LMFDB - Embedded newform 6300.2.v.e.5993.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6300,2,Mod(1457,6300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6300.1457"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6300, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6300.v (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,-8,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$50.3057532734$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 6x^{10} - 24x^{9} + 18x^{8} + 40x^{7} - 82x^{6} + 12x^{5} + 228x^{4} - 284x^{3} + 124x^{2} - 16x + 2 $ x^12 + 6x^10 - 24x^9 + 18x^8 + 40x^7 - 82x^6 + 12x^5 + 228x^4 - 284x^3 + 124x^2 - 16x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 1260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5993.6
Root			$-1.32833 + 3.20687i$ of defining polynomial
Character	$\chi$	$=$	6300.5993
Dual form			6300.2.v.e.1457.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{7} +1.75708i q^{11} +(-1.25153 - 1.25153i) q^{13} +(-1.50555 - 1.50555i) q^{17} +2.37390i q^{19} +(4.16220 - 4.16220i) q^{23} -0.629673 q^{29} +4.08528 q^{31} +(-3.44100 + 3.44100i) q^{37} -1.20261i q^{41} +(-3.05781 - 3.05781i) q^{43} +(-3.51235 - 3.51235i) q^{47} -1.00000i q^{49} +(6.36629 - 6.36629i) q^{53} -10.8093 q^{59} +15.1297 q^{61} +(-4.87916 + 4.87916i) q^{67} -4.26110i q^{71} +(4.06178 + 4.06178i) q^{73} +(1.24244 + 1.24244i) q^{77} +1.49197i q^{79} +(3.25402 - 3.25402i) q^{83} +7.28131 q^{89} -1.76993 q^{91} +(3.07287 - 3.07287i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{13} - 8 q^{17} + 8 q^{23} - 32 q^{29} + 16 q^{31} - 4 q^{37} - 8 q^{43} - 8 q^{47} + 8 q^{53} - 16 q^{59} + 12 q^{73} + 56 q^{83} - 72 q^{89} - 20 q^{97}+O(q^{100}) $ 12 * q + 12 * q^13 - 8 * q^17 + 8 * q^23 - 32 * q^29 + 16 * q^31 - 4 * q^37 - 8 * q^43 - 8 * q^47 + 8 * q^53 - 16 * q^59 + 12 * q^73 + 56 * q^83 - 72 * q^89 - 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times$.

$n$	$2801$	$3151$	$3277$	$3601$
$\chi(n)$	$-1$	$1$	$e\left(\frac{3}{4}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.707107	−	0.707107i	0.267261	−	0.267261i
$7$	0.707107	−	0.707107i	0.267261	−	0.267261i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			1.75708i			0.529779i	0.964279	+	0.264889i	$0.0853354\pi$
$11$			1.75708i			0.529779i	−0.964279	+	0.264889i	$0.914665\pi$
$12$		0			0
$13$	−1.25153	−	1.25153i	−0.347112	−	0.347112i	0.511921	−	0.859033i	$-0.328934\pi$
$13$	−1.25153	−	1.25153i	−0.347112	−	0.347112i	−0.859033	+	0.511921i	$0.828934\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.50555	−	1.50555i	−0.365149	−	0.365149i	0.500556	−	0.865704i	$-0.333129\pi$
$17$	−1.50555	−	1.50555i	−0.365149	−	0.365149i	−0.865704	+	0.500556i	$0.833129\pi$
$18$		0			0
$19$			2.37390i			0.544609i	0.962211	+	0.272305i	$0.0877858\pi$
$19$			2.37390i			0.544609i	−0.962211	+	0.272305i	$0.912214\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.16220	−	4.16220i	0.867879	−	0.867879i	−0.124358	−	0.992237i	$-0.539687\pi$
$23$	4.16220	−	4.16220i	0.867879	−	0.867879i	0.992237	+	0.124358i	$0.0396872\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.629673			−0.116927			−0.0584637	−	0.998290i	$-0.518620\pi$
$29$	−0.629673			−0.116927			−0.0584637	+	0.998290i	$0.518620\pi$
$30$		0			0
$31$	4.08528			0.733738			0.366869	−	0.930273i	$-0.380430\pi$
$31$	4.08528			0.733738			0.366869	+	0.930273i	$0.380430\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.44100	+	3.44100i	−0.565696	+	0.565696i	−0.930920	−	0.365224i	$-0.880993\pi$
$37$	−3.44100	+	3.44100i	−0.565696	+	0.565696i	0.365224	+	0.930920i	$0.380993\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1.20261i		−	0.187815i	−0.995581	−	0.0939077i	$-0.970064\pi$
$41$		−	1.20261i		−	0.187815i	0.995581	−	0.0939077i	$-0.0299359\pi$
$42$		0			0
$43$	−3.05781	−	3.05781i	−0.466312	−	0.466312i	0.434405	−	0.900718i	$-0.356959\pi$
$43$	−3.05781	−	3.05781i	−0.466312	−	0.466312i	−0.900718	+	0.434405i	$0.856959\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.51235	−	3.51235i	−0.512328	−	0.512328i	0.402911	−	0.915239i	$-0.367998\pi$
$47$	−3.51235	−	3.51235i	−0.512328	−	0.512328i	−0.915239	+	0.402911i	$0.867998\pi$
$48$		0			0
$49$		−	1.00000i		−	0.142857i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.36629	−	6.36629i	0.874477	−	0.874477i	−0.118480	−	0.992956i	$-0.537802\pi$
$53$	6.36629	−	6.36629i	0.874477	−	0.874477i	0.992956	+	0.118480i	$0.0378021\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.8093			−1.40725			−0.703625	−	0.710572i	$-0.748436\pi$
$59$	−10.8093			−1.40725			−0.703625	+	0.710572i	$0.748436\pi$
$60$		0			0
$61$	15.1297			1.93716			0.968579	−	0.248708i	$-0.0800059\pi$
$61$	15.1297			1.93716			0.968579	+	0.248708i	$0.0800059\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.87916	+	4.87916i	−0.596084	+	0.596084i	−0.939268	−	0.343184i	$-0.888494\pi$
$67$	−4.87916	+	4.87916i	−0.596084	+	0.596084i	0.343184	+	0.939268i	$0.388494\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.26110i		−	0.505700i	−0.967506	−	0.252850i	$-0.918632\pi$
$71$		−	4.26110i		−	0.505700i	0.967506	−	0.252850i	$-0.0813679\pi$
$72$		0			0
$73$	4.06178	+	4.06178i	0.475395	+	0.475395i	0.903655	−	0.428260i	$-0.140873\pi$
$73$	4.06178	+	4.06178i	0.475395	+	0.475395i	−0.428260	+	0.903655i	$0.640873\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.24244	+	1.24244i	0.141589	+	0.141589i
$78$		0			0
$79$			1.49197i			0.167859i	0.996472	+	0.0839297i	$0.0267471\pi$
$79$			1.49197i			0.167859i	−0.996472	+	0.0839297i	$0.973253\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.25402	−	3.25402i	0.357175	−	0.357175i	−0.505596	−	0.862770i	$-0.668727\pi$
$83$	3.25402	−	3.25402i	0.357175	−	0.357175i	0.862770	+	0.505596i	$0.168727\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.28131			0.771818			0.385909	−	0.922537i	$-0.373888\pi$
$89$	7.28131			0.771818			0.385909	+	0.922537i	$0.373888\pi$
$90$		0			0
$91$	−1.76993			−0.185539
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	3.07287	−	3.07287i	0.312003	−	0.312003i	−0.533682	−	0.845685i	$-0.679192\pi$
$97$	3.07287	−	3.07287i	0.312003	−	0.312003i	0.845685	+	0.533682i	$0.179192\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	2.10374i		−	0.209330i	−0.994508	−	0.104665i	$-0.966623\pi$
$101$		−	2.10374i		−	0.209330i	0.994508	−	0.104665i	$-0.0333770\pi$
$102$		0			0
$103$	−4.84060	−	4.84060i	−0.476959	−	0.476959i	0.427199	−	0.904158i	$-0.359500\pi$
$103$	−4.84060	−	4.84060i	−0.476959	−	0.476959i	−0.904158	+	0.427199i	$0.859500\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.18038	−	6.18038i	−0.597480	−	0.597480i	0.342161	−	0.939641i	$-0.388841\pi$
$107$	−6.18038	−	6.18038i	−0.597480	−	0.597480i	−0.939641	+	0.342161i	$0.888841\pi$
$108$		0			0
$109$		−	14.3345i		−	1.37299i	−0.727133	−	0.686497i	$-0.759147\pi$
$109$		−	14.3345i		−	1.37299i	0.727133	−	0.686497i	$-0.240853\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.952072	+	0.952072i	−0.0895634	+	0.0895634i	−0.750469	−	0.660906i	$-0.770172\pi$
$113$	−0.952072	+	0.952072i	−0.0895634	+	0.0895634i	0.660906	+	0.750469i	$0.270172\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.12916			−0.195180
$120$		0			0
$121$	7.91268			0.719335
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	10.8275	−	10.8275i	0.960782	−	0.960782i	−0.0384774	−	0.999259i	$-0.512251\pi$
$127$	10.8275	−	10.8275i	0.960782	−	0.960782i	0.999259	+	0.0384774i	$0.0122508\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	6.41314i		−	0.560319i	−0.959954	−	0.280159i	$-0.909613\pi$
$131$		−	6.41314i		−	0.560319i	0.959954	−	0.280159i	$-0.0903873\pi$
$132$		0			0
$133$	1.67860	+	1.67860i	0.145553	+	0.145553i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−14.0788	−	14.0788i	−1.20283	−	1.20283i	−0.973301	−	0.229531i	$-0.926281\pi$
$137$	−14.0788	−	14.0788i	−1.20283	−	1.20283i	−0.229531	−	0.973301i	$-0.573719\pi$
$138$		0			0
$139$			4.70894i			0.399407i	0.979856	+	0.199704i	$0.0639979\pi$
$139$			4.70894i			0.399407i	−0.979856	+	0.199704i	$0.936002\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.19904	−	2.19904i	0.183893	−	0.183893i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−15.7568			−1.29085			−0.645423	−	0.763825i	$-0.723319\pi$
$149$	−15.7568			−1.29085			−0.645423	+	0.763825i	$0.723319\pi$
$150$		0			0
$151$	22.1891			1.80572			0.902860	−	0.429934i	$-0.141463\pi$
$151$	22.1891			1.80572			0.902860	+	0.429934i	$0.141463\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	3.54558	−	3.54558i	0.282968	−	0.282968i	−0.551323	−	0.834292i	$-0.685877\pi$
$157$	3.54558	−	3.54558i	0.282968	−	0.282968i	0.834292	+	0.551323i	$0.185877\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	5.88624i		−	0.463901i
$162$		0			0
$163$	6.34258	+	6.34258i	0.496789	+	0.496789i	0.910437	−	0.413648i	$-0.135745\pi$
$163$	6.34258	+	6.34258i	0.496789	+	0.496789i	−0.413648	+	0.910437i	$0.635745\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	16.9692	+	16.9692i	1.31312	+	1.31312i	0.919106	+	0.394010i	$0.128912\pi$
$167$	16.9692	+	16.9692i	1.31312	+	1.31312i	0.394010	+	0.919106i	$0.371088\pi$
$168$		0			0
$169$		−	9.86734i		−	0.759026i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.49802	−	1.49802i	0.113893	−	0.113893i	−0.647864	−	0.761756i	$-0.724337\pi$
$173$	1.49802	−	1.49802i	0.113893	−	0.113893i	0.761756	+	0.647864i	$0.224337\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.93516			0.294128			0.147064	−	0.989127i	$-0.453018\pi$
$179$	3.93516			0.294128			0.147064	+	0.989127i	$0.453018\pi$
$180$		0			0
$181$	−13.0358			−0.968946			−0.484473	−	0.874806i	$-0.660989\pi$
$181$	−13.0358			−0.968946			−0.484473	+	0.874806i	$0.660989\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	2.64536	−	2.64536i	0.193448	−	0.193448i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	15.0935i		−	1.09213i	−0.837743	−	0.546065i	$-0.816125\pi$
$191$		−	15.0935i		−	1.09213i	0.837743	−	0.546065i	$-0.183875\pi$
$192$		0			0
$193$	7.37140	+	7.37140i	0.530605	+	0.530605i	0.920752	−	0.390147i	$-0.127576\pi$
$193$	7.37140	+	7.37140i	0.530605	+	0.530605i	−0.390147	+	0.920752i	$0.627576\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.77438	−	2.77438i	−0.197666	−	0.197666i	0.601333	−	0.798999i	$-0.294637\pi$
$197$	−2.77438	−	2.77438i	−0.197666	−	0.197666i	−0.798999	+	0.601333i	$0.794637\pi$
$198$		0			0
$199$			8.53028i			0.604696i	0.953198	+	0.302348i	$0.0977704\pi$
$199$			8.53028i			0.604696i	−0.953198	+	0.302348i	$0.902230\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.445246	+	0.445246i	−0.0312502	+	0.0312502i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.17112			−0.288522
$210$		0			0
$211$	−7.85634			−0.540853			−0.270426	−	0.962741i	$-0.587165\pi$
$211$	−7.85634			−0.540853			−0.270426	+	0.962741i	$0.587165\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	2.88873	−	2.88873i	0.196100	−	0.196100i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			3.76848i			0.253495i
$222$		0			0
$223$	13.4318	+	13.4318i	0.899462	+	0.899462i	0.995388	−	0.0959263i	$-0.0305813\pi$
$223$	13.4318	+	13.4318i	0.899462	+	0.899462i	−0.0959263	+	0.995388i	$0.530581\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	8.98261	+	8.98261i	0.596197	+	0.596197i	0.939298	−	0.343101i	$-0.111477\pi$
$227$	8.98261	+	8.98261i	0.596197	+	0.596197i	−0.343101	+	0.939298i	$0.611477\pi$
$228$		0			0
$229$		−	26.7648i		−	1.76867i	−0.466853	−	0.884335i	$-0.654612\pi$
$229$		−	26.7648i		−	1.76867i	0.466853	−	0.884335i	$-0.345388\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.92828	+	4.92828i	−0.322862	+	0.322862i	−0.849864	−	0.527002i	$-0.823316\pi$
$233$	−4.92828	+	4.92828i	−0.322862	+	0.322862i	0.527002	+	0.849864i	$0.323316\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−21.1102			−1.36550			−0.682752	−	0.730650i	$-0.739217\pi$
$239$	−21.1102			−1.36550			−0.682752	+	0.730650i	$0.739217\pi$
$240$		0			0
$241$	−25.0446			−1.61326			−0.806631	−	0.591056i	$-0.798711\pi$
$241$	−25.0446			−1.61326			−0.806631	+	0.591056i	$0.798711\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2.97100	−	2.97100i	0.189040	−	0.189040i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	29.5486i		−	1.86509i	−0.361054	−	0.932545i	$-0.617583\pi$
$251$		−	29.5486i		−	1.86509i	0.361054	−	0.932545i	$-0.382417\pi$
$252$		0			0
$253$	7.31331	+	7.31331i	0.459784	+	0.459784i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.20108	−	2.20108i	−0.137300	−	0.137300i	0.635117	−	0.772416i	$-0.280952\pi$
$257$	−2.20108	−	2.20108i	−0.137300	−	0.137300i	−0.772416	+	0.635117i	$0.780952\pi$
$258$		0			0
$259$			4.86630i			0.302377i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.11708	−	2.11708i	0.130545	−	0.130545i	−0.638815	−	0.769360i	$-0.720575\pi$
$263$	2.11708	−	2.11708i	0.130545	−	0.130545i	0.769360	+	0.638815i	$0.220575\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.238843			0.0145625			0.00728127	−	0.999973i	$-0.497682\pi$
$269$	0.238843			0.0145625			0.00728127	+	0.999973i	$0.497682\pi$
$270$		0			0
$271$	9.21426			0.559726			0.279863	−	0.960040i	$-0.409711\pi$
$271$	9.21426			0.559726			0.279863	+	0.960040i	$0.409711\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−23.0749	+	23.0749i	−1.38644	+	1.38644i	−0.553762	+	0.832675i	$0.686808\pi$
$277$	−23.0749	+	23.0749i	−1.38644	+	1.38644i	−0.832675	+	0.553762i	$0.813192\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	24.7376i		−	1.47572i	−0.674953	−	0.737861i	$-0.735836\pi$
$281$		−	24.7376i		−	1.47572i	0.674953	−	0.737861i	$-0.264164\pi$
$282$		0			0
$283$	7.10743	+	7.10743i	0.422493	+	0.422493i	0.886061	−	0.463568i	$-0.153431\pi$
$283$	7.10743	+	7.10743i	0.422493	+	0.422493i	−0.463568	+	0.886061i	$0.653431\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.850371	−	0.850371i	−0.0501958	−	0.0501958i
$288$		0			0
$289$		−	12.4667i		−	0.733333i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.34581	−	9.34581i	0.545988	−	0.545988i	−0.379290	−	0.925278i	$-0.623832\pi$
$293$	9.34581	−	9.34581i	0.545988	−	0.545988i	0.925278	+	0.379290i	$0.123832\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−10.4182			−0.602503
$300$		0			0
$301$	−4.32440			−0.249254
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	8.86325	−	8.86325i	0.505852	−	0.505852i	−0.407398	−	0.913251i	$-0.633564\pi$
$307$	8.86325	−	8.86325i	0.505852	−	0.505852i	0.913251	+	0.407398i	$0.133564\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	27.9652i		−	1.58576i	−0.609375	−	0.792882i	$-0.708580\pi$
$311$		−	27.9652i		−	1.58576i	0.609375	−	0.792882i	$-0.291420\pi$
$312$		0			0
$313$	16.1204	+	16.1204i	0.911178	+	0.911178i	0.996365	−	0.0851869i	$-0.0271487\pi$
$313$	16.1204	+	16.1204i	0.911178	+	0.911178i	−0.0851869	+	0.996365i	$0.527149\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−15.8990	−	15.8990i	−0.892977	−	0.892977i	0.101825	−	0.994802i	$-0.467532\pi$
$317$	−15.8990	−	15.8990i	−0.892977	−	0.892977i	−0.994802	+	0.101825i	$0.967532\pi$
$318$		0			0
$319$		−	1.10638i		−	0.0619457i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.57401	−	3.57401i	0.198863	−	0.198863i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.96721			−0.273851
$330$		0			0
$331$	4.65465			0.255843			0.127921	−	0.991784i	$-0.459169\pi$
$331$	4.65465			0.255843			0.127921	+	0.991784i	$0.459169\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	17.7762	−	17.7762i	0.968329	−	0.968329i	−0.0311844	−	0.999514i	$-0.509928\pi$
$337$	17.7762	−	17.7762i	0.968329	−	0.968329i	0.999514	+	0.0311844i	$0.00992792\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			7.17815i			0.388719i
$342$		0			0
$343$	−0.707107	−	0.707107i	−0.0381802	−	0.0381802i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.28995	+	1.28995i	0.0692481	+	0.0692481i	0.740883	−	0.671635i	$-0.234407\pi$
$347$	1.28995	+	1.28995i	0.0692481	+	0.0692481i	−0.671635	+	0.740883i	$0.734407\pi$
$348$		0			0
$349$			12.1821i			0.652093i	0.945354	+	0.326046i	$0.105717\pi$
$349$			12.1821i			0.652093i	−0.945354	+	0.326046i	$0.894283\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.26875	+	6.26875i	−0.333652	+	0.333652i	−0.853971	−	0.520320i	$-0.825813\pi$
$353$	−6.26875	+	6.26875i	−0.333652	+	0.333652i	0.520320	+	0.853971i	$0.325813\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	29.5533			1.55977			0.779883	−	0.625925i	$-0.215279\pi$
$359$	29.5533			1.55977			0.779883	+	0.625925i	$0.215279\pi$
$360$		0			0
$361$	13.3646			0.703401
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	22.1933	−	22.1933i	1.15848	−	1.15848i	0.173679	−	0.984802i	$-0.444434\pi$
$367$	22.1933	−	22.1933i	1.15848	−	1.15848i	0.984802	−	0.173679i	$-0.0555656\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	9.00329i		−	0.467427i
$372$		0			0
$373$	−17.9117	−	17.9117i	−0.927434	−	0.927434i	0.0701060	−	0.997540i	$-0.477666\pi$
$373$	−17.9117	−	17.9117i	−0.927434	−	0.927434i	−0.997540	+	0.0701060i	$0.977666\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.788055	+	0.788055i	0.0405869	+	0.0405869i
$378$		0			0
$379$			6.65544i			0.341867i	0.985283	+	0.170933i	$0.0546783\pi$
$379$			6.65544i			0.341867i	−0.985283	+	0.170933i	$0.945322\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	24.2538	−	24.2538i	1.23931	−	1.23931i	0.279032	−	0.960282i	$-0.409987\pi$
$383$	24.2538	−	24.2538i	1.23931	−	1.23931i	0.960282	−	0.279032i	$-0.0900134\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	12.8093			0.649459			0.324730	−	0.945807i	$-0.394727\pi$
$389$	12.8093			0.649459			0.324730	+	0.945807i	$0.394727\pi$
$390$		0			0
$391$	−12.5328			−0.633810
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−10.5735	+	10.5735i	−0.530668	+	0.530668i	−0.920771	−	0.390103i	$-0.872439\pi$
$397$	−10.5735	+	10.5735i	−0.530668	+	0.530668i	0.390103	+	0.920771i	$0.372439\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	19.1840i		−	0.958003i	−0.877814	−	0.479001i	$-0.840999\pi$
$401$		−	19.1840i		−	0.958003i	0.877814	−	0.479001i	$-0.159001\pi$
$402$		0			0
$403$	−5.11285	−	5.11285i	−0.254689	−	0.254689i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−6.04609	−	6.04609i	−0.299694	−	0.299694i
$408$		0			0
$409$		−	17.7037i		−	0.875392i	−0.899123	−	0.437696i	$-0.855795\pi$
$409$		−	17.7037i		−	0.875392i	0.899123	−	0.437696i	$-0.144205\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−7.64332	+	7.64332i	−0.376103	+	0.376103i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.3013			−0.600960			−0.300480	−	0.953788i	$-0.597147\pi$
$419$	−12.3013			−0.600960			−0.300480	+	0.953788i	$0.597147\pi$
$420$		0			0
$421$	21.0571			1.02626			0.513131	−	0.858310i	$-0.328486\pi$
$421$	21.0571			1.02626			0.513131	+	0.858310i	$0.328486\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	10.6983	−	10.6983i	0.517727	−	0.517727i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			18.8858i			0.909697i	0.890569	+	0.454848i	$0.150307\pi$
$431$			18.8858i			0.909697i	−0.890569	+	0.454848i	$0.849693\pi$
$432$		0			0
$433$	14.0129	+	14.0129i	0.673415	+	0.673415i	0.958502	−	0.285087i	$-0.0920224\pi$
$433$	14.0129	+	14.0129i	0.673415	+	0.673415i	−0.285087	+	0.958502i	$0.592022\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	9.88064	+	9.88064i	0.472655	+	0.472655i
$438$		0			0
$439$			2.74707i			0.131110i	0.997849	+	0.0655552i	$0.0208819\pi$
$439$			2.74707i			0.131110i	−0.997849	+	0.0655552i	$0.979118\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.91928	+	5.91928i	−0.281233	+	0.281233i	−0.833601	−	0.552367i	$-0.813725\pi$
$443$	−5.91928	+	5.91928i	−0.281233	+	0.281233i	0.552367	+	0.833601i	$0.313725\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−21.8023			−1.02892			−0.514458	−	0.857516i	$-0.672007\pi$
$449$	−21.8023			−1.02892			−0.514458	+	0.857516i	$0.672007\pi$
$450$		0			0
$451$	2.11307			0.0995006
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	10.9740	−	10.9740i	0.513343	−	0.513343i	−0.402206	−	0.915549i	$-0.631757\pi$
$457$	10.9740	−	10.9740i	0.513343	−	0.513343i	0.915549	+	0.402206i	$0.131757\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			29.7346i			1.38488i	0.721475	+	0.692440i	$0.243464\pi$
$461$			29.7346i			1.38488i	−0.721475	+	0.692440i	$0.756536\pi$
$462$		0			0
$463$	7.94451	+	7.94451i	0.369213	+	0.369213i	0.867190	−	0.497977i	$-0.165924\pi$
$463$	7.94451	+	7.94451i	0.369213	+	0.369213i	−0.497977	+	0.867190i	$0.665924\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−14.9720	−	14.9720i	−0.692820	−	0.692820i	0.270031	−	0.962852i	$-0.412966\pi$
$467$	−14.9720	−	14.9720i	−0.692820	−	0.692820i	−0.962852	+	0.270031i	$0.912966\pi$
$468$		0			0
$469$			6.90017i			0.318620i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5.37282	−	5.37282i	0.247042	−	0.247042i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	11.1568			0.509765			0.254882	−	0.966972i	$-0.417963\pi$
$479$	11.1568			0.509765			0.254882	+	0.966972i	$0.417963\pi$
$480$		0			0
$481$	8.61302			0.392720
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−27.2757	+	27.2757i	−1.23598	+	1.23598i	−0.274350	+	0.961630i	$0.588463\pi$
$487$	−27.2757	+	27.2757i	−1.23598	+	1.23598i	−0.961630	+	0.274350i	$0.911537\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	16.1471i		−	0.728707i	−0.931261	−	0.364353i	$-0.881290\pi$
$491$		−	16.1471i		−	0.728707i	0.931261	−	0.364353i	$-0.118710\pi$
$492$		0			0
$493$	0.948003	+	0.948003i	0.0426959	+	0.0426959i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3.01305	−	3.01305i	−0.135154	−	0.135154i
$498$		0			0
$499$		−	30.0186i		−	1.34382i	−0.740633	−	0.671909i	$-0.765474\pi$
$499$		−	30.0186i		−	1.34382i	0.740633	−	0.671909i	$-0.234526\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	8.43925	−	8.43925i	0.376287	−	0.376287i	−0.493473	−	0.869761i	$-0.664273\pi$
$503$	8.43925	−	8.43925i	0.376287	−	0.376287i	0.869761	+	0.493473i	$0.164273\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−25.9643			−1.15085			−0.575424	−	0.817855i	$-0.695163\pi$
$509$	−25.9643			−1.15085			−0.575424	+	0.817855i	$0.695163\pi$
$510$		0			0
$511$	5.74422			0.254109
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	6.17146	−	6.17146i	0.271421	−	0.271421i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	12.2637i		−	0.537284i	−0.963240	−	0.268642i	$-0.913425\pi$
$521$		−	12.2637i		−	0.537284i	0.963240	−	0.268642i	$-0.0865749\pi$
$522$		0			0
$523$	−15.4597	−	15.4597i	−0.676005	−	0.676005i	0.283089	−	0.959094i	$-0.408641\pi$
$523$	−15.4597	−	15.4597i	−0.676005	−	0.676005i	−0.959094	+	0.283089i	$0.908641\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.15058	−	6.15058i	−0.267923	−	0.267923i
$528$		0			0
$529$		−	11.6478i		−	0.506428i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.50510	+	1.50510i	−0.0651930	+	0.0651930i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.75708			0.0756827
$540$		0			0
$541$	39.3958			1.69376			0.846878	−	0.531787i	$-0.178479\pi$
$541$	39.3958			1.69376			0.846878	+	0.531787i	$0.178479\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−14.7712	+	14.7712i	−0.631570	+	0.631570i	−0.948462	−	0.316891i	$-0.897361\pi$
$547$	−14.7712	+	14.7712i	−0.631570	+	0.631570i	0.316891	+	0.948462i	$0.397361\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	1.49478i		−	0.0636797i
$552$		0			0
$553$	1.05498	+	1.05498i	0.0448623	+	0.0448623i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−19.5258	−	19.5258i	−0.827336	−	0.827336i	0.159811	−	0.987148i	$-0.448911\pi$
$557$	−19.5258	−	19.5258i	−0.827336	−	0.827336i	−0.987148	+	0.159811i	$0.948911\pi$
$558$		0			0
$559$			7.65390i			0.323725i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	28.3047	−	28.3047i	1.19290	−	1.19290i	0.216652	−	0.976249i	$-0.430486\pi$
$563$	28.3047	−	28.3047i	1.19290	−	1.19290i	0.976249	−	0.216652i	$-0.0695136\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−36.7109			−1.53900			−0.769502	−	0.638645i	$-0.779495\pi$
$569$	−36.7109			−1.53900			−0.769502	+	0.638645i	$0.779495\pi$
$570$		0			0
$571$	27.0116			1.13040			0.565199	−	0.824955i	$-0.308799\pi$
$571$	27.0116			1.13040			0.565199	+	0.824955i	$0.308799\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	3.45781	−	3.45781i	0.143950	−	0.143950i	−0.631459	−	0.775409i	$-0.717544\pi$
$577$	3.45781	−	3.45781i	0.143950	−	0.143950i	0.775409	+	0.631459i	$0.217544\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	4.60187i		−	0.190918i
$582$		0			0
$583$	11.1861	+	11.1861i	0.463279	+	0.463279i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	31.4297	+	31.4297i	1.29724	+	1.29724i	0.930206	+	0.367038i	$0.119628\pi$
$587$	31.4297	+	31.4297i	1.29724	+	1.29724i	0.367038	+	0.930206i	$0.380372\pi$
$588$		0			0
$589$			9.69803i			0.399600i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−24.2735	+	24.2735i	−0.996793	+	0.996793i	−0.999995	−	0.00320180i	$-0.998981\pi$
$593$	−24.2735	+	24.2735i	−0.996793	+	0.996793i	0.00320180	+	0.999995i	$0.498981\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	4.17790			0.170704			0.0853522	−	0.996351i	$-0.472798\pi$
$599$	4.17790			0.170704			0.0853522	+	0.996351i	$0.472798\pi$
$600$		0			0
$601$	−9.30227			−0.379448			−0.189724	−	0.981838i	$-0.560759\pi$
$601$	−9.30227			−0.379448			−0.189724	+	0.981838i	$0.560759\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−5.98589	+	5.98589i	−0.242960	+	0.242960i	−0.818074	−	0.575114i	$-0.804958\pi$
$607$	−5.98589	+	5.98589i	−0.242960	+	0.242960i	0.575114	+	0.818074i	$0.304958\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			8.79161i			0.355671i
$612$		0			0
$613$	−17.6216	−	17.6216i	−0.711732	−	0.711732i	0.255166	−	0.966897i	$-0.417870\pi$
$613$	−17.6216	−	17.6216i	−0.711732	−	0.711732i	−0.966897	+	0.255166i	$0.917870\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.94438	−	2.94438i	−0.118536	−	0.118536i	0.645350	−	0.763887i	$-0.276711\pi$
$617$	−2.94438	−	2.94438i	−0.118536	−	0.118536i	−0.763887	+	0.645350i	$0.776711\pi$
$618$		0			0
$619$			38.2627i			1.53791i	0.639305	+	0.768953i	$0.279222\pi$
$619$			38.2627i			1.53791i	−0.639305	+	0.768953i	$0.720778\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.14867	−	5.14867i	0.206277	−	0.206277i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	10.3612			0.413126
$630$		0			0
$631$	4.65200			0.185193			0.0925966	−	0.995704i	$-0.470483\pi$
$631$	4.65200			0.185193			0.0925966	+	0.995704i	$0.470483\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.25153	+	1.25153i	−0.0495874	+	0.0495874i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			22.5186i			0.889431i	0.895672	+	0.444715i	$0.146695\pi$
$641$			22.5186i			0.889431i	−0.895672	+	0.444715i	$0.853305\pi$
$642$		0			0
$643$	9.94450	+	9.94450i	0.392173	+	0.392173i	0.875461	−	0.483288i	$-0.160558\pi$
$643$	9.94450	+	9.94450i	0.392173	+	0.392173i	−0.483288	+	0.875461i	$0.660558\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3.43047	+	3.43047i	0.134866	+	0.134866i	0.771317	−	0.636451i	$-0.219598\pi$
$647$	3.43047	+	3.43047i	0.134866	+	0.134866i	−0.636451	+	0.771317i	$0.719598\pi$
$648$		0			0
$649$		−	18.9927i		−	0.745531i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.20047	−	9.20047i	0.360042	−	0.360042i	−0.503786	−	0.863828i	$-0.668060\pi$
$653$	9.20047	−	9.20047i	0.360042	−	0.360042i	0.863828	+	0.503786i	$0.168060\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	34.8844			1.35890			0.679452	−	0.733720i	$-0.262217\pi$
$659$	34.8844			1.35890			0.679452	+	0.733720i	$0.262217\pi$
$660$		0			0
$661$	−9.24236			−0.359486			−0.179743	−	0.983714i	$-0.557527\pi$
$661$	−9.24236			−0.359486			−0.179743	+	0.983714i	$0.557527\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−2.62083	+	2.62083i	−0.101479	+	0.101479i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			26.5840i			1.02626i
$672$		0			0
$673$	−10.6090	−	10.6090i	−0.408949	−	0.408949i	0.472423	−	0.881372i	$-0.343379\pi$
$673$	−10.6090	−	10.6090i	−0.408949	−	0.408949i	−0.881372	+	0.472423i	$0.843379\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−25.1316	−	25.1316i	−0.965887	−	0.965887i	0.0335498	−	0.999437i	$-0.489319\pi$
$677$	−25.1316	−	25.1316i	−0.965887	−	0.965887i	−0.999437	+	0.0335498i	$0.989319\pi$
$678$		0			0
$679$		−	4.34570i		−	0.166773i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9.91527	−	9.91527i	0.379397	−	0.379397i	−0.491487	−	0.870885i	$-0.663547\pi$
$683$	9.91527	−	9.91527i	0.379397	−	0.379397i	0.870885	+	0.491487i	$0.163547\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−15.9352			−0.607083
$690$		0			0
$691$	19.0529			0.724806			0.362403	−	0.932022i	$-0.381956\pi$
$691$	19.0529			0.724806			0.362403	+	0.932022i	$0.381956\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.81058	+	1.81058i	−0.0685806	+	0.0685806i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			21.8614i			0.825692i	0.910801	+	0.412846i	$0.135465\pi$
$701$			21.8614i			0.825692i	−0.910801	+	0.412846i	$0.864535\pi$
$702$		0			0
$703$	−8.16857	−	8.16857i	−0.308083	−	0.308083i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.48757	−	1.48757i	−0.0559457	−	0.0559457i
$708$		0			0
$709$			30.9167i			1.16110i	0.814225	+	0.580550i	$0.197162\pi$
$709$			30.9167i			1.16110i	−0.814225	+	0.580550i	$0.802838\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	17.0037	−	17.0037i	0.636795	−	0.636795i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	27.2125			1.01485			0.507427	−	0.861695i	$-0.330597\pi$
$719$	27.2125			1.01485			0.507427	+	0.861695i	$0.330597\pi$
$720$		0			0
$721$	−6.84564			−0.254945
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−34.8717	+	34.8717i	−1.29332	+	1.29332i	−0.360602	+	0.932720i	$0.617429\pi$
$727$	−34.8717	+	34.8717i	−1.29332	+	1.29332i	−0.932720	+	0.360602i	$0.882571\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			9.20737i			0.340547i
$732$		0			0
$733$	−21.2879	−	21.2879i	−0.786286	−	0.786286i	0.194597	−	0.980883i	$-0.437660\pi$
$733$	−21.2879	−	21.2879i	−0.786286	−	0.786286i	−0.980883	+	0.194597i	$0.937660\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.57306	−	8.57306i	−0.315793	−	0.315793i
$738$		0			0
$739$		−	2.04347i		−	0.0751701i	−0.999293	−	0.0375851i	$-0.988033\pi$
$739$		−	2.04347i		−	0.0751701i	0.999293	−	0.0375851i	$-0.0119665\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−14.7619	+	14.7619i	−0.541561	+	0.541561i	−0.923986	−	0.382426i	$-0.875089\pi$
$743$	−14.7619	+	14.7619i	−0.541561	+	0.541561i	0.382426	+	0.923986i	$0.375089\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−8.74038			−0.319366
$750$		0			0
$751$	14.1494			0.516320			0.258160	−	0.966102i	$-0.416884\pi$
$751$	14.1494			0.516320			0.258160	+	0.966102i	$0.416884\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	21.4200	−	21.4200i	0.778523	−	0.778523i	−0.201057	−	0.979580i	$-0.564438\pi$
$757$	21.4200	−	21.4200i	0.778523	−	0.778523i	0.979580	+	0.201057i	$0.0644376\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	18.1834i		−	0.659146i	−0.944130	−	0.329573i	$-0.893095\pi$
$761$		−	18.1834i		−	0.659146i	0.944130	−	0.329573i	$-0.106905\pi$
$762$		0			0
$763$	−10.1360	−	10.1360i	−0.366948	−	0.366948i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	13.5282	+	13.5282i	0.488473	+	0.488473i
$768$		0			0
$769$			16.3155i			0.588352i	0.955751	+	0.294176i	$0.0950452\pi$
$769$			16.3155i			0.588352i	−0.955751	+	0.294176i	$0.904955\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	2.80152	−	2.80152i	0.100764	−	0.100764i	−0.654928	−	0.755692i	$-0.727301\pi$
$773$	2.80152	−	2.80152i	0.100764	−	0.100764i	0.755692	+	0.654928i	$0.227301\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.85486			0.102286
$780$		0			0
$781$	7.48708			0.267909
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−21.5728	+	21.5728i	−0.768986	+	0.768986i	−0.977928	−	0.208942i	$-0.932998\pi$
$787$	−21.5728	+	21.5728i	−0.768986	+	0.768986i	0.208942	+	0.977928i	$0.432998\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1.34643i			0.0478737i
$792$		0			0
$793$	−18.9353	−	18.9353i	−0.672411	−	0.672411i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	25.7202	+	25.7202i	0.911057	+	0.911057i	0.996355	−	0.0852982i	$-0.0271843\pi$
$797$	25.7202	+	25.7202i	0.911057	+	0.911057i	−0.0852982	+	0.996355i	$0.527184\pi$
$798$		0			0
$799$			10.5760i			0.374152i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−7.13686	+	7.13686i	−0.251854	+	0.251854i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6.42491			−0.225888			−0.112944	−	0.993601i	$-0.536028\pi$
$809$	−6.42491			−0.225888			−0.112944	+	0.993601i	$0.536028\pi$
$810$		0			0
$811$	−29.1628			−1.02405			−0.512023	−	0.858972i	$-0.671104\pi$
$811$	−29.1628			−1.02405			−0.512023	+	0.858972i	$0.671104\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	7.25894	−	7.25894i	0.253958	−	0.253958i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			33.9270i			1.18406i	0.805915	+	0.592031i	$0.201674\pi$
$821$			33.9270i			1.18406i	−0.805915	+	0.592031i	$0.798326\pi$
$822$		0			0
$823$	3.65350	+	3.65350i	0.127353	+	0.127353i	0.767910	−	0.640557i	$-0.221297\pi$
$823$	3.65350	+	3.65350i	0.127353	+	0.127353i	−0.640557	+	0.767910i	$0.721297\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−2.73779	−	2.73779i	−0.0952024	−	0.0952024i	0.657902	−	0.753104i	$-0.271444\pi$
$827$	−2.73779	−	2.73779i	−0.0952024	−	0.0952024i	−0.753104	+	0.657902i	$0.771444\pi$
$828$		0			0
$829$		−	11.3948i		−	0.395759i	−0.980226	−	0.197880i	$-0.936594\pi$
$829$		−	11.3948i		−	0.395759i	0.980226	−	0.197880i	$-0.0634055\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.50555	+	1.50555i	−0.0521641	+	0.0521641i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	42.3828			1.46322			0.731608	−	0.681726i	$-0.238770\pi$
$839$	42.3828			1.46322			0.731608	+	0.681726i	$0.238770\pi$
$840$		0			0
$841$	−28.6035			−0.986328
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	5.59511	−	5.59511i	0.192250	−	0.192250i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			28.6442i			0.981912i
$852$		0			0
$853$	−18.5498	−	18.5498i	−0.635132	−	0.635132i	0.314218	−	0.949351i	$-0.398258\pi$
$853$	−18.5498	−	18.5498i	−0.635132	−	0.635132i	−0.949351	+	0.314218i	$0.898258\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2.99915	+	2.99915i	0.102449	+	0.102449i	0.756473	−	0.654024i	$-0.226921\pi$
$857$	2.99915	+	2.99915i	0.102449	+	0.102449i	−0.654024	+	0.756473i	$0.726921\pi$
$858$		0			0
$859$			9.26686i			0.316181i	0.987425	+	0.158091i	$0.0505338\pi$
$859$			9.26686i			0.316181i	−0.987425	+	0.158091i	$0.949466\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−26.0871	+	26.0871i	−0.888014	+	0.888014i	−0.994332	−	0.106318i	$-0.966094\pi$
$863$	−26.0871	+	26.0871i	−0.888014	+	0.888014i	0.106318	+	0.994332i	$0.466094\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.62150			−0.0889284
$870$		0			0
$871$	12.2128			0.413816
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	18.0469	−	18.0469i	0.609400	−	0.609400i	−0.333389	−	0.942789i	$-0.608192\pi$
$877$	18.0469	−	18.0469i	0.609400	−	0.609400i	0.942789	+	0.333389i	$0.108192\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			4.92641i			0.165975i	0.996551	+	0.0829875i	$0.0264461\pi$
$881$			4.92641i			0.165975i	−0.996551	+	0.0829875i	$0.973554\pi$
$882$		0			0
$883$	−11.2048	−	11.2048i	−0.377072	−	0.377072i	0.492973	−	0.870045i	$-0.335910\pi$
$883$	−11.2048	−	11.2048i	−0.377072	−	0.377072i	−0.870045	+	0.492973i	$0.835910\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.59105	−	1.59105i	−0.0534224	−	0.0534224i	0.679891	−	0.733313i	$-0.262027\pi$
$887$	−1.59105	−	1.59105i	−0.0534224	−	0.0534224i	−0.733313	+	0.679891i	$0.762027\pi$
$888$		0			0
$889$		−	15.3123i		−	0.513560i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	8.33794	−	8.33794i	0.279019	−	0.279019i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.57239			−0.0857940
$900$		0			0
$901$	−19.1695			−0.638628
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−13.5776	+	13.5776i	−0.450836	+	0.450836i	−0.895632	−	0.444796i	$-0.853276\pi$
$907$	−13.5776	+	13.5776i	−0.450836	+	0.450836i	0.444796	+	0.895632i	$0.353276\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	47.0735i		−	1.55961i	−0.626020	−	0.779807i	$-0.715317\pi$
$911$		−	47.0735i		−	1.55961i	0.626020	−	0.779807i	$-0.284683\pi$
$912$		0			0
$913$	5.71756	+	5.71756i	0.189224	+	0.189224i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−4.53477	−	4.53477i	−0.149751	−	0.149751i
$918$		0			0
$919$		−	34.5324i		−	1.13912i	−0.821950	−	0.569560i	$-0.807114\pi$
$919$		−	34.5324i		−	1.13912i	0.821950	−	0.569560i	$-0.192886\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5.33290	+	5.33290i	−0.175534	+	0.175534i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−12.6925			−0.416427			−0.208214	−	0.978083i	$-0.566765\pi$
$929$	−12.6925			−0.416427			−0.208214	+	0.978083i	$0.566765\pi$
$930$		0			0
$931$	2.37390			0.0778013
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−9.16625	+	9.16625i	−0.299448	+	0.299448i	−0.840798	−	0.541349i	$-0.817914\pi$
$937$	−9.16625	+	9.16625i	−0.299448	+	0.299448i	0.541349	+	0.840798i	$0.317914\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			44.4401i			1.44871i	0.689430	+	0.724353i	$0.257861\pi$
$941$			44.4401i			1.44871i	−0.689430	+	0.724353i	$0.742139\pi$
$942$		0			0
$943$	−5.00549	−	5.00549i	−0.163001	−	0.163001i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4.43877	−	4.43877i	−0.144241	−	0.144241i	0.631299	−	0.775540i	$-0.282522\pi$
$947$	−4.43877	−	4.43877i	−0.144241	−	0.144241i	−0.775540	+	0.631299i	$0.782522\pi$
$948$		0			0
$949$		−	10.1669i		−	0.330031i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.98360	+	1.98360i	−0.0642552	+	0.0642552i	−0.738504	−	0.674249i	$-0.764467\pi$
$953$	−1.98360	+	1.98360i	−0.0642552	+	0.0642552i	0.674249	+	0.738504i	$0.264467\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−19.9104			−0.642941
$960$		0			0
$961$	−14.3105			−0.461629
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	4.18629	−	4.18629i	0.134622	−	0.134622i	−0.636585	−	0.771207i	$-0.719653\pi$
$967$	4.18629	−	4.18629i	0.134622	−	0.134622i	0.771207	+	0.636585i	$0.219653\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	15.6434i		−	0.502020i	−0.967984	−	0.251010i	$-0.919237\pi$
$971$		−	15.6434i		−	0.502020i	0.967984	−	0.251010i	$-0.0807628\pi$
$972$		0			0
$973$	3.32972	+	3.32972i	0.106746	+	0.106746i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	43.9954	+	43.9954i	1.40754	+	1.40754i	0.772392	+	0.635146i	$0.219060\pi$
$977$	43.9954	+	43.9954i	1.40754	+	1.40754i	0.635146	+	0.772392i	$0.280940\pi$
$978$		0			0
$979$			12.7938i			0.408893i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−16.4449	+	16.4449i	−0.524510	+	0.524510i	−0.918930	−	0.394420i	$-0.870946\pi$
$983$	−16.4449	+	16.4449i	−0.524510	+	0.524510i	0.394420	+	0.918930i	$0.370946\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−25.4545			−0.809405
$990$		0			0
$991$	−54.9484			−1.74549			−0.872747	−	0.488172i	$-0.837664\pi$
$991$	−54.9484			−1.74549			−0.872747	+	0.488172i	$0.837664\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−11.7296	+	11.7296i	−0.371480	+	0.371480i	−0.868016	−	0.496536i	$-0.834605\pi$
$997$	−11.7296	+	11.7296i	−0.371480	+	0.371480i	0.496536	+	0.868016i	$0.334605\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.v.e.5993.6		12
3.2	odd	2		6300.2.v.f.5993.4		12
5.2	odd	4		6300.2.v.f.1457.6		12
5.3	odd	4		1260.2.v.a.197.1	✓	12
5.4	even	2		1260.2.v.b.953.6	yes	12
15.2	even	4	inner	6300.2.v.e.1457.4		12
15.8	even	4		1260.2.v.b.197.6	yes	12
15.14	odd	2		1260.2.v.a.953.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.v.a.197.1	✓	12	5.3	odd	4
1260.2.v.a.953.1	yes	12	15.14	odd	2
1260.2.v.b.197.6	yes	12	15.8	even	4
1260.2.v.b.953.6	yes	12	5.4	even	2
6300.2.v.e.1457.4		12	15.2	even	4	inner
6300.2.v.e.5993.6		12	1.1	even	1	trivial
6300.2.v.f.1457.6		12	5.2	odd	4
6300.2.v.f.5993.4		12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6300.v (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{7} +1.75708i q^{11} +(-1.25153 - 1.25153i) q^{13} +(-1.50555 - 1.50555i) q^{17} +2.37390i q^{19} +(4.16220 - 4.16220i) q^{23} -0.629673 q^{29} +4.08528 q^{31} +(-3.44100 + 3.44100i) q^{37} -1.20261i q^{41} +(-3.05781 - 3.05781i) q^{43} +(-3.51235 - 3.51235i) q^{47} -1.00000i q^{49} +(6.36629 - 6.36629i) q^{53} -10.8093 q^{59} +15.1297 q^{61} +(-4.87916 + 4.87916i) q^{67} -4.26110i q^{71} +(4.06178 + 4.06178i) q^{73} +(1.24244 + 1.24244i) q^{77} +1.49197i q^{79} +(3.25402 - 3.25402i) q^{83} +7.28131 q^{89} -1.76993 q^{91} +(3.07287 - 3.07287i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{13} - 8 q^{17} + 8 q^{23} - 32 q^{29} + 16 q^{31} - 4 q^{37} - 8 q^{43} - 8 q^{47} + 8 q^{53} - 16 q^{59} + 12 q^{73} + 56 q^{83} - 72 q^{89} - 20 q^{97}+O(q^{100}) \) 12 * q + 12 * q^13 - 8 * q^17 + 8 * q^23 - 32 * q^29 + 16 * q^31 - 4 * q^37 - 8 * q^43 - 8 * q^47 + 8 * q^53 - 16 * q^59 + 12 * q^73 + 56 * q^83 - 72 * q^89 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more