LMFDB - Embedded newform 1575.2.b.d.251.3

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:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			251.3
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1575.251
Dual form			1575.2.b.d.251.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} +(-0.500000 - 2.59808i) q^{7} +2.82843i q^{8} +O(q^{10})$	$q+1.41421i q^{2} +(-0.500000 - 2.59808i) q^{7} +2.82843i q^{8} +1.41421i q^{11} -5.19615i q^{13} +(3.67423 - 0.707107i) q^{14} -4.00000 q^{16} +7.34847 q^{17} +5.19615i q^{19} -2.00000 q^{22} -7.07107i q^{23} +7.34847 q^{26} +1.41421i q^{29} -5.19615i q^{31} +10.3923i q^{34} +8.00000 q^{37} -7.34847 q^{38} +5.00000 q^{43} +10.0000 q^{46} +7.34847 q^{47} +(-6.50000 + 2.59808i) q^{49} +5.65685i q^{53} +(7.34847 - 1.41421i) q^{56} -2.00000 q^{58} +7.34847 q^{59} +5.19615i q^{61} +7.34847 q^{62} -8.00000 q^{64} -7.00000 q^{67} -2.82843i q^{71} -10.3923i q^{73} +11.3137i q^{74} +(3.67423 - 0.707107i) q^{77} +14.0000 q^{79} -7.34847 q^{83} +7.07107i q^{86} -4.00000 q^{88} +14.6969 q^{89} +(-13.5000 + 2.59808i) q^{91} +10.3923i q^{94} +5.19615i q^{97} +(-3.67423 - 9.19239i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{7}+O(q^{10}) $ 4 * q - 2 * q^7	$ 4 q - 2 q^{7} - 16 q^{16} - 8 q^{22} + 32 q^{37} + 20 q^{43} + 40 q^{46} - 26 q^{49} - 8 q^{58} - 32 q^{64} - 28 q^{67} + 56 q^{79} - 16 q^{88} - 54 q^{91}+O(q^{100}) $ 4 * q - 2 * q^7 - 16 * q^16 - 8 * q^22 + 32 * q^37 + 20 * q^43 + 40 * q^46 - 26 * q^49 - 8 * q^58 - 32 * q^64 - 28 * q^67 + 56 * q^79 - 16 * q^88 - 54 * q^91

Display 100 coefficients 10 coefficients

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 $

$2$			1.41421i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$2$			1.41421i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.500000	−	2.59808i	−0.188982	−	0.981981i
$7$	−0.500000	−	2.59808i	−0.188982	−	0.981981i
$8$			2.82843i			1.00000i
$9$		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
$13$		−	5.19615i		−	1.44115i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$		−	5.19615i		−	1.44115i	0.693375	−	0.720577i	$-0.256123\pi$
$14$	3.67423	−	0.707107i	0.981981	−	0.188982i
$15$		0			0
$16$	−4.00000			−1.00000
$17$	7.34847			1.78227			0.891133	−	0.453743i	$-0.149911\pi$
$17$	7.34847			1.78227			0.891133	+	0.453743i	$0.149911\pi$
$18$		0			0
$19$			5.19615i			1.19208i	0.802955	+	0.596040i	$0.203260\pi$
$19$			5.19615i			1.19208i	−0.802955	+	0.596040i	$0.796740\pi$
$20$		0			0
$21$		0			0
$22$	−2.00000			−0.426401
$23$		−	7.07107i		−	1.47442i	−0.675664	−	0.737210i	$-0.736143\pi$
$23$		−	7.07107i		−	1.47442i	0.675664	−	0.737210i	$-0.263857\pi$
$24$		0			0
$25$		0			0
$26$	7.34847			1.44115
$27$		0			0
$28$		0			0
$29$			1.41421i			0.262613i	0.991342	+	0.131306i	$0.0419172\pi$
$29$			1.41421i			0.262613i	−0.991342	+	0.131306i	$0.958083\pi$
$30$		0			0
$31$		−	5.19615i		−	0.933257i	−0.884454	−	0.466628i	$-0.845469\pi$
$31$		−	5.19615i		−	0.933257i	0.884454	−	0.466628i	$-0.154531\pi$
$32$		0			0
$33$		0			0
$34$			10.3923i			1.78227i
$35$		0			0
$36$		0			0
$37$	8.00000			1.31519			0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000			1.31519			0.657596	+	0.753371i	$0.271573\pi$
$38$	−7.34847			−1.19208
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	5.00000			0.762493			0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000			0.762493			0.381246	+	0.924473i	$0.375495\pi$
$44$		0			0
$45$		0			0
$46$	10.0000			1.47442
$47$	7.34847			1.07188			0.535942	−	0.844255i	$-0.319956\pi$
$47$	7.34847			1.07188			0.535942	+	0.844255i	$0.319956\pi$
$48$		0			0
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			5.65685i			0.777029i	0.921443	+	0.388514i	$0.127012\pi$
$53$			5.65685i			0.777029i	−0.921443	+	0.388514i	$0.872988\pi$
$54$		0			0
$55$		0			0
$56$	7.34847	−	1.41421i	0.981981	−	0.188982i
$57$		0			0
$58$	−2.00000			−0.262613
$59$	7.34847			0.956689			0.478345	−	0.878172i	$-0.341237\pi$
$59$	7.34847			0.956689			0.478345	+	0.878172i	$0.341237\pi$
$60$		0			0
$61$			5.19615i			0.665299i	0.943051	+	0.332650i	$0.107943\pi$
$61$			5.19615i			0.665299i	−0.943051	+	0.332650i	$0.892057\pi$
$62$	7.34847			0.933257
$63$		0			0
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	−7.00000			−0.855186			−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000			−0.855186			−0.427593	+	0.903971i	$0.640638\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	2.82843i		−	0.335673i	−0.985815	−	0.167836i	$-0.946322\pi$
$71$		−	2.82843i		−	0.335673i	0.985815	−	0.167836i	$-0.0536780\pi$
$72$		0			0
$73$		−	10.3923i		−	1.21633i	−0.793812	−	0.608164i	$-0.791906\pi$
$73$		−	10.3923i		−	1.21633i	0.793812	−	0.608164i	$-0.208094\pi$
$74$			11.3137i			1.31519i
$75$		0			0
$76$		0			0
$77$	3.67423	−	0.707107i	0.418718	−	0.0805823i
$78$		0			0
$79$	14.0000			1.57512			0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000			1.57512			0.787562	+	0.616236i	$0.211343\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.34847			−0.806599			−0.403300	−	0.915068i	$-0.632137\pi$
$83$	−7.34847			−0.806599			−0.403300	+	0.915068i	$0.632137\pi$
$84$		0			0
$85$		0			0
$86$			7.07107i			0.762493i
$87$		0			0
$88$	−4.00000			−0.426401
$89$	14.6969			1.55787			0.778936	−	0.627103i	$-0.215760\pi$
$89$	14.6969			1.55787			0.778936	+	0.627103i	$0.215760\pi$
$90$		0			0
$91$	−13.5000	+	2.59808i	−1.41518	+	0.272352i
$92$		0			0
$93$		0			0
$94$			10.3923i			1.07188i
$95$		0			0
$96$		0			0
$97$			5.19615i			0.527589i	0.964579	+	0.263795i	$0.0849741\pi$
$97$			5.19615i			0.527589i	−0.964579	+	0.263795i	$0.915026\pi$
$98$	−3.67423	−	9.19239i	−0.371154	−	0.928571i
$99$		0			0
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		−	10.3923i		−	1.02398i	−0.858990	−	0.511992i	$-0.828908\pi$
$103$		−	10.3923i		−	1.02398i	0.858990	−	0.511992i	$-0.171092\pi$
$104$	14.6969			1.44115
$105$		0			0
$106$	−8.00000			−0.777029
$107$			5.65685i			0.546869i	0.961891	+	0.273434i	$0.0881596\pi$
$107$			5.65685i			0.546869i	−0.961891	+	0.273434i	$0.911840\pi$
$108$		0			0
$109$	−1.00000			−0.0957826			−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000			−0.0957826			−0.0478913	+	0.998853i	$0.515250\pi$
$110$		0			0
$111$		0			0
$112$	2.00000	+	10.3923i	0.188982	+	0.981981i
$113$			5.65685i			0.532152i	0.963952	+	0.266076i	$0.0857272\pi$
$113$			5.65685i			0.532152i	−0.963952	+	0.266076i	$0.914273\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$			10.3923i			0.956689i
$119$	−3.67423	−	19.0919i	−0.336817	−	1.75015i
$120$		0			0
$121$	9.00000			0.818182
$122$	−7.34847			−0.665299
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−10.0000			−0.887357			−0.443678	−	0.896186i	$-0.646327\pi$
$127$	−10.0000			−0.887357			−0.443678	+	0.896186i	$0.646327\pi$
$128$		−	11.3137i		−	1.00000i
$129$		0			0
$130$		0			0
$131$	−14.6969			−1.28408			−0.642039	−	0.766672i	$-0.721911\pi$
$131$	−14.6969			−1.28408			−0.642039	+	0.766672i	$0.721911\pi$
$132$		0			0
$133$	13.5000	−	2.59808i	1.17060	−	0.225282i
$134$		−	9.89949i		−	0.855186i
$135$		0			0
$136$			20.7846i			1.78227i
$137$			9.89949i			0.845771i	0.906183	+	0.422885i	$0.138983\pi$
$137$			9.89949i			0.845771i	−0.906183	+	0.422885i	$0.861017\pi$
$138$		0			0
$139$		−	20.7846i		−	1.76293i	−0.472252	−	0.881464i	$-0.656559\pi$
$139$		−	20.7846i		−	1.76293i	0.472252	−	0.881464i	$-0.343441\pi$
$140$		0			0
$141$		0			0
$142$	4.00000			0.335673
$143$	7.34847			0.614510
$144$		0			0
$145$		0			0
$146$	14.6969			1.21633
$147$		0			0
$148$		0			0
$149$			18.3848i			1.50614i	0.657941	+	0.753070i	$0.271428\pi$
$149$			18.3848i			1.50614i	−0.657941	+	0.753070i	$0.728572\pi$
$150$		0			0
$151$	5.00000			0.406894			0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000			0.406894			0.203447	+	0.979086i	$0.434786\pi$
$152$	−14.6969			−1.19208
$153$		0			0
$154$	1.00000	+	5.19615i	0.0805823	+	0.418718i
$155$		0			0
$156$		0			0
$157$			15.5885i			1.24409i	0.782980	+	0.622047i	$0.213699\pi$
$157$			15.5885i			1.24409i	−0.782980	+	0.622047i	$0.786301\pi$
$158$			19.7990i			1.57512i
$159$		0			0
$160$		0			0
$161$	−18.3712	+	3.53553i	−1.44785	+	0.278639i
$162$		0			0
$163$	−7.00000			−0.548282			−0.274141	−	0.961689i	$-0.588394\pi$
$163$	−7.00000			−0.548282			−0.274141	+	0.961689i	$0.588394\pi$
$164$		0			0
$165$		0			0
$166$		−	10.3923i		−	0.806599i
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−14.0000			−1.07692
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−22.0454			−1.67608			−0.838041	−	0.545608i	$-0.816299\pi$
$173$	−22.0454			−1.67608			−0.838041	+	0.545608i	$0.816299\pi$
$174$		0			0
$175$		0			0
$176$		−	5.65685i		−	0.426401i
$177$		0			0
$178$			20.7846i			1.55787i
$179$			18.3848i			1.37414i	0.726590	+	0.687071i	$0.241104\pi$
$179$			18.3848i			1.37414i	−0.726590	+	0.687071i	$0.758896\pi$
$180$		0			0
$181$			5.19615i			0.386227i	0.981176	+	0.193113i	$0.0618586\pi$
$181$			5.19615i			0.386227i	−0.981176	+	0.193113i	$0.938141\pi$
$182$	−3.67423	−	19.0919i	−0.272352	−	1.41518i
$183$		0			0
$184$	20.0000			1.47442
$185$		0			0
$186$		0			0
$187$			10.3923i			0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			9.89949i			0.716302i	0.933664	+	0.358151i	$0.116593\pi$
$191$			9.89949i			0.716302i	−0.933664	+	0.358151i	$0.883407\pi$
$192$		0			0
$193$	−25.0000			−1.79954			−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−25.0000			−1.79954			−0.899770	+	0.436365i	$0.856266\pi$
$194$	−7.34847			−0.527589
$195$		0			0
$196$		0			0
$197$		−	15.5563i		−	1.10834i	−0.832402	−	0.554172i	$-0.813035\pi$
$197$		−	15.5563i		−	1.10834i	0.832402	−	0.554172i	$-0.186965\pi$
$198$		0			0
$199$		−	5.19615i		−	0.368345i	−0.982894	−	0.184173i	$-0.941039\pi$
$199$		−	5.19615i		−	0.368345i	0.982894	−	0.184173i	$-0.0589606\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.67423	−	0.707107i	0.257881	−	0.0496292i
$204$		0			0
$205$		0			0
$206$	14.6969			1.02398
$207$		0			0
$208$			20.7846i			1.44115i
$209$	−7.34847			−0.508304
$210$		0			0
$211$	5.00000			0.344214			0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000			0.344214			0.172107	+	0.985078i	$0.444942\pi$
$212$		0			0
$213$		0			0
$214$	−8.00000			−0.546869
$215$		0			0
$216$		0			0
$217$	−13.5000	+	2.59808i	−0.916440	+	0.176369i
$218$		−	1.41421i		−	0.0957826i
$219$		0			0
$220$		0			0
$221$		−	38.1838i		−	2.56852i
$222$		0			0
$223$		−	5.19615i		−	0.347960i	−0.984749	−	0.173980i	$-0.944337\pi$
$223$		−	5.19615i		−	0.347960i	0.984749	−	0.173980i	$-0.0556628\pi$
$224$		0			0
$225$		0			0
$226$	−8.00000			−0.532152
$227$	−14.6969			−0.975470			−0.487735	−	0.872992i	$-0.662177\pi$
$227$	−14.6969			−0.975470			−0.487735	+	0.872992i	$0.662177\pi$
$228$		0			0
$229$		−	15.5885i		−	1.03011i	−0.857156	−	0.515057i	$-0.827771\pi$
$229$		−	15.5885i		−	1.03011i	0.857156	−	0.515057i	$-0.172229\pi$
$230$		0			0
$231$		0			0
$232$	−4.00000			−0.262613
$233$		−	28.2843i		−	1.85296i	−0.376339	−	0.926482i	$-0.622817\pi$
$233$		−	28.2843i		−	1.85296i	0.376339	−	0.926482i	$-0.377183\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$	27.0000	−	5.19615i	1.75015	−	0.336817i
$239$		−	11.3137i		−	0.731823i	−0.930650	−	0.365911i	$-0.880757\pi$
$239$		−	11.3137i		−	0.731823i	0.930650	−	0.365911i	$-0.119243\pi$
$240$		0			0
$241$		−	5.19615i		−	0.334714i	−0.985896	−	0.167357i	$-0.946477\pi$
$241$		−	5.19615i		−	0.334714i	0.985896	−	0.167357i	$-0.0535232\pi$
$242$			12.7279i			0.818182i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	27.0000			1.71797
$248$	14.6969			0.933257
$249$		0			0
$250$		0			0
$251$	−29.3939			−1.85533			−0.927663	−	0.373420i	$-0.878185\pi$
$251$	−29.3939			−1.85533			−0.927663	+	0.373420i	$0.878185\pi$
$252$		0			0
$253$	10.0000			0.628695
$254$		−	14.1421i		−	0.887357i
$255$		0			0
$256$		0			0
$257$	14.6969			0.916770			0.458385	−	0.888754i	$-0.348428\pi$
$257$	14.6969			0.916770			0.458385	+	0.888754i	$0.348428\pi$
$258$		0			0
$259$	−4.00000	−	20.7846i	−0.248548	−	1.29149i
$260$		0			0
$261$		0			0
$262$		−	20.7846i		−	1.28408i
$263$			14.1421i			0.872041i	0.899937	+	0.436021i	$0.143613\pi$
$263$			14.1421i			0.872041i	−0.899937	+	0.436021i	$0.856387\pi$
$264$		0			0
$265$		0			0
$266$	3.67423	+	19.0919i	0.225282	+	1.17060i
$267$		0			0
$268$		0			0
$269$	−7.34847			−0.448044			−0.224022	−	0.974584i	$-0.571919\pi$
$269$	−7.34847			−0.448044			−0.224022	+	0.974584i	$0.571919\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−29.3939			−1.78227
$273$		0			0
$274$	−14.0000			−0.845771
$275$		0			0
$276$		0			0
$277$	17.0000			1.02143			0.510716	−	0.859750i	$-0.329381\pi$
$277$	17.0000			1.02143			0.510716	+	0.859750i	$0.329381\pi$
$278$	29.3939			1.76293
$279$		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
$283$		−	5.19615i		−	0.308879i	−0.988002	−	0.154440i	$-0.950643\pi$
$283$		−	5.19615i		−	0.308879i	0.988002	−	0.154440i	$-0.0493572\pi$
$284$		0			0
$285$		0			0
$286$			10.3923i			0.614510i
$287$		0			0
$288$		0			0
$289$	37.0000			2.17647
$290$		0			0
$291$		0			0
$292$		0			0
$293$	22.0454			1.28791			0.643953	−	0.765065i	$-0.277293\pi$
$293$	22.0454			1.28791			0.643953	+	0.765065i	$0.277293\pi$
$294$		0			0
$295$		0			0
$296$			22.6274i			1.31519i
$297$		0			0
$298$	−26.0000			−1.50614
$299$	−36.7423			−2.12486
$300$		0			0
$301$	−2.50000	−	12.9904i	−0.144098	−	0.748753i
$302$			7.07107i			0.406894i
$303$		0			0
$304$		−	20.7846i		−	1.19208i
$305$		0			0
$306$		0			0
$307$		−	15.5885i		−	0.889680i	−0.895610	−	0.444840i	$-0.853260\pi$
$307$		−	15.5885i		−	0.889680i	0.895610	−	0.444840i	$-0.146740\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−22.0454			−1.25008			−0.625040	−	0.780593i	$-0.714917\pi$
$311$	−22.0454			−1.25008			−0.625040	+	0.780593i	$0.714917\pi$
$312$		0			0
$313$		−	5.19615i		−	0.293704i	−0.989158	−	0.146852i	$-0.953086\pi$
$313$		−	5.19615i		−	0.293704i	0.989158	−	0.146852i	$-0.0469141\pi$
$314$	−22.0454			−1.24409
$315$		0			0
$316$		0			0
$317$		−	11.3137i		−	0.635441i	−0.948184	−	0.317721i	$-0.897083\pi$
$317$		−	11.3137i		−	0.635441i	0.948184	−	0.317721i	$-0.102917\pi$
$318$		0			0
$319$	−2.00000			−0.111979
$320$		0			0
$321$		0			0
$322$	−5.00000	−	25.9808i	−0.278639	−	1.44785i
$323$			38.1838i			2.12460i
$324$		0			0
$325$		0			0
$326$		−	9.89949i		−	0.548282i
$327$		0			0
$328$		0			0
$329$	−3.67423	−	19.0919i	−0.202567	−	1.05257i
$330$		0			0
$331$	14.0000			0.769510			0.384755	−	0.923019i	$-0.374286\pi$
$331$	14.0000			0.769510			0.384755	+	0.923019i	$0.374286\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−31.0000			−1.68868			−0.844339	−	0.535810i	$-0.820006\pi$
$337$	−31.0000			−1.68868			−0.844339	+	0.535810i	$0.820006\pi$
$338$		−	19.7990i		−	1.07692i
$339$		0			0
$340$		0			0
$341$	7.34847			0.397942
$342$		0			0
$343$	10.0000	+	15.5885i	0.539949	+	0.841698i
$344$			14.1421i			0.762493i
$345$		0			0
$346$		−	31.1769i		−	1.67608i
$347$			18.3848i			0.986947i	0.869761	+	0.493473i	$0.164273\pi$
$347$			18.3848i			0.986947i	−0.869761	+	0.493473i	$0.835727\pi$
$348$		0			0
$349$			31.1769i			1.66886i	0.551112	+	0.834431i	$0.314204\pi$
$349$			31.1769i			1.66886i	−0.551112	+	0.834431i	$0.685796\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−22.0454			−1.17336			−0.586679	−	0.809819i	$-0.699565\pi$
$353$	−22.0454			−1.17336			−0.586679	+	0.809819i	$0.699565\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−26.0000			−1.37414
$359$		−	2.82843i		−	0.149279i	−0.997211	−	0.0746393i	$-0.976219\pi$
$359$		−	2.82843i		−	0.149279i	0.997211	−	0.0746393i	$-0.0237806\pi$
$360$		0			0
$361$	−8.00000			−0.421053
$362$	−7.34847			−0.386227
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	25.9808i		−	1.35618i	−0.734977	−	0.678092i	$-0.762807\pi$
$367$		−	25.9808i		−	1.35618i	0.734977	−	0.678092i	$-0.237193\pi$
$368$			28.2843i			1.47442i
$369$		0			0
$370$		0			0
$371$	14.6969	−	2.82843i	0.763027	−	0.146845i
$372$		0			0
$373$	29.0000			1.50156			0.750782	−	0.660551i	$-0.229677\pi$
$373$	29.0000			1.50156			0.750782	+	0.660551i	$0.229677\pi$
$374$	−14.6969			−0.759961
$375$		0			0
$376$			20.7846i			1.07188i
$377$	7.34847			0.378465
$378$		0			0
$379$	−19.0000			−0.975964			−0.487982	−	0.872854i	$-0.662267\pi$
$379$	−19.0000			−0.975964			−0.487982	+	0.872854i	$0.662267\pi$
$380$		0			0
$381$		0			0
$382$	−14.0000			−0.716302
$383$	29.3939			1.50196			0.750978	−	0.660327i	$-0.229582\pi$
$383$	29.3939			1.50196			0.750978	+	0.660327i	$0.229582\pi$
$384$		0			0
$385$		0			0
$386$		−	35.3553i		−	1.79954i
$387$		0			0
$388$		0			0
$389$			14.1421i			0.717035i	0.933523	+	0.358517i	$0.116718\pi$
$389$			14.1421i			0.717035i	−0.933523	+	0.358517i	$0.883282\pi$
$390$		0			0
$391$		−	51.9615i		−	2.62781i
$392$	−7.34847	−	18.3848i	−0.371154	−	0.928571i
$393$		0			0
$394$	22.0000			1.10834
$395$		0			0
$396$		0			0
$397$		−	25.9808i		−	1.30394i	−0.758246	−	0.651969i	$-0.773943\pi$
$397$		−	25.9808i		−	1.30394i	0.758246	−	0.651969i	$-0.226057\pi$
$398$	7.34847			0.368345
$399$		0			0
$400$		0			0
$401$			31.1127i			1.55369i	0.629689	+	0.776847i	$0.283182\pi$
$401$			31.1127i			1.55369i	−0.629689	+	0.776847i	$0.716818\pi$
$402$		0			0
$403$	−27.0000			−1.34497
$404$		0			0
$405$		0			0
$406$	1.00000	+	5.19615i	0.0496292	+	0.257881i
$407$			11.3137i			0.560800i
$408$		0			0
$409$		−	15.5885i		−	0.770800i	−0.922750	−	0.385400i	$-0.874064\pi$
$409$		−	15.5885i		−	0.770800i	0.922750	−	0.385400i	$-0.125936\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.67423	−	19.0919i	−0.180797	−	0.939450i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		−	10.3923i		−	0.508304i
$419$	−29.3939			−1.43598			−0.717992	−	0.696051i	$-0.754939\pi$
$419$	−29.3939			−1.43598			−0.717992	+	0.696051i	$0.754939\pi$
$420$		0			0
$421$	−4.00000			−0.194948			−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000			−0.194948			−0.0974740	+	0.995238i	$0.531076\pi$
$422$			7.07107i			0.344214i
$423$		0			0
$424$	−16.0000			−0.777029
$425$		0			0
$426$		0			0
$427$	13.5000	−	2.59808i	0.653311	−	0.125730i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	7.07107i		−	0.340601i	−0.985392	−	0.170301i	$-0.945526\pi$
$431$		−	7.07107i		−	0.340601i	0.985392	−	0.170301i	$-0.0544739\pi$
$432$		0			0
$433$			5.19615i			0.249711i	0.992175	+	0.124856i	$0.0398468\pi$
$433$			5.19615i			0.249711i	−0.992175	+	0.124856i	$0.960153\pi$
$434$	−3.67423	−	19.0919i	−0.176369	−	0.916440i
$435$		0			0
$436$		0			0
$437$	36.7423			1.75762
$438$		0			0
$439$			5.19615i			0.247999i	0.992282	+	0.123999i	$0.0395721\pi$
$439$			5.19615i			0.247999i	−0.992282	+	0.123999i	$0.960428\pi$
$440$		0			0
$441$		0			0
$442$	54.0000			2.56852
$443$		−	28.2843i		−	1.34383i	−0.740630	−	0.671913i	$-0.765473\pi$
$443$		−	28.2843i		−	1.34383i	0.740630	−	0.671913i	$-0.234527\pi$
$444$		0			0
$445$		0			0
$446$	7.34847			0.347960
$447$		0			0
$448$	4.00000	+	20.7846i	0.188982	+	0.981981i
$449$		−	28.2843i		−	1.33482i	−0.744692	−	0.667409i	$-0.767403\pi$
$449$		−	28.2843i		−	1.33482i	0.744692	−	0.667409i	$-0.232597\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		−	20.7846i		−	0.975470i
$455$		0			0
$456$		0			0
$457$	14.0000			0.654892			0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000			0.654892			0.327446	+	0.944870i	$0.393812\pi$
$458$	22.0454			1.03011
$459$		0			0
$460$		0			0
$461$	14.6969			0.684505			0.342252	−	0.939608i	$-0.388810\pi$
$461$	14.6969			0.684505			0.342252	+	0.939608i	$0.388810\pi$
$462$		0			0
$463$	14.0000			0.650635			0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000			0.650635			0.325318	+	0.945605i	$0.394529\pi$
$464$		−	5.65685i		−	0.262613i
$465$		0			0
$466$	40.0000			1.85296
$467$	−22.0454			−1.02014			−0.510070	−	0.860133i	$-0.670380\pi$
$467$	−22.0454			−1.02014			−0.510070	+	0.860133i	$0.670380\pi$
$468$		0			0
$469$	3.50000	+	18.1865i	0.161615	+	0.839776i
$470$		0			0
$471$		0			0
$472$			20.7846i			0.956689i
$473$			7.07107i			0.325128i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	16.0000			0.731823
$479$	−36.7423			−1.67880			−0.839400	−	0.543514i	$-0.817094\pi$
$479$	−36.7423			−1.67880			−0.839400	+	0.543514i	$0.817094\pi$
$480$		0			0
$481$		−	41.5692i		−	1.89539i
$482$	7.34847			0.334714
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−1.00000			−0.0453143			−0.0226572	−	0.999743i	$-0.507213\pi$
$487$	−1.00000			−0.0453143			−0.0226572	+	0.999743i	$0.507213\pi$
$488$	−14.6969			−0.665299
$489$		0			0
$490$		0			0
$491$			5.65685i			0.255290i	0.991820	+	0.127645i	$0.0407419\pi$
$491$			5.65685i			0.255290i	−0.991820	+	0.127645i	$0.959258\pi$
$492$		0			0
$493$			10.3923i			0.468046i
$494$			38.1838i			1.71797i
$495$		0			0
$496$			20.7846i			0.933257i
$497$	−7.34847	+	1.41421i	−0.329624	+	0.0634361i
$498$		0			0
$499$	5.00000			0.223831			0.111915	−	0.993718i	$-0.464301\pi$
$499$	5.00000			0.223831			0.111915	+	0.993718i	$0.464301\pi$
$500$		0			0
$501$		0			0
$502$		−	41.5692i		−	1.85533i
$503$	−14.6969			−0.655304			−0.327652	−	0.944798i	$-0.606257\pi$
$503$	−14.6969			−0.655304			−0.327652	+	0.944798i	$0.606257\pi$
$504$		0			0
$505$		0			0
$506$			14.1421i			0.628695i
$507$		0			0
$508$		0			0
$509$	22.0454			0.977146			0.488573	−	0.872523i	$-0.337518\pi$
$509$	22.0454			0.977146			0.488573	+	0.872523i	$0.337518\pi$
$510$		0			0
$511$	−27.0000	+	5.19615i	−1.19441	+	0.229864i
$512$		−	22.6274i		−	1.00000i
$513$		0			0
$514$			20.7846i			0.916770i
$515$		0			0
$516$		0			0
$517$			10.3923i			0.457053i
$518$	29.3939	−	5.65685i	1.29149	−	0.248548i
$519$		0			0
$520$		0			0
$521$	−22.0454			−0.965827			−0.482913	−	0.875668i	$-0.660421\pi$
$521$	−22.0454			−0.965827			−0.482913	+	0.875668i	$0.660421\pi$
$522$		0			0
$523$			15.5885i			0.681636i	0.940129	+	0.340818i	$0.110704\pi$
$523$			15.5885i			0.681636i	−0.940129	+	0.340818i	$0.889296\pi$
$524$		0			0
$525$		0			0
$526$	−20.0000			−0.872041
$527$		−	38.1838i		−	1.66331i
$528$		0			0
$529$	−27.0000			−1.17391
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		−	19.7990i		−	0.855186i
$537$		0			0
$538$		−	10.3923i		−	0.448044i
$539$	−3.67423	−	9.19239i	−0.158260	−	0.395944i
$540$		0			0
$541$	−1.00000			−0.0429934			−0.0214967	−	0.999769i	$-0.506843\pi$
$541$	−1.00000			−0.0429934			−0.0214967	+	0.999769i	$0.506843\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−40.0000			−1.71028			−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−40.0000			−1.71028			−0.855138	+	0.518400i	$0.826528\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7.34847			−0.313055
$552$		0			0
$553$	−7.00000	−	36.3731i	−0.297670	−	1.54674i
$554$			24.0416i			1.02143i
$555$		0			0
$556$		0			0
$557$			9.89949i			0.419455i	0.977760	+	0.209728i	$0.0672577\pi$
$557$			9.89949i			0.419455i	−0.977760	+	0.209728i	$0.932742\pi$
$558$		0			0
$559$		−	25.9808i		−	1.09887i
$560$		0			0
$561$		0			0
$562$	−2.00000			−0.0843649
$563$	−7.34847			−0.309701			−0.154851	−	0.987938i	$-0.549490\pi$
$563$	−7.34847			−0.309701			−0.154851	+	0.987938i	$0.549490\pi$
$564$		0			0
$565$		0			0
$566$	7.34847			0.308879
$567$		0			0
$568$	8.00000			0.335673
$569$			1.41421i			0.0592869i	0.999561	+	0.0296435i	$0.00943719\pi$
$569$			1.41421i			0.0592869i	−0.999561	+	0.0296435i	$0.990563\pi$
$570$		0			0
$571$	29.0000			1.21361			0.606806	−	0.794850i	$-0.292450\pi$
$571$	29.0000			1.21361			0.606806	+	0.794850i	$0.292450\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$			15.5885i			0.648956i	0.945893	+	0.324478i	$0.105189\pi$
$577$			15.5885i			0.648956i	−0.945893	+	0.324478i	$0.894811\pi$
$578$			52.3259i			2.17647i
$579$		0			0
$580$		0			0
$581$	3.67423	+	19.0919i	0.152433	+	0.792065i
$582$		0			0
$583$	−8.00000			−0.331326
$584$	29.3939			1.21633
$585$		0			0
$586$			31.1769i			1.28791i
$587$	29.3939			1.21322			0.606608	−	0.795001i	$-0.292530\pi$
$587$	29.3939			1.21322			0.606608	+	0.795001i	$0.292530\pi$
$588$		0			0
$589$	27.0000			1.11252
$590$		0			0
$591$		0			0
$592$	−32.0000			−1.31519
$593$	−14.6969			−0.603531			−0.301765	−	0.953382i	$-0.597576\pi$
$593$	−14.6969			−0.603531			−0.301765	+	0.953382i	$0.597576\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		−	51.9615i		−	2.12486i
$599$			5.65685i			0.231133i	0.993300	+	0.115566i	$0.0368683\pi$
$599$			5.65685i			0.231133i	−0.993300	+	0.115566i	$0.963132\pi$
$600$		0			0
$601$			36.3731i			1.48369i	0.670572	+	0.741844i	$0.266049\pi$
$601$			36.3731i			1.48369i	−0.670572	+	0.741844i	$0.733951\pi$
$602$	18.3712	−	3.53553i	0.748753	−	0.144098i
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$			41.5692i			1.68724i	0.536939	+	0.843621i	$0.319581\pi$
$607$			41.5692i			1.68724i	−0.536939	+	0.843621i	$0.680419\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	38.1838i		−	1.54475i
$612$		0			0
$613$	−28.0000			−1.13091			−0.565455	−	0.824779i	$-0.691299\pi$
$613$	−28.0000			−1.13091			−0.565455	+	0.824779i	$0.691299\pi$
$614$	22.0454			0.889680
$615$		0			0
$616$	2.00000	+	10.3923i	0.0805823	+	0.418718i
$617$			31.1127i			1.25255i	0.779602	+	0.626275i	$0.215421\pi$
$617$			31.1127i			1.25255i	−0.779602	+	0.626275i	$0.784579\pi$
$618$		0			0
$619$			25.9808i			1.04425i	0.852867	+	0.522127i	$0.174861\pi$
$619$			25.9808i			1.04425i	−0.852867	+	0.522127i	$0.825139\pi$
$620$		0			0
$621$		0			0
$622$		−	31.1769i		−	1.25008i
$623$	−7.34847	−	38.1838i	−0.294410	−	1.52980i
$624$		0			0
$625$		0			0
$626$	7.34847			0.293704
$627$		0			0
$628$		0			0
$629$	58.7878			2.34402
$630$		0			0
$631$	35.0000			1.39333			0.696664	−	0.717398i	$-0.254667\pi$
$631$	35.0000			1.39333			0.696664	+	0.717398i	$0.254667\pi$
$632$			39.5980i			1.57512i
$633$		0			0
$634$	16.0000			0.635441
$635$		0			0
$636$		0			0
$637$	13.5000	+	33.7750i	0.534889	+	1.33821i
$638$		−	2.82843i		−	0.111979i
$639$		0			0
$640$		0			0
$641$		−	11.3137i		−	0.446865i	−0.974719	−	0.223432i	$-0.928274\pi$
$641$		−	11.3137i		−	0.446865i	0.974719	−	0.223432i	$-0.0717262\pi$
$642$		0			0
$643$			20.7846i			0.819665i	0.912161	+	0.409832i	$0.134413\pi$
$643$			20.7846i			0.819665i	−0.912161	+	0.409832i	$0.865587\pi$
$644$		0			0
$645$		0			0
$646$	−54.0000			−2.12460
$647$	−44.0908			−1.73339			−0.866694	−	0.498839i	$-0.833760\pi$
$647$	−44.0908			−1.73339			−0.866694	+	0.498839i	$0.833760\pi$
$648$		0			0
$649$			10.3923i			0.407934i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			1.41421i			0.0553425i	0.999617	+	0.0276712i	$0.00880915\pi$
$653$			1.41421i			0.0553425i	−0.999617	+	0.0276712i	$0.991191\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	27.0000	−	5.19615i	1.05257	−	0.202567i
$659$			22.6274i			0.881439i	0.897645	+	0.440720i	$0.145277\pi$
$659$			22.6274i			0.881439i	−0.897645	+	0.440720i	$0.854723\pi$
$660$		0			0
$661$			10.3923i			0.404214i	0.979363	+	0.202107i	$0.0647788\pi$
$661$			10.3923i			0.404214i	−0.979363	+	0.202107i	$0.935221\pi$
$662$			19.7990i			0.769510i
$663$		0			0
$664$		−	20.7846i		−	0.806599i
$665$		0			0
$666$		0			0
$667$	10.0000			0.387202
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−7.34847			−0.283685
$672$		0			0
$673$	8.00000			0.308377			0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000			0.308377			0.154189	+	0.988041i	$0.450724\pi$
$674$		−	43.8406i		−	1.68868i
$675$		0			0
$676$		0			0
$677$	−7.34847			−0.282425			−0.141212	−	0.989979i	$-0.545100\pi$
$677$	−7.34847			−0.282425			−0.141212	+	0.989979i	$0.545100\pi$
$678$		0			0
$679$	13.5000	−	2.59808i	0.518082	−	0.0997050i
$680$		0			0
$681$		0			0
$682$			10.3923i			0.397942i
$683$			22.6274i			0.865814i	0.901439	+	0.432907i	$0.142512\pi$
$683$			22.6274i			0.865814i	−0.901439	+	0.432907i	$0.857488\pi$
$684$		0			0
$685$		0			0
$686$	−22.0454	+	14.1421i	−0.841698	+	0.539949i
$687$		0			0
$688$	−20.0000			−0.762493
$689$	29.3939			1.11982
$690$		0			0
$691$		−	10.3923i		−	0.395342i	−0.980268	−	0.197671i	$-0.936662\pi$
$691$		−	10.3923i		−	0.395342i	0.980268	−	0.197671i	$-0.0633378\pi$
$692$		0			0
$693$		0			0
$694$	−26.0000			−0.986947
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−44.0908			−1.66886
$699$		0			0
$700$		0			0
$701$			43.8406i			1.65584i	0.560848	+	0.827919i	$0.310475\pi$
$701$			43.8406i			1.65584i	−0.560848	+	0.827919i	$0.689525\pi$
$702$		0			0
$703$			41.5692i			1.56781i
$704$		−	11.3137i		−	0.426401i
$705$		0			0
$706$		−	31.1769i		−	1.17336i
$707$		0			0
$708$		0			0
$709$	−37.0000			−1.38956			−0.694782	−	0.719220i	$-0.744499\pi$
$709$	−37.0000			−1.38956			−0.694782	+	0.719220i	$0.744499\pi$
$710$		0			0
$711$		0			0
$712$			41.5692i			1.55787i
$713$	−36.7423			−1.37601
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	4.00000			0.149279
$719$	−22.0454			−0.822155			−0.411077	−	0.911600i	$-0.634847\pi$
$719$	−22.0454			−0.822155			−0.411077	+	0.911600i	$0.634847\pi$
$720$		0			0
$721$	−27.0000	+	5.19615i	−1.00553	+	0.193515i
$722$		−	11.3137i		−	0.421053i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			25.9808i			0.963573i	0.876289	+	0.481787i	$0.160012\pi$
$727$			25.9808i			0.963573i	−0.876289	+	0.481787i	$0.839988\pi$
$728$	−7.34847	−	38.1838i	−0.272352	−	1.41518i
$729$		0			0
$730$		0			0
$731$	36.7423			1.35896
$732$		0			0
$733$			10.3923i			0.383849i	0.981410	+	0.191924i	$0.0614728\pi$
$733$			10.3923i			0.383849i	−0.981410	+	0.191924i	$0.938527\pi$
$734$	36.7423			1.35618
$735$		0			0
$736$		0			0
$737$		−	9.89949i		−	0.364653i
$738$		0			0
$739$	−10.0000			−0.367856			−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000			−0.367856			−0.183928	+	0.982940i	$0.558881\pi$
$740$		0			0
$741$		0			0
$742$	4.00000	+	20.7846i	0.146845	+	0.763027i
$743$			14.1421i			0.518825i	0.965767	+	0.259412i	$0.0835289\pi$
$743$			14.1421i			0.518825i	−0.965767	+	0.259412i	$0.916471\pi$
$744$		0			0
$745$		0			0
$746$			41.0122i			1.50156i
$747$		0			0
$748$		0			0
$749$	14.6969	−	2.82843i	0.537014	−	0.103348i
$750$		0			0
$751$	−22.0000			−0.802791			−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000			−0.802791			−0.401396	+	0.915905i	$0.631475\pi$
$752$	−29.3939			−1.07188
$753$		0			0
$754$			10.3923i			0.378465i
$755$		0			0
$756$		0			0
$757$	−1.00000			−0.0363456			−0.0181728	−	0.999835i	$-0.505785\pi$
$757$	−1.00000			−0.0363456			−0.0181728	+	0.999835i	$0.505785\pi$
$758$		−	26.8701i		−	0.975964i
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$	0.500000	+	2.59808i	0.0181012	+	0.0940567i
$764$		0			0
$765$		0			0
$766$			41.5692i			1.50196i
$767$		−	38.1838i		−	1.37874i
$768$		0			0
$769$			36.3731i			1.31165i	0.754915	+	0.655823i	$0.227678\pi$
$769$			36.3731i			1.31165i	−0.754915	+	0.655823i	$0.772322\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−29.3939			−1.05722			−0.528612	−	0.848863i	$-0.677287\pi$
$773$	−29.3939			−1.05722			−0.528612	+	0.848863i	$0.677287\pi$
$774$		0			0
$775$		0			0
$776$	−14.6969			−0.527589
$777$		0			0
$778$	−20.0000			−0.717035
$779$		0			0
$780$		0			0
$781$	4.00000			0.143131
$782$	73.4847			2.62781
$783$		0			0
$784$	26.0000	−	10.3923i	0.928571	−	0.371154i
$785$		0			0
$786$		0			0
$787$			5.19615i			0.185223i	0.995702	+	0.0926114i	$0.0295214\pi$
$787$			5.19615i			0.185223i	−0.995702	+	0.0926114i	$0.970479\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	14.6969	−	2.82843i	0.522563	−	0.100567i
$792$		0			0
$793$	27.0000			0.958798
$794$	36.7423			1.30394
$795$		0			0
$796$		0			0
$797$	−29.3939			−1.04118			−0.520592	−	0.853805i	$-0.674289\pi$
$797$	−29.3939			−1.04118			−0.520592	+	0.853805i	$0.674289\pi$
$798$		0			0
$799$	54.0000			1.91038
$800$		0			0
$801$		0			0
$802$	−44.0000			−1.55369
$803$	14.6969			0.518644
$804$		0			0
$805$		0			0
$806$		−	38.1838i		−	1.34497i
$807$		0			0
$808$		0			0
$809$			48.0833i			1.69052i	0.534357	+	0.845259i	$0.320554\pi$
$809$			48.0833i			1.69052i	−0.534357	+	0.845259i	$0.679446\pi$
$810$		0			0
$811$			25.9808i			0.912308i	0.889901	+	0.456154i	$0.150773\pi$
$811$			25.9808i			0.912308i	−0.889901	+	0.456154i	$0.849227\pi$
$812$		0			0
$813$		0			0
$814$	−16.0000			−0.560800
$815$		0			0
$816$		0			0
$817$			25.9808i			0.908952i
$818$	22.0454			0.770800
$819$		0			0
$820$		0			0
$821$		−	15.5563i		−	0.542920i	−0.962450	−	0.271460i	$-0.912493\pi$
$821$		−	15.5563i		−	0.542920i	0.962450	−	0.271460i	$-0.0875065\pi$
$822$		0			0
$823$	11.0000			0.383436			0.191718	−	0.981450i	$-0.438594\pi$
$823$	11.0000			0.383436			0.191718	+	0.981450i	$0.438594\pi$
$824$	29.3939			1.02398
$825$		0			0
$826$	27.0000	−	5.19615i	0.939450	−	0.180797i
$827$		−	2.82843i		−	0.0983540i	−0.998790	−	0.0491770i	$-0.984340\pi$
$827$		−	2.82843i		−	0.0983540i	0.998790	−	0.0491770i	$-0.0156598\pi$
$828$		0			0
$829$			20.7846i			0.721879i	0.932589	+	0.360940i	$0.117544\pi$
$829$			20.7846i			0.721879i	−0.932589	+	0.360940i	$0.882456\pi$
$830$		0			0
$831$		0			0
$832$			41.5692i			1.44115i
$833$	−47.7650	+	19.0919i	−1.65496	+	0.661495i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		−	41.5692i		−	1.43598i
$839$	7.34847			0.253697			0.126849	−	0.991922i	$-0.459514\pi$
$839$	7.34847			0.253697			0.126849	+	0.991922i	$0.459514\pi$
$840$		0			0
$841$	27.0000			0.931034
$842$		−	5.65685i		−	0.194948i
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−4.50000	−	23.3827i	−0.154622	−	0.803439i
$848$		−	22.6274i		−	0.777029i
$849$		0			0
$850$		0			0
$851$		−	56.5685i		−	1.93914i
$852$		0			0
$853$		−	5.19615i		−	0.177913i	−0.996036	−	0.0889564i	$-0.971647\pi$
$853$		−	5.19615i		−	0.177913i	0.996036	−	0.0889564i	$-0.0283532\pi$
$854$	3.67423	+	19.0919i	0.125730	+	0.653311i
$855$		0			0
$856$	−16.0000			−0.546869
$857$	44.0908			1.50611			0.753057	−	0.657956i	$-0.228579\pi$
$857$	44.0908			1.50611			0.753057	+	0.657956i	$0.228579\pi$
$858$		0			0
$859$		−	31.1769i		−	1.06374i	−0.846825	−	0.531871i	$-0.821489\pi$
$859$		−	31.1769i		−	1.06374i	0.846825	−	0.531871i	$-0.178511\pi$
$860$		0			0
$861$		0			0
$862$	10.0000			0.340601
$863$		−	28.2843i		−	0.962808i	−0.876499	−	0.481404i	$-0.840127\pi$
$863$		−	28.2843i		−	0.962808i	0.876499	−	0.481404i	$-0.159873\pi$
$864$		0			0
$865$		0			0
$866$	−7.34847			−0.249711
$867$		0			0
$868$		0			0
$869$			19.7990i			0.671635i
$870$		0			0
$871$			36.3731i			1.23245i
$872$		−	2.82843i		−	0.0957826i
$873$		0			0
$874$			51.9615i			1.75762i
$875$		0			0
$876$		0			0
$877$	−31.0000			−1.04680			−0.523398	−	0.852088i	$-0.675336\pi$
$877$	−31.0000			−1.04680			−0.523398	+	0.852088i	$0.675336\pi$
$878$	−7.34847			−0.247999
$879$		0			0
$880$		0			0
$881$	44.0908			1.48546			0.742729	−	0.669593i	$-0.233531\pi$
$881$	44.0908			1.48546			0.742729	+	0.669593i	$0.233531\pi$
$882$		0			0
$883$	−37.0000			−1.24515			−0.622575	−	0.782560i	$-0.713913\pi$
$883$	−37.0000			−1.24515			−0.622575	+	0.782560i	$0.713913\pi$
$884$		0			0
$885$		0			0
$886$	40.0000			1.34383
$887$	−7.34847			−0.246737			−0.123369	−	0.992361i	$-0.539370\pi$
$887$	−7.34847			−0.246737			−0.123369	+	0.992361i	$0.539370\pi$
$888$		0			0
$889$	5.00000	+	25.9808i	0.167695	+	0.871367i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			38.1838i			1.27777i
$894$		0			0
$895$		0			0
$896$	−29.3939	+	5.65685i	−0.981981	+	0.188982i
$897$		0			0
$898$	40.0000			1.33482
$899$	7.34847			0.245085
$900$		0			0
$901$			41.5692i			1.38487i
$902$		0			0
$903$		0			0
$904$	−16.0000			−0.532152
$905$		0			0
$906$		0			0
$907$	26.0000			0.863316			0.431658	−	0.902037i	$-0.357929\pi$
$907$	26.0000			0.863316			0.431658	+	0.902037i	$0.357929\pi$
$908$		0			0
$909$		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
$913$		−	10.3923i		−	0.343935i
$914$			19.7990i			0.654892i
$915$		0			0
$916$		0			0
$917$	7.34847	+	38.1838i	0.242668	+	1.26094i
$918$		0			0
$919$	−55.0000			−1.81428			−0.907141	−	0.420826i	$-0.861740\pi$
$919$	−55.0000			−1.81428			−0.907141	+	0.420826i	$0.861740\pi$
$920$		0			0
$921$		0			0
$922$			20.7846i			0.684505i
$923$	−14.6969			−0.483756
$924$		0			0
$925$		0			0
$926$			19.7990i			0.650635i
$927$		0			0
$928$		0			0
$929$	36.7423			1.20548			0.602739	−	0.797939i	$-0.294076\pi$
$929$	36.7423			1.20548			0.602739	+	0.797939i	$0.294076\pi$
$930$		0			0
$931$	−13.5000	−	33.7750i	−0.442445	−	1.10693i
$932$		0			0
$933$		0			0
$934$		−	31.1769i		−	1.02014i
$935$		0			0
$936$		0			0
$937$			25.9808i			0.848755i	0.905485	+	0.424377i	$0.139507\pi$
$937$			25.9808i			0.848755i	−0.905485	+	0.424377i	$0.860493\pi$
$938$	−25.7196	+	4.94975i	−0.839776	+	0.161615i
$939$		0			0
$940$		0			0
$941$	51.4393			1.67687			0.838436	−	0.544999i	$-0.183470\pi$
$941$	51.4393			1.67687			0.838436	+	0.544999i	$0.183470\pi$
$942$		0			0
$943$		0			0
$944$	−29.3939			−0.956689
$945$		0			0
$946$	−10.0000			−0.325128
$947$		−	41.0122i		−	1.33272i	−0.745631	−	0.666359i	$-0.767852\pi$
$947$		−	41.0122i		−	1.33272i	0.745631	−	0.666359i	$-0.232148\pi$
$948$		0			0
$949$	−54.0000			−1.75291
$950$		0			0
$951$		0			0
$952$	54.0000	−	10.3923i	1.75015	−	0.336817i
$953$		−	2.82843i		−	0.0916217i	−0.998950	−	0.0458109i	$-0.985413\pi$
$953$		−	2.82843i		−	0.0916217i	0.998950	−	0.0458109i	$-0.0145872\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		−	51.9615i		−	1.67880i
$959$	25.7196	−	4.94975i	0.830531	−	0.159836i
$960$		0			0
$961$	4.00000			0.129032
$962$	58.7878			1.89539
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	14.0000			0.450210			0.225105	−	0.974335i	$-0.427728\pi$
$967$	14.0000			0.450210			0.225105	+	0.974335i	$0.427728\pi$
$968$			25.4558i			0.818182i
$969$		0			0
$970$		0			0
$971$	−7.34847			−0.235824			−0.117912	−	0.993024i	$-0.537620\pi$
$971$	−7.34847			−0.235824			−0.117912	+	0.993024i	$0.537620\pi$
$972$		0			0
$973$	−54.0000	+	10.3923i	−1.73116	+	0.333162i
$974$		−	1.41421i		−	0.0453143i
$975$		0			0
$976$		−	20.7846i		−	0.665299i
$977$			43.8406i			1.40259i	0.712873	+	0.701293i	$0.247393\pi$
$977$			43.8406i			1.40259i	−0.712873	+	0.701293i	$0.752607\pi$
$978$		0			0
$979$			20.7846i			0.664279i
$980$		0			0
$981$		0			0
$982$	−8.00000			−0.255290
$983$	58.7878			1.87504			0.937519	−	0.347934i	$-0.113117\pi$
$983$	58.7878			1.87504			0.937519	+	0.347934i	$0.113117\pi$
$984$		0			0
$985$		0			0
$986$	−14.6969			−0.468046
$987$		0			0
$988$		0			0
$989$		−	35.3553i		−	1.12423i
$990$		0			0
$991$	−37.0000			−1.17534			−0.587672	−	0.809099i	$-0.699955\pi$
$991$	−37.0000			−1.17534			−0.587672	+	0.809099i	$0.699955\pi$
$992$		0			0
$993$		0			0
$994$	−2.00000	−	10.3923i	−0.0634361	−	0.329624i
$995$		0			0
$996$		0			0
$997$		−	10.3923i		−	0.329128i	−0.986366	−	0.164564i	$-0.947378\pi$
$997$		−	10.3923i		−	0.329128i	0.986366	−	0.164564i	$-0.0526216\pi$
$998$			7.07107i			0.223831i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.b.d.251.3	yes	4
3.2	odd	2	inner	1575.2.b.d.251.1	✓	4
5.2	odd	4		1575.2.g.b.1574.3		8
5.3	odd	4		1575.2.g.b.1574.6		8
5.4	even	2		1575.2.b.e.251.2	yes	4
7.6	odd	2	inner	1575.2.b.d.251.4	yes	4
15.2	even	4		1575.2.g.b.1574.7		8
15.8	even	4		1575.2.g.b.1574.2		8
15.14	odd	2		1575.2.b.e.251.4	yes	4
21.20	even	2	inner	1575.2.b.d.251.2	yes	4
35.13	even	4		1575.2.g.b.1574.8		8
35.27	even	4		1575.2.g.b.1574.1		8
35.34	odd	2		1575.2.b.e.251.1	yes	4
105.62	odd	4		1575.2.g.b.1574.5		8
105.83	odd	4		1575.2.g.b.1574.4		8
105.104	even	2		1575.2.b.e.251.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1575.2.b.d.251.1	✓	4	3.2	odd	2	inner
1575.2.b.d.251.2	yes	4	21.20	even	2	inner
1575.2.b.d.251.3	yes	4	1.1	even	1	trivial
1575.2.b.d.251.4	yes	4	7.6	odd	2	inner
1575.2.b.e.251.1	yes	4	35.34	odd	2
1575.2.b.e.251.2	yes	4	5.4	even	2
1575.2.b.e.251.3	yes	4	105.104	even	2
1575.2.b.e.251.4	yes	4	15.14	odd	2
1575.2.g.b.1574.1		8	35.27	even	4
1575.2.g.b.1574.2		8	15.8	even	4
1575.2.g.b.1574.3		8	5.2	odd	4
1575.2.g.b.1574.4		8	105.83	odd	4
1575.2.g.b.1574.5		8	105.62	odd	4
1575.2.g.b.1574.6		8	5.3	odd	4
1575.2.g.b.1574.7		8	15.2	even	4
1575.2.g.b.1574.8		8	35.13	even	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)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(-0.500000 - 2.59808i) q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} +(-0.500000 - 2.59808i) q^{7} +2.82843i q^{8} +1.41421i q^{11} -5.19615i q^{13} +(3.67423 - 0.707107i) q^{14} -4.00000 q^{16} +7.34847 q^{17} +5.19615i q^{19} -2.00000 q^{22} -7.07107i q^{23} +7.34847 q^{26} +1.41421i q^{29} -5.19615i q^{31} +10.3923i q^{34} +8.00000 q^{37} -7.34847 q^{38} +5.00000 q^{43} +10.0000 q^{46} +7.34847 q^{47} +(-6.50000 + 2.59808i) q^{49} +5.65685i q^{53} +(7.34847 - 1.41421i) q^{56} -2.00000 q^{58} +7.34847 q^{59} +5.19615i q^{61} +7.34847 q^{62} -8.00000 q^{64} -7.00000 q^{67} -2.82843i q^{71} -10.3923i q^{73} +11.3137i q^{74} +(3.67423 - 0.707107i) q^{77} +14.0000 q^{79} -7.34847 q^{83} +7.07107i q^{86} -4.00000 q^{88} +14.6969 q^{89} +(-13.5000 + 2.59808i) q^{91} +10.3923i q^{94} +5.19615i q^{97} +(-3.67423 - 9.19239i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{7}+O(q^{10}) \) 4 * q - 2 * q^7	\( 4 q - 2 q^{7} - 16 q^{16} - 8 q^{22} + 32 q^{37} + 20 q^{43} + 40 q^{46} - 26 q^{49} - 8 q^{58} - 32 q^{64} - 28 q^{67} + 56 q^{79} - 16 q^{88} - 54 q^{91}+O(q^{100}) \) 4 * q - 2 * q^7 - 16 * q^16 - 8 * q^22 + 32 * q^37 + 20 * q^43 + 40 * q^46 - 26 * q^49 - 8 * q^58 - 32 * q^64 - 28 * q^67 + 56 * q^79 - 16 * q^88 - 54 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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