LMFDB - Embedded newform 6300.2.ch.f.4301.8

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			4301.8
Character	$\chi$	$=$	6300.4301
Dual form			6300.2.ch.f.1601.8

$q$-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	−0.453137	−	0.891441i	$-0.649695\pi$
$11$	−4.81480	−	2.77982i	−1.45172	−	0.838149i	−0.998579	+	0.0532922i	$0.983029\pi$
$12$		0			0
$13$			6.30365i			1.74832i	0.485639	+	0.874160i	$0.338587\pi$
$13$			6.30365i			1.74832i	−0.485639	+	0.874160i	$0.661413\pi$
$14$		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	$-0.786580\pi$
$29$		−	6.69226i		−	1.24272i	0.783524	−	0.621361i	$-0.213420\pi$
$30$		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	0.842706	−	0.538374i	$-0.180961\pi$
$37$	−0.273071	−	0.472973i	−0.0448926	−	0.0777563i	−0.887599	+	0.460618i	$0.847628\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.53503			−0.552079			−0.276039	−	0.961146i	$-0.589022\pi$
$41$	−3.53503			−0.552079			−0.276039	+	0.961146i	$0.589022\pi$
$42$		0			0
$43$	−6.28394			−0.958291			−0.479146	−	0.877735i	$-0.659053\pi$
$43$	−6.28394			−0.958291			−0.479146	+	0.877735i	$0.659053\pi$
$44$		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	−0.884826	−	0.465922i	$-0.845723\pi$
$59$	−0.298897	+	0.517705i	−0.0389131	+	0.0673994i	0.845913	+	0.533321i	$0.179056\pi$
$60$		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	−0.572649	−	0.819801i	$-0.694084\pi$
$67$	3.46768	−	6.00619i	0.423644	−	0.733773i	0.996293	+	0.0860279i	$0.0274174\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	5.84125i		−	0.693229i	−0.938008	−	0.346614i	$-0.887331\pi$
$71$		−	5.84125i		−	0.693229i	0.938008	−	0.346614i	$-0.112669\pi$
$72$		0			0
$73$	6.98579	+	4.03324i	0.817624	+	0.472056i	0.849597	−	0.527433i	$-0.176845\pi$
$73$	6.98579	+	4.03324i	0.817624	+	0.472056i	−0.0319722	+	0.999489i	$0.510179\pi$
$74$		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	$0.0902736\pi$
$89$	6.81468	+	11.8034i	0.722355	+	1.25116i	−0.237699	+	0.971339i	$0.576393\pi$
$90$		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	$0.0773435\pi$
$97$			4.73923i			0.481196i	−0.970625	+	0.240598i	$0.922657\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.52364	+	2.63903i	−0.151608	+	0.262593i	−0.931819	−	0.362924i	$-0.881778\pi$
$101$	−1.52364	+	2.63903i	−0.151608	+	0.262593i	0.780211	+	0.625517i	$0.215112\pi$
$102$		0			0
$103$	6.02919	−	3.48095i	0.594073	−	0.342988i	−0.172633	−	0.984986i	$-0.555227\pi$
$103$	6.02919	−	3.48095i	0.594073	−	0.342988i	0.766707	+	0.641998i	$0.221894\pi$
$104$		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	$-0.120350\pi$
$127$	20.9470			1.85874			0.929371	+	0.369147i	$0.120350\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.97000	+	8.60829i	0.434231	+	0.752109i	0.997232	−	0.0743462i	$-0.0236870\pi$
$131$	4.97000	+	8.60829i	0.434231	+	0.752109i	−0.563002	+	0.826456i	$0.690354\pi$
$132$		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	−0.161855	−	0.986815i	$-0.551748\pi$
$149$	7.46827	−	4.31181i	0.611825	−	0.353237i	0.773679	+	0.633577i	$0.218414\pi$
$150$		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	0.885256	−	0.465103i	$-0.153983\pi$
$157$	11.5914	+	6.69229i	0.925093	+	0.534103i	0.0398367	+	0.999206i	$0.487316\pi$
$158$		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	0.585993	−	0.810316i	$-0.300705\pi$
$163$	−5.21867	−	9.03900i	−0.408758	−	0.707989i	−0.994751	+	0.102327i	$0.967371\pi$
$164$		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	0.458301	−	0.888797i	$-0.348458\pi$
$179$	−1.10068	−	0.635476i	−0.0822685	−	0.0474977i	−0.540570	+	0.841299i	$0.681791\pi$
$180$		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	−0.403476	−	0.914990i	$-0.632198\pi$
$191$	2.58703	−	1.49362i	0.187191	−	0.108075i	0.590667	+	0.806915i	$0.298865\pi$
$192$		0			0
$193$	3.19070	−	5.52646i	0.229672	−	0.397803i	−0.728039	−	0.685536i	$-0.759568\pi$
$193$	3.19070	−	5.52646i	0.229672	−	0.397803i	0.957711	+	0.287732i	$0.0929014\pi$
$194$		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	$-0.943768\pi$
$223$		−	5.24874i		−	0.351481i	0.984436	−	0.175741i	$-0.0562320\pi$
$224$		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	$-0.752074\pi$
$239$		−	21.7203i		−	1.40497i	0.711698	−	0.702486i	$-0.247926\pi$
$240$		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	$-0.476387\pi$
$251$	2.34842			0.148231			0.0741155	+	0.997250i	$0.476387\pi$
$252$		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	$-0.646009\pi$
$269$	9.10448	−	15.7694i	0.555110	−	0.961479i	0.997895	−	0.0648510i	$-0.0206572\pi$
$270$		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	−0.986374	−	0.164519i	$-0.947393\pi$
$277$	−5.83696	+	10.1099i	−0.350709	+	0.607446i	0.635665	+	0.771965i	$0.280726\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			4.73421i			0.282419i	0.989980	+	0.141210i	$0.0450991\pi$
$281$			4.73421i			0.282419i	−0.989980	+	0.141210i	$0.954901\pi$
$282$		0			0
$283$	8.95022	+	5.16741i	0.532035	+	0.307171i	0.741845	−	0.670572i	$-0.233951\pi$
$283$	8.95022	+	5.16741i	0.532035	+	0.307171i	−0.209810	+	0.977742i	$0.567284\pi$
$284$		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	$0.388905\pi$
$307$			32.9301i			1.87942i	−0.341973	+	0.939710i	$0.611095\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5.79118	−	10.0306i	0.328388	−	0.568784i	−0.653804	−	0.756664i	$-0.726828\pi$
$311$	5.79118	−	10.0306i	0.328388	−	0.568784i	0.982192	+	0.187879i	$0.0601614\pi$
$312$		0			0
$313$	6.29874	−	3.63658i	0.356026	−	0.205551i	−0.311310	−	0.950308i	$-0.600768\pi$
$313$	6.29874	−	3.63658i	0.356026	−	0.205551i	0.667336	+	0.744757i	$0.267435\pi$
$314$		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	$-0.718979\pi$
$337$	−23.3122			−1.26990			−0.634949	+	0.772554i	$0.718979\pi$
$338$		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	−0.723182	−	0.690657i	$-0.757321\pi$
$359$	−9.22065	+	5.32354i	−0.486647	+	0.280966i	0.236535	+	0.971623i	$0.423988\pi$
$360$		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	0.954432	−	0.298428i	$-0.0964623\pi$
$367$	22.4753	+	12.9761i	1.17320	+	0.677349i	0.218770	+	0.975776i	$0.429796\pi$
$368$		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	$-0.872316\pi$
$373$	−15.4208	−	26.7096i	−0.798458	−	1.38297i	0.122163	−	0.992510i	$-0.461017\pi$
$374$		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	$0.0365537\pi$
$389$	31.3469	+	18.0982i	1.58935	+	0.917614i	0.595940	+	0.803029i	$0.296780\pi$
$390$		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	−0.472296	−	0.881440i	$-0.656575\pi$
$397$	1.09398	−	0.631611i	0.0549054	−	0.0316997i	0.527201	+	0.849740i	$0.323241\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−34.1957	+	19.7429i	−1.70765	+	0.985914i	−0.770201	+	0.637801i	$0.779844\pi$
$401$	−34.1957	+	19.7429i	−1.70765	+	0.985914i	−0.937453	+	0.348113i	$0.886823\pi$
$402$		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	$-0.535862\pi$
$419$	−4.60260			−0.224852			−0.112426	+	0.993660i	$0.535862\pi$
$420$		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	0.826175	−	0.563414i	$-0.190512\pi$
$431$	15.5980	+	9.00553i	0.751331	+	0.433781i	−0.0748435	+	0.997195i	$0.523846\pi$
$432$		0			0
$433$			17.5684i			0.844281i	0.906530	+	0.422141i	$0.138721\pi$
$433$			17.5684i			0.844281i	−0.906530	+	0.422141i	$0.861279\pi$
$434$		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	$0.140024\pi$
$449$			18.0471i			0.851696i	−0.904795	+	0.425848i	$0.859976\pi$
$450$		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	0.998481	−	0.0550936i	$-0.0175457\pi$
$457$	9.65257	+	16.7187i	0.451528	+	0.782070i	−0.546953	+	0.837163i	$0.684212\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−22.4305			−1.04469			−0.522347	−	0.852733i	$-0.674944\pi$
$461$	−22.4305			−1.04469			−0.522347	+	0.852733i	$0.674944\pi$
$462$		0			0
$463$	18.4754			0.858625			0.429313	−	0.903156i	$-0.358756\pi$
$463$	18.4754			0.858625			0.429313	+	0.903156i	$0.358756\pi$
$464$		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	−0.798160	−	0.602445i	$-0.794193\pi$
$479$	2.68438	−	4.64948i	0.122652	−	0.212440i	0.920813	+	0.390005i	$0.127527\pi$
$480$		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	$-0.644350\pi$
$487$	12.3457	−	21.3834i	0.559439	−	0.968977i	0.997543	−	0.0700525i	$-0.0223167\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			24.1955i			1.09193i	0.837808	+	0.545964i	$0.183837\pi$
$491$			24.1955i			1.09193i	−0.837808	+	0.545964i	$0.816163\pi$
$492$		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	$-0.989858\pi$
$509$	−11.8972	−	20.6066i	−0.527336	−	0.913373i	0.472156	−	0.881515i	$-0.343476\pi$
$510$		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	−0.695650	−	0.718381i	$-0.744884\pi$
$521$	6.26127	−	10.8448i	0.274311	−	0.475121i	0.969961	+	0.243260i	$0.0782169\pi$
$522$		0			0
$523$	15.3521	−	8.86357i	0.671303	−	0.387577i	−0.125267	−	0.992123i	$-0.539979\pi$
$523$	15.3521	−	8.86357i	0.671303	−	0.387577i	0.796570	+	0.604546i	$0.206646\pi$
$524$		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	$-0.417974\pi$
$547$	11.9209			0.509699			0.254850	+	0.966981i	$0.417974\pi$
$548$		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	−0.128333	−	0.991731i	$-0.540963\pi$
$569$	15.8953	−	9.17714i	0.666364	−	0.384726i	0.794698	+	0.607005i	$0.207629\pi$
$570$		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	0.343561	−	0.939131i	$-0.388367\pi$
$577$	−7.15748	−	4.13238i	−0.297970	−	0.172033i	−0.641531	+	0.767097i	$0.721700\pi$
$578$		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	0.389176	−	0.921163i	$-0.372760\pi$
$599$	−5.23722	−	3.02371i	−0.213987	−	0.123545i	−0.603163	+	0.797618i	$0.706093\pi$
$600$		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	−0.522448	−	0.852671i	$-0.674981\pi$
$607$	−1.11451	+	0.643460i	−0.0452364	+	0.0261172i	0.477211	+	0.878789i	$0.341648\pi$
$608$		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	−0.994930	−	0.100565i	$-0.967935\pi$
$613$	−10.1604	+	17.5983i	−0.410373	+	0.710788i	0.584557	+	0.811353i	$0.301268\pi$
$614$		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	0.641996	−	0.766708i	$-0.278107\pi$
$641$	7.57019	+	4.37065i	0.299005	+	0.172630i	−0.342991	+	0.939339i	$0.611440\pi$
$642$		0			0
$643$		−	3.69796i		−	0.145833i	−0.997338	−	0.0729167i	$-0.976769\pi$
$643$		−	3.69796i		−	0.145833i	0.997338	−	0.0729167i	$-0.0232307\pi$
$644$		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	$-0.874080\pi$
$659$		−	19.7847i		−	0.770704i	0.922770	−	0.385352i	$-0.125920\pi$
$660$		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	$-0.745782\pi$
$673$	−36.1986			−1.39535			−0.697676	+	0.716414i	$0.745782\pi$
$674$		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	$-0.844959\pi$
$701$		−	24.7843i		−	0.936090i	0.883705	−	0.468045i	$-0.155041\pi$
$702$		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	0.743557	−	0.668672i	$-0.233137\pi$
$719$	−5.55881	−	9.62814i	−0.207309	−	0.359069i	−0.950866	+	0.309603i	$0.899804\pi$
$720$		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	$-0.758301\pi$
$727$		−	37.1241i		−	1.37685i	0.725305	−	0.688427i	$-0.241699\pi$
$728$		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	−0.463931	−	0.885871i	$-0.653561\pi$
$733$	1.93012	−	1.11435i	0.0712905	−	0.0411596i	0.535222	+	0.844712i	$0.320228\pi$
$734$		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	$-0.396331\pi$
$757$	17.6064			0.639916			0.319958	+	0.947432i	$0.396331\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	4.88581	+	8.46248i	0.177111	+	0.306765i	0.940890	−	0.338713i	$-0.109992\pi$
$761$	4.88581	+	8.46248i	0.177111	+	0.306765i	−0.763779	+	0.645478i	$0.776658\pi$
$762$		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	−0.117808	−	0.993036i	$-0.537587\pi$
$787$	−29.0833	−	16.7912i	−1.03671	−	0.598543i	−0.918899	+	0.394494i	$0.870920\pi$
$788$		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	0.758232	−	0.651985i	$-0.226063\pi$
$809$	16.2896	+	9.40482i	0.572713	+	0.330656i	−0.185519	+	0.982641i	$0.559397\pi$
$810$		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	0.416247	−	0.909252i	$-0.363345\pi$
$821$	40.4526	−	23.3553i	1.41181	−	0.815106i	0.995558	+	0.0941456i	$0.0300119\pi$
$822$		0			0
$823$	−3.10583	+	5.37945i	−0.108262	+	0.187516i	−0.915066	−	0.403303i	$-0.867862\pi$
$823$	−3.10583	+	5.37945i	−0.108262	+	0.187516i	0.806804	+	0.590819i	$0.201195\pi$
$824$		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	$-0.349436\pi$
$839$	26.3915			0.911136			0.455568	+	0.890201i	$0.349436\pi$
$840$		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	$0.0964719\pi$
$853$			17.4336i			0.596914i	−0.954423	+	0.298457i	$0.903528\pi$
$854$		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	0.907908	−	0.419170i	$-0.137679\pi$
$877$	2.69317	+	4.66471i	0.0909420	+	0.157516i	−0.816966	+	0.576686i	$0.804346\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	16.9730			0.571836			0.285918	−	0.958254i	$-0.407702\pi$
$881$	16.9730			0.571836			0.285918	+	0.958254i	$0.407702\pi$
$882$		0			0
$883$	45.7671			1.54019			0.770094	−	0.637931i	$-0.220210\pi$
$883$	45.7671			1.54019			0.770094	+	0.637931i	$0.220210\pi$
$884$		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	−0.980091	−	0.198550i	$-0.936377\pi$
$907$	−9.57991	+	16.5929i	−0.318096	+	0.550958i	0.661995	+	0.749508i	$0.269710\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			23.0436i			0.763467i	0.924272	+	0.381734i	$0.124673\pi$
$911$			23.0436i			0.763467i	−0.924272	+	0.381734i	$0.875327\pi$
$912$		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	0.644010	−	0.765017i	$-0.277270\pi$
$929$	−10.3789	−	17.9767i	−0.340520	−	0.589797i	−0.984529	+	0.175220i	$0.943936\pi$
$930$		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	$0.0706024\pi$
$937$			13.4680i			0.439980i	−0.975502	+	0.219990i	$0.929398\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	15.4253	−	26.7173i	0.502849	−	0.870961i	−0.497145	−	0.867667i	$-0.665618\pi$
$941$	15.4253	−	26.7173i	0.502849	−	0.870961i	0.999995	−	0.00329321i	$-0.00104826\pi$
$942$		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	$-0.414969\pi$
$967$	16.4169			0.527932			0.263966	+	0.964532i	$0.414969\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	24.5794	+	42.5727i	0.788790	+	1.36622i	0.926709	+	0.375781i	$0.122625\pi$
$971$	24.5794	+	42.5727i	0.788790	+	1.36622i	−0.137918	+	0.990444i	$0.544041\pi$
$972$		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	0.402547	−	0.915399i	$-0.368125\pi$
$997$	−5.96578	−	3.44434i	−0.188938	−	0.109083i	−0.591485	+	0.806316i	$0.701458\pi$
$998$		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.4301.8		32
3.2	odd	2	inner	6300.2.ch.f.4301.7		32
5.2	odd	4		1260.2.dc.a.269.1	yes	32
5.3	odd	4		1260.2.dc.a.269.10	yes	32
5.4	even	2	inner	6300.2.ch.f.4301.10		32
7.5	odd	6	inner	6300.2.ch.f.1601.7		32
15.2	even	4		1260.2.dc.a.269.16	yes	32
15.8	even	4		1260.2.dc.a.269.7	yes	32
15.14	odd	2	inner	6300.2.ch.f.4301.9		32
21.5	even	6	inner	6300.2.ch.f.1601.8		32
35.3	even	12		8820.2.f.a.4409.10		32
35.12	even	12		1260.2.dc.a.89.7	yes	32
35.17	even	12		8820.2.f.a.4409.11		32
35.18	odd	12		8820.2.f.a.4409.23		32
35.19	odd	6	inner	6300.2.ch.f.1601.9		32
35.32	odd	12		8820.2.f.a.4409.22		32
35.33	even	12		1260.2.dc.a.89.16	yes	32
105.17	odd	12		8820.2.f.a.4409.21		32
105.32	even	12		8820.2.f.a.4409.12		32
105.38	odd	12		8820.2.f.a.4409.24		32
105.47	odd	12		1260.2.dc.a.89.10	yes	32
105.53	even	12		8820.2.f.a.4409.9		32
105.68	odd	12		1260.2.dc.a.89.1	✓	32
105.89	even	6	inner	6300.2.ch.f.1601.10		32

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

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