LMFDB - Embedded newform 3344.2.a.ba.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 $ x^7 - x^6 - 14x^5 + 10x^4 + 59x^3 - 27x^2 - 66*x + 30
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.55401$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.99825 q^{3} +0.244850 q^{5} -4.42321 q^{7} +5.98952 q^{9} +O(q^{10})$	$q+2.99825 q^{3} +0.244850 q^{5} -4.42321 q^{7} +5.98952 q^{9} +1.00000 q^{11} -5.89690 q^{13} +0.734121 q^{15} -2.93922 q^{17} -1.00000 q^{19} -13.2619 q^{21} +0.372904 q^{23} -4.94005 q^{25} +8.96335 q^{27} -3.47526 q^{29} -6.37391 q^{31} +2.99825 q^{33} -1.08302 q^{35} +0.926528 q^{37} -17.6804 q^{39} -6.67861 q^{41} -2.12805 q^{43} +1.46653 q^{45} +1.72093 q^{47} +12.5648 q^{49} -8.81252 q^{51} -1.44022 q^{53} +0.244850 q^{55} -2.99825 q^{57} -7.71718 q^{59} +4.16881 q^{61} -26.4929 q^{63} -1.44385 q^{65} +11.3778 q^{67} +1.11806 q^{69} -2.40794 q^{71} +14.4653 q^{73} -14.8115 q^{75} -4.42321 q^{77} -1.67368 q^{79} +8.90583 q^{81} +6.47897 q^{83} -0.719667 q^{85} -10.4197 q^{87} -4.95706 q^{89} +26.0832 q^{91} -19.1106 q^{93} -0.244850 q^{95} +8.24006 q^{97} +5.98952 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9}+O(q^{10}) $ 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9	$ 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9} + 7 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 7 q^{19} - 14 q^{21} - 10 q^{23} + 9 q^{25} + 4 q^{27} - 18 q^{29} - 24 q^{31} - 2 q^{33} - 8 q^{35} - 24 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} - 8 q^{47} + 17 q^{49} + 24 q^{51} + 2 q^{53} + 2 q^{55} + 2 q^{57} + 10 q^{59} + 14 q^{61} - 14 q^{65} - 8 q^{67} - 6 q^{69} - 10 q^{71} - 6 q^{73} - 26 q^{75} - 10 q^{77} - 52 q^{79} - q^{81} + 10 q^{83} - 12 q^{85} - 6 q^{87} - 12 q^{91} + 2 q^{93} - 2 q^{95} - 24 q^{97} + 11 q^{99}+O(q^{100}) $ 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9 + 7 * q^11 - 4 * q^13 - 12 * q^15 + 2 * q^17 - 7 * q^19 - 14 * q^21 - 10 * q^23 + 9 * q^25 + 4 * q^27 - 18 * q^29 - 24 * q^31 - 2 * q^33 - 8 * q^35 - 24 * q^39 - 12 * q^41 - 2 * q^43 - 4 * q^45 - 8 * q^47 + 17 * q^49 + 24 * q^51 + 2 * q^53 + 2 * q^55 + 2 * q^57 + 10 * q^59 + 14 * q^61 - 14 * q^65 - 8 * q^67 - 6 * q^69 - 10 * q^71 - 6 * q^73 - 26 * q^75 - 10 * q^77 - 52 * q^79 - q^81 + 10 * q^83 - 12 * q^85 - 6 * q^87 - 12 * q^91 + 2 * q^93 - 2 * q^95 - 24 * q^97 + 11 * 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
$3$	2.99825	1.73104	0.865521	−	0.500872i	$-0.166987\pi$
$3$	2.99825	1.73104	0.865521	+	0.500872i	$0.166987\pi$
$4$	0	0
$5$	0.244850	0.109500	0.0547500	−	0.998500i	$-0.482564\pi$
$5$	0.244850	0.109500	0.0547500	+	0.998500i	$0.482564\pi$
$6$	0	0
$7$	−4.42321	−1.67182	−0.835908	−	0.548869i	$-0.815058\pi$
$7$	−4.42321	−1.67182	−0.835908	+	0.548869i	$0.815058\pi$
$8$	0	0
$9$	5.98952	1.99651
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.89690	−1.63551	−0.817753	−	0.575570i	$-0.804780\pi$
$13$	−5.89690	−1.63551	−0.817753	+	0.575570i	$0.804780\pi$
$14$	0	0
$15$	0.734121	0.189549
$16$	0	0
$17$	−2.93922	−0.712865	−0.356433	−	0.934321i	$-0.616007\pi$
$17$	−2.93922	−0.712865	−0.356433	+	0.934321i	$0.616007\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−13.2619	−2.89398
$22$	0	0
$23$	0.372904	0.0777558	0.0388779	−	0.999244i	$-0.487622\pi$
$23$	0.372904	0.0777558	0.0388779	+	0.999244i	$0.487622\pi$
$24$	0	0
$25$	−4.94005	−0.988010
$26$	0	0
$27$	8.96335	1.72500
$28$	0	0
$29$	−3.47526	−0.645340	−0.322670	−	0.946512i	$-0.604580\pi$
$29$	−3.47526	−0.645340	−0.322670	+	0.946512i	$0.604580\pi$
$30$	0	0
$31$	−6.37391	−1.14479	−0.572394	−	0.819979i	$-0.693985\pi$
$31$	−6.37391	−1.14479	−0.572394	+	0.819979i	$0.693985\pi$
$32$	0	0
$33$	2.99825	0.521929
$34$	0	0
$35$	−1.08302	−0.183064
$36$	0	0
$37$	0.926528	0.152320	0.0761601	−	0.997096i	$-0.475734\pi$
$37$	0.926528	0.152320	0.0761601	+	0.997096i	$0.475734\pi$
$38$	0	0
$39$	−17.6804	−2.83113
$40$	0	0
$41$	−6.67861	−1.04302	−0.521512	−	0.853244i	$-0.674632\pi$
$41$	−6.67861	−1.04302	−0.521512	+	0.853244i	$0.674632\pi$
$42$	0	0
$43$	−2.12805	−0.324525	−0.162263	−	0.986748i	$-0.551879\pi$
$43$	−2.12805	−0.324525	−0.162263	+	0.986748i	$0.551879\pi$
$44$	0	0
$45$	1.46653	0.218618
$46$	0	0
$47$	1.72093	0.251024	0.125512	−	0.992092i	$-0.459943\pi$
$47$	1.72093	0.251024	0.125512	+	0.992092i	$0.459943\pi$
$48$	0	0
$49$	12.5648	1.79497
$50$	0	0
$51$	−8.81252	−1.23400
$52$	0	0
$53$	−1.44022	−0.197830	−0.0989150	−	0.995096i	$-0.531537\pi$
$53$	−1.44022	−0.197830	−0.0989150	+	0.995096i	$0.531537\pi$
$54$	0	0
$55$	0.244850	0.0330155
$56$	0	0
$57$	−2.99825	−0.397128
$58$	0	0
$59$	−7.71718	−1.00469	−0.502346	−	0.864667i	$-0.667530\pi$
$59$	−7.71718	−1.00469	−0.502346	+	0.864667i	$0.667530\pi$
$60$	0	0
$61$	4.16881	0.533762	0.266881	−	0.963730i	$-0.414007\pi$
$61$	4.16881	0.533762	0.266881	+	0.963730i	$0.414007\pi$
$62$	0	0
$63$	−26.4929	−3.33779
$64$	0	0
$65$	−1.44385	−0.179088
$66$	0	0
$67$	11.3778	1.39002	0.695012	−	0.718998i	$-0.255399\pi$
$67$	11.3778	1.39002	0.695012	+	0.718998i	$0.255399\pi$
$68$	0	0
$69$	1.11806	0.134599
$70$	0	0
$71$	−2.40794	−0.285770	−0.142885	−	0.989739i	$-0.545638\pi$
$71$	−2.40794	−0.285770	−0.142885	+	0.989739i	$0.545638\pi$
$72$	0	0
$73$	14.4653	1.69303	0.846515	−	0.532365i	$-0.178697\pi$
$73$	14.4653	1.69303	0.846515	+	0.532365i	$0.178697\pi$
$74$	0	0
$75$	−14.8115	−1.71029
$76$	0	0
$77$	−4.42321	−0.504072
$78$	0	0
$79$	−1.67368	−0.188304	−0.0941521	−	0.995558i	$-0.530014\pi$
$79$	−1.67368	−0.188304	−0.0941521	+	0.995558i	$0.530014\pi$
$80$	0	0
$81$	8.90583	0.989537
$82$	0	0
$83$	6.47897	0.711159	0.355580	−	0.934646i	$-0.384283\pi$
$83$	6.47897	0.711159	0.355580	+	0.934646i	$0.384283\pi$
$84$	0	0
$85$	−0.719667	−0.0780588
$86$	0	0
$87$	−10.4197	−1.11711
$88$	0	0
$89$	−4.95706	−0.525448	−0.262724	−	0.964871i	$-0.584621\pi$
$89$	−4.95706	−0.525448	−0.262724	+	0.964871i	$0.584621\pi$
$90$	0	0
$91$	26.0832	2.73426
$92$	0	0
$93$	−19.1106	−1.98168
$94$	0	0
$95$	−0.244850	−0.0251210
$96$	0	0
$97$	8.24006	0.836651	0.418325	−	0.908297i	$-0.362617\pi$
$97$	8.24006	0.836651	0.418325	+	0.908297i	$0.362617\pi$
$98$	0	0
$99$	5.98952	0.601970
$100$	0	0
$101$	3.51207	0.349464	0.174732	−	0.984616i	$-0.444094\pi$
$101$	3.51207	0.349464	0.174732	+	0.984616i	$0.444094\pi$
$102$	0	0
$103$	2.43194	0.239626	0.119813	−	0.992796i	$-0.461770\pi$
$103$	2.43194	0.239626	0.119813	+	0.992796i	$0.461770\pi$
$104$	0	0
$105$	−3.24717	−0.316892
$106$	0	0
$107$	−16.9865	−1.64214	−0.821072	−	0.570825i	$-0.806624\pi$
$107$	−16.9865	−1.64214	−0.821072	+	0.570825i	$0.806624\pi$
$108$	0	0
$109$	3.83586	0.367409	0.183704	−	0.982982i	$-0.441191\pi$
$109$	3.83586	0.367409	0.183704	+	0.982982i	$0.441191\pi$
$110$	0	0
$111$	2.77797	0.263673
$112$	0	0
$113$	−10.7909	−1.01512	−0.507562	−	0.861615i	$-0.669453\pi$
$113$	−10.7909	−1.01512	−0.507562	+	0.861615i	$0.669453\pi$
$114$	0	0
$115$	0.0913054	0.00851427
$116$	0	0
$117$	−35.3196	−3.26530
$118$	0	0
$119$	13.0008	1.19178
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−20.0242	−1.80552
$124$	0	0
$125$	−2.43382	−0.217687
$126$	0	0
$127$	12.0969	1.07342	0.536712	−	0.843765i	$-0.319666\pi$
$127$	12.0969	1.07342	0.536712	+	0.843765i	$0.319666\pi$
$128$	0	0
$129$	−6.38045	−0.561767
$130$	0	0
$131$	7.99822	0.698807	0.349404	−	0.936972i	$-0.386384\pi$
$131$	7.99822	0.698807	0.349404	+	0.936972i	$0.386384\pi$
$132$	0	0
$133$	4.42321	0.383541
$134$	0	0
$135$	2.19467	0.188887
$136$	0	0
$137$	15.9191	1.36006	0.680030	−	0.733184i	$-0.261967\pi$
$137$	15.9191	1.36006	0.680030	+	0.733184i	$0.261967\pi$
$138$	0	0
$139$	−14.7278	−1.24920	−0.624599	−	0.780946i	$-0.714737\pi$
$139$	−14.7278	−1.24920	−0.624599	+	0.780946i	$0.714737\pi$
$140$	0	0
$141$	5.15979	0.434533
$142$	0	0
$143$	−5.89690	−0.493124
$144$	0	0
$145$	−0.850917	−0.0706648
$146$	0	0
$147$	37.6724	3.10717
$148$	0	0
$149$	7.35723	0.602727	0.301364	−	0.953509i	$-0.402558\pi$
$149$	7.35723	0.602727	0.301364	+	0.953509i	$0.402558\pi$
$150$	0	0
$151$	−10.5197	−0.856083	−0.428042	−	0.903759i	$-0.640796\pi$
$151$	−10.5197	−0.856083	−0.428042	+	0.903759i	$0.640796\pi$
$152$	0	0
$153$	−17.6045	−1.42324
$154$	0	0
$155$	−1.56065	−0.125354
$156$	0	0
$157$	−16.1669	−1.29026	−0.645131	−	0.764072i	$-0.723197\pi$
$157$	−16.1669	−1.29026	−0.645131	+	0.764072i	$0.723197\pi$
$158$	0	0
$159$	−4.31816	−0.342452
$160$	0	0
$161$	−1.64943	−0.129993
$162$	0	0
$163$	−13.6643	−1.07027	−0.535136	−	0.844766i	$-0.679740\pi$
$163$	−13.6643	−1.07027	−0.535136	+	0.844766i	$0.679740\pi$
$164$	0	0
$165$	0.734121	0.0571513
$166$	0	0
$167$	−10.3060	−0.797501	−0.398750	−	0.917060i	$-0.630556\pi$
$167$	−10.3060	−0.797501	−0.398750	+	0.917060i	$0.630556\pi$
$168$	0	0
$169$	21.7734	1.67488
$170$	0	0
$171$	−5.98952	−0.458030
$172$	0	0
$173$	−25.3276	−1.92562	−0.962811	−	0.270174i	$-0.912919\pi$
$173$	−25.3276	−1.92562	−0.962811	+	0.270174i	$0.912919\pi$
$174$	0	0
$175$	21.8509	1.65177
$176$	0	0
$177$	−23.1381	−1.73916
$178$	0	0
$179$	−22.8968	−1.71139	−0.855695	−	0.517481i	$-0.826870\pi$
$179$	−22.8968	−1.71139	−0.855695	+	0.517481i	$0.826870\pi$
$180$	0	0
$181$	8.05994	0.599091	0.299545	−	0.954082i	$-0.403165\pi$
$181$	8.05994	0.599091	0.299545	+	0.954082i	$0.403165\pi$
$182$	0	0
$183$	12.4992	0.923964
$184$	0	0
$185$	0.226860	0.0166791
$186$	0	0
$187$	−2.93922	−0.214937
$188$	0	0
$189$	−39.6468	−2.88388
$190$	0	0
$191$	17.2143	1.24559	0.622793	−	0.782387i	$-0.285998\pi$
$191$	17.2143	1.24559	0.622793	+	0.782387i	$0.285998\pi$
$192$	0	0
$193$	−8.31588	−0.598590	−0.299295	−	0.954161i	$-0.596751\pi$
$193$	−8.31588	−0.598590	−0.299295	+	0.954161i	$0.596751\pi$
$194$	0	0
$195$	−4.32904	−0.310009
$196$	0	0
$197$	4.38913	0.312713	0.156356	−	0.987701i	$-0.450025\pi$
$197$	4.38913	0.312713	0.156356	+	0.987701i	$0.450025\pi$
$198$	0	0
$199$	−19.8924	−1.41014	−0.705068	−	0.709140i	$-0.749083\pi$
$199$	−19.8924	−1.41014	−0.705068	+	0.709140i	$0.749083\pi$
$200$	0	0
$201$	34.1136	2.40619
$202$	0	0
$203$	15.3718	1.07889
$204$	0	0
$205$	−1.63526	−0.114211
$206$	0	0
$207$	2.23352	0.155240
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−20.8162	−1.43304	−0.716522	−	0.697564i	$-0.754267\pi$
$211$	−20.8162	−1.43304	−0.716522	+	0.697564i	$0.754267\pi$
$212$	0	0
$213$	−7.21962	−0.494680
$214$	0	0
$215$	−0.521053	−0.0355355
$216$	0	0
$217$	28.1931	1.91387
$218$	0	0
$219$	43.3705	2.93071
$220$	0	0
$221$	17.3323	1.16590
$222$	0	0
$223$	−11.4967	−0.769874	−0.384937	−	0.922943i	$-0.625777\pi$
$223$	−11.4967	−0.769874	−0.384937	+	0.922943i	$0.625777\pi$
$224$	0	0
$225$	−29.5885	−1.97257
$226$	0	0
$227$	17.5175	1.16268	0.581339	−	0.813662i	$-0.302529\pi$
$227$	17.5175	1.16268	0.581339	+	0.813662i	$0.302529\pi$
$228$	0	0
$229$	−6.64251	−0.438949	−0.219475	−	0.975618i	$-0.570434\pi$
$229$	−6.64251	−0.438949	−0.219475	+	0.975618i	$0.570434\pi$
$230$	0	0
$231$	−13.2619	−0.872569
$232$	0	0
$233$	−12.5323	−0.821020	−0.410510	−	0.911856i	$-0.634649\pi$
$233$	−12.5323	−0.821020	−0.410510	+	0.911856i	$0.634649\pi$
$234$	0	0
$235$	0.421370	0.0274871
$236$	0	0
$237$	−5.01813	−0.325962
$238$	0	0
$239$	27.3445	1.76877	0.884386	−	0.466757i	$-0.154578\pi$
$239$	27.3445	1.76877	0.884386	+	0.466757i	$0.154578\pi$
$240$	0	0
$241$	23.5681	1.51816	0.759079	−	0.650999i	$-0.225650\pi$
$241$	23.5681	1.51816	0.759079	+	0.650999i	$0.225650\pi$
$242$	0	0
$243$	−0.188120	−0.0120679
$244$	0	0
$245$	3.07648	0.196549
$246$	0	0
$247$	5.89690	0.375211
$248$	0	0
$249$	19.4256	1.23105
$250$	0	0
$251$	2.04497	0.129078	0.0645388	−	0.997915i	$-0.479442\pi$
$251$	2.04497	0.129078	0.0645388	+	0.997915i	$0.479442\pi$
$252$	0	0
$253$	0.372904	0.0234443
$254$	0	0
$255$	−2.15774	−0.135123
$256$	0	0
$257$	21.0705	1.31434	0.657171	−	0.753741i	$-0.271753\pi$
$257$	21.0705	1.31434	0.657171	+	0.753741i	$0.271753\pi$
$258$	0	0
$259$	−4.09823	−0.254651
$260$	0	0
$261$	−20.8152	−1.28843
$262$	0	0
$263$	21.5153	1.32669	0.663345	−	0.748313i	$-0.269136\pi$
$263$	21.5153	1.32669	0.663345	+	0.748313i	$0.269136\pi$
$264$	0	0
$265$	−0.352638	−0.0216624
$266$	0	0
$267$	−14.8625	−0.909572
$268$	0	0
$269$	8.56257	0.522069	0.261034	−	0.965329i	$-0.415936\pi$
$269$	8.56257	0.522069	0.261034	+	0.965329i	$0.415936\pi$
$270$	0	0
$271$	12.5375	0.761602	0.380801	−	0.924657i	$-0.375648\pi$
$271$	12.5375	0.761602	0.380801	+	0.924657i	$0.375648\pi$
$272$	0	0
$273$	78.2041	4.73313
$274$	0	0
$275$	−4.94005	−0.297896
$276$	0	0
$277$	11.2455	0.675679	0.337840	−	0.941204i	$-0.390304\pi$
$277$	11.2455	0.675679	0.337840	+	0.941204i	$0.390304\pi$
$278$	0	0
$279$	−38.1767	−2.28558
$280$	0	0
$281$	−20.3272	−1.21262	−0.606310	−	0.795228i	$-0.707351\pi$
$281$	−20.3272	−1.21262	−0.606310	+	0.795228i	$0.707351\pi$
$282$	0	0
$283$	26.7277	1.58880	0.794400	−	0.607395i	$-0.207786\pi$
$283$	26.7277	1.58880	0.794400	+	0.607395i	$0.207786\pi$
$284$	0	0
$285$	−0.734121	−0.0434856
$286$	0	0
$287$	29.5409	1.74374
$288$	0	0
$289$	−8.36099	−0.491823
$290$	0	0
$291$	24.7058	1.44828
$292$	0	0
$293$	−9.95580	−0.581624	−0.290812	−	0.956780i	$-0.593925\pi$
$293$	−9.95580	−0.581624	−0.290812	+	0.956780i	$0.593925\pi$
$294$	0	0
$295$	−1.88955	−0.110014
$296$	0	0
$297$	8.96335	0.520106
$298$	0	0
$299$	−2.19898	−0.127170
$300$	0	0
$301$	9.41283	0.542547
$302$	0	0
$303$	10.5301	0.604937
$304$	0	0
$305$	1.02073	0.0584469
$306$	0	0
$307$	−3.58808	−0.204783	−0.102391	−	0.994744i	$-0.532649\pi$
$307$	−3.58808	−0.204783	−0.102391	+	0.994744i	$0.532649\pi$
$308$	0	0
$309$	7.29157	0.414803
$310$	0	0
$311$	13.8469	0.785187	0.392594	−	0.919712i	$-0.371578\pi$
$311$	13.8469	0.785187	0.392594	+	0.919712i	$0.371578\pi$
$312$	0	0
$313$	−15.3982	−0.870360	−0.435180	−	0.900344i	$-0.643315\pi$
$313$	−15.3982	−0.870360	−0.435180	+	0.900344i	$0.643315\pi$
$314$	0	0
$315$	−6.48678	−0.365489
$316$	0	0
$317$	−5.96359	−0.334949	−0.167474	−	0.985876i	$-0.553561\pi$
$317$	−5.96359	−0.334949	−0.167474	+	0.985876i	$0.553561\pi$
$318$	0	0
$319$	−3.47526	−0.194577
$320$	0	0
$321$	−50.9297	−2.84262
$322$	0	0
$323$	2.93922	0.163542
$324$	0	0
$325$	29.1310	1.61590
$326$	0	0
$327$	11.5009	0.636000
$328$	0	0
$329$	−7.61204	−0.419665
$330$	0	0
$331$	5.58507	0.306983	0.153492	−	0.988150i	$-0.450948\pi$
$331$	5.58507	0.306983	0.153492	+	0.988150i	$0.450948\pi$
$332$	0	0
$333$	5.54946	0.304109
$334$	0	0
$335$	2.78586	0.152208
$336$	0	0
$337$	−29.0465	−1.58227	−0.791133	−	0.611645i	$-0.790508\pi$
$337$	−29.0465	−1.58227	−0.791133	+	0.611645i	$0.790508\pi$
$338$	0	0
$339$	−32.3539	−1.75722
$340$	0	0
$341$	−6.37391	−0.345166
$342$	0	0
$343$	−24.6142	−1.32904
$344$	0	0
$345$	0.273757	0.0147386
$346$	0	0
$347$	6.25081	0.335561	0.167781	−	0.985824i	$-0.446340\pi$
$347$	6.25081	0.335561	0.167781	+	0.985824i	$0.446340\pi$
$348$	0	0
$349$	7.81752	0.418462	0.209231	−	0.977866i	$-0.432904\pi$
$349$	7.81752	0.418462	0.209231	+	0.977866i	$0.432904\pi$
$350$	0	0
$351$	−52.8560	−2.82124
$352$	0	0
$353$	10.0965	0.537380	0.268690	−	0.963227i	$-0.413409\pi$
$353$	10.0965	0.537380	0.268690	+	0.963227i	$0.413409\pi$
$354$	0	0
$355$	−0.589584	−0.0312919
$356$	0	0
$357$	38.9796	2.06302
$358$	0	0
$359$	−29.3266	−1.54780	−0.773899	−	0.633309i	$-0.781696\pi$
$359$	−29.3266	−1.54780	−0.773899	+	0.633309i	$0.781696\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.99825	0.157368
$364$	0	0
$365$	3.54181	0.185387
$366$	0	0
$367$	13.7848	0.719558	0.359779	−	0.933037i	$-0.382852\pi$
$367$	13.7848	0.719558	0.359779	+	0.933037i	$0.382852\pi$
$368$	0	0
$369$	−40.0017	−2.08241
$370$	0	0
$371$	6.37041	0.330735
$372$	0	0
$373$	16.4169	0.850036	0.425018	−	0.905185i	$-0.360268\pi$
$373$	16.4169	0.850036	0.425018	+	0.905185i	$0.360268\pi$
$374$	0	0
$375$	−7.29720	−0.376826
$376$	0	0
$377$	20.4933	1.05546
$378$	0	0
$379$	−6.93533	−0.356244	−0.178122	−	0.984008i	$-0.557002\pi$
$379$	−6.93533	−0.356244	−0.178122	+	0.984008i	$0.557002\pi$
$380$	0	0
$381$	36.2695	1.85814
$382$	0	0
$383$	−1.47147	−0.0751884	−0.0375942	−	0.999293i	$-0.511969\pi$
$383$	−1.47147	−0.0751884	−0.0375942	+	0.999293i	$0.511969\pi$
$384$	0	0
$385$	−1.08302	−0.0551959
$386$	0	0
$387$	−12.7460	−0.647917
$388$	0	0
$389$	6.79609	0.344575	0.172288	−	0.985047i	$-0.444884\pi$
$389$	6.79609	0.344575	0.172288	+	0.985047i	$0.444884\pi$
$390$	0	0
$391$	−1.09605	−0.0554294
$392$	0	0
$393$	23.9807	1.20967
$394$	0	0
$395$	−0.409801	−0.0206193
$396$	0	0
$397$	−12.3808	−0.621373	−0.310687	−	0.950512i	$-0.600559\pi$
$397$	−12.3808	−0.621373	−0.310687	+	0.950512i	$0.600559\pi$
$398$	0	0
$399$	13.2619	0.663926
$400$	0	0
$401$	−0.0361352	−0.00180451	−0.000902254	−	1.00000i	$-0.500287\pi$
$401$	−0.0361352	−0.00180451	−0.000902254		1.00000i	$0.500287\pi$
$402$	0	0
$403$	37.5863	1.87231
$404$	0	0
$405$	2.18059	0.108354
$406$	0	0
$407$	0.926528	0.0459263
$408$	0	0
$409$	−9.94955	−0.491973	−0.245987	−	0.969273i	$-0.579112\pi$
$409$	−9.94955	−0.491973	−0.245987	+	0.969273i	$0.579112\pi$
$410$	0	0
$411$	47.7295	2.35432
$412$	0	0
$413$	34.1347	1.67966
$414$	0	0
$415$	1.58637	0.0778720
$416$	0	0
$417$	−44.1577	−2.16241
$418$	0	0
$419$	−27.2560	−1.33154	−0.665771	−	0.746156i	$-0.731897\pi$
$419$	−27.2560	−1.33154	−0.665771	+	0.746156i	$0.731897\pi$
$420$	0	0
$421$	16.5342	0.805828	0.402914	−	0.915238i	$-0.367997\pi$
$421$	16.5342	0.805828	0.402914	+	0.915238i	$0.367997\pi$
$422$	0	0
$423$	10.3076	0.501171
$424$	0	0
$425$	14.5199	0.704318
$426$	0	0
$427$	−18.4395	−0.892351
$428$	0	0
$429$	−17.6804	−0.853618
$430$	0	0
$431$	−33.9550	−1.63556	−0.817778	−	0.575533i	$-0.804794\pi$
$431$	−33.9550	−1.63556	−0.817778	+	0.575533i	$0.804794\pi$
$432$	0	0
$433$	−3.55114	−0.170657	−0.0853285	−	0.996353i	$-0.527194\pi$
$433$	−3.55114	−0.170657	−0.0853285	+	0.996353i	$0.527194\pi$
$434$	0	0
$435$	−2.55126	−0.122324
$436$	0	0
$437$	−0.372904	−0.0178384
$438$	0	0
$439$	−6.28232	−0.299839	−0.149919	−	0.988698i	$-0.547901\pi$
$439$	−6.28232	−0.299839	−0.149919	+	0.988698i	$0.547901\pi$
$440$	0	0
$441$	75.2571	3.58367
$442$	0	0
$443$	−26.3969	−1.25415	−0.627077	−	0.778957i	$-0.715749\pi$
$443$	−26.3969	−1.25415	−0.627077	+	0.778957i	$0.715749\pi$
$444$	0	0
$445$	−1.21373	−0.0575366
$446$	0	0
$447$	22.0588	1.04335
$448$	0	0
$449$	−0.649120	−0.0306339	−0.0153169	−	0.999883i	$-0.504876\pi$
$449$	−0.649120	−0.0306339	−0.0153169	+	0.999883i	$0.504876\pi$
$450$	0	0
$451$	−6.67861	−0.314484
$452$	0	0
$453$	−31.5408	−1.48192
$454$	0	0
$455$	6.38647	0.299402
$456$	0	0
$457$	−4.74769	−0.222087	−0.111044	−	0.993816i	$-0.535419\pi$
$457$	−4.74769	−0.222087	−0.111044	+	0.993816i	$0.535419\pi$
$458$	0	0
$459$	−26.3453	−1.22969
$460$	0	0
$461$	13.9711	0.650701	0.325350	−	0.945594i	$-0.394518\pi$
$461$	13.9711	0.650701	0.325350	+	0.945594i	$0.394518\pi$
$462$	0	0
$463$	14.7822	0.686985	0.343492	−	0.939155i	$-0.388390\pi$
$463$	14.7822	0.686985	0.343492	+	0.939155i	$0.388390\pi$
$464$	0	0
$465$	−4.67922	−0.216994
$466$	0	0
$467$	22.5444	1.04323	0.521615	−	0.853181i	$-0.325330\pi$
$467$	22.5444	1.04323	0.521615	+	0.853181i	$0.325330\pi$
$468$	0	0
$469$	−50.3265	−2.32386
$470$	0	0
$471$	−48.4726	−2.23350
$472$	0	0
$473$	−2.12805	−0.0978480
$474$	0	0
$475$	4.94005	0.226665
$476$	0	0
$477$	−8.62625	−0.394969
$478$	0	0
$479$	21.7949	0.995835	0.497918	−	0.867224i	$-0.334098\pi$
$479$	21.7949	0.995835	0.497918	+	0.867224i	$0.334098\pi$
$480$	0	0
$481$	−5.46364	−0.249121
$482$	0	0
$483$	−4.94542	−0.225024
$484$	0	0
$485$	2.01757	0.0916133
$486$	0	0
$487$	5.40569	0.244955	0.122478	−	0.992471i	$-0.460916\pi$
$487$	5.40569	0.244955	0.122478	+	0.992471i	$0.460916\pi$
$488$	0	0
$489$	−40.9691	−1.85269
$490$	0	0
$491$	41.4597	1.87105	0.935524	−	0.353263i	$-0.114928\pi$
$491$	41.4597	1.87105	0.935524	+	0.353263i	$0.114928\pi$
$492$	0	0
$493$	10.2146	0.460040
$494$	0	0
$495$	1.46653	0.0659158
$496$	0	0
$497$	10.6508	0.477755
$498$	0	0
$499$	−18.0596	−0.808459	−0.404229	−	0.914658i	$-0.632460\pi$
$499$	−18.0596	−0.808459	−0.404229	+	0.914658i	$0.632460\pi$
$500$	0	0
$501$	−30.8999	−1.38051
$502$	0	0
$503$	−33.5764	−1.49710	−0.748549	−	0.663080i	$-0.769249\pi$
$503$	−33.5764	−1.49710	−0.748549	+	0.663080i	$0.769249\pi$
$504$	0	0
$505$	0.859929	0.0382664
$506$	0	0
$507$	65.2823	2.89929
$508$	0	0
$509$	−19.8963	−0.881889	−0.440944	−	0.897534i	$-0.645356\pi$
$509$	−19.8963	−0.881889	−0.440944	+	0.897534i	$0.645356\pi$
$510$	0	0
$511$	−63.9829	−2.83044
$512$	0	0
$513$	−8.96335	−0.395742
$514$	0	0
$515$	0.595459	0.0262391
$516$	0	0
$517$	1.72093	0.0756865
$518$	0	0
$519$	−75.9386	−3.33334
$520$	0	0
$521$	−22.1379	−0.969878	−0.484939	−	0.874548i	$-0.661158\pi$
$521$	−22.1379	−0.969878	−0.484939	+	0.874548i	$0.661158\pi$
$522$	0	0
$523$	−16.0313	−0.701001	−0.350500	−	0.936563i	$-0.613988\pi$
$523$	−16.0313	−0.701001	−0.350500	+	0.936563i	$0.613988\pi$
$524$	0	0
$525$	65.5145	2.85929
$526$	0	0
$527$	18.7343	0.816079
$528$	0	0
$529$	−22.8609	−0.993954
$530$	0	0
$531$	−46.2223	−2.00588
$532$	0	0
$533$	39.3831	1.70587
$534$	0	0
$535$	−4.15913	−0.179815
$536$	0	0
$537$	−68.6505	−2.96249
$538$	0	0
$539$	12.5648	0.541204
$540$	0	0
$541$	−30.5422	−1.31311	−0.656556	−	0.754278i	$-0.727987\pi$
$541$	−30.5422	−1.31311	−0.656556	+	0.754278i	$0.727987\pi$
$542$	0	0
$543$	24.1657	1.03705
$544$	0	0
$545$	0.939208	0.0402313
$546$	0	0
$547$	−0.0830197	−0.00354967	−0.00177483	−	0.999998i	$-0.500565\pi$
$547$	−0.0830197	−0.00354967	−0.00177483	+	0.999998i	$0.500565\pi$
$548$	0	0
$549$	24.9692	1.06566
$550$	0	0
$551$	3.47526	0.148051
$552$	0	0
$553$	7.40305	0.314810
$554$	0	0
$555$	0.680184	0.0288722
$556$	0	0
$557$	−6.33324	−0.268348	−0.134174	−	0.990958i	$-0.542838\pi$
$557$	−6.33324	−0.268348	−0.134174	+	0.990958i	$0.542838\pi$
$558$	0	0
$559$	12.5489	0.530763
$560$	0	0
$561$	−8.81252	−0.372065
$562$	0	0
$563$	26.4539	1.11490	0.557451	−	0.830210i	$-0.311780\pi$
$563$	26.4539	1.11490	0.557451	+	0.830210i	$0.311780\pi$
$564$	0	0
$565$	−2.64215	−0.111156
$566$	0	0
$567$	−39.3924	−1.65432
$568$	0	0
$569$	−23.9553	−1.00426	−0.502128	−	0.864793i	$-0.667450\pi$
$569$	−23.9553	−1.00426	−0.502128	+	0.864793i	$0.667450\pi$
$570$	0	0
$571$	−8.33187	−0.348678	−0.174339	−	0.984686i	$-0.555779\pi$
$571$	−8.33187	−0.348678	−0.174339	+	0.984686i	$0.555779\pi$
$572$	0	0
$573$	51.6130	2.15616
$574$	0	0
$575$	−1.84216	−0.0768235
$576$	0	0
$577$	21.7197	0.904203	0.452101	−	0.891967i	$-0.350674\pi$
$577$	21.7197	0.904203	0.452101	+	0.891967i	$0.350674\pi$
$578$	0	0
$579$	−24.9331	−1.03618
$580$	0	0
$581$	−28.6578	−1.18893
$582$	0	0
$583$	−1.44022	−0.0596480
$584$	0	0
$585$	−8.64800	−0.357551
$586$	0	0
$587$	−38.3274	−1.58194	−0.790971	−	0.611853i	$-0.790424\pi$
$587$	−38.3274	−1.58194	−0.790971	+	0.611853i	$0.790424\pi$
$588$	0	0
$589$	6.37391	0.262632
$590$	0	0
$591$	13.1597	0.541319
$592$	0	0
$593$	25.1482	1.03271	0.516357	−	0.856373i	$-0.327288\pi$
$593$	25.1482	1.03271	0.516357	+	0.856373i	$0.327288\pi$
$594$	0	0
$595$	3.18324	0.130500
$596$	0	0
$597$	−59.6425	−2.44100
$598$	0	0
$599$	5.97251	0.244030	0.122015	−	0.992528i	$-0.461064\pi$
$599$	5.97251	0.244030	0.122015	+	0.992528i	$0.461064\pi$
$600$	0	0
$601$	−1.61426	−0.0658469	−0.0329234	−	0.999458i	$-0.510482\pi$
$601$	−1.61426	−0.0658469	−0.0329234	+	0.999458i	$0.510482\pi$
$602$	0	0
$603$	68.1478	2.77519
$604$	0	0
$605$	0.244850	0.00995455
$606$	0	0
$607$	43.1541	1.75157	0.875786	−	0.482700i	$-0.160344\pi$
$607$	43.1541	1.75157	0.875786	+	0.482700i	$0.160344\pi$
$608$	0	0
$609$	46.0886	1.86760
$610$	0	0
$611$	−10.1482	−0.410551
$612$	0	0
$613$	−15.2118	−0.614400	−0.307200	−	0.951645i	$-0.599392\pi$
$613$	−15.2118	−0.614400	−0.307200	+	0.951645i	$0.599392\pi$
$614$	0	0
$615$	−4.90291	−0.197705
$616$	0	0
$617$	−10.1846	−0.410016	−0.205008	−	0.978760i	$-0.565722\pi$
$617$	−10.1846	−0.410016	−0.205008	+	0.978760i	$0.565722\pi$
$618$	0	0
$619$	−23.7636	−0.955139	−0.477569	−	0.878594i	$-0.658482\pi$
$619$	−23.7636	−0.955139	−0.477569	+	0.878594i	$0.658482\pi$
$620$	0	0
$621$	3.34247	0.134129
$622$	0	0
$623$	21.9261	0.878452
$624$	0	0
$625$	24.1043	0.964173
$626$	0	0
$627$	−2.99825	−0.119739
$628$	0	0
$629$	−2.72327	−0.108584
$630$	0	0
$631$	−36.1097	−1.43751	−0.718753	−	0.695266i	$-0.755287\pi$
$631$	−36.1097	−1.43751	−0.718753	+	0.695266i	$0.755287\pi$
$632$	0	0
$633$	−62.4122	−2.48066
$634$	0	0
$635$	2.96192	0.117540
$636$	0	0
$637$	−74.0933	−2.93568
$638$	0	0
$639$	−14.4224	−0.570543
$640$	0	0
$641$	−34.7607	−1.37297	−0.686483	−	0.727146i	$-0.740846\pi$
$641$	−34.7607	−1.37297	−0.686483	+	0.727146i	$0.740846\pi$
$642$	0	0
$643$	32.5548	1.28384	0.641919	−	0.766773i	$-0.278139\pi$
$643$	32.5548	1.28384	0.641919	+	0.766773i	$0.278139\pi$
$644$	0	0
$645$	−1.56225	−0.0615135
$646$	0	0
$647$	0.726588	0.0285651	0.0142826	−	0.999898i	$-0.495454\pi$
$647$	0.726588	0.0285651	0.0142826	+	0.999898i	$0.495454\pi$
$648$	0	0
$649$	−7.71718	−0.302926
$650$	0	0
$651$	84.5302	3.31300
$652$	0	0
$653$	−2.07531	−0.0812132	−0.0406066	−	0.999175i	$-0.512929\pi$
$653$	−2.07531	−0.0812132	−0.0406066	+	0.999175i	$0.512929\pi$
$654$	0	0
$655$	1.95836	0.0765195
$656$	0	0
$657$	86.6400	3.38015
$658$	0	0
$659$	6.83577	0.266284	0.133142	−	0.991097i	$-0.457493\pi$
$659$	6.83577	0.266284	0.133142	+	0.991097i	$0.457493\pi$
$660$	0	0
$661$	26.6223	1.03549	0.517744	−	0.855536i	$-0.326772\pi$
$661$	26.6223	1.03549	0.517744	+	0.855536i	$0.326772\pi$
$662$	0	0
$663$	51.9666	2.01821
$664$	0	0
$665$	1.08302	0.0419978
$666$	0	0
$667$	−1.29594	−0.0501790
$668$	0	0
$669$	−34.4699	−1.33268
$670$	0	0
$671$	4.16881	0.160935
$672$	0	0
$673$	26.9513	1.03890	0.519449	−	0.854502i	$-0.326137\pi$
$673$	26.9513	1.03890	0.519449	+	0.854502i	$0.326137\pi$
$674$	0	0
$675$	−44.2794	−1.70431
$676$	0	0
$677$	−7.83263	−0.301033	−0.150516	−	0.988608i	$-0.548094\pi$
$677$	−7.83263	−0.301033	−0.150516	+	0.988608i	$0.548094\pi$
$678$	0	0
$679$	−36.4475	−1.39873
$680$	0	0
$681$	52.5219	2.01264
$682$	0	0
$683$	−3.91204	−0.149690	−0.0748451	−	0.997195i	$-0.523846\pi$
$683$	−3.91204	−0.149690	−0.0748451	+	0.997195i	$0.523846\pi$
$684$	0	0
$685$	3.89779	0.148927
$686$	0	0
$687$	−19.9159	−0.759840
$688$	0	0
$689$	8.49285	0.323552
$690$	0	0
$691$	34.8112	1.32428	0.662140	−	0.749380i	$-0.269648\pi$
$691$	34.8112	1.32428	0.662140	+	0.749380i	$0.269648\pi$
$692$	0	0
$693$	−26.4929	−1.00638
$694$	0	0
$695$	−3.60610	−0.136787
$696$	0	0
$697$	19.6299	0.743536
$698$	0	0
$699$	−37.5751	−1.42122
$700$	0	0
$701$	−20.0313	−0.756572	−0.378286	−	0.925689i	$-0.623486\pi$
$701$	−20.0313	−0.756572	−0.378286	+	0.925689i	$0.623486\pi$
$702$	0	0
$703$	−0.926528	−0.0349447
$704$	0	0
$705$	1.26337	0.0475814
$706$	0	0
$707$	−15.5346	−0.584240
$708$	0	0
$709$	20.5264	0.770886	0.385443	−	0.922732i	$-0.374049\pi$
$709$	20.5264	0.770886	0.385443	+	0.922732i	$0.374049\pi$
$710$	0	0
$711$	−10.0246	−0.375951
$712$	0	0
$713$	−2.37686	−0.0890139
$714$	0	0
$715$	−1.44385	−0.0539971
$716$	0	0
$717$	81.9859	3.06182
$718$	0	0
$719$	3.14609	0.117329	0.0586647	−	0.998278i	$-0.481316\pi$
$719$	3.14609	0.117329	0.0586647	+	0.998278i	$0.481316\pi$
$720$	0	0
$721$	−10.7570	−0.400611
$722$	0	0
$723$	70.6633	2.62800
$724$	0	0
$725$	17.1680	0.637602
$726$	0	0
$727$	−36.6121	−1.35787	−0.678934	−	0.734199i	$-0.737558\pi$
$727$	−36.6121	−1.35787	−0.678934	+	0.734199i	$0.737558\pi$
$728$	0	0
$729$	−27.2815	−1.01043
$730$	0	0
$731$	6.25482	0.231343
$732$	0	0
$733$	−1.44149	−0.0532428	−0.0266214	−	0.999646i	$-0.508475\pi$
$733$	−1.44149	−0.0532428	−0.0266214	+	0.999646i	$0.508475\pi$
$734$	0	0
$735$	9.22408	0.340235
$736$	0	0
$737$	11.3778	0.419108
$738$	0	0
$739$	27.8090	1.02297	0.511485	−	0.859292i	$-0.329096\pi$
$739$	27.8090	1.02297	0.511485	+	0.859292i	$0.329096\pi$
$740$	0	0
$741$	17.6804	0.649506
$742$	0	0
$743$	−1.15020	−0.0421967	−0.0210983	−	0.999777i	$-0.506716\pi$
$743$	−1.15020	−0.0421967	−0.0210983	+	0.999777i	$0.506716\pi$
$744$	0	0
$745$	1.80141	0.0659987
$746$	0	0
$747$	38.8060	1.41984
$748$	0	0
$749$	75.1347	2.74536
$750$	0	0
$751$	−43.4326	−1.58488	−0.792439	−	0.609951i	$-0.791189\pi$
$751$	−43.4326	−1.58488	−0.792439	+	0.609951i	$0.791189\pi$
$752$	0	0
$753$	6.13135	0.223439
$754$	0	0
$755$	−2.57575	−0.0937412
$756$	0	0
$757$	−22.5876	−0.820962	−0.410481	−	0.911869i	$-0.634639\pi$
$757$	−22.5876	−0.820962	−0.410481	+	0.911869i	$0.634639\pi$
$758$	0	0
$759$	1.11806	0.0405830
$760$	0	0
$761$	23.0477	0.835479	0.417739	−	0.908567i	$-0.362823\pi$
$761$	23.0477	0.835479	0.417739	+	0.908567i	$0.362823\pi$
$762$	0	0
$763$	−16.9668	−0.614239
$764$	0	0
$765$	−4.31046	−0.155845
$766$	0	0
$767$	45.5075	1.64318
$768$	0	0
$769$	−26.7350	−0.964088	−0.482044	−	0.876147i	$-0.660105\pi$
$769$	−26.7350	−0.964088	−0.482044	+	0.876147i	$0.660105\pi$
$770$	0	0
$771$	63.1747	2.27518
$772$	0	0
$773$	28.4074	1.02174	0.510872	−	0.859657i	$-0.329323\pi$
$773$	28.4074	1.02174	0.510872	+	0.859657i	$0.329323\pi$
$774$	0	0
$775$	31.4874	1.13106
$776$	0	0
$777$	−12.2875	−0.440812
$778$	0	0
$779$	6.67861	0.239286
$780$	0	0
$781$	−2.40794	−0.0861630
$782$	0	0
$783$	−31.1500	−1.11321
$784$	0	0
$785$	−3.95847	−0.141284
$786$	0	0
$787$	6.60374	0.235398	0.117699	−	0.993049i	$-0.462448\pi$
$787$	6.60374	0.235398	0.117699	+	0.993049i	$0.462448\pi$
$788$	0	0
$789$	64.5083	2.29656
$790$	0	0
$791$	47.7305	1.69710
$792$	0	0
$793$	−24.5831	−0.872970
$794$	0	0
$795$	−1.05730	−0.0374985
$796$	0	0
$797$	45.9061	1.62608	0.813038	−	0.582210i	$-0.197812\pi$
$797$	45.9061	1.62608	0.813038	+	0.582210i	$0.197812\pi$
$798$	0	0
$799$	−5.05820	−0.178946
$800$	0	0
$801$	−29.6904	−1.04906
$802$	0	0
$803$	14.4653	0.510468
$804$	0	0
$805$	−0.403863	−0.0142343
$806$	0	0
$807$	25.6727	0.903723
$808$	0	0
$809$	−19.0359	−0.669266	−0.334633	−	0.942349i	$-0.608612\pi$
$809$	−19.0359	−0.669266	−0.334633	+	0.942349i	$0.608612\pi$
$810$	0	0
$811$	56.0480	1.96811	0.984056	−	0.177860i	$-0.0569173\pi$
$811$	56.0480	1.96811	0.984056	+	0.177860i	$0.0569173\pi$
$812$	0	0
$813$	37.5908	1.31837
$814$	0	0
$815$	−3.34570	−0.117195
$816$	0	0
$817$	2.12805	0.0744512
$818$	0	0
$819$	156.226	5.45898
$820$	0	0
$821$	3.28817	0.114758	0.0573789	−	0.998352i	$-0.481726\pi$
$821$	3.28817	0.114758	0.0573789	+	0.998352i	$0.481726\pi$
$822$	0	0
$823$	5.84050	0.203587	0.101793	−	0.994806i	$-0.467542\pi$
$823$	5.84050	0.203587	0.101793	+	0.994806i	$0.467542\pi$
$824$	0	0
$825$	−14.8115	−0.515671
$826$	0	0
$827$	14.4956	0.504061	0.252030	−	0.967719i	$-0.418902\pi$
$827$	14.4956	0.504061	0.252030	+	0.967719i	$0.418902\pi$
$828$	0	0
$829$	−52.7194	−1.83102	−0.915511	−	0.402293i	$-0.868213\pi$
$829$	−52.7194	−1.83102	−0.915511	+	0.402293i	$0.868213\pi$
$830$	0	0
$831$	33.7170	1.16963
$832$	0	0
$833$	−36.9306	−1.27957
$834$	0	0
$835$	−2.52342	−0.0873264
$836$	0	0
$837$	−57.1316	−1.97476
$838$	0	0
$839$	28.9506	0.999485	0.499742	−	0.866174i	$-0.333428\pi$
$839$	28.9506	0.999485	0.499742	+	0.866174i	$0.333428\pi$
$840$	0	0
$841$	−16.9226	−0.583536
$842$	0	0
$843$	−60.9462	−2.09910
$844$	0	0
$845$	5.33122	0.183399
$846$	0	0
$847$	−4.42321	−0.151983
$848$	0	0
$849$	80.1365	2.75028
$850$	0	0
$851$	0.345506	0.0118438
$852$	0	0
$853$	−17.2571	−0.590873	−0.295436	−	0.955362i	$-0.595465\pi$
$853$	−17.2571	−0.590873	−0.295436	+	0.955362i	$0.595465\pi$
$854$	0	0
$855$	−1.46653	−0.0501544
$856$	0	0
$857$	43.2863	1.47863	0.739315	−	0.673359i	$-0.235149\pi$
$857$	43.2863	1.47863	0.739315	+	0.673359i	$0.235149\pi$
$858$	0	0
$859$	41.5981	1.41931	0.709654	−	0.704550i	$-0.248851\pi$
$859$	41.5981	1.41931	0.709654	+	0.704550i	$0.248851\pi$
$860$	0	0
$861$	88.5711	3.01850
$862$	0	0
$863$	31.0676	1.05755	0.528777	−	0.848761i	$-0.322651\pi$
$863$	31.0676	1.05755	0.528777	+	0.848761i	$0.322651\pi$
$864$	0	0
$865$	−6.20146	−0.210856
$866$	0	0
$867$	−25.0684	−0.851367
$868$	0	0
$869$	−1.67368	−0.0567758
$870$	0	0
$871$	−67.0939	−2.27339
$872$	0	0
$873$	49.3540	1.67038
$874$	0	0
$875$	10.7653	0.363933
$876$	0	0
$877$	0.708001	0.0239075	0.0119537	−	0.999929i	$-0.496195\pi$
$877$	0.708001	0.0239075	0.0119537	+	0.999929i	$0.496195\pi$
$878$	0	0
$879$	−29.8500	−1.00682
$880$	0	0
$881$	−35.1566	−1.18446	−0.592229	−	0.805770i	$-0.701752\pi$
$881$	−35.1566	−1.18446	−0.592229	+	0.805770i	$0.701752\pi$
$882$	0	0
$883$	1.50768	0.0507374	0.0253687	−	0.999678i	$-0.491924\pi$
$883$	1.50768	0.0507374	0.0253687	+	0.999678i	$0.491924\pi$
$884$	0	0
$885$	−5.66535	−0.190439
$886$	0	0
$887$	−46.0348	−1.54570	−0.772849	−	0.634590i	$-0.781169\pi$
$887$	−46.0348	−1.54570	−0.772849	+	0.634590i	$0.781169\pi$
$888$	0	0
$889$	−53.5070	−1.79457
$890$	0	0
$891$	8.90583	0.298357
$892$	0	0
$893$	−1.72093	−0.0575888
$894$	0	0
$895$	−5.60628	−0.187397
$896$	0	0
$897$	−6.59309	−0.220137
$898$	0	0
$899$	22.1510	0.738777
$900$	0	0
$901$	4.23313	0.141026
$902$	0	0
$903$	28.2221	0.939171
$904$	0	0
$905$	1.97347	0.0656005
$906$	0	0
$907$	−0.994096	−0.0330084	−0.0165042	−	0.999864i	$-0.505254\pi$
$907$	−0.994096	−0.0330084	−0.0165042	+	0.999864i	$0.505254\pi$
$908$	0	0
$909$	21.0356	0.697708
$910$	0	0
$911$	6.86962	0.227601	0.113800	−	0.993504i	$-0.463698\pi$
$911$	6.86962	0.227601	0.113800	+	0.993504i	$0.463698\pi$
$912$	0	0
$913$	6.47897	0.214423
$914$	0	0
$915$	3.06041	0.101174
$916$	0	0
$917$	−35.3778	−1.16828
$918$	0	0
$919$	−12.5820	−0.415043	−0.207521	−	0.978230i	$-0.566540\pi$
$919$	−12.5820	−0.415043	−0.207521	+	0.978230i	$0.566540\pi$
$920$	0	0
$921$	−10.7580	−0.354488
$922$	0	0
$923$	14.1994	0.467379
$924$	0	0
$925$	−4.57709	−0.150494
$926$	0	0
$927$	14.5662	0.478415
$928$	0	0
$929$	43.3197	1.42127	0.710636	−	0.703560i	$-0.248407\pi$
$929$	43.3197	1.42127	0.710636	+	0.703560i	$0.248407\pi$
$930$	0	0
$931$	−12.5648	−0.411794
$932$	0	0
$933$	41.5166	1.35919
$934$	0	0
$935$	−0.719667	−0.0235356
$936$	0	0
$937$	−44.3170	−1.44777	−0.723887	−	0.689918i	$-0.757646\pi$
$937$	−44.3170	−1.44777	−0.723887	+	0.689918i	$0.757646\pi$
$938$	0	0
$939$	−46.1678	−1.50663
$940$	0	0
$941$	58.0333	1.89183	0.945916	−	0.324410i	$-0.105166\pi$
$941$	58.0333	1.89183	0.945916	+	0.324410i	$0.105166\pi$
$942$	0	0
$943$	−2.49048	−0.0811012
$944$	0	0
$945$	−9.70750	−0.315785
$946$	0	0
$947$	−27.2005	−0.883897	−0.441949	−	0.897040i	$-0.645713\pi$
$947$	−27.2005	−0.883897	−0.441949	+	0.897040i	$0.645713\pi$
$948$	0	0
$949$	−85.3002	−2.76896
$950$	0	0
$951$	−17.8804	−0.579810
$952$	0	0
$953$	−7.27158	−0.235550	−0.117775	−	0.993040i	$-0.537576\pi$
$953$	−7.27158	−0.235550	−0.117775	+	0.993040i	$0.537576\pi$
$954$	0	0
$955$	4.21493	0.136392
$956$	0	0
$957$	−10.4197	−0.336822
$958$	0	0
$959$	−70.4135	−2.27377
$960$	0	0
$961$	9.62671	0.310539
$962$	0	0
$963$	−101.741	−3.27855
$964$	0	0
$965$	−2.03614	−0.0655457
$966$	0	0
$967$	−8.85046	−0.284612	−0.142306	−	0.989823i	$-0.545452\pi$
$967$	−8.85046	−0.284612	−0.142306	+	0.989823i	$0.545452\pi$
$968$	0	0
$969$	8.81252	0.283099
$970$	0	0
$971$	37.1779	1.19310	0.596548	−	0.802577i	$-0.296538\pi$
$971$	37.1779	1.19310	0.596548	+	0.802577i	$0.296538\pi$
$972$	0	0
$973$	65.1442	2.08843
$974$	0	0
$975$	87.3420	2.79718
$976$	0	0
$977$	46.5181	1.48825	0.744123	−	0.668043i	$-0.232868\pi$
$977$	46.5181	1.48825	0.744123	+	0.668043i	$0.232868\pi$
$978$	0	0
$979$	−4.95706	−0.158428
$980$	0	0
$981$	22.9750	0.733534
$982$	0	0
$983$	−24.6218	−0.785313	−0.392656	−	0.919685i	$-0.628444\pi$
$983$	−24.6218	−0.785313	−0.392656	+	0.919685i	$0.628444\pi$
$984$	0	0
$985$	1.07468	0.0342421
$986$	0	0
$987$	−22.8228	−0.726459
$988$	0	0
$989$	−0.793560	−0.0252337
$990$	0	0
$991$	−40.2121	−1.27738	−0.638690	−	0.769464i	$-0.720523\pi$
$991$	−40.2121	−1.27738	−0.638690	+	0.769464i	$0.720523\pi$
$992$	0	0
$993$	16.7455	0.531401
$994$	0	0
$995$	−4.87065	−0.154410
$996$	0	0
$997$	21.2470	0.672900	0.336450	−	0.941701i	$-0.390774\pi$
$997$	21.2470	0.672900	0.336450	+	0.941701i	$0.390774\pi$
$998$	0	0
$999$	8.30480	0.262752

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.ba.1.7		7
4.3	odd	2		209.2.a.d.1.7	✓	7
12.11	even	2		1881.2.a.p.1.1		7
20.19	odd	2		5225.2.a.n.1.1		7
44.43	even	2		2299.2.a.q.1.1		7
76.75	even	2		3971.2.a.i.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.d.1.7	✓	7	4.3	odd	2
1881.2.a.p.1.1		7	12.11	even	2
2299.2.a.q.1.1		7	44.43	even	2
3344.2.a.ba.1.7		7	1.1	even	1	trivial
3971.2.a.i.1.1		7	76.75	even	2
5225.2.a.n.1.1		7	20.19	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q+2.99825 q^{3} +0.244850 q^{5} -4.42321 q^{7} +5.98952 q^{9} +O(q^{10})\)	\(q+2.99825 q^{3} +0.244850 q^{5} -4.42321 q^{7} +5.98952 q^{9} +1.00000 q^{11} -5.89690 q^{13} +0.734121 q^{15} -2.93922 q^{17} -1.00000 q^{19} -13.2619 q^{21} +0.372904 q^{23} -4.94005 q^{25} +8.96335 q^{27} -3.47526 q^{29} -6.37391 q^{31} +2.99825 q^{33} -1.08302 q^{35} +0.926528 q^{37} -17.6804 q^{39} -6.67861 q^{41} -2.12805 q^{43} +1.46653 q^{45} +1.72093 q^{47} +12.5648 q^{49} -8.81252 q^{51} -1.44022 q^{53} +0.244850 q^{55} -2.99825 q^{57} -7.71718 q^{59} +4.16881 q^{61} -26.4929 q^{63} -1.44385 q^{65} +11.3778 q^{67} +1.11806 q^{69} -2.40794 q^{71} +14.4653 q^{73} -14.8115 q^{75} -4.42321 q^{77} -1.67368 q^{79} +8.90583 q^{81} +6.47897 q^{83} -0.719667 q^{85} -10.4197 q^{87} -4.95706 q^{89} +26.0832 q^{91} -19.1106 q^{93} -0.244850 q^{95} +8.24006 q^{97} +5.98952 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9}+O(q^{10}) \) 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9	\( 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9} + 7 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 7 q^{19} - 14 q^{21} - 10 q^{23} + 9 q^{25} + 4 q^{27} - 18 q^{29} - 24 q^{31} - 2 q^{33} - 8 q^{35} - 24 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} - 8 q^{47} + 17 q^{49} + 24 q^{51} + 2 q^{53} + 2 q^{55} + 2 q^{57} + 10 q^{59} + 14 q^{61} - 14 q^{65} - 8 q^{67} - 6 q^{69} - 10 q^{71} - 6 q^{73} - 26 q^{75} - 10 q^{77} - 52 q^{79} - q^{81} + 10 q^{83} - 12 q^{85} - 6 q^{87} - 12 q^{91} + 2 q^{93} - 2 q^{95} - 24 q^{97} + 11 q^{99}+O(q^{100}) \) 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9 + 7 * q^11 - 4 * q^13 - 12 * q^15 + 2 * q^17 - 7 * q^19 - 14 * q^21 - 10 * q^23 + 9 * q^25 + 4 * q^27 - 18 * q^29 - 24 * q^31 - 2 * q^33 - 8 * q^35 - 24 * q^39 - 12 * q^41 - 2 * q^43 - 4 * q^45 - 8 * q^47 + 17 * q^49 + 24 * q^51 + 2 * q^53 + 2 * q^55 + 2 * q^57 + 10 * q^59 + 14 * q^61 - 14 * q^65 - 8 * q^67 - 6 * q^69 - 10 * q^71 - 6 * q^73 - 26 * q^75 - 10 * q^77 - 52 * q^79 - q^81 + 10 * q^83 - 12 * q^85 - 6 * q^87 - 12 * q^91 + 2 * q^93 - 2 * q^95 - 24 * q^97 + 11 * q^99

Introduction

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