LMFDB - Embedded newform 6032.2.a.bd.1.11

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 10 x^{10} + 98 x^{9} + 10 x^{8} - 585 x^{7} + 151 x^{6} + 1524 x^{5} - 445 x^{4} + \cdots + 68 $ x^12 - 6x^11 - 10x^10 + 98x^9 + 10x^8 - 585x^7 + 151x^6 + 1524x^5 - 445x^4 - 1567x^3 + 273x^2 + 424*x + 68
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$2.89888$ 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)

$f(q)$	$=$	$q+1.89888 q^{3} -0.496327 q^{5} -2.15081 q^{7} +0.605764 q^{9} +O(q^{10})$	$q+1.89888 q^{3} -0.496327 q^{5} -2.15081 q^{7} +0.605764 q^{9} +1.30075 q^{11} +1.00000 q^{13} -0.942468 q^{15} +1.26604 q^{17} -5.14922 q^{19} -4.08415 q^{21} +3.15784 q^{23} -4.75366 q^{25} -4.54638 q^{27} +1.00000 q^{29} +0.121345 q^{31} +2.46998 q^{33} +1.06751 q^{35} +0.195751 q^{37} +1.89888 q^{39} +3.54277 q^{41} +9.77935 q^{43} -0.300657 q^{45} -2.10244 q^{47} -2.37400 q^{49} +2.40406 q^{51} -3.79652 q^{53} -0.645599 q^{55} -9.77778 q^{57} -14.2773 q^{59} +2.93316 q^{61} -1.30289 q^{63} -0.496327 q^{65} -3.60702 q^{67} +5.99638 q^{69} -10.1815 q^{71} -10.4317 q^{73} -9.02665 q^{75} -2.79768 q^{77} +4.56076 q^{79} -10.4503 q^{81} -12.1666 q^{83} -0.628370 q^{85} +1.89888 q^{87} -8.08758 q^{89} -2.15081 q^{91} +0.230420 q^{93} +2.55570 q^{95} +4.07017 q^{97} +0.787950 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9}+O(q^{10}) $ 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9	$ 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9} - 14 q^{11} + 12 q^{13} - 8 q^{15} + 4 q^{17} - 11 q^{19} - 5 q^{21} - 15 q^{23} + 5 q^{25} - 24 q^{27} + 12 q^{29} - 13 q^{31} - q^{33} - 18 q^{35} - 23 q^{37} - 6 q^{39} - 2 q^{41} - 26 q^{43} - 9 q^{45} - 15 q^{47} + 16 q^{49} - 21 q^{51} + 31 q^{53} - 10 q^{55} - 10 q^{57} - 7 q^{59} + 2 q^{61} + 25 q^{63} + 3 q^{65} - 47 q^{67} - 8 q^{69} - 32 q^{71} - 25 q^{73} - 31 q^{75} - 4 q^{77} - 7 q^{79} + 64 q^{81} - 12 q^{83} + 7 q^{85} - 6 q^{87} + 6 q^{89} - 6 q^{91} + 17 q^{93} - q^{95} - 7 q^{97} - 44 q^{99}+O(q^{100}) $ 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9 - 14 * q^11 + 12 * q^13 - 8 * q^15 + 4 * q^17 - 11 * q^19 - 5 * q^21 - 15 * q^23 + 5 * q^25 - 24 * q^27 + 12 * q^29 - 13 * q^31 - q^33 - 18 * q^35 - 23 * q^37 - 6 * q^39 - 2 * q^41 - 26 * q^43 - 9 * q^45 - 15 * q^47 + 16 * q^49 - 21 * q^51 + 31 * q^53 - 10 * q^55 - 10 * q^57 - 7 * q^59 + 2 * q^61 + 25 * q^63 + 3 * q^65 - 47 * q^67 - 8 * q^69 - 32 * q^71 - 25 * q^73 - 31 * q^75 - 4 * q^77 - 7 * q^79 + 64 * q^81 - 12 * q^83 + 7 * q^85 - 6 * q^87 + 6 * q^89 - 6 * q^91 + 17 * q^93 - q^95 - 7 * q^97 - 44 * 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
$3$	1.89888	1.09632	0.548161	−	0.836373i	$-0.315328\pi$
$3$	1.89888	1.09632	0.548161	+	0.836373i	$0.315328\pi$
$4$	0	0
$5$	−0.496327	−0.221964	−0.110982	−	0.993822i	$-0.535400\pi$
$5$	−0.496327	−0.221964	−0.110982	+	0.993822i	$0.535400\pi$
$6$	0	0
$7$	−2.15081	−0.812931	−0.406466	−	0.913666i	$-0.633239\pi$
$7$	−2.15081	−0.812931	−0.406466	+	0.913666i	$0.633239\pi$
$8$	0	0
$9$	0.605764	0.201921
$10$	0	0
$11$	1.30075	0.392192	0.196096	−	0.980585i	$-0.437174\pi$
$11$	1.30075	0.392192	0.196096	+	0.980585i	$0.437174\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−0.942468	−0.243344
$16$	0	0
$17$	1.26604	0.307060	0.153530	−	0.988144i	$-0.450936\pi$
$17$	1.26604	0.307060	0.153530	+	0.988144i	$0.450936\pi$
$18$	0	0
$19$	−5.14922	−1.18131	−0.590656	−	0.806923i	$-0.701131\pi$
$19$	−5.14922	−1.18131	−0.590656	+	0.806923i	$0.701131\pi$
$20$	0	0
$21$	−4.08415	−0.891234
$22$	0	0
$23$	3.15784	0.658456	0.329228	−	0.944251i	$-0.393212\pi$
$23$	3.15784	0.658456	0.329228	+	0.944251i	$0.393212\pi$
$24$	0	0
$25$	−4.75366	−0.950732
$26$	0	0
$27$	−4.54638	−0.874951
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	0.121345	0.0217942	0.0108971	−	0.999941i	$-0.496531\pi$
$31$	0.121345	0.0217942	0.0108971	+	0.999941i	$0.496531\pi$
$32$	0	0
$33$	2.46998	0.429969
$34$	0	0
$35$	1.06751	0.180442
$36$	0	0
$37$	0.195751	0.0321813	0.0160907	−	0.999871i	$-0.494878\pi$
$37$	0.195751	0.0321813	0.0160907	+	0.999871i	$0.494878\pi$
$38$	0	0
$39$	1.89888	0.304065
$40$	0	0
$41$	3.54277	0.553288	0.276644	−	0.960973i	$-0.410778\pi$
$41$	3.54277	0.553288	0.276644	+	0.960973i	$0.410778\pi$
$42$	0	0
$43$	9.77935	1.49134	0.745669	−	0.666317i	$-0.232130\pi$
$43$	9.77935	1.49134	0.745669	+	0.666317i	$0.232130\pi$
$44$	0	0
$45$	−0.300657	−0.0448193
$46$	0	0
$47$	−2.10244	−0.306672	−0.153336	−	0.988174i	$-0.549002\pi$
$47$	−2.10244	−0.306672	−0.153336	+	0.988174i	$0.549002\pi$
$48$	0	0
$49$	−2.37400	−0.339143
$50$	0	0
$51$	2.40406	0.336636
$52$	0	0
$53$	−3.79652	−0.521493	−0.260746	−	0.965407i	$-0.583969\pi$
$53$	−3.79652	−0.521493	−0.260746	+	0.965407i	$0.583969\pi$
$54$	0	0
$55$	−0.645599	−0.0870526
$56$	0	0
$57$	−9.77778	−1.29510
$58$	0	0
$59$	−14.2773	−1.85874	−0.929371	−	0.369147i	$-0.879650\pi$
$59$	−14.2773	−1.85874	−0.929371	+	0.369147i	$0.879650\pi$
$60$	0	0
$61$	2.93316	0.375552	0.187776	−	0.982212i	$-0.439872\pi$
$61$	2.93316	0.375552	0.187776	+	0.982212i	$0.439872\pi$
$62$	0	0
$63$	−1.30289	−0.164148
$64$	0	0
$65$	−0.496327	−0.0615618
$66$	0	0
$67$	−3.60702	−0.440668	−0.220334	−	0.975424i	$-0.570715\pi$
$67$	−3.60702	−0.440668	−0.220334	+	0.975424i	$0.570715\pi$
$68$	0	0
$69$	5.99638	0.721879
$70$	0	0
$71$	−10.1815	−1.20832	−0.604158	−	0.796864i	$-0.706491\pi$
$71$	−10.1815	−1.20832	−0.604158	+	0.796864i	$0.706491\pi$
$72$	0	0
$73$	−10.4317	−1.22094	−0.610470	−	0.792040i	$-0.709019\pi$
$73$	−10.4317	−1.22094	−0.610470	+	0.792040i	$0.709019\pi$
$74$	0	0
$75$	−9.02665	−1.04231
$76$	0	0
$77$	−2.79768	−0.318825
$78$	0	0
$79$	4.56076	0.513125	0.256563	−	0.966528i	$-0.417410\pi$
$79$	4.56076	0.513125	0.256563	+	0.966528i	$0.417410\pi$
$80$	0	0
$81$	−10.4503	−1.16115
$82$	0	0
$83$	−12.1666	−1.33545	−0.667727	−	0.744406i	$-0.732733\pi$
$83$	−12.1666	−1.33545	−0.667727	+	0.744406i	$0.732733\pi$
$84$	0	0
$85$	−0.628370	−0.0681563
$86$	0	0
$87$	1.89888	0.203582
$88$	0	0
$89$	−8.08758	−0.857282	−0.428641	−	0.903475i	$-0.641007\pi$
$89$	−8.08758	−0.857282	−0.428641	+	0.903475i	$0.641007\pi$
$90$	0	0
$91$	−2.15081	−0.225467
$92$	0	0
$93$	0.230420	0.0238934
$94$	0	0
$95$	2.55570	0.262209
$96$	0	0
$97$	4.07017	0.413263	0.206631	−	0.978419i	$-0.433750\pi$
$97$	4.07017	0.413263	0.206631	+	0.978419i	$0.433750\pi$
$98$	0	0
$99$	0.787950	0.0791919
$100$	0	0
$101$	−1.25628	−0.125004	−0.0625021	−	0.998045i	$-0.519908\pi$
$101$	−1.25628	−0.125004	−0.0625021	+	0.998045i	$0.519908\pi$
$102$	0	0
$103$	8.15734	0.803767	0.401884	−	0.915691i	$-0.368356\pi$
$103$	8.15734	0.803767	0.401884	+	0.915691i	$0.368356\pi$
$104$	0	0
$105$	2.02707	0.197822
$106$	0	0
$107$	−12.7988	−1.23731	−0.618654	−	0.785664i	$-0.712322\pi$
$107$	−12.7988	−1.23731	−0.618654	+	0.785664i	$0.712322\pi$
$108$	0	0
$109$	1.51365	0.144981	0.0724906	−	0.997369i	$-0.476905\pi$
$109$	1.51365	0.144981	0.0724906	+	0.997369i	$0.476905\pi$
$110$	0	0
$111$	0.371709	0.0352811
$112$	0	0
$113$	12.5633	1.18186	0.590929	−	0.806724i	$-0.298762\pi$
$113$	12.5633	1.18186	0.590929	+	0.806724i	$0.298762\pi$
$114$	0	0
$115$	−1.56732	−0.146154
$116$	0	0
$117$	0.605764	0.0560029
$118$	0	0
$119$	−2.72302	−0.249618
$120$	0	0
$121$	−9.30804	−0.846185
$122$	0	0
$123$	6.72731	0.606581
$124$	0	0
$125$	4.84101	0.432993
$126$	0	0
$127$	−13.7944	−1.22406	−0.612028	−	0.790836i	$-0.709646\pi$
$127$	−13.7944	−1.22406	−0.612028	+	0.790836i	$0.709646\pi$
$128$	0	0
$129$	18.5699	1.63499
$130$	0	0
$131$	−13.3584	−1.16713	−0.583564	−	0.812067i	$-0.698342\pi$
$131$	−13.3584	−1.16713	−0.583564	+	0.812067i	$0.698342\pi$
$132$	0	0
$133$	11.0750	0.960326
$134$	0	0
$135$	2.25649	0.194208
$136$	0	0
$137$	8.89994	0.760373	0.380187	−	0.924910i	$-0.375860\pi$
$137$	8.89994	0.760373	0.380187	+	0.924910i	$0.375860\pi$
$138$	0	0
$139$	10.7636	0.912954	0.456477	−	0.889735i	$-0.349111\pi$
$139$	10.7636	0.912954	0.456477	+	0.889735i	$0.349111\pi$
$140$	0	0
$141$	−3.99228	−0.336211
$142$	0	0
$143$	1.30075	0.108774
$144$	0	0
$145$	−0.496327	−0.0412177
$146$	0	0
$147$	−4.50795	−0.371810
$148$	0	0
$149$	1.87983	0.154002	0.0770008	−	0.997031i	$-0.475466\pi$
$149$	1.87983	0.154002	0.0770008	+	0.997031i	$0.475466\pi$
$150$	0	0
$151$	−13.8242	−1.12499	−0.562497	−	0.826799i	$-0.690159\pi$
$151$	−13.8242	−1.12499	−0.562497	+	0.826799i	$0.690159\pi$
$152$	0	0
$153$	0.766921	0.0620019
$154$	0	0
$155$	−0.0602267	−0.00483753
$156$	0	0
$157$	5.79037	0.462122	0.231061	−	0.972939i	$-0.425780\pi$
$157$	5.79037	0.462122	0.231061	+	0.972939i	$0.425780\pi$
$158$	0	0
$159$	−7.20916	−0.571724
$160$	0	0
$161$	−6.79193	−0.535279
$162$	0	0
$163$	6.93307	0.543040	0.271520	−	0.962433i	$-0.412474\pi$
$163$	6.93307	0.543040	0.271520	+	0.962433i	$0.412474\pi$
$164$	0	0
$165$	−1.22592	−0.0954377
$166$	0	0
$167$	−13.0322	−1.00846	−0.504230	−	0.863570i	$-0.668223\pi$
$167$	−13.0322	−1.00846	−0.504230	+	0.863570i	$0.668223\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−3.11921	−0.238532
$172$	0	0
$173$	1.05742	0.0803940	0.0401970	−	0.999192i	$-0.487201\pi$
$173$	1.05742	0.0803940	0.0401970	+	0.999192i	$0.487201\pi$
$174$	0	0
$175$	10.2242	0.772880
$176$	0	0
$177$	−27.1109	−2.03778
$178$	0	0
$179$	−16.7128	−1.24917	−0.624587	−	0.780956i	$-0.714732\pi$
$179$	−16.7128	−1.24917	−0.624587	+	0.780956i	$0.714732\pi$
$180$	0	0
$181$	−0.615016	−0.0457138	−0.0228569	−	0.999739i	$-0.507276\pi$
$181$	−0.615016	−0.0457138	−0.0228569	+	0.999739i	$0.507276\pi$
$182$	0	0
$183$	5.56973	0.411726
$184$	0	0
$185$	−0.0971568	−0.00714311
$186$	0	0
$187$	1.64681	0.120426
$188$	0	0
$189$	9.77841	0.711275
$190$	0	0
$191$	−13.1696	−0.952919	−0.476459	−	0.879197i	$-0.658080\pi$
$191$	−13.1696	−0.952919	−0.476459	+	0.879197i	$0.658080\pi$
$192$	0	0
$193$	−3.76864	−0.271272	−0.135636	−	0.990759i	$-0.543308\pi$
$193$	−3.76864	−0.271272	−0.135636	+	0.990759i	$0.543308\pi$
$194$	0	0
$195$	−0.942468	−0.0674915
$196$	0	0
$197$	16.2371	1.15684	0.578421	−	0.815738i	$-0.303669\pi$
$197$	16.2371	1.15684	0.578421	+	0.815738i	$0.303669\pi$
$198$	0	0
$199$	−7.15358	−0.507104	−0.253552	−	0.967322i	$-0.581599\pi$
$199$	−7.15358	−0.507104	−0.253552	+	0.967322i	$0.581599\pi$
$200$	0	0
$201$	−6.84932	−0.483114
$202$	0	0
$203$	−2.15081	−0.150958
$204$	0	0
$205$	−1.75837	−0.122810
$206$	0	0
$207$	1.91291	0.132956
$208$	0	0
$209$	−6.69787	−0.463301
$210$	0	0
$211$	2.99201	0.205979	0.102989	−	0.994682i	$-0.467159\pi$
$211$	2.99201	0.205979	0.102989	+	0.994682i	$0.467159\pi$
$212$	0	0
$213$	−19.3334	−1.32470
$214$	0	0
$215$	−4.85376	−0.331024
$216$	0	0
$217$	−0.260990	−0.0177172
$218$	0	0
$219$	−19.8086	−1.33854
$220$	0	0
$221$	1.26604	0.0851630
$222$	0	0
$223$	−28.8148	−1.92958	−0.964791	−	0.263019i	$-0.915282\pi$
$223$	−28.8148	−1.92958	−0.964791	+	0.263019i	$0.915282\pi$
$224$	0	0
$225$	−2.87960	−0.191973
$226$	0	0
$227$	−16.2101	−1.07590	−0.537950	−	0.842977i	$-0.680801\pi$
$227$	−16.2101	−1.07590	−0.537950	+	0.842977i	$0.680801\pi$
$228$	0	0
$229$	28.7898	1.90248	0.951241	−	0.308448i	$-0.0998094\pi$
$229$	28.7898	1.90248	0.951241	+	0.308448i	$0.0998094\pi$
$230$	0	0
$231$	−5.31247	−0.349535
$232$	0	0
$233$	−28.7097	−1.88084	−0.940418	−	0.340020i	$-0.889566\pi$
$233$	−28.7097	−1.88084	−0.940418	+	0.340020i	$0.889566\pi$
$234$	0	0
$235$	1.04350	0.0680702
$236$	0	0
$237$	8.66035	0.562550
$238$	0	0
$239$	0.185168	0.0119775	0.00598875	−	0.999982i	$-0.498094\pi$
$239$	0.185168	0.0119775	0.00598875	+	0.999982i	$0.498094\pi$
$240$	0	0
$241$	12.7719	0.822713	0.411356	−	0.911475i	$-0.365055\pi$
$241$	12.7719	0.822713	0.411356	+	0.911475i	$0.365055\pi$
$242$	0	0
$243$	−6.20486	−0.398042
$244$	0	0
$245$	1.17828	0.0752776
$246$	0	0
$247$	−5.14922	−0.327637
$248$	0	0
$249$	−23.1029	−1.46409
$250$	0	0
$251$	−17.7502	−1.12038	−0.560190	−	0.828364i	$-0.689272\pi$
$251$	−17.7502	−1.12038	−0.560190	+	0.828364i	$0.689272\pi$
$252$	0	0
$253$	4.10757	0.258241
$254$	0	0
$255$	−1.19320	−0.0747212
$256$	0	0
$257$	7.80699	0.486987	0.243493	−	0.969903i	$-0.421707\pi$
$257$	7.80699	0.486987	0.243493	+	0.969903i	$0.421707\pi$
$258$	0	0
$259$	−0.421025	−0.0261612
$260$	0	0
$261$	0.605764	0.0374959
$262$	0	0
$263$	4.35962	0.268825	0.134413	−	0.990925i	$-0.457085\pi$
$263$	4.35962	0.268825	0.134413	+	0.990925i	$0.457085\pi$
$264$	0	0
$265$	1.88432	0.115753
$266$	0	0
$267$	−15.3574	−0.939857
$268$	0	0
$269$	7.75167	0.472627	0.236314	−	0.971677i	$-0.424061\pi$
$269$	7.75167	0.472627	0.236314	+	0.971677i	$0.424061\pi$
$270$	0	0
$271$	20.0581	1.21844	0.609220	−	0.793001i	$-0.291483\pi$
$271$	20.0581	1.21844	0.609220	+	0.793001i	$0.291483\pi$
$272$	0	0
$273$	−4.08415	−0.247184
$274$	0	0
$275$	−6.18334	−0.372869
$276$	0	0
$277$	11.7617	0.706690	0.353345	−	0.935493i	$-0.385044\pi$
$277$	11.7617	0.706690	0.353345	+	0.935493i	$0.385044\pi$
$278$	0	0
$279$	0.0735063	0.00440071
$280$	0	0
$281$	−2.10353	−0.125486	−0.0627430	−	0.998030i	$-0.519985\pi$
$281$	−2.10353	−0.125486	−0.0627430	+	0.998030i	$0.519985\pi$
$282$	0	0
$283$	−19.0440	−1.13205	−0.566024	−	0.824389i	$-0.691519\pi$
$283$	−19.0440	−1.13205	−0.566024	+	0.824389i	$0.691519\pi$
$284$	0	0
$285$	4.85298	0.287466
$286$	0	0
$287$	−7.61984	−0.449785
$288$	0	0
$289$	−15.3971	−0.905714
$290$	0	0
$291$	7.72878	0.453069
$292$	0	0
$293$	21.5606	1.25958	0.629792	−	0.776764i	$-0.283140\pi$
$293$	21.5606	1.25958	0.629792	+	0.776764i	$0.283140\pi$
$294$	0	0
$295$	7.08620	0.412574
$296$	0	0
$297$	−5.91372	−0.343149
$298$	0	0
$299$	3.15784	0.182623
$300$	0	0
$301$	−21.0336	−1.21235
$302$	0	0
$303$	−2.38552	−0.137045
$304$	0	0
$305$	−1.45581	−0.0833592
$306$	0	0
$307$	−15.2688	−0.871435	−0.435717	−	0.900084i	$-0.643505\pi$
$307$	−15.2688	−0.871435	−0.435717	+	0.900084i	$0.643505\pi$
$308$	0	0
$309$	15.4899	0.881187
$310$	0	0
$311$	14.0012	0.793936	0.396968	−	0.917832i	$-0.370062\pi$
$311$	14.0012	0.793936	0.396968	+	0.917832i	$0.370062\pi$
$312$	0	0
$313$	3.24267	0.183286	0.0916432	−	0.995792i	$-0.470788\pi$
$313$	3.24267	0.183286	0.0916432	+	0.995792i	$0.470788\pi$
$314$	0	0
$315$	0.646658	0.0364350
$316$	0	0
$317$	5.51529	0.309769	0.154885	−	0.987933i	$-0.450499\pi$
$317$	5.51529	0.309769	0.154885	+	0.987933i	$0.450499\pi$
$318$	0	0
$319$	1.30075	0.0728282
$320$	0	0
$321$	−24.3035	−1.35649
$322$	0	0
$323$	−6.51912	−0.362734
$324$	0	0
$325$	−4.75366	−0.263686
$326$	0	0
$327$	2.87424	0.158946
$328$	0	0
$329$	4.52195	0.249303
$330$	0	0
$331$	−28.9016	−1.58858	−0.794288	−	0.607541i	$-0.792156\pi$
$331$	−28.9016	−1.58858	−0.794288	+	0.607541i	$0.792156\pi$
$332$	0	0
$333$	0.118579	0.00649810
$334$	0	0
$335$	1.79026	0.0978126
$336$	0	0
$337$	−3.01236	−0.164093	−0.0820467	−	0.996628i	$-0.526146\pi$
$337$	−3.01236	−0.164093	−0.0820467	+	0.996628i	$0.526146\pi$
$338$	0	0
$339$	23.8563	1.29570
$340$	0	0
$341$	0.157840	0.00854750
$342$	0	0
$343$	20.1617	1.08863
$344$	0	0
$345$	−2.97617	−0.160231
$346$	0	0
$347$	28.9319	1.55315	0.776573	−	0.630027i	$-0.216956\pi$
$347$	28.9319	1.55315	0.776573	+	0.630027i	$0.216956\pi$
$348$	0	0
$349$	0.191836	0.0102688	0.00513438	−	0.999987i	$-0.498366\pi$
$349$	0.191836	0.0102688	0.00513438	+	0.999987i	$0.498366\pi$
$350$	0	0
$351$	−4.54638	−0.242668
$352$	0	0
$353$	21.1204	1.12412	0.562062	−	0.827095i	$-0.310008\pi$
$353$	21.1204	1.12412	0.562062	+	0.827095i	$0.310008\pi$
$354$	0	0
$355$	5.05333	0.268203
$356$	0	0
$357$	−5.17069	−0.273662
$358$	0	0
$359$	21.1191	1.11462	0.557312	−	0.830303i	$-0.311833\pi$
$359$	21.1191	1.11462	0.557312	+	0.830303i	$0.311833\pi$
$360$	0	0
$361$	7.51450	0.395500
$362$	0	0
$363$	−17.6749	−0.927692
$364$	0	0
$365$	5.17754	0.271005
$366$	0	0
$367$	8.98464	0.468994	0.234497	−	0.972117i	$-0.424656\pi$
$367$	8.98464	0.468994	0.234497	+	0.972117i	$0.424656\pi$
$368$	0	0
$369$	2.14608	0.111721
$370$	0	0
$371$	8.16562	0.423938
$372$	0	0
$373$	−11.5611	−0.598609	−0.299304	−	0.954158i	$-0.596755\pi$
$373$	−11.5611	−0.598609	−0.299304	+	0.954158i	$0.596755\pi$
$374$	0	0
$375$	9.19251	0.474699
$376$	0	0
$377$	1.00000	0.0515026
$378$	0	0
$379$	16.6445	0.854969	0.427484	−	0.904023i	$-0.359400\pi$
$379$	16.6445	0.854969	0.427484	+	0.904023i	$0.359400\pi$
$380$	0	0
$381$	−26.1940	−1.34196
$382$	0	0
$383$	19.5690	0.999929	0.499965	−	0.866046i	$-0.333346\pi$
$383$	19.5690	0.999929	0.499965	+	0.866046i	$0.333346\pi$
$384$	0	0
$385$	1.38856	0.0707678
$386$	0	0
$387$	5.92398	0.301133
$388$	0	0
$389$	1.59270	0.0807533	0.0403767	−	0.999185i	$-0.487144\pi$
$389$	1.59270	0.0807533	0.0403767	+	0.999185i	$0.487144\pi$
$390$	0	0
$391$	3.99795	0.202185
$392$	0	0
$393$	−25.3660	−1.27955
$394$	0	0
$395$	−2.26363	−0.113895
$396$	0	0
$397$	−21.6217	−1.08516	−0.542582	−	0.840003i	$-0.682553\pi$
$397$	−21.6217	−1.08516	−0.542582	+	0.840003i	$0.682553\pi$
$398$	0	0
$399$	21.0302	1.05283
$400$	0	0
$401$	−7.27418	−0.363255	−0.181628	−	0.983367i	$-0.558137\pi$
$401$	−7.27418	−0.363255	−0.181628	+	0.983367i	$0.558137\pi$
$402$	0	0
$403$	0.121345	0.00604461
$404$	0	0
$405$	5.18679	0.257734
$406$	0	0
$407$	0.254624	0.0126213
$408$	0	0
$409$	−6.67553	−0.330084	−0.165042	−	0.986287i	$-0.552776\pi$
$409$	−6.67553	−0.330084	−0.165042	+	0.986287i	$0.552776\pi$
$410$	0	0
$411$	16.9000	0.833614
$412$	0	0
$413$	30.7077	1.51103
$414$	0	0
$415$	6.03860	0.296423
$416$	0	0
$417$	20.4388	1.00089
$418$	0	0
$419$	12.7764	0.624167	0.312083	−	0.950055i	$-0.398973\pi$
$419$	12.7764	0.624167	0.312083	+	0.950055i	$0.398973\pi$
$420$	0	0
$421$	−27.5084	−1.34068	−0.670340	−	0.742054i	$-0.733852\pi$
$421$	−27.5084	−1.34068	−0.670340	+	0.742054i	$0.733852\pi$
$422$	0	0
$423$	−1.27358	−0.0619236
$424$	0	0
$425$	−6.01832	−0.291931
$426$	0	0
$427$	−6.30867	−0.305298
$428$	0	0
$429$	2.46998	0.119252
$430$	0	0
$431$	−15.1876	−0.731561	−0.365780	−	0.930701i	$-0.619198\pi$
$431$	−15.1876	−0.731561	−0.365780	+	0.930701i	$0.619198\pi$
$432$	0	0
$433$	2.06521	0.0992475	0.0496237	−	0.998768i	$-0.484198\pi$
$433$	2.06521	0.0992475	0.0496237	+	0.998768i	$0.484198\pi$
$434$	0	0
$435$	−0.942468	−0.0451879
$436$	0	0
$437$	−16.2604	−0.777842
$438$	0	0
$439$	−32.1119	−1.53262	−0.766309	−	0.642472i	$-0.777909\pi$
$439$	−32.1119	−1.53262	−0.766309	+	0.642472i	$0.777909\pi$
$440$	0	0
$441$	−1.43808	−0.0684802
$442$	0	0
$443$	19.8121	0.941302	0.470651	−	0.882319i	$-0.344019\pi$
$443$	19.8121	0.941302	0.470651	+	0.882319i	$0.344019\pi$
$444$	0	0
$445$	4.01409	0.190286
$446$	0	0
$447$	3.56958	0.168835
$448$	0	0
$449$	−17.5540	−0.828423	−0.414211	−	0.910181i	$-0.635943\pi$
$449$	−17.5540	−0.828423	−0.414211	+	0.910181i	$0.635943\pi$
$450$	0	0
$451$	4.60827	0.216995
$452$	0	0
$453$	−26.2505	−1.23335
$454$	0	0
$455$	1.06751	0.0500455
$456$	0	0
$457$	2.21981	0.103838	0.0519191	−	0.998651i	$-0.483466\pi$
$457$	2.21981	0.103838	0.0519191	+	0.998651i	$0.483466\pi$
$458$	0	0
$459$	−5.75589	−0.268662
$460$	0	0
$461$	−14.3169	−0.666802	−0.333401	−	0.942785i	$-0.608196\pi$
$461$	−14.3169	−0.666802	−0.333401	+	0.942785i	$0.608196\pi$
$462$	0	0
$463$	21.0910	0.980184	0.490092	−	0.871671i	$-0.336963\pi$
$463$	21.0910	0.980184	0.490092	+	0.871671i	$0.336963\pi$
$464$	0	0
$465$	−0.114364	−0.00530348
$466$	0	0
$467$	15.8215	0.732130	0.366065	−	0.930589i	$-0.380705\pi$
$467$	15.8215	0.732130	0.366065	+	0.930589i	$0.380705\pi$
$468$	0	0
$469$	7.75804	0.358233
$470$	0	0
$471$	10.9953	0.506635
$472$	0	0
$473$	12.7205	0.584890
$474$	0	0
$475$	24.4777	1.12311
$476$	0	0
$477$	−2.29980	−0.105300
$478$	0	0
$479$	7.44578	0.340206	0.170103	−	0.985426i	$-0.445590\pi$
$479$	7.44578	0.340206	0.170103	+	0.985426i	$0.445590\pi$
$480$	0	0
$481$	0.195751	0.00892550
$482$	0	0
$483$	−12.8971	−0.586838
$484$	0	0
$485$	−2.02013	−0.0917295
$486$	0	0
$487$	−22.9793	−1.04129	−0.520646	−	0.853773i	$-0.674309\pi$
$487$	−22.9793	−1.04129	−0.520646	+	0.853773i	$0.674309\pi$
$488$	0	0
$489$	13.1651	0.595346
$490$	0	0
$491$	−14.1376	−0.638022	−0.319011	−	0.947751i	$-0.603351\pi$
$491$	−14.1376	−0.638022	−0.319011	+	0.947751i	$0.603351\pi$
$492$	0	0
$493$	1.26604	0.0570195
$494$	0	0
$495$	−0.391081	−0.0175778
$496$	0	0
$497$	21.8984	0.982278
$498$	0	0
$499$	−35.8551	−1.60509	−0.802547	−	0.596589i	$-0.796522\pi$
$499$	−35.8551	−1.60509	−0.802547	+	0.596589i	$0.796522\pi$
$500$	0	0
$501$	−24.7466	−1.10560
$502$	0	0
$503$	38.1883	1.70273	0.851366	−	0.524572i	$-0.175775\pi$
$503$	38.1883	1.70273	0.851366	+	0.524572i	$0.175775\pi$
$504$	0	0
$505$	0.623524	0.0277465
$506$	0	0
$507$	1.89888	0.0843324
$508$	0	0
$509$	20.3125	0.900336	0.450168	−	0.892944i	$-0.351364\pi$
$509$	20.3125	0.900336	0.450168	+	0.892944i	$0.351364\pi$
$510$	0	0
$511$	22.4367	0.992540
$512$	0	0
$513$	23.4103	1.03359
$514$	0	0
$515$	−4.04871	−0.178408
$516$	0	0
$517$	−2.73475	−0.120274
$518$	0	0
$519$	2.00792	0.0881377
$520$	0	0
$521$	35.4369	1.55252	0.776260	−	0.630413i	$-0.217115\pi$
$521$	35.4369	1.55252	0.776260	+	0.630413i	$0.217115\pi$
$522$	0	0
$523$	27.1450	1.18697	0.593484	−	0.804846i	$-0.297752\pi$
$523$	27.1450	1.18697	0.593484	+	0.804846i	$0.297752\pi$
$524$	0	0
$525$	19.4146	0.847325
$526$	0	0
$527$	0.153627	0.00669211
$528$	0	0
$529$	−13.0280	−0.566436
$530$	0	0
$531$	−8.64866	−0.375320
$532$	0	0
$533$	3.54277	0.153454
$534$	0	0
$535$	6.35240	0.274638
$536$	0	0
$537$	−31.7357	−1.36950
$538$	0	0
$539$	−3.08799	−0.133009
$540$	0	0
$541$	27.0912	1.16474	0.582370	−	0.812924i	$-0.302125\pi$
$541$	27.0912	1.16474	0.582370	+	0.812924i	$0.302125\pi$
$542$	0	0
$543$	−1.16785	−0.0501170
$544$	0	0
$545$	−0.751265	−0.0321807
$546$	0	0
$547$	18.5889	0.794802	0.397401	−	0.917645i	$-0.369912\pi$
$547$	18.5889	0.794802	0.397401	+	0.917645i	$0.369912\pi$
$548$	0	0
$549$	1.77680	0.0758320
$550$	0	0
$551$	−5.14922	−0.219364
$552$	0	0
$553$	−9.80934	−0.417136
$554$	0	0
$555$	−0.184490	−0.00783114
$556$	0	0
$557$	−31.0814	−1.31696	−0.658481	−	0.752597i	$-0.728801\pi$
$557$	−31.0814	−1.31696	−0.658481	+	0.752597i	$0.728801\pi$
$558$	0	0
$559$	9.77935	0.413623
$560$	0	0
$561$	3.12709	0.132026
$562$	0	0
$563$	44.3687	1.86992	0.934959	−	0.354755i	$-0.115436\pi$
$563$	44.3687	1.86992	0.934959	+	0.354755i	$0.115436\pi$
$564$	0	0
$565$	−6.23551	−0.262330
$566$	0	0
$567$	22.4767	0.943934
$568$	0	0
$569$	2.38815	0.100117	0.0500583	−	0.998746i	$-0.484059\pi$
$569$	2.38815	0.100117	0.0500583	+	0.998746i	$0.484059\pi$
$570$	0	0
$571$	7.34548	0.307399	0.153699	−	0.988118i	$-0.450881\pi$
$571$	7.34548	0.307399	0.153699	+	0.988118i	$0.450881\pi$
$572$	0	0
$573$	−25.0076	−1.04471
$574$	0	0
$575$	−15.0113	−0.626015
$576$	0	0
$577$	−40.8766	−1.70171	−0.850857	−	0.525397i	$-0.823917\pi$
$577$	−40.8766	−1.70171	−0.850857	+	0.525397i	$0.823917\pi$
$578$	0	0
$579$	−7.15621	−0.297402
$580$	0	0
$581$	26.1680	1.08563
$582$	0	0
$583$	−4.93834	−0.204525
$584$	0	0
$585$	−0.300657	−0.0124306
$586$	0	0
$587$	−18.8054	−0.776183	−0.388092	−	0.921621i	$-0.626866\pi$
$587$	−18.8054	−0.776183	−0.388092	+	0.921621i	$0.626866\pi$
$588$	0	0
$589$	−0.624831	−0.0257457
$590$	0	0
$591$	30.8323	1.26827
$592$	0	0
$593$	40.3406	1.65659	0.828294	−	0.560293i	$-0.189312\pi$
$593$	40.3406	1.65659	0.828294	+	0.560293i	$0.189312\pi$
$594$	0	0
$595$	1.35151	0.0554064
$596$	0	0
$597$	−13.5838	−0.555949
$598$	0	0
$599$	−15.2979	−0.625055	−0.312528	−	0.949909i	$-0.601176\pi$
$599$	−15.2979	−0.625055	−0.312528	+	0.949909i	$0.601176\pi$
$600$	0	0
$601$	12.3760	0.504829	0.252415	−	0.967619i	$-0.418775\pi$
$601$	12.3760	0.504829	0.252415	+	0.967619i	$0.418775\pi$
$602$	0	0
$603$	−2.18501	−0.0889803
$604$	0	0
$605$	4.61983	0.187823
$606$	0	0
$607$	26.7301	1.08494	0.542470	−	0.840075i	$-0.317489\pi$
$607$	26.7301	1.08494	0.542470	+	0.840075i	$0.317489\pi$
$608$	0	0
$609$	−4.08415	−0.165498
$610$	0	0
$611$	−2.10244	−0.0850554
$612$	0	0
$613$	−7.49954	−0.302904	−0.151452	−	0.988465i	$-0.548395\pi$
$613$	−7.49954	−0.302904	−0.151452	+	0.988465i	$0.548395\pi$
$614$	0	0
$615$	−3.33895	−0.134639
$616$	0	0
$617$	−15.0913	−0.607552	−0.303776	−	0.952743i	$-0.598247\pi$
$617$	−15.0913	−0.607552	−0.303776	+	0.952743i	$0.598247\pi$
$618$	0	0
$619$	2.01551	0.0810102	0.0405051	−	0.999179i	$-0.487103\pi$
$619$	2.01551	0.0810102	0.0405051	+	0.999179i	$0.487103\pi$
$620$	0	0
$621$	−14.3567	−0.576116
$622$	0	0
$623$	17.3949	0.696911
$624$	0	0
$625$	21.3656	0.854623
$626$	0	0
$627$	−12.7185	−0.507927
$628$	0	0
$629$	0.247829	0.00988159
$630$	0	0
$631$	−39.0222	−1.55345	−0.776724	−	0.629841i	$-0.783120\pi$
$631$	−39.0222	−1.55345	−0.776724	+	0.629841i	$0.783120\pi$
$632$	0	0
$633$	5.68149	0.225819
$634$	0	0
$635$	6.84654	0.271697
$636$	0	0
$637$	−2.37400	−0.0940613
$638$	0	0
$639$	−6.16756	−0.243985
$640$	0	0
$641$	13.6957	0.540947	0.270473	−	0.962727i	$-0.412820\pi$
$641$	13.6957	0.540947	0.270473	+	0.962727i	$0.412820\pi$
$642$	0	0
$643$	6.49280	0.256051	0.128026	−	0.991771i	$-0.459136\pi$
$643$	6.49280	0.256051	0.128026	+	0.991771i	$0.459136\pi$
$644$	0	0
$645$	−9.21673	−0.362908
$646$	0	0
$647$	14.5122	0.570535	0.285268	−	0.958448i	$-0.407918\pi$
$647$	14.5122	0.570535	0.285268	+	0.958448i	$0.407918\pi$
$648$	0	0
$649$	−18.5712	−0.728984
$650$	0	0
$651$	−0.495590	−0.0194237
$652$	0	0
$653$	8.32348	0.325723	0.162862	−	0.986649i	$-0.447928\pi$
$653$	8.32348	0.325723	0.162862	+	0.986649i	$0.447928\pi$
$654$	0	0
$655$	6.63013	0.259061
$656$	0	0
$657$	−6.31915	−0.246534
$658$	0	0
$659$	−21.8290	−0.850335	−0.425168	−	0.905115i	$-0.639785\pi$
$659$	−21.8290	−0.850335	−0.425168	+	0.905115i	$0.639785\pi$
$660$	0	0
$661$	−40.1820	−1.56290	−0.781450	−	0.623968i	$-0.785520\pi$
$661$	−40.1820	−1.56290	−0.781450	+	0.623968i	$0.785520\pi$
$662$	0	0
$663$	2.40406	0.0933661
$664$	0	0
$665$	−5.49683	−0.213158
$666$	0	0
$667$	3.15784	0.122272
$668$	0	0
$669$	−54.7160	−2.11544
$670$	0	0
$671$	3.81531	0.147289
$672$	0	0
$673$	22.9972	0.886478	0.443239	−	0.896404i	$-0.353829\pi$
$673$	22.9972	0.886478	0.443239	+	0.896404i	$0.353829\pi$
$674$	0	0
$675$	21.6119	0.831844
$676$	0	0
$677$	−41.9421	−1.61197	−0.805983	−	0.591938i	$-0.798363\pi$
$677$	−41.9421	−1.61197	−0.805983	+	0.591938i	$0.798363\pi$
$678$	0	0
$679$	−8.75417	−0.335954
$680$	0	0
$681$	−30.7810	−1.17953
$682$	0	0
$683$	6.12428	0.234339	0.117170	−	0.993112i	$-0.462618\pi$
$683$	6.12428	0.234339	0.117170	+	0.993112i	$0.462618\pi$
$684$	0	0
$685$	−4.41728	−0.168776
$686$	0	0
$687$	54.6685	2.08573
$688$	0	0
$689$	−3.79652	−0.144636
$690$	0	0
$691$	−39.3533	−1.49707	−0.748534	−	0.663096i	$-0.769242\pi$
$691$	−39.3533	−1.49707	−0.748534	+	0.663096i	$0.769242\pi$
$692$	0	0
$693$	−1.69473	−0.0643776
$694$	0	0
$695$	−5.34225	−0.202643
$696$	0	0
$697$	4.48529	0.169892
$698$	0	0
$699$	−54.5165	−2.06200
$700$	0	0
$701$	39.7745	1.50226	0.751130	−	0.660154i	$-0.229509\pi$
$701$	39.7745	1.50226	0.751130	+	0.660154i	$0.229509\pi$
$702$	0	0
$703$	−1.00797	−0.0380162
$704$	0	0
$705$	1.98148	0.0746268
$706$	0	0
$707$	2.70202	0.101620
$708$	0	0
$709$	45.1451	1.69546	0.847729	−	0.530429i	$-0.177969\pi$
$709$	45.1451	1.69546	0.847729	+	0.530429i	$0.177969\pi$
$710$	0	0
$711$	2.76274	0.103611
$712$	0	0
$713$	0.383188	0.0143505
$714$	0	0
$715$	−0.645599	−0.0241440
$716$	0	0
$717$	0.351612	0.0131312
$718$	0	0
$719$	44.4844	1.65899	0.829494	−	0.558516i	$-0.188629\pi$
$719$	44.4844	1.65899	0.829494	+	0.558516i	$0.188629\pi$
$720$	0	0
$721$	−17.5449	−0.653407
$722$	0	0
$723$	24.2524	0.901958
$724$	0	0
$725$	−4.75366	−0.176546
$726$	0	0
$727$	−18.5035	−0.686258	−0.343129	−	0.939288i	$-0.611487\pi$
$727$	−18.5035	−0.686258	−0.343129	+	0.939288i	$0.611487\pi$
$728$	0	0
$729$	19.5687	0.724767
$730$	0	0
$731$	12.3810	0.457929
$732$	0	0
$733$	−35.4243	−1.30842	−0.654212	−	0.756311i	$-0.727000\pi$
$733$	−35.4243	−1.30842	−0.654212	+	0.756311i	$0.727000\pi$
$734$	0	0
$735$	2.23742	0.0825284
$736$	0	0
$737$	−4.69185	−0.172827
$738$	0	0
$739$	−23.2993	−0.857078	−0.428539	−	0.903523i	$-0.640972\pi$
$739$	−23.2993	−0.857078	−0.428539	+	0.903523i	$0.640972\pi$
$740$	0	0
$741$	−9.77778	−0.359196
$742$	0	0
$743$	−12.8205	−0.470339	−0.235170	−	0.971954i	$-0.575565\pi$
$743$	−12.8205	−0.470339	−0.235170	+	0.971954i	$0.575565\pi$
$744$	0	0
$745$	−0.933010	−0.0341828
$746$	0	0
$747$	−7.37007	−0.269657
$748$	0	0
$749$	27.5279	1.00585
$750$	0	0
$751$	12.7936	0.466844	0.233422	−	0.972375i	$-0.425008\pi$
$751$	12.7936	0.466844	0.233422	+	0.972375i	$0.425008\pi$
$752$	0	0
$753$	−33.7055	−1.22830
$754$	0	0
$755$	6.86130	0.249708
$756$	0	0
$757$	45.0619	1.63780	0.818901	−	0.573935i	$-0.194584\pi$
$757$	45.0619	1.63780	0.818901	+	0.573935i	$0.194584\pi$
$758$	0	0
$759$	7.79981	0.283115
$760$	0	0
$761$	1.83904	0.0666653	0.0333327	−	0.999444i	$-0.489388\pi$
$761$	1.83904	0.0666653	0.0333327	+	0.999444i	$0.489388\pi$
$762$	0	0
$763$	−3.25558	−0.117860
$764$	0	0
$765$	−0.380644	−0.0137622
$766$	0	0
$767$	−14.2773	−0.515522
$768$	0	0
$769$	18.5431	0.668681	0.334340	−	0.942452i	$-0.391486\pi$
$769$	18.5431	0.668681	0.334340	+	0.942452i	$0.391486\pi$
$770$	0	0
$771$	14.8246	0.533894
$772$	0	0
$773$	−16.8299	−0.605330	−0.302665	−	0.953097i	$-0.597876\pi$
$773$	−16.8299	−0.605330	−0.302665	+	0.953097i	$0.597876\pi$
$774$	0	0
$775$	−0.576832	−0.0207204
$776$	0	0
$777$	−0.799478	−0.0286811
$778$	0	0
$779$	−18.2425	−0.653606
$780$	0	0
$781$	−13.2436	−0.473892
$782$	0	0
$783$	−4.54638	−0.162474
$784$	0	0
$785$	−2.87392	−0.102575
$786$	0	0
$787$	−18.1768	−0.647933	−0.323966	−	0.946069i	$-0.605016\pi$
$787$	−18.1768	−0.647933	−0.323966	+	0.946069i	$0.605016\pi$
$788$	0	0
$789$	8.27841	0.294719
$790$	0	0
$791$	−27.0214	−0.960769
$792$	0	0
$793$	2.93316	0.104159
$794$	0	0
$795$	3.57810	0.126902
$796$	0	0
$797$	−11.3071	−0.400519	−0.200259	−	0.979743i	$-0.564178\pi$
$797$	−11.3071	−0.400519	−0.200259	+	0.979743i	$0.564178\pi$
$798$	0	0
$799$	−2.66177	−0.0941665
$800$	0	0
$801$	−4.89917	−0.173104
$802$	0	0
$803$	−13.5691	−0.478843
$804$	0	0
$805$	3.37102	0.118813
$806$	0	0
$807$	14.7195	0.518152
$808$	0	0
$809$	23.4117	0.823111	0.411555	−	0.911385i	$-0.364986\pi$
$809$	23.4117	0.823111	0.411555	+	0.911385i	$0.364986\pi$
$810$	0	0
$811$	33.9969	1.19379	0.596896	−	0.802318i	$-0.296400\pi$
$811$	33.9969	1.19379	0.596896	+	0.802318i	$0.296400\pi$
$812$	0	0
$813$	38.0879	1.33580
$814$	0	0
$815$	−3.44107	−0.120535
$816$	0	0
$817$	−50.3561	−1.76174
$818$	0	0
$819$	−1.30289	−0.0455265
$820$	0	0
$821$	23.2751	0.812308	0.406154	−	0.913805i	$-0.366870\pi$
$821$	23.2751	0.812308	0.406154	+	0.913805i	$0.366870\pi$
$822$	0	0
$823$	37.6259	1.31156	0.655778	−	0.754954i	$-0.272341\pi$
$823$	37.6259	1.31156	0.655778	+	0.754954i	$0.272341\pi$
$824$	0	0
$825$	−11.7414	−0.408785
$826$	0	0
$827$	−28.4166	−0.988143	−0.494071	−	0.869421i	$-0.664492\pi$
$827$	−28.4166	−0.988143	−0.494071	+	0.869421i	$0.664492\pi$
$828$	0	0
$829$	12.9631	0.450227	0.225113	−	0.974333i	$-0.427725\pi$
$829$	12.9631	0.450227	0.225113	+	0.974333i	$0.427725\pi$
$830$	0	0
$831$	22.3340	0.774759
$832$	0	0
$833$	−3.00558	−0.104137
$834$	0	0
$835$	6.46822	0.223842
$836$	0	0
$837$	−0.551679	−0.0190688
$838$	0	0
$839$	−7.23449	−0.249762	−0.124881	−	0.992172i	$-0.539855\pi$
$839$	−7.23449	−0.249762	−0.124881	+	0.992172i	$0.539855\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−3.99436	−0.137573
$844$	0	0
$845$	−0.496327	−0.0170742
$846$	0	0
$847$	20.0199	0.687891
$848$	0	0
$849$	−36.1624	−1.24109
$850$	0	0
$851$	0.618152	0.0211900
$852$	0	0
$853$	−24.8075	−0.849391	−0.424696	−	0.905336i	$-0.639619\pi$
$853$	−24.8075	−0.849391	−0.424696	+	0.905336i	$0.639619\pi$
$854$	0	0
$855$	1.54815	0.0529456
$856$	0	0
$857$	13.7553	0.469872	0.234936	−	0.972011i	$-0.424512\pi$
$857$	13.7553	0.469872	0.234936	+	0.972011i	$0.424512\pi$
$858$	0	0
$859$	19.3545	0.660367	0.330184	−	0.943917i	$-0.392889\pi$
$859$	19.3545	0.660367	0.330184	+	0.943917i	$0.392889\pi$
$860$	0	0
$861$	−14.4692	−0.493109
$862$	0	0
$863$	−40.5591	−1.38065	−0.690325	−	0.723500i	$-0.742532\pi$
$863$	−40.5591	−1.38065	−0.690325	+	0.723500i	$0.742532\pi$
$864$	0	0
$865$	−0.524826	−0.0178446
$866$	0	0
$867$	−29.2374	−0.992954
$868$	0	0
$869$	5.93242	0.201244
$870$	0	0
$871$	−3.60702	−0.122219
$872$	0	0
$873$	2.46556	0.0834466
$874$	0	0
$875$	−10.4121	−0.351993
$876$	0	0
$877$	−23.1074	−0.780283	−0.390142	−	0.920755i	$-0.627574\pi$
$877$	−23.1074	−0.780283	−0.390142	+	0.920755i	$0.627574\pi$
$878$	0	0
$879$	40.9411	1.38091
$880$	0	0
$881$	4.35388	0.146686	0.0733430	−	0.997307i	$-0.476633\pi$
$881$	4.35388	0.146686	0.0733430	+	0.997307i	$0.476633\pi$
$882$	0	0
$883$	−51.2050	−1.72319	−0.861593	−	0.507600i	$-0.830533\pi$
$883$	−51.2050	−1.72319	−0.861593	+	0.507600i	$0.830533\pi$
$884$	0	0
$885$	13.4559	0.452314
$886$	0	0
$887$	12.7385	0.427718	0.213859	−	0.976865i	$-0.431397\pi$
$887$	12.7385	0.427718	0.213859	+	0.976865i	$0.431397\pi$
$888$	0	0
$889$	29.6692	0.995073
$890$	0	0
$891$	−13.5933	−0.455393
$892$	0	0
$893$	10.8259	0.362275
$894$	0	0
$895$	8.29501	0.277272
$896$	0	0
$897$	5.99638	0.200213
$898$	0	0
$899$	0.121345	0.00404707
$900$	0	0
$901$	−4.80655	−0.160129
$902$	0	0
$903$	−39.9403	−1.32913
$904$	0	0
$905$	0.305249	0.0101468
$906$	0	0
$907$	−1.52201	−0.0505375	−0.0252687	−	0.999681i	$-0.508044\pi$
$907$	−1.52201	−0.0505375	−0.0252687	+	0.999681i	$0.508044\pi$
$908$	0	0
$909$	−0.761007	−0.0252410
$910$	0	0
$911$	43.4829	1.44065	0.720327	−	0.693635i	$-0.243992\pi$
$911$	43.4829	1.44065	0.720327	+	0.693635i	$0.243992\pi$
$912$	0	0
$913$	−15.8257	−0.523755
$914$	0	0
$915$	−2.76441	−0.0913885
$916$	0	0
$917$	28.7314	0.948795
$918$	0	0
$919$	−26.1390	−0.862246	−0.431123	−	0.902293i	$-0.641882\pi$
$919$	−26.1390	−0.862246	−0.431123	+	0.902293i	$0.641882\pi$
$920$	0	0
$921$	−28.9936	−0.955373
$922$	0	0
$923$	−10.1815	−0.335127
$924$	0	0
$925$	−0.930536	−0.0305958
$926$	0	0
$927$	4.94143	0.162298
$928$	0	0
$929$	57.5117	1.88690	0.943449	−	0.331517i	$-0.107561\pi$
$929$	57.5117	1.88690	0.943449	+	0.331517i	$0.107561\pi$
$930$	0	0
$931$	12.2243	0.400634
$932$	0	0
$933$	26.5867	0.870410
$934$	0	0
$935$	−0.817354	−0.0267303
$936$	0	0
$937$	19.8880	0.649712	0.324856	−	0.945763i	$-0.394684\pi$
$937$	19.8880	0.649712	0.324856	+	0.945763i	$0.394684\pi$
$938$	0	0
$939$	6.15745	0.200941
$940$	0	0
$941$	−35.7430	−1.16519	−0.582594	−	0.812764i	$-0.697962\pi$
$941$	−35.7430	−1.16519	−0.582594	+	0.812764i	$0.697962\pi$
$942$	0	0
$943$	11.1875	0.364315
$944$	0	0
$945$	−4.85329	−0.157878
$946$	0	0
$947$	39.4601	1.28228	0.641141	−	0.767423i	$-0.278461\pi$
$947$	39.4601	1.28228	0.641141	+	0.767423i	$0.278461\pi$
$948$	0	0
$949$	−10.4317	−0.338628
$950$	0	0
$951$	10.4729	0.339607
$952$	0	0
$953$	−18.3855	−0.595564	−0.297782	−	0.954634i	$-0.596247\pi$
$953$	−18.3855	−0.595564	−0.297782	+	0.954634i	$0.596247\pi$
$954$	0	0
$955$	6.53643	0.211514
$956$	0	0
$957$	2.46998	0.0798432
$958$	0	0
$959$	−19.1421	−0.618131
$960$	0	0
$961$	−30.9853	−0.999525
$962$	0	0
$963$	−7.75306	−0.249839
$964$	0	0
$965$	1.87048	0.0602128
$966$	0	0
$967$	34.8380	1.12032	0.560158	−	0.828386i	$-0.310741\pi$
$967$	34.8380	1.12032	0.560158	+	0.828386i	$0.310741\pi$
$968$	0	0
$969$	−12.3791	−0.397673
$970$	0	0
$971$	15.5465	0.498910	0.249455	−	0.968386i	$-0.419748\pi$
$971$	15.5465	0.498910	0.249455	+	0.968386i	$0.419748\pi$
$972$	0	0
$973$	−23.1504	−0.742169
$974$	0	0
$975$	−9.02665	−0.289084
$976$	0	0
$977$	−49.5195	−1.58427	−0.792135	−	0.610346i	$-0.791030\pi$
$977$	−49.5195	−1.58427	−0.792135	+	0.610346i	$0.791030\pi$
$978$	0	0
$979$	−10.5200	−0.336219
$980$	0	0
$981$	0.916914	0.0292748
$982$	0	0
$983$	12.1008	0.385955	0.192978	−	0.981203i	$-0.438185\pi$
$983$	12.1008	0.385955	0.192978	+	0.981203i	$0.438185\pi$
$984$	0	0
$985$	−8.05889	−0.256778
$986$	0	0
$987$	8.58666	0.273316
$988$	0	0
$989$	30.8816	0.981979
$990$	0	0
$991$	−40.8432	−1.29743	−0.648713	−	0.761033i	$-0.724692\pi$
$991$	−40.8432	−1.29743	−0.648713	+	0.761033i	$0.724692\pi$
$992$	0	0
$993$	−54.8808	−1.74159
$994$	0	0
$995$	3.55051	0.112559
$996$	0	0
$997$	−44.5671	−1.41145	−0.705727	−	0.708484i	$-0.749380\pi$
$997$	−44.5671	−1.41145	−0.705727	+	0.708484i	$0.749380\pi$
$998$	0	0
$999$	−0.889960	−0.0281571

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.bd.1.11		12
4.3	odd	2		3016.2.a.j.1.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.j.1.2	✓	12	4.3	odd	2
6032.2.a.bd.1.11		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.89888 q^{3} -0.496327 q^{5} -2.15081 q^{7} +0.605764 q^{9} +O(q^{10})\)	\(q+1.89888 q^{3} -0.496327 q^{5} -2.15081 q^{7} +0.605764 q^{9} +1.30075 q^{11} +1.00000 q^{13} -0.942468 q^{15} +1.26604 q^{17} -5.14922 q^{19} -4.08415 q^{21} +3.15784 q^{23} -4.75366 q^{25} -4.54638 q^{27} +1.00000 q^{29} +0.121345 q^{31} +2.46998 q^{33} +1.06751 q^{35} +0.195751 q^{37} +1.89888 q^{39} +3.54277 q^{41} +9.77935 q^{43} -0.300657 q^{45} -2.10244 q^{47} -2.37400 q^{49} +2.40406 q^{51} -3.79652 q^{53} -0.645599 q^{55} -9.77778 q^{57} -14.2773 q^{59} +2.93316 q^{61} -1.30289 q^{63} -0.496327 q^{65} -3.60702 q^{67} +5.99638 q^{69} -10.1815 q^{71} -10.4317 q^{73} -9.02665 q^{75} -2.79768 q^{77} +4.56076 q^{79} -10.4503 q^{81} -12.1666 q^{83} -0.628370 q^{85} +1.89888 q^{87} -8.08758 q^{89} -2.15081 q^{91} +0.230420 q^{93} +2.55570 q^{95} +4.07017 q^{97} +0.787950 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9}+O(q^{10}) \) 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9	\( 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9} - 14 q^{11} + 12 q^{13} - 8 q^{15} + 4 q^{17} - 11 q^{19} - 5 q^{21} - 15 q^{23} + 5 q^{25} - 24 q^{27} + 12 q^{29} - 13 q^{31} - q^{33} - 18 q^{35} - 23 q^{37} - 6 q^{39} - 2 q^{41} - 26 q^{43} - 9 q^{45} - 15 q^{47} + 16 q^{49} - 21 q^{51} + 31 q^{53} - 10 q^{55} - 10 q^{57} - 7 q^{59} + 2 q^{61} + 25 q^{63} + 3 q^{65} - 47 q^{67} - 8 q^{69} - 32 q^{71} - 25 q^{73} - 31 q^{75} - 4 q^{77} - 7 q^{79} + 64 q^{81} - 12 q^{83} + 7 q^{85} - 6 q^{87} + 6 q^{89} - 6 q^{91} + 17 q^{93} - q^{95} - 7 q^{97} - 44 q^{99}+O(q^{100}) \) 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9 - 14 * q^11 + 12 * q^13 - 8 * q^15 + 4 * q^17 - 11 * q^19 - 5 * q^21 - 15 * q^23 + 5 * q^25 - 24 * q^27 + 12 * q^29 - 13 * q^31 - q^33 - 18 * q^35 - 23 * q^37 - 6 * q^39 - 2 * q^41 - 26 * q^43 - 9 * q^45 - 15 * q^47 + 16 * q^49 - 21 * q^51 + 31 * q^53 - 10 * q^55 - 10 * q^57 - 7 * q^59 + 2 * q^61 + 25 * q^63 + 3 * q^65 - 47 * q^67 - 8 * q^69 - 32 * q^71 - 25 * q^73 - 31 * q^75 - 4 * q^77 - 7 * q^79 + 64 * q^81 - 12 * q^83 + 7 * q^85 - 6 * q^87 + 6 * q^89 - 6 * q^91 + 17 * q^93 - q^95 - 7 * q^97 - 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more