LMFDB - Embedded newform 6012.2.a.g.1.5

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:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 11x^{5} - 7x^{4} + 21x^{3} + 17x^{2} - 4x - 4 $ x^7 - 11x^5 - 7x^4 + 21x^3 + 17x^2 - 4*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 668)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.47685$ 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)

$f(q)$	$=$	$q+1.35423 q^{5} -2.86706 q^{7} +O(q^{10})$	$q+1.35423 q^{5} -2.86706 q^{7} -4.77139 q^{11} -0.418211 q^{13} +4.50887 q^{17} +0.718559 q^{19} +8.56362 q^{23} -3.16605 q^{25} +4.99604 q^{29} -8.79181 q^{31} -3.88267 q^{35} +7.92603 q^{37} -7.19666 q^{41} -2.21710 q^{43} +2.77439 q^{47} +1.22005 q^{49} +4.11728 q^{53} -6.46157 q^{55} -3.85265 q^{59} +11.3485 q^{61} -0.566355 q^{65} +6.07497 q^{67} -10.7226 q^{71} -2.49865 q^{73} +13.6799 q^{77} -16.0341 q^{79} +12.8077 q^{83} +6.10606 q^{85} -16.6573 q^{89} +1.19904 q^{91} +0.973097 q^{95} -9.16191 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{5} - 12 q^{7}+O(q^{10}) $ 7 * q + 2 * q^5 - 12 * q^7	$ 7 q + 2 q^{5} - 12 q^{7} + 7 q^{11} - 9 q^{13} + q^{17} - 11 q^{19} + 19 q^{23} + 3 q^{25} + 5 q^{29} - 13 q^{31} + 7 q^{35} - 26 q^{37} + 2 q^{41} - 24 q^{43} + 11 q^{47} + 19 q^{49} - 4 q^{53} - 4 q^{55} + 4 q^{59} - 5 q^{61} - 13 q^{65} - 42 q^{67} - 9 q^{71} + 27 q^{73} - 12 q^{77} - 8 q^{79} - 16 q^{83} - 27 q^{85} - 9 q^{89} - 2 q^{91} - 10 q^{95} - 8 q^{97}+O(q^{100}) $ 7 * q + 2 * q^5 - 12 * q^7 + 7 * q^11 - 9 * q^13 + q^17 - 11 * q^19 + 19 * q^23 + 3 * q^25 + 5 * q^29 - 13 * q^31 + 7 * q^35 - 26 * q^37 + 2 * q^41 - 24 * q^43 + 11 * q^47 + 19 * q^49 - 4 * q^53 - 4 * q^55 + 4 * q^59 - 5 * q^61 - 13 * q^65 - 42 * q^67 - 9 * q^71 + 27 * q^73 - 12 * q^77 - 8 * q^79 - 16 * q^83 - 27 * q^85 - 9 * q^89 - 2 * q^91 - 10 * q^95 - 8 * 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
$5$	1.35423	0.605631	0.302816	−	0.953049i	$-0.402073\pi$
$5$	1.35423	0.605631	0.302816	+	0.953049i	$0.402073\pi$
$6$	0	0
$7$	−2.86706	−1.08365	−0.541824	−	0.840492i	$-0.682266\pi$
$7$	−2.86706	−1.08365	−0.541824	+	0.840492i	$0.682266\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.77139	−1.43863	−0.719314	−	0.694685i	$-0.755544\pi$
$11$	−4.77139	−1.43863	−0.719314	+	0.694685i	$0.755544\pi$
$12$	0	0
$13$	−0.418211	−0.115991	−0.0579954	−	0.998317i	$-0.518471\pi$
$13$	−0.418211	−0.115991	−0.0579954	+	0.998317i	$0.518471\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.50887	1.09356	0.546781	−	0.837276i	$-0.315853\pi$
$17$	4.50887	1.09356	0.546781	+	0.837276i	$0.315853\pi$
$18$	0	0
$19$	0.718559	0.164849	0.0824244	−	0.996597i	$-0.473734\pi$
$19$	0.718559	0.164849	0.0824244	+	0.996597i	$0.473734\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.56362	1.78564	0.892819	−	0.450415i	$-0.148724\pi$
$23$	8.56362	1.78564	0.892819	+	0.450415i	$0.148724\pi$
$24$	0	0
$25$	−3.16605	−0.633211
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.99604	0.927742	0.463871	−	0.885903i	$-0.346460\pi$
$29$	4.99604	0.927742	0.463871	+	0.885903i	$0.346460\pi$
$30$	0	0
$31$	−8.79181	−1.57906	−0.789528	−	0.613715i	$-0.789674\pi$
$31$	−8.79181	−1.57906	−0.789528	+	0.613715i	$0.789674\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.88267	−0.656291
$36$	0	0
$37$	7.92603	1.30303	0.651516	−	0.758635i	$-0.274133\pi$
$37$	7.92603	1.30303	0.651516	+	0.758635i	$0.274133\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.19666	−1.12393	−0.561965	−	0.827161i	$-0.689954\pi$
$41$	−7.19666	−1.12393	−0.561965	+	0.827161i	$0.689954\pi$
$42$	0	0
$43$	−2.21710	−0.338105	−0.169053	−	0.985607i	$-0.554071\pi$
$43$	−2.21710	−0.338105	−0.169053	+	0.985607i	$0.554071\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.77439	0.404686	0.202343	−	0.979315i	$-0.435144\pi$
$47$	2.77439	0.404686	0.202343	+	0.979315i	$0.435144\pi$
$48$	0	0
$49$	1.22005	0.174292
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.11728	0.565552	0.282776	−	0.959186i	$-0.408745\pi$
$53$	4.11728	0.565552	0.282776	+	0.959186i	$0.408745\pi$
$54$	0	0
$55$	−6.46157	−0.871278
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.85265	−0.501572	−0.250786	−	0.968043i	$-0.580689\pi$
$59$	−3.85265	−0.501572	−0.250786	+	0.968043i	$0.580689\pi$
$60$	0	0
$61$	11.3485	1.45303	0.726515	−	0.687151i	$-0.241139\pi$
$61$	11.3485	1.45303	0.726515	+	0.687151i	$0.241139\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.566355	−0.0702476
$66$	0	0
$67$	6.07497	0.742175	0.371088	−	0.928598i	$-0.378985\pi$
$67$	6.07497	0.742175	0.371088	+	0.928598i	$0.378985\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.7226	−1.27254	−0.636271	−	0.771466i	$-0.719524\pi$
$71$	−10.7226	−1.27254	−0.636271	+	0.771466i	$0.719524\pi$
$72$	0	0
$73$	−2.49865	−0.292444	−0.146222	−	0.989252i	$-0.546711\pi$
$73$	−2.49865	−0.292444	−0.146222	+	0.989252i	$0.546711\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	13.6799	1.55897
$78$	0	0
$79$	−16.0341	−1.80398	−0.901991	−	0.431755i	$-0.857894\pi$
$79$	−16.0341	−1.80398	−0.901991	+	0.431755i	$0.857894\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.8077	1.40583	0.702914	−	0.711275i	$-0.251882\pi$
$83$	12.8077	1.40583	0.702914	+	0.711275i	$0.251882\pi$
$84$	0	0
$85$	6.10606	0.662295
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−16.6573	−1.76566	−0.882832	−	0.469688i	$-0.844366\pi$
$89$	−16.6573	−1.76566	−0.882832	+	0.469688i	$0.844366\pi$
$90$	0	0
$91$	1.19904	0.125693
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.973097	0.0998376
$96$	0	0
$97$	−9.16191	−0.930251	−0.465125	−	0.885245i	$-0.653991\pi$
$97$	−9.16191	−0.930251	−0.465125	+	0.885245i	$0.653991\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.9723	−1.09179	−0.545894	−	0.837854i	$-0.683810\pi$
$101$	−10.9723	−1.09179	−0.545894	+	0.837854i	$0.683810\pi$
$102$	0	0
$103$	2.71575	0.267591	0.133795	−	0.991009i	$-0.457284\pi$
$103$	2.71575	0.267591	0.133795	+	0.991009i	$0.457284\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.0988	−1.07296	−0.536481	−	0.843913i	$-0.680247\pi$
$107$	−11.0988	−1.07296	−0.536481	+	0.843913i	$0.680247\pi$
$108$	0	0
$109$	−12.9660	−1.24191	−0.620957	−	0.783844i	$-0.713256\pi$
$109$	−12.9660	−1.24191	−0.620957	+	0.783844i	$0.713256\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.28578	0.403172	0.201586	−	0.979471i	$-0.435390\pi$
$113$	4.28578	0.403172	0.201586	+	0.979471i	$0.435390\pi$
$114$	0	0
$115$	11.5971	1.08144
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.9272	−1.18504
$120$	0	0
$121$	11.7662	1.06965
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.0587	−0.989124
$126$	0	0
$127$	−8.31241	−0.737607	−0.368804	−	0.929507i	$-0.620233\pi$
$127$	−8.31241	−0.737607	−0.368804	+	0.929507i	$0.620233\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−7.93961	−0.693687	−0.346843	−	0.937923i	$-0.612746\pi$
$131$	−7.93961	−0.693687	−0.346843	+	0.937923i	$0.612746\pi$
$132$	0	0
$133$	−2.06015	−0.178638
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.7937	−1.34935	−0.674673	−	0.738116i	$-0.735715\pi$
$137$	−15.7937	−1.34935	−0.674673	+	0.738116i	$0.735715\pi$
$138$	0	0
$139$	−14.5531	−1.23438	−0.617190	−	0.786814i	$-0.711729\pi$
$139$	−14.5531	−1.23438	−0.617190	+	0.786814i	$0.711729\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.99545	0.166868
$144$	0	0
$145$	6.76580	0.561869
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.65275	−0.463091	−0.231546	−	0.972824i	$-0.574378\pi$
$149$	−5.65275	−0.463091	−0.231546	+	0.972824i	$0.574378\pi$
$150$	0	0
$151$	−11.6857	−0.950965	−0.475483	−	0.879725i	$-0.657727\pi$
$151$	−11.6857	−0.950965	−0.475483	+	0.879725i	$0.657727\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−11.9062	−0.956326
$156$	0	0
$157$	0.423652	0.0338111	0.0169056	−	0.999857i	$-0.494619\pi$
$157$	0.423652	0.0338111	0.0169056	+	0.999857i	$0.494619\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−24.5524	−1.93500
$162$	0	0
$163$	−17.6603	−1.38326	−0.691631	−	0.722251i	$-0.743108\pi$
$163$	−17.6603	−1.38326	−0.691631	+	0.722251i	$0.743108\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−12.8251	−0.986546
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.5573	0.954711	0.477356	−	0.878710i	$-0.341595\pi$
$173$	12.5573	0.954711	0.477356	+	0.878710i	$0.341595\pi$
$174$	0	0
$175$	9.07727	0.686177
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−7.90688	−0.590988	−0.295494	−	0.955345i	$-0.595484\pi$
$179$	−7.90688	−0.590988	−0.295494	+	0.955345i	$0.595484\pi$
$180$	0	0
$181$	5.56740	0.413821	0.206911	−	0.978360i	$-0.433659\pi$
$181$	5.56740	0.413821	0.206911	+	0.978360i	$0.433659\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.7337	0.789157
$186$	0	0
$187$	−21.5136	−1.57323
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.75834	0.633731	0.316866	−	0.948470i	$-0.397370\pi$
$191$	8.75834	0.633731	0.316866	+	0.948470i	$0.397370\pi$
$192$	0	0
$193$	27.6257	1.98854	0.994272	−	0.106882i	$-0.0340868\pi$
$193$	27.6257	1.98854	0.994272	+	0.106882i	$0.0340868\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.9758	1.20948	0.604738	−	0.796424i	$-0.293278\pi$
$197$	16.9758	1.20948	0.604738	+	0.796424i	$0.293278\pi$
$198$	0	0
$199$	−10.2347	−0.725515	−0.362758	−	0.931884i	$-0.618165\pi$
$199$	−10.2347	−0.725515	−0.362758	+	0.931884i	$0.618165\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−14.3240	−1.00534
$204$	0	0
$205$	−9.74595	−0.680687
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.42853	−0.237156
$210$	0	0
$211$	−22.6376	−1.55844	−0.779218	−	0.626754i	$-0.784383\pi$
$211$	−22.6376	−1.55844	−0.779218	+	0.626754i	$0.784383\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.00247	−0.204767
$216$	0	0
$217$	25.2067	1.71114
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.88566	−0.126843
$222$	0	0
$223$	−6.13990	−0.411158	−0.205579	−	0.978641i	$-0.565908\pi$
$223$	−6.13990	−0.411158	−0.205579	+	0.978641i	$0.565908\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−24.6943	−1.63902	−0.819508	−	0.573067i	$-0.805753\pi$
$227$	−24.6943	−1.63902	−0.819508	+	0.573067i	$0.805753\pi$
$228$	0	0
$229$	24.0342	1.58822	0.794112	−	0.607772i	$-0.207936\pi$
$229$	24.0342	1.58822	0.794112	+	0.607772i	$0.207936\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.73738	−0.637917	−0.318958	−	0.947769i	$-0.603333\pi$
$233$	−9.73738	−0.637917	−0.318958	+	0.947769i	$0.603333\pi$
$234$	0	0
$235$	3.75716	0.245090
$236$	0	0
$237$	0	0
$238$	0	0
$239$	25.6278	1.65772	0.828861	−	0.559454i	$-0.188989\pi$
$239$	25.6278	1.65772	0.828861	+	0.559454i	$0.188989\pi$
$240$	0	0
$241$	−1.75012	−0.112735	−0.0563676	−	0.998410i	$-0.517952\pi$
$241$	−1.75012	−0.112735	−0.0563676	+	0.998410i	$0.517952\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.65223	0.105557
$246$	0	0
$247$	−0.300509	−0.0191209
$248$	0	0
$249$	0	0
$250$	0	0
$251$	21.3676	1.34871	0.674356	−	0.738407i	$-0.264422\pi$
$251$	21.3676	1.34871	0.674356	+	0.738407i	$0.264422\pi$
$252$	0	0
$253$	−40.8604	−2.56887
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.0810	0.878346	0.439173	−	0.898403i	$-0.355272\pi$
$257$	14.0810	0.878346	0.439173	+	0.898403i	$0.355272\pi$
$258$	0	0
$259$	−22.7244	−1.41203
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−0.866166	−0.0534101	−0.0267051	−	0.999643i	$-0.508501\pi$
$263$	−0.866166	−0.0534101	−0.0267051	+	0.999643i	$0.508501\pi$
$264$	0	0
$265$	5.57576	0.342516
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−8.41214	−0.512897	−0.256449	−	0.966558i	$-0.582552\pi$
$269$	−8.41214	−0.512897	−0.256449	+	0.966558i	$0.582552\pi$
$270$	0	0
$271$	26.9137	1.63489	0.817447	−	0.576004i	$-0.195389\pi$
$271$	26.9137	1.63489	0.817447	+	0.576004i	$0.195389\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	15.1065	0.910954
$276$	0	0
$277$	−10.6680	−0.640980	−0.320490	−	0.947252i	$-0.603848\pi$
$277$	−10.6680	−0.640980	−0.320490	+	0.947252i	$0.603848\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	29.6393	1.76813	0.884066	−	0.467363i	$-0.154796\pi$
$281$	29.6393	1.76813	0.884066	+	0.467363i	$0.154796\pi$
$282$	0	0
$283$	−5.67000	−0.337046	−0.168523	−	0.985698i	$-0.553900\pi$
$283$	−5.67000	−0.337046	−0.168523	+	0.985698i	$0.553900\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	20.6333	1.21794
$288$	0	0
$289$	3.32991	0.195877
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.41571	0.550072	0.275036	−	0.961434i	$-0.411310\pi$
$293$	9.41571	0.550072	0.275036	+	0.961434i	$0.411310\pi$
$294$	0	0
$295$	−5.21739	−0.303768
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.58140	−0.207118
$300$	0	0
$301$	6.35657	0.366387
$302$	0	0
$303$	0	0
$304$	0	0
$305$	15.3685	0.880001
$306$	0	0
$307$	−0.635313	−0.0362593	−0.0181296	−	0.999836i	$-0.505771\pi$
$307$	−0.635313	−0.0362593	−0.0181296	+	0.999836i	$0.505771\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.73996	−0.382188	−0.191094	−	0.981572i	$-0.561204\pi$
$311$	−6.73996	−0.382188	−0.191094	+	0.981572i	$0.561204\pi$
$312$	0	0
$313$	4.03163	0.227881	0.113941	−	0.993488i	$-0.463653\pi$
$313$	4.03163	0.227881	0.113941	+	0.993488i	$0.463653\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.91054	−0.556631	−0.278316	−	0.960490i	$-0.589776\pi$
$317$	−9.91054	−0.556631	−0.278316	+	0.960490i	$0.589776\pi$
$318$	0	0
$319$	−23.8381	−1.33467
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.23989	0.180272
$324$	0	0
$325$	1.32408	0.0734466
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7.95433	−0.438537
$330$	0	0
$331$	0.413917	0.0227510	0.0113755	−	0.999935i	$-0.496379\pi$
$331$	0.413917	0.0227510	0.0113755	+	0.999935i	$0.496379\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.22692	0.449485
$336$	0	0
$337$	−3.29230	−0.179343	−0.0896714	−	0.995971i	$-0.528582\pi$
$337$	−3.29230	−0.179343	−0.0896714	+	0.995971i	$0.528582\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	41.9491	2.27167
$342$	0	0
$343$	16.5715	0.894776
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.23471	0.334697	0.167348	−	0.985898i	$-0.446480\pi$
$347$	6.23471	0.334697	0.167348	+	0.985898i	$0.446480\pi$
$348$	0	0
$349$	−33.0792	−1.77069	−0.885344	−	0.464937i	$-0.846077\pi$
$349$	−33.0792	−1.77069	−0.885344	+	0.464937i	$0.846077\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.68358	−0.355731	−0.177866	−	0.984055i	$-0.556919\pi$
$353$	−6.68358	−0.355731	−0.177866	+	0.984055i	$0.556919\pi$
$354$	0	0
$355$	−14.5209	−0.770691
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−27.1067	−1.43064	−0.715319	−	0.698798i	$-0.753719\pi$
$359$	−27.1067	−1.43064	−0.715319	+	0.698798i	$0.753719\pi$
$360$	0	0
$361$	−18.4837	−0.972825
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.38375	−0.177114
$366$	0	0
$367$	−21.1521	−1.10413	−0.552065	−	0.833801i	$-0.686160\pi$
$367$	−21.1521	−1.10413	−0.552065	+	0.833801i	$0.686160\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.8045	−0.612859
$372$	0	0
$373$	10.8864	0.563675	0.281837	−	0.959462i	$-0.409056\pi$
$373$	10.8864	0.563675	0.281837	+	0.959462i	$0.409056\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.08940	−0.107609
$378$	0	0
$379$	6.27639	0.322397	0.161198	−	0.986922i	$-0.448464\pi$
$379$	6.27639	0.322397	0.161198	+	0.986922i	$0.448464\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	25.5206	1.30404	0.652021	−	0.758201i	$-0.273921\pi$
$383$	25.5206	1.30404	0.652021	+	0.758201i	$0.273921\pi$
$384$	0	0
$385$	18.5257	0.944159
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10.6289	−0.538907	−0.269453	−	0.963013i	$-0.586843\pi$
$389$	−10.6289	−0.538907	−0.269453	+	0.963013i	$0.586843\pi$
$390$	0	0
$391$	38.6123	1.95271
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−21.7140	−1.09255
$396$	0	0
$397$	−18.8248	−0.944791	−0.472395	−	0.881387i	$-0.656611\pi$
$397$	−18.8248	−0.944791	−0.472395	+	0.881387i	$0.656611\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	26.5164	1.32417	0.662083	−	0.749431i	$-0.269673\pi$
$401$	26.5164	1.32417	0.662083	+	0.749431i	$0.269673\pi$
$402$	0	0
$403$	3.67683	0.183156
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−37.8182	−1.87458
$408$	0	0
$409$	2.68816	0.132921	0.0664605	−	0.997789i	$-0.478829\pi$
$409$	2.68816	0.132921	0.0664605	+	0.997789i	$0.478829\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11.0458	0.543528
$414$	0	0
$415$	17.3446	0.851414
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0.740791	0.0361900	0.0180950	−	0.999836i	$-0.494240\pi$
$419$	0.740791	0.0361900	0.0180950	+	0.999836i	$0.494240\pi$
$420$	0	0
$421$	−21.4364	−1.04474	−0.522372	−	0.852718i	$-0.674953\pi$
$421$	−21.4364	−1.04474	−0.522372	+	0.852718i	$0.674953\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−14.2753	−0.692455
$426$	0	0
$427$	−32.5369	−1.57457
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.8570	0.763803	0.381902	−	0.924203i	$-0.375269\pi$
$431$	15.8570	0.763803	0.381902	+	0.924203i	$0.375269\pi$
$432$	0	0
$433$	−26.4698	−1.27206	−0.636028	−	0.771666i	$-0.719424\pi$
$433$	−26.4698	−1.27206	−0.636028	+	0.771666i	$0.719424\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.15347	0.294360
$438$	0	0
$439$	−0.336616	−0.0160658	−0.00803289	−	0.999968i	$-0.502557\pi$
$439$	−0.336616	−0.0160658	−0.00803289	+	0.999968i	$0.502557\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	23.4538	1.11433	0.557163	−	0.830403i	$-0.311890\pi$
$443$	23.4538	1.11433	0.557163	+	0.830403i	$0.311890\pi$
$444$	0	0
$445$	−22.5578	−1.06934
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−14.0834	−0.664635	−0.332317	−	0.943168i	$-0.607830\pi$
$449$	−14.0834	−0.664635	−0.332317	+	0.943168i	$0.607830\pi$
$450$	0	0
$451$	34.3381	1.61692
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.62377	0.0761237
$456$	0	0
$457$	20.8052	0.973228	0.486614	−	0.873617i	$-0.338232\pi$
$457$	20.8052	0.973228	0.486614	+	0.873617i	$0.338232\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−36.2492	−1.68829	−0.844147	−	0.536111i	$-0.819893\pi$
$461$	−36.2492	−1.68829	−0.844147	+	0.536111i	$0.819893\pi$
$462$	0	0
$463$	−33.2975	−1.54746	−0.773732	−	0.633513i	$-0.781612\pi$
$463$	−33.2975	−1.54746	−0.773732	+	0.633513i	$0.781612\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.0258	1.20433	0.602165	−	0.798371i	$-0.294305\pi$
$467$	26.0258	1.20433	0.602165	+	0.798371i	$0.294305\pi$
$468$	0	0
$469$	−17.4173	−0.804257
$470$	0	0
$471$	0	0
$472$	0	0
$473$	10.5787	0.486407
$474$	0	0
$475$	−2.27500	−0.104384
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.34614	0.107198	0.0535990	−	0.998563i	$-0.482931\pi$
$479$	2.34614	0.107198	0.0535990	+	0.998563i	$0.482931\pi$
$480$	0	0
$481$	−3.31475	−0.151140
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−12.4074	−0.563389
$486$	0	0
$487$	−42.0211	−1.90416	−0.952079	−	0.305853i	$-0.901059\pi$
$487$	−42.0211	−1.90416	−0.952079	+	0.305853i	$0.901059\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−32.2232	−1.45421	−0.727107	−	0.686524i	$-0.759136\pi$
$491$	−32.2232	−1.45421	−0.727107	+	0.686524i	$0.759136\pi$
$492$	0	0
$493$	22.5265	1.01454
$494$	0	0
$495$	0	0
$496$	0	0
$497$	30.7424	1.37899
$498$	0	0
$499$	34.2208	1.53193	0.765966	−	0.642881i	$-0.222261\pi$
$499$	34.2208	1.53193	0.765966	+	0.642881i	$0.222261\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−30.6834	−1.36811	−0.684053	−	0.729432i	$-0.739784\pi$
$503$	−30.6834	−1.36811	−0.684053	+	0.729432i	$0.739784\pi$
$504$	0	0
$505$	−14.8591	−0.661221
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−7.13759	−0.316368	−0.158184	−	0.987410i	$-0.550564\pi$
$509$	−7.13759	−0.316368	−0.158184	+	0.987410i	$0.550564\pi$
$510$	0	0
$511$	7.16378	0.316907
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.67776	0.162061
$516$	0	0
$517$	−13.2377	−0.582192
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.7820	−0.472368	−0.236184	−	0.971708i	$-0.575897\pi$
$521$	−10.7820	−0.472368	−0.236184	+	0.971708i	$0.575897\pi$
$522$	0	0
$523$	20.6564	0.903242	0.451621	−	0.892210i	$-0.350846\pi$
$523$	20.6564	0.903242	0.451621	+	0.892210i	$0.350846\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−39.6411	−1.72679
$528$	0	0
$529$	50.3356	2.18851
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.00972	0.130365
$534$	0	0
$535$	−15.0304	−0.649819
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.82131	−0.250742
$540$	0	0
$541$	2.16307	0.0929979	0.0464989	−	0.998918i	$-0.485194\pi$
$541$	2.16307	0.0929979	0.0464989	+	0.998918i	$0.485194\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−17.5589	−0.752143
$546$	0	0
$547$	−6.66435	−0.284947	−0.142474	−	0.989799i	$-0.545506\pi$
$547$	−6.66435	−0.284947	−0.142474	+	0.989799i	$0.545506\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.58995	0.152937
$552$	0	0
$553$	45.9709	1.95488
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19.0094	−0.805453	−0.402726	−	0.915320i	$-0.631937\pi$
$557$	−19.0094	−0.805453	−0.402726	+	0.915320i	$0.631937\pi$
$558$	0	0
$559$	0.927216	0.0392171
$560$	0	0
$561$	0	0
$562$	0	0
$563$	25.1917	1.06170	0.530851	−	0.847465i	$-0.321872\pi$
$563$	25.1917	1.06170	0.530851	+	0.847465i	$0.321872\pi$
$564$	0	0
$565$	5.80395	0.244174
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−16.4740	−0.690626	−0.345313	−	0.938488i	$-0.612227\pi$
$569$	−16.4740	−0.690626	−0.345313	+	0.938488i	$0.612227\pi$
$570$	0	0
$571$	−8.95731	−0.374852	−0.187426	−	0.982279i	$-0.560014\pi$
$571$	−8.95731	−0.374852	−0.187426	+	0.982279i	$0.560014\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−27.1129	−1.13069
$576$	0	0
$577$	−30.9792	−1.28968	−0.644840	−	0.764318i	$-0.723076\pi$
$577$	−30.9792	−1.28968	−0.644840	+	0.764318i	$0.723076\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−36.7205	−1.52342
$582$	0	0
$583$	−19.6451	−0.813619
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−29.0735	−1.19999	−0.599997	−	0.800002i	$-0.704831\pi$
$587$	−29.0735	−1.19999	−0.599997	+	0.800002i	$0.704831\pi$
$588$	0	0
$589$	−6.31744	−0.260305
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−23.1233	−0.949558	−0.474779	−	0.880105i	$-0.657472\pi$
$593$	−23.1233	−0.949558	−0.474779	+	0.880105i	$0.657472\pi$
$594$	0	0
$595$	−17.5065	−0.717695
$596$	0	0
$597$	0	0
$598$	0	0
$599$	25.4683	1.04060	0.520302	−	0.853982i	$-0.325819\pi$
$599$	25.4683	1.04060	0.520302	+	0.853982i	$0.325819\pi$
$600$	0	0
$601$	24.1588	0.985460	0.492730	−	0.870182i	$-0.335999\pi$
$601$	24.1588	0.985460	0.492730	+	0.870182i	$0.335999\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	15.9341	0.647814
$606$	0	0
$607$	11.8207	0.479785	0.239893	−	0.970799i	$-0.422888\pi$
$607$	11.8207	0.479785	0.239893	+	0.970799i	$0.422888\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.16028	−0.0469398
$612$	0	0
$613$	17.3796	0.701954	0.350977	−	0.936384i	$-0.385849\pi$
$613$	17.3796	0.701954	0.350977	+	0.936384i	$0.385849\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	43.1508	1.73718	0.868592	−	0.495527i	$-0.165025\pi$
$617$	43.1508	1.73718	0.868592	+	0.495527i	$0.165025\pi$
$618$	0	0
$619$	12.1919	0.490033	0.245017	−	0.969519i	$-0.421207\pi$
$619$	12.1919	0.490033	0.245017	+	0.969519i	$0.421207\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	47.7574	1.91336
$624$	0	0
$625$	0.854154	0.0341662
$626$	0	0
$627$	0	0
$628$	0	0
$629$	35.7375	1.42495
$630$	0	0
$631$	9.75685	0.388414	0.194207	−	0.980961i	$-0.437787\pi$
$631$	9.75685	0.388414	0.194207	+	0.980961i	$0.437787\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−11.2569	−0.446718
$636$	0	0
$637$	−0.510236	−0.0202163
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−17.4440	−0.688995	−0.344497	−	0.938787i	$-0.611951\pi$
$641$	−17.4440	−0.688995	−0.344497	+	0.938787i	$0.611951\pi$
$642$	0	0
$643$	34.7748	1.37138	0.685692	−	0.727892i	$-0.259500\pi$
$643$	34.7748	1.37138	0.685692	+	0.727892i	$0.259500\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.8387	−0.622684	−0.311342	−	0.950298i	$-0.600778\pi$
$647$	−15.8387	−0.622684	−0.311342	+	0.950298i	$0.600778\pi$
$648$	0	0
$649$	18.3825	0.721576
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.80735	−0.0707272	−0.0353636	−	0.999375i	$-0.511259\pi$
$653$	−1.80735	−0.0707272	−0.0353636	+	0.999375i	$0.511259\pi$
$654$	0	0
$655$	−10.7521	−0.420118
$656$	0	0
$657$	0	0
$658$	0	0
$659$	27.5477	1.07310	0.536552	−	0.843867i	$-0.319726\pi$
$659$	27.5477	1.07310	0.536552	+	0.843867i	$0.319726\pi$
$660$	0	0
$661$	−23.2023	−0.902463	−0.451231	−	0.892407i	$-0.649015\pi$
$661$	−23.2023	−0.902463	−0.451231	+	0.892407i	$0.649015\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.78993	−0.108189
$666$	0	0
$667$	42.7842	1.65661
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−54.1482	−2.09037
$672$	0	0
$673$	−14.1212	−0.544334	−0.272167	−	0.962250i	$-0.587740\pi$
$673$	−14.1212	−0.544334	−0.272167	+	0.962250i	$0.587740\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.9213	1.41900	0.709500	−	0.704706i	$-0.248921\pi$
$677$	36.9213	1.41900	0.709500	+	0.704706i	$0.248921\pi$
$678$	0	0
$679$	26.2678	1.00806
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−12.6982	−0.485884	−0.242942	−	0.970041i	$-0.578112\pi$
$683$	−12.6982	−0.485884	−0.242942	+	0.970041i	$0.578112\pi$
$684$	0	0
$685$	−21.3883	−0.817207
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.72189	−0.0655988
$690$	0	0
$691$	−13.7331	−0.522431	−0.261216	−	0.965280i	$-0.584123\pi$
$691$	−13.7331	−0.522431	−0.261216	+	0.965280i	$0.584123\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−19.7083	−0.747579
$696$	0	0
$697$	−32.4488	−1.22909
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−28.5116	−1.07687	−0.538433	−	0.842668i	$-0.680984\pi$
$701$	−28.5116	−1.07687	−0.538433	+	0.842668i	$0.680984\pi$
$702$	0	0
$703$	5.69532	0.214803
$704$	0	0
$705$	0	0
$706$	0	0
$707$	31.4584	1.18311
$708$	0	0
$709$	−36.5358	−1.37213	−0.686064	−	0.727541i	$-0.740663\pi$
$709$	−36.5358	−1.37213	−0.686064	+	0.727541i	$0.740663\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−75.2897	−2.81962
$714$	0	0
$715$	2.70230	0.101060
$716$	0	0
$717$	0	0
$718$	0	0
$719$	15.3859	0.573798	0.286899	−	0.957961i	$-0.407376\pi$
$719$	15.3859	0.573798	0.286899	+	0.957961i	$0.407376\pi$
$720$	0	0
$721$	−7.78622	−0.289974
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−15.8177	−0.587456
$726$	0	0
$727$	28.9118	1.07228	0.536140	−	0.844129i	$-0.319882\pi$
$727$	28.9118	1.07228	0.536140	+	0.844129i	$0.319882\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9.99663	−0.369739
$732$	0	0
$733$	19.4252	0.717486	0.358743	−	0.933436i	$-0.383205\pi$
$733$	19.4252	0.717486	0.358743	+	0.933436i	$0.383205\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−28.9860	−1.06771
$738$	0	0
$739$	15.2593	0.561322	0.280661	−	0.959807i	$-0.409446\pi$
$739$	15.2593	0.561322	0.280661	+	0.959807i	$0.409446\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	26.2810	0.964157	0.482079	−	0.876128i	$-0.339882\pi$
$743$	26.2810	0.964157	0.482079	+	0.876128i	$0.339882\pi$
$744$	0	0
$745$	−7.65514	−0.280462
$746$	0	0
$747$	0	0
$748$	0	0
$749$	31.8210	1.16271
$750$	0	0
$751$	13.7150	0.500467	0.250234	−	0.968185i	$-0.419493\pi$
$751$	13.7150	0.500467	0.250234	+	0.968185i	$0.419493\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−15.8251	−0.575934
$756$	0	0
$757$	−25.0559	−0.910673	−0.455337	−	0.890319i	$-0.650481\pi$
$757$	−25.0559	−0.910673	−0.455337	+	0.890319i	$0.650481\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17.1790	0.622738	0.311369	−	0.950289i	$-0.399213\pi$
$761$	17.1790	0.622738	0.311369	+	0.950289i	$0.399213\pi$
$762$	0	0
$763$	37.1742	1.34580
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.61122	0.0581777
$768$	0	0
$769$	−28.4357	−1.02542	−0.512708	−	0.858563i	$-0.671358\pi$
$769$	−28.4357	−1.02542	−0.512708	+	0.858563i	$0.671358\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−8.55279	−0.307623	−0.153811	−	0.988100i	$-0.549155\pi$
$773$	−8.55279	−0.307623	−0.153811	+	0.988100i	$0.549155\pi$
$774$	0	0
$775$	27.8353	0.999875
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.17123	−0.185278
$780$	0	0
$781$	51.1618	1.83071
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.573724	0.0204771
$786$	0	0
$787$	−32.1195	−1.14494	−0.572469	−	0.819926i	$-0.694014\pi$
$787$	−32.1195	−1.14494	−0.572469	+	0.819926i	$0.694014\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.2876	−0.436897
$792$	0	0
$793$	−4.74607	−0.168538
$794$	0	0
$795$	0	0
$796$	0	0
$797$	49.0391	1.73706	0.868528	−	0.495641i	$-0.165067\pi$
$797$	49.0391	1.73706	0.868528	+	0.495641i	$0.165067\pi$
$798$	0	0
$799$	12.5093	0.442549
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11.9220	0.420719
$804$	0	0
$805$	−33.2497	−1.17190
$806$	0	0
$807$	0	0
$808$	0	0
$809$	41.8041	1.46976	0.734878	−	0.678200i	$-0.237240\pi$
$809$	41.8041	1.46976	0.734878	+	0.678200i	$0.237240\pi$
$810$	0	0
$811$	−8.17733	−0.287145	−0.143573	−	0.989640i	$-0.545859\pi$
$811$	−8.17733	−0.287145	−0.143573	+	0.989640i	$0.545859\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−23.9162	−0.837747
$816$	0	0
$817$	−1.59312	−0.0557362
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.3267	0.430206	0.215103	−	0.976591i	$-0.430991\pi$
$821$	12.3267	0.430206	0.215103	+	0.976591i	$0.430991\pi$
$822$	0	0
$823$	−31.1927	−1.08731	−0.543654	−	0.839309i	$-0.682960\pi$
$823$	−31.1927	−1.08731	−0.543654	+	0.839309i	$0.682960\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.7564	−1.13905	−0.569525	−	0.821974i	$-0.692873\pi$
$827$	−32.7564	−1.13905	−0.569525	+	0.821974i	$0.692873\pi$
$828$	0	0
$829$	43.8327	1.52237	0.761187	−	0.648532i	$-0.224617\pi$
$829$	43.8327	1.52237	0.761187	+	0.648532i	$0.224617\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.50103	0.190599
$834$	0	0
$835$	−1.35423	−0.0468652
$836$	0	0
$837$	0	0
$838$	0	0
$839$	19.5611	0.675325	0.337663	−	0.941267i	$-0.390364\pi$
$839$	19.5611	0.675325	0.337663	+	0.941267i	$0.390364\pi$
$840$	0	0
$841$	−4.03958	−0.139296
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−17.3682	−0.597483
$846$	0	0
$847$	−33.7343	−1.15912
$848$	0	0
$849$	0	0
$850$	0	0
$851$	67.8756	2.32674
$852$	0	0
$853$	12.9568	0.443632	0.221816	−	0.975089i	$-0.428802\pi$
$853$	12.9568	0.443632	0.221816	+	0.975089i	$0.428802\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	17.1872	0.587103	0.293551	−	0.955943i	$-0.405163\pi$
$857$	17.1872	0.587103	0.293551	+	0.955943i	$0.405163\pi$
$858$	0	0
$859$	−38.7680	−1.32275	−0.661374	−	0.750056i	$-0.730026\pi$
$859$	−38.7680	−1.32275	−0.661374	+	0.750056i	$0.730026\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5.67623	0.193221	0.0966106	−	0.995322i	$-0.469200\pi$
$863$	5.67623	0.193221	0.0966106	+	0.995322i	$0.469200\pi$
$864$	0	0
$865$	17.0055	0.578203
$866$	0	0
$867$	0	0
$868$	0	0
$869$	76.5051	2.59526
$870$	0	0
$871$	−2.54062	−0.0860855
$872$	0	0
$873$	0	0
$874$	0	0
$875$	31.7061	1.07186
$876$	0	0
$877$	25.7016	0.867882	0.433941	−	0.900941i	$-0.357123\pi$
$877$	25.7016	0.867882	0.433941	+	0.900941i	$0.357123\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−10.0109	−0.337277	−0.168639	−	0.985678i	$-0.553937\pi$
$881$	−10.0109	−0.337277	−0.168639	+	0.985678i	$0.553937\pi$
$882$	0	0
$883$	−21.9040	−0.737128	−0.368564	−	0.929602i	$-0.620150\pi$
$883$	−21.9040	−0.737128	−0.368564	+	0.929602i	$0.620150\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.75242	0.327454	0.163727	−	0.986506i	$-0.447648\pi$
$887$	9.75242	0.327454	0.163727	+	0.986506i	$0.447648\pi$
$888$	0	0
$889$	23.8322	0.799307
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.99356	0.0667119
$894$	0	0
$895$	−10.7078	−0.357921
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−43.9242	−1.46496
$900$	0	0
$901$	18.5643	0.618466
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.53955	0.250623
$906$	0	0
$907$	−15.3826	−0.510770	−0.255385	−	0.966839i	$-0.582202\pi$
$907$	−15.3826	−0.510770	−0.255385	+	0.966839i	$0.582202\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−17.7859	−0.589272	−0.294636	−	0.955610i	$-0.595198\pi$
$911$	−17.7859	−0.589272	−0.294636	+	0.955610i	$0.595198\pi$
$912$	0	0
$913$	−61.1105	−2.02246
$914$	0	0
$915$	0	0
$916$	0	0
$917$	22.7633	0.751712
$918$	0	0
$919$	27.4221	0.904570	0.452285	−	0.891873i	$-0.350609\pi$
$919$	27.4221	0.904570	0.452285	+	0.891873i	$0.350609\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.48431	0.147603
$924$	0	0
$925$	−25.0942	−0.825093
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−24.8349	−0.814805	−0.407403	−	0.913249i	$-0.633565\pi$
$929$	−24.8349	−0.814805	−0.407403	+	0.913249i	$0.633565\pi$
$930$	0	0
$931$	0.876675	0.0287319
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−29.1344	−0.952796
$936$	0	0
$937$	52.8074	1.72514	0.862572	−	0.505935i	$-0.168852\pi$
$937$	52.8074	1.72514	0.862572	+	0.505935i	$0.168852\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−45.4622	−1.48203	−0.741013	−	0.671491i	$-0.765654\pi$
$941$	−45.4622	−1.48203	−0.741013	+	0.671491i	$0.765654\pi$
$942$	0	0
$943$	−61.6295	−2.00693
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.24111	−0.105322	−0.0526610	−	0.998612i	$-0.516770\pi$
$947$	−3.24111	−0.105322	−0.0526610	+	0.998612i	$0.516770\pi$
$948$	0	0
$949$	1.04496	0.0339209
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−33.8475	−1.09643	−0.548214	−	0.836338i	$-0.684692\pi$
$953$	−33.8475	−1.09643	−0.548214	+	0.836338i	$0.684692\pi$
$954$	0	0
$955$	11.8608	0.383807
$956$	0	0
$957$	0	0
$958$	0	0
$959$	45.2815	1.46222
$960$	0	0
$961$	46.2959	1.49342
$962$	0	0
$963$	0	0
$964$	0	0
$965$	37.4117	1.20432
$966$	0	0
$967$	−39.5467	−1.27173	−0.635867	−	0.771798i	$-0.719357\pi$
$967$	−39.5467	−1.27173	−0.635867	+	0.771798i	$0.719357\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	33.2007	1.06546	0.532730	−	0.846285i	$-0.321166\pi$
$971$	33.2007	1.06546	0.532730	+	0.846285i	$0.321166\pi$
$972$	0	0
$973$	41.7247	1.33763
$974$	0	0
$975$	0	0
$976$	0	0
$977$	49.3644	1.57931	0.789653	−	0.613554i	$-0.210261\pi$
$977$	49.3644	1.57931	0.789653	+	0.613554i	$0.210261\pi$
$978$	0	0
$979$	79.4782	2.54013
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−30.4259	−0.970435	−0.485218	−	0.874393i	$-0.661260\pi$
$983$	−30.4259	−0.970435	−0.485218	+	0.874393i	$0.661260\pi$
$984$	0	0
$985$	22.9892	0.732497
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−18.9864	−0.603733
$990$	0	0
$991$	34.9201	1.10928	0.554638	−	0.832092i	$-0.312857\pi$
$991$	34.9201	1.10928	0.554638	+	0.832092i	$0.312857\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−13.8601	−0.439395
$996$	0	0
$997$	−59.0643	−1.87059	−0.935293	−	0.353875i	$-0.884864\pi$
$997$	−59.0643	−1.87059	−0.935293	+	0.353875i	$0.884864\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.g.1.5		7
3.2	odd	2		668.2.a.c.1.5	✓	7
12.11	even	2		2672.2.a.k.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.a.c.1.5	✓	7	3.2	odd	2
2672.2.a.k.1.3		7	12.11	even	2
6012.2.a.g.1.5		7	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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.a (trivial)

\(f(q)\)	\(=\)	\(q+1.35423 q^{5} -2.86706 q^{7} +O(q^{10})\)	\(q+1.35423 q^{5} -2.86706 q^{7} -4.77139 q^{11} -0.418211 q^{13} +4.50887 q^{17} +0.718559 q^{19} +8.56362 q^{23} -3.16605 q^{25} +4.99604 q^{29} -8.79181 q^{31} -3.88267 q^{35} +7.92603 q^{37} -7.19666 q^{41} -2.21710 q^{43} +2.77439 q^{47} +1.22005 q^{49} +4.11728 q^{53} -6.46157 q^{55} -3.85265 q^{59} +11.3485 q^{61} -0.566355 q^{65} +6.07497 q^{67} -10.7226 q^{71} -2.49865 q^{73} +13.6799 q^{77} -16.0341 q^{79} +12.8077 q^{83} +6.10606 q^{85} -16.6573 q^{89} +1.19904 q^{91} +0.973097 q^{95} -9.16191 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{5} - 12 q^{7}+O(q^{10}) \) 7 * q + 2 * q^5 - 12 * q^7	\( 7 q + 2 q^{5} - 12 q^{7} + 7 q^{11} - 9 q^{13} + q^{17} - 11 q^{19} + 19 q^{23} + 3 q^{25} + 5 q^{29} - 13 q^{31} + 7 q^{35} - 26 q^{37} + 2 q^{41} - 24 q^{43} + 11 q^{47} + 19 q^{49} - 4 q^{53} - 4 q^{55} + 4 q^{59} - 5 q^{61} - 13 q^{65} - 42 q^{67} - 9 q^{71} + 27 q^{73} - 12 q^{77} - 8 q^{79} - 16 q^{83} - 27 q^{85} - 9 q^{89} - 2 q^{91} - 10 q^{95} - 8 q^{97}+O(q^{100}) \) 7 * q + 2 * q^5 - 12 * q^7 + 7 * q^11 - 9 * q^13 + q^17 - 11 * q^19 + 19 * q^23 + 3 * q^25 + 5 * q^29 - 13 * q^31 + 7 * q^35 - 26 * q^37 + 2 * q^41 - 24 * q^43 + 11 * q^47 + 19 * q^49 - 4 * q^53 - 4 * q^55 + 4 * q^59 - 5 * q^61 - 13 * q^65 - 42 * q^67 - 9 * q^71 + 27 * q^73 - 12 * q^77 - 8 * q^79 - 16 * q^83 - 27 * q^85 - 9 * q^89 - 2 * q^91 - 10 * q^95 - 8 * q^97

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