LMFDB - Embedded newform 475.2.g.c.18.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [475,2,Mod(18,475)] 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("475.18"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(475, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.g (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$3.79289409601$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 96x^{12} + 2394x^{8} + 9184x^{4} + 2025 $ x^16 + 96x^12 + 2394x^8 + 9184*x^4 + 2025
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			18.4
Root			$-0.492178 + 0.492178i$ of defining polynomial
Character	$\chi$	$=$	475.18
Dual form			475.2.g.c.132.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.492178 - 0.492178i) q^{2} +(1.61999 - 1.61999i) q^{3} -1.51552i q^{4} -1.59464 q^{6} +(-1.73026 + 1.73026i) q^{8} -2.24873i q^{9} -6.27978 q^{11} +(-2.45513 - 2.45513i) q^{12} +(5.09266 - 5.09266i) q^{13} -1.32785 q^{16} +(-1.10678 + 1.10678i) q^{18} -4.35890i q^{19} +(3.09076 + 3.09076i) q^{22} +5.60601i q^{24} -5.01299 q^{26} +(1.21705 + 1.21705i) q^{27} +(4.11406 + 4.11406i) q^{32} +(-10.1732 + 10.1732i) q^{33} -3.40800 q^{36} +(3.80924 + 3.80924i) q^{37} +(-2.14535 + 2.14535i) q^{38} -16.5001i q^{39} +9.51714i q^{44} +(-2.15111 + 2.15111i) q^{48} +7.00000i q^{49} +(-7.71804 - 7.71804i) q^{52} +(10.2892 - 10.2892i) q^{53} -1.19801i q^{54} +(-7.06137 - 7.06137i) q^{57} +11.8083 q^{61} -1.39399i q^{64} +10.0140 q^{66} +(-2.13960 - 2.13960i) q^{67} +(3.89089 + 3.89089i) q^{72} -3.74965i q^{74} -6.60601 q^{76} +(-8.12098 + 8.12098i) q^{78} +10.6894 q^{81} +(10.8657 - 10.8657i) q^{88} +13.3295 q^{96} +(-13.5292 - 13.5292i) q^{97} +(3.44524 - 3.44524i) q^{98} +14.1215i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 64 q^{16} - 48 q^{26} + 208 q^{36} - 208 q^{66} - 144 q^{81} + 304 q^{96}+O(q^{100}) $ 16 * q - 64 * q^16 - 48 * q^26 + 208 * q^36 - 208 * q^66 - 144 * q^81 + 304 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$

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 $

$2$	−0.492178	−	0.492178i	−0.348022	−	0.348022i	0.511350	−	0.859372i	$-0.329145\pi$
$2$	−0.492178	−	0.492178i	−0.348022	−	0.348022i	−0.859372	+	0.511350i	$0.829145\pi$
$3$	1.61999	−	1.61999i	0.935301	−	0.935301i	−0.0627292	−	0.998031i	$-0.519980\pi$
$3$	1.61999	−	1.61999i	0.935301	−	0.935301i	0.998031	+	0.0627292i	$0.0199805\pi$
$4$		−	1.51552i		−	0.757761i
$5$		0			0
$5$		0			0
$6$	−1.59464			−0.651011
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	−1.73026	+	1.73026i	−0.611740	+	0.611740i
$9$		−	2.24873i		−	0.749577i
$10$		0			0
$11$	−6.27978			−1.89342			−0.946712	−	0.322081i	$-0.895618\pi$
$11$	−6.27978			−1.89342			−0.946712	+	0.322081i	$0.895618\pi$
$12$	−2.45513	−	2.45513i	−0.708735	−	0.708735i
$13$	5.09266	−	5.09266i	1.41245	−	1.41245i	0.670921	−	0.741529i	$-0.265899\pi$
$13$	5.09266	−	5.09266i	1.41245	−	1.41245i	0.741529	−	0.670921i	$-0.234101\pi$
$14$		0			0
$15$		0			0
$16$	−1.32785			−0.331963
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$	−1.10678	+	1.10678i	−0.260869	+	0.260869i
$19$		−	4.35890i		−	1.00000i
$19$		−	4.35890i		−	1.00000i
$20$		0			0
$21$		0			0
$22$	3.09076	+	3.09076i	0.658953	+	0.658953i
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$			5.60601i			1.14432i
$25$		0			0
$26$	−5.01299			−0.983127
$27$	1.21705	+	1.21705i	0.234221	+	0.234221i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	4.11406	+	4.11406i	0.727270	+	0.727270i
$33$	−10.1732	+	10.1732i	−1.77092	+	1.77092i
$34$		0			0
$35$		0			0
$36$	−3.40800			−0.568001
$37$	3.80924	+	3.80924i	0.626236	+	0.626236i	0.947119	−	0.320883i	$-0.103980\pi$
$37$	3.80924	+	3.80924i	0.626236	+	0.626236i	−0.320883	+	0.947119i	$0.603980\pi$
$38$	−2.14535	+	2.14535i	−0.348022	+	0.348022i
$39$		−	16.5001i		−	2.64213i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$			9.51714i			1.43476i
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$	−2.15111	+	2.15111i	−0.310486	+	0.310486i
$49$			7.00000i			1.00000i
$50$		0			0
$51$		0			0
$52$	−7.71804	−	7.71804i	−1.07030	−	1.07030i
$53$	10.2892	−	10.2892i	1.41333	−	1.41333i	0.681680	−	0.731651i	$-0.261250\pi$
$53$	10.2892	−	10.2892i	1.41333	−	1.41333i	0.731651	−	0.681680i	$-0.238750\pi$
$54$		−	1.19801i		−	0.163028i
$55$		0			0
$56$		0			0
$57$	−7.06137	−	7.06137i	−0.935301	−	0.935301i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	11.8083			1.51190			0.755948	−	0.654632i	$-0.227176\pi$
$61$	11.8083			1.51190			0.755948	+	0.654632i	$0.227176\pi$
$62$		0			0
$63$		0			0
$64$		−	1.39399i		−	0.174249i
$65$		0			0
$66$	10.0140			1.23264
$67$	−2.13960	−	2.13960i	−0.261393	−	0.261393i	0.564227	−	0.825620i	$-0.309174\pi$
$67$	−2.13960	−	2.13960i	−0.261393	−	0.261393i	−0.825620	+	0.564227i	$0.809174\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	3.89089	+	3.89089i	0.458546	+	0.458546i
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$		−	3.74965i		−	0.435888i
$75$		0			0
$76$	−6.60601			−0.757761
$77$		0			0
$78$	−8.12098	+	8.12098i	−0.919520	+	0.919520i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	10.6894			1.18771
$82$		0			0
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	10.8657	−	10.8657i	1.15828	−	1.15828i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	13.3295			1.36043
$97$	−13.5292	−	13.5292i	−1.37368	−	1.37368i	−0.854920	−	0.518760i	$-0.826394\pi$
$97$	−13.5292	−	13.5292i	−1.37368	−	1.37368i	−0.518760	−	0.854920i	$-0.673606\pi$
$98$	3.44524	−	3.44524i	0.348022	−	0.348022i
$99$			14.1215i			1.41927i
$100$		0			0
$101$	−14.8134			−1.47398			−0.736992	−	0.675901i	$-0.763755\pi$
$101$	−14.8134			−1.47398			−0.736992	+	0.675901i	$0.763755\pi$
$102$		0			0
$103$	−8.04573	+	8.04573i	−0.792769	+	0.792769i	−0.981943	−	0.189175i	$-0.939419\pi$
$103$	−8.04573	+	8.04573i	−0.792769	+	0.792769i	0.189175	+	0.981943i	$0.439419\pi$
$104$			17.6233i			1.72810i
$105$		0			0
$106$	−10.1282			−0.983740
$107$	14.2936	+	14.2936i	1.38182	+	1.38182i	0.841393	+	0.540424i	$0.181736\pi$
$107$	14.2936	+	14.2936i	1.38182	+	1.38182i	0.540424	+	0.841393i	$0.318264\pi$
$108$	1.84446	−	1.84446i	0.177483	−	0.177483i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	12.3419			1.17144
$112$		0			0
$113$	13.3093	−	13.3093i	1.25203	−	1.25203i	0.297223	−	0.954808i	$-0.403940\pi$
$113$	13.3093	−	13.3093i	1.25203	−	1.25203i	0.954808	−	0.297223i	$-0.0960604\pi$
$114$			6.95090i			0.651011i
$115$		0			0
$116$		0			0
$117$	−11.4520	−	11.4520i	−1.05874	−	1.05874i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	28.4356			2.58505
$122$	−5.81177	−	5.81177i	−0.526173	−	0.526173i
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	12.4785	+	12.4785i	1.10728	+	1.10728i	0.993506	+	0.113778i	$0.0362951\pi$
$127$	12.4785	+	12.4785i	1.10728	+	1.10728i	0.113778	+	0.993506i	$0.463705\pi$
$128$	7.54203	−	7.54203i	0.666628	−	0.666628i
$129$		0			0
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$	15.4177	+	15.4177i	1.34194	+	1.34194i
$133$		0			0
$134$			2.10612i			0.181941i
$135$		0			0
$136$		0			0
$137$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$137$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$138$		0			0
$139$			9.28485i			0.787531i	0.919211	+	0.393765i	$0.128828\pi$
$139$			9.28485i			0.787531i	−0.919211	+	0.393765i	$0.871172\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−31.9808	+	31.9808i	−2.67437	+	2.67437i
$144$			2.98599i			0.248832i
$145$		0			0
$146$		0			0
$147$	11.3399	+	11.3399i	0.935301	+	0.935301i
$148$	5.77299	−	5.77299i	0.474537	−	0.474537i
$149$			10.3057i			0.844280i	0.906531	+	0.422140i	$0.138721\pi$
$149$			10.3057i			0.844280i	−0.906531	+	0.422140i	$0.861279\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$	7.54203	+	7.54203i	0.611740	+	0.611740i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−25.0063			−2.00211
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		−	33.3368i		−	2.64378i
$160$		0			0
$161$		0			0
$162$	−5.26108	−	5.26108i	−0.413350	−	0.413350i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.99850	−	5.99850i	−0.464178	−	0.464178i	0.435844	−	0.900022i	$-0.356450\pi$
$167$	−5.99850	−	5.99850i	−0.464178	−	0.464178i	−0.900022	+	0.435844i	$0.856450\pi$
$168$		0			0
$169$		−	38.8704i		−	2.99003i
$170$		0			0
$171$	−9.80199			−0.749577
$172$		0			0
$173$	−11.3406	+	11.3406i	−0.862207	+	0.862207i	−0.991594	−	0.129387i	$-0.958699\pi$
$173$	−11.3406	+	11.3406i	−0.862207	+	0.862207i	0.129387	+	0.991594i	$0.458699\pi$
$174$		0			0
$175$		0			0
$176$	8.33863			0.628548
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	19.1293	−	19.1293i	1.41408	−	1.41408i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−26.1534			−1.89239			−0.946197	−	0.323592i	$-0.895109\pi$
$191$	−26.1534			−1.89239			−0.946197	+	0.323592i	$0.895109\pi$
$192$	−2.25825	−	2.25825i	−0.162975	−	0.162975i
$193$	−5.91069	+	5.91069i	−0.425461	+	0.425461i	−0.887079	−	0.461618i	$-0.847269\pi$
$193$	−5.91069	+	5.91069i	−0.425461	+	0.425461i	0.461618	+	0.887079i	$0.347269\pi$
$194$			13.3175i			0.956142i
$195$		0			0
$196$	10.6087			0.757761
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$	6.95030	−	6.95030i	0.493936	−	0.493936i
$199$		−	8.71780i		−	0.617988i	−0.951064	−	0.308994i	$-0.900008\pi$
$199$		−	8.71780i		−	0.617988i	0.951064	−	0.308994i	$-0.0999924\pi$
$200$		0			0
$201$	−6.93224			−0.488963
$202$	7.29080	+	7.29080i	0.512979	+	0.512979i
$203$		0			0
$204$		0			0
$205$		0			0
$206$	7.91985			0.551802
$207$		0			0
$208$	−6.76231	+	6.76231i	−0.468882	+	0.468882i
$209$			27.3729i			1.89342i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−15.5935	−	15.5935i	−1.07097	−	1.07097i
$213$		0			0
$214$		−	14.0700i		−	0.961806i
$215$		0			0
$216$	−4.21162			−0.286564
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	−6.07439	−	6.07439i	−0.407686	−	0.407686i
$223$	2.75852	−	2.75852i	0.184724	−	0.184724i	−0.608687	−	0.793411i	$-0.708303\pi$
$223$	2.75852	−	2.75852i	0.184724	−	0.184724i	0.793411	+	0.608687i	$0.208303\pi$
$224$		0			0
$225$		0			0
$226$	−13.1010			−0.871469
$227$	−10.3562	−	10.3562i	−0.687365	−	0.687365i	0.274283	−	0.961649i	$-0.411559\pi$
$227$	−10.3562	−	10.3562i	−0.687365	−	0.687365i	−0.961649	+	0.274283i	$0.911559\pi$
$228$	−10.7017	+	10.7017i	−0.708735	+	0.708735i
$229$		−	16.3159i		−	1.07818i	−0.842247	−	0.539092i	$-0.818767\pi$
$229$		−	16.3159i		−	1.07818i	0.842247	−	0.539092i	$-0.181233\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$			11.2729i			0.736930i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			24.0000i			1.55243i	0.630468	+	0.776215i	$0.282863\pi$
$239$			24.0000i			1.55243i	−0.630468	+	0.776215i	$0.717137\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−13.9954	−	13.9954i	−0.899656	−	0.899656i
$243$	13.6656	−	13.6656i	0.876647	−	0.876647i
$244$		−	17.8957i		−	1.14566i
$245$		0			0
$246$		0			0
$247$	−22.1984	−	22.1984i	−1.41245	−	1.41245i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−26.1534			−1.65079			−0.825394	−	0.564557i	$-0.809047\pi$
$251$	−26.1534			−1.65079			−0.825394	+	0.564557i	$0.809047\pi$
$252$		0			0
$253$		0			0
$254$		−	12.2832i		−	0.770718i
$255$		0			0
$256$	−10.2120			−0.638251
$257$	20.0091	+	20.0091i	1.24814	+	1.24814i	0.956542	+	0.291593i	$0.0941854\pi$
$257$	20.0091	+	20.0091i	1.24814	+	1.24814i	0.291593	+	0.956542i	$0.405815\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−5.90613	−	5.90613i	−0.364882	−	0.364882i
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$		−	35.2045i		−	2.16669i
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−3.24260	+	3.24260i	−0.198074	+	0.198074i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−32.9014			−1.99862			−0.999309	−	0.0371559i	$-0.988170\pi$
$271$	−32.9014			−1.99862			−0.999309	+	0.0371559i	$0.988170\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$	4.56979	−	4.56979i	0.274078	−	0.274078i
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		0			0
$286$	31.4804			1.86148
$287$		0			0
$288$	9.25142	−	9.25142i	0.545145	−	0.545145i
$289$			17.0000i			1.00000i
$290$		0			0
$291$	−43.8343			−2.56961
$292$		0			0
$293$	9.15067	−	9.15067i	0.534588	−	0.534588i	−0.387346	−	0.921934i	$-0.626608\pi$
$293$	9.15067	−	9.15067i	0.534588	−	0.534588i	0.921934	+	0.387346i	$0.126608\pi$
$294$		−	11.1625i		−	0.651011i
$295$		0			0
$296$	−13.1820			−0.766186
$297$	−7.64278	−	7.64278i	−0.443479	−	0.443479i
$298$	5.07226	−	5.07226i	0.293828	−	0.293828i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−23.9975	+	23.9975i	−1.37862	+	1.37862i
$304$			5.78798i			0.331963i
$305$		0			0
$306$		0			0
$307$	22.5102	+	22.5102i	1.28473	+	1.28473i	0.937946	+	0.346781i	$0.112725\pi$
$307$	22.5102	+	22.5102i	1.28473	+	1.28473i	0.346781	+	0.937946i	$0.387275\pi$
$308$		0			0
$309$			26.0680i			1.48296i
$310$		0			0
$311$	10.7874			0.611697			0.305848	−	0.952080i	$-0.401060\pi$
$311$	10.7874			0.611697			0.305848	+	0.952080i	$0.401060\pi$
$312$	28.5495	+	28.5495i	1.61630	+	1.61630i
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	19.2154	+	19.2154i	1.07924	+	1.07924i	0.996577	+	0.0826672i	$0.0263439\pi$
$317$	19.2154	+	19.2154i	1.07924	+	1.07924i	0.0826672	+	0.996577i	$0.473656\pi$
$318$	−16.4076	+	16.4076i	−0.920094	+	0.920094i
$319$		0			0
$320$		0			0
$321$	46.3110			2.58483
$322$		0			0
$323$		0			0
$324$		−	16.2000i		−	0.900002i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	8.56597	−	8.56597i	0.469412	−	0.469412i
$334$			5.90465i			0.323088i
$335$		0			0
$336$		0			0
$337$	10.9988	+	10.9988i	0.599142	+	0.599142i	0.940084	−	0.340942i	$-0.110746\pi$
$337$	10.9988	+	10.9988i	0.599142	+	0.599142i	−0.340942	+	0.940084i	$0.610746\pi$
$338$	−19.1311	+	19.1311i	−1.04060	+	1.04060i
$339$		−	43.1218i		−	2.34205i
$340$		0			0
$341$		0			0
$342$	4.82432	+	4.82432i	0.260869	+	0.260869i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	11.1631			0.600134
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$			34.8712i			1.86661i	0.359082	+	0.933306i	$0.383090\pi$
$349$			34.8712i			1.86661i	−0.359082	+	0.933306i	$0.616910\pi$
$350$		0			0
$351$	12.3960			0.661650
$352$	−25.8354	−	25.8354i	−1.37703	−	1.37703i
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			1.77217i			0.0935314i	0.998906	+	0.0467657i	$0.0148914\pi$
$359$			1.77217i			0.0935314i	−0.998906	+	0.0467657i	$0.985109\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$	46.0654	−	46.0654i	2.41780	−	2.41780i
$364$		0			0
$365$		0			0
$366$	−18.8300			−0.984261
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−23.2491	+	23.2491i	−1.20379	+	1.20379i	−0.230790	+	0.973004i	$0.574131\pi$
$373$	−23.2491	+	23.2491i	−1.20379	+	1.20379i	−0.973004	+	0.230790i	$0.925869\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	40.4299			2.07129
$382$	12.8721	+	12.8721i	0.658595	+	0.658595i
$383$	8.38750	−	8.38750i	0.428581	−	0.428581i	−0.459564	−	0.888145i	$-0.651994\pi$
$383$	8.38750	−	8.38750i	0.428581	−	0.428581i	0.888145	+	0.459564i	$0.151994\pi$
$384$		−	24.4360i		−	1.24700i
$385$		0			0
$386$	5.81822			0.296140
$387$		0			0
$388$	−20.5038	+	20.5038i	−1.04092	+	1.04092i
$389$		−	6.00000i		−	0.304212i	−0.988364	−	0.152106i	$-0.951394\pi$
$389$		−	6.00000i		−	0.304212i	0.988364	−	0.152106i	$-0.0486055\pi$
$390$		0			0
$391$		0			0
$392$	−12.1118	−	12.1118i	−0.611740	−	0.611740i
$393$	19.4399	−	19.4399i	0.980612	−	0.980612i
$394$		0			0
$395$		0			0
$396$	21.4015			1.07547
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$	−4.29070	+	4.29070i	−0.215074	+	0.215074i
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$	3.41189	+	3.41189i	0.170170	+	0.170170i
$403$		0			0
$404$			22.4500i			1.11693i
$405$		0			0
$406$		0			0
$407$	−23.9212	−	23.9212i	−1.18573	−	1.18573i
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$	12.1935	+	12.1935i	0.600730	+	0.600730i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	41.9030			2.05447
$417$	15.0414	+	15.0414i	0.736579	+	0.736579i
$418$	13.4723	−	13.4723i	0.658953	−	0.658953i
$419$		−	36.0000i		−	1.75872i	−0.476162	−	0.879358i	$-0.657972\pi$
$419$		−	36.0000i		−	1.75872i	0.476162	−	0.879358i	$-0.342028\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$			35.6060i			1.72918i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	21.6623	−	21.6623i	1.04709	−	1.04709i
$429$			103.617i			5.00268i
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	−1.61606	−	1.61606i	−0.0777527	−	0.0777527i
$433$	−19.5572	+	19.5572i	−0.939858	+	0.939858i	−0.998291	−	0.0584337i	$-0.981389\pi$
$433$	−19.5572	+	19.5572i	−0.939858	+	0.939858i	0.0584337	+	0.998291i	$0.481389\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	15.7411			0.749577
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		−	18.7044i		−	0.887670i
$445$		0			0
$446$	−2.71536			−0.128576
$447$	16.6952	+	16.6952i	0.789656	+	0.789656i
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$	−20.1705	−	20.1705i	−0.948741	−	0.948741i
$453$		0			0
$454$			10.1942i			0.478437i
$455$		0			0
$456$	24.4360			1.14432
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$	−8.03032	+	8.03032i	−0.375232	+	0.375232i
$459$		0			0
$460$		0			0
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$	−17.3558	+	17.3558i	−0.802272	+	0.802272i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−23.1377	−	23.1377i	−1.05940	−	1.05940i
$478$	11.8123	−	11.8123i	0.540280	−	0.540280i
$479$			35.9065i			1.64061i	0.571927	+	0.820305i	$0.306196\pi$
$479$			35.9065i			1.64061i	−0.571927	+	0.820305i	$0.693804\pi$
$480$		0			0
$481$	38.7984			1.76905
$482$		0			0
$483$		0			0
$484$		−	43.0948i		−	1.95885i
$485$		0			0
$486$	−13.4518			−0.610185
$487$	−28.4164	−	28.4164i	−1.28767	−	1.28767i	−0.936200	−	0.351469i	$-0.885682\pi$
$487$	−28.4164	−	28.4164i	−1.28767	−	1.28767i	−0.351469	−	0.936200i	$-0.614318\pi$
$488$	−20.4314	+	20.4314i	−0.924887	+	0.924887i
$489$		0			0
$490$		0			0
$491$	−26.1534			−1.18029			−0.590143	−	0.807299i	$-0.700929\pi$
$491$	−26.1534			−1.18029			−0.590143	+	0.807299i	$0.700929\pi$
$492$		0			0
$493$		0			0
$494$			21.8511i			0.983127i
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			28.3938i			1.27108i	0.772067	+	0.635541i	$0.219223\pi$
$499$			28.3938i			1.27108i	−0.772067	+	0.635541i	$0.780777\pi$
$500$		0			0
$501$	−19.4350			−0.868292
$502$	12.8721	+	12.8721i	0.574510	+	0.574510i
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−62.9696	−	62.9696i	−2.79658	−	2.79658i
$508$	18.9114	−	18.9114i	0.839057	−	0.839057i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	−10.0579	−	10.0579i	−0.444502	−	0.444502i
$513$	5.30498	−	5.30498i	0.234221	−	0.234221i
$514$		−	19.6961i		−	0.868758i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$			36.7432i			1.61285i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	20.0969	−	20.0969i	0.878778	−	0.878778i	−0.114630	−	0.993408i	$-0.536568\pi$
$523$	20.0969	−	20.0969i	0.878778	−	0.878778i	0.993408	+	0.114630i	$0.0365684\pi$
$524$		−	18.1863i		−	0.794471i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	13.5085	−	13.5085i	0.587881	−	0.587881i
$529$		−	23.0000i		−	1.00000i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	7.40412			0.319809
$537$		0			0
$538$		0			0
$539$		−	43.9584i		−	1.89342i
$540$		0			0
$541$	−7.30067			−0.313881			−0.156940	−	0.987608i	$-0.550163\pi$
$541$	−7.30067			−0.313881			−0.156940	+	0.987608i	$0.550163\pi$
$542$	16.1933	+	16.1933i	0.695563	+	0.695563i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	29.8169	+	29.8169i	1.27488	+	1.27488i	0.943497	+	0.331380i	$0.107514\pi$
$547$	29.8169	+	29.8169i	1.27488	+	1.27488i	0.331380	+	0.943497i	$0.392486\pi$
$548$		0			0
$549$		−	26.5537i		−	1.13328i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	14.0714			0.596760
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−16.8570	+	16.8570i	−0.710436	+	0.710436i	−0.966626	−	0.256190i	$-0.917533\pi$
$563$	−16.8570	+	16.8570i	−0.710436	+	0.710436i	0.256190	+	0.966626i	$0.417533\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	46.9635			1.96536			0.982681	−	0.185306i	$-0.0593276\pi$
$571$	46.9635			1.96536			0.982681	+	0.185306i	$0.0593276\pi$
$572$	48.4676	+	48.4676i	2.02653	+	2.02653i
$573$	−42.3682	+	42.3682i	−1.76996	+	1.76996i
$574$		0			0
$575$		0			0
$576$	−3.13471			−0.130613
$577$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$577$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$578$	8.36702	−	8.36702i	0.348022	−	0.348022i
$579$			19.1505i			0.795869i
$580$		0			0
$581$		0			0
$582$	21.5742	+	21.5742i	0.894281	+	0.894281i
$583$	−64.6139	+	64.6139i	−2.67603	+	2.67603i
$584$		0			0
$585$		0			0
$586$	−9.00751			−0.372097
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$	17.1859	−	17.1859i	0.708735	−	0.708735i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−5.05812	−	5.05812i	−0.207887	−	0.207887i
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$			7.52321i			0.308681i
$595$		0			0
$596$	15.6186			0.639763
$597$	−14.1227	−	14.1227i	−0.578005	−	0.578005i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	−4.81137	+	4.81137i	−0.195934	+	0.195934i
$604$		0			0
$605$		0			0
$606$	23.6220			0.959580
$607$	−3.76654	−	3.76654i	−0.152879	−	0.152879i	0.626524	−	0.779402i	$-0.284477\pi$
$607$	−3.76654	−	3.76654i	−0.152879	−	0.152879i	−0.779402	+	0.626524i	$0.784477\pi$
$608$	17.9328	−	17.9328i	0.727270	−	0.727270i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		−	22.1581i		−	0.894227i
$615$		0			0
$616$		0			0
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$	12.8301	−	12.8301i	0.516101	−	0.516101i
$619$		−	0.269629i		−	0.0108373i	−0.999985	−	0.00541866i	$-0.998275\pi$
$619$		−	0.269629i		−	0.0108373i	0.999985	−	0.00541866i	$-0.00172482\pi$
$620$		0			0
$621$		0			0
$622$	−5.30931	−	5.30931i	−0.212884	−	0.212884i
$623$		0			0
$624$			21.9097i			0.877091i
$625$		0			0
$626$		0			0
$627$	44.3438	+	44.3438i	1.77092	+	1.77092i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−42.4559			−1.69014			−0.845071	−	0.534654i	$-0.820442\pi$
$631$	−42.4559			−1.69014			−0.845071	+	0.534654i	$0.820442\pi$
$632$		0			0
$633$		0			0
$634$		−	18.9148i		−	0.751202i
$635$		0			0
$636$	−50.5227			−2.00335
$637$	35.6486	+	35.6486i	1.41245	+	1.41245i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	−22.7933	−	22.7933i	−0.899578	−	0.899578i
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	−18.4955	+	18.4955i	−0.726570	+	0.726570i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$653$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−8.43195			−0.326731
$667$		0			0
$668$	−9.09086	+	9.09086i	−0.351736	+	0.351736i
$669$		−	8.93754i		−	0.345545i
$670$		0			0
$671$	−74.1534			−2.86266
$672$		0			0
$673$	29.7425	−	29.7425i	1.14649	−	1.14649i	0.159251	−	0.987238i	$-0.449092\pi$
$673$	29.7425	−	29.7425i	1.14649	−	1.14649i	0.987238	−	0.159251i	$-0.0509078\pi$
$674$		−	10.8267i		−	0.417029i
$675$		0			0
$676$	−58.9089			−2.26573
$677$	−29.7291	−	29.7291i	−1.14258	−	1.14258i	−0.987976	−	0.154605i	$-0.950590\pi$
$677$	−29.7291	−	29.7291i	−1.14258	−	1.14258i	−0.154605	−	0.987976i	$-0.549410\pi$
$678$	−21.2236	+	21.2236i	−0.815086	+	0.815086i
$679$		0			0
$680$		0			0
$681$	−33.5539			−1.28579
$682$		0			0
$683$	36.2968	−	36.2968i	1.38886	−	1.38886i	0.561137	−	0.827723i	$-0.310364\pi$
$683$	36.2968	−	36.2968i	1.38886	−	1.38886i	0.827723	−	0.561137i	$-0.189636\pi$
$684$			14.8551i			0.568001i
$685$		0			0
$686$		0			0
$687$	−26.4316	−	26.4316i	−1.00843	−	1.00843i
$688$		0			0
$689$		−	104.799i		−	3.99252i
$690$		0			0
$691$	37.4090			1.42311			0.711553	−	0.702632i	$-0.247992\pi$
$691$	37.4090			1.42311			0.711553	+	0.702632i	$0.247992\pi$
$692$	17.1869	+	17.1869i	0.653347	+	0.653347i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	17.1628	−	17.1628i	0.649622	−	0.649622i
$699$		0			0
$700$		0			0
$701$	19.3210			0.729743			0.364871	−	0.931058i	$-0.381113\pi$
$701$	19.3210			0.729743			0.364871	+	0.931058i	$0.381113\pi$
$702$	−6.10104	−	6.10104i	−0.230269	−	0.230269i
$703$	16.6041	−	16.6041i	0.626236	−	0.626236i
$704$			8.75395i			0.329927i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			34.8712i			1.30962i	0.755796	+	0.654808i	$0.227250\pi$
$709$			34.8712i			1.30962i	−0.755796	+	0.654808i	$0.772750\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	38.8797	+	38.8797i	1.45199	+	1.45199i
$718$	0.872220	−	0.872220i	0.0325510	−	0.0325510i
$719$		−	26.8913i		−	1.00288i	−0.865194	−	0.501438i	$-0.832805\pi$
$719$		−	26.8913i		−	1.00288i	0.865194	−	0.501438i	$-0.167195\pi$
$720$		0			0
$721$		0			0
$722$	9.35137	+	9.35137i	0.348022	+	0.348022i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	−45.3447			−1.68290
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$		−	12.2080i		−	0.452147i
$730$		0			0
$731$		0			0
$732$	−28.9909	−	28.9909i	−1.07153	−	1.07153i
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	13.4362	+	13.4362i	0.494928	+	0.494928i
$738$		0			0
$739$		−	8.71780i		−	0.320689i	−0.987061	−	0.160345i	$-0.948739\pi$
$739$		−	8.71780i		−	0.320689i	0.987061	−	0.160345i	$-0.0512606\pi$
$740$		0			0
$741$	−71.9223			−2.64213
$742$		0			0
$743$	35.1583	−	35.1583i	1.28983	−	1.28983i	0.354947	−	0.934887i	$-0.384499\pi$
$743$	35.1583	−	35.1583i	1.28983	−	1.28983i	0.934887	−	0.354947i	$-0.115501\pi$
$744$		0			0
$745$		0			0
$746$	22.8854			0.837893
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−42.3682	+	42.3682i	−1.54398	+	1.54398i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−31.8805			−1.15567			−0.577834	−	0.816154i	$-0.696102\pi$
$761$	−31.8805			−1.15567			−0.577834	+	0.816154i	$0.696102\pi$
$762$	−19.8987	−	19.8987i	−0.720854	−	0.720854i
$763$		0			0
$764$			39.6361i			1.43398i
$765$		0			0
$766$	−8.25627			−0.298311
$767$		0			0
$768$	−16.5434	+	16.5434i	−0.596957	+	0.596957i
$769$			53.9946i			1.94709i	0.228488	+	0.973547i	$0.426622\pi$
$769$			53.9946i			1.94709i	−0.228488	+	0.973547i	$0.573378\pi$
$770$		0			0
$771$	64.8292			2.33477
$772$	8.95779	+	8.95779i	0.322398	+	0.322398i
$773$	23.1528	−	23.1528i	0.832749	−	0.832749i	−0.155143	−	0.987892i	$-0.549584\pi$
$773$	23.1528	−	23.1528i	0.832749	−	0.832749i	0.987892	+	0.155143i	$0.0495838\pi$
$774$		0			0
$775$		0			0
$776$	46.8180			1.68067
$777$		0			0
$778$	−2.95307	+	2.95307i	−0.105873	+	0.105873i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	9.29498i		−	0.331963i
$785$		0			0
$786$	−19.1357			−0.682549
$787$	−39.5368	−	39.5368i	−1.40934	−	1.40934i	−0.763350	−	0.645986i	$-0.776447\pi$
$787$	−39.5368	−	39.5368i	−1.40934	−	1.40934i	−0.645986	−	0.763350i	$-0.723553\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−24.4339	−	24.4339i	−0.868222	−	0.868222i
$793$	60.1356	−	60.1356i	2.13548	−	2.13548i
$794$		0			0
$795$		0			0
$796$	−13.2120			−0.468288
$797$	−32.0061	−	32.0061i	−1.13372	−	1.13372i	−0.989554	−	0.144161i	$-0.953952\pi$
$797$	−32.0061	−	32.0061i	−1.13372	−	1.13372i	−0.144161	−	0.989554i	$-0.546048\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$			10.5060i			0.370517i
$805$		0			0
$806$		0			0
$807$		0			0
$808$	25.6310	−	25.6310i	0.901695	−	0.901695i
$809$		−	52.3068i		−	1.83901i	−0.393080	−	0.919504i	$-0.628590\pi$
$809$		−	52.3068i		−	1.83901i	0.393080	−	0.919504i	$-0.371410\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	−53.3000	+	53.3000i	−1.86931	+	1.86931i
$814$			23.5469i			0.825320i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	42.0000			1.46581			0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000			1.46581			0.732905	+	0.680331i	$0.238164\pi$
$822$		0			0
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		−	27.8424i		−	0.969936i
$825$		0			0
$826$		0			0
$827$	−7.13702	−	7.13702i	−0.248179	−	0.248179i	0.572044	−	0.820223i	$-0.306151\pi$
$827$	−7.13702	−	7.13702i	−0.248179	−	0.248179i	−0.820223	+	0.572044i	$0.806151\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$	−7.09912	−	7.09912i	−0.246118	−	0.246118i
$833$		0			0
$834$		−	14.8060i		−	0.512691i
$835$		0			0
$836$	41.4843			1.43476
$837$		0			0
$838$	−17.7184	+	17.7184i	−0.612072	+	0.612072i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−13.6626	+	13.6626i	−0.469174	+	0.469174i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$	−49.4634			−1.69062
$857$	22.2862	+	22.2862i	0.761282	+	0.761282i	0.976554	−	0.215272i	$-0.0690639\pi$
$857$	22.2862	+	22.2862i	0.761282	+	0.761282i	−0.215272	+	0.976554i	$0.569064\pi$
$858$	50.9980	−	50.9980i	1.74104	−	1.74104i
$859$			4.00000i			0.136478i	0.997669	+	0.0682391i	$0.0217381\pi$
$859$			4.00000i			0.136478i	−0.997669	+	0.0682391i	$0.978262\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−13.4414	+	13.4414i	−0.457550	+	0.457550i	−0.897850	−	0.440301i	$-0.854872\pi$
$863$	−13.4414	+	13.4414i	−0.457550	+	0.457550i	0.440301	+	0.897850i	$0.354872\pi$
$864$			10.0140i			0.340684i
$865$		0			0
$866$	19.2512			0.654182
$867$	27.5398	+	27.5398i	0.935301	+	0.935301i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−21.7925			−0.738409
$872$		0			0
$873$	−30.4235	+	30.4235i	−1.02968	+	1.02968i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	38.4861	+	38.4861i	1.29958	+	1.29958i	0.928666	+	0.370917i	$0.120956\pi$
$877$	38.4861	+	38.4861i	1.29958	+	1.29958i	0.370917	+	0.928666i	$0.379044\pi$
$878$		0			0
$879$		−	29.6480i		−	1.00000i
$880$		0			0
$881$	47.9844			1.61664			0.808318	−	0.588746i	$-0.200378\pi$
$881$	47.9844			1.61664			0.808318	+	0.588746i	$0.200378\pi$
$882$	−7.74743	−	7.74743i	−0.260869	−	0.260869i
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4.45008	+	4.45008i	0.149419	+	0.149419i	0.777858	−	0.628440i	$-0.216306\pi$
$887$	4.45008	+	4.45008i	0.149419	+	0.149419i	−0.628440	+	0.777858i	$0.716306\pi$
$888$	−21.3547	+	21.3547i	−0.716615	+	0.716615i
$889$		0			0
$890$		0			0
$891$	−67.1271			−2.24884
$892$	−4.18060	−	4.18060i	−0.139977	−	0.139977i
$893$		0			0
$894$		−	16.4340i		−	0.549635i
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$			46.0570i			1.53183i
$905$		0			0
$906$		0			0
$907$	−26.7894	−	26.7894i	−0.889528	−	0.889528i	0.104949	−	0.994478i	$-0.466532\pi$
$907$	−26.7894	−	26.7894i	−0.889528	−	0.889528i	−0.994478	+	0.104949i	$0.966532\pi$
$908$	−15.6951	+	15.6951i	−0.520859	+	0.520859i
$909$			33.3113i			1.10487i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	9.37647	+	9.37647i	0.310486	+	0.310486i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−24.7271			−0.817007
$917$		0			0
$918$		0			0
$919$		−	16.0000i		−	0.527791i	−0.964551	−	0.263896i	$-0.914993\pi$
$919$		−	16.0000i		−	0.527791i	0.964551	−	0.263896i	$-0.0850075\pi$
$920$		0			0
$921$	72.9327			2.40321
$922$	8.85920	+	8.85920i	0.291762	+	0.291762i
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	18.0927	+	18.0927i	0.594241	+	0.594241i
$928$		0			0
$929$		−	52.3068i		−	1.71613i	−0.513541	−	0.858065i	$-0.671667\pi$
$929$		−	52.3068i		−	1.71613i	0.513541	−	0.858065i	$-0.328333\pi$
$930$		0			0
$931$	30.5123			1.00000
$932$		0			0
$933$	17.4755	−	17.4755i	0.572121	−	0.572121i
$934$		0			0
$935$		0			0
$936$	39.6300			1.29535
$937$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$937$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$947$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	62.2575			2.01884
$952$		0			0
$953$	−41.7261	+	41.7261i	−1.35164	+	1.35164i	−0.467812	+	0.883828i	$0.654957\pi$
$953$	−41.7261	+	41.7261i	−1.35164	+	1.35164i	−0.883828	+	0.467812i	$0.845043\pi$
$954$			22.7757i			0.737389i
$955$		0			0
$956$	36.3725			1.17637
$957$		0			0
$958$	17.6724	−	17.6724i	0.570968	−	0.570968i
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$	−19.0957	−	19.0957i	−0.615669	−	0.615669i
$963$	32.1425	−	32.1425i	1.03578	−	1.03578i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	−49.2010	+	49.2010i	−1.58138	+	1.58138i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−20.7105	−	20.7105i	−0.664289	−	0.664289i
$973$		0			0
$974$			27.9718i			0.896274i
$975$		0			0
$976$	−15.6797			−0.501894
$977$	43.8652	+	43.8652i	1.40337	+	1.40337i	0.789071	+	0.614302i	$0.210562\pi$
$977$	43.8652	+	43.8652i	1.40337	+	1.40337i	0.614302	+	0.789071i	$0.289438\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	12.8721	+	12.8721i	0.410766	+	0.410766i
$983$	−40.9122	+	40.9122i	−1.30490	+	1.30490i	−0.379846	+	0.925050i	$0.624023\pi$
$983$	−40.9122	+	40.9122i	−1.30490	+	1.30490i	−0.925050	+	0.379846i	$0.875977\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−33.6422	+	33.6422i	−1.07030	+	1.07030i
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$997$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$998$	13.9748	−	13.9748i	0.442364	−	0.442364i
$999$			9.27205i			0.293355i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.g.c.18.4	✓	16
5.2	odd	4	inner	475.2.g.c.132.5	yes	16
5.3	odd	4	inner	475.2.g.c.132.4	yes	16
5.4	even	2	inner	475.2.g.c.18.5	yes	16
19.18	odd	2	inner	475.2.g.c.18.5	yes	16
95.18	even	4	inner	475.2.g.c.132.5	yes	16
95.37	even	4	inner	475.2.g.c.132.4	yes	16
95.94	odd	2	CM	475.2.g.c.18.4	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.g.c.18.4	✓	16	1.1	even	1	trivial
475.2.g.c.18.4	✓	16	95.94	odd	2	CM
475.2.g.c.18.5	yes	16	5.4	even	2	inner
475.2.g.c.18.5	yes	16	19.18	odd	2	inner
475.2.g.c.132.4	yes	16	5.3	odd	4	inner
475.2.g.c.132.4	yes	16	95.37	even	4	inner
475.2.g.c.132.5	yes	16	5.2	odd	4	inner
475.2.g.c.132.5	yes	16	95.18	even	4	inner

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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.492178 - 0.492178i) q^{2} +(1.61999 - 1.61999i) q^{3} -1.51552i q^{4} -1.59464 q^{6} +(-1.73026 + 1.73026i) q^{8} -2.24873i q^{9} -6.27978 q^{11} +(-2.45513 - 2.45513i) q^{12} +(5.09266 - 5.09266i) q^{13} -1.32785 q^{16} +(-1.10678 + 1.10678i) q^{18} -4.35890i q^{19} +(3.09076 + 3.09076i) q^{22} +5.60601i q^{24} -5.01299 q^{26} +(1.21705 + 1.21705i) q^{27} +(4.11406 + 4.11406i) q^{32} +(-10.1732 + 10.1732i) q^{33} -3.40800 q^{36} +(3.80924 + 3.80924i) q^{37} +(-2.14535 + 2.14535i) q^{38} -16.5001i q^{39} +9.51714i q^{44} +(-2.15111 + 2.15111i) q^{48} +7.00000i q^{49} +(-7.71804 - 7.71804i) q^{52} +(10.2892 - 10.2892i) q^{53} -1.19801i q^{54} +(-7.06137 - 7.06137i) q^{57} +11.8083 q^{61} -1.39399i q^{64} +10.0140 q^{66} +(-2.13960 - 2.13960i) q^{67} +(3.89089 + 3.89089i) q^{72} -3.74965i q^{74} -6.60601 q^{76} +(-8.12098 + 8.12098i) q^{78} +10.6894 q^{81} +(10.8657 - 10.8657i) q^{88} +13.3295 q^{96} +(-13.5292 - 13.5292i) q^{97} +(3.44524 - 3.44524i) q^{98} +14.1215i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 64 q^{16} - 48 q^{26} + 208 q^{36} - 208 q^{66} - 144 q^{81} + 304 q^{96}+O(q^{100}) \) 16 * q - 64 * q^16 - 48 * q^26 + 208 * q^36 - 208 * q^66 - 144 * q^81 + 304 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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