LMFDB - Embedded newform 6012.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(1,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, 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("6012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.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:	$48.0060616952$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 31x^{7} + 24x^{6} + 293x^{5} - 101x^{4} - 864x^{3} - 278x^{2} + 24x + 2 $ x^9 - x^8 - 31x^7 + 24x^6 + 293x^5 - 101x^4 - 864x^3 - 278x^2 + 24*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2004)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.25977$ of defining polynomial
Character	$\chi$	$=$	6012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.25977 q^{5} +1.95900 q^{7} +O(q^{10})$	$q-3.25977 q^{5} +1.95900 q^{7} -1.86584 q^{11} +4.83769 q^{13} +4.17274 q^{17} +5.05426 q^{19} +1.26074 q^{23} +5.62609 q^{25} -10.5500 q^{29} -6.26256 q^{31} -6.38588 q^{35} +0.409655 q^{37} +2.65442 q^{41} +0.317178 q^{43} +4.08939 q^{47} -3.16233 q^{49} +7.15844 q^{53} +6.08221 q^{55} +8.72693 q^{59} +9.89631 q^{61} -15.7697 q^{65} -6.55284 q^{67} +10.0764 q^{71} -11.8701 q^{73} -3.65518 q^{77} -2.70687 q^{79} -6.18316 q^{83} -13.6022 q^{85} -14.6820 q^{89} +9.47702 q^{91} -16.4757 q^{95} +10.3373 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{5} + 2 q^{7}+O(q^{10}) $ 9 * q - q^5 + 2 * q^7	$ 9 q - q^{5} + 2 q^{7} + 9 q^{11} + 10 q^{13} - 7 q^{17} - 2 q^{19} + 3 q^{23} + 18 q^{25} - 5 q^{29} + 12 q^{31} + 6 q^{35} + 15 q^{37} - 14 q^{41} + 6 q^{43} + 3 q^{47} + 27 q^{49} - 9 q^{53} + 19 q^{55} + 9 q^{59} + 30 q^{61} - 28 q^{65} + 16 q^{67} + 3 q^{71} + 32 q^{73} - 18 q^{77} + 24 q^{79} + 3 q^{83} + 37 q^{85} - 46 q^{89} + 33 q^{91} - 11 q^{95} + 43 q^{97}+O(q^{100}) $ 9 * q - q^5 + 2 * q^7 + 9 * q^11 + 10 * q^13 - 7 * q^17 - 2 * q^19 + 3 * q^23 + 18 * q^25 - 5 * q^29 + 12 * q^31 + 6 * q^35 + 15 * q^37 - 14 * q^41 + 6 * q^43 + 3 * q^47 + 27 * q^49 - 9 * q^53 + 19 * q^55 + 9 * q^59 + 30 * q^61 - 28 * q^65 + 16 * q^67 + 3 * q^71 + 32 * q^73 - 18 * q^77 + 24 * q^79 + 3 * q^83 + 37 * q^85 - 46 * q^89 + 33 * q^91 - 11 * q^95 + 43 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	−3.25977	−1.45781	−0.728906	−	0.684613i	$-0.759971\pi$
$5$	−3.25977	−1.45781	−0.728906	+	0.684613i	$0.759971\pi$
$6$	0	0
$7$	1.95900	0.740431	0.370216	−	0.928946i	$-0.379284\pi$
$7$	1.95900	0.740431	0.370216	+	0.928946i	$0.379284\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.86584	−0.562572	−0.281286	−	0.959624i	$-0.590761\pi$
$11$	−1.86584	−0.562572	−0.281286	+	0.959624i	$0.590761\pi$
$12$	0	0
$13$	4.83769	1.34173	0.670867	−	0.741578i	$-0.265922\pi$
$13$	4.83769	1.34173	0.670867	+	0.741578i	$0.265922\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.17274	1.01204	0.506019	−	0.862523i	$-0.331117\pi$
$17$	4.17274	1.01204	0.506019	+	0.862523i	$0.331117\pi$
$18$	0	0
$19$	5.05426	1.15953	0.579764	−	0.814785i	$-0.303145\pi$
$19$	5.05426	1.15953	0.579764	+	0.814785i	$0.303145\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.26074	0.262883	0.131441	−	0.991324i	$-0.458039\pi$
$23$	1.26074	0.262883	0.131441	+	0.991324i	$0.458039\pi$
$24$	0	0
$25$	5.62609	1.12522
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−10.5500	−1.95909	−0.979543	−	0.201233i	$-0.935505\pi$
$29$	−10.5500	−1.95909	−0.979543	+	0.201233i	$0.935505\pi$
$30$	0	0
$31$	−6.26256	−1.12479	−0.562394	−	0.826869i	$-0.690120\pi$
$31$	−6.26256	−1.12479	−0.562394	+	0.826869i	$0.690120\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−6.38588	−1.07941
$36$	0	0
$37$	0.409655	0.0673469	0.0336734	−	0.999433i	$-0.489279\pi$
$37$	0.409655	0.0673469	0.0336734	+	0.999433i	$0.489279\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.65442	0.414550	0.207275	−	0.978283i	$-0.433540\pi$
$41$	2.65442	0.414550	0.207275	+	0.978283i	$0.433540\pi$
$42$	0	0
$43$	0.317178	0.0483692	0.0241846	−	0.999708i	$-0.492301\pi$
$43$	0.317178	0.0483692	0.0241846	+	0.999708i	$0.492301\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.08939	0.596499	0.298249	−	0.954488i	$-0.403597\pi$
$47$	4.08939	0.596499	0.298249	+	0.954488i	$0.403597\pi$
$48$	0	0
$49$	−3.16233	−0.451761
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.15844	0.983287	0.491644	−	0.870796i	$-0.336396\pi$
$53$	7.15844	0.983287	0.491644	+	0.870796i	$0.336396\pi$
$54$	0	0
$55$	6.08221	0.820125
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.72693	1.13615	0.568074	−	0.822977i	$-0.307689\pi$
$59$	8.72693	1.13615	0.568074	+	0.822977i	$0.307689\pi$
$60$	0	0
$61$	9.89631	1.26709	0.633546	−	0.773705i	$-0.281599\pi$
$61$	9.89631	1.26709	0.633546	+	0.773705i	$0.281599\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−15.7697	−1.95600
$66$	0	0
$67$	−6.55284	−0.800557	−0.400278	−	0.916394i	$-0.631087\pi$
$67$	−6.55284	−0.800557	−0.400278	+	0.916394i	$0.631087\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.0764	1.19585	0.597927	−	0.801550i	$-0.295991\pi$
$71$	10.0764	1.19585	0.597927	+	0.801550i	$0.295991\pi$
$72$	0	0
$73$	−11.8701	−1.38929	−0.694647	−	0.719351i	$-0.744439\pi$
$73$	−11.8701	−1.38929	−0.694647	+	0.719351i	$0.744439\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.65518	−0.416546
$78$	0	0
$79$	−2.70687	−0.304547	−0.152273	−	0.988338i	$-0.548659\pi$
$79$	−2.70687	−0.304547	−0.152273	+	0.988338i	$0.548659\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.18316	−0.678690	−0.339345	−	0.940662i	$-0.610205\pi$
$83$	−6.18316	−0.678690	−0.339345	+	0.940662i	$0.610205\pi$
$84$	0	0
$85$	−13.6022	−1.47536
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.6820	−1.55629	−0.778147	−	0.628082i	$-0.783840\pi$
$89$	−14.6820	−1.55629	−0.778147	+	0.628082i	$0.783840\pi$
$90$	0	0
$91$	9.47702	0.993461
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−16.4757	−1.69037
$96$	0	0
$97$	10.3373	1.04959	0.524796	−	0.851228i	$-0.324142\pi$
$97$	10.3373	1.04959	0.524796	+	0.851228i	$0.324142\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.40969	−0.737292	−0.368646	−	0.929570i	$-0.620178\pi$
$101$	−7.40969	−0.737292	−0.368646	+	0.929570i	$0.620178\pi$
$102$	0	0
$103$	2.02321	0.199353	0.0996763	−	0.995020i	$-0.468219\pi$
$103$	2.02321	0.199353	0.0996763	+	0.995020i	$0.468219\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.1811	−0.984240	−0.492120	−	0.870527i	$-0.663778\pi$
$107$	−10.1811	−0.984240	−0.492120	+	0.870527i	$0.663778\pi$
$108$	0	0
$109$	17.0121	1.62946	0.814730	−	0.579840i	$-0.196885\pi$
$109$	17.0121	1.62946	0.814730	+	0.579840i	$0.196885\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.94341	−0.276892	−0.138446	−	0.990370i	$-0.544211\pi$
$113$	−2.94341	−0.276892	−0.138446	+	0.990370i	$0.544211\pi$
$114$	0	0
$115$	−4.10973	−0.383234
$116$	0	0
$117$	0	0
$118$	0	0
$119$	8.17438	0.749344
$120$	0	0
$121$	−7.51863	−0.683512
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2.04091	−0.182545
$126$	0	0
$127$	−14.1274	−1.25360	−0.626802	−	0.779179i	$-0.715636\pi$
$127$	−14.1274	−1.25360	−0.626802	+	0.779179i	$0.715636\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.26678	0.722272	0.361136	−	0.932513i	$-0.382389\pi$
$131$	8.26678	0.722272	0.361136	+	0.932513i	$0.382389\pi$
$132$	0	0
$133$	9.90129	0.858550
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.27285	0.535926	0.267963	−	0.963429i	$-0.413650\pi$
$137$	6.27285	0.535926	0.267963	+	0.963429i	$0.413650\pi$
$138$	0	0
$139$	16.3243	1.38461	0.692304	−	0.721606i	$-0.256596\pi$
$139$	16.3243	1.38461	0.692304	+	0.721606i	$0.256596\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−9.02636	−0.754822
$144$	0	0
$145$	34.3906	2.85598
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.806495	0.0660707	0.0330353	−	0.999454i	$-0.489483\pi$
$149$	0.806495	0.0660707	0.0330353	+	0.999454i	$0.489483\pi$
$150$	0	0
$151$	−14.6840	−1.19496	−0.597482	−	0.801882i	$-0.703832\pi$
$151$	−14.6840	−1.19496	−0.597482	+	0.801882i	$0.703832\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	20.4145	1.63973
$156$	0	0
$157$	14.8590	1.18588	0.592938	−	0.805248i	$-0.297968\pi$
$157$	14.8590	1.18588	0.592938	+	0.805248i	$0.297968\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.46979	0.194647
$162$	0	0
$163$	11.6684	0.913942	0.456971	−	0.889482i	$-0.348934\pi$
$163$	11.6684	0.913942	0.456971	+	0.889482i	$0.348934\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	10.4032	0.800248
$170$	0	0
$171$	0	0
$172$	0	0
$173$	24.8278	1.88763	0.943813	−	0.330481i	$-0.107211\pi$
$173$	24.8278	1.88763	0.943813	+	0.330481i	$0.107211\pi$
$174$	0	0
$175$	11.0215	0.833147
$176$	0	0
$177$	0	0
$178$	0	0
$179$	21.8021	1.62957	0.814783	−	0.579766i	$-0.196856\pi$
$179$	21.8021	1.62957	0.814783	+	0.579766i	$0.196856\pi$
$180$	0	0
$181$	18.5477	1.37864	0.689320	−	0.724457i	$-0.257909\pi$
$181$	18.5477	1.37864	0.689320	+	0.724457i	$0.257909\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.33538	−0.0981792
$186$	0	0
$187$	−7.78566	−0.569344
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.90014	0.643992	0.321996	−	0.946741i	$-0.395646\pi$
$191$	8.90014	0.643992	0.321996	+	0.946741i	$0.395646\pi$
$192$	0	0
$193$	22.4480	1.61584	0.807921	−	0.589291i	$-0.200593\pi$
$193$	22.4480	1.61584	0.807921	+	0.589291i	$0.200593\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.49823	0.605474	0.302737	−	0.953074i	$-0.402100\pi$
$197$	8.49823	0.605474	0.302737	+	0.953074i	$0.402100\pi$
$198$	0	0
$199$	3.02067	0.214129	0.107065	−	0.994252i	$-0.465855\pi$
$199$	3.02067	0.214129	0.107065	+	0.994252i	$0.465855\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−20.6674	−1.45057
$204$	0	0
$205$	−8.65279	−0.604337
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−9.43045	−0.652318
$210$	0	0
$211$	16.8920	1.16289	0.581447	−	0.813584i	$-0.302487\pi$
$211$	16.8920	1.16289	0.581447	+	0.813584i	$0.302487\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.03393	−0.0705132
$216$	0	0
$217$	−12.2683	−0.832829
$218$	0	0
$219$	0	0
$220$	0	0
$221$	20.1864	1.35788
$222$	0	0
$223$	0.393347	0.0263405	0.0131702	−	0.999913i	$-0.495808\pi$
$223$	0.393347	0.0263405	0.0131702	+	0.999913i	$0.495808\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.13362	−0.606220	−0.303110	−	0.952956i	$-0.598025\pi$
$227$	−9.13362	−0.606220	−0.303110	+	0.952956i	$0.598025\pi$
$228$	0	0
$229$	−5.86675	−0.387686	−0.193843	−	0.981033i	$-0.562095\pi$
$229$	−5.86675	−0.387686	−0.193843	+	0.981033i	$0.562095\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.0755	−0.856605	−0.428302	−	0.903635i	$-0.640888\pi$
$233$	−13.0755	−0.856605	−0.428302	+	0.903635i	$0.640888\pi$
$234$	0	0
$235$	−13.3305	−0.869584
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.4797	−1.00130	−0.500648	−	0.865651i	$-0.666905\pi$
$239$	−15.4797	−1.00130	−0.500648	+	0.865651i	$0.666905\pi$
$240$	0	0
$241$	9.96592	0.641962	0.320981	−	0.947086i	$-0.395988\pi$
$241$	9.96592	0.641962	0.320981	+	0.947086i	$0.395988\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	10.3085	0.658584
$246$	0	0
$247$	24.4509	1.55578
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.1882	0.769310	0.384655	−	0.923060i	$-0.374320\pi$
$251$	12.1882	0.769310	0.384655	+	0.923060i	$0.374320\pi$
$252$	0	0
$253$	−2.35234	−0.147891
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.91332	0.493619	0.246810	−	0.969064i	$-0.420618\pi$
$257$	7.91332	0.493619	0.246810	+	0.969064i	$0.420618\pi$
$258$	0	0
$259$	0.802513	0.0498658
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−0.615803	−0.0379721	−0.0189860	−	0.999820i	$-0.506044\pi$
$263$	−0.615803	−0.0379721	−0.0189860	+	0.999820i	$0.506044\pi$
$264$	0	0
$265$	−23.3349	−1.43345
$266$	0	0
$267$	0	0
$268$	0	0
$269$	24.5177	1.49487	0.747436	−	0.664334i	$-0.231285\pi$
$269$	24.5177	1.49487	0.747436	+	0.664334i	$0.231285\pi$
$270$	0	0
$271$	26.4785	1.60846	0.804228	−	0.594321i	$-0.202579\pi$
$271$	26.4785	1.60846	0.804228	+	0.594321i	$0.202579\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−10.4974	−0.633017
$276$	0	0
$277$	−5.05737	−0.303868	−0.151934	−	0.988391i	$-0.548550\pi$
$277$	−5.05737	−0.303868	−0.151934	+	0.988391i	$0.548550\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−26.1625	−1.56073	−0.780363	−	0.625326i	$-0.784966\pi$
$281$	−26.1625	−1.56073	−0.780363	+	0.625326i	$0.784966\pi$
$282$	0	0
$283$	−28.9055	−1.71826	−0.859128	−	0.511761i	$-0.828993\pi$
$283$	−28.9055	−1.71826	−0.859128	+	0.511761i	$0.828993\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.20000	0.306946
$288$	0	0
$289$	0.411724	0.0242191
$290$	0	0
$291$	0	0
$292$	0	0
$293$	21.3626	1.24802	0.624009	−	0.781417i	$-0.285503\pi$
$293$	21.3626	1.24802	0.624009	+	0.781417i	$0.285503\pi$
$294$	0	0
$295$	−28.4478	−1.65629
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.09907	0.352719
$300$	0	0
$301$	0.621351	0.0358141
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−32.2597	−1.84718
$306$	0	0
$307$	22.5731	1.28832	0.644158	−	0.764892i	$-0.277208\pi$
$307$	22.5731	1.28832	0.644158	+	0.764892i	$0.277208\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	17.9825	1.01969	0.509846	−	0.860266i	$-0.329702\pi$
$311$	17.9825	1.01969	0.509846	+	0.860266i	$0.329702\pi$
$312$	0	0
$313$	−1.05074	−0.0593913	−0.0296957	−	0.999559i	$-0.509454\pi$
$313$	−1.05074	−0.0593913	−0.0296957	+	0.999559i	$0.509454\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.29356	0.0726537	0.0363269	−	0.999340i	$-0.488434\pi$
$317$	1.29356	0.0726537	0.0363269	+	0.999340i	$0.488434\pi$
$318$	0	0
$319$	19.6846	1.10213
$320$	0	0
$321$	0	0
$322$	0	0
$323$	21.0901	1.17348
$324$	0	0
$325$	27.2173	1.50974
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.01110	0.441666
$330$	0	0
$331$	−2.42036	−0.133035	−0.0665175	−	0.997785i	$-0.521189\pi$
$331$	−2.42036	−0.133035	−0.0665175	+	0.997785i	$0.521189\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	21.3607	1.16706
$336$	0	0
$337$	−9.02901	−0.491842	−0.245921	−	0.969290i	$-0.579090\pi$
$337$	−9.02901	−0.491842	−0.245921	+	0.969290i	$0.579090\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	11.6849	0.632775
$342$	0	0
$343$	−19.9080	−1.07493
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.0918	−1.07858	−0.539292	−	0.842119i	$-0.681308\pi$
$347$	−20.0918	−1.07858	−0.539292	+	0.842119i	$0.681308\pi$
$348$	0	0
$349$	−15.7445	−0.842782	−0.421391	−	0.906879i	$-0.638458\pi$
$349$	−15.7445	−0.842782	−0.421391	+	0.906879i	$0.638458\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.8715	0.844757	0.422379	−	0.906419i	$-0.361195\pi$
$353$	15.8715	0.844757	0.422379	+	0.906419i	$0.361195\pi$
$354$	0	0
$355$	−32.8469	−1.74333
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.1278	1.48453	0.742265	−	0.670107i	$-0.233752\pi$
$359$	28.1278	1.48453	0.742265	+	0.670107i	$0.233752\pi$
$360$	0	0
$361$	6.54557	0.344504
$362$	0	0
$363$	0	0
$364$	0	0
$365$	38.6939	2.02533
$366$	0	0
$367$	−19.0328	−0.993504	−0.496752	−	0.867892i	$-0.665474\pi$
$367$	−19.0328	−0.993504	−0.496752	+	0.867892i	$0.665474\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	14.0234	0.728057
$372$	0	0
$373$	−12.6675	−0.655899	−0.327949	−	0.944695i	$-0.606358\pi$
$373$	−12.6675	−0.655899	−0.327949	+	0.944695i	$0.606358\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−51.0376	−2.62857
$378$	0	0
$379$	8.31841	0.427288	0.213644	−	0.976912i	$-0.431467\pi$
$379$	8.31841	0.427288	0.213644	+	0.976912i	$0.431467\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	29.2821	1.49624	0.748122	−	0.663561i	$-0.230956\pi$
$383$	29.2821	1.49624	0.748122	+	0.663561i	$0.230956\pi$
$384$	0	0
$385$	11.9150	0.607247
$386$	0	0
$387$	0	0
$388$	0	0
$389$	16.3851	0.830756	0.415378	−	0.909649i	$-0.363649\pi$
$389$	16.3851	0.830756	0.415378	+	0.909649i	$0.363649\pi$
$390$	0	0
$391$	5.26074	0.266047
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.82378	0.443972
$396$	0	0
$397$	−13.2130	−0.663140	−0.331570	−	0.943431i	$-0.607578\pi$
$397$	−13.2130	−0.663140	−0.331570	+	0.943431i	$0.607578\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.20672	0.210074	0.105037	−	0.994468i	$-0.466504\pi$
$401$	4.20672	0.210074	0.105037	+	0.994468i	$0.466504\pi$
$402$	0	0
$403$	−30.2963	−1.50917
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.764352	−0.0378875
$408$	0	0
$409$	−22.6454	−1.11974	−0.559871	−	0.828580i	$-0.689149\pi$
$409$	−22.6454	−1.11974	−0.559871	+	0.828580i	$0.689149\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	17.0960	0.841240
$414$	0	0
$415$	20.1557	0.989404
$416$	0	0
$417$	0	0
$418$	0	0
$419$	16.1237	0.787693	0.393846	−	0.919176i	$-0.371144\pi$
$419$	16.1237	0.787693	0.393846	+	0.919176i	$0.371144\pi$
$420$	0	0
$421$	18.3944	0.896487	0.448244	−	0.893911i	$-0.352050\pi$
$421$	18.3944	0.896487	0.448244	+	0.893911i	$0.352050\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	23.4762	1.13876
$426$	0	0
$427$	19.3868	0.938195
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−18.3846	−0.885557	−0.442779	−	0.896631i	$-0.646007\pi$
$431$	−18.3846	−0.885557	−0.442779	+	0.896631i	$0.646007\pi$
$432$	0	0
$433$	15.7250	0.755694	0.377847	−	0.925868i	$-0.376664\pi$
$433$	15.7250	0.755694	0.377847	+	0.925868i	$0.376664\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.37212	0.304820
$438$	0	0
$439$	6.49754	0.310111	0.155055	−	0.987906i	$-0.450444\pi$
$439$	6.49754	0.310111	0.155055	+	0.987906i	$0.450444\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.33495	−0.396005	−0.198003	−	0.980201i	$-0.563445\pi$
$443$	−8.33495	−0.396005	−0.198003	+	0.980201i	$0.563445\pi$
$444$	0	0
$445$	47.8601	2.26879
$446$	0	0
$447$	0	0
$448$	0	0
$449$	37.4466	1.76721	0.883607	−	0.468230i	$-0.155108\pi$
$449$	37.4466	1.76721	0.883607	+	0.468230i	$0.155108\pi$
$450$	0	0
$451$	−4.95272	−0.233215
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−30.8929	−1.44828
$456$	0	0
$457$	−17.9246	−0.838478	−0.419239	−	0.907876i	$-0.637703\pi$
$457$	−17.9246	−0.838478	−0.419239	+	0.907876i	$0.637703\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−38.4054	−1.78872	−0.894359	−	0.447349i	$-0.852368\pi$
$461$	−38.4054	−1.78872	−0.894359	+	0.447349i	$0.852368\pi$
$462$	0	0
$463$	−23.3091	−1.08327	−0.541633	−	0.840615i	$-0.682194\pi$
$463$	−23.3091	−1.08327	−0.541633	+	0.840615i	$0.682194\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.1638	1.11817	0.559084	−	0.829111i	$-0.311153\pi$
$467$	24.1638	1.11817	0.559084	+	0.829111i	$0.311153\pi$
$468$	0	0
$469$	−12.8370	−0.592757
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−0.591804	−0.0272112
$474$	0	0
$475$	28.4357	1.30472
$476$	0	0
$477$	0	0
$478$	0	0
$479$	28.6876	1.31077	0.655384	−	0.755296i	$-0.272507\pi$
$479$	28.6876	1.31077	0.655384	+	0.755296i	$0.272507\pi$
$480$	0	0
$481$	1.98178	0.0903616
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−33.6971	−1.53011
$486$	0	0
$487$	22.6346	1.02567	0.512835	−	0.858487i	$-0.328595\pi$
$487$	22.6346	1.02567	0.512835	+	0.858487i	$0.328595\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−17.7912	−0.802905	−0.401452	−	0.915880i	$-0.631494\pi$
$491$	−17.7912	−0.802905	−0.401452	+	0.915880i	$0.631494\pi$
$492$	0	0
$493$	−44.0224	−1.98267
$494$	0	0
$495$	0	0
$496$	0	0
$497$	19.7397	0.885448
$498$	0	0
$499$	23.0860	1.03347	0.516736	−	0.856145i	$-0.327147\pi$
$499$	23.0860	1.03347	0.516736	+	0.856145i	$0.327147\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−18.9514	−0.845002	−0.422501	−	0.906362i	$-0.638848\pi$
$503$	−18.9514	−0.845002	−0.422501	+	0.906362i	$0.638848\pi$
$504$	0	0
$505$	24.1539	1.07483
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.7388	0.741935	0.370968	−	0.928646i	$-0.379026\pi$
$509$	16.7388	0.741935	0.370968	+	0.928646i	$0.379026\pi$
$510$	0	0
$511$	−23.2535	−1.02868
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.59519	−0.290619
$516$	0	0
$517$	−7.63015	−0.335574
$518$	0	0
$519$	0	0
$520$	0	0
$521$	39.7522	1.74157	0.870787	−	0.491660i	$-0.163610\pi$
$521$	39.7522	1.74157	0.870787	+	0.491660i	$0.163610\pi$
$522$	0	0
$523$	−24.3323	−1.06398	−0.531989	−	0.846751i	$-0.678555\pi$
$523$	−24.3323	−1.06398	−0.531989	+	0.846751i	$0.678555\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−26.1320	−1.13833
$528$	0	0
$529$	−21.4105	−0.930893
$530$	0	0
$531$	0	0
$532$	0	0
$533$	12.8412	0.556216
$534$	0	0
$535$	33.1879	1.43484
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.90041	0.254148
$540$	0	0
$541$	25.4258	1.09314	0.546570	−	0.837413i	$-0.315933\pi$
$541$	25.4258	1.09314	0.546570	+	0.837413i	$0.315933\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−55.4554	−2.37545
$546$	0	0
$547$	24.2511	1.03690	0.518450	−	0.855108i	$-0.326509\pi$
$547$	24.2511	1.03690	0.518450	+	0.855108i	$0.326509\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−53.3225	−2.27161
$552$	0	0
$553$	−5.30276	−0.225496
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.0732	0.426813	0.213407	−	0.976963i	$-0.431544\pi$
$557$	10.0732	0.426813	0.213407	+	0.976963i	$0.431544\pi$
$558$	0	0
$559$	1.53441	0.0648985
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.79902	−0.286545	−0.143272	−	0.989683i	$-0.545762\pi$
$563$	−6.79902	−0.286545	−0.143272	+	0.989683i	$0.545762\pi$
$564$	0	0
$565$	9.59483	0.403657
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−14.1444	−0.592966	−0.296483	−	0.955038i	$-0.595814\pi$
$569$	−14.1444	−0.592966	−0.296483	+	0.955038i	$0.595814\pi$
$570$	0	0
$571$	−8.33250	−0.348704	−0.174352	−	0.984683i	$-0.555783\pi$
$571$	−8.33250	−0.348704	−0.174352	+	0.984683i	$0.555783\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.09305	0.295801
$576$	0	0
$577$	−2.87608	−0.119733	−0.0598665	−	0.998206i	$-0.519067\pi$
$577$	−2.87608	−0.119733	−0.0598665	+	0.998206i	$0.519067\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−12.1128	−0.502524
$582$	0	0
$583$	−13.3565	−0.553170
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.86013	0.241874	0.120937	−	0.992660i	$-0.461410\pi$
$587$	5.86013	0.241874	0.120937	+	0.992660i	$0.461410\pi$
$588$	0	0
$589$	−31.6526	−1.30422
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−33.1853	−1.36276	−0.681378	−	0.731932i	$-0.738619\pi$
$593$	−33.1853	−1.36276	−0.681378	+	0.731932i	$0.738619\pi$
$594$	0	0
$595$	−26.6466	−1.09240
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12.4862	0.510174	0.255087	−	0.966918i	$-0.417896\pi$
$599$	12.4862	0.510174	0.255087	+	0.966918i	$0.417896\pi$
$600$	0	0
$601$	18.1405	0.739964	0.369982	−	0.929039i	$-0.379364\pi$
$601$	18.1405	0.739964	0.369982	+	0.929039i	$0.379364\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	24.5090	0.996433
$606$	0	0
$607$	46.7416	1.89718	0.948591	−	0.316506i	$-0.102510\pi$
$607$	46.7416	1.89718	0.948591	+	0.316506i	$0.102510\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	19.7832	0.800342
$612$	0	0
$613$	−26.1419	−1.05586	−0.527930	−	0.849288i	$-0.677032\pi$
$613$	−26.1419	−1.05586	−0.527930	+	0.849288i	$0.677032\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.8947	0.438606	0.219303	−	0.975657i	$-0.429622\pi$
$617$	10.8947	0.438606	0.219303	+	0.975657i	$0.429622\pi$
$618$	0	0
$619$	16.3051	0.655355	0.327678	−	0.944790i	$-0.393734\pi$
$619$	16.3051	0.655355	0.327678	+	0.944790i	$0.393734\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−28.7621	−1.15233
$624$	0	0
$625$	−21.4776	−0.859102
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.70938	0.0681576
$630$	0	0
$631$	−23.2613	−0.926019	−0.463010	−	0.886353i	$-0.653230\pi$
$631$	−23.2613	−0.926019	−0.463010	+	0.886353i	$0.653230\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	46.0520	1.82752
$636$	0	0
$637$	−15.2984	−0.606143
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.97779	0.236108	0.118054	−	0.993007i	$-0.462334\pi$
$641$	5.97779	0.236108	0.118054	+	0.993007i	$0.462334\pi$
$642$	0	0
$643$	23.0348	0.908403	0.454202	−	0.890899i	$-0.349925\pi$
$643$	23.0348	0.908403	0.454202	+	0.890899i	$0.349925\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.539846	0.0212235	0.0106118	−	0.999944i	$-0.496622\pi$
$647$	0.539846	0.0212235	0.0106118	+	0.999944i	$0.496622\pi$
$648$	0	0
$649$	−16.2831	−0.639166
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−16.6423	−0.651262	−0.325631	−	0.945497i	$-0.605577\pi$
$653$	−16.6423	−0.651262	−0.325631	+	0.945497i	$0.605577\pi$
$654$	0	0
$655$	−26.9478	−1.05294
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−34.1742	−1.33124	−0.665619	−	0.746292i	$-0.731832\pi$
$659$	−34.1742	−1.33124	−0.665619	+	0.746292i	$0.731832\pi$
$660$	0	0
$661$	−30.8130	−1.19849	−0.599243	−	0.800567i	$-0.704532\pi$
$661$	−30.8130	−1.19849	−0.599243	+	0.800567i	$0.704532\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−32.2759	−1.25161
$666$	0	0
$667$	−13.3008	−0.515010
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−18.4649	−0.712831
$672$	0	0
$673$	−21.0355	−0.810859	−0.405430	−	0.914126i	$-0.632878\pi$
$673$	−21.0355	−0.810859	−0.405430	+	0.914126i	$0.632878\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5.32794	−0.204769	−0.102385	−	0.994745i	$-0.532647\pi$
$677$	−5.32794	−0.204769	−0.102385	+	0.994745i	$0.532647\pi$
$678$	0	0
$679$	20.2507	0.777151
$680$	0	0
$681$	0	0
$682$	0	0
$683$	7.97577	0.305185	0.152592	−	0.988289i	$-0.451238\pi$
$683$	7.97577	0.305185	0.152592	+	0.988289i	$0.451238\pi$
$684$	0	0
$685$	−20.4480	−0.781279
$686$	0	0
$687$	0	0
$688$	0	0
$689$	34.6303	1.31931
$690$	0	0
$691$	7.54471	0.287014	0.143507	−	0.989649i	$-0.454162\pi$
$691$	7.54471	0.287014	0.143507	+	0.989649i	$0.454162\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−53.2134	−2.01850
$696$	0	0
$697$	11.0762	0.419540
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−30.7808	−1.16258	−0.581288	−	0.813698i	$-0.697451\pi$
$701$	−30.7808	−1.16258	−0.581288	+	0.813698i	$0.697451\pi$
$702$	0	0
$703$	2.07050	0.0780906
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−14.5156	−0.545914
$708$	0	0
$709$	2.62232	0.0984834	0.0492417	−	0.998787i	$-0.484320\pi$
$709$	2.62232	0.0984834	0.0492417	+	0.998787i	$0.484320\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.89547	−0.295688
$714$	0	0
$715$	29.4238	1.10039
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9.01417	0.336172	0.168086	−	0.985772i	$-0.446241\pi$
$719$	9.01417	0.336172	0.168086	+	0.985772i	$0.446241\pi$
$720$	0	0
$721$	3.96346	0.147607
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−59.3553	−2.20440
$726$	0	0
$727$	−22.0515	−0.817846	−0.408923	−	0.912569i	$-0.634095\pi$
$727$	−22.0515	−0.817846	−0.408923	+	0.912569i	$0.634095\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.32350	0.0489514
$732$	0	0
$733$	51.7621	1.91188	0.955938	−	0.293569i	$-0.0948429\pi$
$733$	51.7621	1.91188	0.955938	+	0.293569i	$0.0948429\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.2266	0.450371
$738$	0	0
$739$	46.4176	1.70750	0.853749	−	0.520684i	$-0.174323\pi$
$739$	46.4176	1.70750	0.853749	+	0.520684i	$0.174323\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	41.9648	1.53954	0.769770	−	0.638321i	$-0.220371\pi$
$743$	41.9648	1.53954	0.769770	+	0.638321i	$0.220371\pi$
$744$	0	0
$745$	−2.62899	−0.0963187
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−19.9447	−0.728762
$750$	0	0
$751$	40.9764	1.49525	0.747625	−	0.664121i	$-0.231194\pi$
$751$	40.9764	1.49525	0.747625	+	0.664121i	$0.231194\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	47.8663	1.74203
$756$	0	0
$757$	−3.14287	−0.114230	−0.0571148	−	0.998368i	$-0.518190\pi$
$757$	−3.14287	−0.114230	−0.0571148	+	0.998368i	$0.518190\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−15.9986	−0.579949	−0.289974	−	0.957034i	$-0.593647\pi$
$761$	−15.9986	−0.579949	−0.289974	+	0.957034i	$0.593647\pi$
$762$	0	0
$763$	33.3266	1.20650
$764$	0	0
$765$	0	0
$766$	0	0
$767$	42.2181	1.52441
$768$	0	0
$769$	−34.4686	−1.24297	−0.621485	−	0.783426i	$-0.713470\pi$
$769$	−34.4686	−1.24297	−0.621485	+	0.783426i	$0.713470\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−5.97582	−0.214935	−0.107468	−	0.994209i	$-0.534274\pi$
$773$	−5.97582	−0.214935	−0.107468	+	0.994209i	$0.534274\pi$
$774$	0	0
$775$	−35.2337	−1.26563
$776$	0	0
$777$	0	0
$778$	0	0
$779$	13.4161	0.480682
$780$	0	0
$781$	−18.8011	−0.672755
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−48.4368	−1.72878
$786$	0	0
$787$	−48.8368	−1.74084	−0.870422	−	0.492306i	$-0.836154\pi$
$787$	−48.8368	−1.74084	−0.870422	+	0.492306i	$0.836154\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.76613	−0.205020
$792$	0	0
$793$	47.8753	1.70010
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−40.1178	−1.42105	−0.710523	−	0.703674i	$-0.751542\pi$
$797$	−40.1178	−1.42105	−0.710523	+	0.703674i	$0.751542\pi$
$798$	0	0
$799$	17.0639	0.603679
$800$	0	0
$801$	0	0
$802$	0	0
$803$	22.1478	0.781578
$804$	0	0
$805$	−8.05094	−0.283758
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5.67223	0.199425	0.0997126	−	0.995016i	$-0.468208\pi$
$809$	5.67223	0.199425	0.0997126	+	0.995016i	$0.468208\pi$
$810$	0	0
$811$	8.49117	0.298165	0.149083	−	0.988825i	$-0.452368\pi$
$811$	8.49117	0.298165	0.149083	+	0.988825i	$0.452368\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−38.0364	−1.33236
$816$	0	0
$817$	1.60310	0.0560854
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8.01425	−0.279699	−0.139850	−	0.990173i	$-0.544662\pi$
$821$	−8.01425	−0.279699	−0.139850	+	0.990173i	$0.544662\pi$
$822$	0	0
$823$	52.3868	1.82609	0.913045	−	0.407858i	$-0.133724\pi$
$823$	52.3868	1.82609	0.913045	+	0.407858i	$0.133724\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−0.346898	−0.0120628	−0.00603141	−	0.999982i	$-0.501920\pi$
$827$	−0.346898	−0.0120628	−0.00603141	+	0.999982i	$0.501920\pi$
$828$	0	0
$829$	22.4031	0.778091	0.389046	−	0.921218i	$-0.372805\pi$
$829$	22.4031	0.778091	0.389046	+	0.921218i	$0.372805\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−13.1956	−0.457199
$834$	0	0
$835$	−3.25977	−0.112809
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−9.27328	−0.320149	−0.160075	−	0.987105i	$-0.551173\pi$
$839$	−9.27328	−0.320149	−0.160075	+	0.987105i	$0.551173\pi$
$840$	0	0
$841$	82.3026	2.83802
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−33.9121	−1.16661
$846$	0	0
$847$	−14.7290	−0.506094
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.516469	0.0177043
$852$	0	0
$853$	−31.6268	−1.08288	−0.541440	−	0.840739i	$-0.682121\pi$
$853$	−31.6268	−1.08288	−0.541440	+	0.840739i	$0.682121\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10.0275	−0.342533	−0.171267	−	0.985225i	$-0.554786\pi$
$857$	−10.0275	−0.342533	−0.171267	+	0.985225i	$0.554786\pi$
$858$	0	0
$859$	−24.1534	−0.824102	−0.412051	−	0.911161i	$-0.635187\pi$
$859$	−24.1534	−0.824102	−0.412051	+	0.911161i	$0.635187\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38.9292	1.32516	0.662582	−	0.748989i	$-0.269460\pi$
$863$	38.9292	1.32516	0.662582	+	0.748989i	$0.269460\pi$
$864$	0	0
$865$	−80.9330	−2.75180
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.05059	0.171330
$870$	0	0
$871$	−31.7006	−1.07413
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.99814	−0.135162
$876$	0	0
$877$	−46.1448	−1.55820	−0.779100	−	0.626900i	$-0.784324\pi$
$877$	−46.1448	−1.55820	−0.779100	+	0.626900i	$0.784324\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	8.41271	0.283431	0.141716	−	0.989907i	$-0.454738\pi$
$881$	8.41271	0.283431	0.141716	+	0.989907i	$0.454738\pi$
$882$	0	0
$883$	−40.4167	−1.36013	−0.680065	−	0.733152i	$-0.738048\pi$
$883$	−40.4167	−1.36013	−0.680065	+	0.733152i	$0.738048\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.8983	0.903156	0.451578	−	0.892232i	$-0.350861\pi$
$887$	26.8983	0.903156	0.451578	+	0.892232i	$0.350861\pi$
$888$	0	0
$889$	−27.6755	−0.928208
$890$	0	0
$891$	0	0
$892$	0	0
$893$	20.6689	0.691657
$894$	0	0
$895$	−71.0699	−2.37560
$896$	0	0
$897$	0	0
$898$	0	0
$899$	66.0700	2.20356
$900$	0	0
$901$	29.8703	0.995123
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−60.4612	−2.00980
$906$	0	0
$907$	−35.2510	−1.17049	−0.585244	−	0.810857i	$-0.699001\pi$
$907$	−35.2510	−1.17049	−0.585244	+	0.810857i	$0.699001\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−7.88805	−0.261343	−0.130671	−	0.991426i	$-0.541713\pi$
$911$	−7.88805	−0.261343	−0.130671	+	0.991426i	$0.541713\pi$
$912$	0	0
$913$	11.5368	0.381813
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.1946	0.534793
$918$	0	0
$919$	−22.3691	−0.737887	−0.368944	−	0.929452i	$-0.620280\pi$
$919$	−22.3691	−0.737887	−0.368944	+	0.929452i	$0.620280\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	48.7467	1.60452
$924$	0	0
$925$	2.30476	0.0757799
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−42.4881	−1.39399	−0.696994	−	0.717077i	$-0.745480\pi$
$929$	−42.4881	−1.39399	−0.696994	+	0.717077i	$0.745480\pi$
$930$	0	0
$931$	−15.9832	−0.523830
$932$	0	0
$933$	0	0
$934$	0	0
$935$	25.3795	0.829997
$936$	0	0
$937$	40.4812	1.32246	0.661232	−	0.750181i	$-0.270034\pi$
$937$	40.4812	1.32246	0.661232	+	0.750181i	$0.270034\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−43.8361	−1.42902	−0.714508	−	0.699627i	$-0.753349\pi$
$941$	−43.8361	−1.42902	−0.714508	+	0.699627i	$0.753349\pi$
$942$	0	0
$943$	3.34653	0.108978
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.4133	1.11828	0.559142	−	0.829072i	$-0.311131\pi$
$947$	34.4133	1.11828	0.559142	+	0.829072i	$0.311131\pi$
$948$	0	0
$949$	−57.4240	−1.86406
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.11149	−0.0683978	−0.0341989	−	0.999415i	$-0.510888\pi$
$953$	−2.11149	−0.0683978	−0.0341989	+	0.999415i	$0.510888\pi$
$954$	0	0
$955$	−29.0124	−0.938819
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12.2885	0.396816
$960$	0	0
$961$	8.21961	0.265149
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−73.1752	−2.35559
$966$	0	0
$967$	24.4443	0.786076	0.393038	−	0.919522i	$-0.371424\pi$
$967$	24.4443	0.786076	0.393038	+	0.919522i	$0.371424\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−25.3151	−0.812401	−0.406200	−	0.913784i	$-0.633147\pi$
$971$	−25.3151	−0.812401	−0.406200	+	0.913784i	$0.633147\pi$
$972$	0	0
$973$	31.9792	1.02521
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−0.758388	−0.0242630	−0.0121315	−	0.999926i	$-0.503862\pi$
$977$	−0.758388	−0.0242630	−0.0121315	+	0.999926i	$0.503862\pi$
$978$	0	0
$979$	27.3944	0.875528
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−13.4057	−0.427575	−0.213787	−	0.976880i	$-0.568580\pi$
$983$	−13.4057	−0.427575	−0.213787	+	0.976880i	$0.568580\pi$
$984$	0	0
$985$	−27.7023	−0.882668
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.399879	0.0127154
$990$	0	0
$991$	8.74067	0.277657	0.138828	−	0.990316i	$-0.455666\pi$
$991$	8.74067	0.277657	0.138828	+	0.990316i	$0.455666\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−9.84667	−0.312161
$996$	0	0
$997$	−54.0544	−1.71192	−0.855960	−	0.517042i	$-0.827033\pi$
$997$	−54.0544	−1.71192	−0.855960	+	0.517042i	$0.827033\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.i.1.2		9
3.2	odd	2		2004.2.a.c.1.8	✓	9
12.11	even	2		8016.2.a.bc.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.c.1.8	✓	9	3.2	odd	2
6012.2.a.i.1.2		9	1.1	even	1	trivial
8016.2.a.bc.1.8		9	12.11	even	2

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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.a (trivial)

\(f(q)\)	\(=\)	\(q-3.25977 q^{5} +1.95900 q^{7} +O(q^{10})\)	\(q-3.25977 q^{5} +1.95900 q^{7} -1.86584 q^{11} +4.83769 q^{13} +4.17274 q^{17} +5.05426 q^{19} +1.26074 q^{23} +5.62609 q^{25} -10.5500 q^{29} -6.26256 q^{31} -6.38588 q^{35} +0.409655 q^{37} +2.65442 q^{41} +0.317178 q^{43} +4.08939 q^{47} -3.16233 q^{49} +7.15844 q^{53} +6.08221 q^{55} +8.72693 q^{59} +9.89631 q^{61} -15.7697 q^{65} -6.55284 q^{67} +10.0764 q^{71} -11.8701 q^{73} -3.65518 q^{77} -2.70687 q^{79} -6.18316 q^{83} -13.6022 q^{85} -14.6820 q^{89} +9.47702 q^{91} -16.4757 q^{95} +10.3373 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{5} + 2 q^{7}+O(q^{10}) \) 9 * q - q^5 + 2 * q^7	\( 9 q - q^{5} + 2 q^{7} + 9 q^{11} + 10 q^{13} - 7 q^{17} - 2 q^{19} + 3 q^{23} + 18 q^{25} - 5 q^{29} + 12 q^{31} + 6 q^{35} + 15 q^{37} - 14 q^{41} + 6 q^{43} + 3 q^{47} + 27 q^{49} - 9 q^{53} + 19 q^{55} + 9 q^{59} + 30 q^{61} - 28 q^{65} + 16 q^{67} + 3 q^{71} + 32 q^{73} - 18 q^{77} + 24 q^{79} + 3 q^{83} + 37 q^{85} - 46 q^{89} + 33 q^{91} - 11 q^{95} + 43 q^{97}+O(q^{100}) \) 9 * q - q^5 + 2 * q^7 + 9 * q^11 + 10 * q^13 - 7 * q^17 - 2 * q^19 + 3 * q^23 + 18 * q^25 - 5 * q^29 + 12 * q^31 + 6 * q^35 + 15 * q^37 - 14 * q^41 + 6 * q^43 + 3 * q^47 + 27 * q^49 - 9 * q^53 + 19 * q^55 + 9 * q^59 + 30 * q^61 - 28 * q^65 + 16 * q^67 + 3 * q^71 + 32 * q^73 - 18 * q^77 + 24 * q^79 + 3 * q^83 + 37 * q^85 - 46 * q^89 + 33 * q^91 - 11 * q^95 + 43 * q^97

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