LMFDB - Embedded newform 3150.2.d.d.3149.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(3149,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, 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("3150.3149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3150.d (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:	$25.1528766367$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.7442857984.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 $ x^8 + 26x^6 + 205x^4 + 540*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3149.6
Root			$1.91681i$ of defining polynomial
Character	$\chi$	$=$	3150.3149
Dual form			3150.2.d.d.3149.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +(1.35539 + 2.27220i) q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +(1.35539 + 2.27220i) q^{7} +1.00000 q^{8} +2.71078i q^{11} -6.54441 q^{13} +(1.35539 + 2.27220i) q^{14} +1.00000 q^{16} +1.53186i q^{17} +2.30177i q^{19} +2.71078i q^{22} -3.83363 q^{23} -6.54441 q^{26} +(1.35539 + 2.27220i) q^{28} +3.83363i q^{29} -3.25519i q^{31} +1.00000 q^{32} +1.53186i q^{34} +3.01255i q^{37} +2.30177i q^{38} -6.54441 q^{41} +0.468142i q^{43} +2.71078i q^{44} -3.83363 q^{46} -9.11980i q^{47} +(-3.32583 + 6.15945i) q^{49} -6.54441 q^{52} -9.25519 q^{53} +(1.35539 + 2.27220i) q^{56} +3.83363i q^{58} -11.1961 q^{59} +4.78705i q^{61} -3.25519i q^{62} +1.00000 q^{64} +13.5570i q^{67} +1.53186i q^{68} +2.30177i q^{71} +11.4216 q^{73} +3.01255i q^{74} +2.30177i q^{76} +(-6.15945 + 3.67417i) q^{77} +12.6768 q^{79} -6.54441 q^{82} -13.0888i q^{83} +0.468142i q^{86} +2.71078i q^{88} +9.60812 q^{89} +(-8.87024 - 14.8702i) q^{91} -3.83363 q^{92} -9.11980i q^{94} -16.9224 q^{97} +(-3.32583 + 6.15945i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^4 + 8 * q^8	$ 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} - 8 q^{13} + 8 q^{16} - 8 q^{23} - 8 q^{26} + 8 q^{32} - 8 q^{41} - 8 q^{46} - 4 q^{49} - 8 q^{52} - 8 q^{53} + 8 q^{64} + 48 q^{73} - 4 q^{77} - 8 q^{79} - 8 q^{82} + 8 q^{89} - 4 q^{91} - 8 q^{92} - 24 q^{97} - 4 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 + 8 * q^8 - 8 * q^13 + 8 * q^16 - 8 * q^23 - 8 * q^26 + 8 * q^32 - 8 * q^41 - 8 * q^46 - 4 * q^49 - 8 * q^52 - 8 * q^53 + 8 * q^64 + 48 * q^73 - 4 * q^77 - 8 * q^79 - 8 * q^82 + 8 * q^89 - 4 * q^91 - 8 * q^92 - 24 * q^97 - 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.35539	+	2.27220i	0.512290	+	0.858813i
$7$	1.35539	+	2.27220i	0.512290	+	0.858813i
$8$	1.00000			0.353553
$9$		0			0
$10$		0			0
$11$			2.71078i			0.817332i	0.912684	+	0.408666i	$0.134006\pi$
$11$			2.71078i			0.817332i	−0.912684	+	0.408666i	$0.865994\pi$
$12$		0			0
$13$	−6.54441			−1.81509			−0.907546	−	0.419952i	$-0.862047\pi$
$13$	−6.54441			−1.81509			−0.907546	+	0.419952i	$0.862047\pi$
$14$	1.35539	+	2.27220i	0.362244	+	0.607272i
$15$		0			0
$16$	1.00000			0.250000
$17$			1.53186i			0.371530i	0.982594	+	0.185765i	$0.0594763\pi$
$17$			1.53186i			0.371530i	−0.982594	+	0.185765i	$0.940524\pi$
$18$		0			0
$19$			2.30177i			0.528062i	0.964514	+	0.264031i	$0.0850521\pi$
$19$			2.30177i			0.528062i	−0.964514	+	0.264031i	$0.914948\pi$
$20$		0			0
$21$		0			0
$22$			2.71078i			0.577941i
$23$	−3.83363			−0.799366			−0.399683	−	0.916653i	$-0.630880\pi$
$23$	−3.83363			−0.799366			−0.399683	+	0.916653i	$0.630880\pi$
$24$		0			0
$25$		0			0
$26$	−6.54441			−1.28346
$27$		0			0
$28$	1.35539	+	2.27220i	0.256145	+	0.429406i
$29$			3.83363i			0.711887i	0.934508	+	0.355943i	$0.115840\pi$
$29$			3.83363i			0.711887i	−0.934508	+	0.355943i	$0.884160\pi$
$30$		0			0
$31$		−	3.25519i		−	0.584650i	−0.956319	−	0.292325i	$-0.905571\pi$
$31$		−	3.25519i		−	0.584650i	0.956319	−	0.292325i	$-0.0944289\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$			1.53186i			0.262711i
$35$		0			0
$36$		0			0
$37$			3.01255i			0.495260i	0.968855	+	0.247630i	$0.0796518\pi$
$37$			3.01255i			0.495260i	−0.968855	+	0.247630i	$0.920348\pi$
$38$			2.30177i			0.373396i
$39$		0			0
$40$		0			0
$41$	−6.54441			−1.02207			−0.511033	−	0.859561i	$-0.670737\pi$
$41$	−6.54441			−1.02207			−0.511033	+	0.859561i	$0.670737\pi$
$42$		0			0
$43$			0.468142i			0.0713910i	0.999363	+	0.0356955i	$0.0113647\pi$
$43$			0.468142i			0.0713910i	−0.999363	+	0.0356955i	$0.988635\pi$
$44$			2.71078i			0.408666i
$45$		0			0
$46$	−3.83363			−0.565237
$47$		−	9.11980i		−	1.33026i	−0.746728	−	0.665130i	$-0.768376\pi$
$47$		−	9.11980i		−	1.33026i	0.746728	−	0.665130i	$-0.231624\pi$
$48$		0			0
$49$	−3.32583	+	6.15945i	−0.475118	+	0.879922i
$50$		0			0
$51$		0			0
$52$	−6.54441			−0.907546
$53$	−9.25519			−1.27130			−0.635649	−	0.771978i	$-0.719268\pi$
$53$	−9.25519			−1.27130			−0.635649	+	0.771978i	$0.719268\pi$
$54$		0			0
$55$		0			0
$56$	1.35539	+	2.27220i	0.181122	+	0.303636i
$57$		0			0
$58$			3.83363i			0.503380i
$59$	−11.1961			−1.45760			−0.728802	−	0.684725i	$-0.759922\pi$
$59$	−11.1961			−1.45760			−0.728802	+	0.684725i	$0.759922\pi$
$60$		0			0
$61$			4.78705i			0.612919i	0.951884	+	0.306459i	$0.0991444\pi$
$61$			4.78705i			0.612919i	−0.951884	+	0.306459i	$0.900856\pi$
$62$		−	3.25519i		−	0.413410i
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$			13.5570i			1.65625i	0.560546	+	0.828123i	$0.310591\pi$
$67$			13.5570i			1.65625i	−0.560546	+	0.828123i	$0.689409\pi$
$68$			1.53186i			0.185765i
$69$		0			0
$70$		0			0
$71$			2.30177i			0.273170i	0.990628	+	0.136585i	$0.0436126\pi$
$71$			2.30177i			0.273170i	−0.990628	+	0.136585i	$0.956387\pi$
$72$		0			0
$73$	11.4216			1.33679			0.668397	−	0.743805i	$-0.266981\pi$
$73$	11.4216			1.33679			0.668397	+	0.743805i	$0.266981\pi$
$74$			3.01255i			0.350202i
$75$		0			0
$76$			2.30177i			0.264031i
$77$	−6.15945	+	3.67417i	−0.701935	+	0.418711i
$78$		0			0
$79$	12.6768			1.42625			0.713123	−	0.701039i	$-0.247280\pi$
$79$	12.6768			1.42625			0.713123	+	0.701039i	$0.247280\pi$
$80$		0			0
$81$		0			0
$82$	−6.54441			−0.722709
$83$		−	13.0888i		−	1.43668i	−0.695690	−	0.718342i	$-0.744901\pi$
$83$		−	13.0888i		−	1.43668i	0.695690	−	0.718342i	$-0.255099\pi$
$84$		0			0
$85$		0			0
$86$			0.468142i			0.0504811i
$87$		0			0
$88$			2.71078i			0.288970i
$89$	9.60812			1.01846			0.509230	−	0.860631i	$-0.329930\pi$
$89$	9.60812			1.01846			0.509230	+	0.860631i	$0.329930\pi$
$90$		0			0
$91$	−8.87024	−	14.8702i	−0.929853	−	1.55882i
$92$	−3.83363			−0.399683
$93$		0			0
$94$		−	9.11980i		−	0.940635i
$95$		0			0
$96$		0			0
$97$	−16.9224			−1.71821			−0.859107	−	0.511796i	$-0.828980\pi$
$97$	−16.9224			−1.71821			−0.859107	+	0.511796i	$0.828980\pi$
$98$	−3.32583	+	6.15945i	−0.335959	+	0.622199i
$99$		0			0
$100$		0			0
$101$	17.7405			1.76524			0.882622	−	0.470084i	$-0.155776\pi$
$101$	17.7405			1.76524			0.882622	+	0.470084i	$0.155776\pi$
$102$		0			0
$103$	−4.05913			−0.399958			−0.199979	−	0.979800i	$-0.564087\pi$
$103$	−4.05913			−0.399958			−0.199979	+	0.979800i	$0.564087\pi$
$104$	−6.54441			−0.641732
$105$		0			0
$106$	−9.25519			−0.898944
$107$	−6.31891			−0.610872			−0.305436	−	0.952213i	$-0.598802\pi$
$107$	−6.31891			−0.610872			−0.305436	+	0.952213i	$0.598802\pi$
$108$		0			0
$109$	12.0251			1.15180			0.575898	−	0.817522i	$-0.304653\pi$
$109$	12.0251			1.15180			0.575898	+	0.817522i	$0.304653\pi$
$110$		0			0
$111$		0			0
$112$	1.35539	+	2.27220i	0.128072	+	0.214703i
$113$	9.06372			0.852643			0.426321	−	0.904572i	$-0.359809\pi$
$113$	9.06372			0.852643			0.426321	+	0.904572i	$0.359809\pi$
$114$		0			0
$115$		0			0
$116$			3.83363i			0.355943i
$117$		0			0
$118$	−11.1961			−1.03068
$119$	−3.48069	+	2.07627i	−0.319075	+	0.190331i
$120$		0			0
$121$	3.65166			0.331969
$122$			4.78705i			0.433399i
$123$		0			0
$124$		−	3.25519i		−	0.292325i
$125$		0			0
$126$		0			0
$127$			21.6332i			1.91964i	0.280619	+	0.959819i	$0.409460\pi$
$127$			21.6332i			1.91964i	−0.280619	+	0.959819i	$0.590540\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$		0			0
$131$	−2.51931			−0.220113			−0.110056	−	0.993925i	$-0.535103\pi$
$131$	−2.51931			−0.220113			−0.110056	+	0.993925i	$0.535103\pi$
$132$		0			0
$133$	−5.23009	+	3.11980i	−0.453506	+	0.270521i
$134$			13.5570i			1.17114i
$135$		0			0
$136$			1.53186i			0.131356i
$137$	1.39646			0.119308			0.0596539	−	0.998219i	$-0.481000\pi$
$137$	1.39646			0.119308			0.0596539	+	0.998219i	$0.481000\pi$
$138$		0			0
$139$			6.13539i			0.520397i	0.965555	+	0.260199i	$0.0837881\pi$
$139$			6.13539i			0.520397i	−0.965555	+	0.260199i	$0.916212\pi$
$140$		0			0
$141$		0			0
$142$			2.30177i			0.193160i
$143$		−	17.7405i		−	1.48353i
$144$		0			0
$145$		0			0
$146$	11.4216			0.945256
$147$		0			0
$148$			3.01255i			0.247630i
$149$			4.65166i			0.381078i	0.981680	+	0.190539i	$0.0610236\pi$
$149$			4.65166i			0.381078i	−0.981680	+	0.190539i	$0.938976\pi$
$150$		0			0
$151$	−3.58794			−0.291982			−0.145991	−	0.989286i	$-0.546637\pi$
$151$	−3.58794			−0.291982			−0.145991	+	0.989286i	$0.546637\pi$
$152$			2.30177i			0.186698i
$153$		0			0
$154$	−6.15945	+	3.67417i	−0.496343	+	0.296073i
$155$		0			0
$156$		0			0
$157$	−13.1228			−1.04732			−0.523658	−	0.851928i	$-0.675433\pi$
$157$	−13.1228			−1.04732			−0.523658	+	0.851928i	$0.675433\pi$
$158$	12.6768			1.00851
$159$		0			0
$160$		0			0
$161$	−5.19606	−	8.71078i	−0.409507	−	0.686506i
$162$		0			0
$163$		−	16.6207i		−	1.30183i	−0.759150	−	0.650916i	$-0.774385\pi$
$163$		−	16.6207i		−	1.30183i	0.759150	−	0.650916i	$-0.225615\pi$
$164$	−6.54441			−0.511033
$165$		0			0
$166$		−	13.0888i		−	1.01589i
$167$			8.62068i			0.667088i	0.942735	+	0.333544i	$0.108245\pi$
$167$			8.62068i			0.667088i	−0.942735	+	0.333544i	$0.891755\pi$
$168$		0			0
$169$	29.8293			2.29456
$170$		0			0
$171$		0			0
$172$			0.468142i			0.0356955i
$173$			10.6517i			0.809830i	0.914354	+	0.404915i	$0.132699\pi$
$173$			10.6517i			0.809830i	−0.914354	+	0.404915i	$0.867301\pi$
$174$		0			0
$175$		0			0
$176$			2.71078i			0.204333i
$177$		0			0
$178$	9.60812			0.720159
$179$		−	23.1961i		−	1.73376i	−0.498521	−	0.866878i	$-0.666123\pi$
$179$		−	23.1961i		−	1.73376i	0.498521	−	0.866878i	$-0.333877\pi$
$180$		0			0
$181$			9.11980i			0.677869i	0.940810	+	0.338935i	$0.110067\pi$
$181$			9.11980i			0.677869i	−0.940810	+	0.338935i	$0.889933\pi$
$182$	−8.87024	−	14.8702i	−0.657506	−	1.10226i
$183$		0			0
$184$	−3.83363			−0.282619
$185$		0			0
$186$		0			0
$187$	−4.15253			−0.303663
$188$		−	9.11980i		−	0.665130i
$189$		0			0
$190$		0			0
$191$			9.69823i			0.701739i	0.936424	+	0.350870i	$0.114114\pi$
$191$			9.69823i			0.701739i	−0.936424	+	0.350870i	$0.885886\pi$
$192$		0			0
$193$			18.1525i			1.30665i	0.757078	+	0.653324i	$0.226626\pi$
$193$			18.1525i			1.30665i	−0.757078	+	0.653324i	$0.773374\pi$
$194$	−16.9224			−1.21496
$195$		0			0
$196$	−3.32583	+	6.15945i	−0.237559	+	0.439961i
$197$	−4.60354			−0.327988			−0.163994	−	0.986461i	$-0.552438\pi$
$197$	−4.60354			−0.327988			−0.163994	+	0.986461i	$0.552438\pi$
$198$		0			0
$199$			11.7405i			0.832260i	0.909305	+	0.416130i	$0.136614\pi$
$199$			11.7405i			0.832260i	−0.909305	+	0.416130i	$0.863386\pi$
$200$		0			0
$201$		0			0
$202$	17.7405			1.24822
$203$	−8.71078	+	5.19606i	−0.611377	+	0.364692i
$204$		0			0
$205$		0			0
$206$	−4.05913			−0.282813
$207$		0			0
$208$	−6.54441			−0.453773
$209$	−6.23960			−0.431602
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$	−9.25519			−0.635649
$213$		0			0
$214$	−6.31891			−0.431952
$215$		0			0
$216$		0			0
$217$	7.39646	−	4.41206i	0.502105	−	0.299510i
$218$	12.0251			0.814443
$219$		0			0
$220$		0			0
$221$		−	10.0251i		−	0.674361i
$222$		0			0
$223$	26.4513			1.77131			0.885654	−	0.464347i	$-0.153711\pi$
$223$	26.4513			1.77131			0.885654	+	0.464347i	$0.153711\pi$
$224$	1.35539	+	2.27220i	0.0905609	+	0.151818i
$225$		0			0
$226$	9.06372			0.602909
$227$			15.0637i			0.999814i	0.866079	+	0.499907i	$0.166632\pi$
$227$			15.0637i			0.999814i	−0.866079	+	0.499907i	$0.833368\pi$
$228$		0			0
$229$			11.3655i			0.751052i	0.926812	+	0.375526i	$0.122538\pi$
$229$			11.3655i			0.751052i	−0.926812	+	0.375526i	$0.877462\pi$
$230$		0			0
$231$		0			0
$232$			3.83363i			0.251690i
$233$	−20.9957			−1.37547			−0.687736	−	0.725961i	$-0.741395\pi$
$233$	−20.9957			−1.37547			−0.687736	+	0.725961i	$0.741395\pi$
$234$		0			0
$235$		0			0
$236$	−11.1961			−0.728802
$237$		0			0
$238$	−3.48069	+	2.07627i	−0.225620	+	0.134584i
$239$			9.69823i			0.627326i	0.949534	+	0.313663i	$0.101556\pi$
$239$			9.69823i			0.627326i	−0.949534	+	0.313663i	$0.898444\pi$
$240$		0			0
$241$			24.7019i			1.59119i	0.605831	+	0.795593i	$0.292841\pi$
$241$			24.7019i			1.59119i	−0.605831	+	0.795593i	$0.707159\pi$
$242$	3.65166			0.234737
$243$		0			0
$244$			4.78705i			0.306459i
$245$		0			0
$246$		0			0
$247$		−	15.0637i		−	0.958481i
$248$		−	3.25519i		−	0.206705i
$249$		0			0
$250$		0			0
$251$	3.60812			0.227743			0.113871	−	0.993495i	$-0.463675\pi$
$251$	3.60812			0.227743			0.113871	+	0.993495i	$0.463675\pi$
$252$		0			0
$253$		−	10.3921i		−	0.653348i
$254$			21.6332i			1.35739i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	28.7983i		−	1.79639i	−0.439598	−	0.898195i	$-0.644879\pi$
$257$		−	28.7983i		−	1.79639i	0.439598	−	0.898195i	$-0.355121\pi$
$258$		0			0
$259$	−6.84513	+	4.08319i	−0.425336	+	0.253717i
$260$		0			0
$261$		0			0
$262$	−2.51931			−0.155643
$263$	−12.7699			−0.787426			−0.393713	−	0.919233i	$-0.628810\pi$
$263$	−12.7699			−0.787426			−0.393713	+	0.919233i	$0.628810\pi$
$264$		0			0
$265$		0			0
$266$	−5.23009	+	3.11980i	−0.320677	+	0.191287i
$267$		0			0
$268$			13.5570i			0.828123i
$269$	8.43716			0.514423			0.257211	−	0.966355i	$-0.417196\pi$
$269$	8.43716			0.514423			0.257211	+	0.966355i	$0.417196\pi$
$270$		0			0
$271$			2.16637i			0.131598i	0.997833	+	0.0657989i	$0.0209596\pi$
$271$			2.16637i			0.131598i	−0.997833	+	0.0657989i	$0.979040\pi$
$272$			1.53186i			0.0928825i
$273$		0			0
$274$	1.39646			0.0843634
$275$		0			0
$276$		0			0
$277$			5.92373i			0.355923i	0.984037	+	0.177961i	$0.0569502\pi$
$277$			5.92373i			0.355923i	−0.984037	+	0.177961i	$0.943050\pi$
$278$			6.13539i			0.367977i
$279$		0			0
$280$		0			0
$281$		−	27.3566i		−	1.63196i	−0.578083	−	0.815978i	$-0.696199\pi$
$281$		−	27.3566i		−	1.63196i	0.578083	−	0.815978i	$-0.303801\pi$
$282$		0			0
$283$	−18.0594			−1.07352			−0.536759	−	0.843735i	$-0.680352\pi$
$283$	−18.0594			−1.07352			−0.536759	+	0.843735i	$0.680352\pi$
$284$			2.30177i			0.136585i
$285$		0			0
$286$		−	17.7405i		−	1.04902i
$287$	−8.87024	−	14.8702i	−0.523594	−	0.877762i
$288$		0			0
$289$	14.6534			0.861965
$290$		0			0
$291$		0			0
$292$	11.4216			0.668397
$293$			29.1139i			1.70085i	0.526094	+	0.850427i	$0.323656\pi$
$293$			29.1139i			1.70085i	−0.526094	+	0.850427i	$0.676344\pi$
$294$		0			0
$295$		0			0
$296$			3.01255i			0.175101i
$297$		0			0
$298$			4.65166i			0.269463i
$299$	25.0888			1.45092
$300$		0			0
$301$	−1.06372	+	0.634516i	−0.0613115	+	0.0365729i
$302$	−3.58794			−0.206463
$303$		0			0
$304$			2.30177i			0.132015i
$305$		0			0
$306$		0			0
$307$	−7.39646			−0.422138			−0.211069	−	0.977471i	$-0.567695\pi$
$307$	−7.39646			−0.422138			−0.211069	+	0.977471i	$0.567695\pi$
$308$	−6.15945	+	3.67417i	−0.350967	+	0.209355i
$309$		0			0
$310$		0			0
$311$	1.97490			0.111986			0.0559931	−	0.998431i	$-0.482168\pi$
$311$	1.97490			0.111986			0.0559931	+	0.998431i	$0.482168\pi$
$312$		0			0
$313$	0.961388			0.0543409			0.0271704	−	0.999631i	$-0.491350\pi$
$313$	0.961388			0.0543409			0.0271704	+	0.999631i	$0.491350\pi$
$314$	−13.1228			−0.740565
$315$		0			0
$316$	12.6768			0.713123
$317$	−17.1620			−0.963916			−0.481958	−	0.876194i	$-0.660074\pi$
$317$	−17.1620			−0.963916			−0.481958	+	0.876194i	$0.660074\pi$
$318$		0			0
$319$	−10.3921			−0.581848
$320$		0			0
$321$		0			0
$322$	−5.19606	−	8.71078i	−0.289565	−	0.485433i
$323$	−3.52598			−0.196191
$324$		0			0
$325$		0			0
$326$		−	16.6207i		−	0.920534i
$327$		0			0
$328$	−6.54441			−0.361355
$329$	20.7220	−	12.3609i	1.14244	−	0.681478i
$330$		0			0
$331$	9.74047			0.535385			0.267692	−	0.963504i	$-0.413739\pi$
$331$	9.74047			0.535385			0.267692	+	0.963504i	$0.413739\pi$
$332$		−	13.0888i		−	0.718342i
$333$		0			0
$334$			8.62068i			0.471702i
$335$		0			0
$336$		0			0
$337$			3.15078i			0.171634i	0.996311	+	0.0858169i	$0.0273500\pi$
$337$			3.15078i			0.171634i	−0.996311	+	0.0858169i	$0.972650\pi$
$338$	29.8293			1.62250
$339$		0			0
$340$		0			0
$341$	8.82412			0.477853
$342$		0			0
$343$	−18.5033	+	0.791511i	−0.999086	+	0.0427376i
$344$			0.468142i			0.0252405i
$345$		0			0
$346$			10.6517i			0.572637i
$347$	30.0594			1.61367			0.806836	−	0.590775i	$-0.201178\pi$
$347$	30.0594			1.61367			0.806836	+	0.590775i	$0.201178\pi$
$348$		0			0
$349$		−	27.5180i		−	1.47301i	−0.676434	−	0.736503i	$-0.736476\pi$
$349$		−	27.5180i		−	1.47301i	0.676434	−	0.736503i	$-0.263524\pi$
$350$		0			0
$351$		0			0
$352$			2.71078i			0.144485i
$353$		−	16.5956i		−	0.883293i	−0.897189	−	0.441647i	$-0.854395\pi$
$353$		−	16.5956i		−	0.883293i	0.897189	−	0.441647i	$-0.145605\pi$
$354$		0			0
$355$		0			0
$356$	9.60812			0.509230
$357$		0			0
$358$		−	23.1961i		−	1.22595i
$359$			15.0076i			0.792073i	0.918235	+	0.396036i	$0.129615\pi$
$359$			15.0076i			0.792073i	−0.918235	+	0.396036i	$0.870385\pi$
$360$		0			0
$361$	13.7019			0.721151
$362$			9.11980i			0.479326i
$363$		0			0
$364$	−8.87024	−	14.8702i	−0.464927	−	0.779412i
$365$		0			0
$366$		0			0
$367$	14.2598			0.744354			0.372177	−	0.928162i	$-0.378611\pi$
$367$	14.2598			0.744354			0.372177	+	0.928162i	$0.378611\pi$
$368$	−3.83363			−0.199842
$369$		0			0
$370$		0			0
$371$	−12.5444	−	21.0297i	−0.651273	−	1.09181i
$372$		0			0
$373$		−	8.98745i		−	0.465352i	−0.972554	−	0.232676i	$-0.925252\pi$
$373$		−	8.98745i		−	0.465352i	0.972554	−	0.232676i	$-0.0747482\pi$
$374$	−4.15253			−0.214722
$375$		0			0
$376$		−	9.11980i		−	0.470318i
$377$		−	25.0888i		−	1.29214i
$378$		0			0
$379$	34.5447			1.77444			0.887220	−	0.461346i	$-0.152633\pi$
$379$	34.5447			1.77444			0.887220	+	0.461346i	$0.152633\pi$
$380$		0			0
$381$		0			0
$382$			9.69823i			0.496205i
$383$		−	2.88020i		−	0.147171i	−0.997289	−	0.0735857i	$-0.976556\pi$
$383$		−	2.88020i		−	0.147171i	0.997289	−	0.0735857i	$-0.0234443\pi$
$384$		0			0
$385$		0			0
$386$			18.1525i			0.923940i
$387$		0			0
$388$	−16.9224			−0.859107
$389$		−	35.1620i		−	1.78279i	−0.453231	−	0.891393i	$-0.649729\pi$
$389$		−	35.1620i		−	1.78279i	0.453231	−	0.891393i	$-0.350271\pi$
$390$		0			0
$391$		−	5.87257i		−	0.296989i
$392$	−3.32583	+	6.15945i	−0.167980	+	0.311099i
$393$		0			0
$394$	−4.60354			−0.231923
$395$		0			0
$396$		0			0
$397$	16.1185			0.808965			0.404482	−	0.914546i	$-0.367452\pi$
$397$	16.1185			0.808965			0.404482	+	0.914546i	$0.367452\pi$
$398$			11.7405i			0.588497i
$399$		0			0
$400$		0			0
$401$		−	16.6256i		−	0.830243i	−0.909766	−	0.415121i	$-0.863739\pi$
$401$		−	16.6256i		−	0.830243i	0.909766	−	0.415121i	$-0.136261\pi$
$402$		0			0
$403$			21.3033i			1.06119i
$404$	17.7405			0.882622
$405$		0			0
$406$	−8.71078	+	5.19606i	−0.432309	+	0.257876i
$407$	−8.16637			−0.404792
$408$		0			0
$409$		−	12.0000i		−	0.593362i	−0.954977	−	0.296681i	$-0.904120\pi$
$409$		−	12.0000i		−	0.593362i	0.954977	−	0.296681i	$-0.0958798\pi$
$410$		0			0
$411$		0			0
$412$	−4.05913			−0.199979
$413$	−15.1751	−	25.4397i	−0.746715	−	1.25181i
$414$		0			0
$415$		0			0
$416$	−6.54441			−0.320866
$417$		0			0
$418$	−6.23960			−0.305189
$419$	36.2849			1.77263			0.886316	−	0.463080i	$-0.153256\pi$
$419$	36.2849			1.77263			0.886316	+	0.463080i	$0.153256\pi$
$420$		0			0
$421$	19.2414			0.937766			0.468883	−	0.883260i	$-0.344657\pi$
$421$	19.2414			0.937766			0.468883	+	0.883260i	$0.344657\pi$
$422$	4.00000			0.194717
$423$		0			0
$424$	−9.25519			−0.449472
$425$		0			0
$426$		0			0
$427$	−10.8772	+	6.48833i	−0.526383	+	0.313992i
$428$	−6.31891			−0.305436
$429$		0			0
$430$		0			0
$431$			17.2974i			0.833188i	0.909093	+	0.416594i	$0.136776\pi$
$431$			17.2974i			0.833188i	−0.909093	+	0.416594i	$0.863224\pi$
$432$		0			0
$433$	31.6473			1.52087			0.760437	−	0.649412i	$-0.224985\pi$
$433$	31.6473			1.52087			0.760437	+	0.649412i	$0.224985\pi$
$434$	7.39646	−	4.41206i	0.355042	−	0.211786i
$435$		0			0
$436$	12.0251			0.575898
$437$		−	8.82412i		−	0.422115i
$438$		0			0
$439$			13.0095i			0.620910i	0.950588	+	0.310455i	$0.100481\pi$
$439$			13.0095i			0.620910i	−0.950588	+	0.310455i	$0.899519\pi$
$440$		0			0
$441$		0			0
$442$		−	10.0251i		−	0.476846i
$443$	3.78551			0.179855			0.0899275	−	0.995948i	$-0.471336\pi$
$443$	3.78551			0.179855			0.0899275	+	0.995948i	$0.471336\pi$
$444$		0			0
$445$		0			0
$446$	26.4513			1.25250
$447$		0			0
$448$	1.35539	+	2.27220i	0.0640362	+	0.107352i
$449$		−	24.9307i		−	1.17655i	−0.808661	−	0.588275i	$-0.799807\pi$
$449$		−	24.9307i		−	1.17655i	0.808661	−	0.588275i	$-0.200193\pi$
$450$		0			0
$451$		−	17.7405i		−	0.835366i
$452$	9.06372			0.426321
$453$		0			0
$454$			15.0637i			0.706975i
$455$		0			0
$456$		0			0
$457$		−	31.3535i		−	1.46666i	−0.679875	−	0.733328i	$-0.737966\pi$
$457$		−	31.3535i		−	1.46666i	0.679875	−	0.733328i	$-0.262034\pi$
$458$			11.3655i			0.531074i
$459$		0			0
$460$		0			0
$461$	9.52598			0.443669			0.221835	−	0.975084i	$-0.428795\pi$
$461$	9.52598			0.443669			0.221835	+	0.975084i	$0.428795\pi$
$462$		0			0
$463$		−	34.0003i		−	1.58013i	−0.613026	−	0.790063i	$-0.710048\pi$
$463$		−	34.0003i		−	1.58013i	0.613026	−	0.790063i	$-0.289952\pi$
$464$			3.83363i			0.177972i
$465$		0			0
$466$	−20.9957			−0.972605
$467$			39.2665i			1.81703i	0.417847	+	0.908517i	$0.362785\pi$
$467$			39.2665i			1.81703i	−0.417847	+	0.908517i	$0.637215\pi$
$468$		0			0
$469$	−30.8042	+	18.3750i	−1.42241	+	0.848478i
$470$		0			0
$471$		0			0
$472$	−11.1961			−0.515341
$473$	−1.26903			−0.0583502
$474$		0			0
$475$		0			0
$476$	−3.48069	+	2.07627i	−0.159537	+	0.0951655i
$477$		0			0
$478$			9.69823i			0.443587i
$479$	−34.0251			−1.55465			−0.777323	−	0.629101i	$-0.783423\pi$
$479$	−34.0251			−1.55465			−0.777323	+	0.629101i	$0.783423\pi$
$480$		0			0
$481$		−	19.7154i		−	0.898944i
$482$			24.7019i			1.12514i
$483$		0			0
$484$	3.65166			0.165984
$485$		0			0
$486$		0			0
$487$		−	12.8091i		−	0.580436i	−0.956961	−	0.290218i	$-0.906272\pi$
$487$		−	12.8091i		−	0.580436i	0.956961	−	0.290218i	$-0.0937278\pi$
$488$			4.78705i			0.216700i
$489$		0			0
$490$		0			0
$491$			11.6471i			0.525625i	0.964847	+	0.262812i	$0.0846500\pi$
$491$			11.6471i			0.525625i	−0.964847	+	0.262812i	$0.915350\pi$
$492$		0			0
$493$	−5.87257			−0.264487
$494$		−	15.0637i		−	0.677749i
$495$		0			0
$496$		−	3.25519i		−	0.146162i
$497$	−5.23009	+	3.11980i	−0.234602	+	0.139942i
$498$		0			0
$499$	−35.5511			−1.59149			−0.795743	−	0.605635i	$-0.792919\pi$
$499$	−35.5511			−1.59149			−0.795743	+	0.605635i	$0.792919\pi$
$500$		0			0
$501$		0			0
$502$	3.60812			0.161038
$503$		−	8.23372i		−	0.367123i	−0.983008	−	0.183562i	$-0.941237\pi$
$503$		−	8.23372i		−	0.367123i	0.983008	−	0.183562i	$-0.0587627\pi$
$504$		0			0
$505$		0			0
$506$		−	10.3921i		−	0.461986i
$507$		0			0
$508$			21.6332i			0.959819i
$509$	−31.0888			−1.37799			−0.688994	−	0.724767i	$-0.741947\pi$
$509$	−31.0888			−1.37799			−0.688994	+	0.724767i	$0.741947\pi$
$510$		0			0
$511$	15.4807	+	25.9521i	0.684826	+	1.14805i
$512$	1.00000			0.0441942
$513$		0			0
$514$		−	28.7983i		−	1.27024i
$515$		0			0
$516$		0			0
$517$	24.7218			1.08726
$518$	−6.84513	+	4.08319i	−0.300758	+	0.179405i
$519$		0			0
$520$		0			0
$521$	1.39363			0.0610561			0.0305281	−	0.999534i	$-0.490281\pi$
$521$	1.39363			0.0610561			0.0305281	+	0.999534i	$0.490281\pi$
$522$		0			0
$523$	20.4853			0.895759			0.447879	−	0.894094i	$-0.352179\pi$
$523$	20.4853			0.895759			0.447879	+	0.894094i	$0.352179\pi$
$524$	−2.51931			−0.110056
$525$		0			0
$526$	−12.7699			−0.556795
$527$	4.98649			0.217215
$528$		0			0
$529$	−8.30331			−0.361013
$530$		0			0
$531$		0			0
$532$	−5.23009	+	3.11980i	−0.226753	+	0.135260i
$533$	42.8293			1.85514
$534$		0			0
$535$		0			0
$536$			13.5570i			0.585572i
$537$		0			0
$538$	8.43716			0.363752
$539$	−16.6969	−	9.01560i	−0.719188	−	0.388329i
$540$		0			0
$541$	−0.0870615			−0.00374306			−0.00187153	−	0.999998i	$-0.500596\pi$
$541$	−0.0870615			−0.00374306			−0.00187153	+	0.999998i	$0.500596\pi$
$542$			2.16637i			0.0930537i
$543$		0			0
$544$			1.53186i			0.0656779i
$545$		0			0
$546$		0			0
$547$		−	5.40443i		−	0.231077i	−0.993303	−	0.115538i	$-0.963141\pi$
$547$		−	5.40443i		−	0.231077i	0.993303	−	0.115538i	$-0.0368593\pi$
$548$	1.39646			0.0596539
$549$		0			0
$550$		0			0
$551$	−8.82412			−0.375920
$552$		0			0
$553$	17.1820	+	28.8042i	0.730652	+	1.22488i
$554$			5.92373i			0.251675i
$555$		0			0
$556$			6.13539i			0.260199i
$557$	0.127431			0.00539940			0.00269970	−	0.999996i	$-0.499141\pi$
$557$	0.127431			0.00539940			0.00269970	+	0.999996i	$0.499141\pi$
$558$		0			0
$559$		−	3.06372i		−	0.129581i
$560$		0			0
$561$		0			0
$562$		−	27.3566i		−	1.15397i
$563$			6.23960i			0.262968i	0.991318	+	0.131484i	$0.0419741\pi$
$563$			6.23960i			0.262968i	−0.991318	+	0.131484i	$0.958026\pi$
$564$		0			0
$565$		0			0
$566$	−18.0594			−0.759092
$567$		0			0
$568$			2.30177i			0.0965801i
$569$			11.6391i			0.487937i	0.969783	+	0.243968i	$0.0784493\pi$
$569$			11.6391i			0.487937i	−0.969783	+	0.243968i	$0.921551\pi$
$570$		0			0
$571$	−35.9181			−1.50313			−0.751563	−	0.659661i	$-0.770700\pi$
$571$	−35.9181			−1.50313			−0.751563	+	0.659661i	$0.770700\pi$
$572$		−	17.7405i		−	0.741766i
$573$		0			0
$574$	−8.87024	−	14.8702i	−0.370237	−	0.620672i
$575$		0			0
$576$		0			0
$577$	7.15687			0.297944			0.148972	−	0.988841i	$-0.452404\pi$
$577$	7.15687			0.297944			0.148972	+	0.988841i	$0.452404\pi$
$578$	14.6534			0.609502
$579$		0			0
$580$		0			0
$581$	29.7405	−	17.7405i	1.23384	−	0.735999i
$582$		0			0
$583$		−	25.0888i		−	1.03907i
$584$	11.4216			0.472628
$585$		0			0
$586$			29.1139i			1.20269i
$587$		−	4.15253i		−	0.171393i	−0.996321	−	0.0856967i	$-0.972688\pi$
$587$		−	4.15253i		−	0.171393i	0.996321	−	0.0856967i	$-0.0273116\pi$
$588$		0			0
$589$	7.49270			0.308731
$590$		0			0
$591$		0			0
$592$			3.01255i			0.123815i
$593$		−	29.7966i		−	1.22360i	−0.791013	−	0.611799i	$-0.790446\pi$
$593$		−	29.7966i		−	1.22360i	0.791013	−	0.611799i	$-0.209554\pi$
$594$		0			0
$595$		0			0
$596$			4.65166i			0.190539i
$597$		0			0
$598$	25.0888			1.02596
$599$		−	35.4249i		−	1.44742i	−0.690104	−	0.723710i	$-0.742435\pi$
$599$		−	35.4249i		−	1.44742i	0.690104	−	0.723710i	$-0.257565\pi$
$600$		0			0
$601$			27.4948i			1.12154i	0.827973	+	0.560768i	$0.189494\pi$
$601$			27.4948i			1.12154i	−0.827973	+	0.560768i	$0.810506\pi$
$602$	−1.06372	+	0.634516i	−0.0433538	+	0.0258609i
$603$		0			0
$604$	−3.58794			−0.145991
$605$		0			0
$606$		0			0
$607$	4.76499			0.193405			0.0967025	−	0.995313i	$-0.469170\pi$
$607$	4.76499			0.193405			0.0967025	+	0.995313i	$0.469170\pi$
$608$			2.30177i			0.0933490i
$609$		0			0
$610$		0			0
$611$			59.6837i			2.41454i
$612$		0			0
$613$		−	10.7981i		−	0.436130i	−0.975934	−	0.218065i	$-0.930026\pi$
$613$		−	10.7981i		−	0.436130i	0.975934	−	0.218065i	$-0.0699744\pi$
$614$	−7.39646			−0.298497
$615$		0			0
$616$	−6.15945	+	3.67417i	−0.248171	+	0.148037i
$617$	34.9025			1.40512			0.702561	−	0.711623i	$-0.252040\pi$
$617$	34.9025			1.40512			0.702561	+	0.711623i	$0.252040\pi$
$618$		0			0
$619$			1.26107i			0.0506866i	0.999679	+	0.0253433i	$0.00806789\pi$
$619$			1.26107i			0.0506866i	−0.999679	+	0.0253433i	$0.991932\pi$
$620$		0			0
$621$		0			0
$622$	1.97490			0.0791861
$623$	13.0228	+	21.8316i	0.521746	+	0.874666i
$624$		0			0
$625$		0			0
$626$	0.961388			0.0384248
$627$		0			0
$628$	−13.1228			−0.523658
$629$	−4.61480			−0.184004
$630$		0			0
$631$	16.2145			0.645489			0.322744	−	0.946486i	$-0.395395\pi$
$631$	16.2145			0.645489			0.322744	+	0.946486i	$0.395395\pi$
$632$	12.6768			0.504254
$633$		0			0
$634$	−17.1620			−0.681592
$635$		0			0
$636$		0			0
$637$	21.7656	−	40.3100i	0.862384	−	1.59714i
$638$	−10.3921			−0.411428
$639$		0			0
$640$		0			0
$641$		−	19.6893i		−	0.777681i	−0.921305	−	0.388840i	$-0.872876\pi$
$641$		−	19.6893i		−	0.777681i	0.921305	−	0.388840i	$-0.127124\pi$
$642$		0			0
$643$	−17.1508			−0.676361			−0.338180	−	0.941081i	$-0.609811\pi$
$643$	−17.1508			−0.676361			−0.338180	+	0.941081i	$0.609811\pi$
$644$	−5.19606	−	8.71078i	−0.204754	−	0.343253i
$645$		0			0
$646$	−3.52598			−0.138728
$647$			29.0578i			1.14238i	0.820817	+	0.571191i	$0.193518\pi$
$647$			29.0578i			1.14238i	−0.820817	+	0.571191i	$0.806482\pi$
$648$		0			0
$649$		−	30.3501i		−	1.19135i
$650$		0			0
$651$		0			0
$652$		−	16.6207i		−	0.650916i
$653$	−9.11183			−0.356574			−0.178287	−	0.983979i	$-0.557055\pi$
$653$	−9.11183			−0.356574			−0.178287	+	0.983979i	$0.557055\pi$
$654$		0			0
$655$		0			0
$656$	−6.54441			−0.255516
$657$		0			0
$658$	20.7220	−	12.3609i	0.807829	−	0.481878i
$659$			37.5539i			1.46289i	0.681899	+	0.731446i	$0.261154\pi$
$659$			37.5539i			1.46289i	−0.681899	+	0.731446i	$0.738846\pi$
$660$		0			0
$661$		−	9.50275i		−	0.369614i	−0.982775	−	0.184807i	$-0.940834\pi$
$661$		−	9.50275i		−	0.369614i	0.982775	−	0.184807i	$-0.0591660\pi$
$662$	9.74047			0.378574
$663$		0			0
$664$		−	13.0888i		−	0.507945i
$665$		0			0
$666$		0			0
$667$		−	14.6967i		−	0.569058i
$668$			8.62068i			0.333544i
$669$		0			0
$670$		0			0
$671$	−12.9767			−0.500958
$672$		0			0
$673$		−	49.6335i		−	1.91323i	−0.291355	−	0.956615i	$-0.594106\pi$
$673$		−	49.6335i		−	1.91323i	0.291355	−	0.956615i	$-0.405894\pi$
$674$			3.15078i			0.121363i
$675$		0			0
$676$	29.8293			1.14728
$677$			37.5312i			1.44244i	0.692706	+	0.721220i	$0.256418\pi$
$677$			37.5312i			1.44244i	−0.692706	+	0.721220i	$0.743582\pi$
$678$		0			0
$679$	−22.9365	−	38.4513i	−0.880224	−	1.47562i
$680$		0			0
$681$		0			0
$682$	8.82412			0.337893
$683$	−9.01560			−0.344972			−0.172486	−	0.985012i	$-0.555180\pi$
$683$	−9.01560			−0.344972			−0.172486	+	0.985012i	$0.555180\pi$
$684$		0			0
$685$		0			0
$686$	−18.5033	+	0.791511i	−0.706461	+	0.0302200i
$687$		0			0
$688$			0.468142i			0.0178478i
$689$	60.5698			2.30752
$690$		0			0
$691$			8.15841i			0.310361i	0.987886	+	0.155180i	$0.0495958\pi$
$691$			8.15841i			0.310361i	−0.987886	+	0.155180i	$0.950404\pi$
$692$			10.6517i			0.404915i
$693$		0			0
$694$	30.0594			1.14104
$695$		0			0
$696$		0			0
$697$		−	10.0251i		−	0.379728i
$698$		−	27.5180i		−	1.04157i
$699$		0			0
$700$		0			0
$701$			34.3440i			1.29716i	0.761148	+	0.648578i	$0.224636\pi$
$701$			34.3440i			1.29716i	−0.761148	+	0.648578i	$0.775364\pi$
$702$		0			0
$703$	−6.93420			−0.261528
$704$			2.71078i			0.102166i
$705$		0			0
$706$		−	16.5956i		−	0.624583i
$707$	24.0453	+	40.3100i	0.904316	+	1.51601i
$708$		0			0
$709$	5.06372			0.190172			0.0950859	−	0.995469i	$-0.469687\pi$
$709$	5.06372			0.190172			0.0950859	+	0.995469i	$0.469687\pi$
$710$		0			0
$711$		0			0
$712$	9.60812			0.360080
$713$			12.4792i			0.467349i
$714$		0			0
$715$		0			0
$716$		−	23.1961i		−	0.866878i
$717$		0			0
$718$			15.0076i			0.560080i
$719$	−18.1274			−0.676039			−0.338020	−	0.941139i	$-0.609757\pi$
$719$	−18.1274			−0.676039			−0.338020	+	0.941139i	$0.609757\pi$
$720$		0			0
$721$	−5.50171	−	9.22317i	−0.204894	−	0.343489i
$722$	13.7019			0.509930
$723$		0			0
$724$			9.11980i			0.338935i
$725$		0			0
$726$		0			0
$727$	−11.1168			−0.412298			−0.206149	−	0.978521i	$-0.566093\pi$
$727$	−11.1168			−0.412298			−0.206149	+	0.978521i	$0.566093\pi$
$728$	−8.87024	−	14.8702i	−0.328753	−	0.551128i
$729$		0			0
$730$		0			0
$731$	−0.717127			−0.0265239
$732$		0			0
$733$	−20.8342			−0.769529			−0.384765	−	0.923015i	$-0.625717\pi$
$733$	−20.8342			−0.769529			−0.384765	+	0.923015i	$0.625717\pi$
$734$	14.2598			0.526338
$735$		0			0
$736$	−3.83363			−0.141309
$737$	−36.7500			−1.35370
$738$		0			0
$739$	−30.9818			−1.13968			−0.569842	−	0.821754i	$-0.692996\pi$
$739$	−30.9818			−1.13968			−0.569842	+	0.821754i	$0.692996\pi$
$740$		0			0
$741$		0			0
$742$	−12.5444	−	21.0297i	−0.460520	−	0.772024i
$743$	45.8449			1.68189			0.840943	−	0.541124i	$-0.182001\pi$
$743$	45.8449			1.68189			0.840943	+	0.541124i	$0.182001\pi$
$744$		0			0
$745$		0			0
$746$		−	8.98745i		−	0.329054i
$747$		0			0
$748$	−4.15253			−0.151832
$749$	−8.56459	−	14.3579i	−0.312943	−	0.524624i
$750$		0			0
$751$	−35.2665			−1.28689			−0.643446	−	0.765492i	$-0.722496\pi$
$751$	−35.2665			−1.28689			−0.643446	+	0.765492i	$0.722496\pi$
$752$		−	9.11980i		−	0.332565i
$753$		0			0
$754$		−	25.0888i		−	0.913681i
$755$		0			0
$756$		0			0
$757$		−	12.3661i		−	0.449452i	−0.974422	−	0.224726i	$-0.927851\pi$
$757$		−	12.3661i		−	0.449452i	0.974422	−	0.224726i	$-0.0721488\pi$
$758$	34.5447			1.25472
$759$		0			0
$760$		0			0
$761$	−1.50580			−0.0545851			−0.0272926	−	0.999627i	$-0.508689\pi$
$761$	−1.50580			−0.0545851			−0.0272926	+	0.999627i	$0.508689\pi$
$762$		0			0
$763$	16.2987	+	27.3235i	0.590053	+	0.989177i
$764$			9.69823i			0.350870i
$765$		0			0
$766$		−	2.88020i		−	0.104066i
$767$	73.2716			2.64569
$768$		0			0
$769$		−	3.76558i		−	0.135790i	−0.997692	−	0.0678951i	$-0.978372\pi$
$769$		−	3.76558i		−	0.135790i	0.997692	−	0.0678951i	$-0.0216283\pi$
$770$		0			0
$771$		0			0
$772$			18.1525i			0.653324i
$773$			15.1911i			0.546388i	0.961959	+	0.273194i	$0.0880800\pi$
$773$			15.1911i			0.546388i	−0.961959	+	0.273194i	$0.911920\pi$
$774$		0			0
$775$		0			0
$776$	−16.9224			−0.607480
$777$		0			0
$778$		−	35.1620i		−	1.26062i
$779$		−	15.0637i		−	0.539714i
$780$		0			0
$781$	−6.23960			−0.223270
$782$		−	5.87257i		−	0.210003i
$783$		0			0
$784$	−3.32583	+	6.15945i	−0.118780	+	0.219980i
$785$		0			0
$786$		0			0
$787$	23.8198			0.849084			0.424542	−	0.905408i	$-0.360435\pi$
$787$	23.8198			0.849084			0.424542	+	0.905408i	$0.360435\pi$
$788$	−4.60354			−0.163994
$789$		0			0
$790$		0			0
$791$	12.2849	+	20.5946i	0.436800	+	0.732260i
$792$		0			0
$793$		−	31.3284i		−	1.11250i
$794$	16.1185			0.572024
$795$		0			0
$796$			11.7405i			0.416130i
$797$			10.6517i			0.377301i	0.982044	+	0.188650i	$0.0604113\pi$
$797$			10.6517i			0.377301i	−0.982044	+	0.188650i	$0.939589\pi$
$798$		0			0
$799$	13.9702			0.494231
$800$		0			0
$801$		0			0
$802$		−	16.6256i		−	0.587070i
$803$			30.9614i			1.09260i
$804$		0			0
$805$		0			0
$806$			21.3033i			0.750377i
$807$		0			0
$808$	17.7405			0.624108
$809$		−	32.8462i		−	1.15481i	−0.816458	−	0.577405i	$-0.804065\pi$
$809$		−	32.8462i		−	1.15481i	0.816458	−	0.577405i	$-0.195935\pi$
$810$		0			0
$811$			26.0734i			0.915562i	0.889065	+	0.457781i	$0.151356\pi$
$811$			26.0734i			0.915562i	−0.889065	+	0.457781i	$0.848644\pi$
$812$	−8.71078	+	5.19606i	−0.305689	+	0.182346i
$813$		0			0
$814$	−8.16637			−0.286231
$815$		0			0
$816$		0			0
$817$	−1.07756			−0.0376989
$818$		−	12.0000i		−	0.419570i
$819$		0			0
$820$		0			0
$821$			14.1352i			0.493321i	0.969102	+	0.246661i	$0.0793333\pi$
$821$			14.1352i			0.493321i	−0.969102	+	0.246661i	$0.920667\pi$
$822$		0			0
$823$			16.9114i			0.589496i	0.955575	+	0.294748i	$0.0952356\pi$
$823$			16.9114i			0.589496i	−0.955575	+	0.294748i	$0.904764\pi$
$824$	−4.05913			−0.141406
$825$		0			0
$826$	−15.1751	−	25.4397i	−0.528008	−	0.885162i
$827$	34.2959			1.19259			0.596293	−	0.802767i	$-0.296640\pi$
$827$	34.2959			1.19259			0.596293	+	0.802767i	$0.296640\pi$
$828$		0			0
$829$			53.4408i			1.85608i	0.372486	+	0.928038i	$0.378505\pi$
$829$			53.4408i			1.85608i	−0.372486	+	0.928038i	$0.621495\pi$
$830$		0			0
$831$		0			0
$832$	−6.54441			−0.226887
$833$	−9.43541	−	5.09469i	−0.326917	−	0.176521i
$834$		0			0
$835$		0			0
$836$	−6.23960			−0.215801
$837$		0			0
$838$	36.2849			1.25344
$839$	14.2898			0.493339			0.246669	−	0.969100i	$-0.420664\pi$
$839$	14.2898			0.493339			0.246669	+	0.969100i	$0.420664\pi$
$840$		0			0
$841$	14.3033			0.493218
$842$	19.2414			0.663101
$843$		0			0
$844$	4.00000			0.137686
$845$		0			0
$846$		0			0
$847$	4.94942	+	8.29731i	0.170064	+	0.285099i
$848$	−9.25519			−0.317825
$849$		0			0
$850$		0			0
$851$		−	11.5490i		−	0.395895i
$852$		0			0
$853$	−47.0118			−1.60966			−0.804828	−	0.593509i	$-0.797742\pi$
$853$	−47.0118			−1.60966			−0.804828	+	0.593509i	$0.797742\pi$
$854$	−10.8772	+	6.48833i	−0.372209	+	0.222026i
$855$		0			0
$856$	−6.31891			−0.215976
$857$		−	7.65929i		−	0.261636i	−0.991406	−	0.130818i	$-0.958240\pi$
$857$		−	7.65929i		−	0.261636i	0.991406	−	0.130818i	$-0.0417604\pi$
$858$		0			0
$859$		−	47.8559i		−	1.63282i	−0.577470	−	0.816412i	$-0.695960\pi$
$859$		−	47.8559i		−	1.63282i	0.577470	−	0.816412i	$-0.304040\pi$
$860$		0			0
$861$		0			0
$862$			17.2974i			0.589153i
$863$	−41.4922			−1.41241			−0.706206	−	0.708007i	$-0.749595\pi$
$863$	−41.4922			−1.41241			−0.706206	+	0.708007i	$0.749595\pi$
$864$		0			0
$865$		0			0
$866$	31.6473			1.07542
$867$		0			0
$868$	7.39646	−	4.41206i	0.251052	−	0.149755i
$869$			34.3639i			1.16572i
$870$		0			0
$871$		−	88.7223i		−	3.00624i
$872$	12.0251			0.407221
$873$		0			0
$874$		−	8.82412i		−	0.298480i
$875$		0			0
$876$		0			0
$877$			17.2925i			0.583927i	0.956429	+	0.291963i	$0.0943085\pi$
$877$			17.2925i			0.583927i	−0.956429	+	0.291963i	$0.905691\pi$
$878$			13.0095i			0.439050i
$879$		0			0
$880$		0			0
$881$	−31.7454			−1.06953			−0.534765	−	0.845001i	$-0.679600\pi$
$881$	−31.7454			−1.06953			−0.534765	+	0.845001i	$0.679600\pi$
$882$		0			0
$883$			2.07601i			0.0698634i	0.999390	+	0.0349317i	$0.0111214\pi$
$883$			2.07601i			0.0698634i	−0.999390	+	0.0349317i	$0.988879\pi$
$884$		−	10.0251i		−	0.337181i
$885$		0			0
$886$	3.78551			0.127177
$887$		−	5.18994i		−	0.174261i	−0.996197	−	0.0871305i	$-0.972230\pi$
$887$		−	5.18994i		−	0.174261i	0.996197	−	0.0871305i	$-0.0277697\pi$
$888$		0			0
$889$	−49.1551	+	29.3215i	−1.64861	+	0.983411i
$890$		0			0
$891$		0			0
$892$	26.4513			0.885654
$893$	20.9917			0.702459
$894$		0			0
$895$		0			0
$896$	1.35539	+	2.27220i	0.0452805	+	0.0759090i
$897$		0			0
$898$		−	24.9307i		−	0.831947i
$899$	12.4792			0.416204
$900$		0			0
$901$		−	14.1776i		−	0.472326i
$902$		−	17.7405i		−	0.590693i
$903$		0			0
$904$	9.06372			0.301455
$905$		0			0
$906$		0			0
$907$			32.9473i			1.09400i	0.837133	+	0.546999i	$0.184230\pi$
$907$			32.9473i			1.09400i	−0.837133	+	0.546999i	$0.815770\pi$
$908$			15.0637i			0.499907i
$909$		0			0
$910$		0			0
$911$			3.02356i			0.100175i	0.998745	+	0.0500875i	$0.0159500\pi$
$911$			3.02356i			0.100175i	−0.998745	+	0.0500875i	$0.984050\pi$
$912$		0			0
$913$	35.4809			1.17425
$914$		−	31.3535i		−	1.03708i
$915$		0			0
$916$			11.3655i			0.375526i
$917$	−3.41465	−	5.72438i	−0.112762	−	0.189036i
$918$		0			0
$919$	−13.9129			−0.458945			−0.229473	−	0.973315i	$-0.573700\pi$
$919$	−13.9129			−0.458945			−0.229473	+	0.973315i	$0.573700\pi$
$920$		0			0
$921$		0			0
$922$	9.52598			0.313721
$923$		−	15.0637i		−	0.495828i
$924$		0			0
$925$		0			0
$926$		−	34.0003i		−	1.11732i
$927$		0			0
$928$			3.83363i			0.125845i
$929$	8.15228			0.267468			0.133734	−	0.991017i	$-0.457303\pi$
$929$	8.15228			0.267468			0.133734	+	0.991017i	$0.457303\pi$
$930$		0			0
$931$	−14.1776	−	7.65529i	−0.464653	−	0.250892i
$932$	−20.9957			−0.687736
$933$		0			0
$934$			39.2665i			1.28484i
$935$		0			0
$936$		0			0
$937$	2.22576			0.0727122			0.0363561	−	0.999339i	$-0.488425\pi$
$937$	2.22576			0.0727122			0.0363561	+	0.999339i	$0.488425\pi$
$938$	−30.8042	+	18.3750i	−1.00579	+	0.599965i
$939$		0			0
$940$		0			0
$941$	−40.6517			−1.32521			−0.662603	−	0.748971i	$-0.730548\pi$
$941$	−40.6517			−1.32521			−0.662603	+	0.748971i	$0.730548\pi$
$942$		0			0
$943$	25.0888			0.817004
$944$	−11.1961			−0.364401
$945$		0			0
$946$	−1.26903			−0.0412598
$947$	34.4874			1.12069			0.560344	−	0.828260i	$-0.310669\pi$
$947$	34.4874			1.12069			0.560344	+	0.828260i	$0.310669\pi$
$948$		0			0
$949$	−74.7474			−2.42640
$950$		0			0
$951$		0			0
$952$	−3.48069	+	2.07627i	−0.112810	+	0.0672922i
$953$	23.9688			0.776426			0.388213	−	0.921570i	$-0.373093\pi$
$953$	23.9688			0.776426			0.388213	+	0.921570i	$0.373093\pi$
$954$		0			0
$955$		0			0
$956$			9.69823i			0.313663i
$957$		0			0
$958$	−34.0251			−1.09930
$959$	1.89275	+	3.17305i	0.0611202	+	0.102463i
$960$		0			0
$961$	20.4037			0.658185
$962$		−	19.7154i		−	0.635649i
$963$		0			0
$964$			24.7019i			0.795593i
$965$		0			0
$966$		0			0
$967$			6.45735i			0.207654i	0.994595	+	0.103827i	$0.0331089\pi$
$967$			6.45735i			0.207654i	−0.994595	+	0.103827i	$0.966891\pi$
$968$	3.65166			0.117369
$969$		0			0
$970$		0			0
$971$	−48.7471			−1.56437			−0.782185	−	0.623046i	$-0.785895\pi$
$971$	−48.7471			−1.56437			−0.782185	+	0.623046i	$0.785895\pi$
$972$		0			0
$973$	−13.9409	+	8.31586i	−0.446924	+	0.266594i
$974$		−	12.8091i		−	0.410430i
$975$		0			0
$976$			4.78705i			0.153230i
$977$	−14.4853			−0.463425			−0.231713	−	0.972784i	$-0.574433\pi$
$977$	−14.4853			−0.463425			−0.231713	+	0.972784i	$0.574433\pi$
$978$		0			0
$979$			26.0455i			0.832419i
$980$		0			0
$981$		0			0
$982$			11.6471i			0.371673i
$983$			46.7784i			1.49200i	0.665947	+	0.745999i	$0.268028\pi$
$983$			46.7784i			1.49200i	−0.665947	+	0.745999i	$0.731972\pi$
$984$		0			0
$985$		0			0
$986$	−5.87257			−0.187021
$987$		0			0
$988$		−	15.0637i		−	0.479241i
$989$		−	1.79468i		−	0.0570676i
$990$		0			0
$991$	−37.1292			−1.17945			−0.589724	−	0.807605i	$-0.700763\pi$
$991$	−37.1292			−1.17945			−0.589724	+	0.807605i	$0.700763\pi$
$992$		−	3.25519i		−	0.103352i
$993$		0			0
$994$	−5.23009	+	3.11980i	−0.165888	+	0.0989540i
$995$		0			0
$996$		0			0
$997$	35.5309			1.12527			0.562637	−	0.826704i	$-0.309787\pi$
$997$	35.5309			1.12527			0.562637	+	0.826704i	$0.309787\pi$
$998$	−35.5511			−1.12535
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.d.d.3149.6		8
3.2	odd	2		3150.2.d.a.3149.6		8
5.2	odd	4		3150.2.b.e.251.5		8
5.3	odd	4		630.2.b.a.251.4	✓	8
5.4	even	2		3150.2.d.c.3149.3		8
7.6	odd	2		3150.2.d.f.3149.4		8
15.2	even	4		3150.2.b.f.251.1		8
15.8	even	4		630.2.b.b.251.8	yes	8
15.14	odd	2		3150.2.d.f.3149.3		8
20.3	even	4		5040.2.f.f.881.2		8
21.20	even	2		3150.2.d.c.3149.4		8
35.13	even	4		630.2.b.b.251.4	yes	8
35.27	even	4		3150.2.b.f.251.5		8
35.34	odd	2		3150.2.d.a.3149.5		8
60.23	odd	4		5040.2.f.i.881.2		8
105.62	odd	4		3150.2.b.e.251.1		8
105.83	odd	4		630.2.b.a.251.8	yes	8
105.104	even	2	inner	3150.2.d.d.3149.5		8
140.83	odd	4		5040.2.f.i.881.1		8
420.83	even	4		5040.2.f.f.881.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.b.a.251.4	✓	8	5.3	odd	4
630.2.b.a.251.8	yes	8	105.83	odd	4
630.2.b.b.251.4	yes	8	35.13	even	4
630.2.b.b.251.8	yes	8	15.8	even	4
3150.2.b.e.251.1		8	105.62	odd	4
3150.2.b.e.251.5		8	5.2	odd	4
3150.2.b.f.251.1		8	15.2	even	4
3150.2.b.f.251.5		8	35.27	even	4
3150.2.d.a.3149.5		8	35.34	odd	2
3150.2.d.a.3149.6		8	3.2	odd	2
3150.2.d.c.3149.3		8	5.4	even	2
3150.2.d.c.3149.4		8	21.20	even	2
3150.2.d.d.3149.5		8	105.104	even	2	inner
3150.2.d.d.3149.6		8	1.1	even	1	trivial
3150.2.d.f.3149.3		8	15.14	odd	2
3150.2.d.f.3149.4		8	7.6	odd	2
5040.2.f.f.881.1		8	420.83	even	4
5040.2.f.f.881.2		8	20.3	even	4
5040.2.f.i.881.1		8	140.83	odd	4
5040.2.f.i.881.2		8	60.23	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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +(1.35539 + 2.27220i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +(1.35539 + 2.27220i) q^{7} +1.00000 q^{8} +2.71078i q^{11} -6.54441 q^{13} +(1.35539 + 2.27220i) q^{14} +1.00000 q^{16} +1.53186i q^{17} +2.30177i q^{19} +2.71078i q^{22} -3.83363 q^{23} -6.54441 q^{26} +(1.35539 + 2.27220i) q^{28} +3.83363i q^{29} -3.25519i q^{31} +1.00000 q^{32} +1.53186i q^{34} +3.01255i q^{37} +2.30177i q^{38} -6.54441 q^{41} +0.468142i q^{43} +2.71078i q^{44} -3.83363 q^{46} -9.11980i q^{47} +(-3.32583 + 6.15945i) q^{49} -6.54441 q^{52} -9.25519 q^{53} +(1.35539 + 2.27220i) q^{56} +3.83363i q^{58} -11.1961 q^{59} +4.78705i q^{61} -3.25519i q^{62} +1.00000 q^{64} +13.5570i q^{67} +1.53186i q^{68} +2.30177i q^{71} +11.4216 q^{73} +3.01255i q^{74} +2.30177i q^{76} +(-6.15945 + 3.67417i) q^{77} +12.6768 q^{79} -6.54441 q^{82} -13.0888i q^{83} +0.468142i q^{86} +2.71078i q^{88} +9.60812 q^{89} +(-8.87024 - 14.8702i) q^{91} -3.83363 q^{92} -9.11980i q^{94} -16.9224 q^{97} +(-3.32583 + 6.15945i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 + 8 * q^4 + 8 * q^8	\( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} - 8 q^{13} + 8 q^{16} - 8 q^{23} - 8 q^{26} + 8 q^{32} - 8 q^{41} - 8 q^{46} - 4 q^{49} - 8 q^{52} - 8 q^{53} + 8 q^{64} + 48 q^{73} - 4 q^{77} - 8 q^{79} - 8 q^{82} + 8 q^{89} - 4 q^{91} - 8 q^{92} - 24 q^{97} - 4 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 + 8 * q^8 - 8 * q^13 + 8 * q^16 - 8 * q^23 - 8 * q^26 + 8 * q^32 - 8 * q^41 - 8 * q^46 - 4 * q^49 - 8 * q^52 - 8 * q^53 + 8 * q^64 + 48 * q^73 - 4 * q^77 - 8 * q^79 - 8 * q^82 + 8 * q^89 - 4 * q^91 - 8 * q^92 - 24 * q^97 - 4 * q^98

Introduction

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