LMFDB - Embedded newform 6300.2.ch.f.4301.9

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 = [32,0,0,0,0,0,0,0,0,0,0] 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:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 1260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4301.9
Character	$\chi$	$=$	6300.4301
Dual form			6300.2.ch.f.1601.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.25359 - 2.32992i) q^{7} +(4.81480 + 2.77982i) q^{11} -6.30365i q^{13} +(2.21467 - 3.83592i) q^{17} +(6.00550 - 3.46728i) q^{19} +(0.968157 - 0.558966i) q^{23} +6.69226i q^{29} +(5.23215 + 3.02078i) q^{31} +(0.273071 + 0.472973i) q^{37} +3.53503 q^{41} +6.28394 q^{43} +(3.02020 + 5.23113i) q^{47} +(-3.85703 - 5.84152i) q^{49} +(-8.83056 - 5.09832i) q^{53} +(0.298897 - 0.517705i) q^{59} +(-4.72271 + 2.72666i) q^{61} +(-3.46768 + 6.00619i) q^{67} +5.84125i q^{71} +(-6.98579 - 4.03324i) q^{73} +(12.5125 - 7.73332i) q^{77} +(-2.87511 - 4.97984i) q^{79} +2.16066 q^{83} +(-6.81468 - 11.8034i) q^{89} +(-14.6870 - 7.90219i) q^{91} -4.73923i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 24 q^{19} + 24 q^{31} - 40 q^{49} - 24 q^{61} - 32 q^{79} - 56 q^{91}+O(q^{100}) $ 32 * q + 24 * q^19 + 24 * q^31 - 40 * q^49 - 24 * q^61 - 32 * q^79 - 56 * q^91

Character values

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

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

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
$7$	1.25359	−	2.32992i	0.473812	−	0.880626i
$7$	1.25359	−	2.32992i	0.473812	−	0.880626i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.81480	+	2.77982i	1.45172	+	0.838149i	0.998579	−	0.0532922i	$-0.0169715\pi$
$11$	4.81480	+	2.77982i	1.45172	+	0.838149i	0.453137	+	0.891441i	$0.350305\pi$
$12$		0			0
$13$		−	6.30365i		−	1.74832i	−0.485639	−	0.874160i	$-0.661413\pi$
$13$		−	6.30365i		−	1.74832i	0.485639	−	0.874160i	$-0.338587\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.21467	−	3.83592i	0.537136	−	0.930347i	−0.461920	−	0.886921i	$-0.652839\pi$
$17$	2.21467	−	3.83592i	0.537136	−	0.930347i	0.999057	−	0.0434259i	$-0.0138273\pi$
$18$		0			0
$19$	6.00550	−	3.46728i	1.37776	−	0.795448i	0.385867	−	0.922554i	$-0.373902\pi$
$19$	6.00550	−	3.46728i	1.37776	−	0.795448i	0.991889	+	0.127106i	$0.0405689\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.968157	−	0.558966i	0.201875	−	0.116552i	−0.395655	−	0.918399i	$-0.629482\pi$
$23$	0.968157	−	0.558966i	0.201875	−	0.116552i	0.597530	+	0.801847i	$0.296149\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$			6.69226i			1.24272i	0.783524	+	0.621361i	$0.213420\pi$
$29$			6.69226i			1.24272i	−0.783524	+	0.621361i	$0.786580\pi$
$30$		0			0
$31$	5.23215	+	3.02078i	0.939721	+	0.542548i	0.889873	−	0.456209i	$-0.150793\pi$
$31$	5.23215	+	3.02078i	0.939721	+	0.542548i	0.0498482	+	0.998757i	$0.484126\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.273071	+	0.472973i	0.0448926	+	0.0777563i	0.887599	−	0.460618i	$-0.152372\pi$
$37$	0.273071	+	0.472973i	0.0448926	+	0.0777563i	−0.842706	+	0.538374i	$0.819039\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.53503			0.552079			0.276039	−	0.961146i	$-0.410978\pi$
$41$	3.53503			0.552079			0.276039	+	0.961146i	$0.410978\pi$
$42$		0			0
$43$	6.28394			0.958291			0.479146	−	0.877735i	$-0.340947\pi$
$43$	6.28394			0.958291			0.479146	+	0.877735i	$0.340947\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.02020	+	5.23113i	0.440541	+	0.763039i	0.997730	−	0.0673467i	$-0.0214534\pi$
$47$	3.02020	+	5.23113i	0.440541	+	0.763039i	−0.557189	+	0.830386i	$0.688120\pi$
$48$		0			0
$49$	−3.85703	−	5.84152i	−0.551004	−	0.834502i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−8.83056	−	5.09832i	−1.21297	−	0.700309i	−0.249565	−	0.968358i	$-0.580288\pi$
$53$	−8.83056	−	5.09832i	−1.21297	−	0.700309i	−0.963405	+	0.268049i	$0.913621\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.298897	−	0.517705i	0.0389131	−	0.0673994i	−0.845913	−	0.533321i	$-0.820944\pi$
$59$	0.298897	−	0.517705i	0.0389131	−	0.0673994i	0.884826	+	0.465922i	$0.154277\pi$
$60$		0			0
$61$	−4.72271	+	2.72666i	−0.604680	+	0.349112i	−0.770881	−	0.636980i	$-0.780184\pi$
$61$	−4.72271	+	2.72666i	−0.604680	+	0.349112i	0.166200	+	0.986092i	$0.446850\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.46768	+	6.00619i	−0.423644	+	0.733773i	−0.996293	−	0.0860279i	$-0.972583\pi$
$67$	−3.46768	+	6.00619i	−0.423644	+	0.733773i	0.572649	+	0.819801i	$0.305916\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			5.84125i			0.693229i	0.938008	+	0.346614i	$0.112669\pi$
$71$			5.84125i			0.693229i	−0.938008	+	0.346614i	$0.887331\pi$
$72$		0			0
$73$	−6.98579	−	4.03324i	−0.817624	−	0.472056i	0.0319722	−	0.999489i	$-0.489821\pi$
$73$	−6.98579	−	4.03324i	−0.817624	−	0.472056i	−0.849597	+	0.527433i	$0.823155\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	12.5125	−	7.73332i	1.42594	−	0.881294i
$78$		0			0
$79$	−2.87511	−	4.97984i	−0.323476	−	0.560276i	0.657727	−	0.753256i	$-0.271518\pi$
$79$	−2.87511	−	4.97984i	−0.323476	−	0.560276i	−0.981203	+	0.192980i	$0.938185\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.16066			0.237164			0.118582	−	0.992944i	$-0.462165\pi$
$83$	2.16066			0.237164			0.118582	+	0.992944i	$0.462165\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.81468	−	11.8034i	−0.722355	−	1.25116i	−0.960053	−	0.279817i	$-0.909726\pi$
$89$	−6.81468	−	11.8034i	−0.722355	−	1.25116i	0.237699	−	0.971339i	$-0.423607\pi$
$90$		0			0
$91$	−14.6870	−	7.90219i	−1.53962	−	0.828375i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	4.73923i		−	0.481196i	−0.970625	−	0.240598i	$-0.922657\pi$
$97$		−	4.73923i		−	0.481196i	0.970625	−	0.240598i	$-0.0773435\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.52364	−	2.63903i	0.151608	−	0.262593i	−0.780211	−	0.625517i	$-0.784888\pi$
$101$	1.52364	−	2.63903i	0.151608	−	0.262593i	0.931819	+	0.362924i	$0.118222\pi$
$102$		0			0
$103$	−6.02919	+	3.48095i	−0.594073	+	0.342988i	−0.766707	−	0.641998i	$-0.778106\pi$
$103$	−6.02919	+	3.48095i	−0.594073	+	0.342988i	0.172633	+	0.984986i	$0.444773\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	10.4101	−	6.01030i	1.00639	−	0.581038i	0.0962550	−	0.995357i	$-0.469314\pi$
$107$	10.4101	−	6.01030i	1.00639	−	0.581038i	0.910132	+	0.414319i	$0.135980\pi$
$108$		0			0
$109$	0.0561512	−	0.0972568i	0.00537831	−	0.00931551i	−0.863324	−	0.504651i	$-0.831621\pi$
$109$	0.0561512	−	0.0972568i	0.00537831	−	0.00931551i	0.868702	+	0.495335i	$0.164955\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			17.1391i			1.61231i	0.591701	+	0.806157i	$0.298457\pi$
$113$			17.1391i			1.61231i	−0.591701	+	0.806157i	$0.701543\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.16109	−	9.96866i	−0.564786	−	0.913826i
$120$		0			0
$121$	9.95485	+	17.2423i	0.904986	+	1.56748i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−20.9470			−1.85874			−0.929371	−	0.369147i	$-0.879650\pi$
$127$	−20.9470			−1.85874			−0.929371	+	0.369147i	$0.879650\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.97000	−	8.60829i	−0.434231	−	0.752109i	0.563002	−	0.826456i	$-0.309646\pi$
$131$	−4.97000	−	8.60829i	−0.434231	−	0.752109i	−0.997232	+	0.0743462i	$0.976313\pi$
$132$		0			0
$133$	−0.550042	−	18.3389i	−0.0476947	−	1.59018i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	16.9720	+	9.79880i	1.45002	+	0.837168i	0.998482	−	0.0550844i	$-0.0175428\pi$
$137$	16.9720	+	9.79880i	1.45002	+	0.837168i	0.451536	+	0.892253i	$0.350876\pi$
$138$		0			0
$139$			17.0888i			1.44945i	0.689038	+	0.724725i	$0.258033\pi$
$139$			17.0888i			1.44945i	−0.689038	+	0.724725i	$0.741967\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	17.5231	−	30.3508i	1.46535	−	2.53806i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.46827	+	4.31181i	−0.611825	+	0.353237i	−0.773679	−	0.633577i	$-0.781586\pi$
$149$	−7.46827	+	4.31181i	−0.611825	+	0.353237i	0.161855	+	0.986815i	$0.448252\pi$
$150$		0			0
$151$	4.52358	−	7.83508i	0.368124	−	0.637610i	−0.621148	−	0.783693i	$-0.713333\pi$
$151$	4.52358	−	7.83508i	0.368124	−	0.637610i	0.989272	+	0.146084i	$0.0466668\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−11.5914	−	6.69229i	−0.925093	−	0.534103i	−0.0398367	−	0.999206i	$-0.512684\pi$
$157$	−11.5914	−	6.69229i	−0.925093	−	0.534103i	−0.885256	+	0.465103i	$0.846017\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.0886733	−	2.95644i	−0.00698843	−	0.233000i
$162$		0			0
$163$	5.21867	+	9.03900i	0.408758	+	0.707989i	0.994751	−	0.102327i	$-0.0326287\pi$
$163$	5.21867	+	9.03900i	0.408758	+	0.707989i	−0.585993	+	0.810316i	$0.699295\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.52462			0.737037			0.368519	−	0.929620i	$-0.379865\pi$
$167$	9.52462			0.737037			0.368519	+	0.929620i	$0.379865\pi$
$168$		0			0
$169$	−26.7361			−2.05662
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.94301	+	13.7577i	0.603896	+	1.04598i	0.992225	+	0.124457i	$0.0397190\pi$
$173$	7.94301	+	13.7577i	0.603896	+	1.04598i	−0.388329	+	0.921521i	$0.626948\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.10068	+	0.635476i	0.0822685	+	0.0474977i	0.540570	−	0.841299i	$-0.318209\pi$
$179$	1.10068	+	0.635476i	0.0822685	+	0.0474977i	−0.458301	+	0.888797i	$0.651542\pi$
$180$		0			0
$181$		−	1.34067i		−	0.0996511i	−0.998758	−	0.0498256i	$-0.984133\pi$
$181$		−	1.34067i		−	0.0996511i	0.998758	−	0.0498256i	$-0.0158665\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	21.3264	−	12.3128i	1.55954	−	0.900400i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.58703	+	1.49362i	−0.187191	+	0.108075i	−0.590667	−	0.806915i	$-0.701135\pi$
$191$	−2.58703	+	1.49362i	−0.187191	+	0.108075i	0.403476	+	0.914990i	$0.367802\pi$
$192$		0			0
$193$	−3.19070	+	5.52646i	−0.229672	+	0.397803i	−0.957711	−	0.287732i	$-0.907099\pi$
$193$	−3.19070	+	5.52646i	−0.229672	+	0.397803i	0.728039	+	0.685536i	$0.240432\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4.10025i			0.292130i	0.989275	+	0.146065i	$0.0466609\pi$
$197$			4.10025i			0.292130i	−0.989275	+	0.146065i	$0.953339\pi$
$198$		0			0
$199$	21.3499	+	12.3264i	1.51346	+	0.873795i	0.999876	+	0.0157546i	$0.00501505\pi$
$199$	21.3499	+	12.3264i	1.51346	+	0.873795i	0.513582	+	0.858041i	$0.328318\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	15.5924	+	8.38935i	1.09437	+	0.588817i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	38.5537			2.66681
$210$		0			0
$211$	1.18464			0.0815538			0.0407769	−	0.999168i	$-0.487017\pi$
$211$	1.18464			0.0815538			0.0407769	+	0.999168i	$0.487017\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	13.5971	−	8.40365i	0.923033	−	0.570477i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−24.1803	−	13.9605i	−1.62654	−	0.939086i
$222$		0			0
$223$			5.24874i			0.351481i	0.984436	+	0.175741i	$0.0562320\pi$
$223$			5.24874i			0.351481i	−0.984436	+	0.175741i	$0.943768\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.0660	−	20.8989i	0.800847	−	1.38711i	−0.118213	−	0.992988i	$-0.537716\pi$
$227$	12.0660	−	20.8989i	0.800847	−	1.38711i	0.919059	−	0.394119i	$-0.128950\pi$
$228$		0			0
$229$	23.8123	−	13.7481i	1.57356	−	0.908497i	0.577837	−	0.816152i	$-0.303897\pi$
$229$	23.8123	−	13.7481i	1.57356	−	0.908497i	0.995727	−	0.0923449i	$-0.0294362\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	10.7893	−	6.22921i	0.706831	−	0.408089i	−0.103055	−	0.994676i	$-0.532862\pi$
$233$	10.7893	−	6.22921i	0.706831	−	0.408089i	0.809887	+	0.586586i	$0.199529\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			21.7203i			1.40497i	0.711698	+	0.702486i	$0.247926\pi$
$239$			21.7203i			1.40497i	−0.711698	+	0.702486i	$0.752074\pi$
$240$		0			0
$241$	−2.74159	−	1.58286i	−0.176601	−	0.101961i	0.409094	−	0.912492i	$-0.365845\pi$
$241$	−2.74159	−	1.58286i	−0.176601	−	0.101961i	−0.585695	+	0.810532i	$0.699178\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−21.8565	−	37.8566i	−1.39070	−	2.40876i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.34842			−0.148231			−0.0741155	−	0.997250i	$-0.523613\pi$
$251$	−2.34842			−0.148231			−0.0741155	+	0.997250i	$0.523613\pi$
$252$		0			0
$253$	6.21531			0.390753
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−13.5326	−	23.4391i	−0.844139	−	1.46209i	−0.886367	−	0.462983i	$-0.846779\pi$
$257$	−13.5326	−	23.4391i	−0.844139	−	1.46209i	0.0422286	−	0.999108i	$-0.486554\pi$
$258$		0			0
$259$	1.44431	−	0.0433195i	0.0897449	−	0.00269174i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−8.51647	−	4.91699i	−0.525148	−	0.303195i	0.213890	−	0.976858i	$-0.431387\pi$
$263$	−8.51647	−	4.91699i	−0.525148	−	0.303195i	−0.739039	+	0.673663i	$0.764720\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−9.10448	+	15.7694i	−0.555110	+	0.961479i	0.442785	+	0.896628i	$0.353991\pi$
$269$	−9.10448	+	15.7694i	−0.555110	+	0.961479i	−0.997895	+	0.0648510i	$0.979343\pi$
$270$		0			0
$271$	−9.74519	+	5.62639i	−0.591978	+	0.341779i	−0.765879	−	0.642984i	$-0.777696\pi$
$271$	−9.74519	+	5.62639i	−0.591978	+	0.341779i	0.173901	+	0.984763i	$0.444363\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	5.83696	−	10.1099i	0.350709	−	0.607446i	−0.635665	−	0.771965i	$-0.719274\pi$
$277$	5.83696	−	10.1099i	0.350709	−	0.607446i	0.986374	+	0.164519i	$0.0526072\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	4.73421i		−	0.282419i	−0.989980	−	0.141210i	$-0.954901\pi$
$281$		−	4.73421i		−	0.282419i	0.989980	−	0.141210i	$-0.0450991\pi$
$282$		0			0
$283$	−8.95022	−	5.16741i	−0.532035	−	0.307171i	0.209810	−	0.977742i	$-0.432716\pi$
$283$	−8.95022	−	5.16741i	−0.532035	−	0.307171i	−0.741845	+	0.670572i	$0.766049\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.43147	−	8.23632i	0.261581	−	0.486175i
$288$		0			0
$289$	−1.30952	−	2.26816i	−0.0770308	−	0.133421i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.661681			−0.0386558			−0.0193279	−	0.999813i	$-0.506153\pi$
$293$	−0.661681			−0.0386558			−0.0193279	+	0.999813i	$0.506153\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.52353	−	6.10293i	−0.203771	−	0.352941i
$300$		0			0
$301$	7.87747	−	14.6411i	0.454050	−	0.843896i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	32.9301i		−	1.87942i	−0.341973	−	0.939710i	$-0.611095\pi$
$307$		−	32.9301i		−	1.87942i	0.341973	−	0.939710i	$-0.388905\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5.79118	+	10.0306i	−0.328388	+	0.568784i	−0.982192	−	0.187879i	$-0.939839\pi$
$311$	−5.79118	+	10.0306i	−0.328388	+	0.568784i	0.653804	+	0.756664i	$0.273172\pi$
$312$		0			0
$313$	−6.29874	+	3.63658i	−0.356026	+	0.205551i	−0.667336	−	0.744757i	$-0.732565\pi$
$313$	−6.29874	+	3.63658i	−0.356026	+	0.205551i	0.311310	+	0.950308i	$0.399232\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.13907	+	3.54439i	−0.344805	+	0.199073i	−0.662395	−	0.749155i	$-0.730460\pi$
$317$	−6.13907	+	3.54439i	−0.344805	+	0.199073i	0.317590	+	0.948228i	$0.397126\pi$
$318$		0			0
$319$	−18.6033	+	32.2219i	−1.04159	+	1.80408i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	30.7155i		−	1.70906i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	15.9742	−	0.479118i	0.880686	−	0.0264146i
$330$		0			0
$331$	−6.61230	−	11.4528i	−0.363445	−	0.629505i	0.625080	−	0.780560i	$-0.285066\pi$
$331$	−6.61230	−	11.4528i	−0.363445	−	0.629505i	−0.988525	+	0.151055i	$0.951733\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	23.3122			1.26990			0.634949	−	0.772554i	$-0.281021\pi$
$337$	23.3122			1.26990			0.634949	+	0.772554i	$0.281021\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	16.7945	+	29.0889i	0.909472	+	1.57525i
$342$		0			0
$343$	−18.4454	+	1.66370i	−0.995957	+	0.0898316i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.22661	−	4.17228i	−0.387945	−	0.223980i	0.293325	−	0.956013i	$-0.405238\pi$
$347$	−7.22661	−	4.17228i	−0.387945	−	0.223980i	−0.681269	+	0.732033i	$0.738572\pi$
$348$		0			0
$349$			3.03784i			0.162612i	0.996689	+	0.0813059i	$0.0259091\pi$
$349$			3.03784i			0.162612i	−0.996689	+	0.0813059i	$0.974091\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−11.7407	+	20.3355i	−0.624895	+	1.08235i	0.363666	+	0.931529i	$0.381525\pi$
$353$	−11.7407	+	20.3355i	−0.624895	+	1.08235i	−0.988561	+	0.150821i	$0.951808\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9.22065	−	5.32354i	0.486647	−	0.280966i	−0.236535	−	0.971623i	$-0.576012\pi$
$359$	9.22065	−	5.32354i	0.486647	−	0.280966i	0.723182	+	0.690657i	$0.242679\pi$
$360$		0			0
$361$	14.5440	−	25.1910i	0.765475	−	1.32584i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−22.4753	−	12.9761i	−1.17320	−	0.677349i	−0.218770	−	0.975776i	$-0.570204\pi$
$367$	−22.4753	−	12.9761i	−1.17320	−	0.677349i	−0.954432	+	0.298428i	$0.903538\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−22.9486	+	14.1833i	−1.19143	+	0.736359i
$372$		0			0
$373$	15.4208	+	26.7096i	0.798458	+	1.38297i	0.920620	+	0.390459i	$0.127684\pi$
$373$	15.4208	+	26.7096i	0.798458	+	1.38297i	−0.122163	+	0.992510i	$0.538983\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	42.1857			2.17268
$378$		0			0
$379$	−10.3992			−0.534169			−0.267084	−	0.963673i	$-0.586060\pi$
$379$	−10.3992			−0.534169			−0.267084	+	0.963673i	$0.586060\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5.78792	+	10.0250i	0.295749	+	0.512252i	0.975159	−	0.221507i	$-0.0710974\pi$
$383$	5.78792	+	10.0250i	0.295749	+	0.512252i	−0.679410	+	0.733759i	$0.737764\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−31.3469	−	18.0982i	−1.58935	−	0.917614i	−0.993413	−	0.114585i	$-0.963446\pi$
$389$	−31.3469	−	18.0982i	−1.58935	−	0.917614i	−0.595940	−	0.803029i	$-0.703220\pi$
$390$		0			0
$391$		−	4.95170i		−	0.250418i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.09398	+	0.631611i	−0.0549054	+	0.0316997i	−0.527201	−	0.849740i	$-0.676759\pi$
$397$	−1.09398	+	0.631611i	−0.0549054	+	0.0316997i	0.472296	+	0.881440i	$0.343425\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	34.1957	−	19.7429i	1.70765	−	0.985914i	0.770201	−	0.637801i	$-0.220156\pi$
$401$	34.1957	−	19.7429i	1.70765	−	0.985914i	0.937453	−	0.348113i	$-0.113177\pi$
$402$		0			0
$403$	19.0420	−	32.9816i	0.948547	−	1.64293i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.03636i			0.150507i
$408$		0			0
$409$	−17.2541	−	9.96169i	−0.853163	−	0.492574i	0.00855400	−	0.999963i	$-0.497277\pi$
$409$	−17.2541	−	9.96169i	−0.853163	−	0.492574i	−0.861717	+	0.507390i	$0.830610\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.831516	−	1.34539i	−0.0409162	−	0.0662025i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.60260			0.224852			0.112426	−	0.993660i	$-0.464138\pi$
$419$	4.60260			0.224852			0.112426	+	0.993660i	$0.464138\pi$
$420$		0			0
$421$	−10.0110			−0.487906			−0.243953	−	0.969787i	$-0.578444\pi$
$421$	−10.0110			−0.487906			−0.243953	+	0.969787i	$0.578444\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.432551	+	14.4216i	0.0209326	+	0.697911i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−15.5980	−	9.00553i	−0.751331	−	0.433781i	0.0748435	−	0.997195i	$-0.476154\pi$
$431$	−15.5980	−	9.00553i	−0.751331	−	0.433781i	−0.826175	+	0.563414i	$0.809488\pi$
$432$		0			0
$433$		−	17.5684i		−	0.844281i	−0.906530	−	0.422141i	$-0.861279\pi$
$433$		−	17.5684i		−	0.844281i	0.906530	−	0.422141i	$-0.138721\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.87618	−	6.71374i	0.185423	−	0.321162i
$438$		0			0
$439$	18.6214	−	10.7511i	0.888751	−	0.513121i	0.0152173	−	0.999884i	$-0.495156\pi$
$439$	18.6214	−	10.7511i	0.888751	−	0.513121i	0.873534	+	0.486764i	$0.161823\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	28.7071	−	16.5740i	1.36391	−	0.787456i	0.373771	−	0.927521i	$-0.378065\pi$
$443$	28.7071	−	16.5740i	1.36391	−	0.787456i	0.990142	+	0.140065i	$0.0447312\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	18.0471i		−	0.851696i	−0.904795	−	0.425848i	$-0.859976\pi$
$449$		−	18.0471i		−	0.851696i	0.904795	−	0.425848i	$-0.140024\pi$
$450$		0			0
$451$	17.0204	+	9.82676i	0.801461	+	0.462724i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−9.65257	−	16.7187i	−0.451528	−	0.782070i	0.546953	−	0.837163i	$-0.315788\pi$
$457$	−9.65257	−	16.7187i	−0.451528	−	0.782070i	−0.998481	+	0.0550936i	$0.982454\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	22.4305			1.04469			0.522347	−	0.852733i	$-0.325056\pi$
$461$	22.4305			1.04469			0.522347	+	0.852733i	$0.325056\pi$
$462$		0			0
$463$	−18.4754			−0.858625			−0.429313	−	0.903156i	$-0.641244\pi$
$463$	−18.4754			−0.858625			−0.429313	+	0.903156i	$0.641244\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5.46276	+	9.46179i	0.252787	+	0.437839i	0.964292	−	0.264841i	$-0.0853196\pi$
$467$	5.46276	+	9.46179i	0.252787	+	0.437839i	−0.711505	+	0.702681i	$0.751986\pi$
$468$		0			0
$469$	9.64689	+	15.6087i	0.445452	+	0.720742i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	30.2559	+	17.4682i	1.39117	+	0.803191i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.68438	+	4.64948i	−0.122652	+	0.212440i	−0.920813	−	0.390005i	$-0.872473\pi$
$479$	−2.68438	+	4.64948i	−0.122652	+	0.212440i	0.798160	+	0.602445i	$0.205807\pi$
$480$		0			0
$481$	2.98146	−	1.72135i	0.135943	−	0.0784866i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−12.3457	+	21.3834i	−0.559439	+	0.968977i	0.438104	+	0.898924i	$0.355650\pi$
$487$	−12.3457	+	21.3834i	−0.559439	+	0.968977i	−0.997543	+	0.0700525i	$0.977683\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	24.1955i		−	1.09193i	−0.837808	−	0.545964i	$-0.816163\pi$
$491$		−	24.1955i		−	1.09193i	0.837808	−	0.545964i	$-0.183837\pi$
$492$		0			0
$493$	25.6710	+	14.8212i	1.15616	+	0.667511i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	13.6096	+	7.32252i	0.610475	+	0.328460i
$498$		0			0
$499$	−16.3119	−	28.2530i	−0.730220	−	1.26478i	−0.956789	−	0.290783i	$-0.906084\pi$
$499$	−16.3119	−	28.2530i	−0.730220	−	1.26478i	0.226569	−	0.973995i	$-0.427249\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4.59339			−0.204809			−0.102405	−	0.994743i	$-0.532654\pi$
$503$	−4.59339			−0.204809			−0.102405	+	0.994743i	$0.532654\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	11.8972	+	20.6066i	0.527336	+	0.913373i	0.999492	+	0.0318582i	$0.0101425\pi$
$509$	11.8972	+	20.6066i	0.527336	+	0.913373i	−0.472156	+	0.881515i	$0.656524\pi$
$510$		0			0
$511$	−18.1544	+	11.2203i	−0.803105	+	0.496356i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			33.5825i			1.47695i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−6.26127	+	10.8448i	−0.274311	+	0.475121i	−0.969961	−	0.243260i	$-0.921783\pi$
$521$	−6.26127	+	10.8448i	−0.274311	+	0.475121i	0.695650	+	0.718381i	$0.255116\pi$
$522$		0			0
$523$	−15.3521	+	8.86357i	−0.671303	+	0.387577i	−0.796570	−	0.604546i	$-0.793354\pi$
$523$	−15.3521	+	8.86357i	−0.671303	+	0.387577i	0.125267	+	0.992123i	$0.460021\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	23.1749	−	13.3801i	1.00952	−	0.582845i
$528$		0			0
$529$	−10.8751	+	18.8363i	−0.472831	+	0.818967i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	22.2836i		−	0.965210i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.33243	−	38.8476i	−0.100465	−	1.67328i
$540$		0			0
$541$	15.1308	+	26.2074i	0.650526	+	1.12674i	0.982995	+	0.183630i	$0.0587848\pi$
$541$	15.1308	+	26.2074i	0.650526	+	1.12674i	−0.332470	+	0.943114i	$0.607882\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−11.9209			−0.509699			−0.254850	−	0.966981i	$-0.582026\pi$
$547$	−11.9209			−0.509699			−0.254850	+	0.966981i	$0.582026\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	23.2039	+	40.1904i	0.988521	+	1.71217i
$552$		0			0
$553$	−15.2068	+	0.456103i	−0.646661	+	0.0193955i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	36.8155	+	21.2555i	1.55992	+	0.900623i	0.997263	+	0.0739349i	$0.0235557\pi$
$557$	36.8155	+	21.2555i	1.55992	+	0.900623i	0.562661	+	0.826688i	$0.309778\pi$
$558$		0			0
$559$		−	39.6118i		−	1.67540i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.61376	+	11.4554i	−0.278737	+	0.482786i	−0.971071	−	0.238791i	$-0.923249\pi$
$563$	−6.61376	+	11.4554i	−0.278737	+	0.482786i	0.692334	+	0.721577i	$0.256582\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−15.8953	+	9.17714i	−0.666364	+	0.384726i	−0.794698	−	0.607005i	$-0.792371\pi$
$569$	−15.8953	+	9.17714i	−0.666364	+	0.384726i	0.128333	+	0.991731i	$0.459037\pi$
$570$		0			0
$571$	−20.7812	+	35.9941i	−0.869667	+	1.50631i	−0.00732921	+	0.999973i	$0.502333\pi$
$571$	−20.7812	+	35.9941i	−0.869667	+	1.50631i	−0.862338	+	0.506334i	$0.831000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	7.15748	+	4.13238i	0.297970	+	0.172033i	0.641531	−	0.767097i	$-0.278300\pi$
$577$	7.15748	+	4.13238i	0.297970	+	0.172033i	−0.343561	+	0.939131i	$0.611633\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.70858	−	5.03417i	0.112371	−	0.208852i
$582$		0			0
$583$	−28.3449	−	49.0948i	−1.17393	−	2.03330i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	15.6801			0.647188			0.323594	−	0.946196i	$-0.395109\pi$
$587$	15.6801			0.647188			0.323594	+	0.946196i	$0.395109\pi$
$588$		0			0
$589$	41.8955			1.72628
$590$		0			0
$591$		0			0
$592$		0			0
$593$	7.49644	+	12.9842i	0.307842	+	0.533198i	0.977890	−	0.209120i	$-0.0670599\pi$
$593$	7.49644	+	12.9842i	0.307842	+	0.533198i	−0.670048	+	0.742318i	$0.733727\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	5.23722	+	3.02371i	0.213987	+	0.123545i	0.603163	−	0.797618i	$-0.293907\pi$
$599$	5.23722	+	3.02371i	0.213987	+	0.123545i	−0.389176	+	0.921163i	$0.627240\pi$
$600$		0			0
$601$		−	7.82990i		−	0.319388i	−0.987167	−	0.159694i	$-0.948949\pi$
$601$		−	7.82990i		−	0.319388i	0.987167	−	0.159694i	$-0.0510508\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	1.11451	−	0.643460i	0.0452364	−	0.0261172i	−0.477211	−	0.878789i	$-0.658352\pi$
$607$	1.11451	−	0.643460i	0.0452364	−	0.0261172i	0.522448	+	0.852671i	$0.325019\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	32.9753	−	19.0383i	1.33404	−	0.770206i
$612$		0			0
$613$	10.1604	−	17.5983i	0.410373	−	0.710788i	−0.584557	−	0.811353i	$-0.698732\pi$
$613$	10.1604	−	17.5983i	0.410373	−	0.710788i	0.994930	+	0.100565i	$0.0320650\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	15.8245i		−	0.637071i	−0.947911	−	0.318536i	$-0.896809\pi$
$617$		−	15.8245i		−	0.637071i	0.947911	−	0.318536i	$-0.103191\pi$
$618$		0			0
$619$	17.0692	+	9.85490i	0.686069	+	0.396102i	0.802138	−	0.597139i	$-0.203696\pi$
$619$	17.0692	+	9.85490i	0.686069	+	0.396102i	−0.116069	+	0.993241i	$0.537029\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−36.0437	+	1.08107i	−1.44406	+	0.0433121i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.41905			0.0964538
$630$		0			0
$631$	42.6898			1.69945			0.849727	−	0.527222i	$-0.176767\pi$
$631$	42.6898			1.69945			0.849727	+	0.527222i	$0.176767\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−36.8229	+	24.3134i	−1.45898	+	0.963332i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7.57019	−	4.37065i	−0.299005	−	0.172630i	0.342991	−	0.939339i	$-0.388560\pi$
$641$	−7.57019	−	4.37065i	−0.299005	−	0.172630i	−0.641996	+	0.766708i	$0.721893\pi$
$642$		0			0
$643$			3.69796i			0.145833i	0.997338	+	0.0729167i	$0.0232307\pi$
$643$			3.69796i			0.145833i	−0.997338	+	0.0729167i	$0.976769\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−15.1365	+	26.2172i	−0.595078	+	1.03071i	0.398458	+	0.917187i	$0.369546\pi$
$647$	−15.1365	+	26.2172i	−0.595078	+	1.03071i	−0.993536	+	0.113519i	$0.963788\pi$
$648$		0			0
$649$	2.87826	−	1.66176i	0.112981	−	0.0652299i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−17.7869	+	10.2693i	−0.696055	+	0.401867i	−0.805876	−	0.592084i	$-0.798305\pi$
$653$	−17.7869	+	10.2693i	−0.696055	+	0.401867i	0.109822	+	0.993951i	$0.464972\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			19.7847i			0.770704i	0.922770	+	0.385352i	$0.125920\pi$
$659$			19.7847i			0.770704i	−0.922770	+	0.385352i	$0.874080\pi$
$660$		0			0
$661$	−17.4583	−	10.0796i	−0.679050	−	0.392049i	0.120447	−	0.992720i	$-0.461567\pi$
$661$	−17.4583	−	10.0796i	−0.679050	−	0.392049i	−0.799497	+	0.600670i	$0.794900\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	3.74075	+	6.47916i	0.144842	+	0.250874i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−30.3185			−1.17043
$672$		0			0
$673$	36.1986			1.39535			0.697676	−	0.716414i	$-0.254218\pi$
$673$	36.1986			1.39535			0.697676	+	0.716414i	$0.254218\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−7.47400	−	12.9454i	−0.287249	−	0.497530i	0.685903	−	0.727693i	$-0.259407\pi$
$677$	−7.47400	−	12.9454i	−0.287249	−	0.497530i	−0.973152	+	0.230163i	$0.926074\pi$
$678$		0			0
$679$	−11.0420	−	5.94104i	−0.423753	−	0.227996i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−35.4758	−	20.4820i	−1.35745	−	0.783721i	−0.368166	−	0.929760i	$-0.620014\pi$
$683$	−35.4758	−	20.4820i	−1.35745	−	0.783721i	−0.989279	+	0.146039i	$0.953348\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−32.1381	+	55.6648i	−1.22436	+	2.12066i
$690$		0			0
$691$	3.82022	−	2.20560i	0.145328	−	0.0839051i	−0.425573	−	0.904924i	$-0.639927\pi$
$691$	3.82022	−	2.20560i	0.145328	−	0.0839051i	0.570901	+	0.821019i	$0.306594\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	7.82892	−	13.5601i	0.296541	−	0.513625i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			24.7843i			0.936090i	0.883705	+	0.468045i	$0.155041\pi$
$701$			24.7843i			0.936090i	−0.883705	+	0.468045i	$0.844959\pi$
$702$		0			0
$703$	3.27986	+	1.89363i	0.123702	+	0.0714195i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.23869	−	6.85821i	−0.159412	−	0.257930i
$708$		0			0
$709$	−22.5209	−	39.0073i	−0.845790	−	1.46495i	−0.884933	−	0.465718i	$-0.845796\pi$
$709$	−22.5209	−	39.0073i	−0.845790	−	1.46495i	0.0391431	−	0.999234i	$-0.487537\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.75405			0.252941
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	5.55881	+	9.62814i	0.207309	+	0.359069i	0.950866	−	0.309603i	$-0.100196\pi$
$719$	5.55881	+	9.62814i	0.207309	+	0.359069i	−0.743557	+	0.668672i	$0.766863\pi$
$720$		0			0
$721$	0.552212	+	18.4112i	0.0205654	+	0.685669i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			37.1241i			1.37685i	0.725305	+	0.688427i	$0.241699\pi$
$727$			37.1241i			1.37685i	−0.725305	+	0.688427i	$0.758301\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	13.9168	−	24.1047i	0.514733	−	0.891544i
$732$		0			0
$733$	−1.93012	+	1.11435i	−0.0712905	+	0.0411596i	−0.535222	−	0.844712i	$-0.679772\pi$
$733$	−1.93012	+	1.11435i	−0.0712905	+	0.0411596i	0.463931	+	0.885871i	$0.346439\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−33.3923	+	19.2791i	−1.23002	+	0.710153i
$738$		0			0
$739$	−5.79538	+	10.0379i	−0.213186	+	0.369250i	−0.952710	−	0.303881i	$-0.901718\pi$
$739$	−5.79538	+	10.0379i	−0.213186	+	0.369250i	0.739524	+	0.673131i	$0.235051\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	15.4284i		−	0.566015i	−0.959118	−	0.283007i	$-0.908668\pi$
$743$		−	15.4284i		−	0.566015i	0.959118	−	0.283007i	$-0.0913321\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.953463	−	31.7892i	−0.0348388	−	1.16155i
$750$		0			0
$751$	−11.7833	−	20.4092i	−0.429977	−	0.744742i	0.566894	−	0.823791i	$-0.308145\pi$
$751$	−11.7833	−	20.4092i	−0.429977	−	0.744742i	−0.996871	+	0.0790489i	$0.974812\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−17.6064			−0.639916			−0.319958	−	0.947432i	$-0.603669\pi$
$757$	−17.6064			−0.639916			−0.319958	+	0.947432i	$0.603669\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.88581	−	8.46248i	−0.177111	−	0.306765i	0.763779	−	0.645478i	$-0.223342\pi$
$761$	−4.88581	−	8.46248i	−0.177111	−	0.306765i	−0.940890	+	0.338713i	$0.890008\pi$
$762$		0			0
$763$	−0.156210	−	0.252748i	−0.00565517	−	0.00915008i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.26343	−	1.88414i	−0.117836	−	0.0680325i
$768$		0			0
$769$		−	29.3473i		−	1.05829i	−0.848531	−	0.529146i	$-0.822512\pi$
$769$		−	29.3473i		−	1.05829i	0.848531	−	0.529146i	$-0.177488\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−14.7610	+	25.5668i	−0.530917	+	0.919575i	0.468432	+	0.883499i	$0.344819\pi$
$773$	−14.7610	+	25.5668i	−0.530917	+	0.919575i	−0.999349	+	0.0360752i	$0.988514\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	21.2296	−	12.2569i	0.760630	−	0.439150i
$780$		0			0
$781$	−16.2376	+	28.1244i	−0.581029	+	1.00637i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	29.0833	+	16.7912i	1.03671	+	0.598543i	0.918899	−	0.394494i	$-0.129080\pi$
$787$	29.0833	+	16.7912i	1.03671	+	0.598543i	0.117808	+	0.993036i	$0.462413\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	39.9328	+	21.4854i	1.41985	+	0.763934i
$792$		0			0
$793$	17.1879	+	29.7703i	0.610360	+	1.05717i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−19.0563			−0.675010			−0.337505	−	0.941324i	$-0.609583\pi$
$797$	−19.0563			−0.675010			−0.337505	+	0.941324i	$0.609583\pi$
$798$		0			0
$799$	26.7549			0.946522
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−22.4234	−	38.8385i	−0.791306	−	1.37058i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−16.2896	−	9.40482i	−0.572713	−	0.330656i	0.185519	−	0.982641i	$-0.440603\pi$
$809$	−16.2896	−	9.40482i	−0.572713	−	0.330656i	−0.758232	+	0.651985i	$0.773937\pi$
$810$		0			0
$811$		−	18.7455i		−	0.658242i	−0.944288	−	0.329121i	$-0.893248\pi$
$811$		−	18.7455i		−	0.658242i	0.944288	−	0.329121i	$-0.106752\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	37.7382	−	21.7882i	1.32029	−	0.762271i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−40.4526	+	23.3553i	−1.41181	+	0.815106i	−0.995558	−	0.0941456i	$-0.969988\pi$
$821$	−40.4526	+	23.3553i	−1.41181	+	0.815106i	−0.416247	+	0.909252i	$0.636655\pi$
$822$		0			0
$823$	3.10583	−	5.37945i	0.108262	−	0.187516i	−0.806804	−	0.590819i	$-0.798805\pi$
$823$	3.10583	−	5.37945i	0.108262	−	0.187516i	0.915066	+	0.403303i	$0.132138\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	35.6188i		−	1.23859i	−0.785159	−	0.619294i	$-0.787419\pi$
$827$		−	35.6188i		−	1.23859i	0.785159	−	0.619294i	$-0.212581\pi$
$828$		0			0
$829$	29.6645	+	17.1268i	1.03029	+	0.594839i	0.917068	−	0.398731i	$-0.130549\pi$
$829$	29.6645	+	17.1268i	1.03029	+	0.594839i	0.113223	+	0.993570i	$0.463883\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−30.9496	+	1.85823i	−1.07234	+	0.0643840i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−26.3915			−0.911136			−0.455568	−	0.890201i	$-0.650564\pi$
$839$	−26.3915			−0.911136			−0.455568	+	0.890201i	$0.650564\pi$
$840$		0			0
$841$	−15.7864			−0.544359
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	52.6524	−	1.57922i	1.80916	−	0.0542626i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.528751	+	0.305275i	0.0181254	+	0.0104647i
$852$		0			0
$853$		−	17.4336i		−	0.596914i	−0.954423	−	0.298457i	$-0.903528\pi$
$853$		−	17.4336i		−	0.596914i	0.954423	−	0.298457i	$-0.0964719\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−20.7957	+	36.0192i	−0.710368	+	1.23039i	0.254351	+	0.967112i	$0.418138\pi$
$857$	−20.7957	+	36.0192i	−0.710368	+	1.23039i	−0.964719	+	0.263282i	$0.915195\pi$
$858$		0			0
$859$	−10.2993	+	5.94630i	−0.351408	+	0.202885i	−0.665305	−	0.746572i	$-0.731698\pi$
$859$	−10.2993	+	5.94630i	−0.351408	+	0.202885i	0.313898	+	0.949457i	$0.398365\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	12.2548	−	7.07530i	0.417157	−	0.240846i	−0.276703	−	0.960956i	$-0.589242\pi$
$863$	12.2548	−	7.07530i	0.417157	−	0.240846i	0.693860	+	0.720110i	$0.255908\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	31.9693i		−	1.08448i
$870$		0			0
$871$	37.8609	+	21.8590i	1.28287	+	0.740665i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−2.69317	−	4.66471i	−0.0909420	−	0.157516i	0.816966	−	0.576686i	$-0.195654\pi$
$877$	−2.69317	−	4.66471i	−0.0909420	−	0.157516i	−0.907908	+	0.419170i	$0.862321\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−16.9730			−0.571836			−0.285918	−	0.958254i	$-0.592298\pi$
$881$	−16.9730			−0.571836			−0.285918	+	0.958254i	$0.592298\pi$
$882$		0			0
$883$	−45.7671			−1.54019			−0.770094	−	0.637931i	$-0.779790\pi$
$883$	−45.7671			−1.54019			−0.770094	+	0.637931i	$0.779790\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−8.03639	−	13.9194i	−0.269836	−	0.467369i	0.698984	−	0.715138i	$-0.253636\pi$
$887$	−8.03639	−	13.9194i	−0.269836	−	0.467369i	−0.968819	+	0.247769i	$0.920303\pi$
$888$		0			0
$889$	−26.2589	+	48.8047i	−0.880694	+	1.63686i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	36.2756	+	20.9437i	1.21392	+	0.700855i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−20.2159	+	35.0149i	−0.674237	+	1.16781i
$900$		0			0
$901$	−39.1135	+	22.5822i	−1.30306	+	0.752323i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	9.57991	−	16.5929i	0.318096	−	0.550958i	−0.661995	−	0.749508i	$-0.730290\pi$
$907$	9.57991	−	16.5929i	0.318096	−	0.550958i	0.980091	+	0.198550i	$0.0636233\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	23.0436i		−	0.763467i	−0.924272	−	0.381734i	$-0.875327\pi$
$911$		−	23.0436i		−	0.763467i	0.924272	−	0.381734i	$-0.124673\pi$
$912$		0			0
$913$	10.4032	+	6.00626i	0.344294	+	0.198778i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−26.2869	+	0.788431i	−0.868071	+	0.0260363i
$918$		0			0
$919$	0.0691922	+	0.119844i	0.00228244	+	0.00395330i	0.867164	−	0.498022i	$-0.165940\pi$
$919$	0.0691922	+	0.119844i	0.00228244	+	0.00395330i	−0.864882	+	0.501975i	$0.832607\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	36.8212			1.21198
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	10.3789	+	17.9767i	0.340520	+	0.589797i	0.984529	−	0.175220i	$-0.0560637\pi$
$929$	10.3789	+	17.9767i	0.340520	+	0.589797i	−0.644010	+	0.765017i	$0.722730\pi$
$930$		0			0
$931$	−43.4176	−	21.7078i	−1.42295	−	0.711446i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	13.4680i		−	0.439980i	−0.975502	−	0.219990i	$-0.929398\pi$
$937$		−	13.4680i		−	0.439980i	0.975502	−	0.219990i	$-0.0706024\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−15.4253	+	26.7173i	−0.502849	+	0.870961i	0.497145	+	0.867667i	$0.334382\pi$
$941$	−15.4253	+	26.7173i	−0.502849	+	0.870961i	−0.999995	+	0.00329321i	$0.998952\pi$
$942$		0			0
$943$	3.42246	−	1.97596i	0.111451	−	0.0643461i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.578340	+	0.333905i	−0.0187935	+	0.0108504i	−0.509367	−	0.860549i	$-0.670121\pi$
$947$	−0.578340	+	0.333905i	−0.0187935	+	0.0108504i	0.490574	+	0.871400i	$0.336787\pi$
$948$		0			0
$949$	−25.4242	+	44.0360i	−0.825304	+	1.42947i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			60.0400i			1.94488i	0.233143	+	0.972442i	$0.425099\pi$
$953$			60.0400i			1.94488i	−0.233143	+	0.972442i	$0.574901\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	44.1063	−	27.2598i	1.42427	−	0.880263i
$960$		0			0
$961$	2.75023	+	4.76354i	0.0887171	+	0.153662i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−16.4169			−0.527932			−0.263966	−	0.964532i	$-0.585031\pi$
$967$	−16.4169			−0.527932			−0.263966	+	0.964532i	$0.585031\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−24.5794	−	42.5727i	−0.788790	−	1.36622i	−0.926709	−	0.375781i	$-0.877375\pi$
$971$	−24.5794	−	42.5727i	−0.788790	−	1.36622i	0.137918	−	0.990444i	$-0.455959\pi$
$972$		0			0
$973$	39.8154	+	21.4223i	1.27642	+	0.686767i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4.45939	−	2.57463i	−0.142668	−	0.0823696i	0.426967	−	0.904267i	$-0.359582\pi$
$977$	−4.45939	−	2.57463i	−0.142668	−	0.0823696i	−0.569635	+	0.821898i	$0.692915\pi$
$978$		0			0
$979$		−	75.7745i		−	2.42176i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	10.7747	−	18.6623i	0.343659	−	0.595235i	−0.641450	−	0.767165i	$-0.721667\pi$
$983$	10.7747	−	18.6623i	0.343659	−	0.595235i	0.985109	+	0.171930i	$0.0550002\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.08384	−	3.51251i	0.193455	−	0.111691i
$990$		0			0
$991$	−3.70856	+	6.42342i	−0.117806	+	0.204047i	−0.918898	−	0.394495i	$-0.870920\pi$
$991$	−3.70856	+	6.42342i	−0.117806	+	0.204047i	0.801092	+	0.598542i	$0.204253\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	5.96578	+	3.44434i	0.188938	+	0.109083i	0.591485	−	0.806316i	$-0.298542\pi$
$997$	5.96578	+	3.44434i	0.188938	+	0.109083i	−0.402547	+	0.915399i	$0.631875\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.ch.f.4301.9		32
3.2	odd	2	inner	6300.2.ch.f.4301.10		32
5.2	odd	4		1260.2.dc.a.269.7	yes	32
5.3	odd	4		1260.2.dc.a.269.16	yes	32
5.4	even	2	inner	6300.2.ch.f.4301.7		32
7.5	odd	6	inner	6300.2.ch.f.1601.10		32
15.2	even	4		1260.2.dc.a.269.10	yes	32
15.8	even	4		1260.2.dc.a.269.1	yes	32
15.14	odd	2	inner	6300.2.ch.f.4301.8		32
21.5	even	6	inner	6300.2.ch.f.1601.9		32
35.3	even	12		8820.2.f.a.4409.21		32
35.12	even	12		1260.2.dc.a.89.1	✓	32
35.17	even	12		8820.2.f.a.4409.24		32
35.18	odd	12		8820.2.f.a.4409.12		32
35.19	odd	6	inner	6300.2.ch.f.1601.8		32
35.32	odd	12		8820.2.f.a.4409.9		32
35.33	even	12		1260.2.dc.a.89.10	yes	32
105.17	odd	12		8820.2.f.a.4409.10		32
105.32	even	12		8820.2.f.a.4409.23		32
105.38	odd	12		8820.2.f.a.4409.11		32
105.47	odd	12		1260.2.dc.a.89.16	yes	32
105.53	even	12		8820.2.f.a.4409.22		32
105.68	odd	12		1260.2.dc.a.89.7	yes	32
105.89	even	6	inner	6300.2.ch.f.1601.7		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.dc.a.89.1	✓	32	35.12	even	12
1260.2.dc.a.89.7	yes	32	105.68	odd	12
1260.2.dc.a.89.10	yes	32	35.33	even	12
1260.2.dc.a.89.16	yes	32	105.47	odd	12
1260.2.dc.a.269.1	yes	32	15.8	even	4
1260.2.dc.a.269.7	yes	32	5.2	odd	4
1260.2.dc.a.269.10	yes	32	15.2	even	4
1260.2.dc.a.269.16	yes	32	5.3	odd	4
6300.2.ch.f.1601.7		32	105.89	even	6	inner
6300.2.ch.f.1601.8		32	35.19	odd	6	inner
6300.2.ch.f.1601.9		32	21.5	even	6	inner
6300.2.ch.f.1601.10		32	7.5	odd	6	inner
6300.2.ch.f.4301.7		32	5.4	even	2	inner
6300.2.ch.f.4301.8		32	15.14	odd	2	inner
6300.2.ch.f.4301.9		32	1.1	even	1	trivial
6300.2.ch.f.4301.10		32	3.2	odd	2	inner
8820.2.f.a.4409.9		32	35.32	odd	12
8820.2.f.a.4409.10		32	105.17	odd	12
8820.2.f.a.4409.11		32	105.38	odd	12
8820.2.f.a.4409.12		32	35.18	odd	12
8820.2.f.a.4409.21		32	35.3	even	12
8820.2.f.a.4409.22		32	105.53	even	12
8820.2.f.a.4409.23		32	105.32	even	12
8820.2.f.a.4409.24		32	35.17	even	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)

\(f(q)\)	\(=\)	\(q+(1.25359 - 2.32992i) q^{7} +(4.81480 + 2.77982i) q^{11} -6.30365i q^{13} +(2.21467 - 3.83592i) q^{17} +(6.00550 - 3.46728i) q^{19} +(0.968157 - 0.558966i) q^{23} +6.69226i q^{29} +(5.23215 + 3.02078i) q^{31} +(0.273071 + 0.472973i) q^{37} +3.53503 q^{41} +6.28394 q^{43} +(3.02020 + 5.23113i) q^{47} +(-3.85703 - 5.84152i) q^{49} +(-8.83056 - 5.09832i) q^{53} +(0.298897 - 0.517705i) q^{59} +(-4.72271 + 2.72666i) q^{61} +(-3.46768 + 6.00619i) q^{67} +5.84125i q^{71} +(-6.98579 - 4.03324i) q^{73} +(12.5125 - 7.73332i) q^{77} +(-2.87511 - 4.97984i) q^{79} +2.16066 q^{83} +(-6.81468 - 11.8034i) q^{89} +(-14.6870 - 7.90219i) q^{91} -4.73923i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 24 q^{19} + 24 q^{31} - 40 q^{49} - 24 q^{61} - 32 q^{79} - 56 q^{91}+O(q^{100}) \) 32 * q + 24 * q^19 + 24 * q^31 - 40 * q^49 - 24 * q^61 - 32 * q^79 - 56 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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