LMFDB - Embedded newform 6300.2.ch.d.1601.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6300,2,Mod(1601,6300)] 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("6300.1601"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6300, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 0, 5])) 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.ch (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0,0,-4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$50.3057532734$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 32 x^{18} - 124 x^{16} + 5094 x^{14} + 61094 x^{12} + 245850 x^{10} + 420152 x^{8} + \cdots + 15876 $ x^20 - 32x^18 - 124x^16 + 5094x^14 + 61094x^12 + 245850x^10 + 420152x^8 + 466066x^6 + 1736173x^4 - 23778*x^2 + 15876
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1601.5
Root			$0.227585 + 0.209112i$ of defining polynomial
Character	$\chi$	$=$	6300.1601
Dual form			6300.2.ch.d.4301.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.651797 + 2.56421i) q^{7} +(-0.362193 + 0.209112i) q^{11} +1.12703i q^{13} +(2.46721 + 4.27334i) q^{17} +(4.27648 + 2.46903i) q^{19} +(6.18653 + 3.57179i) q^{23} +4.95212i q^{29} +(-5.51945 + 3.18666i) q^{31} +(4.42041 - 7.65637i) q^{37} -12.1871 q^{41} +6.05128 q^{43} +(3.85392 - 6.67519i) q^{47} +(-6.15032 - 3.34269i) q^{49} +(-8.93951 + 5.16123i) q^{53} +(-0.569345 - 0.986135i) q^{59} +(10.9401 + 6.31629i) q^{61} +(-5.14561 - 8.91246i) q^{67} -4.87068i q^{71} +(4.00954 - 2.31491i) q^{73} +(-0.300131 - 1.06504i) q^{77} +(2.38171 - 4.12524i) q^{79} -12.1871 q^{83} +(2.33261 - 4.04019i) q^{89} +(-2.88994 - 0.734595i) q^{91} +7.49558i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{7} + 6 q^{31} - 2 q^{37} + 20 q^{43} + 4 q^{49} + 6 q^{61} - 28 q^{67} + 6 q^{73} + 2 q^{79} - 40 q^{91}+O(q^{100}) $ 20 * q - 4 * q^7 + 6 * q^31 - 2 * q^37 + 20 * q^43 + 4 * q^49 + 6 * q^61 - 28 * q^67 + 6 * q^73 + 2 * q^79 - 40 * q^91

Character values

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

$n$	$2801$	$3151$	$3277$	$3601$
$\chi(n)$	$-1$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$7$	−0.651797	+	2.56421i	−0.246356	+	0.969179i
$7$	−0.651797	+	2.56421i	−0.246356	+	0.969179i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.362193	+	0.209112i	−0.109205	+	0.0630497i	−0.553608	−	0.832778i	$-0.686749\pi$
$11$	−0.362193	+	0.209112i	−0.109205	+	0.0630497i	0.444403	+	0.895827i	$0.353416\pi$
$12$		0			0
$13$			1.12703i			0.312582i	0.987711	+	0.156291i	$0.0499538\pi$
$13$			1.12703i			0.312582i	−0.987711	+	0.156291i	$0.950046\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.46721	+	4.27334i	0.598387	+	1.03644i	0.993059	+	0.117615i	$0.0375248\pi$
$17$	2.46721	+	4.27334i	0.598387	+	1.03644i	−0.394672	+	0.918822i	$0.629142\pi$
$18$		0			0
$19$	4.27648	+	2.46903i	0.981091	+	0.566433i	0.902599	−	0.430481i	$-0.141656\pi$
$19$	4.27648	+	2.46903i	0.981091	+	0.566433i	0.0784919	+	0.996915i	$0.474990\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.18653	+	3.57179i	1.28998	+	0.744771i	0.978651	−	0.205531i	$-0.0658923\pi$
$23$	6.18653	+	3.57179i	1.28998	+	0.744771i	0.311330	+	0.950302i	$0.399226\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$			4.95212i			0.919585i	0.888026	+	0.459793i	$0.152076\pi$
$29$			4.95212i			0.919585i	−0.888026	+	0.459793i	$0.847924\pi$
$30$		0			0
$31$	−5.51945	+	3.18666i	−0.991323	+	0.572340i	−0.905669	−	0.423985i	$-0.860631\pi$
$31$	−5.51945	+	3.18666i	−0.991323	+	0.572340i	−0.0856532	+	0.996325i	$0.527298\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.42041	−	7.65637i	0.726711	−	1.25870i	−0.231555	−	0.972822i	$-0.574381\pi$
$37$	4.42041	−	7.65637i	0.726711	−	1.25870i	0.958266	−	0.285878i	$-0.0922853\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−12.1871			−1.90331			−0.951653	−	0.307176i	$-0.900616\pi$
$41$	−12.1871			−1.90331			−0.951653	+	0.307176i	$0.900616\pi$
$42$		0			0
$43$	6.05128			0.922811			0.461406	−	0.887189i	$-0.347345\pi$
$43$	6.05128			0.922811			0.461406	+	0.887189i	$0.347345\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.85392	−	6.67519i	0.562152	−	0.973676i	−0.435156	−	0.900355i	$-0.643307\pi$
$47$	3.85392	−	6.67519i	0.562152	−	0.973676i	0.997308	−	0.0733214i	$-0.0233599\pi$
$48$		0			0
$49$	−6.15032	−	3.34269i	−0.878617	−	0.477527i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−8.93951	+	5.16123i	−1.22794	+	0.708950i	−0.966598	−	0.256298i	$-0.917497\pi$
$53$	−8.93951	+	5.16123i	−1.22794	+	0.708950i	−0.261339	+	0.965247i	$0.584164\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.569345	−	0.986135i	−0.0741224	−	0.128384i	0.826582	−	0.562816i	$-0.190282\pi$
$59$	−0.569345	−	0.986135i	−0.0741224	−	0.128384i	−0.900704	+	0.434433i	$0.856949\pi$
$60$		0			0
$61$	10.9401	+	6.31629i	1.40074	+	0.808719i	0.994469	−	0.105033i	$-0.0334948\pi$
$61$	10.9401	+	6.31629i	1.40074	+	0.808719i	0.406273	+	0.913752i	$0.366828\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−5.14561	−	8.91246i	−0.628636	−	1.08883i	−0.987826	−	0.155565i	$-0.950280\pi$
$67$	−5.14561	−	8.91246i	−0.628636	−	1.08883i	0.359190	−	0.933265i	$-0.383053\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.87068i		−	0.578043i	−0.957323	−	0.289022i	$-0.906670\pi$
$71$		−	4.87068i		−	0.578043i	0.957323	−	0.289022i	$-0.0933299\pi$
$72$		0			0
$73$	4.00954	−	2.31491i	0.469281	−	0.270940i	−0.246657	−	0.969103i	$-0.579332\pi$
$73$	4.00954	−	2.31491i	0.469281	−	0.270940i	0.715939	+	0.698163i	$0.245999\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.300131	−	1.06504i	−0.0342031	−	0.121372i
$78$		0			0
$79$	2.38171	−	4.12524i	0.267963	−	0.464126i	−0.700373	−	0.713777i	$-0.746983\pi$
$79$	2.38171	−	4.12524i	0.267963	−	0.464126i	0.968336	+	0.249652i	$0.0803161\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−12.1871			−1.33771			−0.668854	−	0.743394i	$-0.733215\pi$
$83$	−12.1871			−1.33771			−0.668854	+	0.743394i	$0.733215\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.33261	−	4.04019i	0.247256	−	0.428260i	−0.715508	−	0.698605i	$-0.753805\pi$
$89$	2.33261	−	4.04019i	0.247256	−	0.428260i	0.962763	+	0.270345i	$0.0871379\pi$
$90$		0			0
$91$	−2.88994	−	0.734595i	−0.302948	−	0.0770065i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$			7.49558i			0.761061i	0.924768	+	0.380531i	$0.124259\pi$
$97$			7.49558i			0.761061i	−0.924768	+	0.380531i	$0.875741\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.66219	+	2.87900i	0.165394	+	0.286471i	0.936795	−	0.349878i	$-0.113777\pi$
$101$	1.66219	+	2.87900i	0.165394	+	0.286471i	−0.771401	+	0.636349i	$0.780444\pi$
$102$		0			0
$103$	−0.913253	−	0.527267i	−0.0899855	−	0.0519532i	0.454332	−	0.890832i	$-0.349878\pi$
$103$	−0.913253	−	0.527267i	−0.0899855	−	0.0519532i	−0.544318	+	0.838879i	$0.683211\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.0683306	−	0.0394507i	−0.00660577	−	0.00381384i	0.496693	−	0.867926i	$-0.334547\pi$
$107$	−0.0683306	−	0.0394507i	−0.00660577	−	0.00381384i	−0.503299	+	0.864112i	$0.667881\pi$
$108$		0			0
$109$	4.94437	+	8.56391i	0.473585	+	0.820273i	0.999543	−	0.0302374i	$-0.00962633\pi$
$109$	4.94437	+	8.56391i	0.473585	+	0.820273i	−0.525958	+	0.850511i	$0.676293\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	2.67921i		−	0.252039i	−0.992028	−	0.126019i	$-0.959780\pi$
$113$		−	2.67921i		−	0.252039i	0.992028	−	0.126019i	$-0.0402202\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−12.5659	+	3.54110i	−1.15191	+	0.324612i
$120$		0			0
$121$	−5.41254	+	9.37480i	−0.492049	+	0.852255i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−9.37526			−0.831920			−0.415960	−	0.909383i	$-0.636554\pi$
$127$	−9.37526			−0.831920			−0.415960	+	0.909383i	$0.636554\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−10.7363	+	18.5958i	−0.938036	+	1.62473i	−0.168908	+	0.985632i	$0.554024\pi$
$131$	−10.7363	+	18.5958i	−0.938036	+	1.62473i	−0.769128	+	0.639094i	$0.779309\pi$
$132$		0			0
$133$	−9.11849	+	9.35647i	−0.790673	+	0.811309i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	15.4767	−	8.93547i	1.32226	−	0.763409i	0.338174	−	0.941084i	$-0.390191\pi$
$137$	15.4767	−	8.93547i	1.32226	−	0.763409i	0.984089	+	0.177675i	$0.0568575\pi$
$138$		0			0
$139$			6.19154i			0.525160i	0.964910	+	0.262580i	$0.0845733\pi$
$139$			6.19154i			0.525160i	−0.964910	+	0.262580i	$0.915427\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.235676	−	0.408202i	−0.0197082	−	0.0341356i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.561255	+	0.324041i	0.0459798	+	0.0265465i	0.522814	−	0.852447i	$-0.324882\pi$
$149$	0.561255	+	0.324041i	0.0459798	+	0.0265465i	−0.476834	+	0.878993i	$0.658216\pi$
$150$		0			0
$151$	7.05844	+	12.2256i	0.574407	+	0.994903i	0.996106	+	0.0881666i	$0.0281008\pi$
$151$	7.05844	+	12.2256i	0.574407	+	0.994903i	−0.421698	+	0.906736i	$0.638566\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	5.19873	−	3.00149i	0.414904	−	0.239545i	−0.277991	−	0.960584i	$-0.589668\pi$
$157$	5.19873	−	3.00149i	0.414904	−	0.239545i	0.692895	+	0.721039i	$0.256335\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−13.1912	+	13.5355i	−1.03961	+	1.06674i
$162$		0			0
$163$	−7.16338	+	12.4073i	−0.561079	+	0.971818i	0.436323	+	0.899790i	$0.356280\pi$
$163$	−7.16338	+	12.4073i	−0.561079	+	0.971818i	−0.997403	+	0.0720280i	$0.977053\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	10.6425			0.823544			0.411772	−	0.911287i	$-0.364910\pi$
$167$	10.6425			0.823544			0.411772	+	0.911287i	$0.364910\pi$
$168$		0			0
$169$	11.7298			0.902293
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.15899	+	3.73948i	−0.164145	+	0.284308i	−0.936351	−	0.351064i	$-0.885820\pi$
$173$	−2.15899	+	3.73948i	−0.164145	+	0.284308i	0.772206	+	0.635372i	$0.219153\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−16.8816	+	9.74662i	−1.26179	+	0.728496i	−0.973421	−	0.229022i	$-0.926447\pi$
$179$	−16.8816	+	9.74662i	−1.26179	+	0.728496i	−0.288372	+	0.957519i	$0.593114\pi$
$180$		0			0
$181$			5.30351i			0.394206i	0.980383	+	0.197103i	$0.0631534\pi$
$181$			5.30351i			0.394206i	−0.980383	+	0.197103i	$0.936847\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.78721	−	1.03185i	−0.130694	−	0.0754562i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−18.5745	−	10.7240i	−1.34400	−	0.775961i	−0.356612	−	0.934253i	$-0.616068\pi$
$191$	−18.5745	−	10.7240i	−1.34400	−	0.775961i	−0.987392	+	0.158291i	$0.949402\pi$
$192$		0			0
$193$	9.78531	+	16.9487i	0.704362	+	1.21999i	0.966921	+	0.255075i	$0.0821002\pi$
$193$	9.78531	+	16.9487i	0.704362	+	1.21999i	−0.262559	+	0.964916i	$0.584566\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			5.33697i			0.380244i	0.981761	+	0.190122i	$0.0608883\pi$
$197$			5.33697i			0.380244i	−0.981761	+	0.190122i	$0.939112\pi$
$198$		0			0
$199$	−20.9563	+	12.0991i	−1.48555	+	0.857683i	−0.999865	−	0.0164499i	$-0.994764\pi$
$199$	−20.9563	+	12.0991i	−1.48555	+	0.857683i	−0.485686	+	0.874133i	$0.661430\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−12.6983	−	3.22778i	−0.891243	−	0.226546i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.06521			−0.142854
$210$		0			0
$211$	18.6200			1.28185			0.640926	−	0.767603i	$-0.278551\pi$
$211$	18.6200			1.28185			0.640926	+	0.767603i	$0.278551\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−4.57368	−	16.2301i	−0.310482	−	1.10177i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.81618	+	2.78062i	−0.323972	+	0.187045i
$222$		0			0
$223$			5.73344i			0.383939i	0.981401	+	0.191970i	$0.0614875\pi$
$223$			5.73344i			0.383939i	−0.981401	+	0.191970i	$0.938512\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.59091	−	4.48759i	−0.171965	−	0.297852i	0.767142	−	0.641478i	$-0.221678\pi$
$227$	−2.59091	−	4.48759i	−0.171965	−	0.297852i	−0.939107	+	0.343626i	$0.888345\pi$
$228$		0			0
$229$	−7.74178	−	4.46972i	−0.511591	−	0.295367i	0.221896	−	0.975070i	$-0.428775\pi$
$229$	−7.74178	−	4.46972i	−0.511591	−	0.295367i	−0.733487	+	0.679703i	$0.762109\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−20.4724	−	11.8197i	−1.34119	−	0.774337i	−0.354208	−	0.935167i	$-0.615250\pi$
$233$	−20.4724	−	11.8197i	−1.34119	−	0.774337i	−0.986982	+	0.160830i	$0.948583\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	14.6720i		−	0.949055i	−0.880241	−	0.474527i	$-0.842619\pi$
$239$		−	14.6720i		−	0.949055i	0.880241	−	0.474527i	$-0.157381\pi$
$240$		0			0
$241$	15.1765	−	8.76217i	0.977606	−	0.564421i	0.0760595	−	0.997103i	$-0.475766\pi$
$241$	15.1765	−	8.76217i	0.977606	−	0.564421i	0.901547	+	0.432682i	$0.142433\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−2.78267	+	4.81972i	−0.177057	+	0.306671i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−15.8640			−1.00132			−0.500662	−	0.865643i	$-0.666910\pi$
$251$	−15.8640			−1.00132			−0.500662	+	0.865643i	$0.666910\pi$
$252$		0			0
$253$	−2.98762			−0.187830
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.59575	+	4.49596i	−0.161918	+	0.280451i	−0.935557	−	0.353177i	$-0.885101\pi$
$257$	−2.59575	+	4.49596i	−0.161918	+	0.280451i	0.773638	+	0.633627i	$0.218435\pi$
$258$		0			0
$259$	16.7513	+	16.3253i	1.04088	+	1.01440i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.12333	−	2.38061i	0.254255	−	0.146794i	−0.367456	−	0.930041i	$-0.619771\pi$
$263$	4.12333	−	2.38061i	0.254255	−	0.146794i	0.621711	+	0.783247i	$0.286438\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5.41706	−	9.38263i	−0.330284	−	0.572069i	0.652283	−	0.757975i	$-0.273811\pi$
$269$	−5.41706	−	9.38263i	−0.330284	−	0.572069i	−0.982567	+	0.185906i	$0.940478\pi$
$270$		0			0
$271$	0.474762	+	0.274104i	0.0288397	+	0.0166506i	0.514351	−	0.857580i	$-0.328033\pi$
$271$	0.474762	+	0.274104i	0.0288397	+	0.0166506i	−0.485511	+	0.874231i	$0.661366\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−2.35922	−	4.08629i	−0.141752	−	0.245522i	0.786405	−	0.617712i	$-0.211940\pi$
$277$	−2.35922	−	4.08629i	−0.141752	−	0.245522i	−0.928156	+	0.372190i	$0.878607\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			31.5915i			1.88459i	0.334784	+	0.942295i	$0.391337\pi$
$281$			31.5915i			1.88459i	−0.334784	+	0.942295i	$0.608663\pi$
$282$		0			0
$283$	0.289624	−	0.167214i	0.0172164	−	0.00993987i	−0.491367	−	0.870953i	$-0.663503\pi$
$283$	0.289624	−	0.167214i	0.0172164	−	0.00993987i	0.508583	+	0.861013i	$0.330169\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	7.94352	−	31.2503i	0.468891	−	1.84464i
$288$		0			0
$289$	−3.67428	+	6.36405i	−0.216134	+	0.374356i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−24.6272			−1.43874			−0.719369	−	0.694628i	$-0.755569\pi$
$293$	−24.6272			−1.43874			−0.719369	+	0.694628i	$0.755569\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.02552	+	6.97241i	−0.232802	+	0.403225i
$300$		0			0
$301$	−3.94421	+	15.5167i	−0.227340	+	0.894370i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	23.0923i		−	1.31795i	−0.752167	−	0.658973i	$-0.770991\pi$
$307$		−	23.0923i		−	1.31795i	0.752167	−	0.658973i	$-0.229009\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−0.167345	−	0.289850i	−0.00948927	−	0.0164359i	0.861242	−	0.508195i	$-0.169687\pi$
$311$	−0.167345	−	0.289850i	−0.00948927	−	0.0164359i	−0.870731	+	0.491760i	$0.836354\pi$
$312$		0			0
$313$	10.3133	+	5.95439i	0.582942	+	0.336562i	0.762302	−	0.647222i	$-0.224069\pi$
$313$	10.3133	+	5.95439i	0.582942	+	0.336562i	−0.179360	+	0.983784i	$0.557403\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	16.8665	+	9.73787i	0.947316	+	0.546933i	0.892246	−	0.451550i	$-0.149129\pi$
$317$	16.8665	+	9.73787i	0.947316	+	0.546933i	0.0550697	+	0.998483i	$0.482462\pi$
$318$		0			0
$319$	−1.03555	−	1.79362i	−0.0579796	−	0.100424i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			24.3665i			1.35579i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	14.6046	+	14.2331i	0.805177	+	0.784698i
$330$		0			0
$331$	−12.0359	+	20.8467i	−0.661551	+	1.14584i	0.318657	+	0.947870i	$0.396768\pi$
$331$	−12.0359	+	20.8467i	−0.661551	+	1.14584i	−0.980208	+	0.197970i	$0.936565\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	3.09864			0.168794			0.0843968	−	0.996432i	$-0.473104\pi$
$337$	3.09864			0.168794			0.0843968	+	0.996432i	$0.473104\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.33274	−	2.30837i	0.0721718	−	0.125005i
$342$		0			0
$343$	12.5801	−	13.5919i	0.679262	−	0.733896i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	20.9251	−	12.0811i	1.12332	−	0.648548i	0.181072	−	0.983470i	$-0.442043\pi$
$347$	20.9251	−	12.0811i	1.12332	−	0.648548i	0.942246	+	0.334922i	$0.108710\pi$
$348$		0			0
$349$		−	17.6656i		−	0.945619i	−0.881165	−	0.472810i	$-0.843240\pi$
$349$		−	17.6656i		−	0.945619i	0.881165	−	0.472810i	$-0.156760\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.64565	+	14.9747i	0.460162	+	0.797023i	0.998969	−	0.0454060i	$-0.0144581\pi$
$353$	8.64565	+	14.9747i	0.460162	+	0.797023i	−0.538807	+	0.842429i	$0.681125\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7.44547	−	4.29864i	−0.392957	−	0.226874i	0.290484	−	0.956880i	$-0.406184\pi$
$359$	−7.44547	−	4.29864i	−0.392957	−	0.226874i	−0.683440	+	0.730006i	$0.739517\pi$
$360$		0			0
$361$	2.69218	+	4.66299i	0.141693	+	0.245420i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	12.1984	−	7.04273i	0.636749	−	0.367627i	−0.146612	−	0.989194i	$-0.546837\pi$
$367$	12.1984	−	7.04273i	0.636749	−	0.367627i	0.783361	+	0.621567i	$0.213503\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7.40771	−	26.2868i	−0.384589	−	1.36475i
$372$		0			0
$373$	4.65651	−	8.06531i	0.241105	−	0.417606i	−0.719924	−	0.694053i	$-0.755824\pi$
$373$	4.65651	−	8.06531i	0.241105	−	0.417606i	0.961029	+	0.276447i	$0.0891569\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.58119			−0.287446
$378$		0			0
$379$	−10.1796			−0.522893			−0.261446	−	0.965218i	$-0.584199\pi$
$379$	−10.1796			−0.522893			−0.261446	+	0.965218i	$0.584199\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.37425	+	16.2367i	−0.479002	+	0.829656i	−0.999710	−	0.0240791i	$-0.992335\pi$
$383$	−9.37425	+	16.2367i	−0.479002	+	0.829656i	0.520708	+	0.853735i	$0.325668\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.245202	−	0.141567i	0.0124322	−	0.00717774i	−0.493771	−	0.869592i	$-0.664382\pi$
$389$	0.245202	−	0.141567i	0.0124322	−	0.00717774i	0.506203	+	0.862414i	$0.331049\pi$
$390$		0			0
$391$			35.2495i			1.78264i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−8.24390	−	4.75962i	−0.413750	−	0.238878i	0.278650	−	0.960393i	$-0.410113\pi$
$397$	−8.24390	−	4.75962i	−0.413750	−	0.238878i	−0.692400	+	0.721514i	$0.743446\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6.39612	−	3.69280i	−0.319407	−	0.184410i	0.331721	−	0.943377i	$-0.392371\pi$
$401$	−6.39612	−	3.69280i	−0.319407	−	0.184410i	−0.651128	+	0.758968i	$0.725704\pi$
$402$		0			0
$403$	−3.59146	−	6.22059i	−0.178903	−	0.309870i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.69745i			0.183276i
$408$		0			0
$409$	27.2122	−	15.7110i	1.34556	−	0.776857i	0.357940	−	0.933745i	$-0.383479\pi$
$409$	27.2122	−	15.7110i	1.34556	−	0.776857i	0.987616	+	0.156887i	$0.0501460\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.89975	−	0.817159i	0.142687	−	0.0402098i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−13.3040			−0.649942			−0.324971	−	0.945724i	$-0.605355\pi$
$419$	−13.3040			−0.649942			−0.324971	+	0.945724i	$0.605355\pi$
$420$		0			0
$421$	−11.2491			−0.548249			−0.274124	−	0.961694i	$-0.588388\pi$
$421$	−11.2491			−0.548249			−0.274124	+	0.961694i	$0.588388\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−23.3270	+	23.9359i	−1.12887	+	1.15834i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−33.2897	+	19.2198i	−1.60351	+	0.925786i	−0.612730	+	0.790292i	$0.709929\pi$
$431$	−33.2897	+	19.2198i	−1.60351	+	0.925786i	−0.990778	+	0.135494i	$0.956738\pi$
$432$		0			0
$433$			19.2100i			0.923176i	0.887095	+	0.461588i	$0.152720\pi$
$433$			19.2100i			0.923176i	−0.887095	+	0.461588i	$0.847280\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	17.6377	+	30.5494i	0.843726	+	1.46138i
$438$		0			0
$439$	1.43555	+	0.828815i	0.0685151	+	0.0395572i	0.533866	−	0.845569i	$-0.320739\pi$
$439$	1.43555	+	0.828815i	0.0685151	+	0.0395572i	−0.465351	+	0.885126i	$0.654072\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−17.8422	−	10.3012i	−0.847710	−	0.489426i	0.0121675	−	0.999926i	$-0.496127\pi$
$443$	−17.8422	−	10.3012i	−0.847710	−	0.489426i	−0.859878	+	0.510500i	$0.829460\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			25.8309i			1.21904i	0.792772	+	0.609519i	$0.208637\pi$
$449$			25.8309i			1.21904i	−0.792772	+	0.609519i	$0.791363\pi$
$450$		0			0
$451$	4.41408	−	2.54847i	0.207851	−	0.120003i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−2.28542	+	3.95846i	−0.106907	+	0.185169i	−0.914516	−	0.404550i	$-0.867428\pi$
$457$	−2.28542	+	3.95846i	−0.106907	+	0.185169i	0.807609	+	0.589719i	$0.200762\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−29.8927			−1.39224			−0.696121	−	0.717924i	$-0.745092\pi$
$461$	−29.8927			−1.39224			−0.696121	+	0.717924i	$0.745092\pi$
$462$		0			0
$463$	8.44194			0.392330			0.196165	−	0.980571i	$-0.437151\pi$
$463$	8.44194			0.392330			0.196165	+	0.980571i	$0.437151\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.99480	+	6.91919i	−0.184857	+	0.320182i	−0.943528	−	0.331292i	$-0.892516\pi$
$467$	−3.99480	+	6.91919i	−0.184857	+	0.320182i	0.758671	+	0.651474i	$0.225849\pi$
$468$		0			0
$469$	26.2073	−	7.38529i	1.21014	−	0.341021i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.19173	+	1.26540i	−0.100776	+	0.0581830i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.82985	−	17.0258i	−0.449137	−	0.777928i	0.549193	−	0.835696i	$-0.314935\pi$
$479$	−9.82985	−	17.0258i	−0.449137	−	0.777928i	−0.998330	+	0.0577672i	$0.981602\pi$
$480$		0			0
$481$	8.62897	+	4.98194i	0.393447	+	0.227157i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−13.1510	−	22.7781i	−0.595927	−	1.03218i	−0.993415	−	0.114569i	$-0.963451\pi$
$487$	−13.1510	−	22.7781i	−0.595927	−	1.03218i	0.397488	−	0.917607i	$-0.369882\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	2.03367i		−	0.0917781i	−0.998947	−	0.0458890i	$-0.985388\pi$
$491$		−	2.03367i		−	0.0917781i	0.998947	−	0.0458890i	$-0.0146121\pi$
$492$		0			0
$493$	−21.1621	+	12.2179i	−0.953092	+	0.550268i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12.4894	+	3.17470i	0.560227	+	0.142405i
$498$		0			0
$499$	−13.4706	+	23.3318i	−0.603027	+	1.04447i	0.389332	+	0.921097i	$0.372706\pi$
$499$	−13.4706	+	23.3318i	−0.603027	+	1.04447i	−0.992360	+	0.123377i	$0.960628\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	40.4310			1.80273			0.901365	−	0.433060i	$-0.142566\pi$
$503$	40.4310			1.80273			0.901365	+	0.433060i	$0.142566\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−16.7258	+	28.9699i	−0.741358	+	1.28407i	0.210519	+	0.977590i	$0.432485\pi$
$509$	−16.7258	+	28.9699i	−0.741358	+	1.28407i	−0.951877	+	0.306480i	$0.900849\pi$
$510$		0			0
$511$	3.32250	+	11.7902i	0.146979	+	0.521566i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			3.22361i			0.141774i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.88232	+	4.99232i	0.126277	+	0.218718i	0.922231	−	0.386639i	$-0.126364\pi$
$521$	2.88232	+	4.99232i	0.126277	+	0.218718i	−0.795955	+	0.605356i	$0.793031\pi$
$522$		0			0
$523$	24.9506	+	14.4052i	1.09101	+	0.629896i	0.933846	−	0.357676i	$-0.116431\pi$
$523$	24.9506	+	14.4052i	1.09101	+	0.629896i	0.157166	+	0.987572i	$0.449764\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−27.2353	−	15.7243i	−1.18639	−	0.684962i
$528$		0			0
$529$	14.0154	+	24.2754i	0.609366	+	1.05545i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	13.7352i		−	0.594939i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.92660	−	0.0754091i	0.126058	−	0.00324810i
$540$		0			0
$541$	14.2209	−	24.6313i	0.611404	−	1.05898i	−0.379600	−	0.925151i	$-0.623939\pi$
$541$	14.2209	−	24.6313i	0.611404	−	1.05898i	0.991004	−	0.133832i	$-0.0427281\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	35.3927			1.51328			0.756641	−	0.653831i	$-0.226839\pi$
$547$	35.3927			1.51328			0.756641	+	0.653831i	$0.226839\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−12.2269	+	21.1776i	−0.520884	+	0.902197i
$552$		0			0
$553$	9.02558	+	8.79601i	0.383807	+	0.374045i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	38.2847	−	22.1037i	1.62218	−	0.936564i	0.635839	−	0.771821i	$-0.280654\pi$
$557$	38.2847	−	22.1037i	1.62218	−	0.936564i	0.986337	−	0.164742i	$-0.0526793\pi$
$558$		0			0
$559$			6.81998i			0.288454i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6.11680	+	10.5946i	0.257792	+	0.446510i	0.965650	−	0.259845i	$-0.0836716\pi$
$563$	6.11680	+	10.5946i	0.257792	+	0.446510i	−0.707858	+	0.706355i	$0.750338\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	29.8992	+	17.2623i	1.25344	+	0.723674i	0.971791	−	0.235844i	$-0.0757853\pi$
$569$	29.8992	+	17.2623i	1.25344	+	0.723674i	0.281649	+	0.959518i	$0.409119\pi$
$570$		0			0
$571$	−19.9367	−	34.5314i	−0.834325	−	1.44509i	−0.894579	−	0.446911i	$-0.852524\pi$
$571$	−19.9367	−	34.5314i	−0.834325	−	1.44509i	0.0602534	−	0.998183i	$-0.480809\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	24.2980	−	14.0285i	1.01154	−	0.584013i	0.0998980	−	0.994998i	$-0.468148\pi$
$577$	24.2980	−	14.0285i	1.01154	−	0.584013i	0.911642	+	0.410985i	$0.134815\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.94352	−	31.2503i	0.329553	−	1.29648i
$582$		0			0
$583$	2.15855	−	3.73872i	0.0893981	−	0.154842i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−43.6793			−1.80284			−0.901419	−	0.432948i	$-0.857473\pi$
$587$	−43.6793			−1.80284			−0.901419	+	0.432948i	$0.857473\pi$
$588$		0			0
$589$	−31.4717			−1.29677
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−11.4300	+	19.7973i	−0.469373	+	0.812978i	−0.999387	−	0.0350112i	$-0.988853\pi$
$593$	−11.4300	+	19.7973i	−0.469373	+	0.812978i	0.530014	+	0.847989i	$0.322187\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−10.4810	+	6.05121i	−0.428242	+	0.247246i	−0.698598	−	0.715515i	$-0.746192\pi$
$599$	−10.4810	+	6.05121i	−0.428242	+	0.247246i	0.270355	+	0.962761i	$0.412859\pi$
$600$		0			0
$601$			17.2909i			0.705312i	0.935753	+	0.352656i	$0.114721\pi$
$601$			17.2909i			0.705312i	−0.935753	+	0.352656i	$0.885279\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	11.3538	+	6.55511i	0.460836	+	0.266064i	0.712396	−	0.701778i	$-0.247610\pi$
$607$	11.3538	+	6.55511i	0.460836	+	0.266064i	−0.251560	+	0.967842i	$0.580943\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	7.52314	+	4.34349i	0.304354	+	0.175719i
$612$		0			0
$613$	−19.0721	−	33.0339i	−0.770316	−	1.33423i	−0.937390	−	0.348282i	$-0.886765\pi$
$613$	−19.0721	−	33.0339i	−0.770316	−	1.33423i	0.167074	−	0.985944i	$-0.446568\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	37.4629i		−	1.50820i	−0.656760	−	0.754100i	$-0.728074\pi$
$617$		−	37.4629i		−	1.50820i	0.656760	−	0.754100i	$-0.271926\pi$
$618$		0			0
$619$	−5.81733	+	3.35864i	−0.233818	+	0.134995i	−0.612332	−	0.790601i	$-0.709769\pi$
$619$	−5.81733	+	3.35864i	−0.233818	+	0.134995i	0.378514	+	0.925596i	$0.376435\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	8.83950	+	8.61467i	0.354147	+	0.345140i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	43.6244			1.73942
$630$		0			0
$631$	−16.5402			−0.658457			−0.329228	−	0.944250i	$-0.606789\pi$
$631$	−16.5402			−0.658457			−0.329228	+	0.944250i	$0.606789\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3.76731	−	6.93160i	0.149266	−	0.274640i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	25.9232	−	14.9668i	1.02391	−	0.591152i	0.108673	−	0.994078i	$-0.465340\pi$
$641$	25.9232	−	14.9668i	1.02391	−	0.591152i	0.915233	+	0.402926i	$0.132007\pi$
$642$		0			0
$643$			4.14597i			0.163501i	0.996653	+	0.0817505i	$0.0260511\pi$
$643$			4.14597i			0.163501i	−0.996653	+	0.0817505i	$0.973949\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	24.1476	+	41.8249i	0.949341	+	1.64431i	0.746818	+	0.665028i	$0.231581\pi$
$647$	24.1476	+	41.8249i	0.949341	+	1.64431i	0.202523	+	0.979278i	$0.435086\pi$
$648$		0			0
$649$	0.412426	+	0.238114i	0.0161891	+	0.00934679i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	7.29871	+	4.21391i	0.285621	+	0.164903i	0.635965	−	0.771718i	$-0.280602\pi$
$653$	7.29871	+	4.21391i	0.285621	+	0.164903i	−0.350345	+	0.936621i	$0.613936\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	28.5542i		−	1.11231i	−0.831078	−	0.556156i	$-0.812276\pi$
$659$		−	28.5542i		−	1.11231i	0.831078	−	0.556156i	$-0.187724\pi$
$660$		0			0
$661$	1.75859	−	1.01532i	0.0684011	−	0.0394914i	−0.465409	−	0.885096i	$-0.654093\pi$
$661$	1.75859	−	1.01532i	0.0684011	−	0.0394914i	0.533811	+	0.845604i	$0.320760\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−17.6879	+	30.6364i	−0.684880	+	1.18625i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5.28326			−0.203958
$672$		0			0
$673$	−32.4749			−1.25181			−0.625907	−	0.779898i	$-0.715271\pi$
$673$	−32.4749			−1.25181			−0.625907	+	0.779898i	$0.715271\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.61071	+	4.52189i	−0.100338	+	0.173790i	−0.911824	−	0.410582i	$-0.865326\pi$
$677$	−2.61071	+	4.52189i	−0.100338	+	0.173790i	0.811486	+	0.584372i	$0.198659\pi$
$678$		0			0
$679$	−19.2202	−	4.88560i	−0.737605	−	0.187492i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12.0769	−	6.97259i	0.462109	−	0.266799i	−0.250822	−	0.968033i	$-0.580701\pi$
$683$	12.0769	−	6.97259i	0.462109	−	0.266799i	0.712931	+	0.701235i	$0.247367\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.81686	−	10.0751i	−0.221605	−	0.383831i
$690$		0			0
$691$	17.3513	+	10.0178i	0.660074	+	0.381094i	0.792305	−	0.610125i	$-0.208881\pi$
$691$	17.3513	+	10.0178i	0.660074	+	0.381094i	−0.132231	+	0.991219i	$0.542214\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−30.0682	−	52.0796i	−1.13891	−	1.97266i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	10.0256i		−	0.378663i	−0.981913	−	0.189331i	$-0.939368\pi$
$701$		−	10.0256i		−	0.378663i	0.981913	−	0.189331i	$-0.0606320\pi$
$702$		0			0
$703$	37.8076	−	21.8282i	1.42594	−	0.823267i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8.46577	+	2.38568i	−0.318388	+	0.0897227i
$708$		0			0
$709$	18.5575	−	32.1426i	0.696942	−	1.20714i	−0.272579	−	0.962133i	$-0.587877\pi$
$709$	18.5575	−	32.1426i	0.696942	−	1.20714i	0.969521	−	0.245006i	$-0.0787901\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−45.5283			−1.70505
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	19.7712	−	34.2448i	0.737343	−	1.27712i	−0.216345	−	0.976317i	$-0.569413\pi$
$719$	19.7712	−	34.2448i	0.737343	−	1.27712i	0.953688	−	0.300799i	$-0.0972532\pi$
$720$		0			0
$721$	1.94728	−	1.99810i	0.0725204	−	0.0744131i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			39.5032i			1.46509i	0.680716	+	0.732547i	$0.261669\pi$
$727$			39.5032i			1.46509i	−0.680716	+	0.732547i	$0.738331\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	14.9298	+	25.8592i	0.552198	+	0.956436i
$732$		0			0
$733$	18.1224	+	10.4629i	0.669364	+	0.386458i	0.795836	−	0.605513i	$-0.207032\pi$
$733$	18.1224	+	10.4629i	0.669364	+	0.386458i	−0.126471	+	0.991970i	$0.540365\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.72741	+	2.15202i	0.137301	+	0.0792706i
$738$		0			0
$739$	−3.48447	−	6.03528i	−0.128178	−	0.222011i	0.794793	−	0.606881i	$-0.207580\pi$
$739$	−3.48447	−	6.03528i	−0.128178	−	0.222011i	−0.922971	+	0.384870i	$0.874246\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			3.79105i			0.139080i	0.997579	+	0.0695400i	$0.0221531\pi$
$743$			3.79105i			0.139080i	−0.997579	+	0.0695400i	$0.977847\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	0.145698	−	0.149500i	0.00532367	−	0.00546261i
$750$		0			0
$751$	9.37773	−	16.2427i	0.342198	−	0.592704i	−0.642643	−	0.766166i	$-0.722162\pi$
$751$	9.37773	−	16.2427i	0.342198	−	0.592704i	0.984841	+	0.173462i	$0.0554953\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−13.0853			−0.475593			−0.237797	−	0.971315i	$-0.576425\pi$
$757$	−13.0853			−0.475593			−0.237797	+	0.971315i	$0.576425\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−10.2391	+	17.7347i	−0.371168	+	0.642883i	−0.989746	−	0.142842i	$-0.954376\pi$
$761$	−10.2391	+	17.7347i	−0.371168	+	0.642883i	0.618577	+	0.785724i	$0.287709\pi$
$762$		0			0
$763$	−25.1824	+	7.09647i	−0.911663	+	0.256909i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.11140	−	0.641669i	0.0401305	−	0.0231693i
$768$		0			0
$769$		−	5.99525i		−	0.216194i	−0.994140	−	0.108097i	$-0.965524\pi$
$769$		−	5.99525i		−	0.216194i	0.994140	−	0.108097i	$-0.0344757\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−25.2390	−	43.7153i	−0.907785	−	1.57233i	−0.817134	−	0.576447i	$-0.804439\pi$
$773$	−25.2390	−	43.7153i	−0.907785	−	1.57233i	−0.0906507	−	0.995883i	$-0.528895\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−52.1179	−	30.0903i	−1.86732	−	1.07810i
$780$		0			0
$781$	1.01852	+	1.76412i	0.0364454	+	0.0631254i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−14.7676	+	8.52605i	−0.526407	+	0.303921i	−0.739552	−	0.673100i	$-0.764963\pi$
$787$	−14.7676	+	8.52605i	−0.526407	+	0.303921i	0.213145	+	0.977020i	$0.431629\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6.87005	+	1.74630i	0.244271	+	0.0620913i
$792$		0			0
$793$	−7.11866	+	12.3299i	−0.252791	+	0.437847i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−15.3352			−0.543200			−0.271600	−	0.962410i	$-0.587553\pi$
$797$	−15.3352			−0.543200			−0.271600	+	0.962410i	$0.587553\pi$
$798$		0			0
$799$	38.0338			1.34554
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.968152	+	1.67689i	−0.0341653	+	0.0591761i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−44.7050	+	25.8105i	−1.57175	+	0.907448i	−0.575790	+	0.817598i	$0.695305\pi$
$809$	−44.7050	+	25.8105i	−1.57175	+	0.907448i	−0.995955	+	0.0898500i	$0.971361\pi$
$810$		0			0
$811$			16.3629i			0.574580i	0.957844	+	0.287290i	$0.0927544\pi$
$811$			16.3629i			0.574580i	−0.957844	+	0.287290i	$0.907246\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	25.8782	+	14.9408i	0.905362	+	0.522711i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2.96116	+	1.70963i	0.103345	+	0.0596663i	0.550782	−	0.834649i	$-0.314330\pi$
$821$	2.96116	+	1.70963i	0.103345	+	0.0596663i	−0.447437	+	0.894316i	$0.647663\pi$
$822$		0			0
$823$	−4.36601	−	7.56215i	−0.152190	−	0.263600i	0.779843	−	0.625976i	$-0.215299\pi$
$823$	−4.36601	−	7.56215i	−0.152190	−	0.263600i	−0.932032	+	0.362376i	$0.881966\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			2.46966i			0.0858786i	0.999078	+	0.0429393i	$0.0136722\pi$
$827$			2.46966i			0.0858786i	−0.999078	+	0.0429393i	$0.986328\pi$
$828$		0			0
$829$	−37.0486	+	21.3900i	−1.28675	+	0.742905i	−0.978073	−	0.208261i	$-0.933220\pi$
$829$	−37.0486	+	21.3900i	−1.28675	+	0.742905i	−0.308677	+	0.951167i	$0.599886\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.889715	−	34.5295i	−0.0308268	−	1.19638i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	16.3279			0.563703			0.281852	−	0.959458i	$-0.409051\pi$
$839$	16.3279			0.563703			0.281852	+	0.959458i	$0.409051\pi$
$840$		0			0
$841$	4.47652			0.154363
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−20.5111	−	19.9894i	−0.704768	−	0.686842i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	54.6940	−	31.5776i	1.87489	−	1.08247i
$852$		0			0
$853$			19.2402i			0.658772i	0.944195	+	0.329386i	$0.106842\pi$
$853$			19.2402i			0.658772i	−0.944195	+	0.329386i	$0.893158\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	8.58903	+	14.8766i	0.293396	+	0.508176i	0.974610	−	0.223907i	$-0.0718813\pi$
$857$	8.58903	+	14.8766i	0.293396	+	0.508176i	−0.681215	+	0.732084i	$0.738548\pi$
$858$		0			0
$859$	−23.7587	−	13.7171i	−0.810637	−	0.468021i	0.0365401	−	0.999332i	$-0.488366\pi$
$859$	−23.7587	−	13.7171i	−0.810637	−	0.468021i	−0.847177	+	0.531311i	$0.821700\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.36663	−	1.94373i	−0.114601	−	0.0661652i	0.441604	−	0.897210i	$-0.354410\pi$
$863$	−3.36663	−	1.94373i	−0.114601	−	0.0661652i	−0.556205	+	0.831045i	$0.687743\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1.99218i			0.0675799i
$870$		0			0
$871$	10.0446	−	5.79926i	0.340349	−	0.196500i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	6.71537	−	11.6314i	0.226762	−	0.392763i	−0.730085	−	0.683357i	$-0.760519\pi$
$877$	6.71537	−	11.6314i	0.226762	−	0.392763i	0.956847	+	0.290593i	$0.0938527\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	33.0947			1.11499			0.557494	−	0.830181i	$-0.311763\pi$
$881$	33.0947			1.11499			0.557494	+	0.830181i	$0.311763\pi$
$882$		0			0
$883$	8.30489			0.279482			0.139741	−	0.990188i	$-0.455373\pi$
$883$	8.30489			0.279482			0.139741	+	0.990188i	$0.455373\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10.0972	+	17.4889i	−0.339032	+	0.587221i	−0.984251	−	0.176777i	$-0.943433\pi$
$887$	−10.0972	+	17.4889i	−0.339032	+	0.587221i	0.645219	+	0.763998i	$0.276766\pi$
$888$		0			0
$889$	6.11077	−	24.0401i	0.204949	−	0.806279i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	32.9624	−	19.0309i	1.10305	−	0.636844i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−15.7807	−	27.3330i	−0.526316	−	0.911606i
$900$		0			0
$901$	−44.1114	−	25.4677i	−1.46956	−	0.848453i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	21.0211	+	36.4096i	0.697994	+	1.20896i	0.969161	+	0.246429i	$0.0792572\pi$
$907$	21.0211	+	36.4096i	0.697994	+	1.20896i	−0.271167	+	0.962532i	$0.587409\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			7.92004i			0.262403i	0.991356	+	0.131201i	$0.0418834\pi$
$911$			7.92004i			0.262403i	−0.991356	+	0.131201i	$0.958117\pi$
$912$		0			0
$913$	4.41408	−	2.54847i	0.146085	−	0.0843421i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−40.6857	−	39.6509i	−1.34356	−	1.30939i
$918$		0			0
$919$	1.93406	−	3.34989i	0.0637988	−	0.110503i	−0.832362	−	0.554233i	$-0.813012\pi$
$919$	1.93406	−	3.34989i	0.0637988	−	0.110503i	0.896161	+	0.443730i	$0.146345\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	5.48940			0.180686
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	17.7260	−	30.7024i	0.581573	−	1.00731i	−0.413721	−	0.910404i	$-0.635771\pi$
$929$	17.7260	−	30.7024i	0.581573	−	1.00731i	0.995293	−	0.0969093i	$-0.0308957\pi$
$930$		0			0
$931$	−18.0485	−	29.4802i	−0.591517	−	0.966175i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			38.8730i			1.26992i	0.772543	+	0.634962i	$0.218984\pi$
$937$			38.8730i			1.26992i	−0.772543	+	0.634962i	$0.781016\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	27.5251	+	47.6749i	0.897294	+	1.55416i	0.830940	+	0.556362i	$0.187803\pi$
$941$	27.5251	+	47.6749i	0.897294	+	1.55416i	0.0663534	+	0.997796i	$0.478864\pi$
$942$		0			0
$943$	−75.3958	−	43.5298i	−2.45523	−	1.41753i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−21.3059	−	12.3009i	−0.692347	−	0.399727i	0.112143	−	0.993692i	$-0.464228\pi$
$947$	−21.3059	−	12.3009i	−0.692347	−	0.399727i	−0.804491	+	0.593965i	$0.797562\pi$
$948$		0			0
$949$	2.60897	+	4.51888i	0.0846909	+	0.146689i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	14.3661i		−	0.465363i	−0.972553	−	0.232681i	$-0.925250\pi$
$953$		−	14.3661i		−	0.465363i	0.972553	−	0.232681i	$-0.0747499\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	12.8247	+	45.5096i	0.414132	+	1.46958i
$960$		0			0
$961$	4.80956	−	8.33040i	0.155147	−	0.268723i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−19.5923			−0.630045			−0.315022	−	0.949084i	$-0.602012\pi$
$967$	−19.5923			−0.630045			−0.315022	+	0.949084i	$0.602012\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	20.4512	−	35.4225i	0.656310	−	1.13676i	−0.325254	−	0.945627i	$-0.605450\pi$
$971$	20.4512	−	35.4225i	0.656310	−	1.13676i	0.981564	−	0.191135i	$-0.0612169\pi$
$972$		0			0
$973$	−15.8764	−	4.03563i	−0.508974	−	0.129376i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	18.4860	−	10.6729i	0.591418	−	0.341456i	−0.174240	−	0.984703i	$-0.555747\pi$
$977$	18.4860	−	10.6729i	0.591418	−	0.341456i	0.765658	+	0.643248i	$0.222413\pi$
$978$		0			0
$979$			1.95111i			0.0623576i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	21.4069	+	37.0779i	0.682775	+	1.18260i	0.974131	+	0.225986i	$0.0725604\pi$
$983$	21.4069	+	37.0779i	0.682775	+	1.18260i	−0.291355	+	0.956615i	$0.594106\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	37.4364	+	21.6139i	1.19041	+	0.687283i
$990$		0			0
$991$	−0.0478801	−	0.0829308i	−0.00152096	−	0.00263438i	0.865264	−	0.501317i	$-0.167151\pi$
$991$	−0.0478801	−	0.0829308i	−0.00152096	−	0.00263438i	−0.866785	+	0.498682i	$0.833817\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	51.6419	−	29.8154i	1.63551	−	0.944265i	0.653164	−	0.757217i	$-0.273441\pi$
$997$	51.6419	−	29.8154i	1.63551	−	0.944265i	0.982351	−	0.187048i	$-0.0598920\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.ch.d.1601.5	✓	20
3.2	odd	2	inner	6300.2.ch.d.1601.6	yes	20
5.2	odd	4		6300.2.dd.d.1349.1		40
5.3	odd	4		6300.2.dd.d.1349.19		40
5.4	even	2		6300.2.ch.e.1601.5	yes	20
7.3	odd	6	inner	6300.2.ch.d.4301.6	yes	20
15.2	even	4		6300.2.dd.d.1349.2		40
15.8	even	4		6300.2.dd.d.1349.20		40
15.14	odd	2		6300.2.ch.e.1601.6	yes	20
21.17	even	6	inner	6300.2.ch.d.4301.5	yes	20
35.3	even	12		6300.2.dd.d.4049.1		40
35.17	even	12		6300.2.dd.d.4049.19		40
35.24	odd	6		6300.2.ch.e.4301.6	yes	20
105.17	odd	12		6300.2.dd.d.4049.20		40
105.38	odd	12		6300.2.dd.d.4049.2		40
105.59	even	6		6300.2.ch.e.4301.5	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6300.2.ch.d.1601.5	✓	20	1.1	even	1	trivial
6300.2.ch.d.1601.6	yes	20	3.2	odd	2	inner
6300.2.ch.d.4301.5	yes	20	21.17	even	6	inner
6300.2.ch.d.4301.6	yes	20	7.3	odd	6	inner
6300.2.ch.e.1601.5	yes	20	5.4	even	2
6300.2.ch.e.1601.6	yes	20	15.14	odd	2
6300.2.ch.e.4301.5	yes	20	105.59	even	6
6300.2.ch.e.4301.6	yes	20	35.24	odd	6
6300.2.dd.d.1349.1		40	5.2	odd	4
6300.2.dd.d.1349.2		40	15.2	even	4
6300.2.dd.d.1349.19		40	5.3	odd	4
6300.2.dd.d.1349.20		40	15.8	even	4
6300.2.dd.d.4049.1		40	35.3	even	12
6300.2.dd.d.4049.2		40	105.38	odd	12
6300.2.dd.d.4049.19		40	35.17	even	12
6300.2.dd.d.4049.20		40	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6300.ch (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.651797 + 2.56421i) q^{7} +(-0.362193 + 0.209112i) q^{11} +1.12703i q^{13} +(2.46721 + 4.27334i) q^{17} +(4.27648 + 2.46903i) q^{19} +(6.18653 + 3.57179i) q^{23} +4.95212i q^{29} +(-5.51945 + 3.18666i) q^{31} +(4.42041 - 7.65637i) q^{37} -12.1871 q^{41} +6.05128 q^{43} +(3.85392 - 6.67519i) q^{47} +(-6.15032 - 3.34269i) q^{49} +(-8.93951 + 5.16123i) q^{53} +(-0.569345 - 0.986135i) q^{59} +(10.9401 + 6.31629i) q^{61} +(-5.14561 - 8.91246i) q^{67} -4.87068i q^{71} +(4.00954 - 2.31491i) q^{73} +(-0.300131 - 1.06504i) q^{77} +(2.38171 - 4.12524i) q^{79} -12.1871 q^{83} +(2.33261 - 4.04019i) q^{89} +(-2.88994 - 0.734595i) q^{91} +7.49558i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{7} + 6 q^{31} - 2 q^{37} + 20 q^{43} + 4 q^{49} + 6 q^{61} - 28 q^{67} + 6 q^{73} + 2 q^{79} - 40 q^{91}+O(q^{100}) \) 20 * q - 4 * q^7 + 6 * q^31 - 2 * q^37 + 20 * q^43 + 4 * q^49 + 6 * q^61 - 28 * q^67 + 6 * q^73 + 2 * q^79 - 40 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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