LMFDB - Embedded newform 6300.2.dd.c.4049.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6300,2,Mod(1349,6300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6300, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 3, 5]))

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

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

chi := DirichletCharacter("6300.1349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6300.dd (of order $6$, degree $2$, 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:	$50.3057532734$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 1260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4049.4
Character	$\chi$	$=$	6300.4049
Dual form			6300.2.dd.c.1349.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.390758 + 2.61674i) q^{7} +O(q^{10})$	$q+(-0.390758 + 2.61674i) q^{7} +(2.17681 + 1.25678i) q^{11} -2.11077 q^{13} +(3.89248 + 2.24732i) q^{17} +(-4.89571 + 2.82654i) q^{19} +(-3.99219 - 6.91468i) q^{23} +4.97264i q^{29} +(6.13441 + 3.54171i) q^{31} +(8.09881 - 4.67585i) q^{37} -6.23347 q^{41} +9.47185i q^{43} +(9.04490 - 5.22208i) q^{47} +(-6.69462 - 2.04502i) q^{49} +(-3.74310 + 6.48325i) q^{53} +(0.188210 - 0.325989i) q^{59} +(-8.41210 + 4.85673i) q^{61} +(-3.25263 - 1.87791i) q^{67} +9.01312i q^{71} +(2.46154 - 4.26351i) q^{73} +(-4.13928 + 5.20505i) q^{77} +(-7.97217 - 13.8082i) q^{79} +2.29871i q^{83} +(8.85428 + 15.3361i) q^{89} +(0.824800 - 5.52332i) q^{91} -4.14156 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 24 q^{11} + 12 q^{19} - 12 q^{31} - 16 q^{41} - 44 q^{49} - 28 q^{79} + 40 q^{89} + 20 q^{91}+O(q^{100}) $ 24 * q + 24 * q^11 + 12 * q^19 - 12 * q^31 - 16 * q^41 - 44 * q^49 - 28 * q^79 + 40 * q^89 + 20 * q^91

Display 100 coefficients 10 coefficients

Character values

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

$n$	$2801$	$3151$	$3277$	$3601$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{1}{6}\right)$

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
$7$	−0.390758	+	2.61674i	−0.147693	+	0.989033i
$7$	−0.390758	+	2.61674i	−0.147693	+	0.989033i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.17681	+	1.25678i	0.656334	+	0.378935i	0.790879	−	0.611973i	$-0.209624\pi$
$11$	2.17681	+	1.25678i	0.656334	+	0.378935i	−0.134545	+	0.990908i	$0.542957\pi$
$12$		0			0
$13$	−2.11077			−0.585422			−0.292711	−	0.956201i	$-0.594557\pi$
$13$	−2.11077			−0.585422			−0.292711	+	0.956201i	$0.594557\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.89248	+	2.24732i	0.944064	+	0.545056i	0.891232	−	0.453548i	$-0.149842\pi$
$17$	3.89248	+	2.24732i	0.944064	+	0.545056i	0.0528321	+	0.998603i	$0.483175\pi$
$18$		0			0
$19$	−4.89571	+	2.82654i	−1.12315	+	0.648453i	−0.942204	−	0.335040i	$-0.891250\pi$
$19$	−4.89571	+	2.82654i	−1.12315	+	0.648453i	−0.180949	+	0.983492i	$0.557917\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.99219	−	6.91468i	−0.832429	−	1.44181i	−0.896106	−	0.443839i	$-0.853616\pi$
$23$	−3.99219	−	6.91468i	−0.832429	−	1.44181i	0.0636772	−	0.997971i	$-0.479717\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$			4.97264i			0.923396i	0.887037	+	0.461698i	$0.152760\pi$
$29$			4.97264i			0.923396i	−0.887037	+	0.461698i	$0.847240\pi$
$30$		0			0
$31$	6.13441	+	3.54171i	1.10177	+	0.636109i	0.936686	−	0.350170i	$-0.113876\pi$
$31$	6.13441	+	3.54171i	1.10177	+	0.636109i	0.165087	+	0.986279i	$0.447210\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	8.09881	−	4.67585i	1.33144	−	0.768705i	0.345916	−	0.938266i	$-0.387568\pi$
$37$	8.09881	−	4.67585i	1.33144	−	0.768705i	0.985520	+	0.169561i	$0.0542349\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.23347			−0.973505			−0.486752	−	0.873540i	$-0.661819\pi$
$41$	−6.23347			−0.973505			−0.486752	+	0.873540i	$0.661819\pi$
$42$		0			0
$43$			9.47185i			1.44444i	0.691661	+	0.722222i	$0.256879\pi$
$43$			9.47185i			1.44444i	−0.691661	+	0.722222i	$0.743121\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	9.04490	−	5.22208i	1.31933	−	0.761718i	0.335713	−	0.941964i	$-0.391023\pi$
$47$	9.04490	−	5.22208i	1.31933	−	0.761718i	0.983621	+	0.180246i	$0.0576895\pi$
$48$		0			0
$49$	−6.69462	−	2.04502i	−0.956374	−	0.292146i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.74310	+	6.48325i	−0.514155	+	0.890543i	0.485710	+	0.874120i	$0.338561\pi$
$53$	−3.74310	+	6.48325i	−0.514155	+	0.890543i	−0.999865	+	0.0164226i	$0.994772\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.188210	−	0.325989i	0.0245028	−	0.0424402i	−0.853514	−	0.521070i	$-0.825533\pi$
$59$	0.188210	−	0.325989i	0.0245028	−	0.0424402i	0.878017	+	0.478630i	$0.158866\pi$
$60$		0			0
$61$	−8.41210	+	4.85673i	−1.07706	+	0.621840i	−0.930101	−	0.367303i	$-0.880281\pi$
$61$	−8.41210	+	4.85673i	−1.07706	+	0.621840i	−0.146957	+	0.989143i	$0.546948\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.25263	−	1.87791i	−0.397372	−	0.229423i	0.287977	−	0.957637i	$-0.407017\pi$
$67$	−3.25263	−	1.87791i	−0.397372	−	0.229423i	−0.685350	+	0.728214i	$0.740351\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			9.01312i			1.06966i	0.844959	+	0.534831i	$0.179625\pi$
$71$			9.01312i			1.06966i	−0.844959	+	0.534831i	$0.820375\pi$
$72$		0			0
$73$	2.46154	−	4.26351i	0.288101	−	0.499006i	−0.685255	−	0.728303i	$-0.740309\pi$
$73$	2.46154	−	4.26351i	0.288101	−	0.499006i	0.973357	+	0.229297i	$0.0736427\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.13928	+	5.20505i	−0.471715	+	0.593170i
$78$		0			0
$79$	−7.97217	−	13.8082i	−0.896940	−	1.55355i	−0.831385	−	0.555696i	$-0.812452\pi$
$79$	−7.97217	−	13.8082i	−0.896940	−	1.55355i	−0.0655543	−	0.997849i	$-0.520882\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			2.29871i			0.252316i	0.992010	+	0.126158i	$0.0402646\pi$
$83$			2.29871i			0.252316i	−0.992010	+	0.126158i	$0.959735\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.85428	+	15.3361i	0.938551	+	1.62562i	0.768175	+	0.640239i	$0.221165\pi$
$89$	8.85428	+	15.3361i	0.938551	+	1.62562i	0.170376	+	0.985379i	$0.445502\pi$
$90$		0			0
$91$	0.824800	−	5.52332i	0.0864626	−	0.579002i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−4.14156			−0.420512			−0.210256	−	0.977646i	$-0.567430\pi$
$97$	−4.14156			−0.420512			−0.210256	+	0.977646i	$0.567430\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.28493	+	7.42172i	−0.426366	+	0.738488i	−0.996547	−	0.0830310i	$-0.973540\pi$
$101$	−4.28493	+	7.42172i	−0.426366	+	0.738488i	0.570180	+	0.821519i	$0.306873\pi$
$102$		0			0
$103$	−6.03955	−	10.4608i	−0.595095	−	1.03073i	−0.993533	−	0.113540i	$-0.963781\pi$
$103$	−6.03955	−	10.4608i	−0.595095	−	1.03073i	0.398439	−	0.917195i	$-0.369552\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.246804	+	0.427476i	0.0238594	+	0.0413257i	0.877709	−	0.479195i	$-0.159071\pi$
$107$	0.246804	+	0.427476i	0.0238594	+	0.0413257i	−0.853849	+	0.520521i	$0.825738\pi$
$108$		0			0
$109$	−2.89248	+	5.00993i	−0.277050	+	0.479864i	−0.970650	−	0.240496i	$-0.922690\pi$
$109$	−2.89248	+	5.00993i	−0.277050	+	0.479864i	0.693601	+	0.720360i	$0.256023\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−17.0862			−1.60733			−0.803666	−	0.595081i	$-0.797120\pi$
$113$	−17.0862			−1.60733			−0.803666	+	0.595081i	$0.797120\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−7.40167	+	9.30742i	−0.678510	+	0.853210i
$120$		0			0
$121$	−2.34099	−	4.05471i	−0.212817	−	0.368610i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	5.65596i		−	0.501885i	−0.968002	−	0.250943i	$-0.919259\pi$
$127$		−	5.65596i		−	0.501885i	0.968002	−	0.250943i	$-0.0807405\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.554721	+	0.960805i	0.0484662	+	0.0839460i	0.889241	−	0.457439i	$-0.151233\pi$
$131$	0.554721	+	0.960805i	0.0484662	+	0.0839460i	−0.840775	+	0.541385i	$0.817900\pi$
$132$		0			0
$133$	−5.48327	−	13.9153i	−0.475460	−	1.20661i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.0460100	−	0.0796917i	0.00393090	−	0.00680852i	−0.864053	−	0.503400i	$-0.832082\pi$
$137$	0.0460100	−	0.0796917i	0.00393090	−	0.00680852i	0.867984	+	0.496592i	$0.165415\pi$
$138$		0			0
$139$			9.75255i			0.827200i	0.910459	+	0.413600i	$0.135729\pi$
$139$			9.75255i			0.827200i	−0.910459	+	0.413600i	$0.864271\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.59475	−	2.65278i	−0.384232	−	0.221837i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	12.0337	−	6.94766i	0.985839	−	0.569175i	0.0818114	−	0.996648i	$-0.473929\pi$
$149$	12.0337	−	6.94766i	0.985839	−	0.569175i	0.904028	+	0.427473i	$0.140596\pi$
$150$		0			0
$151$	7.53225	−	13.0462i	0.612966	−	1.06169i	−0.377772	−	0.925899i	$-0.623310\pi$
$151$	7.53225	−	13.0462i	0.612966	−	1.06169i	0.990738	−	0.135789i	$-0.0433571\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.32925	+	2.30233i	−0.106086	+	0.183746i	−0.914181	−	0.405305i	$-0.867165\pi$
$157$	−1.32925	+	2.30233i	−0.106086	+	0.183746i	0.808095	+	0.589052i	$0.200498\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	19.6539	−	7.74454i	1.54894	−	0.610355i
$162$		0			0
$163$	7.95165	−	4.59089i	0.622822	−	0.359586i	−0.155145	−	0.987892i	$-0.549585\pi$
$163$	7.95165	−	4.59089i	0.622822	−	0.359586i	0.777967	+	0.628305i	$0.216251\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			0.0924548i			0.00715437i	0.999994	+	0.00357718i	$0.00113866\pi$
$167$			0.0924548i			0.00715437i	−0.999994	+	0.00357718i	$0.998861\pi$
$168$		0			0
$169$	−8.54466			−0.657281
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.279333	+	0.161273i	−0.0212373	+	0.0122614i	−0.510581	−	0.859830i	$-0.670570\pi$
$173$	−0.279333	+	0.161273i	−0.0212373	+	0.0122614i	0.489344	+	0.872091i	$0.337236\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.53709	−	2.61949i	−0.339118	−	0.195790i	0.320764	−	0.947159i	$-0.396060\pi$
$179$	−4.53709	−	2.61949i	−0.339118	−	0.195790i	−0.659882	+	0.751369i	$0.729394\pi$
$180$		0			0
$181$			20.3013i			1.50898i	0.656310	+	0.754491i	$0.272116\pi$
$181$			20.3013i			1.50898i	−0.656310	+	0.754491i	$0.727884\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	5.64880	+	9.78400i	0.413081	+	0.715477i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−7.45539	+	4.30437i	−0.539453	+	0.311453i	−0.744857	−	0.667224i	$-0.767482\pi$
$191$	−7.45539	+	4.30437i	−0.539453	+	0.311453i	0.205404	+	0.978677i	$0.434149\pi$
$192$		0			0
$193$	−4.27775	−	2.46976i	−0.307920	−	0.177777i	0.338076	−	0.941119i	$-0.390224\pi$
$193$	−4.27775	−	2.46976i	−0.307920	−	0.177777i	−0.645995	+	0.763342i	$0.723557\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−7.78952			−0.554980			−0.277490	−	0.960729i	$-0.589503\pi$
$197$	−7.78952			−0.554980			−0.277490	+	0.960729i	$0.589503\pi$
$198$		0			0
$199$	−13.2832	−	7.66904i	−0.941618	−	0.543644i	−0.0511511	−	0.998691i	$-0.516289\pi$
$199$	−13.2832	−	7.66904i	−0.941618	−	0.543644i	−0.890467	+	0.455047i	$0.849622\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−13.0121	−	1.94310i	−0.913270	−	0.136379i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−14.2094			−0.982885
$210$		0			0
$211$	−17.4987			−1.20466			−0.602329	−	0.798248i	$-0.705761\pi$
$211$	−17.4987			−1.20466			−0.602329	+	0.798248i	$0.705761\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−11.6648	+	14.6682i	−0.791857	+	0.995742i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.21611	−	4.74358i	−0.552676	−	0.319087i
$222$		0			0
$223$	22.4665			1.50447			0.752234	−	0.658897i	$-0.228977\pi$
$223$	22.4665			1.50447			0.752234	+	0.658897i	$0.228977\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.23703	−	3.60095i	−0.413966	−	0.239003i	0.278526	−	0.960429i	$-0.410154\pi$
$227$	−6.23703	−	3.60095i	−0.413966	−	0.239003i	−0.692492	+	0.721425i	$0.743487\pi$
$228$		0			0
$229$	11.9343	−	6.89028i	0.788642	−	0.455323i	−0.0508424	−	0.998707i	$-0.516191\pi$
$229$	11.9343	−	6.89028i	0.788642	−	0.455323i	0.839484	+	0.543384i	$0.182857\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.0286460	+	0.0496162i	0.00187666	+	0.00325047i	0.866962	−	0.498374i	$-0.166069\pi$
$233$	0.0286460	+	0.0496162i	0.00187666	+	0.00325047i	−0.865086	+	0.501624i	$0.832736\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			24.9150i			1.61162i	0.592177	+	0.805808i	$0.298269\pi$
$239$			24.9150i			1.61162i	−0.592177	+	0.805808i	$0.701731\pi$
$240$		0			0
$241$	−18.6765	−	10.7829i	−1.20306	−	0.694586i	−0.241825	−	0.970320i	$-0.577746\pi$
$241$	−18.6765	−	10.7829i	−1.20306	−	0.694586i	−0.961234	+	0.275734i	$0.911079\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	10.3337	−	5.96617i	0.657518	−	0.379618i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	15.6047			0.984962			0.492481	−	0.870323i	$-0.336090\pi$
$251$	15.6047			0.984962			0.492481	+	0.870323i	$0.336090\pi$
$252$		0			0
$253$		−	20.0693i		−	1.26174i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−24.9577	+	14.4093i	−1.55682	+	0.898829i	−0.559258	+	0.828993i	$0.688914\pi$
$257$	−24.9577	+	14.4093i	−1.55682	+	0.898829i	−0.997559	+	0.0698354i	$0.977753\pi$
$258$		0			0
$259$	9.07078	+	23.0196i	0.563631	+	1.43037i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−15.7671	+	27.3093i	−0.972238	+	1.68397i	−0.283474	+	0.958980i	$0.591487\pi$
$263$	−15.7671	+	27.3093i	−0.972238	+	1.68397i	−0.688764	+	0.724986i	$0.741846\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−15.8180	+	27.3975i	−0.964438	+	1.67046i	−0.253320	+	0.967383i	$0.581523\pi$
$269$	−15.8180	+	27.3975i	−0.964438	+	1.67046i	−0.711118	+	0.703073i	$0.751811\pi$
$270$		0			0
$271$	−5.30639	+	3.06364i	−0.322340	+	0.186103i	−0.652435	−	0.757845i	$-0.726253\pi$
$271$	−5.30639	+	3.06364i	−0.322340	+	0.186103i	0.330095	+	0.943948i	$0.392919\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−14.4650	−	8.35137i	−0.869118	−	0.501785i	−0.00206272	−	0.999998i	$-0.500657\pi$
$277$	−14.4650	−	8.35137i	−0.869118	−	0.501785i	−0.867055	+	0.498213i	$0.833990\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			6.24284i			0.372416i	0.982510	+	0.186208i	$0.0596199\pi$
$281$			6.24284i			0.372416i	−0.982510	+	0.186208i	$0.940380\pi$
$282$		0			0
$283$	2.13695	−	3.70131i	0.127028	−	0.220020i	−0.795496	−	0.605959i	$-0.792789\pi$
$283$	2.13695	−	3.70131i	0.127028	−	0.220020i	0.922524	+	0.385940i	$0.126123\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.43578	−	16.3114i	0.143780	−	0.962829i
$288$		0			0
$289$	1.60091	+	2.77286i	0.0941713	+	0.163110i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	19.5954i		−	1.14478i	−0.819983	−	0.572388i	$-0.806017\pi$
$293$		−	19.5954i		−	1.14478i	0.819983	−	0.572388i	$-0.193983\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8.42659	+	14.5953i	0.487322	+	0.844067i
$300$		0			0
$301$	−24.7853	−	3.70120i	−1.42860	−	0.213334i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	7.74134			0.441822			0.220911	−	0.975294i	$-0.429097\pi$
$307$	7.74134			0.441822			0.220911	+	0.975294i	$0.429097\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−7.11222	+	12.3187i	−0.403297	+	0.698531i	−0.994122	−	0.108269i	$-0.965469\pi$
$311$	−7.11222	+	12.3187i	−0.403297	+	0.698531i	0.590825	+	0.806800i	$0.298803\pi$
$312$		0			0
$313$	−11.5078	−	19.9321i	−0.650460	−	1.12663i	−0.983011	−	0.183544i	$-0.941243\pi$
$313$	−11.5078	−	19.9321i	−0.650460	−	1.12663i	0.332552	−	0.943085i	$-0.392090\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.86466	+	3.22969i	0.104730	+	0.181397i	0.913628	−	0.406552i	$-0.133269\pi$
$317$	1.86466	+	3.22969i	0.104730	+	0.181397i	−0.808898	+	0.587949i	$0.799936\pi$
$318$		0			0
$319$	−6.24954	+	10.8245i	−0.349907	+	0.606056i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−25.4086			−1.41377
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	10.1304	+	25.7087i	0.558508	+	1.41737i
$330$		0			0
$331$	15.2830	+	26.4710i	0.840032	+	1.45498i	0.889867	+	0.456220i	$0.150797\pi$
$331$	15.2830	+	26.4710i	0.840032	+	1.45498i	−0.0498354	+	0.998757i	$0.515870\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$			4.06846i			0.221623i	0.993841	+	0.110812i	$0.0353450\pi$
$337$			4.06846i			0.221623i	−0.993841	+	0.110812i	$0.964655\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	8.90231	+	15.4193i	0.482087	+	0.835000i
$342$		0			0
$343$	7.96726	−	16.7189i	0.430192	−	0.902738i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	10.8295	−	18.7572i	0.581358	−	1.00694i	−0.413961	−	0.910295i	$-0.635855\pi$
$347$	10.8295	−	18.7572i	0.581358	−	1.00694i	0.995319	−	0.0966464i	$-0.0308116\pi$
$348$		0			0
$349$		−	3.23900i		−	0.173380i	−0.996235	−	0.0866900i	$-0.972371\pi$
$349$		−	3.23900i		−	0.173380i	0.996235	−	0.0866900i	$-0.0276290\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4.29127	+	2.47757i	0.228401	+	0.131868i	0.609834	−	0.792529i	$-0.291236\pi$
$353$	4.29127	+	2.47757i	0.228401	+	0.131868i	−0.381433	+	0.924396i	$0.624569\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−20.2757	+	11.7062i	−1.07011	+	0.617828i	−0.928211	−	0.372053i	$-0.878654\pi$
$359$	−20.2757	+	11.7062i	−1.07011	+	0.617828i	−0.141898	+	0.989881i	$0.545320\pi$
$360$		0			0
$361$	6.47865	−	11.2214i	0.340982	−	0.590598i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−7.04887	+	12.2090i	−0.367948	+	0.637304i	−0.989245	−	0.146271i	$-0.953273\pi$
$367$	−7.04887	+	12.2090i	−0.367948	+	0.637304i	0.621297	+	0.783575i	$0.286606\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−15.5023	−	12.3281i	−0.804839	−	0.640043i
$372$		0			0
$373$	−19.7144	+	11.3821i	−1.02077	+	0.589345i	−0.914328	−	0.404974i	$-0.867281\pi$
$373$	−19.7144	+	11.3821i	−1.02077	+	0.589345i	−0.106447	+	0.994318i	$0.533947\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	10.4961i		−	0.540576i
$378$		0			0
$379$	−23.4861			−1.20640			−0.603200	−	0.797590i	$-0.706108\pi$
$379$	−23.4861			−1.20640			−0.603200	+	0.797590i	$0.706108\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	21.3039	−	12.2998i	1.08858	−	0.628490i	0.155379	−	0.987855i	$-0.450340\pi$
$383$	21.3039	−	12.2998i	1.08858	−	0.628490i	0.933197	+	0.359365i	$0.117007\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	20.4376	+	11.7997i	1.03623	+	0.598267i	0.918763	−	0.394810i	$-0.129189\pi$
$389$	20.4376	+	11.7997i	1.03623	+	0.598267i	0.117466	+	0.993077i	$0.462523\pi$
$390$		0			0
$391$		−	35.8869i		−	1.81488i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	10.5312	+	18.2406i	0.528545	+	0.915467i	0.999446	+	0.0332809i	$0.0105956\pi$
$397$	10.5312	+	18.2406i	0.528545	+	0.915467i	−0.470901	+	0.882186i	$0.656071\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11.8724	+	6.85456i	−0.592881	+	0.342300i	−0.766236	−	0.642559i	$-0.777873\pi$
$401$	−11.8724	+	6.85456i	−0.592881	+	0.342300i	0.173355	+	0.984859i	$0.444539\pi$
$402$		0			0
$403$	−12.9483	−	7.47572i	−0.645002	−	0.372392i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	23.5061			1.16516
$408$		0			0
$409$	−11.8683	−	6.85219i	−0.586851	−	0.338819i	0.177000	−	0.984211i	$-0.443361\pi$
$409$	−11.8683	−	6.85219i	−0.586851	−	0.338819i	−0.763852	+	0.645392i	$0.776694\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.779483	+	0.619878i	0.0383558	+	0.0305022i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.10776			0.102971			0.0514855	−	0.998674i	$-0.483604\pi$
$419$	2.10776			0.102971			0.0514855	+	0.998674i	$0.483604\pi$
$420$		0			0
$421$	12.3265			0.600756			0.300378	−	0.953820i	$-0.402887\pi$
$421$	12.3265			0.600756			0.300378	+	0.953820i	$0.402887\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−9.42168	−	23.9100i	−0.455947	−	1.15709i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2.82519	−	1.63112i	−0.136085	−	0.0785685i	0.430412	−	0.902633i	$-0.358368\pi$
$431$	−2.82519	−	1.63112i	−0.136085	−	0.0785685i	−0.566497	+	0.824064i	$0.691702\pi$
$432$		0			0
$433$	19.4147			0.933013			0.466506	−	0.884518i	$-0.345512\pi$
$433$	19.4147			0.933013			0.466506	+	0.884518i	$0.345512\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	39.0892	+	22.5682i	1.86989	+	1.07958i
$438$		0			0
$439$	−4.76645	+	2.75191i	−0.227490	+	0.131342i	−0.609414	−	0.792852i	$-0.708595\pi$
$439$	−4.76645	+	2.75191i	−0.227490	+	0.131342i	0.381923	+	0.924194i	$0.375262\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13.3532	−	23.1284i	−0.634430	−	1.09886i	−0.986636	−	0.162942i	$-0.947902\pi$
$443$	−13.3532	−	23.1284i	−0.634430	−	1.09886i	0.352206	−	0.935922i	$-0.385432\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	2.96668i		−	0.140006i	−0.997547	−	0.0700031i	$-0.977699\pi$
$449$		−	2.96668i		−	0.140006i	0.997547	−	0.0700031i	$-0.0223009\pi$
$450$		0			0
$451$	−13.5691	−	7.83413i	−0.638944	−	0.368895i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−23.9939	+	13.8529i	−1.12239	+	0.648010i	−0.942009	−	0.335587i	$-0.891065\pi$
$457$	−23.9939	+	13.8529i	−1.12239	+	0.648010i	−0.180377	+	0.983598i	$0.557732\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	39.3720			1.83374			0.916868	−	0.399190i	$-0.130709\pi$
$461$	39.3720			1.83374			0.916868	+	0.399190i	$0.130709\pi$
$462$		0			0
$463$			9.64212i			0.448107i	0.974577	+	0.224054i	$0.0719291\pi$
$463$			9.64212i			0.448107i	−0.974577	+	0.224054i	$0.928071\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.517993	−	0.299063i	0.0239698	−	0.0138390i	−0.487967	−	0.872862i	$-0.662261\pi$
$467$	0.517993	−	0.299063i	0.0239698	−	0.0138390i	0.511937	+	0.859023i	$0.328928\pi$
$468$		0			0
$469$	6.18498	−	7.77747i	0.285596	−	0.359130i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−11.9041	+	20.6185i	−0.547350	+	0.948037i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.28344	−	16.0794i	0.424171	−	0.734686i	−0.572171	−	0.820134i	$-0.693899\pi$
$479$	9.28344	−	16.0794i	0.424171	−	0.734686i	0.996343	+	0.0854479i	$0.0272321\pi$
$480$		0			0
$481$	−17.0947	+	9.86963i	−0.779451	+	0.450016i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	0.764981	+	0.441662i	0.0346646	+	0.0200136i	0.517232	−	0.855845i	$-0.326962\pi$
$487$	0.764981	+	0.441662i	0.0346646	+	0.0200136i	−0.482568	+	0.875859i	$0.660296\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	6.63816i		−	0.299576i	−0.988718	−	0.149788i	$-0.952141\pi$
$491$		−	6.63816i		−	0.299576i	0.988718	−	0.149788i	$-0.0478591\pi$
$492$		0			0
$493$	−11.1751	+	19.3559i	−0.503302	+	0.871745i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−23.5850	−	3.52195i	−1.05793	−	0.157981i
$498$		0			0
$499$	−15.9390	−	27.6071i	−0.713527	−	1.23587i	−0.963525	−	0.267619i	$-0.913763\pi$
$499$	−15.9390	−	27.6071i	−0.713527	−	1.23587i	0.249997	−	0.968247i	$-0.419570\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			25.2668i			1.12659i	0.826256	+	0.563294i	$0.190466\pi$
$503$			25.2668i			1.12659i	−0.826256	+	0.563294i	$0.809534\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.33523	+	16.1691i	0.413777	+	0.716683i	0.995299	−	0.0968479i	$-0.0308760\pi$
$509$	9.33523	+	16.1691i	0.413777	+	0.716683i	−0.581522	+	0.813530i	$0.697543\pi$
$510$		0			0
$511$	10.1946	+	8.10720i	0.450983	+	0.358641i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	26.2521			1.15457
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3.86329	+	6.69142i	−0.169254	+	0.293156i	−0.938158	−	0.346208i	$-0.887469\pi$
$521$	−3.86329	+	6.69142i	−0.169254	+	0.293156i	0.768904	+	0.639364i	$0.220802\pi$
$522$		0			0
$523$	−6.93885	−	12.0184i	−0.303415	−	0.525530i	0.673492	−	0.739194i	$-0.264793\pi$
$523$	−6.93885	−	12.0184i	−0.303415	−	0.525530i	−0.976907	+	0.213664i	$0.931460\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	15.9187	+	27.5720i	0.693430	+	1.20106i
$528$		0			0
$529$	−20.3752	+	35.2908i	−0.885877	+	1.53438i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	13.1574			0.569911
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−12.0028	−	12.8653i	−0.516996	−	0.554148i
$540$		0			0
$541$	17.0839	+	29.5902i	0.734495	+	1.27218i	0.954945	+	0.296784i	$0.0959143\pi$
$541$	17.0839	+	29.5902i	0.734495	+	1.27218i	−0.220449	+	0.975398i	$0.570752\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			17.3564i			0.742105i	0.928612	+	0.371052i	$0.121003\pi$
$547$			17.3564i			0.742105i	−0.928612	+	0.371052i	$0.878997\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−14.0554	−	24.3446i	−0.598779	−	1.03712i
$552$		0			0
$553$	39.2476	−	15.4654i	1.66898	−	0.657656i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−10.5686	+	18.3054i	−0.447807	+	0.775624i	−0.998243	−	0.0592533i	$-0.981128\pi$
$557$	−10.5686	+	18.3054i	−0.447807	+	0.775624i	0.550436	+	0.834877i	$0.314461\pi$
$558$		0			0
$559$		−	19.9929i		−	0.845609i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6.84471	+	3.95179i	0.288470	+	0.166548i	0.637252	−	0.770656i	$-0.280071\pi$
$563$	6.84471	+	3.95179i	0.288470	+	0.166548i	−0.348782	+	0.937204i	$0.613405\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−28.8526	+	16.6581i	−1.20956	+	0.698343i	−0.962664	−	0.270698i	$-0.912745\pi$
$569$	−28.8526	+	16.6581i	−1.20956	+	0.698343i	−0.246901	+	0.969041i	$0.579412\pi$
$570$		0			0
$571$	14.2121	−	24.6160i	0.594756	−	1.03015i	−0.398825	−	0.917027i	$-0.630582\pi$
$571$	14.2121	−	24.6160i	0.594756	−	1.03015i	0.993581	−	0.113121i	$-0.0360846\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	11.9944	−	20.7750i	0.499335	−	0.864874i	−0.500664	−	0.865642i	$-0.666911\pi$
$577$	11.9944	−	20.7750i	0.499335	−	0.864874i	1.00000		0.000767262i	$0.000244227\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6.01511	−	0.898239i	−0.249549	−	0.0372652i
$582$		0			0
$583$	−16.2961	+	9.40855i	−0.674915	+	0.389662i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	29.3522i		−	1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$587$		−	29.3522i		−	1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$588$		0			0
$589$	−40.0431			−1.64995
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−21.8028	+	12.5879i	−0.895335	+	0.516922i	−0.875684	−	0.482885i	$-0.839589\pi$
$593$	−21.8028	+	12.5879i	−0.895335	+	0.516922i	−0.0196513	+	0.999807i	$0.506256\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	13.1996	+	7.62078i	0.539320	+	0.311377i	0.744803	−	0.667284i	$-0.232543\pi$
$599$	13.1996	+	7.62078i	0.539320	+	0.311377i	−0.205483	+	0.978661i	$0.565877\pi$
$600$		0			0
$601$			35.1651i			1.43441i	0.696861	+	0.717206i	$0.254579\pi$
$601$			35.1651i			1.43441i	−0.696861	+	0.717206i	$0.745421\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	12.5220	+	21.6887i	0.508252	+	0.880318i	0.999954	+	0.00955496i	$0.00304148\pi$
$607$	12.5220	+	21.6887i	0.508252	+	0.880318i	−0.491702	+	0.870763i	$0.663625\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−19.0917	+	11.0226i	−0.772367	+	0.445926i
$612$		0			0
$613$	10.3744	+	5.98969i	0.419020	+	0.241921i	0.694658	−	0.719340i	$-0.255556\pi$
$613$	10.3744	+	5.98969i	0.419020	+	0.241921i	−0.275638	+	0.961262i	$0.588889\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	25.8763			1.04174			0.520871	−	0.853635i	$-0.325607\pi$
$617$	25.8763			1.04174			0.520871	+	0.853635i	$0.325607\pi$
$618$		0			0
$619$	−1.57389	−	0.908684i	−0.0632598	−	0.0365231i	0.468036	−	0.883709i	$-0.344962\pi$
$619$	−1.57389	−	0.908684i	−0.0632598	−	0.0365231i	−0.531296	+	0.847186i	$0.678295\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−43.5903	+	17.1766i	−1.74641	+	0.688166i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	42.0325			1.67595
$630$		0			0
$631$	0.129106			0.00513962			0.00256981	−	0.999997i	$-0.499182\pi$
$631$	0.129106			0.00513962			0.00256981	+	0.999997i	$0.499182\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	14.1308	+	4.31657i	0.559882	+	0.171029i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	28.7294	+	16.5869i	1.13474	+	0.655145i	0.945124	−	0.326713i	$-0.105941\pi$
$641$	28.7294	+	16.5869i	1.13474	+	0.655145i	0.189621	+	0.981857i	$0.439274\pi$
$642$		0			0
$643$	2.44358			0.0963653			0.0481827	−	0.998839i	$-0.484657\pi$
$643$	2.44358			0.0963653			0.0481827	+	0.998839i	$0.484657\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−5.23026	−	3.01969i	−0.205623	−	0.118716i	0.393653	−	0.919259i	$-0.371211\pi$
$647$	−5.23026	−	3.01969i	−0.205623	−	0.118716i	−0.599275	+	0.800543i	$0.704545\pi$
$648$		0			0
$649$	0.819395	−	0.473078i	0.0321641	−	0.0185699i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−17.9294	−	31.0547i	−0.701633	−	1.21526i	−0.967893	−	0.251362i	$-0.919121\pi$
$653$	−17.9294	−	31.0547i	−0.701633	−	1.21526i	0.266260	−	0.963901i	$-0.414212\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	20.3494i		−	0.792702i	−0.918099	−	0.396351i	$-0.870276\pi$
$659$		−	20.3494i		−	0.792702i	0.918099	−	0.396351i	$-0.129724\pi$
$660$		0			0
$661$	6.75143	+	3.89794i	0.262600	+	0.151612i	0.625520	−	0.780208i	$-0.284887\pi$
$661$	6.75143	+	3.89794i	0.262600	+	0.151612i	−0.362920	+	0.931820i	$0.618220\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	34.3842	−	19.8517i	1.33136	−	0.768662i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−24.4154			−0.942547
$672$		0			0
$673$			16.3705i			0.631035i	0.948920	+	0.315517i	$0.102178\pi$
$673$			16.3705i			0.631035i	−0.948920	+	0.315517i	$0.897822\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−7.22825	+	4.17323i	−0.277804	+	0.160390i	−0.632429	−	0.774618i	$-0.717942\pi$
$677$	−7.22825	+	4.17323i	−0.277804	+	0.160390i	0.354625	+	0.935009i	$0.384609\pi$
$678$		0			0
$679$	1.61835	−	10.8374i	0.0621066	−	0.415900i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	5.40196	−	9.35647i	0.206700	−	0.358015i	−0.743973	−	0.668210i	$-0.767061\pi$
$683$	5.40196	−	9.35647i	0.206700	−	0.358015i	0.950673	+	0.310194i	$0.100394\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7.90083	−	13.6846i	0.300998	−	0.521343i
$690$		0			0
$691$	−3.92473	+	2.26594i	−0.149304	+	0.0862006i	−0.572791	−	0.819702i	$-0.694139\pi$
$691$	−3.92473	+	2.26594i	−0.149304	+	0.0862006i	0.423487	+	0.905902i	$0.360806\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−24.2636	−	14.0086i	−0.919051	−	0.530614i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			33.7665i			1.27534i	0.770309	+	0.637671i	$0.220102\pi$
$701$			33.7665i			1.27534i	−0.770309	+	0.637671i	$0.779898\pi$
$702$		0			0
$703$	−26.4329	+	45.7832i	−0.996937	+	1.72675i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−17.7463	−	14.1126i	−0.667418	−	0.530760i
$708$		0			0
$709$	0.0130869	+	0.0226672i	0.000491489	+	0.000851284i	0.866271	−	0.499574i	$-0.166510\pi$
$709$	0.0130869	+	0.0226672i	0.000491489	+	0.000851284i	−0.865780	+	0.500426i	$0.833177\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	56.5566i		−	2.11806i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−14.2861	−	24.7442i	−0.532780	−	0.922803i	−0.999267	−	0.0382745i	$-0.987814\pi$
$719$	−14.2861	−	24.7442i	−0.532780	−	0.922803i	0.466487	−	0.884528i	$-0.345519\pi$
$720$		0			0
$721$	29.7332	−	11.7163i	1.10732	−	0.436337i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−33.4387			−1.24017			−0.620087	−	0.784533i	$-0.712903\pi$
$727$	−33.4387			−1.24017			−0.620087	+	0.784533i	$0.712903\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−21.2863	+	36.8690i	−0.787302	+	1.36365i
$732$		0			0
$733$	10.0900	+	17.4764i	0.372684	+	0.645507i	0.989977	−	0.141226i	$-0.0451043\pi$
$733$	10.0900	+	17.4764i	0.372684	+	0.645507i	−0.617294	+	0.786733i	$0.711771\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.72025	−	8.17571i	−0.173873	−	0.301156i
$738$		0			0
$739$	4.84939	−	8.39940i	0.178388	−	0.308977i	−0.762941	−	0.646469i	$-0.776245\pi$
$739$	4.84939	−	8.39940i	0.178388	−	0.308977i	0.941328	+	0.337492i	$0.109579\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	44.3997			1.62887			0.814434	−	0.580256i	$-0.197048\pi$
$743$	44.3997			1.62887			0.814434	+	0.580256i	$0.197048\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.21503	+	0.478780i	−0.0443963	+	0.0174942i
$750$		0			0
$751$	−11.1840	−	19.3713i	−0.408111	−	0.706869i	0.586567	−	0.809901i	$-0.300479\pi$
$751$	−11.1840	−	19.3713i	−0.408111	−	0.706869i	−0.994678	+	0.103032i	$0.967146\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	50.9667i		−	1.85242i	−0.377011	−	0.926209i	$-0.623048\pi$
$757$		−	50.9667i		−	1.85242i	0.377011	−	0.926209i	$-0.376952\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	19.6328	+	34.0050i	0.711688	+	1.23268i	0.964223	+	0.265092i	$0.0854025\pi$
$761$	19.6328	+	34.0050i	0.711688	+	1.23268i	−0.252535	+	0.967588i	$0.581264\pi$
$762$		0			0
$763$	−11.9794	−	9.52654i	−0.433683	−	0.344884i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.397267	+	0.688087i	−0.0143445	+	0.0248454i
$768$		0			0
$769$		−	3.86290i		−	0.139300i	−0.997572	−	0.0696498i	$-0.977812\pi$
$769$		−	3.86290i		−	0.139300i	0.997572	−	0.0696498i	$-0.0221882\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	41.9601	+	24.2257i	1.50920	+	0.871337i	0.999942	+	0.0107237i	$0.00341351\pi$
$773$	41.9601	+	24.2257i	1.50920	+	0.871337i	0.509258	+	0.860614i	$0.329920\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	30.5173	−	17.6192i	1.09339	−	0.631272i
$780$		0			0
$781$	−11.3275	+	19.6199i	−0.405332	+	0.702055i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	9.09803	−	15.7582i	0.324310	−	0.561721i	−0.657063	−	0.753836i	$-0.728201\pi$
$787$	9.09803	−	15.7582i	0.324310	−	0.561721i	0.981372	+	0.192115i	$0.0615347\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6.67656	−	44.7100i	0.237391	−	1.58970i
$792$		0			0
$793$	17.7560	−	10.2514i	0.630534	−	0.364039i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			30.3824i			1.07620i	0.842881	+	0.538100i	$0.180858\pi$
$797$			30.3824i			1.07620i	−0.842881	+	0.538100i	$0.819142\pi$
$798$		0			0
$799$	46.9427			1.66071
$800$		0			0
$801$		0			0
$802$		0			0
$803$	10.7166	−	6.18724i	0.378181	−	0.218343i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−44.8140	−	25.8734i	−1.57558	−	0.909660i	−0.995466	−	0.0951229i	$-0.969676\pi$
$809$	−44.8140	−	25.8734i	−1.57558	−	0.909660i	−0.580112	−	0.814537i	$-0.696991\pi$
$810$		0			0
$811$		−	19.2500i		−	0.675958i	−0.941154	−	0.337979i	$-0.890257\pi$
$811$		−	19.2500i		−	0.675958i	0.941154	−	0.337979i	$-0.109743\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−26.7726	−	46.3714i	−0.936653	−	1.62233i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	44.2808	−	25.5656i	1.54541	−	0.892244i	0.546929	−	0.837179i	$-0.315797\pi$
$821$	44.2808	−	25.5656i	1.54541	−	0.892244i	0.998483	−	0.0550649i	$-0.0175366\pi$
$822$		0			0
$823$	−36.5195	−	21.0846i	−1.27299	−	0.734962i	−0.297441	−	0.954740i	$-0.596133\pi$
$823$	−36.5195	−	21.0846i	−1.27299	−	0.734962i	−0.975550	+	0.219778i	$0.929467\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	5.16301			0.179536			0.0897678	−	0.995963i	$-0.471388\pi$
$827$	5.16301			0.179536			0.0897678	+	0.995963i	$0.471388\pi$
$828$		0			0
$829$	29.2982	+	16.9153i	1.01757	+	0.587494i	0.913399	−	0.407066i	$-0.133448\pi$
$829$	29.2982	+	16.9153i	1.01757	+	0.587494i	0.104170	+	0.994559i	$0.466781\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−21.4628	−	23.0052i	−0.743642	−	0.797082i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−28.5450			−0.985483			−0.492741	−	0.870176i	$-0.664005\pi$
$839$	−28.5450			−0.985483			−0.492741	+	0.870176i	$0.664005\pi$
$840$		0			0
$841$	4.27283			0.147339
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	11.5249	−	4.54134i	0.395999	−	0.156042i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−64.6640	−	37.3338i	−2.21665	−	1.27978i
$852$		0			0
$853$	−22.2041			−0.760252			−0.380126	−	0.924935i	$-0.624119\pi$
$853$	−22.2041			−0.760252			−0.380126	+	0.924935i	$0.624119\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−14.5811	−	8.41843i	−0.498083	−	0.287568i	0.229839	−	0.973229i	$-0.426180\pi$
$857$	−14.5811	−	8.41843i	−0.498083	−	0.287568i	−0.727921	+	0.685661i	$0.759513\pi$
$858$		0			0
$859$	41.7832	−	24.1236i	1.42562	−	0.823085i	0.428853	−	0.903374i	$-0.358918\pi$
$859$	41.7832	−	24.1236i	1.42562	−	0.823085i	0.996772	+	0.0802893i	$0.0255844\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−7.40921	−	12.8331i	−0.252213	−	0.436845i	0.711922	−	0.702258i	$-0.247825\pi$
$863$	−7.40921	−	12.8331i	−0.252213	−	0.436845i	−0.964135	+	0.265413i	$0.914492\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	40.0772i		−	1.35953i
$870$		0			0
$871$	6.86555	+	3.96383i	0.232630	+	0.134309i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	10.5951	−	6.11711i	0.357773	−	0.206560i	−0.310331	−	0.950629i	$-0.600440\pi$
$877$	10.5951	−	6.11711i	0.357773	−	0.206560i	0.668103	+	0.744069i	$0.267106\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	39.7904			1.34057			0.670286	−	0.742103i	$-0.266172\pi$
$881$	39.7904			1.34057			0.670286	+	0.742103i	$0.266172\pi$
$882$		0			0
$883$		−	34.7168i		−	1.16831i	−0.811641	−	0.584156i	$-0.801426\pi$
$883$		−	34.7168i		−	1.16831i	0.811641	−	0.584156i	$-0.198574\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.818112	−	0.472337i	0.0274695	−	0.0158595i	−0.486202	−	0.873846i	$-0.661618\pi$
$887$	0.818112	−	0.472337i	0.0274695	−	0.0158595i	0.513672	+	0.857987i	$0.328285\pi$
$888$		0			0
$889$	14.8001	+	2.21011i	0.496381	+	0.0741248i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−29.5208	+	51.1315i	−0.987876	+	1.71105i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−17.6116	+	30.5042i	−0.587381	+	1.01737i
$900$		0			0
$901$	−29.1399	+	16.8239i	−0.970790	+	0.560486i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	21.4568	+	12.3881i	0.712462	+	0.411340i	0.811972	−	0.583697i	$-0.198394\pi$
$907$	21.4568	+	12.3881i	0.712462	+	0.411340i	−0.0995103	+	0.995037i	$0.531728\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	14.5204i		−	0.481083i	−0.970639	−	0.240542i	$-0.922675\pi$
$911$		−	14.5204i		−	0.481083i	0.970639	−	0.240542i	$-0.0773251\pi$
$912$		0			0
$913$	−2.88898	+	5.00386i	−0.0956112	+	0.165603i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.73094	+	1.07612i	−0.0901835	+	0.0355365i
$918$		0			0
$919$	3.49682	+	6.05667i	0.115350	+	0.199791i	0.917919	−	0.396767i	$-0.129868\pi$
$919$	3.49682	+	6.05667i	0.115350	+	0.199791i	−0.802570	+	0.596558i	$0.796535\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	19.0246i		−	0.626203i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−7.38153	−	12.7852i	−0.242180	−	0.419468i	0.719155	−	0.694850i	$-0.244529\pi$
$929$	−7.38153	−	12.7852i	−0.242180	−	0.419468i	−0.961335	+	0.275381i	$0.911196\pi$
$930$		0			0
$931$	38.5552	−	8.91076i	1.26360	−	0.292038i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−48.8385			−1.59549			−0.797743	−	0.602998i	$-0.793973\pi$
$937$	−48.8385			−1.59549			−0.797743	+	0.602998i	$0.793973\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−14.0092	+	24.2646i	−0.456686	+	0.791004i	−0.998783	−	0.0493121i	$-0.984297\pi$
$941$	−14.0092	+	24.2646i	−0.456686	+	0.791004i	0.542097	+	0.840316i	$0.317630\pi$
$942$		0			0
$943$	24.8852	+	43.1024i	0.810374	+	1.40361i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.28478	−	2.22531i	−0.0417499	−	0.0723129i	0.844395	−	0.535720i	$-0.179960\pi$
$947$	−1.28478	−	2.22531i	−0.0417499	−	0.0723129i	−0.886145	+	0.463408i	$0.846627\pi$
$948$		0			0
$949$	−5.19574	+	8.99928i	−0.168661	+	0.292129i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	12.9268			0.418740			0.209370	−	0.977837i	$-0.432859\pi$
$953$	12.9268			0.418740			0.209370	+	0.977837i	$0.432859\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	0.190553	+	0.151536i	0.00615329	+	0.00489336i
$960$		0			0
$961$	9.58735	+	16.6058i	0.309269	+	0.535670i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$			28.4481i			0.914829i	0.889254	+	0.457414i	$0.151224\pi$
$967$			28.4481i			0.914829i	−0.889254	+	0.457414i	$0.848776\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−15.4575	−	26.7732i	−0.496054	−	0.859191i	0.503935	−	0.863741i	$-0.331885\pi$
$971$	−15.4575	−	26.7732i	−0.496054	−	0.859191i	−0.999990	+	0.00455008i	$0.998552\pi$
$972$		0			0
$973$	−25.5198	−	3.81089i	−0.818129	−	0.122172i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−25.2992	+	43.8195i	−0.809393	+	1.40191i	0.103892	+	0.994589i	$0.466870\pi$
$977$	−25.2992	+	43.8195i	−0.809393	+	1.40191i	−0.913285	+	0.407321i	$0.866463\pi$
$978$		0			0
$979$			44.5116i			1.42260i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	20.0779	+	11.5920i	0.640385	+	0.369726i	0.784763	−	0.619796i	$-0.212785\pi$
$983$	20.0779	+	11.5920i	0.640385	+	0.369726i	−0.144378	+	0.989523i	$0.546118\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	65.4948	−	37.8134i	2.08261	−	1.20240i
$990$		0			0
$991$	8.15946	−	14.1326i	0.259194	−	0.448937i	−0.706832	−	0.707381i	$-0.749876\pi$
$991$	8.15946	−	14.1326i	0.259194	−	0.448937i	0.966026	+	0.258444i	$0.0832098\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−27.5686	+	47.7502i	−0.873106	+	1.51226i	−0.0143390	+	0.999897i	$0.504564\pi$
$997$	−27.5686	+	47.7502i	−0.873106	+	1.51226i	−0.858767	+	0.512367i	$0.828769\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.dd.c.4049.4		24
3.2	odd	2		6300.2.dd.b.4049.4		24
5.2	odd	4		1260.2.cg.a.521.2	yes	12
5.3	odd	4		6300.2.ch.c.4301.5		12
5.4	even	2	inner	6300.2.dd.c.4049.9		24
7.5	odd	6		6300.2.dd.b.1349.9		24
15.2	even	4		1260.2.cg.b.521.2	yes	12
15.8	even	4		6300.2.ch.b.4301.5		12
15.14	odd	2		6300.2.dd.b.4049.9		24
21.5	even	6	inner	6300.2.dd.c.1349.9		24
35.12	even	12		1260.2.cg.b.341.2	yes	12
35.17	even	12		8820.2.d.a.881.3		12
35.19	odd	6		6300.2.dd.b.1349.4		24
35.32	odd	12		8820.2.d.b.881.3		12
35.33	even	12		6300.2.ch.b.1601.5		12
105.17	odd	12		8820.2.d.b.881.10		12
105.32	even	12		8820.2.d.a.881.10		12
105.47	odd	12		1260.2.cg.a.341.2	✓	12
105.68	odd	12		6300.2.ch.c.1601.5		12
105.89	even	6	inner	6300.2.dd.c.1349.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.cg.a.341.2	✓	12	105.47	odd	12
1260.2.cg.a.521.2	yes	12	5.2	odd	4
1260.2.cg.b.341.2	yes	12	35.12	even	12
1260.2.cg.b.521.2	yes	12	15.2	even	4
6300.2.ch.b.1601.5		12	35.33	even	12
6300.2.ch.b.4301.5		12	15.8	even	4
6300.2.ch.c.1601.5		12	105.68	odd	12
6300.2.ch.c.4301.5		12	5.3	odd	4
6300.2.dd.b.1349.4		24	35.19	odd	6
6300.2.dd.b.1349.9		24	7.5	odd	6
6300.2.dd.b.4049.4		24	3.2	odd	2
6300.2.dd.b.4049.9		24	15.14	odd	2
6300.2.dd.c.1349.4		24	105.89	even	6	inner
6300.2.dd.c.1349.9		24	21.5	even	6	inner
6300.2.dd.c.4049.4		24	1.1	even	1	trivial
6300.2.dd.c.4049.9		24	5.4	even	2	inner
8820.2.d.a.881.3		12	35.17	even	12
8820.2.d.a.881.10		12	105.32	even	12
8820.2.d.b.881.3		12	35.32	odd	12
8820.2.d.b.881.10		12	105.17	odd	12

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:	\( N \)	\(=\)	\( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6300.dd (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.390758 + 2.61674i) q^{7} +O(q^{10})\)	\(q+(-0.390758 + 2.61674i) q^{7} +(2.17681 + 1.25678i) q^{11} -2.11077 q^{13} +(3.89248 + 2.24732i) q^{17} +(-4.89571 + 2.82654i) q^{19} +(-3.99219 - 6.91468i) q^{23} +4.97264i q^{29} +(6.13441 + 3.54171i) q^{31} +(8.09881 - 4.67585i) q^{37} -6.23347 q^{41} +9.47185i q^{43} +(9.04490 - 5.22208i) q^{47} +(-6.69462 - 2.04502i) q^{49} +(-3.74310 + 6.48325i) q^{53} +(0.188210 - 0.325989i) q^{59} +(-8.41210 + 4.85673i) q^{61} +(-3.25263 - 1.87791i) q^{67} +9.01312i q^{71} +(2.46154 - 4.26351i) q^{73} +(-4.13928 + 5.20505i) q^{77} +(-7.97217 - 13.8082i) q^{79} +2.29871i q^{83} +(8.85428 + 15.3361i) q^{89} +(0.824800 - 5.52332i) q^{91} -4.14156 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 24 q^{11} + 12 q^{19} - 12 q^{31} - 16 q^{41} - 44 q^{49} - 28 q^{79} + 40 q^{89} + 20 q^{91}+O(q^{100}) \) 24 * q + 24 * q^11 + 12 * q^19 - 12 * q^31 - 16 * q^41 - 44 * q^49 - 28 * q^79 + 40 * q^89 + 20 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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