LMFDB - Embedded newform 6032.2.a.bc.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:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5x^{10} - 9x^{9} + 65x^{8} + 19x^{7} - 298x^{6} + 17x^{5} + 541x^{4} - 60x^{3} - 287x^{2} + 16x + 16 $ x^11 - 5x^10 - 9x^9 + 65x^8 + 19x^7 - 298x^6 + 17x^5 + 541x^4 - 60x^3 - 287x^2 + 16x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.86331$ 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)

$f(q)$	$=$	$q-0.863310 q^{3} -1.89826 q^{5} -2.78049 q^{7} -2.25470 q^{9} +O(q^{10})$	$q-0.863310 q^{3} -1.89826 q^{5} -2.78049 q^{7} -2.25470 q^{9} +4.90008 q^{11} +1.00000 q^{13} +1.63879 q^{15} +1.74104 q^{17} -7.28448 q^{19} +2.40042 q^{21} +1.60350 q^{23} -1.39661 q^{25} +4.53643 q^{27} -1.00000 q^{29} +1.09072 q^{31} -4.23029 q^{33} +5.27808 q^{35} -9.64819 q^{37} -0.863310 q^{39} -0.176851 q^{41} -5.66566 q^{43} +4.28000 q^{45} -5.00856 q^{47} +0.731101 q^{49} -1.50306 q^{51} +0.212328 q^{53} -9.30162 q^{55} +6.28876 q^{57} +10.4166 q^{59} -13.5394 q^{61} +6.26915 q^{63} -1.89826 q^{65} +1.20978 q^{67} -1.38431 q^{69} +14.7798 q^{71} -2.26565 q^{73} +1.20571 q^{75} -13.6246 q^{77} -1.66653 q^{79} +2.84774 q^{81} -4.39244 q^{83} -3.30494 q^{85} +0.863310 q^{87} -12.4250 q^{89} -2.78049 q^{91} -0.941631 q^{93} +13.8278 q^{95} -14.5465 q^{97} -11.0482 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 6 q^{3} - 2 q^{5} + 3 q^{7} + 11 q^{9}+O(q^{10}) $ 11 * q + 6 * q^3 - 2 * q^5 + 3 * q^7 + 11 * q^9	$ 11 q + 6 q^{3} - 2 q^{5} + 3 q^{7} + 11 q^{9} + 13 q^{11} + 11 q^{13} + 8 q^{15} - 6 q^{17} + 12 q^{19} + q^{21} + 13 q^{23} + 11 q^{25} + 24 q^{27} - 11 q^{29} + 11 q^{31} + 17 q^{33} + 4 q^{35} + 11 q^{37} + 6 q^{39} - 9 q^{41} + 30 q^{43} - 16 q^{45} + q^{47} - 4 q^{49} + 13 q^{51} - 9 q^{53} + q^{55} + 2 q^{57} + 9 q^{59} - 5 q^{61} + 6 q^{63} - 2 q^{65} + 25 q^{67} + 26 q^{71} + 10 q^{73} + 41 q^{75} - 8 q^{77} + 14 q^{79} + 3 q^{81} + 6 q^{83} + 19 q^{85} - 6 q^{87} - 11 q^{89} + 3 q^{91} - 3 q^{93} + 31 q^{95} + 12 q^{97} + 35 q^{99}+O(q^{100}) $ 11 * q + 6 * q^3 - 2 * q^5 + 3 * q^7 + 11 * q^9 + 13 * q^11 + 11 * q^13 + 8 * q^15 - 6 * q^17 + 12 * q^19 + q^21 + 13 * q^23 + 11 * q^25 + 24 * q^27 - 11 * q^29 + 11 * q^31 + 17 * q^33 + 4 * q^35 + 11 * q^37 + 6 * q^39 - 9 * q^41 + 30 * q^43 - 16 * q^45 + q^47 - 4 * q^49 + 13 * q^51 - 9 * q^53 + q^55 + 2 * q^57 + 9 * q^59 - 5 * q^61 + 6 * q^63 - 2 * q^65 + 25 * q^67 + 26 * q^71 + 10 * q^73 + 41 * q^75 - 8 * q^77 + 14 * q^79 + 3 * q^81 + 6 * q^83 + 19 * q^85 - 6 * q^87 - 11 * q^89 + 3 * q^91 - 3 * q^93 + 31 * q^95 + 12 * q^97 + 35 * 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
$3$	−0.863310	−0.498432	−0.249216	−	0.968448i	$-0.580173\pi$
$3$	−0.863310	−0.498432	−0.249216	+	0.968448i	$0.580173\pi$
$4$	0	0
$5$	−1.89826	−0.848927	−0.424464	−	0.905445i	$-0.639537\pi$
$5$	−1.89826	−0.848927	−0.424464	+	0.905445i	$0.639537\pi$
$6$	0	0
$7$	−2.78049	−1.05092	−0.525462	−	0.850817i	$-0.676108\pi$
$7$	−2.78049	−1.05092	−0.525462	+	0.850817i	$0.676108\pi$
$8$	0	0
$9$	−2.25470	−0.751565
$10$	0	0
$11$	4.90008	1.47743	0.738715	−	0.674018i	$-0.235433\pi$
$11$	4.90008	1.47743	0.738715	+	0.674018i	$0.235433\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.63879	0.423133
$16$	0	0
$17$	1.74104	0.422263	0.211132	−	0.977458i	$-0.432285\pi$
$17$	1.74104	0.422263	0.211132	+	0.977458i	$0.432285\pi$
$18$	0	0
$19$	−7.28448	−1.67117	−0.835587	−	0.549358i	$-0.814872\pi$
$19$	−7.28448	−1.67117	−0.835587	+	0.549358i	$0.814872\pi$
$20$	0	0
$21$	2.40042	0.523815
$22$	0	0
$23$	1.60350	0.334352	0.167176	−	0.985927i	$-0.446535\pi$
$23$	1.60350	0.334352	0.167176	+	0.985927i	$0.446535\pi$
$24$	0	0
$25$	−1.39661	−0.279322
$26$	0	0
$27$	4.53643	0.873037
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	1.09072	0.195899	0.0979496	−	0.995191i	$-0.468772\pi$
$31$	1.09072	0.195899	0.0979496	+	0.995191i	$0.468772\pi$
$32$	0	0
$33$	−4.23029	−0.736399
$34$	0	0
$35$	5.27808	0.892159
$36$	0	0
$37$	−9.64819	−1.58615	−0.793077	−	0.609122i	$-0.791522\pi$
$37$	−9.64819	−1.58615	−0.793077	+	0.609122i	$0.791522\pi$
$38$	0	0
$39$	−0.863310	−0.138240
$40$	0	0
$41$	−0.176851	−0.0276195	−0.0138098	−	0.999905i	$-0.504396\pi$
$41$	−0.176851	−0.0276195	−0.0138098	+	0.999905i	$0.504396\pi$
$42$	0	0
$43$	−5.66566	−0.864005	−0.432003	−	0.901872i	$-0.642193\pi$
$43$	−5.66566	−0.864005	−0.432003	+	0.901872i	$0.642193\pi$
$44$	0	0
$45$	4.28000	0.638024
$46$	0	0
$47$	−5.00856	−0.730574	−0.365287	−	0.930895i	$-0.619029\pi$
$47$	−5.00856	−0.730574	−0.365287	+	0.930895i	$0.619029\pi$
$48$	0	0
$49$	0.731101	0.104443
$50$	0	0
$51$	−1.50306	−0.210470
$52$	0	0
$53$	0.212328	0.0291655	0.0145828	−	0.999894i	$-0.495358\pi$
$53$	0.212328	0.0291655	0.0145828	+	0.999894i	$0.495358\pi$
$54$	0	0
$55$	−9.30162	−1.25423
$56$	0	0
$57$	6.28876	0.832967
$58$	0	0
$59$	10.4166	1.35613	0.678063	−	0.735004i	$-0.262820\pi$
$59$	10.4166	1.35613	0.678063	+	0.735004i	$0.262820\pi$
$60$	0	0
$61$	−13.5394	−1.73355	−0.866774	−	0.498702i	$-0.833810\pi$
$61$	−13.5394	−1.73355	−0.866774	+	0.498702i	$0.833810\pi$
$62$	0	0
$63$	6.26915	0.789838
$64$	0	0
$65$	−1.89826	−0.235450
$66$	0	0
$67$	1.20978	0.147798	0.0738991	−	0.997266i	$-0.476456\pi$
$67$	1.20978	0.147798	0.0738991	+	0.997266i	$0.476456\pi$
$68$	0	0
$69$	−1.38431	−0.166652
$70$	0	0
$71$	14.7798	1.75404	0.877021	−	0.480452i	$-0.159527\pi$
$71$	14.7798	1.75404	0.877021	+	0.480452i	$0.159527\pi$
$72$	0	0
$73$	−2.26565	−0.265174	−0.132587	−	0.991171i	$-0.542328\pi$
$73$	−2.26565	−0.265174	−0.132587	+	0.991171i	$0.542328\pi$
$74$	0	0
$75$	1.20571	0.139223
$76$	0	0
$77$	−13.6246	−1.55267
$78$	0	0
$79$	−1.66653	−0.187499	−0.0937495	−	0.995596i	$-0.529885\pi$
$79$	−1.66653	−0.187499	−0.0937495	+	0.995596i	$0.529885\pi$
$80$	0	0
$81$	2.84774	0.316415
$82$	0	0
$83$	−4.39244	−0.482132	−0.241066	−	0.970509i	$-0.577497\pi$
$83$	−4.39244	−0.482132	−0.241066	+	0.970509i	$0.577497\pi$
$84$	0	0
$85$	−3.30494	−0.358471
$86$	0	0
$87$	0.863310	0.0925566
$88$	0	0
$89$	−12.4250	−1.31705	−0.658525	−	0.752559i	$-0.728819\pi$
$89$	−12.4250	−1.31705	−0.658525	+	0.752559i	$0.728819\pi$
$90$	0	0
$91$	−2.78049	−0.291474
$92$	0	0
$93$	−0.941631	−0.0976426
$94$	0	0
$95$	13.8278	1.41871
$96$	0	0
$97$	−14.5465	−1.47698	−0.738489	−	0.674266i	$-0.764460\pi$
$97$	−14.5465	−1.47698	−0.738489	+	0.674266i	$0.764460\pi$
$98$	0	0
$99$	−11.0482	−1.11038
$100$	0	0
$101$	−4.59994	−0.457711	−0.228855	−	0.973460i	$-0.573498\pi$
$101$	−4.59994	−0.457711	−0.228855	+	0.973460i	$0.573498\pi$
$102$	0	0
$103$	1.93445	0.190607	0.0953037	−	0.995448i	$-0.469618\pi$
$103$	1.93445	0.190607	0.0953037	+	0.995448i	$0.469618\pi$
$104$	0	0
$105$	−4.55662	−0.444681
$106$	0	0
$107$	19.6579	1.90040	0.950199	−	0.311642i	$-0.100879\pi$
$107$	19.6579	1.90040	0.950199	+	0.311642i	$0.100879\pi$
$108$	0	0
$109$	6.32337	0.605669	0.302835	−	0.953043i	$-0.402067\pi$
$109$	6.32337	0.605669	0.302835	+	0.953043i	$0.402067\pi$
$110$	0	0
$111$	8.32939	0.790590
$112$	0	0
$113$	−4.32124	−0.406508	−0.203254	−	0.979126i	$-0.565152\pi$
$113$	−4.32124	−0.406508	−0.203254	+	0.979126i	$0.565152\pi$
$114$	0	0
$115$	−3.04385	−0.283841
$116$	0	0
$117$	−2.25470	−0.208447
$118$	0	0
$119$	−4.84093	−0.443767
$120$	0	0
$121$	13.0108	1.18280
$122$	0	0
$123$	0.152677	0.0137665
$124$	0	0
$125$	12.1424	1.08605
$126$	0	0
$127$	−2.01874	−0.179134	−0.0895672	−	0.995981i	$-0.528548\pi$
$127$	−2.01874	−0.179134	−0.0895672	+	0.995981i	$0.528548\pi$
$128$	0	0
$129$	4.89122	0.430648
$130$	0	0
$131$	−5.54050	−0.484076	−0.242038	−	0.970267i	$-0.577816\pi$
$131$	−5.54050	−0.484076	−0.242038	+	0.970267i	$0.577816\pi$
$132$	0	0
$133$	20.2544	1.75628
$134$	0	0
$135$	−8.61133	−0.741145
$136$	0	0
$137$	19.2738	1.64667	0.823336	−	0.567554i	$-0.192110\pi$
$137$	19.2738	1.64667	0.823336	+	0.567554i	$0.192110\pi$
$138$	0	0
$139$	2.63223	0.223263	0.111631	−	0.993750i	$-0.464392\pi$
$139$	2.63223	0.223263	0.111631	+	0.993750i	$0.464392\pi$
$140$	0	0
$141$	4.32394	0.364142
$142$	0	0
$143$	4.90008	0.409765
$144$	0	0
$145$	1.89826	0.157642
$146$	0	0
$147$	−0.631167	−0.0520578
$148$	0	0
$149$	−1.77360	−0.145299	−0.0726496	−	0.997358i	$-0.523145\pi$
$149$	−1.77360	−0.145299	−0.0726496	+	0.997358i	$0.523145\pi$
$150$	0	0
$151$	−20.9580	−1.70554	−0.852768	−	0.522289i	$-0.825078\pi$
$151$	−20.9580	−1.70554	−0.852768	+	0.522289i	$0.825078\pi$
$152$	0	0
$153$	−3.92551	−0.317359
$154$	0	0
$155$	−2.07047	−0.166304
$156$	0	0
$157$	10.8439	0.865439	0.432719	−	0.901529i	$-0.357554\pi$
$157$	10.8439	0.865439	0.432719	+	0.901529i	$0.357554\pi$
$158$	0	0
$159$	−0.183305	−0.0145370
$160$	0	0
$161$	−4.45850	−0.351379
$162$	0	0
$163$	19.3948	1.51912	0.759558	−	0.650439i	$-0.225415\pi$
$163$	19.3948	1.51912	0.759558	+	0.650439i	$0.225415\pi$
$164$	0	0
$165$	8.03019	0.625149
$166$	0	0
$167$	17.5133	1.35522	0.677611	−	0.735420i	$-0.263015\pi$
$167$	17.5133	1.35522	0.677611	+	0.735420i	$0.263015\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	16.4243	1.25600
$172$	0	0
$173$	−15.5217	−1.18009	−0.590047	−	0.807369i	$-0.700891\pi$
$173$	−15.5217	−1.18009	−0.590047	+	0.807369i	$0.700891\pi$
$174$	0	0
$175$	3.88326	0.293547
$176$	0	0
$177$	−8.99276	−0.675937
$178$	0	0
$179$	20.4640	1.52955	0.764776	−	0.644296i	$-0.222849\pi$
$179$	20.4640	1.52955	0.764776	+	0.644296i	$0.222849\pi$
$180$	0	0
$181$	−2.91692	−0.216813	−0.108406	−	0.994107i	$-0.534575\pi$
$181$	−2.91692	−0.216813	−0.108406	+	0.994107i	$0.534575\pi$
$182$	0	0
$183$	11.6887	0.864056
$184$	0	0
$185$	18.3148	1.34653
$186$	0	0
$187$	8.53122	0.623865
$188$	0	0
$189$	−12.6135	−0.917496
$190$	0	0
$191$	−14.2690	−1.03247	−0.516233	−	0.856448i	$-0.672666\pi$
$191$	−14.2690	−1.03247	−0.516233	+	0.856448i	$0.672666\pi$
$192$	0	0
$193$	26.8306	1.93131	0.965653	−	0.259834i	$-0.0836679\pi$
$193$	26.8306	1.93131	0.965653	+	0.259834i	$0.0836679\pi$
$194$	0	0
$195$	1.63879	0.117356
$196$	0	0
$197$	−12.4966	−0.890346	−0.445173	−	0.895445i	$-0.646858\pi$
$197$	−12.4966	−0.890346	−0.445173	+	0.895445i	$0.646858\pi$
$198$	0	0
$199$	−21.7030	−1.53848	−0.769242	−	0.638958i	$-0.779366\pi$
$199$	−21.7030	−1.53848	−0.769242	+	0.638958i	$0.779366\pi$
$200$	0	0
$201$	−1.04442	−0.0736675
$202$	0	0
$203$	2.78049	0.195152
$204$	0	0
$205$	0.335709	0.0234470
$206$	0	0
$207$	−3.61540	−0.251287
$208$	0	0
$209$	−35.6945	−2.46904
$210$	0	0
$211$	2.66956	0.183780	0.0918900	−	0.995769i	$-0.470709\pi$
$211$	2.66956	0.183780	0.0918900	+	0.995769i	$0.470709\pi$
$212$	0	0
$213$	−12.7596	−0.874272
$214$	0	0
$215$	10.7549	0.733478
$216$	0	0
$217$	−3.03273	−0.205875
$218$	0	0
$219$	1.95596	0.132171
$220$	0	0
$221$	1.74104	0.117115
$222$	0	0
$223$	17.2084	1.15236	0.576180	−	0.817323i	$-0.304543\pi$
$223$	17.2084	1.15236	0.576180	+	0.817323i	$0.304543\pi$
$224$	0	0
$225$	3.14893	0.209929
$226$	0	0
$227$	11.8656	0.787545	0.393772	−	0.919208i	$-0.371170\pi$
$227$	11.8656	0.787545	0.393772	+	0.919208i	$0.371170\pi$
$228$	0	0
$229$	16.5346	1.09264	0.546320	−	0.837577i	$-0.316028\pi$
$229$	16.5346	1.09264	0.546320	+	0.837577i	$0.316028\pi$
$230$	0	0
$231$	11.7623	0.773900
$232$	0	0
$233$	19.8056	1.29751	0.648753	−	0.760999i	$-0.275291\pi$
$233$	19.8056	1.29751	0.648753	+	0.760999i	$0.275291\pi$
$234$	0	0
$235$	9.50755	0.620204
$236$	0	0
$237$	1.43873	0.0934556
$238$	0	0
$239$	−5.24855	−0.339500	−0.169750	−	0.985487i	$-0.554296\pi$
$239$	−5.24855	−0.339500	−0.169750	+	0.985487i	$0.554296\pi$
$240$	0	0
$241$	17.5913	1.13315	0.566577	−	0.824009i	$-0.308267\pi$
$241$	17.5913	1.13315	0.566577	+	0.824009i	$0.308267\pi$
$242$	0	0
$243$	−16.0678	−1.03075
$244$	0	0
$245$	−1.38782	−0.0886645
$246$	0	0
$247$	−7.28448	−0.463500
$248$	0	0
$249$	3.79203	0.240310
$250$	0	0
$251$	27.8744	1.75942	0.879709	−	0.475513i	$-0.157737\pi$
$251$	27.8744	1.75942	0.879709	+	0.475513i	$0.157737\pi$
$252$	0	0
$253$	7.85726	0.493982
$254$	0	0
$255$	2.85319	0.178674
$256$	0	0
$257$	26.8787	1.67665	0.838325	−	0.545172i	$-0.183535\pi$
$257$	26.8787	1.67665	0.838325	+	0.545172i	$0.183535\pi$
$258$	0	0
$259$	26.8267	1.66693
$260$	0	0
$261$	2.25470	0.139562
$262$	0	0
$263$	9.45301	0.582897	0.291449	−	0.956586i	$-0.405863\pi$
$263$	9.45301	0.582897	0.291449	+	0.956586i	$0.405863\pi$
$264$	0	0
$265$	−0.403054	−0.0247594
$266$	0	0
$267$	10.7266	0.656460
$268$	0	0
$269$	−18.8010	−1.14632	−0.573160	−	0.819444i	$-0.694283\pi$
$269$	−18.8010	−1.14632	−0.573160	+	0.819444i	$0.694283\pi$
$270$	0	0
$271$	10.5205	0.639077	0.319539	−	0.947573i	$-0.396472\pi$
$271$	10.5205	0.639077	0.319539	+	0.947573i	$0.396472\pi$
$272$	0	0
$273$	2.40042	0.145280
$274$	0	0
$275$	−6.84351	−0.412679
$276$	0	0
$277$	−13.7057	−0.823495	−0.411748	−	0.911298i	$-0.635081\pi$
$277$	−13.7057	−0.823495	−0.411748	+	0.911298i	$0.635081\pi$
$278$	0	0
$279$	−2.45924	−0.147231
$280$	0	0
$281$	12.9043	0.769807	0.384904	−	0.922957i	$-0.374235\pi$
$281$	12.9043	0.769807	0.384904	+	0.922957i	$0.374235\pi$
$282$	0	0
$283$	24.0665	1.43060	0.715302	−	0.698815i	$-0.246289\pi$
$283$	24.0665	1.43060	0.715302	+	0.698815i	$0.246289\pi$
$284$	0	0
$285$	−11.9377	−0.707129
$286$	0	0
$287$	0.491732	0.0290260
$288$	0	0
$289$	−13.9688	−0.821694
$290$	0	0
$291$	12.5582	0.736173
$292$	0	0
$293$	16.5302	0.965702	0.482851	−	0.875702i	$-0.339601\pi$
$293$	16.5302	0.965702	0.482851	+	0.875702i	$0.339601\pi$
$294$	0	0
$295$	−19.7734	−1.15125
$296$	0	0
$297$	22.2289	1.28985
$298$	0	0
$299$	1.60350	0.0927326
$300$	0	0
$301$	15.7533	0.908004
$302$	0	0
$303$	3.97117	0.228138
$304$	0	0
$305$	25.7014	1.47166
$306$	0	0
$307$	−19.8091	−1.13057	−0.565283	−	0.824897i	$-0.691233\pi$
$307$	−19.8091	−1.13057	−0.565283	+	0.824897i	$0.691233\pi$
$308$	0	0
$309$	−1.67003	−0.0950049
$310$	0	0
$311$	6.65451	0.377343	0.188671	−	0.982040i	$-0.439582\pi$
$311$	6.65451	0.377343	0.188671	+	0.982040i	$0.439582\pi$
$312$	0	0
$313$	−18.4444	−1.04254	−0.521268	−	0.853393i	$-0.674541\pi$
$313$	−18.4444	−1.04254	−0.521268	+	0.853393i	$0.674541\pi$
$314$	0	0
$315$	−11.9005	−0.670516
$316$	0	0
$317$	2.60277	0.146186	0.0730931	−	0.997325i	$-0.476713\pi$
$317$	2.60277	0.146186	0.0730931	+	0.997325i	$0.476713\pi$
$318$	0	0
$319$	−4.90008	−0.274352
$320$	0	0
$321$	−16.9708	−0.947220
$322$	0	0
$323$	−12.6825	−0.705676
$324$	0	0
$325$	−1.39661	−0.0774700
$326$	0	0
$327$	−5.45903	−0.301885
$328$	0	0
$329$	13.9262	0.767778
$330$	0	0
$331$	9.60152	0.527747	0.263874	−	0.964557i	$-0.415000\pi$
$331$	9.60152	0.527747	0.263874	+	0.964557i	$0.415000\pi$
$332$	0	0
$333$	21.7537	1.19210
$334$	0	0
$335$	−2.29648	−0.125470
$336$	0	0
$337$	−24.1793	−1.31713	−0.658565	−	0.752524i	$-0.728836\pi$
$337$	−24.1793	−1.31713	−0.658565	+	0.752524i	$0.728836\pi$
$338$	0	0
$339$	3.73057	0.202617
$340$	0	0
$341$	5.34462	0.289427
$342$	0	0
$343$	17.4306	0.941163
$344$	0	0
$345$	2.62779	0.141475
$346$	0	0
$347$	24.0472	1.29092	0.645462	−	0.763793i	$-0.276665\pi$
$347$	24.0472	1.29092	0.645462	+	0.763793i	$0.276665\pi$
$348$	0	0
$349$	32.2147	1.72441	0.862207	−	0.506555i	$-0.169081\pi$
$349$	32.2147	1.72441	0.862207	+	0.506555i	$0.169081\pi$
$350$	0	0
$351$	4.53643	0.242137
$352$	0	0
$353$	−22.2197	−1.18263	−0.591317	−	0.806439i	$-0.701392\pi$
$353$	−22.2197	−1.18263	−0.591317	+	0.806439i	$0.701392\pi$
$354$	0	0
$355$	−28.0559	−1.48905
$356$	0	0
$357$	4.17922	0.221188
$358$	0	0
$359$	−5.00227	−0.264010	−0.132005	−	0.991249i	$-0.542141\pi$
$359$	−5.00227	−0.264010	−0.132005	+	0.991249i	$0.542141\pi$
$360$	0	0
$361$	34.0636	1.79282
$362$	0	0
$363$	−11.2323	−0.589545
$364$	0	0
$365$	4.30079	0.225114
$366$	0	0
$367$	12.8773	0.672188	0.336094	−	0.941828i	$-0.390894\pi$
$367$	12.8773	0.672188	0.336094	+	0.941828i	$0.390894\pi$
$368$	0	0
$369$	0.398745	0.0207579
$370$	0	0
$371$	−0.590375	−0.0306508
$372$	0	0
$373$	34.5671	1.78982	0.894908	−	0.446251i	$-0.147241\pi$
$373$	34.5671	1.78982	0.894908	+	0.446251i	$0.147241\pi$
$374$	0	0
$375$	−10.4827	−0.541323
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	−18.1425	−0.931919	−0.465960	−	0.884806i	$-0.654291\pi$
$379$	−18.1425	−0.931919	−0.465960	+	0.884806i	$0.654291\pi$
$380$	0	0
$381$	1.74280	0.0892864
$382$	0	0
$383$	−26.0269	−1.32991	−0.664956	−	0.746883i	$-0.731550\pi$
$383$	−26.0269	−1.32991	−0.664956	+	0.746883i	$0.731550\pi$
$384$	0	0
$385$	25.8630	1.31810
$386$	0	0
$387$	12.7743	0.649356
$388$	0	0
$389$	−11.5475	−0.585482	−0.292741	−	0.956192i	$-0.594567\pi$
$389$	−11.5475	−0.585482	−0.292741	+	0.956192i	$0.594567\pi$
$390$	0	0
$391$	2.79175	0.141185
$392$	0	0
$393$	4.78317	0.241279
$394$	0	0
$395$	3.16350	0.159173
$396$	0	0
$397$	−6.02119	−0.302195	−0.151098	−	0.988519i	$-0.548281\pi$
$397$	−6.02119	−0.302195	−0.151098	+	0.988519i	$0.548281\pi$
$398$	0	0
$399$	−17.4858	−0.875386
$400$	0	0
$401$	−5.93556	−0.296408	−0.148204	−	0.988957i	$-0.547349\pi$
$401$	−5.93556	−0.296408	−0.148204	+	0.988957i	$0.547349\pi$
$402$	0	0
$403$	1.09072	0.0543327
$404$	0	0
$405$	−5.40574	−0.268614
$406$	0	0
$407$	−47.2769	−2.34343
$408$	0	0
$409$	−17.4578	−0.863232	−0.431616	−	0.902057i	$-0.642057\pi$
$409$	−17.4578	−0.863232	−0.431616	+	0.902057i	$0.642057\pi$
$410$	0	0
$411$	−16.6393	−0.820755
$412$	0	0
$413$	−28.9632	−1.42519
$414$	0	0
$415$	8.33798	0.409295
$416$	0	0
$417$	−2.27243	−0.111281
$418$	0	0
$419$	−23.9068	−1.16792	−0.583962	−	0.811781i	$-0.698498\pi$
$419$	−23.9068	−1.16792	−0.583962	+	0.811781i	$0.698498\pi$
$420$	0	0
$421$	18.9848	0.925263	0.462632	−	0.886551i	$-0.346905\pi$
$421$	18.9848	0.925263	0.462632	+	0.886551i	$0.346905\pi$
$422$	0	0
$423$	11.2928	0.549074
$424$	0	0
$425$	−2.43155	−0.117948
$426$	0	0
$427$	37.6462	1.82183
$428$	0	0
$429$	−4.23029	−0.204240
$430$	0	0
$431$	−17.8651	−0.860534	−0.430267	−	0.902702i	$-0.641581\pi$
$431$	−17.8651	−0.860534	−0.430267	+	0.902702i	$0.641581\pi$
$432$	0	0
$433$	−26.7168	−1.28393	−0.641963	−	0.766735i	$-0.721880\pi$
$433$	−26.7168	−1.28393	−0.641963	+	0.766735i	$0.721880\pi$
$434$	0	0
$435$	−1.63879	−0.0785738
$436$	0	0
$437$	−11.6806	−0.558760
$438$	0	0
$439$	−8.39629	−0.400733	−0.200366	−	0.979721i	$-0.564213\pi$
$439$	−8.39629	−0.400733	−0.200366	+	0.979721i	$0.564213\pi$
$440$	0	0
$441$	−1.64841	−0.0784957
$442$	0	0
$443$	28.5643	1.35713	0.678565	−	0.734541i	$-0.262602\pi$
$443$	28.5643	1.35713	0.678565	+	0.734541i	$0.262602\pi$
$444$	0	0
$445$	23.5859	1.11808
$446$	0	0
$447$	1.53117	0.0724218
$448$	0	0
$449$	30.4259	1.43589	0.717943	−	0.696102i	$-0.245084\pi$
$449$	30.4259	1.43589	0.717943	+	0.696102i	$0.245084\pi$
$450$	0	0
$451$	−0.866585	−0.0408059
$452$	0	0
$453$	18.0932	0.850095
$454$	0	0
$455$	5.27808	0.247440
$456$	0	0
$457$	−11.1582	−0.521956	−0.260978	−	0.965345i	$-0.584045\pi$
$457$	−11.1582	−0.521956	−0.260978	+	0.965345i	$0.584045\pi$
$458$	0	0
$459$	7.89810	0.368652
$460$	0	0
$461$	16.9273	0.788382	0.394191	−	0.919028i	$-0.371025\pi$
$461$	16.9273	0.788382	0.394191	+	0.919028i	$0.371025\pi$
$462$	0	0
$463$	−32.4467	−1.50792	−0.753962	−	0.656918i	$-0.771860\pi$
$463$	−32.4467	−1.50792	−0.753962	+	0.656918i	$0.771860\pi$
$464$	0	0
$465$	1.78746	0.0828914
$466$	0	0
$467$	−23.2433	−1.07557	−0.537787	−	0.843081i	$-0.680740\pi$
$467$	−23.2433	−1.07557	−0.537787	+	0.843081i	$0.680740\pi$
$468$	0	0
$469$	−3.36378	−0.155325
$470$	0	0
$471$	−9.36167	−0.431363
$472$	0	0
$473$	−27.7622	−1.27651
$474$	0	0
$475$	10.1736	0.466796
$476$	0	0
$477$	−0.478735	−0.0219198
$478$	0	0
$479$	6.66306	0.304443	0.152221	−	0.988346i	$-0.451357\pi$
$479$	6.66306	0.304443	0.152221	+	0.988346i	$0.451357\pi$
$480$	0	0
$481$	−9.64819	−0.439920
$482$	0	0
$483$	3.84907	0.175139
$484$	0	0
$485$	27.6131	1.25385
$486$	0	0
$487$	−22.3919	−1.01467	−0.507337	−	0.861748i	$-0.669370\pi$
$487$	−22.3919	−1.01467	−0.507337	+	0.861748i	$0.669370\pi$
$488$	0	0
$489$	−16.7437	−0.757177
$490$	0	0
$491$	15.2637	0.688842	0.344421	−	0.938815i	$-0.388075\pi$
$491$	15.2637	0.688842	0.344421	+	0.938815i	$0.388075\pi$
$492$	0	0
$493$	−1.74104	−0.0784124
$494$	0	0
$495$	20.9723	0.942636
$496$	0	0
$497$	−41.0951	−1.84337
$498$	0	0
$499$	−21.8830	−0.979618	−0.489809	−	0.871830i	$-0.662933\pi$
$499$	−21.8830	−0.979618	−0.489809	+	0.871830i	$0.662933\pi$
$500$	0	0
$501$	−15.1194	−0.675487
$502$	0	0
$503$	−8.64196	−0.385326	−0.192663	−	0.981265i	$-0.561712\pi$
$503$	−8.64196	−0.385326	−0.192663	+	0.981265i	$0.561712\pi$
$504$	0	0
$505$	8.73187	0.388563
$506$	0	0
$507$	−0.863310	−0.0383410
$508$	0	0
$509$	1.86160	0.0825140	0.0412570	−	0.999149i	$-0.486864\pi$
$509$	1.86160	0.0825140	0.0412570	+	0.999149i	$0.486864\pi$
$510$	0	0
$511$	6.29961	0.278678
$512$	0	0
$513$	−33.0455	−1.45900
$514$	0	0
$515$	−3.67210	−0.161812
$516$	0	0
$517$	−24.5423	−1.07937
$518$	0	0
$519$	13.4000	0.588197
$520$	0	0
$521$	37.8417	1.65788	0.828938	−	0.559341i	$-0.188946\pi$
$521$	37.8417	1.65788	0.828938	+	0.559341i	$0.188946\pi$
$522$	0	0
$523$	27.9374	1.22162	0.610808	−	0.791779i	$-0.290845\pi$
$523$	27.9374	1.22162	0.610808	+	0.791779i	$0.290845\pi$
$524$	0	0
$525$	−3.35246	−0.146313
$526$	0	0
$527$	1.89899	0.0827211
$528$	0	0
$529$	−20.4288	−0.888209
$530$	0	0
$531$	−23.4863	−1.01922
$532$	0	0
$533$	−0.176851	−0.00766027
$534$	0	0
$535$	−37.3158	−1.61330
$536$	0	0
$537$	−17.6668	−0.762379
$538$	0	0
$539$	3.58245	0.154307
$540$	0	0
$541$	−28.8890	−1.24204	−0.621018	−	0.783796i	$-0.713281\pi$
$541$	−28.8890	−1.24204	−0.621018	+	0.783796i	$0.713281\pi$
$542$	0	0
$543$	2.51821	0.108067
$544$	0	0
$545$	−12.0034	−0.514169
$546$	0	0
$547$	−11.5679	−0.494607	−0.247303	−	0.968938i	$-0.579544\pi$
$547$	−11.5679	−0.494607	−0.247303	+	0.968938i	$0.579544\pi$
$548$	0	0
$549$	30.5273	1.30287
$550$	0	0
$551$	7.28448	0.310329
$552$	0	0
$553$	4.63376	0.197047
$554$	0	0
$555$	−15.8113	−0.671154
$556$	0	0
$557$	25.2689	1.07068	0.535340	−	0.844637i	$-0.320184\pi$
$557$	25.2689	1.07068	0.535340	+	0.844637i	$0.320184\pi$
$558$	0	0
$559$	−5.66566	−0.239632
$560$	0	0
$561$	−7.36509	−0.310954
$562$	0	0
$563$	11.0204	0.464454	0.232227	−	0.972662i	$-0.425399\pi$
$563$	11.0204	0.464454	0.232227	+	0.972662i	$0.425399\pi$
$564$	0	0
$565$	8.20283	0.345096
$566$	0	0
$567$	−7.91809	−0.332529
$568$	0	0
$569$	−14.5426	−0.609658	−0.304829	−	0.952407i	$-0.598599\pi$
$569$	−14.5426	−0.609658	−0.304829	+	0.952407i	$0.598599\pi$
$570$	0	0
$571$	18.7328	0.783944	0.391972	−	0.919977i	$-0.371793\pi$
$571$	18.7328	0.783944	0.391972	+	0.919977i	$0.371793\pi$
$572$	0	0
$573$	12.3186	0.514615
$574$	0	0
$575$	−2.23946	−0.0933920
$576$	0	0
$577$	13.5178	0.562753	0.281377	−	0.959597i	$-0.409209\pi$
$577$	13.5178	0.562753	0.281377	+	0.959597i	$0.409209\pi$
$578$	0	0
$579$	−23.1631	−0.962626
$580$	0	0
$581$	12.2131	0.506685
$582$	0	0
$583$	1.04043	0.0430900
$584$	0	0
$585$	4.28000	0.176956
$586$	0	0
$587$	31.6401	1.30592	0.652962	−	0.757390i	$-0.273526\pi$
$587$	31.6401	1.30592	0.652962	+	0.757390i	$0.273526\pi$
$588$	0	0
$589$	−7.94533	−0.327382
$590$	0	0
$591$	10.7884	0.443777
$592$	0	0
$593$	−10.9588	−0.450024	−0.225012	−	0.974356i	$-0.572242\pi$
$593$	−10.9588	−0.450024	−0.225012	+	0.974356i	$0.572242\pi$
$594$	0	0
$595$	9.18934	0.376726
$596$	0	0
$597$	18.7364	0.766830
$598$	0	0
$599$	−27.3475	−1.11739	−0.558694	−	0.829374i	$-0.688697\pi$
$599$	−27.3475	−1.11739	−0.558694	+	0.829374i	$0.688697\pi$
$600$	0	0
$601$	29.6660	1.21010	0.605050	−	0.796188i	$-0.293153\pi$
$601$	29.6660	1.21010	0.605050	+	0.796188i	$0.293153\pi$
$602$	0	0
$603$	−2.72769	−0.111080
$604$	0	0
$605$	−24.6978	−1.00411
$606$	0	0
$607$	−10.4939	−0.425933	−0.212967	−	0.977059i	$-0.568313\pi$
$607$	−10.4939	−0.425933	−0.212967	+	0.977059i	$0.568313\pi$
$608$	0	0
$609$	−2.40042	−0.0972700
$610$	0	0
$611$	−5.00856	−0.202625
$612$	0	0
$613$	41.7147	1.68484	0.842421	−	0.538821i	$-0.181130\pi$
$613$	41.7147	1.68484	0.842421	+	0.538821i	$0.181130\pi$
$614$	0	0
$615$	−0.289821	−0.0116867
$616$	0	0
$617$	−12.7079	−0.511601	−0.255801	−	0.966730i	$-0.582339\pi$
$617$	−12.7079	−0.511601	−0.255801	+	0.966730i	$0.582339\pi$
$618$	0	0
$619$	28.5339	1.14687	0.573437	−	0.819250i	$-0.305610\pi$
$619$	28.5339	1.14687	0.573437	+	0.819250i	$0.305610\pi$
$620$	0	0
$621$	7.27415	0.291902
$622$	0	0
$623$	34.5476	1.38412
$624$	0	0
$625$	−16.0664	−0.642657
$626$	0	0
$627$	30.8154	1.23065
$628$	0	0
$629$	−16.7979	−0.669775
$630$	0	0
$631$	−9.71260	−0.386652	−0.193326	−	0.981135i	$-0.561928\pi$
$631$	−9.71260	−0.386652	−0.193326	+	0.981135i	$0.561928\pi$
$632$	0	0
$633$	−2.30466	−0.0916019
$634$	0	0
$635$	3.83210	0.152072
$636$	0	0
$637$	0.731101	0.0289673
$638$	0	0
$639$	−33.3240	−1.31828
$640$	0	0
$641$	39.8848	1.57536	0.787678	−	0.616087i	$-0.211283\pi$
$641$	39.8848	1.57536	0.787678	+	0.616087i	$0.211283\pi$
$642$	0	0
$643$	25.1095	0.990223	0.495111	−	0.868830i	$-0.335127\pi$
$643$	25.1095	0.990223	0.495111	+	0.868830i	$0.335127\pi$
$644$	0	0
$645$	−9.28481	−0.365589
$646$	0	0
$647$	−20.7174	−0.814485	−0.407243	−	0.913320i	$-0.633510\pi$
$647$	−20.7174	−0.814485	−0.407243	+	0.913320i	$0.633510\pi$
$648$	0	0
$649$	51.0422	2.00358
$650$	0	0
$651$	2.61819	0.102615
$652$	0	0
$653$	3.48099	0.136222	0.0681108	−	0.997678i	$-0.478303\pi$
$653$	3.48099	0.136222	0.0681108	+	0.997678i	$0.478303\pi$
$654$	0	0
$655$	10.5173	0.410945
$656$	0	0
$657$	5.10835	0.199296
$658$	0	0
$659$	20.9082	0.814469	0.407234	−	0.913324i	$-0.366493\pi$
$659$	20.9082	0.814469	0.407234	+	0.913324i	$0.366493\pi$
$660$	0	0
$661$	42.6254	1.65793	0.828967	−	0.559298i	$-0.188929\pi$
$661$	42.6254	1.65793	0.828967	+	0.559298i	$0.188929\pi$
$662$	0	0
$663$	−1.50306	−0.0583738
$664$	0	0
$665$	−38.4481	−1.49095
$666$	0	0
$667$	−1.60350	−0.0620876
$668$	0	0
$669$	−14.8562	−0.574373
$670$	0	0
$671$	−66.3443	−2.56119
$672$	0	0
$673$	−22.5316	−0.868530	−0.434265	−	0.900785i	$-0.642992\pi$
$673$	−22.5316	−0.868530	−0.434265	+	0.900785i	$0.642992\pi$
$674$	0	0
$675$	−6.33563	−0.243859
$676$	0	0
$677$	−24.2545	−0.932175	−0.466088	−	0.884739i	$-0.654337\pi$
$677$	−24.2545	−0.932175	−0.466088	+	0.884739i	$0.654337\pi$
$678$	0	0
$679$	40.4464	1.55219
$680$	0	0
$681$	−10.2437	−0.392538
$682$	0	0
$683$	−28.7917	−1.10168	−0.550841	−	0.834610i	$-0.685693\pi$
$683$	−28.7917	−1.10168	−0.550841	+	0.834610i	$0.685693\pi$
$684$	0	0
$685$	−36.5867	−1.39791
$686$	0	0
$687$	−14.2745	−0.544607
$688$	0	0
$689$	0.212328	0.00808906
$690$	0	0
$691$	32.9400	1.25310	0.626548	−	0.779383i	$-0.284467\pi$
$691$	32.9400	1.25310	0.626548	+	0.779383i	$0.284467\pi$
$692$	0	0
$693$	30.7193	1.16693
$694$	0	0
$695$	−4.99665	−0.189534
$696$	0	0
$697$	−0.307904	−0.0116627
$698$	0	0
$699$	−17.0984	−0.646719
$700$	0	0
$701$	27.5319	1.03986	0.519932	−	0.854208i	$-0.325957\pi$
$701$	27.5319	1.03986	0.519932	+	0.854208i	$0.325957\pi$
$702$	0	0
$703$	70.2821	2.65074
$704$	0	0
$705$	−8.20796	−0.309130
$706$	0	0
$707$	12.7901	0.481020
$708$	0	0
$709$	32.1569	1.20768	0.603839	−	0.797107i	$-0.293637\pi$
$709$	32.1569	1.20768	0.603839	+	0.797107i	$0.293637\pi$
$710$	0	0
$711$	3.75751	0.140918
$712$	0	0
$713$	1.74897	0.0654993
$714$	0	0
$715$	−9.30162	−0.347861
$716$	0	0
$717$	4.53113	0.169218
$718$	0	0
$719$	19.3925	0.723218	0.361609	−	0.932330i	$-0.382227\pi$
$719$	19.3925	0.723218	0.361609	+	0.932330i	$0.382227\pi$
$720$	0	0
$721$	−5.37872	−0.200314
$722$	0	0
$723$	−15.1867	−0.564801
$724$	0	0
$725$	1.39661	0.0518688
$726$	0	0
$727$	−4.67303	−0.173313	−0.0866565	−	0.996238i	$-0.527618\pi$
$727$	−4.67303	−0.173313	−0.0866565	+	0.996238i	$0.527618\pi$
$728$	0	0
$729$	5.32827	0.197343
$730$	0	0
$731$	−9.86412	−0.364838
$732$	0	0
$733$	9.22639	0.340784	0.170392	−	0.985376i	$-0.445497\pi$
$733$	9.22639	0.340784	0.170392	+	0.985376i	$0.445497\pi$
$734$	0	0
$735$	1.19812	0.0441933
$736$	0	0
$737$	5.92802	0.218362
$738$	0	0
$739$	−51.6435	−1.89974	−0.949869	−	0.312649i	$-0.898783\pi$
$739$	−51.6435	−1.89974	−0.949869	+	0.312649i	$0.898783\pi$
$740$	0	0
$741$	6.28876	0.231024
$742$	0	0
$743$	39.8257	1.46106	0.730532	−	0.682879i	$-0.239272\pi$
$743$	39.8257	1.46106	0.730532	+	0.682879i	$0.239272\pi$
$744$	0	0
$745$	3.36676	0.123348
$746$	0	0
$747$	9.90360	0.362354
$748$	0	0
$749$	−54.6585	−1.99718
$750$	0	0
$751$	−6.78795	−0.247696	−0.123848	−	0.992301i	$-0.539523\pi$
$751$	−6.78795	−0.247696	−0.123848	+	0.992301i	$0.539523\pi$
$752$	0	0
$753$	−24.0643	−0.876951
$754$	0	0
$755$	39.7837	1.44788
$756$	0	0
$757$	1.12088	0.0407391	0.0203695	−	0.999793i	$-0.493516\pi$
$757$	1.12088	0.0407391	0.0203695	+	0.999793i	$0.493516\pi$
$758$	0	0
$759$	−6.78325	−0.246216
$760$	0	0
$761$	5.79102	0.209924	0.104962	−	0.994476i	$-0.466528\pi$
$761$	5.79102	0.209924	0.104962	+	0.994476i	$0.466528\pi$
$762$	0	0
$763$	−17.5820	−0.636513
$764$	0	0
$765$	7.45163	0.269414
$766$	0	0
$767$	10.4166	0.376122
$768$	0	0
$769$	−21.5510	−0.777149	−0.388575	−	0.921417i	$-0.627032\pi$
$769$	−21.5510	−0.777149	−0.388575	+	0.921417i	$0.627032\pi$
$770$	0	0
$771$	−23.2047	−0.835696
$772$	0	0
$773$	13.3142	0.478878	0.239439	−	0.970911i	$-0.423037\pi$
$773$	13.3142	0.478878	0.239439	+	0.970911i	$0.423037\pi$
$774$	0	0
$775$	−1.52331	−0.0547190
$776$	0	0
$777$	−23.1597	−0.830851
$778$	0	0
$779$	1.28827	0.0461570
$780$	0	0
$781$	72.4223	2.59147
$782$	0	0
$783$	−4.53643	−0.162119
$784$	0	0
$785$	−20.5846	−0.734695
$786$	0	0
$787$	31.9476	1.13881	0.569404	−	0.822058i	$-0.307174\pi$
$787$	31.9476	1.13881	0.569404	+	0.822058i	$0.307174\pi$
$788$	0	0
$789$	−8.16088	−0.290535
$790$	0	0
$791$	12.0151	0.427209
$792$	0	0
$793$	−13.5394	−0.480799
$794$	0	0
$795$	0.347961	0.0123409
$796$	0	0
$797$	43.7225	1.54873	0.774366	−	0.632738i	$-0.218069\pi$
$797$	43.7225	1.54873	0.774366	+	0.632738i	$0.218069\pi$
$798$	0	0
$799$	−8.72009	−0.308495
$800$	0	0
$801$	28.0146	0.989848
$802$	0	0
$803$	−11.1019	−0.391776
$804$	0	0
$805$	8.46339	0.298295
$806$	0	0
$807$	16.2311	0.571363
$808$	0	0
$809$	−19.6699	−0.691556	−0.345778	−	0.938316i	$-0.612385\pi$
$809$	−19.6699	−0.691556	−0.345778	+	0.938316i	$0.612385\pi$
$810$	0	0
$811$	−8.08351	−0.283850	−0.141925	−	0.989877i	$-0.545329\pi$
$811$	−8.08351	−0.283850	−0.141925	+	0.989877i	$0.545329\pi$
$812$	0	0
$813$	−9.08249	−0.318537
$814$	0	0
$815$	−36.8163	−1.28962
$816$	0	0
$817$	41.2714	1.44390
$818$	0	0
$819$	6.26915	0.219062
$820$	0	0
$821$	3.69254	0.128871	0.0644354	−	0.997922i	$-0.479475\pi$
$821$	3.69254	0.128871	0.0644354	+	0.997922i	$0.479475\pi$
$822$	0	0
$823$	11.8428	0.412813	0.206406	−	0.978466i	$-0.433823\pi$
$823$	11.8428	0.412813	0.206406	+	0.978466i	$0.433823\pi$
$824$	0	0
$825$	5.90807	0.205693
$826$	0	0
$827$	−10.5017	−0.365179	−0.182590	−	0.983189i	$-0.558448\pi$
$827$	−10.5017	−0.365179	−0.182590	+	0.983189i	$0.558448\pi$
$828$	0	0
$829$	−21.3738	−0.742341	−0.371171	−	0.928565i	$-0.621044\pi$
$829$	−21.3738	−0.742341	−0.371171	+	0.928565i	$0.621044\pi$
$830$	0	0
$831$	11.8323	0.410457
$832$	0	0
$833$	1.27287	0.0441025
$834$	0	0
$835$	−33.2449	−1.15049
$836$	0	0
$837$	4.94798	0.171027
$838$	0	0
$839$	−31.8560	−1.09979	−0.549896	−	0.835233i	$-0.685333\pi$
$839$	−31.8560	−1.09979	−0.549896	+	0.835233i	$0.685333\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−11.1404	−0.383697
$844$	0	0
$845$	−1.89826	−0.0653021
$846$	0	0
$847$	−36.1763	−1.24303
$848$	0	0
$849$	−20.7769	−0.713060
$850$	0	0
$851$	−15.4708	−0.530334
$852$	0	0
$853$	−8.10697	−0.277578	−0.138789	−	0.990322i	$-0.544321\pi$
$853$	−8.10697	−0.277578	−0.138789	+	0.990322i	$0.544321\pi$
$854$	0	0
$855$	−31.1775	−1.06625
$856$	0	0
$857$	−10.0324	−0.342699	−0.171349	−	0.985210i	$-0.554813\pi$
$857$	−10.0324	−0.342699	−0.171349	+	0.985210i	$0.554813\pi$
$858$	0	0
$859$	−31.9558	−1.09032	−0.545160	−	0.838332i	$-0.683531\pi$
$859$	−31.9558	−1.09032	−0.545160	+	0.838332i	$0.683531\pi$
$860$	0	0
$861$	−0.424517	−0.0144675
$862$	0	0
$863$	−5.74024	−0.195400	−0.0977000	−	0.995216i	$-0.531149\pi$
$863$	−5.74024	−0.195400	−0.0977000	+	0.995216i	$0.531149\pi$
$864$	0	0
$865$	29.4642	1.00181
$866$	0	0
$867$	12.0594	0.409559
$868$	0	0
$869$	−8.16612	−0.277017
$870$	0	0
$871$	1.20978	0.0409919
$872$	0	0
$873$	32.7980	1.11004
$874$	0	0
$875$	−33.7618	−1.14136
$876$	0	0
$877$	33.5065	1.13143	0.565717	−	0.824600i	$-0.308600\pi$
$877$	33.5065	1.13143	0.565717	+	0.824600i	$0.308600\pi$
$878$	0	0
$879$	−14.2707	−0.481337
$880$	0	0
$881$	16.4464	0.554093	0.277047	−	0.960856i	$-0.410644\pi$
$881$	16.4464	0.554093	0.277047	+	0.960856i	$0.410644\pi$
$882$	0	0
$883$	58.1865	1.95813	0.979065	−	0.203546i	$-0.0652467\pi$
$883$	58.1865	1.95813	0.979065	+	0.203546i	$0.0652467\pi$
$884$	0	0
$885$	17.0706	0.573821
$886$	0	0
$887$	−41.4855	−1.39295	−0.696473	−	0.717583i	$-0.745249\pi$
$887$	−41.4855	−1.39295	−0.696473	+	0.717583i	$0.745249\pi$
$888$	0	0
$889$	5.61308	0.188257
$890$	0	0
$891$	13.9541	0.467481
$892$	0	0
$893$	36.4847	1.22092
$894$	0	0
$895$	−38.8460	−1.29848
$896$	0	0
$897$	−1.38431	−0.0462209
$898$	0	0
$899$	−1.09072	−0.0363776
$900$	0	0
$901$	0.369671	0.0123155
$902$	0	0
$903$	−13.6000	−0.452579
$904$	0	0
$905$	5.53707	0.184058
$906$	0	0
$907$	−44.6277	−1.48184	−0.740920	−	0.671593i	$-0.765610\pi$
$907$	−44.6277	−1.48184	−0.740920	+	0.671593i	$0.765610\pi$
$908$	0	0
$909$	10.3715	0.343999
$910$	0	0
$911$	−55.7830	−1.84817	−0.924087	−	0.382181i	$-0.875173\pi$
$911$	−55.7830	−1.84817	−0.924087	+	0.382181i	$0.875173\pi$
$912$	0	0
$913$	−21.5233	−0.712317
$914$	0	0
$915$	−22.1882	−0.733521
$916$	0	0
$917$	15.4053	0.508727
$918$	0	0
$919$	33.6091	1.10866	0.554332	−	0.832296i	$-0.312974\pi$
$919$	33.6091	1.10866	0.554332	+	0.832296i	$0.312974\pi$
$920$	0	0
$921$	17.1014	0.563511
$922$	0	0
$923$	14.7798	0.486484
$924$	0	0
$925$	13.4748	0.443048
$926$	0	0
$927$	−4.36160	−0.143254
$928$	0	0
$929$	2.73310	0.0896701	0.0448350	−	0.998994i	$-0.485724\pi$
$929$	2.73310	0.0896701	0.0448350	+	0.998994i	$0.485724\pi$
$930$	0	0
$931$	−5.32569	−0.174542
$932$	0	0
$933$	−5.74491	−0.188080
$934$	0	0
$935$	−16.1945	−0.529616
$936$	0	0
$937$	−3.81133	−0.124511	−0.0622553	−	0.998060i	$-0.519829\pi$
$937$	−3.81133	−0.124511	−0.0622553	+	0.998060i	$0.519829\pi$
$938$	0	0
$939$	15.9232	0.519634
$940$	0	0
$941$	−37.0039	−1.20629	−0.603147	−	0.797630i	$-0.706087\pi$
$941$	−37.0039	−1.20629	−0.603147	+	0.797630i	$0.706087\pi$
$942$	0	0
$943$	−0.283580	−0.00923464
$944$	0	0
$945$	23.9437	0.778888
$946$	0	0
$947$	−1.04837	−0.0340675	−0.0170338	−	0.999855i	$-0.505422\pi$
$947$	−1.04837	−0.0340675	−0.0170338	+	0.999855i	$0.505422\pi$
$948$	0	0
$949$	−2.26565	−0.0735461
$950$	0	0
$951$	−2.24700	−0.0728639
$952$	0	0
$953$	3.31199	0.107286	0.0536429	−	0.998560i	$-0.482917\pi$
$953$	3.31199	0.107286	0.0536429	+	0.998560i	$0.482917\pi$
$954$	0	0
$955$	27.0862	0.876490
$956$	0	0
$957$	4.23029	0.136746
$958$	0	0
$959$	−53.5905	−1.73053
$960$	0	0
$961$	−29.8103	−0.961623
$962$	0	0
$963$	−44.3225	−1.42827
$964$	0	0
$965$	−50.9314	−1.63954
$966$	0	0
$967$	12.9979	0.417984	0.208992	−	0.977917i	$-0.432982\pi$
$967$	12.9979	0.417984	0.208992	+	0.977917i	$0.432982\pi$
$968$	0	0
$969$	10.9490	0.351732
$970$	0	0
$971$	37.5144	1.20389	0.601947	−	0.798536i	$-0.294392\pi$
$971$	37.5144	1.20389	0.601947	+	0.798536i	$0.294392\pi$
$972$	0	0
$973$	−7.31888	−0.234632
$974$	0	0
$975$	1.20571	0.0386136
$976$	0	0
$977$	4.31649	0.138097	0.0690484	−	0.997613i	$-0.478004\pi$
$977$	4.31649	0.138097	0.0690484	+	0.997613i	$0.478004\pi$
$978$	0	0
$979$	−60.8836	−1.94585
$980$	0	0
$981$	−14.2573	−0.455200
$982$	0	0
$983$	18.6893	0.596096	0.298048	−	0.954551i	$-0.403664\pi$
$983$	18.6893	0.596096	0.298048	+	0.954551i	$0.403664\pi$
$984$	0	0
$985$	23.7218	0.755839
$986$	0	0
$987$	−12.0227	−0.382685
$988$	0	0
$989$	−9.08486	−0.288882
$990$	0	0
$991$	19.8119	0.629344	0.314672	−	0.949200i	$-0.398105\pi$
$991$	19.8119	0.629344	0.314672	+	0.949200i	$0.398105\pi$
$992$	0	0
$993$	−8.28909	−0.263046
$994$	0	0
$995$	41.1979	1.30606
$996$	0	0
$997$	−9.05091	−0.286645	−0.143323	−	0.989676i	$-0.545779\pi$
$997$	−9.05091	−0.286645	−0.143323	+	0.989676i	$0.545779\pi$
$998$	0	0
$999$	−43.7684	−1.38477

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.bc.1.4		11
4.3	odd	2		3016.2.a.i.1.8	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.i.1.8	✓	11	4.3	odd	2
6032.2.a.bc.1.4		11	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)

\(f(q)\)	\(=\)	\(q-0.863310 q^{3} -1.89826 q^{5} -2.78049 q^{7} -2.25470 q^{9} +O(q^{10})\)	\(q-0.863310 q^{3} -1.89826 q^{5} -2.78049 q^{7} -2.25470 q^{9} +4.90008 q^{11} +1.00000 q^{13} +1.63879 q^{15} +1.74104 q^{17} -7.28448 q^{19} +2.40042 q^{21} +1.60350 q^{23} -1.39661 q^{25} +4.53643 q^{27} -1.00000 q^{29} +1.09072 q^{31} -4.23029 q^{33} +5.27808 q^{35} -9.64819 q^{37} -0.863310 q^{39} -0.176851 q^{41} -5.66566 q^{43} +4.28000 q^{45} -5.00856 q^{47} +0.731101 q^{49} -1.50306 q^{51} +0.212328 q^{53} -9.30162 q^{55} +6.28876 q^{57} +10.4166 q^{59} -13.5394 q^{61} +6.26915 q^{63} -1.89826 q^{65} +1.20978 q^{67} -1.38431 q^{69} +14.7798 q^{71} -2.26565 q^{73} +1.20571 q^{75} -13.6246 q^{77} -1.66653 q^{79} +2.84774 q^{81} -4.39244 q^{83} -3.30494 q^{85} +0.863310 q^{87} -12.4250 q^{89} -2.78049 q^{91} -0.941631 q^{93} +13.8278 q^{95} -14.5465 q^{97} -11.0482 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 6 q^{3} - 2 q^{5} + 3 q^{7} + 11 q^{9}+O(q^{10}) \) 11 * q + 6 * q^3 - 2 * q^5 + 3 * q^7 + 11 * q^9	\( 11 q + 6 q^{3} - 2 q^{5} + 3 q^{7} + 11 q^{9} + 13 q^{11} + 11 q^{13} + 8 q^{15} - 6 q^{17} + 12 q^{19} + q^{21} + 13 q^{23} + 11 q^{25} + 24 q^{27} - 11 q^{29} + 11 q^{31} + 17 q^{33} + 4 q^{35} + 11 q^{37} + 6 q^{39} - 9 q^{41} + 30 q^{43} - 16 q^{45} + q^{47} - 4 q^{49} + 13 q^{51} - 9 q^{53} + q^{55} + 2 q^{57} + 9 q^{59} - 5 q^{61} + 6 q^{63} - 2 q^{65} + 25 q^{67} + 26 q^{71} + 10 q^{73} + 41 q^{75} - 8 q^{77} + 14 q^{79} + 3 q^{81} + 6 q^{83} + 19 q^{85} - 6 q^{87} - 11 q^{89} + 3 q^{91} - 3 q^{93} + 31 q^{95} + 12 q^{97} + 35 q^{99}+O(q^{100}) \) 11 * q + 6 * q^3 - 2 * q^5 + 3 * q^7 + 11 * q^9 + 13 * q^11 + 11 * q^13 + 8 * q^15 - 6 * q^17 + 12 * q^19 + q^21 + 13 * q^23 + 11 * q^25 + 24 * q^27 - 11 * q^29 + 11 * q^31 + 17 * q^33 + 4 * q^35 + 11 * q^37 + 6 * q^39 - 9 * q^41 + 30 * q^43 - 16 * q^45 + q^47 - 4 * q^49 + 13 * q^51 - 9 * q^53 + q^55 + 2 * q^57 + 9 * q^59 - 5 * q^61 + 6 * q^63 - 2 * q^65 + 25 * q^67 + 26 * q^71 + 10 * q^73 + 41 * q^75 - 8 * q^77 + 14 * q^79 + 3 * q^81 + 6 * q^83 + 19 * q^85 - 6 * q^87 - 11 * q^89 + 3 * q^91 - 3 * q^93 + 31 * q^95 + 12 * q^97 + 35 * q^99

Introduction

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