0.0821679
$366$	−12.8362	−0.670960
$367$	−7.82576	−0.408501	−0.204251	−	0.978919i	$-0.565476\pi$
$367$	−7.82576	−0.408501	−0.204251	+	0.978919i	$0.565476\pi$
$368$	−10.3368	−0.538844
$369$	2.26760	0.118047
$370$	−11.5006	−0.597888
$371$	−13.6453	−0.708428
$372$	−23.5582	−1.22143
$373$	−28.7912	−1.49075	−0.745375	−	0.666646i	$-0.767729\pi$
$373$	−28.7912	−1.49075	−0.745375	+	0.666646i	$0.767729\pi$
$374$	−39.6032	−2.04783
$375$	5.03461	0.259986
$376$	−14.8992	−0.768367
$377$	−35.4290	−1.82469
$378$	−2.34889	−0.120814
$379$	−35.9642	−1.84736	−0.923678	−	0.383169i	$-0.874833\pi$
$379$	−35.9642	−1.84736	−0.923678	+	0.383169i	$0.874833\pi$
$380$	1.81952	0.0933392
$381$	18.4302	0.944207
$382$	−23.0230	−1.17796
$383$	19.9642	1.02012	0.510061	−	0.860138i	$-0.329623\pi$
$383$	19.9642	1.02012	0.510061	+	0.860138i	$0.329623\pi$
$384$	−20.3068	−1.03628
$385$	−1.93078	−0.0984019
$386$	−34.9054	−1.77664
$387$	1.30221	0.0661951
$388$	54.1509	2.74910
$389$	−7.03461	−0.356669	−0.178334	−	0.983970i	$-0.557071\pi$
$389$	−7.03461	−0.356669	−0.178334	+	0.983970i	$0.557071\pi$
$390$	−6.96539	−0.352706
$391$	−34.9296	−1.76647
$392$	3.56399	0.180009
$393$	14.6107	0.737011
$394$	21.9066	1.10364
$395$	−3.13963	−0.157972
$396$	−13.1280	−0.659705
$397$	−26.9988	−1.35503	−0.677516	−	0.735508i	$-0.736943\pi$
$397$	−26.9988	−1.35503	−0.677516	+	0.735508i	$0.736943\pi$
$398$	−19.6390	−0.984416
$399$	1.00000	0.0500626
$400$	−6.32636	−0.316318
$401$	19.7145	0.984495	0.492247	−	0.870455i	$-0.336176\pi$
$401$	19.7145	0.984495	0.492247	+	0.870455i	$0.336176\pi$
$402$	15.1038	0.753310
$403$	−38.3944	−1.91256
$404$	−42.1447	−2.09678
$405$	−0.517304	−0.0257050
$406$	14.5173	0.720482
$407$	−35.3264	−1.75106
$408$	16.0996	0.797050
$409$	−19.1280	−0.945817	−0.472909	−	0.881111i	$-0.656796\pi$
$409$	−19.1280	−0.945817	−0.472909	+	0.881111i	$0.656796\pi$
$410$	−2.75535	−0.136077
$411$	−3.39558	−0.167491
$412$	54.3944	2.67982
$413$	−6.06922	−0.298647
$414$	−18.1626	−0.892642
$415$	−3.23300	−0.158702
$416$	−22.8604	−1.12082
$417$	2.96539	0.145216
$418$	8.76700	0.428808
$419$	13.0409	0.637087	0.318544	−	0.947908i	$-0.396806\pi$
$419$	13.0409	0.637087	0.318544	+	0.947908i	$0.396806\pi$
$420$	1.81952	0.0887833
$421$	−25.1521	−1.22584	−0.612920	−	0.790145i	$-0.710005\pi$
$421$	−25.1521	−1.22584	−0.612920	+	0.790145i	$0.710005\pi$
$422$	−31.7900	−1.54751
$423$	−4.18048	−0.203262
$424$	48.6316	2.36176
$425$	−21.3777	−1.03697
$426$	2.22555	0.107828
$427$	5.46479	0.264460
$428$	−51.0618	−2.46816
$429$	−21.3956	−1.03299
$430$	−1.58231	−0.0763056
$431$	22.4815	1.08290	0.541448	−	0.840734i	$-0.317876\pi$
$431$	22.4815	1.08290	0.541448	+	0.840734i	$0.317876\pi$
$432$	1.33682	0.0643178
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	15.7324	0.755179
$435$	3.19719	0.153294
$436$	−22.8604	−1.09481
$437$	7.73240	0.369891
$438$	7.12797	0.340588
$439$	−33.4889	−1.59834	−0.799170	−	0.601105i	$-0.794727\pi$
$439$	−33.4889	−1.59834	−0.799170	+	0.601105i	$0.794727\pi$
$440$	6.88129	0.328053
$441$	1.00000	0.0476190
$442$	60.8246	2.89313
$443$	23.6966	1.12586	0.562930	−	0.826505i	$-0.309674\pi$
$443$	23.6966	1.12586	0.562930	+	0.826505i	$0.309674\pi$
$444$	33.2906	1.57990
$445$	4.03581	0.191316
$446$	33.2664	1.57521
$447$	−10.3610	−0.490057
$448$	12.0409	0.568877
$449$	7.38933	0.348724	0.174362	−	0.984682i	$-0.444214\pi$
$449$	7.38933	0.348724	0.174362	+	0.984682i	$0.444214\pi$
$450$	−11.1159	−0.524008
$451$	−8.46360	−0.398535
$452$	−7.66943	−0.360739
$453$	13.8950	0.652843
$454$	−46.5686	−2.18557
$455$	2.96539	0.139020
$456$	−3.56399	−0.166899
$457$	−18.2676	−0.854522	−0.427261	−	0.904128i	$-0.640522\pi$
$457$	−18.2676	−0.854522	−0.427261	+	0.904128i	$0.640522\pi$
$458$	−43.1280	−2.01524
$459$	4.51730	0.210850
$460$	14.0692	0.655981
$461$	−32.8425	−1.52963	−0.764813	−	0.644252i	$-0.777169\pi$
$461$	−32.8425	−1.52963	−0.764813	+	0.644252i	$0.777169\pi$
$462$	8.76700	0.407878
$463$	−13.9308	−0.647418	−0.323709	−	0.946157i	$-0.604930\pi$
$463$	−13.9308	−0.647418	−0.323709	+	0.946157i	$0.604930\pi$
$464$	−8.26219	−0.383563
$465$	3.46479	0.160676
$466$	21.3839	0.990591
$467$	−6.78491	−0.313968	−0.156984	−	0.987601i	$-0.550177\pi$
$467$	−6.78491	−0.313968	−0.156984	+	0.987601i	$0.550177\pi$
$468$	20.1626	0.932016
$469$	−6.43018	−0.296918
$470$	5.07968	0.234308
$471$	−8.06922	−0.371810
$472$	21.6306	0.995629
$473$	−4.86037	−0.223480
$474$	14.2559	0.654797
$475$	4.73240	0.217137
$476$	−15.8887	−0.728259
$477$	13.6453	0.624774
$478$	−47.5224	−2.17362
$479$	3.11007	0.142103	0.0710514	−	0.997473i	$-0.477365\pi$
$479$	3.11007	0.142103	0.0710514	+	0.997473i	$0.477365\pi$
$480$	2.06297	0.0941613
$481$	54.2559	2.47386
$482$	17.3714	0.791247
$483$	7.73240	0.351836
$484$	10.3085	0.468566
$485$	−7.96419	−0.361635
$486$	2.34889	0.106548
$487$	−35.7900	−1.62180	−0.810899	−	0.585186i	$-0.801021\pi$
$487$	−35.7900	−1.62180	−0.810899	+	0.585186i	$0.801021\pi$
$488$	−19.4764	−0.881657
$489$	−8.76700	−0.396458
$490$	−1.21509	−0.0548923
$491$	−15.2330	−0.687455	−0.343728	−	0.939069i	$-0.611690\pi$
$491$	−15.2330	−0.687455	−0.343728	+	0.939069i	$0.611690\pi$
$492$	7.97585	0.359579
$493$	−27.9191	−1.25741
$494$	−13.4648	−0.605810
$495$	1.93078	0.0867823
$496$	−8.95374	−0.402035
$497$	−0.947489	−0.0425007
$498$	14.6799	0.657821
$499$	28.1867	1.26181	0.630906	−	0.775860i	$-0.282683\pi$
$499$	28.1867	1.26181	0.630906	+	0.775860i	$0.282683\pi$
$500$	17.7082	0.791937
$501$	−14.5686	−0.650878
$502$	53.4352	2.38493
$503$	6.47224	0.288583	0.144291	−	0.989535i	$-0.453910\pi$
$503$	6.47224	0.288583	0.144291	+	0.989535i	$0.453910\pi$
$504$	−3.56399	−0.158753
$505$	6.19839	0.275825
$506$	67.7900	3.01363
$507$	19.8604	0.882030
$508$	64.8246	2.87612
$509$	−43.6274	−1.93375	−0.966875	−	0.255252i	$-0.917842\pi$
$509$	−43.6274	−1.93375	−0.966875	+	0.255252i	$0.917842\pi$
$510$	−5.48894	−0.243054
$511$	−3.03461	−0.134243
$512$	−14.8600	−0.656723
$513$	−1.00000	−0.0441511
$514$	46.1026	2.03350
$515$	−8.00000	−0.352522
$516$	4.58027	0.201635
$517$	15.6032	0.686229
$518$	−22.2318	−0.976809
$519$	−3.80161	−0.166872
$520$	−10.5686	−0.463465
$521$	38.9537	1.70659	0.853297	−	0.521425i	$-0.174599\pi$
$521$	38.9537	1.70659	0.853297	+	0.521425i	$0.174599\pi$
$522$	−14.5173	−0.635405
$523$	37.7658	1.65138	0.825692	−	0.564122i	$-0.190785\pi$
$523$	37.7658	1.65138	0.825692	+	0.564122i	$0.190785\pi$
$524$	51.3902	2.24499
$525$	4.73240	0.206539
$526$	−43.9191	−1.91496
$527$	−30.2559	−1.31797
$528$	−4.98954	−0.217142
$529$	36.7900	1.59956
$530$	−16.5803	−0.720201
$531$	6.06922	0.263382
$532$	3.51730	0.152494
$533$	12.9988	0.563041
$534$	−18.3252	−0.793007
$535$	7.50986	0.324680
$536$	22.9171	0.989868
$537$	−3.55191	−0.153276
$538$	−46.1026	−1.98763
$539$	−3.73240	−0.160766
$540$	−1.81952	−0.0782995
$541$	20.6527	0.887930	0.443965	−	0.896044i	$-0.353571\pi$
$541$	20.6527	0.887930	0.443965	+	0.896044i	$0.353571\pi$
$542$	−26.9296	−1.15672
$543$	19.3956	0.832344
$544$	−18.0147	−0.772373
$545$	3.36217	0.144019
$546$	−13.4648	−0.576240
$547$	10.7912	0.461396	0.230698	−	0.973025i	$-0.425899\pi$
$547$	10.7912	0.461396	0.230698	+	0.973025i	$0.425899\pi$
$548$	−11.9433	−0.510191
$549$	−5.46479	−0.233232
$550$	41.4889	1.76909
$551$	6.18048	0.263297
$552$	−27.5582	−1.17295
$553$	−6.06922	−0.258089
$554$	−8.60442	−0.365567
$555$	−4.89618	−0.207831
$556$	10.4302	0.442338
$557$	38.8246	1.64505	0.822525	−	0.568729i	$-0.192565\pi$
$557$	38.8246	1.64505	0.822525	+	0.568729i	$0.192565\pi$
$558$	−15.7324	−0.666005
$559$	7.46479	0.315727
$560$	0.691542	0.0292230
$561$	−16.8604	−0.711845
$562$	54.5298	2.30020
$563$	−1.56982	−0.0661598	−0.0330799	−	0.999453i	$-0.510532\pi$
$563$	−1.56982	−0.0661598	−0.0330799	+	0.999453i	$0.510532\pi$
$564$	−14.7040	−0.619152
$565$	1.12797	0.0474542
$566$	70.2201	2.95157
$567$	−1.00000	−0.0419961
$568$	3.37684	0.141689
$569$	16.2139	0.679722	0.339861	−	0.940476i	$-0.389620\pi$
$569$	16.2139	0.679722	0.339861	+	0.940476i	$0.389620\pi$
$570$	1.21509	0.0508946
$571$	46.2559	1.93575	0.967876	−	0.251430i	$-0.0809007\pi$
$571$	46.2559	1.93575	0.967876	+	0.251430i	$0.0809007\pi$
$572$	−75.2547	−3.14656
$573$	−9.80161	−0.409468
$574$	−5.32636	−0.222318
$575$	36.5928	1.52602
$576$	−12.0409	−0.501702
$577$	1.86157	0.0774981	0.0387490	−	0.999249i	$-0.487663\pi$
$577$	1.86157	0.0774981	0.0387490	+	0.999249i	$0.487663\pi$
$578$	8.00042	0.332774
$579$	−14.8604	−0.617576
$580$	11.2455	0.466943
$581$	−6.24970	−0.259281
$582$	36.1626	1.49899
$583$	−50.9296	−2.10929
$584$	10.8153	0.447540
$585$	−2.96539	−0.122604
$586$	33.5131	1.38441
$587$	11.9579	0.493557	0.246779	−	0.969072i	$-0.420628\pi$
$587$	11.9579	0.493557	0.246779	+	0.969072i	$0.420628\pi$
$588$	3.51730	0.145051
$589$	6.69779	0.275978
$590$	−7.37466	−0.303610
$591$	9.32636	0.383635
$592$	12.6527	0.520024
$593$	−16.8425	−0.691637	−0.345819	−	0.938301i	$-0.612399\pi$
$593$	−16.8425	−0.691637	−0.345819	+	0.938301i	$0.612399\pi$
$594$	−8.76700	−0.359715
$595$	2.33682	0.0958003
$596$	−36.4427	−1.49275
$597$	−8.36097	−0.342192
$598$	−104.115	−4.25758
$599$	−35.8771	−1.46590	−0.732949	−	0.680284i	$-0.761857\pi$
$599$	−35.8771	−1.46590	−0.732949	+	0.680284i	$0.761857\pi$
$600$	−16.8662	−0.688560
$601$	20.0692	0.818640	0.409320	−	0.912391i	$-0.365766\pi$
$601$	20.0692	0.818640	0.409320	+	0.912391i	$0.365766\pi$
$602$	−3.05876	−0.124666
$603$	6.43018	0.261857
$604$	48.8729	1.98861
$605$	−1.51611	−0.0616385
$606$	−28.1447	−1.14330
$607$	16.5352	0.671143	0.335572	−	0.942015i	$-0.391071\pi$
$607$	16.5352	0.671143	0.335572	+	0.942015i	$0.391071\pi$
$608$	3.98793	0.161732
$609$	6.18048	0.250446
$610$	6.64023	0.268855
$611$	−23.9642	−0.969488
$612$	15.8887	0.642264
$613$	24.1984	0.977364	0.488682	−	0.872462i	$-0.337478\pi$
$613$	24.1984	0.977364	0.488682	+	0.872462i	$0.337478\pi$
$614$	−11.1972	−0.451882
$615$	−1.17304	−0.0473016
$616$	13.3022	0.535961
$617$	−3.93078	−0.158247	−0.0791237	−	0.996865i	$-0.525212\pi$
$617$	−3.93078	−0.158247	−0.0791237	+	0.996865i	$0.525212\pi$
$618$	36.3252	1.46121
$619$	−5.03461	−0.202358	−0.101179	−	0.994868i	$-0.532261\pi$
$619$	−5.03461	−0.202358	−0.101179	+	0.994868i	$0.532261\pi$
$620$	12.1867	0.489431
$621$	−7.73240	−0.310290
$622$	26.1805	1.04974
$623$	7.80161	0.312565
$624$	7.66318	0.306773
$625$	21.0576	0.842302
$626$	−48.8362	−1.95189
$627$	3.73240	0.149058
$628$	−28.3819	−1.13256
$629$	42.7553	1.70477
$630$	1.21509	0.0484104
$631$	−23.3714	−0.930402	−0.465201	−	0.885205i	$-0.654018\pi$
$631$	−23.3714	−0.930402	−0.465201	+	0.885205i	$0.654018\pi$
$632$	21.6306	0.860419
$633$	−13.5340	−0.537929
$634$	−15.7744	−0.626483
$635$	−9.53401	−0.378346
$636$	47.9946	1.90311
$637$	5.73240	0.227126
$638$	54.1843	2.14518
$639$	0.947489	0.0374821
$640$	10.5048	0.415239
$641$	−13.1459	−0.519231	−0.259615	−	0.965712i	$-0.583596\pi$
$641$	−13.1459	−0.519231	−0.259615	+	0.965712i	$0.583596\pi$
$642$	−34.0996	−1.34580
$643$	6.96539	0.274688	0.137344	−	0.990523i	$-0.456143\pi$
$643$	6.96539	0.274688	0.137344	+	0.990523i	$0.456143\pi$
$644$	27.1972	1.07172
$645$	−0.673639	−0.0265245
$646$	−10.6107	−0.417471
$647$	22.6107	0.888917	0.444459	−	0.895799i	$-0.353396\pi$
$647$	22.6107	0.888917	0.444459	+	0.895799i	$0.353396\pi$
$648$	3.56399	0.140007
$649$	−22.6527	−0.889197
$650$	−63.7207	−2.49933
$651$	6.69779	0.262507
$652$	−30.8362	−1.20764
$653$	−30.8604	−1.20766	−0.603830	−	0.797113i	$-0.706359\pi$
$653$	−30.8604	−1.20766	−0.603830	+	0.797113i	$0.706359\pi$
$654$	−15.2664	−0.596964
$655$	−7.55816	−0.295322
$656$	3.03138	0.118355
$657$	3.03461	0.118391
$658$	9.81952	0.382805
$659$	16.4481	0.640727	0.320363	−	0.947295i	$-0.396195\pi$
$659$	16.4481	0.640727	0.320363	+	0.947295i	$0.396195\pi$
$660$	6.79115	0.264345
$661$	−35.9883	−1.39978	−0.699892	−	0.714249i	$-0.746769\pi$
$661$	−35.9883	−1.39978	−0.699892	+	0.714249i	$0.746769\pi$
$662$	24.4994	0.952196
$663$	25.8950	1.00568
$664$	22.2738	0.864393
$665$	−0.517304	−0.0200602
$666$	22.2318	0.861465
$667$	47.7900	1.85043
$668$	−51.2423	−1.98262
$669$	14.1626	0.547557
$670$	−7.81327	−0.301853
$671$	20.3968	0.787409
$672$	3.98793	0.153838
$673$	−2.99880	−0.115595	−0.0577977	−	0.998328i	$-0.518408\pi$
$673$	−2.99880	−0.115595	−0.0577977	+	0.998328i	$0.518408\pi$
$674$	47.9883	1.84844
$675$	−4.73240	−0.182150
$676$	69.8550	2.68673
$677$	−14.7670	−0.567542	−0.283771	−	0.958892i	$-0.591586\pi$
$677$	−14.7670	−0.567542	−0.283771	+	0.958892i	$0.591586\pi$
$678$	−5.12173	−0.196699
$679$	−15.3956	−0.590828
$680$	−8.32839	−0.319379
$681$	−19.8258	−0.759725
$682$	58.7195	2.24849
$683$	−40.4123	−1.54633	−0.773166	−	0.634203i	$-0.781328\pi$
$683$	−40.4123	−1.54633	−0.773166	+	0.634203i	$0.781328\pi$
$684$	−3.51730	−0.134487
$685$	1.75655	0.0671142
$686$	−2.34889	−0.0896812
$687$	−18.3610	−0.700515
$688$	1.74082	0.0663682
$689$	78.2201	2.97995
$690$	9.39558	0.357683
$691$	−27.3264	−1.03954	−0.519772	−	0.854305i	$-0.673983\pi$
$691$	−27.3264	−1.03954	−0.519772	+	0.854305i	$0.673983\pi$
$692$	−13.3714	−0.508305
$693$	3.73240	0.141782
$694$	−35.7807	−1.35822
$695$	−1.53401	−0.0581883
$696$	−22.0272	−0.834938
$697$	10.2435	0.387999
$698$	−1.09456	−0.0414298
$699$	9.10382	0.344338
$700$	16.6453	0.629132
$701$	12.4660	0.470834	0.235417	−	0.971894i	$-0.424354\pi$
$701$	12.4660	0.470834	0.235417	+	0.971894i	$0.424354\pi$
$702$	13.4648	0.508196
$703$	−9.46479	−0.356971
$704$	44.9412	1.69379
$705$	2.16258	0.0814475
$706$	24.8666	0.935867
$707$	11.9821	0.450633
$708$	21.3473	0.802280
$709$	−19.4048	−0.728764	−0.364382	−	0.931250i	$-0.618720\pi$
$709$	−19.4048	−0.728764	−0.364382	+	0.931250i	$0.618720\pi$
$710$	−1.15129	−0.0432070
$711$	6.06922	0.227613
$712$	−27.8048	−1.04203
$713$	51.7900	1.93955
$714$	−10.6107	−0.397094
$715$	11.0680	0.413920
$716$	−12.4932	−0.466891
$717$	−20.2318	−0.755570
$718$	−53.2306	−1.98655
$719$	28.3189	1.05612	0.528059	−	0.849208i	$-0.322920\pi$
$719$	28.3189	1.05612	0.528059	+	0.849208i	$0.322920\pi$
$720$	−0.691542	−0.0257723
$721$	−15.4648	−0.575939
$722$	2.34889	0.0874168
$723$	7.39558	0.275044
$724$	68.2201	2.53538
$725$	29.2485	1.08626
$726$	6.88410	0.255493
$727$	26.6527	0.988495	0.494247	−	0.869321i	$-0.335444\pi$
$727$	26.6527	0.988495	0.494247	+	0.869321i	$0.335444\pi$
$728$	−20.4302	−0.757193
$729$	1.00000	0.0370370
$730$	−3.68733	−0.136474
$731$	5.88249	0.217572
$732$	−19.2213	−0.710441
$733$	−19.7565	−0.729725	−0.364862	−	0.931061i	$-0.618884\pi$
$733$	−19.7565	−0.729725	−0.364862	+	0.931061i	$0.618884\pi$
$734$	−18.3819	−0.678488
$735$	−0.517304	−0.0190810
$736$	30.8362	1.13664
$737$	−24.0000	−0.884051
$738$	5.32636	0.196066
$739$	36.0934	1.32772	0.663858	−	0.747859i	$-0.268918\pi$
$739$	36.0934	1.32772	0.663858	+	0.747859i	$0.268918\pi$
$740$	−17.2213	−0.633069
$741$	−5.73240	−0.210585
$742$	−32.0513	−1.17664
$743$	−33.1700	−1.21689	−0.608445	−	0.793596i	$-0.708206\pi$
$743$	−33.1700	−1.21689	−0.608445	+	0.793596i	$0.708206\pi$
$744$	−23.8708	−0.875147
$745$	5.35977	0.196367
$746$	−67.6274	−2.47601
$747$	6.24970	0.228665
$748$	−59.3030	−2.16833
$749$	14.5173	0.530451
$750$	11.8258	0.431816
$751$	36.5477	1.33364	0.666822	−	0.745217i	$-0.267654\pi$
$751$	36.5477	1.33364	0.666822	+	0.745217i	$0.267654\pi$
$752$	−5.58855	−0.203794
$753$	22.7491	0.829023
$754$	−83.2189	−3.03066
$755$	−7.18793	−0.261595
$756$	−3.51730	−0.127923
$757$	−44.7912	−1.62796	−0.813981	−	0.580891i	$-0.802704\pi$
$757$	−44.7912	−1.62796	−0.813981	+	0.580891i	$0.802704\pi$
$758$	−84.4761	−3.06831
$759$	28.8604	1.04757
$760$	1.84366	0.0668767
$761$	−10.7250	−0.388779	−0.194390	−	0.980924i	$-0.562273\pi$
$761$	−10.7250	−0.388779	−0.194390	+	0.980924i	$0.562273\pi$
$762$	43.2906	1.56825
$763$	6.49940	0.235294
$764$	−34.4753	−1.24727
$765$	−2.33682	−0.0844879
$766$	46.8938	1.69434
$767$	34.7912	1.25624
$768$	−23.6169	−0.852202
$769$	16.0692	0.579471	0.289735	−	0.957107i	$-0.406433\pi$
$769$	16.0692	0.579471	0.289735	+	0.957107i	$0.406433\pi$
$770$	−4.53521	−0.163438
$771$	19.6274	0.706863
$772$	−52.2684	−1.88118
$773$	9.05876	0.325821	0.162910	−	0.986641i	$-0.447912\pi$
$773$	9.05876	0.325821	0.162910	+	0.986641i	$0.447912\pi$
$774$	3.05876	0.109945
$775$	31.6966	1.13858
$776$	54.8696	1.96970
$777$	−9.46479	−0.339548
$778$	−16.5236	−0.592398
$779$	−2.26760	−0.0812453
$780$	−10.4302	−0.373461
$781$	−3.53640	−0.126543
$782$	−82.0459	−2.93396
$783$	−6.18048	−0.220872
$784$	1.33682	0.0477436
$785$	4.17424	0.148985
$786$	34.3189	1.22412
$787$	−37.7658	−1.34621	−0.673103	−	0.739549i	$-0.735039\pi$
$787$	−37.7658	−1.34621	−0.673103	+	0.739549i	$0.735039\pi$
$788$	32.8036	1.16858
$789$	−18.6978	−0.665659
$790$	−7.37466	−0.262379
$791$	2.18048	0.0775291
$792$	−13.3022	−0.472674
$793$	−31.3264	−1.11243
$794$	−63.4173	−2.25060
$795$	−7.05876	−0.250348
$796$	−29.4081	−1.04234
$797$	30.7312	1.08855	0.544277	−	0.838905i	$-0.316804\pi$
$797$	30.7312	1.08855	0.544277	+	0.838905i	$0.316804\pi$
$798$	2.34889	0.0831500
$799$	−18.8845	−0.668086
$800$	18.8724	0.667242
$801$	−7.80161	−0.275656
$802$	46.3073	1.63517
$803$	−11.3264	−0.399699
$804$	22.6169	0.797637
$805$	−4.00000	−0.140981
$806$	−90.1843	−3.17661
$807$	−19.6274	−0.690916
$808$	−42.7040	−1.50232
$809$	40.1175	1.41046	0.705228	−	0.708980i	$-0.250844\pi$
$809$	40.1175	1.41046	0.705228	+	0.708980i	$0.250844\pi$
$810$	−1.21509	−0.0426940
$811$	−40.3252	−1.41601	−0.708004	−	0.706208i	$-0.750404\pi$
$811$	−40.3252	−1.41601	−0.708004	+	0.706208i	$0.750404\pi$
$812$	21.7386	0.762877
$813$	−11.4648	−0.402088
$814$	−82.9779	−2.90837
$815$	4.53521	0.158861
$816$	6.03882	0.211401
$817$	−1.30221	−0.0455586
$818$	−44.9296	−1.57093
$819$	−5.73240	−0.200306
$820$	−4.12594	−0.144084
$821$	23.3598	0.815262	0.407631	−	0.913147i	$-0.366355\pi$
$821$	23.3598	0.815262	0.407631	+	0.913147i	$0.366355\pi$
$822$	−7.97585	−0.278190
$823$	−14.3010	−0.498502	−0.249251	−	0.968439i	$-0.580184\pi$
$823$	−14.3010	−0.498502	−0.249251	+	0.968439i	$0.580184\pi$
$824$	55.1163	1.92007
$825$	17.6632	0.614953
$826$	−14.2559	−0.496028
$827$	6.73984	0.234367	0.117184	−	0.993110i	$-0.462613\pi$
$827$	6.73984	0.234367	0.117184	+	0.993110i	$0.462613\pi$
$828$	−27.1972	−0.945168
$829$	−28.6044	−0.993473	−0.496736	−	0.867901i	$-0.665468\pi$
$829$	−28.6044	−0.993473	−0.496736	+	0.867901i	$0.665468\pi$
$830$	−7.59396	−0.263590
$831$	−3.66318	−0.127074
$832$	−69.0230	−2.39294
$833$	4.51730	0.156515
$834$	6.96539	0.241192
$835$	7.53640	0.260808
$836$	13.1280	0.454040
$837$	−6.69779	−0.231509
$838$	30.6316	1.05815
$839$	20.1384	0.695256	0.347628	−	0.937633i	$-0.386987\pi$
$839$	20.1384	0.695256	0.347628	+	0.937633i	$0.386987\pi$
$840$	1.84366	0.0636125
$841$	9.19839	0.317186
$842$	−59.0797	−2.03602
$843$	23.2151	0.799570
$844$	−47.6032	−1.63857
$845$	−10.2738	−0.353431
$846$	−9.81952	−0.337602
$847$	−2.93078	−0.100703
$848$	18.2413	0.626408
$849$	29.8950	1.02599
$850$	−50.2139	−1.72232
$851$	−73.1855	−2.50877
$852$	3.33261	0.114173
$853$	10.3252	0.353527	0.176763	−	0.984253i	$-0.443437\pi$
$853$	10.3252	0.353527	0.176763	+	0.984253i	$0.443437\pi$
$854$	12.8362	0.439246
$855$	0.517304	0.0176914
$856$	−51.7395	−1.76842
$857$	−15.2664	−0.521490	−0.260745	−	0.965408i	$-0.583968\pi$
$857$	−15.2664	−0.521490	−0.260745	+	0.965408i	$0.583968\pi$
$858$	−50.2559	−1.71571
$859$	−8.67364	−0.295941	−0.147970	−	0.988992i	$-0.547274\pi$
$859$	−8.67364	−0.295941	−0.147970	+	0.988992i	$0.547274\pi$
$860$	−2.36939	−0.0807957
$861$	−2.26760	−0.0772797
$862$	52.8067	1.79860
$863$	−6.55311	−0.223070	−0.111535	−	0.993760i	$-0.535577\pi$
$863$	−6.55311	−0.223070	−0.111535	+	0.993760i	$0.535577\pi$
$864$	−3.98793	−0.135672
$865$	1.96659	0.0668661
$866$	42.2801	1.43674
$867$	3.40604	0.115675
$868$	23.5582	0.799616
$869$	−22.6527	−0.768441
$870$	7.50986	0.254608
$871$	36.8604	1.24897
$872$	−23.1638	−0.784425
$873$	15.3956	0.521061
$874$	18.1626	0.614358
$875$	−5.03461	−0.170201
$876$	10.6736	0.360629
$877$	30.9988	1.04676	0.523378	−	0.852101i	$-0.324672\pi$
$877$	30.9988	1.04676	0.523378	+	0.852101i	$0.324672\pi$
$878$	−78.6620	−2.65471
$879$	14.2676	0.481234
$880$	2.58111	0.0870092
$881$	8.51730	0.286955	0.143478	−	0.989654i	$-0.454171\pi$
$881$	8.51730	0.286955	0.143478	+	0.989654i	$0.454171\pi$
$882$	2.34889	0.0790914
$883$	23.3264	0.784995	0.392497	−	0.919753i	$-0.371611\pi$
$883$	23.3264	0.784995	0.392497	+	0.919753i	$0.371611\pi$
$884$	91.0805	3.06337
$885$	−3.13963	−0.105538
$886$	55.6608	1.86996
$887$	19.9642	0.670332	0.335166	−	0.942159i	$-0.391208\pi$
$887$	19.9642	0.670332	0.335166	+	0.942159i	$0.391208\pi$
$888$	33.7324	1.13199
$889$	−18.4302	−0.618129
$890$	9.47968	0.317760
$891$	−3.73240	−0.125040
$892$	49.8141	1.66790
$893$	4.18048	0.139895
$894$	−24.3368	−0.813945
$895$	1.83742	0.0614181
$896$	20.3068	0.678404
$897$	−44.3252	−1.47997
$898$	17.3568	0.579202
$899$	41.3956	1.38062
$900$	−16.6453	−0.554843
$901$	61.6399	2.05352
$902$	−19.8801	−0.661935
$903$	−1.30221	−0.0433349
$904$	−7.77122	−0.258467
$905$	−10.0334	−0.333522
$906$	32.6378	1.08432
$907$	7.86157	0.261039	0.130520	−	0.991446i	$-0.458335\pi$
$907$	7.86157	0.261039	0.130520	+	0.991446i	$0.458335\pi$
$908$	−69.7332	−2.31418
$909$	−11.9821	−0.397421
$910$	6.96539	0.230900
$911$	−32.4123	−1.07387	−0.536933	−	0.843625i	$-0.680417\pi$
$911$	−32.4123	−1.07387	−0.536933	+	0.843625i	$0.680417\pi$
$912$	−1.33682	−0.0442665
$913$	−23.3264	−0.771990
$914$	−42.9087	−1.41929
$915$	2.82696	0.0934564
$916$	−64.5811	−2.13382
$917$	−14.6107	−0.482487
$918$	10.6107	0.350204
$919$	53.8592	1.77665	0.888325	−	0.459215i	$-0.151869\pi$
$919$	53.8592	1.77665	0.888325	+	0.459215i	$0.151869\pi$
$920$	14.2559	0.470005
$921$	−4.76700	−0.157078
$922$	−77.1435	−2.54059
$923$	5.43138	0.178776
$924$	13.1280	0.431879
$925$	−44.7912	−1.47272
$926$	−32.7219	−1.07531
$927$	15.4648	0.507930
$928$	24.6473	0.809088
$929$	45.7028	1.49946	0.749731	−	0.661743i	$-0.230183\pi$
$929$	45.7028	1.49946	0.749731	+	0.661743i	$0.230183\pi$
$930$	8.13843	0.266870
$931$	−1.00000	−0.0327737
$932$	32.0209	1.04888
$933$	11.1459	0.364900
$934$	−15.9370	−0.521476
$935$	8.72194	0.285238
$936$	20.4302	0.667781
$937$	13.0680	0.426914	0.213457	−	0.976953i	$-0.431528\pi$
$937$	13.0680	0.426914	0.213457	+	0.976953i	$0.431528\pi$
$938$	−15.1038	−0.493157
$939$	−20.7912	−0.678494
$940$	7.60646	0.248095
$941$	50.6954	1.65262	0.826311	−	0.563214i	$-0.190435\pi$
$941$	50.6954	1.65262	0.826311	+	0.563214i	$0.190435\pi$
$942$	−18.9537	−0.617546
$943$	−17.5340	−0.570986
$944$	8.11345	0.264070
$945$	0.517304	0.0168279
$946$	−11.4165	−0.371182
$947$	−23.7807	−0.772769	−0.386384	−	0.922338i	$-0.626276\pi$
$947$	−23.7807	−0.772769	−0.386384	+	0.922338i	$0.626276\pi$
$948$	21.3473	0.693327
$949$	17.3956	0.564684
$950$	11.1159	0.360647
$951$	−6.71569	−0.217771
$952$	−16.0996	−0.521791
$953$	5.32011	0.172335	0.0861677	−	0.996281i	$-0.472538\pi$
$953$	5.32011	0.172335	0.0861677	+	0.996281i	$0.472538\pi$
$954$	32.0513	1.03770
$955$	5.07041	0.164075
$956$	−71.1614	−2.30152
$957$	23.0680	0.745683
$958$	7.30523	0.236021
$959$	3.39558	0.109649
$960$	6.22878	0.201033
$961$	13.8604	0.447109
$962$	127.441	4.10888
$963$	−14.5173	−0.467814
$964$	26.0125	0.837806
$965$	7.68733	0.247464
$966$	18.1626	0.584371
$967$	−3.41973	−0.109971	−0.0549855	−	0.998487i	$-0.517511\pi$
$967$	−3.41973	−0.109971	−0.0549855	+	0.998487i	$0.517511\pi$
$968$	10.4453	0.335724
$969$	−4.51730	−0.145117
$970$	−18.7070	−0.600647
$971$	29.1855	0.936608	0.468304	−	0.883567i	$-0.344865\pi$
$971$	29.1855	0.936608	0.468304	+	0.883567i	$0.344865\pi$
$972$	3.51730	0.112818
$973$	−2.96539	−0.0950661
$974$	−84.0668	−2.69367
$975$	−27.1280	−0.868790
$976$	−7.30544	−0.233842
$977$	−3.93703	−0.125957	−0.0629784	−	0.998015i	$-0.520060\pi$
$977$	−3.93703	−0.125957	−0.0629784	+	0.998015i	$0.520060\pi$
$978$	−20.5928	−0.658484
$979$	29.1187	0.930638
$980$	−1.81952	−0.0581223
$981$	−6.49940	−0.207510
$982$	−35.7807	−1.14181
$983$	44.8604	1.43082	0.715412	−	0.698703i	$-0.246239\pi$
$983$	44.8604	1.43082	0.715412	+	0.698703i	$0.246239\pi$
$984$	8.08171	0.257635
$985$	−4.82456	−0.153723
$986$	−65.5791	−2.08846
$987$	4.18048	0.133066
$988$	−20.1626	−0.641457
$989$	−10.0692	−0.320182
$990$	4.53521	0.144138
$991$	−42.7912	−1.35931	−0.679653	−	0.733534i	$-0.737870\pi$
$991$	−42.7912	−1.35931	−0.679653	+	0.733534i	$0.737870\pi$
$992$	26.7103	0.848052
$993$	10.4302	0.330992
$994$	−2.22555	−0.0705902
$995$	4.32516	0.137117
$996$	21.9821	0.696529
$997$	19.7207	0.624562	0.312281	−	0.949990i	$-0.398907\pi$
$997$	19.7207	0.624562	0.312281	+	0.949990i	$0.398907\pi$
$998$	66.2076	2.09577
$999$	9.46479	0.299453

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	399.2.a.e.1.3	✓	3
3.2	odd	2		1197.2.a.m.1.1		3
4.3	odd	2		6384.2.a.bu.1.2		3
5.4	even	2		9975.2.a.x.1.1		3
7.6	odd	2		2793.2.a.w.1.3		3
19.18	odd	2		7581.2.a.l.1.1		3
21.20	even	2		8379.2.a.bq.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.e.1.3	✓	3	1.1	even	1	trivial
1197.2.a.m.1.1		3	3.2	odd	2
2793.2.a.w.1.3		3	7.6	odd	2
6384.2.a.bu.1.2		3	4.3	odd	2
7581.2.a.l.1.1		3	19.18	odd	2
8379.2.a.bq.1.1		3	21.20	even	2
9975.2.a.x.1.1		3	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 399 = 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	399.a (trivial)

\(f(q)\)	\(=\)	\(q+2.34889 q^{2} +1.00000 q^{3} +3.51730 q^{4} -0.517304 q^{5} +2.34889 q^{6} -1.00000 q^{7} +3.56399 q^{8} +1.00000 q^{9} -1.21509 q^{10} -3.73240 q^{11} +3.51730 q^{12} +5.73240 q^{13} -2.34889 q^{14} -0.517304 q^{15} +1.33682 q^{16} +4.51730 q^{17} +2.34889 q^{18} -1.00000 q^{19} -1.81952 q^{20} -1.00000 q^{21} -8.76700 q^{22} -7.73240 q^{23} +3.56399 q^{24} -4.73240 q^{25} +13.4648 q^{26} +1.00000 q^{27} -3.51730 q^{28} -6.18048 q^{29} -1.21509 q^{30} -6.69779 q^{31} -3.98793 q^{32} -3.73240 q^{33} +10.6107 q^{34} +0.517304 q^{35} +3.51730 q^{36} +9.46479 q^{37} -2.34889 q^{38} +5.73240 q^{39} -1.84366 q^{40} +2.26760 q^{41} -2.34889 q^{42} +1.30221 q^{43} -13.1280 q^{44} -0.517304 q^{45} -18.1626 q^{46} -4.18048 q^{47} +1.33682 q^{48} +1.00000 q^{49} -11.1159 q^{50} +4.51730 q^{51} +20.1626 q^{52} +13.6453 q^{53} +2.34889 q^{54} +1.93078 q^{55} -3.56399 q^{56} -1.00000 q^{57} -14.5173 q^{58} +6.06922 q^{59} -1.81952 q^{60} -5.46479 q^{61} -15.7324 q^{62} -1.00000 q^{63} -12.0409 q^{64} -2.96539 q^{65} -8.76700 q^{66} +6.43018 q^{67} +15.8887 q^{68} -7.73240 q^{69} +1.21509 q^{70} +0.947489 q^{71} +3.56399 q^{72} +3.03461 q^{73} +22.2318 q^{74} -4.73240 q^{75} -3.51730 q^{76} +3.73240 q^{77} +13.4648 q^{78} +6.06922 q^{79} -0.691542 q^{80} +1.00000 q^{81} +5.32636 q^{82} +6.24970 q^{83} -3.51730 q^{84} -2.33682 q^{85} +3.05876 q^{86} -6.18048 q^{87} -13.3022 q^{88} -7.80161 q^{89} -1.21509 q^{90} -5.73240 q^{91} -27.1972 q^{92} -6.69779 q^{93} -9.81952 q^{94} +0.517304 q^{95} -3.98793 q^{96} +15.3956 q^{97} +2.34889 q^{98} -3.73240 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{3} + 9 q^{4} + q^{6} - 3 q^{7} - 9 q^{8} + 3 q^{9} + 10 q^{10} + 4 q^{11} + 9 q^{12} + 2 q^{13} - q^{14} + 13 q^{16} + 12 q^{17} + q^{18} - 3 q^{19} - 16 q^{20} - 3 q^{21} - 8 q^{22}+ \cdots + 4 q^{99}+O(q^{100}) \) 3 * q + q^2 + 3 * q^3 + 9 * q^4 + q^6 - 3 * q^7 - 9 * q^8 + 3 * q^9 + 10 * q^10 + 4 * q^11 + 9 * q^12 + 2 * q^13 - q^14 + 13 * q^16 + 12 * q^17 + q^18 - 3 * q^19 - 16 * q^20 - 3 * q^21 - 8 * q^22 - 8 * q^23 - 9 * q^24 + q^25 + 10 * q^26 + 3 * q^27 - 9 * q^28 - 8 * q^29 + 10 * q^30 - 8 * q^31 - 27 * q^32 + 4 * q^33 - 6 * q^34 + 9 * q^36 - 2 * q^37 - q^38 + 2 * q^39 + 14 * q^40 + 22 * q^41 - q^42 + 16 * q^43 - 12 * q^46 - 2 * q^47 + 13 * q^48 + 3 * q^49 - 9 * q^50 + 12 * q^51 + 18 * q^52 + q^54 + 12 * q^55 + 9 * q^56 - 3 * q^57 - 42 * q^58 + 12 * q^59 - 16 * q^60 + 14 * q^61 - 32 * q^62 - 3 * q^63 + 29 * q^64 - 12 * q^65 - 8 * q^66 - 8 * q^67 + 52 * q^68 - 8 * q^69 - 10 * q^70 - 26 * q^71 - 9 * q^72 + 6 * q^73 + 18 * q^74 + q^75 - 9 * q^76 - 4 * q^77 + 10 * q^78 + 12 * q^79 - 52 * q^80 + 3 * q^81 - 2 * q^82 + 2 * q^83 - 9 * q^84 - 16 * q^85 - 24 * q^86 - 8 * q^87 - 52 * q^88 - 2 * q^89 + 10 * q^90 - 2 * q^91 - 36 * q^92 - 8 * q^93 - 40 * q^94 - 27 * q^96 + 22 * q^97 + q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more