LMFDB - Embedded newform 9200.2.a.dc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, 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("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.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:	$73.4623698596$
Analytic rank:	$0$
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} - 3x^{6} - 7x^{5} + 24x^{4} + x^{3} - 35x^{2} + 17x - 2 $ x^7 - 3x^6 - 7x^5 + 24x^4 + x^3 - 35x^2 + 17*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.40334$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.40334 q^{3} +1.57117 q^{7} -1.03063 q^{9} +O(q^{10})$	$q-1.40334 q^{3} +1.57117 q^{7} -1.03063 q^{9} -4.35401 q^{11} -0.964590 q^{13} +0.300242 q^{17} +8.62443 q^{19} -2.20489 q^{21} -1.00000 q^{23} +5.65635 q^{27} +4.76644 q^{29} +5.59148 q^{31} +6.11017 q^{33} +4.38462 q^{37} +1.35365 q^{39} -6.62014 q^{41} -1.72988 q^{43} -0.687333 q^{47} -4.53143 q^{49} -0.421342 q^{51} -8.05208 q^{53} -12.1030 q^{57} -5.74620 q^{59} -13.6547 q^{61} -1.61929 q^{63} +6.49053 q^{67} +1.40334 q^{69} -9.89977 q^{71} -6.35994 q^{73} -6.84088 q^{77} +6.95266 q^{79} -4.84592 q^{81} -0.185320 q^{83} -6.68895 q^{87} -1.64878 q^{89} -1.51553 q^{91} -7.84676 q^{93} +9.21689 q^{97} +4.48736 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 7 * q + 3 * q^3 + 4 * q^7 + 2 * q^9	$ 7 q + 3 q^{3} + 4 q^{7} + 2 q^{9} + 7 q^{11} + 7 q^{13} + 7 q^{19} - 6 q^{21} - 7 q^{23} - 11 q^{29} + 10 q^{31} + 19 q^{33} + 19 q^{37} + 24 q^{39} - 16 q^{41} + 6 q^{43} + 6 q^{47} - 17 q^{49} + 7 q^{51} + 15 q^{53} + 8 q^{57} + 11 q^{59} + 5 q^{61} + 13 q^{63} + 9 q^{67} - 3 q^{69} + 14 q^{71} + 10 q^{73} + 6 q^{77} + 32 q^{79} - 5 q^{81} + q^{83} + 10 q^{87} - 24 q^{89} + 7 q^{91} + 26 q^{93} - 7 q^{97} + 61 q^{99}+O(q^{100}) $ 7 * q + 3 * q^3 + 4 * q^7 + 2 * q^9 + 7 * q^11 + 7 * q^13 + 7 * q^19 - 6 * q^21 - 7 * q^23 - 11 * q^29 + 10 * q^31 + 19 * q^33 + 19 * q^37 + 24 * q^39 - 16 * q^41 + 6 * q^43 + 6 * q^47 - 17 * q^49 + 7 * q^51 + 15 * q^53 + 8 * q^57 + 11 * q^59 + 5 * q^61 + 13 * q^63 + 9 * q^67 - 3 * q^69 + 14 * q^71 + 10 * q^73 + 6 * q^77 + 32 * q^79 - 5 * q^81 + q^83 + 10 * q^87 - 24 * q^89 + 7 * q^91 + 26 * q^93 - 7 * q^97 + 61 * 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
$3$	−1.40334	−0.810221	−0.405110	−	0.914268i	$-0.632767\pi$
$3$	−1.40334	−0.810221	−0.405110	+	0.914268i	$0.632767\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.57117	0.593846	0.296923	−	0.954901i	$-0.404040\pi$
$7$	1.57117	0.593846	0.296923	+	0.954901i	$0.404040\pi$
$8$	0	0
$9$	−1.03063	−0.343543
$10$	0	0
$11$	−4.35401	−1.31278	−0.656391	−	0.754421i	$-0.727918\pi$
$11$	−4.35401	−1.31278	−0.656391	+	0.754421i	$0.727918\pi$
$12$	0	0
$13$	−0.964590	−0.267529	−0.133765	−	0.991013i	$-0.542707\pi$
$13$	−0.964590	−0.267529	−0.133765	+	0.991013i	$0.542707\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.300242	0.0728193	0.0364096	−	0.999337i	$-0.488408\pi$
$17$	0.300242	0.0728193	0.0364096	+	0.999337i	$0.488408\pi$
$18$	0	0
$19$	8.62443	1.97858	0.989290	−	0.145963i	$-0.0466280\pi$
$19$	8.62443	1.97858	0.989290	+	0.145963i	$0.0466280\pi$
$20$	0	0
$21$	−2.20489	−0.481146
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.65635	1.08857
$28$	0	0
$29$	4.76644	0.885106	0.442553	−	0.896742i	$-0.354073\pi$
$29$	4.76644	0.885106	0.442553	+	0.896742i	$0.354073\pi$
$30$	0	0
$31$	5.59148	1.00426	0.502129	−	0.864792i	$-0.332550\pi$
$31$	5.59148	1.00426	0.502129	+	0.864792i	$0.332550\pi$
$32$	0	0
$33$	6.11017	1.06364
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.38462	0.720827	0.360413	−	0.932793i	$-0.382636\pi$
$37$	4.38462	0.720827	0.360413	+	0.932793i	$0.382636\pi$
$38$	0	0
$39$	1.35365	0.216758
$40$	0	0
$41$	−6.62014	−1.03389	−0.516946	−	0.856018i	$-0.672931\pi$
$41$	−6.62014	−1.03389	−0.516946	+	0.856018i	$0.672931\pi$
$42$	0	0
$43$	−1.72988	−0.263804	−0.131902	−	0.991263i	$-0.542108\pi$
$43$	−1.72988	−0.263804	−0.131902	+	0.991263i	$0.542108\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.687333	−0.100258	−0.0501289	−	0.998743i	$-0.515963\pi$
$47$	−0.687333	−0.100258	−0.0501289	+	0.998743i	$0.515963\pi$
$48$	0	0
$49$	−4.53143	−0.647347
$50$	0	0
$51$	−0.421342	−0.0589997
$52$	0	0
$53$	−8.05208	−1.10604	−0.553019	−	0.833168i	$-0.686524\pi$
$53$	−8.05208	−1.10604	−0.553019	+	0.833168i	$0.686524\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−12.1030	−1.60309
$58$	0	0
$59$	−5.74620	−0.748091	−0.374046	−	0.927410i	$-0.622030\pi$
$59$	−5.74620	−0.748091	−0.374046	+	0.927410i	$0.622030\pi$
$60$	0	0
$61$	−13.6547	−1.74831	−0.874153	−	0.485651i	$-0.838583\pi$
$61$	−13.6547	−1.74831	−0.874153	+	0.485651i	$0.838583\pi$
$62$	0	0
$63$	−1.61929	−0.204011
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.49053	0.792945	0.396472	−	0.918047i	$-0.370234\pi$
$67$	6.49053	0.792945	0.396472	+	0.918047i	$0.370234\pi$
$68$	0	0
$69$	1.40334	0.168943
$70$	0	0
$71$	−9.89977	−1.17489	−0.587443	−	0.809265i	$-0.699866\pi$
$71$	−9.89977	−1.17489	−0.587443	+	0.809265i	$0.699866\pi$
$72$	0	0
$73$	−6.35994	−0.744375	−0.372187	−	0.928158i	$-0.621392\pi$
$73$	−6.35994	−0.744375	−0.372187	+	0.928158i	$0.621392\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.84088	−0.779591
$78$	0	0
$79$	6.95266	0.782236	0.391118	−	0.920341i	$-0.372088\pi$
$79$	6.95266	0.782236	0.391118	+	0.920341i	$0.372088\pi$
$80$	0	0
$81$	−4.84592	−0.538436
$82$	0	0
$83$	−0.185320	−0.0203415	−0.0101707	−	0.999948i	$-0.503238\pi$
$83$	−0.185320	−0.0203415	−0.0101707	+	0.999948i	$0.503238\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.68895	−0.717131
$88$	0	0
$89$	−1.64878	−0.174770	−0.0873852	−	0.996175i	$-0.527851\pi$
$89$	−1.64878	−0.174770	−0.0873852	+	0.996175i	$0.527851\pi$
$90$	0	0
$91$	−1.51553	−0.158871
$92$	0	0
$93$	−7.84676	−0.813671
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.21689	0.935834	0.467917	−	0.883773i	$-0.345005\pi$
$97$	9.21689	0.935834	0.467917	+	0.883773i	$0.345005\pi$
$98$	0	0
$99$	4.48736	0.450997
$100$	0	0
$101$	−0.719780	−0.0716208	−0.0358104	−	0.999359i	$-0.511401\pi$
$101$	−0.719780	−0.0716208	−0.0358104	+	0.999359i	$0.511401\pi$
$102$	0	0
$103$	9.16137	0.902696	0.451348	−	0.892348i	$-0.350943\pi$
$103$	9.16137	0.902696	0.451348	+	0.892348i	$0.350943\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.7486	−1.23245	−0.616226	−	0.787570i	$-0.711339\pi$
$107$	−12.7486	−1.23245	−0.616226	+	0.787570i	$0.711339\pi$
$108$	0	0
$109$	−3.38019	−0.323763	−0.161882	−	0.986810i	$-0.551756\pi$
$109$	−3.38019	−0.323763	−0.161882	+	0.986810i	$0.551756\pi$
$110$	0	0
$111$	−6.15312	−0.584029
$112$	0	0
$113$	15.8547	1.49148	0.745741	−	0.666237i	$-0.232096\pi$
$113$	15.8547	1.49148	0.745741	+	0.666237i	$0.232096\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.994133	0.0919076
$118$	0	0
$119$	0.471730	0.0432434
$120$	0	0
$121$	7.95739	0.723399
$122$	0	0
$123$	9.29032	0.837680
$124$	0	0
$125$	0	0
$126$	0	0
$127$	15.2222	1.35075	0.675377	−	0.737473i	$-0.263981\pi$
$127$	15.2222	1.35075	0.675377	+	0.737473i	$0.263981\pi$
$128$	0	0
$129$	2.42762	0.213740
$130$	0	0
$131$	12.8695	1.12441	0.562207	−	0.826997i	$-0.309953\pi$
$131$	12.8695	1.12441	0.562207	+	0.826997i	$0.309953\pi$
$132$	0	0
$133$	13.5504	1.17497
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.7433	−1.43048	−0.715238	−	0.698881i	$-0.753682\pi$
$137$	−16.7433	−1.43048	−0.715238	+	0.698881i	$0.753682\pi$
$138$	0	0
$139$	11.8293	1.00335	0.501676	−	0.865056i	$-0.332717\pi$
$139$	11.8293	1.00335	0.501676	+	0.865056i	$0.332717\pi$
$140$	0	0
$141$	0.964565	0.0812310
$142$	0	0
$143$	4.19983	0.351208
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.35915	0.524494
$148$	0	0
$149$	1.46021	0.119625	0.0598125	−	0.998210i	$-0.480950\pi$
$149$	1.46021	0.119625	0.0598125	+	0.998210i	$0.480950\pi$
$150$	0	0
$151$	−17.7897	−1.44771	−0.723854	−	0.689953i	$-0.757631\pi$
$151$	−17.7897	−1.44771	−0.723854	+	0.689953i	$0.757631\pi$
$152$	0	0
$153$	−0.309437	−0.0250165
$154$	0	0
$155$	0	0
$156$	0	0
$157$	20.0359	1.59904	0.799518	−	0.600642i	$-0.205088\pi$
$157$	20.0359	1.59904	0.799518	+	0.600642i	$0.205088\pi$
$158$	0	0
$159$	11.2998	0.896135
$160$	0	0
$161$	−1.57117	−0.123825
$162$	0	0
$163$	22.9076	1.79426	0.897130	−	0.441766i	$-0.145648\pi$
$163$	22.9076	1.79426	0.897130	+	0.441766i	$0.145648\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.826887	−0.0639864	−0.0319932	−	0.999488i	$-0.510185\pi$
$167$	−0.826887	−0.0639864	−0.0319932	+	0.999488i	$0.510185\pi$
$168$	0	0
$169$	−12.0696	−0.928428
$170$	0	0
$171$	−8.88858	−0.679727
$172$	0	0
$173$	8.03904	0.611197	0.305599	−	0.952160i	$-0.401143\pi$
$173$	8.03904	0.611197	0.305599	+	0.952160i	$0.401143\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.06389	0.606119
$178$	0	0
$179$	21.6941	1.62149	0.810747	−	0.585397i	$-0.199061\pi$
$179$	21.6941	1.62149	0.810747	+	0.585397i	$0.199061\pi$
$180$	0	0
$181$	−14.9499	−1.11122	−0.555609	−	0.831444i	$-0.687515\pi$
$181$	−14.9499	−1.11122	−0.555609	+	0.831444i	$0.687515\pi$
$182$	0	0
$183$	19.1622	1.41651
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.30725	−0.0955959
$188$	0	0
$189$	8.88709	0.646441
$190$	0	0
$191$	7.86406	0.569024	0.284512	−	0.958673i	$-0.408169\pi$
$191$	7.86406	0.569024	0.284512	+	0.958673i	$0.408169\pi$
$192$	0	0
$193$	−15.1864	−1.09314	−0.546571	−	0.837413i	$-0.684067\pi$
$193$	−15.1864	−1.09314	−0.546571	+	0.837413i	$0.684067\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.2911	0.875707	0.437853	−	0.899046i	$-0.355739\pi$
$197$	12.2911	0.875707	0.437853	+	0.899046i	$0.355739\pi$
$198$	0	0
$199$	4.37123	0.309869	0.154934	−	0.987925i	$-0.450483\pi$
$199$	4.37123	0.309869	0.154934	+	0.987925i	$0.450483\pi$
$200$	0	0
$201$	−9.10845	−0.642460
$202$	0	0
$203$	7.48888	0.525617
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.03063	0.0716336
$208$	0	0
$209$	−37.5508	−2.59745
$210$	0	0
$211$	8.93960	0.615427	0.307714	−	0.951479i	$-0.400436\pi$
$211$	8.93960	0.615427	0.307714	+	0.951479i	$0.400436\pi$
$212$	0	0
$213$	13.8928	0.951917
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.78516	0.596375
$218$	0	0
$219$	8.92518	0.603108
$220$	0	0
$221$	−0.289610	−0.0194813
$222$	0	0
$223$	11.9586	0.800806	0.400403	−	0.916339i	$-0.368870\pi$
$223$	11.9586	0.800806	0.400403	+	0.916339i	$0.368870\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.85445	−0.123084	−0.0615419	−	0.998104i	$-0.519602\pi$
$227$	−1.85445	−0.123084	−0.0615419	+	0.998104i	$0.519602\pi$
$228$	0	0
$229$	−16.5190	−1.09161	−0.545804	−	0.837913i	$-0.683776\pi$
$229$	−16.5190	−1.09161	−0.545804	+	0.837913i	$0.683776\pi$
$230$	0	0
$231$	9.60011	0.631641
$232$	0	0
$233$	24.5906	1.61099	0.805493	−	0.592606i	$-0.201901\pi$
$233$	24.5906	1.61099	0.805493	+	0.592606i	$0.201901\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−9.75697	−0.633783
$238$	0	0
$239$	−27.1169	−1.75405	−0.877025	−	0.480446i	$-0.840475\pi$
$239$	−27.1169	−1.75405	−0.877025	+	0.480446i	$0.840475\pi$
$240$	0	0
$241$	−12.8855	−0.830030	−0.415015	−	0.909815i	$-0.636224\pi$
$241$	−12.8855	−0.830030	−0.415015	+	0.909815i	$0.636224\pi$
$242$	0	0
$243$	−10.1686	−0.652314
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.31904	−0.529328
$248$	0	0
$249$	0.260067	0.0164811
$250$	0	0
$251$	12.4610	0.786533	0.393267	−	0.919424i	$-0.371345\pi$
$251$	12.4610	0.786533	0.393267	+	0.919424i	$0.371345\pi$
$252$	0	0
$253$	4.35401	0.273734
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.5073	0.842564	0.421282	−	0.906930i	$-0.361580\pi$
$257$	13.5073	0.842564	0.421282	+	0.906930i	$0.361580\pi$
$258$	0	0
$259$	6.88898	0.428060
$260$	0	0
$261$	−4.91243	−0.304072
$262$	0	0
$263$	−27.2579	−1.68080	−0.840399	−	0.541969i	$-0.817679\pi$
$263$	−27.2579	−1.68080	−0.840399	+	0.541969i	$0.817679\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.31380	0.141603
$268$	0	0
$269$	−17.8409	−1.08778	−0.543889	−	0.839157i	$-0.683049\pi$
$269$	−17.8409	−1.08778	−0.543889	+	0.839157i	$0.683049\pi$
$270$	0	0
$271$	−2.31906	−0.140873	−0.0704364	−	0.997516i	$-0.522439\pi$
$271$	−2.31906	−0.140873	−0.0704364	+	0.997516i	$0.522439\pi$
$272$	0	0
$273$	2.12681	0.128721
$274$	0	0
$275$	0	0
$276$	0	0
$277$	32.0762	1.92727	0.963637	−	0.267215i	$-0.0861034\pi$
$277$	32.0762	1.92727	0.963637	+	0.267215i	$0.0861034\pi$
$278$	0	0
$279$	−5.76273	−0.345006
$280$	0	0
$281$	12.9448	0.772221	0.386110	−	0.922453i	$-0.373818\pi$
$281$	12.9448	0.772221	0.386110	+	0.922453i	$0.373818\pi$
$282$	0	0
$283$	22.6070	1.34385	0.671923	−	0.740621i	$-0.265468\pi$
$283$	22.6070	1.34385	0.671923	+	0.740621i	$0.265468\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.4014	−0.613972
$288$	0	0
$289$	−16.9099	−0.994697
$290$	0	0
$291$	−12.9345	−0.758232
$292$	0	0
$293$	5.61612	0.328097	0.164049	−	0.986452i	$-0.447545\pi$
$293$	5.61612	0.328097	0.164049	+	0.986452i	$0.447545\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−24.6278	−1.42905
$298$	0	0
$299$	0.964590	0.0557837
$300$	0	0
$301$	−2.71794	−0.156659
$302$	0	0
$303$	1.01010	0.0580286
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2.42467	−0.138383	−0.0691915	−	0.997603i	$-0.522042\pi$
$307$	−2.42467	−0.138383	−0.0691915	+	0.997603i	$0.522042\pi$
$308$	0	0
$309$	−12.8565	−0.731383
$310$	0	0
$311$	−3.57886	−0.202938	−0.101469	−	0.994839i	$-0.532354\pi$
$311$	−3.57886	−0.202938	−0.101469	+	0.994839i	$0.532354\pi$
$312$	0	0
$313$	−5.51000	−0.311443	−0.155722	−	0.987801i	$-0.549770\pi$
$313$	−5.51000	−0.311443	−0.155722	+	0.987801i	$0.549770\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.88368	0.330461	0.165230	−	0.986255i	$-0.447163\pi$
$317$	5.88368	0.330461	0.165230	+	0.986255i	$0.447163\pi$
$318$	0	0
$319$	−20.7531	−1.16195
$320$	0	0
$321$	17.8906	0.998558
$322$	0	0
$323$	2.58941	0.144079
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.74356	0.262320
$328$	0	0
$329$	−1.07992	−0.0595377
$330$	0	0
$331$	18.6138	1.02311	0.511553	−	0.859252i	$-0.329070\pi$
$331$	18.6138	1.02311	0.511553	+	0.859252i	$0.329070\pi$
$332$	0	0
$333$	−4.51891	−0.247635
$334$	0	0
$335$	0	0
$336$	0	0
$337$	29.9571	1.63187	0.815933	−	0.578146i	$-0.196224\pi$
$337$	29.9571	1.63187	0.815933	+	0.578146i	$0.196224\pi$
$338$	0	0
$339$	−22.2495	−1.20843
$340$	0	0
$341$	−24.3453	−1.31837
$342$	0	0
$343$	−18.1178	−0.978270
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.7363	1.43528	0.717640	−	0.696415i	$-0.245222\pi$
$347$	26.7363	1.43528	0.717640	+	0.696415i	$0.245222\pi$
$348$	0	0
$349$	28.7392	1.53837	0.769186	−	0.639025i	$-0.220662\pi$
$349$	28.7392	1.53837	0.769186	+	0.639025i	$0.220662\pi$
$350$	0	0
$351$	−5.45606	−0.291223
$352$	0	0
$353$	−0.662874	−0.0352812	−0.0176406	−	0.999844i	$-0.505615\pi$
$353$	−0.662874	−0.0352812	−0.0176406	+	0.999844i	$0.505615\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.661999	−0.0350367
$358$	0	0
$359$	16.0576	0.847485	0.423743	−	0.905783i	$-0.360716\pi$
$359$	16.0576	0.847485	0.423743	+	0.905783i	$0.360716\pi$
$360$	0	0
$361$	55.3808	2.91478
$362$	0	0
$363$	−11.1669	−0.586113
$364$	0	0
$365$	0	0
$366$	0	0
$367$	27.5855	1.43995	0.719976	−	0.693999i	$-0.244153\pi$
$367$	27.5855	1.43995	0.719976	+	0.693999i	$0.244153\pi$
$368$	0	0
$369$	6.82290	0.355186
$370$	0	0
$371$	−12.6512	−0.656817
$372$	0	0
$373$	33.1014	1.71393	0.856964	−	0.515376i	$-0.172348\pi$
$373$	33.1014	1.71393	0.856964	+	0.515376i	$0.172348\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.59766	−0.236791
$378$	0	0
$379$	−2.17121	−0.111527	−0.0557637	−	0.998444i	$-0.517759\pi$
$379$	−2.17121	−0.111527	−0.0557637	+	0.998444i	$0.517759\pi$
$380$	0	0
$381$	−21.3620	−1.09441
$382$	0	0
$383$	−5.52673	−0.282403	−0.141201	−	0.989981i	$-0.545097\pi$
$383$	−5.52673	−0.282403	−0.141201	+	0.989981i	$0.545097\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.78286	0.0906281
$388$	0	0
$389$	17.1036	0.867185	0.433593	−	0.901109i	$-0.357246\pi$
$389$	17.1036	0.867185	0.433593	+	0.901109i	$0.357246\pi$
$390$	0	0
$391$	−0.300242	−0.0151839
$392$	0	0
$393$	−18.0603	−0.911023
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−12.7419	−0.639497	−0.319748	−	0.947503i	$-0.603598\pi$
$397$	−12.7419	−0.639497	−0.319748	+	0.947503i	$0.603598\pi$
$398$	0	0
$399$	−19.0159	−0.951987
$400$	0	0
$401$	14.1997	0.709098	0.354549	−	0.935038i	$-0.384634\pi$
$401$	14.1997	0.709098	0.354549	+	0.935038i	$0.384634\pi$
$402$	0	0
$403$	−5.39348	−0.268668
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−19.0907	−0.946289
$408$	0	0
$409$	−0.0235143	−0.00116271	−0.000581354	−	1.00000i	$-0.500185\pi$
$409$	−0.0235143	−0.00116271	−0.000581354		1.00000i	$0.500185\pi$
$410$	0	0
$411$	23.4966	1.15900
$412$	0	0
$413$	−9.02825	−0.444251
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−16.6006	−0.812936
$418$	0	0
$419$	20.2763	0.990562	0.495281	−	0.868733i	$-0.335065\pi$
$419$	20.2763	0.990562	0.495281	+	0.868733i	$0.335065\pi$
$420$	0	0
$421$	12.5320	0.610771	0.305386	−	0.952229i	$-0.401215\pi$
$421$	12.5320	0.610771	0.305386	+	0.952229i	$0.401215\pi$
$422$	0	0
$423$	0.708385	0.0344429
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−21.4539	−1.03822
$428$	0	0
$429$	−5.89380	−0.284556
$430$	0	0
$431$	22.8635	1.10130	0.550648	−	0.834737i	$-0.314381\pi$
$431$	22.8635	1.10130	0.550648	+	0.834737i	$0.314381\pi$
$432$	0	0
$433$	23.7997	1.14374	0.571871	−	0.820343i	$-0.306218\pi$
$433$	23.7997	1.14374	0.571871	+	0.820343i	$0.306218\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.62443	−0.412562
$438$	0	0
$439$	−4.66686	−0.222737	−0.111368	−	0.993779i	$-0.535523\pi$
$439$	−4.66686	−0.222737	−0.111368	+	0.993779i	$0.535523\pi$
$440$	0	0
$441$	4.67022	0.222391
$442$	0	0
$443$	−13.6616	−0.649082	−0.324541	−	0.945872i	$-0.605210\pi$
$443$	−13.6616	−0.649082	−0.324541	+	0.945872i	$0.605210\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−2.04917	−0.0969226
$448$	0	0
$449$	−25.8921	−1.22192	−0.610961	−	0.791661i	$-0.709217\pi$
$449$	−25.8921	−1.22192	−0.610961	+	0.791661i	$0.709217\pi$
$450$	0	0
$451$	28.8241	1.35727
$452$	0	0
$453$	24.9651	1.17296
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−12.6856	−0.593405	−0.296703	−	0.954970i	$-0.595887\pi$
$457$	−12.6856	−0.593405	−0.296703	+	0.954970i	$0.595887\pi$
$458$	0	0
$459$	1.69827	0.0792686
$460$	0	0
$461$	26.5769	1.23781	0.618904	−	0.785466i	$-0.287577\pi$
$461$	26.5769	1.23781	0.618904	+	0.785466i	$0.287577\pi$
$462$	0	0
$463$	14.6461	0.680664	0.340332	−	0.940305i	$-0.389461\pi$
$463$	14.6461	0.680664	0.340332	+	0.940305i	$0.389461\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.7601	0.590467	0.295233	−	0.955425i	$-0.404603\pi$
$467$	12.7601	0.590467	0.295233	+	0.955425i	$0.404603\pi$
$468$	0	0
$469$	10.1977	0.470887
$470$	0	0
$471$	−28.1172	−1.29557
$472$	0	0
$473$	7.53192	0.346318
$474$	0	0
$475$	0	0
$476$	0	0
$477$	8.29870	0.379971
$478$	0	0
$479$	−10.2315	−0.467488	−0.233744	−	0.972298i	$-0.575098\pi$
$479$	−10.2315	−0.467488	−0.233744	+	0.972298i	$0.575098\pi$
$480$	0	0
$481$	−4.22936	−0.192842
$482$	0	0
$483$	2.20489	0.100326
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−21.8299	−0.989207	−0.494603	−	0.869119i	$-0.664687\pi$
$487$	−21.8299	−0.989207	−0.494603	+	0.869119i	$0.664687\pi$
$488$	0	0
$489$	−32.1472	−1.45375
$490$	0	0
$491$	38.1315	1.72085	0.860426	−	0.509575i	$-0.170198\pi$
$491$	38.1315	1.72085	0.860426	+	0.509575i	$0.170198\pi$
$492$	0	0
$493$	1.43108	0.0644527
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−15.5542	−0.697702
$498$	0	0
$499$	−6.36523	−0.284947	−0.142474	−	0.989799i	$-0.545506\pi$
$499$	−6.36523	−0.284947	−0.142474	+	0.989799i	$0.545506\pi$
$500$	0	0
$501$	1.16041	0.0518431
$502$	0	0
$503$	−21.3160	−0.950433	−0.475217	−	0.879869i	$-0.657630\pi$
$503$	−21.3160	−0.950433	−0.475217	+	0.879869i	$0.657630\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	16.9377	0.752232
$508$	0	0
$509$	−25.5122	−1.13081	−0.565405	−	0.824814i	$-0.691280\pi$
$509$	−25.5122	−1.13081	−0.565405	+	0.824814i	$0.691280\pi$
$510$	0	0
$511$	−9.99254	−0.442044
$512$	0	0
$513$	48.7828	2.15381
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.99266	0.131617
$518$	0	0
$519$	−11.2815	−0.495205
$520$	0	0
$521$	−40.7247	−1.78418	−0.892091	−	0.451856i	$-0.850762\pi$
$521$	−40.7247	−1.78418	−0.892091	+	0.451856i	$0.850762\pi$
$522$	0	0
$523$	23.3437	1.02075	0.510374	−	0.859953i	$-0.329507\pi$
$523$	23.3437	1.02075	0.510374	+	0.859953i	$0.329507\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.67879	0.0731294
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	5.92219	0.257001
$532$	0	0
$533$	6.38571	0.276596
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−30.4443	−1.31377
$538$	0	0
$539$	19.7299	0.849826
$540$	0	0
$541$	1.23303	0.0530123	0.0265061	−	0.999649i	$-0.491562\pi$
$541$	1.23303	0.0530123	0.0265061	+	0.999649i	$0.491562\pi$
$542$	0	0
$543$	20.9798	0.900331
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−1.56470	−0.0669018	−0.0334509	−	0.999440i	$-0.510650\pi$
$547$	−1.56470	−0.0669018	−0.0334509	+	0.999440i	$0.510650\pi$
$548$	0	0
$549$	14.0729	0.600618
$550$	0	0
$551$	41.1078	1.75125
$552$	0	0
$553$	10.9238	0.464528
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−22.4857	−0.952749	−0.476375	−	0.879242i	$-0.658049\pi$
$557$	−22.4857	−0.952749	−0.476375	+	0.879242i	$0.658049\pi$
$558$	0	0
$559$	1.66863	0.0705753
$560$	0	0
$561$	1.83453	0.0774537
$562$	0	0
$563$	7.68774	0.324000	0.162000	−	0.986791i	$-0.448206\pi$
$563$	7.68774	0.324000	0.162000	+	0.986791i	$0.448206\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−7.61376	−0.319748
$568$	0	0
$569$	28.1828	1.18149	0.590743	−	0.806860i	$-0.298835\pi$
$569$	28.1828	1.18149	0.590743	+	0.806860i	$0.298835\pi$
$570$	0	0
$571$	−31.1444	−1.30335	−0.651676	−	0.758497i	$-0.725934\pi$
$571$	−31.1444	−1.30335	−0.651676	+	0.758497i	$0.725934\pi$
$572$	0	0
$573$	−11.0360	−0.461035
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.20837	0.0503051	0.0251525	−	0.999684i	$-0.491993\pi$
$577$	1.20837	0.0503051	0.0251525	+	0.999684i	$0.491993\pi$
$578$	0	0
$579$	21.3118	0.885687
$580$	0	0
$581$	−0.291168	−0.0120797
$582$	0	0
$583$	35.0588	1.45199
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13.4162	−0.553746	−0.276873	−	0.960906i	$-0.589298\pi$
$587$	−13.4162	−0.553746	−0.276873	+	0.960906i	$0.589298\pi$
$588$	0	0
$589$	48.2233	1.98701
$590$	0	0
$591$	−17.2487	−0.709516
$592$	0	0
$593$	5.37465	0.220711	0.110355	−	0.993892i	$-0.464801\pi$
$593$	5.37465	0.220711	0.110355	+	0.993892i	$0.464801\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.13434	−0.251062
$598$	0	0
$599$	18.7001	0.764064	0.382032	−	0.924149i	$-0.375224\pi$
$599$	18.7001	0.764064	0.382032	+	0.924149i	$0.375224\pi$
$600$	0	0
$601$	18.4418	0.752255	0.376128	−	0.926568i	$-0.377255\pi$
$601$	18.4418	0.752255	0.376128	+	0.926568i	$0.377255\pi$
$602$	0	0
$603$	−6.68932	−0.272410
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−15.6501	−0.635218	−0.317609	−	0.948222i	$-0.602880\pi$
$607$	−15.6501	−0.635218	−0.317609	+	0.948222i	$0.602880\pi$
$608$	0	0
$609$	−10.5095	−0.425865
$610$	0	0
$611$	0.662995	0.0268219
$612$	0	0
$613$	−2.18552	−0.0882725	−0.0441363	−	0.999026i	$-0.514054\pi$
$613$	−2.18552	−0.0882725	−0.0441363	+	0.999026i	$0.514054\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4.07539	−0.164069	−0.0820346	−	0.996629i	$-0.526142\pi$
$617$	−4.07539	−0.164069	−0.0820346	+	0.996629i	$0.526142\pi$
$618$	0	0
$619$	0.0289543	0.00116377	0.000581885	−	1.00000i	$-0.499815\pi$
$619$	0.0289543	0.00116377	0.000581885		1.00000i	$0.499815\pi$
$620$	0	0
$621$	−5.65635	−0.226982
$622$	0	0
$623$	−2.59051	−0.103787
$624$	0	0
$625$	0	0
$626$	0	0
$627$	52.6967	2.10450
$628$	0	0
$629$	1.31644	0.0524901
$630$	0	0
$631$	−33.3679	−1.32836	−0.664178	−	0.747575i	$-0.731218\pi$
$631$	−33.3679	−1.32836	−0.664178	+	0.747575i	$0.731218\pi$
$632$	0	0
$633$	−12.5453	−0.498632
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.37097	0.173184
$638$	0	0
$639$	10.2030	0.403624
$640$	0	0
$641$	1.56295	0.0617327	0.0308663	−	0.999524i	$-0.490173\pi$
$641$	1.56295	0.0617327	0.0308663	+	0.999524i	$0.490173\pi$
$642$	0	0
$643$	−33.3037	−1.31337	−0.656684	−	0.754166i	$-0.728042\pi$
$643$	−33.3037	−1.31337	−0.656684	+	0.754166i	$0.728042\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.54230	0.0606343	0.0303171	−	0.999540i	$-0.490348\pi$
$647$	1.54230	0.0606343	0.0303171	+	0.999540i	$0.490348\pi$
$648$	0	0
$649$	25.0190	0.982081
$650$	0	0
$651$	−12.3286	−0.483195
$652$	0	0
$653$	13.1195	0.513407	0.256703	−	0.966490i	$-0.417364\pi$
$653$	13.1195	0.513407	0.256703	+	0.966490i	$0.417364\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.55473	0.255724
$658$	0	0
$659$	−7.38016	−0.287490	−0.143745	−	0.989615i	$-0.545915\pi$
$659$	−7.38016	−0.287490	−0.143745	+	0.989615i	$0.545915\pi$
$660$	0	0
$661$	−25.0266	−0.973421	−0.486710	−	0.873563i	$-0.661803\pi$
$661$	−25.0266	−0.973421	−0.486710	+	0.873563i	$0.661803\pi$
$662$	0	0
$663$	0.406422	0.0157841
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.76644	−0.184557
$668$	0	0
$669$	−16.7820	−0.648830
$670$	0	0
$671$	59.4527	2.29515
$672$	0	0
$673$	−35.2500	−1.35879	−0.679395	−	0.733773i	$-0.737757\pi$
$673$	−35.2500	−1.35879	−0.679395	+	0.733773i	$0.737757\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.04850	0.232463	0.116231	−	0.993222i	$-0.462919\pi$
$677$	6.04850	0.232463	0.116231	+	0.993222i	$0.462919\pi$
$678$	0	0
$679$	14.4813	0.555741
$680$	0	0
$681$	2.60242	0.0997251
$682$	0	0
$683$	32.9347	1.26021	0.630105	−	0.776510i	$-0.283012\pi$
$683$	32.9347	1.26021	0.630105	+	0.776510i	$0.283012\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	23.1819	0.884443
$688$	0	0
$689$	7.76696	0.295897
$690$	0	0
$691$	−11.7152	−0.445668	−0.222834	−	0.974856i	$-0.571531\pi$
$691$	−11.7152	−0.445668	−0.222834	+	0.974856i	$0.571531\pi$
$692$	0	0
$693$	7.05040	0.267823
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.98764	−0.0752872
$698$	0	0
$699$	−34.5091	−1.30525
$700$	0	0
$701$	43.8861	1.65756	0.828779	−	0.559577i	$-0.189036\pi$
$701$	43.8861	1.65756	0.828779	+	0.559577i	$0.189036\pi$
$702$	0	0
$703$	37.8148	1.42621
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.13090	−0.0425317
$708$	0	0
$709$	30.2394	1.13567	0.567833	−	0.823144i	$-0.307782\pi$
$709$	30.2394	1.13567	0.567833	+	0.823144i	$0.307782\pi$
$710$	0	0
$711$	−7.16561	−0.268731
$712$	0	0
$713$	−5.59148	−0.209402
$714$	0	0
$715$	0	0
$716$	0	0
$717$	38.0544	1.42117
$718$	0	0
$719$	−27.5338	−1.02684	−0.513419	−	0.858138i	$-0.671621\pi$
$719$	−27.5338	−1.02684	−0.513419	+	0.858138i	$0.671621\pi$
$720$	0	0
$721$	14.3941	0.536063
$722$	0	0
$723$	18.0828	0.672507
$724$	0	0
$725$	0	0
$726$	0	0
$727$	33.7995	1.25356	0.626778	−	0.779198i	$-0.284373\pi$
$727$	33.7995	1.25356	0.626778	+	0.779198i	$0.284373\pi$
$728$	0	0
$729$	28.8078	1.06695
$730$	0	0
$731$	−0.519382	−0.0192100
$732$	0	0
$733$	−48.3133	−1.78449	−0.892246	−	0.451550i	$-0.850871\pi$
$733$	−48.3133	−1.78449	−0.892246	+	0.451550i	$0.850871\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−28.2598	−1.04096
$738$	0	0
$739$	22.7004	0.835048	0.417524	−	0.908666i	$-0.362898\pi$
$739$	22.7004	0.835048	0.417524	+	0.908666i	$0.362898\pi$
$740$	0	0
$741$	11.6745	0.428872
$742$	0	0
$743$	20.9953	0.770242	0.385121	−	0.922866i	$-0.374160\pi$
$743$	20.9953	0.770242	0.385121	+	0.922866i	$0.374160\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.190995	0.00698816
$748$	0	0
$749$	−20.0302	−0.731887
$750$	0	0
$751$	21.3909	0.780565	0.390283	−	0.920695i	$-0.372377\pi$
$751$	21.3909	0.780565	0.390283	+	0.920695i	$0.372377\pi$
$752$	0	0
$753$	−17.4871	−0.637266
$754$	0	0
$755$	0	0
$756$	0	0
$757$	54.7364	1.98943	0.994715	−	0.102676i	$-0.0327404\pi$
$757$	54.7364	1.98943	0.994715	+	0.102676i	$0.0327404\pi$
$758$	0	0
$759$	−6.11017	−0.221785
$760$	0	0
$761$	7.87321	0.285404	0.142702	−	0.989766i	$-0.454421\pi$
$761$	7.87321	0.285404	0.142702	+	0.989766i	$0.454421\pi$
$762$	0	0
$763$	−5.31084	−0.192265
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.54272	0.200136
$768$	0	0
$769$	32.7134	1.17968	0.589838	−	0.807522i	$-0.299192\pi$
$769$	32.7134	1.17968	0.589838	+	0.807522i	$0.299192\pi$
$770$	0	0
$771$	−18.9554	−0.682663
$772$	0	0
$773$	−2.88974	−0.103937	−0.0519684	−	0.998649i	$-0.516550\pi$
$773$	−2.88974	−0.103937	−0.0519684	+	0.998649i	$0.516550\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−9.66760	−0.346823
$778$	0	0
$779$	−57.0949	−2.04564
$780$	0	0
$781$	43.1037	1.54237
$782$	0	0
$783$	26.9607	0.963496
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−46.4472	−1.65566	−0.827832	−	0.560976i	$-0.810426\pi$
$787$	−46.4472	−1.65566	−0.827832	+	0.560976i	$0.810426\pi$
$788$	0	0
$789$	38.2523	1.36182
$790$	0	0
$791$	24.9104	0.885710
$792$	0	0
$793$	13.1712	0.467723
$794$	0	0
$795$	0	0
$796$	0	0
$797$	45.6713	1.61776	0.808880	−	0.587974i	$-0.200074\pi$
$797$	45.6713	1.61776	0.808880	+	0.587974i	$0.200074\pi$
$798$	0	0
$799$	−0.206366	−0.00730070
$800$	0	0
$801$	1.69928	0.0600411
$802$	0	0
$803$	27.6912	0.977202
$804$	0	0
$805$	0	0
$806$	0	0
$807$	25.0369	0.881340
$808$	0	0
$809$	37.7080	1.32574	0.662872	−	0.748733i	$-0.269337\pi$
$809$	37.7080	1.32574	0.662872	+	0.748733i	$0.269337\pi$
$810$	0	0
$811$	−9.02541	−0.316925	−0.158462	−	0.987365i	$-0.550654\pi$
$811$	−9.02541	−0.316925	−0.158462	+	0.987365i	$0.550654\pi$
$812$	0	0
$813$	3.25444	0.114138
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−14.9192	−0.521958
$818$	0	0
$819$	1.56195	0.0545790
$820$	0	0
$821$	−34.2481	−1.19527	−0.597634	−	0.801769i	$-0.703892\pi$
$821$	−34.2481	−1.19527	−0.597634	+	0.801769i	$0.703892\pi$
$822$	0	0
$823$	−10.1842	−0.354999	−0.177499	−	0.984121i	$-0.556801\pi$
$823$	−10.1842	−0.354999	−0.177499	+	0.984121i	$0.556801\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−34.0710	−1.18477	−0.592383	−	0.805656i	$-0.701813\pi$
$827$	−34.0710	−1.18477	−0.592383	+	0.805656i	$0.701813\pi$
$828$	0	0
$829$	−24.5131	−0.851376	−0.425688	−	0.904870i	$-0.639968\pi$
$829$	−24.5131	−0.851376	−0.425688	+	0.904870i	$0.639968\pi$
$830$	0	0
$831$	−45.0140	−1.56152
$832$	0	0
$833$	−1.36052	−0.0471393
$834$	0	0
$835$	0	0
$836$	0	0
$837$	31.6274	1.09320
$838$	0	0
$839$	−19.2362	−0.664107	−0.332053	−	0.943261i	$-0.607741\pi$
$839$	−19.2362	−0.664107	−0.332053	+	0.943261i	$0.607741\pi$
$840$	0	0
$841$	−6.28105	−0.216588
$842$	0	0
$843$	−18.1660	−0.625669
$844$	0	0
$845$	0	0
$846$	0	0
$847$	12.5024	0.429588
$848$	0	0
$849$	−31.7254	−1.08881
$850$	0	0
$851$	−4.38462	−0.150303
$852$	0	0
$853$	34.8964	1.19483	0.597415	−	0.801932i	$-0.296195\pi$
$853$	34.8964	1.19483	0.597415	+	0.801932i	$0.296195\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41.9040	1.43141	0.715706	−	0.698401i	$-0.246105\pi$
$857$	41.9040	1.43141	0.715706	+	0.698401i	$0.246105\pi$
$858$	0	0
$859$	19.1286	0.652658	0.326329	−	0.945256i	$-0.394188\pi$
$859$	19.1286	0.652658	0.326329	+	0.945256i	$0.394188\pi$
$860$	0	0
$861$	14.5967	0.497453
$862$	0	0
$863$	0.509621	0.0173477	0.00867385	−	0.999962i	$-0.497239\pi$
$863$	0.509621	0.0173477	0.00867385	+	0.999962i	$0.497239\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	23.7303	0.805924
$868$	0	0
$869$	−30.2719	−1.02691
$870$	0	0
$871$	−6.26070	−0.212136
$872$	0	0
$873$	−9.49919	−0.321499
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.6710	0.529171	0.264586	−	0.964362i	$-0.414765\pi$
$877$	15.6710	0.529171	0.264586	+	0.964362i	$0.414765\pi$
$878$	0	0
$879$	−7.88135	−0.265831
$880$	0	0
$881$	−2.00846	−0.0676667	−0.0338333	−	0.999427i	$-0.510772\pi$
$881$	−2.00846	−0.0676667	−0.0338333	+	0.999427i	$0.510772\pi$
$882$	0	0
$883$	−16.7848	−0.564854	−0.282427	−	0.959289i	$-0.591139\pi$
$883$	−16.7848	−0.564854	−0.282427	+	0.959289i	$0.591139\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	51.7294	1.73690	0.868452	−	0.495774i	$-0.165116\pi$
$887$	51.7294	1.73690	0.868452	+	0.495774i	$0.165116\pi$
$888$	0	0
$889$	23.9167	0.802140
$890$	0	0
$891$	21.0992	0.706849
$892$	0	0
$893$	−5.92786	−0.198368
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.35365	−0.0451971
$898$	0	0
$899$	26.6514	0.888875
$900$	0	0
$901$	−2.41757	−0.0805409
$902$	0	0
$903$	3.81420	0.126929
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−29.0849	−0.965747	−0.482874	−	0.875690i	$-0.660407\pi$
$907$	−29.0849	−0.965747	−0.482874	+	0.875690i	$0.660407\pi$
$908$	0	0
$909$	0.741825	0.0246048
$910$	0	0
$911$	−26.3517	−0.873070	−0.436535	−	0.899687i	$-0.643795\pi$
$911$	−26.3517	−0.873070	−0.436535	+	0.899687i	$0.643795\pi$
$912$	0	0
$913$	0.806883	0.0267039
$914$	0	0
$915$	0	0
$916$	0	0
$917$	20.2202	0.667728
$918$	0	0
$919$	46.0148	1.51789	0.758945	−	0.651155i	$-0.225715\pi$
$919$	46.0148	1.51789	0.758945	+	0.651155i	$0.225715\pi$
$920$	0	0
$921$	3.40264	0.112121
$922$	0	0
$923$	9.54922	0.314316
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−9.44196	−0.310115
$928$	0	0
$929$	26.1770	0.858838	0.429419	−	0.903105i	$-0.358718\pi$
$929$	26.1770	0.858838	0.429419	+	0.903105i	$0.358718\pi$
$930$	0	0
$931$	−39.0810	−1.28083
$932$	0	0
$933$	5.02237	0.164425
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−44.4429	−1.45189	−0.725944	−	0.687754i	$-0.758597\pi$
$937$	−44.4429	−1.45189	−0.725944	+	0.687754i	$0.758597\pi$
$938$	0	0
$939$	7.73242	0.252338
$940$	0	0
$941$	7.07346	0.230588	0.115294	−	0.993331i	$-0.463219\pi$
$941$	7.07346	0.230588	0.115294	+	0.993331i	$0.463219\pi$
$942$	0	0
$943$	6.62014	0.215581
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14.4533	0.469670	0.234835	−	0.972035i	$-0.424545\pi$
$947$	14.4533	0.469670	0.234835	+	0.972035i	$0.424545\pi$
$948$	0	0
$949$	6.13473	0.199142
$950$	0	0
$951$	−8.25683	−0.267746
$952$	0	0
$953$	−32.3773	−1.04880	−0.524402	−	0.851471i	$-0.675711\pi$
$953$	−32.3773	−1.04880	−0.524402	+	0.851471i	$0.675711\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	29.1237	0.941437
$958$	0	0
$959$	−26.3065	−0.849482
$960$	0	0
$961$	0.264608	0.00853575
$962$	0	0
$963$	13.1390	0.423400
$964$	0	0
$965$	0	0
$966$	0	0
$967$	25.8993	0.832865	0.416433	−	0.909167i	$-0.363280\pi$
$967$	25.8993	0.832865	0.416433	+	0.909167i	$0.363280\pi$
$968$	0	0
$969$	−3.63383	−0.116736
$970$	0	0
$971$	25.2153	0.809197	0.404599	−	0.914494i	$-0.367411\pi$
$971$	25.2153	0.809197	0.404599	+	0.914494i	$0.367411\pi$
$972$	0	0
$973$	18.5859	0.595837
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.64235	−0.180515	−0.0902573	−	0.995918i	$-0.528769\pi$
$977$	−5.64235	−0.180515	−0.0902573	+	0.995918i	$0.528769\pi$
$978$	0	0
$979$	7.17880	0.229436
$980$	0	0
$981$	3.48371	0.111226
$982$	0	0
$983$	37.0322	1.18114	0.590571	−	0.806986i	$-0.298903\pi$
$983$	37.0322	1.18114	0.590571	+	0.806986i	$0.298903\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.51549	0.0482387
$988$	0	0
$989$	1.72988	0.0550070
$990$	0	0
$991$	−7.14822	−0.227071	−0.113535	−	0.993534i	$-0.536218\pi$
$991$	−7.14822	−0.227071	−0.113535	+	0.993534i	$0.536218\pi$
$992$	0	0
$993$	−26.1215	−0.828942
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.96521	0.157250	0.0786249	−	0.996904i	$-0.474947\pi$
$997$	4.96521	0.157250	0.0786249	+	0.996904i	$0.474947\pi$
$998$	0	0
$999$	24.8010	0.784667

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.dc.1.2		7
4.3	odd	2		4600.2.a.bh.1.6		7
5.2	odd	4		1840.2.e.g.369.10		14
5.3	odd	4		1840.2.e.g.369.5		14
5.4	even	2		9200.2.a.cz.1.6		7
20.3	even	4		920.2.e.b.369.10	yes	14
20.7	even	4		920.2.e.b.369.5	✓	14
20.19	odd	2		4600.2.a.bi.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.b.369.5	✓	14	20.7	even	4
920.2.e.b.369.10	yes	14	20.3	even	4
1840.2.e.g.369.5		14	5.3	odd	4
1840.2.e.g.369.10		14	5.2	odd	4
4600.2.a.bh.1.6		7	4.3	odd	2
4600.2.a.bi.1.2		7	20.19	odd	2
9200.2.a.cz.1.6		7	5.4	even	2
9200.2.a.dc.1.2		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:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q-1.40334 q^{3} +1.57117 q^{7} -1.03063 q^{9} +O(q^{10})\)	\(q-1.40334 q^{3} +1.57117 q^{7} -1.03063 q^{9} -4.35401 q^{11} -0.964590 q^{13} +0.300242 q^{17} +8.62443 q^{19} -2.20489 q^{21} -1.00000 q^{23} +5.65635 q^{27} +4.76644 q^{29} +5.59148 q^{31} +6.11017 q^{33} +4.38462 q^{37} +1.35365 q^{39} -6.62014 q^{41} -1.72988 q^{43} -0.687333 q^{47} -4.53143 q^{49} -0.421342 q^{51} -8.05208 q^{53} -12.1030 q^{57} -5.74620 q^{59} -13.6547 q^{61} -1.61929 q^{63} +6.49053 q^{67} +1.40334 q^{69} -9.89977 q^{71} -6.35994 q^{73} -6.84088 q^{77} +6.95266 q^{79} -4.84592 q^{81} -0.185320 q^{83} -6.68895 q^{87} -1.64878 q^{89} -1.51553 q^{91} -7.84676 q^{93} +9.21689 q^{97} +4.48736 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 7 * q + 3 * q^3 + 4 * q^7 + 2 * q^9	\( 7 q + 3 q^{3} + 4 q^{7} + 2 q^{9} + 7 q^{11} + 7 q^{13} + 7 q^{19} - 6 q^{21} - 7 q^{23} - 11 q^{29} + 10 q^{31} + 19 q^{33} + 19 q^{37} + 24 q^{39} - 16 q^{41} + 6 q^{43} + 6 q^{47} - 17 q^{49} + 7 q^{51} + 15 q^{53} + 8 q^{57} + 11 q^{59} + 5 q^{61} + 13 q^{63} + 9 q^{67} - 3 q^{69} + 14 q^{71} + 10 q^{73} + 6 q^{77} + 32 q^{79} - 5 q^{81} + q^{83} + 10 q^{87} - 24 q^{89} + 7 q^{91} + 26 q^{93} - 7 q^{97} + 61 q^{99}+O(q^{100}) \) 7 * q + 3 * q^3 + 4 * q^7 + 2 * q^9 + 7 * q^11 + 7 * q^13 + 7 * q^19 - 6 * q^21 - 7 * q^23 - 11 * q^29 + 10 * q^31 + 19 * q^33 + 19 * q^37 + 24 * q^39 - 16 * q^41 + 6 * q^43 + 6 * q^47 - 17 * q^49 + 7 * q^51 + 15 * q^53 + 8 * q^57 + 11 * q^59 + 5 * q^61 + 13 * q^63 + 9 * q^67 - 3 * q^69 + 14 * q^71 + 10 * q^73 + 6 * q^77 + 32 * q^79 - 5 * q^81 + q^83 + 10 * q^87 - 24 * q^89 + 7 * q^91 + 26 * q^93 - 7 * q^97 + 61 * q^99

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