LMFDB - Embedded newform 8046.2.a.m.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, 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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
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} - 5 x^{11} - 21 x^{10} + 116 x^{9} + 106 x^{8} - 774 x^{7} - 63 x^{6} + 2013 x^{5} - 417 x^{4} + \cdots - 375 $ x^12 - 5x^11 - 21x^10 + 116x^9 + 106x^8 - 774x^7 - 63x^6 + 2013x^5 - 417x^4 - 2249x^3 + 761x^2 + 910*x - 375
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.06735$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.06735 q^{5} +2.41300 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.06735 q^{5} +2.41300 q^{7} +1.00000 q^{8} -1.06735 q^{10} -2.87774 q^{11} +6.31805 q^{13} +2.41300 q^{14} +1.00000 q^{16} -5.80297 q^{17} +2.51725 q^{19} -1.06735 q^{20} -2.87774 q^{22} -4.87648 q^{23} -3.86076 q^{25} +6.31805 q^{26} +2.41300 q^{28} -5.18081 q^{29} -7.91286 q^{31} +1.00000 q^{32} -5.80297 q^{34} -2.57552 q^{35} -4.64638 q^{37} +2.51725 q^{38} -1.06735 q^{40} -10.6771 q^{41} +11.1447 q^{43} -2.87774 q^{44} -4.87648 q^{46} -8.19941 q^{47} -1.17744 q^{49} -3.86076 q^{50} +6.31805 q^{52} -6.76900 q^{53} +3.07157 q^{55} +2.41300 q^{56} -5.18081 q^{58} +6.78939 q^{59} -11.8514 q^{61} -7.91286 q^{62} +1.00000 q^{64} -6.74360 q^{65} +12.4327 q^{67} -5.80297 q^{68} -2.57552 q^{70} -3.88117 q^{71} -1.97024 q^{73} -4.64638 q^{74} +2.51725 q^{76} -6.94399 q^{77} +2.26625 q^{79} -1.06735 q^{80} -10.6771 q^{82} +2.68059 q^{83} +6.19382 q^{85} +11.1447 q^{86} -2.87774 q^{88} +12.9074 q^{89} +15.2455 q^{91} -4.87648 q^{92} -8.19941 q^{94} -2.68680 q^{95} +6.69563 q^{97} -1.17744 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} - 5 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 - 5 * q^5 - 6 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} - 5 q^{5} - 6 q^{7} + 12 q^{8} - 5 q^{10} - 10 q^{11} - q^{13} - 6 q^{14} + 12 q^{16} - 6 q^{17} - 10 q^{19} - 5 q^{20} - 10 q^{22} - 15 q^{23} + 7 q^{25} - q^{26} - 6 q^{28} - 33 q^{29} - 6 q^{31} + 12 q^{32} - 6 q^{34} - 16 q^{35} - 13 q^{37} - 10 q^{38} - 5 q^{40} - 20 q^{41} - 11 q^{43} - 10 q^{44} - 15 q^{46} - 15 q^{47} + 2 q^{49} + 7 q^{50} - q^{52} - 4 q^{53} - 17 q^{55} - 6 q^{56} - 33 q^{58} - 10 q^{59} - 12 q^{61} - 6 q^{62} + 12 q^{64} - 40 q^{65} - 19 q^{67} - 6 q^{68} - 16 q^{70} - 47 q^{71} - 2 q^{73} - 13 q^{74} - 10 q^{76} + 6 q^{77} - 15 q^{79} - 5 q^{80} - 20 q^{82} - 18 q^{83} - 25 q^{85} - 11 q^{86} - 10 q^{88} - 24 q^{89} - 3 q^{91} - 15 q^{92} - 15 q^{94} + 3 q^{95} - 25 q^{97} + 2 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 - 5 * q^5 - 6 * q^7 + 12 * q^8 - 5 * q^10 - 10 * q^11 - q^13 - 6 * q^14 + 12 * q^16 - 6 * q^17 - 10 * q^19 - 5 * q^20 - 10 * q^22 - 15 * q^23 + 7 * q^25 - q^26 - 6 * q^28 - 33 * q^29 - 6 * q^31 + 12 * q^32 - 6 * q^34 - 16 * q^35 - 13 * q^37 - 10 * q^38 - 5 * q^40 - 20 * q^41 - 11 * q^43 - 10 * q^44 - 15 * q^46 - 15 * q^47 + 2 * q^49 + 7 * q^50 - q^52 - 4 * q^53 - 17 * q^55 - 6 * q^56 - 33 * q^58 - 10 * q^59 - 12 * q^61 - 6 * q^62 + 12 * q^64 - 40 * q^65 - 19 * q^67 - 6 * q^68 - 16 * q^70 - 47 * q^71 - 2 * q^73 - 13 * q^74 - 10 * q^76 + 6 * q^77 - 15 * q^79 - 5 * q^80 - 20 * q^82 - 18 * q^83 - 25 * q^85 - 11 * q^86 - 10 * q^88 - 24 * q^89 - 3 * q^91 - 15 * q^92 - 15 * q^94 + 3 * q^95 - 25 * q^97 + 2 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.06735	−0.477335	−0.238668	−	0.971101i	$-0.576711\pi$
$5$	−1.06735	−0.477335	−0.238668	+	0.971101i	$0.576711\pi$
$6$	0	0
$7$	2.41300	0.912028	0.456014	−	0.889973i	$-0.349277\pi$
$7$	2.41300	0.912028	0.456014	+	0.889973i	$0.349277\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.06735	−0.337527
$11$	−2.87774	−0.867672	−0.433836	−	0.900992i	$-0.642840\pi$
$11$	−2.87774	−0.867672	−0.433836	+	0.900992i	$0.642840\pi$
$12$	0	0
$13$	6.31805	1.75231	0.876156	−	0.482027i	$-0.160099\pi$
$13$	6.31805	1.75231	0.876156	+	0.482027i	$0.160099\pi$
$14$	2.41300	0.644901
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.80297	−1.40743	−0.703714	−	0.710484i	$-0.748476\pi$
$17$	−5.80297	−1.40743	−0.703714	+	0.710484i	$0.748476\pi$
$18$	0	0
$19$	2.51725	0.577498	0.288749	−	0.957405i	$-0.406761\pi$
$19$	2.51725	0.577498	0.288749	+	0.957405i	$0.406761\pi$
$20$	−1.06735	−0.238668
$21$	0	0
$22$	−2.87774	−0.613537
$23$	−4.87648	−1.01682	−0.508408	−	0.861116i	$-0.669766\pi$
$23$	−4.87648	−1.01682	−0.508408	+	0.861116i	$0.669766\pi$
$24$	0	0
$25$	−3.86076	−0.772151
$26$	6.31805	1.23907
$27$	0	0
$28$	2.41300	0.456014
$29$	−5.18081	−0.962052	−0.481026	−	0.876706i	$-0.659736\pi$
$29$	−5.18081	−0.962052	−0.481026	+	0.876706i	$0.659736\pi$
$30$	0	0
$31$	−7.91286	−1.42119	−0.710596	−	0.703600i	$-0.751575\pi$
$31$	−7.91286	−1.42119	−0.710596	+	0.703600i	$0.751575\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.80297	−0.995201
$35$	−2.57552	−0.435343
$36$	0	0
$37$	−4.64638	−0.763860	−0.381930	−	0.924191i	$-0.624740\pi$
$37$	−4.64638	−0.763860	−0.381930	+	0.924191i	$0.624740\pi$
$38$	2.51725	0.408353
$39$	0	0
$40$	−1.06735	−0.168763
$41$	−10.6771	−1.66749	−0.833744	−	0.552152i	$-0.813807\pi$
$41$	−10.6771	−1.66749	−0.833744	+	0.552152i	$0.813807\pi$
$42$	0	0
$43$	11.1447	1.69955	0.849776	−	0.527144i	$-0.176737\pi$
$43$	11.1447	1.69955	0.849776	+	0.527144i	$0.176737\pi$
$44$	−2.87774	−0.433836
$45$	0	0
$46$	−4.87648	−0.718998
$47$	−8.19941	−1.19601	−0.598004	−	0.801493i	$-0.704039\pi$
$47$	−8.19941	−1.19601	−0.598004	+	0.801493i	$0.704039\pi$
$48$	0	0
$49$	−1.17744	−0.168206
$50$	−3.86076	−0.545993
$51$	0	0
$52$	6.31805	0.876156
$53$	−6.76900	−0.929794	−0.464897	−	0.885365i	$-0.653909\pi$
$53$	−6.76900	−0.929794	−0.464897	+	0.885365i	$0.653909\pi$
$54$	0	0
$55$	3.07157	0.414170
$56$	2.41300	0.322450
$57$	0	0
$58$	−5.18081	−0.680274
$59$	6.78939	0.883903	0.441951	−	0.897039i	$-0.354286\pi$
$59$	6.78939	0.883903	0.441951	+	0.897039i	$0.354286\pi$
$60$	0	0
$61$	−11.8514	−1.51742	−0.758711	−	0.651428i	$-0.774170\pi$
$61$	−11.8514	−1.51742	−0.758711	+	0.651428i	$0.774170\pi$
$62$	−7.91286	−1.00493
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.74360	−0.836440
$66$	0	0
$67$	12.4327	1.51890	0.759448	−	0.650568i	$-0.225469\pi$
$67$	12.4327	1.51890	0.759448	+	0.650568i	$0.225469\pi$
$68$	−5.80297	−0.703714
$69$	0	0
$70$	−2.57552	−0.307834
$71$	−3.88117	−0.460610	−0.230305	−	0.973118i	$-0.573972\pi$
$71$	−3.88117	−0.460610	−0.230305	+	0.973118i	$0.573972\pi$
$72$	0	0
$73$	−1.97024	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$73$	−1.97024	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$74$	−4.64638	−0.540130
$75$	0	0
$76$	2.51725	0.288749
$77$	−6.94399	−0.791341
$78$	0	0
$79$	2.26625	0.254973	0.127487	−	0.991840i	$-0.459309\pi$
$79$	2.26625	0.254973	0.127487	+	0.991840i	$0.459309\pi$
$80$	−1.06735	−0.119334
$81$	0	0
$82$	−10.6771	−1.17909
$83$	2.68059	0.294233	0.147117	−	0.989119i	$-0.453001\pi$
$83$	2.68059	0.294233	0.147117	+	0.989119i	$0.453001\pi$
$84$	0	0
$85$	6.19382	0.671814
$86$	11.1447	1.20176
$87$	0	0
$88$	−2.87774	−0.306768
$89$	12.9074	1.36818	0.684091	−	0.729397i	$-0.260199\pi$
$89$	12.9074	1.36818	0.684091	+	0.729397i	$0.260199\pi$
$90$	0	0
$91$	15.2455	1.59816
$92$	−4.87648	−0.508408
$93$	0	0
$94$	−8.19941	−0.845705
$95$	−2.68680	−0.275660
$96$	0	0
$97$	6.69563	0.679839	0.339919	−	0.940455i	$-0.389600\pi$
$97$	6.69563	0.679839	0.339919	+	0.940455i	$0.389600\pi$
$98$	−1.17744	−0.118939
$99$	0	0
$100$	−3.86076	−0.386076
$101$	9.08647	0.904137	0.452069	−	0.891983i	$-0.350686\pi$
$101$	9.08647	0.904137	0.452069	+	0.891983i	$0.350686\pi$
$102$	0	0
$103$	−12.3461	−1.21650	−0.608251	−	0.793745i	$-0.708129\pi$
$103$	−12.3461	−1.21650	−0.608251	+	0.793745i	$0.708129\pi$
$104$	6.31805	0.619536
$105$	0	0
$106$	−6.76900	−0.657464
$107$	4.76521	0.460670	0.230335	−	0.973111i	$-0.426018\pi$
$107$	4.76521	0.460670	0.230335	+	0.973111i	$0.426018\pi$
$108$	0	0
$109$	−5.74157	−0.549943	−0.274971	−	0.961452i	$-0.588668\pi$
$109$	−5.74157	−0.549943	−0.274971	+	0.961452i	$0.588668\pi$
$110$	3.07157	0.292863
$111$	0	0
$112$	2.41300	0.228007
$113$	−13.1555	−1.23756	−0.618781	−	0.785563i	$-0.712373\pi$
$113$	−13.1555	−1.23756	−0.618781	+	0.785563i	$0.712373\pi$
$114$	0	0
$115$	5.20493	0.485362
$116$	−5.18081	−0.481026
$117$	0	0
$118$	6.78939	0.625014
$119$	−14.0026	−1.28361
$120$	0	0
$121$	−2.71860	−0.247145
$122$	−11.8514	−1.07298
$123$	0	0
$124$	−7.91286	−0.710596
$125$	9.45756	0.845910
$126$	0	0
$127$	3.03628	0.269426	0.134713	−	0.990885i	$-0.456989\pi$
$127$	3.03628	0.269426	0.134713	+	0.990885i	$0.456989\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−6.74360	−0.591453
$131$	20.9274	1.82843	0.914217	−	0.405226i	$-0.132807\pi$
$131$	20.9274	1.82843	0.914217	+	0.405226i	$0.132807\pi$
$132$	0	0
$133$	6.07413	0.526694
$134$	12.4327	1.07402
$135$	0	0
$136$	−5.80297	−0.497601
$137$	5.40932	0.462150	0.231075	−	0.972936i	$-0.425776\pi$
$137$	5.40932	0.462150	0.231075	+	0.972936i	$0.425776\pi$
$138$	0	0
$139$	−11.5243	−0.977475	−0.488738	−	0.872431i	$-0.662542\pi$
$139$	−11.5243	−0.977475	−0.488738	+	0.872431i	$0.662542\pi$
$140$	−2.57552	−0.217671
$141$	0	0
$142$	−3.88117	−0.325701
$143$	−18.1817	−1.52043
$144$	0	0
$145$	5.52976	0.459221
$146$	−1.97024	−0.163058
$147$	0	0
$148$	−4.64638	−0.381930
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−18.4343	−1.50016	−0.750080	−	0.661347i	$-0.769985\pi$
$151$	−18.4343	−1.50016	−0.750080	+	0.661347i	$0.769985\pi$
$152$	2.51725	0.204176
$153$	0	0
$154$	−6.94399	−0.559562
$155$	8.44582	0.678385
$156$	0	0
$157$	−18.8174	−1.50179	−0.750896	−	0.660421i	$-0.770378\pi$
$157$	−18.8174	−1.50179	−0.750896	+	0.660421i	$0.770378\pi$
$158$	2.26625	0.180293
$159$	0	0
$160$	−1.06735	−0.0843817
$161$	−11.7669	−0.927365
$162$	0	0
$163$	−18.6680	−1.46219	−0.731097	−	0.682273i	$-0.760991\pi$
$163$	−18.6680	−1.46219	−0.731097	+	0.682273i	$0.760991\pi$
$164$	−10.6771	−0.833744
$165$	0	0
$166$	2.68059	0.208054
$167$	10.7650	0.833017	0.416509	−	0.909132i	$-0.363254\pi$
$167$	10.7650	0.833017	0.416509	+	0.909132i	$0.363254\pi$
$168$	0	0
$169$	26.9178	2.07060
$170$	6.19382	0.475045
$171$	0	0
$172$	11.1447	0.849776
$173$	14.3128	1.08818	0.544092	−	0.839025i	$-0.316874\pi$
$173$	14.3128	1.08818	0.544092	+	0.839025i	$0.316874\pi$
$174$	0	0
$175$	−9.31600	−0.704223
$176$	−2.87774	−0.216918
$177$	0	0
$178$	12.9074	0.967451
$179$	14.8159	1.10739	0.553696	−	0.832719i	$-0.313217\pi$
$179$	14.8159	1.10739	0.553696	+	0.832719i	$0.313217\pi$
$180$	0	0
$181$	12.4854	0.928030	0.464015	−	0.885827i	$-0.346408\pi$
$181$	12.4854	0.928030	0.464015	+	0.885827i	$0.346408\pi$
$182$	15.2455	1.13007
$183$	0	0
$184$	−4.87648	−0.359499
$185$	4.95933	0.364617
$186$	0	0
$187$	16.6995	1.22119
$188$	−8.19941	−0.598004
$189$	0	0
$190$	−2.68680	−0.194921
$191$	−0.208714	−0.0151020	−0.00755100	−	0.999971i	$-0.502404\pi$
$191$	−0.208714	−0.0151020	−0.00755100	+	0.999971i	$0.502404\pi$
$192$	0	0
$193$	20.4582	1.47261	0.736307	−	0.676648i	$-0.236568\pi$
$193$	20.4582	1.47261	0.736307	+	0.676648i	$0.236568\pi$
$194$	6.69563	0.480718
$195$	0	0
$196$	−1.17744	−0.0841028
$197$	−24.9769	−1.77953	−0.889765	−	0.456419i	$-0.849132\pi$
$197$	−24.9769	−1.77953	−0.889765	+	0.456419i	$0.849132\pi$
$198$	0	0
$199$	−18.3928	−1.30383	−0.651916	−	0.758291i	$-0.726035\pi$
$199$	−18.3928	−1.30383	−0.651916	+	0.758291i	$0.726035\pi$
$200$	−3.86076	−0.272997
$201$	0	0
$202$	9.08647	0.639322
$203$	−12.5013	−0.877418
$204$	0	0
$205$	11.3963	0.795950
$206$	−12.3461	−0.860197
$207$	0	0
$208$	6.31805	0.438078
$209$	−7.24401	−0.501079
$210$	0	0
$211$	−5.11885	−0.352396	−0.176198	−	0.984355i	$-0.556380\pi$
$211$	−5.11885	−0.352396	−0.176198	+	0.984355i	$0.556380\pi$
$212$	−6.76900	−0.464897
$213$	0	0
$214$	4.76521	0.325743
$215$	−11.8953	−0.811256
$216$	0	0
$217$	−19.0937	−1.29617
$218$	−5.74157	−0.388868
$219$	0	0
$220$	3.07157	0.207085
$221$	−36.6635	−2.46625
$222$	0	0
$223$	−24.0494	−1.61047	−0.805233	−	0.592958i	$-0.797960\pi$
$223$	−24.0494	−1.61047	−0.805233	+	0.592958i	$0.797960\pi$
$224$	2.41300	0.161225
$225$	0	0
$226$	−13.1555	−0.875089
$227$	−16.4469	−1.09162	−0.545809	−	0.837910i	$-0.683777\pi$
$227$	−16.4469	−1.09162	−0.545809	+	0.837910i	$0.683777\pi$
$228$	0	0
$229$	3.67491	0.242845	0.121423	−	0.992601i	$-0.461254\pi$
$229$	3.67491	0.242845	0.121423	+	0.992601i	$0.461254\pi$
$230$	5.20493	0.343203
$231$	0	0
$232$	−5.18081	−0.340137
$233$	1.22524	0.0802679	0.0401340	−	0.999194i	$-0.487222\pi$
$233$	1.22524	0.0802679	0.0401340	+	0.999194i	$0.487222\pi$
$234$	0	0
$235$	8.75167	0.570896
$236$	6.78939	0.441951
$237$	0	0
$238$	−14.0026	−0.907651
$239$	19.6354	1.27011	0.635055	−	0.772467i	$-0.280978\pi$
$239$	19.6354	1.27011	0.635055	+	0.772467i	$0.280978\pi$
$240$	0	0
$241$	−21.6153	−1.39237	−0.696183	−	0.717865i	$-0.745120\pi$
$241$	−21.6153	−1.39237	−0.696183	+	0.717865i	$0.745120\pi$
$242$	−2.71860	−0.174758
$243$	0	0
$244$	−11.8514	−0.758711
$245$	1.25674	0.0802904
$246$	0	0
$247$	15.9041	1.01196
$248$	−7.91286	−0.502467
$249$	0	0
$250$	9.45756	0.598149
$251$	−22.9313	−1.44741	−0.723704	−	0.690110i	$-0.757562\pi$
$251$	−22.9313	−1.44741	−0.723704	+	0.690110i	$0.757562\pi$
$252$	0	0
$253$	14.0333	0.882263
$254$	3.03628	0.190513
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.7438	0.857313	0.428656	−	0.903468i	$-0.358987\pi$
$257$	13.7438	0.857313	0.428656	+	0.903468i	$0.358987\pi$
$258$	0	0
$259$	−11.2117	−0.696661
$260$	−6.74360	−0.418220
$261$	0	0
$262$	20.9274	1.29290
$263$	−3.02740	−0.186678	−0.0933388	−	0.995634i	$-0.529754\pi$
$263$	−3.02740	−0.186678	−0.0933388	+	0.995634i	$0.529754\pi$
$264$	0	0
$265$	7.22492	0.443823
$266$	6.07413	0.372429
$267$	0	0
$268$	12.4327	0.759448
$269$	30.7276	1.87350	0.936748	−	0.350005i	$-0.113820\pi$
$269$	30.7276	1.87350	0.936748	+	0.350005i	$0.113820\pi$
$270$	0	0
$271$	−12.7435	−0.774111	−0.387056	−	0.922056i	$-0.626508\pi$
$271$	−12.7435	−0.774111	−0.387056	+	0.922056i	$0.626508\pi$
$272$	−5.80297	−0.351857
$273$	0	0
$274$	5.40932	0.326789
$275$	11.1103	0.669974
$276$	0	0
$277$	10.0878	0.606117	0.303059	−	0.952972i	$-0.401992\pi$
$277$	10.0878	0.606117	0.303059	+	0.952972i	$0.401992\pi$
$278$	−11.5243	−0.691179
$279$	0	0
$280$	−2.57552	−0.153917
$281$	−20.1407	−1.20149	−0.600746	−	0.799440i	$-0.705130\pi$
$281$	−20.1407	−1.20149	−0.600746	+	0.799440i	$0.705130\pi$
$282$	0	0
$283$	−9.42116	−0.560030	−0.280015	−	0.959996i	$-0.590339\pi$
$283$	−9.42116	−0.560030	−0.280015	+	0.959996i	$0.590339\pi$
$284$	−3.88117	−0.230305
$285$	0	0
$286$	−18.1817	−1.07511
$287$	−25.7639	−1.52079
$288$	0	0
$289$	16.6745	0.980852
$290$	5.52976	0.324718
$291$	0	0
$292$	−1.97024	−0.115299
$293$	9.52070	0.556205	0.278103	−	0.960551i	$-0.410294\pi$
$293$	9.52070	0.556205	0.278103	+	0.960551i	$0.410294\pi$
$294$	0	0
$295$	−7.24668	−0.421918
$296$	−4.64638	−0.270065
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−30.8099	−1.78178
$300$	0	0
$301$	26.8922	1.55004
$302$	−18.4343	−1.06077
$303$	0	0
$304$	2.51725	0.144374
$305$	12.6497	0.724318
$306$	0	0
$307$	4.01472	0.229132	0.114566	−	0.993416i	$-0.463452\pi$
$307$	4.01472	0.229132	0.114566	+	0.993416i	$0.463452\pi$
$308$	−6.94399	−0.395670
$309$	0	0
$310$	8.44582	0.479690
$311$	−12.5289	−0.710450	−0.355225	−	0.934781i	$-0.615596\pi$
$311$	−12.5289	−0.710450	−0.355225	+	0.934781i	$0.615596\pi$
$312$	0	0
$313$	−17.9628	−1.01532	−0.507660	−	0.861558i	$-0.669489\pi$
$313$	−17.9628	−1.01532	−0.507660	+	0.861558i	$0.669489\pi$
$314$	−18.8174	−1.06193
$315$	0	0
$316$	2.26625	0.127487
$317$	0.635891	0.0357152	0.0178576	−	0.999841i	$-0.494315\pi$
$317$	0.635891	0.0357152	0.0178576	+	0.999841i	$0.494315\pi$
$318$	0	0
$319$	14.9090	0.834746
$320$	−1.06735	−0.0596669
$321$	0	0
$322$	−11.7669	−0.655746
$323$	−14.6076	−0.812786
$324$	0	0
$325$	−24.3925	−1.35305
$326$	−18.6680	−1.03393
$327$	0	0
$328$	−10.6771	−0.589546
$329$	−19.7852	−1.09079
$330$	0	0
$331$	−3.11325	−0.171120	−0.0855599	−	0.996333i	$-0.527268\pi$
$331$	−3.11325	−0.171120	−0.0855599	+	0.996333i	$0.527268\pi$
$332$	2.68059	0.147117
$333$	0	0
$334$	10.7650	0.589032
$335$	−13.2701	−0.725022
$336$	0	0
$337$	−33.9883	−1.85146	−0.925731	−	0.378182i	$-0.876549\pi$
$337$	−33.9883	−1.85146	−0.925731	+	0.378182i	$0.876549\pi$
$338$	26.9178	1.46413
$339$	0	0
$340$	6.19382	0.335907
$341$	22.7712	1.23313
$342$	0	0
$343$	−19.7321	−1.06544
$344$	11.1447	0.600882
$345$	0	0
$346$	14.3128	0.769463
$347$	25.5691	1.37262	0.686309	−	0.727310i	$-0.259230\pi$
$347$	25.5691	1.37262	0.686309	+	0.727310i	$0.259230\pi$
$348$	0	0
$349$	−6.64078	−0.355473	−0.177736	−	0.984078i	$-0.556877\pi$
$349$	−6.64078	−0.355473	−0.177736	+	0.984078i	$0.556877\pi$
$350$	−9.31600	−0.497961
$351$	0	0
$352$	−2.87774	−0.153384
$353$	4.63793	0.246852	0.123426	−	0.992354i	$-0.460612\pi$
$353$	4.63793	0.246852	0.123426	+	0.992354i	$0.460612\pi$
$354$	0	0
$355$	4.14258	0.219866
$356$	12.9074	0.684091
$357$	0	0
$358$	14.8159	0.783045
$359$	14.7744	0.779761	0.389881	−	0.920865i	$-0.372516\pi$
$359$	14.7744	0.779761	0.389881	+	0.920865i	$0.372516\pi$
$360$	0	0
$361$	−12.6634	−0.666496
$362$	12.4854	0.656216
$363$	0	0
$364$	15.2455	0.799079
$365$	2.10294	0.110073
$366$	0	0
$367$	6.92697	0.361585	0.180792	−	0.983521i	$-0.442134\pi$
$367$	6.92697	0.361585	0.180792	+	0.983521i	$0.442134\pi$
$368$	−4.87648	−0.254204
$369$	0	0
$370$	4.95933	0.257823
$371$	−16.3336	−0.847998
$372$	0	0
$373$	7.15412	0.370426	0.185213	−	0.982698i	$-0.440702\pi$
$373$	7.15412	0.370426	0.185213	+	0.982698i	$0.440702\pi$
$374$	16.6995	0.863508
$375$	0	0
$376$	−8.19941	−0.422853
$377$	−32.7326	−1.68582
$378$	0	0
$379$	25.6854	1.31937	0.659684	−	0.751543i	$-0.270690\pi$
$379$	25.6854	1.31937	0.659684	+	0.751543i	$0.270690\pi$
$380$	−2.68680	−0.137830
$381$	0	0
$382$	−0.208714	−0.0106787
$383$	23.5765	1.20471	0.602353	−	0.798230i	$-0.294230\pi$
$383$	23.5765	1.20471	0.602353	+	0.798230i	$0.294230\pi$
$384$	0	0
$385$	7.41169	0.377735
$386$	20.4582	1.04130
$387$	0	0
$388$	6.69563	0.339919
$389$	−16.7592	−0.849724	−0.424862	−	0.905258i	$-0.639677\pi$
$389$	−16.7592	−0.849724	−0.424862	+	0.905258i	$0.639677\pi$
$390$	0	0
$391$	28.2981	1.43110
$392$	−1.17744	−0.0594697
$393$	0	0
$394$	−24.9769	−1.25832
$395$	−2.41889	−0.121708
$396$	0	0
$397$	−5.12235	−0.257083	−0.128542	−	0.991704i	$-0.541030\pi$
$397$	−5.12235	−0.257083	−0.128542	+	0.991704i	$0.541030\pi$
$398$	−18.3928	−0.921949
$399$	0	0
$400$	−3.86076	−0.193038
$401$	19.0659	0.952106	0.476053	−	0.879417i	$-0.342067\pi$
$401$	19.0659	0.952106	0.476053	+	0.879417i	$0.342067\pi$
$402$	0	0
$403$	−49.9939	−2.49037
$404$	9.08647	0.452069
$405$	0	0
$406$	−12.5013	−0.620428
$407$	13.3711	0.662780
$408$	0	0
$409$	15.0451	0.743933	0.371966	−	0.928246i	$-0.378684\pi$
$409$	15.0451	0.743933	0.371966	+	0.928246i	$0.378684\pi$
$410$	11.3963	0.562822
$411$	0	0
$412$	−12.3461	−0.608251
$413$	16.3828	0.806144
$414$	0	0
$415$	−2.86114	−0.140448
$416$	6.31805	0.309768
$417$	0	0
$418$	−7.24401	−0.354316
$419$	12.6427	0.617637	0.308818	−	0.951121i	$-0.400066\pi$
$419$	12.6427	0.617637	0.308818	+	0.951121i	$0.400066\pi$
$420$	0	0
$421$	−8.70185	−0.424102	−0.212051	−	0.977259i	$-0.568014\pi$
$421$	−8.70185	−0.424102	−0.212051	+	0.977259i	$0.568014\pi$
$422$	−5.11885	−0.249182
$423$	0	0
$424$	−6.76900	−0.328732
$425$	22.4039	1.08675
$426$	0	0
$427$	−28.5975	−1.38393
$428$	4.76521	0.230335
$429$	0	0
$430$	−11.8953	−0.573644
$431$	−13.9905	−0.673897	−0.336948	−	0.941523i	$-0.609395\pi$
$431$	−13.9905	−0.673897	−0.336948	+	0.941523i	$0.609395\pi$
$432$	0	0
$433$	23.4355	1.12624	0.563120	−	0.826375i	$-0.309601\pi$
$433$	23.4355	1.12624	0.563120	+	0.826375i	$0.309601\pi$
$434$	−19.0937	−0.916528
$435$	0	0
$436$	−5.74157	−0.274971
$437$	−12.2753	−0.587209
$438$	0	0
$439$	−17.3808	−0.829540	−0.414770	−	0.909926i	$-0.636138\pi$
$439$	−17.3808	−0.829540	−0.414770	+	0.909926i	$0.636138\pi$
$440$	3.07157	0.146431
$441$	0	0
$442$	−36.6635	−1.74390
$443$	−14.3666	−0.682579	−0.341290	−	0.939958i	$-0.610864\pi$
$443$	−14.3666	−0.682579	−0.341290	+	0.939958i	$0.610864\pi$
$444$	0	0
$445$	−13.7768	−0.653081
$446$	−24.0494	−1.13877
$447$	0	0
$448$	2.41300	0.114003
$449$	4.35069	0.205322	0.102661	−	0.994716i	$-0.467264\pi$
$449$	4.35069	0.205322	0.102661	+	0.994716i	$0.467264\pi$
$450$	0	0
$451$	30.7260	1.44683
$452$	−13.1555	−0.618781
$453$	0	0
$454$	−16.4469	−0.771890
$455$	−16.2723	−0.762857
$456$	0	0
$457$	−28.9116	−1.35243	−0.676214	−	0.736705i	$-0.736380\pi$
$457$	−28.9116	−1.35243	−0.676214	+	0.736705i	$0.736380\pi$
$458$	3.67491	0.171717
$459$	0	0
$460$	5.20493	0.242681
$461$	3.48477	0.162302	0.0811509	−	0.996702i	$-0.474140\pi$
$461$	3.48477	0.162302	0.0811509	+	0.996702i	$0.474140\pi$
$462$	0	0
$463$	−17.3353	−0.805640	−0.402820	−	0.915279i	$-0.631970\pi$
$463$	−17.3353	−0.805640	−0.402820	+	0.915279i	$0.631970\pi$
$464$	−5.18081	−0.240513
$465$	0	0
$466$	1.22524	0.0567580
$467$	−21.8269	−1.01003	−0.505013	−	0.863111i	$-0.668512\pi$
$467$	−21.8269	−1.01003	−0.505013	+	0.863111i	$0.668512\pi$
$468$	0	0
$469$	30.0001	1.38527
$470$	8.75167	0.403685
$471$	0	0
$472$	6.78939	0.312507
$473$	−32.0716	−1.47465
$474$	0	0
$475$	−9.71850	−0.445916
$476$	−14.0026	−0.641806
$477$	0	0
$478$	19.6354	0.898103
$479$	32.2298	1.47262	0.736308	−	0.676647i	$-0.236567\pi$
$479$	32.2298	1.47262	0.736308	+	0.676647i	$0.236567\pi$
$480$	0	0
$481$	−29.3561	−1.33852
$482$	−21.6153	−0.984551
$483$	0	0
$484$	−2.71860	−0.123573
$485$	−7.14661	−0.324511
$486$	0	0
$487$	36.3486	1.64711	0.823556	−	0.567235i	$-0.191987\pi$
$487$	36.3486	1.64711	0.823556	+	0.567235i	$0.191987\pi$
$488$	−11.8514	−0.536489
$489$	0	0
$490$	1.25674	0.0567739
$491$	6.27108	0.283010	0.141505	−	0.989938i	$-0.454806\pi$
$491$	6.27108	0.283010	0.141505	+	0.989938i	$0.454806\pi$
$492$	0	0
$493$	30.0641	1.35402
$494$	15.9041	0.715561
$495$	0	0
$496$	−7.91286	−0.355298
$497$	−9.36526	−0.420089
$498$	0	0
$499$	31.2490	1.39890	0.699449	−	0.714683i	$-0.253429\pi$
$499$	31.2490	1.39890	0.699449	+	0.714683i	$0.253429\pi$
$500$	9.45756	0.422955
$501$	0	0
$502$	−22.9313	−1.02347
$503$	−7.83456	−0.349326	−0.174663	−	0.984628i	$-0.555884\pi$
$503$	−7.83456	−0.349326	−0.174663	+	0.984628i	$0.555884\pi$
$504$	0	0
$505$	−9.69847	−0.431576
$506$	14.0333	0.623854
$507$	0	0
$508$	3.03628	0.134713
$509$	13.6569	0.605331	0.302665	−	0.953097i	$-0.402124\pi$
$509$	13.6569	0.605331	0.302665	+	0.953097i	$0.402124\pi$
$510$	0	0
$511$	−4.75418	−0.210312
$512$	1.00000	0.0441942
$513$	0	0
$514$	13.7438	0.606212
$515$	13.1777	0.580679
$516$	0	0
$517$	23.5958	1.03774
$518$	−11.2117	−0.492614
$519$	0	0
$520$	−6.74360	−0.295726
$521$	−21.5222	−0.942903	−0.471452	−	0.881892i	$-0.656270\pi$
$521$	−21.5222	−0.942903	−0.471452	+	0.881892i	$0.656270\pi$
$522$	0	0
$523$	17.3077	0.756815	0.378407	−	0.925639i	$-0.376472\pi$
$523$	17.3077	0.756815	0.378407	+	0.925639i	$0.376472\pi$
$524$	20.9274	0.914217
$525$	0	0
$526$	−3.02740	−0.132001
$527$	45.9181	2.00022
$528$	0	0
$529$	0.780061	0.0339157
$530$	7.22492	0.313831
$531$	0	0
$532$	6.07413	0.263347
$533$	−67.4587	−2.92196
$534$	0	0
$535$	−5.08617	−0.219894
$536$	12.4327	0.537011
$537$	0	0
$538$	30.7276	1.32476
$539$	3.38837	0.145947
$540$	0	0
$541$	34.3916	1.47861	0.739306	−	0.673370i	$-0.235154\pi$
$541$	34.3916	1.47861	0.739306	+	0.673370i	$0.235154\pi$
$542$	−12.7435	−0.547379
$543$	0	0
$544$	−5.80297	−0.248800
$545$	6.12829	0.262507
$546$	0	0
$547$	−1.48789	−0.0636177	−0.0318089	−	0.999494i	$-0.510127\pi$
$547$	−1.48789	−0.0636177	−0.0318089	+	0.999494i	$0.510127\pi$
$548$	5.40932	0.231075
$549$	0	0
$550$	11.1103	0.473743
$551$	−13.0414	−0.555583
$552$	0	0
$553$	5.46846	0.232543
$554$	10.0878	0.428590
$555$	0	0
$556$	−11.5243	−0.488738
$557$	−27.4929	−1.16491	−0.582457	−	0.812862i	$-0.697908\pi$
$557$	−27.4929	−1.16491	−0.582457	+	0.812862i	$0.697908\pi$
$558$	0	0
$559$	70.4128	2.97815
$560$	−2.57552	−0.108836
$561$	0	0
$562$	−20.1407	−0.849584
$563$	1.46719	0.0618345	0.0309173	−	0.999522i	$-0.490157\pi$
$563$	1.46719	0.0618345	0.0309173	+	0.999522i	$0.490157\pi$
$564$	0	0
$565$	14.0415	0.590732
$566$	−9.42116	−0.396001
$567$	0	0
$568$	−3.88117	−0.162850
$569$	34.4730	1.44518	0.722592	−	0.691274i	$-0.242950\pi$
$569$	34.4730	1.44518	0.722592	+	0.691274i	$0.242950\pi$
$570$	0	0
$571$	46.4755	1.94494	0.972470	−	0.233028i	$-0.0748633\pi$
$571$	46.4755	1.94494	0.972470	+	0.233028i	$0.0748633\pi$
$572$	−18.1817	−0.760216
$573$	0	0
$574$	−25.7639	−1.07536
$575$	18.8269	0.785136
$576$	0	0
$577$	11.1971	0.466140	0.233070	−	0.972460i	$-0.425123\pi$
$577$	11.1971	0.466140	0.233070	+	0.972460i	$0.425123\pi$
$578$	16.6745	0.693567
$579$	0	0
$580$	5.52976	0.229611
$581$	6.46826	0.268349
$582$	0	0
$583$	19.4795	0.806756
$584$	−1.97024	−0.0815289
$585$	0	0
$586$	9.52070	0.393296
$587$	−13.7039	−0.565622	−0.282811	−	0.959176i	$-0.591267\pi$
$587$	−13.7039	−0.565622	−0.282811	+	0.959176i	$0.591267\pi$
$588$	0	0
$589$	−19.9187	−0.820735
$590$	−7.24668	−0.298341
$591$	0	0
$592$	−4.64638	−0.190965
$593$	−45.7421	−1.87840	−0.939201	−	0.343367i	$-0.888432\pi$
$593$	−45.7421	−1.87840	−0.939201	+	0.343367i	$0.888432\pi$
$594$	0	0
$595$	14.9457	0.612713
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	−30.8099	−1.25991
$599$	−27.8465	−1.13778	−0.568889	−	0.822414i	$-0.692627\pi$
$599$	−27.8465	−1.13778	−0.568889	+	0.822414i	$0.692627\pi$
$600$	0	0
$601$	−7.82327	−0.319118	−0.159559	−	0.987188i	$-0.551007\pi$
$601$	−7.82327	−0.319118	−0.159559	+	0.987188i	$0.551007\pi$
$602$	26.8922	1.09604
$603$	0	0
$604$	−18.4343	−0.750080
$605$	2.90171	0.117971
$606$	0	0
$607$	−7.46791	−0.303113	−0.151557	−	0.988449i	$-0.548429\pi$
$607$	−7.46791	−0.303113	−0.151557	+	0.988449i	$0.548429\pi$
$608$	2.51725	0.102088
$609$	0	0
$610$	12.6497	0.512170
$611$	−51.8043	−2.09578
$612$	0	0
$613$	26.6402	1.07599	0.537993	−	0.842949i	$-0.319183\pi$
$613$	26.6402	1.07599	0.537993	+	0.842949i	$0.319183\pi$
$614$	4.01472	0.162021
$615$	0	0
$616$	−6.94399	−0.279781
$617$	16.0512	0.646198	0.323099	−	0.946365i	$-0.395275\pi$
$617$	16.0512	0.646198	0.323099	+	0.946365i	$0.395275\pi$
$618$	0	0
$619$	−42.0240	−1.68909	−0.844544	−	0.535487i	$-0.820128\pi$
$619$	−42.0240	−1.68909	−0.844544	+	0.535487i	$0.820128\pi$
$620$	8.44582	0.339192
$621$	0	0
$622$	−12.5289	−0.502364
$623$	31.1455	1.24782
$624$	0	0
$625$	9.20922	0.368369
$626$	−17.9628	−0.717939
$627$	0	0
$628$	−18.8174	−0.750896
$629$	26.9628	1.07508
$630$	0	0
$631$	−23.5755	−0.938524	−0.469262	−	0.883059i	$-0.655480\pi$
$631$	−23.5755	−0.938524	−0.469262	+	0.883059i	$0.655480\pi$
$632$	2.26625	0.0901467
$633$	0	0
$634$	0.635891	0.0252545
$635$	−3.24079	−0.128607
$636$	0	0
$637$	−7.43912	−0.294749
$638$	14.9090	0.590254
$639$	0	0
$640$	−1.06735	−0.0421909
$641$	11.1542	0.440564	0.220282	−	0.975436i	$-0.429302\pi$
$641$	11.1542	0.440564	0.220282	+	0.975436i	$0.429302\pi$
$642$	0	0
$643$	8.17816	0.322515	0.161258	−	0.986912i	$-0.448445\pi$
$643$	8.17816	0.322515	0.161258	+	0.986912i	$0.448445\pi$
$644$	−11.7669	−0.463682
$645$	0	0
$646$	−14.6076	−0.574726
$647$	2.23055	0.0876921	0.0438461	−	0.999038i	$-0.486039\pi$
$647$	2.23055	0.0876921	0.0438461	+	0.999038i	$0.486039\pi$
$648$	0	0
$649$	−19.5381	−0.766938
$650$	−24.3925	−0.956751
$651$	0	0
$652$	−18.6680	−0.731097
$653$	−13.8175	−0.540721	−0.270361	−	0.962759i	$-0.587143\pi$
$653$	−13.8175	−0.540721	−0.270361	+	0.962759i	$0.587143\pi$
$654$	0	0
$655$	−22.3369	−0.872775
$656$	−10.6771	−0.416872
$657$	0	0
$658$	−19.7852	−0.771306
$659$	17.9428	0.698951	0.349475	−	0.936946i	$-0.386360\pi$
$659$	17.9428	0.698951	0.349475	+	0.936946i	$0.386360\pi$
$660$	0	0
$661$	6.74759	0.262451	0.131225	−	0.991353i	$-0.458109\pi$
$661$	6.74759	0.262451	0.131225	+	0.991353i	$0.458109\pi$
$662$	−3.11325	−0.121000
$663$	0	0
$664$	2.68059	0.104027
$665$	−6.48324	−0.251409
$666$	0	0
$667$	25.2641	0.978230
$668$	10.7650	0.416509
$669$	0	0
$670$	−13.2701	−0.512668
$671$	34.1054	1.31662
$672$	0	0
$673$	29.2188	1.12630	0.563152	−	0.826353i	$-0.309589\pi$
$673$	29.2188	1.12630	0.563152	+	0.826353i	$0.309589\pi$
$674$	−33.9883	−1.30918
$675$	0	0
$676$	26.9178	1.03530
$677$	41.1604	1.58192	0.790961	−	0.611866i	$-0.209581\pi$
$677$	41.1604	1.58192	0.790961	+	0.611866i	$0.209581\pi$
$678$	0	0
$679$	16.1566	0.620032
$680$	6.19382	0.237522
$681$	0	0
$682$	22.7712	0.871953
$683$	−25.1460	−0.962184	−0.481092	−	0.876670i	$-0.659760\pi$
$683$	−25.1460	−0.962184	−0.481092	+	0.876670i	$0.659760\pi$
$684$	0	0
$685$	−5.77366	−0.220600
$686$	−19.7321	−0.753377
$687$	0	0
$688$	11.1447	0.424888
$689$	−42.7669	−1.62929
$690$	0	0
$691$	9.33945	0.355290	0.177645	−	0.984095i	$-0.443152\pi$
$691$	9.33945	0.355290	0.177645	+	0.984095i	$0.443152\pi$
$692$	14.3128	0.544092
$693$	0	0
$694$	25.5691	0.970588
$695$	12.3005	0.466583
$696$	0	0
$697$	61.9591	2.34687
$698$	−6.64078	−0.251357
$699$	0	0
$700$	−9.31600	−0.352112
$701$	−20.7946	−0.785400	−0.392700	−	0.919667i	$-0.628459\pi$
$701$	−20.7946	−0.785400	−0.392700	+	0.919667i	$0.628459\pi$
$702$	0	0
$703$	−11.6961	−0.441127
$704$	−2.87774	−0.108459
$705$	0	0
$706$	4.63793	0.174551
$707$	21.9256	0.824598
$708$	0	0
$709$	−34.3030	−1.28828	−0.644139	−	0.764909i	$-0.722784\pi$
$709$	−34.3030	−1.28828	−0.644139	+	0.764909i	$0.722784\pi$
$710$	4.14258	0.155468
$711$	0	0
$712$	12.9074	0.483725
$713$	38.5869	1.44509
$714$	0	0
$715$	19.4063	0.725756
$716$	14.8159	0.553696
$717$	0	0
$718$	14.7744	0.551374
$719$	−42.1754	−1.57288	−0.786438	−	0.617670i	$-0.788077\pi$
$719$	−42.1754	−1.57288	−0.786438	+	0.617670i	$0.788077\pi$
$720$	0	0
$721$	−29.7912	−1.10948
$722$	−12.6634	−0.471284
$723$	0	0
$724$	12.4854	0.464015
$725$	20.0018	0.742850
$726$	0	0
$727$	−30.1726	−1.11904	−0.559520	−	0.828817i	$-0.689015\pi$
$727$	−30.1726	−1.11904	−0.559520	+	0.828817i	$0.689015\pi$
$728$	15.2455	0.565034
$729$	0	0
$730$	2.10294	0.0778332
$731$	−64.6724	−2.39200
$732$	0	0
$733$	21.2752	0.785816	0.392908	−	0.919578i	$-0.371469\pi$
$733$	21.2752	0.785816	0.392908	+	0.919578i	$0.371469\pi$
$734$	6.92697	0.255679
$735$	0	0
$736$	−4.87648	−0.179749
$737$	−35.7781	−1.31790
$738$	0	0
$739$	−27.0373	−0.994583	−0.497292	−	0.867583i	$-0.665672\pi$
$739$	−27.0373	−0.994583	−0.497292	+	0.867583i	$0.665672\pi$
$740$	4.95933	0.182309
$741$	0	0
$742$	−16.3336	−0.599625
$743$	−14.0456	−0.515281	−0.257641	−	0.966241i	$-0.582945\pi$
$743$	−14.0456	−0.515281	−0.257641	+	0.966241i	$0.582945\pi$
$744$	0	0
$745$	1.06735	0.0391048
$746$	7.15412	0.261931
$747$	0	0
$748$	16.6995	0.610593
$749$	11.4984	0.420144
$750$	0	0
$751$	35.1905	1.28412	0.642060	−	0.766655i	$-0.278080\pi$
$751$	35.1905	1.28412	0.642060	+	0.766655i	$0.278080\pi$
$752$	−8.19941	−0.299002
$753$	0	0
$754$	−32.7326	−1.19205
$755$	19.6759	0.716079
$756$	0	0
$757$	12.0391	0.437567	0.218783	−	0.975773i	$-0.429791\pi$
$757$	12.0391	0.437567	0.218783	+	0.975773i	$0.429791\pi$
$758$	25.6854	0.932935
$759$	0	0
$760$	−2.68680	−0.0974605
$761$	45.8827	1.66325	0.831623	−	0.555341i	$-0.187412\pi$
$761$	45.8827	1.66325	0.831623	+	0.555341i	$0.187412\pi$
$762$	0	0
$763$	−13.8544	−0.501563
$764$	−0.208714	−0.00755100
$765$	0	0
$766$	23.5765	0.851855
$767$	42.8957	1.54887
$768$	0	0
$769$	−5.55195	−0.200208	−0.100104	−	0.994977i	$-0.531918\pi$
$769$	−5.55195	−0.200208	−0.100104	+	0.994977i	$0.531918\pi$
$770$	7.41169	0.267099
$771$	0	0
$772$	20.4582	0.736307
$773$	−10.5156	−0.378219	−0.189110	−	0.981956i	$-0.560560\pi$
$773$	−10.5156	−0.378219	−0.189110	+	0.981956i	$0.560560\pi$
$774$	0	0
$775$	30.5496	1.09738
$776$	6.69563	0.240359
$777$	0	0
$778$	−16.7592	−0.600846
$779$	−26.8770	−0.962970
$780$	0	0
$781$	11.1690	0.399659
$782$	28.2981	1.01194
$783$	0	0
$784$	−1.17744	−0.0420514
$785$	20.0848	0.716858
$786$	0	0
$787$	−39.3725	−1.40348	−0.701739	−	0.712434i	$-0.747593\pi$
$787$	−39.3725	−1.40348	−0.701739	+	0.712434i	$0.747593\pi$
$788$	−24.9769	−0.889765
$789$	0	0
$790$	−2.41889	−0.0860604
$791$	−31.7441	−1.12869
$792$	0	0
$793$	−74.8780	−2.65900
$794$	−5.12235	−0.181785
$795$	0	0
$796$	−18.3928	−0.651916
$797$	−19.0023	−0.673097	−0.336549	−	0.941666i	$-0.609260\pi$
$797$	−19.0023	−0.673097	−0.336549	+	0.941666i	$0.609260\pi$
$798$	0	0
$799$	47.5810	1.68329
$800$	−3.86076	−0.136498
$801$	0	0
$802$	19.0659	0.673241
$803$	5.66983	0.200084
$804$	0	0
$805$	12.5595	0.442664
$806$	−49.9939	−1.76096
$807$	0	0
$808$	9.08647	0.319661
$809$	−8.93922	−0.314286	−0.157143	−	0.987576i	$-0.550228\pi$
$809$	−8.93922	−0.314286	−0.157143	+	0.987576i	$0.550228\pi$
$810$	0	0
$811$	5.23231	0.183731	0.0918657	−	0.995771i	$-0.470717\pi$
$811$	5.23231	0.183731	0.0918657	+	0.995771i	$0.470717\pi$
$812$	−12.5013	−0.438709
$813$	0	0
$814$	13.3711	0.468656
$815$	19.9254	0.697957
$816$	0	0
$817$	28.0541	0.981487
$818$	15.0451	0.526040
$819$	0	0
$820$	11.3963	0.397975
$821$	−38.6144	−1.34765	−0.673826	−	0.738890i	$-0.735350\pi$
$821$	−38.6144	−1.34765	−0.673826	+	0.738890i	$0.735350\pi$
$822$	0	0
$823$	−54.1702	−1.88825	−0.944127	−	0.329583i	$-0.893092\pi$
$823$	−54.1702	−1.88825	−0.944127	+	0.329583i	$0.893092\pi$
$824$	−12.3461	−0.430098
$825$	0	0
$826$	16.3828	0.570030
$827$	29.8399	1.03764	0.518818	−	0.854885i	$-0.326372\pi$
$827$	29.8399	1.03764	0.518818	+	0.854885i	$0.326372\pi$
$828$	0	0
$829$	29.4296	1.02213	0.511067	−	0.859541i	$-0.329251\pi$
$829$	29.4296	1.02213	0.511067	+	0.859541i	$0.329251\pi$
$830$	−2.86114	−0.0993116
$831$	0	0
$832$	6.31805	0.219039
$833$	6.83265	0.236737
$834$	0	0
$835$	−11.4900	−0.397628
$836$	−7.24401	−0.250539
$837$	0	0
$838$	12.6427	0.436735
$839$	−6.91988	−0.238901	−0.119450	−	0.992840i	$-0.538113\pi$
$839$	−6.91988	−0.238901	−0.119450	+	0.992840i	$0.538113\pi$
$840$	0	0
$841$	−2.15922	−0.0744558
$842$	−8.70185	−0.299885
$843$	0	0
$844$	−5.11885	−0.176198
$845$	−28.7308	−0.988370
$846$	0	0
$847$	−6.55997	−0.225403
$848$	−6.76900	−0.232449
$849$	0	0
$850$	22.4039	0.768446
$851$	22.6580	0.776705
$852$	0	0
$853$	−30.8776	−1.05723	−0.528614	−	0.848862i	$-0.677288\pi$
$853$	−30.8776	−1.05723	−0.528614	+	0.848862i	$0.677288\pi$
$854$	−28.5975	−0.978586
$855$	0	0
$856$	4.76521	0.162872
$857$	−24.3599	−0.832119	−0.416059	−	0.909337i	$-0.636589\pi$
$857$	−24.3599	−0.832119	−0.416059	+	0.909337i	$0.636589\pi$
$858$	0	0
$859$	46.2628	1.57847	0.789233	−	0.614093i	$-0.210478\pi$
$859$	46.2628	1.57847	0.789233	+	0.614093i	$0.210478\pi$
$860$	−11.8953	−0.405628
$861$	0	0
$862$	−13.9905	−0.476517
$863$	−51.5003	−1.75309	−0.876546	−	0.481318i	$-0.840158\pi$
$863$	−51.5003	−1.75309	−0.876546	+	0.481318i	$0.840158\pi$
$864$	0	0
$865$	−15.2769	−0.519429
$866$	23.4355	0.796371
$867$	0	0
$868$	−19.0937	−0.648083
$869$	−6.52169	−0.221233
$870$	0	0
$871$	78.5504	2.66158
$872$	−5.74157	−0.194434
$873$	0	0
$874$	−12.2753	−0.415220
$875$	22.8211	0.771493
$876$	0	0
$877$	−6.71660	−0.226803	−0.113402	−	0.993549i	$-0.536175\pi$
$877$	−6.71660	−0.226803	−0.113402	+	0.993549i	$0.536175\pi$
$878$	−17.3808	−0.586574
$879$	0	0
$880$	3.07157	0.103543
$881$	−15.0720	−0.507789	−0.253894	−	0.967232i	$-0.581712\pi$
$881$	−15.0720	−0.507789	−0.253894	+	0.967232i	$0.581712\pi$
$882$	0	0
$883$	−1.56028	−0.0525077	−0.0262539	−	0.999655i	$-0.508358\pi$
$883$	−1.56028	−0.0525077	−0.0262539	+	0.999655i	$0.508358\pi$
$884$	−36.6635	−1.23313
$885$	0	0
$886$	−14.3666	−0.482656
$887$	3.52972	0.118517	0.0592583	−	0.998243i	$-0.481126\pi$
$887$	3.52972	0.118517	0.0592583	+	0.998243i	$0.481126\pi$
$888$	0	0
$889$	7.32654	0.245724
$890$	−13.7768	−0.461798
$891$	0	0
$892$	−24.0494	−0.805233
$893$	−20.6400	−0.690692
$894$	0	0
$895$	−15.8138	−0.528597
$896$	2.41300	0.0806126
$897$	0	0
$898$	4.35069	0.145184
$899$	40.9950	1.36726
$900$	0	0
$901$	39.2803	1.30862
$902$	30.7260	1.02306
$903$	0	0
$904$	−13.1555	−0.437544
$905$	−13.3263	−0.442981
$906$	0	0
$907$	0.131286	0.00435928	0.00217964	−	0.999998i	$-0.499306\pi$
$907$	0.131286	0.00435928	0.00217964	+	0.999998i	$0.499306\pi$
$908$	−16.4469	−0.545809
$909$	0	0
$910$	−16.2723	−0.539421
$911$	0.346696	0.0114866	0.00574328	−	0.999984i	$-0.498172\pi$
$911$	0.346696	0.0114866	0.00574328	+	0.999984i	$0.498172\pi$
$912$	0	0
$913$	−7.71405	−0.255298
$914$	−28.9116	−0.956311
$915$	0	0
$916$	3.67491	0.121423
$917$	50.4977	1.66758
$918$	0	0
$919$	52.9227	1.74576	0.872880	−	0.487935i	$-0.162250\pi$
$919$	52.9227	1.74576	0.872880	+	0.487935i	$0.162250\pi$
$920$	5.20493	0.171601
$921$	0	0
$922$	3.48477	0.114765
$923$	−24.5215	−0.807133
$924$	0	0
$925$	17.9385	0.589815
$926$	−17.3353	−0.569673
$927$	0	0
$928$	−5.18081	−0.170068
$929$	29.1343	0.955864	0.477932	−	0.878397i	$-0.341387\pi$
$929$	29.1343	0.955864	0.477932	+	0.878397i	$0.341387\pi$
$930$	0	0
$931$	−2.96391	−0.0971383
$932$	1.22524	0.0401340
$933$	0	0
$934$	−21.8269	−0.714197
$935$	−17.8242	−0.582915
$936$	0	0
$937$	55.7638	1.82172	0.910862	−	0.412711i	$-0.135418\pi$
$937$	55.7638	1.82172	0.910862	+	0.412711i	$0.135418\pi$
$938$	30.0001	0.979537
$939$	0	0
$940$	8.75167	0.285448
$941$	43.0006	1.40178	0.700889	−	0.713270i	$-0.252787\pi$
$941$	43.0006	1.40178	0.700889	+	0.713270i	$0.252787\pi$
$942$	0	0
$943$	52.0668	1.69553
$944$	6.78939	0.220976
$945$	0	0
$946$	−32.0716	−1.04274
$947$	7.81340	0.253901	0.126951	−	0.991909i	$-0.459481\pi$
$947$	7.81340	0.253901	0.126951	+	0.991909i	$0.459481\pi$
$948$	0	0
$949$	−12.4481	−0.404081
$950$	−9.71850	−0.315310
$951$	0	0
$952$	−14.0026	−0.453826
$953$	17.0448	0.552135	0.276067	−	0.961138i	$-0.410969\pi$
$953$	17.0448	0.552135	0.276067	+	0.961138i	$0.410969\pi$
$954$	0	0
$955$	0.222771	0.00720871
$956$	19.6354	0.635055
$957$	0	0
$958$	32.2298	1.04130
$959$	13.0527	0.421493
$960$	0	0
$961$	31.6134	1.01979
$962$	−29.3561	−0.946477
$963$	0	0
$964$	−21.6153	−0.696183
$965$	−21.8361	−0.702930
$966$	0	0
$967$	30.5292	0.981753	0.490876	−	0.871229i	$-0.336677\pi$
$967$	30.5292	0.981753	0.490876	+	0.871229i	$0.336677\pi$
$968$	−2.71860	−0.0873791
$969$	0	0
$970$	−7.14661	−0.229464
$971$	41.4714	1.33088	0.665440	−	0.746451i	$-0.268244\pi$
$971$	41.4714	1.33088	0.665440	+	0.746451i	$0.268244\pi$
$972$	0	0
$973$	−27.8080	−0.891485
$974$	36.3486	1.16468
$975$	0	0
$976$	−11.8514	−0.379355
$977$	−56.3272	−1.80207	−0.901033	−	0.433750i	$-0.857190\pi$
$977$	−56.3272	−1.80207	−0.901033	+	0.433750i	$0.857190\pi$
$978$	0	0
$979$	−37.1442	−1.18713
$980$	1.25674	0.0401452
$981$	0	0
$982$	6.27108	0.200118
$983$	−43.9298	−1.40114	−0.700571	−	0.713583i	$-0.747071\pi$
$983$	−43.9298	−1.40114	−0.700571	+	0.713583i	$0.747071\pi$
$984$	0	0
$985$	26.6592	0.849432
$986$	30.0641	0.957436
$987$	0	0
$988$	15.9041	0.505978
$989$	−54.3469	−1.72813
$990$	0	0
$991$	−39.1494	−1.24362	−0.621810	−	0.783168i	$-0.713602\pi$
$991$	−39.1494	−1.24362	−0.621810	+	0.783168i	$0.713602\pi$
$992$	−7.91286	−0.251234
$993$	0	0
$994$	−9.36526	−0.297048
$995$	19.6316	0.622365
$996$	0	0
$997$	−30.2386	−0.957666	−0.478833	−	0.877906i	$-0.658940\pi$
$997$	−30.2386	−0.957666	−0.478833	+	0.877906i	$0.658940\pi$
$998$	31.2490	0.989170
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.m.1.6	yes	12
3.2	odd	2		8046.2.a.l.1.7	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.l.1.7	✓	12	3.2	odd	2
8046.2.a.m.1.6	yes	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.06735 q^{5} +2.41300 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.06735 q^{5} +2.41300 q^{7} +1.00000 q^{8} -1.06735 q^{10} -2.87774 q^{11} +6.31805 q^{13} +2.41300 q^{14} +1.00000 q^{16} -5.80297 q^{17} +2.51725 q^{19} -1.06735 q^{20} -2.87774 q^{22} -4.87648 q^{23} -3.86076 q^{25} +6.31805 q^{26} +2.41300 q^{28} -5.18081 q^{29} -7.91286 q^{31} +1.00000 q^{32} -5.80297 q^{34} -2.57552 q^{35} -4.64638 q^{37} +2.51725 q^{38} -1.06735 q^{40} -10.6771 q^{41} +11.1447 q^{43} -2.87774 q^{44} -4.87648 q^{46} -8.19941 q^{47} -1.17744 q^{49} -3.86076 q^{50} +6.31805 q^{52} -6.76900 q^{53} +3.07157 q^{55} +2.41300 q^{56} -5.18081 q^{58} +6.78939 q^{59} -11.8514 q^{61} -7.91286 q^{62} +1.00000 q^{64} -6.74360 q^{65} +12.4327 q^{67} -5.80297 q^{68} -2.57552 q^{70} -3.88117 q^{71} -1.97024 q^{73} -4.64638 q^{74} +2.51725 q^{76} -6.94399 q^{77} +2.26625 q^{79} -1.06735 q^{80} -10.6771 q^{82} +2.68059 q^{83} +6.19382 q^{85} +11.1447 q^{86} -2.87774 q^{88} +12.9074 q^{89} +15.2455 q^{91} -4.87648 q^{92} -8.19941 q^{94} -2.68680 q^{95} +6.69563 q^{97} -1.17744 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} - 5 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 - 5 * q^5 - 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} - 5 q^{5} - 6 q^{7} + 12 q^{8} - 5 q^{10} - 10 q^{11} - q^{13} - 6 q^{14} + 12 q^{16} - 6 q^{17} - 10 q^{19} - 5 q^{20} - 10 q^{22} - 15 q^{23} + 7 q^{25} - q^{26} - 6 q^{28} - 33 q^{29} - 6 q^{31} + 12 q^{32} - 6 q^{34} - 16 q^{35} - 13 q^{37} - 10 q^{38} - 5 q^{40} - 20 q^{41} - 11 q^{43} - 10 q^{44} - 15 q^{46} - 15 q^{47} + 2 q^{49} + 7 q^{50} - q^{52} - 4 q^{53} - 17 q^{55} - 6 q^{56} - 33 q^{58} - 10 q^{59} - 12 q^{61} - 6 q^{62} + 12 q^{64} - 40 q^{65} - 19 q^{67} - 6 q^{68} - 16 q^{70} - 47 q^{71} - 2 q^{73} - 13 q^{74} - 10 q^{76} + 6 q^{77} - 15 q^{79} - 5 q^{80} - 20 q^{82} - 18 q^{83} - 25 q^{85} - 11 q^{86} - 10 q^{88} - 24 q^{89} - 3 q^{91} - 15 q^{92} - 15 q^{94} + 3 q^{95} - 25 q^{97} + 2 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 - 5 * q^5 - 6 * q^7 + 12 * q^8 - 5 * q^10 - 10 * q^11 - q^13 - 6 * q^14 + 12 * q^16 - 6 * q^17 - 10 * q^19 - 5 * q^20 - 10 * q^22 - 15 * q^23 + 7 * q^25 - q^26 - 6 * q^28 - 33 * q^29 - 6 * q^31 + 12 * q^32 - 6 * q^34 - 16 * q^35 - 13 * q^37 - 10 * q^38 - 5 * q^40 - 20 * q^41 - 11 * q^43 - 10 * q^44 - 15 * q^46 - 15 * q^47 + 2 * q^49 + 7 * q^50 - q^52 - 4 * q^53 - 17 * q^55 - 6 * q^56 - 33 * q^58 - 10 * q^59 - 12 * q^61 - 6 * q^62 + 12 * q^64 - 40 * q^65 - 19 * q^67 - 6 * q^68 - 16 * q^70 - 47 * q^71 - 2 * q^73 - 13 * q^74 - 10 * q^76 + 6 * q^77 - 15 * q^79 - 5 * q^80 - 20 * q^82 - 18 * q^83 - 25 * q^85 - 11 * q^86 - 10 * q^88 - 24 * q^89 - 3 * q^91 - 15 * q^92 - 15 * q^94 + 3 * q^95 - 25 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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