LMFDB - Embedded newform 7436.2.a.v.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7436,2,Mod(1,7436)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7436, 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("7436.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7436 = 2^{2} \cdot 11 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7436.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:	$59.3767589430$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 23 x^{10} + 46 x^{9} + 182 x^{8} - 372 x^{7} - 575 x^{6} + 1224 x^{5} + 624 x^{4} + \cdots + 36 $ x^12 - 2x^11 - 23x^10 + 46x^9 + 182x^8 - 372x^7 - 575x^6 + 1224x^5 + 624x^4 - 1452x^3 - 84x^2 + 288*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 572)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.98897$ of defining polynomial
Character	$\chi$	$=$	7436.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.98897 q^{3} +2.64929 q^{5} +0.190541 q^{7} +0.955984 q^{9} +O(q^{10})$	$q-1.98897 q^{3} +2.64929 q^{5} +0.190541 q^{7} +0.955984 q^{9} +1.00000 q^{11} -5.26935 q^{15} +3.47690 q^{17} +8.16009 q^{19} -0.378979 q^{21} -3.29846 q^{23} +2.01875 q^{25} +4.06548 q^{27} +4.66676 q^{29} -2.17098 q^{31} -1.98897 q^{33} +0.504798 q^{35} -7.49962 q^{37} +1.82368 q^{41} +9.77479 q^{43} +2.53268 q^{45} +12.0728 q^{47} -6.96369 q^{49} -6.91544 q^{51} -2.84991 q^{53} +2.64929 q^{55} -16.2301 q^{57} -7.40983 q^{59} +10.6188 q^{61} +0.182154 q^{63} +10.3382 q^{67} +6.56053 q^{69} -12.8822 q^{71} -10.7491 q^{73} -4.01521 q^{75} +0.190541 q^{77} +1.21943 q^{79} -10.9540 q^{81} +7.54640 q^{83} +9.21132 q^{85} -9.28202 q^{87} -9.23167 q^{89} +4.31801 q^{93} +21.6184 q^{95} +8.90666 q^{97} +0.955984 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{3} + 8 q^{7} + 14 q^{9}+O(q^{10}) $ 12 * q + 2 * q^3 + 8 * q^7 + 14 * q^9	$ 12 q + 2 q^{3} + 8 q^{7} + 14 q^{9} + 12 q^{11} + 8 q^{15} + 22 q^{19} + 10 q^{21} + 10 q^{23} + 20 q^{25} - 4 q^{27} + 8 q^{29} + 28 q^{31} + 2 q^{33} - 8 q^{35} - 16 q^{37} + 16 q^{41} + 10 q^{43} + 12 q^{47} + 14 q^{49} - 22 q^{51} + 8 q^{53} + 36 q^{57} + 32 q^{59} - 6 q^{61} - 22 q^{63} + 48 q^{67} + 10 q^{69} + 20 q^{71} + 16 q^{73} + 6 q^{75} + 8 q^{77} - 16 q^{79} + 4 q^{81} + 10 q^{83} + 42 q^{85} - 52 q^{87} - 16 q^{89} - 4 q^{93} - 12 q^{95} + 20 q^{97} + 14 q^{99}+O(q^{100}) $ 12 * q + 2 * q^3 + 8 * q^7 + 14 * q^9 + 12 * q^11 + 8 * q^15 + 22 * q^19 + 10 * q^21 + 10 * q^23 + 20 * q^25 - 4 * q^27 + 8 * q^29 + 28 * q^31 + 2 * q^33 - 8 * q^35 - 16 * q^37 + 16 * q^41 + 10 * q^43 + 12 * q^47 + 14 * q^49 - 22 * q^51 + 8 * q^53 + 36 * q^57 + 32 * q^59 - 6 * q^61 - 22 * q^63 + 48 * q^67 + 10 * q^69 + 20 * q^71 + 16 * q^73 + 6 * q^75 + 8 * q^77 - 16 * q^79 + 4 * q^81 + 10 * q^83 + 42 * q^85 - 52 * q^87 - 16 * q^89 - 4 * q^93 - 12 * q^95 + 20 * q^97 + 14 * 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
$3$	−1.98897	−1.14833	−0.574165	−	0.818740i	$-0.694673\pi$
$3$	−1.98897	−1.14833	−0.574165	+	0.818740i	$0.694673\pi$
$4$	0	0
$5$	2.64929	1.18480	0.592400	−	0.805644i	$-0.298181\pi$
$5$	2.64929	1.18480	0.592400	+	0.805644i	$0.298181\pi$
$6$	0	0
$7$	0.190541	0.0720177	0.0360089	−	0.999351i	$-0.488536\pi$
$7$	0.190541	0.0720177	0.0360089	+	0.999351i	$0.488536\pi$
$8$	0	0
$9$	0.955984	0.318661
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−5.26935	−1.36054
$16$	0	0
$17$	3.47690	0.843272	0.421636	−	0.906765i	$-0.361456\pi$
$17$	3.47690	0.843272	0.421636	+	0.906765i	$0.361456\pi$
$18$	0	0
$19$	8.16009	1.87205	0.936026	−	0.351931i	$-0.114475\pi$
$19$	8.16009	1.87205	0.936026	+	0.351931i	$0.114475\pi$
$20$	0	0
$21$	−0.378979	−0.0827001
$22$	0	0
$23$	−3.29846	−0.687777	−0.343889	−	0.939010i	$-0.611744\pi$
$23$	−3.29846	−0.687777	−0.343889	+	0.939010i	$0.611744\pi$
$24$	0	0
$25$	2.01875	0.403749
$26$	0	0
$27$	4.06548	0.782401
$28$	0	0
$29$	4.66676	0.866595	0.433297	−	0.901251i	$-0.357350\pi$
$29$	4.66676	0.866595	0.433297	+	0.901251i	$0.357350\pi$
$30$	0	0
$31$	−2.17098	−0.389920	−0.194960	−	0.980811i	$-0.562458\pi$
$31$	−2.17098	−0.389920	−0.194960	+	0.980811i	$0.562458\pi$
$32$	0	0
$33$	−1.98897	−0.346234
$34$	0	0
$35$	0.504798	0.0853265
$36$	0	0
$37$	−7.49962	−1.23293	−0.616465	−	0.787382i	$-0.711436\pi$
$37$	−7.49962	−1.23293	−0.616465	+	0.787382i	$0.711436\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.82368	0.284810	0.142405	−	0.989808i	$-0.454516\pi$
$41$	1.82368	0.284810	0.142405	+	0.989808i	$0.454516\pi$
$42$	0	0
$43$	9.77479	1.49064	0.745321	−	0.666706i	$-0.232296\pi$
$43$	9.77479	1.49064	0.745321	+	0.666706i	$0.232296\pi$
$44$	0	0
$45$	2.53268	0.377550
$46$	0	0
$47$	12.0728	1.76099	0.880497	−	0.474051i	$-0.157209\pi$
$47$	12.0728	1.76099	0.880497	+	0.474051i	$0.157209\pi$
$48$	0	0
$49$	−6.96369	−0.994813
$50$	0	0
$51$	−6.91544	−0.968355
$52$	0	0
$53$	−2.84991	−0.391466	−0.195733	−	0.980657i	$-0.562709\pi$
$53$	−2.84991	−0.391466	−0.195733	+	0.980657i	$0.562709\pi$
$54$	0	0
$55$	2.64929	0.357230
$56$	0	0
$57$	−16.2301	−2.14973
$58$	0	0
$59$	−7.40983	−0.964678	−0.482339	−	0.875985i	$-0.660213\pi$
$59$	−7.40983	−0.964678	−0.482339	+	0.875985i	$0.660213\pi$
$60$	0	0
$61$	10.6188	1.35960	0.679801	−	0.733397i	$-0.262066\pi$
$61$	10.6188	1.35960	0.679801	+	0.733397i	$0.262066\pi$
$62$	0	0
$63$	0.182154	0.0229493
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.3382	1.26301	0.631507	−	0.775370i	$-0.282437\pi$
$67$	10.3382	1.26301	0.631507	+	0.775370i	$0.282437\pi$
$68$	0	0
$69$	6.56053	0.789795
$70$	0	0
$71$	−12.8822	−1.52884	−0.764420	−	0.644719i	$-0.776975\pi$
$71$	−12.8822	−1.52884	−0.764420	+	0.644719i	$0.776975\pi$
$72$	0	0
$73$	−10.7491	−1.25809	−0.629045	−	0.777369i	$-0.716554\pi$
$73$	−10.7491	−1.25809	−0.629045	+	0.777369i	$0.716554\pi$
$74$	0	0
$75$	−4.01521	−0.463637
$76$	0	0
$77$	0.190541	0.0217142
$78$	0	0
$79$	1.21943	0.137197	0.0685985	−	0.997644i	$-0.478147\pi$
$79$	1.21943	0.137197	0.0685985	+	0.997644i	$0.478147\pi$
$80$	0	0
$81$	−10.9540	−1.21712
$82$	0	0
$83$	7.54640	0.828326	0.414163	−	0.910203i	$-0.364074\pi$
$83$	7.54640	0.828326	0.414163	+	0.910203i	$0.364074\pi$
$84$	0	0
$85$	9.21132	0.999108
$86$	0	0
$87$	−9.28202	−0.995137
$88$	0	0
$89$	−9.23167	−0.978556	−0.489278	−	0.872128i	$-0.662740\pi$
$89$	−9.23167	−0.978556	−0.489278	+	0.872128i	$0.662740\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.31801	0.447757
$94$	0	0
$95$	21.6184	2.21801
$96$	0	0
$97$	8.90666	0.904334	0.452167	−	0.891933i	$-0.350651\pi$
$97$	8.90666	0.904334	0.452167	+	0.891933i	$0.350651\pi$
$98$	0	0
$99$	0.955984	0.0960800
$100$	0	0
$101$	0.547925	0.0545206	0.0272603	−	0.999628i	$-0.491322\pi$
$101$	0.547925	0.0545206	0.0272603	+	0.999628i	$0.491322\pi$
$102$	0	0
$103$	−4.10775	−0.404749	−0.202374	−	0.979308i	$-0.564866\pi$
$103$	−4.10775	−0.404749	−0.202374	+	0.979308i	$0.564866\pi$
$104$	0	0
$105$	−1.00403	−0.0979830
$106$	0	0
$107$	18.9714	1.83403	0.917015	−	0.398852i	$-0.130591\pi$
$107$	18.9714	1.83403	0.917015	+	0.398852i	$0.130591\pi$
$108$	0	0
$109$	−3.76372	−0.360499	−0.180250	−	0.983621i	$-0.557691\pi$
$109$	−3.76372	−0.360499	−0.180250	+	0.983621i	$0.557691\pi$
$110$	0	0
$111$	14.9165	1.41581
$112$	0	0
$113$	7.49801	0.705353	0.352677	−	0.935745i	$-0.385272\pi$
$113$	7.49801	0.705353	0.352677	+	0.935745i	$0.385272\pi$
$114$	0	0
$115$	−8.73859	−0.814878
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.662492	0.0607305
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−3.62723	−0.327056
$124$	0	0
$125$	−7.89821	−0.706438
$126$	0	0
$127$	−8.70907	−0.772805	−0.386403	−	0.922330i	$-0.626282\pi$
$127$	−8.70907	−0.772805	−0.386403	+	0.922330i	$0.626282\pi$
$128$	0	0
$129$	−19.4417	−1.71175
$130$	0	0
$131$	−0.0899845	−0.00786198	−0.00393099	−	0.999992i	$-0.501251\pi$
$131$	−0.0899845	−0.00786198	−0.00393099	+	0.999992i	$0.501251\pi$
$132$	0	0
$133$	1.55483	0.134821
$134$	0	0
$135$	10.7706	0.926988
$136$	0	0
$137$	0.0332830	0.00284356	0.00142178	−	0.999999i	$-0.499547\pi$
$137$	0.0332830	0.00284356	0.00142178	+	0.999999i	$0.499547\pi$
$138$	0	0
$139$	9.31180	0.789817	0.394908	−	0.918720i	$-0.370776\pi$
$139$	9.31180	0.789817	0.394908	+	0.918720i	$0.370776\pi$
$140$	0	0
$141$	−24.0123	−2.02220
$142$	0	0
$143$	0	0
$144$	0	0
$145$	12.3636	1.02674
$146$	0	0
$147$	13.8505	1.14237
$148$	0	0
$149$	5.98202	0.490066	0.245033	−	0.969515i	$-0.421201\pi$
$149$	5.98202	0.490066	0.245033	+	0.969515i	$0.421201\pi$
$150$	0	0
$151$	16.7393	1.36223	0.681113	−	0.732179i	$-0.261496\pi$
$151$	16.7393	1.36223	0.681113	+	0.732179i	$0.261496\pi$
$152$	0	0
$153$	3.32386	0.268718
$154$	0	0
$155$	−5.75157	−0.461977
$156$	0	0
$157$	−23.9964	−1.91512	−0.957560	−	0.288235i	$-0.906932\pi$
$157$	−23.9964	−1.91512	−0.957560	+	0.288235i	$0.906932\pi$
$158$	0	0
$159$	5.66838	0.449532
$160$	0	0
$161$	−0.628492	−0.0495321
$162$	0	0
$163$	−5.04481	−0.395140	−0.197570	−	0.980289i	$-0.563305\pi$
$163$	−5.04481	−0.395140	−0.197570	+	0.980289i	$0.563305\pi$
$164$	0	0
$165$	−5.26935	−0.410218
$166$	0	0
$167$	−8.19019	−0.633776	−0.316888	−	0.948463i	$-0.602638\pi$
$167$	−8.19019	−0.633776	−0.316888	+	0.948463i	$0.602638\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	7.80091	0.596551
$172$	0	0
$173$	−1.33773	−0.101706	−0.0508530	−	0.998706i	$-0.516194\pi$
$173$	−1.33773	−0.101706	−0.0508530	+	0.998706i	$0.516194\pi$
$174$	0	0
$175$	0.384654	0.0290771
$176$	0	0
$177$	14.7379	1.10777
$178$	0	0
$179$	11.5655	0.864445	0.432222	−	0.901767i	$-0.357730\pi$
$179$	11.5655	0.864445	0.432222	+	0.901767i	$0.357730\pi$
$180$	0	0
$181$	18.5975	1.38234	0.691170	−	0.722693i	$-0.257096\pi$
$181$	18.5975	1.38234	0.691170	+	0.722693i	$0.257096\pi$
$182$	0	0
$183$	−21.1205	−1.56127
$184$	0	0
$185$	−19.8687	−1.46078
$186$	0	0
$187$	3.47690	0.254256
$188$	0	0
$189$	0.774640	0.0563468
$190$	0	0
$191$	5.25241	0.380051	0.190025	−	0.981779i	$-0.439143\pi$
$191$	5.25241	0.380051	0.190025	+	0.981779i	$0.439143\pi$
$192$	0	0
$193$	−3.89604	−0.280443	−0.140222	−	0.990120i	$-0.544781\pi$
$193$	−3.89604	−0.280443	−0.140222	+	0.990120i	$0.544781\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−20.2033	−1.43943	−0.719715	−	0.694270i	$-0.755727\pi$
$197$	−20.2033	−1.43943	−0.719715	+	0.694270i	$0.755727\pi$
$198$	0	0
$199$	−11.3664	−0.805744	−0.402872	−	0.915256i	$-0.631988\pi$
$199$	−11.3664	−0.805744	−0.402872	+	0.915256i	$0.631988\pi$
$200$	0	0
$201$	−20.5624	−1.45036
$202$	0	0
$203$	0.889208	0.0624102
$204$	0	0
$205$	4.83145	0.337443
$206$	0	0
$207$	−3.15328	−0.219168
$208$	0	0
$209$	8.16009	0.564445
$210$	0	0
$211$	3.27496	0.225458	0.112729	−	0.993626i	$-0.464041\pi$
$211$	3.27496	0.225458	0.112729	+	0.993626i	$0.464041\pi$
$212$	0	0
$213$	25.6223	1.75561
$214$	0	0
$215$	25.8963	1.76611
$216$	0	0
$217$	−0.413661	−0.0280812
$218$	0	0
$219$	21.3796	1.44470
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−16.6233	−1.11318	−0.556588	−	0.830788i	$-0.687890\pi$
$223$	−16.6233	−1.11318	−0.556588	+	0.830788i	$0.687890\pi$
$224$	0	0
$225$	1.92989	0.128659
$226$	0	0
$227$	15.9807	1.06068	0.530340	−	0.847785i	$-0.322064\pi$
$227$	15.9807	1.06068	0.530340	+	0.847785i	$0.322064\pi$
$228$	0	0
$229$	−0.675351	−0.0446284	−0.0223142	−	0.999751i	$-0.507103\pi$
$229$	−0.675351	−0.0446284	−0.0223142	+	0.999751i	$0.507103\pi$
$230$	0	0
$231$	−0.378979	−0.0249350
$232$	0	0
$233$	1.05661	0.0692206	0.0346103	−	0.999401i	$-0.488981\pi$
$233$	1.05661	0.0692206	0.0346103	+	0.999401i	$0.488981\pi$
$234$	0	0
$235$	31.9843	2.08642
$236$	0	0
$237$	−2.42541	−0.157547
$238$	0	0
$239$	12.9873	0.840080	0.420040	−	0.907506i	$-0.362016\pi$
$239$	12.9873	0.840080	0.420040	+	0.907506i	$0.362016\pi$
$240$	0	0
$241$	7.05294	0.454320	0.227160	−	0.973857i	$-0.427056\pi$
$241$	7.05294	0.454320	0.227160	+	0.973857i	$0.427056\pi$
$242$	0	0
$243$	9.59079	0.615250
$244$	0	0
$245$	−18.4489	−1.17865
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−15.0095	−0.951191
$250$	0	0
$251$	9.08618	0.573515	0.286757	−	0.958003i	$-0.407423\pi$
$251$	9.08618	0.573515	0.286757	+	0.958003i	$0.407423\pi$
$252$	0	0
$253$	−3.29846	−0.207373
$254$	0	0
$255$	−18.3210	−1.14731
$256$	0	0
$257$	−1.93492	−0.120697	−0.0603484	−	0.998177i	$-0.519221\pi$
$257$	−1.93492	−0.120697	−0.0603484	+	0.998177i	$0.519221\pi$
$258$	0	0
$259$	−1.42899	−0.0887928
$260$	0	0
$261$	4.46135	0.276150
$262$	0	0
$263$	−13.3251	−0.821662	−0.410831	−	0.911712i	$-0.634761\pi$
$263$	−13.3251	−0.821662	−0.410831	+	0.911712i	$0.634761\pi$
$264$	0	0
$265$	−7.55025	−0.463808
$266$	0	0
$267$	18.3615	1.12370
$268$	0	0
$269$	−16.2778	−0.992476	−0.496238	−	0.868186i	$-0.665286\pi$
$269$	−16.2778	−0.992476	−0.496238	+	0.868186i	$0.665286\pi$
$270$	0	0
$271$	−0.119934	−0.00728549	−0.00364275	−	0.999993i	$-0.501160\pi$
$271$	−0.119934	−0.00728549	−0.00364275	+	0.999993i	$0.501160\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.01875	0.121735
$276$	0	0
$277$	23.8942	1.43566	0.717831	−	0.696217i	$-0.245135\pi$
$277$	23.8942	1.43566	0.717831	+	0.696217i	$0.245135\pi$
$278$	0	0
$279$	−2.07543	−0.124253
$280$	0	0
$281$	24.2316	1.44554	0.722768	−	0.691091i	$-0.242870\pi$
$281$	24.2316	1.44554	0.722768	+	0.691091i	$0.242870\pi$
$282$	0	0
$283$	−15.8983	−0.945054	−0.472527	−	0.881316i	$-0.656658\pi$
$283$	−15.8983	−0.945054	−0.472527	+	0.881316i	$0.656658\pi$
$284$	0	0
$285$	−42.9983	−2.54700
$286$	0	0
$287$	0.347485	0.0205114
$288$	0	0
$289$	−4.91116	−0.288892
$290$	0	0
$291$	−17.7150	−1.03847
$292$	0	0
$293$	−18.3667	−1.07300	−0.536498	−	0.843901i	$-0.680253\pi$
$293$	−18.3667	−1.07300	−0.536498	+	0.843901i	$0.680253\pi$
$294$	0	0
$295$	−19.6308	−1.14295
$296$	0	0
$297$	4.06548	0.235903
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.86250	0.107353
$302$	0	0
$303$	−1.08980	−0.0626076
$304$	0	0
$305$	28.1324	1.61085
$306$	0	0
$307$	20.2829	1.15760	0.578802	−	0.815468i	$-0.303520\pi$
$307$	20.2829	1.15760	0.578802	+	0.815468i	$0.303520\pi$
$308$	0	0
$309$	8.17018	0.464785
$310$	0	0
$311$	−27.6230	−1.56636	−0.783179	−	0.621797i	$-0.786403\pi$
$311$	−27.6230	−1.56636	−0.783179	+	0.621797i	$0.786403\pi$
$312$	0	0
$313$	−33.5385	−1.89571	−0.947853	−	0.318708i	$-0.896751\pi$
$313$	−33.5385	−1.89571	−0.947853	+	0.318708i	$0.896751\pi$
$314$	0	0
$315$	0.482579	0.0271903
$316$	0	0
$317$	27.6904	1.55525	0.777625	−	0.628728i	$-0.216424\pi$
$317$	27.6904	1.55525	0.777625	+	0.628728i	$0.216424\pi$
$318$	0	0
$319$	4.66676	0.261288
$320$	0	0
$321$	−37.7334	−2.10607
$322$	0	0
$323$	28.3718	1.57865
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.48591	0.413972
$328$	0	0
$329$	2.30036	0.126823
$330$	0	0
$331$	−7.47881	−0.411073	−0.205536	−	0.978649i	$-0.565894\pi$
$331$	−7.47881	−0.411073	−0.205536	+	0.978649i	$0.565894\pi$
$332$	0	0
$333$	−7.16952	−0.392887
$334$	0	0
$335$	27.3890	1.49642
$336$	0	0
$337$	19.2210	1.04704	0.523518	−	0.852015i	$-0.324619\pi$
$337$	19.2210	1.04704	0.523518	+	0.852015i	$0.324619\pi$
$338$	0	0
$339$	−14.9133	−0.809978
$340$	0	0
$341$	−2.17098	−0.117565
$342$	0	0
$343$	−2.66066	−0.143662
$344$	0	0
$345$	17.3808	0.935749
$346$	0	0
$347$	9.10138	0.488587	0.244294	−	0.969701i	$-0.421444\pi$
$347$	9.10138	0.488587	0.244294	+	0.969701i	$0.421444\pi$
$348$	0	0
$349$	12.2569	0.656094	0.328047	−	0.944661i	$-0.393610\pi$
$349$	12.2569	0.656094	0.328047	+	0.944661i	$0.393610\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	31.5140	1.67732	0.838662	−	0.544653i	$-0.183339\pi$
$353$	31.5140	1.67732	0.838662	+	0.544653i	$0.183339\pi$
$354$	0	0
$355$	−34.1288	−1.81137
$356$	0	0
$357$	−1.31767	−0.0697387
$358$	0	0
$359$	28.7885	1.51940	0.759699	−	0.650275i	$-0.225346\pi$
$359$	28.7885	1.51940	0.759699	+	0.650275i	$0.225346\pi$
$360$	0	0
$361$	47.5870	2.50458
$362$	0	0
$363$	−1.98897	−0.104394
$364$	0	0
$365$	−28.4776	−1.49058
$366$	0	0
$367$	−36.5926	−1.91012	−0.955059	−	0.296414i	$-0.904209\pi$
$367$	−36.5926	−1.91012	−0.955059	+	0.296414i	$0.904209\pi$
$368$	0	0
$369$	1.74341	0.0907581
$370$	0	0
$371$	−0.543025	−0.0281925
$372$	0	0
$373$	−18.6950	−0.967993	−0.483996	−	0.875070i	$-0.660815\pi$
$373$	−18.6950	−0.967993	−0.483996	+	0.875070i	$0.660815\pi$
$374$	0	0
$375$	15.7093	0.811223
$376$	0	0
$377$	0	0
$378$	0	0
$379$	24.8585	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$379$	24.8585	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$380$	0	0
$381$	17.3220	0.887435
$382$	0	0
$383$	−17.8547	−0.912334	−0.456167	−	0.889894i	$-0.650778\pi$
$383$	−17.8547	−0.912334	−0.456167	+	0.889894i	$0.650778\pi$
$384$	0	0
$385$	0.504798	0.0257269
$386$	0	0
$387$	9.34455	0.475010
$388$	0	0
$389$	−17.2194	−0.873060	−0.436530	−	0.899690i	$-0.643793\pi$
$389$	−17.2194	−0.873060	−0.436530	+	0.899690i	$0.643793\pi$
$390$	0	0
$391$	−11.4684	−0.579984
$392$	0	0
$393$	0.178976	0.00902815
$394$	0	0
$395$	3.23064	0.162551
$396$	0	0
$397$	12.7807	0.641446	0.320723	−	0.947173i	$-0.396074\pi$
$397$	12.7807	0.641446	0.320723	+	0.947173i	$0.396074\pi$
$398$	0	0
$399$	−3.09250	−0.154819
$400$	0	0
$401$	−16.6920	−0.833559	−0.416779	−	0.909008i	$-0.636841\pi$
$401$	−16.6920	−0.833559	−0.416779	+	0.909008i	$0.636841\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−29.0205	−1.44204
$406$	0	0
$407$	−7.49962	−0.371743
$408$	0	0
$409$	30.1946	1.49303	0.746514	−	0.665370i	$-0.231726\pi$
$409$	30.1946	1.49303	0.746514	+	0.665370i	$0.231726\pi$
$410$	0	0
$411$	−0.0661987	−0.00326534
$412$	0	0
$413$	−1.41188	−0.0694739
$414$	0	0
$415$	19.9926	0.981399
$416$	0	0
$417$	−18.5209	−0.906970
$418$	0	0
$419$	16.6396	0.812899	0.406450	−	0.913673i	$-0.366767\pi$
$419$	16.6396	0.812899	0.406450	+	0.913673i	$0.366767\pi$
$420$	0	0
$421$	13.8517	0.675088	0.337544	−	0.941310i	$-0.390404\pi$
$421$	13.8517	0.675088	0.337544	+	0.941310i	$0.390404\pi$
$422$	0	0
$423$	11.5414	0.561161
$424$	0	0
$425$	7.01898	0.340470
$426$	0	0
$427$	2.02332	0.0979153
$428$	0	0
$429$	0	0
$430$	0	0
$431$	19.6823	0.948065	0.474033	−	0.880507i	$-0.342798\pi$
$431$	19.6823	0.948065	0.474033	+	0.880507i	$0.342798\pi$
$432$	0	0
$433$	33.2885	1.59974	0.799871	−	0.600173i	$-0.204901\pi$
$433$	33.2885	1.59974	0.799871	+	0.600173i	$0.204901\pi$
$434$	0	0
$435$	−24.5908	−1.17904
$436$	0	0
$437$	−26.9158	−1.28755
$438$	0	0
$439$	30.9472	1.47703	0.738515	−	0.674237i	$-0.235527\pi$
$439$	30.9472	1.47703	0.738515	+	0.674237i	$0.235527\pi$
$440$	0	0
$441$	−6.65718	−0.317009
$442$	0	0
$443$	−18.5217	−0.879991	−0.439995	−	0.898000i	$-0.645020\pi$
$443$	−18.5217	−0.879991	−0.439995	+	0.898000i	$0.645020\pi$
$444$	0	0
$445$	−24.4574	−1.15939
$446$	0	0
$447$	−11.8980	−0.562757
$448$	0	0
$449$	2.71565	0.128159	0.0640796	−	0.997945i	$-0.479589\pi$
$449$	2.71565	0.128159	0.0640796	+	0.997945i	$0.479589\pi$
$450$	0	0
$451$	1.82368	0.0858735
$452$	0	0
$453$	−33.2939	−1.56428
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.12070	−0.286314	−0.143157	−	0.989700i	$-0.545725\pi$
$457$	−6.12070	−0.286314	−0.143157	+	0.989700i	$0.545725\pi$
$458$	0	0
$459$	14.1353	0.659777
$460$	0	0
$461$	−19.2607	−0.897061	−0.448531	−	0.893767i	$-0.648053\pi$
$461$	−19.2607	−0.897061	−0.448531	+	0.893767i	$0.648053\pi$
$462$	0	0
$463$	−1.62227	−0.0753935	−0.0376967	−	0.999289i	$-0.512002\pi$
$463$	−1.62227	−0.0753935	−0.0376967	+	0.999289i	$0.512002\pi$
$464$	0	0
$465$	11.4397	0.530502
$466$	0	0
$467$	16.6992	0.772746	0.386373	−	0.922343i	$-0.373728\pi$
$467$	16.6992	0.772746	0.386373	+	0.922343i	$0.373728\pi$
$468$	0	0
$469$	1.96985	0.0909594
$470$	0	0
$471$	47.7280	2.19919
$472$	0	0
$473$	9.77479	0.449445
$474$	0	0
$475$	16.4731	0.755839
$476$	0	0
$477$	−2.72447	−0.124745
$478$	0	0
$479$	25.7989	1.17878	0.589390	−	0.807849i	$-0.299368\pi$
$479$	25.7989	1.17878	0.589390	+	0.807849i	$0.299368\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	1.25005	0.0568792
$484$	0	0
$485$	23.5963	1.07145
$486$	0	0
$487$	6.32278	0.286513	0.143256	−	0.989686i	$-0.454243\pi$
$487$	6.32278	0.286513	0.143256	+	0.989686i	$0.454243\pi$
$488$	0	0
$489$	10.0339	0.453751
$490$	0	0
$491$	16.2080	0.731458	0.365729	−	0.930721i	$-0.380820\pi$
$491$	16.2080	0.731458	0.365729	+	0.930721i	$0.380820\pi$
$492$	0	0
$493$	16.2258	0.730775
$494$	0	0
$495$	2.53268	0.113836
$496$	0	0
$497$	−2.45459	−0.110104
$498$	0	0
$499$	26.6520	1.19311	0.596554	−	0.802573i	$-0.296536\pi$
$499$	26.6520	1.19311	0.596554	+	0.802573i	$0.296536\pi$
$500$	0	0
$501$	16.2900	0.727784
$502$	0	0
$503$	11.4111	0.508796	0.254398	−	0.967100i	$-0.418123\pi$
$503$	11.4111	0.508796	0.254398	+	0.967100i	$0.418123\pi$
$504$	0	0
$505$	1.45161	0.0645959
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.36245	0.414983	0.207492	−	0.978237i	$-0.433470\pi$
$509$	9.36245	0.414983	0.207492	+	0.978237i	$0.433470\pi$
$510$	0	0
$511$	−2.04815	−0.0906047
$512$	0	0
$513$	33.1746	1.46470
$514$	0	0
$515$	−10.8826	−0.479546
$516$	0	0
$517$	12.0728	0.530960
$518$	0	0
$519$	2.66070	0.116792
$520$	0	0
$521$	15.9092	0.696995	0.348497	−	0.937310i	$-0.386692\pi$
$521$	15.9092	0.696995	0.348497	+	0.937310i	$0.386692\pi$
$522$	0	0
$523$	−24.1469	−1.05587	−0.527936	−	0.849284i	$-0.677034\pi$
$523$	−24.1469	−1.05587	−0.527936	+	0.849284i	$0.677034\pi$
$524$	0	0
$525$	−0.765063	−0.0333901
$526$	0	0
$527$	−7.54830	−0.328809
$528$	0	0
$529$	−12.1201	−0.526962
$530$	0	0
$531$	−7.08369	−0.307406
$532$	0	0
$533$	0	0
$534$	0	0
$535$	50.2607	2.17296
$536$	0	0
$537$	−23.0033	−0.992667
$538$	0	0
$539$	−6.96369	−0.299948
$540$	0	0
$541$	0.761896	0.0327565	0.0163782	−	0.999866i	$-0.494786\pi$
$541$	0.761896	0.0327565	0.0163782	+	0.999866i	$0.494786\pi$
$542$	0	0
$543$	−36.9897	−1.58738
$544$	0	0
$545$	−9.97119	−0.427119
$546$	0	0
$547$	26.3836	1.12808	0.564040	−	0.825747i	$-0.309246\pi$
$547$	26.3836	1.12808	0.564040	+	0.825747i	$0.309246\pi$
$548$	0	0
$549$	10.1514	0.433252
$550$	0	0
$551$	38.0811	1.62231
$552$	0	0
$553$	0.232352	0.00988062
$554$	0	0
$555$	39.5181	1.67745
$556$	0	0
$557$	−9.29976	−0.394043	−0.197022	−	0.980399i	$-0.563127\pi$
$557$	−9.29976	−0.394043	−0.197022	+	0.980399i	$0.563127\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−6.91544	−0.291970
$562$	0	0
$563$	8.19630	0.345433	0.172716	−	0.984972i	$-0.444746\pi$
$563$	8.19630	0.345433	0.172716	+	0.984972i	$0.444746\pi$
$564$	0	0
$565$	19.8644	0.835702
$566$	0	0
$567$	−2.08719	−0.0876539
$568$	0	0
$569$	16.3614	0.685906	0.342953	−	0.939353i	$-0.388573\pi$
$569$	16.3614	0.685906	0.342953	+	0.939353i	$0.388573\pi$
$570$	0	0
$571$	13.2551	0.554710	0.277355	−	0.960768i	$-0.410542\pi$
$571$	13.2551	0.554710	0.277355	+	0.960768i	$0.410542\pi$
$572$	0	0
$573$	−10.4469	−0.436424
$574$	0	0
$575$	−6.65876	−0.277689
$576$	0	0
$577$	40.1367	1.67091	0.835456	−	0.549557i	$-0.185204\pi$
$577$	40.1367	1.67091	0.835456	+	0.549557i	$0.185204\pi$
$578$	0	0
$579$	7.74909	0.322041
$580$	0	0
$581$	1.43790	0.0596541
$582$	0	0
$583$	−2.84991	−0.118031
$584$	0	0
$585$	0	0
$586$	0	0
$587$	38.0170	1.56913	0.784566	−	0.620046i	$-0.212886\pi$
$587$	38.0170	1.56913	0.784566	+	0.620046i	$0.212886\pi$
$588$	0	0
$589$	−17.7154	−0.729951
$590$	0	0
$591$	40.1838	1.65294
$592$	0	0
$593$	−21.9059	−0.899569	−0.449785	−	0.893137i	$-0.648499\pi$
$593$	−21.9059	−0.899569	−0.449785	+	0.893137i	$0.648499\pi$
$594$	0	0
$595$	1.75513	0.0719535
$596$	0	0
$597$	22.6074	0.925260
$598$	0	0
$599$	26.4065	1.07894	0.539470	−	0.842005i	$-0.318625\pi$
$599$	26.4065	1.07894	0.539470	+	0.842005i	$0.318625\pi$
$600$	0	0
$601$	30.4633	1.24262	0.621312	−	0.783564i	$-0.286600\pi$
$601$	30.4633	1.24262	0.621312	+	0.783564i	$0.286600\pi$
$602$	0	0
$603$	9.88318	0.402474
$604$	0	0
$605$	2.64929	0.107709
$606$	0	0
$607$	−26.4114	−1.07200	−0.536002	−	0.844217i	$-0.680066\pi$
$607$	−26.4114	−1.07200	−0.536002	+	0.844217i	$0.680066\pi$
$608$	0	0
$609$	−1.76860	−0.0716675
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−27.4108	−1.10711	−0.553555	−	0.832813i	$-0.686729\pi$
$613$	−27.4108	−1.10711	−0.553555	+	0.832813i	$0.686729\pi$
$614$	0	0
$615$	−9.60958	−0.387496
$616$	0	0
$617$	−39.0798	−1.57329	−0.786647	−	0.617403i	$-0.788185\pi$
$617$	−39.0798	−1.57329	−0.786647	+	0.617403i	$0.788185\pi$
$618$	0	0
$619$	19.5639	0.786339	0.393170	−	0.919466i	$-0.371379\pi$
$619$	19.5639	0.786339	0.393170	+	0.919466i	$0.371379\pi$
$620$	0	0
$621$	−13.4098	−0.538118
$622$	0	0
$623$	−1.75901	−0.0704733
$624$	0	0
$625$	−31.0184	−1.24074
$626$	0	0
$627$	−16.2301	−0.648169
$628$	0	0
$629$	−26.0754	−1.03970
$630$	0	0
$631$	11.8368	0.471214	0.235607	−	0.971848i	$-0.424292\pi$
$631$	11.8368	0.471214	0.235607	+	0.971848i	$0.424292\pi$
$632$	0	0
$633$	−6.51379	−0.258900
$634$	0	0
$635$	−23.0729	−0.915619
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−12.3152	−0.487182
$640$	0	0
$641$	−11.0021	−0.434558	−0.217279	−	0.976110i	$-0.569718\pi$
$641$	−11.0021	−0.434558	−0.217279	+	0.976110i	$0.569718\pi$
$642$	0	0
$643$	15.0785	0.594637	0.297318	−	0.954778i	$-0.403908\pi$
$643$	15.0785	0.594637	0.297318	+	0.954778i	$0.403908\pi$
$644$	0	0
$645$	−51.5068	−2.02808
$646$	0	0
$647$	−19.1000	−0.750900	−0.375450	−	0.926843i	$-0.622512\pi$
$647$	−19.1000	−0.750900	−0.375450	+	0.926843i	$0.622512\pi$
$648$	0	0
$649$	−7.40983	−0.290861
$650$	0	0
$651$	0.822758	0.0322464
$652$	0	0
$653$	26.1313	1.02260	0.511299	−	0.859403i	$-0.329165\pi$
$653$	26.1313	1.02260	0.511299	+	0.859403i	$0.329165\pi$
$654$	0	0
$655$	−0.238395	−0.00931487
$656$	0	0
$657$	−10.2760	−0.400905
$658$	0	0
$659$	−46.9540	−1.82907	−0.914535	−	0.404507i	$-0.867443\pi$
$659$	−46.9540	−1.82907	−0.914535	+	0.404507i	$0.867443\pi$
$660$	0	0
$661$	16.7213	0.650383	0.325191	−	0.945648i	$-0.394571\pi$
$661$	16.7213	0.650383	0.325191	+	0.945648i	$0.394571\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.11920	0.159736
$666$	0	0
$667$	−15.3931	−0.596024
$668$	0	0
$669$	33.0631	1.27829
$670$	0	0
$671$	10.6188	0.409935
$672$	0	0
$673$	15.7450	0.606926	0.303463	−	0.952843i	$-0.401857\pi$
$673$	15.7450	0.606926	0.303463	+	0.952843i	$0.401857\pi$
$674$	0	0
$675$	8.20716	0.315894
$676$	0	0
$677$	29.8791	1.14835	0.574174	−	0.818733i	$-0.305323\pi$
$677$	29.8791	1.14835	0.574174	+	0.818733i	$0.305323\pi$
$678$	0	0
$679$	1.69708	0.0651281
$680$	0	0
$681$	−31.7852	−1.21801
$682$	0	0
$683$	4.85016	0.185586	0.0927930	−	0.995685i	$-0.470421\pi$
$683$	4.85016	0.185586	0.0927930	+	0.995685i	$0.470421\pi$
$684$	0	0
$685$	0.0881763	0.00336904
$686$	0	0
$687$	1.34325	0.0512482
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−22.8112	−0.867778	−0.433889	−	0.900966i	$-0.642859\pi$
$691$	−22.8112	−0.867778	−0.433889	+	0.900966i	$0.642859\pi$
$692$	0	0
$693$	0.182154	0.00691946
$694$	0	0
$695$	24.6697	0.935774
$696$	0	0
$697$	6.34074	0.240173
$698$	0	0
$699$	−2.10156	−0.0794881
$700$	0	0
$701$	50.4218	1.90441	0.952203	−	0.305466i	$-0.0988121\pi$
$701$	50.4218	1.90441	0.952203	+	0.305466i	$0.0988121\pi$
$702$	0	0
$703$	−61.1976	−2.30811
$704$	0	0
$705$	−63.6156	−2.39590
$706$	0	0
$707$	0.104402	0.00392645
$708$	0	0
$709$	24.1469	0.906854	0.453427	−	0.891293i	$-0.350201\pi$
$709$	24.1469	0.906854	0.453427	+	0.891293i	$0.350201\pi$
$710$	0	0
$711$	1.16576	0.0437194
$712$	0	0
$713$	7.16091	0.268178
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−25.8313	−0.964689
$718$	0	0
$719$	14.8940	0.555453	0.277726	−	0.960660i	$-0.410419\pi$
$719$	14.8940	0.555453	0.277726	+	0.960660i	$0.410419\pi$
$720$	0	0
$721$	−0.782695	−0.0291491
$722$	0	0
$723$	−14.0281	−0.521709
$724$	0	0
$725$	9.42099	0.349887
$726$	0	0
$727$	−19.1909	−0.711752	−0.355876	−	0.934533i	$-0.615817\pi$
$727$	−19.1909	−0.711752	−0.355876	+	0.934533i	$0.615817\pi$
$728$	0	0
$729$	13.7864	0.510607
$730$	0	0
$731$	33.9860	1.25702
$732$	0	0
$733$	−22.7418	−0.839988	−0.419994	−	0.907527i	$-0.637968\pi$
$733$	−22.7418	−0.839988	−0.419994	+	0.907527i	$0.637968\pi$
$734$	0	0
$735$	36.6941	1.35348
$736$	0	0
$737$	10.3382	0.380813
$738$	0	0
$739$	20.4492	0.752235	0.376117	−	0.926572i	$-0.377259\pi$
$739$	20.4492	0.752235	0.376117	+	0.926572i	$0.377259\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	48.0651	1.76334	0.881669	−	0.471869i	$-0.156420\pi$
$743$	48.0651	1.76334	0.881669	+	0.471869i	$0.156420\pi$
$744$	0	0
$745$	15.8481	0.580630
$746$	0	0
$747$	7.21425	0.263955
$748$	0	0
$749$	3.61482	0.132083
$750$	0	0
$751$	6.58747	0.240380	0.120190	−	0.992751i	$-0.461650\pi$
$751$	6.58747	0.240380	0.120190	+	0.992751i	$0.461650\pi$
$752$	0	0
$753$	−18.0721	−0.658584
$754$	0	0
$755$	44.3473	1.61396
$756$	0	0
$757$	3.23728	0.117661	0.0588304	−	0.998268i	$-0.481263\pi$
$757$	3.23728	0.117661	0.0588304	+	0.998268i	$0.481263\pi$
$758$	0	0
$759$	6.56053	0.238132
$760$	0	0
$761$	−41.9276	−1.51988	−0.759938	−	0.649996i	$-0.774771\pi$
$761$	−41.9276	−1.51988	−0.759938	+	0.649996i	$0.774771\pi$
$762$	0	0
$763$	−0.717143	−0.0259623
$764$	0	0
$765$	8.80588	0.318377
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−4.16872	−0.150328	−0.0751640	−	0.997171i	$-0.523948\pi$
$769$	−4.16872	−0.150328	−0.0751640	+	0.997171i	$0.523948\pi$
$770$	0	0
$771$	3.84848	0.138600
$772$	0	0
$773$	−22.0159	−0.791857	−0.395929	−	0.918281i	$-0.629577\pi$
$773$	−22.0159	−0.791857	−0.395929	+	0.918281i	$0.629577\pi$
$774$	0	0
$775$	−4.38266	−0.157430
$776$	0	0
$777$	2.84220	0.101963
$778$	0	0
$779$	14.8813	0.533180
$780$	0	0
$781$	−12.8822	−0.460963
$782$	0	0
$783$	18.9726	0.678025
$784$	0	0
$785$	−63.5734	−2.26903
$786$	0	0
$787$	−4.15976	−0.148279	−0.0741397	−	0.997248i	$-0.523621\pi$
$787$	−4.15976	−0.148279	−0.0741397	+	0.997248i	$0.523621\pi$
$788$	0	0
$789$	26.5032	0.943539
$790$	0	0
$791$	1.42868	0.0507979
$792$	0	0
$793$	0	0
$794$	0	0
$795$	15.0172	0.532605
$796$	0	0
$797$	47.3724	1.67802	0.839009	−	0.544117i	$-0.183135\pi$
$797$	47.3724	1.67802	0.839009	+	0.544117i	$0.183135\pi$
$798$	0	0
$799$	41.9758	1.48500
$800$	0	0
$801$	−8.82534	−0.311828
$802$	0	0
$803$	−10.7491	−0.379328
$804$	0	0
$805$	−1.66506	−0.0586856
$806$	0	0
$807$	32.3760	1.13969
$808$	0	0
$809$	−48.3590	−1.70021	−0.850106	−	0.526611i	$-0.823462\pi$
$809$	−48.3590	−1.70021	−0.850106	+	0.526611i	$0.823462\pi$
$810$	0	0
$811$	12.6854	0.445446	0.222723	−	0.974882i	$-0.428506\pi$
$811$	12.6854	0.445446	0.222723	+	0.974882i	$0.428506\pi$
$812$	0	0
$813$	0.238545	0.00836615
$814$	0	0
$815$	−13.3652	−0.468161
$816$	0	0
$817$	79.7631	2.79056
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−5.48837	−0.191546	−0.0957728	−	0.995403i	$-0.530532\pi$
$821$	−5.48837	−0.191546	−0.0957728	+	0.995403i	$0.530532\pi$
$822$	0	0
$823$	−19.2895	−0.672391	−0.336196	−	0.941792i	$-0.609140\pi$
$823$	−19.2895	−0.672391	−0.336196	+	0.941792i	$0.609140\pi$
$824$	0	0
$825$	−4.01521	−0.139792
$826$	0	0
$827$	−15.3225	−0.532815	−0.266407	−	0.963861i	$-0.585837\pi$
$827$	−15.3225	−0.532815	−0.266407	+	0.963861i	$0.585837\pi$
$828$	0	0
$829$	−0.246410	−0.00855818	−0.00427909	−	0.999991i	$-0.501362\pi$
$829$	−0.246410	−0.00855818	−0.00427909	+	0.999991i	$0.501362\pi$
$830$	0	0
$831$	−47.5247	−1.64861
$832$	0	0
$833$	−24.2121	−0.838899
$834$	0	0
$835$	−21.6982	−0.750897
$836$	0	0
$837$	−8.82609	−0.305074
$838$	0	0
$839$	−4.12856	−0.142534	−0.0712668	−	0.997457i	$-0.522704\pi$
$839$	−4.12856	−0.142534	−0.0712668	+	0.997457i	$0.522704\pi$
$840$	0	0
$841$	−7.22140	−0.249014
$842$	0	0
$843$	−48.1958	−1.65995
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.190541	0.00654706
$848$	0	0
$849$	31.6211	1.08523
$850$	0	0
$851$	24.7372	0.847982
$852$	0	0
$853$	−0.871461	−0.0298382	−0.0149191	−	0.999889i	$-0.504749\pi$
$853$	−0.871461	−0.0298382	−0.0149191	+	0.999889i	$0.504749\pi$
$854$	0	0
$855$	20.6669	0.706793
$856$	0	0
$857$	−18.4913	−0.631652	−0.315826	−	0.948817i	$-0.602282\pi$
$857$	−18.4913	−0.631652	−0.315826	+	0.948817i	$0.602282\pi$
$858$	0	0
$859$	16.3878	0.559146	0.279573	−	0.960124i	$-0.409807\pi$
$859$	16.3878	0.559146	0.279573	+	0.960124i	$0.409807\pi$
$860$	0	0
$861$	−0.691135	−0.0235538
$862$	0	0
$863$	−8.73163	−0.297228	−0.148614	−	0.988895i	$-0.547481\pi$
$863$	−8.73163	−0.297228	−0.148614	+	0.988895i	$0.547481\pi$
$864$	0	0
$865$	−3.54404	−0.120501
$866$	0	0
$867$	9.76813	0.331743
$868$	0	0
$869$	1.21943	0.0413665
$870$	0	0
$871$	0	0
$872$	0	0
$873$	8.51463	0.288176
$874$	0	0
$875$	−1.50493	−0.0508760
$876$	0	0
$877$	−24.9404	−0.842178	−0.421089	−	0.907019i	$-0.638352\pi$
$877$	−24.9404	−0.842178	−0.421089	+	0.907019i	$0.638352\pi$
$878$	0	0
$879$	36.5308	1.23215
$880$	0	0
$881$	−53.8711	−1.81496	−0.907481	−	0.420093i	$-0.861997\pi$
$881$	−53.8711	−1.81496	−0.907481	+	0.420093i	$0.861997\pi$
$882$	0	0
$883$	−4.59752	−0.154719	−0.0773595	−	0.997003i	$-0.524649\pi$
$883$	−4.59752	−0.154719	−0.0773595	+	0.997003i	$0.524649\pi$
$884$	0	0
$885$	39.0450	1.31248
$886$	0	0
$887$	20.0063	0.671747	0.335873	−	0.941907i	$-0.390969\pi$
$887$	20.0063	0.671747	0.335873	+	0.941907i	$0.390969\pi$
$888$	0	0
$889$	−1.65944	−0.0556557
$890$	0	0
$891$	−10.9540	−0.366974
$892$	0	0
$893$	98.5148	3.29667
$894$	0	0
$895$	30.6403	1.02419
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−10.1315	−0.337903
$900$	0	0
$901$	−9.90887	−0.330112
$902$	0	0
$903$	−3.70444	−0.123276
$904$	0	0
$905$	49.2701	1.63779
$906$	0	0
$907$	−28.8631	−0.958384	−0.479192	−	0.877710i	$-0.659070\pi$
$907$	−28.8631	−0.958384	−0.479192	+	0.877710i	$0.659070\pi$
$908$	0	0
$909$	0.523808	0.0173736
$910$	0	0
$911$	24.5946	0.814855	0.407428	−	0.913237i	$-0.366426\pi$
$911$	24.5946	0.814855	0.407428	+	0.913237i	$0.366426\pi$
$912$	0	0
$913$	7.54640	0.249750
$914$	0	0
$915$	−55.9543	−1.84979
$916$	0	0
$917$	−0.0171457	−0.000566202 0
$918$	0	0
$919$	−46.5427	−1.53530	−0.767651	−	0.640869i	$-0.778574\pi$
$919$	−46.5427	−1.53530	−0.767651	+	0.640869i	$0.778574\pi$
$920$	0	0
$921$	−40.3419	−1.32931
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−15.1398	−0.497795
$926$	0	0
$927$	−3.92695	−0.128978
$928$	0	0
$929$	47.6660	1.56387	0.781935	−	0.623360i	$-0.214233\pi$
$929$	47.6660	1.56387	0.781935	+	0.623360i	$0.214233\pi$
$930$	0	0
$931$	−56.8243	−1.86234
$932$	0	0
$933$	54.9412	1.79869
$934$	0	0
$935$	9.21132	0.301242
$936$	0	0
$937$	−44.3595	−1.44916	−0.724581	−	0.689190i	$-0.757967\pi$
$937$	−44.3595	−1.44916	−0.724581	+	0.689190i	$0.757967\pi$
$938$	0	0
$939$	66.7068	2.17690
$940$	0	0
$941$	45.1446	1.47167	0.735836	−	0.677159i	$-0.236789\pi$
$941$	45.1446	1.47167	0.735836	+	0.677159i	$0.236789\pi$
$942$	0	0
$943$	−6.01533	−0.195886
$944$	0	0
$945$	2.05225	0.0667596
$946$	0	0
$947$	1.89948	0.0617248	0.0308624	−	0.999524i	$-0.490175\pi$
$947$	1.89948	0.0617248	0.0308624	+	0.999524i	$0.490175\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−55.0754	−1.78594
$952$	0	0
$953$	50.5256	1.63669	0.818343	−	0.574731i	$-0.194893\pi$
$953$	50.5256	1.63669	0.818343	+	0.574731i	$0.194893\pi$
$954$	0	0
$955$	13.9152	0.450284
$956$	0	0
$957$	−9.28202	−0.300045
$958$	0	0
$959$	0.00634177	0.000204786 0
$960$	0	0
$961$	−26.2868	−0.847962
$962$	0	0
$963$	18.1363	0.584435
$964$	0	0
$965$	−10.3217	−0.332269
$966$	0	0
$967$	−4.86827	−0.156553	−0.0782765	−	0.996932i	$-0.524942\pi$
$967$	−4.86827	−0.156553	−0.0782765	+	0.996932i	$0.524942\pi$
$968$	0	0
$969$	−56.4306	−1.81281
$970$	0	0
$971$	−59.9096	−1.92259	−0.961295	−	0.275522i	$-0.911149\pi$
$971$	−59.9096	−1.92259	−0.961295	+	0.275522i	$0.911149\pi$
$972$	0	0
$973$	1.77428	0.0568808
$974$	0	0
$975$	0	0
$976$	0	0
$977$	25.7420	0.823561	0.411780	−	0.911283i	$-0.364907\pi$
$977$	25.7420	0.823561	0.411780	+	0.911283i	$0.364907\pi$
$978$	0	0
$979$	−9.23167	−0.295046
$980$	0	0
$981$	−3.59806	−0.114877
$982$	0	0
$983$	24.9613	0.796142	0.398071	−	0.917355i	$-0.369680\pi$
$983$	24.9613	0.796142	0.398071	+	0.917355i	$0.369680\pi$
$984$	0	0
$985$	−53.5246	−1.70543
$986$	0	0
$987$	−4.57533	−0.145634
$988$	0	0
$989$	−32.2418	−1.02523
$990$	0	0
$991$	−42.1060	−1.33754	−0.668770	−	0.743469i	$-0.733179\pi$
$991$	−42.1060	−1.33754	−0.668770	+	0.743469i	$0.733179\pi$
$992$	0	0
$993$	14.8751	0.472047
$994$	0	0
$995$	−30.1130	−0.954645
$996$	0	0
$997$	−49.1450	−1.55644	−0.778218	−	0.627994i	$-0.783876\pi$
$997$	−49.1450	−1.55644	−0.778218	+	0.627994i	$0.783876\pi$
$998$	0	0
$999$	−30.4895	−0.964647

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7436.2.a.v.1.3		12
13.6	odd	12		572.2.p.a.309.10	✓	24
13.11	odd	12		572.2.p.a.485.10	yes	24
13.12	even	2		7436.2.a.u.1.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.p.a.309.10	✓	24	13.6	odd	12
572.2.p.a.485.10	yes	24	13.11	odd	12
7436.2.a.u.1.3		12	13.12	even	2
7436.2.a.v.1.3		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7436 = 2^{2} \cdot 11 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7436.a (trivial)

\(f(q)\)	\(=\)	\(q-1.98897 q^{3} +2.64929 q^{5} +0.190541 q^{7} +0.955984 q^{9} +O(q^{10})\)	\(q-1.98897 q^{3} +2.64929 q^{5} +0.190541 q^{7} +0.955984 q^{9} +1.00000 q^{11} -5.26935 q^{15} +3.47690 q^{17} +8.16009 q^{19} -0.378979 q^{21} -3.29846 q^{23} +2.01875 q^{25} +4.06548 q^{27} +4.66676 q^{29} -2.17098 q^{31} -1.98897 q^{33} +0.504798 q^{35} -7.49962 q^{37} +1.82368 q^{41} +9.77479 q^{43} +2.53268 q^{45} +12.0728 q^{47} -6.96369 q^{49} -6.91544 q^{51} -2.84991 q^{53} +2.64929 q^{55} -16.2301 q^{57} -7.40983 q^{59} +10.6188 q^{61} +0.182154 q^{63} +10.3382 q^{67} +6.56053 q^{69} -12.8822 q^{71} -10.7491 q^{73} -4.01521 q^{75} +0.190541 q^{77} +1.21943 q^{79} -10.9540 q^{81} +7.54640 q^{83} +9.21132 q^{85} -9.28202 q^{87} -9.23167 q^{89} +4.31801 q^{93} +21.6184 q^{95} +8.90666 q^{97} +0.955984 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{3} + 8 q^{7} + 14 q^{9}+O(q^{10}) \) 12 * q + 2 * q^3 + 8 * q^7 + 14 * q^9	\( 12 q + 2 q^{3} + 8 q^{7} + 14 q^{9} + 12 q^{11} + 8 q^{15} + 22 q^{19} + 10 q^{21} + 10 q^{23} + 20 q^{25} - 4 q^{27} + 8 q^{29} + 28 q^{31} + 2 q^{33} - 8 q^{35} - 16 q^{37} + 16 q^{41} + 10 q^{43} + 12 q^{47} + 14 q^{49} - 22 q^{51} + 8 q^{53} + 36 q^{57} + 32 q^{59} - 6 q^{61} - 22 q^{63} + 48 q^{67} + 10 q^{69} + 20 q^{71} + 16 q^{73} + 6 q^{75} + 8 q^{77} - 16 q^{79} + 4 q^{81} + 10 q^{83} + 42 q^{85} - 52 q^{87} - 16 q^{89} - 4 q^{93} - 12 q^{95} + 20 q^{97} + 14 q^{99}+O(q^{100}) \) 12 * q + 2 * q^3 + 8 * q^7 + 14 * q^9 + 12 * q^11 + 8 * q^15 + 22 * q^19 + 10 * q^21 + 10 * q^23 + 20 * q^25 - 4 * q^27 + 8 * q^29 + 28 * q^31 + 2 * q^33 - 8 * q^35 - 16 * q^37 + 16 * q^41 + 10 * q^43 + 12 * q^47 + 14 * q^49 - 22 * q^51 + 8 * q^53 + 36 * q^57 + 32 * q^59 - 6 * q^61 - 22 * q^63 + 48 * q^67 + 10 * q^69 + 20 * q^71 + 16 * q^73 + 6 * q^75 + 8 * q^77 - 16 * q^79 + 4 * q^81 + 10 * q^83 + 42 * q^85 - 52 * q^87 - 16 * q^89 - 4 * q^93 - 12 * q^95 + 20 * q^97 + 14 * q^99

Introduction

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