LMFDB - Embedded newform 4016.2.a.i.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 17 x^{10} + 49 x^{9} + 106 x^{8} - 277 x^{7} - 317 x^{6} + 644 x^{5} + 537 x^{4} + \cdots + 104 $ x^12 - 3x^11 - 17x^10 + 49x^9 + 106x^8 - 277x^7 - 317x^6 + 644x^5 + 537x^4 - 601x^3 - 462x^2 + 136*x + 104
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-2.54171$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.54171 q^{3} +1.59809 q^{5} -2.13408 q^{7} +3.46027 q^{9} +O(q^{10})$	$q+2.54171 q^{3} +1.59809 q^{5} -2.13408 q^{7} +3.46027 q^{9} -5.04529 q^{11} -1.30452 q^{13} +4.06188 q^{15} +1.38632 q^{17} -3.89363 q^{19} -5.42421 q^{21} -7.19413 q^{23} -2.44611 q^{25} +1.16987 q^{27} -3.67098 q^{29} -1.32640 q^{31} -12.8236 q^{33} -3.41046 q^{35} -1.51613 q^{37} -3.31572 q^{39} -8.98934 q^{41} -6.23204 q^{43} +5.52983 q^{45} +6.82639 q^{47} -2.44568 q^{49} +3.52361 q^{51} +2.91183 q^{53} -8.06282 q^{55} -9.89647 q^{57} +9.42567 q^{59} +11.4693 q^{61} -7.38451 q^{63} -2.08475 q^{65} -0.381738 q^{67} -18.2854 q^{69} +1.60686 q^{71} +13.1431 q^{73} -6.21728 q^{75} +10.7671 q^{77} -1.64917 q^{79} -7.40734 q^{81} -0.795405 q^{83} +2.21546 q^{85} -9.33055 q^{87} -0.551016 q^{89} +2.78397 q^{91} -3.37133 q^{93} -6.22238 q^{95} -7.66229 q^{97} -17.4580 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) $ 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9	$ 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9} - 10 q^{11} + 3 q^{13} - 11 q^{15} + 2 q^{17} - 15 q^{19} + 3 q^{21} - 20 q^{23} - 3 q^{25} - 15 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{33} - 16 q^{35} + 5 q^{37} - 21 q^{39} - 21 q^{43} + 10 q^{45} - 27 q^{47} - 13 q^{49} - 19 q^{51} + 22 q^{53} - 24 q^{55} + q^{57} - 23 q^{59} + 4 q^{61} - 21 q^{63} - q^{65} - 26 q^{67} + 10 q^{69} - 23 q^{71} - 8 q^{73} - 16 q^{75} + 22 q^{77} - 37 q^{79} - 20 q^{81} - 30 q^{83} + 2 q^{85} - 16 q^{87} + 3 q^{89} - 8 q^{91} + 20 q^{93} - 33 q^{95} - 4 q^{97} - 15 q^{99}+O(q^{100}) $ 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9 - 10 * q^11 + 3 * q^13 - 11 * q^15 + 2 * q^17 - 15 * q^19 + 3 * q^21 - 20 * q^23 - 3 * q^25 - 15 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^33 - 16 * q^35 + 5 * q^37 - 21 * q^39 - 21 * q^43 + 10 * q^45 - 27 * q^47 - 13 * q^49 - 19 * q^51 + 22 * q^53 - 24 * q^55 + q^57 - 23 * q^59 + 4 * q^61 - 21 * q^63 - q^65 - 26 * q^67 + 10 * q^69 - 23 * q^71 - 8 * q^73 - 16 * q^75 + 22 * q^77 - 37 * q^79 - 20 * q^81 - 30 * q^83 + 2 * q^85 - 16 * q^87 + 3 * q^89 - 8 * q^91 + 20 * q^93 - 33 * q^95 - 4 * q^97 - 15 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.54171	1.46745	0.733727	−	0.679444i	$-0.237779\pi$
$3$	2.54171	1.46745	0.733727	+	0.679444i	$0.237779\pi$
$4$	0	0
$5$	1.59809	0.714688	0.357344	−	0.933973i	$-0.383682\pi$
$5$	1.59809	0.714688	0.357344	+	0.933973i	$0.383682\pi$
$6$	0	0
$7$	−2.13408	−0.806608	−0.403304	−	0.915066i	$-0.632138\pi$
$7$	−2.13408	−0.806608	−0.403304	+	0.915066i	$0.632138\pi$
$8$	0	0
$9$	3.46027	1.15342
$10$	0	0
$11$	−5.04529	−1.52121	−0.760605	−	0.649215i	$-0.775098\pi$
$11$	−5.04529	−1.52121	−0.760605	+	0.649215i	$0.775098\pi$
$12$	0	0
$13$	−1.30452	−0.361810	−0.180905	−	0.983501i	$-0.557903\pi$
$13$	−1.30452	−0.361810	−0.180905	+	0.983501i	$0.557903\pi$
$14$	0	0
$15$	4.06188	1.04877
$16$	0	0
$17$	1.38632	0.336231	0.168116	−	0.985767i	$-0.446232\pi$
$17$	1.38632	0.336231	0.168116	+	0.985767i	$0.446232\pi$
$18$	0	0
$19$	−3.89363	−0.893260	−0.446630	−	0.894719i	$-0.647376\pi$
$19$	−3.89363	−0.893260	−0.446630	+	0.894719i	$0.647376\pi$
$20$	0	0
$21$	−5.42421	−1.18366
$22$	0	0
$23$	−7.19413	−1.50008	−0.750039	−	0.661393i	$-0.769966\pi$
$23$	−7.19413	−1.50008	−0.750039	+	0.661393i	$0.769966\pi$
$24$	0	0
$25$	−2.44611	−0.489221
$26$	0	0
$27$	1.16987	0.225141
$28$	0	0
$29$	−3.67098	−0.681684	−0.340842	−	0.940121i	$-0.610712\pi$
$29$	−3.67098	−0.681684	−0.340842	+	0.940121i	$0.610712\pi$
$30$	0	0
$31$	−1.32640	−0.238229	−0.119115	−	0.992881i	$-0.538006\pi$
$31$	−1.32640	−0.238229	−0.119115	+	0.992881i	$0.538006\pi$
$32$	0	0
$33$	−12.8236	−2.23231
$34$	0	0
$35$	−3.41046	−0.576473
$36$	0	0
$37$	−1.51613	−0.249250	−0.124625	−	0.992204i	$-0.539773\pi$
$37$	−1.51613	−0.249250	−0.124625	+	0.992204i	$0.539773\pi$
$38$	0	0
$39$	−3.31572	−0.530940
$40$	0	0
$41$	−8.98934	−1.40390	−0.701949	−	0.712227i	$-0.747687\pi$
$41$	−8.98934	−1.40390	−0.701949	+	0.712227i	$0.747687\pi$
$42$	0	0
$43$	−6.23204	−0.950377	−0.475188	−	0.879884i	$-0.657620\pi$
$43$	−6.23204	−0.950377	−0.475188	+	0.879884i	$0.657620\pi$
$44$	0	0
$45$	5.52983	0.824338
$46$	0	0
$47$	6.82639	0.995731	0.497866	−	0.867254i	$-0.334117\pi$
$47$	6.82639	0.995731	0.497866	+	0.867254i	$0.334117\pi$
$48$	0	0
$49$	−2.44568	−0.349384
$50$	0	0
$51$	3.52361	0.493404
$52$	0	0
$53$	2.91183	0.399970	0.199985	−	0.979799i	$-0.435911\pi$
$53$	2.91183	0.399970	0.199985	+	0.979799i	$0.435911\pi$
$54$	0	0
$55$	−8.06282	−1.08719
$56$	0	0
$57$	−9.89647	−1.31082
$58$	0	0
$59$	9.42567	1.22712	0.613559	−	0.789649i	$-0.289737\pi$
$59$	9.42567	1.22712	0.613559	+	0.789649i	$0.289737\pi$
$60$	0	0
$61$	11.4693	1.46849	0.734244	−	0.678886i	$-0.237537\pi$
$61$	11.4693	1.46849	0.734244	+	0.678886i	$0.237537\pi$
$62$	0	0
$63$	−7.38451	−0.930360
$64$	0	0
$65$	−2.08475	−0.258581
$66$	0	0
$67$	−0.381738	−0.0466367	−0.0233184	−	0.999728i	$-0.507423\pi$
$67$	−0.381738	−0.0466367	−0.0233184	+	0.999728i	$0.507423\pi$
$68$	0	0
$69$	−18.2854	−2.20130
$70$	0	0
$71$	1.60686	0.190699	0.0953497	−	0.995444i	$-0.469603\pi$
$71$	1.60686	0.190699	0.0953497	+	0.995444i	$0.469603\pi$
$72$	0	0
$73$	13.1431	1.53828	0.769142	−	0.639078i	$-0.220684\pi$
$73$	13.1431	1.53828	0.769142	+	0.639078i	$0.220684\pi$
$74$	0	0
$75$	−6.21728	−0.717910
$76$	0	0
$77$	10.7671	1.22702
$78$	0	0
$79$	−1.64917	−0.185546	−0.0927732	−	0.995687i	$-0.529573\pi$
$79$	−1.64917	−0.185546	−0.0927732	+	0.995687i	$0.529573\pi$
$80$	0	0
$81$	−7.40734	−0.823038
$82$	0	0
$83$	−0.795405	−0.0873070	−0.0436535	−	0.999047i	$-0.513900\pi$
$83$	−0.795405	−0.0873070	−0.0436535	+	0.999047i	$0.513900\pi$
$84$	0	0
$85$	2.21546	0.240301
$86$	0	0
$87$	−9.33055	−1.00034
$88$	0	0
$89$	−0.551016	−0.0584076	−0.0292038	−	0.999573i	$-0.509297\pi$
$89$	−0.551016	−0.0584076	−0.0292038	+	0.999573i	$0.509297\pi$
$90$	0	0
$91$	2.78397	0.291839
$92$	0	0
$93$	−3.37133	−0.349590
$94$	0	0
$95$	−6.22238	−0.638403
$96$	0	0
$97$	−7.66229	−0.777988	−0.388994	−	0.921240i	$-0.627177\pi$
$97$	−7.66229	−0.777988	−0.388994	+	0.921240i	$0.627177\pi$
$98$	0	0
$99$	−17.4580	−1.75460
$100$	0	0
$101$	8.08961	0.804946	0.402473	−	0.915432i	$-0.368151\pi$
$101$	8.08961	0.804946	0.402473	+	0.915432i	$0.368151\pi$
$102$	0	0
$103$	16.8314	1.65845	0.829223	−	0.558918i	$-0.188783\pi$
$103$	16.8314	1.65845	0.829223	+	0.558918i	$0.188783\pi$
$104$	0	0
$105$	−8.66839	−0.845948
$106$	0	0
$107$	−10.9663	−1.06015	−0.530075	−	0.847951i	$-0.677836\pi$
$107$	−10.9663	−1.06015	−0.530075	+	0.847951i	$0.677836\pi$
$108$	0	0
$109$	17.8382	1.70859	0.854296	−	0.519786i	$-0.173988\pi$
$109$	17.8382	1.70859	0.854296	+	0.519786i	$0.173988\pi$
$110$	0	0
$111$	−3.85356	−0.365763
$112$	0	0
$113$	4.50047	0.423369	0.211684	−	0.977338i	$-0.432105\pi$
$113$	4.50047	0.423369	0.211684	+	0.977338i	$0.432105\pi$
$114$	0	0
$115$	−11.4969	−1.07209
$116$	0	0
$117$	−4.51401	−0.417320
$118$	0	0
$119$	−2.95852	−0.271207
$120$	0	0
$121$	14.4549	1.31408
$122$	0	0
$123$	−22.8482	−2.06016
$124$	0	0
$125$	−11.8996	−1.06433
$126$	0	0
$127$	−20.7104	−1.83775	−0.918877	−	0.394543i	$-0.870903\pi$
$127$	−20.7104	−1.83775	−0.918877	+	0.394543i	$0.870903\pi$
$128$	0	0
$129$	−15.8400	−1.39463
$130$	0	0
$131$	−8.27198	−0.722727	−0.361363	−	0.932425i	$-0.617689\pi$
$131$	−8.27198	−0.722727	−0.361363	+	0.932425i	$0.617689\pi$
$132$	0	0
$133$	8.30934	0.720511
$134$	0	0
$135$	1.86956	0.160906
$136$	0	0
$137$	21.0707	1.80019	0.900096	−	0.435692i	$-0.143496\pi$
$137$	21.0707	1.80019	0.900096	+	0.435692i	$0.143496\pi$
$138$	0	0
$139$	−6.97751	−0.591825	−0.295912	−	0.955215i	$-0.595624\pi$
$139$	−6.97751	−0.591825	−0.295912	+	0.955215i	$0.595624\pi$
$140$	0	0
$141$	17.3507	1.46119
$142$	0	0
$143$	6.58170	0.550389
$144$	0	0
$145$	−5.86656	−0.487191
$146$	0	0
$147$	−6.21621	−0.512704
$148$	0	0
$149$	−1.52562	−0.124984	−0.0624920	−	0.998045i	$-0.519905\pi$
$149$	−1.52562	−0.124984	−0.0624920	+	0.998045i	$0.519905\pi$
$150$	0	0
$151$	3.20585	0.260889	0.130444	−	0.991456i	$-0.458360\pi$
$151$	3.20585	0.260889	0.130444	+	0.991456i	$0.458360\pi$
$152$	0	0
$153$	4.79703	0.387817
$154$	0	0
$155$	−2.11971	−0.170259
$156$	0	0
$157$	−2.55587	−0.203981	−0.101990	−	0.994785i	$-0.532521\pi$
$157$	−2.55587	−0.203981	−0.101990	+	0.994785i	$0.532521\pi$
$158$	0	0
$159$	7.40101	0.586938
$160$	0	0
$161$	15.3529	1.20998
$162$	0	0
$163$	8.41086	0.658790	0.329395	−	0.944192i	$-0.393155\pi$
$163$	8.41086	0.658790	0.329395	+	0.944192i	$0.393155\pi$
$164$	0	0
$165$	−20.4933	−1.59540
$166$	0	0
$167$	−20.2059	−1.56358	−0.781788	−	0.623544i	$-0.785692\pi$
$167$	−20.2059	−1.56358	−0.781788	+	0.623544i	$0.785692\pi$
$168$	0	0
$169$	−11.2982	−0.869093
$170$	0	0
$171$	−13.4730	−1.03031
$172$	0	0
$173$	21.5098	1.63536	0.817682	−	0.575671i	$-0.195259\pi$
$173$	21.5098	1.63536	0.817682	+	0.575671i	$0.195259\pi$
$174$	0	0
$175$	5.22019	0.394610
$176$	0	0
$177$	23.9573	1.80074
$178$	0	0
$179$	1.80625	0.135006	0.0675029	−	0.997719i	$-0.478497\pi$
$179$	1.80625	0.135006	0.0675029	+	0.997719i	$0.478497\pi$
$180$	0	0
$181$	−18.3324	−1.36264	−0.681318	−	0.731988i	$-0.738593\pi$
$181$	−18.3324	−1.36264	−0.681318	+	0.731988i	$0.738593\pi$
$182$	0	0
$183$	29.1515	2.15494
$184$	0	0
$185$	−2.42291	−0.178136
$186$	0	0
$187$	−6.99437	−0.511479
$188$	0	0
$189$	−2.49660	−0.181601
$190$	0	0
$191$	15.4495	1.11789	0.558944	−	0.829206i	$-0.311207\pi$
$191$	15.4495	1.11789	0.558944	+	0.829206i	$0.311207\pi$
$192$	0	0
$193$	−11.1402	−0.801890	−0.400945	−	0.916102i	$-0.631318\pi$
$193$	−11.1402	−0.801890	−0.400945	+	0.916102i	$0.631318\pi$
$194$	0	0
$195$	−5.29882	−0.379456
$196$	0	0
$197$	−4.47432	−0.318782	−0.159391	−	0.987216i	$-0.550953\pi$
$197$	−4.47432	−0.318782	−0.159391	+	0.987216i	$0.550953\pi$
$198$	0	0
$199$	9.09932	0.645034	0.322517	−	0.946564i	$-0.395471\pi$
$199$	9.09932	0.645034	0.322517	+	0.946564i	$0.395471\pi$
$200$	0	0
$201$	−0.970265	−0.0684372
$202$	0	0
$203$	7.83418	0.549852
$204$	0	0
$205$	−14.3658	−1.00335
$206$	0	0
$207$	−24.8936	−1.73023
$208$	0	0
$209$	19.6445	1.35884
$210$	0	0
$211$	0.452599	0.0311582	0.0155791	−	0.999879i	$-0.495041\pi$
$211$	0.452599	0.0311582	0.0155791	+	0.999879i	$0.495041\pi$
$212$	0	0
$213$	4.08417	0.279843
$214$	0	0
$215$	−9.95936	−0.679223
$216$	0	0
$217$	2.83066	0.192157
$218$	0	0
$219$	33.4059	2.25736
$220$	0	0
$221$	−1.80849	−0.121652
$222$	0	0
$223$	10.3626	0.693932	0.346966	−	0.937878i	$-0.387212\pi$
$223$	10.3626	0.693932	0.346966	+	0.937878i	$0.387212\pi$
$224$	0	0
$225$	−8.46418	−0.564279
$226$	0	0
$227$	−28.9433	−1.92103	−0.960516	−	0.278224i	$-0.910254\pi$
$227$	−28.9433	−1.92103	−0.960516	+	0.278224i	$0.910254\pi$
$228$	0	0
$229$	−15.8134	−1.04498	−0.522489	−	0.852646i	$-0.674997\pi$
$229$	−15.8134	−1.04498	−0.522489	+	0.852646i	$0.674997\pi$
$230$	0	0
$231$	27.3667	1.80060
$232$	0	0
$233$	−1.37375	−0.0899975	−0.0449987	−	0.998987i	$-0.514328\pi$
$233$	−1.37375	−0.0899975	−0.0449987	+	0.998987i	$0.514328\pi$
$234$	0	0
$235$	10.9092	0.711637
$236$	0	0
$237$	−4.19171	−0.272281
$238$	0	0
$239$	6.55257	0.423850	0.211925	−	0.977286i	$-0.432027\pi$
$239$	6.55257	0.423850	0.211925	+	0.977286i	$0.432027\pi$
$240$	0	0
$241$	12.1044	0.779714	0.389857	−	0.920875i	$-0.372524\pi$
$241$	12.1044	0.779714	0.389857	+	0.920875i	$0.372524\pi$
$242$	0	0
$243$	−22.3369	−1.43291
$244$	0	0
$245$	−3.90843	−0.249700
$246$	0	0
$247$	5.07934	0.323191
$248$	0	0
$249$	−2.02169	−0.128119
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	36.2964	2.28194
$254$	0	0
$255$	5.63105	0.352630
$256$	0	0
$257$	6.95959	0.434127	0.217064	−	0.976157i	$-0.430352\pi$
$257$	6.95959	0.434127	0.217064	+	0.976157i	$0.430352\pi$
$258$	0	0
$259$	3.23555	0.201047
$260$	0	0
$261$	−12.7026	−0.786270
$262$	0	0
$263$	−5.55435	−0.342496	−0.171248	−	0.985228i	$-0.554780\pi$
$263$	−5.55435	−0.342496	−0.171248	+	0.985228i	$0.554780\pi$
$264$	0	0
$265$	4.65337	0.285854
$266$	0	0
$267$	−1.40052	−0.0857105
$268$	0	0
$269$	3.82631	0.233294	0.116647	−	0.993173i	$-0.462785\pi$
$269$	3.82631	0.233294	0.116647	+	0.993173i	$0.462785\pi$
$270$	0	0
$271$	−9.73772	−0.591524	−0.295762	−	0.955262i	$-0.595574\pi$
$271$	−9.73772	−0.591524	−0.295762	+	0.955262i	$0.595574\pi$
$272$	0	0
$273$	7.07602	0.428260
$274$	0	0
$275$	12.3413	0.744208
$276$	0	0
$277$	0.183259	0.0110110	0.00550548	−	0.999985i	$-0.498248\pi$
$277$	0.183259	0.0110110	0.00550548	+	0.999985i	$0.498248\pi$
$278$	0	0
$279$	−4.58971	−0.274779
$280$	0	0
$281$	−22.9547	−1.36937	−0.684683	−	0.728841i	$-0.740059\pi$
$281$	−22.9547	−1.36937	−0.684683	+	0.728841i	$0.740059\pi$
$282$	0	0
$283$	2.52082	0.149847	0.0749235	−	0.997189i	$-0.476129\pi$
$283$	2.52082	0.149847	0.0749235	+	0.997189i	$0.476129\pi$
$284$	0	0
$285$	−15.8155	−0.936827
$286$	0	0
$287$	19.1840	1.13240
$288$	0	0
$289$	−15.0781	−0.886948
$290$	0	0
$291$	−19.4753	−1.14166
$292$	0	0
$293$	−23.5994	−1.37869	−0.689344	−	0.724434i	$-0.742101\pi$
$293$	−23.5994	−1.37869	−0.689344	+	0.724434i	$0.742101\pi$
$294$	0	0
$295$	15.0631	0.877006
$296$	0	0
$297$	−5.90232	−0.342488
$298$	0	0
$299$	9.38492	0.542744
$300$	0	0
$301$	13.2997	0.766581
$302$	0	0
$303$	20.5614	1.18122
$304$	0	0
$305$	18.3289	1.04951
$306$	0	0
$307$	−7.97689	−0.455265	−0.227632	−	0.973747i	$-0.573098\pi$
$307$	−7.97689	−0.455265	−0.227632	+	0.973747i	$0.573098\pi$
$308$	0	0
$309$	42.7805	2.43369
$310$	0	0
$311$	10.3344	0.586012	0.293006	−	0.956111i	$-0.405344\pi$
$311$	10.3344	0.586012	0.293006	+	0.956111i	$0.405344\pi$
$312$	0	0
$313$	−3.34392	−0.189010	−0.0945048	−	0.995524i	$-0.530127\pi$
$313$	−3.34392	−0.189010	−0.0945048	+	0.995524i	$0.530127\pi$
$314$	0	0
$315$	−11.8011	−0.664917
$316$	0	0
$317$	5.85428	0.328809	0.164405	−	0.986393i	$-0.447430\pi$
$317$	5.85428	0.328809	0.164405	+	0.986393i	$0.447430\pi$
$318$	0	0
$319$	18.5211	1.03698
$320$	0	0
$321$	−27.8731	−1.55572
$322$	0	0
$323$	−5.39781	−0.300342
$324$	0	0
$325$	3.19100	0.177005
$326$	0	0
$327$	45.3395	2.50728
$328$	0	0
$329$	−14.5681	−0.803165
$330$	0	0
$331$	−3.39239	−0.186463	−0.0932313	−	0.995644i	$-0.529720\pi$
$331$	−3.39239	−0.186463	−0.0932313	+	0.995644i	$0.529720\pi$
$332$	0	0
$333$	−5.24622	−0.287491
$334$	0	0
$335$	−0.610052	−0.0333307
$336$	0	0
$337$	21.6012	1.17669	0.588347	−	0.808608i	$-0.299779\pi$
$337$	21.6012	1.17669	0.588347	+	0.808608i	$0.299779\pi$
$338$	0	0
$339$	11.4389	0.621274
$340$	0	0
$341$	6.69208	0.362397
$342$	0	0
$343$	20.1579	1.08842
$344$	0	0
$345$	−29.2217	−1.57324
$346$	0	0
$347$	−25.1731	−1.35136	−0.675680	−	0.737195i	$-0.736150\pi$
$347$	−25.1731	−1.35136	−0.675680	+	0.737195i	$0.736150\pi$
$348$	0	0
$349$	19.1278	1.02389	0.511945	−	0.859019i	$-0.328925\pi$
$349$	19.1278	1.02389	0.511945	+	0.859019i	$0.328925\pi$
$350$	0	0
$351$	−1.52612	−0.0814584
$352$	0	0
$353$	−20.2057	−1.07544	−0.537721	−	0.843123i	$-0.680714\pi$
$353$	−20.2057	−1.07544	−0.537721	+	0.843123i	$0.680714\pi$
$354$	0	0
$355$	2.56791	0.136291
$356$	0	0
$357$	−7.51969	−0.397984
$358$	0	0
$359$	−8.81237	−0.465099	−0.232549	−	0.972585i	$-0.574707\pi$
$359$	−8.81237	−0.465099	−0.232549	+	0.972585i	$0.574707\pi$
$360$	0	0
$361$	−3.83963	−0.202086
$362$	0	0
$363$	36.7401	1.92836
$364$	0	0
$365$	21.0039	1.09939
$366$	0	0
$367$	23.3718	1.22000	0.610000	−	0.792401i	$-0.291169\pi$
$367$	23.3718	1.22000	0.610000	+	0.792401i	$0.291169\pi$
$368$	0	0
$369$	−31.1055	−1.61929
$370$	0	0
$371$	−6.21409	−0.322619
$372$	0	0
$373$	−20.5810	−1.06565	−0.532823	−	0.846227i	$-0.678869\pi$
$373$	−20.5810	−1.06565	−0.532823	+	0.846227i	$0.678869\pi$
$374$	0	0
$375$	−30.2452	−1.56185
$376$	0	0
$377$	4.78888	0.246640
$378$	0	0
$379$	−15.6588	−0.804340	−0.402170	−	0.915565i	$-0.631744\pi$
$379$	−15.6588	−0.804340	−0.402170	+	0.915565i	$0.631744\pi$
$380$	0	0
$381$	−52.6399	−2.69682
$382$	0	0
$383$	−17.8618	−0.912697	−0.456349	−	0.889801i	$-0.650843\pi$
$383$	−17.8618	−0.912697	−0.456349	+	0.889801i	$0.650843\pi$
$384$	0	0
$385$	17.2067	0.876937
$386$	0	0
$387$	−21.5645	−1.09619
$388$	0	0
$389$	0.193458	0.00980869	0.00490435	−	0.999988i	$-0.498439\pi$
$389$	0.193458	0.00980869	0.00490435	+	0.999988i	$0.498439\pi$
$390$	0	0
$391$	−9.97335	−0.504374
$392$	0	0
$393$	−21.0250	−1.06057
$394$	0	0
$395$	−2.63553	−0.132608
$396$	0	0
$397$	−6.91134	−0.346870	−0.173435	−	0.984845i	$-0.555487\pi$
$397$	−6.91134	−0.346870	−0.173435	+	0.984845i	$0.555487\pi$
$398$	0	0
$399$	21.1199	1.05732
$400$	0	0
$401$	0.0625622	0.00312421	0.00156210	−	0.999999i	$-0.499503\pi$
$401$	0.0625622	0.00312421	0.00156210	+	0.999999i	$0.499503\pi$
$402$	0	0
$403$	1.73033	0.0861937
$404$	0	0
$405$	−11.8376	−0.588216
$406$	0	0
$407$	7.64931	0.379162
$408$	0	0
$409$	−37.7758	−1.86789	−0.933946	−	0.357413i	$-0.883659\pi$
$409$	−37.7758	−1.86789	−0.933946	+	0.357413i	$0.883659\pi$
$410$	0	0
$411$	53.5555	2.64170
$412$	0	0
$413$	−20.1152	−0.989803
$414$	0	0
$415$	−1.27113	−0.0623973
$416$	0	0
$417$	−17.7348	−0.868476
$418$	0	0
$419$	9.50796	0.464494	0.232247	−	0.972657i	$-0.425392\pi$
$419$	9.50796	0.464494	0.232247	+	0.972657i	$0.425392\pi$
$420$	0	0
$421$	3.46623	0.168934	0.0844670	−	0.996426i	$-0.473081\pi$
$421$	3.46623	0.168934	0.0844670	+	0.996426i	$0.473081\pi$
$422$	0	0
$423$	23.6211	1.14850
$424$	0	0
$425$	−3.39108	−0.164492
$426$	0	0
$427$	−24.4764	−1.18449
$428$	0	0
$429$	16.7287	0.807671
$430$	0	0
$431$	−2.25683	−0.108707	−0.0543537	−	0.998522i	$-0.517310\pi$
$431$	−2.25683	−0.108707	−0.0543537	+	0.998522i	$0.517310\pi$
$432$	0	0
$433$	−30.6218	−1.47159	−0.735794	−	0.677205i	$-0.763191\pi$
$433$	−30.6218	−1.47159	−0.735794	+	0.677205i	$0.763191\pi$
$434$	0	0
$435$	−14.9111	−0.714931
$436$	0	0
$437$	28.0113	1.33996
$438$	0	0
$439$	18.8888	0.901515	0.450757	−	0.892646i	$-0.351154\pi$
$439$	18.8888	0.901515	0.450757	+	0.892646i	$0.351154\pi$
$440$	0	0
$441$	−8.46273	−0.402987
$442$	0	0
$443$	−25.3435	−1.20411	−0.602053	−	0.798456i	$-0.705650\pi$
$443$	−25.3435	−1.20411	−0.602053	+	0.798456i	$0.705650\pi$
$444$	0	0
$445$	−0.880575	−0.0417432
$446$	0	0
$447$	−3.87769	−0.183408
$448$	0	0
$449$	25.3032	1.19413	0.597067	−	0.802192i	$-0.296333\pi$
$449$	25.3032	1.19413	0.597067	+	0.802192i	$0.296333\pi$
$450$	0	0
$451$	45.3538	2.13563
$452$	0	0
$453$	8.14834	0.382842
$454$	0	0
$455$	4.44903	0.208574
$456$	0	0
$457$	−25.4337	−1.18974	−0.594869	−	0.803823i	$-0.702796\pi$
$457$	−25.4337	−1.18974	−0.594869	+	0.803823i	$0.702796\pi$
$458$	0	0
$459$	1.62181	0.0756996
$460$	0	0
$461$	9.02340	0.420261	0.210131	−	0.977673i	$-0.432611\pi$
$461$	9.02340	0.420261	0.210131	+	0.977673i	$0.432611\pi$
$462$	0	0
$463$	12.7729	0.593606	0.296803	−	0.954939i	$-0.404080\pi$
$463$	12.7729	0.593606	0.296803	+	0.954939i	$0.404080\pi$
$464$	0	0
$465$	−5.38769	−0.249848
$466$	0	0
$467$	10.6856	0.494469	0.247235	−	0.968956i	$-0.420478\pi$
$467$	10.6856	0.494469	0.247235	+	0.968956i	$0.420478\pi$
$468$	0	0
$469$	0.814661	0.0376175
$470$	0	0
$471$	−6.49627	−0.299332
$472$	0	0
$473$	31.4424	1.44572
$474$	0	0
$475$	9.52423	0.437002
$476$	0	0
$477$	10.0757	0.461335
$478$	0	0
$479$	1.54775	0.0707187	0.0353594	−	0.999375i	$-0.488742\pi$
$479$	1.54775	0.0707187	0.0353594	+	0.999375i	$0.488742\pi$
$480$	0	0
$481$	1.97783	0.0901813
$482$	0	0
$483$	39.0225	1.77558
$484$	0	0
$485$	−12.2450	−0.556019
$486$	0	0
$487$	−24.1077	−1.09242	−0.546212	−	0.837647i	$-0.683931\pi$
$487$	−24.1077	−1.09242	−0.546212	+	0.837647i	$0.683931\pi$
$488$	0	0
$489$	21.3779	0.966744
$490$	0	0
$491$	12.8426	0.579579	0.289790	−	0.957090i	$-0.406415\pi$
$491$	12.8426	0.579579	0.289790	+	0.957090i	$0.406415\pi$
$492$	0	0
$493$	−5.08914	−0.229204
$494$	0	0
$495$	−27.8995	−1.25399
$496$	0	0
$497$	−3.42918	−0.153820
$498$	0	0
$499$	−31.2582	−1.39931	−0.699656	−	0.714480i	$-0.746663\pi$
$499$	−31.2582	−1.39931	−0.699656	+	0.714480i	$0.746663\pi$
$500$	0	0
$501$	−51.3574	−2.29448
$502$	0	0
$503$	−40.4896	−1.80534	−0.902671	−	0.430331i	$-0.858397\pi$
$503$	−40.4896	−1.80534	−0.902671	+	0.430331i	$0.858397\pi$
$504$	0	0
$505$	12.9279	0.575285
$506$	0	0
$507$	−28.7167	−1.27536
$508$	0	0
$509$	11.6633	0.516969	0.258484	−	0.966015i	$-0.416777\pi$
$509$	11.6633	0.516969	0.258484	+	0.966015i	$0.416777\pi$
$510$	0	0
$511$	−28.0485	−1.24079
$512$	0	0
$513$	−4.55504	−0.201110
$514$	0	0
$515$	26.8981	1.18527
$516$	0	0
$517$	−34.4411	−1.51472
$518$	0	0
$519$	54.6717	2.39982
$520$	0	0
$521$	−21.9667	−0.962380	−0.481190	−	0.876616i	$-0.659795\pi$
$521$	−21.9667	−0.962380	−0.481190	+	0.876616i	$0.659795\pi$
$522$	0	0
$523$	24.0482	1.05155	0.525777	−	0.850623i	$-0.323775\pi$
$523$	24.0482	1.05155	0.525777	+	0.850623i	$0.323775\pi$
$524$	0	0
$525$	13.2682	0.579072
$526$	0	0
$527$	−1.83882	−0.0801001
$528$	0	0
$529$	28.7554	1.25024
$530$	0	0
$531$	32.6154	1.41539
$532$	0	0
$533$	11.7268	0.507945
$534$	0	0
$535$	−17.5251	−0.757676
$536$	0	0
$537$	4.59097	0.198115
$538$	0	0
$539$	12.3392	0.531486
$540$	0	0
$541$	26.8332	1.15365	0.576824	−	0.816868i	$-0.304292\pi$
$541$	26.8332	1.15365	0.576824	+	0.816868i	$0.304292\pi$
$542$	0	0
$543$	−46.5955	−1.99961
$544$	0	0
$545$	28.5071	1.22111
$546$	0	0
$547$	13.0363	0.557391	0.278696	−	0.960379i	$-0.410098\pi$
$547$	13.0363	0.557391	0.278696	+	0.960379i	$0.410098\pi$
$548$	0	0
$549$	39.6867	1.69379
$550$	0	0
$551$	14.2934	0.608921
$552$	0	0
$553$	3.51947	0.149663
$554$	0	0
$555$	−6.15833	−0.261407
$556$	0	0
$557$	43.3232	1.83566	0.917831	−	0.396972i	$-0.129939\pi$
$557$	43.3232	1.83566	0.917831	+	0.396972i	$0.129939\pi$
$558$	0	0
$559$	8.12985	0.343856
$560$	0	0
$561$	−17.7776	−0.750572
$562$	0	0
$563$	11.3743	0.479368	0.239684	−	0.970851i	$-0.422956\pi$
$563$	11.3743	0.479368	0.239684	+	0.970851i	$0.422956\pi$
$564$	0	0
$565$	7.19216	0.302577
$566$	0	0
$567$	15.8079	0.663869
$568$	0	0
$569$	−35.3860	−1.48346	−0.741729	−	0.670700i	$-0.765994\pi$
$569$	−35.3860	−1.48346	−0.741729	+	0.670700i	$0.765994\pi$
$570$	0	0
$571$	−27.3697	−1.14539	−0.572693	−	0.819770i	$-0.694101\pi$
$571$	−27.3697	−1.14539	−0.572693	+	0.819770i	$0.694101\pi$
$572$	0	0
$573$	39.2681	1.64045
$574$	0	0
$575$	17.5976	0.733870
$576$	0	0
$577$	41.8609	1.74269	0.871347	−	0.490667i	$-0.163247\pi$
$577$	41.8609	1.74269	0.871347	+	0.490667i	$0.163247\pi$
$578$	0	0
$579$	−28.3151	−1.17674
$580$	0	0
$581$	1.69746	0.0704225
$582$	0	0
$583$	−14.6910	−0.608439
$584$	0	0
$585$	−7.21379	−0.298254
$586$	0	0
$587$	6.28935	0.259589	0.129795	−	0.991541i	$-0.458568\pi$
$587$	6.28935	0.259589	0.129795	+	0.991541i	$0.458568\pi$
$588$	0	0
$589$	5.16453	0.212801
$590$	0	0
$591$	−11.3724	−0.467798
$592$	0	0
$593$	−30.2028	−1.24028	−0.620141	−	0.784491i	$-0.712925\pi$
$593$	−30.2028	−1.24028	−0.620141	+	0.784491i	$0.712925\pi$
$594$	0	0
$595$	−4.72798	−0.193828
$596$	0	0
$597$	23.1278	0.946558
$598$	0	0
$599$	−22.5651	−0.921984	−0.460992	−	0.887404i	$-0.652506\pi$
$599$	−22.5651	−0.921984	−0.460992	+	0.887404i	$0.652506\pi$
$600$	0	0
$601$	−17.8820	−0.729422	−0.364711	−	0.931121i	$-0.618832\pi$
$601$	−17.8820	−0.729422	−0.364711	+	0.931121i	$0.618832\pi$
$602$	0	0
$603$	−1.32092	−0.0537919
$604$	0	0
$605$	23.1002	0.939159
$606$	0	0
$607$	−1.12709	−0.0457472	−0.0228736	−	0.999738i	$-0.507282\pi$
$607$	−1.12709	−0.0457472	−0.0228736	+	0.999738i	$0.507282\pi$
$608$	0	0
$609$	19.9122	0.806882
$610$	0	0
$611$	−8.90520	−0.360266
$612$	0	0
$613$	−7.96703	−0.321785	−0.160893	−	0.986972i	$-0.551437\pi$
$613$	−7.96703	−0.321785	−0.160893	+	0.986972i	$0.551437\pi$
$614$	0	0
$615$	−36.5136	−1.47237
$616$	0	0
$617$	13.1850	0.530809	0.265405	−	0.964137i	$-0.414494\pi$
$617$	13.1850	0.530809	0.265405	+	0.964137i	$0.414494\pi$
$618$	0	0
$619$	33.1297	1.33160	0.665798	−	0.746132i	$-0.268091\pi$
$619$	33.1297	1.33160	0.665798	+	0.746132i	$0.268091\pi$
$620$	0	0
$621$	−8.41619	−0.337730
$622$	0	0
$623$	1.17592	0.0471121
$624$	0	0
$625$	−6.78604	−0.271442
$626$	0	0
$627$	49.9305	1.99403
$628$	0	0
$629$	−2.10184	−0.0838058
$630$	0	0
$631$	−19.8604	−0.790631	−0.395316	−	0.918545i	$-0.629365\pi$
$631$	−19.8604	−0.790631	−0.395316	+	0.918545i	$0.629365\pi$
$632$	0	0
$633$	1.15037	0.0457232
$634$	0	0
$635$	−33.0972	−1.31342
$636$	0	0
$637$	3.19046	0.126410
$638$	0	0
$639$	5.56017	0.219957
$640$	0	0
$641$	−11.7746	−0.465070	−0.232535	−	0.972588i	$-0.574702\pi$
$641$	−11.7746	−0.465070	−0.232535	+	0.972588i	$0.574702\pi$
$642$	0	0
$643$	−7.43202	−0.293090	−0.146545	−	0.989204i	$-0.546815\pi$
$643$	−7.43202	−0.293090	−0.146545	+	0.989204i	$0.546815\pi$
$644$	0	0
$645$	−25.3138	−0.996729
$646$	0	0
$647$	−25.7841	−1.01368	−0.506838	−	0.862041i	$-0.669186\pi$
$647$	−25.7841	−1.01368	−0.506838	+	0.862041i	$0.669186\pi$
$648$	0	0
$649$	−47.5552	−1.86670
$650$	0	0
$651$	7.19470	0.281982
$652$	0	0
$653$	−9.13174	−0.357352	−0.178676	−	0.983908i	$-0.557181\pi$
$653$	−9.13174	−0.357352	−0.178676	+	0.983908i	$0.557181\pi$
$654$	0	0
$655$	−13.2194	−0.516524
$656$	0	0
$657$	45.4787	1.77429
$658$	0	0
$659$	35.0069	1.36367	0.681837	−	0.731504i	$-0.261181\pi$
$659$	35.0069	1.36367	0.681837	+	0.731504i	$0.261181\pi$
$660$	0	0
$661$	−20.5112	−0.797795	−0.398897	−	0.916996i	$-0.630607\pi$
$661$	−20.5112	−0.797795	−0.398897	+	0.916996i	$0.630607\pi$
$662$	0	0
$663$	−4.59664	−0.178519
$664$	0	0
$665$	13.2791	0.514941
$666$	0	0
$667$	26.4095	1.02258
$668$	0	0
$669$	26.3387	1.01831
$670$	0	0
$671$	−57.8657	−2.23388
$672$	0	0
$673$	−28.3968	−1.09462	−0.547308	−	0.836931i	$-0.684347\pi$
$673$	−28.3968	−1.09462	−0.547308	+	0.836931i	$0.684347\pi$
$674$	0	0
$675$	−2.86162	−0.110144
$676$	0	0
$677$	−29.4454	−1.13168	−0.565839	−	0.824516i	$-0.691447\pi$
$677$	−29.4454	−1.13168	−0.565839	+	0.824516i	$0.691447\pi$
$678$	0	0
$679$	16.3520	0.627531
$680$	0	0
$681$	−73.5653	−2.81903
$682$	0	0
$683$	16.9146	0.647219	0.323609	−	0.946191i	$-0.395104\pi$
$683$	16.9146	0.647219	0.323609	+	0.946191i	$0.395104\pi$
$684$	0	0
$685$	33.6729	1.28658
$686$	0	0
$687$	−40.1930	−1.53346
$688$	0	0
$689$	−3.79855	−0.144713
$690$	0	0
$691$	−22.6037	−0.859885	−0.429942	−	0.902856i	$-0.641466\pi$
$691$	−22.6037	−0.859885	−0.429942	+	0.902856i	$0.641466\pi$
$692$	0	0
$693$	37.2569	1.41527
$694$	0	0
$695$	−11.1507	−0.422970
$696$	0	0
$697$	−12.4621	−0.472035
$698$	0	0
$699$	−3.49167	−0.132067
$700$	0	0
$701$	23.9476	0.904487	0.452243	−	0.891895i	$-0.350624\pi$
$701$	23.9476	0.904487	0.452243	+	0.891895i	$0.350624\pi$
$702$	0	0
$703$	5.90325	0.222645
$704$	0	0
$705$	27.7280	1.04430
$706$	0	0
$707$	−17.2639	−0.649276
$708$	0	0
$709$	−36.1930	−1.35926	−0.679629	−	0.733556i	$-0.737859\pi$
$709$	−36.1930	−1.35926	−0.679629	+	0.733556i	$0.737859\pi$
$710$	0	0
$711$	−5.70658	−0.214014
$712$	0	0
$713$	9.54231	0.357362
$714$	0	0
$715$	10.5182	0.393357
$716$	0	0
$717$	16.6547	0.621981
$718$	0	0
$719$	22.7318	0.847754	0.423877	−	0.905720i	$-0.360669\pi$
$719$	22.7318	0.847754	0.423877	+	0.905720i	$0.360669\pi$
$720$	0	0
$721$	−35.9196	−1.33772
$722$	0	0
$723$	30.7659	1.14419
$724$	0	0
$725$	8.97960	0.333494
$726$	0	0
$727$	−19.1774	−0.711250	−0.355625	−	0.934629i	$-0.615732\pi$
$727$	−19.1774	−0.711250	−0.355625	+	0.934629i	$0.615732\pi$
$728$	0	0
$729$	−34.5518	−1.27970
$730$	0	0
$731$	−8.63958	−0.319547
$732$	0	0
$733$	−41.1975	−1.52166	−0.760832	−	0.648949i	$-0.775209\pi$
$733$	−41.1975	−1.52166	−0.760832	+	0.648949i	$0.775209\pi$
$734$	0	0
$735$	−9.93407	−0.366424
$736$	0	0
$737$	1.92598	0.0709443
$738$	0	0
$739$	−12.3856	−0.455611	−0.227806	−	0.973707i	$-0.573155\pi$
$739$	−12.3856	−0.455611	−0.227806	+	0.973707i	$0.573155\pi$
$740$	0	0
$741$	12.9102	0.474268
$742$	0	0
$743$	−32.9274	−1.20799	−0.603994	−	0.796989i	$-0.706425\pi$
$743$	−32.9274	−1.20799	−0.603994	+	0.796989i	$0.706425\pi$
$744$	0	0
$745$	−2.43808	−0.0893245
$746$	0	0
$747$	−2.75232	−0.100702
$748$	0	0
$749$	23.4030	0.855125
$750$	0	0
$751$	−29.8582	−1.08954	−0.544770	−	0.838586i	$-0.683383\pi$
$751$	−29.8582	−1.08954	−0.544770	+	0.838586i	$0.683383\pi$
$752$	0	0
$753$	−2.54171	−0.0926249
$754$	0	0
$755$	5.12325	0.186454
$756$	0	0
$757$	44.9563	1.63397	0.816983	−	0.576662i	$-0.195645\pi$
$757$	44.9563	1.63397	0.816983	+	0.576662i	$0.195645\pi$
$758$	0	0
$759$	92.2548	3.34864
$760$	0	0
$761$	33.4840	1.21379	0.606897	−	0.794781i	$-0.292414\pi$
$761$	33.4840	1.21379	0.606897	+	0.794781i	$0.292414\pi$
$762$	0	0
$763$	−38.0683	−1.37816
$764$	0	0
$765$	7.66610	0.277168
$766$	0	0
$767$	−12.2960	−0.443984
$768$	0	0
$769$	−35.2470	−1.27104	−0.635520	−	0.772084i	$-0.719214\pi$
$769$	−35.2470	−1.27104	−0.635520	+	0.772084i	$0.719214\pi$
$770$	0	0
$771$	17.6892	0.637062
$772$	0	0
$773$	−40.6216	−1.46106	−0.730528	−	0.682883i	$-0.760726\pi$
$773$	−40.6216	−1.46106	−0.730528	+	0.682883i	$0.760726\pi$
$774$	0	0
$775$	3.24452	0.116547
$776$	0	0
$777$	8.22382	0.295028
$778$	0	0
$779$	35.0012	1.25405
$780$	0	0
$781$	−8.10707	−0.290094
$782$	0	0
$783$	−4.29457	−0.153475
$784$	0	0
$785$	−4.08451	−0.145783
$786$	0	0
$787$	−33.6719	−1.20027	−0.600137	−	0.799897i	$-0.704887\pi$
$787$	−33.6719	−1.20027	−0.600137	+	0.799897i	$0.704887\pi$
$788$	0	0
$789$	−14.1175	−0.502598
$790$	0	0
$791$	−9.60438	−0.341493
$792$	0	0
$793$	−14.9619	−0.531314
$794$	0	0
$795$	11.8275	0.419478
$796$	0	0
$797$	−10.5850	−0.374940	−0.187470	−	0.982270i	$-0.560029\pi$
$797$	−10.5850	−0.374940	−0.187470	+	0.982270i	$0.560029\pi$
$798$	0	0
$799$	9.46355	0.334796
$800$	0	0
$801$	−1.90667	−0.0673687
$802$	0	0
$803$	−66.3107	−2.34006
$804$	0	0
$805$	24.5353	0.864755
$806$	0	0
$807$	9.72535	0.342349
$808$	0	0
$809$	36.1819	1.27209	0.636045	−	0.771652i	$-0.280569\pi$
$809$	36.1819	1.27209	0.636045	+	0.771652i	$0.280569\pi$
$810$	0	0
$811$	9.59789	0.337027	0.168514	−	0.985699i	$-0.446103\pi$
$811$	9.59789	0.337027	0.168514	+	0.985699i	$0.446103\pi$
$812$	0	0
$813$	−24.7504	−0.868035
$814$	0	0
$815$	13.4413	0.470829
$816$	0	0
$817$	24.2653	0.848934
$818$	0	0
$819$	9.63327	0.336614
$820$	0	0
$821$	23.5397	0.821540	0.410770	−	0.911739i	$-0.365260\pi$
$821$	23.5397	0.821540	0.410770	+	0.911739i	$0.365260\pi$
$822$	0	0
$823$	50.7477	1.76895	0.884476	−	0.466586i	$-0.154516\pi$
$823$	50.7477	1.76895	0.884476	+	0.466586i	$0.154516\pi$
$824$	0	0
$825$	31.3680	1.09209
$826$	0	0
$827$	23.6898	0.823773	0.411887	−	0.911235i	$-0.364870\pi$
$827$	23.6898	0.823773	0.411887	+	0.911235i	$0.364870\pi$
$828$	0	0
$829$	−27.6980	−0.961992	−0.480996	−	0.876723i	$-0.659725\pi$
$829$	−27.6980	−0.961992	−0.480996	+	0.876723i	$0.659725\pi$
$830$	0	0
$831$	0.465790	0.0161581
$832$	0	0
$833$	−3.39050	−0.117474
$834$	0	0
$835$	−32.2908	−1.11747
$836$	0	0
$837$	−1.55172	−0.0536352
$838$	0	0
$839$	25.4722	0.879397	0.439698	−	0.898146i	$-0.355085\pi$
$839$	25.4722	0.879397	0.439698	+	0.898146i	$0.355085\pi$
$840$	0	0
$841$	−15.5239	−0.535307
$842$	0	0
$843$	−58.3442	−2.00948
$844$	0	0
$845$	−18.0556	−0.621131
$846$	0	0
$847$	−30.8480	−1.05995
$848$	0	0
$849$	6.40718	0.219894
$850$	0	0
$851$	10.9072	0.373895
$852$	0	0
$853$	51.5546	1.76520	0.882598	−	0.470128i	$-0.155792\pi$
$853$	51.5546	1.76520	0.882598	+	0.470128i	$0.155792\pi$
$854$	0	0
$855$	−21.5311	−0.736348
$856$	0	0
$857$	−2.96362	−0.101235	−0.0506176	−	0.998718i	$-0.516119\pi$
$857$	−2.96362	−0.101235	−0.0506176	+	0.998718i	$0.516119\pi$
$858$	0	0
$859$	30.4356	1.03845	0.519225	−	0.854637i	$-0.326221\pi$
$859$	30.4356	1.03845	0.519225	+	0.854637i	$0.326221\pi$
$860$	0	0
$861$	48.7601	1.66174
$862$	0	0
$863$	34.0957	1.16063	0.580315	−	0.814392i	$-0.302929\pi$
$863$	34.0957	1.16063	0.580315	+	0.814392i	$0.302929\pi$
$864$	0	0
$865$	34.3747	1.16877
$866$	0	0
$867$	−38.3242	−1.30156
$868$	0	0
$869$	8.32055	0.282255
$870$	0	0
$871$	0.497987	0.0168736
$872$	0	0
$873$	−26.5136	−0.897349
$874$	0	0
$875$	25.3946	0.858496
$876$	0	0
$877$	−3.76496	−0.127134	−0.0635669	−	0.997978i	$-0.520248\pi$
$877$	−3.76496	−0.127134	−0.0635669	+	0.997978i	$0.520248\pi$
$878$	0	0
$879$	−59.9826	−2.02316
$880$	0	0
$881$	22.0494	0.742864	0.371432	−	0.928460i	$-0.378867\pi$
$881$	22.0494	0.742864	0.371432	+	0.928460i	$0.378867\pi$
$882$	0	0
$883$	17.7325	0.596746	0.298373	−	0.954449i	$-0.403556\pi$
$883$	17.7325	0.596746	0.298373	+	0.954449i	$0.403556\pi$
$884$	0	0
$885$	38.2859	1.28697
$886$	0	0
$887$	−0.336822	−0.0113094	−0.00565468	−	0.999984i	$-0.501800\pi$
$887$	−0.336822	−0.0113094	−0.00565468	+	0.999984i	$0.501800\pi$
$888$	0	0
$889$	44.1978	1.48235
$890$	0	0
$891$	37.3722	1.25201
$892$	0	0
$893$	−26.5795	−0.889448
$894$	0	0
$895$	2.88656	0.0964870
$896$	0	0
$897$	23.8537	0.796452
$898$	0	0
$899$	4.86920	0.162397
$900$	0	0
$901$	4.03672	0.134483
$902$	0	0
$903$	33.8039	1.12492
$904$	0	0
$905$	−29.2968	−0.973859
$906$	0	0
$907$	−12.9599	−0.430327	−0.215164	−	0.976578i	$-0.569029\pi$
$907$	−12.9599	−0.430327	−0.215164	+	0.976578i	$0.569029\pi$
$908$	0	0
$909$	27.9922	0.928443
$910$	0	0
$911$	−24.0769	−0.797702	−0.398851	−	0.917016i	$-0.630591\pi$
$911$	−24.0769	−0.797702	−0.398851	+	0.917016i	$0.630591\pi$
$912$	0	0
$913$	4.01304	0.132812
$914$	0	0
$915$	46.5867	1.54011
$916$	0	0
$917$	17.6531	0.582957
$918$	0	0
$919$	−14.3872	−0.474592	−0.237296	−	0.971437i	$-0.576261\pi$
$919$	−14.3872	−0.474592	−0.237296	+	0.971437i	$0.576261\pi$
$920$	0	0
$921$	−20.2749	−0.668081
$922$	0	0
$923$	−2.09619	−0.0689969
$924$	0	0
$925$	3.70861	0.121938
$926$	0	0
$927$	58.2412	1.91289
$928$	0	0
$929$	2.60076	0.0853282	0.0426641	−	0.999089i	$-0.486415\pi$
$929$	2.60076	0.0853282	0.0426641	+	0.999089i	$0.486415\pi$
$930$	0	0
$931$	9.52260	0.312091
$932$	0	0
$933$	26.2671	0.859945
$934$	0	0
$935$	−11.1776	−0.365548
$936$	0	0
$937$	−26.2275	−0.856815	−0.428408	−	0.903586i	$-0.640925\pi$
$937$	−26.2275	−0.856815	−0.428408	+	0.903586i	$0.640925\pi$
$938$	0	0
$939$	−8.49926	−0.277363
$940$	0	0
$941$	−25.8090	−0.841350	−0.420675	−	0.907211i	$-0.638207\pi$
$941$	−25.8090	−0.841350	−0.420675	+	0.907211i	$0.638207\pi$
$942$	0	0
$943$	64.6704	2.10596
$944$	0	0
$945$	−3.98979	−0.129788
$946$	0	0
$947$	49.7228	1.61577	0.807886	−	0.589338i	$-0.200611\pi$
$947$	49.7228	1.61577	0.807886	+	0.589338i	$0.200611\pi$
$948$	0	0
$949$	−17.1455	−0.556567
$950$	0	0
$951$	14.8799	0.482513
$952$	0	0
$953$	−7.91565	−0.256413	−0.128207	−	0.991747i	$-0.540922\pi$
$953$	−7.91565	−0.256413	−0.128207	+	0.991747i	$0.540922\pi$
$954$	0	0
$955$	24.6897	0.798941
$956$	0	0
$957$	47.0753	1.52173
$958$	0	0
$959$	−44.9666	−1.45205
$960$	0	0
$961$	−29.2407	−0.943247
$962$	0	0
$963$	−37.9463	−1.22280
$964$	0	0
$965$	−17.8031	−0.573101
$966$	0	0
$967$	48.7013	1.56613	0.783065	−	0.621940i	$-0.213655\pi$
$967$	48.7013	1.56613	0.783065	+	0.621940i	$0.213655\pi$
$968$	0	0
$969$	−13.7197	−0.440739
$970$	0	0
$971$	−20.5260	−0.658711	−0.329356	−	0.944206i	$-0.606832\pi$
$971$	−20.5260	−0.658711	−0.329356	+	0.944206i	$0.606832\pi$
$972$	0	0
$973$	14.8906	0.477371
$974$	0	0
$975$	8.11060	0.259747
$976$	0	0
$977$	24.7385	0.791455	0.395727	−	0.918368i	$-0.370492\pi$
$977$	24.7385	0.791455	0.395727	+	0.918368i	$0.370492\pi$
$978$	0	0
$979$	2.78004	0.0888503
$980$	0	0
$981$	61.7251	1.97073
$982$	0	0
$983$	46.6117	1.48668	0.743341	−	0.668913i	$-0.233240\pi$
$983$	46.6117	1.48668	0.743341	+	0.668913i	$0.233240\pi$
$984$	0	0
$985$	−7.15037	−0.227830
$986$	0	0
$987$	−37.0278	−1.17861
$988$	0	0
$989$	44.8341	1.42564
$990$	0	0
$991$	−39.2400	−1.24650	−0.623249	−	0.782023i	$-0.714188\pi$
$991$	−39.2400	−1.24650	−0.623249	+	0.782023i	$0.714188\pi$
$992$	0	0
$993$	−8.62246	−0.273625
$994$	0	0
$995$	14.5415	0.460998
$996$	0	0
$997$	−46.4562	−1.47128	−0.735642	−	0.677371i	$-0.763119\pi$
$997$	−46.4562	−1.47128	−0.735642	+	0.677371i	$0.763119\pi$
$998$	0	0
$999$	−1.77367	−0.0561166

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.i.1.12		12
4.3	odd	2		2008.2.a.b.1.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.b.1.1	✓	12	4.3	odd	2
4016.2.a.i.1.12		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+2.54171 q^{3} +1.59809 q^{5} -2.13408 q^{7} +3.46027 q^{9} +O(q^{10})\)	\(q+2.54171 q^{3} +1.59809 q^{5} -2.13408 q^{7} +3.46027 q^{9} -5.04529 q^{11} -1.30452 q^{13} +4.06188 q^{15} +1.38632 q^{17} -3.89363 q^{19} -5.42421 q^{21} -7.19413 q^{23} -2.44611 q^{25} +1.16987 q^{27} -3.67098 q^{29} -1.32640 q^{31} -12.8236 q^{33} -3.41046 q^{35} -1.51613 q^{37} -3.31572 q^{39} -8.98934 q^{41} -6.23204 q^{43} +5.52983 q^{45} +6.82639 q^{47} -2.44568 q^{49} +3.52361 q^{51} +2.91183 q^{53} -8.06282 q^{55} -9.89647 q^{57} +9.42567 q^{59} +11.4693 q^{61} -7.38451 q^{63} -2.08475 q^{65} -0.381738 q^{67} -18.2854 q^{69} +1.60686 q^{71} +13.1431 q^{73} -6.21728 q^{75} +10.7671 q^{77} -1.64917 q^{79} -7.40734 q^{81} -0.795405 q^{83} +2.21546 q^{85} -9.33055 q^{87} -0.551016 q^{89} +2.78397 q^{91} -3.37133 q^{93} -6.22238 q^{95} -7.66229 q^{97} -17.4580 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) \) 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9	\( 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9} - 10 q^{11} + 3 q^{13} - 11 q^{15} + 2 q^{17} - 15 q^{19} + 3 q^{21} - 20 q^{23} - 3 q^{25} - 15 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{33} - 16 q^{35} + 5 q^{37} - 21 q^{39} - 21 q^{43} + 10 q^{45} - 27 q^{47} - 13 q^{49} - 19 q^{51} + 22 q^{53} - 24 q^{55} + q^{57} - 23 q^{59} + 4 q^{61} - 21 q^{63} - q^{65} - 26 q^{67} + 10 q^{69} - 23 q^{71} - 8 q^{73} - 16 q^{75} + 22 q^{77} - 37 q^{79} - 20 q^{81} - 30 q^{83} + 2 q^{85} - 16 q^{87} + 3 q^{89} - 8 q^{91} + 20 q^{93} - 33 q^{95} - 4 q^{97} - 15 q^{99}+O(q^{100}) \) 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9 - 10 * q^11 + 3 * q^13 - 11 * q^15 + 2 * q^17 - 15 * q^19 + 3 * q^21 - 20 * q^23 - 3 * q^25 - 15 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^33 - 16 * q^35 + 5 * q^37 - 21 * q^39 - 21 * q^43 + 10 * q^45 - 27 * q^47 - 13 * q^49 - 19 * q^51 + 22 * q^53 - 24 * q^55 + q^57 - 23 * q^59 + 4 * q^61 - 21 * q^63 - q^65 - 26 * q^67 + 10 * q^69 - 23 * q^71 - 8 * q^73 - 16 * q^75 + 22 * q^77 - 37 * q^79 - 20 * q^81 - 30 * q^83 + 2 * q^85 - 16 * q^87 + 3 * q^89 - 8 * q^91 + 20 * q^93 - 33 * q^95 - 4 * q^97 - 15 * q^99

Introduction

L-functions

Modular forms

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Groups

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