LMFDB - Embedded newform 3042.2.b.i.1351.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1351,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("3042.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$24.2904922949$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3042.1351
Dual form			3042.2.b.i.1351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -0.267949i q^{5} -0.732051i q^{7} +1.00000i q^{8} -0.267949 q^{10} +4.73205i q^{11} -0.732051 q^{14} +1.00000 q^{16} -2.26795 q^{17} +1.26795i q^{19} +0.267949i q^{20} +4.73205 q^{22} -6.19615 q^{23} +4.92820 q^{25} +0.732051i q^{28} -2.46410 q^{29} -5.46410i q^{31} -1.00000i q^{32} +2.26795i q^{34} -0.196152 q^{35} -10.4641i q^{37} +1.26795 q^{38} +0.267949 q^{40} -11.3923i q^{41} -7.66025 q^{43} -4.73205i q^{44} +6.19615i q^{46} -8.19615i q^{47} +6.46410 q^{49} -4.92820i q^{50} -0.464102 q^{53} +1.26795 q^{55} +0.732051 q^{56} +2.46410i q^{58} +8.00000i q^{59} +1.19615 q^{61} -5.46410 q^{62} -1.00000 q^{64} -11.1244i q^{67} +2.26795 q^{68} +0.196152i q^{70} -1.26795i q^{71} +9.73205i q^{73} -10.4641 q^{74} -1.26795i q^{76} +3.46410 q^{77} -9.46410 q^{79} -0.267949i q^{80} -11.3923 q^{82} -10.1962i q^{83} +0.607695i q^{85} +7.66025i q^{86} -4.73205 q^{88} -2.53590i q^{89} +6.19615 q^{92} -8.19615 q^{94} +0.339746 q^{95} -6.00000i q^{97} -6.46410i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 8 q^{10} + 4 q^{14} + 4 q^{16} - 16 q^{17} + 12 q^{22} - 4 q^{23} - 8 q^{25} + 4 q^{29} + 20 q^{35} + 12 q^{38} + 8 q^{40} + 4 q^{43} + 12 q^{49} + 12 q^{53} + 12 q^{55} - 4 q^{56} - 16 q^{61}+ \cdots + 36 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 - 8 * q^10 + 4 * q^14 + 4 * q^16 - 16 * q^17 + 12 * q^22 - 4 * q^23 - 8 * q^25 + 4 * q^29 + 20 * q^35 + 12 * q^38 + 8 * q^40 + 4 * q^43 + 12 * q^49 + 12 * q^53 + 12 * q^55 - 4 * q^56 - 16 * q^61 - 8 * q^62 - 4 * q^64 + 16 * q^68 - 28 * q^74 - 24 * q^79 - 4 * q^82 - 12 * q^88 + 4 * q^92 - 12 * q^94 + 36 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	− 0.267949i	− 0.119831i	−0.998203	−	0.0599153i	$-0.980917\pi$
$5$	− 0.267949i	− 0.119831i	0.998203	−	0.0599153i	$-0.0190830\pi$
$6$	0	0
$7$	− 0.732051i	− 0.276689i	−0.990384	−	0.138345i	$-0.955822\pi$
$7$	− 0.732051i	− 0.276689i	0.990384	−	0.138345i	$-0.0441781\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	−0.267949	−0.0847330
$11$	4.73205i	1.42677i	0.700774	+	0.713384i	$0.252838\pi$
$11$	4.73205i	1.42677i	−0.700774	+	0.713384i	$0.747162\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−0.732051	−0.195649
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.26795	−0.550058	−0.275029	−	0.961436i	$-0.588688\pi$
$17$	−2.26795	−0.550058	−0.275029	+	0.961436i	$0.588688\pi$
$18$	0	0
$19$	1.26795i	0.290887i	0.989367	+	0.145444i	$0.0464610\pi$
$19$	1.26795i	0.290887i	−0.989367	+	0.145444i	$0.953539\pi$
$20$	0.267949i	0.0599153i
$21$	0	0
$22$	4.73205	1.00888
$23$	−6.19615	−1.29199	−0.645994	−	0.763343i	$-0.723557\pi$
$23$	−6.19615	−1.29199	−0.645994	+	0.763343i	$0.723557\pi$
$24$	0	0
$25$	4.92820	0.985641
$26$	0	0
$27$	0	0
$28$	0.732051i	0.138345i
$29$	−2.46410	−0.457572	−0.228786	−	0.973477i	$-0.573476\pi$
$29$	−2.46410	−0.457572	−0.228786	+	0.973477i	$0.573476\pi$
$30$	0	0
$31$	− 5.46410i	− 0.981382i	−0.871334	−	0.490691i	$-0.836744\pi$
$31$	− 5.46410i	− 0.981382i	0.871334	−	0.490691i	$-0.163256\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	2.26795i	0.388950i
$35$	−0.196152	−0.0331558
$36$	0	0
$37$	− 10.4641i	− 1.72029i	−0.510052	−	0.860144i	$-0.670374\pi$
$37$	− 10.4641i	− 1.72029i	0.510052	−	0.860144i	$-0.329626\pi$
$38$	1.26795	0.205689
$39$	0	0
$40$	0.267949	0.0423665
$41$	− 11.3923i	− 1.77918i	−0.456761	−	0.889590i	$-0.650990\pi$
$41$	− 11.3923i	− 1.77918i	0.456761	−	0.889590i	$-0.349010\pi$
$42$	0	0
$43$	−7.66025	−1.16818	−0.584089	−	0.811690i	$-0.698548\pi$
$43$	−7.66025	−1.16818	−0.584089	+	0.811690i	$0.698548\pi$
$44$	− 4.73205i	− 0.713384i
$45$	0	0
$46$	6.19615i	0.913573i
$47$	− 8.19615i	− 1.19553i	−0.801671	−	0.597766i	$-0.796055\pi$
$47$	− 8.19615i	− 1.19553i	0.801671	−	0.597766i	$-0.203945\pi$
$48$	0	0
$49$	6.46410	0.923443
$50$	− 4.92820i	− 0.696953i
$51$	0	0
$52$	0	0
$53$	−0.464102	−0.0637493	−0.0318746	−	0.999492i	$-0.510148\pi$
$53$	−0.464102	−0.0637493	−0.0318746	+	0.999492i	$0.510148\pi$
$54$	0	0
$55$	1.26795	0.170970
$56$	0.732051	0.0978244
$57$	0	0
$58$	2.46410i	0.323552i
$59$	8.00000i	1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$	8.00000i	1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$	0	0
$61$	1.19615	0.153152	0.0765758	−	0.997064i	$-0.475601\pi$
$61$	1.19615	0.153152	0.0765758	+	0.997064i	$0.475601\pi$
$62$	−5.46410	−0.693942
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 11.1244i	− 1.35906i	−0.733649	−	0.679528i	$-0.762185\pi$
$67$	− 11.1244i	− 1.35906i	0.733649	−	0.679528i	$-0.237815\pi$
$68$	2.26795	0.275029
$69$	0	0
$70$	0.196152i	0.0234447i
$71$	− 1.26795i	− 0.150478i	−0.997166	−	0.0752389i	$-0.976028\pi$
$71$	− 1.26795i	− 0.150478i	0.997166	−	0.0752389i	$-0.0239720\pi$
$72$	0	0
$73$	9.73205i	1.13905i	0.821974	+	0.569525i	$0.192873\pi$
$73$	9.73205i	1.13905i	−0.821974	+	0.569525i	$0.807127\pi$
$74$	−10.4641	−1.21643
$75$	0	0
$76$	− 1.26795i	− 0.145444i
$77$	3.46410	0.394771
$78$	0	0
$79$	−9.46410	−1.06479	−0.532397	−	0.846495i	$-0.678709\pi$
$79$	−9.46410	−1.06479	−0.532397	+	0.846495i	$0.678709\pi$
$80$	− 0.267949i	− 0.0299576i
$81$	0	0
$82$	−11.3923	−1.25807
$83$	− 10.1962i	− 1.11917i	−0.828772	−	0.559587i	$-0.810960\pi$
$83$	− 10.1962i	− 1.11917i	0.828772	−	0.559587i	$-0.189040\pi$
$84$	0	0
$85$	0.607695i	0.0659138i
$86$	7.66025i	0.826026i
$87$	0	0
$88$	−4.73205	−0.504438
$89$	− 2.53590i	− 0.268805i	−0.990927	−	0.134402i	$-0.957089\pi$
$89$	− 2.53590i	− 0.268805i	0.990927	−	0.134402i	$-0.0429115\pi$
$90$	0	0
$91$	0	0
$92$	6.19615	0.645994
$93$	0	0
$94$	−8.19615	−0.845369
$95$	0.339746	0.0348572
$96$	0	0
$97$	− 6.00000i	− 0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$	− 6.00000i	− 0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$	− 6.46410i	− 0.652973i
$99$	0	0
$100$	−4.92820	−0.492820
$101$	−11.9282	−1.18690	−0.593450	−	0.804871i	$-0.702235\pi$
$101$	−11.9282	−1.18690	−0.593450	+	0.804871i	$0.702235\pi$
$102$	0	0
$103$	18.7321	1.84572	0.922862	−	0.385131i	$-0.125844\pi$
$103$	18.7321	1.84572	0.922862	+	0.385131i	$0.125844\pi$
$104$	0	0
$105$	0	0
$106$	0.464102i	0.0450775i
$107$	0.196152	0.0189628	0.00948139	−	0.999955i	$-0.496982\pi$
$107$	0.196152	0.0189628	0.00948139	+	0.999955i	$0.496982\pi$
$108$	0	0
$109$	− 5.46410i	− 0.523366i	−0.965154	−	0.261683i	$-0.915723\pi$
$109$	− 5.46410i	− 0.523366i	0.965154	−	0.261683i	$-0.0842775\pi$
$110$	− 1.26795i	− 0.120894i
$111$	0	0
$112$	− 0.732051i	− 0.0691723i
$113$	18.6603	1.75541	0.877705	−	0.479202i	$-0.159074\pi$
$113$	18.6603	1.75541	0.877705	+	0.479202i	$0.159074\pi$
$114$	0	0
$115$	1.66025i	0.154819i
$116$	2.46410	0.228786
$117$	0	0
$118$	8.00000	0.736460
$119$	1.66025i	0.152195i
$120$	0	0
$121$	−11.3923	−1.03566
$122$	− 1.19615i	− 0.108295i
$123$	0	0
$124$	5.46410i	0.490691i
$125$	− 2.66025i	− 0.237940i
$126$	0	0
$127$	−17.8564	−1.58450	−0.792250	−	0.610197i	$-0.791090\pi$
$127$	−17.8564	−1.58450	−0.792250	+	0.610197i	$0.791090\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−13.4641	−1.17636	−0.588182	−	0.808729i	$-0.700156\pi$
$131$	−13.4641	−1.17636	−0.588182	+	0.808729i	$0.700156\pi$
$132$	0	0
$133$	0.928203	0.0804854
$134$	−11.1244	−0.960998
$135$	0	0
$136$	− 2.26795i	− 0.194475i
$137$	− 1.92820i	− 0.164738i	−0.996602	−	0.0823688i	$-0.973751\pi$
$137$	− 1.92820i	− 0.164738i	0.996602	−	0.0823688i	$-0.0262485\pi$
$138$	0	0
$139$	−9.85641	−0.836009	−0.418005	−	0.908445i	$-0.637270\pi$
$139$	−9.85641	−0.836009	−0.418005	+	0.908445i	$0.637270\pi$
$140$	0.196152	0.0165779
$141$	0	0
$142$	−1.26795	−0.106404
$143$	0	0
$144$	0	0
$145$	0.660254i	0.0548311i
$146$	9.73205	0.805430
$147$	0	0
$148$	10.4641i	0.860144i
$149$	− 2.80385i	− 0.229700i	−0.993383	−	0.114850i	$-0.963361\pi$
$149$	− 2.80385i	− 0.229700i	0.993383	−	0.114850i	$-0.0366388\pi$
$150$	0	0
$151$	− 3.26795i	− 0.265942i	−0.991120	−	0.132971i	$-0.957548\pi$
$151$	− 3.26795i	− 0.265942i	0.991120	−	0.132971i	$-0.0424517\pi$
$152$	−1.26795	−0.102844
$153$	0	0
$154$	− 3.46410i	− 0.279145i
$155$	−1.46410	−0.117599
$156$	0	0
$157$	−23.5885	−1.88256	−0.941282	−	0.337622i	$-0.890378\pi$
$157$	−23.5885	−1.88256	−0.941282	+	0.337622i	$0.890378\pi$
$158$	9.46410i	0.752923i
$159$	0	0
$160$	−0.267949	−0.0211832
$161$	4.53590i	0.357479i
$162$	0	0
$163$	6.53590i	0.511931i	0.966686	+	0.255966i	$0.0823934\pi$
$163$	6.53590i	0.511931i	−0.966686	+	0.255966i	$0.917607\pi$
$164$	11.3923i	0.889590i
$165$	0	0
$166$	−10.1962	−0.791375
$167$	− 2.53590i	− 0.196234i	−0.995175	−	0.0981169i	$-0.968718\pi$
$167$	− 2.53590i	− 0.196234i	0.995175	−	0.0981169i	$-0.0312819\pi$
$168$	0	0
$169$	0	0
$170$	0.607695	0.0466081
$171$	0	0
$172$	7.66025	0.584089
$173$	−16.3923	−1.24628	−0.623142	−	0.782109i	$-0.714144\pi$
$173$	−16.3923	−1.24628	−0.623142	+	0.782109i	$0.714144\pi$
$174$	0	0
$175$	− 3.60770i	− 0.272716i
$176$	4.73205i	0.356692i
$177$	0	0
$178$	−2.53590	−0.190074
$179$	22.0526	1.64829	0.824143	−	0.566382i	$-0.191657\pi$
$179$	22.0526	1.64829	0.824143	+	0.566382i	$0.191657\pi$
$180$	0	0
$181$	−8.80385	−0.654385	−0.327192	−	0.944958i	$-0.606103\pi$
$181$	−8.80385	−0.654385	−0.327192	+	0.944958i	$0.606103\pi$
$182$	0	0
$183$	0	0
$184$	− 6.19615i	− 0.456786i
$185$	−2.80385	−0.206143
$186$	0	0
$187$	− 10.7321i	− 0.784805i
$188$	8.19615i	0.597766i
$189$	0	0
$190$	− 0.339746i	− 0.0246478i
$191$	−6.92820	−0.501307	−0.250654	−	0.968077i	$-0.580646\pi$
$191$	−6.92820	−0.501307	−0.250654	+	0.968077i	$0.580646\pi$
$192$	0	0
$193$	8.26795i	0.595140i	0.954700	+	0.297570i	$0.0961762\pi$
$193$	8.26795i	0.595140i	−0.954700	+	0.297570i	$0.903824\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	−6.46410	−0.461722
$197$	− 9.85641i	− 0.702240i	−0.936330	−	0.351120i	$-0.885801\pi$
$197$	− 9.85641i	− 0.702240i	0.936330	−	0.351120i	$-0.114199\pi$
$198$	0	0
$199$	3.80385	0.269648	0.134824	−	0.990870i	$-0.456953\pi$
$199$	3.80385	0.269648	0.134824	+	0.990870i	$0.456953\pi$
$200$	4.92820i	0.348477i
$201$	0	0
$202$	11.9282i	0.839265i
$203$	1.80385i	0.126605i
$204$	0	0
$205$	−3.05256	−0.213200
$206$	− 18.7321i	− 1.30512i
$207$	0	0
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−4.39230	−0.302379	−0.151189	−	0.988505i	$-0.548310\pi$
$211$	−4.39230	−0.302379	−0.151189	+	0.988505i	$0.548310\pi$
$212$	0.464102	0.0318746
$213$	0	0
$214$	− 0.196152i	− 0.0134087i
$215$	2.05256i	0.139983i
$216$	0	0
$217$	−4.00000	−0.271538
$218$	−5.46410	−0.370076
$219$	0	0
$220$	−1.26795	−0.0854851
$221$	0	0
$222$	0	0
$223$	− 13.0718i	− 0.875352i	−0.899133	−	0.437676i	$-0.855802\pi$
$223$	− 13.0718i	− 0.875352i	0.899133	−	0.437676i	$-0.144198\pi$
$224$	−0.732051	−0.0489122
$225$	0	0
$226$	− 18.6603i	− 1.24126i
$227$	1.80385i	0.119726i	0.998207	+	0.0598628i	$0.0190663\pi$
$227$	1.80385i	0.119726i	−0.998207	+	0.0598628i	$0.980934\pi$
$228$	0	0
$229$	− 15.8564i	− 1.04782i	−0.851773	−	0.523910i	$-0.824473\pi$
$229$	− 15.8564i	− 1.04782i	0.851773	−	0.523910i	$-0.175527\pi$
$230$	1.66025	0.109474
$231$	0	0
$232$	− 2.46410i	− 0.161776i
$233$	19.8564	1.30084	0.650418	−	0.759576i	$-0.274594\pi$
$233$	19.8564	1.30084	0.650418	+	0.759576i	$0.274594\pi$
$234$	0	0
$235$	−2.19615	−0.143261
$236$	− 8.00000i	− 0.520756i
$237$	0	0
$238$	1.66025	0.107618
$239$	9.66025i	0.624870i	0.949939	+	0.312435i	$0.101145\pi$
$239$	9.66025i	0.624870i	−0.949939	+	0.312435i	$0.898855\pi$
$240$	0	0
$241$	17.5885i	1.13297i	0.824071	+	0.566486i	$0.191698\pi$
$241$	17.5885i	1.13297i	−0.824071	+	0.566486i	$0.808302\pi$
$242$	11.3923i	0.732325i
$243$	0	0
$244$	−1.19615	−0.0765758
$245$	− 1.73205i	− 0.110657i
$246$	0	0
$247$	0	0
$248$	5.46410	0.346971
$249$	0	0
$250$	−2.66025	−0.168249
$251$	6.53590	0.412542	0.206271	−	0.978495i	$-0.433867\pi$
$251$	6.53590	0.412542	0.206271	+	0.978495i	$0.433867\pi$
$252$	0	0
$253$	− 29.3205i	− 1.84336i
$254$	17.8564i	1.12041i
$255$	0	0
$256$	1.00000	0.0625000
$257$	26.6603	1.66302	0.831510	−	0.555509i	$-0.187477\pi$
$257$	26.6603	1.66302	0.831510	+	0.555509i	$0.187477\pi$
$258$	0	0
$259$	−7.66025	−0.475985
$260$	0	0
$261$	0	0
$262$	13.4641i	0.831815i
$263$	−28.0526	−1.72979	−0.864897	−	0.501949i	$-0.832617\pi$
$263$	−28.0526	−1.72979	−0.864897	+	0.501949i	$0.832617\pi$
$264$	0	0
$265$	0.124356i	0.00763911i
$266$	− 0.928203i	− 0.0569118i
$267$	0	0
$268$	11.1244i	0.679528i
$269$	1.46410	0.0892679	0.0446339	−	0.999003i	$-0.485788\pi$
$269$	1.46410	0.0892679	0.0446339	+	0.999003i	$0.485788\pi$
$270$	0	0
$271$	− 5.85641i	− 0.355751i	−0.984053	−	0.177876i	$-0.943078\pi$
$271$	− 5.85641i	− 0.355751i	0.984053	−	0.177876i	$-0.0569225\pi$
$272$	−2.26795	−0.137515
$273$	0	0
$274$	−1.92820	−0.116487
$275$	23.3205i	1.40628i
$276$	0	0
$277$	2.26795	0.136268	0.0681339	−	0.997676i	$-0.478295\pi$
$277$	2.26795	0.136268	0.0681339	+	0.997676i	$0.478295\pi$
$278$	9.85641i	0.591148i
$279$	0	0
$280$	− 0.196152i	− 0.0117223i
$281$	− 22.3205i	− 1.33153i	−0.746162	−	0.665765i	$-0.768105\pi$
$281$	− 22.3205i	− 1.33153i	0.746162	−	0.665765i	$-0.231895\pi$
$282$	0	0
$283$	−8.33975	−0.495746	−0.247873	−	0.968792i	$-0.579732\pi$
$283$	−8.33975	−0.495746	−0.247873	+	0.968792i	$0.579732\pi$
$284$	1.26795i	0.0752389i
$285$	0	0
$286$	0	0
$287$	−8.33975	−0.492280
$288$	0	0
$289$	−11.8564	−0.697436
$290$	0.660254	0.0387715
$291$	0	0
$292$	− 9.73205i	− 0.569525i
$293$	− 14.5167i	− 0.848072i	−0.905645	−	0.424036i	$-0.860613\pi$
$293$	− 14.5167i	− 0.848072i	0.905645	−	0.424036i	$-0.139387\pi$
$294$	0	0
$295$	2.14359	0.124805
$296$	10.4641	0.608214
$297$	0	0
$298$	−2.80385	−0.162423
$299$	0	0
$300$	0	0
$301$	5.60770i	0.323222i
$302$	−3.26795	−0.188049
$303$	0	0
$304$	1.26795i	0.0727219i
$305$	− 0.320508i	− 0.0183522i
$306$	0	0
$307$	− 8.58846i	− 0.490169i	−0.969502	−	0.245085i	$-0.921184\pi$
$307$	− 8.58846i	− 0.490169i	0.969502	−	0.245085i	$-0.0788157\pi$
$308$	−3.46410	−0.197386
$309$	0	0
$310$	1.46410i	0.0831554i
$311$	15.6603	0.888012	0.444006	−	0.896024i	$-0.353557\pi$
$311$	15.6603	0.888012	0.444006	+	0.896024i	$0.353557\pi$
$312$	0	0
$313$	13.4641	0.761036	0.380518	−	0.924774i	$-0.375746\pi$
$313$	13.4641	0.761036	0.380518	+	0.924774i	$0.375746\pi$
$314$	23.5885i	1.33117i
$315$	0	0
$316$	9.46410	0.532397
$317$	− 3.33975i	− 0.187579i	−0.995592	−	0.0937894i	$-0.970102\pi$
$317$	− 3.33975i	− 0.187579i	0.995592	−	0.0937894i	$-0.0298980\pi$
$318$	0	0
$319$	− 11.6603i	− 0.652849i
$320$	0.267949i	0.0149788i
$321$	0	0
$322$	4.53590	0.252776
$323$	− 2.87564i	− 0.160005i
$324$	0	0
$325$	0	0
$326$	6.53590	0.361990
$327$	0	0
$328$	11.3923	0.629035
$329$	−6.00000	−0.330791
$330$	0	0
$331$	20.0000i	1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$	20.0000i	1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$	10.1962i	0.559587i
$333$	0	0
$334$	−2.53590	−0.138758
$335$	−2.98076	−0.162856
$336$	0	0
$337$	−6.85641	−0.373492	−0.186746	−	0.982408i	$-0.559794\pi$
$337$	−6.85641	−0.373492	−0.186746	+	0.982408i	$0.559794\pi$
$338$	0	0
$339$	0	0
$340$	− 0.607695i	− 0.0329569i
$341$	25.8564	1.40020
$342$	0	0
$343$	− 9.85641i	− 0.532196i
$344$	− 7.66025i	− 0.413013i
$345$	0	0
$346$	16.3923i	0.881256i
$347$	−8.87564	−0.476470	−0.238235	−	0.971208i	$-0.576569\pi$
$347$	−8.87564	−0.476470	−0.238235	+	0.971208i	$0.576569\pi$
$348$	0	0
$349$	19.3205i	1.03420i	0.855924	+	0.517102i	$0.172989\pi$
$349$	19.3205i	1.03420i	−0.855924	+	0.517102i	$0.827011\pi$
$350$	−3.60770	−0.192839
$351$	0	0
$352$	4.73205	0.252219
$353$	− 19.7846i	− 1.05303i	−0.850166	−	0.526514i	$-0.823499\pi$
$353$	− 19.7846i	− 1.05303i	0.850166	−	0.526514i	$-0.176501\pi$
$354$	0	0
$355$	−0.339746	−0.0180318
$356$	2.53590i	0.134402i
$357$	0	0
$358$	− 22.0526i	− 1.16551i
$359$	23.1244i	1.22046i	0.792226	+	0.610228i	$0.208922\pi$
$359$	23.1244i	1.22046i	−0.792226	+	0.610228i	$0.791078\pi$
$360$	0	0
$361$	17.3923	0.915384
$362$	8.80385i	0.462720i
$363$	0	0
$364$	0	0
$365$	2.60770	0.136493
$366$	0	0
$367$	−14.7321	−0.769007	−0.384503	−	0.923124i	$-0.625627\pi$
$367$	−14.7321	−0.769007	−0.384503	+	0.923124i	$0.625627\pi$
$368$	−6.19615	−0.322997
$369$	0	0
$370$	2.80385i	0.145765i
$371$	0.339746i	0.0176387i
$372$	0	0
$373$	−10.2679	−0.531654	−0.265827	−	0.964021i	$-0.585645\pi$
$373$	−10.2679	−0.531654	−0.265827	+	0.964021i	$0.585645\pi$
$374$	−10.7321	−0.554941
$375$	0	0
$376$	8.19615	0.422684
$377$	0	0
$378$	0	0
$379$	1.46410i	0.0752058i	0.999293	+	0.0376029i	$0.0119722\pi$
$379$	1.46410i	0.0752058i	−0.999293	+	0.0376029i	$0.988028\pi$
$380$	−0.339746	−0.0174286
$381$	0	0
$382$	6.92820i	0.354478i
$383$	− 5.46410i	− 0.279203i	−0.990208	−	0.139601i	$-0.955418\pi$
$383$	− 5.46410i	− 0.279203i	0.990208	−	0.139601i	$-0.0445821\pi$
$384$	0	0
$385$	− 0.928203i	− 0.0473056i
$386$	8.26795	0.420828
$387$	0	0
$388$	6.00000i	0.304604i
$389$	−29.7846	−1.51014	−0.755070	−	0.655644i	$-0.772397\pi$
$389$	−29.7846	−1.51014	−0.755070	+	0.655644i	$0.772397\pi$
$390$	0	0
$391$	14.0526	0.710668
$392$	6.46410i	0.326486i
$393$	0	0
$394$	−9.85641	−0.496559
$395$	2.53590i	0.127595i
$396$	0	0
$397$	0.392305i	0.0196892i	0.999952	+	0.00984461i	$0.00313369\pi$
$397$	0.392305i	0.0196892i	−0.999952	+	0.00984461i	$0.996866\pi$
$398$	− 3.80385i	− 0.190670i
$399$	0	0
$400$	4.92820	0.246410
$401$	21.9282i	1.09504i	0.836792	+	0.547521i	$0.184428\pi$
$401$	21.9282i	1.09504i	−0.836792	+	0.547521i	$0.815572\pi$
$402$	0	0
$403$	0	0
$404$	11.9282	0.593450
$405$	0	0
$406$	1.80385	0.0895235
$407$	49.5167	2.45445
$408$	0	0
$409$	14.2679i	0.705505i	0.935717	+	0.352752i	$0.114754\pi$
$409$	14.2679i	0.705505i	−0.935717	+	0.352752i	$0.885246\pi$
$410$	3.05256i	0.150755i
$411$	0	0
$412$	−18.7321	−0.922862
$413$	5.85641	0.288175
$414$	0	0
$415$	−2.73205	−0.134111
$416$	0	0
$417$	0	0
$418$	6.00000i	0.293470i
$419$	10.5359	0.514712	0.257356	−	0.966317i	$-0.417149\pi$
$419$	10.5359	0.514712	0.257356	+	0.966317i	$0.417149\pi$
$420$	0	0
$421$	32.7128i	1.59432i	0.603765	+	0.797162i	$0.293667\pi$
$421$	32.7128i	1.59432i	−0.603765	+	0.797162i	$0.706333\pi$
$422$	4.39230i	0.213814i
$423$	0	0
$424$	− 0.464102i	− 0.0225388i
$425$	−11.1769	−0.542160
$426$	0	0
$427$	− 0.875644i	− 0.0423754i
$428$	−0.196152	−0.00948139
$429$	0	0
$430$	2.05256	0.0989832
$431$	11.1244i	0.535841i	0.963441	+	0.267921i	$0.0863365\pi$
$431$	11.1244i	0.535841i	−0.963441	+	0.267921i	$0.913663\pi$
$432$	0	0
$433$	14.8564	0.713953	0.356977	−	0.934113i	$-0.383808\pi$
$433$	14.8564	0.713953	0.356977	+	0.934113i	$0.383808\pi$
$434$	4.00000i	0.192006i
$435$	0	0
$436$	5.46410i	0.261683i
$437$	− 7.85641i	− 0.375823i
$438$	0	0
$439$	17.6603	0.842878	0.421439	−	0.906857i	$-0.361525\pi$
$439$	17.6603	0.842878	0.421439	+	0.906857i	$0.361525\pi$
$440$	1.26795i	0.0604471i
$441$	0	0
$442$	0	0
$443$	−36.3923	−1.72905	−0.864525	−	0.502589i	$-0.832381\pi$
$443$	−36.3923	−1.72905	−0.864525	+	0.502589i	$0.832381\pi$
$444$	0	0
$445$	−0.679492	−0.0322110
$446$	−13.0718	−0.618968
$447$	0	0
$448$	0.732051i	0.0345861i
$449$	− 23.3205i	− 1.10056i	−0.834979	−	0.550281i	$-0.814520\pi$
$449$	− 23.3205i	− 1.10056i	0.834979	−	0.550281i	$-0.185480\pi$
$450$	0	0
$451$	53.9090	2.53847
$452$	−18.6603	−0.877705
$453$	0	0
$454$	1.80385	0.0846588
$455$	0	0
$456$	0	0
$457$	18.6603i	0.872890i	0.899731	+	0.436445i	$0.143763\pi$
$457$	18.6603i	0.872890i	−0.899731	+	0.436445i	$0.856237\pi$
$458$	−15.8564	−0.740921
$459$	0	0
$460$	− 1.66025i	− 0.0774097i
$461$	− 25.7321i	− 1.19846i	−0.800577	−	0.599231i	$-0.795473\pi$
$461$	− 25.7321i	− 1.19846i	0.800577	−	0.599231i	$-0.204527\pi$
$462$	0	0
$463$	28.0526i	1.30371i	0.758342	+	0.651856i	$0.226010\pi$
$463$	28.0526i	1.30371i	−0.758342	+	0.651856i	$0.773990\pi$
$464$	−2.46410	−0.114393
$465$	0	0
$466$	− 19.8564i	− 0.919830i
$467$	−12.5885	−0.582524	−0.291262	−	0.956643i	$-0.594075\pi$
$467$	−12.5885	−0.582524	−0.291262	+	0.956643i	$0.594075\pi$
$468$	0	0
$469$	−8.14359	−0.376036
$470$	2.19615i	0.101301i
$471$	0	0
$472$	−8.00000	−0.368230
$473$	− 36.2487i	− 1.66672i
$474$	0	0
$475$	6.24871i	0.286711i
$476$	− 1.66025i	− 0.0760976i
$477$	0	0
$478$	9.66025	0.441850
$479$	26.5359i	1.21246i	0.795291	+	0.606228i	$0.207318\pi$
$479$	26.5359i	1.21246i	−0.795291	+	0.606228i	$0.792682\pi$
$480$	0	0
$481$	0	0
$482$	17.5885	0.801132
$483$	0	0
$484$	11.3923	0.517832
$485$	−1.60770	−0.0730017
$486$	0	0
$487$	21.1244i	0.957236i	0.878023	+	0.478618i	$0.158862\pi$
$487$	21.1244i	0.957236i	−0.878023	+	0.478618i	$0.841138\pi$
$488$	1.19615i	0.0541473i
$489$	0	0
$490$	−1.73205	−0.0782461
$491$	5.26795	0.237739	0.118870	−	0.992910i	$-0.462073\pi$
$491$	5.26795	0.237739	0.118870	+	0.992910i	$0.462073\pi$
$492$	0	0
$493$	5.58846	0.251691
$494$	0	0
$495$	0	0
$496$	− 5.46410i	− 0.245345i
$497$	−0.928203	−0.0416356
$498$	0	0
$499$	32.0000i	1.43252i	0.697835	+	0.716258i	$0.254147\pi$
$499$	32.0000i	1.43252i	−0.697835	+	0.716258i	$0.745853\pi$
$500$	2.66025i	0.118970i
$501$	0	0
$502$	− 6.53590i	− 0.291711i
$503$	10.9808	0.489608	0.244804	−	0.969573i	$-0.421276\pi$
$503$	10.9808	0.489608	0.244804	+	0.969573i	$0.421276\pi$
$504$	0	0
$505$	3.19615i	0.142227i
$506$	−29.3205	−1.30346
$507$	0	0
$508$	17.8564	0.792250
$509$	− 10.2679i	− 0.455119i	−0.973764	−	0.227559i	$-0.926925\pi$
$509$	− 10.2679i	− 0.455119i	0.973764	−	0.227559i	$-0.0730746\pi$
$510$	0	0
$511$	7.12436	0.315163
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	− 26.6603i	− 1.17593i
$515$	− 5.01924i	− 0.221174i
$516$	0	0
$517$	38.7846	1.70575
$518$	7.66025i	0.336572i
$519$	0	0
$520$	0	0
$521$	17.4449	0.764273	0.382137	−	0.924106i	$-0.375188\pi$
$521$	17.4449	0.764273	0.382137	+	0.924106i	$0.375188\pi$
$522$	0	0
$523$	36.4449	1.59362	0.796811	−	0.604228i	$-0.206518\pi$
$523$	36.4449	1.59362	0.796811	+	0.604228i	$0.206518\pi$
$524$	13.4641	0.588182
$525$	0	0
$526$	28.0526i	1.22315i
$527$	12.3923i	0.539817i
$528$	0	0
$529$	15.3923	0.669231
$530$	0.124356	0.00540166
$531$	0	0
$532$	−0.928203	−0.0402427
$533$	0	0
$534$	0	0
$535$	− 0.0525589i	− 0.00227232i
$536$	11.1244	0.480499
$537$	0	0
$538$	− 1.46410i	− 0.0631219i
$539$	30.5885i	1.31754i
$540$	0	0
$541$	− 40.3205i	− 1.73351i	−0.498731	−	0.866757i	$-0.666200\pi$
$541$	− 40.3205i	− 1.73351i	0.498731	−	0.866757i	$-0.333800\pi$
$542$	−5.85641	−0.251554
$543$	0	0
$544$	2.26795i	0.0972375i
$545$	−1.46410	−0.0627152
$546$	0	0
$547$	6.19615	0.264928	0.132464	−	0.991188i	$-0.457711\pi$
$547$	6.19615	0.264928	0.132464	+	0.991188i	$0.457711\pi$
$548$	1.92820i	0.0823688i
$549$	0	0
$550$	23.3205	0.994390
$551$	− 3.12436i	− 0.133102i
$552$	0	0
$553$	6.92820i	0.294617i
$554$	− 2.26795i	− 0.0963559i
$555$	0	0
$556$	9.85641	0.418005
$557$	− 30.3731i	− 1.28695i	−0.765468	−	0.643474i	$-0.777492\pi$
$557$	− 30.3731i	− 1.28695i	0.765468	−	0.643474i	$-0.222508\pi$
$558$	0	0
$559$	0	0
$560$	−0.196152	−0.00828895
$561$	0	0
$562$	−22.3205	−0.941534
$563$	21.0718	0.888070	0.444035	−	0.896009i	$-0.353547\pi$
$563$	21.0718	0.888070	0.444035	+	0.896009i	$0.353547\pi$
$564$	0	0
$565$	− 5.00000i	− 0.210352i
$566$	8.33975i	0.350546i
$567$	0	0
$568$	1.26795	0.0532020
$569$	−38.6410	−1.61992	−0.809958	−	0.586488i	$-0.800510\pi$
$569$	−38.6410	−1.61992	−0.809958	+	0.586488i	$0.800510\pi$
$570$	0	0
$571$	24.0526	1.00657	0.503284	−	0.864121i	$-0.332125\pi$
$571$	24.0526	1.00657	0.503284	+	0.864121i	$0.332125\pi$
$572$	0	0
$573$	0	0
$574$	8.33975i	0.348094i
$575$	−30.5359	−1.27343
$576$	0	0
$577$	0.267949i	0.0111549i	0.999984	+	0.00557744i	$0.00177536\pi$
$577$	0.267949i	0.0111549i	−0.999984	+	0.00557744i	$0.998225\pi$
$578$	11.8564i	0.493161i
$579$	0	0
$580$	− 0.660254i	− 0.0274156i
$581$	−7.46410	−0.309663
$582$	0	0
$583$	− 2.19615i	− 0.0909553i
$584$	−9.73205	−0.402715
$585$	0	0
$586$	−14.5167	−0.599678
$587$	− 16.0000i	− 0.660391i	−0.943913	−	0.330195i	$-0.892885\pi$
$587$	− 16.0000i	− 0.660391i	0.943913	−	0.330195i	$-0.107115\pi$
$588$	0	0
$589$	6.92820	0.285472
$590$	− 2.14359i	− 0.0882503i
$591$	0	0
$592$	− 10.4641i	− 0.430072i
$593$	− 36.8564i	− 1.51351i	−0.653698	−	0.756756i	$-0.726783\pi$
$593$	− 36.8564i	− 1.51351i	0.653698	−	0.756756i	$-0.273217\pi$
$594$	0	0
$595$	0.444864	0.0182376
$596$	2.80385i	0.114850i
$597$	0	0
$598$	0	0
$599$	9.46410	0.386693	0.193346	−	0.981131i	$-0.438066\pi$
$599$	9.46410	0.386693	0.193346	+	0.981131i	$0.438066\pi$
$600$	0	0
$601$	−5.92820	−0.241816	−0.120908	−	0.992664i	$-0.538581\pi$
$601$	−5.92820	−0.241816	−0.120908	+	0.992664i	$0.538581\pi$
$602$	5.60770	0.228553
$603$	0	0
$604$	3.26795i	0.132971i
$605$	3.05256i	0.124104i
$606$	0	0
$607$	0.784610	0.0318463	0.0159232	−	0.999873i	$-0.494931\pi$
$607$	0.784610	0.0318463	0.0159232	+	0.999873i	$0.494931\pi$
$608$	1.26795	0.0514221
$609$	0	0
$610$	−0.320508	−0.0129770
$611$	0	0
$612$	0	0
$613$	11.3923i	0.460131i	0.973175	+	0.230065i	$0.0738940\pi$
$613$	11.3923i	0.460131i	−0.973175	+	0.230065i	$0.926106\pi$
$614$	−8.58846	−0.346602
$615$	0	0
$616$	3.46410i	0.139573i
$617$	− 35.2487i	− 1.41906i	−0.704675	−	0.709530i	$-0.748907\pi$
$617$	− 35.2487i	− 1.41906i	0.704675	−	0.709530i	$-0.251093\pi$
$618$	0	0
$619$	− 10.5359i	− 0.423474i	−0.977327	−	0.211737i	$-0.932088\pi$
$619$	− 10.5359i	− 0.423474i	0.977327	−	0.211737i	$-0.0679119\pi$
$620$	1.46410	0.0587997
$621$	0	0
$622$	− 15.6603i	− 0.627919i
$623$	−1.85641	−0.0743754
$624$	0	0
$625$	23.9282	0.957128
$626$	− 13.4641i	− 0.538134i
$627$	0	0
$628$	23.5885	0.941282
$629$	23.7321i	0.946259i
$630$	0	0
$631$	− 47.7128i	− 1.89942i	−0.313135	−	0.949709i	$-0.601379\pi$
$631$	− 47.7128i	− 1.89942i	0.313135	−	0.949709i	$-0.398621\pi$
$632$	− 9.46410i	− 0.376462i
$633$	0	0
$634$	−3.33975	−0.132638
$635$	4.78461i	0.189871i
$636$	0	0
$637$	0	0
$638$	−11.6603	−0.461634
$639$	0	0
$640$	0.267949	0.0105916
$641$	25.9808	1.02618	0.513089	−	0.858335i	$-0.328501\pi$
$641$	25.9808	1.02618	0.513089	+	0.858335i	$0.328501\pi$
$642$	0	0
$643$	13.8564i	0.546443i	0.961951	+	0.273222i	$0.0880892\pi$
$643$	13.8564i	0.546443i	−0.961951	+	0.273222i	$0.911911\pi$
$644$	− 4.53590i	− 0.178739i
$645$	0	0
$646$	−2.87564	−0.113141
$647$	26.2487	1.03194	0.515972	−	0.856606i	$-0.327431\pi$
$647$	26.2487	1.03194	0.515972	+	0.856606i	$0.327431\pi$
$648$	0	0
$649$	−37.8564	−1.48599
$650$	0	0
$651$	0	0
$652$	− 6.53590i	− 0.255966i
$653$	10.5359	0.412302	0.206151	−	0.978520i	$-0.433906\pi$
$653$	10.5359	0.412302	0.206151	+	0.978520i	$0.433906\pi$
$654$	0	0
$655$	3.60770i	0.140964i
$656$	− 11.3923i	− 0.444795i
$657$	0	0
$658$	6.00000i	0.233904i
$659$	−38.2487	−1.48996	−0.744979	−	0.667088i	$-0.767541\pi$
$659$	−38.2487	−1.48996	−0.744979	+	0.667088i	$0.767541\pi$
$660$	0	0
$661$	9.39230i	0.365318i	0.983176	+	0.182659i	$0.0584705\pi$
$661$	9.39230i	0.365318i	−0.983176	+	0.182659i	$0.941530\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	10.1962	0.395687
$665$	− 0.248711i	− 0.00964461i
$666$	0	0
$667$	15.2679	0.591177
$668$	2.53590i	0.0981169i
$669$	0	0
$670$	2.98076i	0.115157i
$671$	5.66025i	0.218512i
$672$	0	0
$673$	−14.0718	−0.542428	−0.271214	−	0.962519i	$-0.587425\pi$
$673$	−14.0718	−0.542428	−0.271214	+	0.962519i	$0.587425\pi$
$674$	6.85641i	0.264099i
$675$	0	0
$676$	0	0
$677$	38.5359	1.48105	0.740527	−	0.672026i	$-0.234576\pi$
$677$	38.5359	1.48105	0.740527	+	0.672026i	$0.234576\pi$
$678$	0	0
$679$	−4.39230	−0.168561
$680$	−0.607695	−0.0233040
$681$	0	0
$682$	− 25.8564i	− 0.990093i
$683$	− 37.8564i	− 1.44854i	−0.689519	−	0.724268i	$-0.742178\pi$
$683$	− 37.8564i	− 1.44854i	0.689519	−	0.724268i	$-0.257822\pi$
$684$	0	0
$685$	−0.516660	−0.0197406
$686$	−9.85641	−0.376319
$687$	0	0
$688$	−7.66025	−0.292044
$689$	0	0
$690$	0	0
$691$	26.3397i	1.00201i	0.865444	+	0.501006i	$0.167036\pi$
$691$	26.3397i	1.00201i	−0.865444	+	0.501006i	$0.832964\pi$
$692$	16.3923	0.623142
$693$	0	0
$694$	8.87564i	0.336915i
$695$	2.64102i	0.100179i
$696$	0	0
$697$	25.8372i	0.978653i
$698$	19.3205	0.731292
$699$	0	0
$700$	3.60770i	0.136358i
$701$	31.3205	1.18296	0.591480	−	0.806320i	$-0.298544\pi$
$701$	31.3205	1.18296	0.591480	+	0.806320i	$0.298544\pi$
$702$	0	0
$703$	13.2679	0.500410
$704$	− 4.73205i	− 0.178346i
$705$	0	0
$706$	−19.7846	−0.744604
$707$	8.73205i	0.328403i
$708$	0	0
$709$	− 40.8564i	− 1.53439i	−0.641411	−	0.767197i	$-0.721651\pi$
$709$	− 40.8564i	− 1.53439i	0.641411	−	0.767197i	$-0.278349\pi$
$710$	0.339746i	0.0127504i
$711$	0	0
$712$	2.53590	0.0950368
$713$	33.8564i	1.26793i
$714$	0	0
$715$	0	0
$716$	−22.0526	−0.824143
$717$	0	0
$718$	23.1244	0.862993
$719$	−22.5359	−0.840447	−0.420224	−	0.907421i	$-0.638048\pi$
$719$	−22.5359	−0.840447	−0.420224	+	0.907421i	$0.638048\pi$
$720$	0	0
$721$	− 13.7128i	− 0.510692i
$722$	− 17.3923i	− 0.647275i
$723$	0	0
$724$	8.80385	0.327192
$725$	−12.1436	−0.451002
$726$	0	0
$727$	−20.9808	−0.778133	−0.389067	−	0.921210i	$-0.627202\pi$
$727$	−20.9808	−0.778133	−0.389067	+	0.921210i	$0.627202\pi$
$728$	0	0
$729$	0	0
$730$	− 2.60770i	− 0.0965151i
$731$	17.3731	0.642566
$732$	0	0
$733$	19.0000i	0.701781i	0.936416	+	0.350891i	$0.114121\pi$
$733$	19.0000i	0.701781i	−0.936416	+	0.350891i	$0.885879\pi$
$734$	14.7321i	0.543770i
$735$	0	0
$736$	6.19615i	0.228393i
$737$	52.6410	1.93906
$738$	0	0
$739$	10.9282i	0.402000i	0.979591	+	0.201000i	$0.0644192\pi$
$739$	10.9282i	0.402000i	−0.979591	+	0.201000i	$0.935581\pi$
$740$	2.80385	0.103071
$741$	0	0
$742$	0.339746	0.0124725
$743$	27.6077i	1.01283i	0.862290	+	0.506414i	$0.169029\pi$
$743$	27.6077i	1.01283i	−0.862290	+	0.506414i	$0.830971\pi$
$744$	0	0
$745$	−0.751289	−0.0275251
$746$	10.2679i	0.375936i
$747$	0	0
$748$	10.7321i	0.392403i
$749$	− 0.143594i	− 0.00524679i
$750$	0	0
$751$	−15.9090	−0.580526	−0.290263	−	0.956947i	$-0.593743\pi$
$751$	−15.9090	−0.580526	−0.290263	+	0.956947i	$0.593743\pi$
$752$	− 8.19615i	− 0.298883i
$753$	0	0
$754$	0	0
$755$	−0.875644	−0.0318680
$756$	0	0
$757$	7.07180	0.257029	0.128514	−	0.991708i	$-0.458979\pi$
$757$	7.07180	0.257029	0.128514	+	0.991708i	$0.458979\pi$
$758$	1.46410	0.0531786
$759$	0	0
$760$	0.339746i	0.0123239i
$761$	− 23.3205i	− 0.845368i	−0.906277	−	0.422684i	$-0.861088\pi$
$761$	− 23.3205i	− 0.845368i	0.906277	−	0.422684i	$-0.138912\pi$
$762$	0	0
$763$	−4.00000	−0.144810
$764$	6.92820	0.250654
$765$	0	0
$766$	−5.46410	−0.197426
$767$	0	0
$768$	0	0
$769$	− 16.1436i	− 0.582153i	−0.956700	−	0.291076i	$-0.905987\pi$
$769$	− 16.1436i	− 0.582153i	0.956700	−	0.291076i	$-0.0940134\pi$
$770$	−0.928203	−0.0334501
$771$	0	0
$772$	− 8.26795i	− 0.297570i
$773$	35.0718i	1.26144i	0.776009	+	0.630722i	$0.217241\pi$
$773$	35.0718i	1.26144i	−0.776009	+	0.630722i	$0.782759\pi$
$774$	0	0
$775$	− 26.9282i	− 0.967290i
$776$	6.00000	0.215387
$777$	0	0
$778$	29.7846i	1.06783i
$779$	14.4449	0.517541
$780$	0	0
$781$	6.00000	0.214697
$782$	− 14.0526i	− 0.502518i
$783$	0	0
$784$	6.46410	0.230861
$785$	6.32051i	0.225589i
$786$	0	0
$787$	− 39.3205i	− 1.40162i	−0.713346	−	0.700812i	$-0.752821\pi$
$787$	− 39.3205i	− 1.40162i	0.713346	−	0.700812i	$-0.247179\pi$
$788$	9.85641i	0.351120i
$789$	0	0
$790$	2.53590	0.0902232
$791$	− 13.6603i	− 0.485703i
$792$	0	0
$793$	0	0
$794$	0.392305	0.0139224
$795$	0	0
$796$	−3.80385	−0.134824
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	18.5885i	0.657612i
$800$	− 4.92820i	− 0.174238i
$801$	0	0
$802$	21.9282	0.774312
$803$	−46.0526	−1.62516
$804$	0	0
$805$	1.21539	0.0428369
$806$	0	0
$807$	0	0
$808$	− 11.9282i	− 0.419633i
$809$	22.4115	0.787948	0.393974	−	0.919122i	$-0.371100\pi$
$809$	22.4115	0.787948	0.393974	+	0.919122i	$0.371100\pi$
$810$	0	0
$811$	− 45.1769i	− 1.58638i	−0.608977	−	0.793188i	$-0.708420\pi$
$811$	− 45.1769i	− 1.58638i	0.608977	−	0.793188i	$-0.291580\pi$
$812$	− 1.80385i	− 0.0633026i
$813$	0	0
$814$	− 49.5167i	− 1.73556i
$815$	1.75129	0.0613450
$816$	0	0
$817$	− 9.71281i	− 0.339808i
$818$	14.2679	0.498867
$819$	0	0
$820$	3.05256	0.106600
$821$	− 12.9282i	− 0.451197i	−0.974220	−	0.225599i	$-0.927566\pi$
$821$	− 12.9282i	− 0.451197i	0.974220	−	0.225599i	$-0.0724338\pi$
$822$	0	0
$823$	41.5692	1.44901	0.724506	−	0.689269i	$-0.242068\pi$
$823$	41.5692	1.44901	0.724506	+	0.689269i	$0.242068\pi$
$824$	18.7321i	0.652562i
$825$	0	0
$826$	− 5.85641i	− 0.203770i
$827$	33.4641i	1.16366i	0.813310	+	0.581830i	$0.197663\pi$
$827$	33.4641i	1.16366i	−0.813310	+	0.581830i	$0.802337\pi$
$828$	0	0
$829$	−12.1244	−0.421096	−0.210548	−	0.977583i	$-0.567525\pi$
$829$	−12.1244	−0.421096	−0.210548	+	0.977583i	$0.567525\pi$
$830$	2.73205i	0.0948309i
$831$	0	0
$832$	0	0
$833$	−14.6603	−0.507948
$834$	0	0
$835$	−0.679492	−0.0235148
$836$	6.00000	0.207514
$837$	0	0
$838$	− 10.5359i	− 0.363957i
$839$	14.1436i	0.488291i	0.969739	+	0.244146i	$0.0785075\pi$
$839$	14.1436i	0.488291i	−0.969739	+	0.244146i	$0.921493\pi$
$840$	0	0
$841$	−22.9282	−0.790628
$842$	32.7128	1.12736
$843$	0	0
$844$	4.39230	0.151189
$845$	0	0
$846$	0	0
$847$	8.33975i	0.286557i
$848$	−0.464102	−0.0159373
$849$	0	0
$850$	11.1769i	0.383365i
$851$	64.8372i	2.22259i
$852$	0	0
$853$	− 8.17691i	− 0.279972i	−0.990153	−	0.139986i	$-0.955294\pi$
$853$	− 8.17691i	− 0.279972i	0.990153	−	0.139986i	$-0.0447058\pi$
$854$	−0.875644	−0.0299639
$855$	0	0
$856$	0.196152i	0.00670435i
$857$	−19.4449	−0.664224	−0.332112	−	0.943240i	$-0.607761\pi$
$857$	−19.4449	−0.664224	−0.332112	+	0.943240i	$0.607761\pi$
$858$	0	0
$859$	−22.8756	−0.780507	−0.390253	−	0.920707i	$-0.627613\pi$
$859$	−22.8756	−0.780507	−0.390253	+	0.920707i	$0.627613\pi$
$860$	− 2.05256i	− 0.0699917i
$861$	0	0
$862$	11.1244	0.378897
$863$	− 7.12436i	− 0.242516i	−0.992621	−	0.121258i	$-0.961307\pi$
$863$	− 7.12436i	− 0.242516i	0.992621	−	0.121258i	$-0.0386928\pi$
$864$	0	0
$865$	4.39230i	0.149343i
$866$	− 14.8564i	− 0.504841i
$867$	0	0
$868$	4.00000	0.135769
$869$	− 44.7846i	− 1.51921i
$870$	0	0
$871$	0	0
$872$	5.46410	0.185038
$873$	0	0
$874$	−7.85641	−0.265747
$875$	−1.94744	−0.0658355
$876$	0	0
$877$	10.0718i	0.340100i	0.985435	+	0.170050i	$0.0543930\pi$
$877$	10.0718i	0.340100i	−0.985435	+	0.170050i	$0.945607\pi$
$878$	− 17.6603i	− 0.596005i
$879$	0	0
$880$	1.26795	0.0427426
$881$	51.8372	1.74644	0.873219	−	0.487327i	$-0.162028\pi$
$881$	51.8372	1.74644	0.873219	+	0.487327i	$0.162028\pi$
$882$	0	0
$883$	−29.0718	−0.978344	−0.489172	−	0.872187i	$-0.662701\pi$
$883$	−29.0718	−0.978344	−0.489172	+	0.872187i	$0.662701\pi$
$884$	0	0
$885$	0	0
$886$	36.3923i	1.22262i
$887$	−10.1436	−0.340589	−0.170294	−	0.985393i	$-0.554472\pi$
$887$	−10.1436	−0.340589	−0.170294	+	0.985393i	$0.554472\pi$
$888$	0	0
$889$	13.0718i	0.438414i
$890$	0.679492i	0.0227766i
$891$	0	0
$892$	13.0718i	0.437676i
$893$	10.3923	0.347765
$894$	0	0
$895$	− 5.90897i	− 0.197515i
$896$	0.732051	0.0244561
$897$	0	0
$898$	−23.3205	−0.778215
$899$	13.4641i	0.449053i
$900$	0	0
$901$	1.05256	0.0350658
$902$	− 53.9090i	− 1.79497i
$903$	0	0
$904$	18.6603i	0.620631i
$905$	2.35898i	0.0784153i
$906$	0	0
$907$	−15.6077	−0.518245	−0.259123	−	0.965844i	$-0.583433\pi$
$907$	−15.6077	−0.518245	−0.259123	+	0.965844i	$0.583433\pi$
$908$	− 1.80385i	− 0.0598628i
$909$	0	0
$910$	0	0
$911$	9.46410	0.313560	0.156780	−	0.987634i	$-0.449889\pi$
$911$	9.46410	0.313560	0.156780	+	0.987634i	$0.449889\pi$
$912$	0	0
$913$	48.2487	1.59680
$914$	18.6603	0.617226
$915$	0	0
$916$	15.8564i	0.523910i
$917$	9.85641i	0.325487i
$918$	0	0
$919$	−57.9615	−1.91197	−0.955987	−	0.293409i	$-0.905210\pi$
$919$	−57.9615	−1.91197	−0.955987	+	0.293409i	$0.905210\pi$
$920$	−1.66025	−0.0547370
$921$	0	0
$922$	−25.7321	−0.847440
$923$	0	0
$924$	0	0
$925$	− 51.5692i	− 1.69559i
$926$	28.0526	0.921864
$927$	0	0
$928$	2.46410i	0.0808881i
$929$	− 9.24871i	− 0.303440i	−0.988423	−	0.151720i	$-0.951519\pi$
$929$	− 9.24871i	− 0.303440i	0.988423	−	0.151720i	$-0.0484813\pi$
$930$	0	0
$931$	8.19615i	0.268618i
$932$	−19.8564	−0.650418
$933$	0	0
$934$	12.5885i	0.411907i
$935$	−2.87564	−0.0940436
$936$	0	0
$937$	43.2487	1.41287	0.706437	−	0.707776i	$-0.250301\pi$
$937$	43.2487	1.41287	0.706437	+	0.707776i	$0.250301\pi$
$938$	8.14359i	0.265898i
$939$	0	0
$940$	2.19615	0.0716306
$941$	56.6410i	1.84644i	0.384267	+	0.923222i	$0.374454\pi$
$941$	56.6410i	1.84644i	−0.384267	+	0.923222i	$0.625546\pi$
$942$	0	0
$943$	70.5885i	2.29868i
$944$	8.00000i	0.260378i
$945$	0	0
$946$	−36.2487	−1.17855
$947$	34.9282i	1.13501i	0.823369	+	0.567507i	$0.192092\pi$
$947$	34.9282i	1.13501i	−0.823369	+	0.567507i	$0.807908\pi$
$948$	0	0
$949$	0	0
$950$	6.24871	0.202735
$951$	0	0
$952$	−1.66025	−0.0538091
$953$	−41.5692	−1.34656	−0.673280	−	0.739388i	$-0.735115\pi$
$953$	−41.5692	−1.34656	−0.673280	+	0.739388i	$0.735115\pi$
$954$	0	0
$955$	1.85641i	0.0600719i
$956$	− 9.66025i	− 0.312435i
$957$	0	0
$958$	26.5359	0.857336
$959$	−1.41154	−0.0455811
$960$	0	0
$961$	1.14359	0.0368901
$962$	0	0
$963$	0	0
$964$	− 17.5885i	− 0.566486i
$965$	2.21539	0.0713159
$966$	0	0
$967$	− 18.8756i	− 0.607000i	−0.952831	−	0.303500i	$-0.901845\pi$
$967$	− 18.8756i	− 0.607000i	0.952831	−	0.303500i	$-0.0981552\pi$
$968$	− 11.3923i	− 0.366163i
$969$	0	0
$970$	1.60770i	0.0516200i
$971$	18.2487	0.585629	0.292815	−	0.956169i	$-0.405408\pi$
$971$	18.2487	0.585629	0.292815	+	0.956169i	$0.405408\pi$
$972$	0	0
$973$	7.21539i	0.231315i
$974$	21.1244	0.676868
$975$	0	0
$976$	1.19615	0.0382879
$977$	− 32.0718i	− 1.02607i	−0.858368	−	0.513034i	$-0.828522\pi$
$977$	− 32.0718i	− 1.02607i	0.858368	−	0.513034i	$-0.171478\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	1.73205i	0.0553283i
$981$	0	0
$982$	− 5.26795i	− 0.168107i
$983$	− 20.7846i	− 0.662926i	−0.943468	−	0.331463i	$-0.892458\pi$
$983$	− 20.7846i	− 0.662926i	0.943468	−	0.331463i	$-0.107542\pi$
$984$	0	0
$985$	−2.64102	−0.0841498
$986$	− 5.58846i	− 0.177973i
$987$	0	0
$988$	0	0
$989$	47.4641	1.50927
$990$	0	0
$991$	−8.58846	−0.272821	−0.136411	−	0.990652i	$-0.543557\pi$
$991$	−8.58846	−0.272821	−0.136411	+	0.990652i	$0.543557\pi$
$992$	−5.46410	−0.173485
$993$	0	0
$994$	0.928203i	0.0294408i
$995$	− 1.01924i	− 0.0323120i
$996$	0	0
$997$	38.6603	1.22438	0.612191	−	0.790710i	$-0.290288\pi$
$997$	38.6603	1.22438	0.612191	+	0.790710i	$0.290288\pi$
$998$	32.0000	1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.b.i.1351.2		4
3.2	odd	2		1014.2.b.e.337.3		4
13.5	odd	4		3042.2.a.p.1.2		2
13.8	odd	4		3042.2.a.y.1.1		2
13.9	even	3		234.2.l.c.127.1		4
13.10	even	6		234.2.l.c.199.1		4
13.12	even	2	inner	3042.2.b.i.1351.3		4
39.2	even	12		1014.2.e.g.529.1		4
39.5	even	4		1014.2.a.k.1.1		2
39.8	even	4		1014.2.a.i.1.2		2
39.11	even	12		1014.2.e.i.529.2		4
39.17	odd	6		1014.2.i.a.361.1		4
39.20	even	12		1014.2.e.i.991.2		4
39.23	odd	6		78.2.i.a.43.2	✓	4
39.29	odd	6		1014.2.i.a.823.1		4
39.32	even	12		1014.2.e.g.991.1		4
39.35	odd	6		78.2.i.a.49.2	yes	4
39.38	odd	2		1014.2.b.e.337.2		4
52.23	odd	6		1872.2.by.h.433.2		4
52.35	odd	6		1872.2.by.h.1297.1		4
156.23	even	6		624.2.bv.e.433.1		4
156.35	even	6		624.2.bv.e.49.2		4
156.47	odd	4		8112.2.a.bj.1.2		2
156.83	odd	4		8112.2.a.bp.1.1		2
195.23	even	12		1950.2.y.g.199.1		4
195.62	even	12		1950.2.y.b.199.2		4
195.74	odd	6		1950.2.bc.d.751.1		4
195.113	even	12		1950.2.y.b.49.2		4
195.152	even	12		1950.2.y.g.49.1		4
195.179	odd	6		1950.2.bc.d.901.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.i.a.43.2	✓	4	39.23	odd	6
78.2.i.a.49.2	yes	4	39.35	odd	6
234.2.l.c.127.1		4	13.9	even	3
234.2.l.c.199.1		4	13.10	even	6
624.2.bv.e.49.2		4	156.35	even	6
624.2.bv.e.433.1		4	156.23	even	6
1014.2.a.i.1.2		2	39.8	even	4
1014.2.a.k.1.1		2	39.5	even	4
1014.2.b.e.337.2		4	39.38	odd	2
1014.2.b.e.337.3		4	3.2	odd	2
1014.2.e.g.529.1		4	39.2	even	12
1014.2.e.g.991.1		4	39.32	even	12
1014.2.e.i.529.2		4	39.11	even	12
1014.2.e.i.991.2		4	39.20	even	12
1014.2.i.a.361.1		4	39.17	odd	6
1014.2.i.a.823.1		4	39.29	odd	6
1872.2.by.h.433.2		4	52.23	odd	6
1872.2.by.h.1297.1		4	52.35	odd	6
1950.2.y.b.49.2		4	195.113	even	12
1950.2.y.b.199.2		4	195.62	even	12
1950.2.y.g.49.1		4	195.152	even	12
1950.2.y.g.199.1		4	195.23	even	12
1950.2.bc.d.751.1		4	195.74	odd	6
1950.2.bc.d.901.1		4	195.179	odd	6
3042.2.a.p.1.2		2	13.5	odd	4
3042.2.a.y.1.1		2	13.8	odd	4
3042.2.b.i.1351.2		4	1.1	even	1	trivial
3042.2.b.i.1351.3		4	13.12	even	2	inner
8112.2.a.bj.1.2		2	156.47	odd	4
8112.2.a.bp.1.1		2	156.83	odd	4

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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -0.267949i q^{5} -0.732051i q^{7} +1.00000i q^{8} -0.267949 q^{10} +4.73205i q^{11} -0.732051 q^{14} +1.00000 q^{16} -2.26795 q^{17} +1.26795i q^{19} +0.267949i q^{20} +4.73205 q^{22} -6.19615 q^{23} +4.92820 q^{25} +0.732051i q^{28} -2.46410 q^{29} -5.46410i q^{31} -1.00000i q^{32} +2.26795i q^{34} -0.196152 q^{35} -10.4641i q^{37} +1.26795 q^{38} +0.267949 q^{40} -11.3923i q^{41} -7.66025 q^{43} -4.73205i q^{44} +6.19615i q^{46} -8.19615i q^{47} +6.46410 q^{49} -4.92820i q^{50} -0.464102 q^{53} +1.26795 q^{55} +0.732051 q^{56} +2.46410i q^{58} +8.00000i q^{59} +1.19615 q^{61} -5.46410 q^{62} -1.00000 q^{64} -11.1244i q^{67} +2.26795 q^{68} +0.196152i q^{70} -1.26795i q^{71} +9.73205i q^{73} -10.4641 q^{74} -1.26795i q^{76} +3.46410 q^{77} -9.46410 q^{79} -0.267949i q^{80} -11.3923 q^{82} -10.1962i q^{83} +0.607695i q^{85} +7.66025i q^{86} -4.73205 q^{88} -2.53590i q^{89} +6.19615 q^{92} -8.19615 q^{94} +0.339746 q^{95} -6.00000i q^{97} -6.46410i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 8 q^{10} + 4 q^{14} + 4 q^{16} - 16 q^{17} + 12 q^{22} - 4 q^{23} - 8 q^{25} + 4 q^{29} + 20 q^{35} + 12 q^{38} + 8 q^{40} + 4 q^{43} + 12 q^{49} + 12 q^{53} + 12 q^{55} - 4 q^{56} - 16 q^{61}+ \cdots + 36 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 - 8 * q^10 + 4 * q^14 + 4 * q^16 - 16 * q^17 + 12 * q^22 - 4 * q^23 - 8 * q^25 + 4 * q^29 + 20 * q^35 + 12 * q^38 + 8 * q^40 + 4 * q^43 + 12 * q^49 + 12 * q^53 + 12 * q^55 - 4 * q^56 - 16 * q^61 - 8 * q^62 - 4 * q^64 + 16 * q^68 - 28 * q^74 - 24 * q^79 - 4 * q^82 - 12 * q^88 + 4 * q^92 - 12 * q^94 + 36 * q^95

Introduction

L-functions

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