LMFDB - Embedded newform 3969.2.a.z.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, 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("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.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:	$31.6926245622$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.574857.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 9x^{2} + 3x - 3 $ x^5 - 2x^4 - 5x^3 + 9x^2 + 3x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.670333$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.670333 q^{2} -1.55065 q^{4} +1.42494 q^{5} -2.38012 q^{8} +0.955182 q^{10} -4.93077 q^{11} +2.75460 q^{13} +1.50584 q^{16} -1.11968 q^{17} +4.01505 q^{19} -2.20958 q^{20} -3.30526 q^{22} +5.43661 q^{23} -2.96955 q^{25} +1.84650 q^{26} -6.81109 q^{29} +2.50584 q^{31} +5.76965 q^{32} -0.750557 q^{34} -1.41957 q^{37} +2.69142 q^{38} -3.39152 q^{40} -0.248768 q^{41} +0.996627 q^{43} +7.64592 q^{44} +3.64434 q^{46} -9.47579 q^{47} -1.99059 q^{50} -4.27144 q^{52} +0.820458 q^{53} -7.02604 q^{55} -4.56570 q^{58} -6.58407 q^{59} +0.0752645 q^{61} +1.67974 q^{62} +0.855913 q^{64} +3.92514 q^{65} -12.5877 q^{67} +1.73623 q^{68} +0.0804951 q^{71} -10.6910 q^{73} -0.951587 q^{74} -6.22595 q^{76} -1.84491 q^{79} +2.14572 q^{80} -0.166758 q^{82} +14.4717 q^{83} -1.59547 q^{85} +0.668072 q^{86} +11.7358 q^{88} -13.5258 q^{89} -8.43030 q^{92} -6.35193 q^{94} +5.72119 q^{95} -5.40319 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 3 q^{8} + 7 q^{10} - 4 q^{11} + 8 q^{13} - 2 q^{16} - 12 q^{17} - q^{19} - 5 q^{20} + q^{22} - 3 q^{23} + q^{25} - 11 q^{26} - 7 q^{29} + 3 q^{31} + 2 q^{32} - 3 q^{34}+ \cdots + 12 q^{97}+O(q^{100}) $ 5 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 3 * q^8 + 7 * q^10 - 4 * q^11 + 8 * q^13 - 2 * q^16 - 12 * q^17 - q^19 - 5 * q^20 + q^22 - 3 * q^23 + q^25 - 11 * q^26 - 7 * q^29 + 3 * q^31 + 2 * q^32 - 3 * q^34 - 20 * q^38 + 3 * q^40 - 5 * q^41 + 7 * q^43 + 10 * q^44 - 3 * q^46 - 27 * q^47 - 19 * q^50 + 10 * q^52 + 21 * q^53 + 2 * q^55 + 10 * q^58 - 30 * q^59 + 14 * q^61 - 6 * q^62 - 25 * q^64 + 11 * q^65 + 2 * q^67 - 27 * q^68 - 3 * q^71 - 15 * q^73 + 36 * q^74 - 5 * q^76 + 4 * q^79 - 20 * q^80 + 5 * q^82 - 9 * q^83 + 6 * q^85 + 8 * q^86 + 18 * q^88 - 28 * q^89 - 27 * q^92 + 3 * q^94 + 14 * q^95 + 12 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.670333	0.473997	0.236998	−	0.971510i	$-0.423836\pi$
$2$	0.670333	0.473997	0.236998	+	0.971510i	$0.423836\pi$
$3$	0	0
$3$	0	0
$4$	−1.55065	−0.775327
$5$	1.42494	0.637251	0.318626	−	0.947881i	$-0.396779\pi$
$5$	1.42494	0.637251	0.318626	+	0.947881i	$0.396779\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.38012	−0.841499
$9$	0	0
$10$	0.955182	0.302055
$11$	−4.93077	−1.48668	−0.743342	−	0.668911i	$-0.766761\pi$
$11$	−4.93077	−1.48668	−0.743342	+	0.668911i	$0.766761\pi$
$12$	0	0
$13$	2.75460	0.763990	0.381995	−	0.924164i	$-0.375237\pi$
$13$	2.75460	0.763990	0.381995	+	0.924164i	$0.375237\pi$
$14$	0	0
$15$	0	0
$16$	1.50584	0.376459
$17$	−1.11968	−0.271562	−0.135781	−	0.990739i	$-0.543354\pi$
$17$	−1.11968	−0.271562	−0.135781	+	0.990739i	$0.543354\pi$
$18$	0	0
$19$	4.01505	0.921115	0.460557	−	0.887630i	$-0.347650\pi$
$19$	4.01505	0.921115	0.460557	+	0.887630i	$0.347650\pi$
$20$	−2.20958	−0.494078
$21$	0	0
$22$	−3.30526	−0.704684
$23$	5.43661	1.13361	0.566806	−	0.823851i	$-0.308179\pi$
$23$	5.43661	1.13361	0.566806	+	0.823851i	$0.308179\pi$
$24$	0	0
$25$	−2.96955	−0.593911
$26$	1.84650	0.362129
$27$	0	0
$28$	0	0
$29$	−6.81109	−1.26479	−0.632394	−	0.774647i	$-0.717928\pi$
$29$	−6.81109	−1.26479	−0.632394	+	0.774647i	$0.717928\pi$
$30$	0	0
$31$	2.50584	0.450061	0.225031	−	0.974352i	$-0.427752\pi$
$31$	2.50584	0.450061	0.225031	+	0.974352i	$0.427752\pi$
$32$	5.76965	1.01994
$33$	0	0
$34$	−0.750557	−0.128719
$35$	0	0
$36$	0	0
$37$	−1.41957	−0.233376	−0.116688	−	0.993169i	$-0.537228\pi$
$37$	−1.41957	−0.233376	−0.116688	+	0.993169i	$0.537228\pi$
$38$	2.69142	0.436605
$39$	0	0
$40$	−3.39152	−0.536247
$41$	−0.248768	−0.0388511	−0.0194256	−	0.999811i	$-0.506184\pi$
$41$	−0.248768	−0.0388511	−0.0194256	+	0.999811i	$0.506184\pi$
$42$	0	0
$43$	0.996627	0.151984	0.0759921	−	0.997108i	$-0.475788\pi$
$43$	0.996627	0.151984	0.0759921	+	0.997108i	$0.475788\pi$
$44$	7.64592	1.15267
$45$	0	0
$46$	3.64434	0.537328
$47$	−9.47579	−1.38219	−0.691093	−	0.722766i	$-0.742871\pi$
$47$	−9.47579	−1.38219	−0.691093	+	0.722766i	$0.742871\pi$
$48$	0	0
$49$	0	0
$50$	−1.99059	−0.281512
$51$	0	0
$52$	−4.27144	−0.592342
$53$	0.820458	0.112699	0.0563493	−	0.998411i	$-0.482054\pi$
$53$	0.820458	0.112699	0.0563493	+	0.998411i	$0.482054\pi$
$54$	0	0
$55$	−7.02604	−0.947392
$56$	0	0
$57$	0	0
$58$	−4.56570	−0.599506
$59$	−6.58407	−0.857173	−0.428586	−	0.903501i	$-0.640988\pi$
$59$	−6.58407	−0.857173	−0.428586	+	0.903501i	$0.640988\pi$
$60$	0	0
$61$	0.0752645	0.00963663	0.00481831	−	0.999988i	$-0.498466\pi$
$61$	0.0752645	0.00963663	0.00481831	+	0.999988i	$0.498466\pi$
$62$	1.67974	0.213328
$63$	0	0
$64$	0.855913	0.106989
$65$	3.92514	0.486854
$66$	0	0
$67$	−12.5877	−1.53783	−0.768916	−	0.639350i	$-0.779204\pi$
$67$	−12.5877	−1.53783	−0.768916	+	0.639350i	$0.779204\pi$
$68$	1.73623	0.210549
$69$	0	0
$70$	0	0
$71$	0.0804951	0.00955301	0.00477651	−	0.999989i	$-0.498480\pi$
$71$	0.0804951	0.00955301	0.00477651	+	0.999989i	$0.498480\pi$
$72$	0	0
$73$	−10.6910	−1.25129	−0.625644	−	0.780109i	$-0.715164\pi$
$73$	−10.6910	−1.25129	−0.625644	+	0.780109i	$0.715164\pi$
$74$	−0.951587	−0.110620
$75$	0	0
$76$	−6.22595	−0.714165
$77$	0	0
$78$	0	0
$79$	−1.84491	−0.207569	−0.103785	−	0.994600i	$-0.533095\pi$
$79$	−1.84491	−0.207569	−0.103785	+	0.994600i	$0.533095\pi$
$80$	2.14572	0.239899
$81$	0	0
$82$	−0.166758	−0.0184153
$83$	14.4717	1.58847	0.794236	−	0.607610i	$-0.207872\pi$
$83$	14.4717	1.58847	0.794236	+	0.607610i	$0.207872\pi$
$84$	0	0
$85$	−1.59547	−0.173053
$86$	0.668072	0.0720400
$87$	0	0
$88$	11.7358	1.25104
$89$	−13.5258	−1.43374	−0.716868	−	0.697209i	$-0.754425\pi$
$89$	−13.5258	−1.43374	−0.716868	+	0.697209i	$0.754425\pi$
$90$	0	0
$91$	0	0
$92$	−8.43030	−0.878920
$93$	0	0
$94$	−6.35193	−0.655152
$95$	5.72119	0.586982
$96$	0	0
$97$	−5.40319	−0.548611	−0.274306	−	0.961643i	$-0.588448\pi$
$97$	−5.40319	−0.548611	−0.274306	+	0.961643i	$0.588448\pi$
$98$	0	0
$99$	0	0
$100$	4.60475	0.460475
$101$	−5.13540	−0.510991	−0.255496	−	0.966810i	$-0.582239\pi$
$101$	−5.13540	−0.510991	−0.255496	+	0.966810i	$0.582239\pi$
$102$	0	0
$103$	−14.2112	−1.40027	−0.700137	−	0.714009i	$-0.746878\pi$
$103$	−14.2112	−1.40027	−0.700137	+	0.714009i	$0.746878\pi$
$104$	−6.55629	−0.642897
$105$	0	0
$106$	0.549980	0.0534188
$107$	−7.66030	−0.740549	−0.370274	−	0.928922i	$-0.620736\pi$
$107$	−7.66030	−0.740549	−0.370274	+	0.928922i	$0.620736\pi$
$108$	0	0
$109$	1.69879	0.162714	0.0813572	−	0.996685i	$-0.474075\pi$
$109$	1.69879	0.162714	0.0813572	+	0.996685i	$0.474075\pi$
$110$	−4.70979	−0.449061
$111$	0	0
$112$	0	0
$113$	0.600703	0.0565093	0.0282547	−	0.999601i	$-0.491005\pi$
$113$	0.600703	0.0565093	0.0282547	+	0.999601i	$0.491005\pi$
$114$	0	0
$115$	7.74683	0.722395
$116$	10.5617	0.980625
$117$	0	0
$118$	−4.41352	−0.406297
$119$	0	0
$120$	0	0
$121$	13.3125	1.21023
$122$	0.0504522	0.00456773
$123$	0	0
$124$	−3.88569	−0.348945
$125$	−11.3561	−1.01572
$126$	0	0
$127$	7.25977	0.644200	0.322100	−	0.946706i	$-0.395611\pi$
$127$	7.25977	0.644200	0.322100	+	0.946706i	$0.395611\pi$
$128$	−10.9656	−0.969227
$129$	0	0
$130$	2.63115	0.230767
$131$	−20.4530	−1.78698	−0.893492	−	0.449079i	$-0.851752\pi$
$131$	−20.4530	−1.78698	−0.893492	+	0.449079i	$0.851752\pi$
$132$	0	0
$133$	0	0
$134$	−8.43794	−0.728927
$135$	0	0
$136$	2.66497	0.228519
$137$	12.2116	1.04331	0.521655	−	0.853157i	$-0.325315\pi$
$137$	12.2116	1.04331	0.521655	+	0.853157i	$0.325315\pi$
$138$	0	0
$139$	2.48183	0.210506	0.105253	−	0.994445i	$-0.466435\pi$
$139$	2.48183	0.210506	0.105253	+	0.994445i	$0.466435\pi$
$140$	0	0
$141$	0	0
$142$	0.0539585	0.00452810
$143$	−13.5823	−1.13581
$144$	0	0
$145$	−9.70538	−0.805988
$146$	−7.16654	−0.593107
$147$	0	0
$148$	2.20127	0.180943
$149$	−8.55593	−0.700929	−0.350465	−	0.936576i	$-0.613976\pi$
$149$	−8.55593	−0.700929	−0.350465	+	0.936576i	$0.613976\pi$
$150$	0	0
$151$	−17.6592	−1.43709	−0.718544	−	0.695482i	$-0.755191\pi$
$151$	−17.6592	−1.43709	−0.718544	+	0.695482i	$0.755191\pi$
$152$	−9.55629	−0.775117
$153$	0	0
$154$	0	0
$155$	3.57066	0.286802
$156$	0	0
$157$	6.32149	0.504510	0.252255	−	0.967661i	$-0.418828\pi$
$157$	6.32149	0.504510	0.252255	+	0.967661i	$0.418828\pi$
$158$	−1.23671	−0.0983871
$159$	0	0
$160$	8.22139	0.649958
$161$	0	0
$162$	0	0
$163$	8.02267	0.628384	0.314192	−	0.949359i	$-0.398266\pi$
$163$	8.02267	0.628384	0.314192	+	0.949359i	$0.398266\pi$
$164$	0.385754	0.0301223
$165$	0	0
$166$	9.70083	0.752930
$167$	−2.12076	−0.164109	−0.0820545	−	0.996628i	$-0.526148\pi$
$167$	−2.12076	−0.164109	−0.0820545	+	0.996628i	$0.526148\pi$
$168$	0	0
$169$	−5.41215	−0.416319
$170$	−1.06950	−0.0820267
$171$	0	0
$172$	−1.54542	−0.117837
$173$	−18.2881	−1.39042	−0.695208	−	0.718808i	$-0.744688\pi$
$173$	−18.2881	−1.39042	−0.695208	+	0.718808i	$0.744688\pi$
$174$	0	0
$175$	0	0
$176$	−7.42494	−0.559676
$177$	0	0
$178$	−9.06681	−0.679586
$179$	−7.62551	−0.569958	−0.284979	−	0.958534i	$-0.591987\pi$
$179$	−7.62551	−0.569958	−0.284979	+	0.958534i	$0.591987\pi$
$180$	0	0
$181$	15.5305	1.15438	0.577188	−	0.816611i	$-0.304150\pi$
$181$	15.5305	1.15438	0.577188	+	0.816611i	$0.304150\pi$
$182$	0	0
$183$	0	0
$184$	−12.9398	−0.953933
$185$	−2.02280	−0.148719
$186$	0	0
$187$	5.52088	0.403727
$188$	14.6937	1.07165
$189$	0	0
$190$	3.83510	0.278227
$191$	14.8325	1.07324	0.536620	−	0.843824i	$-0.319701\pi$
$191$	14.8325	1.07324	0.536620	+	0.843824i	$0.319701\pi$
$192$	0	0
$193$	16.5677	1.19257	0.596286	−	0.802772i	$-0.296642\pi$
$193$	16.5677	1.19257	0.596286	+	0.802772i	$0.296642\pi$
$194$	−3.62194	−0.260040
$195$	0	0
$196$	0	0
$197$	−4.03740	−0.287653	−0.143826	−	0.989603i	$-0.545941\pi$
$197$	−4.03740	−0.287653	−0.143826	+	0.989603i	$0.545941\pi$
$198$	0	0
$199$	25.2814	1.79215	0.896076	−	0.443901i	$-0.146406\pi$
$199$	25.2814	1.79215	0.896076	+	0.443901i	$0.146406\pi$
$200$	7.06789	0.499775
$201$	0	0
$202$	−3.44243	−0.242208
$203$	0	0
$204$	0	0
$205$	−0.354480	−0.0247579
$206$	−9.52625	−0.663725
$207$	0	0
$208$	4.14798	0.287611
$209$	−19.7973	−1.36941
$210$	0	0
$211$	7.52493	0.518037	0.259019	−	0.965872i	$-0.416601\pi$
$211$	7.52493	0.518037	0.259019	+	0.965872i	$0.416601\pi$
$212$	−1.27225	−0.0873782
$213$	0	0
$214$	−5.13495	−0.351018
$215$	1.42013	0.0968521
$216$	0	0
$217$	0	0
$218$	1.13875	0.0771261
$219$	0	0
$220$	10.8950	0.734538
$221$	−3.08427	−0.207471
$222$	0	0
$223$	−12.9846	−0.869513	−0.434757	−	0.900548i	$-0.643166\pi$
$223$	−12.9846	−0.869513	−0.434757	+	0.900548i	$0.643166\pi$
$224$	0	0
$225$	0	0
$226$	0.402671	0.0267852
$227$	−28.9665	−1.92257	−0.961286	−	0.275551i	$-0.911140\pi$
$227$	−28.9665	−1.92257	−0.961286	+	0.275551i	$0.911140\pi$
$228$	0	0
$229$	15.4358	1.02003	0.510013	−	0.860167i	$-0.329640\pi$
$229$	15.4358	1.02003	0.510013	+	0.860167i	$0.329640\pi$
$230$	5.19295	0.342413
$231$	0	0
$232$	16.2112	1.06432
$233$	4.94648	0.324055	0.162027	−	0.986786i	$-0.448197\pi$
$233$	4.94648	0.324055	0.162027	+	0.986786i	$0.448197\pi$
$234$	0	0
$235$	−13.5024	−0.880800
$236$	10.2096	0.664589
$237$	0	0
$238$	0	0
$239$	−13.0346	−0.843141	−0.421571	−	0.906796i	$-0.638521\pi$
$239$	−13.0346	−0.843141	−0.421571	+	0.906796i	$0.638521\pi$
$240$	0	0
$241$	14.5825	0.939339	0.469670	−	0.882842i	$-0.344373\pi$
$241$	14.5825	0.939339	0.469670	+	0.882842i	$0.344373\pi$
$242$	8.92382	0.573645
$243$	0	0
$244$	−0.116709	−0.00747154
$245$	0	0
$246$	0	0
$247$	11.0599	0.703722
$248$	−5.96419	−0.378726
$249$	0	0
$250$	−7.61238	−0.481449
$251$	−14.0715	−0.888187	−0.444094	−	0.895980i	$-0.646474\pi$
$251$	−14.0715	−0.888187	−0.444094	+	0.895980i	$0.646474\pi$
$252$	0	0
$253$	−26.8067	−1.68532
$254$	4.86646	0.305349
$255$	0	0
$256$	−9.06240	−0.566400
$257$	−8.36215	−0.521617	−0.260808	−	0.965391i	$-0.583989\pi$
$257$	−8.36215	−0.521617	−0.260808	+	0.965391i	$0.583989\pi$
$258$	0	0
$259$	0	0
$260$	−6.08653	−0.377471
$261$	0	0
$262$	−13.7103	−0.847025
$263$	3.27066	0.201678	0.100839	−	0.994903i	$-0.467847\pi$
$263$	3.27066	0.201678	0.100839	+	0.994903i	$0.467847\pi$
$264$	0	0
$265$	1.16910	0.0718173
$266$	0	0
$267$	0	0
$268$	19.5192	1.19232
$269$	15.3870	0.938161	0.469081	−	0.883155i	$-0.344585\pi$
$269$	15.3870	0.938161	0.469081	+	0.883155i	$0.344585\pi$
$270$	0	0
$271$	−8.12617	−0.493630	−0.246815	−	0.969063i	$-0.579384\pi$
$271$	−8.12617	−0.493630	−0.246815	+	0.969063i	$0.579384\pi$
$272$	−1.68605	−0.102232
$273$	0	0
$274$	8.18585	0.494525
$275$	14.6422	0.882958
$276$	0	0
$277$	12.8457	0.771826	0.385913	−	0.922535i	$-0.373887\pi$
$277$	12.8457	0.771826	0.385913	+	0.922535i	$0.373887\pi$
$278$	1.66365	0.0997793
$279$	0	0
$280$	0	0
$281$	1.44816	0.0863901	0.0431951	−	0.999067i	$-0.486246\pi$
$281$	1.44816	0.0863901	0.0431951	+	0.999067i	$0.486246\pi$
$282$	0	0
$283$	−17.4385	−1.03661	−0.518306	−	0.855195i	$-0.673437\pi$
$283$	−17.4385	−1.03661	−0.518306	+	0.855195i	$0.673437\pi$
$284$	−0.124820	−0.00740671
$285$	0	0
$286$	−9.10468	−0.538371
$287$	0	0
$288$	0	0
$289$	−15.7463	−0.926254
$290$	−6.50584	−0.382036
$291$	0	0
$292$	16.5781	0.970158
$293$	1.80010	0.105163	0.0525814	−	0.998617i	$-0.483255\pi$
$293$	1.80010	0.105163	0.0525814	+	0.998617i	$0.483255\pi$
$294$	0	0
$295$	−9.38189	−0.546235
$296$	3.37875	0.196386
$297$	0	0
$298$	−5.73532	−0.332238
$299$	14.9757	0.866068
$300$	0	0
$301$	0	0
$302$	−11.8376	−0.681175
$303$	0	0
$304$	6.04600	0.346762
$305$	0.107247	0.00614095
$306$	0	0
$307$	1.06478	0.0607699	0.0303850	−	0.999538i	$-0.490327\pi$
$307$	1.06478	0.0607699	0.0303850	+	0.999538i	$0.490327\pi$
$308$	0	0
$309$	0	0
$310$	2.39353	0.135943
$311$	−16.9293	−0.959970	−0.479985	−	0.877277i	$-0.659358\pi$
$311$	−16.9293	−0.959970	−0.479985	+	0.877277i	$0.659358\pi$
$312$	0	0
$313$	−8.27856	−0.467932	−0.233966	−	0.972245i	$-0.575170\pi$
$313$	−8.27856	−0.467932	−0.233966	+	0.972245i	$0.575170\pi$
$314$	4.23750	0.239136
$315$	0	0
$316$	2.86082	0.160934
$317$	6.54741	0.367739	0.183870	−	0.982951i	$-0.441138\pi$
$317$	6.54741	0.367739	0.183870	+	0.982951i	$0.441138\pi$
$318$	0	0
$319$	33.5840	1.88034
$320$	1.21962	0.0681790
$321$	0	0
$322$	0	0
$323$	−4.49556	−0.250140
$324$	0	0
$325$	−8.17995	−0.453742
$326$	5.37786	0.297852
$327$	0	0
$328$	0.592099	0.0326932
$329$	0	0
$330$	0	0
$331$	−26.7258	−1.46899	−0.734493	−	0.678617i	$-0.762580\pi$
$331$	−26.7258	−1.46899	−0.734493	+	0.678617i	$0.762580\pi$
$332$	−22.4405	−1.23158
$333$	0	0
$334$	−1.42161	−0.0777872
$335$	−17.9367	−0.979985
$336$	0	0
$337$	9.52328	0.518766	0.259383	−	0.965775i	$-0.416481\pi$
$337$	9.52328	0.518766	0.259383	+	0.965775i	$0.416481\pi$
$338$	−3.62794	−0.197334
$339$	0	0
$340$	2.47403	0.134173
$341$	−12.3557	−0.669099
$342$	0	0
$343$	0	0
$344$	−2.37209	−0.127895
$345$	0	0
$346$	−12.2591	−0.659053
$347$	−18.7031	−1.00404	−0.502018	−	0.864857i	$-0.667409\pi$
$347$	−18.7031	−1.00404	−0.502018	+	0.864857i	$0.667409\pi$
$348$	0	0
$349$	30.1084	1.61167	0.805834	−	0.592142i	$-0.201718\pi$
$349$	30.1084	1.61167	0.805834	+	0.592142i	$0.201718\pi$
$350$	0	0
$351$	0	0
$352$	−28.4488	−1.51633
$353$	6.25933	0.333150	0.166575	−	0.986029i	$-0.446729\pi$
$353$	6.25933	0.333150	0.166575	+	0.986029i	$0.446729\pi$
$354$	0	0
$355$	0.114700	0.00608767
$356$	20.9739	1.11161
$357$	0	0
$358$	−5.11163	−0.270158
$359$	10.1951	0.538077	0.269038	−	0.963129i	$-0.413294\pi$
$359$	10.1951	0.538077	0.269038	+	0.963129i	$0.413294\pi$
$360$	0	0
$361$	−2.87941	−0.151548
$362$	10.4106	0.547171
$363$	0	0
$364$	0	0
$365$	−15.2340	−0.797385
$366$	0	0
$367$	−28.6557	−1.49581	−0.747906	−	0.663804i	$-0.768941\pi$
$367$	−28.6557	−1.49581	−0.747906	+	0.663804i	$0.768941\pi$
$368$	8.18664	0.426758
$369$	0	0
$370$	−1.35595	−0.0704926
$371$	0	0
$372$	0	0
$373$	−16.0734	−0.832249	−0.416124	−	0.909308i	$-0.636612\pi$
$373$	−16.0734	−0.832249	−0.416124	+	0.909308i	$0.636612\pi$
$374$	3.70083	0.191365
$375$	0	0
$376$	22.5535	1.16311
$377$	−18.7619	−0.966286
$378$	0	0
$379$	−1.01893	−0.0523388	−0.0261694	−	0.999658i	$-0.508331\pi$
$379$	−1.01893	−0.0523388	−0.0261694	+	0.999658i	$0.508331\pi$
$380$	−8.87158	−0.455103
$381$	0	0
$382$	9.94270	0.508713
$383$	−11.5865	−0.592044	−0.296022	−	0.955181i	$-0.595660\pi$
$383$	−11.5865	−0.592044	−0.296022	+	0.955181i	$0.595660\pi$
$384$	0	0
$385$	0	0
$386$	11.1059	0.565275
$387$	0	0
$388$	8.37848	0.425353
$389$	17.8135	0.903181	0.451590	−	0.892225i	$-0.350857\pi$
$389$	17.8135	0.903181	0.451590	+	0.892225i	$0.350857\pi$
$390$	0	0
$391$	−6.08726	−0.307846
$392$	0	0
$393$	0	0
$394$	−2.70640	−0.136346
$395$	−2.62889	−0.132274
$396$	0	0
$397$	13.0846	0.656696	0.328348	−	0.944557i	$-0.393508\pi$
$397$	13.0846	0.656696	0.328348	+	0.944557i	$0.393508\pi$
$398$	16.9470	0.849474
$399$	0	0
$400$	−4.47166	−0.223583
$401$	14.1033	0.704285	0.352143	−	0.935946i	$-0.385453\pi$
$401$	14.1033	0.704285	0.352143	+	0.935946i	$0.385453\pi$
$402$	0	0
$403$	6.90259	0.343842
$404$	7.96323	0.396185
$405$	0	0
$406$	0	0
$407$	6.99960	0.346957
$408$	0	0
$409$	−2.64599	−0.130836	−0.0654179	−	0.997858i	$-0.520838\pi$
$409$	−2.64599	−0.130836	−0.0654179	+	0.997858i	$0.520838\pi$
$410$	−0.237619	−0.0117352
$411$	0	0
$412$	22.0367	1.08567
$413$	0	0
$414$	0	0
$415$	20.6212	1.01226
$416$	15.8931	0.779224
$417$	0	0
$418$	−13.2708	−0.649094
$419$	−33.5134	−1.63724	−0.818619	−	0.574337i	$-0.805260\pi$
$419$	−33.5134	−1.63724	−0.818619	+	0.574337i	$0.805260\pi$
$420$	0	0
$421$	4.83901	0.235839	0.117919	−	0.993023i	$-0.462378\pi$
$421$	4.83901	0.235839	0.117919	+	0.993023i	$0.462378\pi$
$422$	5.04421	0.245548
$423$	0	0
$424$	−1.95279	−0.0948358
$425$	3.32495	0.161284
$426$	0	0
$427$	0	0
$428$	11.8785	0.574167
$429$	0	0
$430$	0.951960	0.0459076
$431$	−35.3285	−1.70172	−0.850858	−	0.525396i	$-0.823917\pi$
$431$	−35.3285	−1.70172	−0.850858	+	0.525396i	$0.823917\pi$
$432$	0	0
$433$	5.47404	0.263066	0.131533	−	0.991312i	$-0.458010\pi$
$433$	5.47404	0.263066	0.131533	+	0.991312i	$0.458010\pi$
$434$	0	0
$435$	0	0
$436$	−2.63423	−0.126157
$437$	21.8282	1.04419
$438$	0	0
$439$	6.39812	0.305365	0.152683	−	0.988275i	$-0.451209\pi$
$439$	6.39812	0.305365	0.152683	+	0.988275i	$0.451209\pi$
$440$	16.7228	0.797229
$441$	0	0
$442$	−2.06749	−0.0983404
$443$	−6.38682	−0.303447	−0.151723	−	0.988423i	$-0.548482\pi$
$443$	−6.38682	−0.303447	−0.151723	+	0.988423i	$0.548482\pi$
$444$	0	0
$445$	−19.2735	−0.913650
$446$	−8.70400	−0.412146
$447$	0	0
$448$	0	0
$449$	−11.7460	−0.554327	−0.277163	−	0.960823i	$-0.589394\pi$
$449$	−11.7460	−0.554327	−0.277163	+	0.960823i	$0.589394\pi$
$450$	0	0
$451$	1.22662	0.0577593
$452$	−0.931482	−0.0438132
$453$	0	0
$454$	−19.4172	−0.911293
$455$	0	0
$456$	0	0
$457$	10.5224	0.492217	0.246108	−	0.969242i	$-0.420848\pi$
$457$	10.5224	0.492217	0.246108	+	0.969242i	$0.420848\pi$
$458$	10.3471	0.483489
$459$	0	0
$460$	−12.0127	−0.560093
$461$	7.08555	0.330007	0.165004	−	0.986293i	$-0.447236\pi$
$461$	7.08555	0.330007	0.165004	+	0.986293i	$0.447236\pi$
$462$	0	0
$463$	−32.7521	−1.52212	−0.761059	−	0.648683i	$-0.775320\pi$
$463$	−32.7521	−1.52212	−0.761059	+	0.648683i	$0.775320\pi$
$464$	−10.2564	−0.476141
$465$	0	0
$466$	3.31579	0.153601
$467$	−3.92431	−0.181596	−0.0907978	−	0.995869i	$-0.528942\pi$
$467$	−3.92431	−0.181596	−0.0907978	+	0.995869i	$0.528942\pi$
$468$	0	0
$469$	0	0
$470$	−9.05111	−0.417497
$471$	0	0
$472$	15.6709	0.721310
$473$	−4.91414	−0.225952
$474$	0	0
$475$	−11.9229	−0.547060
$476$	0	0
$477$	0	0
$478$	−8.73755	−0.399646
$479$	16.0865	0.735010	0.367505	−	0.930022i	$-0.380212\pi$
$479$	16.0865	0.735010	0.367505	+	0.930022i	$0.380212\pi$
$480$	0	0
$481$	−3.91036	−0.178297
$482$	9.77510	0.445244
$483$	0	0
$484$	−20.6431	−0.938324
$485$	−7.69921	−0.349603
$486$	0	0
$487$	3.50344	0.158756	0.0793781	−	0.996845i	$-0.474707\pi$
$487$	3.50344	0.158756	0.0793781	+	0.996845i	$0.474707\pi$
$488$	−0.179138	−0.00810921
$489$	0	0
$490$	0	0
$491$	41.1093	1.85524	0.927618	−	0.373531i	$-0.121853\pi$
$491$	41.1093	1.85524	0.927618	+	0.373531i	$0.121853\pi$
$492$	0	0
$493$	7.62624	0.343468
$494$	7.41379	0.333562
$495$	0	0
$496$	3.77338	0.169430
$497$	0	0
$498$	0	0
$499$	11.8297	0.529571	0.264785	−	0.964307i	$-0.414699\pi$
$499$	11.8297	0.529571	0.264785	+	0.964307i	$0.414699\pi$
$500$	17.6094	0.787517
$501$	0	0
$502$	−9.43261	−0.420998
$503$	21.8595	0.974665	0.487332	−	0.873217i	$-0.337970\pi$
$503$	21.8595	0.974665	0.487332	+	0.873217i	$0.337970\pi$
$504$	0	0
$505$	−7.31762	−0.325630
$506$	−17.9694	−0.798837
$507$	0	0
$508$	−11.2574	−0.499466
$509$	16.8966	0.748930	0.374465	−	0.927241i	$-0.377826\pi$
$509$	16.8966	0.748930	0.374465	+	0.927241i	$0.377826\pi$
$510$	0	0
$511$	0	0
$512$	15.8563	0.700756
$513$	0	0
$514$	−5.60542	−0.247245
$515$	−20.2501	−0.892326
$516$	0	0
$517$	46.7230	2.05487
$518$	0	0
$519$	0	0
$520$	−9.34230	−0.409687
$521$	34.4932	1.51117	0.755587	−	0.655048i	$-0.227352\pi$
$521$	34.4932	1.51117	0.755587	+	0.655048i	$0.227352\pi$
$522$	0	0
$523$	−1.99123	−0.0870704	−0.0435352	−	0.999052i	$-0.513862\pi$
$523$	−1.99123	−0.0870704	−0.0435352	+	0.999052i	$0.513862\pi$
$524$	31.7155	1.38550
$525$	0	0
$526$	2.19243	0.0955945
$527$	−2.80573	−0.122220
$528$	0	0
$529$	6.55673	0.285075
$530$	0.783687	0.0340412
$531$	0	0
$532$	0	0
$533$	−0.685259	−0.0296819
$534$	0	0
$535$	−10.9154	−0.471916
$536$	29.9602	1.29408
$537$	0	0
$538$	10.3144	0.444685
$539$	0	0
$540$	0	0
$541$	30.1363	1.29566	0.647830	−	0.761785i	$-0.275677\pi$
$541$	30.1363	1.29566	0.647830	+	0.761785i	$0.275677\pi$
$542$	−5.44724	−0.233979
$543$	0	0
$544$	−6.46015	−0.276977
$545$	2.42067	0.103690
$546$	0	0
$547$	−15.3614	−0.656806	−0.328403	−	0.944538i	$-0.606510\pi$
$547$	−15.3614	−0.656806	−0.328403	+	0.944538i	$0.606510\pi$
$548$	−18.9360	−0.808906
$549$	0	0
$550$	9.81514	0.418519
$551$	−27.3469	−1.16502
$552$	0	0
$553$	0	0
$554$	8.61092	0.365843
$555$	0	0
$556$	−3.84846	−0.163211
$557$	23.2823	0.986504	0.493252	−	0.869886i	$-0.335808\pi$
$557$	23.2823	0.986504	0.493252	+	0.869886i	$0.335808\pi$
$558$	0	0
$559$	2.74531	0.116114
$560$	0	0
$561$	0	0
$562$	0.970751	0.0409487
$563$	4.55885	0.192133	0.0960663	−	0.995375i	$-0.469374\pi$
$563$	4.55885	0.192133	0.0960663	+	0.995375i	$0.469374\pi$
$564$	0	0
$565$	0.855964	0.0360107
$566$	−11.6896	−0.491351
$567$	0	0
$568$	−0.191588	−0.00803885
$569$	18.1995	0.762963	0.381482	−	0.924376i	$-0.375414\pi$
$569$	18.1995	0.762963	0.381482	+	0.924376i	$0.375414\pi$
$570$	0	0
$571$	−17.0455	−0.713332	−0.356666	−	0.934232i	$-0.616087\pi$
$571$	−17.0455	−0.713332	−0.356666	+	0.934232i	$0.616087\pi$
$572$	21.0615	0.880625
$573$	0	0
$574$	0	0
$575$	−16.1443	−0.673264
$576$	0	0
$577$	11.4095	0.474982	0.237491	−	0.971390i	$-0.423675\pi$
$577$	11.4095	0.474982	0.237491	+	0.971390i	$0.423675\pi$
$578$	−10.5553	−0.439041
$579$	0	0
$580$	15.0497	0.624904
$581$	0	0
$582$	0	0
$583$	−4.04549	−0.167547
$584$	25.4459	1.05296
$585$	0	0
$586$	1.20666	0.0498468
$587$	−5.05089	−0.208473	−0.104236	−	0.994553i	$-0.533240\pi$
$587$	−5.05089	−0.208473	−0.104236	+	0.994553i	$0.533240\pi$
$588$	0	0
$589$	10.0610	0.414558
$590$	−6.28899	−0.258913
$591$	0	0
$592$	−2.13765	−0.0878567
$593$	19.9778	0.820391	0.410196	−	0.911998i	$-0.365460\pi$
$593$	19.9778	0.820391	0.410196	+	0.911998i	$0.365460\pi$
$594$	0	0
$595$	0	0
$596$	13.2673	0.543449
$597$	0	0
$598$	10.0387	0.410513
$599$	4.39321	0.179502	0.0897508	−	0.995964i	$-0.471393\pi$
$599$	4.39321	0.179502	0.0897508	+	0.995964i	$0.471393\pi$
$600$	0	0
$601$	−24.3556	−0.993487	−0.496743	−	0.867897i	$-0.665471\pi$
$601$	−24.3556	−0.993487	−0.496743	+	0.867897i	$0.665471\pi$
$602$	0	0
$603$	0	0
$604$	27.3834	1.11421
$605$	18.9695	0.771221
$606$	0	0
$607$	13.1256	0.532752	0.266376	−	0.963869i	$-0.414174\pi$
$607$	13.1256	0.532752	0.266376	+	0.963869i	$0.414174\pi$
$608$	23.1654	0.939481
$609$	0	0
$610$	0.0718913	0.00291079
$611$	−26.1021	−1.05598
$612$	0	0
$613$	46.4806	1.87733	0.938667	−	0.344825i	$-0.112062\pi$
$613$	46.4806	1.87733	0.938667	+	0.344825i	$0.112062\pi$
$614$	0.713754	0.0288048
$615$	0	0
$616$	0	0
$617$	−28.3897	−1.14293	−0.571463	−	0.820628i	$-0.693624\pi$
$617$	−28.3897	−1.14293	−0.571463	+	0.820628i	$0.693624\pi$
$618$	0	0
$619$	31.9212	1.28302	0.641511	−	0.767114i	$-0.278308\pi$
$619$	31.9212	1.28302	0.641511	+	0.767114i	$0.278308\pi$
$620$	−5.53686	−0.222366
$621$	0	0
$622$	−11.3482	−0.455023
$623$	0	0
$624$	0	0
$625$	−1.33399	−0.0533594
$626$	−5.54939	−0.221798
$627$	0	0
$628$	−9.80244	−0.391160
$629$	1.58947	0.0633762
$630$	0	0
$631$	38.7184	1.54135	0.770677	−	0.637226i	$-0.219918\pi$
$631$	38.7184	1.54135	0.770677	+	0.637226i	$0.219918\pi$
$632$	4.39112	0.174669
$633$	0	0
$634$	4.38895	0.174307
$635$	10.3447	0.410517
$636$	0	0
$637$	0	0
$638$	22.5124	0.891276
$639$	0	0
$640$	−15.6252	−0.617641
$641$	−40.4001	−1.59571	−0.797854	−	0.602851i	$-0.794032\pi$
$641$	−40.4001	−1.59571	−0.797854	+	0.602851i	$0.794032\pi$
$642$	0	0
$643$	−12.5471	−0.494809	−0.247405	−	0.968912i	$-0.579578\pi$
$643$	−12.5471	−0.494809	−0.247405	+	0.968912i	$0.579578\pi$
$644$	0	0
$645$	0	0
$646$	−3.01352	−0.118565
$647$	−34.5548	−1.35849	−0.679245	−	0.733912i	$-0.737692\pi$
$647$	−34.5548	−1.35849	−0.679245	+	0.733912i	$0.737692\pi$
$648$	0	0
$649$	32.4646	1.27435
$650$	−5.48329	−0.215072
$651$	0	0
$652$	−12.4404	−0.487203
$653$	−22.2944	−0.872446	−0.436223	−	0.899839i	$-0.643684\pi$
$653$	−22.2944	−0.872446	−0.436223	+	0.899839i	$0.643684\pi$
$654$	0	0
$655$	−29.1442	−1.13876
$656$	−0.374605	−0.0146259
$657$	0	0
$658$	0	0
$659$	−7.14986	−0.278519	−0.139259	−	0.990256i	$-0.544472\pi$
$659$	−7.14986	−0.278519	−0.139259	+	0.990256i	$0.544472\pi$
$660$	0	0
$661$	42.9060	1.66885	0.834425	−	0.551122i	$-0.185800\pi$
$661$	42.9060	1.66885	0.834425	+	0.551122i	$0.185800\pi$
$662$	−17.9152	−0.696294
$663$	0	0
$664$	−34.4443	−1.33670
$665$	0	0
$666$	0	0
$667$	−37.0293	−1.43378
$668$	3.28856	0.127238
$669$	0	0
$670$	−12.0235	−0.464510
$671$	−0.371112	−0.0143266
$672$	0	0
$673$	37.6541	1.45146	0.725729	−	0.687980i	$-0.241503\pi$
$673$	37.6541	1.45146	0.725729	+	0.687980i	$0.241503\pi$
$674$	6.38376	0.245893
$675$	0	0
$676$	8.39238	0.322784
$677$	−26.3616	−1.01316	−0.506580	−	0.862193i	$-0.669090\pi$
$677$	−26.3616	−1.01316	−0.506580	+	0.862193i	$0.669090\pi$
$678$	0	0
$679$	0	0
$680$	3.79741	0.145624
$681$	0	0
$682$	−8.28244	−0.317151
$683$	−3.93175	−0.150444	−0.0752222	−	0.997167i	$-0.523967\pi$
$683$	−3.93175	−0.150444	−0.0752222	+	0.997167i	$0.523967\pi$
$684$	0	0
$685$	17.4008	0.664850
$686$	0	0
$687$	0	0
$688$	1.50076	0.0572158
$689$	2.26004	0.0861006
$690$	0	0
$691$	19.9010	0.757072	0.378536	−	0.925587i	$-0.376428\pi$
$691$	19.9010	0.757072	0.378536	+	0.925587i	$0.376428\pi$
$692$	28.3585	1.07803
$693$	0	0
$694$	−12.5373	−0.475910
$695$	3.53645	0.134145
$696$	0	0
$697$	0.278541	0.0105505
$698$	20.1827	0.763925
$699$	0	0
$700$	0	0
$701$	43.7908	1.65396	0.826979	−	0.562234i	$-0.190058\pi$
$701$	43.7908	1.65396	0.826979	+	0.562234i	$0.190058\pi$
$702$	0	0
$703$	−5.69965	−0.214966
$704$	−4.22031	−0.159059
$705$	0	0
$706$	4.19583	0.157912
$707$	0	0
$708$	0	0
$709$	44.6344	1.67628	0.838139	−	0.545457i	$-0.183644\pi$
$709$	44.6344	1.67628	0.838139	+	0.545457i	$0.183644\pi$
$710$	0.0768875	0.00288554
$711$	0	0
$712$	32.1931	1.20649
$713$	13.6233	0.510195
$714$	0	0
$715$	−19.3540	−0.723798
$716$	11.8245	0.441904
$717$	0	0
$718$	6.83411	0.255047
$719$	39.0192	1.45517	0.727586	−	0.686016i	$-0.240642\pi$
$719$	39.0192	1.45517	0.727586	+	0.686016i	$0.240642\pi$
$720$	0	0
$721$	0	0
$722$	−1.93016	−0.0718332
$723$	0	0
$724$	−24.0825	−0.895019
$725$	20.2259	0.751171
$726$	0	0
$727$	22.5107	0.834877	0.417439	−	0.908705i	$-0.362928\pi$
$727$	22.5107	0.834877	0.417439	+	0.908705i	$0.362928\pi$
$728$	0	0
$729$	0	0
$730$	−10.2119	−0.377958
$731$	−1.11590	−0.0412731
$732$	0	0
$733$	−0.897039	−0.0331329	−0.0165664	−	0.999863i	$-0.505274\pi$
$733$	−0.897039	−0.0331329	−0.0165664	+	0.999863i	$0.505274\pi$
$734$	−19.2088	−0.709011
$735$	0	0
$736$	31.3673	1.15622
$737$	62.0671	2.28627
$738$	0	0
$739$	−3.58063	−0.131716	−0.0658578	−	0.997829i	$-0.520978\pi$
$739$	−3.58063	−0.131716	−0.0658578	+	0.997829i	$0.520978\pi$
$740$	3.13667	0.115306
$741$	0	0
$742$	0	0
$743$	49.5928	1.81938	0.909691	−	0.415286i	$-0.136318\pi$
$743$	49.5928	1.81938	0.909691	+	0.415286i	$0.136318\pi$
$744$	0	0
$745$	−12.1917	−0.446668
$746$	−10.7745	−0.394483
$747$	0	0
$748$	−8.56098	−0.313020
$749$	0	0
$750$	0	0
$751$	−42.9030	−1.56555	−0.782776	−	0.622304i	$-0.786197\pi$
$751$	−42.9030	−1.56555	−0.782776	+	0.622304i	$0.786197\pi$
$752$	−14.2690	−0.520337
$753$	0	0
$754$	−12.5767	−0.458016
$755$	−25.1633	−0.915786
$756$	0	0
$757$	13.8029	0.501677	0.250838	−	0.968029i	$-0.419294\pi$
$757$	13.8029	0.501677	0.250838	+	0.968029i	$0.419294\pi$
$758$	−0.683021	−0.0248084
$759$	0	0
$760$	−13.6171	−0.493945
$761$	40.7197	1.47609	0.738044	−	0.674752i	$-0.235749\pi$
$761$	40.7197	1.47609	0.738044	+	0.674752i	$0.235749\pi$
$762$	0	0
$763$	0	0
$764$	−23.0001	−0.832113
$765$	0	0
$766$	−7.76683	−0.280627
$767$	−18.1365	−0.654871
$768$	0	0
$769$	−11.1476	−0.401994	−0.200997	−	0.979592i	$-0.564418\pi$
$769$	−11.1476	−0.401994	−0.200997	+	0.979592i	$0.564418\pi$
$770$	0	0
$771$	0	0
$772$	−25.6908	−0.924633
$773$	0.925662	0.0332937	0.0166469	−	0.999861i	$-0.494701\pi$
$773$	0.925662	0.0332937	0.0166469	+	0.999861i	$0.494701\pi$
$774$	0	0
$775$	−7.44121	−0.267296
$776$	12.8602	0.461656
$777$	0	0
$778$	11.9410	0.428105
$779$	−0.998817	−0.0357863
$780$	0	0
$781$	−0.396903	−0.0142023
$782$	−4.08049	−0.145918
$783$	0	0
$784$	0	0
$785$	9.00772	0.321499
$786$	0	0
$787$	23.0240	0.820716	0.410358	−	0.911925i	$-0.365404\pi$
$787$	23.0240	0.820716	0.410358	+	0.911925i	$0.365404\pi$
$788$	6.26061	0.223025
$789$	0	0
$790$	−1.76223	−0.0626973
$791$	0	0
$792$	0	0
$793$	0.207324	0.00736228
$794$	8.77102	0.311272
$795$	0	0
$796$	−39.2027	−1.38950
$797$	−22.7851	−0.807089	−0.403544	−	0.914960i	$-0.632222\pi$
$797$	−22.7851	−0.807089	−0.403544	+	0.914960i	$0.632222\pi$
$798$	0	0
$799$	10.6098	0.375349
$800$	−17.1333	−0.605753
$801$	0	0
$802$	9.45390	0.333829
$803$	52.7150	1.86027
$804$	0	0
$805$	0	0
$806$	4.62703	0.162980
$807$	0	0
$808$	12.2229	0.429999
$809$	−13.4751	−0.473758	−0.236879	−	0.971539i	$-0.576124\pi$
$809$	−13.4751	−0.473758	−0.236879	+	0.971539i	$0.576124\pi$
$810$	0	0
$811$	−30.7348	−1.07924	−0.539622	−	0.841907i	$-0.681433\pi$
$811$	−30.7348	−1.07924	−0.539622	+	0.841907i	$0.681433\pi$
$812$	0	0
$813$	0	0
$814$	4.69206	0.164457
$815$	11.4318	0.400439
$816$	0	0
$817$	4.00150	0.139995
$818$	−1.77369	−0.0620158
$819$	0	0
$820$	0.549675	0.0191955
$821$	−16.9864	−0.592829	−0.296414	−	0.955059i	$-0.595791\pi$
$821$	−16.9864	−0.592829	−0.296414	+	0.955059i	$0.595791\pi$
$822$	0	0
$823$	−18.5831	−0.647768	−0.323884	−	0.946097i	$-0.604989\pi$
$823$	−18.5831	−0.647768	−0.323884	+	0.946097i	$0.604989\pi$
$824$	33.8244	1.17833
$825$	0	0
$826$	0	0
$827$	14.5419	0.505670	0.252835	−	0.967509i	$-0.418637\pi$
$827$	14.5419	0.505670	0.252835	+	0.967509i	$0.418637\pi$
$828$	0	0
$829$	−9.57433	−0.332530	−0.166265	−	0.986081i	$-0.553171\pi$
$829$	−9.57433	−0.332530	−0.166265	+	0.986081i	$0.553171\pi$
$830$	13.8231	0.479806
$831$	0	0
$832$	2.35770	0.0817386
$833$	0	0
$834$	0	0
$835$	−3.02195	−0.104579
$836$	30.6987	1.06174
$837$	0	0
$838$	−22.4651	−0.776045
$839$	−42.4606	−1.46590	−0.732952	−	0.680281i	$-0.761858\pi$
$839$	−42.4606	−1.46590	−0.732952	+	0.680281i	$0.761858\pi$
$840$	0	0
$841$	17.3910	0.599690
$842$	3.24375	0.111787
$843$	0	0
$844$	−11.6686	−0.401648
$845$	−7.71198	−0.265300
$846$	0	0
$847$	0	0
$848$	1.23548	0.0424264
$849$	0	0
$850$	2.22882	0.0764479
$851$	−7.71767	−0.264558
$852$	0	0
$853$	−14.2808	−0.488965	−0.244482	−	0.969654i	$-0.578618\pi$
$853$	−14.2808	−0.488965	−0.244482	+	0.969654i	$0.578618\pi$
$854$	0	0
$855$	0	0
$856$	18.2324	0.623171
$857$	34.7790	1.18803	0.594013	−	0.804455i	$-0.297543\pi$
$857$	34.7790	1.18803	0.594013	+	0.804455i	$0.297543\pi$
$858$	0	0
$859$	−12.6486	−0.431564	−0.215782	−	0.976442i	$-0.569230\pi$
$859$	−12.6486	−0.431564	−0.215782	+	0.976442i	$0.569230\pi$
$860$	−2.20213	−0.0750921
$861$	0	0
$862$	−23.6819	−0.806608
$863$	−26.4797	−0.901379	−0.450690	−	0.892681i	$-0.648822\pi$
$863$	−26.4797	−0.901379	−0.450690	+	0.892681i	$0.648822\pi$
$864$	0	0
$865$	−26.0594	−0.886045
$866$	3.66943	0.124692
$867$	0	0
$868$	0	0
$869$	9.09686	0.308590
$870$	0	0
$871$	−34.6741	−1.17489
$872$	−4.04332	−0.136924
$873$	0	0
$874$	14.6322	0.494941
$875$	0	0
$876$	0	0
$877$	28.4534	0.960805	0.480402	−	0.877048i	$-0.340491\pi$
$877$	28.4534	0.960805	0.480402	+	0.877048i	$0.340491\pi$
$878$	4.28887	0.144742
$879$	0	0
$880$	−10.5801	−0.356654
$881$	−20.3637	−0.686071	−0.343036	−	0.939322i	$-0.611455\pi$
$881$	−20.3637	−0.686071	−0.343036	+	0.939322i	$0.611455\pi$
$882$	0	0
$883$	49.1950	1.65554	0.827772	−	0.561065i	$-0.189608\pi$
$883$	49.1950	1.65554	0.827772	+	0.561065i	$0.189608\pi$
$884$	4.78264	0.160858
$885$	0	0
$886$	−4.28129	−0.143833
$887$	−4.21692	−0.141590	−0.0707952	−	0.997491i	$-0.522554\pi$
$887$	−4.21692	−0.141590	−0.0707952	+	0.997491i	$0.522554\pi$
$888$	0	0
$889$	0	0
$890$	−12.9196	−0.433067
$891$	0	0
$892$	20.1346	0.674157
$893$	−38.0457	−1.27315
$894$	0	0
$895$	−10.8659	−0.363206
$896$	0	0
$897$	0	0
$898$	−7.87371	−0.262749
$899$	−17.0675	−0.569233
$900$	0	0
$901$	−0.918649	−0.0306046
$902$	0.822244	0.0273777
$903$	0	0
$904$	−1.42974	−0.0475526
$905$	22.1301	0.735628
$906$	0	0
$907$	47.9851	1.59332	0.796659	−	0.604429i	$-0.206599\pi$
$907$	47.9851	1.59332	0.796659	+	0.604429i	$0.206599\pi$
$908$	44.9170	1.49062
$909$	0	0
$910$	0	0
$911$	25.7335	0.852587	0.426294	−	0.904585i	$-0.359819\pi$
$911$	25.7335	0.852587	0.426294	+	0.904585i	$0.359819\pi$
$912$	0	0
$913$	−71.3565	−2.36155
$914$	7.05351	0.233309
$915$	0	0
$916$	−23.9356	−0.790854
$917$	0	0
$918$	0	0
$919$	−2.26957	−0.0748661	−0.0374330	−	0.999299i	$-0.511918\pi$
$919$	−2.26957	−0.0748661	−0.0374330	+	0.999299i	$0.511918\pi$
$920$	−18.4384	−0.607895
$921$	0	0
$922$	4.74968	0.156422
$923$	0.221732	0.00729840
$924$	0	0
$925$	4.21550	0.138605
$926$	−21.9548	−0.721479
$927$	0	0
$928$	−39.2976	−1.29001
$929$	45.8496	1.50428	0.752138	−	0.659006i	$-0.229023\pi$
$929$	45.8496	1.50428	0.752138	+	0.659006i	$0.229023\pi$
$930$	0	0
$931$	0	0
$932$	−7.67028	−0.251248
$933$	0	0
$934$	−2.63060	−0.0860757
$935$	7.86691	0.257275
$936$	0	0
$937$	−56.2075	−1.83622	−0.918110	−	0.396325i	$-0.870285\pi$
$937$	−56.2075	−1.83622	−0.918110	+	0.396325i	$0.870285\pi$
$938$	0	0
$939$	0	0
$940$	20.9376	0.682908
$941$	−35.2803	−1.15011	−0.575053	−	0.818116i	$-0.695018\pi$
$941$	−35.2803	−1.15011	−0.575053	+	0.818116i	$0.695018\pi$
$942$	0	0
$943$	−1.35246	−0.0440421
$944$	−9.91453	−0.322691
$945$	0	0
$946$	−3.29411	−0.107101
$947$	−50.7130	−1.64795	−0.823976	−	0.566625i	$-0.808249\pi$
$947$	−50.7130	−1.64795	−0.823976	+	0.566625i	$0.808249\pi$
$948$	0	0
$949$	−29.4495	−0.955972
$950$	−7.99230	−0.259305
$951$	0	0
$952$	0	0
$953$	25.9988	0.842184	0.421092	−	0.907018i	$-0.361647\pi$
$953$	25.9988	0.842184	0.421092	+	0.907018i	$0.361647\pi$
$954$	0	0
$955$	21.1354	0.683924
$956$	20.2122	0.653710
$957$	0	0
$958$	10.7833	0.348392
$959$	0	0
$960$	0	0
$961$	−24.7208	−0.797445
$962$	−2.62125	−0.0845123
$963$	0	0
$964$	−22.6124	−0.728295
$965$	23.6080	0.759968
$966$	0	0
$967$	25.9621	0.834885	0.417442	−	0.908703i	$-0.362927\pi$
$967$	25.9621	0.834885	0.417442	+	0.908703i	$0.362927\pi$
$968$	−31.6854	−1.01841
$969$	0	0
$970$	−5.16103	−0.165711
$971$	7.94412	0.254939	0.127469	−	0.991843i	$-0.459315\pi$
$971$	7.94412	0.254939	0.127469	+	0.991843i	$0.459315\pi$
$972$	0	0
$973$	0	0
$974$	2.34847	0.0752499
$975$	0	0
$976$	0.113336	0.00362779
$977$	−52.2548	−1.67178	−0.835889	−	0.548898i	$-0.815048\pi$
$977$	−52.2548	−1.67178	−0.835889	+	0.548898i	$0.815048\pi$
$978$	0	0
$979$	66.6929	2.13151
$980$	0	0
$981$	0	0
$982$	27.5569	0.879376
$983$	−38.8379	−1.23874	−0.619369	−	0.785100i	$-0.712611\pi$
$983$	−38.8379	−1.23874	−0.619369	+	0.785100i	$0.712611\pi$
$984$	0	0
$985$	−5.75304	−0.183307
$986$	5.11212	0.162803
$987$	0	0
$988$	−17.1500	−0.545615
$989$	5.41827	0.172291
$990$	0	0
$991$	30.9378	0.982771	0.491385	−	0.870942i	$-0.336491\pi$
$991$	30.9378	0.982771	0.491385	+	0.870942i	$0.336491\pi$
$992$	14.4578	0.459036
$993$	0	0
$994$	0	0
$995$	36.0244	1.14205
$996$	0	0
$997$	47.0670	1.49063	0.745313	−	0.666714i	$-0.232300\pi$
$997$	47.0670	1.49063	0.745313	+	0.666714i	$0.232300\pi$
$998$	7.92985	0.251015
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.z.1.4		5
3.2	odd	2		3969.2.a.bc.1.2		5
7.2	even	3		567.2.e.f.487.2		10
7.4	even	3		567.2.e.f.163.2		10
7.6	odd	2		3969.2.a.ba.1.4		5
9.2	odd	6		1323.2.f.e.442.4		10
9.4	even	3		441.2.f.e.295.2		10
9.5	odd	6		1323.2.f.e.883.4		10
9.7	even	3		441.2.f.e.148.2		10
21.2	odd	6		567.2.e.e.487.4		10
21.11	odd	6		567.2.e.e.163.4		10
21.20	even	2		3969.2.a.bb.1.2		5
63.2	odd	6		189.2.g.b.172.4		10
63.4	even	3		63.2.g.b.16.2	yes	10
63.5	even	6		1323.2.h.f.802.2		10
63.11	odd	6		189.2.h.b.37.2		10
63.13	odd	6		441.2.f.f.295.2		10
63.16	even	3		63.2.g.b.4.2	✓	10
63.20	even	6		1323.2.f.f.442.4		10
63.23	odd	6		189.2.h.b.46.2		10
63.25	even	3		63.2.h.b.58.4	yes	10
63.31	odd	6		441.2.g.f.79.2		10
63.32	odd	6		189.2.g.b.100.4		10
63.34	odd	6		441.2.f.f.148.2		10
63.38	even	6		1323.2.h.f.226.2		10
63.40	odd	6		441.2.h.f.214.4		10
63.41	even	6		1323.2.f.f.883.4		10
63.47	even	6		1323.2.g.f.361.4		10
63.52	odd	6		441.2.h.f.373.4		10
63.58	even	3		63.2.h.b.25.4	yes	10
63.59	even	6		1323.2.g.f.667.4		10
63.61	odd	6		441.2.g.f.67.2		10
252.11	even	6		3024.2.q.i.2305.5		10
252.23	even	6		3024.2.q.i.2881.5		10
252.67	odd	6		1008.2.t.i.961.4		10
252.79	odd	6		1008.2.t.i.193.4		10
252.95	even	6		3024.2.t.i.289.1		10
252.151	odd	6		1008.2.q.i.625.1		10
252.191	even	6		3024.2.t.i.1873.1		10
252.247	odd	6		1008.2.q.i.529.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.g.b.4.2	✓	10	63.16	even	3
63.2.g.b.16.2	yes	10	63.4	even	3
63.2.h.b.25.4	yes	10	63.58	even	3
63.2.h.b.58.4	yes	10	63.25	even	3
189.2.g.b.100.4		10	63.32	odd	6
189.2.g.b.172.4		10	63.2	odd	6
189.2.h.b.37.2		10	63.11	odd	6
189.2.h.b.46.2		10	63.23	odd	6
441.2.f.e.148.2		10	9.7	even	3
441.2.f.e.295.2		10	9.4	even	3
441.2.f.f.148.2		10	63.34	odd	6
441.2.f.f.295.2		10	63.13	odd	6
441.2.g.f.67.2		10	63.61	odd	6
441.2.g.f.79.2		10	63.31	odd	6
441.2.h.f.214.4		10	63.40	odd	6
441.2.h.f.373.4		10	63.52	odd	6
567.2.e.e.163.4		10	21.11	odd	6
567.2.e.e.487.4		10	21.2	odd	6
567.2.e.f.163.2		10	7.4	even	3
567.2.e.f.487.2		10	7.2	even	3
1008.2.q.i.529.1		10	252.247	odd	6
1008.2.q.i.625.1		10	252.151	odd	6
1008.2.t.i.193.4		10	252.79	odd	6
1008.2.t.i.961.4		10	252.67	odd	6
1323.2.f.e.442.4		10	9.2	odd	6
1323.2.f.e.883.4		10	9.5	odd	6
1323.2.f.f.442.4		10	63.20	even	6
1323.2.f.f.883.4		10	63.41	even	6
1323.2.g.f.361.4		10	63.47	even	6
1323.2.g.f.667.4		10	63.59	even	6
1323.2.h.f.226.2		10	63.38	even	6
1323.2.h.f.802.2		10	63.5	even	6
3024.2.q.i.2305.5		10	252.11	even	6
3024.2.q.i.2881.5		10	252.23	even	6
3024.2.t.i.289.1		10	252.95	even	6
3024.2.t.i.1873.1		10	252.191	even	6
3969.2.a.z.1.4		5	1.1	even	1	trivial
3969.2.a.ba.1.4		5	7.6	odd	2
3969.2.a.bb.1.2		5	21.20	even	2
3969.2.a.bc.1.2		5	3.2	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:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q+0.670333 q^{2} -1.55065 q^{4} +1.42494 q^{5} -2.38012 q^{8} +0.955182 q^{10} -4.93077 q^{11} +2.75460 q^{13} +1.50584 q^{16} -1.11968 q^{17} +4.01505 q^{19} -2.20958 q^{20} -3.30526 q^{22} +5.43661 q^{23} -2.96955 q^{25} +1.84650 q^{26} -6.81109 q^{29} +2.50584 q^{31} +5.76965 q^{32} -0.750557 q^{34} -1.41957 q^{37} +2.69142 q^{38} -3.39152 q^{40} -0.248768 q^{41} +0.996627 q^{43} +7.64592 q^{44} +3.64434 q^{46} -9.47579 q^{47} -1.99059 q^{50} -4.27144 q^{52} +0.820458 q^{53} -7.02604 q^{55} -4.56570 q^{58} -6.58407 q^{59} +0.0752645 q^{61} +1.67974 q^{62} +0.855913 q^{64} +3.92514 q^{65} -12.5877 q^{67} +1.73623 q^{68} +0.0804951 q^{71} -10.6910 q^{73} -0.951587 q^{74} -6.22595 q^{76} -1.84491 q^{79} +2.14572 q^{80} -0.166758 q^{82} +14.4717 q^{83} -1.59547 q^{85} +0.668072 q^{86} +11.7358 q^{88} -13.5258 q^{89} -8.43030 q^{92} -6.35193 q^{94} +5.72119 q^{95} -5.40319 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 3 q^{8} + 7 q^{10} - 4 q^{11} + 8 q^{13} - 2 q^{16} - 12 q^{17} - q^{19} - 5 q^{20} + q^{22} - 3 q^{23} + q^{25} - 11 q^{26} - 7 q^{29} + 3 q^{31} + 2 q^{32} - 3 q^{34}+ \cdots + 12 q^{97}+O(q^{100}) \) 5 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 3 * q^8 + 7 * q^10 - 4 * q^11 + 8 * q^13 - 2 * q^16 - 12 * q^17 - q^19 - 5 * q^20 + q^22 - 3 * q^23 + q^25 - 11 * q^26 - 7 * q^29 + 3 * q^31 + 2 * q^32 - 3 * q^34 - 20 * q^38 + 3 * q^40 - 5 * q^41 + 7 * q^43 + 10 * q^44 - 3 * q^46 - 27 * q^47 - 19 * q^50 + 10 * q^52 + 21 * q^53 + 2 * q^55 + 10 * q^58 - 30 * q^59 + 14 * q^61 - 6 * q^62 - 25 * q^64 + 11 * q^65 + 2 * q^67 - 27 * q^68 - 3 * q^71 - 15 * q^73 + 36 * q^74 - 5 * q^76 + 4 * q^79 - 20 * q^80 + 5 * q^82 - 9 * q^83 + 6 * q^85 + 8 * q^86 + 18 * q^88 - 28 * q^89 - 27 * q^92 + 3 * q^94 + 14 * q^95 + 12 * q^97

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