LMFDB - Embedded newform 1575.2.b.h.251.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(251,1575)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1575.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$12.5764383184$
Analytic rank:	$0$
Dimension:	$16$
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} - 24x^{14} + 184x^{12} - 240x^{10} + 228x^{8} + 912x^{6} + 976x^{4} + 480x^{2} + 100 $ x^16 - 24x^14 + 184x^12 - 240x^10 + 228x^8 + 912x^6 + 976x^4 + 480*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 315)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			251.8
Root			$1.40513 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1575.251
Dual form			1575.2.b.h.251.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.741964i q^{2} +1.44949 q^{4} +(2.33441 + 1.24519i) q^{7} -2.55940i q^{8} +1.41421i q^{11} -5.54048i q^{13} +(0.923889 - 1.73205i) q^{14} +1.00000 q^{16} -6.07445 q^{17} -7.12018i q^{19} +1.04930 q^{22} -4.78529i q^{23} -4.11084 q^{26} +(3.38371 + 1.80490i) q^{28} +5.51399i q^{29} -1.30658i q^{31} -5.86076i q^{32} +4.50702i q^{34} +2.57024 q^{37} -5.28291 q^{38} +5.95862 q^{41} +6.76742 q^{43} +2.04989i q^{44} -3.55051 q^{46} +7.83542 q^{47} +(3.89898 + 5.81360i) q^{49} -8.03087i q^{52} -9.90408i q^{53} +(3.18695 - 5.97469i) q^{56} +4.09118 q^{58} +1.84778 q^{59} +11.6272i q^{61} -0.969433 q^{62} -2.34847 q^{64} -7.23907 q^{67} -8.80486 q^{68} +8.34242i q^{71} +0.559702i q^{73} -1.90702i q^{74} -10.3206i q^{76} +(-1.76097 + 3.30136i) q^{77} +4.00000 q^{79} -4.42108i q^{82} -7.83542 q^{83} -5.02118i q^{86} +3.61953 q^{88} +12.3325 q^{89} +(6.89898 - 12.9338i) q^{91} -6.93623i q^{92} -5.81360i q^{94} +5.54048i q^{97} +(4.31348 - 2.89290i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} + 16 q^{16} - 96 q^{46} - 16 q^{49} + 80 q^{64} + 64 q^{79} + 32 q^{91}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^16 - 96 * q^46 - 16 * q^49 + 80 * q^64 + 64 * q^79 + 32 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$1226$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$		−	0.741964i		−	0.524648i	−0.964980	−	0.262324i	$-0.915511\pi$
$2$		−	0.741964i		−	0.524648i	0.964980	−	0.262324i	$-0.0844889\pi$
$3$		0			0
$3$		0			0
$4$	1.44949			0.724745
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.33441	+	1.24519i	0.882326	+	0.470639i
$7$	2.33441	+	1.24519i	0.882326	+	0.470639i
$8$		−	2.55940i		−	0.904883i
$9$		0			0
$10$		0			0
$11$			1.41421i			0.426401i	0.977008	+	0.213201i	$0.0683888\pi$
$11$			1.41421i			0.426401i	−0.977008	+	0.213201i	$0.931611\pi$
$12$		0			0
$13$		−	5.54048i		−	1.53665i	−0.640058	−	0.768327i	$-0.721090\pi$
$13$		−	5.54048i		−	1.53665i	0.640058	−	0.768327i	$-0.278910\pi$
$14$	0.923889	−	1.73205i	0.246920	−	0.462910i
$15$		0			0
$16$	1.00000			0.250000
$17$	−6.07445			−1.47327			−0.736636	−	0.676290i	$-0.763587\pi$
$17$	−6.07445			−1.47327			−0.736636	+	0.676290i	$0.763587\pi$
$18$		0			0
$19$		−	7.12018i		−	1.63348i	−0.577005	−	0.816740i	$-0.695779\pi$
$19$		−	7.12018i		−	1.63348i	0.577005	−	0.816740i	$-0.304221\pi$
$20$		0			0
$21$		0			0
$22$	1.04930			0.223710
$23$		−	4.78529i		−	0.997801i	−0.866659	−	0.498901i	$-0.833737\pi$
$23$		−	4.78529i		−	0.997801i	0.866659	−	0.498901i	$-0.166263\pi$
$24$		0			0
$25$		0			0
$26$	−4.11084			−0.806201
$27$		0			0
$28$	3.38371	+	1.80490i	0.639461	+	0.341094i
$29$			5.51399i			1.02392i	0.859009	+	0.511961i	$0.171081\pi$
$29$			5.51399i			1.02392i	−0.859009	+	0.511961i	$0.828919\pi$
$30$		0			0
$31$		−	1.30658i		−	0.234668i	−0.993092	−	0.117334i	$-0.962565\pi$
$31$		−	1.30658i		−	0.234668i	0.993092	−	0.117334i	$-0.0374348\pi$
$32$		−	5.86076i		−	1.03605i
$33$		0			0
$34$			4.50702i			0.772948i
$35$		0			0
$36$		0			0
$37$	2.57024			0.422545			0.211272	−	0.977427i	$-0.432239\pi$
$37$	2.57024			0.422545			0.211272	+	0.977427i	$0.432239\pi$
$38$	−5.28291			−0.857002
$39$		0			0
$40$		0			0
$41$	5.95862			0.930579			0.465290	−	0.885158i	$-0.345950\pi$
$41$	5.95862			0.930579			0.465290	+	0.885158i	$0.345950\pi$
$42$		0			0
$43$	6.76742			1.03202			0.516011	−	0.856582i	$-0.327416\pi$
$43$	6.76742			1.03202			0.516011	+	0.856582i	$0.327416\pi$
$44$			2.04989i			0.309032i
$45$		0			0
$46$	−3.55051			−0.523494
$47$	7.83542			1.14291			0.571457	−	0.820632i	$-0.306378\pi$
$47$	7.83542			1.14291			0.571457	+	0.820632i	$0.306378\pi$
$48$		0			0
$49$	3.89898	+	5.81360i	0.556997	+	0.830514i
$50$		0			0
$51$		0			0
$52$		−	8.03087i		−	1.11368i
$53$		−	9.90408i		−	1.36043i	−0.733013	−	0.680215i	$-0.761886\pi$
$53$		−	9.90408i		−	1.36043i	0.733013	−	0.680215i	$-0.238114\pi$
$54$		0			0
$55$		0			0
$56$	3.18695	−	5.97469i	0.425874	−	0.798402i
$57$		0			0
$58$	4.09118			0.537198
$59$	1.84778			0.240560			0.120280	−	0.992740i	$-0.461621\pi$
$59$	1.84778			0.240560			0.120280	+	0.992740i	$0.461621\pi$
$60$		0			0
$61$			11.6272i			1.48871i	0.667784	+	0.744355i	$0.267243\pi$
$61$			11.6272i			1.48871i	−0.667784	+	0.744355i	$0.732757\pi$
$62$	−0.969433			−0.123118
$63$		0			0
$64$	−2.34847			−0.293559
$65$		0			0
$66$		0			0
$67$	−7.23907			−0.884393			−0.442196	−	0.896918i	$-0.645801\pi$
$67$	−7.23907			−0.884393			−0.442196	+	0.896918i	$0.645801\pi$
$68$	−8.80486			−1.06775
$69$		0			0
$70$		0			0
$71$			8.34242i			0.990063i	0.868875	+	0.495031i	$0.164843\pi$
$71$			8.34242i			0.990063i	−0.868875	+	0.495031i	$0.835157\pi$
$72$		0			0
$73$			0.559702i			0.0655082i	0.999463	+	0.0327541i	$0.0104278\pi$
$73$			0.559702i			0.0655082i	−0.999463	+	0.0327541i	$0.989572\pi$
$74$		−	1.90702i		−	0.221687i
$75$		0			0
$76$		−	10.3206i		−	1.18386i
$77$	−1.76097	+	3.30136i	−0.200681	+	0.376225i
$78$		0			0
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$		0			0
$82$		−	4.42108i		−	0.488226i
$83$	−7.83542			−0.860050			−0.430025	−	0.902817i	$-0.641495\pi$
$83$	−7.83542			−0.860050			−0.430025	+	0.902817i	$0.641495\pi$
$84$		0			0
$85$		0			0
$86$		−	5.02118i		−	0.541448i
$87$		0			0
$88$	3.61953			0.385844
$89$	12.3325			1.30724			0.653622	−	0.756821i	$-0.273249\pi$
$89$	12.3325			1.30724			0.653622	+	0.756821i	$0.273249\pi$
$90$		0			0
$91$	6.89898	−	12.9338i	0.723210	−	1.35583i
$92$		−	6.93623i		−	0.723152i
$93$		0			0
$94$		−	5.81360i		−	0.599627i
$95$		0			0
$96$		0			0
$97$			5.54048i			0.562551i	0.959627	+	0.281275i	$0.0907574\pi$
$97$			5.54048i			0.562551i	−0.959627	+	0.281275i	$0.909243\pi$
$98$	4.31348	−	2.89290i	0.435727	−	0.292227i
$99$		0			0
$100$		0			0
$101$	−5.95862			−0.592904			−0.296452	−	0.955048i	$-0.595804\pi$
$101$	−5.95862			−0.592904			−0.296452	+	0.955048i	$0.595804\pi$
$102$		0			0
$103$			4.98078i			0.490771i	0.969426	+	0.245385i	$0.0789145\pi$
$103$			4.98078i			0.490771i	−0.969426	+	0.245385i	$0.921085\pi$
$104$	−14.1803			−1.39049
$105$		0			0
$106$	−7.34847			−0.713746
$107$			1.81743i			0.175698i	0.996134	+	0.0878489i	$0.0279993\pi$
$107$			1.81743i			0.175698i	−0.996134	+	0.0878489i	$0.972001\pi$
$108$		0			0
$109$	−3.10102			−0.297024			−0.148512	−	0.988911i	$-0.547448\pi$
$109$	−3.10102			−0.297024			−0.148512	+	0.988911i	$0.547448\pi$
$110$		0			0
$111$		0			0
$112$	2.33441	+	1.24519i	0.220581	+	0.117660i
$113$		−	0.333505i		−	0.0313735i	−0.999877	−	0.0156868i	$-0.995007\pi$
$113$		−	0.333505i		−	0.0313735i	0.999877	−	0.0156868i	$-0.00499346\pi$
$114$		0			0
$115$		0			0
$116$			7.99247i			0.742082i
$117$		0			0
$118$		−	1.37099i		−	0.126209i
$119$	−14.1803	−	7.56388i	−1.29991	−	0.693380i
$120$		0			0
$121$	9.00000			0.818182
$122$	8.62696			0.781049
$123$		0			0
$124$		−	1.89387i		−	0.170075i
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		−	9.97903i		−	0.882030i
$129$		0			0
$130$		0			0
$131$	−6.37389			−0.556890			−0.278445	−	0.960452i	$-0.589819\pi$
$131$	−6.37389			−0.556890			−0.278445	+	0.960452i	$0.589819\pi$
$132$		0			0
$133$	8.86601	−	16.6214i	0.768781	−	1.44126i
$134$			5.37113i			0.463995i
$135$		0			0
$136$			15.5469i			1.33314i
$137$			3.30136i			0.282054i	0.990006	+	0.141027i	$0.0450405\pi$
$137$			3.30136i			0.282054i	−0.990006	+	0.141027i	$0.954960\pi$
$138$		0			0
$139$			1.30658i			0.110822i	0.998464	+	0.0554112i	$0.0176470\pi$
$139$			1.30658i			0.110822i	−0.998464	+	0.0554112i	$0.982353\pi$
$140$		0			0
$141$		0			0
$142$	6.18977			0.519434
$143$	7.83542			0.655231
$144$		0			0
$145$		0			0
$146$	0.415279			0.0343687
$147$		0			0
$148$	3.72553			0.306237
$149$		−	9.89949i		−	0.810998i	−0.914095	−	0.405499i	$-0.867098\pi$
$149$		−	9.89949i		−	0.810998i	0.914095	−	0.405499i	$-0.132902\pi$
$150$		0			0
$151$	−10.0000			−0.813788			−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000			−0.813788			−0.406894	+	0.913475i	$0.633388\pi$
$152$	−18.2234			−1.47811
$153$		0			0
$154$	2.44949	+	1.30658i	0.197386	+	0.105287i
$155$		0			0
$156$		0			0
$157$			13.0117i			1.03844i	0.854640	+	0.519221i	$0.173778\pi$
$157$			13.0117i			1.03844i	−0.854640	+	0.519221i	$0.826222\pi$
$158$		−	2.96786i		−	0.236110i
$159$		0			0
$160$		0			0
$161$	5.95862	−	11.1708i	0.469605	−	0.880386i
$162$		0			0
$163$	−17.7320			−1.38888			−0.694439	−	0.719551i	$-0.744348\pi$
$163$	−17.7320			−1.38888			−0.694439	+	0.719551i	$0.744348\pi$
$164$	8.63695			0.674433
$165$		0			0
$166$			5.81360i			0.451223i
$167$	−5.10502			−0.395038			−0.197519	−	0.980299i	$-0.563288\pi$
$167$	−5.10502			−0.395038			−0.197519	+	0.980299i	$0.563288\pi$
$168$		0			0
$169$	−17.6969			−1.36130
$170$		0			0
$171$		0			0
$172$	9.80930			0.747952
$173$	1.76097			0.133884			0.0669421	−	0.997757i	$-0.478676\pi$
$173$	1.76097			0.133884			0.0669421	+	0.997757i	$0.478676\pi$
$174$		0			0
$175$		0			0
$176$			1.41421i			0.106600i
$177$		0			0
$178$		−	9.15028i		−	0.685842i
$179$			0.142865i			0.0106782i	0.999986	+	0.00533910i	$0.00169950\pi$
$179$			0.142865i			0.0106782i	−0.999986	+	0.00533910i	$0.998301\pi$
$180$		0			0
$181$			3.20045i			0.237887i	0.992901	+	0.118944i	$0.0379508\pi$
$181$			3.20045i			0.237887i	−0.992901	+	0.118944i	$0.962049\pi$
$182$	−9.59640	−	5.11879i	−0.711332	−	0.379430i
$183$		0			0
$184$	−12.2474			−0.902894
$185$		0			0
$186$		0			0
$187$		−	8.59057i		−	0.628205i
$188$	11.3574			0.828321
$189$		0			0
$190$		0			0
$191$		−	22.4846i		−	1.62693i	−0.581617	−	0.813463i	$-0.697580\pi$
$191$		−	22.4846i		−	1.62693i	0.581617	−	0.813463i	$-0.302420\pi$
$192$		0			0
$193$	−6.76742			−0.487129			−0.243565	−	0.969885i	$-0.578317\pi$
$193$	−6.76742			−0.487129			−0.243565	+	0.969885i	$0.578317\pi$
$194$	4.11084			0.295141
$195$		0			0
$196$	5.65153	+	8.42676i	0.403681	+	0.601911i
$197$			13.5389i			0.964610i	0.876003	+	0.482305i	$0.160200\pi$
$197$			13.5389i			0.964610i	−0.876003	+	0.482305i	$0.839800\pi$
$198$		0			0
$199$			7.12018i			0.504736i	0.967631	+	0.252368i	$0.0812094\pi$
$199$			7.12018i			0.504736i	−0.967631	+	0.252368i	$0.918791\pi$
$200$		0			0
$201$		0			0
$202$			4.42108i			0.311066i
$203$	−6.86599	+	12.8719i	−0.481898	+	0.903433i
$204$		0			0
$205$		0			0
$206$	3.69556			0.257482
$207$		0			0
$208$		−	5.54048i		−	0.384163i
$209$	10.0695			0.696519
$210$		0			0
$211$	−10.0000			−0.688428			−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000			−0.688428			−0.344214	+	0.938891i	$0.611855\pi$
$212$		−	14.3559i		−	0.985965i
$213$		0			0
$214$	1.34847			0.0921795
$215$		0			0
$216$		0			0
$217$	1.62694	−	3.05009i	0.110444	−	0.207054i
$218$			2.30084i			0.155833i
$219$		0			0
$220$		0			0
$221$			33.6554i			2.26391i
$222$		0			0
$223$			18.5521i			1.24234i	0.783675	+	0.621171i	$0.213343\pi$
$223$			18.5521i			1.24234i	−0.783675	+	0.621171i	$0.786657\pi$
$224$	7.29778	−	13.6814i	0.487604	−	0.914129i
$225$		0			0
$226$	−0.247449			−0.0164600
$227$	18.4013			1.22133			0.610667	−	0.791887i	$-0.290901\pi$
$227$	18.4013			1.22133			0.610667	+	0.791887i	$0.290901\pi$
$228$		0			0
$229$		−	5.81360i		−	0.384174i	−0.981378	−	0.192087i	$-0.938474\pi$
$229$		−	5.81360i		−	0.384174i	0.981378	−	0.192087i	$-0.0615255\pi$
$230$		0			0
$231$		0			0
$232$	14.1125			0.926530
$233$		−	2.63435i		−	0.172582i	−0.996270	−	0.0862910i	$-0.972499\pi$
$233$		−	2.63435i		−	0.172582i	0.996270	−	0.0862910i	$-0.0275015\pi$
$234$		0			0
$235$		0			0
$236$	2.67834			0.174345
$237$		0			0
$238$	−5.61212	+	10.5213i	−0.363780	+	0.681992i
$239$			13.9993i			0.905538i	0.891628	+	0.452769i	$0.149564\pi$
$239$			13.9993i			0.905538i	−0.891628	+	0.452769i	$0.850436\pi$
$240$		0			0
$241$		−	20.0540i		−	1.29179i	−0.763427	−	0.645894i	$-0.776485\pi$
$241$		−	20.0540i		−	1.29179i	0.763427	−	0.645894i	$-0.223515\pi$
$242$		−	6.67767i		−	0.429257i
$243$		0			0
$244$			16.8535i			1.07894i
$245$		0			0
$246$		0			0
$247$	−39.4492			−2.51009
$248$	−3.34405			−0.212347
$249$		0			0
$250$		0			0
$251$	−10.0695			−0.635578			−0.317789	−	0.948161i	$-0.602940\pi$
$251$	−10.0695			−0.635578			−0.317789	+	0.948161i	$0.602940\pi$
$252$		0			0
$253$	6.76742			0.425464
$254$		0			0
$255$		0			0
$256$	−12.1010			−0.756314
$257$	13.1183			0.818299			0.409150	−	0.912467i	$-0.365825\pi$
$257$	13.1183			0.818299			0.409150	+	0.912467i	$0.365825\pi$
$258$		0			0
$259$	6.00000	+	3.20045i	0.372822	+	0.198866i
$260$		0			0
$261$		0			0
$262$			4.72920i			0.292171i
$263$			28.2283i			1.74063i	0.492493	+	0.870316i	$0.336086\pi$
$263$			28.2283i			1.74063i	−0.492493	+	0.870316i	$0.663914\pi$
$264$		0			0
$265$		0			0
$266$	−12.3325	−	6.57826i	−0.756155	−	0.403339i
$267$		0			0
$268$	−10.4930			−0.640959
$269$	−22.4020			−1.36587			−0.682936	−	0.730478i	$-0.739297\pi$
$269$	−22.4020			−1.36587			−0.682936	+	0.730478i	$0.739297\pi$
$270$		0			0
$271$		−	10.3206i		−	0.626933i	−0.949599	−	0.313467i	$-0.898510\pi$
$271$		−	10.3206i		−	0.626933i	0.949599	−	0.313467i	$-0.101490\pi$
$272$	−6.07445			−0.368318
$273$		0			0
$274$	2.44949			0.147979
$275$		0			0
$276$		0			0
$277$	7.23907			0.434953			0.217477	−	0.976066i	$-0.430217\pi$
$277$	7.23907			0.434953			0.217477	+	0.976066i	$0.430217\pi$
$278$	0.969433			0.0581427
$279$		0			0
$280$		0			0
$281$			1.41421i			0.0843649i	0.999110	+	0.0421825i	$0.0134311\pi$
$281$			1.41421i			0.0843649i	−0.999110	+	0.0421825i	$0.986569\pi$
$282$		0			0
$283$			18.5521i			1.10281i	0.834238	+	0.551405i	$0.185908\pi$
$283$			18.5521i			1.10281i	−0.834238	+	0.551405i	$0.814092\pi$
$284$			12.0922i			0.717543i
$285$		0			0
$286$		−	5.81360i		−	0.343765i
$287$	13.9099	+	7.41964i	0.821074	+	0.437967i
$288$		0			0
$289$	19.8990			1.17053
$290$		0			0
$291$		0			0
$292$			0.811283i			0.0474767i
$293$	14.7014			0.858866			0.429433	−	0.903099i	$-0.358713\pi$
$293$	14.7014			0.858866			0.429433	+	0.903099i	$0.358713\pi$
$294$		0			0
$295$		0			0
$296$		−	6.57826i		−	0.382353i
$297$		0			0
$298$	−7.34507			−0.425488
$299$	−26.5128			−1.53327
$300$		0			0
$301$	15.7980	+	8.42676i	0.910579	+	0.485710i
$302$			7.41964i			0.426952i
$303$		0			0
$304$		−	7.12018i		−	0.408370i
$305$		0			0
$306$		0			0
$307$		−	24.6523i		−	1.40698i	−0.710704	−	0.703491i	$-0.751624\pi$
$307$		−	24.6523i		−	1.40698i	0.710704	−	0.703491i	$-0.248376\pi$
$308$	−2.55251	+	4.78529i	−0.145443	+	0.272667i
$309$		0			0
$310$		0			0
$311$	18.2911			1.03719			0.518597	−	0.855019i	$-0.326454\pi$
$311$	18.2911			1.03719			0.518597	+	0.855019i	$0.326454\pi$
$312$		0			0
$313$			14.1311i			0.798734i	0.916791	+	0.399367i	$0.130770\pi$
$313$			14.1311i			0.798734i	−0.916791	+	0.399367i	$0.869230\pi$
$314$	9.65417			0.544817
$315$		0			0
$316$	5.79796			0.326161
$317$			3.30136i			0.185423i	0.995693	+	0.0927114i	$0.0295534\pi$
$317$			3.30136i			0.185423i	−0.995693	+	0.0927114i	$0.970447\pi$
$318$		0			0
$319$	−7.79796			−0.436602
$320$		0			0
$321$		0			0
$322$	−8.28836	−	4.42108i	−0.461892	−	0.246377i
$323$			43.2512i			2.40656i
$324$		0			0
$325$		0			0
$326$			13.1565i			0.728672i
$327$		0			0
$328$		−	15.2505i		−	0.842066i
$329$	18.2911	+	9.75663i	1.00842	+	0.537900i
$330$		0			0
$331$	31.3939			1.72556			0.862782	−	0.505577i	$-0.168720\pi$
$331$	31.3939			1.72556			0.862782	+	0.505577i	$0.168720\pi$
$332$	−11.3574			−0.623317
$333$		0			0
$334$			3.78774i			0.207256i
$335$		0			0
$336$		0			0
$337$	−22.4008			−1.22025			−0.610126	−	0.792304i	$-0.708881\pi$
$337$	−22.4008			−1.22025			−0.610126	+	0.792304i	$0.708881\pi$
$338$			13.1305i			0.714204i
$339$		0			0
$340$		0			0
$341$	1.84778			0.100063
$342$		0			0
$343$	1.86277	+	18.4263i	0.100580	+	0.994929i
$344$		−	17.3205i		−	0.933859i
$345$		0			0
$346$		−	1.30658i		−	0.0702420i
$347$			37.1319i			1.99334i	0.0815179	+	0.996672i	$0.474023\pi$
$347$			37.1319i			1.99334i	−0.0815179	+	0.996672i	$0.525977\pi$
$348$		0			0
$349$		−	25.8676i		−	1.38466i	−0.721582	−	0.692329i	$-0.756585\pi$
$349$		−	25.8676i		−	1.38466i	0.721582	−	0.692329i	$-0.243415\pi$
$350$		0			0
$351$		0			0
$352$	8.28836			0.441771
$353$	1.76097			0.0937271			0.0468635	−	0.998901i	$-0.485077\pi$
$353$	1.76097			0.0937271			0.0468635	+	0.998901i	$0.485077\pi$
$354$		0			0
$355$		0			0
$356$	17.8758			0.947418
$357$		0			0
$358$	0.106000			0.00560229
$359$		−	9.89949i		−	0.522475i	−0.965275	−	0.261238i	$-0.915869\pi$
$359$		−	9.89949i		−	0.522475i	0.965275	−	0.261238i	$-0.0841306\pi$
$360$		0			0
$361$	−31.6969			−1.66826
$362$	2.37462			0.124807
$363$		0			0
$364$	10.0000	−	18.7474i	0.524142	−	0.982630i
$365$		0			0
$366$		0			0
$367$		−	3.60979i		−	0.188430i	−0.995552	−	0.0942149i	$-0.969966\pi$
$367$		−	3.60979i		−	0.188430i	0.995552	−	0.0942149i	$-0.0300341\pi$
$368$		−	4.78529i		−	0.249450i
$369$		0			0
$370$		0			0
$371$	12.3325	−	23.1202i	0.640272	−	1.20034i
$372$		0			0
$373$	−32.2102			−1.66778			−0.833889	−	0.551932i	$-0.813891\pi$
$373$	−32.2102			−1.66778			−0.833889	+	0.551932i	$0.813891\pi$
$374$	−6.37389			−0.329586
$375$		0			0
$376$		−	20.0540i		−	1.03420i
$377$	30.5502			1.57341
$378$		0			0
$379$	17.5959			0.903842			0.451921	−	0.892058i	$-0.350739\pi$
$379$	17.5959			0.903842			0.451921	+	0.892058i	$0.350739\pi$
$380$		0			0
$381$		0			0
$382$	−16.6827			−0.853562
$383$	26.2367			1.34063			0.670316	−	0.742076i	$-0.266159\pi$
$383$	26.2367			1.34063			0.670316	+	0.742076i	$0.266159\pi$
$384$		0			0
$385$		0			0
$386$			5.02118i			0.255571i
$387$		0			0
$388$			8.03087i			0.407706i
$389$			19.3704i			0.982118i	0.871126	+	0.491059i	$0.163390\pi$
$389$			19.3704i			0.982118i	−0.871126	+	0.491059i	$0.836610\pi$
$390$		0			0
$391$			29.0680i			1.47003i
$392$	14.8793	−	9.97903i	0.751519	−	0.504017i
$393$		0			0
$394$	10.0454			0.506080
$395$		0			0
$396$		0			0
$397$			31.3122i			1.57151i	0.618535	+	0.785757i	$0.287726\pi$
$397$			31.3122i			1.57151i	−0.618535	+	0.785757i	$0.712274\pi$
$398$	5.28291			0.264809
$399$		0			0
$400$		0			0
$401$		−	22.4846i		−	1.12282i	−0.827536	−	0.561412i	$-0.810258\pi$
$401$		−	22.4846i		−	1.12282i	0.827536	−	0.561412i	$-0.189742\pi$
$402$		0			0
$403$	−7.23907			−0.360604
$404$	−8.63695			−0.429704
$405$		0			0
$406$	9.55051	+	5.09432i	0.473984	+	0.252827i
$407$			3.63487i			0.180174i
$408$		0			0
$409$			14.8276i			0.733180i	0.930383	+	0.366590i	$0.119475\pi$
$409$			14.8276i			0.733180i	−0.930383	+	0.366590i	$0.880525\pi$
$410$		0			0
$411$		0			0
$412$			7.21959i			0.355684i
$413$	4.31348	+	2.30084i	0.212253	+	0.113217i
$414$		0			0
$415$		0			0
$416$	−32.4714			−1.59204
$417$		0			0
$418$		−	7.47117i		−	0.365427i
$419$	16.4433			0.803310			0.401655	−	0.915791i	$-0.368435\pi$
$419$	16.4433			0.803310			0.401655	+	0.915791i	$0.368435\pi$
$420$		0			0
$421$	−12.6969			−0.618811			−0.309405	−	0.950930i	$-0.600130\pi$
$421$	−12.6969			−0.618811			−0.309405	+	0.950930i	$0.600130\pi$
$422$			7.41964i			0.361182i
$423$		0			0
$424$	−25.3485			−1.23103
$425$		0			0
$426$		0			0
$427$	−14.4781	+	27.1427i	−0.700646	+	1.31353i
$428$			2.63435i			0.127336i
$429$		0			0
$430$		0			0
$431$			37.6123i			1.81172i	0.423576	+	0.905861i	$0.360775\pi$
$431$			37.6123i			1.81172i	−0.423576	+	0.905861i	$0.639225\pi$
$432$		0			0
$433$			1.93069i			0.0927829i	0.998923	+	0.0463915i	$0.0147722\pi$
$433$			1.93069i			0.0927829i	−0.998923	+	0.0463915i	$0.985228\pi$
$434$	−2.26306	−	1.20713i	−0.108630	−	0.0579442i
$435$		0			0
$436$	−4.49490			−0.215267
$437$	−34.0721			−1.62989
$438$		0			0
$439$			30.3746i			1.44970i	0.688907	+	0.724850i	$0.258091\pi$
$439$			30.3746i			1.44970i	−0.688907	+	0.724850i	$0.741909\pi$
$440$		0			0
$441$		0			0
$442$	24.9711			1.18775
$443$			4.78529i			0.227356i	0.993518	+	0.113678i	$0.0362632\pi$
$443$			4.78529i			0.227356i	−0.993518	+	0.113678i	$0.963737\pi$
$444$		0			0
$445$		0			0
$446$	13.7650			0.651792
$447$		0			0
$448$	−5.48230	−	2.92430i	−0.259014	−	0.138160i
$449$		−	5.22826i		−	0.246737i	−0.992361	−	0.123368i	$-0.960630\pi$
$449$		−	5.22826i		−	0.246737i	0.992361	−	0.123368i	$-0.0393697\pi$
$450$		0			0
$451$			8.42676i			0.396800i
$452$		−	0.483412i		−	0.0227378i
$453$		0			0
$454$		−	13.6531i		−	0.640770i
$455$		0			0
$456$		0			0
$457$	4.66883			0.218399			0.109199	−	0.994020i	$-0.465171\pi$
$457$	4.66883			0.218399			0.109199	+	0.994020i	$0.465171\pi$
$458$	−4.31348			−0.201556
$459$		0			0
$460$		0			0
$461$	−40.6931			−1.89527			−0.947633	−	0.319361i	$-0.896532\pi$
$461$	−40.6931			−1.89527			−0.947633	+	0.319361i	$0.896532\pi$
$462$		0			0
$463$	−32.2102			−1.49693			−0.748466	−	0.663173i	$-0.769209\pi$
$463$	−32.2102			−1.49693			−0.748466	+	0.663173i	$0.769209\pi$
$464$			5.51399i			0.255981i
$465$		0			0
$466$	−1.95459			−0.0905447
$467$	−0.791539			−0.0366280			−0.0183140	−	0.999832i	$-0.505830\pi$
$467$	−0.791539			−0.0366280			−0.0183140	+	0.999832i	$0.505830\pi$
$468$		0			0
$469$	−16.8990	−	9.01405i	−0.780322	−	0.416230i
$470$		0			0
$471$		0			0
$472$		−	4.72920i		−	0.217679i
$473$			9.57058i			0.440056i
$474$		0			0
$475$		0			0
$476$	−20.5542	−	10.9638i	−0.942099	−	0.502523i
$477$		0			0
$478$	10.3870			0.475088
$479$	38.4300			1.75591			0.877956	−	0.478740i	$-0.158907\pi$
$479$	38.4300			1.75591			0.877956	+	0.478740i	$0.158907\pi$
$480$		0			0
$481$		−	14.2404i		−	0.649304i
$482$	−14.8793			−0.677734
$483$		0			0
$484$	13.0454			0.592973
$485$		0			0
$486$		0			0
$487$	15.1618			0.687046			0.343523	−	0.939144i	$-0.388380\pi$
$487$	15.1618			0.687046			0.343523	+	0.939144i	$0.388380\pi$
$488$	29.7586			1.34711
$489$		0			0
$490$		0			0
$491$			26.8701i			1.21263i	0.795225	+	0.606314i	$0.207353\pi$
$491$			26.8701i			1.21263i	−0.795225	+	0.606314i	$0.792647\pi$
$492$		0			0
$493$		−	33.4945i		−	1.50852i
$494$			29.2699i			1.31691i
$495$		0			0
$496$		−	1.30658i		−	0.0586670i
$497$	−10.3879	+	19.4747i	−0.465963	+	0.873558i
$498$		0			0
$499$	10.0000			0.447661			0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000			0.447661			0.223831	+	0.974628i	$0.428144\pi$
$500$		0			0
$501$		0			0
$502$			7.47117i			0.333455i
$503$	−20.7759			−0.926350			−0.463175	−	0.886267i	$-0.653290\pi$
$503$	−20.7759			−0.926350			−0.463175	+	0.886267i	$0.653290\pi$
$504$		0			0
$505$		0			0
$506$		−	5.02118i		−	0.223219i
$507$		0			0
$508$		0			0
$509$	14.1803			0.628530			0.314265	−	0.949335i	$-0.398242\pi$
$509$	14.1803			0.628530			0.314265	+	0.949335i	$0.398242\pi$
$510$		0			0
$511$	−0.696938	+	1.30658i	−0.0308307	+	0.0577996i
$512$		−	10.9795i		−	0.485232i
$513$		0			0
$514$		−	9.73333i		−	0.429319i
$515$		0			0
$516$		0			0
$517$			11.0810i			0.487340i
$518$	2.37462	−	4.45178i	0.104335	−	0.195600i
$519$		0			0
$520$		0			0
$521$	26.9281			1.17974			0.589870	−	0.807498i	$-0.299179\pi$
$521$	26.9281			1.17974			0.589870	+	0.807498i	$0.299179\pi$
$522$		0			0
$523$		−	34.3623i		−	1.50256i	−0.659985	−	0.751279i	$-0.729437\pi$
$523$		−	34.3623i		−	1.50256i	0.659985	−	0.751279i	$-0.270563\pi$
$524$	−9.23889			−0.403603
$525$		0			0
$526$	20.9444			0.913219
$527$			7.93674i			0.345730i
$528$		0			0
$529$	0.101021			0.00439220
$530$		0			0
$531$		0			0
$532$	12.8512	−	24.0926i	0.557170	−	1.04455i
$533$		−	33.0136i		−	1.42998i
$534$		0			0
$535$		0			0
$536$			18.5276i			0.800272i
$537$		0			0
$538$			16.6214i			0.716601i
$539$	−8.22167	+	5.51399i	−0.354133	+	0.237504i
$540$		0			0
$541$	21.1010			0.907204			0.453602	−	0.891204i	$-0.350139\pi$
$541$	21.1010			0.907204			0.453602	+	0.891204i	$0.350139\pi$
$542$	−7.65753			−0.328919
$543$		0			0
$544$			35.6009i			1.52638i
$545$		0			0
$546$		0			0
$547$	−10.4930			−0.448646			−0.224323	−	0.974515i	$-0.572017\pi$
$547$	−10.4930			−0.448646			−0.224323	+	0.974515i	$0.572017\pi$
$548$			4.78529i			0.204417i
$549$		0			0
$550$		0			0
$551$	39.2606			1.67256
$552$		0			0
$553$	9.33766	+	4.98078i	0.397078	+	0.211804i
$554$		−	5.37113i		−	0.228197i
$555$		0			0
$556$			1.89387i			0.0803180i
$557$		−	29.7122i		−	1.25895i	−0.777022	−	0.629474i	$-0.783270\pi$
$557$		−	29.7122i		−	1.25895i	0.777022	−	0.629474i	$-0.216730\pi$
$558$		0			0
$559$		−	37.4948i		−	1.58586i
$560$		0			0
$561$		0			0
$562$	1.04930			0.0442618
$563$	26.2367			1.10574			0.552872	−	0.833266i	$-0.313532\pi$
$563$	26.2367			1.10574			0.552872	+	0.833266i	$0.313532\pi$
$564$		0			0
$565$		0			0
$566$	13.7650			0.578587
$567$		0			0
$568$	21.3516			0.895891
$569$		−	39.1694i		−	1.64207i	−0.570881	−	0.821033i	$-0.693398\pi$
$569$		−	39.1694i		−	1.64207i	0.570881	−	0.821033i	$-0.306602\pi$
$570$		0			0
$571$	−33.3939			−1.39749			−0.698745	−	0.715371i	$-0.746258\pi$
$571$	−33.3939			−1.39749			−0.698745	+	0.715371i	$0.746258\pi$
$572$	11.3574			0.474875
$573$		0			0
$574$	5.50510	−	10.3206i	0.229779	−	0.430775i
$575$		0			0
$576$		0			0
$577$		−	26.3314i		−	1.09619i	−0.836416	−	0.548096i	$-0.815353\pi$
$577$		−	26.3314i		−	1.09619i	0.836416	−	0.548096i	$-0.184647\pi$
$578$		−	14.7643i		−	0.614115i
$579$		0			0
$580$		0			0
$581$	−18.2911	−	9.75663i	−0.758844	−	0.404773i
$582$		0			0
$583$	14.0065			0.580089
$584$	1.43250			0.0592773
$585$		0			0
$586$		−	10.9079i		−	0.450602i
$587$	3.52194			0.145366			0.0726831	−	0.997355i	$-0.476844\pi$
$587$	3.52194			0.145366			0.0726831	+	0.997355i	$0.476844\pi$
$588$		0			0
$589$	−9.30306			−0.383326
$590$		0			0
$591$		0			0
$592$	2.57024			0.105636
$593$	−47.1904			−1.93788			−0.968940	−	0.247298i	$-0.920457\pi$
$593$	−47.1904			−1.93788			−0.968940	+	0.247298i	$0.920457\pi$
$594$		0			0
$595$		0			0
$596$		−	14.3492i		−	0.587767i
$597$		0			0
$598$			19.6715i			0.804429i
$599$			10.8851i			0.444754i	0.974961	+	0.222377i	$0.0713815\pi$
$599$			10.8851i			0.444754i	−0.974961	+	0.222377i	$0.928618\pi$
$600$		0			0
$601$			19.4667i			0.794062i	0.917805	+	0.397031i	$0.129959\pi$
$601$			19.4667i			0.794062i	−0.917805	+	0.397031i	$0.870041\pi$
$602$	6.25235	−	11.7215i	0.254827	−	0.477733i
$603$		0			0
$604$	−14.4949			−0.589789
$605$		0			0
$606$		0			0
$607$			17.4327i			0.707573i	0.935326	+	0.353786i	$0.115106\pi$
$607$			17.4327i			0.707573i	−0.935326	+	0.353786i	$0.884894\pi$
$608$	−41.7296			−1.69236
$609$		0			0
$610$		0			0
$611$		−	43.4120i		−	1.75626i
$612$		0			0
$613$	17.7320			0.716189			0.358095	−	0.933685i	$-0.383426\pi$
$613$	17.7320			0.716189			0.358095	+	0.933685i	$0.383426\pi$
$614$	−18.2911			−0.738170
$615$		0			0
$616$	8.44949	+	4.50702i	0.340440	+	0.181593i
$617$		−	15.8398i		−	0.637686i	−0.947808	−	0.318843i	$-0.896706\pi$
$617$		−	15.8398i		−	0.637686i	0.947808	−	0.318843i	$-0.103294\pi$
$618$		0			0
$619$			6.53289i			0.262579i	0.991344	+	0.131289i	$0.0419117\pi$
$619$			6.53289i			0.262579i	−0.991344	+	0.131289i	$0.958088\pi$
$620$		0			0
$621$		0			0
$622$		−	13.5714i		−	0.544162i
$623$	28.7892	+	15.3564i	1.15341	+	0.615240i
$624$		0			0
$625$		0			0
$626$	10.4847			0.419054
$627$		0			0
$628$			18.8603i			0.752606i
$629$	−15.6128			−0.622523
$630$		0			0
$631$	5.79796			0.230813			0.115407	−	0.993318i	$-0.463183\pi$
$631$	5.79796			0.230813			0.115407	+	0.993318i	$0.463183\pi$
$632$		−	10.2376i		−	0.407229i
$633$		0			0
$634$	2.44949			0.0972817
$635$		0			0
$636$		0			0
$637$	32.2102	−	21.6022i	1.27621	−	0.855911i
$638$			5.78580i			0.229062i
$639$		0			0
$640$		0			0
$641$			32.2412i			1.27345i	0.771091	+	0.636725i	$0.219711\pi$
$641$			32.2412i			1.27345i	−0.771091	+	0.636725i	$0.780289\pi$
$642$		0			0
$643$			18.5521i			0.731625i	0.930689	+	0.365812i	$0.119209\pi$
$643$			18.5521i			0.731625i	−0.930689	+	0.365812i	$0.880791\pi$
$644$	8.63695	−	16.1920i	0.340344	−	0.638055i
$645$		0			0
$646$	32.0908			1.26260
$647$	31.3417			1.23217			0.616085	−	0.787680i	$-0.288718\pi$
$647$	31.3417			1.23217			0.616085	+	0.787680i	$0.288718\pi$
$648$		0			0
$649$			2.61315i			0.102575i
$650$		0			0
$651$		0			0
$652$	−25.7024			−1.00658
$653$		−	32.6801i		−	1.27887i	−0.768845	−	0.639436i	$-0.779168\pi$
$653$		−	32.6801i		−	1.27887i	0.768845	−	0.639436i	$-0.220832\pi$
$654$		0			0
$655$		0			0
$656$	5.95862			0.232645
$657$		0			0
$658$	7.23907	−	13.5714i	0.282208	−	0.529066i
$659$			3.25702i			0.126876i	0.997986	+	0.0634378i	$0.0202064\pi$
$659$			3.25702i			0.126876i	−0.997986	+	0.0634378i	$0.979794\pi$
$660$		0			0
$661$		−	18.0281i		−	0.701212i	−0.936523	−	0.350606i	$-0.885976\pi$
$661$		−	18.0281i		−	0.701212i	0.936523	−	0.350606i	$-0.114024\pi$
$662$		−	23.2931i		−	0.905313i
$663$		0			0
$664$			20.0540i			0.778244i
$665$		0			0
$666$		0			0
$667$	26.3860			1.02167
$668$	−7.39967			−0.286302
$669$		0			0
$670$		0			0
$671$	−16.4433			−0.634788
$672$		0			0
$673$	22.1888			0.855317			0.427659	−	0.903940i	$-0.359339\pi$
$673$	22.1888			0.855317			0.427659	+	0.903940i	$0.359339\pi$
$674$			16.6206i			0.640202i
$675$		0			0
$676$	−25.6515			−0.986597
$677$	−44.4600			−1.70874			−0.854369	−	0.519667i	$-0.826056\pi$
$677$	−44.4600			−1.70874			−0.854369	+	0.519667i	$0.826056\pi$
$678$		0			0
$679$	−6.89898	+	12.9338i	−0.264759	+	0.496353i
$680$		0			0
$681$		0			0
$682$		−	1.37099i		−	0.0524977i
$683$		−	25.2605i		−	0.966565i	−0.875465	−	0.483282i	$-0.839444\pi$
$683$		−	25.2605i		−	0.966565i	0.875465	−	0.483282i	$-0.160556\pi$
$684$		0			0
$685$		0			0
$686$	13.6717	−	1.38211i	0.521987	−	0.0527690i
$687$		0			0
$688$	6.76742			0.258005
$689$	−54.8734			−2.09051
$690$		0			0
$691$		−	12.9338i		−	0.492024i	−0.969267	−	0.246012i	$-0.920880\pi$
$691$		−	12.9338i		−	0.492024i	0.969267	−	0.246012i	$-0.0791203\pi$
$692$	2.55251			0.0970319
$693$		0			0
$694$	27.5505			1.04580
$695$		0			0
$696$		0			0
$697$	−36.1953			−1.37100
$698$	−19.1928			−0.726458
$699$		0			0
$700$		0			0
$701$		−	20.9275i		−	0.790420i	−0.918591	−	0.395210i	$-0.870672\pi$
$701$		−	20.9275i		−	0.790420i	0.918591	−	0.395210i	$-0.129328\pi$
$702$		0			0
$703$		−	18.3006i		−	0.690218i
$704$		−	3.32124i		−	0.125174i
$705$		0			0
$706$		−	1.30658i		−	0.0491737i
$707$	−13.9099	−	7.41964i	−0.523135	−	0.279044i
$708$		0			0
$709$	13.7980			0.518193			0.259097	−	0.965851i	$-0.416575\pi$
$709$	13.7980			0.518193			0.259097	+	0.965851i	$0.416575\pi$
$710$		0			0
$711$		0			0
$712$		−	31.5638i		−	1.18290i
$713$	−6.25235			−0.234152
$714$		0			0
$715$		0			0
$716$			0.207081i			0.00773897i
$717$		0			0
$718$	−7.34507			−0.274115
$719$	−5.54334			−0.206732			−0.103366	−	0.994643i	$-0.532961\pi$
$719$	−5.54334			−0.206732			−0.103366	+	0.994643i	$0.532961\pi$
$720$		0			0
$721$	−6.20204	+	11.6272i	−0.230976	+	0.433020i
$722$			23.5180i			0.875249i
$723$		0			0
$724$			4.63902i			0.172408i
$725$		0			0
$726$		0			0
$727$			7.21959i			0.267760i	0.990998	+	0.133880i	$0.0427436\pi$
$727$			7.21959i			0.267760i	−0.990998	+	0.133880i	$0.957256\pi$
$728$	−33.1027	−	17.6572i	−1.22687	−	0.654420i
$729$		0			0
$730$		0			0
$731$	−41.1084			−1.52045
$732$		0			0
$733$			6.65989i			0.245989i	0.992407	+	0.122994i	$0.0392497\pi$
$733$			6.65989i			0.245989i	−0.992407	+	0.122994i	$0.960750\pi$
$734$	−2.67834			−0.0988592
$735$		0			0
$736$	−28.0454			−1.03377
$737$		−	10.2376i		−	0.377106i
$738$		0			0
$739$	−37.3939			−1.37556			−0.687778	−	0.725921i	$-0.741414\pi$
$739$	−37.3939			−1.37556			−0.687778	+	0.725921i	$0.741414\pi$
$740$		0			0
$741$		0			0
$742$	−17.1544	−	9.15028i	−0.629757	−	0.335917i
$743$		−	28.2283i		−	1.03560i	−0.855503	−	0.517798i	$-0.826752\pi$
$743$		−	28.2283i		−	1.03560i	0.855503	−	0.517798i	$-0.173248\pi$
$744$		0			0
$745$		0			0
$746$			23.8988i			0.874996i
$747$		0			0
$748$		−	12.4519i		−	0.455288i
$749$	−2.26306	+	4.24264i	−0.0826903	+	0.155023i
$750$		0			0
$751$	−25.1918			−0.919263			−0.459632	−	0.888110i	$-0.652019\pi$
$751$	−25.1918			−0.919263			−0.459632	+	0.888110i	$0.652019\pi$
$752$	7.83542			0.285729
$753$		0			0
$754$		−	22.6671i		−	0.825488i
$755$		0			0
$756$		0			0
$757$	24.2874			0.882742			0.441371	−	0.897325i	$-0.354492\pi$
$757$	24.2874			0.882742			0.441371	+	0.897325i	$0.354492\pi$
$758$		−	13.0555i		−	0.474198i
$759$		0			0
$760$		0			0
$761$	−44.3886			−1.60909			−0.804544	−	0.593894i	$-0.797590\pi$
$761$	−44.3886			−1.60909			−0.804544	+	0.593894i	$0.797590\pi$
$762$		0			0
$763$	−7.23907	−	3.86137i	−0.262072	−	0.139791i
$764$		−	32.5911i		−	1.17911i
$765$		0			0
$766$		−	19.4667i		−	0.703359i
$767$		−	10.2376i		−	0.369658i
$768$		0			0
$769$			19.4667i			0.701986i	0.936378	+	0.350993i	$0.114156\pi$
$769$			19.4667i			0.701986i	−0.936378	+	0.350993i	$0.885844\pi$
$770$		0			0
$771$		0			0
$772$	−9.80930			−0.353045
$773$	−17.4318			−0.626979			−0.313490	−	0.949592i	$-0.601498\pi$
$773$	−17.4318			−0.626979			−0.313490	+	0.949592i	$0.601498\pi$
$774$		0			0
$775$		0			0
$776$	14.1803			0.509043
$777$		0			0
$778$	14.3721			0.515266
$779$		−	42.4264i		−	1.52008i
$780$		0			0
$781$	−11.7980			−0.422164
$782$	21.5674			0.771249
$783$		0			0
$784$	3.89898	+	5.81360i	0.139249	+	0.207629i
$785$		0			0
$786$		0			0
$787$			40.4625i			1.44233i	0.692762	+	0.721166i	$0.256393\pi$
$787$			40.4625i			1.44233i	−0.692762	+	0.721166i	$0.743607\pi$
$788$			19.6246i			0.699096i
$789$		0			0
$790$		0			0
$791$	0.415279	−	0.778539i	0.0147656	−	0.0276817i
$792$		0			0
$793$	64.4203			2.28763
$794$	23.2325			0.824491
$795$		0			0
$796$			10.3206i			0.365805i
$797$	15.4930			0.548789			0.274394	−	0.961617i	$-0.411523\pi$
$797$	15.4930			0.548789			0.274394	+	0.961617i	$0.411523\pi$
$798$		0			0
$799$	−47.5959			−1.68382
$800$		0			0
$801$		0			0
$802$	−16.6827			−0.589087
$803$	−0.791539			−0.0279328
$804$		0			0
$805$		0			0
$806$			5.37113i			0.189190i
$807$		0			0
$808$			15.2505i			0.536509i
$809$			8.62815i			0.303349i	0.988430	+	0.151675i	$0.0484666\pi$
$809$			8.62815i			0.303349i	−0.988430	+	0.151675i	$0.951533\pi$
$810$		0			0
$811$		−	10.3206i		−	0.362406i	−0.983446	−	0.181203i	$-0.942001\pi$
$811$		−	10.3206i		−	0.362406i	0.983446	−	0.181203i	$-0.0579991\pi$
$812$	−9.95218	+	18.6577i	−0.349253	+	0.654758i
$813$		0			0
$814$	2.69694			0.0945276
$815$		0			0
$816$		0			0
$817$		−	48.1852i		−	1.68579i
$818$	11.0016			0.384661
$819$		0			0
$820$		0			0
$821$		−	27.8557i		−	0.972170i	−0.873912	−	0.486085i	$-0.838425\pi$
$821$		−	27.8557i		−	0.972170i	0.873912	−	0.486085i	$-0.161575\pi$
$822$		0			0
$823$	−14.4781			−0.504676			−0.252338	−	0.967639i	$-0.581199\pi$
$823$	−14.4781			−0.504676			−0.252338	+	0.967639i	$0.581199\pi$
$824$	12.7478			0.444090
$825$		0			0
$826$	1.70714	−	3.20045i	0.0593991	−	0.111358i
$827$		−	10.7210i		−	0.372806i	−0.982473	−	0.186403i	$-0.940317\pi$
$827$		−	10.7210i		−	0.372806i	0.982473	−	0.186403i	$-0.0596829\pi$
$828$		0			0
$829$			17.4408i			0.605744i	0.953031	+	0.302872i	$0.0979455\pi$
$829$			17.4408i			0.605744i	−0.953031	+	0.302872i	$0.902055\pi$
$830$		0			0
$831$		0			0
$832$			13.0117i			0.451098i
$833$	−23.6842	−	35.3144i	−0.820608	−	1.22357i
$834$		0			0
$835$		0			0
$836$	14.5956			0.504798
$837$		0			0
$838$		−	12.2004i		−	0.421455i
$839$	28.3606			0.979116			0.489558	−	0.871971i	$-0.337158\pi$
$839$	28.3606			0.979116			0.489558	+	0.871971i	$0.337158\pi$
$840$		0			0
$841$	−1.40408			−0.0484166
$842$			9.42067i			0.324658i
$843$		0			0
$844$	−14.4949			−0.498935
$845$		0			0
$846$		0			0
$847$	21.0097	+	11.2068i	0.721903	+	0.385069i
$848$		−	9.90408i		−	0.340108i
$849$		0			0
$850$		0			0
$851$		−	12.2993i		−	0.421616i
$852$		0			0
$853$		−	34.0542i		−	1.16599i	−0.812475	−	0.582996i	$-0.801880\pi$
$853$		−	34.0542i		−	1.16599i	0.812475	−	0.582996i	$-0.198120\pi$
$854$	20.1389	+	10.7423i	0.689139	+	0.367592i
$855$		0			0
$856$	4.65153			0.158986
$857$	15.4930			0.529229			0.264615	−	0.964354i	$-0.414755\pi$
$857$	15.4930			0.529229			0.264615	+	0.964354i	$0.414755\pi$
$858$		0			0
$859$		−	30.3746i		−	1.03637i	−0.855269	−	0.518184i	$-0.826608\pi$
$859$		−	30.3746i		−	1.03637i	0.855269	−	0.518184i	$-0.173392\pi$
$860$		0			0
$861$		0			0
$862$	27.9070			0.950515
$863$			37.7989i			1.28669i	0.765577	+	0.643345i	$0.222454\pi$
$863$			37.7989i			1.28669i	−0.765577	+	0.643345i	$0.777546\pi$
$864$		0			0
$865$		0			0
$866$	1.43250			0.0486783
$867$		0			0
$868$	2.35824	−	4.42108i	0.0800438	−	0.150061i
$869$			5.65685i			0.191896i
$870$		0			0
$871$			40.1079i			1.35900i
$872$			7.93674i			0.268772i
$873$		0			0
$874$			25.2803i			0.855118i
$875$		0			0
$876$		0			0
$877$	14.4781			0.488892			0.244446	−	0.969663i	$-0.421394\pi$
$877$	14.4781			0.488892			0.244446	+	0.969663i	$0.421394\pi$
$878$	22.5368			0.760582
$879$		0			0
$880$		0			0
$881$	36.9975			1.24648			0.623239	−	0.782031i	$-0.285816\pi$
$881$	36.9975			1.24648			0.623239	+	0.782031i	$0.285816\pi$
$882$		0			0
$883$	−14.0065			−0.471356			−0.235678	−	0.971831i	$-0.575731\pi$
$883$	−14.0065			−0.471356			−0.235678	+	0.971831i	$0.575731\pi$
$884$			48.7832i			1.64075i
$885$		0			0
$886$	3.55051			0.119282
$887$	−30.5502			−1.02577			−0.512887	−	0.858456i	$-0.671424\pi$
$887$	−30.5502			−1.02577			−0.512887	+	0.858456i	$0.671424\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$			26.8911i			0.900382i
$893$		−	55.7896i		−	1.86693i
$894$		0			0
$895$		0			0
$896$	12.4258	−	23.2952i	0.415118	−	0.778238i
$897$		0			0
$898$	−3.87918			−0.129450
$899$	7.20445			0.240282
$900$		0			0
$901$			60.1619i			2.00428i
$902$	6.25235			0.208180
$903$		0			0
$904$	−0.853572			−0.0283894
$905$		0			0
$906$		0			0
$907$	22.4008			0.743808			0.371904	−	0.928271i	$-0.378705\pi$
$907$	22.4008			0.743808			0.371904	+	0.928271i	$0.378705\pi$
$908$	26.6724			0.885156
$909$		0			0
$910$		0			0
$911$			26.8701i			0.890245i	0.895470	+	0.445122i	$0.146840\pi$
$911$			26.8701i			0.890245i	−0.895470	+	0.445122i	$0.853160\pi$
$912$		0			0
$913$		−	11.0810i		−	0.366726i
$914$		−	3.46410i		−	0.114582i
$915$		0			0
$916$		−	8.42676i		−	0.278428i
$917$	−14.8793	−	7.93674i	−0.491358	−	0.262094i
$918$		0			0
$919$	10.0000			0.329870			0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000			0.329870			0.164935	+	0.986304i	$0.447259\pi$
$920$		0			0
$921$		0			0
$922$			30.1928i			0.994347i
$923$	46.2210			1.52138
$924$		0			0
$925$		0			0
$926$			23.8988i			0.785362i
$927$		0			0
$928$	32.3162			1.06083
$929$	−8.63695			−0.283369			−0.141685	−	0.989912i	$-0.545252\pi$
$929$	−8.63695			−0.283369			−0.141685	+	0.989912i	$0.545252\pi$
$930$		0			0
$931$	41.3939	−	27.7614i	1.35663	−	0.909844i
$932$		−	3.81846i		−	0.125078i
$933$		0			0
$934$			0.587293i			0.0192168i
$935$		0			0
$936$		0			0
$937$		−	8.03087i		−	0.262357i	−0.991359	−	0.131179i	$-0.958124\pi$
$937$		−	8.03087i		−	0.262357i	0.991359	−	0.131179i	$-0.0418761\pi$
$938$	−6.68810	+	12.5384i	−0.218374	+	0.409394i
$939$		0			0
$940$		0			0
$941$	36.1670			1.17901			0.589505	−	0.807765i	$-0.299323\pi$
$941$	36.1670			1.17901			0.589505	+	0.807765i	$0.299323\pi$
$942$		0			0
$943$		−	28.5137i		−	0.928534i
$944$	1.84778			0.0601401
$945$		0			0
$946$	7.10102			0.230874
$947$			13.0218i			0.423153i	0.977361	+	0.211577i	$0.0678598\pi$
$947$			13.0218i			0.423153i	−0.977361	+	0.211577i	$0.932140\pi$
$948$		0			0
$949$	3.10102			0.100663
$950$		0			0
$951$		0			0
$952$	−19.3590	+	36.2930i	−0.627428	+	1.17626i
$953$		−	27.7112i		−	0.897654i	−0.893619	−	0.448827i	$-0.851842\pi$
$953$		−	27.7112i		−	0.897654i	0.893619	−	0.448827i	$-0.148158\pi$
$954$		0			0
$955$		0			0
$956$			20.2918i			0.656284i
$957$		0			0
$958$		−	28.5137i		−	0.921236i
$959$	−4.11084	+	7.70674i	−0.132746	+	0.248864i
$960$		0			0
$961$	29.2929			0.944931
$962$	−10.5658			−0.340656
$963$		0			0
$964$		−	29.0680i		−	0.936217i
$965$		0			0
$966$		0			0
$967$	−42.7031			−1.37324			−0.686620	−	0.727017i	$-0.740906\pi$
$967$	−42.7031			−1.37324			−0.686620	+	0.727017i	$0.740906\pi$
$968$		−	23.0346i		−	0.740359i
$969$		0			0
$970$		0			0
$971$	−40.2778			−1.29258			−0.646288	−	0.763093i	$-0.723680\pi$
$971$	−40.2778			−1.29258			−0.646288	+	0.763093i	$0.723680\pi$
$972$		0			0
$973$	−1.62694	+	3.05009i	−0.0521574	+	0.0977815i
$974$		−	11.2495i		−	0.360457i
$975$		0			0
$976$			11.6272i			0.372178i
$977$			34.3139i			1.09780i	0.835888	+	0.548900i	$0.184953\pi$
$977$			34.3139i			1.09780i	−0.835888	+	0.548900i	$0.815047\pi$
$978$		0			0
$979$			17.4408i			0.557411i
$980$		0			0
$981$		0			0
$982$	19.9366			0.636203
$983$	−54.8480			−1.74938			−0.874689	−	0.484684i	$-0.838935\pi$
$983$	−54.8480			−1.74938			−0.874689	+	0.484684i	$0.838935\pi$
$984$		0			0
$985$		0			0
$986$	−24.8517			−0.791439
$987$		0			0
$988$	−57.1812			−1.81918
$989$		−	32.3840i		−	1.02975i
$990$		0			0
$991$	23.7980			0.755967			0.377984	−	0.925812i	$-0.376618\pi$
$991$	23.7980			0.755967			0.377984	+	0.925812i	$0.376618\pi$
$992$	−7.65753			−0.243127
$993$		0			0
$994$	14.4495	+	7.70747i	0.458310	+	0.244466i
$995$		0			0
$996$		0			0
$997$			4.16950i			0.132049i	0.997818	+	0.0660246i	$0.0210316\pi$
$997$			4.16950i			0.132049i	−0.997818	+	0.0660246i	$0.978968\pi$
$998$		−	7.41964i		−	0.234865i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.b.h.251.8		16
3.2	odd	2	inner	1575.2.b.h.251.12		16
5.2	odd	4		315.2.g.a.314.9	yes	16
5.3	odd	4		315.2.g.a.314.6	yes	16
5.4	even	2	inner	1575.2.b.h.251.9		16
7.6	odd	2	inner	1575.2.b.h.251.7		16
15.2	even	4		315.2.g.a.314.8	yes	16
15.8	even	4		315.2.g.a.314.11	yes	16
15.14	odd	2	inner	1575.2.b.h.251.5		16
20.3	even	4		5040.2.k.g.1889.3		16
20.7	even	4		5040.2.k.g.1889.2		16
21.20	even	2	inner	1575.2.b.h.251.11		16
35.13	even	4		315.2.g.a.314.7	yes	16
35.27	even	4		315.2.g.a.314.12	yes	16
35.34	odd	2	inner	1575.2.b.h.251.10		16
60.23	odd	4		5040.2.k.g.1889.13		16
60.47	odd	4		5040.2.k.g.1889.16		16
105.62	odd	4		315.2.g.a.314.5	✓	16
105.83	odd	4		315.2.g.a.314.10	yes	16
105.104	even	2	inner	1575.2.b.h.251.6		16
140.27	odd	4		5040.2.k.g.1889.15		16
140.83	odd	4		5040.2.k.g.1889.14		16
420.83	even	4		5040.2.k.g.1889.4		16
420.167	even	4		5040.2.k.g.1889.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.g.a.314.5	✓	16	105.62	odd	4
315.2.g.a.314.6	yes	16	5.3	odd	4
315.2.g.a.314.7	yes	16	35.13	even	4
315.2.g.a.314.8	yes	16	15.2	even	4
315.2.g.a.314.9	yes	16	5.2	odd	4
315.2.g.a.314.10	yes	16	105.83	odd	4
315.2.g.a.314.11	yes	16	15.8	even	4
315.2.g.a.314.12	yes	16	35.27	even	4
1575.2.b.h.251.5		16	15.14	odd	2	inner
1575.2.b.h.251.6		16	105.104	even	2	inner
1575.2.b.h.251.7		16	7.6	odd	2	inner
1575.2.b.h.251.8		16	1.1	even	1	trivial
1575.2.b.h.251.9		16	5.4	even	2	inner
1575.2.b.h.251.10		16	35.34	odd	2	inner
1575.2.b.h.251.11		16	21.20	even	2	inner
1575.2.b.h.251.12		16	3.2	odd	2	inner
5040.2.k.g.1889.1		16	420.167	even	4
5040.2.k.g.1889.2		16	20.7	even	4
5040.2.k.g.1889.3		16	20.3	even	4
5040.2.k.g.1889.4		16	420.83	even	4
5040.2.k.g.1889.13		16	60.23	odd	4
5040.2.k.g.1889.14		16	140.83	odd	4
5040.2.k.g.1889.15		16	140.27	odd	4
5040.2.k.g.1889.16		16	60.47	odd	4

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.741964i q^{2} +1.44949 q^{4} +(2.33441 + 1.24519i) q^{7} -2.55940i q^{8} +1.41421i q^{11} -5.54048i q^{13} +(0.923889 - 1.73205i) q^{14} +1.00000 q^{16} -6.07445 q^{17} -7.12018i q^{19} +1.04930 q^{22} -4.78529i q^{23} -4.11084 q^{26} +(3.38371 + 1.80490i) q^{28} +5.51399i q^{29} -1.30658i q^{31} -5.86076i q^{32} +4.50702i q^{34} +2.57024 q^{37} -5.28291 q^{38} +5.95862 q^{41} +6.76742 q^{43} +2.04989i q^{44} -3.55051 q^{46} +7.83542 q^{47} +(3.89898 + 5.81360i) q^{49} -8.03087i q^{52} -9.90408i q^{53} +(3.18695 - 5.97469i) q^{56} +4.09118 q^{58} +1.84778 q^{59} +11.6272i q^{61} -0.969433 q^{62} -2.34847 q^{64} -7.23907 q^{67} -8.80486 q^{68} +8.34242i q^{71} +0.559702i q^{73} -1.90702i q^{74} -10.3206i q^{76} +(-1.76097 + 3.30136i) q^{77} +4.00000 q^{79} -4.42108i q^{82} -7.83542 q^{83} -5.02118i q^{86} +3.61953 q^{88} +12.3325 q^{89} +(6.89898 - 12.9338i) q^{91} -6.93623i q^{92} -5.81360i q^{94} +5.54048i q^{97} +(4.31348 - 2.89290i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} + 16 q^{16} - 96 q^{46} - 16 q^{49} + 80 q^{64} + 64 q^{79} + 32 q^{91}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^16 - 96 * q^46 - 16 * q^49 + 80 * q^64 + 64 * q^79 + 32 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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