LMFDB - Embedded newform 7396.2.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7396,2,Mod(1,7396)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7396, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7396.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7396 = 2^{2} \cdot 43^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7396.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:	$59.0573573349$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.26624689.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 18x^{4} + 7x^{3} + 96x^{2} - 2x - 139 $ x^6 - x^5 - 18x^4 + 7x^3 + 96x^2 - 2x - 139
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 172)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.41164$ of defining polynomial
Character	$\chi$	$=$	7396.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.445042 q^{3} -0.411642 q^{5} +2.98873 q^{7} -2.80194 q^{9} +O(q^{10})$	$q+0.445042 q^{3} -0.411642 q^{5} +2.98873 q^{7} -2.80194 q^{9} +1.13205 q^{11} +3.31525 q^{13} -0.183198 q^{15} +2.90560 q^{17} +4.84542 q^{19} +1.33011 q^{21} +7.11918 q^{23} -4.83055 q^{25} -2.58211 q^{27} +1.22563 q^{29} -1.65361 q^{31} +0.503809 q^{33} -1.23029 q^{35} -1.26184 q^{37} +1.47542 q^{39} +3.62241 q^{41} +1.15339 q^{45} +5.70512 q^{47} +1.93252 q^{49} +1.29311 q^{51} -11.0057 q^{53} -0.465998 q^{55} +2.15641 q^{57} +9.76677 q^{59} -1.43459 q^{61} -8.37424 q^{63} -1.36469 q^{65} -3.45975 q^{67} +3.16833 q^{69} +4.26552 q^{71} +13.5605 q^{73} -2.14980 q^{75} +3.38339 q^{77} +3.59704 q^{79} +7.25667 q^{81} -2.20982 q^{83} -1.19607 q^{85} +0.545458 q^{87} -18.4966 q^{89} +9.90838 q^{91} -0.735924 q^{93} -1.99457 q^{95} -9.51384 q^{97} -3.17193 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} + 5 q^{5} - 8 q^{9}+O(q^{10}) $ 6 * q + 2 * q^3 + 5 * q^5 - 8 * q^9	$ 6 q + 2 q^{3} + 5 q^{5} - 8 q^{9} - 3 q^{11} + 5 q^{13} + 4 q^{15} - 9 q^{17} + 3 q^{19} + 7 q^{21} + 11 q^{25} - 4 q^{27} + 4 q^{29} + 6 q^{31} - q^{33} - 8 q^{35} - 12 q^{37} - 3 q^{39} - 3 q^{41} - 9 q^{45} + 18 q^{47} + 12 q^{49} + 4 q^{51} - 4 q^{53} + 24 q^{55} + 15 q^{57} + 23 q^{59} - 10 q^{61} + 7 q^{63} + 25 q^{65} + 25 q^{67} + 9 q^{71} + 37 q^{73} - q^{75} + 39 q^{77} + 7 q^{79} - 10 q^{81} + 4 q^{83} - 35 q^{85} + 6 q^{87} + 13 q^{89} + 9 q^{91} + 23 q^{93} - 40 q^{95} - 51 q^{97} + 4 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 + 5 * q^5 - 8 * q^9 - 3 * q^11 + 5 * q^13 + 4 * q^15 - 9 * q^17 + 3 * q^19 + 7 * q^21 + 11 * q^25 - 4 * q^27 + 4 * q^29 + 6 * q^31 - q^33 - 8 * q^35 - 12 * q^37 - 3 * q^39 - 3 * q^41 - 9 * q^45 + 18 * q^47 + 12 * q^49 + 4 * q^51 - 4 * q^53 + 24 * q^55 + 15 * q^57 + 23 * q^59 - 10 * q^61 + 7 * q^63 + 25 * q^65 + 25 * q^67 + 9 * q^71 + 37 * q^73 - q^75 + 39 * q^77 + 7 * q^79 - 10 * q^81 + 4 * q^83 - 35 * q^85 + 6 * q^87 + 13 * q^89 + 9 * q^91 + 23 * q^93 - 40 * q^95 - 51 * q^97 + 4 * 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
$3$	0.445042	0.256945	0.128473	−	0.991713i	$-0.458993\pi$
$3$	0.445042	0.256945	0.128473	+	0.991713i	$0.458993\pi$
$4$	0	0
$5$	−0.411642	−0.184092	−0.0920459	−	0.995755i	$-0.529341\pi$
$5$	−0.411642	−0.184092	−0.0920459	+	0.995755i	$0.529341\pi$
$6$	0	0
$7$	2.98873	1.12963	0.564817	−	0.825216i	$-0.308947\pi$
$7$	2.98873	1.12963	0.564817	+	0.825216i	$0.308947\pi$
$8$	0	0
$9$	−2.80194	−0.933979
$10$	0	0
$11$	1.13205	0.341326	0.170663	−	0.985329i	$-0.445409\pi$
$11$	1.13205	0.341326	0.170663	+	0.985329i	$0.445409\pi$
$12$	0	0
$13$	3.31525	0.919484	0.459742	−	0.888053i	$-0.347942\pi$
$13$	3.31525	0.919484	0.459742	+	0.888053i	$0.347942\pi$
$14$	0	0
$15$	−0.183198	−0.0473015
$16$	0	0
$17$	2.90560	0.704712	0.352356	−	0.935866i	$-0.385381\pi$
$17$	2.90560	0.704712	0.352356	+	0.935866i	$0.385381\pi$
$18$	0	0
$19$	4.84542	1.11161	0.555807	−	0.831311i	$-0.312409\pi$
$19$	4.84542	1.11161	0.555807	+	0.831311i	$0.312409\pi$
$20$	0	0
$21$	1.33011	0.290254
$22$	0	0
$23$	7.11918	1.48445	0.742226	−	0.670150i	$-0.233770\pi$
$23$	7.11918	1.48445	0.742226	+	0.670150i	$0.233770\pi$
$24$	0	0
$25$	−4.83055	−0.966110
$26$	0	0
$27$	−2.58211	−0.496926
$28$	0	0
$29$	1.22563	0.227595	0.113797	−	0.993504i	$-0.463699\pi$
$29$	1.22563	0.227595	0.113797	+	0.993504i	$0.463699\pi$
$30$	0	0
$31$	−1.65361	−0.296997	−0.148498	−	0.988913i	$-0.547444\pi$
$31$	−1.65361	−0.296997	−0.148498	+	0.988913i	$0.547444\pi$
$32$	0	0
$33$	0.503809	0.0877019
$34$	0	0
$35$	−1.23029	−0.207956
$36$	0	0
$37$	−1.26184	−0.207446	−0.103723	−	0.994606i	$-0.533076\pi$
$37$	−1.26184	−0.207446	−0.103723	+	0.994606i	$0.533076\pi$
$38$	0	0
$39$	1.47542	0.236257
$40$	0	0
$41$	3.62241	0.565726	0.282863	−	0.959160i	$-0.408716\pi$
$41$	3.62241	0.565726	0.282863	+	0.959160i	$0.408716\pi$
$42$	0	0
$43$	0	0
$43$	0	0
$44$	0	0
$45$	1.15339	0.171938
$46$	0	0
$47$	5.70512	0.832177	0.416088	−	0.909324i	$-0.363401\pi$
$47$	5.70512	0.832177	0.416088	+	0.909324i	$0.363401\pi$
$48$	0	0
$49$	1.93252	0.276074
$50$	0	0
$51$	1.29311	0.181072
$52$	0	0
$53$	−11.0057	−1.51175	−0.755876	−	0.654715i	$-0.772789\pi$
$53$	−11.0057	−1.51175	−0.755876	+	0.654715i	$0.772789\pi$
$54$	0	0
$55$	−0.465998	−0.0628352
$56$	0	0
$57$	2.15641	0.285624
$58$	0	0
$59$	9.76677	1.27153	0.635763	−	0.771885i	$-0.280686\pi$
$59$	9.76677	1.27153	0.635763	+	0.771885i	$0.280686\pi$
$60$	0	0
$61$	−1.43459	−0.183680	−0.0918401	−	0.995774i	$-0.529275\pi$
$61$	−1.43459	−0.183680	−0.0918401	+	0.995774i	$0.529275\pi$
$62$	0	0
$63$	−8.37424	−1.05506
$64$	0	0
$65$	−1.36469	−0.169269
$66$	0	0
$67$	−3.45975	−0.422675	−0.211338	−	0.977413i	$-0.567782\pi$
$67$	−3.45975	−0.422675	−0.211338	+	0.977413i	$0.567782\pi$
$68$	0	0
$69$	3.16833	0.381423
$70$	0	0
$71$	4.26552	0.506224	0.253112	−	0.967437i	$-0.418546\pi$
$71$	4.26552	0.506224	0.253112	+	0.967437i	$0.418546\pi$
$72$	0	0
$73$	13.5605	1.58713	0.793566	−	0.608484i	$-0.208222\pi$
$73$	13.5605	1.58713	0.793566	+	0.608484i	$0.208222\pi$
$74$	0	0
$75$	−2.14980	−0.248237
$76$	0	0
$77$	3.38339	0.385573
$78$	0	0
$79$	3.59704	0.404699	0.202350	−	0.979313i	$-0.435142\pi$
$79$	3.59704	0.404699	0.202350	+	0.979313i	$0.435142\pi$
$80$	0	0
$81$	7.25667	0.806296
$82$	0	0
$83$	−2.20982	−0.242559	−0.121279	−	0.992618i	$-0.538700\pi$
$83$	−2.20982	−0.242559	−0.121279	+	0.992618i	$0.538700\pi$
$84$	0	0
$85$	−1.19607	−0.129732
$86$	0	0
$87$	0.545458	0.0584793
$88$	0	0
$89$	−18.4966	−1.96064	−0.980319	−	0.197418i	$-0.936744\pi$
$89$	−18.4966	−1.96064	−0.980319	+	0.197418i	$0.936744\pi$
$90$	0	0
$91$	9.90838	1.03868
$92$	0	0
$93$	−0.735924	−0.0763118
$94$	0	0
$95$	−1.99457	−0.204639
$96$	0	0
$97$	−9.51384	−0.965984	−0.482992	−	0.875625i	$-0.660450\pi$
$97$	−9.51384	−0.965984	−0.482992	+	0.875625i	$0.660450\pi$
$98$	0	0
$99$	−3.17193	−0.318791
$100$	0	0
$101$	3.94749	0.392790	0.196395	−	0.980525i	$-0.437077\pi$
$101$	3.94749	0.392790	0.196395	+	0.980525i	$0.437077\pi$
$102$	0	0
$103$	−2.36738	−0.233265	−0.116632	−	0.993175i	$-0.537210\pi$
$103$	−2.36738	−0.233265	−0.116632	+	0.993175i	$0.537210\pi$
$104$	0	0
$105$	−0.547529	−0.0534334
$106$	0	0
$107$	−3.21277	−0.310590	−0.155295	−	0.987868i	$-0.549633\pi$
$107$	−3.21277	−0.310590	−0.155295	+	0.987868i	$0.549633\pi$
$108$	0	0
$109$	−4.97891	−0.476893	−0.238447	−	0.971156i	$-0.576638\pi$
$109$	−4.97891	−0.476893	−0.238447	+	0.971156i	$0.576638\pi$
$110$	0	0
$111$	−0.561573	−0.0533022
$112$	0	0
$113$	−11.4378	−1.07598	−0.537989	−	0.842952i	$-0.680816\pi$
$113$	−11.4378	−1.07598	−0.537989	+	0.842952i	$0.680816\pi$
$114$	0	0
$115$	−2.93055	−0.273275
$116$	0	0
$117$	−9.28911	−0.858779
$118$	0	0
$119$	8.68406	0.796067
$120$	0	0
$121$	−9.71847	−0.883497
$122$	0	0
$123$	1.61212	0.145360
$124$	0	0
$125$	4.04666	0.361945
$126$	0	0
$127$	9.53079	0.845721	0.422861	−	0.906195i	$-0.361026\pi$
$127$	9.53079	0.845721	0.422861	+	0.906195i	$0.361026\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.0158	−1.31193	−0.655967	−	0.754790i	$-0.727739\pi$
$131$	−15.0158	−1.31193	−0.655967	+	0.754790i	$0.727739\pi$
$132$	0	0
$133$	14.4816	1.25572
$134$	0	0
$135$	1.06290	0.0914800
$136$	0	0
$137$	−2.40116	−0.205145	−0.102573	−	0.994726i	$-0.532707\pi$
$137$	−2.40116	−0.205145	−0.102573	+	0.994726i	$0.532707\pi$
$138$	0	0
$139$	18.6639	1.58305	0.791525	−	0.611137i	$-0.209287\pi$
$139$	18.6639	1.58305	0.791525	+	0.611137i	$0.209287\pi$
$140$	0	0
$141$	2.53902	0.213824
$142$	0	0
$143$	3.75302	0.313843
$144$	0	0
$145$	−0.504522	−0.0418983
$146$	0	0
$147$	0.860052	0.0709359
$148$	0	0
$149$	21.7072	1.77833	0.889163	−	0.457590i	$-0.151287\pi$
$149$	21.7072	1.77833	0.889163	+	0.457590i	$0.151287\pi$
$150$	0	0
$151$	12.6844	1.03224	0.516121	−	0.856515i	$-0.327375\pi$
$151$	12.6844	1.03224	0.516121	+	0.856515i	$0.327375\pi$
$152$	0	0
$153$	−8.14131	−0.658186
$154$	0	0
$155$	0.680693	0.0546746
$156$	0	0
$157$	20.1618	1.60909	0.804545	−	0.593891i	$-0.202409\pi$
$157$	20.1618	1.60909	0.804545	+	0.593891i	$0.202409\pi$
$158$	0	0
$159$	−4.89800	−0.388437
$160$	0	0
$161$	21.2773	1.67689
$162$	0	0
$163$	13.8295	1.08321	0.541604	−	0.840634i	$-0.317817\pi$
$163$	13.8295	1.08321	0.541604	+	0.840634i	$0.317817\pi$
$164$	0	0
$165$	−0.207389	−0.0161452
$166$	0	0
$167$	11.9360	0.923632	0.461816	−	0.886976i	$-0.347198\pi$
$167$	11.9360	0.923632	0.461816	+	0.886976i	$0.347198\pi$
$168$	0	0
$169$	−2.00914	−0.154549
$170$	0	0
$171$	−13.5766	−1.03822
$172$	0	0
$173$	−11.6203	−0.883474	−0.441737	−	0.897145i	$-0.645638\pi$
$173$	−11.6203	−0.883474	−0.441737	+	0.897145i	$0.645638\pi$
$174$	0	0
$175$	−14.4372	−1.09135
$176$	0	0
$177$	4.34662	0.326712
$178$	0	0
$179$	20.5632	1.53697	0.768484	−	0.639869i	$-0.221011\pi$
$179$	20.5632	1.53697	0.768484	+	0.639869i	$0.221011\pi$
$180$	0	0
$181$	−19.8552	−1.47582	−0.737911	−	0.674898i	$-0.764188\pi$
$181$	−19.8552	−1.47582	−0.737911	+	0.674898i	$0.764188\pi$
$182$	0	0
$183$	−0.638452	−0.0471957
$184$	0	0
$185$	0.519428	0.0381891
$186$	0	0
$187$	3.28928	0.240536
$188$	0	0
$189$	−7.71722	−0.561345
$190$	0	0
$191$	−7.45301	−0.539281	−0.269640	−	0.962961i	$-0.586905\pi$
$191$	−7.45301	−0.539281	−0.269640	+	0.962961i	$0.586905\pi$
$192$	0	0
$193$	−16.6768	−1.20042	−0.600210	−	0.799842i	$-0.704916\pi$
$193$	−16.6768	−1.20042	−0.600210	+	0.799842i	$0.704916\pi$
$194$	0	0
$195$	−0.607346	−0.0434929
$196$	0	0
$197$	9.04029	0.644094	0.322047	−	0.946724i	$-0.395629\pi$
$197$	9.04029	0.644094	0.322047	+	0.946724i	$0.395629\pi$
$198$	0	0
$199$	−4.37265	−0.309969	−0.154985	−	0.987917i	$-0.549533\pi$
$199$	−4.37265	−0.309969	−0.154985	+	0.987917i	$0.549533\pi$
$200$	0	0
$201$	−1.53973	−0.108604
$202$	0	0
$203$	3.66309	0.257099
$204$	0	0
$205$	−1.49114	−0.104145
$206$	0	0
$207$	−19.9475	−1.38645
$208$	0	0
$209$	5.48525	0.379422
$210$	0	0
$211$	13.8724	0.955018	0.477509	−	0.878627i	$-0.341540\pi$
$211$	13.8724	0.955018	0.477509	+	0.878627i	$0.341540\pi$
$212$	0	0
$213$	1.89833	0.130072
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.94219	−0.335498
$218$	0	0
$219$	6.03497	0.407806
$220$	0	0
$221$	9.63278	0.647971
$222$	0	0
$223$	23.9713	1.60524	0.802618	−	0.596493i	$-0.203440\pi$
$223$	23.9713	1.60524	0.802618	+	0.596493i	$0.203440\pi$
$224$	0	0
$225$	13.5349	0.902327
$226$	0	0
$227$	−15.0045	−0.995882	−0.497941	−	0.867211i	$-0.665910\pi$
$227$	−15.0045	−0.995882	−0.497941	+	0.867211i	$0.665910\pi$
$228$	0	0
$229$	17.1474	1.13313	0.566567	−	0.824016i	$-0.308271\pi$
$229$	17.1474	1.13313	0.566567	+	0.824016i	$0.308271\pi$
$230$	0	0
$231$	1.50575	0.0990711
$232$	0	0
$233$	25.7010	1.68373	0.841864	−	0.539689i	$-0.181458\pi$
$233$	25.7010	1.68373	0.841864	+	0.539689i	$0.181458\pi$
$234$	0	0
$235$	−2.34846	−0.153197
$236$	0	0
$237$	1.60084	0.103985
$238$	0	0
$239$	−21.1341	−1.36705	−0.683526	−	0.729926i	$-0.739554\pi$
$239$	−21.1341	−1.36705	−0.683526	+	0.729926i	$0.739554\pi$
$240$	0	0
$241$	19.3239	1.24476	0.622382	−	0.782713i	$-0.286165\pi$
$241$	19.3239	1.24476	0.622382	+	0.782713i	$0.286165\pi$
$242$	0	0
$243$	10.9758	0.704100
$244$	0	0
$245$	−0.795506	−0.0508230
$246$	0	0
$247$	16.0637	1.02211
$248$	0	0
$249$	−0.983460	−0.0623243
$250$	0	0
$251$	25.4283	1.60502	0.802510	−	0.596639i	$-0.203498\pi$
$251$	25.4283	1.60502	0.802510	+	0.596639i	$0.203498\pi$
$252$	0	0
$253$	8.05926	0.506681
$254$	0	0
$255$	−0.532300	−0.0333339
$256$	0	0
$257$	13.4695	0.840204	0.420102	−	0.907477i	$-0.361994\pi$
$257$	13.4695	0.840204	0.420102	+	0.907477i	$0.361994\pi$
$258$	0	0
$259$	−3.77131	−0.234338
$260$	0	0
$261$	−3.43415	−0.212569
$262$	0	0
$263$	30.7827	1.89815	0.949073	−	0.315057i	$-0.102024\pi$
$263$	30.7827	1.89815	0.949073	+	0.315057i	$0.102024\pi$
$264$	0	0
$265$	4.53041	0.278301
$266$	0	0
$267$	−8.23177	−0.503776
$268$	0	0
$269$	−28.8048	−1.75626	−0.878128	−	0.478425i	$-0.841208\pi$
$269$	−28.8048	−1.75626	−0.878128	+	0.478425i	$0.841208\pi$
$270$	0	0
$271$	−20.6052	−1.25168	−0.625840	−	0.779952i	$-0.715244\pi$
$271$	−20.6052	−1.25168	−0.625840	+	0.779952i	$0.715244\pi$
$272$	0	0
$273$	4.40965	0.266884
$274$	0	0
$275$	−5.46842	−0.329758
$276$	0	0
$277$	−6.17188	−0.370832	−0.185416	−	0.982660i	$-0.559363\pi$
$277$	−6.17188	−0.370832	−0.185416	+	0.982660i	$0.559363\pi$
$278$	0	0
$279$	4.63330	0.277389
$280$	0	0
$281$	16.8563	1.00556	0.502781	−	0.864414i	$-0.332310\pi$
$281$	16.8563	1.00556	0.502781	+	0.864414i	$0.332310\pi$
$282$	0	0
$283$	−8.41171	−0.500024	−0.250012	−	0.968243i	$-0.580435\pi$
$283$	−8.41171	−0.500024	−0.250012	+	0.968243i	$0.580435\pi$
$284$	0	0
$285$	−0.887669	−0.0525810
$286$	0	0
$287$	10.8264	0.639063
$288$	0	0
$289$	−8.55748	−0.503381
$290$	0	0
$291$	−4.23406	−0.248205
$292$	0	0
$293$	−4.58356	−0.267774	−0.133887	−	0.990997i	$-0.542746\pi$
$293$	−4.58356	−0.267774	−0.133887	+	0.990997i	$0.542746\pi$
$294$	0	0
$295$	−4.02041	−0.234077
$296$	0	0
$297$	−2.92307	−0.169614
$298$	0	0
$299$	23.6018	1.36493
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1.75680	0.100925
$304$	0	0
$305$	0.590536	0.0338140
$306$	0	0
$307$	1.81686	0.103694	0.0518468	−	0.998655i	$-0.483489\pi$
$307$	1.81686	0.103694	0.0518468	+	0.998655i	$0.483489\pi$
$308$	0	0
$309$	−1.05358	−0.0599362
$310$	0	0
$311$	−17.9922	−1.02024	−0.510121	−	0.860103i	$-0.670399\pi$
$311$	−17.9922	−1.02024	−0.510121	+	0.860103i	$0.670399\pi$
$312$	0	0
$313$	−26.5250	−1.49928	−0.749640	−	0.661845i	$-0.769774\pi$
$313$	−26.5250	−1.49928	−0.749640	+	0.661845i	$0.769774\pi$
$314$	0	0
$315$	3.44719	0.194227
$316$	0	0
$317$	29.7968	1.67356	0.836778	−	0.547542i	$-0.184436\pi$
$317$	29.7968	1.67356	0.836778	+	0.547542i	$0.184436\pi$
$318$	0	0
$319$	1.38748	0.0776838
$320$	0	0
$321$	−1.42982	−0.0798045
$322$	0	0
$323$	14.0788	0.783368
$324$	0	0
$325$	−16.0145	−0.888323
$326$	0	0
$327$	−2.21582	−0.122535
$328$	0	0
$329$	17.0511	0.940056
$330$	0	0
$331$	6.55262	0.360165	0.180082	−	0.983652i	$-0.442364\pi$
$331$	6.55262	0.360165	0.180082	+	0.983652i	$0.442364\pi$
$332$	0	0
$333$	3.53561	0.193750
$334$	0	0
$335$	1.42418	0.0778110
$336$	0	0
$337$	−9.06955	−0.494050	−0.247025	−	0.969009i	$-0.579453\pi$
$337$	−9.06955	−0.494050	−0.247025	+	0.969009i	$0.579453\pi$
$338$	0	0
$339$	−5.09030	−0.276467
$340$	0	0
$341$	−1.87196	−0.101372
$342$	0	0
$343$	−15.1453	−0.817772
$344$	0	0
$345$	−1.30422	−0.0702167
$346$	0	0
$347$	−17.9354	−0.962823	−0.481412	−	0.876495i	$-0.659876\pi$
$347$	−17.9354	−0.962823	−0.481412	+	0.876495i	$0.659876\pi$
$348$	0	0
$349$	7.47591	0.400177	0.200088	−	0.979778i	$-0.435877\pi$
$349$	7.47591	0.400177	0.200088	+	0.979778i	$0.435877\pi$
$350$	0	0
$351$	−8.56032	−0.456916
$352$	0	0
$353$	−6.47638	−0.344703	−0.172351	−	0.985036i	$-0.555136\pi$
$353$	−6.47638	−0.344703	−0.172351	+	0.985036i	$0.555136\pi$
$354$	0	0
$355$	−1.75586	−0.0931916
$356$	0	0
$357$	3.86477	0.204545
$358$	0	0
$359$	24.3937	1.28745	0.643724	−	0.765258i	$-0.277388\pi$
$359$	24.3937	1.28745	0.643724	+	0.765258i	$0.277388\pi$
$360$	0	0
$361$	4.47805	0.235687
$362$	0	0
$363$	−4.32512	−0.227010
$364$	0	0
$365$	−5.58205	−0.292178
$366$	0	0
$367$	2.47185	0.129029	0.0645147	−	0.997917i	$-0.479450\pi$
$367$	2.47185	0.129029	0.0645147	+	0.997917i	$0.479450\pi$
$368$	0	0
$369$	−10.1498	−0.528376
$370$	0	0
$371$	−32.8931	−1.70773
$372$	0	0
$373$	19.8301	1.02676	0.513382	−	0.858160i	$-0.328392\pi$
$373$	19.8301	1.02676	0.513382	+	0.858160i	$0.328392\pi$
$374$	0	0
$375$	1.80093	0.0929999
$376$	0	0
$377$	4.06328	0.209270
$378$	0	0
$379$	−37.1980	−1.91073	−0.955365	−	0.295427i	$-0.904538\pi$
$379$	−37.1980	−1.91073	−0.955365	+	0.295427i	$0.904538\pi$
$380$	0	0
$381$	4.24160	0.217304
$382$	0	0
$383$	20.7733	1.06147	0.530733	−	0.847539i	$-0.321917\pi$
$383$	20.7733	1.06147	0.530733	+	0.847539i	$0.321917\pi$
$384$	0	0
$385$	−1.39274	−0.0709808
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−20.9842	−1.06394	−0.531970	−	0.846763i	$-0.678548\pi$
$389$	−20.9842	−1.06394	−0.531970	+	0.846763i	$0.678548\pi$
$390$	0	0
$391$	20.6855	1.04611
$392$	0	0
$393$	−6.68264	−0.337095
$394$	0	0
$395$	−1.48069	−0.0745018
$396$	0	0
$397$	−20.0544	−1.00650	−0.503251	−	0.864140i	$-0.667863\pi$
$397$	−20.0544	−1.00650	−0.503251	+	0.864140i	$0.667863\pi$
$398$	0	0
$399$	6.44494	0.322651
$400$	0	0
$401$	−8.85226	−0.442061	−0.221030	−	0.975267i	$-0.570942\pi$
$401$	−8.85226	−0.442061	−0.221030	+	0.975267i	$0.570942\pi$
$402$	0	0
$403$	−5.48211	−0.273084
$404$	0	0
$405$	−2.98715	−0.148433
$406$	0	0
$407$	−1.42847	−0.0708066
$408$	0	0
$409$	−25.7868	−1.27508	−0.637538	−	0.770419i	$-0.720047\pi$
$409$	−25.7868	−1.27508	−0.637538	+	0.770419i	$0.720047\pi$
$410$	0	0
$411$	−1.06862	−0.0527110
$412$	0	0
$413$	29.1903	1.43636
$414$	0	0
$415$	0.909652	0.0446531
$416$	0	0
$417$	8.30621	0.406757
$418$	0	0
$419$	−10.0977	−0.493304	−0.246652	−	0.969104i	$-0.579330\pi$
$419$	−10.0977	−0.493304	−0.246652	+	0.969104i	$0.579330\pi$
$420$	0	0
$421$	2.77057	0.135029	0.0675146	−	0.997718i	$-0.478493\pi$
$421$	2.77057	0.135029	0.0675146	+	0.997718i	$0.478493\pi$
$422$	0	0
$423$	−15.9854	−0.777236
$424$	0	0
$425$	−14.0357	−0.680829
$426$	0	0
$427$	−4.28760	−0.207491
$428$	0	0
$429$	1.67025	0.0806405
$430$	0	0
$431$	37.1361	1.78878	0.894391	−	0.447286i	$-0.147609\pi$
$431$	37.1361	1.78878	0.894391	+	0.447286i	$0.147609\pi$
$432$	0	0
$433$	−10.4658	−0.502957	−0.251478	−	0.967863i	$-0.580917\pi$
$433$	−10.4658	−0.502957	−0.251478	+	0.967863i	$0.580917\pi$
$434$	0	0
$435$	−0.224533	−0.0107656
$436$	0	0
$437$	34.4954	1.65014
$438$	0	0
$439$	−16.9466	−0.808816	−0.404408	−	0.914579i	$-0.632522\pi$
$439$	−16.9466	−0.808816	−0.404408	+	0.914579i	$0.632522\pi$
$440$	0	0
$441$	−5.41480	−0.257848
$442$	0	0
$443$	11.7823	0.559792	0.279896	−	0.960030i	$-0.409700\pi$
$443$	11.7823	0.559792	0.279896	+	0.960030i	$0.409700\pi$
$444$	0	0
$445$	7.61398	0.360937
$446$	0	0
$447$	9.66063	0.456932
$448$	0	0
$449$	30.9249	1.45943	0.729717	−	0.683749i	$-0.239652\pi$
$449$	30.9249	1.45943	0.729717	+	0.683749i	$0.239652\pi$
$450$	0	0
$451$	4.10075	0.193097
$452$	0	0
$453$	5.64509	0.265230
$454$	0	0
$455$	−4.07870	−0.191213
$456$	0	0
$457$	−18.9173	−0.884914	−0.442457	−	0.896790i	$-0.645893\pi$
$457$	−18.9173	−0.884914	−0.442457	+	0.896790i	$0.645893\pi$
$458$	0	0
$459$	−7.50257	−0.350190
$460$	0	0
$461$	−20.8267	−0.969996	−0.484998	−	0.874515i	$-0.661180\pi$
$461$	−20.8267	−0.969996	−0.484998	+	0.874515i	$0.661180\pi$
$462$	0	0
$463$	25.0444	1.16391	0.581955	−	0.813221i	$-0.302288\pi$
$463$	25.0444	1.16391	0.581955	+	0.813221i	$0.302288\pi$
$464$	0	0
$465$	0.302937	0.0140484
$466$	0	0
$467$	−6.44116	−0.298061	−0.149031	−	0.988833i	$-0.547615\pi$
$467$	−6.44116	−0.298061	−0.149031	+	0.988833i	$0.547615\pi$
$468$	0	0
$469$	−10.3403	−0.477468
$470$	0	0
$471$	8.97286	0.413448
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−23.4060	−1.07394
$476$	0	0
$477$	30.8373	1.41194
$478$	0	0
$479$	16.9258	0.773361	0.386680	−	0.922214i	$-0.373622\pi$
$479$	16.9258	0.773361	0.386680	+	0.922214i	$0.373622\pi$
$480$	0	0
$481$	−4.18332	−0.190743
$482$	0	0
$483$	9.46930	0.430868
$484$	0	0
$485$	3.91629	0.177830
$486$	0	0
$487$	16.2197	0.734983	0.367492	−	0.930027i	$-0.380217\pi$
$487$	16.2197	0.734983	0.367492	+	0.930027i	$0.380217\pi$
$488$	0	0
$489$	6.15470	0.278325
$490$	0	0
$491$	−13.7852	−0.622115	−0.311058	−	0.950391i	$-0.600683\pi$
$491$	−13.7852	−0.622115	−0.311058	+	0.950391i	$0.600683\pi$
$492$	0	0
$493$	3.56120	0.160389
$494$	0	0
$495$	1.30570	0.0586868
$496$	0	0
$497$	12.7485	0.571848
$498$	0	0
$499$	−5.05165	−0.226143	−0.113072	−	0.993587i	$-0.536069\pi$
$499$	−5.05165	−0.226143	−0.113072	+	0.993587i	$0.536069\pi$
$500$	0	0
$501$	5.31200	0.237323
$502$	0	0
$503$	20.8774	0.930876	0.465438	−	0.885080i	$-0.345897\pi$
$503$	20.8774	0.930876	0.465438	+	0.885080i	$0.345897\pi$
$504$	0	0
$505$	−1.62495	−0.0723093
$506$	0	0
$507$	−0.894152	−0.0397107
$508$	0	0
$509$	24.6278	1.09161	0.545803	−	0.837913i	$-0.316225\pi$
$509$	24.6278	1.09161	0.545803	+	0.837913i	$0.316225\pi$
$510$	0	0
$511$	40.5286	1.79288
$512$	0	0
$513$	−12.5114	−0.552391
$514$	0	0
$515$	0.974511	0.0429421
$516$	0	0
$517$	6.45847	0.284043
$518$	0	0
$519$	−5.17151	−0.227004
$520$	0	0
$521$	3.32524	0.145681	0.0728406	−	0.997344i	$-0.476794\pi$
$521$	3.32524	0.145681	0.0728406	+	0.997344i	$0.476794\pi$
$522$	0	0
$523$	4.41262	0.192951	0.0964753	−	0.995335i	$-0.469243\pi$
$523$	4.41262	0.192951	0.0964753	+	0.995335i	$0.469243\pi$
$524$	0	0
$525$	−6.42517	−0.280417
$526$	0	0
$527$	−4.80472	−0.209297
$528$	0	0
$529$	27.6827	1.20360
$530$	0	0
$531$	−27.3659	−1.18758
$532$	0	0
$533$	12.0092	0.520176
$534$	0	0
$535$	1.32251	0.0571770
$536$	0	0
$537$	9.15150	0.394917
$538$	0	0
$539$	2.18771	0.0942312
$540$	0	0
$541$	−27.2197	−1.17027	−0.585134	−	0.810937i	$-0.698958\pi$
$541$	−27.2197	−1.17027	−0.585134	+	0.810937i	$0.698958\pi$
$542$	0	0
$543$	−8.83637	−0.379205
$544$	0	0
$545$	2.04953	0.0877921
$546$	0	0
$547$	−28.2610	−1.20835	−0.604177	−	0.796850i	$-0.706498\pi$
$547$	−28.2610	−1.20835	−0.604177	+	0.796850i	$0.706498\pi$
$548$	0	0
$549$	4.01963	0.171553
$550$	0	0
$551$	5.93871	0.252997
$552$	0	0
$553$	10.7506	0.457162
$554$	0	0
$555$	0.231167	0.00981249
$556$	0	0
$557$	28.9588	1.22702	0.613512	−	0.789686i	$-0.289756\pi$
$557$	28.9588	1.22702	0.613512	+	0.789686i	$0.289756\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.46387	0.0618046
$562$	0	0
$563$	16.1506	0.680668	0.340334	−	0.940305i	$-0.389460\pi$
$563$	16.1506	0.680668	0.340334	+	0.940305i	$0.389460\pi$
$564$	0	0
$565$	4.70828	0.198079
$566$	0	0
$567$	21.6882	0.910820
$568$	0	0
$569$	−25.9291	−1.08700	−0.543501	−	0.839408i	$-0.682902\pi$
$569$	−25.9291	−1.08700	−0.543501	+	0.839408i	$0.682902\pi$
$570$	0	0
$571$	37.3821	1.56439	0.782196	−	0.623032i	$-0.214100\pi$
$571$	37.3821	1.56439	0.782196	+	0.623032i	$0.214100\pi$
$572$	0	0
$573$	−3.31690	−0.138566
$574$	0	0
$575$	−34.3896	−1.43414
$576$	0	0
$577$	31.4866	1.31080	0.655402	−	0.755281i	$-0.272499\pi$
$577$	31.4866	1.31080	0.655402	+	0.755281i	$0.272499\pi$
$578$	0	0
$579$	−7.42186	−0.308442
$580$	0	0
$581$	−6.60455	−0.274003
$582$	0	0
$583$	−12.4590	−0.515999
$584$	0	0
$585$	3.82379	0.158094
$586$	0	0
$587$	−28.6520	−1.18259	−0.591297	−	0.806454i	$-0.701384\pi$
$587$	−28.6520	−1.18259	−0.591297	+	0.806454i	$0.701384\pi$
$588$	0	0
$589$	−8.01241	−0.330146
$590$	0	0
$591$	4.02331	0.165497
$592$	0	0
$593$	35.2954	1.44941	0.724704	−	0.689060i	$-0.241976\pi$
$593$	35.2954	1.44941	0.724704	+	0.689060i	$0.241976\pi$
$594$	0	0
$595$	−3.57472	−0.146549
$596$	0	0
$597$	−1.94601	−0.0796450
$598$	0	0
$599$	−3.93716	−0.160868	−0.0804340	−	0.996760i	$-0.525631\pi$
$599$	−3.93716	−0.160868	−0.0804340	+	0.996760i	$0.525631\pi$
$600$	0	0
$601$	−13.5242	−0.551662	−0.275831	−	0.961206i	$-0.588953\pi$
$601$	−13.5242	−0.551662	−0.275831	+	0.961206i	$0.588953\pi$
$602$	0	0
$603$	9.69399	0.394770
$604$	0	0
$605$	4.00053	0.162644
$606$	0	0
$607$	−22.1771	−0.900140	−0.450070	−	0.892993i	$-0.648601\pi$
$607$	−22.1771	−0.900140	−0.450070	+	0.892993i	$0.648601\pi$
$608$	0	0
$609$	1.63023	0.0660602
$610$	0	0
$611$	18.9139	0.765173
$612$	0	0
$613$	3.83361	0.154838	0.0774191	−	0.996999i	$-0.475332\pi$
$613$	3.83361	0.154838	0.0774191	+	0.996999i	$0.475332\pi$
$614$	0	0
$615$	−0.663618	−0.0267596
$616$	0	0
$617$	17.0179	0.685114	0.342557	−	0.939497i	$-0.388707\pi$
$617$	17.0179	0.685114	0.342557	+	0.939497i	$0.388707\pi$
$618$	0	0
$619$	−24.6821	−0.992058	−0.496029	−	0.868306i	$-0.665209\pi$
$619$	−24.6821	−0.992058	−0.496029	+	0.868306i	$0.665209\pi$
$620$	0	0
$621$	−18.3825	−0.737663
$622$	0	0
$623$	−55.2815	−2.21481
$624$	0	0
$625$	22.4870	0.899479
$626$	0	0
$627$	2.44116	0.0974907
$628$	0	0
$629$	−3.66642	−0.146190
$630$	0	0
$631$	−38.9346	−1.54996	−0.774981	−	0.631984i	$-0.782241\pi$
$631$	−38.9346	−1.54996	−0.774981	+	0.631984i	$0.782241\pi$
$632$	0	0
$633$	6.17381	0.245387
$634$	0	0
$635$	−3.92327	−0.155690
$636$	0	0
$637$	6.40678	0.253846
$638$	0	0
$639$	−11.9517	−0.472802
$640$	0	0
$641$	46.7031	1.84466	0.922331	−	0.386402i	$-0.126282\pi$
$641$	46.7031	1.84466	0.922331	+	0.386402i	$0.126282\pi$
$642$	0	0
$643$	−7.82005	−0.308393	−0.154196	−	0.988040i	$-0.549279\pi$
$643$	−7.82005	−0.308393	−0.154196	+	0.988040i	$0.549279\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	38.2463	1.50362	0.751809	−	0.659381i	$-0.229182\pi$
$647$	38.2463	1.50362	0.751809	+	0.659381i	$0.229182\pi$
$648$	0	0
$649$	11.0565	0.434004
$650$	0	0
$651$	−2.19948	−0.0862044
$652$	0	0
$653$	9.54678	0.373594	0.186797	−	0.982399i	$-0.440189\pi$
$653$	9.54678	0.373594	0.186797	+	0.982399i	$0.440189\pi$
$654$	0	0
$655$	6.18111	0.241516
$656$	0	0
$657$	−37.9956	−1.48235
$658$	0	0
$659$	20.0009	0.779125	0.389562	−	0.921000i	$-0.372626\pi$
$659$	20.0009	0.779125	0.389562	+	0.921000i	$0.372626\pi$
$660$	0	0
$661$	43.4485	1.68995	0.844975	−	0.534805i	$-0.179615\pi$
$661$	43.4485	1.68995	0.844975	+	0.534805i	$0.179615\pi$
$662$	0	0
$663$	4.28699	0.166493
$664$	0	0
$665$	−5.96125	−0.231167
$666$	0	0
$667$	8.72551	0.337853
$668$	0	0
$669$	10.6682	0.412458
$670$	0	0
$671$	−1.62402	−0.0626947
$672$	0	0
$673$	10.6194	0.409350	0.204675	−	0.978830i	$-0.434386\pi$
$673$	10.6194	0.409350	0.204675	+	0.978830i	$0.434386\pi$
$674$	0	0
$675$	12.4730	0.480086
$676$	0	0
$677$	21.1014	0.810992	0.405496	−	0.914097i	$-0.367099\pi$
$677$	21.1014	0.810992	0.405496	+	0.914097i	$0.367099\pi$
$678$	0	0
$679$	−28.4343	−1.09121
$680$	0	0
$681$	−6.67762	−0.255887
$682$	0	0
$683$	4.87806	0.186654	0.0933269	−	0.995636i	$-0.470250\pi$
$683$	4.87806	0.186654	0.0933269	+	0.995636i	$0.470250\pi$
$684$	0	0
$685$	0.988418	0.0377655
$686$	0	0
$687$	7.63132	0.291153
$688$	0	0
$689$	−36.4867	−1.39003
$690$	0	0
$691$	−27.5112	−1.04657	−0.523287	−	0.852156i	$-0.675294\pi$
$691$	−27.5112	−1.04657	−0.523287	+	0.852156i	$0.675294\pi$
$692$	0	0
$693$	−9.48005	−0.360117
$694$	0	0
$695$	−7.68283	−0.291426
$696$	0	0
$697$	10.5253	0.398673
$698$	0	0
$699$	11.4380	0.432626
$700$	0	0
$701$	−41.7043	−1.57515	−0.787575	−	0.616219i	$-0.788664\pi$
$701$	−41.7043	−1.57515	−0.787575	+	0.616219i	$0.788664\pi$
$702$	0	0
$703$	−6.11416	−0.230600
$704$	0	0
$705$	−1.04516	−0.0393632
$706$	0	0
$707$	11.7980	0.443709
$708$	0	0
$709$	45.0266	1.69101	0.845504	−	0.533969i	$-0.179300\pi$
$709$	45.0266	1.69101	0.845504	+	0.533969i	$0.179300\pi$
$710$	0	0
$711$	−10.0787	−0.377981
$712$	0	0
$713$	−11.7723	−0.440877
$714$	0	0
$715$	−1.54490	−0.0577760
$716$	0	0
$717$	−9.40557	−0.351257
$718$	0	0
$719$	7.11934	0.265507	0.132753	−	0.991149i	$-0.457618\pi$
$719$	7.11934	0.265507	0.132753	+	0.991149i	$0.457618\pi$
$720$	0	0
$721$	−7.07546	−0.263504
$722$	0	0
$723$	8.59997	0.319836
$724$	0	0
$725$	−5.92049	−0.219881
$726$	0	0
$727$	−17.3854	−0.644790	−0.322395	−	0.946605i	$-0.604488\pi$
$727$	−17.3854	−0.644790	−0.322395	+	0.946605i	$0.604488\pi$
$728$	0	0
$729$	−16.8853	−0.625381
$730$	0	0
$731$	0	0
$732$	0	0
$733$	10.5166	0.388441	0.194221	−	0.980958i	$-0.437782\pi$
$733$	10.5166	0.388441	0.194221	+	0.980958i	$0.437782\pi$
$734$	0	0
$735$	−0.354033	−0.0130587
$736$	0	0
$737$	−3.91660	−0.144270
$738$	0	0
$739$	4.87607	0.179369	0.0896845	−	0.995970i	$-0.471414\pi$
$739$	4.87607	0.179369	0.0896845	+	0.995970i	$0.471414\pi$
$740$	0	0
$741$	7.14904	0.262627
$742$	0	0
$743$	38.7814	1.42275	0.711376	−	0.702811i	$-0.248072\pi$
$743$	38.7814	1.42275	0.711376	+	0.702811i	$0.248072\pi$
$744$	0	0
$745$	−8.93560	−0.327375
$746$	0	0
$747$	6.19176	0.226545
$748$	0	0
$749$	−9.60210	−0.350853
$750$	0	0
$751$	−0.336010	−0.0122612	−0.00613059	−	0.999981i	$-0.501951\pi$
$751$	−0.336010	−0.0122612	−0.00613059	+	0.999981i	$0.501951\pi$
$752$	0	0
$753$	11.3167	0.412402
$754$	0	0
$755$	−5.22143	−0.190027
$756$	0	0
$757$	−9.19851	−0.334325	−0.167163	−	0.985929i	$-0.553461\pi$
$757$	−9.19851	−0.334325	−0.167163	+	0.985929i	$0.553461\pi$
$758$	0	0
$759$	3.58671	0.130189
$760$	0	0
$761$	−14.4775	−0.524810	−0.262405	−	0.964958i	$-0.584516\pi$
$761$	−14.4775	−0.524810	−0.262405	+	0.964958i	$0.584516\pi$
$762$	0	0
$763$	−14.8806	−0.538715
$764$	0	0
$765$	3.35130	0.121167
$766$	0	0
$767$	32.3792	1.16915
$768$	0	0
$769$	−11.3026	−0.407581	−0.203790	−	0.979015i	$-0.565326\pi$
$769$	−11.3026	−0.407581	−0.203790	+	0.979015i	$0.565326\pi$
$770$	0	0
$771$	5.99449	0.215886
$772$	0	0
$773$	18.9181	0.680438	0.340219	−	0.940346i	$-0.389499\pi$
$773$	18.9181	0.680438	0.340219	+	0.940346i	$0.389499\pi$
$774$	0	0
$775$	7.98783	0.286931
$776$	0	0
$777$	−1.67839	−0.0602120
$778$	0	0
$779$	17.5521	0.628869
$780$	0	0
$781$	4.82877	0.172787
$782$	0	0
$783$	−3.16472	−0.113098
$784$	0	0
$785$	−8.29945	−0.296220
$786$	0	0
$787$	−20.0618	−0.715127	−0.357564	−	0.933889i	$-0.616392\pi$
$787$	−20.0618	−0.715127	−0.357564	+	0.933889i	$0.616392\pi$
$788$	0	0
$789$	13.6996	0.487719
$790$	0	0
$791$	−34.1845	−1.21546
$792$	0	0
$793$	−4.75601	−0.168891
$794$	0	0
$795$	2.01622	0.0715080
$796$	0	0
$797$	−20.1820	−0.714882	−0.357441	−	0.933936i	$-0.616351\pi$
$797$	−20.1820	−0.714882	−0.357441	+	0.933936i	$0.616351\pi$
$798$	0	0
$799$	16.5768	0.586445
$800$	0	0
$801$	51.8264	1.83120
$802$	0	0
$803$	15.3511	0.541729
$804$	0	0
$805$	−8.75863	−0.308701
$806$	0	0
$807$	−12.8193	−0.451261
$808$	0	0
$809$	2.58215	0.0907837	0.0453918	−	0.998969i	$-0.485546\pi$
$809$	2.58215	0.0907837	0.0453918	+	0.998969i	$0.485546\pi$
$810$	0	0
$811$	33.7281	1.18435	0.592176	−	0.805809i	$-0.298269\pi$
$811$	33.7281	1.18435	0.592176	+	0.805809i	$0.298269\pi$
$812$	0	0
$813$	−9.17019	−0.321613
$814$	0	0
$815$	−5.69279	−0.199410
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−27.7627	−0.970106
$820$	0	0
$821$	−37.4588	−1.30732	−0.653660	−	0.756788i	$-0.726767\pi$
$821$	−37.4588	−1.30732	−0.653660	+	0.756788i	$0.726767\pi$
$822$	0	0
$823$	−39.2010	−1.36646	−0.683231	−	0.730203i	$-0.739426\pi$
$823$	−39.2010	−1.36646	−0.683231	+	0.730203i	$0.739426\pi$
$824$	0	0
$825$	−2.43368	−0.0847297
$826$	0	0
$827$	34.0443	1.18384	0.591918	−	0.805998i	$-0.298371\pi$
$827$	34.0443	1.18384	0.591918	+	0.805998i	$0.298371\pi$
$828$	0	0
$829$	−47.4514	−1.64805	−0.824027	−	0.566550i	$-0.808278\pi$
$829$	−47.4514	−1.64805	−0.824027	+	0.566550i	$0.808278\pi$
$830$	0	0
$831$	−2.74674	−0.0952835
$832$	0	0
$833$	5.61513	0.194553
$834$	0	0
$835$	−4.91334	−0.170033
$836$	0	0
$837$	4.26979	0.147585
$838$	0	0
$839$	25.4516	0.878688	0.439344	−	0.898319i	$-0.355211\pi$
$839$	25.4516	0.878688	0.439344	+	0.898319i	$0.355211\pi$
$840$	0	0
$841$	−27.4978	−0.948201
$842$	0	0
$843$	7.50176	0.258374
$844$	0	0
$845$	0.827046	0.0284512
$846$	0	0
$847$	−29.0459	−0.998029
$848$	0	0
$849$	−3.74356	−0.128479
$850$	0	0
$851$	−8.98330	−0.307943
$852$	0	0
$853$	−20.8798	−0.714911	−0.357455	−	0.933930i	$-0.616356\pi$
$853$	−20.8798	−0.714911	−0.357455	+	0.933930i	$0.616356\pi$
$854$	0	0
$855$	5.58867	0.191129
$856$	0	0
$857$	11.4648	0.391630	0.195815	−	0.980641i	$-0.437265\pi$
$857$	11.4648	0.391630	0.195815	+	0.980641i	$0.437265\pi$
$858$	0	0
$859$	2.00539	0.0684230	0.0342115	−	0.999415i	$-0.489108\pi$
$859$	2.00539	0.0684230	0.0342115	+	0.999415i	$0.489108\pi$
$860$	0	0
$861$	4.81821	0.164204
$862$	0	0
$863$	3.32262	0.113103	0.0565516	−	0.998400i	$-0.481989\pi$
$863$	3.32262	0.113103	0.0565516	+	0.998400i	$0.481989\pi$
$864$	0	0
$865$	4.78339	0.162640
$866$	0	0
$867$	−3.80844	−0.129341
$868$	0	0
$869$	4.07203	0.138134
$870$	0	0
$871$	−11.4699	−0.388643
$872$	0	0
$873$	26.6572	0.902209
$874$	0	0
$875$	12.0944	0.408865
$876$	0	0
$877$	33.1732	1.12018	0.560090	−	0.828432i	$-0.310767\pi$
$877$	33.1732	1.12018	0.560090	+	0.828432i	$0.310767\pi$
$878$	0	0
$879$	−2.03988	−0.0688033
$880$	0	0
$881$	6.38676	0.215175	0.107588	−	0.994196i	$-0.465687\pi$
$881$	6.38676	0.215175	0.107588	+	0.994196i	$0.465687\pi$
$882$	0	0
$883$	−41.7092	−1.40363	−0.701813	−	0.712361i	$-0.747626\pi$
$883$	−41.7092	−1.40363	−0.701813	+	0.712361i	$0.747626\pi$
$884$	0	0
$885$	−1.78925	−0.0601450
$886$	0	0
$887$	−28.1469	−0.945080	−0.472540	−	0.881309i	$-0.656663\pi$
$887$	−28.1469	−0.945080	−0.472540	+	0.881309i	$0.656663\pi$
$888$	0	0
$889$	28.4850	0.955356
$890$	0	0
$891$	8.21490	0.275210
$892$	0	0
$893$	27.6437	0.925060
$894$	0	0
$895$	−8.46469	−0.282943
$896$	0	0
$897$	10.5038	0.350712
$898$	0	0
$899$	−2.02672	−0.0675948
$900$	0	0
$901$	−31.9782	−1.06535
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.17321	0.271687
$906$	0	0
$907$	−45.0185	−1.49482	−0.747408	−	0.664365i	$-0.768702\pi$
$907$	−45.0185	−1.49482	−0.747408	+	0.664365i	$0.768702\pi$
$908$	0	0
$909$	−11.0606	−0.366857
$910$	0	0
$911$	−23.6227	−0.782655	−0.391327	−	0.920252i	$-0.627984\pi$
$911$	−23.6227	−0.782655	−0.391327	+	0.920252i	$0.627984\pi$
$912$	0	0
$913$	−2.50162	−0.0827915
$914$	0	0
$915$	0.262813	0.00868834
$916$	0	0
$917$	−44.8781	−1.48201
$918$	0	0
$919$	−29.7213	−0.980414	−0.490207	−	0.871606i	$-0.663079\pi$
$919$	−29.7213	−0.980414	−0.490207	+	0.871606i	$0.663079\pi$
$920$	0	0
$921$	0.808577	0.0266435
$922$	0	0
$923$	14.1412	0.465464
$924$	0	0
$925$	6.09540	0.200416
$926$	0	0
$927$	6.63324	0.217864
$928$	0	0
$929$	−3.31012	−0.108601	−0.0543007	−	0.998525i	$-0.517293\pi$
$929$	−3.31012	−0.108601	−0.0543007	+	0.998525i	$0.517293\pi$
$930$	0	0
$931$	9.36386	0.306888
$932$	0	0
$933$	−8.00727	−0.262146
$934$	0	0
$935$	−1.35401	−0.0442807
$936$	0	0
$937$	1.15310	0.0376701	0.0188350	−	0.999823i	$-0.494004\pi$
$937$	1.15310	0.0376701	0.0188350	+	0.999823i	$0.494004\pi$
$938$	0	0
$939$	−11.8047	−0.385233
$940$	0	0
$941$	−45.7464	−1.49129	−0.745645	−	0.666344i	$-0.767858\pi$
$941$	−45.7464	−1.49129	−0.745645	+	0.666344i	$0.767858\pi$
$942$	0	0
$943$	25.7886	0.839792
$944$	0	0
$945$	3.17673	0.103339
$946$	0	0
$947$	−38.1369	−1.23928	−0.619642	−	0.784885i	$-0.712722\pi$
$947$	−38.1369	−1.23928	−0.619642	+	0.784885i	$0.712722\pi$
$948$	0	0
$949$	44.9563	1.45934
$950$	0	0
$951$	13.2608	0.430012
$952$	0	0
$953$	21.7434	0.704337	0.352168	−	0.935937i	$-0.385444\pi$
$953$	21.7434	0.704337	0.352168	+	0.935937i	$0.385444\pi$
$954$	0	0
$955$	3.06797	0.0992772
$956$	0	0
$957$	0.617486	0.0199605
$958$	0	0
$959$	−7.17643	−0.231739
$960$	0	0
$961$	−28.2656	−0.911793
$962$	0	0
$963$	9.00197	0.290084
$964$	0	0
$965$	6.86485	0.220987
$966$	0	0
$967$	−21.1648	−0.680615	−0.340307	−	0.940314i	$-0.610531\pi$
$967$	−21.1648	−0.680615	−0.340307	+	0.940314i	$0.610531\pi$
$968$	0	0
$969$	6.26567	0.201282
$970$	0	0
$971$	32.3135	1.03699	0.518495	−	0.855081i	$-0.326493\pi$
$971$	32.3135	1.03699	0.518495	+	0.855081i	$0.326493\pi$
$972$	0	0
$973$	55.7814	1.78827
$974$	0	0
$975$	−7.12711	−0.228250
$976$	0	0
$977$	29.9700	0.958826	0.479413	−	0.877589i	$-0.340850\pi$
$977$	29.9700	0.958826	0.479413	+	0.877589i	$0.340850\pi$
$978$	0	0
$979$	−20.9391	−0.669216
$980$	0	0
$981$	13.9506	0.445408
$982$	0	0
$983$	−8.59558	−0.274156	−0.137078	−	0.990560i	$-0.543771\pi$
$983$	−8.59558	−0.274156	−0.137078	+	0.990560i	$0.543771\pi$
$984$	0	0
$985$	−3.72136	−0.118572
$986$	0	0
$987$	7.58844	0.241543
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−23.6280	−0.750569	−0.375284	−	0.926910i	$-0.622455\pi$
$991$	−23.6280	−0.750569	−0.375284	+	0.926910i	$0.622455\pi$
$992$	0	0
$993$	2.91619	0.0925425
$994$	0	0
$995$	1.79997	0.0570627
$996$	0	0
$997$	−33.7260	−1.06811	−0.534057	−	0.845449i	$-0.679333\pi$
$997$	−33.7260	−1.06811	−0.534057	+	0.845449i	$0.679333\pi$
$998$	0	0
$999$	3.25821	0.103085

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7396.2.a.j.1.3		6
43.4	even	7		172.2.i.b.145.1	yes	12
43.11	even	7		172.2.i.b.121.1	✓	12
43.42	odd	2		7396.2.a.i.1.4		6
172.11	odd	14		688.2.u.e.465.1		12
172.47	odd	14		688.2.u.e.145.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
172.2.i.b.121.1	✓	12	43.11	even	7
172.2.i.b.145.1	yes	12	43.4	even	7
688.2.u.e.145.1		12	172.47	odd	14
688.2.u.e.465.1		12	172.11	odd	14
7396.2.a.i.1.4		6	43.42	odd	2
7396.2.a.j.1.3		6	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:	\( N \)	\(=\)	\( 7396 = 2^{2} \cdot 43^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7396.a (trivial)

\(f(q)\)	\(=\)	\(q+0.445042 q^{3} -0.411642 q^{5} +2.98873 q^{7} -2.80194 q^{9} +O(q^{10})\)	\(q+0.445042 q^{3} -0.411642 q^{5} +2.98873 q^{7} -2.80194 q^{9} +1.13205 q^{11} +3.31525 q^{13} -0.183198 q^{15} +2.90560 q^{17} +4.84542 q^{19} +1.33011 q^{21} +7.11918 q^{23} -4.83055 q^{25} -2.58211 q^{27} +1.22563 q^{29} -1.65361 q^{31} +0.503809 q^{33} -1.23029 q^{35} -1.26184 q^{37} +1.47542 q^{39} +3.62241 q^{41} +1.15339 q^{45} +5.70512 q^{47} +1.93252 q^{49} +1.29311 q^{51} -11.0057 q^{53} -0.465998 q^{55} +2.15641 q^{57} +9.76677 q^{59} -1.43459 q^{61} -8.37424 q^{63} -1.36469 q^{65} -3.45975 q^{67} +3.16833 q^{69} +4.26552 q^{71} +13.5605 q^{73} -2.14980 q^{75} +3.38339 q^{77} +3.59704 q^{79} +7.25667 q^{81} -2.20982 q^{83} -1.19607 q^{85} +0.545458 q^{87} -18.4966 q^{89} +9.90838 q^{91} -0.735924 q^{93} -1.99457 q^{95} -9.51384 q^{97} -3.17193 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} + 5 q^{5} - 8 q^{9}+O(q^{10}) \) 6 * q + 2 * q^3 + 5 * q^5 - 8 * q^9	\( 6 q + 2 q^{3} + 5 q^{5} - 8 q^{9} - 3 q^{11} + 5 q^{13} + 4 q^{15} - 9 q^{17} + 3 q^{19} + 7 q^{21} + 11 q^{25} - 4 q^{27} + 4 q^{29} + 6 q^{31} - q^{33} - 8 q^{35} - 12 q^{37} - 3 q^{39} - 3 q^{41} - 9 q^{45} + 18 q^{47} + 12 q^{49} + 4 q^{51} - 4 q^{53} + 24 q^{55} + 15 q^{57} + 23 q^{59} - 10 q^{61} + 7 q^{63} + 25 q^{65} + 25 q^{67} + 9 q^{71} + 37 q^{73} - q^{75} + 39 q^{77} + 7 q^{79} - 10 q^{81} + 4 q^{83} - 35 q^{85} + 6 q^{87} + 13 q^{89} + 9 q^{91} + 23 q^{93} - 40 q^{95} - 51 q^{97} + 4 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 + 5 * q^5 - 8 * q^9 - 3 * q^11 + 5 * q^13 + 4 * q^15 - 9 * q^17 + 3 * q^19 + 7 * q^21 + 11 * q^25 - 4 * q^27 + 4 * q^29 + 6 * q^31 - q^33 - 8 * q^35 - 12 * q^37 - 3 * q^39 - 3 * q^41 - 9 * q^45 + 18 * q^47 + 12 * q^49 + 4 * q^51 - 4 * q^53 + 24 * q^55 + 15 * q^57 + 23 * q^59 - 10 * q^61 + 7 * q^63 + 25 * q^65 + 25 * q^67 + 9 * q^71 + 37 * q^73 - q^75 + 39 * q^77 + 7 * q^79 - 10 * q^81 + 4 * q^83 - 35 * q^85 + 6 * q^87 + 13 * q^89 + 9 * q^91 + 23 * q^93 - 40 * q^95 - 51 * q^97 + 4 * q^99

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