LMFDB - Embedded newform 9196.2.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9196,2,Mod(1,9196)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9196, 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("9196.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9196 = 2^{2} \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9196.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:	$73.4304296988$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 6 x^{19} - 26 x^{18} + 208 x^{17} + 185 x^{16} - 2910 x^{15} + 687 x^{14} + 21067 x^{13} + \cdots - 3520 $ x^20 - 6x^19 - 26x^18 + 208x^17 + 185x^16 - 2910x^15 + 687x^14 + 21067x^13 - 16530x^12 - 83962x^11 + 93139x^10 + 180932x^9 - 242362x^8 - 194923x^7 + 303963x^6 + 96332x^5 - 176108x^4 - 24984x^3 + 44976x^2 + 3840*x - 3520
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 836)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.45218$ of defining polynomial
Character	$\chi$	$=$	9196.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.45218 q^{3} +0.520700 q^{5} +0.586094 q^{7} +3.01318 q^{9} -3.56613 q^{13} -1.27685 q^{15} +3.94008 q^{17} -1.00000 q^{19} -1.43721 q^{21} -5.71492 q^{23} -4.72887 q^{25} -0.0323087 q^{27} -9.77619 q^{29} -0.656936 q^{31} +0.305179 q^{35} +3.51752 q^{37} +8.74478 q^{39} +11.2712 q^{41} -10.5252 q^{43} +1.56896 q^{45} +10.1084 q^{47} -6.65649 q^{49} -9.66177 q^{51} -10.5248 q^{53} +2.45218 q^{57} +8.47821 q^{59} +5.34925 q^{61} +1.76600 q^{63} -1.85688 q^{65} -2.38499 q^{67} +14.0140 q^{69} +10.4904 q^{71} +4.46532 q^{73} +11.5960 q^{75} +6.22693 q^{79} -8.96030 q^{81} -0.430250 q^{83} +2.05160 q^{85} +23.9730 q^{87} -16.3343 q^{89} -2.09009 q^{91} +1.61092 q^{93} -0.520700 q^{95} +13.3949 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 6 q^{3} + 2 q^{5} + 4 q^{7} + 28 q^{9} - 5 q^{13} + 14 q^{15} - 6 q^{17} - 20 q^{19} - q^{21} + 18 q^{23} + 44 q^{25} + 24 q^{27} + q^{29} + 23 q^{31} - 36 q^{35} + q^{37} + 21 q^{39} + 6 q^{41}+ \cdots + 9 q^{97}+O(q^{100}) $ 20 * q + 6 * q^3 + 2 * q^5 + 4 * q^7 + 28 * q^9 - 5 * q^13 + 14 * q^15 - 6 * q^17 - 20 * q^19 - q^21 + 18 * q^23 + 44 * q^25 + 24 * q^27 + q^29 + 23 * q^31 - 36 * q^35 + q^37 + 21 * q^39 + 6 * q^41 - 8 * q^43 - 15 * q^45 + 34 * q^47 + 42 * q^49 + 14 * q^51 - 10 * q^53 - 6 * q^57 + 52 * q^59 - 2 * q^61 + 59 * q^63 + 25 * q^65 + 14 * q^67 + 15 * q^69 + 88 * q^71 - 13 * q^73 + 15 * q^75 - 2 * q^79 + 48 * q^81 - 2 * q^83 + 11 * q^85 + 37 * q^87 + 46 * q^89 - 4 * q^91 + 2 * q^93 - 2 * q^95 + 9 * q^97

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
$3$	−2.45218	−1.41577	−0.707883	−	0.706330i	$-0.750349\pi$
$3$	−2.45218	−1.41577	−0.707883	+	0.706330i	$0.750349\pi$
$4$	0	0
$5$	0.520700	0.232864	0.116432	−	0.993199i	$-0.462854\pi$
$5$	0.520700	0.232864	0.116432	+	0.993199i	$0.462854\pi$
$6$	0	0
$7$	0.586094	0.221523	0.110761	−	0.993847i	$-0.464671\pi$
$7$	0.586094	0.221523	0.110761	+	0.993847i	$0.464671\pi$
$8$	0	0
$9$	3.01318	1.00439
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.56613	−0.989066	−0.494533	−	0.869159i	$-0.664661\pi$
$13$	−3.56613	−0.989066	−0.494533	+	0.869159i	$0.664661\pi$
$14$	0	0
$15$	−1.27685	−0.329681
$16$	0	0
$17$	3.94008	0.955609	0.477804	−	0.878466i	$-0.341433\pi$
$17$	3.94008	0.955609	0.477804	+	0.878466i	$0.341433\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.43721	−0.313624
$22$	0	0
$23$	−5.71492	−1.19164	−0.595822	−	0.803117i	$-0.703173\pi$
$23$	−5.71492	−1.19164	−0.595822	+	0.803117i	$0.703173\pi$
$24$	0	0
$25$	−4.72887	−0.945774
$26$	0	0
$27$	−0.0323087	−0.00621781
$28$	0	0
$29$	−9.77619	−1.81539	−0.907696	−	0.419628i	$-0.862161\pi$
$29$	−9.77619	−1.81539	−0.907696	+	0.419628i	$0.862161\pi$
$30$	0	0
$31$	−0.656936	−0.117989	−0.0589946	−	0.998258i	$-0.518789\pi$
$31$	−0.656936	−0.117989	−0.0589946	+	0.998258i	$0.518789\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.305179	0.0515847
$36$	0	0
$37$	3.51752	0.578276	0.289138	−	0.957287i	$-0.406631\pi$
$37$	3.51752	0.578276	0.289138	+	0.957287i	$0.406631\pi$
$38$	0	0
$39$	8.74478	1.40029
$40$	0	0
$41$	11.2712	1.76026	0.880129	−	0.474734i	$-0.157456\pi$
$41$	11.2712	1.76026	0.880129	+	0.474734i	$0.157456\pi$
$42$	0	0
$43$	−10.5252	−1.60508	−0.802540	−	0.596598i	$-0.796519\pi$
$43$	−10.5252	−1.60508	−0.802540	+	0.596598i	$0.796519\pi$
$44$	0	0
$45$	1.56896	0.233887
$46$	0	0
$47$	10.1084	1.47447	0.737234	−	0.675637i	$-0.236131\pi$
$47$	10.1084	1.47447	0.737234	+	0.675637i	$0.236131\pi$
$48$	0	0
$49$	−6.65649	−0.950928
$50$	0	0
$51$	−9.66177	−1.35292
$52$	0	0
$53$	−10.5248	−1.44569	−0.722844	−	0.691011i	$-0.757166\pi$
$53$	−10.5248	−1.44569	−0.722844	+	0.691011i	$0.757166\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.45218	0.324799
$58$	0	0
$59$	8.47821	1.10377	0.551884	−	0.833921i	$-0.313909\pi$
$59$	8.47821	1.10377	0.551884	+	0.833921i	$0.313909\pi$
$60$	0	0
$61$	5.34925	0.684901	0.342451	−	0.939536i	$-0.388743\pi$
$61$	5.34925	0.684901	0.342451	+	0.939536i	$0.388743\pi$
$62$	0	0
$63$	1.76600	0.222495
$64$	0	0
$65$	−1.85688	−0.230318
$66$	0	0
$67$	−2.38499	−0.291373	−0.145686	−	0.989331i	$-0.546539\pi$
$67$	−2.38499	−0.291373	−0.145686	+	0.989331i	$0.546539\pi$
$68$	0	0
$69$	14.0140	1.68709
$70$	0	0
$71$	10.4904	1.24498	0.622490	−	0.782628i	$-0.286121\pi$
$71$	10.4904	1.24498	0.622490	+	0.782628i	$0.286121\pi$
$72$	0	0
$73$	4.46532	0.522627	0.261313	−	0.965254i	$-0.415844\pi$
$73$	4.46532	0.522627	0.261313	+	0.965254i	$0.415844\pi$
$74$	0	0
$75$	11.5960	1.33899
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.22693	0.700584	0.350292	−	0.936640i	$-0.386082\pi$
$79$	6.22693	0.700584	0.350292	+	0.936640i	$0.386082\pi$
$80$	0	0
$81$	−8.96030	−0.995589
$82$	0	0
$83$	−0.430250	−0.0472261	−0.0236130	−	0.999721i	$-0.507517\pi$
$83$	−0.430250	−0.0472261	−0.0236130	+	0.999721i	$0.507517\pi$
$84$	0	0
$85$	2.05160	0.222527
$86$	0	0
$87$	23.9730	2.57017
$88$	0	0
$89$	−16.3343	−1.73143	−0.865714	−	0.500540i	$-0.833135\pi$
$89$	−16.3343	−1.73143	−0.865714	+	0.500540i	$0.833135\pi$
$90$	0	0
$91$	−2.09009	−0.219100
$92$	0	0
$93$	1.61092	0.167045
$94$	0	0
$95$	−0.520700	−0.0534227
$96$	0	0
$97$	13.3949	1.36005	0.680023	−	0.733191i	$-0.261970\pi$
$97$	13.3949	1.36005	0.680023	+	0.733191i	$0.261970\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−18.4916	−1.83999	−0.919993	−	0.391936i	$-0.871805\pi$
$101$	−18.4916	−1.83999	−0.919993	+	0.391936i	$0.871805\pi$
$102$	0	0
$103$	16.1344	1.58977	0.794887	−	0.606758i	$-0.207530\pi$
$103$	16.1344	1.58977	0.794887	+	0.606758i	$0.207530\pi$
$104$	0	0
$105$	−0.748353	−0.0730318
$106$	0	0
$107$	−14.3626	−1.38848	−0.694242	−	0.719742i	$-0.744260\pi$
$107$	−14.3626	−1.38848	−0.694242	+	0.719742i	$0.744260\pi$
$108$	0	0
$109$	1.93081	0.184938	0.0924692	−	0.995716i	$-0.470524\pi$
$109$	1.93081	0.184938	0.0924692	+	0.995716i	$0.470524\pi$
$110$	0	0
$111$	−8.62557	−0.818703
$112$	0	0
$113$	−11.0377	−1.03834	−0.519169	−	0.854671i	$-0.673759\pi$
$113$	−11.0377	−1.03834	−0.519169	+	0.854671i	$0.673759\pi$
$114$	0	0
$115$	−2.97576	−0.277491
$116$	0	0
$117$	−10.7454	−0.993410
$118$	0	0
$119$	2.30925	0.211689
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−27.6389	−2.49211
$124$	0	0
$125$	−5.06583	−0.453101
$126$	0	0
$127$	5.71470	0.507098	0.253549	−	0.967323i	$-0.418402\pi$
$127$	5.71470	0.507098	0.253549	+	0.967323i	$0.418402\pi$
$128$	0	0
$129$	25.8097	2.27242
$130$	0	0
$131$	−0.655087	−0.0572352	−0.0286176	−	0.999590i	$-0.509111\pi$
$131$	−0.655087	−0.0572352	−0.0286176	+	0.999590i	$0.509111\pi$
$132$	0	0
$133$	−0.586094	−0.0508208
$134$	0	0
$135$	−0.0168231	−0.00144791
$136$	0	0
$137$	−1.22032	−0.104259	−0.0521296	−	0.998640i	$-0.516601\pi$
$137$	−1.22032	−0.104259	−0.0521296	+	0.998640i	$0.516601\pi$
$138$	0	0
$139$	14.1265	1.19820	0.599099	−	0.800675i	$-0.295526\pi$
$139$	14.1265	1.19820	0.599099	+	0.800675i	$0.295526\pi$
$140$	0	0
$141$	−24.7877	−2.08750
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−5.09046	−0.422740
$146$	0	0
$147$	16.3229	1.34629
$148$	0	0
$149$	0.642009	0.0525954	0.0262977	−	0.999654i	$-0.491628\pi$
$149$	0.642009	0.0525954	0.0262977	+	0.999654i	$0.491628\pi$
$150$	0	0
$151$	4.65871	0.379120	0.189560	−	0.981869i	$-0.439294\pi$
$151$	4.65871	0.379120	0.189560	+	0.981869i	$0.439294\pi$
$152$	0	0
$153$	11.8721	0.959806
$154$	0	0
$155$	−0.342067	−0.0274755
$156$	0	0
$157$	−11.8542	−0.946064	−0.473032	−	0.881045i	$-0.656841\pi$
$157$	−11.8542	−0.946064	−0.473032	+	0.881045i	$0.656841\pi$
$158$	0	0
$159$	25.8086	2.04676
$160$	0	0
$161$	−3.34948	−0.263976
$162$	0	0
$163$	−17.8842	−1.40080	−0.700400	−	0.713751i	$-0.746995\pi$
$163$	−17.8842	−1.40080	−0.700400	+	0.713751i	$0.746995\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.89669	−0.146770	−0.0733851	−	0.997304i	$-0.523380\pi$
$167$	−1.89669	−0.146770	−0.0733851	+	0.997304i	$0.523380\pi$
$168$	0	0
$169$	−0.282722	−0.0217478
$170$	0	0
$171$	−3.01318	−0.230423
$172$	0	0
$173$	−3.93706	−0.299329	−0.149665	−	0.988737i	$-0.547819\pi$
$173$	−3.93706	−0.299329	−0.149665	+	0.988737i	$0.547819\pi$
$174$	0	0
$175$	−2.77156	−0.209510
$176$	0	0
$177$	−20.7901	−1.56268
$178$	0	0
$179$	13.6414	1.01961	0.509805	−	0.860290i	$-0.329718\pi$
$179$	13.6414	1.01961	0.509805	+	0.860290i	$0.329718\pi$
$180$	0	0
$181$	10.8817	0.808832	0.404416	−	0.914575i	$-0.367475\pi$
$181$	10.8817	0.808832	0.404416	+	0.914575i	$0.367475\pi$
$182$	0	0
$183$	−13.1173	−0.969659
$184$	0	0
$185$	1.83157	0.134660
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−0.0189359	−0.00137738
$190$	0	0
$191$	10.7140	0.775236	0.387618	−	0.921820i	$-0.373298\pi$
$191$	10.7140	0.775236	0.387618	+	0.921820i	$0.373298\pi$
$192$	0	0
$193$	18.7181	1.34736	0.673679	−	0.739024i	$-0.264713\pi$
$193$	18.7181	1.34736	0.673679	+	0.739024i	$0.264713\pi$
$194$	0	0
$195$	4.55341	0.326076
$196$	0	0
$197$	25.9523	1.84902	0.924512	−	0.381154i	$-0.124473\pi$
$197$	25.9523	1.84902	0.924512	+	0.381154i	$0.124473\pi$
$198$	0	0
$199$	−6.01887	−0.426667	−0.213333	−	0.976979i	$-0.568432\pi$
$199$	−6.01887	−0.426667	−0.213333	+	0.976979i	$0.568432\pi$
$200$	0	0
$201$	5.84842	0.412516
$202$	0	0
$203$	−5.72976	−0.402150
$204$	0	0
$205$	5.86889	0.409901
$206$	0	0
$207$	−17.2201	−1.19688
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−15.8843	−1.09352	−0.546760	−	0.837289i	$-0.684139\pi$
$211$	−15.8843	−1.09352	−0.546760	+	0.837289i	$0.684139\pi$
$212$	0	0
$213$	−25.7243	−1.76260
$214$	0	0
$215$	−5.48048	−0.373766
$216$	0	0
$217$	−0.385026	−0.0261373
$218$	0	0
$219$	−10.9498	−0.739917
$220$	0	0
$221$	−14.0508	−0.945160
$222$	0	0
$223$	−15.5144	−1.03892	−0.519461	−	0.854494i	$-0.673867\pi$
$223$	−15.5144	−1.03892	−0.519461	+	0.854494i	$0.673867\pi$
$224$	0	0
$225$	−14.2489	−0.949928
$226$	0	0
$227$	−1.59384	−0.105787	−0.0528935	−	0.998600i	$-0.516844\pi$
$227$	−1.59384	−0.105787	−0.0528935	+	0.998600i	$0.516844\pi$
$228$	0	0
$229$	20.8659	1.37886	0.689429	−	0.724354i	$-0.257862\pi$
$229$	20.8659	1.37886	0.689429	+	0.724354i	$0.257862\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.6149	−1.15399	−0.576996	−	0.816747i	$-0.695775\pi$
$233$	−17.6149	−1.15399	−0.576996	+	0.816747i	$0.695775\pi$
$234$	0	0
$235$	5.26347	0.343351
$236$	0	0
$237$	−15.2695	−0.991863
$238$	0	0
$239$	−0.802519	−0.0519106	−0.0259553	−	0.999663i	$-0.508263\pi$
$239$	−0.802519	−0.0519106	−0.0259553	+	0.999663i	$0.508263\pi$
$240$	0	0
$241$	19.1726	1.23502	0.617509	−	0.786564i	$-0.288142\pi$
$241$	19.1726	1.23502	0.617509	+	0.786564i	$0.288142\pi$
$242$	0	0
$243$	22.0692	1.41574
$244$	0	0
$245$	−3.46604	−0.221437
$246$	0	0
$247$	3.56613	0.226907
$248$	0	0
$249$	1.05505	0.0668610
$250$	0	0
$251$	0.442602	0.0279368	0.0139684	−	0.999902i	$-0.495554\pi$
$251$	0.442602	0.0279368	0.0139684	+	0.999902i	$0.495554\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−5.03088	−0.315046
$256$	0	0
$257$	−24.1880	−1.50880	−0.754402	−	0.656412i	$-0.772073\pi$
$257$	−24.1880	−1.50880	−0.754402	+	0.656412i	$0.772073\pi$
$258$	0	0
$259$	2.06159	0.128101
$260$	0	0
$261$	−29.4574	−1.82337
$262$	0	0
$263$	−13.5494	−0.835492	−0.417746	−	0.908564i	$-0.637180\pi$
$263$	−13.5494	−0.835492	−0.417746	+	0.908564i	$0.637180\pi$
$264$	0	0
$265$	−5.48025	−0.336649
$266$	0	0
$267$	40.0545	2.45129
$268$	0	0
$269$	18.3177	1.11685	0.558425	−	0.829555i	$-0.311406\pi$
$269$	18.3177	1.11685	0.558425	+	0.829555i	$0.311406\pi$
$270$	0	0
$271$	−14.0859	−0.855659	−0.427830	−	0.903859i	$-0.640722\pi$
$271$	−14.0859	−0.855659	−0.427830	+	0.903859i	$0.640722\pi$
$272$	0	0
$273$	5.12526	0.310195
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−26.4707	−1.59047	−0.795235	−	0.606301i	$-0.792653\pi$
$277$	−26.4707	−1.59047	−0.795235	+	0.606301i	$0.792653\pi$
$278$	0	0
$279$	−1.97946	−0.118507
$280$	0	0
$281$	−18.3868	−1.09686	−0.548431	−	0.836196i	$-0.684775\pi$
$281$	−18.3868	−1.09686	−0.548431	+	0.836196i	$0.684775\pi$
$282$	0	0
$283$	29.5117	1.75429	0.877143	−	0.480229i	$-0.159446\pi$
$283$	29.5117	1.75429	0.877143	+	0.480229i	$0.159446\pi$
$284$	0	0
$285$	1.27685	0.0756340
$286$	0	0
$287$	6.60595	0.389937
$288$	0	0
$289$	−1.47581	−0.0868121
$290$	0	0
$291$	−32.8467	−1.92550
$292$	0	0
$293$	0.0316771	0.00185060	0.000925299	−	1.00000i	$-0.499705\pi$
$293$	0.0316771	0.00185060	0.000925299		1.00000i	$0.499705\pi$
$294$	0	0
$295$	4.41461	0.257028
$296$	0	0
$297$	0	0
$298$	0	0
$299$	20.3801	1.17861
$300$	0	0
$301$	−6.16876	−0.355562
$302$	0	0
$303$	45.3447	2.60499
$304$	0	0
$305$	2.78535	0.159489
$306$	0	0
$307$	21.2743	1.21419	0.607095	−	0.794629i	$-0.292335\pi$
$307$	21.2743	1.21419	0.607095	+	0.794629i	$0.292335\pi$
$308$	0	0
$309$	−39.5645	−2.25075
$310$	0	0
$311$	−1.91647	−0.108673	−0.0543365	−	0.998523i	$-0.517304\pi$
$311$	−1.91647	−0.108673	−0.0543365	+	0.998523i	$0.517304\pi$
$312$	0	0
$313$	7.08754	0.400611	0.200306	−	0.979733i	$-0.435806\pi$
$313$	7.08754	0.400611	0.200306	+	0.979733i	$0.435806\pi$
$314$	0	0
$315$	0.919558	0.0518112
$316$	0	0
$317$	28.4258	1.59655	0.798277	−	0.602290i	$-0.205745\pi$
$317$	28.4258	1.59655	0.798277	+	0.602290i	$0.205745\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	35.2196	1.96577
$322$	0	0
$323$	−3.94008	−0.219232
$324$	0	0
$325$	16.8638	0.935433
$326$	0	0
$327$	−4.73470	−0.261830
$328$	0	0
$329$	5.92449	0.326628
$330$	0	0
$331$	12.8389	0.705691	0.352845	−	0.935682i	$-0.385214\pi$
$331$	12.8389	0.705691	0.352845	+	0.935682i	$0.385214\pi$
$332$	0	0
$333$	10.5989	0.580816
$334$	0	0
$335$	−1.24186	−0.0678503
$336$	0	0
$337$	−9.05943	−0.493499	−0.246749	−	0.969079i	$-0.579362\pi$
$337$	−9.05943	−0.493499	−0.246749	+	0.969079i	$0.579362\pi$
$338$	0	0
$339$	27.0664	1.47004
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−8.00398	−0.432174
$344$	0	0
$345$	7.29709	0.392862
$346$	0	0
$347$	27.8527	1.49521	0.747605	−	0.664143i	$-0.231204\pi$
$347$	27.8527	1.49521	0.747605	+	0.664143i	$0.231204\pi$
$348$	0	0
$349$	−32.4159	−1.73519	−0.867593	−	0.497275i	$-0.834334\pi$
$349$	−32.4159	−1.73519	−0.867593	+	0.497275i	$0.834334\pi$
$350$	0	0
$351$	0.115217	0.00614983
$352$	0	0
$353$	23.8089	1.26722	0.633609	−	0.773653i	$-0.281573\pi$
$353$	23.8089	1.26722	0.633609	+	0.773653i	$0.281573\pi$
$354$	0	0
$355$	5.46235	0.289911
$356$	0	0
$357$	−5.66270	−0.299702
$358$	0	0
$359$	1.73025	0.0913192	0.0456596	−	0.998957i	$-0.485461\pi$
$359$	1.73025	0.0913192	0.0456596	+	0.998957i	$0.485461\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.32509	0.121701
$366$	0	0
$367$	−1.03496	−0.0540244	−0.0270122	−	0.999635i	$-0.508599\pi$
$367$	−1.03496	−0.0540244	−0.0270122	+	0.999635i	$0.508599\pi$
$368$	0	0
$369$	33.9620	1.76799
$370$	0	0
$371$	−6.16850	−0.320253
$372$	0	0
$373$	6.12290	0.317032	0.158516	−	0.987356i	$-0.449329\pi$
$373$	6.12290	0.317032	0.158516	+	0.987356i	$0.449329\pi$
$374$	0	0
$375$	12.4223	0.641485
$376$	0	0
$377$	34.8632	1.79554
$378$	0	0
$379$	13.4004	0.688335	0.344167	−	0.938908i	$-0.388161\pi$
$379$	13.4004	0.688335	0.344167	+	0.938908i	$0.388161\pi$
$380$	0	0
$381$	−14.0135	−0.717931
$382$	0	0
$383$	29.8753	1.52655	0.763277	−	0.646071i	$-0.223589\pi$
$383$	29.8753	1.52655	0.763277	+	0.646071i	$0.223589\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−31.7143	−1.61213
$388$	0	0
$389$	31.3583	1.58993	0.794963	−	0.606658i	$-0.207490\pi$
$389$	31.3583	1.58993	0.794963	+	0.606658i	$0.207490\pi$
$390$	0	0
$391$	−22.5172	−1.13874
$392$	0	0
$393$	1.60639	0.0810316
$394$	0	0
$395$	3.24236	0.163141
$396$	0	0
$397$	0.821202	0.0412150	0.0206075	−	0.999788i	$-0.493440\pi$
$397$	0.821202	0.0412150	0.0206075	+	0.999788i	$0.493440\pi$
$398$	0	0
$399$	1.43721	0.0719503
$400$	0	0
$401$	18.5139	0.924539	0.462269	−	0.886740i	$-0.347035\pi$
$401$	18.5139	0.924539	0.462269	+	0.886740i	$0.347035\pi$
$402$	0	0
$403$	2.34272	0.116699
$404$	0	0
$405$	−4.66563	−0.231837
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−5.39924	−0.266975	−0.133488	−	0.991050i	$-0.542618\pi$
$409$	−5.39924	−0.266975	−0.133488	+	0.991050i	$0.542618\pi$
$410$	0	0
$411$	2.99245	0.147607
$412$	0	0
$413$	4.96902	0.244510
$414$	0	0
$415$	−0.224031	−0.0109973
$416$	0	0
$417$	−34.6408	−1.69637
$418$	0	0
$419$	6.28062	0.306829	0.153414	−	0.988162i	$-0.450973\pi$
$419$	6.28062	0.306829	0.153414	+	0.988162i	$0.450973\pi$
$420$	0	0
$421$	7.67563	0.374087	0.187044	−	0.982352i	$-0.440109\pi$
$421$	7.67563	0.374087	0.187044	+	0.982352i	$0.440109\pi$
$422$	0	0
$423$	30.4585	1.48094
$424$	0	0
$425$	−18.6321	−0.903790
$426$	0	0
$427$	3.13516	0.151721
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.46664	0.166982	0.0834910	−	0.996509i	$-0.473393\pi$
$431$	3.46664	0.166982	0.0834910	+	0.996509i	$0.473393\pi$
$432$	0	0
$433$	−0.501246	−0.0240884	−0.0120442	−	0.999927i	$-0.503834\pi$
$433$	−0.501246	−0.0240884	−0.0120442	+	0.999927i	$0.503834\pi$
$434$	0	0
$435$	12.4827	0.598501
$436$	0	0
$437$	5.71492	0.273382
$438$	0	0
$439$	−12.7463	−0.608348	−0.304174	−	0.952616i	$-0.598380\pi$
$439$	−12.7463	−0.608348	−0.304174	+	0.952616i	$0.598380\pi$
$440$	0	0
$441$	−20.0572	−0.955104
$442$	0	0
$443$	0.821111	0.0390122	0.0195061	−	0.999810i	$-0.493791\pi$
$443$	0.821111	0.0390122	0.0195061	+	0.999810i	$0.493791\pi$
$444$	0	0
$445$	−8.50525	−0.403187
$446$	0	0
$447$	−1.57432	−0.0744628
$448$	0	0
$449$	20.2411	0.955235	0.477617	−	0.878568i	$-0.341501\pi$
$449$	20.2411	0.955235	0.477617	+	0.878568i	$0.341501\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−11.4240	−0.536745
$454$	0	0
$455$	−1.08831	−0.0510207
$456$	0	0
$457$	−22.3012	−1.04321	−0.521604	−	0.853188i	$-0.674666\pi$
$457$	−22.3012	−1.04321	−0.521604	+	0.853188i	$0.674666\pi$
$458$	0	0
$459$	−0.127299	−0.00594179
$460$	0	0
$461$	−25.9697	−1.20953	−0.604766	−	0.796403i	$-0.706733\pi$
$461$	−25.9697	−1.20953	−0.604766	+	0.796403i	$0.706733\pi$
$462$	0	0
$463$	−3.83882	−0.178405	−0.0892026	−	0.996014i	$-0.528432\pi$
$463$	−3.83882	−0.178405	−0.0892026	+	0.996014i	$0.528432\pi$
$464$	0	0
$465$	0.838809	0.0388988
$466$	0	0
$467$	19.6234	0.908062	0.454031	−	0.890986i	$-0.349986\pi$
$467$	19.6234	0.908062	0.454031	+	0.890986i	$0.349986\pi$
$468$	0	0
$469$	−1.39783	−0.0645457
$470$	0	0
$471$	29.0685	1.33941
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.72887	0.216975
$476$	0	0
$477$	−31.7130	−1.45204
$478$	0	0
$479$	5.07599	0.231928	0.115964	−	0.993253i	$-0.463004\pi$
$479$	5.07599	0.231928	0.115964	+	0.993253i	$0.463004\pi$
$480$	0	0
$481$	−12.5439	−0.571953
$482$	0	0
$483$	8.21351	0.373728
$484$	0	0
$485$	6.97472	0.316706
$486$	0	0
$487$	19.1188	0.866357	0.433178	−	0.901308i	$-0.357392\pi$
$487$	19.1188	0.866357	0.433178	+	0.901308i	$0.357392\pi$
$488$	0	0
$489$	43.8553	1.98320
$490$	0	0
$491$	17.5199	0.790660	0.395330	−	0.918539i	$-0.370630\pi$
$491$	17.5199	0.790660	0.395330	+	0.918539i	$0.370630\pi$
$492$	0	0
$493$	−38.5189	−1.73480
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.14835	0.275791
$498$	0	0
$499$	−4.78582	−0.214243	−0.107121	−	0.994246i	$-0.534163\pi$
$499$	−4.78582	−0.214243	−0.107121	+	0.994246i	$0.534163\pi$
$500$	0	0
$501$	4.65102	0.207792
$502$	0	0
$503$	15.1151	0.673949	0.336975	−	0.941514i	$-0.390596\pi$
$503$	15.1151	0.673949	0.336975	+	0.941514i	$0.390596\pi$
$504$	0	0
$505$	−9.62859	−0.428467
$506$	0	0
$507$	0.693284	0.0307898
$508$	0	0
$509$	−8.99352	−0.398631	−0.199315	−	0.979935i	$-0.563872\pi$
$509$	−8.99352	−0.398631	−0.199315	+	0.979935i	$0.563872\pi$
$510$	0	0
$511$	2.61710	0.115774
$512$	0	0
$513$	0.0323087	0.00142646
$514$	0	0
$515$	8.40120	0.370201
$516$	0	0
$517$	0	0
$518$	0	0
$519$	9.65437	0.423780
$520$	0	0
$521$	20.2618	0.887688	0.443844	−	0.896104i	$-0.353614\pi$
$521$	20.2618	0.887688	0.443844	+	0.896104i	$0.353614\pi$
$522$	0	0
$523$	−25.8442	−1.13009	−0.565045	−	0.825060i	$-0.691141\pi$
$523$	−25.8442	−1.13009	−0.565045	+	0.825060i	$0.691141\pi$
$524$	0	0
$525$	6.79636	0.296617
$526$	0	0
$527$	−2.58838	−0.112752
$528$	0	0
$529$	9.66031	0.420013
$530$	0	0
$531$	25.5463	1.10862
$532$	0	0
$533$	−40.1944	−1.74101
$534$	0	0
$535$	−7.47860	−0.323328
$536$	0	0
$537$	−33.4512	−1.44353
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6.47571	−0.278413	−0.139206	−	0.990263i	$-0.544455\pi$
$541$	−6.47571	−0.278413	−0.139206	+	0.990263i	$0.544455\pi$
$542$	0	0
$543$	−26.6839	−1.14512
$544$	0	0
$545$	1.00538	0.0430656
$546$	0	0
$547$	40.0938	1.71429	0.857144	−	0.515078i	$-0.172237\pi$
$547$	40.0938	1.71429	0.857144	+	0.515078i	$0.172237\pi$
$548$	0	0
$549$	16.1182	0.687909
$550$	0	0
$551$	9.77619	0.416480
$552$	0	0
$553$	3.64956	0.155195
$554$	0	0
$555$	−4.49134	−0.190647
$556$	0	0
$557$	−10.4559	−0.443031	−0.221516	−	0.975157i	$-0.571100\pi$
$557$	−10.4559	−0.443031	−0.221516	+	0.975157i	$0.571100\pi$
$558$	0	0
$559$	37.5343	1.58753
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.80512	−0.0760768	−0.0380384	−	0.999276i	$-0.512111\pi$
$563$	−1.80512	−0.0760768	−0.0380384	+	0.999276i	$0.512111\pi$
$564$	0	0
$565$	−5.74733	−0.241792
$566$	0	0
$567$	−5.25157	−0.220545
$568$	0	0
$569$	0.181632	0.00761443	0.00380721	−	0.999993i	$-0.498788\pi$
$569$	0.181632	0.00761443	0.00380721	+	0.999993i	$0.498788\pi$
$570$	0	0
$571$	−29.3911	−1.22998	−0.614989	−	0.788536i	$-0.710839\pi$
$571$	−29.3911	−1.22998	−0.614989	+	0.788536i	$0.710839\pi$
$572$	0	0
$573$	−26.2726	−1.09755
$574$	0	0
$575$	27.0251	1.12703
$576$	0	0
$577$	−32.2896	−1.34423	−0.672117	−	0.740445i	$-0.734615\pi$
$577$	−32.2896	−1.34423	−0.672117	+	0.740445i	$0.734615\pi$
$578$	0	0
$579$	−45.9001	−1.90754
$580$	0	0
$581$	−0.252167	−0.0104616
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−5.59512	−0.231330
$586$	0	0
$587$	3.58558	0.147993	0.0739963	−	0.997259i	$-0.476425\pi$
$587$	3.58558	0.147993	0.0739963	+	0.997259i	$0.476425\pi$
$588$	0	0
$589$	0.656936	0.0270686
$590$	0	0
$591$	−63.6396	−2.61778
$592$	0	0
$593$	30.3828	1.24767	0.623835	−	0.781556i	$-0.285574\pi$
$593$	30.3828	1.24767	0.623835	+	0.781556i	$0.285574\pi$
$594$	0	0
$595$	1.20243	0.0492948
$596$	0	0
$597$	14.7593	0.604060
$598$	0	0
$599$	19.9076	0.813403	0.406701	−	0.913561i	$-0.366679\pi$
$599$	19.9076	0.813403	0.406701	+	0.913561i	$0.366679\pi$
$600$	0	0
$601$	47.4452	1.93533	0.967665	−	0.252240i	$-0.0811674\pi$
$601$	47.4452	1.93533	0.967665	+	0.252240i	$0.0811674\pi$
$602$	0	0
$603$	−7.18639	−0.292653
$604$	0	0
$605$	0	0
$606$	0	0
$607$	14.5260	0.589591	0.294795	−	0.955560i	$-0.404748\pi$
$607$	14.5260	0.589591	0.294795	+	0.955560i	$0.404748\pi$
$608$	0	0
$609$	14.0504	0.569351
$610$	0	0
$611$	−36.0480	−1.45835
$612$	0	0
$613$	−23.7397	−0.958838	−0.479419	−	0.877586i	$-0.659153\pi$
$613$	−23.7397	−0.958838	−0.479419	+	0.877586i	$0.659153\pi$
$614$	0	0
$615$	−14.3916	−0.580324
$616$	0	0
$617$	11.8261	0.476102	0.238051	−	0.971253i	$-0.423491\pi$
$617$	11.8261	0.476102	0.238051	+	0.971253i	$0.423491\pi$
$618$	0	0
$619$	44.1910	1.77619	0.888094	−	0.459663i	$-0.152030\pi$
$619$	44.1910	1.77619	0.888094	+	0.459663i	$0.152030\pi$
$620$	0	0
$621$	0.184642	0.00740941
$622$	0	0
$623$	−9.57340	−0.383550
$624$	0	0
$625$	21.0066	0.840263
$626$	0	0
$627$	0	0
$628$	0	0
$629$	13.8593	0.552605
$630$	0	0
$631$	37.5180	1.49357	0.746783	−	0.665068i	$-0.231597\pi$
$631$	37.5180	1.49357	0.746783	+	0.665068i	$0.231597\pi$
$632$	0	0
$633$	38.9511	1.54817
$634$	0	0
$635$	2.97565	0.118085
$636$	0	0
$637$	23.7379	0.940531
$638$	0	0
$639$	31.6094	1.25045
$640$	0	0
$641$	25.6798	1.01429	0.507145	−	0.861861i	$-0.330701\pi$
$641$	25.6798	1.01429	0.507145	+	0.861861i	$0.330701\pi$
$642$	0	0
$643$	−4.99182	−0.196858	−0.0984291	−	0.995144i	$-0.531382\pi$
$643$	−4.99182	−0.196858	−0.0984291	+	0.995144i	$0.531382\pi$
$644$	0	0
$645$	13.4391	0.529165
$646$	0	0
$647$	13.7462	0.540419	0.270210	−	0.962802i	$-0.412907\pi$
$647$	13.7462	0.540419	0.270210	+	0.962802i	$0.412907\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0.944153	0.0370043
$652$	0	0
$653$	−8.24860	−0.322793	−0.161396	−	0.986890i	$-0.551600\pi$
$653$	−8.24860	−0.322793	−0.161396	+	0.986890i	$0.551600\pi$
$654$	0	0
$655$	−0.341104	−0.0133280
$656$	0	0
$657$	13.4548	0.524922
$658$	0	0
$659$	16.7449	0.652289	0.326144	−	0.945320i	$-0.394250\pi$
$659$	16.7449	0.652289	0.326144	+	0.945320i	$0.394250\pi$
$660$	0	0
$661$	37.3254	1.45179	0.725895	−	0.687806i	$-0.241426\pi$
$661$	37.3254	1.45179	0.725895	+	0.687806i	$0.241426\pi$
$662$	0	0
$663$	34.4551	1.33813
$664$	0	0
$665$	−0.305179	−0.0118343
$666$	0	0
$667$	55.8701	2.16330
$668$	0	0
$669$	38.0441	1.47087
$670$	0	0
$671$	0	0
$672$	0	0
$673$	31.9717	1.23242	0.616209	−	0.787583i	$-0.288668\pi$
$673$	31.9717	1.23242	0.616209	+	0.787583i	$0.288668\pi$
$674$	0	0
$675$	0.152784	0.00588064
$676$	0	0
$677$	−34.2136	−1.31493	−0.657467	−	0.753483i	$-0.728372\pi$
$677$	−34.2136	−1.31493	−0.657467	+	0.753483i	$0.728372\pi$
$678$	0	0
$679$	7.85066	0.301281
$680$	0	0
$681$	3.90838	0.149770
$682$	0	0
$683$	−0.710355	−0.0271810	−0.0135905	−	0.999908i	$-0.504326\pi$
$683$	−0.710355	−0.0271810	−0.0135905	+	0.999908i	$0.504326\pi$
$684$	0	0
$685$	−0.635422	−0.0242782
$686$	0	0
$687$	−51.1669	−1.95214
$688$	0	0
$689$	37.5327	1.42988
$690$	0	0
$691$	41.4320	1.57615	0.788074	−	0.615581i	$-0.211079\pi$
$691$	41.4320	1.57615	0.788074	+	0.615581i	$0.211079\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.35570	0.279017
$696$	0	0
$697$	44.4092	1.68212
$698$	0	0
$699$	43.1949	1.63378
$700$	0	0
$701$	0.404165	0.0152651	0.00763255	−	0.999971i	$-0.497570\pi$
$701$	0.404165	0.0152651	0.00763255	+	0.999971i	$0.497570\pi$
$702$	0	0
$703$	−3.51752	−0.132666
$704$	0	0
$705$	−12.9070	−0.486104
$706$	0	0
$707$	−10.8378	−0.407598
$708$	0	0
$709$	49.1993	1.84772	0.923859	−	0.382733i	$-0.125017\pi$
$709$	49.1993	1.84772	0.923859	+	0.382733i	$0.125017\pi$
$710$	0	0
$711$	18.7628	0.703661
$712$	0	0
$713$	3.75434	0.140601
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.96792	0.0734933
$718$	0	0
$719$	15.7532	0.587496	0.293748	−	0.955883i	$-0.405097\pi$
$719$	15.7532	0.587496	0.293748	+	0.955883i	$0.405097\pi$
$720$	0	0
$721$	9.45629	0.352171
$722$	0	0
$723$	−47.0147	−1.74849
$724$	0	0
$725$	46.2303	1.71695
$726$	0	0
$727$	41.6876	1.54611	0.773053	−	0.634341i	$-0.218729\pi$
$727$	41.6876	1.54611	0.773053	+	0.634341i	$0.218729\pi$
$728$	0	0
$729$	−27.2366	−1.00876
$730$	0	0
$731$	−41.4701	−1.53383
$732$	0	0
$733$	−15.1128	−0.558203	−0.279101	−	0.960262i	$-0.590037\pi$
$733$	−15.1128	−0.558203	−0.279101	+	0.960262i	$0.590037\pi$
$734$	0	0
$735$	8.49934	0.313503
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−17.4779	−0.642936	−0.321468	−	0.946920i	$-0.604176\pi$
$739$	−17.4779	−0.642936	−0.321468	+	0.946920i	$0.604176\pi$
$740$	0	0
$741$	−8.74478	−0.321248
$742$	0	0
$743$	−4.25658	−0.156159	−0.0780794	−	0.996947i	$-0.524879\pi$
$743$	−4.25658	−0.156159	−0.0780794	+	0.996947i	$0.524879\pi$
$744$	0	0
$745$	0.334294	0.0122476
$746$	0	0
$747$	−1.29642	−0.0474335
$748$	0	0
$749$	−8.41782	−0.307580
$750$	0	0
$751$	19.9941	0.729595	0.364798	−	0.931087i	$-0.381138\pi$
$751$	19.9941	0.729595	0.364798	+	0.931087i	$0.381138\pi$
$752$	0	0
$753$	−1.08534	−0.0395520
$754$	0	0
$755$	2.42579	0.0882835
$756$	0	0
$757$	−42.0100	−1.52688	−0.763440	−	0.645879i	$-0.776491\pi$
$757$	−42.0100	−1.52688	−0.763440	+	0.645879i	$0.776491\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	9.73673	0.352956	0.176478	−	0.984305i	$-0.443530\pi$
$761$	9.73673	0.352956	0.176478	+	0.984305i	$0.443530\pi$
$762$	0	0
$763$	1.13164	0.0409680
$764$	0	0
$765$	6.18182	0.223504
$766$	0	0
$767$	−30.2344	−1.09170
$768$	0	0
$769$	34.9177	1.25916	0.629582	−	0.776934i	$-0.283226\pi$
$769$	34.9177	1.25916	0.629582	+	0.776934i	$0.283226\pi$
$770$	0	0
$771$	59.3132	2.13611
$772$	0	0
$773$	−33.0979	−1.19045	−0.595225	−	0.803559i	$-0.702937\pi$
$773$	−33.0979	−1.19045	−0.595225	+	0.803559i	$0.702937\pi$
$774$	0	0
$775$	3.10657	0.111591
$776$	0	0
$777$	−5.05539	−0.181361
$778$	0	0
$779$	−11.2712	−0.403831
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0.315856	0.0112878
$784$	0	0
$785$	−6.17246	−0.220304
$786$	0	0
$787$	−4.13672	−0.147458	−0.0737291	−	0.997278i	$-0.523490\pi$
$787$	−4.13672	−0.147458	−0.0737291	+	0.997278i	$0.523490\pi$
$788$	0	0
$789$	33.2255	1.18286
$790$	0	0
$791$	−6.46912	−0.230015
$792$	0	0
$793$	−19.0761	−0.677413
$794$	0	0
$795$	13.4386	0.476616
$796$	0	0
$797$	50.8157	1.79999	0.899993	−	0.435905i	$-0.143572\pi$
$797$	50.8157	1.79999	0.899993	+	0.435905i	$0.143572\pi$
$798$	0	0
$799$	39.8280	1.40901
$800$	0	0
$801$	−49.2180	−1.73903
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−1.74407	−0.0614705
$806$	0	0
$807$	−44.9182	−1.58120
$808$	0	0
$809$	−15.0597	−0.529471	−0.264735	−	0.964321i	$-0.585285\pi$
$809$	−15.0597	−0.529471	−0.264735	+	0.964321i	$0.585285\pi$
$810$	0	0
$811$	16.5147	0.579908	0.289954	−	0.957041i	$-0.406360\pi$
$811$	16.5147	0.579908	0.289954	+	0.957041i	$0.406360\pi$
$812$	0	0
$813$	34.5412	1.21141
$814$	0	0
$815$	−9.31231	−0.326196
$816$	0	0
$817$	10.5252	0.368231
$818$	0	0
$819$	−6.29779	−0.220063
$820$	0	0
$821$	−5.89178	−0.205624	−0.102812	−	0.994701i	$-0.532784\pi$
$821$	−5.89178	−0.205624	−0.102812	+	0.994701i	$0.532784\pi$
$822$	0	0
$823$	−29.5668	−1.03063	−0.515316	−	0.857000i	$-0.672326\pi$
$823$	−29.5668	−1.03063	−0.515316	+	0.857000i	$0.672326\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0.837957	0.0291386	0.0145693	−	0.999894i	$-0.495362\pi$
$827$	0.837957	0.0291386	0.0145693	+	0.999894i	$0.495362\pi$
$828$	0	0
$829$	−53.3043	−1.85134	−0.925668	−	0.378337i	$-0.876496\pi$
$829$	−53.3043	−1.85134	−0.925668	+	0.378337i	$0.876496\pi$
$830$	0	0
$831$	64.9109	2.25173
$832$	0	0
$833$	−26.2271	−0.908715
$834$	0	0
$835$	−0.987607	−0.0341775
$836$	0	0
$837$	0.0212247	0.000733635 0
$838$	0	0
$839$	−21.3985	−0.738758	−0.369379	−	0.929279i	$-0.620430\pi$
$839$	−21.3985	−0.738758	−0.369379	+	0.929279i	$0.620430\pi$
$840$	0	0
$841$	66.5739	2.29565
$842$	0	0
$843$	45.0876	1.55290
$844$	0	0
$845$	−0.147213	−0.00506429
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−72.3678	−2.48366
$850$	0	0
$851$	−20.1023	−0.689099
$852$	0	0
$853$	5.46110	0.186984	0.0934922	−	0.995620i	$-0.470197\pi$
$853$	5.46110	0.186984	0.0934922	+	0.995620i	$0.470197\pi$
$854$	0	0
$855$	−1.56896	−0.0536573
$856$	0	0
$857$	−18.5240	−0.632767	−0.316384	−	0.948631i	$-0.602469\pi$
$857$	−18.5240	−0.632767	−0.316384	+	0.948631i	$0.602469\pi$
$858$	0	0
$859$	5.03468	0.171781	0.0858905	−	0.996305i	$-0.472626\pi$
$859$	5.03468	0.171781	0.0858905	+	0.996305i	$0.472626\pi$
$860$	0	0
$861$	−16.1990	−0.552059
$862$	0	0
$863$	57.9672	1.97323	0.986613	−	0.163079i	$-0.0521426\pi$
$863$	57.9672	1.97323	0.986613	+	0.163079i	$0.0521426\pi$
$864$	0	0
$865$	−2.05003	−0.0697031
$866$	0	0
$867$	3.61894	0.122906
$868$	0	0
$869$	0	0
$870$	0	0
$871$	8.50518	0.288187
$872$	0	0
$873$	40.3612	1.36602
$874$	0	0
$875$	−2.96905	−0.100372
$876$	0	0
$877$	−30.9803	−1.04613	−0.523065	−	0.852293i	$-0.675211\pi$
$877$	−30.9803	−1.04613	−0.523065	+	0.852293i	$0.675211\pi$
$878$	0	0
$879$	−0.0776779	−0.00262001
$880$	0	0
$881$	−32.5662	−1.09718	−0.548592	−	0.836090i	$-0.684836\pi$
$881$	−32.5662	−1.09718	−0.548592	+	0.836090i	$0.684836\pi$
$882$	0	0
$883$	22.0122	0.740769	0.370385	−	0.928879i	$-0.379226\pi$
$883$	22.0122	0.740769	0.370385	+	0.928879i	$0.379226\pi$
$884$	0	0
$885$	−10.8254	−0.363892
$886$	0	0
$887$	13.4632	0.452048	0.226024	−	0.974122i	$-0.427427\pi$
$887$	13.4632	0.452048	0.226024	+	0.974122i	$0.427427\pi$
$888$	0	0
$889$	3.34935	0.112334
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−10.1084	−0.338266
$894$	0	0
$895$	7.10310	0.237430
$896$	0	0
$897$	−49.9757	−1.66864
$898$	0	0
$899$	6.42233	0.214197
$900$	0	0
$901$	−41.4684	−1.38151
$902$	0	0
$903$	15.1269	0.503392
$904$	0	0
$905$	5.66611	0.188348
$906$	0	0
$907$	−7.52753	−0.249948	−0.124974	−	0.992160i	$-0.539885\pi$
$907$	−7.52753	−0.249948	−0.124974	+	0.992160i	$0.539885\pi$
$908$	0	0
$909$	−55.7185	−1.84807
$910$	0	0
$911$	32.7854	1.08623	0.543115	−	0.839658i	$-0.317245\pi$
$911$	32.7854	1.08623	0.543115	+	0.839658i	$0.317245\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−6.83018	−0.225799
$916$	0	0
$917$	−0.383942	−0.0126789
$918$	0	0
$919$	−57.4814	−1.89614	−0.948068	−	0.318069i	$-0.896966\pi$
$919$	−57.4814	−1.89614	−0.948068	+	0.318069i	$0.896966\pi$
$920$	0	0
$921$	−52.1685	−1.71901
$922$	0	0
$923$	−37.4101	−1.23137
$924$	0	0
$925$	−16.6339	−0.546918
$926$	0	0
$927$	48.6159	1.59676
$928$	0	0
$929$	7.88261	0.258620	0.129310	−	0.991604i	$-0.458724\pi$
$929$	7.88261	0.258620	0.129310	+	0.991604i	$0.458724\pi$
$930$	0	0
$931$	6.65649	0.218158
$932$	0	0
$933$	4.69953	0.153856
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16.3210	0.533184	0.266592	−	0.963809i	$-0.414102\pi$
$937$	16.3210	0.533184	0.266592	+	0.963809i	$0.414102\pi$
$938$	0	0
$939$	−17.3799	−0.567172
$940$	0	0
$941$	25.5088	0.831563	0.415782	−	0.909465i	$-0.363508\pi$
$941$	25.5088	0.831563	0.415782	+	0.909465i	$0.363508\pi$
$942$	0	0
$943$	−64.4137	−2.09760
$944$	0	0
$945$	−0.00985993	−0.000320744 0
$946$	0	0
$947$	58.7890	1.91039	0.955193	−	0.295985i	$-0.0956478\pi$
$947$	58.7890	1.91039	0.955193	+	0.295985i	$0.0956478\pi$
$948$	0	0
$949$	−15.9239	−0.516912
$950$	0	0
$951$	−69.7052	−2.26035
$952$	0	0
$953$	−18.9467	−0.613744	−0.306872	−	0.951751i	$-0.599282\pi$
$953$	−18.9467	−0.613744	−0.306872	+	0.951751i	$0.599282\pi$
$954$	0	0
$955$	5.57877	0.180525
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−0.715223	−0.0230958
$960$	0	0
$961$	−30.5684	−0.986079
$962$	0	0
$963$	−43.2770	−1.39458
$964$	0	0
$965$	9.74651	0.313751
$966$	0	0
$967$	48.7523	1.56777	0.783884	−	0.620908i	$-0.213236\pi$
$967$	48.7523	1.56777	0.783884	+	0.620908i	$0.213236\pi$
$968$	0	0
$969$	9.66177	0.310381
$970$	0	0
$971$	15.3032	0.491102	0.245551	−	0.969384i	$-0.421031\pi$
$971$	15.3032	0.491102	0.245551	+	0.969384i	$0.421031\pi$
$972$	0	0
$973$	8.27948	0.265428
$974$	0	0
$975$	−41.3530	−1.32435
$976$	0	0
$977$	−28.3023	−0.905472	−0.452736	−	0.891645i	$-0.649552\pi$
$977$	−28.3023	−0.905472	−0.452736	+	0.891645i	$0.649552\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	5.81788	0.185751
$982$	0	0
$983$	−0.619163	−0.0197482	−0.00987411	−	0.999951i	$-0.503143\pi$
$983$	−0.619163	−0.0197482	−0.00987411	+	0.999951i	$0.503143\pi$
$984$	0	0
$985$	13.5134	0.430571
$986$	0	0
$987$	−14.5279	−0.462429
$988$	0	0
$989$	60.1508	1.91268
$990$	0	0
$991$	41.1874	1.30836	0.654180	−	0.756339i	$-0.273014\pi$
$991$	41.1874	1.30836	0.654180	+	0.756339i	$0.273014\pi$
$992$	0	0
$993$	−31.4833	−0.999093
$994$	0	0
$995$	−3.13403	−0.0993554
$996$	0	0
$997$	−11.0829	−0.351000	−0.175500	−	0.984479i	$-0.556154\pi$
$997$	−11.0829	−0.351000	−0.175500	+	0.984479i	$0.556154\pi$
$998$	0	0
$999$	−0.113646	−0.00359561

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.x.1.3		20
11.5	even	5		836.2.j.c.685.2	yes	40
11.9	even	5		836.2.j.c.609.2	✓	40
11.10	odd	2		9196.2.a.w.1.3		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.j.c.609.2	✓	40	11.9	even	5
836.2.j.c.685.2	yes	40	11.5	even	5
9196.2.a.w.1.3		20	11.10	odd	2
9196.2.a.x.1.3		20	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:	\( N \)	\(=\)	\( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9196.a (trivial)

\(f(q)\)	\(=\)	\(q-2.45218 q^{3} +0.520700 q^{5} +0.586094 q^{7} +3.01318 q^{9} -3.56613 q^{13} -1.27685 q^{15} +3.94008 q^{17} -1.00000 q^{19} -1.43721 q^{21} -5.71492 q^{23} -4.72887 q^{25} -0.0323087 q^{27} -9.77619 q^{29} -0.656936 q^{31} +0.305179 q^{35} +3.51752 q^{37} +8.74478 q^{39} +11.2712 q^{41} -10.5252 q^{43} +1.56896 q^{45} +10.1084 q^{47} -6.65649 q^{49} -9.66177 q^{51} -10.5248 q^{53} +2.45218 q^{57} +8.47821 q^{59} +5.34925 q^{61} +1.76600 q^{63} -1.85688 q^{65} -2.38499 q^{67} +14.0140 q^{69} +10.4904 q^{71} +4.46532 q^{73} +11.5960 q^{75} +6.22693 q^{79} -8.96030 q^{81} -0.430250 q^{83} +2.05160 q^{85} +23.9730 q^{87} -16.3343 q^{89} -2.09009 q^{91} +1.61092 q^{93} -0.520700 q^{95} +13.3949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{3} + 2 q^{5} + 4 q^{7} + 28 q^{9} - 5 q^{13} + 14 q^{15} - 6 q^{17} - 20 q^{19} - q^{21} + 18 q^{23} + 44 q^{25} + 24 q^{27} + q^{29} + 23 q^{31} - 36 q^{35} + q^{37} + 21 q^{39} + 6 q^{41}+ \cdots + 9 q^{97}+O(q^{100}) \) 20 * q + 6 * q^3 + 2 * q^5 + 4 * q^7 + 28 * q^9 - 5 * q^13 + 14 * q^15 - 6 * q^17 - 20 * q^19 - q^21 + 18 * q^23 + 44 * q^25 + 24 * q^27 + q^29 + 23 * q^31 - 36 * q^35 + q^37 + 21 * q^39 + 6 * q^41 - 8 * q^43 - 15 * q^45 + 34 * q^47 + 42 * q^49 + 14 * q^51 - 10 * q^53 - 6 * q^57 + 52 * q^59 - 2 * q^61 + 59 * q^63 + 25 * q^65 + 14 * q^67 + 15 * q^69 + 88 * q^71 - 13 * q^73 + 15 * q^75 - 2 * q^79 + 48 * q^81 - 2 * q^83 + 11 * q^85 + 37 * q^87 + 46 * q^89 - 4 * q^91 + 2 * q^93 - 2 * q^95 + 9 * q^97

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