LMFDB - Embedded newform 6021.2.a.t.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6021,2,Mod(1,6021)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6021, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6021.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6021 = 3^{3} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6021.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:	$48.0779270570$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	6021.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.968595 q^{2} -1.06182 q^{4} +2.59691 q^{5} +3.83305 q^{7} +2.96567 q^{8} +O(q^{10})$	\(q-0.968595 q^{2} -1.06182 q^{4} +2.59691 q^{5} +3.83305 q^{7} +2.96567 q^{8} -2.51535 q^{10} +0.279425 q^{11} -4.21771 q^{13} -3.71267 q^{14} -0.748885 q^{16} +7.81149 q^{17} +5.87888 q^{19} -2.75746 q^{20} -0.270650 q^{22} +3.65142 q^{23} +1.74394 q^{25} +4.08526 q^{26} -4.07002 q^{28} +0.344370 q^{29} +6.17801 q^{31} -5.20597 q^{32} -7.56617 q^{34} +9.95408 q^{35} -3.88762 q^{37} -5.69426 q^{38} +7.70157 q^{40} +2.01212 q^{41} -3.50845 q^{43} -0.296700 q^{44} -3.53675 q^{46} -10.9567 q^{47} +7.69225 q^{49} -1.68917 q^{50} +4.47847 q^{52} +2.30159 q^{53} +0.725642 q^{55} +11.3675 q^{56} -0.333555 q^{58} +9.48835 q^{59} +13.9438 q^{61} -5.98399 q^{62} +6.54025 q^{64} -10.9530 q^{65} +7.74509 q^{67} -8.29442 q^{68} -9.64147 q^{70} -2.43428 q^{71} +3.28858 q^{73} +3.76553 q^{74} -6.24233 q^{76} +1.07105 q^{77} -10.8574 q^{79} -1.94479 q^{80} -1.94893 q^{82} -13.0778 q^{83} +20.2857 q^{85} +3.39826 q^{86} +0.828682 q^{88} +13.3036 q^{89} -16.1667 q^{91} -3.87716 q^{92} +10.6126 q^{94} +15.2669 q^{95} -18.4797 q^{97} -7.45068 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 46 q^{4} + 16 q^{7}+O(q^{10}) $ 40 * q + 46 * q^4 + 16 * q^7	$ 40 q + 46 q^{4} + 16 q^{7} + 22 q^{10} + 14 q^{13} + 50 q^{16} + 64 q^{19} + 12 q^{22} + 40 q^{25} + 48 q^{28} + 54 q^{31} + 32 q^{34} + 24 q^{37} + 40 q^{40} + 24 q^{43} + 52 q^{46} + 64 q^{49} + 18 q^{52} + 36 q^{55} + 8 q^{58} + 58 q^{61} + 120 q^{64} + 52 q^{67} - 30 q^{70} + 50 q^{73} + 112 q^{76} + 60 q^{79} + 50 q^{82} + 38 q^{85} + 16 q^{88} + 118 q^{91} + 44 q^{94} + 38 q^{97}+O(q^{100}) $ 40 * q + 46 * q^4 + 16 * q^7 + 22 * q^10 + 14 * q^13 + 50 * q^16 + 64 * q^19 + 12 * q^22 + 40 * q^25 + 48 * q^28 + 54 * q^31 + 32 * q^34 + 24 * q^37 + 40 * q^40 + 24 * q^43 + 52 * q^46 + 64 * q^49 + 18 * q^52 + 36 * q^55 + 8 * q^58 + 58 * q^61 + 120 * q^64 + 52 * q^67 - 30 * q^70 + 50 * q^73 + 112 * q^76 + 60 * q^79 + 50 * q^82 + 38 * q^85 + 16 * q^88 + 118 * q^91 + 44 * q^94 + 38 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.968595	−0.684900	−0.342450	−	0.939536i	$-0.611257\pi$
$2$	−0.968595	−0.684900	−0.342450	+	0.939536i	$0.611257\pi$
$3$	0	0
$3$	0	0
$4$	−1.06182	−0.530912
$5$	2.59691	1.16137	0.580687	−	0.814127i	$-0.302784\pi$
$5$	2.59691	1.16137	0.580687	+	0.814127i	$0.302784\pi$
$6$	0	0
$7$	3.83305	1.44876	0.724378	−	0.689403i	$-0.242127\pi$
$7$	3.83305	1.44876	0.724378	+	0.689403i	$0.242127\pi$
$8$	2.96567	1.04852
$9$	0	0
$10$	−2.51535	−0.795425
$11$	0.279425	0.0842498	0.0421249	−	0.999112i	$-0.486587\pi$
$11$	0.279425	0.0842498	0.0421249	+	0.999112i	$0.486587\pi$
$12$	0	0
$13$	−4.21771	−1.16978	−0.584892	−	0.811111i	$-0.698863\pi$
$13$	−4.21771	−1.16978	−0.584892	+	0.811111i	$0.698863\pi$
$14$	−3.71267	−0.992253
$15$	0	0
$16$	−0.748885	−0.187221
$17$	7.81149	1.89456	0.947282	−	0.320402i	$-0.103818\pi$
$17$	7.81149	1.89456	0.947282	+	0.320402i	$0.103818\pi$
$18$	0	0
$19$	5.87888	1.34871	0.674354	−	0.738408i	$-0.264422\pi$
$19$	5.87888	1.34871	0.674354	+	0.738408i	$0.264422\pi$
$20$	−2.75746	−0.616587
$21$	0	0
$22$	−0.270650	−0.0577027
$23$	3.65142	0.761373	0.380687	−	0.924704i	$-0.375688\pi$
$23$	3.65142	0.761373	0.380687	+	0.924704i	$0.375688\pi$
$24$	0	0
$25$	1.74394	0.348788
$26$	4.08526	0.801185
$27$	0	0
$28$	−4.07002	−0.769161
$29$	0.344370	0.0639478	0.0319739	−	0.999489i	$-0.489821\pi$
$29$	0.344370	0.0639478	0.0319739	+	0.999489i	$0.489821\pi$
$30$	0	0
$31$	6.17801	1.10960	0.554802	−	0.831983i	$-0.312794\pi$
$31$	6.17801	1.10960	0.554802	+	0.831983i	$0.312794\pi$
$32$	−5.20597	−0.920294
$33$	0	0
$34$	−7.56617	−1.29759
$35$	9.95408	1.68255
$36$	0	0
$37$	−3.88762	−0.639121	−0.319560	−	0.947566i	$-0.603535\pi$
$37$	−3.88762	−0.639121	−0.319560	+	0.947566i	$0.603535\pi$
$38$	−5.69426	−0.923731
$39$	0	0
$40$	7.70157	1.21773
$41$	2.01212	0.314241	0.157120	−	0.987579i	$-0.449779\pi$
$41$	2.01212	0.314241	0.157120	+	0.987579i	$0.449779\pi$
$42$	0	0
$43$	−3.50845	−0.535033	−0.267516	−	0.963553i	$-0.586203\pi$
$43$	−3.50845	−0.535033	−0.267516	+	0.963553i	$0.586203\pi$
$44$	−0.296700	−0.0447292
$45$	0	0
$46$	−3.53675	−0.521465
$47$	−10.9567	−1.59820	−0.799100	−	0.601199i	$-0.794690\pi$
$47$	−10.9567	−1.59820	−0.799100	+	0.601199i	$0.794690\pi$
$48$	0	0
$49$	7.69225	1.09889
$50$	−1.68917	−0.238885
$51$	0	0
$52$	4.47847	0.621051
$53$	2.30159	0.316148	0.158074	−	0.987427i	$-0.449472\pi$
$53$	2.30159	0.316148	0.158074	+	0.987427i	$0.449472\pi$
$54$	0	0
$55$	0.725642	0.0978455
$56$	11.3675	1.51905
$57$	0	0
$58$	−0.333555	−0.0437979
$59$	9.48835	1.23528	0.617639	−	0.786462i	$-0.288089\pi$
$59$	9.48835	1.23528	0.617639	+	0.786462i	$0.288089\pi$
$60$	0	0
$61$	13.9438	1.78531	0.892657	−	0.450736i	$-0.148838\pi$
$61$	13.9438	1.78531	0.892657	+	0.450736i	$0.148838\pi$
$62$	−5.98399	−0.759968
$63$	0	0
$64$	6.54025	0.817531
$65$	−10.9530	−1.35855
$66$	0	0
$67$	7.74509	0.946214	0.473107	−	0.881005i	$-0.343132\pi$
$67$	7.74509	0.946214	0.473107	+	0.881005i	$0.343132\pi$
$68$	−8.29442	−1.00585
$69$	0	0
$70$	−9.64147	−1.15238
$71$	−2.43428	−0.288895	−0.144448	−	0.989512i	$-0.546141\pi$
$71$	−2.43428	−0.288895	−0.144448	+	0.989512i	$0.546141\pi$
$72$	0	0
$73$	3.28858	0.384900	0.192450	−	0.981307i	$-0.438357\pi$
$73$	3.28858	0.384900	0.192450	+	0.981307i	$0.438357\pi$
$74$	3.76553	0.437734
$75$	0	0
$76$	−6.24233	−0.716045
$77$	1.07105	0.122057
$78$	0	0
$79$	−10.8574	−1.22155	−0.610775	−	0.791804i	$-0.709142\pi$
$79$	−10.8574	−1.22155	−0.610775	+	0.791804i	$0.709142\pi$
$80$	−1.94479	−0.217434
$81$	0	0
$82$	−1.94893	−0.215224
$83$	−13.0778	−1.43548	−0.717739	−	0.696312i	$-0.754823\pi$
$83$	−13.0778	−1.43548	−0.717739	+	0.696312i	$0.754823\pi$
$84$	0	0
$85$	20.2857	2.20030
$86$	3.39826	0.366444
$87$	0	0
$88$	0.828682	0.0883378
$89$	13.3036	1.41018	0.705089	−	0.709119i	$-0.250907\pi$
$89$	13.3036	1.41018	0.705089	+	0.709119i	$0.250907\pi$
$90$	0	0
$91$	−16.1667	−1.69473
$92$	−3.87716	−0.404222
$93$	0	0
$94$	10.6126	1.09461
$95$	15.2669	1.56635
$96$	0	0
$97$	−18.4797	−1.87633	−0.938164	−	0.346191i	$-0.887475\pi$
$97$	−18.4797	−1.87633	−0.938164	+	0.346191i	$0.887475\pi$
$98$	−7.45068	−0.752632
$99$	0	0
$100$	−1.85176	−0.185176
$101$	13.8803	1.38115	0.690573	−	0.723263i	$-0.257359\pi$
$101$	13.8803	1.38115	0.690573	+	0.723263i	$0.257359\pi$
$102$	0	0
$103$	10.1623	1.00132	0.500659	−	0.865645i	$-0.333091\pi$
$103$	10.1623	1.00132	0.500659	+	0.865645i	$0.333091\pi$
$104$	−12.5083	−1.22654
$105$	0	0
$106$	−2.22931	−0.216530
$107$	−0.205628	−0.0198788	−0.00993942	−	0.999951i	$-0.503164\pi$
$107$	−0.205628	−0.0198788	−0.00993942	+	0.999951i	$0.503164\pi$
$108$	0	0
$109$	16.1871	1.55045	0.775223	−	0.631688i	$-0.217638\pi$
$109$	16.1871	1.55045	0.775223	+	0.631688i	$0.217638\pi$
$110$	−0.702853	−0.0670144
$111$	0	0
$112$	−2.87051	−0.271238
$113$	−10.3268	−0.971464	−0.485732	−	0.874108i	$-0.661447\pi$
$113$	−10.3268	−0.971464	−0.485732	+	0.874108i	$0.661447\pi$
$114$	0	0
$115$	9.48240	0.884238
$116$	−0.365660	−0.0339506
$117$	0	0
$118$	−9.19038	−0.846043
$119$	29.9418	2.74476
$120$	0	0
$121$	−10.9219	−0.992902
$122$	−13.5059	−1.22276
$123$	0	0
$124$	−6.55996	−0.589101
$125$	−8.45569	−0.756300
$126$	0	0
$127$	1.64226	0.145727	0.0728634	−	0.997342i	$-0.476786\pi$
$127$	1.64226	0.145727	0.0728634	+	0.997342i	$0.476786\pi$
$128$	4.07708	0.360367
$129$	0	0
$130$	10.6090	0.930475
$131$	−10.1120	−0.883487	−0.441744	−	0.897141i	$-0.645640\pi$
$131$	−10.1120	−0.883487	−0.441744	+	0.897141i	$0.645640\pi$
$132$	0	0
$133$	22.5340	1.95395
$134$	−7.50186	−0.648062
$135$	0	0
$136$	23.1663	1.98649
$137$	−22.6264	−1.93310	−0.966552	−	0.256472i	$-0.917440\pi$
$137$	−22.6264	−1.93310	−0.966552	+	0.256472i	$0.917440\pi$
$138$	0	0
$139$	−1.22592	−0.103981	−0.0519905	−	0.998648i	$-0.516557\pi$
$139$	−1.22592	−0.103981	−0.0519905	+	0.998648i	$0.516557\pi$
$140$	−10.5695	−0.893283
$141$	0	0
$142$	2.35783	0.197865
$143$	−1.17853	−0.0985540
$144$	0	0
$145$	0.894297	0.0742673
$146$	−3.18531	−0.263618
$147$	0	0
$148$	4.12796	0.339317
$149$	−11.5691	−0.947774	−0.473887	−	0.880586i	$-0.657149\pi$
$149$	−11.5691	−0.947774	−0.473887	+	0.880586i	$0.657149\pi$
$150$	0	0
$151$	17.7080	1.44105	0.720527	−	0.693427i	$-0.243900\pi$
$151$	17.7080	1.44105	0.720527	+	0.693427i	$0.243900\pi$
$152$	17.4348	1.41415
$153$	0	0
$154$	−1.03741	−0.0835972
$155$	16.0437	1.28866
$156$	0	0
$157$	12.2831	0.980297	0.490149	−	0.871639i	$-0.336943\pi$
$157$	12.2831	0.980297	0.490149	+	0.871639i	$0.336943\pi$
$158$	10.5164	0.836640
$159$	0	0
$160$	−13.5194	−1.06880
$161$	13.9961	1.10304
$162$	0	0
$163$	−15.7793	−1.23593	−0.617963	−	0.786207i	$-0.712042\pi$
$163$	−15.7793	−1.23593	−0.617963	+	0.786207i	$0.712042\pi$
$164$	−2.13652	−0.166834
$165$	0	0
$166$	12.6671	0.983159
$167$	−12.7424	−0.986038	−0.493019	−	0.870019i	$-0.664107\pi$
$167$	−12.7424	−0.986038	−0.493019	+	0.870019i	$0.664107\pi$
$168$	0	0
$169$	4.78911	0.368393
$170$	−19.6487	−1.50698
$171$	0	0
$172$	3.72535	0.284055
$173$	−1.19895	−0.0911542	−0.0455771	−	0.998961i	$-0.514513\pi$
$173$	−1.19895	−0.0911542	−0.0455771	+	0.998961i	$0.514513\pi$
$174$	0	0
$175$	6.68460	0.505308
$176$	−0.209257	−0.0157734
$177$	0	0
$178$	−12.8858	−0.965831
$179$	4.30207	0.321552	0.160776	−	0.986991i	$-0.448600\pi$
$179$	4.30207	0.321552	0.160776	+	0.986991i	$0.448600\pi$
$180$	0	0
$181$	−22.7934	−1.69422	−0.847109	−	0.531419i	$-0.821659\pi$
$181$	−22.7934	−1.69422	−0.847109	+	0.531419i	$0.821659\pi$
$182$	15.6590	1.16072
$183$	0	0
$184$	10.8289	0.798316
$185$	−10.0958	−0.742258
$186$	0	0
$187$	2.18272	0.159617
$188$	11.6341	0.848502
$189$	0	0
$190$	−14.7875	−1.07280
$191$	−17.1285	−1.23938	−0.619688	−	0.784849i	$-0.712741\pi$
$191$	−17.1285	−1.23938	−0.619688	+	0.784849i	$0.712741\pi$
$192$	0	0
$193$	9.09237	0.654483	0.327242	−	0.944941i	$-0.393881\pi$
$193$	9.09237	0.654483	0.327242	+	0.944941i	$0.393881\pi$
$194$	17.8993	1.28510
$195$	0	0
$196$	−8.16781	−0.583415
$197$	−23.2646	−1.65754	−0.828769	−	0.559592i	$-0.810958\pi$
$197$	−23.2646	−1.65754	−0.828769	+	0.559592i	$0.810958\pi$
$198$	0	0
$199$	15.5966	1.10562	0.552808	−	0.833309i	$-0.313556\pi$
$199$	15.5966	1.10562	0.552808	+	0.833309i	$0.313556\pi$
$200$	5.17194	0.365712
$201$	0	0
$202$	−13.4444	−0.945947
$203$	1.31998	0.0926448
$204$	0	0
$205$	5.22530	0.364951
$206$	−9.84312	−0.685802
$207$	0	0
$208$	3.15858	0.219008
$209$	1.64271	0.113628
$210$	0	0
$211$	−0.599109	−0.0412444	−0.0206222	−	0.999787i	$-0.506565\pi$
$211$	−0.599109	−0.0412444	−0.0206222	+	0.999787i	$0.506565\pi$
$212$	−2.44388	−0.167847
$213$	0	0
$214$	0.199171	0.0136150
$215$	−9.11112	−0.621373
$216$	0	0
$217$	23.6806	1.60754
$218$	−15.6788	−1.06190
$219$	0	0
$220$	−0.770503	−0.0519473
$221$	−32.9466	−2.21623
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	−19.9547	−1.33328
$225$	0	0
$226$	10.0025	0.665356
$227$	−14.0856	−0.934893	−0.467446	−	0.884021i	$-0.654826\pi$
$227$	−14.0856	−0.934893	−0.467446	+	0.884021i	$0.654826\pi$
$228$	0	0
$229$	16.8497	1.11346	0.556731	−	0.830693i	$-0.312055\pi$
$229$	16.8497	1.11346	0.556731	+	0.830693i	$0.312055\pi$
$230$	−9.18461	−0.605615
$231$	0	0
$232$	1.02129	0.0670507
$233$	10.3461	0.677795	0.338898	−	0.940823i	$-0.389946\pi$
$233$	10.3461	0.677795	0.338898	+	0.940823i	$0.389946\pi$
$234$	0	0
$235$	−28.4536	−1.85611
$236$	−10.0750	−0.655824
$237$	0	0
$238$	−29.0015	−1.87989
$239$	−18.5150	−1.19763	−0.598817	−	0.800886i	$-0.704362\pi$
$239$	−18.5150	−1.19763	−0.598817	+	0.800886i	$0.704362\pi$
$240$	0	0
$241$	−11.8214	−0.761485	−0.380742	−	0.924681i	$-0.624332\pi$
$241$	−11.8214	−0.761485	−0.380742	+	0.924681i	$0.624332\pi$
$242$	10.5789	0.680039
$243$	0	0
$244$	−14.8058	−0.947844
$245$	19.9761	1.27622
$246$	0	0
$247$	−24.7954	−1.57770
$248$	18.3219	1.16344
$249$	0	0
$250$	8.19015	0.517990
$251$	−1.59886	−0.100919	−0.0504595	−	0.998726i	$-0.516069\pi$
$251$	−1.59886	−0.100919	−0.0504595	+	0.998726i	$0.516069\pi$
$252$	0	0
$253$	1.02030	0.0641456
$254$	−1.59068	−0.0998083
$255$	0	0
$256$	−17.0295	−1.06435
$257$	−3.48255	−0.217236	−0.108618	−	0.994084i	$-0.534642\pi$
$257$	−3.48255	−0.217236	−0.108618	+	0.994084i	$0.534642\pi$
$258$	0	0
$259$	−14.9014	−0.925930
$260$	11.6302	0.721273
$261$	0	0
$262$	9.79441	0.605101
$263$	14.9846	0.923988	0.461994	−	0.886883i	$-0.347134\pi$
$263$	14.9846	0.923988	0.461994	+	0.886883i	$0.347134\pi$
$264$	0	0
$265$	5.97703	0.367166
$266$	−21.8264	−1.33826
$267$	0	0
$268$	−8.22392	−0.502356
$269$	−26.8597	−1.63766	−0.818832	−	0.574033i	$-0.805378\pi$
$269$	−26.8597	−1.63766	−0.818832	+	0.574033i	$0.805378\pi$
$270$	0	0
$271$	−14.6214	−0.888189	−0.444094	−	0.895980i	$-0.646474\pi$
$271$	−14.6214	−0.888189	−0.444094	+	0.895980i	$0.646474\pi$
$272$	−5.84991	−0.354703
$273$	0	0
$274$	21.9158	1.32398
$275$	0.487300	0.0293853
$276$	0	0
$277$	25.0790	1.50685	0.753427	−	0.657532i	$-0.228399\pi$
$277$	25.0790	1.50685	0.753427	+	0.657532i	$0.228399\pi$
$278$	1.18742	0.0712167
$279$	0	0
$280$	29.5205	1.76419
$281$	−0.0765969	−0.00456939	−0.00228469	−	0.999997i	$-0.500727\pi$
$281$	−0.0765969	−0.00456939	−0.00228469	+	0.999997i	$0.500727\pi$
$282$	0	0
$283$	−32.5579	−1.93536	−0.967682	−	0.252172i	$-0.918855\pi$
$283$	−32.5579	−1.93536	−0.967682	+	0.252172i	$0.918855\pi$
$284$	2.58477	0.153378
$285$	0	0
$286$	1.14152	0.0674997
$287$	7.71256	0.455258
$288$	0	0
$289$	44.0193	2.58937
$290$	−0.866211	−0.0508657
$291$	0	0
$292$	−3.49189	−0.204348
$293$	4.17548	0.243934	0.121967	−	0.992534i	$-0.461080\pi$
$293$	4.17548	0.243934	0.121967	+	0.992534i	$0.461080\pi$
$294$	0	0
$295$	24.6404	1.43462
$296$	−11.5294	−0.670132
$297$	0	0
$298$	11.2057	0.649130
$299$	−15.4006	−0.890642
$300$	0	0
$301$	−13.4480	−0.775132
$302$	−17.1518	−0.986978
$303$	0	0
$304$	−4.40261	−0.252507
$305$	36.2107	2.07342
$306$	0	0
$307$	−2.81450	−0.160632	−0.0803161	−	0.996769i	$-0.525593\pi$
$307$	−2.81450	−0.160632	−0.0803161	+	0.996769i	$0.525593\pi$
$308$	−1.13727	−0.0648017
$309$	0	0
$310$	−15.5399	−0.882606
$311$	4.88213	0.276840	0.138420	−	0.990374i	$-0.455798\pi$
$311$	4.88213	0.276840	0.138420	+	0.990374i	$0.455798\pi$
$312$	0	0
$313$	32.2842	1.82481	0.912406	−	0.409285i	$-0.134222\pi$
$313$	32.2842	1.82481	0.912406	+	0.409285i	$0.134222\pi$
$314$	−11.8973	−0.671406
$315$	0	0
$316$	11.5286	0.648535
$317$	10.8407	0.608874	0.304437	−	0.952532i	$-0.401532\pi$
$317$	10.8407	0.608874	0.304437	+	0.952532i	$0.401532\pi$
$318$	0	0
$319$	0.0962255	0.00538759
$320$	16.9844	0.949459
$321$	0	0
$322$	−13.5565	−0.755475
$323$	45.9228	2.55521
$324$	0	0
$325$	−7.35544	−0.408006
$326$	15.2837	0.846487
$327$	0	0
$328$	5.96728	0.329488
$329$	−41.9976	−2.31540
$330$	0	0
$331$	22.3166	1.22663	0.613316	−	0.789838i	$-0.289835\pi$
$331$	22.3166	1.22663	0.613316	+	0.789838i	$0.289835\pi$
$332$	13.8863	0.762112
$333$	0	0
$334$	12.3422	0.675338
$335$	20.1133	1.09891
$336$	0	0
$337$	3.38194	0.184226	0.0921130	−	0.995749i	$-0.470638\pi$
$337$	3.38194	0.184226	0.0921130	+	0.995749i	$0.470638\pi$
$338$	−4.63871	−0.252312
$339$	0	0
$340$	−21.5398	−1.16816
$341$	1.72629	0.0934839
$342$	0	0
$343$	2.65342	0.143271
$344$	−10.4049	−0.560994
$345$	0	0
$346$	1.16129	0.0624315
$347$	−12.9715	−0.696349	−0.348174	−	0.937430i	$-0.613198\pi$
$347$	−12.9715	−0.696349	−0.348174	+	0.937430i	$0.613198\pi$
$348$	0	0
$349$	2.48281	0.132902	0.0664508	−	0.997790i	$-0.478832\pi$
$349$	2.48281	0.132902	0.0664508	+	0.997790i	$0.478832\pi$
$350$	−6.47467	−0.346086
$351$	0	0
$352$	−1.45468	−0.0775346
$353$	−13.9288	−0.741357	−0.370678	−	0.928761i	$-0.620875\pi$
$353$	−13.9288	−0.741357	−0.370678	+	0.928761i	$0.620875\pi$
$354$	0	0
$355$	−6.32159	−0.335515
$356$	−14.1261	−0.748679
$357$	0	0
$358$	−4.16696	−0.220231
$359$	15.4633	0.816123	0.408062	−	0.912954i	$-0.366205\pi$
$359$	15.4633	0.816123	0.408062	+	0.912954i	$0.366205\pi$
$360$	0	0
$361$	15.5613	0.819014
$362$	22.0776	1.16037
$363$	0	0
$364$	17.1662	0.899752
$365$	8.54015	0.447012
$366$	0	0
$367$	−21.2307	−1.10823	−0.554116	−	0.832439i	$-0.686944\pi$
$367$	−21.2307	−1.10823	−0.554116	+	0.832439i	$0.686944\pi$
$368$	−2.73449	−0.142545
$369$	0	0
$370$	9.77874	0.508372
$371$	8.82211	0.458021
$372$	0	0
$373$	4.17460	0.216153	0.108076	−	0.994143i	$-0.465531\pi$
$373$	4.17460	0.216153	0.108076	+	0.994143i	$0.465531\pi$
$374$	−2.11418	−0.109321
$375$	0	0
$376$	−32.4939	−1.67575
$377$	−1.45245	−0.0748051
$378$	0	0
$379$	33.6289	1.72740	0.863700	−	0.504007i	$-0.168141\pi$
$379$	33.6289	1.72740	0.863700	+	0.504007i	$0.168141\pi$
$380$	−16.2108	−0.831595
$381$	0	0
$382$	16.5906	0.848848
$383$	20.7352	1.05952	0.529761	−	0.848147i	$-0.322282\pi$
$383$	20.7352	1.05952	0.529761	+	0.848147i	$0.322282\pi$
$384$	0	0
$385$	2.78142	0.141754
$386$	−8.80683	−0.448256
$387$	0	0
$388$	19.6222	0.996164
$389$	−11.5006	−0.583101	−0.291551	−	0.956555i	$-0.594171\pi$
$389$	−11.5006	−0.583101	−0.291551	+	0.956555i	$0.594171\pi$
$390$	0	0
$391$	28.5230	1.44247
$392$	22.8127	1.15221
$393$	0	0
$394$	22.5340	1.13525
$395$	−28.1956	−1.41868
$396$	0	0
$397$	16.3105	0.818598	0.409299	−	0.912400i	$-0.365773\pi$
$397$	16.3105	0.818598	0.409299	+	0.912400i	$0.365773\pi$
$398$	−15.1068	−0.757236
$399$	0	0
$400$	−1.30601	−0.0653005
$401$	11.9080	0.594656	0.297328	−	0.954775i	$-0.403905\pi$
$401$	11.9080	0.594656	0.297328	+	0.954775i	$0.403905\pi$
$402$	0	0
$403$	−26.0571	−1.29800
$404$	−14.7385	−0.733266
$405$	0	0
$406$	−1.27853	−0.0634524
$407$	−1.08630	−0.0538458
$408$	0	0
$409$	34.4660	1.70424	0.852118	−	0.523350i	$-0.175318\pi$
$409$	34.4660	1.70424	0.852118	+	0.523350i	$0.175318\pi$
$410$	−5.06120	−0.249955
$411$	0	0
$412$	−10.7905	−0.531611
$413$	36.3693	1.78962
$414$	0	0
$415$	−33.9619	−1.66713
$416$	21.9573	1.07654
$417$	0	0
$418$	−1.59112	−0.0778241
$419$	11.4060	0.557222	0.278611	−	0.960404i	$-0.410126\pi$
$419$	11.4060	0.557222	0.278611	+	0.960404i	$0.410126\pi$
$420$	0	0
$421$	−21.6448	−1.05490	−0.527452	−	0.849585i	$-0.676853\pi$
$421$	−21.6448	−1.05490	−0.527452	+	0.849585i	$0.676853\pi$
$422$	0.580294	0.0282483
$423$	0	0
$424$	6.82576	0.331488
$425$	13.6228	0.660801
$426$	0	0
$427$	53.4471	2.58648
$428$	0.218341	0.0105539
$429$	0	0
$430$	8.82498	0.425579
$431$	−8.25992	−0.397866	−0.198933	−	0.980013i	$-0.563748\pi$
$431$	−8.25992	−0.397866	−0.198933	+	0.980013i	$0.563748\pi$
$432$	0	0
$433$	29.0084	1.39405	0.697027	−	0.717045i	$-0.254506\pi$
$433$	29.0084	1.39405	0.697027	+	0.717045i	$0.254506\pi$
$434$	−22.9369	−1.10101
$435$	0	0
$436$	−17.1879	−0.823149
$437$	21.4663	1.02687
$438$	0	0
$439$	−34.9072	−1.66603	−0.833015	−	0.553250i	$-0.813387\pi$
$439$	−34.9072	−1.66603	−0.833015	+	0.553250i	$0.813387\pi$
$440$	2.15201	0.102593
$441$	0	0
$442$	31.9119	1.51790
$443$	19.9880	0.949658	0.474829	−	0.880078i	$-0.342510\pi$
$443$	19.9880	0.949658	0.474829	+	0.880078i	$0.342510\pi$
$444$	0	0
$445$	34.5482	1.63774
$446$	−0.968595	−0.0458643
$447$	0	0
$448$	25.0691	1.18440
$449$	30.2848	1.42923	0.714615	−	0.699518i	$-0.246602\pi$
$449$	30.2848	1.42923	0.714615	+	0.699518i	$0.246602\pi$
$450$	0	0
$451$	0.562237	0.0264747
$452$	10.9652	0.515761
$453$	0	0
$454$	13.6432	0.640308
$455$	−41.9834	−1.96821
$456$	0	0
$457$	−36.9715	−1.72945	−0.864727	−	0.502242i	$-0.832508\pi$
$457$	−36.9715	−1.72945	−0.864727	+	0.502242i	$0.832508\pi$
$458$	−16.3206	−0.762611
$459$	0	0
$460$	−10.0686	−0.469452
$461$	6.39808	0.297988	0.148994	−	0.988838i	$-0.452396\pi$
$461$	6.39808	0.297988	0.148994	+	0.988838i	$0.452396\pi$
$462$	0	0
$463$	−1.86454	−0.0866526	−0.0433263	−	0.999061i	$-0.513796\pi$
$463$	−1.86454	−0.0866526	−0.0433263	+	0.999061i	$0.513796\pi$
$464$	−0.257893	−0.0119724
$465$	0	0
$466$	−10.0212	−0.464222
$467$	−1.10699	−0.0512253	−0.0256127	−	0.999672i	$-0.508154\pi$
$467$	−1.10699	−0.0512253	−0.0256127	+	0.999672i	$0.508154\pi$
$468$	0	0
$469$	29.6873	1.37083
$470$	27.5600	1.27125
$471$	0	0
$472$	28.1393	1.29522
$473$	−0.980348	−0.0450764
$474$	0	0
$475$	10.2524	0.470413
$476$	−31.7929	−1.45722
$477$	0	0
$478$	17.9335	0.820261
$479$	−0.446128	−0.0203841	−0.0101921	−	0.999948i	$-0.503244\pi$
$479$	−0.446128	−0.0203841	−0.0101921	+	0.999948i	$0.503244\pi$
$480$	0	0
$481$	16.3969	0.747633
$482$	11.4502	0.521541
$483$	0	0
$484$	11.5971	0.527143
$485$	−47.9901	−2.17912
$486$	0	0
$487$	20.8368	0.944207	0.472103	−	0.881543i	$-0.343495\pi$
$487$	20.8368	0.944207	0.472103	+	0.881543i	$0.343495\pi$
$488$	41.3525	1.87194
$489$	0	0
$490$	−19.3487	−0.874087
$491$	−14.8583	−0.670546	−0.335273	−	0.942121i	$-0.608829\pi$
$491$	−14.8583	−0.670546	−0.335273	+	0.942121i	$0.608829\pi$
$492$	0	0
$493$	2.69004	0.121153
$494$	24.0167	1.08056
$495$	0	0
$496$	−4.62662	−0.207741
$497$	−9.33069	−0.418539
$498$	0	0
$499$	14.5122	0.649655	0.324827	−	0.945773i	$-0.394694\pi$
$499$	14.5122	0.649655	0.324827	+	0.945773i	$0.394694\pi$
$500$	8.97845	0.401529
$501$	0	0
$502$	1.54865	0.0691195
$503$	35.7947	1.59601	0.798004	−	0.602652i	$-0.205889\pi$
$503$	35.7947	1.59601	0.798004	+	0.602652i	$0.205889\pi$
$504$	0	0
$505$	36.0460	1.60403
$506$	−0.988255	−0.0439333
$507$	0	0
$508$	−1.74379	−0.0773680
$509$	23.6601	1.04871	0.524357	−	0.851498i	$-0.324306\pi$
$509$	23.6601	1.04871	0.524357	+	0.851498i	$0.324306\pi$
$510$	0	0
$511$	12.6053	0.557625
$512$	8.34056	0.368604
$513$	0	0
$514$	3.37318	0.148785
$515$	26.3905	1.16290
$516$	0	0
$517$	−3.06158	−0.134648
$518$	14.4335	0.634169
$519$	0	0
$520$	−32.4830	−1.42447
$521$	10.1878	0.446337	0.223169	−	0.974780i	$-0.428360\pi$
$521$	10.1878	0.446337	0.223169	+	0.974780i	$0.428360\pi$
$522$	0	0
$523$	9.95142	0.435145	0.217573	−	0.976044i	$-0.430186\pi$
$523$	9.95142	0.435145	0.217573	+	0.976044i	$0.430186\pi$
$524$	10.7371	0.469054
$525$	0	0
$526$	−14.5140	−0.632840
$527$	48.2594	2.10221
$528$	0	0
$529$	−9.66715	−0.420311
$530$	−5.78932	−0.251472
$531$	0	0
$532$	−23.9272	−1.03737
$533$	−8.48655	−0.367593
$534$	0	0
$535$	−0.533998	−0.0230867
$536$	22.9694	0.992126
$537$	0	0
$538$	26.0162	1.12164
$539$	2.14941	0.0925815
$540$	0	0
$541$	17.9026	0.769693	0.384846	−	0.922981i	$-0.374254\pi$
$541$	17.9026	0.769693	0.384846	+	0.922981i	$0.374254\pi$
$542$	14.1622	0.608321
$543$	0	0
$544$	−40.6663	−1.74356
$545$	42.0365	1.80065
$546$	0	0
$547$	31.6637	1.35384	0.676920	−	0.736056i	$-0.263314\pi$
$547$	31.6637	1.35384	0.676920	+	0.736056i	$0.263314\pi$
$548$	24.0252	1.02631
$549$	0	0
$550$	−0.471997	−0.0201260
$551$	2.02451	0.0862469
$552$	0	0
$553$	−41.6168	−1.76973
$554$	−24.2914	−1.03204
$555$	0	0
$556$	1.30171	0.0552048
$557$	34.6270	1.46719	0.733596	−	0.679586i	$-0.237841\pi$
$557$	34.6270	1.46719	0.733596	+	0.679586i	$0.237841\pi$
$558$	0	0
$559$	14.7976	0.625873
$560$	−7.45446	−0.315009
$561$	0	0
$562$	0.0741914	0.00312957
$563$	8.74442	0.368533	0.184267	−	0.982876i	$-0.441009\pi$
$563$	8.74442	0.368533	0.184267	+	0.982876i	$0.441009\pi$
$564$	0	0
$565$	−26.8178	−1.12823
$566$	31.5354	1.32553
$567$	0	0
$568$	−7.21925	−0.302913
$569$	−38.0651	−1.59577	−0.797885	−	0.602810i	$-0.794048\pi$
$569$	−38.0651	−1.59577	−0.797885	+	0.602810i	$0.794048\pi$
$570$	0	0
$571$	−0.333485	−0.0139559	−0.00697797	−	0.999976i	$-0.502221\pi$
$571$	−0.333485	−0.0139559	−0.00697797	+	0.999976i	$0.502221\pi$
$572$	1.25140	0.0523235
$573$	0	0
$574$	−7.47035	−0.311806
$575$	6.36785	0.265558
$576$	0	0
$577$	−37.4906	−1.56075	−0.780377	−	0.625309i	$-0.784973\pi$
$577$	−37.4906	−1.56075	−0.780377	+	0.625309i	$0.784973\pi$
$578$	−42.6369	−1.77346
$579$	0	0
$580$	−0.949585	−0.0394294
$581$	−50.1279	−2.07966
$582$	0	0
$583$	0.643122	0.0266354
$584$	9.75284	0.403576
$585$	0	0
$586$	−4.04435	−0.167071
$587$	−35.9748	−1.48484	−0.742420	−	0.669935i	$-0.766322\pi$
$587$	−35.9748	−1.48484	−0.742420	+	0.669935i	$0.766322\pi$
$588$	0	0
$589$	36.3198	1.49653
$590$	−23.8666	−0.982571
$591$	0	0
$592$	2.91138	0.119657
$593$	−20.1843	−0.828870	−0.414435	−	0.910079i	$-0.636021\pi$
$593$	−20.1843	−0.828870	−0.414435	+	0.910079i	$0.636021\pi$
$594$	0	0
$595$	77.7561	3.18769
$596$	12.2843	0.503184
$597$	0	0
$598$	14.9170	0.610001
$599$	22.2460	0.908945	0.454472	−	0.890761i	$-0.349828\pi$
$599$	22.2460	0.908945	0.454472	+	0.890761i	$0.349828\pi$
$600$	0	0
$601$	22.5518	0.919908	0.459954	−	0.887943i	$-0.347866\pi$
$601$	22.5518	0.919908	0.459954	+	0.887943i	$0.347866\pi$
$602$	13.0257	0.530888
$603$	0	0
$604$	−18.8027	−0.765072
$605$	−28.3632	−1.15313
$606$	0	0
$607$	−27.1625	−1.10249	−0.551246	−	0.834343i	$-0.685848\pi$
$607$	−27.1625	−1.10249	−0.551246	+	0.834343i	$0.685848\pi$
$608$	−30.6053	−1.24121
$609$	0	0
$610$	−35.0735	−1.42008
$611$	46.2122	1.86955
$612$	0	0
$613$	−11.7705	−0.475407	−0.237703	−	0.971338i	$-0.576395\pi$
$613$	−11.7705	−0.475407	−0.237703	+	0.971338i	$0.576395\pi$
$614$	2.72612	0.110017
$615$	0	0
$616$	3.17638	0.127980
$617$	−20.4030	−0.821392	−0.410696	−	0.911772i	$-0.634714\pi$
$617$	−20.4030	−0.821392	−0.410696	+	0.911772i	$0.634714\pi$
$618$	0	0
$619$	18.9606	0.762092	0.381046	−	0.924556i	$-0.375564\pi$
$619$	18.9606	0.762092	0.381046	+	0.924556i	$0.375564\pi$
$620$	−17.0356	−0.684167
$621$	0	0
$622$	−4.72881	−0.189608
$623$	50.9933	2.04300
$624$	0	0
$625$	−30.6784	−1.22713
$626$	−31.2704	−1.24981
$627$	0	0
$628$	−13.0425	−0.520451
$629$	−30.3681	−1.21085
$630$	0	0
$631$	20.5896	0.819660	0.409830	−	0.912162i	$-0.365588\pi$
$631$	20.5896	0.819660	0.409830	+	0.912162i	$0.365588\pi$
$632$	−32.1994	−1.28082
$633$	0	0
$634$	−10.5002	−0.417018
$635$	4.26479	0.169243
$636$	0	0
$637$	−32.4437	−1.28547
$638$	−0.0932036	−0.00368996
$639$	0	0
$640$	10.5878	0.418520
$641$	−27.3686	−1.08099	−0.540497	−	0.841346i	$-0.681764\pi$
$641$	−27.3686	−1.08099	−0.540497	+	0.841346i	$0.681764\pi$
$642$	0	0
$643$	−5.76684	−0.227422	−0.113711	−	0.993514i	$-0.536274\pi$
$643$	−5.76684	−0.227422	−0.113711	+	0.993514i	$0.536274\pi$
$644$	−14.8613	−0.585619
$645$	0	0
$646$	−44.4806	−1.75007
$647$	−6.84468	−0.269092	−0.134546	−	0.990907i	$-0.542958\pi$
$647$	−6.84468	−0.269092	−0.134546	+	0.990907i	$0.542958\pi$
$648$	0	0
$649$	2.65128	0.104072
$650$	7.12444	0.279444
$651$	0	0
$652$	16.7548	0.656168
$653$	17.9894	0.703981	0.351990	−	0.936004i	$-0.385505\pi$
$653$	17.9894	0.703981	0.351990	+	0.936004i	$0.385505\pi$
$654$	0	0
$655$	−26.2599	−1.02606
$656$	−1.50685	−0.0588326
$657$	0	0
$658$	40.6786	1.58582
$659$	26.1281	1.01781	0.508903	−	0.860824i	$-0.330051\pi$
$659$	26.1281	1.01781	0.508903	+	0.860824i	$0.330051\pi$
$660$	0	0
$661$	−18.5886	−0.723015	−0.361507	−	0.932369i	$-0.617738\pi$
$661$	−18.5886	−0.723015	−0.361507	+	0.932369i	$0.617738\pi$
$662$	−21.6158	−0.840120
$663$	0	0
$664$	−38.7845	−1.50513
$665$	58.5188	2.26926
$666$	0	0
$667$	1.25744	0.0486882
$668$	13.5302	0.523499
$669$	0	0
$670$	−19.4817	−0.752642
$671$	3.89623	0.150412
$672$	0	0
$673$	−3.87472	−0.149359	−0.0746797	−	0.997208i	$-0.523793\pi$
$673$	−3.87472	−0.149359	−0.0746797	+	0.997208i	$0.523793\pi$
$674$	−3.27573	−0.126176
$675$	0	0
$676$	−5.08518	−0.195584
$677$	−40.6699	−1.56307	−0.781535	−	0.623861i	$-0.785563\pi$
$677$	−40.6699	−1.56307	−0.781535	+	0.623861i	$0.785563\pi$
$678$	0	0
$679$	−70.8335	−2.71834
$680$	60.1607	2.30706
$681$	0	0
$682$	−1.67208	−0.0640272
$683$	10.8518	0.415232	0.207616	−	0.978210i	$-0.433430\pi$
$683$	10.8518	0.415232	0.207616	+	0.978210i	$0.433430\pi$
$684$	0	0
$685$	−58.7587	−2.24505
$686$	−2.57009	−0.0981266
$687$	0	0
$688$	2.62742	0.100170
$689$	−9.70745	−0.369825
$690$	0	0
$691$	−3.81049	−0.144958	−0.0724789	−	0.997370i	$-0.523091\pi$
$691$	−3.81049	−0.144958	−0.0724789	+	0.997370i	$0.523091\pi$
$692$	1.27307	0.0483948
$693$	0	0
$694$	12.5642	0.476930
$695$	−3.18360	−0.120761
$696$	0	0
$697$	15.7177	0.595349
$698$	−2.40484	−0.0910244
$699$	0	0
$700$	−7.09787	−0.268274
$701$	−25.6747	−0.969719	−0.484859	−	0.874592i	$-0.661129\pi$
$701$	−25.6747	−0.969719	−0.484859	+	0.874592i	$0.661129\pi$
$702$	0	0
$703$	−22.8549	−0.861987
$704$	1.82751	0.0688768
$705$	0	0
$706$	13.4914	0.507756
$707$	53.2040	2.00094
$708$	0	0
$709$	25.6741	0.964212	0.482106	−	0.876113i	$-0.339872\pi$
$709$	25.6741	0.964212	0.482106	+	0.876113i	$0.339872\pi$
$710$	6.12307	0.229795
$711$	0	0
$712$	39.4540	1.47860
$713$	22.5585	0.844822
$714$	0	0
$715$	−3.06055	−0.114458
$716$	−4.56804	−0.170716
$717$	0	0
$718$	−14.9777	−0.558963
$719$	−20.3685	−0.759616	−0.379808	−	0.925065i	$-0.624010\pi$
$719$	−20.3685	−0.759616	−0.379808	+	0.925065i	$0.624010\pi$
$720$	0	0
$721$	38.9524	1.45066
$722$	−15.0726	−0.560943
$723$	0	0
$724$	24.2025	0.899480
$725$	0.600560	0.0223042
$726$	0	0
$727$	1.70458	0.0632196	0.0316098	−	0.999500i	$-0.489937\pi$
$727$	1.70458	0.0632196	0.0316098	+	0.999500i	$0.489937\pi$
$728$	−47.9450	−1.77696
$729$	0	0
$730$	−8.27195	−0.306159
$731$	−27.4062	−1.01365
$732$	0	0
$733$	−47.1692	−1.74223	−0.871116	−	0.491077i	$-0.836603\pi$
$733$	−47.1692	−1.74223	−0.871116	+	0.491077i	$0.836603\pi$
$734$	20.5639	0.759028
$735$	0	0
$736$	−19.0092	−0.700687
$737$	2.16417	0.0797183
$738$	0	0
$739$	0.633784	0.0233141	0.0116571	−	0.999932i	$-0.496289\pi$
$739$	0.633784	0.0233141	0.0116571	+	0.999932i	$0.496289\pi$
$740$	10.7199	0.394073
$741$	0	0
$742$	−8.54505	−0.313699
$743$	32.8949	1.20680	0.603398	−	0.797440i	$-0.293813\pi$
$743$	32.8949	1.20680	0.603398	+	0.797440i	$0.293813\pi$
$744$	0	0
$745$	−30.0438	−1.10072
$746$	−4.04350	−0.148043
$747$	0	0
$748$	−2.31767	−0.0847423
$749$	−0.788183	−0.0287996
$750$	0	0
$751$	−44.7494	−1.63293	−0.816465	−	0.577395i	$-0.804069\pi$
$751$	−44.7494	−1.63293	−0.816465	+	0.577395i	$0.804069\pi$
$752$	8.20531	0.299217
$753$	0	0
$754$	1.40684	0.0512340
$755$	45.9860	1.67360
$756$	0	0
$757$	−11.6428	−0.423164	−0.211582	−	0.977360i	$-0.567862\pi$
$757$	−11.6428	−0.423164	−0.211582	+	0.977360i	$0.567862\pi$
$758$	−32.5728	−1.18310
$759$	0	0
$760$	45.2766	1.64236
$761$	31.4947	1.14168	0.570840	−	0.821061i	$-0.306618\pi$
$761$	31.4947	1.14168	0.570840	+	0.821061i	$0.306618\pi$
$762$	0	0
$763$	62.0460	2.24622
$764$	18.1874	0.657999
$765$	0	0
$766$	−20.0841	−0.725667
$767$	−40.0192	−1.44501
$768$	0	0
$769$	−37.6004	−1.35591	−0.677953	−	0.735105i	$-0.737133\pi$
$769$	−37.6004	−1.35591	−0.677953	+	0.735105i	$0.737133\pi$
$770$	−2.69407	−0.0970875
$771$	0	0
$772$	−9.65449	−0.347473
$773$	−24.9059	−0.895803	−0.447901	−	0.894083i	$-0.647828\pi$
$773$	−24.9059	−0.895803	−0.447901	+	0.894083i	$0.647828\pi$
$774$	0	0
$775$	10.7741	0.387016
$776$	−54.8046	−1.96737
$777$	0	0
$778$	11.1394	0.399366
$779$	11.8290	0.423819
$780$	0	0
$781$	−0.680198	−0.0243394
$782$	−27.6272	−0.987948
$783$	0	0
$784$	−5.76061	−0.205736
$785$	31.8981	1.13849
$786$	0	0
$787$	4.88103	0.173990	0.0869950	−	0.996209i	$-0.472274\pi$
$787$	4.88103	0.173990	0.0869950	+	0.996209i	$0.472274\pi$
$788$	24.7029	0.880006
$789$	0	0
$790$	27.3101	0.971652
$791$	−39.5831	−1.40741
$792$	0	0
$793$	−58.8108	−2.08843
$794$	−15.7982	−0.560658
$795$	0	0
$796$	−16.5609	−0.586984
$797$	−7.46655	−0.264479	−0.132239	−	0.991218i	$-0.542217\pi$
$797$	−7.46655	−0.264479	−0.132239	+	0.991218i	$0.542217\pi$
$798$	0	0
$799$	−85.5881	−3.02789
$800$	−9.07889	−0.320987
$801$	0	0
$802$	−11.5340	−0.407280
$803$	0.918913	0.0324277
$804$	0	0
$805$	36.3465	1.28105
$806$	25.2388	0.888998
$807$	0	0
$808$	41.1645	1.44816
$809$	12.1097	0.425755	0.212878	−	0.977079i	$-0.431716\pi$
$809$	12.1097	0.425755	0.212878	+	0.977079i	$0.431716\pi$
$810$	0	0
$811$	42.3624	1.48754	0.743772	−	0.668433i	$-0.233035\pi$
$811$	42.3624	1.48754	0.743772	+	0.668433i	$0.233035\pi$
$812$	−1.40159	−0.0491862
$813$	0	0
$814$	1.05218	0.0368790
$815$	−40.9773	−1.43537
$816$	0	0
$817$	−20.6257	−0.721603
$818$	−33.3836	−1.16723
$819$	0	0
$820$	−5.54834	−0.193757
$821$	54.0230	1.88541	0.942707	−	0.333622i	$-0.108271\pi$
$821$	54.0230	1.88541	0.942707	+	0.333622i	$0.108271\pi$
$822$	0	0
$823$	−12.6032	−0.439318	−0.219659	−	0.975577i	$-0.570495\pi$
$823$	−12.6032	−0.439318	−0.219659	+	0.975577i	$0.570495\pi$
$824$	30.1379	1.04990
$825$	0	0
$826$	−35.2271	−1.22571
$827$	52.8054	1.83622	0.918111	−	0.396323i	$-0.129714\pi$
$827$	52.8054	1.83622	0.918111	+	0.396323i	$0.129714\pi$
$828$	0	0
$829$	17.9346	0.622895	0.311447	−	0.950263i	$-0.399186\pi$
$829$	17.9346	0.622895	0.311447	+	0.950263i	$0.399186\pi$
$830$	32.8954	1.14181
$831$	0	0
$832$	−27.5849	−0.956334
$833$	60.0879	2.08192
$834$	0	0
$835$	−33.0909	−1.14516
$836$	−1.74426	−0.0603266
$837$	0	0
$838$	−11.0478	−0.381641
$839$	−24.6836	−0.852171	−0.426085	−	0.904683i	$-0.640108\pi$
$839$	−24.6836	−0.852171	−0.426085	+	0.904683i	$0.640108\pi$
$840$	0	0
$841$	−28.8814	−0.995911
$842$	20.9651	0.722505
$843$	0	0
$844$	0.636148	0.0218971
$845$	12.4369	0.427842
$846$	0	0
$847$	−41.8642	−1.43847
$848$	−1.72363	−0.0591896
$849$	0	0
$850$	−13.1949	−0.452583
$851$	−14.1953	−0.486609
$852$	0	0
$853$	−4.12152	−0.141118	−0.0705591	−	0.997508i	$-0.522478\pi$
$853$	−4.12152	−0.141118	−0.0705591	+	0.997508i	$0.522478\pi$
$854$	−51.7686	−1.77148
$855$	0	0
$856$	−0.609825	−0.0208434
$857$	45.3681	1.54974	0.774872	−	0.632118i	$-0.217814\pi$
$857$	45.3681	1.54974	0.774872	+	0.632118i	$0.217814\pi$
$858$	0	0
$859$	18.9719	0.647314	0.323657	−	0.946174i	$-0.395088\pi$
$859$	18.9719	0.647314	0.323657	+	0.946174i	$0.395088\pi$
$860$	9.67439	0.329894
$861$	0	0
$862$	8.00052	0.272499
$863$	29.6919	1.01072	0.505361	−	0.862908i	$-0.331359\pi$
$863$	29.6919	1.01072	0.505361	+	0.862908i	$0.331359\pi$
$864$	0	0
$865$	−3.11356	−0.105864
$866$	−28.0974	−0.954788
$867$	0	0
$868$	−25.1446	−0.853464
$869$	−3.03382	−0.102915
$870$	0	0
$871$	−32.6666	−1.10686
$872$	48.0056	1.62568
$873$	0	0
$874$	−20.7921	−0.703304
$875$	−32.4111	−1.09569
$876$	0	0
$877$	7.31589	0.247040	0.123520	−	0.992342i	$-0.460582\pi$
$877$	7.31589	0.247040	0.123520	+	0.992342i	$0.460582\pi$
$878$	33.8110	1.14106
$879$	0	0
$880$	−0.543422	−0.0183188
$881$	−28.7219	−0.967666	−0.483833	−	0.875160i	$-0.660756\pi$
$881$	−28.7219	−0.967666	−0.483833	+	0.875160i	$0.660756\pi$
$882$	0	0
$883$	−41.0086	−1.38005	−0.690024	−	0.723786i	$-0.742400\pi$
$883$	−41.0086	−1.38005	−0.690024	+	0.723786i	$0.742400\pi$
$884$	34.9835	1.17662
$885$	0	0
$886$	−19.3603	−0.650421
$887$	−30.2230	−1.01479	−0.507394	−	0.861714i	$-0.669391\pi$
$887$	−30.2230	−1.01479	−0.507394	+	0.861714i	$0.669391\pi$
$888$	0	0
$889$	6.29485	0.211122
$890$	−33.4632	−1.12169
$891$	0	0
$892$	−1.06182	−0.0355525
$893$	−64.4132	−2.15550
$894$	0	0
$895$	11.1721	0.373442
$896$	15.6277	0.522083
$897$	0	0
$898$	−29.3337	−0.978879
$899$	2.12752	0.0709567
$900$	0	0
$901$	17.9788	0.598962
$902$	−0.544580	−0.0181325
$903$	0	0
$904$	−30.6259	−1.01860
$905$	−59.1923	−1.96762
$906$	0	0
$907$	−8.76180	−0.290931	−0.145465	−	0.989363i	$-0.546468\pi$
$907$	−8.76180	−0.290931	−0.145465	+	0.989363i	$0.546468\pi$
$908$	14.9564	0.496345
$909$	0	0
$910$	40.6650	1.34803
$911$	26.7725	0.887012	0.443506	−	0.896271i	$-0.353735\pi$
$911$	26.7725	0.887012	0.443506	+	0.896271i	$0.353735\pi$
$912$	0	0
$913$	−3.65427	−0.120939
$914$	35.8104	1.18450
$915$	0	0
$916$	−17.8915	−0.591150
$917$	−38.7597	−1.27996
$918$	0	0
$919$	−6.85577	−0.226151	−0.113076	−	0.993586i	$-0.536070\pi$
$919$	−6.85577	−0.226151	−0.113076	+	0.993586i	$0.536070\pi$
$920$	28.1217	0.927143
$921$	0	0
$922$	−6.19715	−0.204092
$923$	10.2671	0.337945
$924$	0	0
$925$	−6.77977	−0.222918
$926$	1.80599	0.0593484
$927$	0	0
$928$	−1.79278	−0.0588508
$929$	−5.50786	−0.180707	−0.0903534	−	0.995910i	$-0.528800\pi$
$929$	−5.50786	−0.180707	−0.0903534	+	0.995910i	$0.528800\pi$
$930$	0	0
$931$	45.2218	1.48209
$932$	−10.9857	−0.359849
$933$	0	0
$934$	1.07222	0.0350842
$935$	5.66834	0.185375
$936$	0	0
$937$	−11.3917	−0.372150	−0.186075	−	0.982536i	$-0.559577\pi$
$937$	−11.3917	−0.372150	−0.186075	+	0.982536i	$0.559577\pi$
$938$	−28.7550	−0.938884
$939$	0	0
$940$	30.2127	0.985428
$941$	−5.83177	−0.190110	−0.0950552	−	0.995472i	$-0.530303\pi$
$941$	−5.83177	−0.190110	−0.0950552	+	0.995472i	$0.530303\pi$
$942$	0	0
$943$	7.34710	0.239254
$944$	−7.10569	−0.231271
$945$	0	0
$946$	0.949560	0.0308729
$947$	36.5449	1.18755	0.593774	−	0.804632i	$-0.297637\pi$
$947$	36.5449	1.18755	0.593774	+	0.804632i	$0.297637\pi$
$948$	0	0
$949$	−13.8703	−0.450249
$950$	−9.93044	−0.322186
$951$	0	0
$952$	88.7974	2.87794
$953$	−36.0226	−1.16688	−0.583442	−	0.812154i	$-0.698295\pi$
$953$	−36.0226	−1.16688	−0.583442	+	0.812154i	$0.698295\pi$
$954$	0	0
$955$	−44.4812	−1.43938
$956$	19.6596	0.635838
$957$	0	0
$958$	0.432118	0.0139611
$959$	−86.7280	−2.80059
$960$	0	0
$961$	7.16782	0.231220
$962$	−15.8819	−0.512054
$963$	0	0
$964$	12.5523	0.404281
$965$	23.6121	0.760099
$966$	0	0
$967$	40.9791	1.31780	0.658900	−	0.752231i	$-0.271022\pi$
$967$	40.9791	1.31780	0.658900	+	0.752231i	$0.271022\pi$
$968$	−32.3908	−1.04108
$969$	0	0
$970$	46.4830	1.49248
$971$	32.1444	1.03156	0.515782	−	0.856720i	$-0.327501\pi$
$971$	32.1444	1.03156	0.515782	+	0.856720i	$0.327501\pi$
$972$	0	0
$973$	−4.69900	−0.150643
$974$	−20.1825	−0.646687
$975$	0	0
$976$	−10.4423	−0.334249
$977$	27.9039	0.892723	0.446362	−	0.894853i	$-0.352719\pi$
$977$	27.9039	0.892723	0.446362	+	0.894853i	$0.352719\pi$
$978$	0	0
$979$	3.71735	0.118807
$980$	−21.2111	−0.677562
$981$	0	0
$982$	14.3917	0.459257
$983$	−37.2209	−1.18716	−0.593582	−	0.804774i	$-0.702287\pi$
$983$	−37.2209	−1.18716	−0.593582	+	0.804774i	$0.702287\pi$
$984$	0	0
$985$	−60.4162	−1.92502
$986$	−2.60556	−0.0829779
$987$	0	0
$988$	26.3284	0.837617
$989$	−12.8108	−0.407360
$990$	0	0
$991$	−28.4080	−0.902409	−0.451204	−	0.892421i	$-0.649005\pi$
$991$	−28.4080	−0.902409	−0.451204	+	0.892421i	$0.649005\pi$
$992$	−32.1625	−1.02116
$993$	0	0
$994$	9.03767	0.286657
$995$	40.5030	1.28403
$996$	0	0
$997$	46.1015	1.46005	0.730024	−	0.683422i	$-0.239509\pi$
$997$	46.1015	1.46005	0.730024	+	0.683422i	$0.239509\pi$
$998$	−14.0564	−0.444949
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6021.2.a.t.1.16	✓	40
3.2	odd	2	inner	6021.2.a.t.1.25	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6021.2.a.t.1.16	✓	40	1.1	even	1	trivial
6021.2.a.t.1.25	yes	40	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 6021 = 3^{3} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6021.a (trivial)

\(f(q)\)	\(=\)	\(q-0.968595 q^{2} -1.06182 q^{4} +2.59691 q^{5} +3.83305 q^{7} +2.96567 q^{8} +O(q^{10})\)	\(q-0.968595 q^{2} -1.06182 q^{4} +2.59691 q^{5} +3.83305 q^{7} +2.96567 q^{8} -2.51535 q^{10} +0.279425 q^{11} -4.21771 q^{13} -3.71267 q^{14} -0.748885 q^{16} +7.81149 q^{17} +5.87888 q^{19} -2.75746 q^{20} -0.270650 q^{22} +3.65142 q^{23} +1.74394 q^{25} +4.08526 q^{26} -4.07002 q^{28} +0.344370 q^{29} +6.17801 q^{31} -5.20597 q^{32} -7.56617 q^{34} +9.95408 q^{35} -3.88762 q^{37} -5.69426 q^{38} +7.70157 q^{40} +2.01212 q^{41} -3.50845 q^{43} -0.296700 q^{44} -3.53675 q^{46} -10.9567 q^{47} +7.69225 q^{49} -1.68917 q^{50} +4.47847 q^{52} +2.30159 q^{53} +0.725642 q^{55} +11.3675 q^{56} -0.333555 q^{58} +9.48835 q^{59} +13.9438 q^{61} -5.98399 q^{62} +6.54025 q^{64} -10.9530 q^{65} +7.74509 q^{67} -8.29442 q^{68} -9.64147 q^{70} -2.43428 q^{71} +3.28858 q^{73} +3.76553 q^{74} -6.24233 q^{76} +1.07105 q^{77} -10.8574 q^{79} -1.94479 q^{80} -1.94893 q^{82} -13.0778 q^{83} +20.2857 q^{85} +3.39826 q^{86} +0.828682 q^{88} +13.3036 q^{89} -16.1667 q^{91} -3.87716 q^{92} +10.6126 q^{94} +15.2669 q^{95} -18.4797 q^{97} -7.45068 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 46 q^{4} + 16 q^{7}+O(q^{10}) \) 40 * q + 46 * q^4 + 16 * q^7	\( 40 q + 46 q^{4} + 16 q^{7} + 22 q^{10} + 14 q^{13} + 50 q^{16} + 64 q^{19} + 12 q^{22} + 40 q^{25} + 48 q^{28} + 54 q^{31} + 32 q^{34} + 24 q^{37} + 40 q^{40} + 24 q^{43} + 52 q^{46} + 64 q^{49} + 18 q^{52} + 36 q^{55} + 8 q^{58} + 58 q^{61} + 120 q^{64} + 52 q^{67} - 30 q^{70} + 50 q^{73} + 112 q^{76} + 60 q^{79} + 50 q^{82} + 38 q^{85} + 16 q^{88} + 118 q^{91} + 44 q^{94} + 38 q^{97}+O(q^{100}) \) 40 * q + 46 * q^4 + 16 * q^7 + 22 * q^10 + 14 * q^13 + 50 * q^16 + 64 * q^19 + 12 * q^22 + 40 * q^25 + 48 * q^28 + 54 * q^31 + 32 * q^34 + 24 * q^37 + 40 * q^40 + 24 * q^43 + 52 * q^46 + 64 * q^49 + 18 * q^52 + 36 * q^55 + 8 * q^58 + 58 * q^61 + 120 * q^64 + 52 * q^67 - 30 * q^70 + 50 * q^73 + 112 * q^76 + 60 * q^79 + 50 * q^82 + 38 * q^85 + 16 * q^88 + 118 * q^91 + 44 * q^94 + 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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