LMFDB - Embedded newform 6032.2.a.z.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$48.1657624992$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 17x^{8} + 47x^{7} + 104x^{6} - 235x^{5} - 283x^{4} + 364x^{3} + 330x^{2} + 12x - 8 $ x^10 - 3x^9 - 17x^8 + 47x^7 + 104x^6 - 235x^5 - 283x^4 + 364x^3 + 330x^2 + 12*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.02548$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.02548 q^{3} -3.01540 q^{5} +3.00262 q^{7} +1.10258 q^{9} +O(q^{10})$	$q-2.02548 q^{3} -3.01540 q^{5} +3.00262 q^{7} +1.10258 q^{9} -3.74558 q^{11} -1.00000 q^{13} +6.10764 q^{15} +1.06592 q^{17} -1.48031 q^{19} -6.08175 q^{21} +4.25737 q^{23} +4.09263 q^{25} +3.84320 q^{27} -1.00000 q^{29} -7.29243 q^{31} +7.58659 q^{33} -9.05410 q^{35} +7.39773 q^{37} +2.02548 q^{39} +10.0578 q^{41} -4.85699 q^{43} -3.32471 q^{45} -3.52129 q^{47} +2.01572 q^{49} -2.15899 q^{51} +10.0557 q^{53} +11.2944 q^{55} +2.99834 q^{57} +2.35043 q^{59} -8.53386 q^{61} +3.31062 q^{63} +3.01540 q^{65} -4.35843 q^{67} -8.62322 q^{69} +2.86989 q^{71} +11.7590 q^{73} -8.28955 q^{75} -11.2465 q^{77} -9.18923 q^{79} -11.0921 q^{81} +0.299467 q^{83} -3.21416 q^{85} +2.02548 q^{87} +5.16495 q^{89} -3.00262 q^{91} +14.7707 q^{93} +4.46372 q^{95} -7.20186 q^{97} -4.12978 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9}+O(q^{10}) $ 10 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 13 * q^9	$ 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9} - 14 q^{11} - 10 q^{13} - 7 q^{15} + 5 q^{17} - 11 q^{19} - 7 q^{23} + 10 q^{25} - 21 q^{27} - 10 q^{29} - 5 q^{31} + 5 q^{33} - 11 q^{35} + 8 q^{37} + 3 q^{39} + 14 q^{41} - 35 q^{43} + 7 q^{45} - 7 q^{49} - 20 q^{51} - 11 q^{53} - 8 q^{55} + 4 q^{57} - 23 q^{59} - 8 q^{61} - 43 q^{63} - 4 q^{65} - 27 q^{67} + 10 q^{69} - 3 q^{71} + 7 q^{73} - 23 q^{75} + 2 q^{77} - 9 q^{79} - 6 q^{81} - 48 q^{83} - 6 q^{85} + 3 q^{87} + 20 q^{89} - 3 q^{91} - 11 q^{93} - 11 q^{95} + q^{97} - 54 q^{99}+O(q^{100}) $ 10 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 13 * q^9 - 14 * q^11 - 10 * q^13 - 7 * q^15 + 5 * q^17 - 11 * q^19 - 7 * q^23 + 10 * q^25 - 21 * q^27 - 10 * q^29 - 5 * q^31 + 5 * q^33 - 11 * q^35 + 8 * q^37 + 3 * q^39 + 14 * q^41 - 35 * q^43 + 7 * q^45 - 7 * q^49 - 20 * q^51 - 11 * q^53 - 8 * q^55 + 4 * q^57 - 23 * q^59 - 8 * q^61 - 43 * q^63 - 4 * q^65 - 27 * q^67 + 10 * q^69 - 3 * q^71 + 7 * q^73 - 23 * q^75 + 2 * q^77 - 9 * q^79 - 6 * q^81 - 48 * q^83 - 6 * q^85 + 3 * q^87 + 20 * q^89 - 3 * q^91 - 11 * q^93 - 11 * q^95 + q^97 - 54 * 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
$3$	−2.02548	−1.16941	−0.584706	−	0.811245i	$-0.698790\pi$
$3$	−2.02548	−1.16941	−0.584706	+	0.811245i	$0.698790\pi$
$4$	0	0
$5$	−3.01540	−1.34853	−0.674264	−	0.738491i	$-0.735539\pi$
$5$	−3.01540	−1.34853	−0.674264	+	0.738491i	$0.735539\pi$
$6$	0	0
$7$	3.00262	1.13488	0.567442	−	0.823414i	$-0.307933\pi$
$7$	3.00262	1.13488	0.567442	+	0.823414i	$0.307933\pi$
$8$	0	0
$9$	1.10258	0.367525
$10$	0	0
$11$	−3.74558	−1.12933	−0.564667	−	0.825319i	$-0.690995\pi$
$11$	−3.74558	−1.12933	−0.564667	+	0.825319i	$0.690995\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	6.10764	1.57698
$16$	0	0
$17$	1.06592	0.258523	0.129261	−	0.991611i	$-0.458739\pi$
$17$	1.06592	0.258523	0.129261	+	0.991611i	$0.458739\pi$
$18$	0	0
$19$	−1.48031	−0.339606	−0.169803	−	0.985478i	$-0.554313\pi$
$19$	−1.48031	−0.339606	−0.169803	+	0.985478i	$0.554313\pi$
$20$	0	0
$21$	−6.08175	−1.32715
$22$	0	0
$23$	4.25737	0.887723	0.443861	−	0.896095i	$-0.353608\pi$
$23$	4.25737	0.887723	0.443861	+	0.896095i	$0.353608\pi$
$24$	0	0
$25$	4.09263	0.818526
$26$	0	0
$27$	3.84320	0.739624
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−7.29243	−1.30976	−0.654880	−	0.755733i	$-0.727281\pi$
$31$	−7.29243	−1.30976	−0.654880	+	0.755733i	$0.727281\pi$
$32$	0	0
$33$	7.58659	1.32066
$34$	0	0
$35$	−9.05410	−1.53042
$36$	0	0
$37$	7.39773	1.21618	0.608090	−	0.793868i	$-0.291936\pi$
$37$	7.39773	1.21618	0.608090	+	0.793868i	$0.291936\pi$
$38$	0	0
$39$	2.02548	0.324337
$40$	0	0
$41$	10.0578	1.57077	0.785385	−	0.619007i	$-0.212465\pi$
$41$	10.0578	1.57077	0.785385	+	0.619007i	$0.212465\pi$
$42$	0	0
$43$	−4.85699	−0.740684	−0.370342	−	0.928895i	$-0.620760\pi$
$43$	−4.85699	−0.740684	−0.370342	+	0.928895i	$0.620760\pi$
$44$	0	0
$45$	−3.32471	−0.495618
$46$	0	0
$47$	−3.52129	−0.513633	−0.256817	−	0.966460i	$-0.582674\pi$
$47$	−3.52129	−0.513633	−0.256817	+	0.966460i	$0.582674\pi$
$48$	0	0
$49$	2.01572	0.287961
$50$	0	0
$51$	−2.15899	−0.302319
$52$	0	0
$53$	10.0557	1.38126	0.690628	−	0.723210i	$-0.257334\pi$
$53$	10.0557	1.38126	0.690628	+	0.723210i	$0.257334\pi$
$54$	0	0
$55$	11.2944	1.52294
$56$	0	0
$57$	2.99834	0.397140
$58$	0	0
$59$	2.35043	0.306000	0.153000	−	0.988226i	$-0.451107\pi$
$59$	2.35043	0.306000	0.153000	+	0.988226i	$0.451107\pi$
$60$	0	0
$61$	−8.53386	−1.09265	−0.546325	−	0.837573i	$-0.683974\pi$
$61$	−8.53386	−1.09265	−0.546325	+	0.837573i	$0.683974\pi$
$62$	0	0
$63$	3.31062	0.417098
$64$	0	0
$65$	3.01540	0.374014
$66$	0	0
$67$	−4.35843	−0.532468	−0.266234	−	0.963908i	$-0.585779\pi$
$67$	−4.35843	−0.532468	−0.266234	+	0.963908i	$0.585779\pi$
$68$	0	0
$69$	−8.62322	−1.03811
$70$	0	0
$71$	2.86989	0.340593	0.170297	−	0.985393i	$-0.445527\pi$
$71$	2.86989	0.340593	0.170297	+	0.985393i	$0.445527\pi$
$72$	0	0
$73$	11.7590	1.37628	0.688142	−	0.725576i	$-0.258427\pi$
$73$	11.7590	1.37628	0.688142	+	0.725576i	$0.258427\pi$
$74$	0	0
$75$	−8.28955	−0.957195
$76$	0	0
$77$	−11.2465	−1.28166
$78$	0	0
$79$	−9.18923	−1.03387	−0.516935	−	0.856025i	$-0.672927\pi$
$79$	−9.18923	−1.03387	−0.516935	+	0.856025i	$0.672927\pi$
$80$	0	0
$81$	−11.0921	−1.23245
$82$	0	0
$83$	0.299467	0.0328708	0.0164354	−	0.999865i	$-0.494768\pi$
$83$	0.299467	0.0328708	0.0164354	+	0.999865i	$0.494768\pi$
$84$	0	0
$85$	−3.21416	−0.348625
$86$	0	0
$87$	2.02548	0.217154
$88$	0	0
$89$	5.16495	0.547484	0.273742	−	0.961803i	$-0.411739\pi$
$89$	5.16495	0.547484	0.273742	+	0.961803i	$0.411739\pi$
$90$	0	0
$91$	−3.00262	−0.314760
$92$	0	0
$93$	14.7707	1.53165
$94$	0	0
$95$	4.46372	0.457968
$96$	0	0
$97$	−7.20186	−0.731238	−0.365619	−	0.930765i	$-0.619143\pi$
$97$	−7.20186	−0.731238	−0.365619	+	0.930765i	$0.619143\pi$
$98$	0	0
$99$	−4.12978	−0.415059
$100$	0	0
$101$	19.0651	1.89705	0.948524	−	0.316705i	$-0.102576\pi$
$101$	19.0651	1.89705	0.948524	+	0.316705i	$0.102576\pi$
$102$	0	0
$103$	13.6127	1.34130	0.670652	−	0.741772i	$-0.266014\pi$
$103$	13.6127	1.34130	0.670652	+	0.741772i	$0.266014\pi$
$104$	0	0
$105$	18.3389	1.78969
$106$	0	0
$107$	0.550091	0.0531793	0.0265897	−	0.999646i	$-0.491535\pi$
$107$	0.550091	0.0531793	0.0265897	+	0.999646i	$0.491535\pi$
$108$	0	0
$109$	−5.98726	−0.573476	−0.286738	−	0.958009i	$-0.592571\pi$
$109$	−5.98726	−0.573476	−0.286738	+	0.958009i	$0.592571\pi$
$110$	0	0
$111$	−14.9840	−1.42222
$112$	0	0
$113$	14.2933	1.34460	0.672302	−	0.740277i	$-0.265306\pi$
$113$	14.2933	1.34460	0.672302	+	0.740277i	$0.265306\pi$
$114$	0	0
$115$	−12.8377	−1.19712
$116$	0	0
$117$	−1.10258	−0.101933
$118$	0	0
$119$	3.20054	0.293393
$120$	0	0
$121$	3.02933	0.275394
$122$	0	0
$123$	−20.3720	−1.83688
$124$	0	0
$125$	2.73608	0.244722
$126$	0	0
$127$	−2.18977	−0.194311	−0.0971553	−	0.995269i	$-0.530974\pi$
$127$	−2.18977	−0.194311	−0.0971553	+	0.995269i	$0.530974\pi$
$128$	0	0
$129$	9.83774	0.866165
$130$	0	0
$131$	5.61405	0.490502	0.245251	−	0.969460i	$-0.421130\pi$
$131$	5.61405	0.490502	0.245251	+	0.969460i	$0.421130\pi$
$132$	0	0
$133$	−4.44480	−0.385413
$134$	0	0
$135$	−11.5888	−0.997403
$136$	0	0
$137$	14.7017	1.25605	0.628027	−	0.778191i	$-0.283863\pi$
$137$	14.7017	1.25605	0.628027	+	0.778191i	$0.283863\pi$
$138$	0	0
$139$	2.11458	0.179356	0.0896780	−	0.995971i	$-0.471416\pi$
$139$	2.11458	0.179356	0.0896780	+	0.995971i	$0.471416\pi$
$140$	0	0
$141$	7.13232	0.600649
$142$	0	0
$143$	3.74558	0.313221
$144$	0	0
$145$	3.01540	0.250415
$146$	0	0
$147$	−4.08281	−0.336745
$148$	0	0
$149$	−20.9385	−1.71535	−0.857675	−	0.514193i	$-0.828092\pi$
$149$	−20.9385	−1.71535	−0.857675	+	0.514193i	$0.828092\pi$
$150$	0	0
$151$	−16.3741	−1.33251	−0.666253	−	0.745726i	$-0.732103\pi$
$151$	−16.3741	−1.33251	−0.666253	+	0.745726i	$0.732103\pi$
$152$	0	0
$153$	1.17525	0.0950136
$154$	0	0
$155$	21.9896	1.76625
$156$	0	0
$157$	6.95770	0.555285	0.277643	−	0.960684i	$-0.410447\pi$
$157$	6.95770	0.555285	0.277643	+	0.960684i	$0.410447\pi$
$158$	0	0
$159$	−20.3676	−1.61526
$160$	0	0
$161$	12.7833	1.00746
$162$	0	0
$163$	−2.90498	−0.227536	−0.113768	−	0.993507i	$-0.536292\pi$
$163$	−2.90498	−0.227536	−0.113768	+	0.993507i	$0.536292\pi$
$164$	0	0
$165$	−22.8766	−1.78094
$166$	0	0
$167$	−5.23079	−0.404771	−0.202386	−	0.979306i	$-0.564869\pi$
$167$	−5.23079	−0.404771	−0.202386	+	0.979306i	$0.564869\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.63215	−0.124814
$172$	0	0
$173$	−10.8847	−0.827552	−0.413776	−	0.910379i	$-0.635790\pi$
$173$	−10.8847	−0.827552	−0.413776	+	0.910379i	$0.635790\pi$
$174$	0	0
$175$	12.2886	0.928932
$176$	0	0
$177$	−4.76075	−0.357840
$178$	0	0
$179$	−23.0543	−1.72316	−0.861580	−	0.507623i	$-0.830524\pi$
$179$	−23.0543	−1.72316	−0.861580	+	0.507623i	$0.830524\pi$
$180$	0	0
$181$	−20.7692	−1.54376	−0.771881	−	0.635767i	$-0.780684\pi$
$181$	−20.7692	−1.54376	−0.771881	+	0.635767i	$0.780684\pi$
$182$	0	0
$183$	17.2852	1.27776
$184$	0	0
$185$	−22.3071	−1.64005
$186$	0	0
$187$	−3.99247	−0.291958
$188$	0	0
$189$	11.5397	0.839387
$190$	0	0
$191$	9.94962	0.719929	0.359965	−	0.932966i	$-0.382789\pi$
$191$	9.94962	0.719929	0.359965	+	0.932966i	$0.382789\pi$
$192$	0	0
$193$	6.00585	0.432311	0.216155	−	0.976359i	$-0.430648\pi$
$193$	6.00585	0.432311	0.216155	+	0.976359i	$0.430648\pi$
$194$	0	0
$195$	−6.10764	−0.437377
$196$	0	0
$197$	11.5389	0.822115	0.411058	−	0.911609i	$-0.365159\pi$
$197$	11.5389	0.822115	0.411058	+	0.911609i	$0.365159\pi$
$198$	0	0
$199$	−17.2057	−1.21968	−0.609840	−	0.792524i	$-0.708766\pi$
$199$	−17.2057	−1.21968	−0.609840	+	0.792524i	$0.708766\pi$
$200$	0	0
$201$	8.82793	0.622674
$202$	0	0
$203$	−3.00262	−0.210743
$204$	0	0
$205$	−30.3284	−2.11823
$206$	0	0
$207$	4.69407	0.326261
$208$	0	0
$209$	5.54461	0.383528
$210$	0	0
$211$	−7.04104	−0.484725	−0.242363	−	0.970186i	$-0.577922\pi$
$211$	−7.04104	−0.484725	−0.242363	+	0.970186i	$0.577922\pi$
$212$	0	0
$213$	−5.81291	−0.398294
$214$	0	0
$215$	14.6458	0.998833
$216$	0	0
$217$	−21.8964	−1.48642
$218$	0	0
$219$	−23.8176	−1.60944
$220$	0	0
$221$	−1.06592	−0.0717013
$222$	0	0
$223$	19.4359	1.30152	0.650762	−	0.759282i	$-0.274449\pi$
$223$	19.4359	1.30152	0.650762	+	0.759282i	$0.274449\pi$
$224$	0	0
$225$	4.51244	0.300829
$226$	0	0
$227$	13.6876	0.908477	0.454239	−	0.890880i	$-0.349911\pi$
$227$	13.6876	0.908477	0.454239	+	0.890880i	$0.349911\pi$
$228$	0	0
$229$	9.23036	0.609959	0.304980	−	0.952359i	$-0.401350\pi$
$229$	9.23036	0.609959	0.304980	+	0.952359i	$0.401350\pi$
$230$	0	0
$231$	22.7797	1.49879
$232$	0	0
$233$	−16.8571	−1.10435	−0.552173	−	0.833729i	$-0.686201\pi$
$233$	−16.8571	−1.10435	−0.552173	+	0.833729i	$0.686201\pi$
$234$	0	0
$235$	10.6181	0.692649
$236$	0	0
$237$	18.6126	1.20902
$238$	0	0
$239$	−5.62603	−0.363917	−0.181959	−	0.983306i	$-0.558244\pi$
$239$	−5.62603	−0.363917	−0.181959	+	0.983306i	$0.558244\pi$
$240$	0	0
$241$	24.0405	1.54858	0.774292	−	0.632828i	$-0.218106\pi$
$241$	24.0405	1.54858	0.774292	+	0.632828i	$0.218106\pi$
$242$	0	0
$243$	10.9372	0.701619
$244$	0	0
$245$	−6.07821	−0.388323
$246$	0	0
$247$	1.48031	0.0941898
$248$	0	0
$249$	−0.606566	−0.0384395
$250$	0	0
$251$	−13.0204	−0.821842	−0.410921	−	0.911671i	$-0.634793\pi$
$251$	−13.0204	−0.821842	−0.410921	+	0.911671i	$0.634793\pi$
$252$	0	0
$253$	−15.9463	−1.00254
$254$	0	0
$255$	6.51023	0.407686
$256$	0	0
$257$	−16.5067	−1.02966	−0.514829	−	0.857293i	$-0.672145\pi$
$257$	−16.5067	−1.02966	−0.514829	+	0.857293i	$0.672145\pi$
$258$	0	0
$259$	22.2126	1.38022
$260$	0	0
$261$	−1.10258	−0.0682477
$262$	0	0
$263$	−7.50908	−0.463030	−0.231515	−	0.972831i	$-0.574368\pi$
$263$	−7.50908	−0.463030	−0.231515	+	0.972831i	$0.574368\pi$
$264$	0	0
$265$	−30.3219	−1.86266
$266$	0	0
$267$	−10.4615	−0.640234
$268$	0	0
$269$	−14.7828	−0.901323	−0.450662	−	0.892695i	$-0.648812\pi$
$269$	−14.7828	−0.901323	−0.450662	+	0.892695i	$0.648812\pi$
$270$	0	0
$271$	−23.3333	−1.41740	−0.708699	−	0.705511i	$-0.750718\pi$
$271$	−23.3333	−1.41740	−0.708699	+	0.705511i	$0.750718\pi$
$272$	0	0
$273$	6.08175	0.368084
$274$	0	0
$275$	−15.3293	−0.924389
$276$	0	0
$277$	16.7539	1.00664	0.503321	−	0.864100i	$-0.332111\pi$
$277$	16.7539	1.00664	0.503321	+	0.864100i	$0.332111\pi$
$278$	0	0
$279$	−8.04046	−0.481370
$280$	0	0
$281$	−11.8411	−0.706381	−0.353191	−	0.935551i	$-0.614903\pi$
$281$	−11.8411	−0.706381	−0.353191	+	0.935551i	$0.614903\pi$
$282$	0	0
$283$	−14.1832	−0.843101	−0.421550	−	0.906805i	$-0.638514\pi$
$283$	−14.1832	−0.843101	−0.421550	+	0.906805i	$0.638514\pi$
$284$	0	0
$285$	−9.04118	−0.535554
$286$	0	0
$287$	30.1999	1.78264
$288$	0	0
$289$	−15.8638	−0.933166
$290$	0	0
$291$	14.5872	0.855119
$292$	0	0
$293$	0.977197	0.0570885	0.0285442	−	0.999593i	$-0.490913\pi$
$293$	0.977197	0.0570885	0.0285442	+	0.999593i	$0.490913\pi$
$294$	0	0
$295$	−7.08748	−0.412649
$296$	0	0
$297$	−14.3950	−0.835282
$298$	0	0
$299$	−4.25737	−0.246210
$300$	0	0
$301$	−14.5837	−0.840590
$302$	0	0
$303$	−38.6160	−2.21843
$304$	0	0
$305$	25.7330	1.47347
$306$	0	0
$307$	15.8077	0.902194	0.451097	−	0.892475i	$-0.351033\pi$
$307$	15.8077	0.902194	0.451097	+	0.892475i	$0.351033\pi$
$308$	0	0
$309$	−27.5724	−1.56854
$310$	0	0
$311$	7.34913	0.416731	0.208365	−	0.978051i	$-0.433186\pi$
$311$	7.34913	0.416731	0.208365	+	0.978051i	$0.433186\pi$
$312$	0	0
$313$	−21.0496	−1.18980	−0.594898	−	0.803801i	$-0.702807\pi$
$313$	−21.0496	−1.18980	−0.594898	+	0.803801i	$0.702807\pi$
$314$	0	0
$315$	−9.98283	−0.562469
$316$	0	0
$317$	−22.5584	−1.26701	−0.633504	−	0.773740i	$-0.718384\pi$
$317$	−22.5584	−1.26701	−0.633504	+	0.773740i	$0.718384\pi$
$318$	0	0
$319$	3.74558	0.209712
$320$	0	0
$321$	−1.11420	−0.0621885
$322$	0	0
$323$	−1.57788	−0.0877958
$324$	0	0
$325$	−4.09263	−0.227018
$326$	0	0
$327$	12.1271	0.670630
$328$	0	0
$329$	−10.5731	−0.582914
$330$	0	0
$331$	−19.4522	−1.06919	−0.534594	−	0.845109i	$-0.679535\pi$
$331$	−19.4522	−1.06919	−0.534594	+	0.845109i	$0.679535\pi$
$332$	0	0
$333$	8.15656	0.446977
$334$	0	0
$335$	13.1424	0.718047
$336$	0	0
$337$	−24.8466	−1.35348	−0.676742	−	0.736221i	$-0.736609\pi$
$337$	−24.8466	−1.35348	−0.676742	+	0.736221i	$0.736609\pi$
$338$	0	0
$339$	−28.9509	−1.57240
$340$	0	0
$341$	27.3144	1.47916
$342$	0	0
$343$	−14.9659	−0.808082
$344$	0	0
$345$	26.0025	1.39993
$346$	0	0
$347$	14.5650	0.781888	0.390944	−	0.920414i	$-0.372149\pi$
$347$	14.5650	0.781888	0.390944	+	0.920414i	$0.372149\pi$
$348$	0	0
$349$	−29.2080	−1.56347	−0.781733	−	0.623613i	$-0.785664\pi$
$349$	−29.2080	−1.56347	−0.781733	+	0.623613i	$0.785664\pi$
$350$	0	0
$351$	−3.84320	−0.205135
$352$	0	0
$353$	6.88171	0.366276	0.183138	−	0.983087i	$-0.441374\pi$
$353$	6.88171	0.366276	0.183138	+	0.983087i	$0.441374\pi$
$354$	0	0
$355$	−8.65386	−0.459299
$356$	0	0
$357$	−6.48263	−0.343097
$358$	0	0
$359$	15.2457	0.804636	0.402318	−	0.915500i	$-0.368205\pi$
$359$	15.2457	0.804636	0.402318	+	0.915500i	$0.368205\pi$
$360$	0	0
$361$	−16.8087	−0.884668
$362$	0	0
$363$	−6.13586	−0.322049
$364$	0	0
$365$	−35.4580	−1.85596
$366$	0	0
$367$	7.37060	0.384742	0.192371	−	0.981322i	$-0.438382\pi$
$367$	7.37060	0.384742	0.192371	+	0.981322i	$0.438382\pi$
$368$	0	0
$369$	11.0895	0.577298
$370$	0	0
$371$	30.1934	1.56756
$372$	0	0
$373$	−28.2887	−1.46473	−0.732366	−	0.680911i	$-0.761584\pi$
$373$	−28.2887	−1.46473	−0.732366	+	0.680911i	$0.761584\pi$
$374$	0	0
$375$	−5.54188	−0.286181
$376$	0	0
$377$	1.00000	0.0515026
$378$	0	0
$379$	7.43895	0.382113	0.191057	−	0.981579i	$-0.438809\pi$
$379$	7.43895	0.382113	0.191057	+	0.981579i	$0.438809\pi$
$380$	0	0
$381$	4.43534	0.227229
$382$	0	0
$383$	−11.1290	−0.568665	−0.284332	−	0.958726i	$-0.591772\pi$
$383$	−11.1290	−0.568665	−0.284332	+	0.958726i	$0.591772\pi$
$384$	0	0
$385$	33.9128	1.72836
$386$	0	0
$387$	−5.35520	−0.272220
$388$	0	0
$389$	13.7396	0.696624	0.348312	−	0.937379i	$-0.386755\pi$
$389$	13.7396	0.696624	0.348312	+	0.937379i	$0.386755\pi$
$390$	0	0
$391$	4.53800	0.229496
$392$	0	0
$393$	−11.3711	−0.573599
$394$	0	0
$395$	27.7092	1.39420
$396$	0	0
$397$	35.4474	1.77905	0.889526	−	0.456884i	$-0.151034\pi$
$397$	35.4474	1.77905	0.889526	+	0.456884i	$0.151034\pi$
$398$	0	0
$399$	9.00287	0.450707
$400$	0	0
$401$	12.2943	0.613950	0.306975	−	0.951718i	$-0.400683\pi$
$401$	12.2943	0.613950	0.306975	+	0.951718i	$0.400683\pi$
$402$	0	0
$403$	7.29243	0.363262
$404$	0	0
$405$	33.4470	1.66199
$406$	0	0
$407$	−27.7088	−1.37347
$408$	0	0
$409$	−36.3188	−1.79585	−0.897923	−	0.440152i	$-0.854925\pi$
$409$	−36.3188	−1.79585	−0.897923	+	0.440152i	$0.854925\pi$
$410$	0	0
$411$	−29.7781	−1.46885
$412$	0	0
$413$	7.05744	0.347274
$414$	0	0
$415$	−0.903013	−0.0443272
$416$	0	0
$417$	−4.28303	−0.209741
$418$	0	0
$419$	−31.3171	−1.52994	−0.764970	−	0.644066i	$-0.777246\pi$
$419$	−31.3171	−1.52994	−0.764970	+	0.644066i	$0.777246\pi$
$420$	0	0
$421$	20.5265	1.00040	0.500201	−	0.865909i	$-0.333259\pi$
$421$	20.5265	1.00040	0.500201	+	0.865909i	$0.333259\pi$
$422$	0	0
$423$	−3.88249	−0.188773
$424$	0	0
$425$	4.36240	0.211607
$426$	0	0
$427$	−25.6239	−1.24003
$428$	0	0
$429$	−7.58659	−0.366284
$430$	0	0
$431$	−40.0447	−1.92889	−0.964443	−	0.264292i	$-0.914862\pi$
$431$	−40.0447	−1.92889	−0.964443	+	0.264292i	$0.914862\pi$
$432$	0	0
$433$	−23.8756	−1.14739	−0.573694	−	0.819070i	$-0.694490\pi$
$433$	−23.8756	−1.14739	−0.573694	+	0.819070i	$0.694490\pi$
$434$	0	0
$435$	−6.10764	−0.292839
$436$	0	0
$437$	−6.30222	−0.301476
$438$	0	0
$439$	−21.5767	−1.02980	−0.514899	−	0.857251i	$-0.672171\pi$
$439$	−21.5767	−1.02980	−0.514899	+	0.857251i	$0.672171\pi$
$440$	0	0
$441$	2.22249	0.105833
$442$	0	0
$443$	−20.7312	−0.984969	−0.492485	−	0.870321i	$-0.663911\pi$
$443$	−20.7312	−0.984969	−0.492485	+	0.870321i	$0.663911\pi$
$444$	0	0
$445$	−15.5744	−0.738297
$446$	0	0
$447$	42.4106	2.00595
$448$	0	0
$449$	4.02056	0.189742	0.0948709	−	0.995490i	$-0.469756\pi$
$449$	4.02056	0.189742	0.0948709	+	0.995490i	$0.469756\pi$
$450$	0	0
$451$	−37.6724	−1.77392
$452$	0	0
$453$	33.1655	1.55825
$454$	0	0
$455$	9.05410	0.424463
$456$	0	0
$457$	−22.4975	−1.05239	−0.526194	−	0.850365i	$-0.676381\pi$
$457$	−22.4975	−1.05239	−0.526194	+	0.850365i	$0.676381\pi$
$458$	0	0
$459$	4.09653	0.191209
$460$	0	0
$461$	23.2011	1.08058	0.540292	−	0.841477i	$-0.318314\pi$
$461$	23.2011	1.08058	0.540292	+	0.841477i	$0.318314\pi$
$462$	0	0
$463$	−6.00788	−0.279210	−0.139605	−	0.990207i	$-0.544583\pi$
$463$	−6.00788	−0.279210	−0.139605	+	0.990207i	$0.544583\pi$
$464$	0	0
$465$	−44.5395	−2.06547
$466$	0	0
$467$	−7.00255	−0.324039	−0.162020	−	0.986788i	$-0.551801\pi$
$467$	−7.00255	−0.324039	−0.162020	+	0.986788i	$0.551801\pi$
$468$	0	0
$469$	−13.0867	−0.604289
$470$	0	0
$471$	−14.0927	−0.649357
$472$	0	0
$473$	18.1922	0.836479
$474$	0	0
$475$	−6.05836	−0.277976
$476$	0	0
$477$	11.0872	0.507647
$478$	0	0
$479$	−32.9906	−1.50738	−0.753690	−	0.657230i	$-0.771728\pi$
$479$	−32.9906	−1.50738	−0.753690	+	0.657230i	$0.771728\pi$
$480$	0	0
$481$	−7.39773	−0.337308
$482$	0	0
$483$	−25.8923	−1.17814
$484$	0	0
$485$	21.7165	0.986095
$486$	0	0
$487$	26.7959	1.21424	0.607120	−	0.794610i	$-0.292325\pi$
$487$	26.7959	1.21424	0.607120	+	0.794610i	$0.292325\pi$
$488$	0	0
$489$	5.88398	0.266083
$490$	0	0
$491$	−20.0278	−0.903841	−0.451920	−	0.892058i	$-0.649261\pi$
$491$	−20.0278	−0.903841	−0.451920	+	0.892058i	$0.649261\pi$
$492$	0	0
$493$	−1.06592	−0.0480064
$494$	0	0
$495$	12.4529	0.559718
$496$	0	0
$497$	8.61719	0.386534
$498$	0	0
$499$	−1.93352	−0.0865564	−0.0432782	−	0.999063i	$-0.513780\pi$
$499$	−1.93352	−0.0865564	−0.0432782	+	0.999063i	$0.513780\pi$
$500$	0	0
$501$	10.5949	0.473344
$502$	0	0
$503$	4.47807	0.199667	0.0998337	−	0.995004i	$-0.468169\pi$
$503$	4.47807	0.199667	0.0998337	+	0.995004i	$0.468169\pi$
$504$	0	0
$505$	−57.4889	−2.55822
$506$	0	0
$507$	−2.02548	−0.0899548
$508$	0	0
$509$	5.07754	0.225058	0.112529	−	0.993648i	$-0.464105\pi$
$509$	5.07754	0.225058	0.112529	+	0.993648i	$0.464105\pi$
$510$	0	0
$511$	35.3077	1.56192
$512$	0	0
$513$	−5.68912	−0.251181
$514$	0	0
$515$	−41.0479	−1.80879
$516$	0	0
$517$	13.1893	0.580063
$518$	0	0
$519$	22.0468	0.967749
$520$	0	0
$521$	28.6374	1.25463	0.627314	−	0.778766i	$-0.284154\pi$
$521$	28.6374	1.25463	0.627314	+	0.778766i	$0.284154\pi$
$522$	0	0
$523$	17.2184	0.752907	0.376453	−	0.926436i	$-0.377143\pi$
$523$	17.2184	0.752907	0.376453	+	0.926436i	$0.377143\pi$
$524$	0	0
$525$	−24.8904	−1.08630
$526$	0	0
$527$	−7.77312	−0.338602
$528$	0	0
$529$	−4.87481	−0.211948
$530$	0	0
$531$	2.59153	0.112463
$532$	0	0
$533$	−10.0578	−0.435653
$534$	0	0
$535$	−1.65874	−0.0717138
$536$	0	0
$537$	46.6961	2.01508
$538$	0	0
$539$	−7.55005	−0.325204
$540$	0	0
$541$	−12.4453	−0.535067	−0.267534	−	0.963549i	$-0.586209\pi$
$541$	−12.4453	−0.535067	−0.267534	+	0.963549i	$0.586209\pi$
$542$	0	0
$543$	42.0676	1.80529
$544$	0	0
$545$	18.0540	0.773348
$546$	0	0
$547$	23.9280	1.02309	0.511543	−	0.859258i	$-0.329074\pi$
$547$	23.9280	1.02309	0.511543	+	0.859258i	$0.329074\pi$
$548$	0	0
$549$	−9.40923	−0.401576
$550$	0	0
$551$	1.48031	0.0630633
$552$	0	0
$553$	−27.5918	−1.17332
$554$	0	0
$555$	45.1827	1.91790
$556$	0	0
$557$	3.94656	0.167221	0.0836106	−	0.996499i	$-0.473355\pi$
$557$	3.94656	0.167221	0.0836106	+	0.996499i	$0.473355\pi$
$558$	0	0
$559$	4.85699	0.205429
$560$	0	0
$561$	8.08667	0.341420
$562$	0	0
$563$	−15.7630	−0.664333	−0.332167	−	0.943221i	$-0.607780\pi$
$563$	−15.7630	−0.664333	−0.332167	+	0.943221i	$0.607780\pi$
$564$	0	0
$565$	−43.1001	−1.81323
$566$	0	0
$567$	−33.3052	−1.39869
$568$	0	0
$569$	10.5527	0.442394	0.221197	−	0.975229i	$-0.429004\pi$
$569$	10.5527	0.442394	0.221197	+	0.975229i	$0.429004\pi$
$570$	0	0
$571$	−18.7529	−0.784785	−0.392392	−	0.919798i	$-0.628352\pi$
$571$	−18.7529	−0.784785	−0.392392	+	0.919798i	$0.628352\pi$
$572$	0	0
$573$	−20.1528	−0.841894
$574$	0	0
$575$	17.4238	0.726624
$576$	0	0
$577$	−24.9305	−1.03787	−0.518934	−	0.854814i	$-0.673671\pi$
$577$	−24.9305	−1.03787	−0.518934	+	0.854814i	$0.673671\pi$
$578$	0	0
$579$	−12.1647	−0.505549
$580$	0	0
$581$	0.899186	0.0373045
$582$	0	0
$583$	−37.6644	−1.55990
$584$	0	0
$585$	3.32471	0.137460
$586$	0	0
$587$	−18.6383	−0.769285	−0.384643	−	0.923066i	$-0.625675\pi$
$587$	−18.6383	−0.769285	−0.384643	+	0.923066i	$0.625675\pi$
$588$	0	0
$589$	10.7951	0.444802
$590$	0	0
$591$	−23.3719	−0.961392
$592$	0	0
$593$	20.3093	0.834002	0.417001	−	0.908906i	$-0.363081\pi$
$593$	20.3093	0.834002	0.417001	+	0.908906i	$0.363081\pi$
$594$	0	0
$595$	−9.65090	−0.395649
$596$	0	0
$597$	34.8498	1.42631
$598$	0	0
$599$	13.5246	0.552598	0.276299	−	0.961072i	$-0.410892\pi$
$599$	13.5246	0.552598	0.276299	+	0.961072i	$0.410892\pi$
$600$	0	0
$601$	−1.42090	−0.0579596	−0.0289798	−	0.999580i	$-0.509226\pi$
$601$	−1.42090	−0.0579596	−0.0289798	+	0.999580i	$0.509226\pi$
$602$	0	0
$603$	−4.80551	−0.195695
$604$	0	0
$605$	−9.13465	−0.371376
$606$	0	0
$607$	16.2075	0.657844	0.328922	−	0.944357i	$-0.393315\pi$
$607$	16.2075	0.657844	0.328922	+	0.944357i	$0.393315\pi$
$608$	0	0
$609$	6.08175	0.246445
$610$	0	0
$611$	3.52129	0.142456
$612$	0	0
$613$	−38.5135	−1.55555	−0.777773	−	0.628546i	$-0.783650\pi$
$613$	−38.5135	−1.55555	−0.777773	+	0.628546i	$0.783650\pi$
$614$	0	0
$615$	61.4296	2.47708
$616$	0	0
$617$	−4.00132	−0.161087	−0.0805435	−	0.996751i	$-0.525666\pi$
$617$	−4.00132	−0.161087	−0.0805435	+	0.996751i	$0.525666\pi$
$618$	0	0
$619$	22.7343	0.913767	0.456884	−	0.889526i	$-0.348966\pi$
$619$	22.7343	0.913767	0.456884	+	0.889526i	$0.348966\pi$
$620$	0	0
$621$	16.3619	0.656581
$622$	0	0
$623$	15.5084	0.621330
$624$	0	0
$625$	−28.7135	−1.14854
$626$	0	0
$627$	−11.2305	−0.448503
$628$	0	0
$629$	7.88536	0.314410
$630$	0	0
$631$	28.5503	1.13657	0.568285	−	0.822832i	$-0.307607\pi$
$631$	28.5503	1.13657	0.568285	+	0.822832i	$0.307607\pi$
$632$	0	0
$633$	14.2615	0.566844
$634$	0	0
$635$	6.60303	0.262033
$636$	0	0
$637$	−2.01572	−0.0798659
$638$	0	0
$639$	3.16427	0.125177
$640$	0	0
$641$	−0.636234	−0.0251297	−0.0125649	−	0.999921i	$-0.504000\pi$
$641$	−0.636234	−0.0251297	−0.0125649	+	0.999921i	$0.504000\pi$
$642$	0	0
$643$	12.5495	0.494906	0.247453	−	0.968900i	$-0.420406\pi$
$643$	12.5495	0.494906	0.247453	+	0.968900i	$0.420406\pi$
$644$	0	0
$645$	−29.6647	−1.16805
$646$	0	0
$647$	−5.63795	−0.221651	−0.110825	−	0.993840i	$-0.535349\pi$
$647$	−5.63795	−0.221651	−0.110825	+	0.993840i	$0.535349\pi$
$648$	0	0
$649$	−8.80371	−0.345576
$650$	0	0
$651$	44.3508	1.73824
$652$	0	0
$653$	14.1845	0.555082	0.277541	−	0.960714i	$-0.410480\pi$
$653$	14.1845	0.555082	0.277541	+	0.960714i	$0.410480\pi$
$654$	0	0
$655$	−16.9286	−0.661455
$656$	0	0
$657$	12.9652	0.505819
$658$	0	0
$659$	−8.85776	−0.345050	−0.172525	−	0.985005i	$-0.555192\pi$
$659$	−8.85776	−0.345050	−0.172525	+	0.985005i	$0.555192\pi$
$660$	0	0
$661$	25.8063	1.00375	0.501874	−	0.864941i	$-0.332644\pi$
$661$	25.8063	1.00375	0.501874	+	0.864941i	$0.332644\pi$
$662$	0	0
$663$	2.15899	0.0838483
$664$	0	0
$665$	13.4029	0.519740
$666$	0	0
$667$	−4.25737	−0.164846
$668$	0	0
$669$	−39.3671	−1.52202
$670$	0	0
$671$	31.9642	1.23397
$672$	0	0
$673$	22.4531	0.865502	0.432751	−	0.901513i	$-0.357543\pi$
$673$	22.4531	0.865502	0.432751	+	0.901513i	$0.357543\pi$
$674$	0	0
$675$	15.7288	0.605401
$676$	0	0
$677$	−25.9655	−0.997935	−0.498968	−	0.866621i	$-0.666287\pi$
$677$	−25.9655	−0.997935	−0.498968	+	0.866621i	$0.666287\pi$
$678$	0	0
$679$	−21.6244	−0.829870
$680$	0	0
$681$	−27.7240	−1.06238
$682$	0	0
$683$	1.34024	0.0512830	0.0256415	−	0.999671i	$-0.491837\pi$
$683$	1.34024	0.0512830	0.0256415	+	0.999671i	$0.491837\pi$
$684$	0	0
$685$	−44.3316	−1.69382
$686$	0	0
$687$	−18.6959	−0.713294
$688$	0	0
$689$	−10.0557	−0.383092
$690$	0	0
$691$	−0.00242332	−9.21875e−5 0	−4.60938e−5	−	1.00000i	$-0.500015\pi$
$691$	−0.00242332	−9.21875e−5 0	−4.60938e−5		1.00000i	$0.500015\pi$
$692$	0	0
$693$	−12.4002	−0.471043
$694$	0	0
$695$	−6.37629	−0.241866
$696$	0	0
$697$	10.7208	0.406080
$698$	0	0
$699$	34.1438	1.29144
$700$	0	0
$701$	31.9517	1.20680	0.603400	−	0.797438i	$-0.293812\pi$
$701$	31.9517	1.20680	0.603400	+	0.797438i	$0.293812\pi$
$702$	0	0
$703$	−10.9509	−0.413022
$704$	0	0
$705$	−21.5068	−0.809992
$706$	0	0
$707$	57.2452	2.15293
$708$	0	0
$709$	3.67724	0.138102	0.0690509	−	0.997613i	$-0.478003\pi$
$709$	3.67724	0.138102	0.0690509	+	0.997613i	$0.478003\pi$
$710$	0	0
$711$	−10.1318	−0.379973
$712$	0	0
$713$	−31.0466	−1.16270
$714$	0	0
$715$	−11.2944	−0.422387
$716$	0	0
$717$	11.3954	0.425570
$718$	0	0
$719$	21.1445	0.788555	0.394278	−	0.918991i	$-0.370995\pi$
$719$	21.1445	0.788555	0.394278	+	0.918991i	$0.370995\pi$
$720$	0	0
$721$	40.8739	1.52222
$722$	0	0
$723$	−48.6936	−1.81093
$724$	0	0
$725$	−4.09263	−0.151996
$726$	0	0
$727$	47.5978	1.76530	0.882651	−	0.470028i	$-0.155756\pi$
$727$	47.5978	1.76530	0.882651	+	0.470028i	$0.155756\pi$
$728$	0	0
$729$	11.1231	0.411968
$730$	0	0
$731$	−5.17714	−0.191484
$732$	0	0
$733$	9.33916	0.344950	0.172475	−	0.985014i	$-0.444824\pi$
$733$	9.33916	0.344950	0.172475	+	0.985014i	$0.444824\pi$
$734$	0	0
$735$	12.3113	0.454110
$736$	0	0
$737$	16.3248	0.601333
$738$	0	0
$739$	51.9522	1.91109	0.955545	−	0.294844i	$-0.0952677\pi$
$739$	51.9522	1.91109	0.955545	+	0.294844i	$0.0952677\pi$
$740$	0	0
$741$	−2.99834	−0.110147
$742$	0	0
$743$	−24.4451	−0.896802	−0.448401	−	0.893833i	$-0.648006\pi$
$743$	−24.4451	−0.896802	−0.448401	+	0.893833i	$0.648006\pi$
$744$	0	0
$745$	63.1380	2.31320
$746$	0	0
$747$	0.330185	0.0120809
$748$	0	0
$749$	1.65171	0.0603523
$750$	0	0
$751$	−8.05843	−0.294056	−0.147028	−	0.989132i	$-0.546971\pi$
$751$	−8.05843	−0.294056	−0.147028	+	0.989132i	$0.546971\pi$
$752$	0	0
$753$	26.3726	0.961072
$754$	0	0
$755$	49.3745	1.79692
$756$	0	0
$757$	15.9012	0.577941	0.288970	−	0.957338i	$-0.406687\pi$
$757$	15.9012	0.577941	0.288970	+	0.957338i	$0.406687\pi$
$758$	0	0
$759$	32.2989	1.17238
$760$	0	0
$761$	−17.0527	−0.618160	−0.309080	−	0.951036i	$-0.600021\pi$
$761$	−17.0527	−0.618160	−0.309080	+	0.951036i	$0.600021\pi$
$762$	0	0
$763$	−17.9775	−0.650828
$764$	0	0
$765$	−3.54386	−0.128128
$766$	0	0
$767$	−2.35043	−0.0848691
$768$	0	0
$769$	−32.6886	−1.17878	−0.589391	−	0.807848i	$-0.700632\pi$
$769$	−32.6886	−1.17878	−0.589391	+	0.807848i	$0.700632\pi$
$770$	0	0
$771$	33.4340	1.20409
$772$	0	0
$773$	47.6851	1.71511	0.857557	−	0.514389i	$-0.171981\pi$
$773$	47.6851	1.71511	0.857557	+	0.514389i	$0.171981\pi$
$774$	0	0
$775$	−29.8452	−1.07207
$776$	0	0
$777$	−44.9912	−1.61405
$778$	0	0
$779$	−14.8887	−0.533443
$780$	0	0
$781$	−10.7494	−0.384643
$782$	0	0
$783$	−3.84320	−0.137345
$784$	0	0
$785$	−20.9802	−0.748817
$786$	0	0
$787$	12.4120	0.442441	0.221221	−	0.975224i	$-0.428996\pi$
$787$	12.4120	0.442441	0.221221	+	0.975224i	$0.428996\pi$
$788$	0	0
$789$	15.2095	0.541473
$790$	0	0
$791$	42.9174	1.52597
$792$	0	0
$793$	8.53386	0.303046
$794$	0	0
$795$	61.4165	2.17822
$796$	0	0
$797$	−12.8965	−0.456819	−0.228409	−	0.973565i	$-0.573352\pi$
$797$	−12.8965	−0.456819	−0.228409	+	0.973565i	$0.573352\pi$
$798$	0	0
$799$	−3.75340	−0.132786
$800$	0	0
$801$	5.69475	0.201214
$802$	0	0
$803$	−44.0441	−1.55428
$804$	0	0
$805$	−38.5466	−1.35859
$806$	0	0
$807$	29.9423	1.05402
$808$	0	0
$809$	−26.9536	−0.947637	−0.473818	−	0.880623i	$-0.657125\pi$
$809$	−26.9536	−0.947637	−0.473818	+	0.880623i	$0.657125\pi$
$810$	0	0
$811$	−16.4558	−0.577843	−0.288921	−	0.957353i	$-0.593297\pi$
$811$	−16.4558	−0.577843	−0.288921	+	0.957353i	$0.593297\pi$
$812$	0	0
$813$	47.2612	1.65752
$814$	0	0
$815$	8.75968	0.306838
$816$	0	0
$817$	7.18984	0.251541
$818$	0	0
$819$	−3.31062	−0.115682
$820$	0	0
$821$	−21.5271	−0.751302	−0.375651	−	0.926761i	$-0.622581\pi$
$821$	−21.5271	−0.751302	−0.375651	+	0.926761i	$0.622581\pi$
$822$	0	0
$823$	−22.3146	−0.777838	−0.388919	−	0.921272i	$-0.627151\pi$
$823$	−22.3146	−0.777838	−0.388919	+	0.921272i	$0.627151\pi$
$824$	0	0
$825$	31.0491	1.08099
$826$	0	0
$827$	−33.3351	−1.15918	−0.579588	−	0.814910i	$-0.696787\pi$
$827$	−33.3351	−1.15918	−0.579588	+	0.814910i	$0.696787\pi$
$828$	0	0
$829$	42.7848	1.48598	0.742989	−	0.669304i	$-0.233407\pi$
$829$	42.7848	1.48598	0.742989	+	0.669304i	$0.233407\pi$
$830$	0	0
$831$	−33.9346	−1.17718
$832$	0	0
$833$	2.14859	0.0744443
$834$	0	0
$835$	15.7729	0.545845
$836$	0	0
$837$	−28.0263	−0.968730
$838$	0	0
$839$	10.7009	0.369435	0.184718	−	0.982792i	$-0.440863\pi$
$839$	10.7009	0.369435	0.184718	+	0.982792i	$0.440863\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	23.9840	0.826051
$844$	0	0
$845$	−3.01540	−0.103733
$846$	0	0
$847$	9.09594	0.312540
$848$	0	0
$849$	28.7277	0.985933
$850$	0	0
$851$	31.4949	1.07963
$852$	0	0
$853$	46.4852	1.59162	0.795812	−	0.605544i	$-0.207044\pi$
$853$	46.4852	1.59162	0.795812	+	0.605544i	$0.207044\pi$
$854$	0	0
$855$	4.92159	0.168315
$856$	0	0
$857$	−27.1414	−0.927133	−0.463566	−	0.886062i	$-0.653430\pi$
$857$	−27.1414	−0.927133	−0.463566	+	0.886062i	$0.653430\pi$
$858$	0	0
$859$	−25.7915	−0.879994	−0.439997	−	0.897999i	$-0.645020\pi$
$859$	−25.7915	−0.879994	−0.439997	+	0.897999i	$0.645020\pi$
$860$	0	0
$861$	−61.1693	−2.08464
$862$	0	0
$863$	−49.5047	−1.68516	−0.842580	−	0.538571i	$-0.818964\pi$
$863$	−49.5047	−1.68516	−0.842580	+	0.538571i	$0.818964\pi$
$864$	0	0
$865$	32.8218	1.11598
$866$	0	0
$867$	32.1319	1.09126
$868$	0	0
$869$	34.4190	1.16758
$870$	0	0
$871$	4.35843	0.147680
$872$	0	0
$873$	−7.94060	−0.268749
$874$	0	0
$875$	8.21541	0.277731
$876$	0	0
$877$	−28.8185	−0.973131	−0.486566	−	0.873644i	$-0.661751\pi$
$877$	−28.8185	−0.973131	−0.486566	+	0.873644i	$0.661751\pi$
$878$	0	0
$879$	−1.97930	−0.0667600
$880$	0	0
$881$	40.0982	1.35094	0.675472	−	0.737386i	$-0.263940\pi$
$881$	40.0982	1.35094	0.675472	+	0.737386i	$0.263940\pi$
$882$	0	0
$883$	56.6345	1.90590	0.952952	−	0.303121i	$-0.0980286\pi$
$883$	56.6345	1.90590	0.952952	+	0.303121i	$0.0980286\pi$
$884$	0	0
$885$	14.3556	0.482557
$886$	0	0
$887$	49.6772	1.66800	0.833999	−	0.551767i	$-0.186046\pi$
$887$	49.6772	1.66800	0.833999	+	0.551767i	$0.186046\pi$
$888$	0	0
$889$	−6.57504	−0.220520
$890$	0	0
$891$	41.5461	1.39185
$892$	0	0
$893$	5.21260	0.174433
$894$	0	0
$895$	69.5179	2.32373
$896$	0	0
$897$	8.62322	0.287921
$898$	0	0
$899$	7.29243	0.243216
$900$	0	0
$901$	10.7185	0.357086
$902$	0	0
$903$	29.5390	0.982996
$904$	0	0
$905$	62.6274	2.08181
$906$	0	0
$907$	−13.9518	−0.463264	−0.231632	−	0.972804i	$-0.574406\pi$
$907$	−13.9518	−0.463264	−0.231632	+	0.972804i	$0.574406\pi$
$908$	0	0
$909$	21.0207	0.697213
$910$	0	0
$911$	−1.43577	−0.0475691	−0.0237845	−	0.999717i	$-0.507572\pi$
$911$	−1.43577	−0.0475691	−0.0237845	+	0.999717i	$0.507572\pi$
$912$	0	0
$913$	−1.12168	−0.0371221
$914$	0	0
$915$	−52.1217	−1.72309
$916$	0	0
$917$	16.8568	0.556662
$918$	0	0
$919$	−57.4247	−1.89427	−0.947133	−	0.320843i	$-0.896034\pi$
$919$	−57.4247	−1.89427	−0.947133	+	0.320843i	$0.896034\pi$
$920$	0	0
$921$	−32.0183	−1.05504
$922$	0	0
$923$	−2.86989	−0.0944636
$924$	0	0
$925$	30.2762	0.995475
$926$	0	0
$927$	15.0091	0.492963
$928$	0	0
$929$	−16.3337	−0.535890	−0.267945	−	0.963434i	$-0.586345\pi$
$929$	−16.3337	−0.535890	−0.267945	+	0.963434i	$0.586345\pi$
$930$	0	0
$931$	−2.98389	−0.0977932
$932$	0	0
$933$	−14.8855	−0.487330
$934$	0	0
$935$	12.0389	0.393714
$936$	0	0
$937$	21.0420	0.687413	0.343706	−	0.939077i	$-0.388317\pi$
$937$	21.0420	0.687413	0.343706	+	0.939077i	$0.388317\pi$
$938$	0	0
$939$	42.6356	1.39136
$940$	0	0
$941$	56.3274	1.83622	0.918111	−	0.396324i	$-0.129714\pi$
$941$	56.3274	1.83622	0.918111	+	0.396324i	$0.129714\pi$
$942$	0	0
$943$	42.8199	1.39441
$944$	0	0
$945$	−34.7967	−1.13194
$946$	0	0
$947$	−19.2122	−0.624313	−0.312157	−	0.950031i	$-0.601051\pi$
$947$	−19.2122	−0.624313	−0.312157	+	0.950031i	$0.601051\pi$
$948$	0	0
$949$	−11.7590	−0.381712
$950$	0	0
$951$	45.6917	1.48165
$952$	0	0
$953$	51.6808	1.67410	0.837052	−	0.547123i	$-0.184277\pi$
$953$	51.6808	1.67410	0.837052	+	0.547123i	$0.184277\pi$
$954$	0	0
$955$	−30.0021	−0.970844
$956$	0	0
$957$	−7.58659	−0.245240
$958$	0	0
$959$	44.1437	1.42548
$960$	0	0
$961$	22.1796	0.715471
$962$	0	0
$963$	0.606517	0.0195447
$964$	0	0
$965$	−18.1100	−0.582983
$966$	0	0
$967$	23.7860	0.764905	0.382453	−	0.923975i	$-0.375080\pi$
$967$	23.7860	0.764905	0.382453	+	0.923975i	$0.375080\pi$
$968$	0	0
$969$	3.19598	0.102670
$970$	0	0
$971$	32.7702	1.05165	0.525823	−	0.850594i	$-0.323757\pi$
$971$	32.7702	1.05165	0.525823	+	0.850594i	$0.323757\pi$
$972$	0	0
$973$	6.34926	0.203548
$974$	0	0
$975$	8.28955	0.265478
$976$	0	0
$977$	1.33970	0.0428608	0.0214304	−	0.999770i	$-0.493178\pi$
$977$	1.33970	0.0428608	0.0214304	+	0.999770i	$0.493178\pi$
$978$	0	0
$979$	−19.3457	−0.618292
$980$	0	0
$981$	−6.60141	−0.210767
$982$	0	0
$983$	25.2047	0.803907	0.401953	−	0.915660i	$-0.368331\pi$
$983$	25.2047	0.803907	0.401953	+	0.915660i	$0.368331\pi$
$984$	0	0
$985$	−34.7945	−1.10865
$986$	0	0
$987$	21.4156	0.681667
$988$	0	0
$989$	−20.6780	−0.657522
$990$	0	0
$991$	−20.0852	−0.638028	−0.319014	−	0.947750i	$-0.603352\pi$
$991$	−20.0852	−0.638028	−0.319014	+	0.947750i	$0.603352\pi$
$992$	0	0
$993$	39.4000	1.25032
$994$	0	0
$995$	51.8821	1.64477
$996$	0	0
$997$	−38.0106	−1.20381	−0.601903	−	0.798569i	$-0.705591\pi$
$997$	−38.0106	−1.20381	−0.601903	+	0.798569i	$0.705591\pi$
$998$	0	0
$999$	28.4310	0.899516

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.z.1.4		10
4.3	odd	2		3016.2.a.h.1.7	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.h.1.7	✓	10	4.3	odd	2
6032.2.a.z.1.4		10	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:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q-2.02548 q^{3} -3.01540 q^{5} +3.00262 q^{7} +1.10258 q^{9} +O(q^{10})\)	\(q-2.02548 q^{3} -3.01540 q^{5} +3.00262 q^{7} +1.10258 q^{9} -3.74558 q^{11} -1.00000 q^{13} +6.10764 q^{15} +1.06592 q^{17} -1.48031 q^{19} -6.08175 q^{21} +4.25737 q^{23} +4.09263 q^{25} +3.84320 q^{27} -1.00000 q^{29} -7.29243 q^{31} +7.58659 q^{33} -9.05410 q^{35} +7.39773 q^{37} +2.02548 q^{39} +10.0578 q^{41} -4.85699 q^{43} -3.32471 q^{45} -3.52129 q^{47} +2.01572 q^{49} -2.15899 q^{51} +10.0557 q^{53} +11.2944 q^{55} +2.99834 q^{57} +2.35043 q^{59} -8.53386 q^{61} +3.31062 q^{63} +3.01540 q^{65} -4.35843 q^{67} -8.62322 q^{69} +2.86989 q^{71} +11.7590 q^{73} -8.28955 q^{75} -11.2465 q^{77} -9.18923 q^{79} -11.0921 q^{81} +0.299467 q^{83} -3.21416 q^{85} +2.02548 q^{87} +5.16495 q^{89} -3.00262 q^{91} +14.7707 q^{93} +4.46372 q^{95} -7.20186 q^{97} -4.12978 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9}+O(q^{10}) \) 10 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 13 * q^9	\( 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9} - 14 q^{11} - 10 q^{13} - 7 q^{15} + 5 q^{17} - 11 q^{19} - 7 q^{23} + 10 q^{25} - 21 q^{27} - 10 q^{29} - 5 q^{31} + 5 q^{33} - 11 q^{35} + 8 q^{37} + 3 q^{39} + 14 q^{41} - 35 q^{43} + 7 q^{45} - 7 q^{49} - 20 q^{51} - 11 q^{53} - 8 q^{55} + 4 q^{57} - 23 q^{59} - 8 q^{61} - 43 q^{63} - 4 q^{65} - 27 q^{67} + 10 q^{69} - 3 q^{71} + 7 q^{73} - 23 q^{75} + 2 q^{77} - 9 q^{79} - 6 q^{81} - 48 q^{83} - 6 q^{85} + 3 q^{87} + 20 q^{89} - 3 q^{91} - 11 q^{93} - 11 q^{95} + q^{97} - 54 q^{99}+O(q^{100}) \) 10 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 13 * q^9 - 14 * q^11 - 10 * q^13 - 7 * q^15 + 5 * q^17 - 11 * q^19 - 7 * q^23 + 10 * q^25 - 21 * q^27 - 10 * q^29 - 5 * q^31 + 5 * q^33 - 11 * q^35 + 8 * q^37 + 3 * q^39 + 14 * q^41 - 35 * q^43 + 7 * q^45 - 7 * q^49 - 20 * q^51 - 11 * q^53 - 8 * q^55 + 4 * q^57 - 23 * q^59 - 8 * q^61 - 43 * q^63 - 4 * q^65 - 27 * q^67 + 10 * q^69 - 3 * q^71 + 7 * q^73 - 23 * q^75 + 2 * q^77 - 9 * q^79 - 6 * q^81 - 48 * q^83 - 6 * q^85 + 3 * q^87 + 20 * q^89 - 3 * q^91 - 11 * q^93 - 11 * q^95 + q^97 - 54 * q^99

Introduction

L-functions

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