LMFDB - Embedded newform 9196.2.a.u.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9196,2,Mod(1,9196)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [14,0,-2,0,-6,0,-4,0,12,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$73.4304296988$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 25 x^{12} + 52 x^{11} + 222 x^{10} - 492 x^{9} - 800 x^{8} + 1984 x^{7} + 854 x^{6} + \cdots + 13 $ x^14 - 2x^13 - 25x^12 + 52x^11 + 222x^10 - 492x^9 - 800x^8 + 1984x^7 + 854x^6 - 2946x^5 + 300x^4 + 878x^3 - 194x^2 - 52*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.280760$ 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)

$f(q)$	$=$	$q+0.280760 q^{3} +1.39648 q^{5} +1.54858 q^{7} -2.92117 q^{9} -1.91235 q^{13} +0.392077 q^{15} +0.813559 q^{17} +1.00000 q^{19} +0.434779 q^{21} +2.02620 q^{23} -3.04983 q^{25} -1.66243 q^{27} -8.17401 q^{29} +9.56752 q^{31} +2.16257 q^{35} -10.7516 q^{37} -0.536912 q^{39} +1.31964 q^{41} -1.86077 q^{43} -4.07937 q^{45} +3.67963 q^{47} -4.60190 q^{49} +0.228415 q^{51} +8.16070 q^{53} +0.280760 q^{57} -6.79179 q^{59} -0.315351 q^{61} -4.52367 q^{63} -2.67057 q^{65} +3.73653 q^{67} +0.568876 q^{69} -13.6592 q^{71} -15.4134 q^{73} -0.856270 q^{75} +5.29213 q^{79} +8.29678 q^{81} +8.84752 q^{83} +1.13612 q^{85} -2.29493 q^{87} +7.43563 q^{89} -2.96143 q^{91} +2.68617 q^{93} +1.39648 q^{95} -5.01854 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 2 q^{3} - 6 q^{5} - 4 q^{7} + 12 q^{9} - 4 q^{13} + 2 q^{15} - 8 q^{17} + 14 q^{19} - 18 q^{21} - 2 q^{23} + 8 q^{25} + 10 q^{27} + 4 q^{29} - 2 q^{31} - 24 q^{35} - 10 q^{37} - 24 q^{39} - 4 q^{41}+ \cdots - 20 q^{97}+O(q^{100}) $ 14 * q - 2 * q^3 - 6 * q^5 - 4 * q^7 + 12 * q^9 - 4 * q^13 + 2 * q^15 - 8 * q^17 + 14 * q^19 - 18 * q^21 - 2 * q^23 + 8 * q^25 + 10 * q^27 + 4 * q^29 - 2 * q^31 - 24 * q^35 - 10 * q^37 - 24 * q^39 - 4 * q^41 - 32 * q^43 - 20 * q^45 - 4 * q^47 - 14 * q^49 - 22 * q^51 + 6 * q^53 - 2 * q^57 - 6 * q^59 + 4 * q^61 + 16 * q^63 - 54 * q^65 + 6 * q^67 + 4 * q^69 - 6 * q^71 + 24 * q^73 - 10 * q^75 - 48 * q^79 + 2 * q^81 + 40 * q^85 - 28 * q^87 + 2 * q^89 + 28 * q^91 + 34 * q^93 - 6 * q^95 - 20 * 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
$3$	0.280760	0.162097	0.0810484	−	0.996710i	$-0.474173\pi$
$3$	0.280760	0.162097	0.0810484	+	0.996710i	$0.474173\pi$
$4$	0	0
$5$	1.39648	0.624527	0.312263	−	0.949996i	$-0.398913\pi$
$5$	1.39648	0.624527	0.312263	+	0.949996i	$0.398913\pi$
$6$	0	0
$7$	1.54858	0.585308	0.292654	−	0.956218i	$-0.405462\pi$
$7$	1.54858	0.585308	0.292654	+	0.956218i	$0.405462\pi$
$8$	0	0
$9$	−2.92117	−0.973725
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.91235	−0.530391	−0.265196	−	0.964195i	$-0.585437\pi$
$13$	−1.91235	−0.530391	−0.265196	+	0.964195i	$0.585437\pi$
$14$	0	0
$15$	0.392077	0.101234
$16$	0	0
$17$	0.813559	0.197317	0.0986585	−	0.995121i	$-0.468545\pi$
$17$	0.813559	0.197317	0.0986585	+	0.995121i	$0.468545\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.434779	0.0948765
$22$	0	0
$23$	2.02620	0.422492	0.211246	−	0.977433i	$-0.432248\pi$
$23$	2.02620	0.422492	0.211246	+	0.977433i	$0.432248\pi$
$24$	0	0
$25$	−3.04983	−0.609966
$26$	0	0
$27$	−1.66243	−0.319934
$28$	0	0
$29$	−8.17401	−1.51788	−0.758938	−	0.651163i	$-0.774281\pi$
$29$	−8.17401	−1.51788	−0.758938	+	0.651163i	$0.774281\pi$
$30$	0	0
$31$	9.56752	1.71838	0.859189	−	0.511659i	$-0.170969\pi$
$31$	9.56752	1.71838	0.859189	+	0.511659i	$0.170969\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.16257	0.365540
$36$	0	0
$37$	−10.7516	−1.76755	−0.883777	−	0.467908i	$-0.845008\pi$
$37$	−10.7516	−1.76755	−0.883777	+	0.467908i	$0.845008\pi$
$38$	0	0
$39$	−0.536912	−0.0859747
$40$	0	0
$41$	1.31964	0.206094	0.103047	−	0.994677i	$-0.467141\pi$
$41$	1.31964	0.206094	0.103047	+	0.994677i	$0.467141\pi$
$42$	0	0
$43$	−1.86077	−0.283765	−0.141883	−	0.989883i	$-0.545316\pi$
$43$	−1.86077	−0.283765	−0.141883	+	0.989883i	$0.545316\pi$
$44$	0	0
$45$	−4.07937	−0.608117
$46$	0	0
$47$	3.67963	0.536730	0.268365	−	0.963317i	$-0.413517\pi$
$47$	3.67963	0.536730	0.268365	+	0.963317i	$0.413517\pi$
$48$	0	0
$49$	−4.60190	−0.657415
$50$	0	0
$51$	0.228415	0.0319844
$52$	0	0
$53$	8.16070	1.12096	0.560479	−	0.828168i	$-0.310617\pi$
$53$	8.16070	1.12096	0.560479	+	0.828168i	$0.310617\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.280760	0.0371875
$58$	0	0
$59$	−6.79179	−0.884216	−0.442108	−	0.896962i	$-0.645769\pi$
$59$	−6.79179	−0.884216	−0.442108	+	0.896962i	$0.645769\pi$
$60$	0	0
$61$	−0.315351	−0.0403765	−0.0201883	−	0.999796i	$-0.506427\pi$
$61$	−0.315351	−0.0403765	−0.0201883	+	0.999796i	$0.506427\pi$
$62$	0	0
$63$	−4.52367	−0.569929
$64$	0	0
$65$	−2.67057	−0.331244
$66$	0	0
$67$	3.73653	0.456490	0.228245	−	0.973604i	$-0.426701\pi$
$67$	3.73653	0.456490	0.228245	+	0.973604i	$0.426701\pi$
$68$	0	0
$69$	0.568876	0.0684846
$70$	0	0
$71$	−13.6592	−1.62105	−0.810525	−	0.585705i	$-0.800818\pi$
$71$	−13.6592	−1.62105	−0.810525	+	0.585705i	$0.800818\pi$
$72$	0	0
$73$	−15.4134	−1.80400	−0.902002	−	0.431731i	$-0.857903\pi$
$73$	−15.4134	−1.80400	−0.902002	+	0.431731i	$0.857903\pi$
$74$	0	0
$75$	−0.856270	−0.0988736
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.29213	0.595411	0.297706	−	0.954658i	$-0.403779\pi$
$79$	5.29213	0.595411	0.297706	+	0.954658i	$0.403779\pi$
$80$	0	0
$81$	8.29678	0.921864
$82$	0	0
$83$	8.84752	0.971141	0.485571	−	0.874197i	$-0.338612\pi$
$83$	8.84752	0.971141	0.485571	+	0.874197i	$0.338612\pi$
$84$	0	0
$85$	1.13612	0.123230
$86$	0	0
$87$	−2.29493	−0.246043
$88$	0	0
$89$	7.43563	0.788175	0.394088	−	0.919073i	$-0.371061\pi$
$89$	7.43563	0.788175	0.394088	+	0.919073i	$0.371061\pi$
$90$	0	0
$91$	−2.96143	−0.310442
$92$	0	0
$93$	2.68617	0.278543
$94$	0	0
$95$	1.39648	0.143276
$96$	0	0
$97$	−5.01854	−0.509555	−0.254778	−	0.967000i	$-0.582002\pi$
$97$	−5.01854	−0.509555	−0.254778	+	0.967000i	$0.582002\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−19.5446	−1.94476	−0.972381	−	0.233399i	$-0.925015\pi$
$101$	−19.5446	−1.94476	−0.972381	+	0.233399i	$0.925015\pi$
$102$	0	0
$103$	8.70528	0.857757	0.428878	−	0.903362i	$-0.358909\pi$
$103$	8.70528	0.857757	0.428878	+	0.903362i	$0.358909\pi$
$104$	0	0
$105$	0.607161	0.0592529
$106$	0	0
$107$	11.5632	1.11786	0.558929	−	0.829216i	$-0.311213\pi$
$107$	11.5632	1.11786	0.558929	+	0.829216i	$0.311213\pi$
$108$	0	0
$109$	−3.97367	−0.380609	−0.190304	−	0.981725i	$-0.560947\pi$
$109$	−3.97367	−0.380609	−0.190304	+	0.981725i	$0.560947\pi$
$110$	0	0
$111$	−3.01862	−0.286515
$112$	0	0
$113$	−14.1577	−1.33185	−0.665925	−	0.746019i	$-0.731963\pi$
$113$	−14.1577	−1.33185	−0.665925	+	0.746019i	$0.731963\pi$
$114$	0	0
$115$	2.82956	0.263858
$116$	0	0
$117$	5.58632	0.516455
$118$	0	0
$119$	1.25986	0.115491
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0.370503	0.0334071
$124$	0	0
$125$	−11.2415	−1.00547
$126$	0	0
$127$	−17.1116	−1.51841	−0.759204	−	0.650853i	$-0.774411\pi$
$127$	−17.1116	−1.51841	−0.759204	+	0.650853i	$0.774411\pi$
$128$	0	0
$129$	−0.522430	−0.0459974
$130$	0	0
$131$	−6.30045	−0.550473	−0.275236	−	0.961377i	$-0.588756\pi$
$131$	−6.30045	−0.550473	−0.275236	+	0.961377i	$0.588756\pi$
$132$	0	0
$133$	1.54858	0.134279
$134$	0	0
$135$	−2.32155	−0.199808
$136$	0	0
$137$	1.63865	0.139999	0.0699996	−	0.997547i	$-0.477700\pi$
$137$	1.63865	0.139999	0.0699996	+	0.997547i	$0.477700\pi$
$138$	0	0
$139$	−8.26174	−0.700751	−0.350376	−	0.936609i	$-0.613946\pi$
$139$	−8.26174	−0.700751	−0.350376	+	0.936609i	$0.613946\pi$
$140$	0	0
$141$	1.03309	0.0870022
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−11.4149	−0.947953
$146$	0	0
$147$	−1.29203	−0.106565
$148$	0	0
$149$	16.3438	1.33894	0.669470	−	0.742839i	$-0.266521\pi$
$149$	16.3438	1.33894	0.669470	+	0.742839i	$0.266521\pi$
$150$	0	0
$151$	−8.89783	−0.724095	−0.362047	−	0.932160i	$-0.617922\pi$
$151$	−8.89783	−0.724095	−0.362047	+	0.932160i	$0.617922\pi$
$152$	0	0
$153$	−2.37655	−0.192132
$154$	0	0
$155$	13.3609	1.07317
$156$	0	0
$157$	16.1817	1.29144	0.645719	−	0.763575i	$-0.276558\pi$
$157$	16.1817	1.29144	0.645719	+	0.763575i	$0.276558\pi$
$158$	0	0
$159$	2.29120	0.181704
$160$	0	0
$161$	3.13773	0.247288
$162$	0	0
$163$	−15.3690	−1.20379	−0.601896	−	0.798575i	$-0.705588\pi$
$163$	−15.3690	−1.20379	−0.601896	+	0.798575i	$0.705588\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.2424	−1.64379	−0.821894	−	0.569640i	$-0.807083\pi$
$167$	−21.2424	−1.64379	−0.821894	+	0.569640i	$0.807083\pi$
$168$	0	0
$169$	−9.34290	−0.718685
$170$	0	0
$171$	−2.92117	−0.223388
$172$	0	0
$173$	−10.8900	−0.827949	−0.413974	−	0.910289i	$-0.635860\pi$
$173$	−10.8900	−0.827949	−0.413974	+	0.910289i	$0.635860\pi$
$174$	0	0
$175$	−4.72291	−0.357018
$176$	0	0
$177$	−1.90686	−0.143329
$178$	0	0
$179$	21.9372	1.63966	0.819832	−	0.572604i	$-0.194067\pi$
$179$	21.9372	1.63966	0.819832	+	0.572604i	$0.194067\pi$
$180$	0	0
$181$	16.1819	1.20279	0.601394	−	0.798952i	$-0.294612\pi$
$181$	16.1819	1.20279	0.601394	+	0.798952i	$0.294612\pi$
$182$	0	0
$183$	−0.0885378	−0.00654490
$184$	0	0
$185$	−15.0145	−1.10388
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2.57440	−0.187260
$190$	0	0
$191$	−6.36343	−0.460442	−0.230221	−	0.973138i	$-0.573945\pi$
$191$	−6.36343	−0.460442	−0.230221	+	0.973138i	$0.573945\pi$
$192$	0	0
$193$	−6.13627	−0.441698	−0.220849	−	0.975308i	$-0.570883\pi$
$193$	−6.13627	−0.441698	−0.220849	+	0.975308i	$0.570883\pi$
$194$	0	0
$195$	−0.749789	−0.0536935
$196$	0	0
$197$	−17.3529	−1.23634	−0.618172	−	0.786043i	$-0.712126\pi$
$197$	−17.3529	−1.23634	−0.618172	+	0.786043i	$0.712126\pi$
$198$	0	0
$199$	10.0901	0.715271	0.357636	−	0.933861i	$-0.383583\pi$
$199$	10.0901	0.715271	0.357636	+	0.933861i	$0.383583\pi$
$200$	0	0
$201$	1.04907	0.0739955
$202$	0	0
$203$	−12.6581	−0.888424
$204$	0	0
$205$	1.84286	0.128711
$206$	0	0
$207$	−5.91889	−0.411391
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−13.7559	−0.946995	−0.473497	−	0.880795i	$-0.657009\pi$
$211$	−13.7559	−0.946995	−0.473497	+	0.880795i	$0.657009\pi$
$212$	0	0
$213$	−3.83496	−0.262767
$214$	0	0
$215$	−2.59854	−0.177219
$216$	0	0
$217$	14.8161	1.00578
$218$	0	0
$219$	−4.32747	−0.292423
$220$	0	0
$221$	−1.55581	−0.104655
$222$	0	0
$223$	17.3651	1.16285	0.581425	−	0.813600i	$-0.302495\pi$
$223$	17.3651	1.16285	0.581425	+	0.813600i	$0.302495\pi$
$224$	0	0
$225$	8.90909	0.593939
$226$	0	0
$227$	9.42256	0.625398	0.312699	−	0.949852i	$-0.398767\pi$
$227$	9.42256	0.625398	0.312699	+	0.949852i	$0.398767\pi$
$228$	0	0
$229$	−1.12627	−0.0744262	−0.0372131	−	0.999307i	$-0.511848\pi$
$229$	−1.12627	−0.0744262	−0.0372131	+	0.999307i	$0.511848\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.06620	−0.462922	−0.231461	−	0.972844i	$-0.574351\pi$
$233$	−7.06620	−0.462922	−0.231461	+	0.972844i	$0.574351\pi$
$234$	0	0
$235$	5.13855	0.335202
$236$	0	0
$237$	1.48582	0.0965143
$238$	0	0
$239$	−5.99198	−0.387589	−0.193794	−	0.981042i	$-0.562079\pi$
$239$	−5.99198	−0.387589	−0.193794	+	0.981042i	$0.562079\pi$
$240$	0	0
$241$	8.55077	0.550803	0.275402	−	0.961329i	$-0.411189\pi$
$241$	8.55077	0.550803	0.275402	+	0.961329i	$0.411189\pi$
$242$	0	0
$243$	7.31668	0.469366
$244$	0	0
$245$	−6.42649	−0.410573
$246$	0	0
$247$	−1.91235	−0.121680
$248$	0	0
$249$	2.48403	0.157419
$250$	0	0
$251$	16.1775	1.02111	0.510557	−	0.859844i	$-0.329439\pi$
$251$	16.1775	1.02111	0.510557	+	0.859844i	$0.329439\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0.318977	0.0199751
$256$	0	0
$257$	−7.89892	−0.492721	−0.246361	−	0.969178i	$-0.579235\pi$
$257$	−7.89892	−0.492721	−0.246361	+	0.969178i	$0.579235\pi$
$258$	0	0
$259$	−16.6497	−1.03456
$260$	0	0
$261$	23.8777	1.47799
$262$	0	0
$263$	−27.1918	−1.67672	−0.838359	−	0.545119i	$-0.816485\pi$
$263$	−27.1918	−1.67672	−0.838359	+	0.545119i	$0.816485\pi$
$264$	0	0
$265$	11.3963	0.700069
$266$	0	0
$267$	2.08763	0.127761
$268$	0	0
$269$	−13.4995	−0.823077	−0.411539	−	0.911392i	$-0.635008\pi$
$269$	−13.4995	−0.823077	−0.411539	+	0.911392i	$0.635008\pi$
$270$	0	0
$271$	−30.0892	−1.82779	−0.913895	−	0.405952i	$-0.866940\pi$
$271$	−30.0892	−1.82779	−0.913895	+	0.405952i	$0.866940\pi$
$272$	0	0
$273$	−0.831450	−0.0503217
$274$	0	0
$275$	0	0
$276$	0	0
$277$	22.5348	1.35399	0.676993	−	0.735990i	$-0.263283\pi$
$277$	22.5348	1.35399	0.676993	+	0.735990i	$0.263283\pi$
$278$	0	0
$279$	−27.9484	−1.67323
$280$	0	0
$281$	−8.55599	−0.510408	−0.255204	−	0.966887i	$-0.582143\pi$
$281$	−8.55599	−0.510408	−0.255204	+	0.966887i	$0.582143\pi$
$282$	0	0
$283$	−24.5265	−1.45795	−0.728974	−	0.684541i	$-0.760002\pi$
$283$	−24.5265	−1.45795	−0.728974	+	0.684541i	$0.760002\pi$
$284$	0	0
$285$	0.392077	0.0232246
$286$	0	0
$287$	2.04357	0.120628
$288$	0	0
$289$	−16.3381	−0.961066
$290$	0	0
$291$	−1.40900	−0.0825973
$292$	0	0
$293$	−13.0556	−0.762719	−0.381359	−	0.924427i	$-0.624544\pi$
$293$	−13.0556	−0.762719	−0.381359	+	0.924427i	$0.624544\pi$
$294$	0	0
$295$	−9.48463	−0.552216
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.87482	−0.224086
$300$	0	0
$301$	−2.88155	−0.166090
$302$	0	0
$303$	−5.48734	−0.315240
$304$	0	0
$305$	−0.440382	−0.0252162
$306$	0	0
$307$	−7.02305	−0.400827	−0.200413	−	0.979711i	$-0.564229\pi$
$307$	−7.02305	−0.400827	−0.200413	+	0.979711i	$0.564229\pi$
$308$	0	0
$309$	2.44409	0.139040
$310$	0	0
$311$	13.8113	0.783169	0.391585	−	0.920142i	$-0.371927\pi$
$311$	13.8113	0.783169	0.391585	+	0.920142i	$0.371927\pi$
$312$	0	0
$313$	−19.8394	−1.12139	−0.560696	−	0.828022i	$-0.689466\pi$
$313$	−19.8394	−1.12139	−0.560696	+	0.828022i	$0.689466\pi$
$314$	0	0
$315$	−6.31723	−0.355936
$316$	0	0
$317$	−3.38250	−0.189980	−0.0949901	−	0.995478i	$-0.530282\pi$
$317$	−3.38250	−0.189980	−0.0949901	+	0.995478i	$0.530282\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	3.24648	0.181201
$322$	0	0
$323$	0.813559	0.0452676
$324$	0	0
$325$	5.83236	0.323521
$326$	0	0
$327$	−1.11565	−0.0616954
$328$	0	0
$329$	5.69820	0.314152
$330$	0	0
$331$	17.8084	0.978836	0.489418	−	0.872049i	$-0.337209\pi$
$331$	17.8084	0.978836	0.489418	+	0.872049i	$0.337209\pi$
$332$	0	0
$333$	31.4073	1.72111
$334$	0	0
$335$	5.21800	0.285090
$336$	0	0
$337$	28.1729	1.53468	0.767338	−	0.641243i	$-0.221581\pi$
$337$	28.1729	1.53468	0.767338	+	0.641243i	$0.221581\pi$
$338$	0	0
$339$	−3.97493	−0.215888
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−17.9665	−0.970098
$344$	0	0
$345$	0.794427	0.0427705
$346$	0	0
$347$	26.3141	1.41261	0.706307	−	0.707905i	$-0.250360\pi$
$347$	26.3141	1.41261	0.706307	+	0.707905i	$0.250360\pi$
$348$	0	0
$349$	5.26265	0.281703	0.140852	−	0.990031i	$-0.455016\pi$
$349$	5.26265	0.281703	0.140852	+	0.990031i	$0.455016\pi$
$350$	0	0
$351$	3.17915	0.169690
$352$	0	0
$353$	14.7064	0.782742	0.391371	−	0.920233i	$-0.372001\pi$
$353$	14.7064	0.782742	0.391371	+	0.920233i	$0.372001\pi$
$354$	0	0
$355$	−19.0749	−1.01239
$356$	0	0
$357$	0.353718	0.0187207
$358$	0	0
$359$	13.4174	0.708143	0.354072	−	0.935218i	$-0.384797\pi$
$359$	13.4174	0.708143	0.354072	+	0.935218i	$0.384797\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−21.5246	−1.12665
$366$	0	0
$367$	−4.65868	−0.243181	−0.121591	−	0.992580i	$-0.538799\pi$
$367$	−4.65868	−0.243181	−0.121591	+	0.992580i	$0.538799\pi$
$368$	0	0
$369$	−3.85491	−0.200678
$370$	0	0
$371$	12.6375	0.656106
$372$	0	0
$373$	−6.59506	−0.341479	−0.170740	−	0.985316i	$-0.554616\pi$
$373$	−6.59506	−0.341479	−0.170740	+	0.985316i	$0.554616\pi$
$374$	0	0
$375$	−3.15615	−0.162983
$376$	0	0
$377$	15.6316	0.805068
$378$	0	0
$379$	−23.5367	−1.20900	−0.604501	−	0.796605i	$-0.706627\pi$
$379$	−23.5367	−1.20900	−0.604501	+	0.796605i	$0.706627\pi$
$380$	0	0
$381$	−4.80424	−0.246129
$382$	0	0
$383$	−21.7688	−1.11234	−0.556168	−	0.831070i	$-0.687729\pi$
$383$	−21.7688	−1.11234	−0.556168	+	0.831070i	$0.687729\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.43564	0.276309
$388$	0	0
$389$	9.18188	0.465540	0.232770	−	0.972532i	$-0.425221\pi$
$389$	9.18188	0.465540	0.232770	+	0.972532i	$0.425221\pi$
$390$	0	0
$391$	1.64844	0.0833650
$392$	0	0
$393$	−1.76891	−0.0892298
$394$	0	0
$395$	7.39038	0.371850
$396$	0	0
$397$	26.2997	1.31994	0.659972	−	0.751290i	$-0.270568\pi$
$397$	26.2997	1.31994	0.659972	+	0.751290i	$0.270568\pi$
$398$	0	0
$399$	0.434779	0.0217662
$400$	0	0
$401$	29.8018	1.48823	0.744115	−	0.668052i	$-0.232872\pi$
$401$	29.8018	1.48823	0.744115	+	0.668052i	$0.232872\pi$
$402$	0	0
$403$	−18.2965	−0.911413
$404$	0	0
$405$	11.5863	0.575729
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−3.57964	−0.177002	−0.0885009	−	0.996076i	$-0.528208\pi$
$409$	−3.57964	−0.177002	−0.0885009	+	0.996076i	$0.528208\pi$
$410$	0	0
$411$	0.460067	0.0226934
$412$	0	0
$413$	−10.5176	−0.517538
$414$	0	0
$415$	12.3554	0.606504
$416$	0	0
$417$	−2.31956	−0.113590
$418$	0	0
$419$	−2.47979	−0.121146	−0.0605728	−	0.998164i	$-0.519293\pi$
$419$	−2.47979	−0.121146	−0.0605728	+	0.998164i	$0.519293\pi$
$420$	0	0
$421$	29.9072	1.45759	0.728795	−	0.684732i	$-0.240081\pi$
$421$	29.9072	1.45759	0.728795	+	0.684732i	$0.240081\pi$
$422$	0	0
$423$	−10.7489	−0.522627
$424$	0	0
$425$	−2.48122	−0.120357
$426$	0	0
$427$	−0.488345	−0.0236327
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7.05631	0.339890	0.169945	−	0.985454i	$-0.445641\pi$
$431$	7.05631	0.339890	0.169945	+	0.985454i	$0.445641\pi$
$432$	0	0
$433$	−30.9744	−1.48854	−0.744268	−	0.667881i	$-0.767201\pi$
$433$	−30.9744	−1.48854	−0.744268	+	0.667881i	$0.767201\pi$
$434$	0	0
$435$	−3.20484	−0.153660
$436$	0	0
$437$	2.02620	0.0969264
$438$	0	0
$439$	3.40633	0.162575	0.0812876	−	0.996691i	$-0.474097\pi$
$439$	3.40633	0.162575	0.0812876	+	0.996691i	$0.474097\pi$
$440$	0	0
$441$	13.4430	0.640141
$442$	0	0
$443$	−4.06793	−0.193273	−0.0966365	−	0.995320i	$-0.530808\pi$
$443$	−4.06793	−0.193273	−0.0966365	+	0.995320i	$0.530808\pi$
$444$	0	0
$445$	10.3837	0.492236
$446$	0	0
$447$	4.58869	0.217038
$448$	0	0
$449$	−15.8124	−0.746234	−0.373117	−	0.927784i	$-0.621711\pi$
$449$	−15.8124	−0.746234	−0.373117	+	0.927784i	$0.621711\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−2.49815	−0.117373
$454$	0	0
$455$	−4.13559	−0.193879
$456$	0	0
$457$	26.9233	1.25942	0.629710	−	0.776830i	$-0.283174\pi$
$457$	26.9233	1.25942	0.629710	+	0.776830i	$0.283174\pi$
$458$	0	0
$459$	−1.35248	−0.0631285
$460$	0	0
$461$	−30.8729	−1.43789	−0.718947	−	0.695065i	$-0.755375\pi$
$461$	−30.8729	−1.43789	−0.718947	+	0.695065i	$0.755375\pi$
$462$	0	0
$463$	30.4823	1.41663	0.708316	−	0.705895i	$-0.249455\pi$
$463$	30.4823	1.41663	0.708316	+	0.705895i	$0.249455\pi$
$464$	0	0
$465$	3.75120	0.173958
$466$	0	0
$467$	16.6342	0.769738	0.384869	−	0.922971i	$-0.374247\pi$
$467$	16.6342	0.769738	0.384869	+	0.922971i	$0.374247\pi$
$468$	0	0
$469$	5.78631	0.267187
$470$	0	0
$471$	4.54316	0.209338
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−3.04983	−0.139936
$476$	0	0
$477$	−23.8388	−1.09151
$478$	0	0
$479$	−28.6404	−1.30861	−0.654306	−	0.756230i	$-0.727039\pi$
$479$	−28.6404	−1.30861	−0.654306	+	0.756230i	$0.727039\pi$
$480$	0	0
$481$	20.5609	0.937496
$482$	0	0
$483$	0.880950	0.0400846
$484$	0	0
$485$	−7.00831	−0.318231
$486$	0	0
$487$	28.1115	1.27385	0.636926	−	0.770925i	$-0.280206\pi$
$487$	28.1115	1.27385	0.636926	+	0.770925i	$0.280206\pi$
$488$	0	0
$489$	−4.31499	−0.195131
$490$	0	0
$491$	−13.7752	−0.621667	−0.310833	−	0.950464i	$-0.600608\pi$
$491$	−13.7752	−0.621667	−0.310833	+	0.950464i	$0.600608\pi$
$492$	0	0
$493$	−6.65004	−0.299503
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−21.1524	−0.948813
$498$	0	0
$499$	18.9349	0.847644	0.423822	−	0.905745i	$-0.360688\pi$
$499$	18.9349	0.847644	0.423822	+	0.905745i	$0.360688\pi$
$500$	0	0
$501$	−5.96402	−0.266453
$502$	0	0
$503$	−17.8005	−0.793686	−0.396843	−	0.917886i	$-0.629894\pi$
$503$	−17.8005	−0.793686	−0.396843	+	0.917886i	$0.629894\pi$
$504$	0	0
$505$	−27.2937	−1.21456
$506$	0	0
$507$	−2.62311	−0.116496
$508$	0	0
$509$	−9.20783	−0.408130	−0.204065	−	0.978957i	$-0.565415\pi$
$509$	−9.20783	−0.408130	−0.204065	+	0.978957i	$0.565415\pi$
$510$	0	0
$511$	−23.8689	−1.05590
$512$	0	0
$513$	−1.66243	−0.0733980
$514$	0	0
$515$	12.1568	0.535692
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−3.05746	−0.134208
$520$	0	0
$521$	−17.5391	−0.768404	−0.384202	−	0.923249i	$-0.625523\pi$
$521$	−17.5391	−0.768404	−0.384202	+	0.923249i	$0.625523\pi$
$522$	0	0
$523$	17.0616	0.746053	0.373027	−	0.927821i	$-0.378320\pi$
$523$	17.0616	0.746053	0.373027	+	0.927821i	$0.378320\pi$
$524$	0	0
$525$	−1.32600	−0.0578715
$526$	0	0
$527$	7.78374	0.339065
$528$	0	0
$529$	−18.8945	−0.821500
$530$	0	0
$531$	19.8400	0.860983
$532$	0	0
$533$	−2.52362	−0.109310
$534$	0	0
$535$	16.1478	0.698132
$536$	0	0
$537$	6.15909	0.265784
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−29.1641	−1.25386	−0.626932	−	0.779074i	$-0.715690\pi$
$541$	−29.1641	−1.25386	−0.626932	+	0.779074i	$0.715690\pi$
$542$	0	0
$543$	4.54322	0.194968
$544$	0	0
$545$	−5.54917	−0.237700
$546$	0	0
$547$	5.77207	0.246796	0.123398	−	0.992357i	$-0.460621\pi$
$547$	5.77207	0.246796	0.123398	+	0.992357i	$0.460621\pi$
$548$	0	0
$549$	0.921194	0.0393156
$550$	0	0
$551$	−8.17401	−0.348224
$552$	0	0
$553$	8.19528	0.348499
$554$	0	0
$555$	−4.21546	−0.178936
$556$	0	0
$557$	−24.3211	−1.03052	−0.515259	−	0.857034i	$-0.672304\pi$
$557$	−24.3211	−1.03052	−0.515259	+	0.857034i	$0.672304\pi$
$558$	0	0
$559$	3.55845	0.150507
$560$	0	0
$561$	0	0
$562$	0	0
$563$	34.1863	1.44078	0.720390	−	0.693569i	$-0.243963\pi$
$563$	34.1863	1.44078	0.720390	+	0.693569i	$0.243963\pi$
$564$	0	0
$565$	−19.7711	−0.831775
$566$	0	0
$567$	12.8482	0.539574
$568$	0	0
$569$	21.2140	0.889339	0.444669	−	0.895695i	$-0.353321\pi$
$569$	21.2140	0.889339	0.444669	+	0.895695i	$0.353321\pi$
$570$	0	0
$571$	0.929659	0.0389050	0.0194525	−	0.999811i	$-0.493808\pi$
$571$	0.929659	0.0389050	0.0194525	+	0.999811i	$0.493808\pi$
$572$	0	0
$573$	−1.78660	−0.0746361
$574$	0	0
$575$	−6.17958	−0.257706
$576$	0	0
$577$	−22.9321	−0.954676	−0.477338	−	0.878720i	$-0.658398\pi$
$577$	−22.9321	−0.954676	−0.477338	+	0.878720i	$0.658398\pi$
$578$	0	0
$579$	−1.72282	−0.0715979
$580$	0	0
$581$	13.7011	0.568417
$582$	0	0
$583$	0	0
$584$	0	0
$585$	7.80120	0.322540
$586$	0	0
$587$	−16.1694	−0.667383	−0.333692	−	0.942682i	$-0.608294\pi$
$587$	−16.1694	−0.667383	−0.333692	+	0.942682i	$0.608294\pi$
$588$	0	0
$589$	9.56752	0.394223
$590$	0	0
$591$	−4.87200	−0.200407
$592$	0	0
$593$	−5.39959	−0.221734	−0.110867	−	0.993835i	$-0.535363\pi$
$593$	−5.39959	−0.221734	−0.110867	+	0.993835i	$0.535363\pi$
$594$	0	0
$595$	1.75937	0.0721273
$596$	0	0
$597$	2.83291	0.115943
$598$	0	0
$599$	−27.7206	−1.13263	−0.566316	−	0.824188i	$-0.691632\pi$
$599$	−27.7206	−1.13263	−0.566316	+	0.824188i	$0.691632\pi$
$600$	0	0
$601$	25.2903	1.03161	0.515807	−	0.856705i	$-0.327492\pi$
$601$	25.2903	1.03161	0.515807	+	0.856705i	$0.327492\pi$
$602$	0	0
$603$	−10.9151	−0.444495
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0.454023	0.0184282	0.00921412	−	0.999958i	$-0.497067\pi$
$607$	0.454023	0.0184282	0.00921412	+	0.999958i	$0.497067\pi$
$608$	0	0
$609$	−3.55388	−0.144011
$610$	0	0
$611$	−7.03676	−0.284677
$612$	0	0
$613$	−9.41112	−0.380111	−0.190056	−	0.981773i	$-0.560867\pi$
$613$	−9.41112	−0.380111	−0.190056	+	0.981773i	$0.560867\pi$
$614$	0	0
$615$	0.517401	0.0208636
$616$	0	0
$617$	−3.03058	−0.122006	−0.0610032	−	0.998138i	$-0.519430\pi$
$617$	−3.03058	−0.122006	−0.0610032	+	0.998138i	$0.519430\pi$
$618$	0	0
$619$	30.5630	1.22843	0.614215	−	0.789139i	$-0.289473\pi$
$619$	30.5630	1.22843	0.614215	+	0.789139i	$0.289473\pi$
$620$	0	0
$621$	−3.36842	−0.135170
$622$	0	0
$623$	11.5147	0.461325
$624$	0	0
$625$	−0.449359	−0.0179744
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−8.74707	−0.348769
$630$	0	0
$631$	−28.7212	−1.14337	−0.571686	−	0.820473i	$-0.693710\pi$
$631$	−28.7212	−1.14337	−0.571686	+	0.820473i	$0.693710\pi$
$632$	0	0
$633$	−3.86210	−0.153505
$634$	0	0
$635$	−23.8960	−0.948286
$636$	0	0
$637$	8.80047	0.348687
$638$	0	0
$639$	39.9009	1.57846
$640$	0	0
$641$	−13.4178	−0.529973	−0.264986	−	0.964252i	$-0.585367\pi$
$641$	−13.4178	−0.529973	−0.264986	+	0.964252i	$0.585367\pi$
$642$	0	0
$643$	−28.0617	−1.10664	−0.553322	−	0.832967i	$-0.686640\pi$
$643$	−28.0617	−1.10664	−0.553322	+	0.832967i	$0.686640\pi$
$644$	0	0
$645$	−0.729565	−0.0287266
$646$	0	0
$647$	25.8426	1.01598	0.507988	−	0.861364i	$-0.330389\pi$
$647$	25.8426	1.01598	0.507988	+	0.861364i	$0.330389\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	4.15975	0.163034
$652$	0	0
$653$	−26.1957	−1.02512	−0.512558	−	0.858653i	$-0.671302\pi$
$653$	−26.1957	−1.02512	−0.512558	+	0.858653i	$0.671302\pi$
$654$	0	0
$655$	−8.79847	−0.343785
$656$	0	0
$657$	45.0253	1.75660
$658$	0	0
$659$	−14.1081	−0.549573	−0.274787	−	0.961505i	$-0.588607\pi$
$659$	−14.1081	−0.549573	−0.274787	+	0.961505i	$0.588607\pi$
$660$	0	0
$661$	−9.37691	−0.364719	−0.182360	−	0.983232i	$-0.558374\pi$
$661$	−9.37691	−0.364719	−0.182360	+	0.983232i	$0.558374\pi$
$662$	0	0
$663$	−0.436810	−0.0169643
$664$	0	0
$665$	2.16257	0.0838607
$666$	0	0
$667$	−16.5622	−0.641291
$668$	0	0
$669$	4.87541	0.188494
$670$	0	0
$671$	0	0
$672$	0	0
$673$	29.3059	1.12966	0.564830	−	0.825208i	$-0.308942\pi$
$673$	29.3059	1.12966	0.564830	+	0.825208i	$0.308942\pi$
$674$	0	0
$675$	5.07013	0.195149
$676$	0	0
$677$	41.3932	1.59087	0.795435	−	0.606039i	$-0.207242\pi$
$677$	41.3932	1.59087	0.795435	+	0.606039i	$0.207242\pi$
$678$	0	0
$679$	−7.77160	−0.298247
$680$	0	0
$681$	2.64548	0.101375
$682$	0	0
$683$	−5.71863	−0.218817	−0.109409	−	0.993997i	$-0.534896\pi$
$683$	−5.71863	−0.218817	−0.109409	+	0.993997i	$0.534896\pi$
$684$	0	0
$685$	2.28835	0.0874333
$686$	0	0
$687$	−0.316212	−0.0120642
$688$	0	0
$689$	−15.6062	−0.594547
$690$	0	0
$691$	4.60103	0.175032	0.0875158	−	0.996163i	$-0.472107\pi$
$691$	4.60103	0.175032	0.0875158	+	0.996163i	$0.472107\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.5374	−0.437638
$696$	0	0
$697$	1.07361	0.0406658
$698$	0	0
$699$	−1.98391	−0.0750382
$700$	0	0
$701$	32.1896	1.21578	0.607892	−	0.794020i	$-0.292015\pi$
$701$	32.1896	1.21578	0.607892	+	0.794020i	$0.292015\pi$
$702$	0	0
$703$	−10.7516	−0.405505
$704$	0	0
$705$	1.44270	0.0543352
$706$	0	0
$707$	−30.2664	−1.13828
$708$	0	0
$709$	−2.40894	−0.0904698	−0.0452349	−	0.998976i	$-0.514404\pi$
$709$	−2.40894	−0.0904698	−0.0452349	+	0.998976i	$0.514404\pi$
$710$	0	0
$711$	−15.4592	−0.579767
$712$	0	0
$713$	19.3857	0.726001
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.68231	−0.0628269
$718$	0	0
$719$	21.8792	0.815958	0.407979	−	0.912991i	$-0.366234\pi$
$719$	21.8792	0.815958	0.407979	+	0.912991i	$0.366234\pi$
$720$	0	0
$721$	13.4808	0.502052
$722$	0	0
$723$	2.40071	0.0892834
$724$	0	0
$725$	24.9294	0.925853
$726$	0	0
$727$	8.08443	0.299835	0.149917	−	0.988699i	$-0.452099\pi$
$727$	8.08443	0.299835	0.149917	+	0.988699i	$0.452099\pi$
$728$	0	0
$729$	−22.8361	−0.845782
$730$	0	0
$731$	−1.51385	−0.0559917
$732$	0	0
$733$	−26.9676	−0.996073	−0.498036	−	0.867156i	$-0.665945\pi$
$733$	−26.9676	−0.996073	−0.498036	+	0.867156i	$0.665945\pi$
$734$	0	0
$735$	−1.80430	−0.0665526
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−51.8537	−1.90747	−0.953735	−	0.300650i	$-0.902797\pi$
$739$	−51.8537	−1.90747	−0.953735	+	0.300650i	$0.902797\pi$
$740$	0	0
$741$	−0.536912	−0.0197240
$742$	0	0
$743$	17.8998	0.656681	0.328340	−	0.944559i	$-0.393511\pi$
$743$	17.8998	0.656681	0.328340	+	0.944559i	$0.393511\pi$
$744$	0	0
$745$	22.8239	0.836203
$746$	0	0
$747$	−25.8451	−0.945624
$748$	0	0
$749$	17.9065	0.654291
$750$	0	0
$751$	−38.2873	−1.39712	−0.698562	−	0.715550i	$-0.746176\pi$
$751$	−38.2873	−1.39712	−0.698562	+	0.715550i	$0.746176\pi$
$752$	0	0
$753$	4.54199	0.165519
$754$	0	0
$755$	−12.4257	−0.452216
$756$	0	0
$757$	−8.08613	−0.293896	−0.146948	−	0.989144i	$-0.546945\pi$
$757$	−8.08613	−0.293896	−0.146948	+	0.989144i	$0.546945\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−32.4610	−1.17671	−0.588354	−	0.808603i	$-0.700224\pi$
$761$	−32.4610	−1.17671	−0.588354	+	0.808603i	$0.700224\pi$
$762$	0	0
$763$	−6.15354	−0.222773
$764$	0	0
$765$	−3.31881	−0.119992
$766$	0	0
$767$	12.9883	0.468981
$768$	0	0
$769$	−23.3940	−0.843610	−0.421805	−	0.906687i	$-0.638603\pi$
$769$	−23.3940	−0.843610	−0.421805	+	0.906687i	$0.638603\pi$
$770$	0	0
$771$	−2.21770	−0.0798685
$772$	0	0
$773$	−21.1569	−0.760962	−0.380481	−	0.924789i	$-0.624242\pi$
$773$	−21.1569	−0.760962	−0.380481	+	0.924789i	$0.624242\pi$
$774$	0	0
$775$	−29.1793	−1.04815
$776$	0	0
$777$	−4.67457	−0.167699
$778$	0	0
$779$	1.31964	0.0472811
$780$	0	0
$781$	0	0
$782$	0	0
$783$	13.5887	0.485620
$784$	0	0
$785$	22.5974	0.806537
$786$	0	0
$787$	−4.51618	−0.160985	−0.0804923	−	0.996755i	$-0.525649\pi$
$787$	−4.51618	−0.160985	−0.0804923	+	0.996755i	$0.525649\pi$
$788$	0	0
$789$	−7.63436	−0.271790
$790$	0	0
$791$	−21.9244	−0.779542
$792$	0	0
$793$	0.603062	0.0214154
$794$	0	0
$795$	3.19962	0.113479
$796$	0	0
$797$	9.76784	0.345995	0.172997	−	0.984922i	$-0.444655\pi$
$797$	9.76784	0.345995	0.172997	+	0.984922i	$0.444655\pi$
$798$	0	0
$799$	2.99360	0.105906
$800$	0	0
$801$	−21.7208	−0.767465
$802$	0	0
$803$	0	0
$804$	0	0
$805$	4.38180	0.154438
$806$	0	0
$807$	−3.79011	−0.133418
$808$	0	0
$809$	−1.22447	−0.0430499	−0.0215250	−	0.999768i	$-0.506852\pi$
$809$	−1.22447	−0.0430499	−0.0215250	+	0.999768i	$0.506852\pi$
$810$	0	0
$811$	−3.21197	−0.112787	−0.0563937	−	0.998409i	$-0.517960\pi$
$811$	−3.21197	−0.112787	−0.0563937	+	0.998409i	$0.517960\pi$
$812$	0	0
$813$	−8.44784	−0.296279
$814$	0	0
$815$	−21.4625	−0.751800
$816$	0	0
$817$	−1.86077	−0.0651002
$818$	0	0
$819$	8.65085	0.302285
$820$	0	0
$821$	50.7554	1.77138	0.885689	−	0.464280i	$-0.153687\pi$
$821$	50.7554	1.77138	0.885689	+	0.464280i	$0.153687\pi$
$822$	0	0
$823$	−29.7434	−1.03679	−0.518394	−	0.855142i	$-0.673470\pi$
$823$	−29.7434	−1.03679	−0.518394	+	0.855142i	$0.673470\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−18.6503	−0.648534	−0.324267	−	0.945966i	$-0.605118\pi$
$827$	−18.6503	−0.648534	−0.324267	+	0.945966i	$0.605118\pi$
$828$	0	0
$829$	−9.35960	−0.325072	−0.162536	−	0.986703i	$-0.551967\pi$
$829$	−9.35960	−0.325072	−0.162536	+	0.986703i	$0.551967\pi$
$830$	0	0
$831$	6.32687	0.219477
$832$	0	0
$833$	−3.74392	−0.129719
$834$	0	0
$835$	−29.6647	−1.02659
$836$	0	0
$837$	−15.9053	−0.549768
$838$	0	0
$839$	−23.1854	−0.800449	−0.400225	−	0.916417i	$-0.631068\pi$
$839$	−23.1854	−0.800449	−0.400225	+	0.916417i	$0.631068\pi$
$840$	0	0
$841$	37.8144	1.30394
$842$	0	0
$843$	−2.40218	−0.0827354
$844$	0	0
$845$	−13.0472	−0.448838
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−6.88605	−0.236329
$850$	0	0
$851$	−21.7849	−0.746778
$852$	0	0
$853$	−39.2780	−1.34485	−0.672427	−	0.740164i	$-0.734748\pi$
$853$	−39.2780	−1.34485	−0.672427	+	0.740164i	$0.734748\pi$
$854$	0	0
$855$	−4.07937	−0.139512
$856$	0	0
$857$	−2.60116	−0.0888538	−0.0444269	−	0.999013i	$-0.514146\pi$
$857$	−2.60116	−0.0888538	−0.0444269	+	0.999013i	$0.514146\pi$
$858$	0	0
$859$	−7.87368	−0.268647	−0.134323	−	0.990938i	$-0.542886\pi$
$859$	−7.87368	−0.268647	−0.134323	+	0.990938i	$0.542886\pi$
$860$	0	0
$861$	0.573752	0.0195534
$862$	0	0
$863$	52.0501	1.77181	0.885903	−	0.463871i	$-0.153540\pi$
$863$	52.0501	1.77181	0.885903	+	0.463871i	$0.153540\pi$
$864$	0	0
$865$	−15.2077	−0.517076
$866$	0	0
$867$	−4.58709	−0.155786
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−7.14557	−0.242118
$872$	0	0
$873$	14.6600	0.496167
$874$	0	0
$875$	−17.4083	−0.588508
$876$	0	0
$877$	−35.3885	−1.19499	−0.597493	−	0.801874i	$-0.703836\pi$
$877$	−35.3885	−1.19499	−0.597493	+	0.801874i	$0.703836\pi$
$878$	0	0
$879$	−3.66550	−0.123634
$880$	0	0
$881$	−54.2450	−1.82756	−0.913780	−	0.406210i	$-0.866850\pi$
$881$	−54.2450	−1.82756	−0.913780	+	0.406210i	$0.866850\pi$
$882$	0	0
$883$	23.6535	0.796005	0.398003	−	0.917384i	$-0.369704\pi$
$883$	23.6535	0.796005	0.398003	+	0.917384i	$0.369704\pi$
$884$	0	0
$885$	−2.66290	−0.0895125
$886$	0	0
$887$	−58.4570	−1.96279	−0.981397	−	0.191990i	$-0.938506\pi$
$887$	−58.4570	−1.96279	−0.981397	+	0.191990i	$0.938506\pi$
$888$	0	0
$889$	−26.4986	−0.888735
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.67963	0.123134
$894$	0	0
$895$	30.6350	1.02401
$896$	0	0
$897$	−1.08789	−0.0363237
$898$	0	0
$899$	−78.2050	−2.60828
$900$	0	0
$901$	6.63921	0.221184
$902$	0	0
$903$	−0.809024	−0.0269226
$904$	0	0
$905$	22.5977	0.751173
$906$	0	0
$907$	−0.0872244	−0.00289624	−0.00144812	−	0.999999i	$-0.500461\pi$
$907$	−0.0872244	−0.00289624	−0.00144812	+	0.999999i	$0.500461\pi$
$908$	0	0
$909$	57.0932	1.89366
$910$	0	0
$911$	13.5146	0.447759	0.223880	−	0.974617i	$-0.428128\pi$
$911$	13.5146	0.447759	0.223880	+	0.974617i	$0.428128\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−0.123642	−0.00408746
$916$	0	0
$917$	−9.75674	−0.322196
$918$	0	0
$919$	−18.4582	−0.608879	−0.304440	−	0.952532i	$-0.598469\pi$
$919$	−18.4582	−0.608879	−0.304440	+	0.952532i	$0.598469\pi$
$920$	0	0
$921$	−1.97179	−0.0649727
$922$	0	0
$923$	26.1212	0.859791
$924$	0	0
$925$	32.7906	1.07815
$926$	0	0
$927$	−25.4296	−0.835219
$928$	0	0
$929$	51.8030	1.69960	0.849801	−	0.527103i	$-0.176722\pi$
$929$	51.8030	1.69960	0.849801	+	0.527103i	$0.176722\pi$
$930$	0	0
$931$	−4.60190	−0.150821
$932$	0	0
$933$	3.87767	0.126949
$934$	0	0
$935$	0	0
$936$	0	0
$937$	11.5185	0.376295	0.188147	−	0.982141i	$-0.439752\pi$
$937$	11.5185	0.376295	0.188147	+	0.982141i	$0.439752\pi$
$938$	0	0
$939$	−5.57012	−0.181774
$940$	0	0
$941$	−40.2273	−1.31137	−0.655687	−	0.755033i	$-0.727621\pi$
$941$	−40.2273	−1.31137	−0.655687	+	0.755033i	$0.727621\pi$
$942$	0	0
$943$	2.67386	0.0870730
$944$	0	0
$945$	−3.59511	−0.116949
$946$	0	0
$947$	15.1209	0.491362	0.245681	−	0.969351i	$-0.420988\pi$
$947$	15.1209	0.491362	0.245681	+	0.969351i	$0.420988\pi$
$948$	0	0
$949$	29.4759	0.956829
$950$	0	0
$951$	−0.949670	−0.0307952
$952$	0	0
$953$	37.9503	1.22933	0.614666	−	0.788787i	$-0.289291\pi$
$953$	37.9503	1.22933	0.614666	+	0.788787i	$0.289291\pi$
$954$	0	0
$955$	−8.88644	−0.287558
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.53758	0.0819427
$960$	0	0
$961$	60.5374	1.95282
$962$	0	0
$963$	−33.7781	−1.08849
$964$	0	0
$965$	−8.56920	−0.275852
$966$	0	0
$967$	51.8117	1.66615	0.833076	−	0.553159i	$-0.186578\pi$
$967$	51.8117	1.66615	0.833076	+	0.553159i	$0.186578\pi$
$968$	0	0
$969$	0.228415	0.00733774
$970$	0	0
$971$	54.4061	1.74597	0.872987	−	0.487744i	$-0.162180\pi$
$971$	54.4061	1.74597	0.872987	+	0.487744i	$0.162180\pi$
$972$	0	0
$973$	−12.7939	−0.410155
$974$	0	0
$975$	1.63749	0.0524417
$976$	0	0
$977$	15.4380	0.493906	0.246953	−	0.969027i	$-0.420571\pi$
$977$	15.4380	0.493906	0.246953	+	0.969027i	$0.420571\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	11.6078	0.370608
$982$	0	0
$983$	2.96466	0.0945580	0.0472790	−	0.998882i	$-0.484945\pi$
$983$	2.96466	0.0945580	0.0472790	+	0.998882i	$0.484945\pi$
$984$	0	0
$985$	−24.2331	−0.772130
$986$	0	0
$987$	1.59983	0.0509230
$988$	0	0
$989$	−3.77030	−0.119889
$990$	0	0
$991$	30.0556	0.954747	0.477374	−	0.878700i	$-0.341589\pi$
$991$	30.0556	0.954747	0.477374	+	0.878700i	$0.341589\pi$
$992$	0	0
$993$	4.99987	0.158666
$994$	0	0
$995$	14.0907	0.446706
$996$	0	0
$997$	21.9787	0.696074	0.348037	−	0.937481i	$-0.386848\pi$
$997$	21.9787	0.696074	0.348037	+	0.937481i	$0.386848\pi$
$998$	0	0
$999$	17.8738	0.565501

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.u.1.9	✓	14
11.10	odd	2		9196.2.a.v.1.9	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9196.2.a.u.1.9	✓	14	1.1	even	1	trivial
9196.2.a.v.1.9	yes	14	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9196.a (trivial)

\(f(q)\)	\(=\)	\(q+0.280760 q^{3} +1.39648 q^{5} +1.54858 q^{7} -2.92117 q^{9} -1.91235 q^{13} +0.392077 q^{15} +0.813559 q^{17} +1.00000 q^{19} +0.434779 q^{21} +2.02620 q^{23} -3.04983 q^{25} -1.66243 q^{27} -8.17401 q^{29} +9.56752 q^{31} +2.16257 q^{35} -10.7516 q^{37} -0.536912 q^{39} +1.31964 q^{41} -1.86077 q^{43} -4.07937 q^{45} +3.67963 q^{47} -4.60190 q^{49} +0.228415 q^{51} +8.16070 q^{53} +0.280760 q^{57} -6.79179 q^{59} -0.315351 q^{61} -4.52367 q^{63} -2.67057 q^{65} +3.73653 q^{67} +0.568876 q^{69} -13.6592 q^{71} -15.4134 q^{73} -0.856270 q^{75} +5.29213 q^{79} +8.29678 q^{81} +8.84752 q^{83} +1.13612 q^{85} -2.29493 q^{87} +7.43563 q^{89} -2.96143 q^{91} +2.68617 q^{93} +1.39648 q^{95} -5.01854 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{3} - 6 q^{5} - 4 q^{7} + 12 q^{9} - 4 q^{13} + 2 q^{15} - 8 q^{17} + 14 q^{19} - 18 q^{21} - 2 q^{23} + 8 q^{25} + 10 q^{27} + 4 q^{29} - 2 q^{31} - 24 q^{35} - 10 q^{37} - 24 q^{39} - 4 q^{41}+ \cdots - 20 q^{97}+O(q^{100}) \) 14 * q - 2 * q^3 - 6 * q^5 - 4 * q^7 + 12 * q^9 - 4 * q^13 + 2 * q^15 - 8 * q^17 + 14 * q^19 - 18 * q^21 - 2 * q^23 + 8 * q^25 + 10 * q^27 + 4 * q^29 - 2 * q^31 - 24 * q^35 - 10 * q^37 - 24 * q^39 - 4 * q^41 - 32 * q^43 - 20 * q^45 - 4 * q^47 - 14 * q^49 - 22 * q^51 + 6 * q^53 - 2 * q^57 - 6 * q^59 + 4 * q^61 + 16 * q^63 - 54 * q^65 + 6 * q^67 + 4 * q^69 - 6 * q^71 + 24 * q^73 - 10 * q^75 - 48 * q^79 + 2 * q^81 + 40 * q^85 - 28 * q^87 + 2 * q^89 + 28 * q^91 + 34 * q^93 - 6 * q^95 - 20 * q^97

Introduction

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Database

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