LMFDB - Embedded newform 6300.2.ch.f.4301.5

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 = [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			4301.5
Character	$\chi$	$=$	6300.4301
Dual form			6300.2.ch.f.1601.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.29877 - 2.30504i) q^{7} +(1.33175 + 0.768889i) q^{11} -0.219560i q^{13} +(-0.713072 + 1.23508i) q^{17} +(-3.80115 + 2.19460i) q^{19} +(0.673823 - 0.389032i) q^{23} -6.82176i q^{29} +(5.32371 + 3.07365i) q^{31} +(2.99736 + 5.19158i) q^{37} +8.90761 q^{41} -1.91655 q^{43} +(-5.60192 - 9.70282i) q^{47} +(-3.62640 + 5.98742i) q^{49} +(-4.18095 - 2.41388i) q^{53} +(-0.336280 + 0.582453i) q^{59} +(9.64133 - 5.56643i) q^{61} +(5.65124 - 9.78824i) q^{67} +8.48310i q^{71} +(-1.88874 - 1.09046i) q^{73} +(0.0426786 - 4.06835i) q^{77} +(-3.19731 - 5.53790i) q^{79} -5.09349 q^{83} +(-2.97819 - 5.15838i) q^{89} +(-0.506094 + 0.285157i) q^{91} -12.7593i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−1.29877	−	2.30504i	−0.490888	−	0.871223i
$7$	−1.29877	−	2.30504i	−0.490888	−	0.871223i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.33175	+	0.768889i	0.401539	+	0.231829i	0.687148	−	0.726518i	$-0.258862\pi$
$11$	1.33175	+	0.768889i	0.401539	+	0.231829i	−0.285609	+	0.958346i	$0.592196\pi$
$12$		0			0
$13$		−	0.219560i		−	0.0608949i	−0.999536	−	0.0304475i	$-0.990307\pi$
$13$		−	0.219560i		−	0.0608949i	0.999536	−	0.0304475i	$-0.00969322\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.713072	+	1.23508i	−0.172945	+	0.299550i	−0.939448	−	0.342691i	$-0.888662\pi$
$17$	−0.713072	+	1.23508i	−0.172945	+	0.299550i	0.766503	+	0.642241i	$0.221995\pi$
$18$		0			0
$19$	−3.80115	+	2.19460i	−0.872044	+	0.503475i	−0.868027	−	0.496517i	$-0.834612\pi$
$19$	−3.80115	+	2.19460i	−0.872044	+	0.503475i	−0.00401693	+	0.999992i	$0.501279\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.673823	−	0.389032i	0.140502	−	0.0811187i	−0.428101	−	0.903731i	$-0.640817\pi$
$23$	0.673823	−	0.389032i	0.140502	−	0.0811187i	0.568603	+	0.822612i	$0.307484\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	6.82176i		−	1.26677i	−0.773837	−	0.633385i	$-0.781665\pi$
$29$		−	6.82176i		−	1.26677i	0.773837	−	0.633385i	$-0.218335\pi$
$30$		0			0
$31$	5.32371	+	3.07365i	0.956167	+	0.552043i	0.894991	−	0.446084i	$-0.147182\pi$
$31$	5.32371	+	3.07365i	0.956167	+	0.552043i	0.0611759	+	0.998127i	$0.480515\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.99736	+	5.19158i	0.492763	+	0.853490i	0.999965	−	0.00833661i	$-0.00265366\pi$
$37$	2.99736	+	5.19158i	0.492763	+	0.853490i	−0.507202	+	0.861827i	$0.669320\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.90761			1.39113			0.695567	−	0.718461i	$-0.255153\pi$
$41$	8.90761			1.39113			0.695567	+	0.718461i	$0.255153\pi$
$42$		0			0
$43$	−1.91655			−0.292271			−0.146135	−	0.989265i	$-0.546684\pi$
$43$	−1.91655			−0.292271			−0.146135	+	0.989265i	$0.546684\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.60192	−	9.70282i	−0.817125	−	1.41530i	−0.907792	−	0.419420i	$-0.862233\pi$
$47$	−5.60192	−	9.70282i	−0.817125	−	1.41530i	0.0906675	−	0.995881i	$-0.471100\pi$
$48$		0			0
$49$	−3.62640	+	5.98742i	−0.518058	+	0.855346i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.18095	−	2.41388i	−0.574298	−	0.331571i	0.184566	−	0.982820i	$-0.440912\pi$
$53$	−4.18095	−	2.41388i	−0.574298	−	0.331571i	−0.758864	+	0.651249i	$0.774245\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.336280	+	0.582453i	−0.0437799	+	0.0758290i	−0.887085	−	0.461606i	$-0.847273\pi$
$59$	−0.336280	+	0.582453i	−0.0437799	+	0.0758290i	0.843305	+	0.537435i	$0.180607\pi$
$60$		0			0
$61$	9.64133	−	5.56643i	1.23445	−	0.712708i	0.266493	−	0.963837i	$-0.414135\pi$
$61$	9.64133	−	5.56643i	1.23445	−	0.712708i	0.967954	+	0.251129i	$0.0808019\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.65124	−	9.78824i	0.690409	−	1.19582i	−0.281295	−	0.959621i	$-0.590764\pi$
$67$	5.65124	−	9.78824i	0.690409	−	1.19582i	0.971704	−	0.236202i	$-0.0759027\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			8.48310i			1.00676i	0.864065	+	0.503380i	$0.167910\pi$
$71$			8.48310i			1.00676i	−0.864065	+	0.503380i	$0.832090\pi$
$72$		0			0
$73$	−1.88874	−	1.09046i	−0.221060	−	0.127629i	0.385381	−	0.922758i	$-0.374070\pi$
$73$	−1.88874	−	1.09046i	−0.221060	−	0.127629i	−0.606441	+	0.795129i	$0.707403\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.0426786	−	4.06835i	0.00486368	−	0.463632i
$78$		0			0
$79$	−3.19731	−	5.53790i	−0.359725	−	0.623063i	0.628189	−	0.778060i	$-0.283796\pi$
$79$	−3.19731	−	5.53790i	−0.359725	−	0.623063i	−0.987915	+	0.154998i	$0.950463\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.09349			−0.559083			−0.279542	−	0.960134i	$-0.590182\pi$
$83$	−5.09349			−0.559083			−0.279542	+	0.960134i	$0.590182\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.97819	−	5.15838i	−0.315688	−	0.546788i	0.663896	−	0.747825i	$-0.268902\pi$
$89$	−2.97819	−	5.15838i	−0.315688	−	0.546788i	−0.979584	+	0.201038i	$0.935569\pi$
$90$		0			0
$91$	−0.506094	+	0.285157i	−0.0530530	+	0.0298926i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	12.7593i		−	1.29551i	−0.761850	−	0.647754i	$-0.775709\pi$
$97$		−	12.7593i		−	1.29551i	0.761850	−	0.647754i	$-0.224291\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.56102	+	2.70377i	−0.155328	+	0.269036i	−0.933178	−	0.359414i	$-0.882977\pi$
$101$	−1.56102	+	2.70377i	−0.155328	+	0.269036i	0.777851	+	0.628449i	$0.216310\pi$
$102$		0			0
$103$	11.0477	−	6.37840i	1.08856	−	0.628483i	0.155370	−	0.987856i	$-0.450343\pi$
$103$	11.0477	−	6.37840i	1.08856	−	0.628483i	0.933194	+	0.359374i	$0.117010\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.7690	+	6.21746i	−1.04107	+	0.601064i	−0.920137	−	0.391596i	$-0.871923\pi$
$107$	−10.7690	+	6.21746i	−1.04107	+	0.601064i	−0.120937	+	0.992660i	$0.538590\pi$
$108$		0			0
$109$	−5.28468	+	9.15334i	−0.506181	+	0.876731i	0.493794	+	0.869579i	$0.335610\pi$
$109$	−5.28468	+	9.15334i	−0.506181	+	0.876731i	−0.999974	+	0.00715162i	$0.997724\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	9.57684i		−	0.900913i	−0.892798	−	0.450457i	$-0.851261\pi$
$113$		−	9.57684i		−	0.900913i	0.892798	−	0.450457i	$-0.148739\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.77301	+	0.0395804i	0.345872	+	0.00362833i
$120$		0			0
$121$	−4.31762	−	7.47834i	−0.392511	−	0.679849i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	10.0099			0.888231			0.444115	−	0.895970i	$-0.353518\pi$
$127$	10.0099			0.888231			0.444115	+	0.895970i	$0.353518\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.57471	−	9.65568i	−0.487065	−	0.843621i	0.512825	−	0.858493i	$-0.328599\pi$
$131$	−5.57471	−	9.65568i	−0.487065	−	0.843621i	−0.999889	+	0.0148725i	$0.995266\pi$
$132$		0			0
$133$	9.99544	+	5.91153i	0.866714	+	0.512595i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.2553	−	5.92092i	−0.876173	−	0.505859i	−0.00677833	−	0.999977i	$-0.502158\pi$
$137$	−10.2553	−	5.92092i	−0.876173	−	0.505859i	−0.869395	+	0.494118i	$0.835491\pi$
$138$		0			0
$139$		−	11.6201i		−	0.985601i	−0.870142	−	0.492800i	$-0.835973\pi$
$139$		−	11.6201i		−	0.985601i	0.870142	−	0.492800i	$-0.164027\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.168817	−	0.292400i	0.0141172	−	0.0244517i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.72940	+	5.61727i	−0.797064	+	0.460185i	−0.842443	−	0.538785i	$-0.818884\pi$
$149$	−9.72940	+	5.61727i	−0.797064	+	0.460185i	0.0453796	+	0.998970i	$0.485550\pi$
$150$		0			0
$151$	−4.73025	+	8.19303i	−0.384942	+	0.666739i	−0.991761	−	0.128101i	$-0.959112\pi$
$151$	−4.73025	+	8.19303i	−0.384942	+	0.666739i	0.606819	+	0.794840i	$0.292445\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	5.35668	+	3.09268i	0.427510	+	0.246823i	0.698285	−	0.715820i	$-0.253947\pi$
$157$	5.35668	+	3.09268i	0.427510	+	0.246823i	−0.270775	+	0.962643i	$0.587280\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.77187	−	1.04793i	−0.139643	−	0.0825881i
$162$		0			0
$163$	−9.25512	−	16.0303i	−0.724917	−	1.25559i	−0.959008	−	0.283378i	$-0.908545\pi$
$163$	−9.25512	−	16.0303i	−0.724917	−	1.25559i	0.234091	−	0.972215i	$-0.424789\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−13.3700			−1.03460			−0.517302	−	0.855803i	$-0.673064\pi$
$167$	−13.3700			−1.03460			−0.517302	+	0.855803i	$0.673064\pi$
$168$		0			0
$169$	12.9518			0.996292
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.41850	+	5.92101i	0.259904	+	0.450166i	0.966216	−	0.257734i	$-0.0829759\pi$
$173$	3.41850	+	5.92101i	0.259904	+	0.450166i	−0.706312	+	0.707900i	$0.749643\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.83830	+	5.68015i	0.735349	+	0.424554i	0.820376	−	0.571825i	$-0.193764\pi$
$179$	9.83830	+	5.68015i	0.735349	+	0.424554i	−0.0850268	+	0.996379i	$0.527098\pi$
$180$		0			0
$181$		−	2.24531i		−	0.166893i	−0.996512	−	0.0834464i	$-0.973407\pi$
$181$		−	2.24531i		−	0.166893i	0.996512	−	0.0834464i	$-0.0265927\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.89927	+	1.09655i	−0.138889	+	0.0801874i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10.4953	+	6.05945i	−0.759412	+	0.438447i	−0.829085	−	0.559123i	$-0.811138\pi$
$191$	−10.4953	+	6.05945i	−0.759412	+	0.438447i	0.0696725	+	0.997570i	$0.477805\pi$
$192$		0			0
$193$	−7.34156	+	12.7160i	−0.528457	+	0.915314i	0.470993	+	0.882137i	$0.343896\pi$
$193$	−7.34156	+	12.7160i	−0.528457	+	0.915314i	−0.999449	+	0.0331770i	$0.989437\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	14.8181i		−	1.05575i	−0.849322	−	0.527874i	$-0.822989\pi$
$197$		−	14.8181i		−	1.05575i	0.849322	−	0.527874i	$-0.177011\pi$
$198$		0			0
$199$	0.953199	+	0.550330i	0.0675705	+	0.0390119i	0.533405	−	0.845860i	$-0.320912\pi$
$199$	0.953199	+	0.550330i	0.0675705	+	0.0390119i	−0.465834	+	0.884872i	$0.654246\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−15.7244	+	8.85988i	−1.10364	+	0.621842i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.74960			−0.466880
$210$		0			0
$211$	−7.28575			−0.501572			−0.250786	−	0.968043i	$-0.580689\pi$
$211$	−7.28575			−0.501572			−0.250786	+	0.968043i	$0.580689\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.170608	−	16.2633i	0.0115817	−	1.10403i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.271173	+	0.156562i	0.0182411	+	0.0105315i
$222$		0			0
$223$		−	4.06241i		−	0.272039i	−0.990706	−	0.136020i	$-0.956569\pi$
$223$		−	4.06241i		−	0.272039i	0.990706	−	0.136020i	$-0.0434310\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.78218	−	4.81888i	0.184660	−	0.319840i	−0.758802	−	0.651321i	$-0.774215\pi$
$227$	2.78218	−	4.81888i	0.184660	−	0.319840i	0.943462	+	0.331481i	$0.107548\pi$
$228$		0			0
$229$	10.5980	−	6.11877i	0.700337	−	0.404340i	−0.107136	−	0.994244i	$-0.534168\pi$
$229$	10.5980	−	6.11877i	0.700337	−	0.404340i	0.807473	+	0.589905i	$0.200835\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13.0410	−	7.52920i	0.854341	−	0.493254i	−0.00777208	−	0.999970i	$-0.502474\pi$
$233$	13.0410	−	7.52920i	0.854341	−	0.493254i	0.862113	+	0.506716i	$0.169141\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			26.7865i			1.73268i	0.499457	+	0.866339i	$0.333533\pi$
$239$			26.7865i			1.73268i	−0.499457	+	0.866339i	$0.666467\pi$
$240$		0			0
$241$	−17.2888	−	9.98167i	−1.11367	−	0.642976i	−0.173890	−	0.984765i	$-0.555634\pi$
$241$	−17.2888	−	9.98167i	−1.11367	−	0.642976i	−0.939777	+	0.341789i	$0.888967\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.481845	+	0.834579i	0.0306590	+	0.0531030i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−26.2579			−1.65739			−0.828693	−	0.559703i	$-0.810915\pi$
$251$	−26.2579			−1.65739			−0.828693	+	0.559703i	$0.810915\pi$
$252$		0			0
$253$	1.19649			0.0752226
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−15.2397	−	26.3959i	−0.950626	−	1.64653i	−0.744075	−	0.668096i	$-0.767110\pi$
$257$	−15.2397	−	26.3959i	−0.950626	−	1.64653i	−0.206550	−	0.978436i	$-0.566224\pi$
$258$		0			0
$259$	8.07392	−	13.6517i	0.501689	−	0.848274i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−21.3550	−	12.3293i	−1.31681	−	0.760258i	−0.333593	−	0.942717i	$-0.608261\pi$
$263$	−21.3550	−	12.3293i	−1.31681	−	0.760258i	−0.983213	+	0.182459i	$0.941594\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	11.7169	−	20.2942i	0.714389	−	1.23736i	−0.248805	−	0.968554i	$-0.580038\pi$
$269$	11.7169	−	20.2942i	0.714389	−	1.23736i	0.963195	−	0.268805i	$-0.0866288\pi$
$270$		0			0
$271$	−21.4850	+	12.4044i	−1.30512	+	0.753512i	−0.981278	−	0.192599i	$-0.938308\pi$
$271$	−21.4850	+	12.4044i	−1.30512	+	0.753512i	−0.323843	+	0.946111i	$0.604975\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	13.5798	−	23.5209i	0.815930	−	1.41323i	−0.0927281	−	0.995691i	$-0.529559\pi$
$277$	13.5798	−	23.5209i	0.815930	−	1.41323i	0.908658	−	0.417541i	$-0.137108\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	16.0353i		−	0.956584i	−0.878201	−	0.478292i	$-0.841256\pi$
$281$		−	16.0353i		−	0.956584i	0.878201	−	0.478292i	$-0.158744\pi$
$282$		0			0
$283$	−3.07926	−	1.77781i	−0.183043	−	0.105680i	0.405679	−	0.914016i	$-0.367035\pi$
$283$	−3.07926	−	1.77781i	−0.183043	−	0.105680i	−0.588721	+	0.808336i	$0.700369\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−11.5689	−	20.5324i	−0.682891	−	1.21199i
$288$		0			0
$289$	7.48306	+	12.9610i	0.440180	+	0.762414i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.76778			−0.570640			−0.285320	−	0.958432i	$-0.592100\pi$
$293$	−9.76778			−0.570640			−0.285320	+	0.958432i	$0.592100\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.0854157	−	0.147944i	−0.00493972	−	0.00855584i
$300$		0			0
$301$	2.48915	+	4.41772i	0.143472	+	0.254633i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	12.6916i		−	0.724348i	−0.932111	−	0.362174i	$-0.882035\pi$
$307$		−	12.6916i		−	0.724348i	0.932111	−	0.362174i	$-0.117965\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	13.9104	−	24.0936i	0.788788	−	1.36622i	−0.137922	−	0.990443i	$-0.544042\pi$
$311$	13.9104	−	24.0936i	0.788788	−	1.36622i	0.926710	−	0.375777i	$-0.122624\pi$
$312$		0			0
$313$	28.0036	−	16.1679i	1.58286	−	0.913862i	0.588416	−	0.808559i	$-0.299752\pi$
$313$	28.0036	−	16.1679i	1.58286	−	0.913862i	0.994440	−	0.105304i	$-0.0335814\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−24.8095	+	14.3238i	−1.39344	+	0.804504i	−0.993695	−	0.112121i	$-0.964236\pi$
$317$	−24.8095	+	14.3238i	−1.39344	+	0.804504i	−0.399748	+	0.916625i	$0.630902\pi$
$318$		0			0
$319$	5.24518	−	9.08491i	0.293673	−	0.508657i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	6.25962i		−	0.348294i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−15.0898	+	25.5144i	−0.831926	+	1.40665i
$330$		0			0
$331$	4.06936	+	7.04834i	0.223672	+	0.387412i	0.955920	−	0.293626i	$-0.0948621\pi$
$331$	4.06936	+	7.04834i	0.223672	+	0.387412i	−0.732248	+	0.681038i	$0.761529\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	15.0409			0.819329			0.409665	−	0.912236i	$-0.365646\pi$
$337$	15.0409			0.819329			0.409665	+	0.912236i	$0.365646\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.72659	+	8.18669i	0.255959	+	0.443334i
$342$		0			0
$343$	18.5111	+	0.582736i	0.999505	+	0.0314648i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5.15301	−	2.97509i	−0.276628	−	0.159711i	0.355268	−	0.934765i	$-0.384390\pi$
$347$	−5.15301	−	2.97509i	−0.276628	−	0.159711i	−0.631896	+	0.775053i	$0.717723\pi$
$348$		0			0
$349$		−	7.16980i		−	0.383790i	−0.981415	−	0.191895i	$-0.938537\pi$
$349$		−	7.16980i		−	0.383790i	0.981415	−	0.191895i	$-0.0614634\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.34317	−	14.4508i	0.444062	−	0.769138i	−0.553924	−	0.832567i	$-0.686870\pi$
$353$	8.34317	−	14.4508i	0.444062	−	0.769138i	0.997986	+	0.0634289i	$0.0202036\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	15.7020	−	9.06556i	0.828720	−	0.478462i	−0.0246942	−	0.999695i	$-0.507861\pi$
$359$	15.7020	−	9.06556i	0.828720	−	0.478462i	0.853414	+	0.521233i	$0.174528\pi$
$360$		0			0
$361$	0.132498	−	0.229494i	0.00697360	−	0.0120786i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	3.41987	+	1.97447i	0.178516	+	0.103066i	0.586595	−	0.809880i	$-0.300468\pi$
$367$	3.41987	+	1.97447i	0.178516	+	0.103066i	−0.408079	+	0.912947i	$0.633801\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.133987	+	12.7723i	−0.00695624	+	0.663106i
$372$		0			0
$373$	−12.3660	−	21.4186i	−0.640288	−	1.10901i	−0.985368	−	0.170439i	$-0.945482\pi$
$373$	−12.3660	−	21.4186i	−0.640288	−	1.10901i	0.345080	−	0.938573i	$-0.387852\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.49778			−0.0771398
$378$		0			0
$379$	−2.75630			−0.141581			−0.0707907	−	0.997491i	$-0.522552\pi$
$379$	−2.75630			−0.141581			−0.0707907	+	0.997491i	$0.522552\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2.27053	+	3.93267i	0.116019	+	0.200950i	0.918186	−	0.396148i	$-0.129653\pi$
$383$	2.27053	+	3.93267i	0.116019	+	0.200950i	−0.802168	+	0.597099i	$0.796320\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	19.1660	+	11.0655i	0.971754	+	0.561043i	0.899771	−	0.436363i	$-0.143734\pi$
$389$	19.1660	+	11.0655i	0.971754	+	0.561043i	0.0719838	+	0.997406i	$0.477067\pi$
$390$		0			0
$391$			1.10963i			0.0561164i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−16.8808	+	9.74615i	−0.847224	+	0.489145i	−0.859713	−	0.510777i	$-0.829358\pi$
$397$	−16.8808	+	9.74615i	−0.847224	+	0.489145i	0.0124890	+	0.999922i	$0.496025\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	24.7858	−	14.3101i	1.23775	−	0.714612i	0.269112	−	0.963109i	$-0.413270\pi$
$401$	24.7858	−	14.3101i	1.23775	−	0.714612i	0.968633	+	0.248496i	$0.0799364\pi$
$402$		0			0
$403$	0.674849	−	1.16887i	0.0336166	−	0.0582257i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			9.21855i			0.456946i
$408$		0			0
$409$	21.8809	+	12.6329i	1.08194	+	0.624659i	0.931419	−	0.363948i	$-0.118571\pi$
$409$	21.8809	+	12.6329i	1.08194	+	0.624659i	0.150521	+	0.988607i	$0.451905\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.77933	+	0.0186658i	0.0875549	+	0.000918485i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11.7457			−0.573816			−0.286908	−	0.957958i	$-0.592627\pi$
$419$	−11.7457			−0.573816			−0.286908	+	0.957958i	$0.592627\pi$
$420$		0			0
$421$	9.60230			0.467988			0.233994	−	0.972238i	$-0.424820\pi$
$421$	9.60230			0.467988			0.233994	+	0.972238i	$0.424820\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−25.3527	−	14.9941i	−1.22690	−	0.725618i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12.2719	+	7.08518i	0.591116	+	0.341281i	0.765539	−	0.643390i	$-0.222473\pi$
$431$	12.2719	+	7.08518i	0.591116	+	0.341281i	−0.174423	+	0.984671i	$0.555806\pi$
$432$		0			0
$433$			26.9828i			1.29671i	0.761338	+	0.648356i	$0.224543\pi$
$433$			26.9828i			1.29671i	−0.761338	+	0.648356i	$0.775457\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.70753	+	2.95754i	−0.0816824	+	0.141478i
$438$		0			0
$439$	−4.67427	+	2.69869i	−0.223091	+	0.128802i	−0.607381	−	0.794411i	$-0.707780\pi$
$439$	−4.67427	+	2.69869i	−0.223091	+	0.128802i	0.384290	+	0.923213i	$0.374446\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.05292	+	2.33995i	−0.192560	+	0.111175i	−0.593180	−	0.805070i	$-0.702128\pi$
$443$	−4.05292	+	2.33995i	−0.192560	+	0.111175i	0.400620	+	0.916244i	$0.368795\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	8.77899i		−	0.414306i	−0.978309	−	0.207153i	$-0.933580\pi$
$449$		−	8.77899i		−	0.414306i	0.978309	−	0.207153i	$-0.0664198\pi$
$450$		0			0
$451$	11.8627	+	6.84896i	0.558595	+	0.322505i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	18.8884	+	32.7157i	0.883562	+	1.53037i	0.847353	+	0.531030i	$0.178195\pi$
$457$	18.8884	+	32.7157i	0.883562	+	1.53037i	0.0362085	+	0.999344i	$0.488472\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8.85808			−0.412562			−0.206281	−	0.978493i	$-0.566136\pi$
$461$	−8.85808			−0.412562			−0.206281	+	0.978493i	$0.566136\pi$
$462$		0			0
$463$	−31.8659			−1.48094			−0.740468	−	0.672092i	$-0.765396\pi$
$463$	−31.8659			−1.48094			−0.740468	+	0.672092i	$0.765396\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−17.1590	−	29.7202i	−0.794022	−	1.37529i	−0.923458	−	0.383698i	$-0.874650\pi$
$467$	−17.1590	−	29.7202i	−0.794022	−	1.37529i	0.129437	−	0.991588i	$-0.458683\pi$
$468$		0			0
$469$	−29.9019	−	0.313683i	−1.38074	−	0.0144845i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.55237	−	1.47361i	−0.117358	−	0.0677568i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.4399	−	21.5465i	0.568392	−	0.984484i	−0.428333	−	0.903621i	$-0.640899\pi$
$479$	12.4399	−	21.5465i	0.568392	−	0.984484i	0.996725	−	0.0808634i	$-0.0257678\pi$
$480$		0			0
$481$	1.13986	−	0.658099i	0.0519732	−	0.0300067i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	19.4122	−	33.6229i	0.879649	−	1.52360i	0.0279233	−	0.999610i	$-0.491111\pi$
$487$	19.4122	−	33.6229i	0.879649	−	1.52360i	0.851726	−	0.523987i	$-0.175556\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	23.4000i		−	1.05603i	−0.849235	−	0.528014i	$-0.822937\pi$
$491$		−	23.4000i		−	1.05603i	0.849235	−	0.528014i	$-0.177063\pi$
$492$		0			0
$493$	8.42540	+	4.86441i	0.379461	+	0.219082i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	19.5539	−	11.0176i	0.877111	−	0.494206i
$498$		0			0
$499$	−1.80878	−	3.13291i	−0.0809723	−	0.140248i	0.822696	−	0.568482i	$-0.192469\pi$
$499$	−1.80878	−	3.13291i	−0.0809723	−	0.140248i	−0.903668	+	0.428234i	$0.859136\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−9.87647			−0.440370			−0.220185	−	0.975458i	$-0.570666\pi$
$503$	−9.87647			−0.440370			−0.220185	+	0.975458i	$0.570666\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−11.3323	−	19.6282i	−0.502297	−	0.870003i	−0.999996	−	0.00265410i	$-0.999155\pi$
$509$	−11.3323	−	19.6282i	−0.502297	−	0.870003i	0.497700	−	0.867349i	$-0.334178\pi$
$510$		0			0
$511$	−0.0605281	+	5.76987i	−0.00267761	+	0.255244i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	17.2290i		−	0.757732i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−6.82105	+	11.8144i	−0.298836	+	0.517598i	−0.975870	−	0.218353i	$-0.929932\pi$
$521$	−6.82105	+	11.8144i	−0.298836	+	0.517598i	0.677034	+	0.735952i	$0.263265\pi$
$522$		0			0
$523$	9.25298	−	5.34221i	0.404605	−	0.233599i	−0.283864	−	0.958864i	$-0.591616\pi$
$523$	9.25298	−	5.34221i	0.404605	−	0.233599i	0.688469	+	0.725266i	$0.258283\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7.59238	+	4.38346i	−0.330729	+	0.190947i
$528$		0			0
$529$	−11.1973	+	19.3943i	−0.486840	+	0.843231i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	1.95575i		−	0.0847130i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−9.43314	+	5.18547i	−0.406314	+	0.223354i
$540$		0			0
$541$	6.29078	+	10.8959i	0.270462	+	0.468453i	0.968980	−	0.247139i	$-0.0794904\pi$
$541$	6.29078	+	10.8959i	0.270462	+	0.468453i	−0.698519	+	0.715592i	$0.746157\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	36.2103			1.54824			0.774119	−	0.633040i	$-0.218193\pi$
$547$	36.2103			1.54824			0.774119	+	0.633040i	$0.218193\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	14.9710	+	25.9305i	0.637786	+	1.10468i
$552$		0			0
$553$	−8.61251	+	14.5624i	−0.366241	+	0.619255i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	16.4081	+	9.47324i	0.695235	+	0.401394i	0.805570	−	0.592500i	$-0.201859\pi$
$557$	16.4081	+	9.47324i	0.695235	+	0.401394i	−0.110335	+	0.993894i	$0.535192\pi$
$558$		0			0
$559$			0.420797i			0.0177978i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−15.0368	+	26.0445i	−0.633725	+	1.09764i	0.353058	+	0.935601i	$0.385142\pi$
$563$	−15.0368	+	26.0445i	−0.633725	+	1.09764i	−0.986784	+	0.162043i	$0.948192\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−12.9184	+	7.45847i	−0.541569	+	0.312675i	−0.745715	−	0.666265i	$-0.767892\pi$
$569$	−12.9184	+	7.45847i	−0.541569	+	0.312675i	0.204145	+	0.978941i	$0.434558\pi$
$570$		0			0
$571$	4.63417	−	8.02662i	0.193934	−	0.335904i	−0.752616	−	0.658459i	$-0.771209\pi$
$571$	4.63417	−	8.02662i	0.193934	−	0.335904i	0.946551	+	0.322555i	$0.104542\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	4.27659	+	2.46909i	0.178037	+	0.102790i	0.586370	−	0.810043i	$-0.300556\pi$
$577$	4.27659	+	2.46909i	0.178037	+	0.102790i	−0.408333	+	0.912833i	$0.633890\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	6.61526	+	11.7407i	0.274447	+	0.487086i
$582$		0			0
$583$	−3.71200	−	6.42938i	−0.153735	−	0.266278i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−24.6187			−1.01612			−0.508061	−	0.861321i	$-0.669638\pi$
$587$	−24.6187			−1.01612			−0.508061	+	0.861321i	$0.669638\pi$
$588$		0			0
$589$	−26.9817			−1.11176
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6.15800	+	10.6660i	0.252879	+	0.437999i	0.964317	−	0.264750i	$-0.0852893\pi$
$593$	6.15800	+	10.6660i	0.252879	+	0.437999i	−0.711438	+	0.702748i	$0.751956\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.3011	+	7.10203i	0.502608	+	0.290181i	0.729790	−	0.683671i	$-0.239618\pi$
$599$	12.3011	+	7.10203i	0.502608	+	0.290181i	−0.227182	+	0.973852i	$0.572951\pi$
$600$		0			0
$601$		−	21.1744i		−	0.863722i	−0.901940	−	0.431861i	$-0.857857\pi$
$601$		−	21.1744i		−	0.863722i	0.901940	−	0.431861i	$-0.142143\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−10.1345	+	5.85117i	−0.411348	+	0.237492i	−0.691369	−	0.722502i	$-0.742992\pi$
$607$	−10.1345	+	5.85117i	−0.411348	+	0.237492i	0.280021	+	0.959994i	$0.409659\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.13035	+	1.22996i	−0.0861846	+	0.0497587i
$612$		0			0
$613$	−5.51744	+	9.55649i	−0.222847	+	0.385983i	−0.955671	−	0.294435i	$-0.904869\pi$
$613$	−5.51744	+	9.55649i	−0.222847	+	0.385983i	0.732824	+	0.680418i	$0.238202\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	2.14075i		−	0.0861833i	−0.999071	−	0.0430917i	$-0.986279\pi$
$617$		−	2.14075i		−	0.0861833i	0.999071	−	0.0430917i	$-0.0137208\pi$
$618$		0			0
$619$	23.3766	+	13.4965i	0.939585	+	0.542470i	0.889830	−	0.456292i	$-0.150823\pi$
$619$	23.3766	+	13.4965i	0.939585	+	0.542470i	0.0497549	+	0.998761i	$0.484156\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−8.02229	+	13.5644i	−0.321406	+	0.543446i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−8.54933			−0.340884
$630$		0			0
$631$	24.0872			0.958895			0.479448	−	0.877571i	$-0.340837\pi$
$631$	24.0872			0.958895			0.479448	+	0.877571i	$0.340837\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.31460	+	0.796212i	0.0520862	+	0.0315471i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.634430	+	0.366288i	0.0250585	+	0.0144675i	0.512477	−	0.858701i	$-0.328728\pi$
$641$	0.634430	+	0.366288i	0.0250585	+	0.0144675i	−0.487418	+	0.873169i	$0.662061\pi$
$642$		0			0
$643$		−	35.1704i		−	1.38699i	−0.720463	−	0.693493i	$-0.756071\pi$
$643$		−	35.1704i		−	1.38699i	0.720463	−	0.693493i	$-0.243929\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−14.2676	+	24.7123i	−0.560919	+	0.971539i	0.436498	+	0.899705i	$0.356219\pi$
$647$	−14.2676	+	24.7123i	−0.560919	+	0.971539i	−0.997417	+	0.0718342i	$0.977115\pi$
$648$		0			0
$649$	−0.895684	+	0.517123i	−0.0351587	+	0.0202989i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	42.3967	−	24.4778i	1.65911	−	0.957889i	0.685988	−	0.727613i	$-0.259370\pi$
$653$	42.3967	−	24.4778i	1.65911	−	0.957889i	0.973125	−	0.230276i	$-0.0739629\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	51.1673i		−	1.99320i	−0.0824134	−	0.996598i	$-0.526263\pi$
$659$		−	51.1673i		−	1.99320i	0.0824134	−	0.996598i	$-0.473737\pi$
$660$		0			0
$661$	37.8824	+	21.8714i	1.47345	+	0.850699i	0.999553	−	0.0298803i	$-0.00951262\pi$
$661$	37.8824	+	21.8714i	1.47345	+	0.850699i	0.473900	+	0.880579i	$0.342846\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−2.65388	−	4.59666i	−0.102759	−	0.177983i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	17.1199			0.660905
$672$		0			0
$673$	46.7354			1.80152			0.900758	−	0.434322i	$-0.143012\pi$
$673$	46.7354			1.80152			0.900758	+	0.434322i	$0.143012\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.35435	+	2.34581i	0.0520520	+	0.0901566i	0.890877	−	0.454244i	$-0.150091\pi$
$677$	1.35435	+	2.34581i	0.0520520	+	0.0901566i	−0.838825	+	0.544401i	$0.816757\pi$
$678$		0			0
$679$	−29.4106	+	16.5713i	−1.12868	+	0.635949i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−24.6266	−	14.2182i	−0.942312	−	0.544044i	−0.0516279	−	0.998666i	$-0.516441\pi$
$683$	−24.6266	−	14.2182i	−0.942312	−	0.544044i	−0.890684	+	0.454622i	$0.849774\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.529990	+	0.917969i	−0.0201910	+	0.0349718i
$690$		0			0
$691$	39.1304	−	22.5920i	1.48859	−	0.859439i	0.488676	−	0.872465i	$-0.337480\pi$
$691$	39.1304	−	22.5920i	1.48859	−	0.859439i	0.999915	+	0.0130268i	$0.00414668\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−6.35176	+	11.0016i	−0.240590	+	0.416715i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			36.4833i			1.37796i	0.724782	+	0.688978i	$0.241940\pi$
$701$			36.4833i			1.37796i	−0.724782	+	0.688978i	$0.758060\pi$
$702$		0			0
$703$	−22.7868	−	13.1560i	−0.859422	−	0.496187i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	8.25971	+	0.0866475i	0.310638	+	0.00325872i
$708$		0			0
$709$	−18.6520	−	32.3062i	−0.700490	−	1.21328i	−0.968295	−	0.249811i	$-0.919632\pi$
$709$	−18.6520	−	32.3062i	−0.700490	−	1.21328i	0.267805	−	0.963473i	$-0.413702\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4.78299			0.179124
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−11.1031	−	19.2311i	−0.414074	−	0.717198i	0.581256	−	0.813720i	$-0.302561\pi$
$719$	−11.1031	−	19.2311i	−0.414074	−	0.717198i	−0.995331	+	0.0965227i	$0.969228\pi$
$720$		0			0
$721$	−29.0509	−	17.1813i	−1.08191	−	0.639867i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			16.0910i			0.596783i	0.954443	+	0.298392i	$0.0964502\pi$
$727$			16.0910i			0.596783i	−0.954443	+	0.298392i	$0.903550\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.36664	−	2.36708i	0.0505469	−	0.0875498i
$732$		0			0
$733$	−2.74704	+	1.58600i	−0.101464	+	0.0585803i	−0.549873	−	0.835248i	$-0.685324\pi$
$733$	−2.74704	+	1.58600i	−0.101464	+	0.0585803i	0.448409	+	0.893828i	$0.351991\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	15.0521	−	8.69035i	0.554452	−	0.320113i
$738$		0			0
$739$	−20.7122	+	35.8746i	−0.761912	+	1.31967i	0.179952	+	0.983675i	$0.442406\pi$
$739$	−20.7122	+	35.8746i	−0.761912	+	1.31967i	−0.941864	+	0.335995i	$0.890928\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	14.7140i		−	0.539805i	−0.962888	−	0.269903i	$-0.913008\pi$
$743$		−	14.7140i		−	0.539805i	0.962888	−	0.269903i	$-0.0869915\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	28.3178	+	16.7478i	1.03471	+	0.611952i
$750$		0			0
$751$	−17.6294	−	30.5350i	−0.643307	−	1.11424i	−0.984690	−	0.174316i	$-0.944229\pi$
$751$	−17.6294	−	30.5350i	−0.643307	−	1.11424i	0.341383	−	0.939924i	$-0.389105\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−32.1067			−1.16694			−0.583470	−	0.812135i	$-0.698305\pi$
$757$	−32.1067			−1.16694			−0.583470	+	0.812135i	$0.698305\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11.8054	−	20.4476i	−0.427947	−	0.741226i	0.568744	−	0.822515i	$-0.307430\pi$
$761$	−11.8054	−	20.4476i	−0.427947	−	0.741226i	−0.996691	+	0.0812889i	$0.974096\pi$
$762$		0			0
$763$	27.9624	+	0.293336i	1.01231	+	0.0106195i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	0.127883	+	0.0738335i	0.00461760	+	0.00266597i
$768$		0			0
$769$		−	16.3200i		−	0.588516i	−0.955726	−	0.294258i	$-0.904928\pi$
$769$		−	16.3200i		−	0.588516i	0.955726	−	0.294258i	$-0.0950724\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	22.2492	−	38.5368i	0.800249	−	1.38607i	−0.119203	−	0.992870i	$-0.538034\pi$
$773$	22.2492	−	38.5368i	0.800249	−	1.38607i	0.919452	−	0.393202i	$-0.128633\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−33.8592	+	19.5486i	−1.21313	+	0.700401i
$780$		0			0
$781$	−6.52256	+	11.2974i	−0.233396	+	0.404253i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−14.9208	−	8.61451i	−0.531868	−	0.307074i	0.209909	−	0.977721i	$-0.432683\pi$
$787$	−14.9208	−	8.61451i	−0.531868	−	0.307074i	−0.741777	+	0.670647i	$0.766017\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−22.0750	+	12.4381i	−0.784896	+	0.442248i
$792$		0			0
$793$	−1.22216	−	2.11685i	−0.0434003	−	0.0751715i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	35.5457			1.25909			0.629547	−	0.776962i	$-0.283240\pi$
$797$	35.5457			1.25909			0.629547	+	0.776962i	$0.283240\pi$
$798$		0			0
$799$	15.9783			0.565272
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.67689	−	2.90446i	−0.0591761	−	0.102496i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	20.9699	+	12.1070i	0.737262	+	0.425659i	0.821073	−	0.570823i	$-0.193376\pi$
$809$	20.9699	+	12.1070i	0.737262	+	0.425659i	−0.0838107	+	0.996482i	$0.526709\pi$
$810$		0			0
$811$		−	10.7691i		−	0.378155i	−0.981962	−	0.189077i	$-0.939450\pi$
$811$		−	10.7691i		−	0.378155i	0.981962	−	0.189077i	$-0.0605497\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	7.28509	−	4.20605i	0.254873	−	0.147151i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−28.1730	+	16.2657i	−0.983244	+	0.567676i	−0.903248	−	0.429119i	$-0.858824\pi$
$821$	−28.1730	+	16.2657i	−0.983244	+	0.567676i	−0.0799960	+	0.996795i	$0.525491\pi$
$822$		0			0
$823$	−14.4636	+	25.0517i	−0.504169	+	0.873246i	0.495819	+	0.868426i	$0.334868\pi$
$823$	−14.4636	+	25.0517i	−0.504169	+	0.873246i	−0.999988	+	0.00482067i	$0.998466\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			22.1968i			0.771857i	0.922529	+	0.385928i	$0.126119\pi$
$827$			22.1968i			0.771857i	−0.922529	+	0.385928i	$0.873881\pi$
$828$		0			0
$829$	−10.6202	−	6.13160i	−0.368856	−	0.212959i	0.304102	−	0.952639i	$-0.401643\pi$
$829$	−10.6202	−	6.13160i	−0.368856	−	0.212959i	−0.672959	+	0.739680i	$0.734977\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.80904	−	8.74835i	−0.166623	−	0.303112i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−53.7542			−1.85580			−0.927901	−	0.372826i	$-0.878389\pi$
$839$	−53.7542			−1.85580			−0.927901	+	0.372826i	$0.878389\pi$
$840$		0			0
$841$	−17.5364			−0.604704
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−11.6303	+	19.6649i	−0.399621	+	0.675694i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.03938	+	2.33214i	0.138468	+	0.0799446i
$852$		0			0
$853$			50.9882i			1.74580i	0.487896	+	0.872902i	$0.337764\pi$
$853$			50.9882i			1.74580i	−0.487896	+	0.872902i	$0.662236\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18.9075	−	32.7487i	0.645867	−	1.11868i	−0.338233	−	0.941062i	$-0.609829\pi$
$857$	18.9075	−	32.7487i	0.645867	−	1.11868i	0.984100	−	0.177613i	$-0.0568374\pi$
$858$		0			0
$859$	14.5633	−	8.40811i	0.496892	−	0.286881i	−0.230537	−	0.973064i	$-0.574048\pi$
$859$	14.5633	−	8.40811i	0.496892	−	0.286881i	0.727429	+	0.686183i	$0.240715\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−45.9362	+	26.5213i	−1.56369	+	0.902795i	−0.566808	+	0.823850i	$0.691822\pi$
$863$	−45.9362	+	26.5213i	−1.56369	+	0.902795i	−0.996879	+	0.0789449i	$0.974845\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	9.83350i		−	0.333579i
$870$		0			0
$871$	−2.14910	−	1.24078i	−0.0728196	−	0.0420424i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.523776	+	0.907207i	0.0176867	+	0.0306342i	0.874733	−	0.484605i	$-0.161037\pi$
$877$	0.523776	+	0.907207i	0.0176867	+	0.0306342i	−0.857047	+	0.515239i	$0.827703\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−9.68108			−0.326164			−0.163082	−	0.986613i	$-0.552143\pi$
$881$	−9.68108			−0.326164			−0.163082	+	0.986613i	$0.552143\pi$
$882$		0			0
$883$	−48.8068			−1.64248			−0.821240	−	0.570583i	$-0.806717\pi$
$883$	−48.8068			−1.64248			−0.821240	+	0.570583i	$0.806717\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	20.0214	+	34.6780i	0.672252	+	1.16437i	0.977264	+	0.212026i	$0.0680062\pi$
$887$	20.0214	+	34.6780i	0.672252	+	1.16437i	−0.305012	+	0.952349i	$0.598660\pi$
$888$		0			0
$889$	−13.0005	−	23.0731i	−0.436022	−	0.773847i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	42.5875	+	24.5879i	1.42514	+	0.822803i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	20.9677	−	36.3171i	0.699312	−	1.21124i
$900$		0			0
$901$	5.96264	−	3.44253i	0.198644	−	0.114687i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−9.95564	+	17.2437i	−0.330572	+	0.572567i	−0.982624	−	0.185607i	$-0.940575\pi$
$907$	−9.95564	+	17.2437i	−0.330572	+	0.572567i	0.652052	+	0.758174i	$0.273908\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			3.57388i			0.118408i	0.998246	+	0.0592040i	$0.0188562\pi$
$911$			3.57388i			0.118408i	−0.998246	+	0.0592040i	$0.981144\pi$
$912$		0			0
$913$	−6.78328	−	3.91633i	−0.224494	−	0.129612i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−15.0165	+	25.3904i	−0.495887	+	0.838465i
$918$		0			0
$919$	6.37661	+	11.0446i	0.210345	+	0.364328i	0.951822	−	0.306650i	$-0.0992080\pi$
$919$	6.37661	+	11.0446i	0.210345	+	0.364328i	−0.741478	+	0.670978i	$0.765875\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.86255			0.0613065
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	12.0062	+	20.7953i	0.393910	+	0.682272i	0.992961	−	0.118439i	$-0.0377889\pi$
$929$	12.0062	+	20.7953i	0.393910	+	0.682272i	−0.599052	+	0.800710i	$0.704456\pi$
$930$		0			0
$931$	0.644549	−	30.7176i	0.0211242	−	1.00673i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			14.2712i			0.466219i	0.972450	+	0.233110i	$0.0748901\pi$
$937$			14.2712i			0.466219i	−0.972450	+	0.233110i	$0.925110\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3.24431	−	5.61931i	0.105761	−	0.183184i	−0.808288	−	0.588788i	$-0.799605\pi$
$941$	3.24431	−	5.61931i	0.105761	−	0.183184i	0.914049	+	0.405604i	$0.132939\pi$
$942$		0			0
$943$	6.00215	−	3.46534i	0.195457	−	0.112847i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	9.92760	−	5.73171i	0.322604	−	0.186255i	−0.329949	−	0.943999i	$-0.607031\pi$
$947$	9.92760	−	5.73171i	0.322604	−	0.186255i	0.652553	+	0.757743i	$0.273698\pi$
$948$		0			0
$949$	−0.239422	+	0.414690i	−0.00777195	+	0.0134614i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			37.3915i			1.21123i	0.795758	+	0.605614i	$0.207072\pi$
$953$			37.3915i			1.21123i	−0.795758	+	0.605614i	$0.792928\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−0.328652	+	31.3289i	−0.0106127	+	1.01166i
$960$		0			0
$961$	3.39462	+	5.87965i	0.109504	+	0.189666i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−23.8585			−0.767238			−0.383619	−	0.923491i	$-0.625322\pi$
$967$	−23.8585			−0.767238			−0.383619	+	0.923491i	$0.625322\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−9.98661	−	17.2973i	−0.320486	−	0.555097i	0.660103	−	0.751175i	$-0.270513\pi$
$971$	−9.98661	−	17.2973i	−0.320486	−	0.555097i	−0.980588	+	0.196078i	$0.937179\pi$
$972$		0			0
$973$	−26.7847	+	15.0918i	−0.858678	+	0.483820i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.99389	+	1.15117i	0.0637901	+	0.0368292i	0.531556	−	0.847023i	$-0.321608\pi$
$977$	1.99389	+	1.15117i	0.0637901	+	0.0368292i	−0.467766	+	0.883852i	$0.654941\pi$
$978$		0			0
$979$		−	9.15960i		−	0.292742i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−19.1598	+	33.1857i	−0.611101	+	1.05846i	0.379954	+	0.925006i	$0.375940\pi$
$983$	−19.1598	+	33.1857i	−0.611101	+	1.05846i	−0.991055	+	0.133453i	$0.957393\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.29141	+	0.745598i	−0.0410646	+	0.0237086i
$990$		0			0
$991$	−13.0540	+	22.6101i	−0.414673	+	0.718234i	−0.995394	−	0.0958682i	$-0.969437\pi$
$991$	−13.0540	+	22.6101i	−0.414673	+	0.718234i	0.580721	+	0.814102i	$0.302771\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	16.9099	+	9.76295i	0.535543	+	0.309196i	0.743271	−	0.668991i	$-0.233274\pi$
$997$	16.9099	+	9.76295i	0.535543	+	0.309196i	−0.207728	+	0.978187i	$0.566607\pi$
$998$		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.4301.5		32
3.2	odd	2	inner	6300.2.ch.f.4301.6		32
5.2	odd	4		1260.2.dc.a.269.4	yes	32
5.3	odd	4		1260.2.dc.a.269.15	yes	32
5.4	even	2	inner	6300.2.ch.f.4301.11		32
7.5	odd	6	inner	6300.2.ch.f.1601.6		32
15.2	even	4		1260.2.dc.a.269.13	yes	32
15.8	even	4		1260.2.dc.a.269.2	yes	32
15.14	odd	2	inner	6300.2.ch.f.4301.12		32
21.5	even	6	inner	6300.2.ch.f.1601.5		32
35.3	even	12		8820.2.f.a.4409.17		32
35.12	even	12		1260.2.dc.a.89.2	✓	32
35.17	even	12		8820.2.f.a.4409.20		32
35.18	odd	12		8820.2.f.a.4409.15		32
35.19	odd	6	inner	6300.2.ch.f.1601.12		32
35.32	odd	12		8820.2.f.a.4409.14		32
35.33	even	12		1260.2.dc.a.89.13	yes	32
105.17	odd	12		8820.2.f.a.4409.13		32
105.32	even	12		8820.2.f.a.4409.19		32
105.38	odd	12		8820.2.f.a.4409.16		32
105.47	odd	12		1260.2.dc.a.89.15	yes	32
105.53	even	12		8820.2.f.a.4409.18		32
105.68	odd	12		1260.2.dc.a.89.4	yes	32
105.89	even	6	inner	6300.2.ch.f.1601.11		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.dc.a.89.2	✓	32	35.12	even	12
1260.2.dc.a.89.4	yes	32	105.68	odd	12
1260.2.dc.a.89.13	yes	32	35.33	even	12
1260.2.dc.a.89.15	yes	32	105.47	odd	12
1260.2.dc.a.269.2	yes	32	15.8	even	4
1260.2.dc.a.269.4	yes	32	5.2	odd	4
1260.2.dc.a.269.13	yes	32	15.2	even	4
1260.2.dc.a.269.15	yes	32	5.3	odd	4
6300.2.ch.f.1601.5		32	21.5	even	6	inner
6300.2.ch.f.1601.6		32	7.5	odd	6	inner
6300.2.ch.f.1601.11		32	105.89	even	6	inner
6300.2.ch.f.1601.12		32	35.19	odd	6	inner
6300.2.ch.f.4301.5		32	1.1	even	1	trivial
6300.2.ch.f.4301.6		32	3.2	odd	2	inner
6300.2.ch.f.4301.11		32	5.4	even	2	inner
6300.2.ch.f.4301.12		32	15.14	odd	2	inner
8820.2.f.a.4409.13		32	105.17	odd	12
8820.2.f.a.4409.14		32	35.32	odd	12
8820.2.f.a.4409.15		32	35.18	odd	12
8820.2.f.a.4409.16		32	105.38	odd	12
8820.2.f.a.4409.17		32	35.3	even	12
8820.2.f.a.4409.18		32	105.53	even	12
8820.2.f.a.4409.19		32	105.32	even	12
8820.2.f.a.4409.20		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)

\(f(q)\)	\(=\)	\(q+(-1.29877 - 2.30504i) q^{7} +(1.33175 + 0.768889i) q^{11} -0.219560i q^{13} +(-0.713072 + 1.23508i) q^{17} +(-3.80115 + 2.19460i) q^{19} +(0.673823 - 0.389032i) q^{23} -6.82176i q^{29} +(5.32371 + 3.07365i) q^{31} +(2.99736 + 5.19158i) q^{37} +8.90761 q^{41} -1.91655 q^{43} +(-5.60192 - 9.70282i) q^{47} +(-3.62640 + 5.98742i) q^{49} +(-4.18095 - 2.41388i) q^{53} +(-0.336280 + 0.582453i) q^{59} +(9.64133 - 5.56643i) q^{61} +(5.65124 - 9.78824i) q^{67} +8.48310i q^{71} +(-1.88874 - 1.09046i) q^{73} +(0.0426786 - 4.06835i) q^{77} +(-3.19731 - 5.53790i) q^{79} -5.09349 q^{83} +(-2.97819 - 5.15838i) q^{89} +(-0.506094 + 0.285157i) q^{91} -12.7593i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 24 q^{19} + 24 q^{31} - 40 q^{49} - 24 q^{61} - 32 q^{79} - 56 q^{91}+O(q^{100}) \) 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

Related objects

Downloads

Learn more