LMFDB - Embedded newform 4012.2.a.h.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 16 x^{13} + 123 x^{12} - 76 x^{11} - 630 x^{10} + 937 x^{9} + 1083 x^{8} - 2602 x^{7} + \cdots - 2 $ x^15 - 5x^14 - 16x^13 + 123x^12 - 76x^11 - 630x^10 + 937x^9 + 1083x^8 - 2602x^7 - 249x^6 + 2736x^5 - 801x^4 - 900x^3 + 429x^2 - 36x - 2
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			1.8
Root			$-0.0375146$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.249278 q^{3} -1.04042 q^{5} -3.44893 q^{7} -2.93786 q^{9} +O(q^{10})$	$q+0.249278 q^{3} -1.04042 q^{5} -3.44893 q^{7} -2.93786 q^{9} +2.84117 q^{11} +5.81393 q^{13} -0.259354 q^{15} +1.00000 q^{17} -2.54718 q^{19} -0.859743 q^{21} +5.98742 q^{23} -3.91752 q^{25} -1.48018 q^{27} +8.08678 q^{29} -8.87396 q^{31} +0.708241 q^{33} +3.58835 q^{35} -10.3119 q^{37} +1.44928 q^{39} +6.62343 q^{41} +1.69804 q^{43} +3.05661 q^{45} -8.91580 q^{47} +4.89514 q^{49} +0.249278 q^{51} +11.2452 q^{53} -2.95601 q^{55} -0.634955 q^{57} -1.00000 q^{59} -14.2013 q^{61} +10.1325 q^{63} -6.04894 q^{65} +3.56596 q^{67} +1.49253 q^{69} -0.493696 q^{71} +2.55701 q^{73} -0.976552 q^{75} -9.79900 q^{77} +7.81372 q^{79} +8.44461 q^{81} -6.96035 q^{83} -1.04042 q^{85} +2.01586 q^{87} -4.89646 q^{89} -20.0518 q^{91} -2.21208 q^{93} +2.65014 q^{95} -7.86250 q^{97} -8.34695 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) $ 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9	$ 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) $ 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9 - 12 * q^11 - 10 * q^13 - 13 * q^15 + 15 * q^17 - 4 * q^21 - 21 * q^23 + 4 * q^25 - 37 * q^27 + 23 * q^29 - 31 * q^31 - 11 * q^33 - 35 * q^35 - 10 * q^37 - 2 * q^39 - 15 * q^41 - 3 * q^43 + 15 * q^45 - 47 * q^47 + 18 * q^49 - q^51 - 7 * q^53 - 20 * q^55 - 30 * q^57 - 15 * q^59 + q^61 + 19 * q^63 + 11 * q^65 - 20 * q^67 - 24 * q^69 - 13 * q^71 - 24 * q^73 - 10 * q^75 + 21 * q^77 - 34 * q^79 - 21 * q^81 - 40 * q^83 + q^85 - 60 * q^87 - 46 * q^89 - 19 * q^91 + 33 * q^93 - 4 * q^95 - 5 * q^97 - 74 * 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
$3$	0.249278	0.143921	0.0719604	−	0.997407i	$-0.477074\pi$
$3$	0.249278	0.143921	0.0719604	+	0.997407i	$0.477074\pi$
$4$	0	0
$5$	−1.04042	−0.465291	−0.232645	−	0.972562i	$-0.574738\pi$
$5$	−1.04042	−0.465291	−0.232645	+	0.972562i	$0.574738\pi$
$6$	0	0
$7$	−3.44893	−1.30357	−0.651787	−	0.758402i	$-0.725980\pi$
$7$	−3.44893	−1.30357	−0.651787	+	0.758402i	$0.725980\pi$
$8$	0	0
$9$	−2.93786	−0.979287
$10$	0	0
$11$	2.84117	0.856644	0.428322	−	0.903626i	$-0.359105\pi$
$11$	2.84117	0.856644	0.428322	+	0.903626i	$0.359105\pi$
$12$	0	0
$13$	5.81393	1.61249	0.806246	−	0.591580i	$-0.201496\pi$
$13$	5.81393	1.61249	0.806246	+	0.591580i	$0.201496\pi$
$14$	0	0
$15$	−0.259354	−0.0669650
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.54718	−0.584362	−0.292181	−	0.956363i	$-0.594381\pi$
$19$	−2.54718	−0.584362	−0.292181	+	0.956363i	$0.594381\pi$
$20$	0	0
$21$	−0.859743	−0.187611
$22$	0	0
$23$	5.98742	1.24846	0.624232	−	0.781239i	$-0.285412\pi$
$23$	5.98742	1.24846	0.624232	+	0.781239i	$0.285412\pi$
$24$	0	0
$25$	−3.91752	−0.783504
$26$	0	0
$27$	−1.48018	−0.284860
$28$	0	0
$29$	8.08678	1.50168	0.750838	−	0.660486i	$-0.229650\pi$
$29$	8.08678	1.50168	0.750838	+	0.660486i	$0.229650\pi$
$30$	0	0
$31$	−8.87396	−1.59381	−0.796906	−	0.604104i	$-0.793531\pi$
$31$	−8.87396	−1.59381	−0.796906	+	0.604104i	$0.793531\pi$
$32$	0	0
$33$	0.708241	0.123289
$34$	0	0
$35$	3.58835	0.606541
$36$	0	0
$37$	−10.3119	−1.69526	−0.847632	−	0.530585i	$-0.821972\pi$
$37$	−10.3119	−1.69526	−0.847632	+	0.530585i	$0.821972\pi$
$38$	0	0
$39$	1.44928	0.232071
$40$	0	0
$41$	6.62343	1.03441	0.517203	−	0.855863i	$-0.326973\pi$
$41$	6.62343	1.03441	0.517203	+	0.855863i	$0.326973\pi$
$42$	0	0
$43$	1.69804	0.258949	0.129474	−	0.991583i	$-0.458671\pi$
$43$	1.69804	0.258949	0.129474	+	0.991583i	$0.458671\pi$
$44$	0	0
$45$	3.05661	0.455653
$46$	0	0
$47$	−8.91580	−1.30050	−0.650251	−	0.759719i	$-0.725336\pi$
$47$	−8.91580	−1.30050	−0.650251	+	0.759719i	$0.725336\pi$
$48$	0	0
$49$	4.89514	0.699306
$50$	0	0
$51$	0.249278	0.0349059
$52$	0	0
$53$	11.2452	1.54465	0.772325	−	0.635227i	$-0.219094\pi$
$53$	11.2452	1.54465	0.772325	+	0.635227i	$0.219094\pi$
$54$	0	0
$55$	−2.95601	−0.398589
$56$	0	0
$57$	−0.634955	−0.0841018
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−14.2013	−1.81829	−0.909146	−	0.416478i	$-0.863264\pi$
$61$	−14.2013	−1.81829	−0.909146	+	0.416478i	$0.863264\pi$
$62$	0	0
$63$	10.1325	1.27657
$64$	0	0
$65$	−6.04894	−0.750278
$66$	0	0
$67$	3.56596	0.435652	0.217826	−	0.975988i	$-0.430104\pi$
$67$	3.56596	0.435652	0.217826	+	0.975988i	$0.430104\pi$
$68$	0	0
$69$	1.49253	0.179680
$70$	0	0
$71$	−0.493696	−0.0585910	−0.0292955	−	0.999571i	$-0.509326\pi$
$71$	−0.493696	−0.0585910	−0.0292955	+	0.999571i	$0.509326\pi$
$72$	0	0
$73$	2.55701	0.299275	0.149637	−	0.988741i	$-0.452189\pi$
$73$	2.55701	0.299275	0.149637	+	0.988741i	$0.452189\pi$
$74$	0	0
$75$	−0.976552	−0.112763
$76$	0	0
$77$	−9.79900	−1.11670
$78$	0	0
$79$	7.81372	0.879112	0.439556	−	0.898215i	$-0.355136\pi$
$79$	7.81372	0.879112	0.439556	+	0.898215i	$0.355136\pi$
$80$	0	0
$81$	8.44461	0.938290
$82$	0	0
$83$	−6.96035	−0.763998	−0.381999	−	0.924163i	$-0.624764\pi$
$83$	−6.96035	−0.763998	−0.381999	+	0.924163i	$0.624764\pi$
$84$	0	0
$85$	−1.04042	−0.112850
$86$	0	0
$87$	2.01586	0.216122
$88$	0	0
$89$	−4.89646	−0.519024	−0.259512	−	0.965740i	$-0.583562\pi$
$89$	−4.89646	−0.519024	−0.259512	+	0.965740i	$0.583562\pi$
$90$	0	0
$91$	−20.0518	−2.10200
$92$	0	0
$93$	−2.21208	−0.229382
$94$	0	0
$95$	2.65014	0.271898
$96$	0	0
$97$	−7.86250	−0.798316	−0.399158	−	0.916882i	$-0.630697\pi$
$97$	−7.86250	−0.798316	−0.399158	+	0.916882i	$0.630697\pi$
$98$	0	0
$99$	−8.34695	−0.838900
$100$	0	0
$101$	−1.42383	−0.141677	−0.0708383	−	0.997488i	$-0.522567\pi$
$101$	−1.42383	−0.141677	−0.0708383	+	0.997488i	$0.522567\pi$
$102$	0	0
$103$	6.45046	0.635583	0.317791	−	0.948161i	$-0.397059\pi$
$103$	6.45046	0.635583	0.317791	+	0.948161i	$0.397059\pi$
$104$	0	0
$105$	0.894496	0.0872938
$106$	0	0
$107$	−2.17414	−0.210182	−0.105091	−	0.994463i	$-0.533513\pi$
$107$	−2.17414	−0.210182	−0.105091	+	0.994463i	$0.533513\pi$
$108$	0	0
$109$	−8.08201	−0.774116	−0.387058	−	0.922055i	$-0.626509\pi$
$109$	−8.08201	−0.774116	−0.387058	+	0.922055i	$0.626509\pi$
$110$	0	0
$111$	−2.57053	−0.243984
$112$	0	0
$113$	2.67420	0.251568	0.125784	−	0.992058i	$-0.459855\pi$
$113$	2.67420	0.251568	0.125784	+	0.992058i	$0.459855\pi$
$114$	0	0
$115$	−6.22944	−0.580899
$116$	0	0
$117$	−17.0805	−1.57909
$118$	0	0
$119$	−3.44893	−0.316163
$120$	0	0
$121$	−2.92777	−0.266161
$122$	0	0
$123$	1.65108	0.148873
$124$	0	0
$125$	9.27798	0.829848
$126$	0	0
$127$	−12.5848	−1.11672	−0.558360	−	0.829599i	$-0.688569\pi$
$127$	−12.5848	−1.11672	−0.558360	+	0.829599i	$0.688569\pi$
$128$	0	0
$129$	0.423284	0.0372681
$130$	0	0
$131$	−20.8991	−1.82596	−0.912982	−	0.408000i	$-0.866226\pi$
$131$	−20.8991	−1.82596	−0.912982	+	0.408000i	$0.866226\pi$
$132$	0	0
$133$	8.78504	0.761759
$134$	0	0
$135$	1.54001	0.132543
$136$	0	0
$137$	−4.02768	−0.344108	−0.172054	−	0.985087i	$-0.555040\pi$
$137$	−4.02768	−0.344108	−0.172054	+	0.985087i	$0.555040\pi$
$138$	0	0
$139$	−17.0028	−1.44216	−0.721081	−	0.692851i	$-0.756354\pi$
$139$	−17.0028	−1.44216	−0.721081	+	0.692851i	$0.756354\pi$
$140$	0	0
$141$	−2.22251	−0.187169
$142$	0	0
$143$	16.5183	1.38133
$144$	0	0
$145$	−8.41366	−0.698716
$146$	0	0
$147$	1.22025	0.100645
$148$	0	0
$149$	−16.8556	−1.38086	−0.690432	−	0.723397i	$-0.742579\pi$
$149$	−16.8556	−1.38086	−0.690432	+	0.723397i	$0.742579\pi$
$150$	0	0
$151$	3.25175	0.264624	0.132312	−	0.991208i	$-0.457760\pi$
$151$	3.25175	0.264624	0.132312	+	0.991208i	$0.457760\pi$
$152$	0	0
$153$	−2.93786	−0.237512
$154$	0	0
$155$	9.23267	0.741586
$156$	0	0
$157$	−21.5577	−1.72049	−0.860247	−	0.509877i	$-0.829691\pi$
$157$	−21.5577	−1.72049	−0.860247	+	0.509877i	$0.829691\pi$
$158$	0	0
$159$	2.80319	0.222307
$160$	0	0
$161$	−20.6502	−1.62747
$162$	0	0
$163$	−21.5989	−1.69176	−0.845879	−	0.533375i	$-0.820924\pi$
$163$	−21.5989	−1.69176	−0.845879	+	0.533375i	$0.820924\pi$
$164$	0	0
$165$	−0.736869	−0.0573652
$166$	0	0
$167$	−16.5695	−1.28219	−0.641093	−	0.767464i	$-0.721519\pi$
$167$	−16.5695	−1.28219	−0.641093	+	0.767464i	$0.721519\pi$
$168$	0	0
$169$	20.8017	1.60013
$170$	0	0
$171$	7.48325	0.572258
$172$	0	0
$173$	−6.10980	−0.464519	−0.232260	−	0.972654i	$-0.574612\pi$
$173$	−6.10980	−0.464519	−0.232260	+	0.972654i	$0.574612\pi$
$174$	0	0
$175$	13.5113	1.02136
$176$	0	0
$177$	−0.249278	−0.0187369
$178$	0	0
$179$	−9.99417	−0.746999	−0.373500	−	0.927630i	$-0.621842\pi$
$179$	−9.99417	−0.746999	−0.373500	+	0.927630i	$0.621842\pi$
$180$	0	0
$181$	−7.13659	−0.530459	−0.265229	−	0.964185i	$-0.585448\pi$
$181$	−7.13659	−0.530459	−0.265229	+	0.964185i	$0.585448\pi$
$182$	0	0
$183$	−3.54007	−0.261690
$184$	0	0
$185$	10.7287	0.788790
$186$	0	0
$187$	2.84117	0.207767
$188$	0	0
$189$	5.10504	0.371337
$190$	0	0
$191$	4.15050	0.300320	0.150160	−	0.988662i	$-0.452021\pi$
$191$	4.15050	0.300320	0.150160	+	0.988662i	$0.452021\pi$
$192$	0	0
$193$	9.18306	0.661011	0.330505	−	0.943804i	$-0.392781\pi$
$193$	9.18306	0.661011	0.330505	+	0.943804i	$0.392781\pi$
$194$	0	0
$195$	−1.50787	−0.107981
$196$	0	0
$197$	14.3502	1.02241	0.511203	−	0.859460i	$-0.329200\pi$
$197$	14.3502	1.02241	0.511203	+	0.859460i	$0.329200\pi$
$198$	0	0
$199$	−2.72892	−0.193448	−0.0967241	−	0.995311i	$-0.530836\pi$
$199$	−2.72892	−0.193448	−0.0967241	+	0.995311i	$0.530836\pi$
$200$	0	0
$201$	0.888916	0.0626993
$202$	0	0
$203$	−27.8907	−1.95755
$204$	0	0
$205$	−6.89116	−0.481300
$206$	0	0
$207$	−17.5902	−1.22260
$208$	0	0
$209$	−7.23695	−0.500590
$210$	0	0
$211$	16.0509	1.10499	0.552494	−	0.833517i	$-0.313676\pi$
$211$	16.0509	1.10499	0.552494	+	0.833517i	$0.313676\pi$
$212$	0	0
$213$	−0.123068	−0.00843245
$214$	0	0
$215$	−1.76668	−0.120486
$216$	0	0
$217$	30.6057	2.07765
$218$	0	0
$219$	0.637405	0.0430719
$220$	0	0
$221$	5.81393	0.391087
$222$	0	0
$223$	−12.8903	−0.863196	−0.431598	−	0.902066i	$-0.642050\pi$
$223$	−12.8903	−0.863196	−0.431598	+	0.902066i	$0.642050\pi$
$224$	0	0
$225$	11.5091	0.767276
$226$	0	0
$227$	−17.9805	−1.19341	−0.596705	−	0.802461i	$-0.703524\pi$
$227$	−17.9805	−1.19341	−0.596705	+	0.802461i	$0.703524\pi$
$228$	0	0
$229$	6.29999	0.416315	0.208158	−	0.978095i	$-0.433253\pi$
$229$	6.29999	0.416315	0.208158	+	0.978095i	$0.433253\pi$
$230$	0	0
$231$	−2.44267	−0.160716
$232$	0	0
$233$	−26.8813	−1.76105	−0.880526	−	0.473998i	$-0.842810\pi$
$233$	−26.8813	−1.76105	−0.880526	+	0.473998i	$0.842810\pi$
$234$	0	0
$235$	9.27619	0.605112
$236$	0	0
$237$	1.94779	0.126522
$238$	0	0
$239$	−10.1452	−0.656241	−0.328121	−	0.944636i	$-0.606415\pi$
$239$	−10.1452	−0.656241	−0.328121	+	0.944636i	$0.606415\pi$
$240$	0	0
$241$	14.1015	0.908358	0.454179	−	0.890910i	$-0.349933\pi$
$241$	14.1015	0.908358	0.454179	+	0.890910i	$0.349933\pi$
$242$	0	0
$243$	6.54559	0.419900
$244$	0	0
$245$	−5.09301	−0.325381
$246$	0	0
$247$	−14.8091	−0.942280
$248$	0	0
$249$	−1.73506	−0.109955
$250$	0	0
$251$	−4.42397	−0.279239	−0.139619	−	0.990205i	$-0.544588\pi$
$251$	−4.42397	−0.279239	−0.139619	+	0.990205i	$0.544588\pi$
$252$	0	0
$253$	17.0113	1.06949
$254$	0	0
$255$	−0.259354	−0.0162414
$256$	0	0
$257$	16.1817	1.00938	0.504692	−	0.863299i	$-0.331606\pi$
$257$	16.1817	1.00938	0.504692	+	0.863299i	$0.331606\pi$
$258$	0	0
$259$	35.5650	2.20990
$260$	0	0
$261$	−23.7578	−1.47057
$262$	0	0
$263$	5.80924	0.358213	0.179106	−	0.983830i	$-0.442679\pi$
$263$	5.80924	0.358213	0.179106	+	0.983830i	$0.442679\pi$
$264$	0	0
$265$	−11.6998	−0.718712
$266$	0	0
$267$	−1.22058	−0.0746983
$268$	0	0
$269$	3.06966	0.187161	0.0935803	−	0.995612i	$-0.470169\pi$
$269$	3.06966	0.187161	0.0935803	+	0.995612i	$0.470169\pi$
$270$	0	0
$271$	13.2694	0.806056	0.403028	−	0.915188i	$-0.367958\pi$
$271$	13.2694	0.806056	0.403028	+	0.915188i	$0.367958\pi$
$272$	0	0
$273$	−4.99848	−0.302522
$274$	0	0
$275$	−11.1303	−0.671185
$276$	0	0
$277$	15.6648	0.941207	0.470603	−	0.882345i	$-0.344036\pi$
$277$	15.6648	0.941207	0.470603	+	0.882345i	$0.344036\pi$
$278$	0	0
$279$	26.0705	1.56080
$280$	0	0
$281$	−17.4588	−1.04151	−0.520753	−	0.853708i	$-0.674349\pi$
$281$	−17.4588	−1.04151	−0.520753	+	0.853708i	$0.674349\pi$
$282$	0	0
$283$	9.28927	0.552190	0.276095	−	0.961130i	$-0.410960\pi$
$283$	9.28927	0.552190	0.276095	+	0.961130i	$0.410960\pi$
$284$	0	0
$285$	0.660621	0.0391318
$286$	0	0
$287$	−22.8438	−1.34843
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−1.95995	−0.114894
$292$	0	0
$293$	−21.2861	−1.24355	−0.621773	−	0.783197i	$-0.713587\pi$
$293$	−21.2861	−1.24355	−0.621773	+	0.783197i	$0.713587\pi$
$294$	0	0
$295$	1.04042	0.0605757
$296$	0	0
$297$	−4.20543	−0.244024
$298$	0	0
$299$	34.8104	2.01314
$300$	0	0
$301$	−5.85643	−0.337559
$302$	0	0
$303$	−0.354930	−0.0203902
$304$	0	0
$305$	14.7754	0.846034
$306$	0	0
$307$	5.20305	0.296954	0.148477	−	0.988916i	$-0.452563\pi$
$307$	5.20305	0.296954	0.148477	+	0.988916i	$0.452563\pi$
$308$	0	0
$309$	1.60796	0.0914736
$310$	0	0
$311$	−3.13135	−0.177563	−0.0887813	−	0.996051i	$-0.528297\pi$
$311$	−3.13135	−0.177563	−0.0887813	+	0.996051i	$0.528297\pi$
$312$	0	0
$313$	29.7541	1.68180	0.840901	−	0.541190i	$-0.182026\pi$
$313$	29.7541	1.68180	0.840901	+	0.541190i	$0.182026\pi$
$314$	0	0
$315$	−10.5421	−0.593978
$316$	0	0
$317$	29.0981	1.63431	0.817156	−	0.576417i	$-0.195550\pi$
$317$	29.0981	1.63431	0.817156	+	0.576417i	$0.195550\pi$
$318$	0	0
$319$	22.9759	1.28640
$320$	0	0
$321$	−0.541965	−0.0302495
$322$	0	0
$323$	−2.54718	−0.141729
$324$	0	0
$325$	−22.7762	−1.26340
$326$	0	0
$327$	−2.01467	−0.111411
$328$	0	0
$329$	30.7500	1.69530
$330$	0	0
$331$	21.4654	1.17984	0.589922	−	0.807460i	$-0.299159\pi$
$331$	21.4654	1.17984	0.589922	+	0.807460i	$0.299159\pi$
$332$	0	0
$333$	30.2949	1.66015
$334$	0	0
$335$	−3.71011	−0.202705
$336$	0	0
$337$	−25.9130	−1.41157	−0.705787	−	0.708424i	$-0.749406\pi$
$337$	−25.9130	−1.41157	−0.705787	+	0.708424i	$0.749406\pi$
$338$	0	0
$339$	0.666620	0.0362058
$340$	0	0
$341$	−25.2124	−1.36533
$342$	0	0
$343$	7.25952	0.391977
$344$	0	0
$345$	−1.55286	−0.0836034
$346$	0	0
$347$	31.9508	1.71521	0.857603	−	0.514312i	$-0.171953\pi$
$347$	31.9508	1.71521	0.857603	+	0.514312i	$0.171953\pi$
$348$	0	0
$349$	−8.06065	−0.431477	−0.215738	−	0.976451i	$-0.569216\pi$
$349$	−8.06065	−0.431477	−0.215738	+	0.976451i	$0.569216\pi$
$350$	0	0
$351$	−8.60565	−0.459335
$352$	0	0
$353$	6.62998	0.352878	0.176439	−	0.984312i	$-0.443542\pi$
$353$	6.62998	0.352878	0.176439	+	0.984312i	$0.443542\pi$
$354$	0	0
$355$	0.513652	0.0272618
$356$	0	0
$357$	−0.859743	−0.0455024
$358$	0	0
$359$	8.46707	0.446875	0.223437	−	0.974718i	$-0.428272\pi$
$359$	8.46707	0.446875	0.223437	+	0.974718i	$0.428272\pi$
$360$	0	0
$361$	−12.5119	−0.658521
$362$	0	0
$363$	−0.729828	−0.0383060
$364$	0	0
$365$	−2.66036	−0.139250
$366$	0	0
$367$	21.3831	1.11619	0.558094	−	0.829778i	$-0.311533\pi$
$367$	21.3831	1.11619	0.558094	+	0.829778i	$0.311533\pi$
$368$	0	0
$369$	−19.4587	−1.01298
$370$	0	0
$371$	−38.7840	−2.01357
$372$	0	0
$373$	27.5670	1.42737	0.713683	−	0.700469i	$-0.247026\pi$
$373$	27.5670	1.42737	0.713683	+	0.700469i	$0.247026\pi$
$374$	0	0
$375$	2.31280	0.119432
$376$	0	0
$377$	47.0159	2.42144
$378$	0	0
$379$	−22.8980	−1.17619	−0.588096	−	0.808791i	$-0.700122\pi$
$379$	−22.8980	−1.17619	−0.588096	+	0.808791i	$0.700122\pi$
$380$	0	0
$381$	−3.13711	−0.160719
$382$	0	0
$383$	−8.16250	−0.417084	−0.208542	−	0.978013i	$-0.566872\pi$
$383$	−8.16250	−0.417084	−0.208542	+	0.978013i	$0.566872\pi$
$384$	0	0
$385$	10.1951	0.519590
$386$	0	0
$387$	−4.98860	−0.253585
$388$	0	0
$389$	34.3007	1.73912	0.869558	−	0.493830i	$-0.164404\pi$
$389$	34.3007	1.73912	0.869558	+	0.493830i	$0.164404\pi$
$390$	0	0
$391$	5.98742	0.302797
$392$	0	0
$393$	−5.20969	−0.262794
$394$	0	0
$395$	−8.12956	−0.409043
$396$	0	0
$397$	−31.0014	−1.55592	−0.777959	−	0.628315i	$-0.783745\pi$
$397$	−31.0014	−1.55592	−0.777959	+	0.628315i	$0.783745\pi$
$398$	0	0
$399$	2.18992	0.109633
$400$	0	0
$401$	−25.4956	−1.27319	−0.636595	−	0.771198i	$-0.719658\pi$
$401$	−25.4956	−1.27319	−0.636595	+	0.771198i	$0.719658\pi$
$402$	0	0
$403$	−51.5926	−2.57001
$404$	0	0
$405$	−8.78595	−0.436577
$406$	0	0
$407$	−29.2978	−1.45224
$408$	0	0
$409$	−20.6162	−1.01940	−0.509702	−	0.860351i	$-0.670244\pi$
$409$	−20.6162	−1.01940	−0.509702	+	0.860351i	$0.670244\pi$
$410$	0	0
$411$	−1.00401	−0.0495243
$412$	0	0
$413$	3.44893	0.169711
$414$	0	0
$415$	7.24170	0.355481
$416$	0	0
$417$	−4.23843	−0.207557
$418$	0	0
$419$	−35.6793	−1.74305	−0.871525	−	0.490352i	$-0.836868\pi$
$419$	−35.6793	−1.74305	−0.871525	+	0.490352i	$0.836868\pi$
$420$	0	0
$421$	10.0142	0.488062	0.244031	−	0.969767i	$-0.421530\pi$
$421$	10.0142	0.488062	0.244031	+	0.969767i	$0.421530\pi$
$422$	0	0
$423$	26.1934	1.27356
$424$	0	0
$425$	−3.91752	−0.190028
$426$	0	0
$427$	48.9794	2.37028
$428$	0	0
$429$	4.11766	0.198802
$430$	0	0
$431$	−25.7481	−1.24024	−0.620121	−	0.784506i	$-0.712916\pi$
$431$	−25.7481	−1.24024	−0.620121	+	0.784506i	$0.712916\pi$
$432$	0	0
$433$	30.9745	1.48854	0.744270	−	0.667878i	$-0.232797\pi$
$433$	30.9745	1.48854	0.744270	+	0.667878i	$0.232797\pi$
$434$	0	0
$435$	−2.09734	−0.100560
$436$	0	0
$437$	−15.2510	−0.729555
$438$	0	0
$439$	−10.0142	−0.477950	−0.238975	−	0.971026i	$-0.576811\pi$
$439$	−10.0142	−0.477950	−0.238975	+	0.971026i	$0.576811\pi$
$440$	0	0
$441$	−14.3812	−0.684821
$442$	0	0
$443$	−8.18134	−0.388707	−0.194354	−	0.980932i	$-0.562261\pi$
$443$	−8.18134	−0.388707	−0.194354	+	0.980932i	$0.562261\pi$
$444$	0	0
$445$	5.09438	0.241497
$446$	0	0
$447$	−4.20173	−0.198735
$448$	0	0
$449$	5.78747	0.273128	0.136564	−	0.990631i	$-0.456394\pi$
$449$	5.78747	0.273128	0.136564	+	0.990631i	$0.456394\pi$
$450$	0	0
$451$	18.8183	0.886118
$452$	0	0
$453$	0.810590	0.0380848
$454$	0	0
$455$	20.8624	0.978043
$456$	0	0
$457$	8.29410	0.387982	0.193991	−	0.981003i	$-0.437857\pi$
$457$	8.29410	0.387982	0.193991	+	0.981003i	$0.437857\pi$
$458$	0	0
$459$	−1.48018	−0.0690888
$460$	0	0
$461$	29.2632	1.36292	0.681462	−	0.731854i	$-0.261345\pi$
$461$	29.2632	1.36292	0.681462	+	0.731854i	$0.261345\pi$
$462$	0	0
$463$	−31.6995	−1.47320	−0.736600	−	0.676328i	$-0.763570\pi$
$463$	−31.6995	−1.47320	−0.736600	+	0.676328i	$0.763570\pi$
$464$	0	0
$465$	2.30150	0.106730
$466$	0	0
$467$	4.74012	0.219347	0.109673	−	0.993968i	$-0.465020\pi$
$467$	4.74012	0.219347	0.109673	+	0.993968i	$0.465020\pi$
$468$	0	0
$469$	−12.2988	−0.567904
$470$	0	0
$471$	−5.37387	−0.247615
$472$	0	0
$473$	4.82442	0.221827
$474$	0	0
$475$	9.97862	0.457850
$476$	0	0
$477$	−33.0369	−1.51266
$478$	0	0
$479$	−0.672549	−0.0307295	−0.0153648	−	0.999882i	$-0.504891\pi$
$479$	−0.672549	−0.0307295	−0.0153648	+	0.999882i	$0.504891\pi$
$480$	0	0
$481$	−59.9525	−2.73360
$482$	0	0
$483$	−5.14765	−0.234226
$484$	0	0
$485$	8.18031	0.371449
$486$	0	0
$487$	−0.226239	−0.0102519	−0.00512593	−	0.999987i	$-0.501632\pi$
$487$	−0.226239	−0.0102519	−0.00512593	+	0.999987i	$0.501632\pi$
$488$	0	0
$489$	−5.38414	−0.243479
$490$	0	0
$491$	−10.2098	−0.460761	−0.230381	−	0.973101i	$-0.573997\pi$
$491$	−10.2098	−0.460761	−0.230381	+	0.973101i	$0.573997\pi$
$492$	0	0
$493$	8.08678	0.364210
$494$	0	0
$495$	8.68435	0.390333
$496$	0	0
$497$	1.70273	0.0763777
$498$	0	0
$499$	−39.5678	−1.77130	−0.885649	−	0.464356i	$-0.846286\pi$
$499$	−39.5678	−1.77130	−0.885649	+	0.464356i	$0.846286\pi$
$500$	0	0
$501$	−4.13041	−0.184533
$502$	0	0
$503$	−14.6610	−0.653702	−0.326851	−	0.945076i	$-0.605988\pi$
$503$	−14.6610	−0.653702	−0.326851	+	0.945076i	$0.605988\pi$
$504$	0	0
$505$	1.48139	0.0659208
$506$	0	0
$507$	5.18541	0.230292
$508$	0	0
$509$	28.3492	1.25656	0.628278	−	0.777989i	$-0.283760\pi$
$509$	28.3492	1.25656	0.628278	+	0.777989i	$0.283760\pi$
$510$	0	0
$511$	−8.81894	−0.390127
$512$	0	0
$513$	3.77027	0.166462
$514$	0	0
$515$	−6.71120	−0.295731
$516$	0	0
$517$	−25.3313	−1.11407
$518$	0	0
$519$	−1.52304	−0.0668539
$520$	0	0
$521$	−6.49998	−0.284769	−0.142385	−	0.989811i	$-0.545477\pi$
$521$	−6.49998	−0.284769	−0.142385	+	0.989811i	$0.545477\pi$
$522$	0	0
$523$	−9.79656	−0.428374	−0.214187	−	0.976793i	$-0.568710\pi$
$523$	−9.79656	−0.428374	−0.214187	+	0.976793i	$0.568710\pi$
$524$	0	0
$525$	3.36806	0.146994
$526$	0	0
$527$	−8.87396	−0.386556
$528$	0	0
$529$	12.8492	0.558661
$530$	0	0
$531$	2.93786	0.127492
$532$	0	0
$533$	38.5081	1.66797
$534$	0	0
$535$	2.26202	0.0977957
$536$	0	0
$537$	−2.49133	−0.107509
$538$	0	0
$539$	13.9079	0.599056
$540$	0	0
$541$	−2.59458	−0.111550	−0.0557748	−	0.998443i	$-0.517763\pi$
$541$	−2.59458	−0.111550	−0.0557748	+	0.998443i	$0.517763\pi$
$542$	0	0
$543$	−1.77900	−0.0763440
$544$	0	0
$545$	8.40870	0.360189
$546$	0	0
$547$	−39.8235	−1.70273	−0.851364	−	0.524575i	$-0.824224\pi$
$547$	−39.8235	−1.70273	−0.851364	+	0.524575i	$0.824224\pi$
$548$	0	0
$549$	41.7215	1.78063
$550$	0	0
$551$	−20.5984	−0.877523
$552$	0	0
$553$	−26.9490	−1.14599
$554$	0	0
$555$	2.67443	0.113523
$556$	0	0
$557$	29.4863	1.24938	0.624688	−	0.780874i	$-0.285226\pi$
$557$	29.4863	1.24938	0.624688	+	0.780874i	$0.285226\pi$
$558$	0	0
$559$	9.87228	0.417553
$560$	0	0
$561$	0.708241	0.0299019
$562$	0	0
$563$	−0.935898	−0.0394434	−0.0197217	−	0.999806i	$-0.506278\pi$
$563$	−0.935898	−0.0394434	−0.0197217	+	0.999806i	$0.506278\pi$
$564$	0	0
$565$	−2.78230	−0.117052
$566$	0	0
$567$	−29.1249	−1.22313
$568$	0	0
$569$	2.54121	0.106533	0.0532666	−	0.998580i	$-0.483037\pi$
$569$	2.54121	0.106533	0.0532666	+	0.998580i	$0.483037\pi$
$570$	0	0
$571$	6.55683	0.274395	0.137197	−	0.990544i	$-0.456191\pi$
$571$	6.55683	0.274395	0.137197	+	0.990544i	$0.456191\pi$
$572$	0	0
$573$	1.03463	0.0432222
$574$	0	0
$575$	−23.4559	−0.978177
$576$	0	0
$577$	−12.1753	−0.506865	−0.253432	−	0.967353i	$-0.581560\pi$
$577$	−12.1753	−0.506865	−0.253432	+	0.967353i	$0.581560\pi$
$578$	0	0
$579$	2.28913	0.0951332
$580$	0	0
$581$	24.0058	0.995928
$582$	0	0
$583$	31.9496	1.32322
$584$	0	0
$585$	17.7709	0.734737
$586$	0	0
$587$	−13.8835	−0.573034	−0.286517	−	0.958075i	$-0.592497\pi$
$587$	−13.8835	−0.573034	−0.286517	+	0.958075i	$0.592497\pi$
$588$	0	0
$589$	22.6035	0.931363
$590$	0	0
$591$	3.57718	0.147145
$592$	0	0
$593$	−35.8959	−1.47407	−0.737033	−	0.675856i	$-0.763774\pi$
$593$	−35.8959	−1.47407	−0.737033	+	0.675856i	$0.763774\pi$
$594$	0	0
$595$	3.58835	0.147108
$596$	0	0
$597$	−0.680261	−0.0278412
$598$	0	0
$599$	20.0658	0.819865	0.409932	−	0.912116i	$-0.365552\pi$
$599$	20.0658	0.819865	0.409932	+	0.912116i	$0.365552\pi$
$600$	0	0
$601$	44.7616	1.82586	0.912931	−	0.408113i	$-0.133813\pi$
$601$	44.7616	1.82586	0.912931	+	0.408113i	$0.133813\pi$
$602$	0	0
$603$	−10.4763	−0.426628
$604$	0	0
$605$	3.04611	0.123842
$606$	0	0
$607$	37.0512	1.50386	0.751931	−	0.659242i	$-0.229123\pi$
$607$	37.0512	1.50386	0.751931	+	0.659242i	$0.229123\pi$
$608$	0	0
$609$	−6.95255	−0.281732
$610$	0	0
$611$	−51.8358	−2.09705
$612$	0	0
$613$	−29.0623	−1.17382	−0.586908	−	0.809654i	$-0.699655\pi$
$613$	−29.0623	−1.17382	−0.586908	+	0.809654i	$0.699655\pi$
$614$	0	0
$615$	−1.71782	−0.0692690
$616$	0	0
$617$	18.8690	0.759637	0.379818	−	0.925061i	$-0.375987\pi$
$617$	18.8690	0.759637	0.379818	+	0.925061i	$0.375987\pi$
$618$	0	0
$619$	1.52263	0.0611996	0.0305998	−	0.999532i	$-0.490258\pi$
$619$	1.52263	0.0611996	0.0305998	+	0.999532i	$0.490258\pi$
$620$	0	0
$621$	−8.86245	−0.355638
$622$	0	0
$623$	16.8876	0.676586
$624$	0	0
$625$	9.93459	0.397384
$626$	0	0
$627$	−1.80401	−0.0720453
$628$	0	0
$629$	−10.3119	−0.411162
$630$	0	0
$631$	20.3778	0.811229	0.405615	−	0.914044i	$-0.367058\pi$
$631$	20.3778	0.811229	0.405615	+	0.914044i	$0.367058\pi$
$632$	0	0
$633$	4.00113	0.159031
$634$	0	0
$635$	13.0935	0.519599
$636$	0	0
$637$	28.4600	1.12763
$638$	0	0
$639$	1.45041	0.0573774
$640$	0	0
$641$	39.8381	1.57351	0.786756	−	0.617265i	$-0.211759\pi$
$641$	39.8381	1.57351	0.786756	+	0.617265i	$0.211759\pi$
$642$	0	0
$643$	−21.8330	−0.861009	−0.430504	−	0.902589i	$-0.641664\pi$
$643$	−21.8330	−0.861009	−0.430504	+	0.902589i	$0.641664\pi$
$644$	0	0
$645$	−0.440394	−0.0173405
$646$	0	0
$647$	42.2607	1.66144	0.830720	−	0.556690i	$-0.187929\pi$
$647$	42.2607	1.66144	0.830720	+	0.556690i	$0.187929\pi$
$648$	0	0
$649$	−2.84117	−0.111526
$650$	0	0
$651$	7.62933	0.299017
$652$	0	0
$653$	7.24400	0.283480	0.141740	−	0.989904i	$-0.454730\pi$
$653$	7.24400	0.283480	0.141740	+	0.989904i	$0.454730\pi$
$654$	0	0
$655$	21.7439	0.849604
$656$	0	0
$657$	−7.51213	−0.293076
$658$	0	0
$659$	−17.4704	−0.680551	−0.340275	−	0.940326i	$-0.610520\pi$
$659$	−17.4704	−0.680551	−0.340275	+	0.940326i	$0.610520\pi$
$660$	0	0
$661$	15.8009	0.614584	0.307292	−	0.951615i	$-0.400577\pi$
$661$	15.8009	0.614584	0.307292	+	0.951615i	$0.400577\pi$
$662$	0	0
$663$	1.44928	0.0562855
$664$	0	0
$665$	−9.14015	−0.354440
$666$	0	0
$667$	48.4189	1.87479
$668$	0	0
$669$	−3.21326	−0.124232
$670$	0	0
$671$	−40.3483	−1.55763
$672$	0	0
$673$	−45.2357	−1.74371	−0.871855	−	0.489764i	$-0.837083\pi$
$673$	−45.2357	−1.74371	−0.871855	+	0.489764i	$0.837083\pi$
$674$	0	0
$675$	5.79863	0.223189
$676$	0	0
$677$	16.7671	0.644414	0.322207	−	0.946669i	$-0.395575\pi$
$677$	16.7671	0.644414	0.322207	+	0.946669i	$0.395575\pi$
$678$	0	0
$679$	27.1172	1.04066
$680$	0	0
$681$	−4.48215	−0.171757
$682$	0	0
$683$	−16.1239	−0.616962	−0.308481	−	0.951231i	$-0.599821\pi$
$683$	−16.1239	−0.616962	−0.308481	+	0.951231i	$0.599821\pi$
$684$	0	0
$685$	4.19049	0.160110
$686$	0	0
$687$	1.57045	0.0599164
$688$	0	0
$689$	65.3789	2.49074
$690$	0	0
$691$	−27.3636	−1.04096	−0.520481	−	0.853873i	$-0.674247\pi$
$691$	−27.3636	−1.04096	−0.520481	+	0.853873i	$0.674247\pi$
$692$	0	0
$693$	28.7881	1.09357
$694$	0	0
$695$	17.6901	0.671025
$696$	0	0
$697$	6.62343	0.250880
$698$	0	0
$699$	−6.70091	−0.253452
$700$	0	0
$701$	27.7459	1.04795	0.523974	−	0.851734i	$-0.324449\pi$
$701$	27.7459	1.04795	0.523974	+	0.851734i	$0.324449\pi$
$702$	0	0
$703$	26.2662	0.990648
$704$	0	0
$705$	2.31235	0.0870881
$706$	0	0
$707$	4.91070	0.184686
$708$	0	0
$709$	2.22376	0.0835151	0.0417575	−	0.999128i	$-0.486704\pi$
$709$	2.22376	0.0835151	0.0417575	+	0.999128i	$0.486704\pi$
$710$	0	0
$711$	−22.9556	−0.860903
$712$	0	0
$713$	−53.1322	−1.98982
$714$	0	0
$715$	−17.1860	−0.642721
$716$	0	0
$717$	−2.52899	−0.0944468
$718$	0	0
$719$	13.9492	0.520219	0.260109	−	0.965579i	$-0.416241\pi$
$719$	13.9492	0.520219	0.260109	+	0.965579i	$0.416241\pi$
$720$	0	0
$721$	−22.2472	−0.828530
$722$	0	0
$723$	3.51520	0.130732
$724$	0	0
$725$	−31.6801	−1.17657
$726$	0	0
$727$	8.17769	0.303294	0.151647	−	0.988435i	$-0.451542\pi$
$727$	8.17769	0.303294	0.151647	+	0.988435i	$0.451542\pi$
$728$	0	0
$729$	−23.7021	−0.877857
$730$	0	0
$731$	1.69804	0.0628043
$732$	0	0
$733$	−11.1594	−0.412181	−0.206090	−	0.978533i	$-0.566074\pi$
$733$	−11.1594	−0.412181	−0.206090	+	0.978533i	$0.566074\pi$
$734$	0	0
$735$	−1.26958	−0.0468290
$736$	0	0
$737$	10.1315	0.373198
$738$	0	0
$739$	30.0168	1.10418	0.552092	−	0.833783i	$-0.313830\pi$
$739$	30.0168	1.10418	0.552092	+	0.833783i	$0.313830\pi$
$740$	0	0
$741$	−3.69158	−0.135614
$742$	0	0
$743$	51.9106	1.90441	0.952207	−	0.305454i	$-0.0988081\pi$
$743$	51.9106	1.90441	0.952207	+	0.305454i	$0.0988081\pi$
$744$	0	0
$745$	17.5369	0.642503
$746$	0	0
$747$	20.4485	0.748173
$748$	0	0
$749$	7.49846	0.273988
$750$	0	0
$751$	−3.48958	−0.127337	−0.0636683	−	0.997971i	$-0.520280\pi$
$751$	−3.48958	−0.127337	−0.0636683	+	0.997971i	$0.520280\pi$
$752$	0	0
$753$	−1.10280	−0.0401882
$754$	0	0
$755$	−3.38319	−0.123127
$756$	0	0
$757$	46.7307	1.69846	0.849229	−	0.528025i	$-0.177067\pi$
$757$	46.7307	1.69846	0.849229	+	0.528025i	$0.177067\pi$
$758$	0	0
$759$	4.24053	0.153922
$760$	0	0
$761$	19.0711	0.691326	0.345663	−	0.938359i	$-0.387654\pi$
$761$	19.0711	0.691326	0.345663	+	0.938359i	$0.387654\pi$
$762$	0	0
$763$	27.8743	1.00912
$764$	0	0
$765$	3.05661	0.110512
$766$	0	0
$767$	−5.81393	−0.209929
$768$	0	0
$769$	12.6110	0.454765	0.227383	−	0.973806i	$-0.426983\pi$
$769$	12.6110	0.454765	0.227383	+	0.973806i	$0.426983\pi$
$770$	0	0
$771$	4.03373	0.145271
$772$	0	0
$773$	−6.87900	−0.247420	−0.123710	−	0.992318i	$-0.539479\pi$
$773$	−6.87900	−0.247420	−0.123710	+	0.992318i	$0.539479\pi$
$774$	0	0
$775$	34.7640	1.24876
$776$	0	0
$777$	8.86557	0.318051
$778$	0	0
$779$	−16.8710	−0.604468
$780$	0	0
$781$	−1.40267	−0.0501916
$782$	0	0
$783$	−11.9699	−0.427768
$784$	0	0
$785$	22.4291	0.800530
$786$	0	0
$787$	−49.2280	−1.75479	−0.877394	−	0.479771i	$-0.840720\pi$
$787$	−49.2280	−1.75479	−0.877394	+	0.479771i	$0.840720\pi$
$788$	0	0
$789$	1.44811	0.0515543
$790$	0	0
$791$	−9.22314	−0.327937
$792$	0	0
$793$	−82.5654	−2.93198
$794$	0	0
$795$	−2.91650	−0.103438
$796$	0	0
$797$	28.6492	1.01481	0.507403	−	0.861709i	$-0.330605\pi$
$797$	28.6492	1.01481	0.507403	+	0.861709i	$0.330605\pi$
$798$	0	0
$799$	−8.91580	−0.315418
$800$	0	0
$801$	14.3851	0.508273
$802$	0	0
$803$	7.26488	0.256372
$804$	0	0
$805$	21.4849	0.757244
$806$	0	0
$807$	0.765200	0.0269363
$808$	0	0
$809$	−16.0163	−0.563103	−0.281551	−	0.959546i	$-0.590849\pi$
$809$	−16.0163	−0.563103	−0.281551	+	0.959546i	$0.590849\pi$
$810$	0	0
$811$	23.7495	0.833956	0.416978	−	0.908917i	$-0.363089\pi$
$811$	23.7495	0.833956	0.416978	+	0.908917i	$0.363089\pi$
$812$	0	0
$813$	3.30776	0.116008
$814$	0	0
$815$	22.4720	0.787159
$816$	0	0
$817$	−4.32521	−0.151320
$818$	0	0
$819$	58.9095	2.05847
$820$	0	0
$821$	−17.2906	−0.603447	−0.301723	−	0.953395i	$-0.597562\pi$
$821$	−17.2906	−0.603447	−0.301723	+	0.953395i	$0.597562\pi$
$822$	0	0
$823$	−19.4925	−0.679466	−0.339733	−	0.940522i	$-0.610337\pi$
$823$	−19.4925	−0.679466	−0.339733	+	0.940522i	$0.610337\pi$
$824$	0	0
$825$	−2.77455	−0.0965974
$826$	0	0
$827$	−9.88841	−0.343854	−0.171927	−	0.985110i	$-0.554999\pi$
$827$	−9.88841	−0.343854	−0.171927	+	0.985110i	$0.554999\pi$
$828$	0	0
$829$	5.45726	0.189539	0.0947693	−	0.995499i	$-0.469789\pi$
$829$	5.45726	0.189539	0.0947693	+	0.995499i	$0.469789\pi$
$830$	0	0
$831$	3.90489	0.135459
$832$	0	0
$833$	4.89514	0.169607
$834$	0	0
$835$	17.2393	0.596589
$836$	0	0
$837$	13.1350	0.454014
$838$	0	0
$839$	21.1251	0.729321	0.364660	−	0.931141i	$-0.381185\pi$
$839$	21.1251	0.729321	0.364660	+	0.931141i	$0.381185\pi$
$840$	0	0
$841$	36.3959	1.25503
$842$	0	0
$843$	−4.35210	−0.149894
$844$	0	0
$845$	−21.6426	−0.744527
$846$	0	0
$847$	10.0977	0.346960
$848$	0	0
$849$	2.31561	0.0794715
$850$	0	0
$851$	−61.7416	−2.11647
$852$	0	0
$853$	−18.2537	−0.624995	−0.312497	−	0.949919i	$-0.601166\pi$
$853$	−18.2537	−0.624995	−0.312497	+	0.949919i	$0.601166\pi$
$854$	0	0
$855$	−7.78573	−0.266266
$856$	0	0
$857$	−42.8438	−1.46352	−0.731758	−	0.681564i	$-0.761300\pi$
$857$	−42.8438	−1.46352	−0.731758	+	0.681564i	$0.761300\pi$
$858$	0	0
$859$	−6.93954	−0.236774	−0.118387	−	0.992968i	$-0.537772\pi$
$859$	−6.93954	−0.236774	−0.118387	+	0.992968i	$0.537772\pi$
$860$	0	0
$861$	−5.69445	−0.194066
$862$	0	0
$863$	−12.4033	−0.422214	−0.211107	−	0.977463i	$-0.567707\pi$
$863$	−12.4033	−0.422214	−0.211107	+	0.977463i	$0.567707\pi$
$864$	0	0
$865$	6.35677	0.216137
$866$	0	0
$867$	0.249278	0.00846593
$868$	0	0
$869$	22.2001	0.753086
$870$	0	0
$871$	20.7322	0.702485
$872$	0	0
$873$	23.0989	0.781780
$874$	0	0
$875$	−31.9992	−1.08177
$876$	0	0
$877$	−1.48637	−0.0501912	−0.0250956	−	0.999685i	$-0.507989\pi$
$877$	−1.48637	−0.0501912	−0.0250956	+	0.999685i	$0.507989\pi$
$878$	0	0
$879$	−5.30615	−0.178972
$880$	0	0
$881$	21.4824	0.723759	0.361880	−	0.932225i	$-0.382135\pi$
$881$	21.4824	0.723759	0.361880	+	0.932225i	$0.382135\pi$
$882$	0	0
$883$	38.4176	1.29285	0.646427	−	0.762976i	$-0.276262\pi$
$883$	38.4176	1.29285	0.646427	+	0.762976i	$0.276262\pi$
$884$	0	0
$885$	0.259354	0.00871810
$886$	0	0
$887$	−39.2277	−1.31714	−0.658568	−	0.752521i	$-0.728837\pi$
$887$	−39.2277	−1.31714	−0.658568	+	0.752521i	$0.728837\pi$
$888$	0	0
$889$	43.4041	1.45573
$890$	0	0
$891$	23.9925	0.803780
$892$	0	0
$893$	22.7101	0.759964
$894$	0	0
$895$	10.3981	0.347572
$896$	0	0
$897$	8.67747	0.289732
$898$	0	0
$899$	−71.7618	−2.39339
$900$	0	0
$901$	11.2452	0.374633
$902$	0	0
$903$	−1.45988	−0.0485817
$904$	0	0
$905$	7.42506	0.246817
$906$	0	0
$907$	−47.2900	−1.57024	−0.785120	−	0.619344i	$-0.787399\pi$
$907$	−47.2900	−1.57024	−0.785120	+	0.619344i	$0.787399\pi$
$908$	0	0
$909$	4.18302	0.138742
$910$	0	0
$911$	−16.9128	−0.560346	−0.280173	−	0.959950i	$-0.590392\pi$
$911$	−16.9128	−0.560346	−0.280173	+	0.959950i	$0.590392\pi$
$912$	0	0
$913$	−19.7755	−0.654475
$914$	0	0
$915$	3.68317	0.121762
$916$	0	0
$917$	72.0796	2.38028
$918$	0	0
$919$	−15.1885	−0.501023	−0.250511	−	0.968114i	$-0.580599\pi$
$919$	−15.1885	−0.501023	−0.250511	+	0.968114i	$0.580599\pi$
$920$	0	0
$921$	1.29701	0.0427378
$922$	0	0
$923$	−2.87031	−0.0944775
$924$	0	0
$925$	40.3970	1.32825
$926$	0	0
$927$	−18.9506	−0.622418
$928$	0	0
$929$	33.4085	1.09610	0.548049	−	0.836446i	$-0.315371\pi$
$929$	33.4085	1.09610	0.548049	+	0.836446i	$0.315371\pi$
$930$	0	0
$931$	−12.4688	−0.408648
$932$	0	0
$933$	−0.780577	−0.0255550
$934$	0	0
$935$	−2.95601	−0.0966720
$936$	0	0
$937$	−55.6376	−1.81760	−0.908801	−	0.417230i	$-0.863001\pi$
$937$	−55.6376	−1.81760	−0.908801	+	0.417230i	$0.863001\pi$
$938$	0	0
$939$	7.41704	0.242046
$940$	0	0
$941$	14.0075	0.456632	0.228316	−	0.973587i	$-0.426678\pi$
$941$	14.0075	0.456632	0.228316	+	0.973587i	$0.426678\pi$
$942$	0	0
$943$	39.6573	1.29142
$944$	0	0
$945$	−5.31139	−0.172780
$946$	0	0
$947$	−42.2127	−1.37173	−0.685865	−	0.727729i	$-0.740576\pi$
$947$	−42.2127	−1.37173	−0.685865	+	0.727729i	$0.740576\pi$
$948$	0	0
$949$	14.8662	0.482579
$950$	0	0
$951$	7.25351	0.235211
$952$	0	0
$953$	−2.90324	−0.0940451	−0.0470226	−	0.998894i	$-0.514973\pi$
$953$	−2.90324	−0.0940451	−0.0470226	+	0.998894i	$0.514973\pi$
$954$	0	0
$955$	−4.31827	−0.139736
$956$	0	0
$957$	5.72738	0.185140
$958$	0	0
$959$	13.8912	0.448571
$960$	0	0
$961$	47.7473	1.54023
$962$	0	0
$963$	6.38732	0.205828
$964$	0	0
$965$	−9.55425	−0.307562
$966$	0	0
$967$	−35.4672	−1.14055	−0.570274	−	0.821454i	$-0.693163\pi$
$967$	−35.4672	−1.14055	−0.570274	+	0.821454i	$0.693163\pi$
$968$	0	0
$969$	−0.634955	−0.0203977
$970$	0	0
$971$	2.02451	0.0649696	0.0324848	−	0.999472i	$-0.489658\pi$
$971$	2.02451	0.0649696	0.0324848	+	0.999472i	$0.489658\pi$
$972$	0	0
$973$	58.6416	1.87996
$974$	0	0
$975$	−5.67760	−0.181829
$976$	0	0
$977$	21.2634	0.680276	0.340138	−	0.940375i	$-0.389526\pi$
$977$	21.2634	0.680276	0.340138	+	0.940375i	$0.389526\pi$
$978$	0	0
$979$	−13.9117	−0.444619
$980$	0	0
$981$	23.7438	0.758082
$982$	0	0
$983$	−15.1386	−0.482848	−0.241424	−	0.970420i	$-0.577614\pi$
$983$	−15.1386	−0.482848	−0.241424	+	0.970420i	$0.577614\pi$
$984$	0	0
$985$	−14.9302	−0.475716
$986$	0	0
$987$	7.66530	0.243989
$988$	0	0
$989$	10.1669	0.323288
$990$	0	0
$991$	−12.4791	−0.396411	−0.198205	−	0.980161i	$-0.563511\pi$
$991$	−12.4791	−0.396411	−0.198205	+	0.980161i	$0.563511\pi$
$992$	0	0
$993$	5.35085	0.169804
$994$	0	0
$995$	2.83923	0.0900097
$996$	0	0
$997$	−56.4245	−1.78698	−0.893490	−	0.449082i	$-0.851751\pi$
$997$	−56.4245	−1.78698	−0.893490	+	0.449082i	$0.851751\pi$
$998$	0	0
$999$	15.2634	0.482913

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.h.1.8	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.h.1.8	✓	15	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q+0.249278 q^{3} -1.04042 q^{5} -3.44893 q^{7} -2.93786 q^{9} +O(q^{10})\)	\(q+0.249278 q^{3} -1.04042 q^{5} -3.44893 q^{7} -2.93786 q^{9} +2.84117 q^{11} +5.81393 q^{13} -0.259354 q^{15} +1.00000 q^{17} -2.54718 q^{19} -0.859743 q^{21} +5.98742 q^{23} -3.91752 q^{25} -1.48018 q^{27} +8.08678 q^{29} -8.87396 q^{31} +0.708241 q^{33} +3.58835 q^{35} -10.3119 q^{37} +1.44928 q^{39} +6.62343 q^{41} +1.69804 q^{43} +3.05661 q^{45} -8.91580 q^{47} +4.89514 q^{49} +0.249278 q^{51} +11.2452 q^{53} -2.95601 q^{55} -0.634955 q^{57} -1.00000 q^{59} -14.2013 q^{61} +10.1325 q^{63} -6.04894 q^{65} +3.56596 q^{67} +1.49253 q^{69} -0.493696 q^{71} +2.55701 q^{73} -0.976552 q^{75} -9.79900 q^{77} +7.81372 q^{79} +8.44461 q^{81} -6.96035 q^{83} -1.04042 q^{85} +2.01586 q^{87} -4.89646 q^{89} -20.0518 q^{91} -2.21208 q^{93} +2.65014 q^{95} -7.86250 q^{97} -8.34695 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) \) 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) \) 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9 - 12 * q^11 - 10 * q^13 - 13 * q^15 + 15 * q^17 - 4 * q^21 - 21 * q^23 + 4 * q^25 - 37 * q^27 + 23 * q^29 - 31 * q^31 - 11 * q^33 - 35 * q^35 - 10 * q^37 - 2 * q^39 - 15 * q^41 - 3 * q^43 + 15 * q^45 - 47 * q^47 + 18 * q^49 - q^51 - 7 * q^53 - 20 * q^55 - 30 * q^57 - 15 * q^59 + q^61 + 19 * q^63 + 11 * q^65 - 20 * q^67 - 24 * q^69 - 13 * q^71 - 24 * q^73 - 10 * q^75 + 21 * q^77 - 34 * q^79 - 21 * q^81 - 40 * q^83 + q^85 - 60 * q^87 - 46 * q^89 - 19 * q^91 + 33 * q^93 - 4 * q^95 - 5 * q^97 - 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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