LMFDB - Embedded newform 3150.2.b.e.251.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(251,3150)]

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, 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("3150.251");

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.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:	$25.1528766367$
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^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			251.5
Root			$1.91681i$ of defining polynomial
Character	$\chi$	$=$	3150.251
Dual form			3150.2.b.e.251.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +(-2.27220 + 1.35539i) q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +(-2.27220 + 1.35539i) q^{7} -1.00000i q^{8} +2.71078i q^{11} +6.54441i q^{13} +(-1.35539 - 2.27220i) q^{14} +1.00000 q^{16} -1.53186 q^{17} -2.30177i q^{19} -2.71078 q^{22} +3.83363i q^{23} -6.54441 q^{26} +(2.27220 - 1.35539i) q^{28} -3.83363i q^{29} -3.25519i q^{31} +1.00000i q^{32} -1.53186i q^{34} -3.01255 q^{37} +2.30177 q^{38} -6.54441 q^{41} +0.468142 q^{43} -2.71078i q^{44} -3.83363 q^{46} +9.11980 q^{47} +(3.32583 - 6.15945i) q^{49} -6.54441i q^{52} +9.25519i q^{53} +(1.35539 + 2.27220i) q^{56} +3.83363 q^{58} +11.1961 q^{59} +4.78705i q^{61} +3.25519 q^{62} -1.00000 q^{64} -13.5570 q^{67} +1.53186 q^{68} +2.30177i q^{71} -11.4216i q^{73} -3.01255i q^{74} +2.30177i q^{76} +(-3.67417 - 6.15945i) q^{77} -12.6768 q^{79} -6.54441i q^{82} -13.0888 q^{83} +0.468142i q^{86} +2.71078 q^{88} -9.60812 q^{89} +(-8.87024 - 14.8702i) q^{91} -3.83363i q^{92} +9.11980i q^{94} -16.9224i q^{97} +(6.15945 + 3.32583i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} + 4 q^{7}+O(q^{10}) $ 8 * q - 8 * q^4 + 4 * q^7	$ 8 q - 8 q^{4} + 4 q^{7} + 8 q^{16} - 8 q^{26} - 4 q^{28} + 8 q^{37} + 8 q^{38} - 8 q^{41} + 16 q^{43} - 8 q^{46} + 40 q^{47} + 4 q^{49} + 8 q^{58} - 40 q^{62} - 8 q^{64} - 32 q^{67} - 52 q^{77} + 8 q^{79} - 16 q^{83} - 8 q^{89} - 4 q^{91} + 4 q^{98}+O(q^{100}) $ 8 * q - 8 * q^4 + 4 * q^7 + 8 * q^16 - 8 * q^26 - 4 * q^28 + 8 * q^37 + 8 * q^38 - 8 * q^41 + 16 * q^43 - 8 * q^46 + 40 * q^47 + 4 * q^49 + 8 * q^58 - 40 * q^62 - 8 * q^64 - 32 * q^67 - 52 * q^77 + 8 * q^79 - 16 * q^83 - 8 * q^89 - 4 * q^91 + 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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.27220	+	1.35539i	−0.858813	+	0.512290i
$7$	−2.27220	+	1.35539i	−0.858813	+	0.512290i
$8$		−	1.00000i		−	0.353553i
$9$		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
$13$			6.54441i			1.81509i	0.419952	+	0.907546i	$0.362047\pi$
$13$			6.54441i			1.81509i	−0.419952	+	0.907546i	$0.637953\pi$
$14$	−1.35539	−	2.27220i	−0.362244	−	0.607272i
$15$		0			0
$16$	1.00000			0.250000
$17$	−1.53186			−0.371530			−0.185765	−	0.982594i	$-0.559476\pi$
$17$	−1.53186			−0.371530			−0.185765	+	0.982594i	$0.559476\pi$
$18$		0			0
$19$		−	2.30177i		−	0.528062i	−0.964514	−	0.264031i	$-0.914948\pi$
$19$		−	2.30177i		−	0.528062i	0.964514	−	0.264031i	$-0.0850521\pi$
$20$		0			0
$21$		0			0
$22$	−2.71078			−0.577941
$23$			3.83363i			0.799366i	0.916653	+	0.399683i	$0.130880\pi$
$23$			3.83363i			0.799366i	−0.916653	+	0.399683i	$0.869120\pi$
$24$		0			0
$25$		0			0
$26$	−6.54441			−1.28346
$27$		0			0
$28$	2.27220	−	1.35539i	0.429406	−	0.256145i
$29$		−	3.83363i		−	0.711887i	−0.934508	−	0.355943i	$-0.884160\pi$
$29$		−	3.83363i		−	0.711887i	0.934508	−	0.355943i	$-0.115840\pi$
$30$		0			0
$31$		−	3.25519i		−	0.584650i	−0.956319	−	0.292325i	$-0.905571\pi$
$31$		−	3.25519i		−	0.584650i	0.956319	−	0.292325i	$-0.0944289\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$		−	1.53186i		−	0.262711i
$35$		0			0
$36$		0			0
$37$	−3.01255			−0.495260			−0.247630	−	0.968855i	$-0.579652\pi$
$37$	−3.01255			−0.495260			−0.247630	+	0.968855i	$0.579652\pi$
$38$	2.30177			0.373396
$39$		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
$43$	0.468142			0.0713910			0.0356955	−	0.999363i	$-0.488635\pi$
$43$	0.468142			0.0713910			0.0356955	+	0.999363i	$0.488635\pi$
$44$		−	2.71078i		−	0.408666i
$45$		0			0
$46$	−3.83363			−0.565237
$47$	9.11980			1.33026			0.665130	−	0.746728i	$-0.268376\pi$
$47$	9.11980			1.33026			0.665130	+	0.746728i	$0.268376\pi$
$48$		0			0
$49$	3.32583	−	6.15945i	0.475118	−	0.879922i
$50$		0			0
$51$		0			0
$52$		−	6.54441i		−	0.907546i
$53$			9.25519i			1.27130i	0.771978	+	0.635649i	$0.219268\pi$
$53$			9.25519i			1.27130i	−0.771978	+	0.635649i	$0.780732\pi$
$54$		0			0
$55$		0			0
$56$	1.35539	+	2.27220i	0.181122	+	0.303636i
$57$		0			0
$58$	3.83363			0.503380
$59$	11.1961			1.45760			0.728802	−	0.684725i	$-0.240078\pi$
$59$	11.1961			1.45760			0.728802	+	0.684725i	$0.240078\pi$
$60$		0			0
$61$			4.78705i			0.612919i	0.951884	+	0.306459i	$0.0991444\pi$
$61$			4.78705i			0.612919i	−0.951884	+	0.306459i	$0.900856\pi$
$62$	3.25519			0.413410
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$	−13.5570			−1.65625			−0.828123	−	0.560546i	$-0.810591\pi$
$67$	−13.5570			−1.65625			−0.828123	+	0.560546i	$0.810591\pi$
$68$	1.53186			0.185765
$69$		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
$73$		−	11.4216i		−	1.33679i	−0.743805	−	0.668397i	$-0.766981\pi$
$73$		−	11.4216i		−	1.33679i	0.743805	−	0.668397i	$-0.233019\pi$
$74$		−	3.01255i		−	0.350202i
$75$		0			0
$76$			2.30177i			0.264031i
$77$	−3.67417	−	6.15945i	−0.418711	−	0.701935i
$78$		0			0
$79$	−12.6768			−1.42625			−0.713123	−	0.701039i	$-0.752720\pi$
$79$	−12.6768			−1.42625			−0.713123	+	0.701039i	$0.752720\pi$
$80$		0			0
$81$		0			0
$82$		−	6.54441i		−	0.722709i
$83$	−13.0888			−1.43668			−0.718342	−	0.695690i	$-0.755099\pi$
$83$	−13.0888			−1.43668			−0.718342	+	0.695690i	$0.755099\pi$
$84$		0			0
$85$		0			0
$86$			0.468142i			0.0504811i
$87$		0			0
$88$	2.71078			0.288970
$89$	−9.60812			−1.01846			−0.509230	−	0.860631i	$-0.670070\pi$
$89$	−9.60812			−1.01846			−0.509230	+	0.860631i	$0.670070\pi$
$90$		0			0
$91$	−8.87024	−	14.8702i	−0.929853	−	1.55882i
$92$		−	3.83363i		−	0.399683i
$93$		0			0
$94$			9.11980i			0.940635i
$95$		0			0
$96$		0			0
$97$		−	16.9224i		−	1.71821i	−0.511796	−	0.859107i	$-0.671020\pi$
$97$		−	16.9224i		−	1.71821i	0.511796	−	0.859107i	$-0.328980\pi$
$98$	6.15945	+	3.32583i	0.622199	+	0.335959i
$99$		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
$103$			4.05913i			0.399958i	0.979800	+	0.199979i	$0.0640874\pi$
$103$			4.05913i			0.399958i	−0.979800	+	0.199979i	$0.935913\pi$
$104$	6.54441			0.641732
$105$		0			0
$106$	−9.25519			−0.898944
$107$		−	6.31891i		−	0.610872i	−0.952213	−	0.305436i	$-0.901198\pi$
$107$		−	6.31891i		−	0.610872i	0.952213	−	0.305436i	$-0.0988022\pi$
$108$		0			0
$109$	−12.0251			−1.15180			−0.575898	−	0.817522i	$-0.695347\pi$
$109$	−12.0251			−1.15180			−0.575898	+	0.817522i	$0.695347\pi$
$110$		0			0
$111$		0			0
$112$	−2.27220	+	1.35539i	−0.214703	+	0.128072i
$113$		−	9.06372i		−	0.852643i	−0.904572	−	0.426321i	$-0.859809\pi$
$113$		−	9.06372i		−	0.852643i	0.904572	−	0.426321i	$-0.140191\pi$
$114$		0			0
$115$		0			0
$116$			3.83363i			0.355943i
$117$		0			0
$118$			11.1961i			1.03068i
$119$	3.48069	−	2.07627i	0.319075	−	0.190331i
$120$		0			0
$121$	3.65166			0.331969
$122$	−4.78705			−0.433399
$123$		0			0
$124$			3.25519i			0.292325i
$125$		0			0
$126$		0			0
$127$	−21.6332			−1.91964			−0.959819	−	0.280619i	$-0.909460\pi$
$127$	−21.6332			−1.91964			−0.959819	+	0.280619i	$0.909460\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		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
$133$	3.11980	+	5.23009i	0.270521	+	0.453506i
$134$		−	13.5570i		−	1.17114i
$135$		0			0
$136$			1.53186i			0.131356i
$137$			1.39646i			0.119308i	0.998219	+	0.0596539i	$0.0189997\pi$
$137$			1.39646i			0.119308i	−0.998219	+	0.0596539i	$0.981000\pi$
$138$		0			0
$139$		−	6.13539i		−	0.520397i	−0.965555	−	0.260199i	$-0.916212\pi$
$139$		−	6.13539i		−	0.520397i	0.965555	−	0.260199i	$-0.0837881\pi$
$140$		0			0
$141$		0			0
$142$	−2.30177			−0.193160
$143$	−17.7405			−1.48353
$144$		0			0
$145$		0			0
$146$	11.4216			0.945256
$147$		0			0
$148$	3.01255			0.247630
$149$		−	4.65166i		−	0.381078i	−0.981680	−	0.190539i	$-0.938976\pi$
$149$		−	4.65166i		−	0.381078i	0.981680	−	0.190539i	$-0.0610236\pi$
$150$		0			0
$151$	−3.58794			−0.291982			−0.145991	−	0.989286i	$-0.546637\pi$
$151$	−3.58794			−0.291982			−0.145991	+	0.989286i	$0.546637\pi$
$152$	−2.30177			−0.186698
$153$		0			0
$154$	6.15945	−	3.67417i	0.496343	−	0.296073i
$155$		0			0
$156$		0			0
$157$		−	13.1228i		−	1.04732i	−0.851928	−	0.523658i	$-0.824567\pi$
$157$		−	13.1228i		−	1.04732i	0.851928	−	0.523658i	$-0.175433\pi$
$158$		−	12.6768i		−	1.00851i
$159$		0			0
$160$		0			0
$161$	−5.19606	−	8.71078i	−0.409507	−	0.686506i
$162$		0			0
$163$	−16.6207			−1.30183			−0.650916	−	0.759150i	$-0.725615\pi$
$163$	−16.6207			−1.30183			−0.650916	+	0.759150i	$0.725615\pi$
$164$	6.54441			0.511033
$165$		0			0
$166$		−	13.0888i		−	1.01589i
$167$	−8.62068			−0.667088			−0.333544	−	0.942735i	$-0.608245\pi$
$167$	−8.62068			−0.667088			−0.333544	+	0.942735i	$0.608245\pi$
$168$		0			0
$169$	−29.8293			−2.29456
$170$		0			0
$171$		0			0
$172$	−0.468142			−0.0356955
$173$	10.6517			0.809830			0.404915	−	0.914354i	$-0.367301\pi$
$173$	10.6517			0.809830			0.404915	+	0.914354i	$0.367301\pi$
$174$		0			0
$175$		0			0
$176$			2.71078i			0.204333i
$177$		0			0
$178$		−	9.60812i		−	0.720159i
$179$			23.1961i			1.73376i	0.498521	+	0.866878i	$0.333877\pi$
$179$			23.1961i			1.73376i	−0.498521	+	0.866878i	$0.666123\pi$
$180$		0			0
$181$			9.11980i			0.677869i	0.940810	+	0.338935i	$0.110067\pi$
$181$			9.11980i			0.677869i	−0.940810	+	0.338935i	$0.889933\pi$
$182$	14.8702	−	8.87024i	1.10226	−	0.657506i
$183$		0			0
$184$	3.83363			0.282619
$185$		0			0
$186$		0			0
$187$		−	4.15253i		−	0.303663i
$188$	−9.11980			−0.665130
$189$		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
$193$	18.1525			1.30665			0.653324	−	0.757078i	$-0.273374\pi$
$193$	18.1525			1.30665			0.653324	+	0.757078i	$0.273374\pi$
$194$	16.9224			1.21496
$195$		0			0
$196$	−3.32583	+	6.15945i	−0.237559	+	0.439961i
$197$		−	4.60354i		−	0.327988i	−0.986461	−	0.163994i	$-0.947562\pi$
$197$		−	4.60354i		−	0.327988i	0.986461	−	0.163994i	$-0.0524378\pi$
$198$		0			0
$199$		−	11.7405i		−	0.832260i	−0.909305	−	0.416130i	$-0.863386\pi$
$199$		−	11.7405i		−	0.832260i	0.909305	−	0.416130i	$-0.136614\pi$
$200$		0			0
$201$		0			0
$202$			17.7405i			1.24822i
$203$	5.19606	+	8.71078i	0.364692	+	0.611377i
$204$		0			0
$205$		0			0
$206$	−4.05913			−0.282813
$207$		0			0
$208$			6.54441i			0.453773i
$209$	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	$0.456034\pi$
$212$		−	9.25519i		−	0.635649i
$213$		0			0
$214$	6.31891			0.431952
$215$		0			0
$216$		0			0
$217$	4.41206	+	7.39646i	0.299510	+	0.502105i
$218$		−	12.0251i		−	0.814443i
$219$		0			0
$220$		0			0
$221$		−	10.0251i		−	0.674361i
$222$		0			0
$223$		−	26.4513i		−	1.77131i	−0.464347	−	0.885654i	$-0.653711\pi$
$223$		−	26.4513i		−	1.77131i	0.464347	−	0.885654i	$-0.346289\pi$
$224$	−1.35539	−	2.27220i	−0.0905609	−	0.151818i
$225$		0			0
$226$	9.06372			0.602909
$227$	−15.0637			−0.999814			−0.499907	−	0.866079i	$-0.666632\pi$
$227$	−15.0637			−0.999814			−0.499907	+	0.866079i	$0.666632\pi$
$228$		0			0
$229$		−	11.3655i		−	0.751052i	−0.926812	−	0.375526i	$-0.877462\pi$
$229$		−	11.3655i		−	0.751052i	0.926812	−	0.375526i	$-0.122538\pi$
$230$		0			0
$231$		0			0
$232$	−3.83363			−0.251690
$233$			20.9957i			1.37547i	0.725961	+	0.687736i	$0.241395\pi$
$233$			20.9957i			1.37547i	−0.725961	+	0.687736i	$0.758605\pi$
$234$		0			0
$235$		0			0
$236$	−11.1961			−0.728802
$237$		0			0
$238$	2.07627	+	3.48069i	0.134584	+	0.225620i
$239$		−	9.69823i		−	0.627326i	−0.949534	−	0.313663i	$-0.898444\pi$
$239$		−	9.69823i		−	0.627326i	0.949534	−	0.313663i	$-0.101556\pi$
$240$		0			0
$241$			24.7019i			1.59119i	0.605831	+	0.795593i	$0.292841\pi$
$241$			24.7019i			1.59119i	−0.605831	+	0.795593i	$0.707159\pi$
$242$			3.65166i			0.234737i
$243$		0			0
$244$		−	4.78705i		−	0.306459i
$245$		0			0
$246$		0			0
$247$	15.0637			0.958481
$248$	−3.25519			−0.206705
$249$		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
$253$	−10.3921			−0.653348
$254$		−	21.6332i		−	1.35739i
$255$		0			0
$256$	1.00000			0.0625000
$257$	28.7983			1.79639			0.898195	−	0.439598i	$-0.144879\pi$
$257$	28.7983			1.79639			0.898195	+	0.439598i	$0.144879\pi$
$258$		0			0
$259$	6.84513	−	4.08319i	0.425336	−	0.253717i
$260$		0			0
$261$		0			0
$262$		−	2.51931i		−	0.155643i
$263$			12.7699i			0.787426i	0.919233	+	0.393713i	$0.128810\pi$
$263$			12.7699i			0.787426i	−0.919233	+	0.393713i	$0.871190\pi$
$264$		0			0
$265$		0			0
$266$	−5.23009	+	3.11980i	−0.320677	+	0.191287i
$267$		0			0
$268$	13.5570			0.828123
$269$	−8.43716			−0.514423			−0.257211	−	0.966355i	$-0.582804\pi$
$269$	−8.43716			−0.514423			−0.257211	+	0.966355i	$0.582804\pi$
$270$		0			0
$271$			2.16637i			0.131598i	0.997833	+	0.0657989i	$0.0209596\pi$
$271$			2.16637i			0.131598i	−0.997833	+	0.0657989i	$0.979040\pi$
$272$	−1.53186			−0.0928825
$273$		0			0
$274$	−1.39646			−0.0843634
$275$		0			0
$276$		0			0
$277$	−5.92373			−0.355923			−0.177961	−	0.984037i	$-0.556950\pi$
$277$	−5.92373			−0.355923			−0.177961	+	0.984037i	$0.556950\pi$
$278$	6.13539			0.367977
$279$		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
$283$			18.0594i			1.07352i	0.843735	+	0.536759i	$0.180352\pi$
$283$			18.0594i			1.07352i	−0.843735	+	0.536759i	$0.819648\pi$
$284$		−	2.30177i		−	0.136585i
$285$		0			0
$286$		−	17.7405i		−	1.04902i
$287$	14.8702	−	8.87024i	0.877762	−	0.523594i
$288$		0			0
$289$	−14.6534			−0.861965
$290$		0			0
$291$		0			0
$292$			11.4216i			0.668397i
$293$	29.1139			1.70085			0.850427	−	0.526094i	$-0.176344\pi$
$293$	29.1139			1.70085			0.850427	+	0.526094i	$0.176344\pi$
$294$		0			0
$295$		0			0
$296$			3.01255i			0.175101i
$297$		0			0
$298$	4.65166			0.269463
$299$	−25.0888			−1.45092
$300$		0			0
$301$	−1.06372	+	0.634516i	−0.0613115	+	0.0365729i
$302$		−	3.58794i		−	0.206463i
$303$		0			0
$304$		−	2.30177i		−	0.132015i
$305$		0			0
$306$		0			0
$307$		−	7.39646i		−	0.422138i	−0.977471	−	0.211069i	$-0.932305\pi$
$307$		−	7.39646i		−	0.422138i	0.977471	−	0.211069i	$-0.0676945\pi$
$308$	3.67417	+	6.15945i	0.209355	+	0.350967i
$309$		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
$313$		−	0.961388i		−	0.0543409i	−0.999631	−	0.0271704i	$-0.991350\pi$
$313$		−	0.961388i		−	0.0543409i	0.999631	−	0.0271704i	$-0.00864968\pi$
$314$	13.1228			0.740565
$315$		0			0
$316$	12.6768			0.713123
$317$		−	17.1620i		−	0.963916i	−0.876194	−	0.481958i	$-0.839926\pi$
$317$		−	17.1620i		−	0.963916i	0.876194	−	0.481958i	$-0.160074\pi$
$318$		0			0
$319$	10.3921			0.581848
$320$		0			0
$321$		0			0
$322$	8.71078	−	5.19606i	0.485433	−	0.289565i
$323$			3.52598i			0.196191i
$324$		0			0
$325$		0			0
$326$		−	16.6207i		−	0.920534i
$327$		0			0
$328$			6.54441i			0.361355i
$329$	−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	$0.413739\pi$
$332$	13.0888			0.718342
$333$		0			0
$334$		−	8.62068i		−	0.471702i
$335$		0			0
$336$		0			0
$337$	−3.15078			−0.171634			−0.0858169	−	0.996311i	$-0.527350\pi$
$337$	−3.15078			−0.171634			−0.0858169	+	0.996311i	$0.527350\pi$
$338$		−	29.8293i		−	1.62250i
$339$		0			0
$340$		0			0
$341$	8.82412			0.477853
$342$		0			0
$343$	0.791511	+	18.5033i	0.0427376	+	0.999086i
$344$		−	0.468142i		−	0.0252405i
$345$		0			0
$346$			10.6517i			0.572637i
$347$			30.0594i			1.61367i	0.590775	+	0.806836i	$0.298822\pi$
$347$			30.0594i			1.61367i	−0.590775	+	0.806836i	$0.701178\pi$
$348$		0			0
$349$			27.5180i			1.47301i	0.676434	+	0.736503i	$0.263524\pi$
$349$			27.5180i			1.47301i	−0.676434	+	0.736503i	$0.736476\pi$
$350$		0			0
$351$		0			0
$352$	−2.71078			−0.144485
$353$	−16.5956			−0.883293			−0.441647	−	0.897189i	$-0.645605\pi$
$353$	−16.5956			−0.883293			−0.441647	+	0.897189i	$0.645605\pi$
$354$		0			0
$355$		0			0
$356$	9.60812			0.509230
$357$		0			0
$358$	−23.1961			−1.22595
$359$		−	15.0076i		−	0.792073i	−0.918235	−	0.396036i	$-0.870385\pi$
$359$		−	15.0076i		−	0.792073i	0.918235	−	0.396036i	$-0.129615\pi$
$360$		0			0
$361$	13.7019			0.721151
$362$	−9.11980			−0.479326
$363$		0			0
$364$	8.87024	+	14.8702i	0.464927	+	0.779412i
$365$		0			0
$366$		0			0
$367$			14.2598i			0.744354i	0.928162	+	0.372177i	$0.121389\pi$
$367$			14.2598i			0.744354i	−0.928162	+	0.372177i	$0.878611\pi$
$368$			3.83363i			0.199842i
$369$		0			0
$370$		0			0
$371$	−12.5444	−	21.0297i	−0.651273	−	1.09181i
$372$		0			0
$373$	−8.98745			−0.465352			−0.232676	−	0.972554i	$-0.574748\pi$
$373$	−8.98745			−0.465352			−0.232676	+	0.972554i	$0.574748\pi$
$374$	4.15253			0.214722
$375$		0			0
$376$		−	9.11980i		−	0.470318i
$377$	25.0888			1.29214
$378$		0			0
$379$	−34.5447			−1.77444			−0.887220	−	0.461346i	$-0.847367\pi$
$379$	−34.5447			−1.77444			−0.887220	+	0.461346i	$0.847367\pi$
$380$		0			0
$381$		0			0
$382$	−9.69823			−0.496205
$383$	−2.88020			−0.147171			−0.0735857	−	0.997289i	$-0.523444\pi$
$383$	−2.88020			−0.147171			−0.0735857	+	0.997289i	$0.523444\pi$
$384$		0			0
$385$		0			0
$386$			18.1525i			0.923940i
$387$		0			0
$388$			16.9224i			0.859107i
$389$			35.1620i			1.78279i	0.453231	+	0.891393i	$0.350271\pi$
$389$			35.1620i			1.78279i	−0.453231	+	0.891393i	$0.649729\pi$
$390$		0			0
$391$		−	5.87257i		−	0.296989i
$392$	−6.15945	−	3.32583i	−0.311099	−	0.167980i
$393$		0			0
$394$	4.60354			0.231923
$395$		0			0
$396$		0			0
$397$			16.1185i			0.808965i	0.914546	+	0.404482i	$0.132548\pi$
$397$			16.1185i			0.808965i	−0.914546	+	0.404482i	$0.867452\pi$
$398$	11.7405			0.588497
$399$		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
$403$	21.3033			1.06119
$404$	−17.7405			−0.882622
$405$		0			0
$406$	−8.71078	+	5.19606i	−0.432309	+	0.257876i
$407$		−	8.16637i		−	0.404792i
$408$		0			0
$409$			12.0000i			0.593362i	0.954977	+	0.296681i	$0.0958798\pi$
$409$			12.0000i			0.593362i	−0.954977	+	0.296681i	$0.904120\pi$
$410$		0			0
$411$		0			0
$412$		−	4.05913i		−	0.199979i
$413$	−25.4397	+	15.1751i	−1.25181	+	0.746715i
$414$		0			0
$415$		0			0
$416$	−6.54441			−0.320866
$417$		0			0
$418$			6.23960i			0.305189i
$419$	−36.2849			−1.77263			−0.886316	−	0.463080i	$-0.846744\pi$
$419$	−36.2849			−1.77263			−0.886316	+	0.463080i	$0.846744\pi$
$420$		0			0
$421$	19.2414			0.937766			0.468883	−	0.883260i	$-0.344657\pi$
$421$	19.2414			0.937766			0.468883	+	0.883260i	$0.344657\pi$
$422$			4.00000i			0.194717i
$423$		0			0
$424$	9.25519			0.449472
$425$		0			0
$426$		0			0
$427$	−6.48833	−	10.8772i	−0.313992	−	0.526383i
$428$			6.31891i			0.305436i
$429$		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
$433$		−	31.6473i		−	1.52087i	−0.649412	−	0.760437i	$-0.724985\pi$
$433$		−	31.6473i		−	1.52087i	0.649412	−	0.760437i	$-0.275015\pi$
$434$	−7.39646	+	4.41206i	−0.355042	+	0.211786i
$435$		0			0
$436$	12.0251			0.575898
$437$	8.82412			0.422115
$438$		0			0
$439$		−	13.0095i		−	0.620910i	−0.950588	−	0.310455i	$-0.899519\pi$
$439$		−	13.0095i		−	0.620910i	0.950588	−	0.310455i	$-0.100481\pi$
$440$		0			0
$441$		0			0
$442$	10.0251			0.476846
$443$		−	3.78551i		−	0.179855i	−0.995948	−	0.0899275i	$-0.971336\pi$
$443$		−	3.78551i		−	0.179855i	0.995948	−	0.0899275i	$-0.0286635\pi$
$444$		0			0
$445$		0			0
$446$	26.4513			1.25250
$447$		0			0
$448$	2.27220	−	1.35539i	0.107352	−	0.0640362i
$449$			24.9307i			1.17655i	0.808661	+	0.588275i	$0.200193\pi$
$449$			24.9307i			1.17655i	−0.808661	+	0.588275i	$0.799807\pi$
$450$		0			0
$451$		−	17.7405i		−	0.835366i
$452$			9.06372i			0.426321i
$453$		0			0
$454$		−	15.0637i		−	0.706975i
$455$		0			0
$456$		0			0
$457$	31.3535			1.46666			0.733328	−	0.679875i	$-0.237966\pi$
$457$	31.3535			1.46666			0.733328	+	0.679875i	$0.237966\pi$
$458$	11.3655			0.531074
$459$		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
$463$	−34.0003			−1.58013			−0.790063	−	0.613026i	$-0.789952\pi$
$463$	−34.0003			−1.58013			−0.790063	+	0.613026i	$0.789952\pi$
$464$		−	3.83363i		−	0.177972i
$465$		0			0
$466$	−20.9957			−0.972605
$467$	−39.2665			−1.81703			−0.908517	−	0.417847i	$-0.862785\pi$
$467$	−39.2665			−1.81703			−0.908517	+	0.417847i	$0.862785\pi$
$468$		0			0
$469$	30.8042	−	18.3750i	1.42241	−	0.848478i
$470$		0			0
$471$		0			0
$472$		−	11.1961i		−	0.515341i
$473$			1.26903i			0.0583502i
$474$		0			0
$475$		0			0
$476$	−3.48069	+	2.07627i	−0.159537	+	0.0951655i
$477$		0			0
$478$	9.69823			0.443587
$479$	34.0251			1.55465			0.777323	−	0.629101i	$-0.216577\pi$
$479$	34.0251			1.55465			0.777323	+	0.629101i	$0.216577\pi$
$480$		0			0
$481$		−	19.7154i		−	0.898944i
$482$	−24.7019			−1.12514
$483$		0			0
$484$	−3.65166			−0.165984
$485$		0			0
$486$		0			0
$487$	12.8091			0.580436			0.290218	−	0.956961i	$-0.406272\pi$
$487$	12.8091			0.580436			0.290218	+	0.956961i	$0.406272\pi$
$488$	4.78705			0.216700
$489$		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
$493$			5.87257i			0.264487i
$494$			15.0637i			0.677749i
$495$		0			0
$496$		−	3.25519i		−	0.146162i
$497$	−3.11980	−	5.23009i	−0.139942	−	0.234602i
$498$		0			0
$499$	35.5511			1.59149			0.795743	−	0.605635i	$-0.207081\pi$
$499$	35.5511			1.59149			0.795743	+	0.605635i	$0.207081\pi$
$500$		0			0
$501$		0			0
$502$			3.60812i			0.161038i
$503$	−8.23372			−0.367123			−0.183562	−	0.983008i	$-0.558763\pi$
$503$	−8.23372			−0.367123			−0.183562	+	0.983008i	$0.558763\pi$
$504$		0			0
$505$		0			0
$506$		−	10.3921i		−	0.461986i
$507$		0			0
$508$	21.6332			0.959819
$509$	31.0888			1.37799			0.688994	−	0.724767i	$-0.258053\pi$
$509$	31.0888			1.37799			0.688994	+	0.724767i	$0.258053\pi$
$510$		0			0
$511$	15.4807	+	25.9521i	0.684826	+	1.14805i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$			28.7983i			1.27024i
$515$		0			0
$516$		0			0
$517$			24.7218i			1.08726i
$518$	4.08319	+	6.84513i	0.179405	+	0.300758i
$519$		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
$523$		−	20.4853i		−	0.895759i	−0.894094	−	0.447879i	$-0.852179\pi$
$523$		−	20.4853i		−	0.895759i	0.894094	−	0.447879i	$-0.147821\pi$
$524$	2.51931			0.110056
$525$		0			0
$526$	−12.7699			−0.556795
$527$			4.98649i			0.217215i
$528$		0			0
$529$	8.30331			0.361013
$530$		0			0
$531$		0			0
$532$	−3.11980	−	5.23009i	−0.135260	−	0.226753i
$533$		−	42.8293i		−	1.85514i
$534$		0			0
$535$		0			0
$536$			13.5570i			0.585572i
$537$		0			0
$538$		−	8.43716i		−	0.363752i
$539$	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	$0.500596\pi$
$542$	−2.16637			−0.0930537
$543$		0			0
$544$		−	1.53186i		−	0.0656779i
$545$		0			0
$546$		0			0
$547$	5.40443			0.231077			0.115538	−	0.993303i	$-0.463141\pi$
$547$	5.40443			0.231077			0.115538	+	0.993303i	$0.463141\pi$
$548$		−	1.39646i		−	0.0596539i
$549$		0			0
$550$		0			0
$551$	−8.82412			−0.375920
$552$		0			0
$553$	28.8042	−	17.1820i	1.22488	−	0.730652i
$554$		−	5.92373i		−	0.251675i
$555$		0			0
$556$			6.13539i			0.260199i
$557$			0.127431i			0.00539940i	0.999996	+	0.00269970i	$0.000859343\pi$
$557$			0.127431i			0.00539940i	−0.999996	+	0.00269970i	$0.999141\pi$
$558$		0			0
$559$			3.06372i			0.129581i
$560$		0			0
$561$		0			0
$562$	27.3566			1.15397
$563$	6.23960			0.262968			0.131484	−	0.991318i	$-0.458026\pi$
$563$	6.23960			0.262968			0.131484	+	0.991318i	$0.458026\pi$
$564$		0			0
$565$		0			0
$566$	−18.0594			−0.759092
$567$		0			0
$568$	2.30177			0.0965801
$569$		−	11.6391i		−	0.487937i	−0.969783	−	0.243968i	$-0.921551\pi$
$569$		−	11.6391i		−	0.487937i	0.969783	−	0.243968i	$-0.0784493\pi$
$570$		0			0
$571$	−35.9181			−1.50313			−0.751563	−	0.659661i	$-0.770700\pi$
$571$	−35.9181			−1.50313			−0.751563	+	0.659661i	$0.770700\pi$
$572$	17.7405			0.741766
$573$		0			0
$574$	8.87024	+	14.8702i	0.370237	+	0.620672i
$575$		0			0
$576$		0			0
$577$			7.15687i			0.297944i	0.988841	+	0.148972i	$0.0475965\pi$
$577$			7.15687i			0.297944i	−0.988841	+	0.148972i	$0.952404\pi$
$578$		−	14.6534i		−	0.609502i
$579$		0			0
$580$		0			0
$581$	29.7405	−	17.7405i	1.23384	−	0.735999i
$582$		0			0
$583$	−25.0888			−1.03907
$584$	−11.4216			−0.472628
$585$		0			0
$586$			29.1139i			1.20269i
$587$	4.15253			0.171393			0.0856967	−	0.996321i	$-0.472688\pi$
$587$	4.15253			0.171393			0.0856967	+	0.996321i	$0.472688\pi$
$588$		0			0
$589$	−7.49270			−0.308731
$590$		0			0
$591$		0			0
$592$	−3.01255			−0.123815
$593$	−29.7966			−1.22360			−0.611799	−	0.791013i	$-0.709554\pi$
$593$	−29.7966			−1.22360			−0.611799	+	0.791013i	$0.709554\pi$
$594$		0			0
$595$		0			0
$596$			4.65166i			0.190539i
$597$		0			0
$598$		−	25.0888i		−	1.02596i
$599$			35.4249i			1.44742i	0.690104	+	0.723710i	$0.257565\pi$
$599$			35.4249i			1.44742i	−0.690104	+	0.723710i	$0.742435\pi$
$600$		0			0
$601$			27.4948i			1.12154i	0.827973	+	0.560768i	$0.189494\pi$
$601$			27.4948i			1.12154i	−0.827973	+	0.560768i	$0.810506\pi$
$602$	−0.634516	−	1.06372i	−0.0258609	−	0.0433538i
$603$		0			0
$604$	3.58794			0.145991
$605$		0			0
$606$		0			0
$607$			4.76499i			0.193405i	0.995313	+	0.0967025i	$0.0308296\pi$
$607$			4.76499i			0.193405i	−0.995313	+	0.0967025i	$0.969170\pi$
$608$	2.30177			0.0933490
$609$		0			0
$610$		0			0
$611$			59.6837i			2.41454i
$612$		0			0
$613$	−10.7981			−0.436130			−0.218065	−	0.975934i	$-0.569974\pi$
$613$	−10.7981			−0.436130			−0.218065	+	0.975934i	$0.569974\pi$
$614$	7.39646			0.298497
$615$		0			0
$616$	−6.15945	+	3.67417i	−0.248171	+	0.148037i
$617$			34.9025i			1.40512i	0.711623	+	0.702561i	$0.247960\pi$
$617$			34.9025i			1.40512i	−0.711623	+	0.702561i	$0.752040\pi$
$618$		0			0
$619$		−	1.26107i		−	0.0506866i	−0.999679	−	0.0253433i	$-0.991932\pi$
$619$		−	1.26107i		−	0.0506866i	0.999679	−	0.0253433i	$-0.00806789\pi$
$620$		0			0
$621$		0			0
$622$			1.97490i			0.0791861i
$623$	21.8316	−	13.0228i	0.874666	−	0.521746i
$624$		0			0
$625$		0			0
$626$	0.961388			0.0384248
$627$		0			0
$628$			13.1228i			0.523658i
$629$	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	$0.395395\pi$
$632$			12.6768i			0.504254i
$633$		0			0
$634$	17.1620			0.681592
$635$		0			0
$636$		0			0
$637$	40.3100	+	21.7656i	1.59714	+	0.862384i
$638$			10.3921i			0.411428i
$639$		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
$643$			17.1508i			0.676361i	0.941081	+	0.338180i	$0.109811\pi$
$643$			17.1508i			0.676361i	−0.941081	+	0.338180i	$0.890189\pi$
$644$	5.19606	+	8.71078i	0.204754	+	0.343253i
$645$		0			0
$646$	−3.52598			−0.138728
$647$	−29.0578			−1.14238			−0.571191	−	0.820817i	$-0.693518\pi$
$647$	−29.0578			−1.14238			−0.571191	+	0.820817i	$0.693518\pi$
$648$		0			0
$649$			30.3501i			1.19135i
$650$		0			0
$651$		0			0
$652$	16.6207			0.650916
$653$			9.11183i			0.356574i	0.983979	+	0.178287i	$0.0570555\pi$
$653$			9.11183i			0.356574i	−0.983979	+	0.178287i	$0.942945\pi$
$654$		0			0
$655$		0			0
$656$	−6.54441			−0.255516
$657$		0			0
$658$	−12.3609	−	20.7220i	−0.481878	−	0.807829i
$659$		−	37.5539i		−	1.46289i	−0.681899	−	0.731446i	$-0.738846\pi$
$659$		−	37.5539i		−	1.46289i	0.681899	−	0.731446i	$-0.261154\pi$
$660$		0			0
$661$		−	9.50275i		−	0.369614i	−0.982775	−	0.184807i	$-0.940834\pi$
$661$		−	9.50275i		−	0.369614i	0.982775	−	0.184807i	$-0.0591660\pi$
$662$			9.74047i			0.378574i
$663$		0			0
$664$			13.0888i			0.507945i
$665$		0			0
$666$		0			0
$667$	14.6967			0.569058
$668$	8.62068			0.333544
$669$		0			0
$670$		0			0
$671$	−12.9767			−0.500958
$672$		0			0
$673$	−49.6335			−1.91323			−0.956615	−	0.291355i	$-0.905894\pi$
$673$	−49.6335			−1.91323			−0.956615	+	0.291355i	$0.905894\pi$
$674$		−	3.15078i		−	0.121363i
$675$		0			0
$676$	29.8293			1.14728
$677$	−37.5312			−1.44244			−0.721220	−	0.692706i	$-0.756418\pi$
$677$	−37.5312			−1.44244			−0.721220	+	0.692706i	$0.756418\pi$
$678$		0			0
$679$	22.9365	+	38.4513i	0.880224	+	1.47562i
$680$		0			0
$681$		0			0
$682$			8.82412i			0.337893i
$683$			9.01560i			0.344972i	0.985012	+	0.172486i	$0.0551800\pi$
$683$			9.01560i			0.344972i	−0.985012	+	0.172486i	$0.944820\pi$
$684$		0			0
$685$		0			0
$686$	−18.5033	+	0.791511i	−0.706461	+	0.0302200i
$687$		0			0
$688$	0.468142			0.0178478
$689$	−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	$0.950404\pi$
$692$	−10.6517			−0.404915
$693$		0			0
$694$	−30.0594			−1.14104
$695$		0			0
$696$		0			0
$697$	10.0251			0.379728
$698$	−27.5180			−1.04157
$699$		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
$703$			6.93420i			0.261528i
$704$		−	2.71078i		−	0.102166i
$705$		0			0
$706$		−	16.5956i		−	0.624583i
$707$	−40.3100	+	24.0453i	−1.51601	+	0.904316i
$708$		0			0
$709$	−5.06372			−0.190172			−0.0950859	−	0.995469i	$-0.530313\pi$
$709$	−5.06372			−0.190172			−0.0950859	+	0.995469i	$0.530313\pi$
$710$		0			0
$711$		0			0
$712$			9.60812i			0.360080i
$713$	12.4792			0.467349
$714$		0			0
$715$		0			0
$716$		−	23.1961i		−	0.866878i
$717$		0			0
$718$	15.0076			0.560080
$719$	18.1274			0.676039			0.338020	−	0.941139i	$-0.390243\pi$
$719$	18.1274			0.676039			0.338020	+	0.941139i	$0.390243\pi$
$720$		0			0
$721$	−5.50171	−	9.22317i	−0.204894	−	0.343489i
$722$			13.7019i			0.509930i
$723$		0			0
$724$		−	9.11980i		−	0.338935i
$725$		0			0
$726$		0			0
$727$		−	11.1168i		−	0.412298i	−0.978521	−	0.206149i	$-0.933907\pi$
$727$		−	11.1168i		−	0.412298i	0.978521	−	0.206149i	$-0.0660931\pi$
$728$	−14.8702	+	8.87024i	−0.551128	+	0.328753i
$729$		0			0
$730$		0			0
$731$	−0.717127			−0.0265239
$732$		0			0
$733$			20.8342i			0.769529i	0.923015	+	0.384765i	$0.125717\pi$
$733$			20.8342i			0.769529i	−0.923015	+	0.384765i	$0.874283\pi$
$734$	−14.2598			−0.526338
$735$		0			0
$736$	−3.83363			−0.141309
$737$		−	36.7500i		−	1.35370i
$738$		0			0
$739$	30.9818			1.13968			0.569842	−	0.821754i	$-0.307004\pi$
$739$	30.9818			1.13968			0.569842	+	0.821754i	$0.307004\pi$
$740$		0			0
$741$		0			0
$742$	21.0297	−	12.5444i	0.772024	−	0.460520i
$743$		−	45.8449i		−	1.68189i	−0.541124	−	0.840943i	$-0.682001\pi$
$743$		−	45.8449i		−	1.68189i	0.541124	−	0.840943i	$-0.317999\pi$
$744$		0			0
$745$		0			0
$746$		−	8.98745i		−	0.329054i
$747$		0			0
$748$			4.15253i			0.151832i
$749$	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	$0.722496\pi$
$752$	9.11980			0.332565
$753$		0			0
$754$			25.0888i			0.913681i
$755$		0			0
$756$		0			0
$757$	12.3661			0.449452			0.224726	−	0.974422i	$-0.427851\pi$
$757$	12.3661			0.449452			0.224726	+	0.974422i	$0.427851\pi$
$758$		−	34.5447i		−	1.25472i
$759$		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
$763$	27.3235	−	16.2987i	0.989177	−	0.590053i
$764$		−	9.69823i		−	0.350870i
$765$		0			0
$766$		−	2.88020i		−	0.104066i
$767$			73.2716i			2.64569i
$768$		0			0
$769$			3.76558i			0.135790i	0.997692	+	0.0678951i	$0.0216283\pi$
$769$			3.76558i			0.135790i	−0.997692	+	0.0678951i	$0.978372\pi$
$770$		0			0
$771$		0			0
$772$	−18.1525			−0.653324
$773$	15.1911			0.546388			0.273194	−	0.961959i	$-0.411920\pi$
$773$	15.1911			0.546388			0.273194	+	0.961959i	$0.411920\pi$
$774$		0			0
$775$		0			0
$776$	−16.9224			−0.607480
$777$		0			0
$778$	−35.1620			−1.26062
$779$			15.0637i			0.539714i
$780$		0			0
$781$	−6.23960			−0.223270
$782$	5.87257			0.210003
$783$		0			0
$784$	3.32583	−	6.15945i	0.118780	−	0.219980i
$785$		0			0
$786$		0			0
$787$			23.8198i			0.849084i	0.905408	+	0.424542i	$0.139565\pi$
$787$			23.8198i			0.849084i	−0.905408	+	0.424542i	$0.860435\pi$
$788$			4.60354i			0.163994i
$789$		0			0
$790$		0			0
$791$	12.2849	+	20.5946i	0.436800	+	0.732260i
$792$		0			0
$793$	−31.3284			−1.11250
$794$	−16.1185			−0.572024
$795$		0			0
$796$			11.7405i			0.416130i
$797$	−10.6517			−0.377301			−0.188650	−	0.982044i	$-0.560411\pi$
$797$	−10.6517			−0.377301			−0.188650	+	0.982044i	$0.560411\pi$
$798$		0			0
$799$	−13.9702			−0.494231
$800$		0			0
$801$		0			0
$802$	16.6256			0.587070
$803$	30.9614			1.09260
$804$		0			0
$805$		0			0
$806$			21.3033i			0.750377i
$807$		0			0
$808$		−	17.7405i		−	0.624108i
$809$			32.8462i			1.15481i	0.816458	+	0.577405i	$0.195935\pi$
$809$			32.8462i			1.15481i	−0.816458	+	0.577405i	$0.804065\pi$
$810$		0			0
$811$			26.0734i			0.915562i	0.889065	+	0.457781i	$0.151356\pi$
$811$			26.0734i			0.915562i	−0.889065	+	0.457781i	$0.848644\pi$
$812$	−5.19606	−	8.71078i	−0.182346	−	0.305689i
$813$		0			0
$814$	8.16637			0.286231
$815$		0			0
$816$		0			0
$817$		−	1.07756i		−	0.0376989i
$818$	−12.0000			−0.419570
$819$		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
$823$	16.9114			0.589496			0.294748	−	0.955575i	$-0.404764\pi$
$823$	16.9114			0.589496			0.294748	+	0.955575i	$0.404764\pi$
$824$	4.05913			0.141406
$825$		0			0
$826$	−15.1751	−	25.4397i	−0.528008	−	0.885162i
$827$			34.2959i			1.19259i	0.802767	+	0.596293i	$0.203360\pi$
$827$			34.2959i			1.19259i	−0.802767	+	0.596293i	$0.796640\pi$
$828$		0			0
$829$		−	53.4408i		−	1.85608i	−0.372486	−	0.928038i	$-0.621495\pi$
$829$		−	53.4408i		−	1.85608i	0.372486	−	0.928038i	$-0.378505\pi$
$830$		0			0
$831$		0			0
$832$		−	6.54441i		−	0.226887i
$833$	−5.09469	+	9.43541i	−0.176521	+	0.326917i
$834$		0			0
$835$		0			0
$836$	−6.23960			−0.215801
$837$		0			0
$838$		−	36.2849i		−	1.25344i
$839$	−14.2898			−0.493339			−0.246669	−	0.969100i	$-0.579336\pi$
$839$	−14.2898			−0.493339			−0.246669	+	0.969100i	$0.579336\pi$
$840$		0			0
$841$	14.3033			0.493218
$842$			19.2414i			0.663101i
$843$		0			0
$844$	−4.00000			−0.137686
$845$		0			0
$846$		0			0
$847$	−8.29731	+	4.94942i	−0.285099	+	0.170064i
$848$			9.25519i			0.317825i
$849$		0			0
$850$		0			0
$851$		−	11.5490i		−	0.395895i
$852$		0			0
$853$			47.0118i			1.60966i	0.593509	+	0.804828i	$0.297742\pi$
$853$			47.0118i			1.60966i	−0.593509	+	0.804828i	$0.702258\pi$
$854$	10.8772	−	6.48833i	0.372209	−	0.222026i
$855$		0			0
$856$	−6.31891			−0.215976
$857$	7.65929			0.261636			0.130818	−	0.991406i	$-0.458240\pi$
$857$	7.65929			0.261636			0.130818	+	0.991406i	$0.458240\pi$
$858$		0			0
$859$			47.8559i			1.63282i	0.577470	+	0.816412i	$0.304040\pi$
$859$			47.8559i			1.63282i	−0.577470	+	0.816412i	$0.695960\pi$
$860$		0			0
$861$		0			0
$862$	−17.2974			−0.589153
$863$			41.4922i			1.41241i	0.708007	+	0.706206i	$0.249595\pi$
$863$			41.4922i			1.41241i	−0.708007	+	0.706206i	$0.750405\pi$
$864$		0			0
$865$		0			0
$866$	31.6473			1.07542
$867$		0			0
$868$	−4.41206	−	7.39646i	−0.149755	−	0.251052i
$869$		−	34.3639i		−	1.16572i
$870$		0			0
$871$		−	88.7223i		−	3.00624i
$872$			12.0251i			0.407221i
$873$		0			0
$874$			8.82412i			0.298480i
$875$		0			0
$876$		0			0
$877$	−17.2925			−0.583927			−0.291963	−	0.956429i	$-0.594309\pi$
$877$	−17.2925			−0.583927			−0.291963	+	0.956429i	$0.594309\pi$
$878$	13.0095			0.439050
$879$		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
$883$	2.07601			0.0698634			0.0349317	−	0.999390i	$-0.488879\pi$
$883$	2.07601			0.0698634			0.0349317	+	0.999390i	$0.488879\pi$
$884$			10.0251i			0.337181i
$885$		0			0
$886$	3.78551			0.127177
$887$	5.18994			0.174261			0.0871305	−	0.996197i	$-0.472230\pi$
$887$	5.18994			0.174261			0.0871305	+	0.996197i	$0.472230\pi$
$888$		0			0
$889$	49.1551	−	29.3215i	1.64861	−	0.983411i
$890$		0			0
$891$		0			0
$892$			26.4513i			0.885654i
$893$		−	20.9917i		−	0.702459i
$894$		0			0
$895$		0			0
$896$	1.35539	+	2.27220i	0.0452805	+	0.0759090i
$897$		0			0
$898$	−24.9307			−0.831947
$899$	−12.4792			−0.416204
$900$		0			0
$901$		−	14.1776i		−	0.472326i
$902$	17.7405			0.590693
$903$		0			0
$904$	−9.06372			−0.301455
$905$		0			0
$906$		0			0
$907$	−32.9473			−1.09400			−0.546999	−	0.837133i	$-0.684230\pi$
$907$	−32.9473			−1.09400			−0.546999	+	0.837133i	$0.684230\pi$
$908$	15.0637			0.499907
$909$		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
$913$		−	35.4809i		−	1.17425i
$914$			31.3535i			1.03708i
$915$		0			0
$916$			11.3655i			0.375526i
$917$	5.72438	−	3.41465i	0.189036	−	0.112762i
$918$		0			0
$919$	13.9129			0.458945			0.229473	−	0.973315i	$-0.426300\pi$
$919$	13.9129			0.458945			0.229473	+	0.973315i	$0.426300\pi$
$920$		0			0
$921$		0			0
$922$			9.52598i			0.313721i
$923$	−15.0637			−0.495828
$924$		0			0
$925$		0			0
$926$		−	34.0003i		−	1.11732i
$927$		0			0
$928$	3.83363			0.125845
$929$	−8.15228			−0.267468			−0.133734	−	0.991017i	$-0.542697\pi$
$929$	−8.15228			−0.267468			−0.133734	+	0.991017i	$0.542697\pi$
$930$		0			0
$931$	−14.1776	−	7.65529i	−0.464653	−	0.250892i
$932$		−	20.9957i		−	0.687736i
$933$		0			0
$934$		−	39.2665i		−	1.28484i
$935$		0			0
$936$		0			0
$937$			2.22576i			0.0727122i	0.999339	+	0.0363561i	$0.0115751\pi$
$937$			2.22576i			0.0727122i	−0.999339	+	0.0363561i	$0.988425\pi$
$938$	18.3750	+	30.8042i	0.599965	+	1.00579i
$939$		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
$943$		−	25.0888i		−	0.817004i
$944$	11.1961			0.364401
$945$		0			0
$946$	−1.26903			−0.0412598
$947$			34.4874i			1.12069i	0.828260	+	0.560344i	$0.189331\pi$
$947$			34.4874i			1.12069i	−0.828260	+	0.560344i	$0.810669\pi$
$948$		0			0
$949$	74.7474			2.42640
$950$		0			0
$951$		0			0
$952$	−2.07627	−	3.48069i	−0.0672922	−	0.112810i
$953$		−	23.9688i		−	0.776426i	−0.921570	−	0.388213i	$-0.873093\pi$
$953$		−	23.9688i		−	0.776426i	0.921570	−	0.388213i	$-0.126907\pi$
$954$		0			0
$955$		0			0
$956$			9.69823i			0.313663i
$957$		0			0
$958$			34.0251i			1.09930i
$959$	−1.89275	−	3.17305i	−0.0611202	−	0.102463i
$960$		0			0
$961$	20.4037			0.658185
$962$	19.7154			0.635649
$963$		0			0
$964$		−	24.7019i		−	0.795593i
$965$		0			0
$966$		0			0
$967$	−6.45735			−0.207654			−0.103827	−	0.994595i	$-0.533109\pi$
$967$	−6.45735			−0.207654			−0.103827	+	0.994595i	$0.533109\pi$
$968$		−	3.65166i		−	0.117369i
$969$		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
$973$	8.31586	+	13.9409i	0.266594	+	0.446924i
$974$			12.8091i			0.410430i
$975$		0			0
$976$			4.78705i			0.153230i
$977$		−	14.4853i		−	0.463425i	−0.972784	−	0.231713i	$-0.925567\pi$
$977$		−	14.4853i		−	0.463425i	0.972784	−	0.231713i	$-0.0744329\pi$
$978$		0			0
$979$		−	26.0455i		−	0.832419i
$980$		0			0
$981$		0			0
$982$	−11.6471			−0.371673
$983$	46.7784			1.49200			0.745999	−	0.665947i	$-0.231972\pi$
$983$	46.7784			1.49200			0.745999	+	0.665947i	$0.231972\pi$
$984$		0			0
$985$		0			0
$986$	−5.87257			−0.187021
$987$		0			0
$988$	−15.0637			−0.479241
$989$			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	$0.700763\pi$
$992$	3.25519			0.103352
$993$		0			0
$994$	5.23009	−	3.11980i	0.165888	−	0.0989540i
$995$		0			0
$996$		0			0
$997$			35.5309i			1.12527i	0.826704	+	0.562637i	$0.190213\pi$
$997$			35.5309i			1.12527i	−0.826704	+	0.562637i	$0.809787\pi$
$998$			35.5511i			1.12535i
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.b.a.251.4	✓	8	5.4	even	2
630.2.b.a.251.8	yes	8	105.104	even	2
630.2.b.b.251.4	yes	8	35.34	odd	2
630.2.b.b.251.8	yes	8	15.14	odd	2
3150.2.b.e.251.1		8	21.20	even	2	inner
3150.2.b.e.251.5		8	1.1	even	1	trivial
3150.2.b.f.251.1		8	3.2	odd	2
3150.2.b.f.251.5		8	7.6	odd	2
3150.2.d.a.3149.5		8	35.27	even	4
3150.2.d.a.3149.6		8	15.8	even	4
3150.2.d.c.3149.3		8	5.2	odd	4
3150.2.d.c.3149.4		8	105.83	odd	4
3150.2.d.d.3149.5		8	105.62	odd	4
3150.2.d.d.3149.6		8	5.3	odd	4
3150.2.d.f.3149.3		8	15.2	even	4
3150.2.d.f.3149.4		8	35.13	even	4
5040.2.f.f.881.1		8	420.419	odd	2
5040.2.f.f.881.2		8	20.19	odd	2
5040.2.f.i.881.1		8	140.139	even	2
5040.2.f.i.881.2		8	60.59	even	2

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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-2.27220 + 1.35539i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-2.27220 + 1.35539i) q^{7} -1.00000i q^{8} +2.71078i q^{11} +6.54441i q^{13} +(-1.35539 - 2.27220i) q^{14} +1.00000 q^{16} -1.53186 q^{17} -2.30177i q^{19} -2.71078 q^{22} +3.83363i q^{23} -6.54441 q^{26} +(2.27220 - 1.35539i) q^{28} -3.83363i q^{29} -3.25519i q^{31} +1.00000i q^{32} -1.53186i q^{34} -3.01255 q^{37} +2.30177 q^{38} -6.54441 q^{41} +0.468142 q^{43} -2.71078i q^{44} -3.83363 q^{46} +9.11980 q^{47} +(3.32583 - 6.15945i) q^{49} -6.54441i q^{52} +9.25519i q^{53} +(1.35539 + 2.27220i) q^{56} +3.83363 q^{58} +11.1961 q^{59} +4.78705i q^{61} +3.25519 q^{62} -1.00000 q^{64} -13.5570 q^{67} +1.53186 q^{68} +2.30177i q^{71} -11.4216i q^{73} -3.01255i q^{74} +2.30177i q^{76} +(-3.67417 - 6.15945i) q^{77} -12.6768 q^{79} -6.54441i q^{82} -13.0888 q^{83} +0.468142i q^{86} +2.71078 q^{88} -9.60812 q^{89} +(-8.87024 - 14.8702i) q^{91} -3.83363i q^{92} +9.11980i q^{94} -16.9224i q^{97} +(6.15945 + 3.32583i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} + 4 q^{7}+O(q^{10}) \) 8 * q - 8 * q^4 + 4 * q^7	\( 8 q - 8 q^{4} + 4 q^{7} + 8 q^{16} - 8 q^{26} - 4 q^{28} + 8 q^{37} + 8 q^{38} - 8 q^{41} + 16 q^{43} - 8 q^{46} + 40 q^{47} + 4 q^{49} + 8 q^{58} - 40 q^{62} - 8 q^{64} - 32 q^{67} - 52 q^{77} + 8 q^{79} - 16 q^{83} - 8 q^{89} - 4 q^{91} + 4 q^{98}+O(q^{100}) \) 8 * q - 8 * q^4 + 4 * q^7 + 8 * q^16 - 8 * q^26 - 4 * q^28 + 8 * q^37 + 8 * q^38 - 8 * q^41 + 16 * q^43 - 8 * q^46 + 40 * q^47 + 4 * q^49 + 8 * q^58 - 40 * q^62 - 8 * q^64 - 32 * q^67 - 52 * q^77 + 8 * q^79 - 16 * q^83 - 8 * q^89 - 4 * q^91 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more