LMFDB - Embedded newform 1575.2.g.b.1574.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1574,1575)]

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, 1, 1]))

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

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

chi := DirichletCharacter("1575.1574");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.g (of order $2$, degree $1$, not 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:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1574.2
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1575.1574
Dual form			1575.2.g.b.1574.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +(-2.59808 + 0.500000i) q^{7} +2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +(-2.59808 + 0.500000i) q^{7} +2.82843 q^{8} -1.41421i q^{11} +5.19615 q^{13} +(3.67423 - 0.707107i) q^{14} -4.00000 q^{16} +7.34847i q^{17} -5.19615i q^{19} +2.00000i q^{22} -7.07107 q^{23} -7.34847 q^{26} +1.41421i q^{29} -5.19615i q^{31} -10.3923i q^{34} -8.00000i q^{37} +7.34847i q^{38} +5.00000i q^{43} +10.0000 q^{46} +7.34847i q^{47} +(6.50000 - 2.59808i) q^{49} +5.65685 q^{53} +(-7.34847 + 1.41421i) q^{56} -2.00000i q^{58} +7.34847 q^{59} +5.19615i q^{61} +7.34847i q^{62} +8.00000 q^{64} +7.00000i q^{67} +2.82843i q^{71} +10.3923 q^{73} +11.3137i q^{74} +(0.707107 + 3.67423i) q^{77} -14.0000 q^{79} +7.34847i q^{83} -7.07107i q^{86} -4.00000i q^{88} +14.6969 q^{89} +(-13.5000 + 2.59808i) q^{91} -10.3923i q^{94} +5.19615 q^{97} +(-9.19239 + 3.67423i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 32 q^{16} + 80 q^{46} + 52 q^{49} + 64 q^{64} - 112 q^{79} - 108 q^{91}+O(q^{100}) $ 8 * q - 32 * q^16 + 80 * q^46 + 52 * q^49 + 64 * q^64 - 112 * q^79 - 108 * 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 $

$2$	−1.41421			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.41421			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.59808	+	0.500000i	−0.981981	+	0.188982i
$7$	−2.59808	+	0.500000i	−0.981981	+	0.188982i
$8$	2.82843			1.00000
$9$		0			0
$10$		0			0
$11$		−	1.41421i		−	0.426401i	−0.977008	−	0.213201i	$-0.931611\pi$
$11$		−	1.41421i		−	0.426401i	0.977008	−	0.213201i	$-0.0683888\pi$
$12$		0			0
$13$	5.19615			1.44115			0.720577	−	0.693375i	$-0.243877\pi$
$13$	5.19615			1.44115			0.720577	+	0.693375i	$0.243877\pi$
$14$	3.67423	−	0.707107i	0.981981	−	0.188982i
$15$		0			0
$16$	−4.00000			−1.00000
$17$			7.34847i			1.78227i	0.453743	+	0.891133i	$0.350089\pi$
$17$			7.34847i			1.78227i	−0.453743	+	0.891133i	$0.649911\pi$
$18$		0			0
$19$		−	5.19615i		−	1.19208i	−0.802955	−	0.596040i	$-0.796740\pi$
$19$		−	5.19615i		−	1.19208i	0.802955	−	0.596040i	$-0.203260\pi$
$20$		0			0
$21$		0			0
$22$			2.00000i			0.426401i
$23$	−7.07107			−1.47442			−0.737210	−	0.675664i	$-0.763857\pi$
$23$	−7.07107			−1.47442			−0.737210	+	0.675664i	$0.763857\pi$
$24$		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
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$			7.34847i			1.19208i
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$			5.00000i			0.762493i	0.924473	+	0.381246i	$0.124505\pi$
$43$			5.00000i			0.762493i	−0.924473	+	0.381246i	$0.875495\pi$
$44$		0			0
$45$		0			0
$46$	10.0000			1.47442
$47$			7.34847i			1.07188i	0.844255	+	0.535942i	$0.180044\pi$
$47$			7.34847i			1.07188i	−0.844255	+	0.535942i	$0.819956\pi$
$48$		0			0
$49$	6.50000	−	2.59808i	0.928571	−	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.65685			0.777029			0.388514	−	0.921443i	$-0.372988\pi$
$53$	5.65685			0.777029			0.388514	+	0.921443i	$0.372988\pi$
$54$		0			0
$55$		0			0
$56$	−7.34847	+	1.41421i	−0.981981	+	0.188982i
$57$		0			0
$58$		−	2.00000i		−	0.262613i
$59$	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	$0.892057\pi$
$62$			7.34847i			0.933257i
$63$		0			0
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$			7.00000i			0.855186i	0.903971	+	0.427593i	$0.140638\pi$
$67$			7.00000i			0.855186i	−0.903971	+	0.427593i	$0.859362\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2.82843i			0.335673i	0.985815	+	0.167836i	$0.0536780\pi$
$71$			2.82843i			0.335673i	−0.985815	+	0.167836i	$0.946322\pi$
$72$		0			0
$73$	10.3923			1.21633			0.608164	−	0.793812i	$-0.291906\pi$
$73$	10.3923			1.21633			0.608164	+	0.793812i	$0.291906\pi$
$74$			11.3137i			1.31519i
$75$		0			0
$76$		0			0
$77$	0.707107	+	3.67423i	0.0805823	+	0.418718i
$78$		0			0
$79$	−14.0000			−1.57512			−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000			−1.57512			−0.787562	+	0.616236i	$0.788657\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			7.34847i			0.806599i	0.915068	+	0.403300i	$0.132137\pi$
$83$			7.34847i			0.806599i	−0.915068	+	0.403300i	$0.867863\pi$
$84$		0			0
$85$		0			0
$86$		−	7.07107i		−	0.762493i
$87$		0			0
$88$		−	4.00000i		−	0.426401i
$89$	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
$97$	5.19615			0.527589			0.263795	−	0.964579i	$-0.415026\pi$
$97$	5.19615			0.527589			0.263795	+	0.964579i	$0.415026\pi$
$98$	−9.19239	+	3.67423i	−0.928571	+	0.371154i
$99$		0			0
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	10.3923			1.02398			0.511992	−	0.858990i	$-0.328908\pi$
$103$	10.3923			1.02398			0.511992	+	0.858990i	$0.328908\pi$
$104$	14.6969			1.44115
$105$		0			0
$106$	−8.00000			−0.777029
$107$	−5.65685			−0.546869			−0.273434	−	0.961891i	$-0.588160\pi$
$107$	−5.65685			−0.546869			−0.273434	+	0.961891i	$0.588160\pi$
$108$		0			0
$109$	1.00000			0.0957826			0.0478913	−	0.998853i	$-0.484750\pi$
$109$	1.00000			0.0957826			0.0478913	+	0.998853i	$0.484750\pi$
$110$		0			0
$111$		0			0
$112$	10.3923	−	2.00000i	0.981981	−	0.188982i
$113$	5.65685			0.532152			0.266076	−	0.963952i	$-0.414273\pi$
$113$	5.65685			0.532152			0.266076	+	0.963952i	$0.414273\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−10.3923			−0.956689
$119$	−3.67423	−	19.0919i	−0.336817	−	1.75015i
$120$		0			0
$121$	9.00000			0.818182
$122$		−	7.34847i		−	0.665299i
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$			10.0000i			0.887357i	0.896186	+	0.443678i	$0.146327\pi$
$127$			10.0000i			0.887357i	−0.896186	+	0.443678i	$0.853673\pi$
$128$	−11.3137			−1.00000
$129$		0			0
$130$		0			0
$131$	14.6969			1.28408			0.642039	−	0.766672i	$-0.278089\pi$
$131$	14.6969			1.28408			0.642039	+	0.766672i	$0.278089\pi$
$132$		0			0
$133$	2.59808	+	13.5000i	0.225282	+	1.17060i
$134$		−	9.89949i		−	0.855186i
$135$		0			0
$136$			20.7846i			1.78227i
$137$	−9.89949			−0.845771			−0.422885	−	0.906183i	$-0.638983\pi$
$137$	−9.89949			−0.845771			−0.422885	+	0.906183i	$0.638983\pi$
$138$		0			0
$139$			20.7846i			1.76293i	0.472252	+	0.881464i	$0.343441\pi$
$139$			20.7846i			1.76293i	−0.472252	+	0.881464i	$0.656559\pi$
$140$		0			0
$141$		0			0
$142$		−	4.00000i		−	0.335673i
$143$		−	7.34847i		−	0.614510i
$144$		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	$0.434786\pi$
$152$		−	14.6969i		−	1.19208i
$153$		0			0
$154$	−1.00000	−	5.19615i	−0.0805823	−	0.418718i
$155$		0			0
$156$		0			0
$157$	15.5885			1.24409			0.622047	−	0.782980i	$-0.286301\pi$
$157$	15.5885			1.24409			0.622047	+	0.782980i	$0.286301\pi$
$158$	19.7990			1.57512
$159$		0			0
$160$		0			0
$161$	18.3712	−	3.53553i	1.44785	−	0.278639i
$162$		0			0
$163$		−	7.00000i		−	0.548282i	−0.961689	−	0.274141i	$-0.911606\pi$
$163$		−	7.00000i		−	0.548282i	0.961689	−	0.274141i	$-0.0883936\pi$
$164$		0			0
$165$		0			0
$166$		−	10.3923i		−	0.806599i
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	14.0000			1.07692
$170$		0			0
$171$		0			0
$172$		0			0
$173$			22.0454i			1.67608i	0.545608	+	0.838041i	$0.316299\pi$
$173$			22.0454i			1.67608i	−0.545608	+	0.838041i	$0.683701\pi$
$174$		0			0
$175$		0			0
$176$			5.65685i			0.426401i
$177$		0			0
$178$	−20.7846			−1.55787
$179$			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	$0.938141\pi$
$182$	19.0919	−	3.67423i	1.41518	−	0.272352i
$183$		0			0
$184$	−20.0000			−1.47442
$185$		0			0
$186$		0			0
$187$	10.3923			0.759961
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	9.89949i		−	0.716302i	−0.933664	−	0.358151i	$-0.883407\pi$
$191$		−	9.89949i		−	0.716302i	0.933664	−	0.358151i	$-0.116593\pi$
$192$		0			0
$193$		−	25.0000i		−	1.79954i	−0.436365	−	0.899770i	$-0.643734\pi$
$193$		−	25.0000i		−	1.79954i	0.436365	−	0.899770i	$-0.356266\pi$
$194$	−7.34847			−0.527589
$195$		0			0
$196$		0			0
$197$	15.5563			1.10834			0.554172	−	0.832402i	$-0.313035\pi$
$197$	15.5563			1.10834			0.554172	+	0.832402i	$0.313035\pi$
$198$		0			0
$199$			5.19615i			0.368345i	0.982894	+	0.184173i	$0.0589606\pi$
$199$			5.19615i			0.368345i	−0.982894	+	0.184173i	$0.941039\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.707107	−	3.67423i	−0.0496292	−	0.257881i
$204$		0			0
$205$		0			0
$206$	−14.6969			−1.02398
$207$		0			0
$208$	−20.7846			−1.44115
$209$	−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
$217$	2.59808	+	13.5000i	0.176369	+	0.916440i
$218$	−1.41421			−0.0957826
$219$		0			0
$220$		0			0
$221$			38.1838i			2.56852i
$222$		0			0
$223$	5.19615			0.347960			0.173980	−	0.984749i	$-0.444337\pi$
$223$	5.19615			0.347960			0.173980	+	0.984749i	$0.444337\pi$
$224$		0			0
$225$		0			0
$226$	−8.00000			−0.532152
$227$		−	14.6969i		−	0.975470i	−0.872992	−	0.487735i	$-0.837823\pi$
$227$		−	14.6969i		−	0.975470i	0.872992	−	0.487735i	$-0.162177\pi$
$228$		0			0
$229$			15.5885i			1.03011i	0.857156	+	0.515057i	$0.172229\pi$
$229$			15.5885i			1.03011i	−0.857156	+	0.515057i	$0.827771\pi$
$230$		0			0
$231$		0			0
$232$			4.00000i			0.262613i
$233$	−28.2843			−1.85296			−0.926482	−	0.376339i	$-0.877183\pi$
$233$	−28.2843			−1.85296			−0.926482	+	0.376339i	$0.877183\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$	5.19615	+	27.0000i	0.336817	+	1.75015i
$239$		−	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	$-0.0535232\pi$
$242$	−12.7279			−0.818182
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	27.0000i		−	1.71797i
$248$		−	14.6969i		−	0.933257i
$249$		0			0
$250$		0			0
$251$	29.3939			1.85533			0.927663	−	0.373420i	$-0.121815\pi$
$251$	29.3939			1.85533			0.927663	+	0.373420i	$0.121815\pi$
$252$		0			0
$253$			10.0000i			0.628695i
$254$		−	14.1421i		−	0.887357i
$255$		0			0
$256$		0			0
$257$			14.6969i			0.916770i	0.888754	+	0.458385i	$0.151572\pi$
$257$			14.6969i			0.916770i	−0.888754	+	0.458385i	$0.848428\pi$
$258$		0			0
$259$	4.00000	+	20.7846i	0.248548	+	1.29149i
$260$		0			0
$261$		0			0
$262$	−20.7846			−1.28408
$263$	14.1421			0.872041			0.436021	−	0.899937i	$-0.356387\pi$
$263$	14.1421			0.872041			0.436021	+	0.899937i	$0.356387\pi$
$264$		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			$\pi$
$272$		−	29.3939i		−	1.78227i
$273$		0			0
$274$	14.0000			0.845771
$275$		0			0
$276$		0			0
$277$		−	17.0000i		−	1.02143i	−0.859750	−	0.510716i	$-0.829381\pi$
$277$		−	17.0000i		−	1.02143i	0.859750	−	0.510716i	$-0.170619\pi$
$278$		−	29.3939i		−	1.76293i
$279$		0			0
$280$		0			0
$281$		−	1.41421i		−	0.0843649i	−0.999110	−	0.0421825i	$-0.986569\pi$
$281$		−	1.41421i		−	0.0843649i	0.999110	−	0.0421825i	$-0.0134311\pi$
$282$		0			0
$283$	5.19615			0.308879			0.154440	−	0.988002i	$-0.450643\pi$
$283$	5.19615			0.308879			0.154440	+	0.988002i	$0.450643\pi$
$284$		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
$293$		−	22.0454i		−	1.28791i	−0.765065	−	0.643953i	$-0.777293\pi$
$293$		−	22.0454i		−	1.28791i	0.765065	−	0.643953i	$-0.222707\pi$
$294$		0			0
$295$		0			0
$296$		−	22.6274i		−	1.31519i
$297$		0			0
$298$		−	26.0000i		−	1.50614i
$299$	−36.7423			−2.12486
$300$		0			0
$301$	−2.50000	−	12.9904i	−0.144098	−	0.748753i
$302$	−7.07107			−0.406894
$303$		0			0
$304$			20.7846i			1.19208i
$305$		0			0
$306$		0			0
$307$	−15.5885			−0.889680			−0.444840	−	0.895610i	$-0.646740\pi$
$307$	−15.5885			−0.889680			−0.444840	+	0.895610i	$0.646740\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	22.0454			1.25008			0.625040	−	0.780593i	$-0.285083\pi$
$311$	22.0454			1.25008			0.625040	+	0.780593i	$0.285083\pi$
$312$		0			0
$313$	5.19615			0.293704			0.146852	−	0.989158i	$-0.453086\pi$
$313$	5.19615			0.293704			0.146852	+	0.989158i	$0.453086\pi$
$314$	−22.0454			−1.24409
$315$		0			0
$316$		0			0
$317$	11.3137			0.635441			0.317721	−	0.948184i	$-0.397083\pi$
$317$	11.3137			0.635441			0.317721	+	0.948184i	$0.397083\pi$
$318$		0			0
$319$	2.00000			0.111979
$320$		0			0
$321$		0			0
$322$	−25.9808	+	5.00000i	−1.44785	+	0.278639i
$323$	38.1838			2.12460
$324$		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
$337$			31.0000i			1.68868i	0.535810	+	0.844339i	$0.320006\pi$
$337$			31.0000i			1.68868i	−0.535810	+	0.844339i	$0.679994\pi$
$338$	−19.7990			−1.07692
$339$		0			0
$340$		0			0
$341$	−7.34847			−0.397942
$342$		0			0
$343$	−15.5885	+	10.0000i	−0.841698	+	0.539949i
$344$			14.1421i			0.762493i
$345$		0			0
$346$		−	31.1769i		−	1.67608i
$347$	−18.3848			−0.986947			−0.493473	−	0.869761i	$-0.664273\pi$
$347$	−18.3848			−0.986947			−0.493473	+	0.869761i	$0.664273\pi$
$348$		0			0
$349$		−	31.1769i		−	1.66886i	−0.551112	−	0.834431i	$-0.685796\pi$
$349$		−	31.1769i		−	1.66886i	0.551112	−	0.834431i	$-0.314204\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			22.0454i			1.17336i	0.809819	+	0.586679i	$0.199565\pi$
$353$			22.0454i			1.17336i	−0.809819	+	0.586679i	$0.800435\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		−	26.0000i		−	1.37414i
$359$		−	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
$362$		−	7.34847i		−	0.386227i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−25.9808			−1.35618			−0.678092	−	0.734977i	$-0.737193\pi$
$367$	−25.9808			−1.35618			−0.678092	+	0.734977i	$0.737193\pi$
$368$	28.2843			1.47442
$369$		0			0
$370$		0			0
$371$	−14.6969	+	2.82843i	−0.763027	+	0.146845i
$372$		0			0
$373$			29.0000i			1.50156i	0.660551	+	0.750782i	$0.270323\pi$
$373$			29.0000i			1.50156i	−0.660551	+	0.750782i	$0.729677\pi$
$374$	−14.6969			−0.759961
$375$		0			0
$376$			20.7846i			1.07188i
$377$			7.34847i			0.378465i
$378$		0			0
$379$	19.0000			0.975964			0.487982	−	0.872854i	$-0.337733\pi$
$379$	19.0000			0.975964			0.487982	+	0.872854i	$0.337733\pi$
$380$		0			0
$381$		0			0
$382$			14.0000i			0.716302i
$383$		−	29.3939i		−	1.50196i	−0.660327	−	0.750978i	$-0.729582\pi$
$383$		−	29.3939i		−	1.50196i	0.660327	−	0.750978i	$-0.270418\pi$
$384$		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
$392$	18.3848	−	7.34847i	0.928571	−	0.371154i
$393$		0			0
$394$	−22.0000			−1.10834
$395$		0			0
$396$		0			0
$397$	−25.9808			−1.30394			−0.651969	−	0.758246i	$-0.726057\pi$
$397$	−25.9808			−1.30394			−0.651969	+	0.758246i	$0.726057\pi$
$398$		−	7.34847i		−	0.368345i
$399$		0			0
$400$		0			0
$401$		−	31.1127i		−	1.55369i	−0.629689	−	0.776847i	$-0.716818\pi$
$401$		−	31.1127i		−	1.55369i	0.629689	−	0.776847i	$-0.283182\pi$
$402$		0			0
$403$		−	27.0000i		−	1.34497i
$404$		0			0
$405$		0			0
$406$	1.00000	+	5.19615i	0.0496292	+	0.257881i
$407$	−11.3137			−0.560800
$408$		0			0
$409$			15.5885i			0.770800i	0.922750	+	0.385400i	$0.125936\pi$
$409$			15.5885i			0.770800i	−0.922750	+	0.385400i	$0.874064\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−19.0919	+	3.67423i	−0.939450	+	0.180797i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	10.3923			0.508304
$419$	−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	$0.531076\pi$
$422$	−7.07107			−0.344214
$423$		0			0
$424$	16.0000			0.777029
$425$		0			0
$426$		0			0
$427$	−2.59808	−	13.5000i	−0.125730	−	0.653311i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			7.07107i			0.340601i	0.985392	+	0.170301i	$0.0544739\pi$
$431$			7.07107i			0.340601i	−0.985392	+	0.170301i	$0.945526\pi$
$432$		0			0
$433$	−5.19615			−0.249711			−0.124856	−	0.992175i	$-0.539847\pi$
$433$	−5.19615			−0.249711			−0.124856	+	0.992175i	$0.539847\pi$
$434$	−3.67423	−	19.0919i	−0.176369	−	0.916440i
$435$		0			0
$436$		0			0
$437$			36.7423i			1.75762i
$438$		0			0
$439$		−	5.19615i		−	0.247999i	−0.992282	−	0.123999i	$-0.960428\pi$
$439$		−	5.19615i		−	0.247999i	0.992282	−	0.123999i	$-0.0395721\pi$
$440$		0			0
$441$		0			0
$442$		−	54.0000i		−	2.56852i
$443$	−28.2843			−1.34383			−0.671913	−	0.740630i	$-0.734527\pi$
$443$	−28.2843			−1.34383			−0.671913	+	0.740630i	$0.734527\pi$
$444$		0			0
$445$		0			0
$446$	−7.34847			−0.347960
$447$		0			0
$448$	−20.7846	+	4.00000i	−0.981981	+	0.188982i
$449$		−	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
$457$		−	14.0000i		−	0.654892i	−0.944870	−	0.327446i	$-0.893812\pi$
$457$		−	14.0000i		−	0.654892i	0.944870	−	0.327446i	$-0.106188\pi$
$458$		−	22.0454i		−	1.03011i
$459$		0			0
$460$		0			0
$461$	−14.6969			−0.684505			−0.342252	−	0.939608i	$-0.611190\pi$
$461$	−14.6969			−0.684505			−0.342252	+	0.939608i	$0.611190\pi$
$462$		0			0
$463$			14.0000i			0.650635i	0.945605	+	0.325318i	$0.105471\pi$
$463$			14.0000i			0.650635i	−0.945605	+	0.325318i	$0.894529\pi$
$464$		−	5.65685i		−	0.262613i
$465$		0			0
$466$	40.0000			1.85296
$467$		−	22.0454i		−	1.02014i	−0.860133	−	0.510070i	$-0.829620\pi$
$467$		−	22.0454i		−	1.02014i	0.860133	−	0.510070i	$-0.170380\pi$
$468$		0			0
$469$	−3.50000	−	18.1865i	−0.161615	−	0.839776i
$470$		0			0
$471$		0			0
$472$	20.7846			0.956689
$473$	7.07107			0.325128
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$			16.0000i			0.731823i
$479$	−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
$482$			7.34847i			0.334714i
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$			1.00000i			0.0453143i	0.999743	+	0.0226572i	$0.00721262\pi$
$487$			1.00000i			0.0453143i	−0.999743	+	0.0226572i	$0.992787\pi$
$488$			14.6969i			0.665299i
$489$		0			0
$490$		0			0
$491$		−	5.65685i		−	0.255290i	−0.991820	−	0.127645i	$-0.959258\pi$
$491$		−	5.65685i		−	0.255290i	0.991820	−	0.127645i	$-0.0407419\pi$
$492$		0			0
$493$	−10.3923			−0.468046
$494$			38.1838i			1.71797i
$495$		0			0
$496$			20.7846i			0.933257i
$497$	−1.41421	−	7.34847i	−0.0634361	−	0.329624i
$498$		0			0
$499$	−5.00000			−0.223831			−0.111915	−	0.993718i	$-0.535699\pi$
$499$	−5.00000			−0.223831			−0.111915	+	0.993718i	$0.535699\pi$
$500$		0			0
$501$		0			0
$502$	−41.5692			−1.85533
$503$			14.6969i			0.655304i	0.944798	+	0.327652i	$0.106257\pi$
$503$			14.6969i			0.655304i	−0.944798	+	0.327652i	$0.893743\pi$
$504$		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
$512$	22.6274			1.00000
$513$		0			0
$514$		−	20.7846i		−	0.916770i
$515$		0			0
$516$		0			0
$517$	10.3923			0.457053
$518$	−5.65685	−	29.3939i	−0.248548	−	1.29149i
$519$		0			0
$520$		0			0
$521$	22.0454			0.965827			0.482913	−	0.875668i	$-0.339579\pi$
$521$	22.0454			0.965827			0.482913	+	0.875668i	$0.339579\pi$
$522$		0			0
$523$	−15.5885			−0.681636			−0.340818	−	0.940129i	$-0.610704\pi$
$523$	−15.5885			−0.681636			−0.340818	+	0.940129i	$0.610704\pi$
$524$		0			0
$525$		0			0
$526$	−20.0000			−0.872041
$527$	38.1838			1.66331
$528$		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
$538$	10.3923			0.448044
$539$	−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
$547$			40.0000i			1.71028i	0.518400	+	0.855138i	$0.326528\pi$
$547$			40.0000i			1.71028i	−0.518400	+	0.855138i	$0.673472\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	7.34847			0.313055
$552$		0			0
$553$	36.3731	−	7.00000i	1.54674	−	0.297670i
$554$			24.0416i			1.02143i
$555$		0			0
$556$		0			0
$557$	−9.89949			−0.419455			−0.209728	−	0.977760i	$-0.567258\pi$
$557$	−9.89949			−0.419455			−0.209728	+	0.977760i	$0.567258\pi$
$558$		0			0
$559$			25.9808i			1.09887i
$560$		0			0
$561$		0			0
$562$			2.00000i			0.0843649i
$563$			7.34847i			0.309701i	0.987938	+	0.154851i	$0.0494896\pi$
$563$			7.34847i			0.309701i	−0.987938	+	0.154851i	$0.950510\pi$
$564$		0			0
$565$		0			0
$566$	−7.34847			−0.308879
$567$		0			0
$568$			8.00000i			0.335673i
$569$			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
$577$	15.5885			0.648956			0.324478	−	0.945893i	$-0.394811\pi$
$577$	15.5885			0.648956			0.324478	+	0.945893i	$0.394811\pi$
$578$	52.3259			2.17647
$579$		0			0
$580$		0			0
$581$	−3.67423	−	19.0919i	−0.152433	−	0.792065i
$582$		0			0
$583$		−	8.00000i		−	0.331326i
$584$	29.3939			1.21633
$585$		0			0
$586$			31.1769i			1.28791i
$587$			29.3939i			1.21322i	0.795001	+	0.606608i	$0.207470\pi$
$587$			29.3939i			1.21322i	−0.795001	+	0.606608i	$0.792530\pi$
$588$		0			0
$589$	−27.0000			−1.11252
$590$		0			0
$591$		0			0
$592$			32.0000i			1.31519i
$593$			14.6969i			0.603531i	0.953382	+	0.301765i	$0.0975760\pi$
$593$			14.6969i			0.603531i	−0.953382	+	0.301765i	$0.902424\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	51.9615			2.12486
$599$			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	$0.733951\pi$
$602$	3.53553	+	18.3712i	0.144098	+	0.748753i
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	41.5692			1.68724			0.843621	−	0.536939i	$-0.180419\pi$
$607$	41.5692			1.68724			0.843621	+	0.536939i	$0.180419\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			38.1838i			1.54475i
$612$		0			0
$613$		−	28.0000i		−	1.13091i	−0.824779	−	0.565455i	$-0.808701\pi$
$613$		−	28.0000i		−	1.13091i	0.824779	−	0.565455i	$-0.191299\pi$
$614$	22.0454			0.889680
$615$		0			0
$616$	2.00000	+	10.3923i	0.0805823	+	0.418718i
$617$	−31.1127			−1.25255			−0.626275	−	0.779602i	$-0.715421\pi$
$617$	−31.1127			−1.25255			−0.626275	+	0.779602i	$0.715421\pi$
$618$		0			0
$619$		−	25.9808i		−	1.04425i	−0.852867	−	0.522127i	$-0.825139\pi$
$619$		−	25.9808i		−	1.04425i	0.852867	−	0.522127i	$-0.174861\pi$
$620$		0			0
$621$		0			0
$622$	−31.1769			−1.25008
$623$	−38.1838	+	7.34847i	−1.52980	+	0.294410i
$624$		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	$0.254667\pi$
$632$	−39.5980			−1.57512
$633$		0			0
$634$	−16.0000			−0.635441
$635$		0			0
$636$		0			0
$637$	33.7750	−	13.5000i	1.33821	−	0.534889i
$638$	−2.82843			−0.111979
$639$		0			0
$640$		0			0
$641$			11.3137i			0.446865i	0.974719	+	0.223432i	$0.0717262\pi$
$641$			11.3137i			0.446865i	−0.974719	+	0.223432i	$0.928274\pi$
$642$		0			0
$643$	−20.7846			−0.819665			−0.409832	−	0.912161i	$-0.634413\pi$
$643$	−20.7846			−0.819665			−0.409832	+	0.912161i	$0.634413\pi$
$644$		0			0
$645$		0			0
$646$	−54.0000			−2.12460
$647$		−	44.0908i		−	1.73339i	−0.498839	−	0.866694i	$-0.666240\pi$
$647$		−	44.0908i		−	1.73339i	0.498839	−	0.866694i	$-0.333760\pi$
$648$		0			0
$649$		−	10.3923i		−	0.407934i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.41421			0.0553425			0.0276712	−	0.999617i	$-0.491191\pi$
$653$	1.41421			0.0553425			0.0276712	+	0.999617i	$0.491191\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	5.19615	+	27.0000i	0.202567	+	1.05257i
$659$			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	$0.935221\pi$
$662$	−19.7990			−0.769510
$663$		0			0
$664$			20.7846i			0.806599i
$665$		0			0
$666$		0			0
$667$		−	10.0000i		−	0.387202i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7.34847			0.283685
$672$		0			0
$673$			8.00000i			0.308377i	0.988041	+	0.154189i	$0.0492764\pi$
$673$			8.00000i			0.308377i	−0.988041	+	0.154189i	$0.950724\pi$
$674$		−	43.8406i		−	1.68868i
$675$		0			0
$676$		0			0
$677$		−	7.34847i		−	0.282425i	−0.989979	−	0.141212i	$-0.954900\pi$
$677$		−	7.34847i		−	0.282425i	0.989979	−	0.141212i	$-0.0451000\pi$
$678$		0			0
$679$	−13.5000	+	2.59808i	−0.518082	+	0.0997050i
$680$		0			0
$681$		0			0
$682$	10.3923			0.397942
$683$	22.6274			0.865814			0.432907	−	0.901439i	$-0.357488\pi$
$683$	22.6274			0.865814			0.432907	+	0.901439i	$0.357488\pi$
$684$		0			0
$685$		0			0
$686$	22.0454	−	14.1421i	0.841698	−	0.539949i
$687$		0			0
$688$		−	20.0000i		−	0.762493i
$689$	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
$698$			44.0908i			1.66886i
$699$		0			0
$700$		0			0
$701$		−	43.8406i		−	1.65584i	−0.560848	−	0.827919i	$-0.689525\pi$
$701$		−	43.8406i		−	1.65584i	0.560848	−	0.827919i	$-0.310475\pi$
$702$		0			0
$703$	−41.5692			−1.56781
$704$		−	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	$-0.255501\pi$
$709$	37.0000			1.38956			0.694782	+	0.719220i	$0.255501\pi$
$710$		0			0
$711$		0			0
$712$	41.5692			1.55787
$713$			36.7423i			1.37601i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$			4.00000i			0.149279i
$719$	−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
$722$	11.3137			0.421053
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	25.9808			0.963573			0.481787	−	0.876289i	$-0.339988\pi$
$727$	25.9808			0.963573			0.481787	+	0.876289i	$0.339988\pi$
$728$	−38.1838	+	7.34847i	−1.41518	+	0.272352i
$729$		0			0
$730$		0			0
$731$	−36.7423			−1.35896
$732$		0			0
$733$	−10.3923			−0.383849			−0.191924	−	0.981410i	$-0.561473\pi$
$733$	−10.3923			−0.383849			−0.191924	+	0.981410i	$0.561473\pi$
$734$	36.7423			1.35618
$735$		0			0
$736$		0			0
$737$	9.89949			0.364653
$738$		0			0
$739$	10.0000			0.367856			0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000			0.367856			0.183928	+	0.982940i	$0.441119\pi$
$740$		0			0
$741$		0			0
$742$	20.7846	−	4.00000i	0.763027	−	0.146845i
$743$	14.1421			0.518825			0.259412	−	0.965767i	$-0.416471\pi$
$743$	14.1421			0.518825			0.259412	+	0.965767i	$0.416471\pi$
$744$		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	$0.631475\pi$
$752$		−	29.3939i		−	1.07188i
$753$		0			0
$754$		−	10.3923i		−	0.378465i
$755$		0			0
$756$		0			0
$757$			1.00000i			0.0363456i	0.999835	+	0.0181728i	$0.00578490\pi$
$757$			1.00000i			0.0363456i	−0.999835	+	0.0181728i	$0.994215\pi$
$758$	−26.8701			−0.975964
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$	−2.59808	+	0.500000i	−0.0940567	+	0.0181012i
$764$		0			0
$765$		0			0
$766$			41.5692i			1.50196i
$767$	38.1838			1.37874
$768$		0			0
$769$		−	36.3731i		−	1.31165i	−0.754915	−	0.655823i	$-0.772322\pi$
$769$		−	36.3731i		−	1.31165i	0.754915	−	0.655823i	$-0.227678\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			29.3939i			1.05722i	0.848863	+	0.528612i	$0.177287\pi$
$773$			29.3939i			1.05722i	−0.848863	+	0.528612i	$0.822713\pi$
$774$		0			0
$775$		0			0
$776$	14.6969			0.527589
$777$		0			0
$778$		−	20.0000i		−	0.717035i
$779$		0			0
$780$		0			0
$781$	4.00000			0.143131
$782$			73.4847i			2.62781i
$783$		0			0
$784$	−26.0000	+	10.3923i	−0.928571	+	0.371154i
$785$		0			0
$786$		0			0
$787$	5.19615			0.185223			0.0926114	−	0.995702i	$-0.470479\pi$
$787$	5.19615			0.185223			0.0926114	+	0.995702i	$0.470479\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−14.6969	+	2.82843i	−0.522563	+	0.100567i
$792$		0			0
$793$			27.0000i			0.958798i
$794$	36.7423			1.30394
$795$		0			0
$796$		0			0
$797$		−	29.3939i		−	1.04118i	−0.853805	−	0.520592i	$-0.825711\pi$
$797$		−	29.3939i		−	1.04118i	0.853805	−	0.520592i	$-0.174289\pi$
$798$		0			0
$799$	−54.0000			−1.91038
$800$		0			0
$801$		0			0
$802$			44.0000i			1.55369i
$803$		−	14.6969i		−	0.518644i
$804$		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
$817$	25.9808			0.908952
$818$		−	22.0454i		−	0.770800i
$819$		0			0
$820$		0			0
$821$			15.5563i			0.542920i	0.962450	+	0.271460i	$0.0875065\pi$
$821$			15.5563i			0.542920i	−0.962450	+	0.271460i	$0.912493\pi$
$822$		0			0
$823$			11.0000i			0.383436i	0.981450	+	0.191718i	$0.0614059\pi$
$823$			11.0000i			0.383436i	−0.981450	+	0.191718i	$0.938594\pi$
$824$	29.3939			1.02398
$825$		0			0
$826$	27.0000	−	5.19615i	0.939450	−	0.180797i
$827$	2.82843			0.0983540			0.0491770	−	0.998790i	$-0.484340\pi$
$827$	2.82843			0.0983540			0.0491770	+	0.998790i	$0.484340\pi$
$828$		0			0
$829$		−	20.7846i		−	0.721879i	−0.932589	−	0.360940i	$-0.882456\pi$
$829$		−	20.7846i		−	0.721879i	0.932589	−	0.360940i	$-0.117544\pi$
$830$		0			0
$831$		0			0
$832$	41.5692			1.44115
$833$	19.0919	+	47.7650i	0.661495	+	1.65496i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	41.5692			1.43598
$839$	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
$842$	5.65685			0.194948
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−23.3827	+	4.50000i	−0.803439	+	0.154622i
$848$	−22.6274			−0.777029
$849$		0			0
$850$		0			0
$851$			56.5685i			1.93914i
$852$		0			0
$853$	5.19615			0.177913			0.0889564	−	0.996036i	$-0.471647\pi$
$853$	5.19615			0.177913			0.0889564	+	0.996036i	$0.471647\pi$
$854$	3.67423	+	19.0919i	0.125730	+	0.653311i
$855$		0			0
$856$	−16.0000			−0.546869
$857$			44.0908i			1.50611i	0.657956	+	0.753057i	$0.271421\pi$
$857$			44.0908i			1.50611i	−0.657956	+	0.753057i	$0.728579\pi$
$858$		0			0
$859$			31.1769i			1.06374i	0.846825	+	0.531871i	$0.178511\pi$
$859$			31.1769i			1.06374i	−0.846825	+	0.531871i	$0.821489\pi$
$860$		0			0
$861$		0			0
$862$		−	10.0000i		−	0.340601i
$863$	−28.2843			−0.962808			−0.481404	−	0.876499i	$-0.659873\pi$
$863$	−28.2843			−0.962808			−0.481404	+	0.876499i	$0.659873\pi$
$864$		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
$872$	2.82843			0.0957826
$873$		0			0
$874$		−	51.9615i		−	1.75762i
$875$		0			0
$876$		0			0
$877$			31.0000i			1.04680i	0.852088	+	0.523398i	$0.175336\pi$
$877$			31.0000i			1.04680i	−0.852088	+	0.523398i	$0.824664\pi$
$878$			7.34847i			0.247999i
$879$		0			0
$880$		0			0
$881$	−44.0908			−1.48546			−0.742729	−	0.669593i	$-0.766469\pi$
$881$	−44.0908			−1.48546			−0.742729	+	0.669593i	$0.766469\pi$
$882$		0			0
$883$		−	37.0000i		−	1.24515i	−0.782560	−	0.622575i	$-0.786087\pi$
$883$		−	37.0000i		−	1.24515i	0.782560	−	0.622575i	$-0.213913\pi$
$884$		0			0
$885$		0			0
$886$	40.0000			1.34383
$887$		−	7.34847i		−	0.246737i	−0.992361	−	0.123369i	$-0.960630\pi$
$887$		−	7.34847i		−	0.246737i	0.992361	−	0.123369i	$-0.0393698\pi$
$888$		0			0
$889$	−5.00000	−	25.9808i	−0.167695	−	0.871367i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	38.1838			1.27777
$894$		0			0
$895$		0			0
$896$	29.3939	−	5.65685i	0.981981	−	0.188982i
$897$		0			0
$898$			40.0000i			1.33482i
$899$	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
$907$		−	26.0000i		−	0.863316i	−0.902037	−	0.431658i	$-0.857929\pi$
$907$		−	26.0000i		−	0.863316i	0.902037	−	0.431658i	$-0.142071\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	26.8701i		−	0.890245i	−0.895470	−	0.445122i	$-0.853160\pi$
$911$		−	26.8701i		−	0.890245i	0.895470	−	0.445122i	$-0.146840\pi$
$912$		0			0
$913$	10.3923			0.343935
$914$			19.7990i			0.654892i
$915$		0			0
$916$		0			0
$917$	−38.1838	+	7.34847i	−1.26094	+	0.242668i
$918$		0			0
$919$	55.0000			1.81428			0.907141	−	0.420826i	$-0.138260\pi$
$919$	55.0000			1.81428			0.907141	+	0.420826i	$0.138260\pi$
$920$		0			0
$921$		0			0
$922$	20.7846			0.684505
$923$			14.6969i			0.483756i
$924$		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
$937$	25.9808			0.848755			0.424377	−	0.905485i	$-0.360493\pi$
$937$	25.9808			0.848755			0.424377	+	0.905485i	$0.360493\pi$
$938$	4.94975	+	25.7196i	0.161615	+	0.839776i
$939$		0			0
$940$		0			0
$941$	−51.4393			−1.67687			−0.838436	−	0.544999i	$-0.816530\pi$
$941$	−51.4393			−1.67687			−0.838436	+	0.544999i	$0.816530\pi$
$942$		0			0
$943$		0			0
$944$	−29.3939			−0.956689
$945$		0			0
$946$	−10.0000			−0.325128
$947$	41.0122			1.33272			0.666359	−	0.745631i	$-0.267852\pi$
$947$	41.0122			1.33272			0.666359	+	0.745631i	$0.267852\pi$
$948$		0			0
$949$	54.0000			1.75291
$950$		0			0
$951$		0			0
$952$	−10.3923	−	54.0000i	−0.336817	−	1.75015i
$953$	−2.82843			−0.0916217			−0.0458109	−	0.998950i	$-0.514587\pi$
$953$	−2.82843			−0.0916217			−0.0458109	+	0.998950i	$0.514587\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$	51.9615			1.67880
$959$	25.7196	−	4.94975i	0.830531	−	0.159836i
$960$		0			0
$961$	4.00000			0.129032
$962$			58.7878i			1.89539i
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	14.0000i		−	0.450210i	−0.974335	−	0.225105i	$-0.927728\pi$
$967$		−	14.0000i		−	0.450210i	0.974335	−	0.225105i	$-0.0722725\pi$
$968$	25.4558			0.818182
$969$		0			0
$970$		0			0
$971$	7.34847			0.235824			0.117912	−	0.993024i	$-0.462380\pi$
$971$	7.34847			0.235824			0.117912	+	0.993024i	$0.462380\pi$
$972$		0			0
$973$	−10.3923	−	54.0000i	−0.333162	−	1.73116i
$974$		−	1.41421i		−	0.0453143i
$975$		0			0
$976$		−	20.7846i		−	0.665299i
$977$	−43.8406			−1.40259			−0.701293	−	0.712873i	$-0.747393\pi$
$977$	−43.8406			−1.40259			−0.701293	+	0.712873i	$0.747393\pi$
$978$		0			0
$979$		−	20.7846i		−	0.664279i
$980$		0			0
$981$		0			0
$982$			8.00000i			0.255290i
$983$		−	58.7878i		−	1.87504i	−0.347934	−	0.937519i	$-0.613117\pi$
$983$		−	58.7878i		−	1.87504i	0.347934	−	0.937519i	$-0.386883\pi$
$984$		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
$997$	−10.3923			−0.329128			−0.164564	−	0.986366i	$-0.552622\pi$
$997$	−10.3923			−0.329128			−0.164564	+	0.986366i	$0.552622\pi$
$998$	7.07107			0.223831
$999$		0			0

Twists

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

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

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +(-2.59808 + 0.500000i) q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +(-2.59808 + 0.500000i) q^{7} +2.82843 q^{8} -1.41421i q^{11} +5.19615 q^{13} +(3.67423 - 0.707107i) q^{14} -4.00000 q^{16} +7.34847i q^{17} -5.19615i q^{19} +2.00000i q^{22} -7.07107 q^{23} -7.34847 q^{26} +1.41421i q^{29} -5.19615i q^{31} -10.3923i q^{34} -8.00000i q^{37} +7.34847i q^{38} +5.00000i q^{43} +10.0000 q^{46} +7.34847i q^{47} +(6.50000 - 2.59808i) q^{49} +5.65685 q^{53} +(-7.34847 + 1.41421i) q^{56} -2.00000i q^{58} +7.34847 q^{59} +5.19615i q^{61} +7.34847i q^{62} +8.00000 q^{64} +7.00000i q^{67} +2.82843i q^{71} +10.3923 q^{73} +11.3137i q^{74} +(0.707107 + 3.67423i) q^{77} -14.0000 q^{79} +7.34847i q^{83} -7.07107i q^{86} -4.00000i q^{88} +14.6969 q^{89} +(-13.5000 + 2.59808i) q^{91} -10.3923i q^{94} +5.19615 q^{97} +(-9.19239 + 3.67423i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 32 q^{16} + 80 q^{46} + 52 q^{49} + 64 q^{64} - 112 q^{79} - 108 q^{91}+O(q^{100}) \) 8 * q - 32 * q^16 + 80 * q^46 + 52 * q^49 + 64 * q^64 - 112 * q^79 - 108 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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