LMFDB - Embedded newform 3960.2.f.d.3761.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3960,2,Mod(3761,3960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3960.3761");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3960.f (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:	$31.6207592004$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} + 2 x^{10} + 8 x^{9} + 2 x^{8} + 12 x^{7} + 40 x^{6} + 4 x^{5} + 49 x^{4} - 56 x^{3} + \cdots + 2 $ x^12 - 4x^11 + 2x^10 + 8x^9 + 2x^8 + 12x^7 + 40x^6 + 4x^5 + 49x^4 - 56x^3 + 34x^2 - 12*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3761.7
Root			$0.343932 - 0.142462i$ of defining polynomial
Character	$\chi$	$=$	3960.3761
Dual form			3960.2.f.d.3761.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} -4.60668i q^{7} +O(q^{10})$	$q+1.00000i q^{5} -4.60668i q^{7} +(2.38159 - 2.30825i) q^{11} +4.20374i q^{13} -6.65334 q^{17} +7.35714i q^{19} +2.38494i q^{23} -1.00000 q^{25} -2.56985 q^{29} -3.29455 q^{31} +4.60668 q^{35} +5.15225 q^{37} -3.69872 q^{41} +10.3397i q^{43} -6.86252i q^{47} -14.2215 q^{49} +9.94953i q^{53} +(2.30825 + 2.38159i) q^{55} -8.67992i q^{59} +13.2998i q^{61} -4.20374 q^{65} +13.8347 q^{67} +13.8088i q^{71} -1.11541i q^{73} +(-10.6334 - 10.9712i) q^{77} +4.14669i q^{79} +10.8521 q^{83} -6.65334i q^{85} +0.932418i q^{89} +19.3653 q^{91} -7.35714 q^{95} +17.2301 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 4 q^{11} + 32 q^{17} - 12 q^{25} - 24 q^{29} + 16 q^{35} + 16 q^{37} - 16 q^{41} - 36 q^{49} - 12 q^{55} - 8 q^{65} + 48 q^{67} + 4 q^{77} + 56 q^{83} + 88 q^{91} + 8 q^{97}+O(q^{100}) $ 12 * q + 4 * q^11 + 32 * q^17 - 12 * q^25 - 24 * q^29 + 16 * q^35 + 16 * q^37 - 16 * q^41 - 36 * q^49 - 12 * q^55 - 8 * q^65 + 48 * q^67 + 4 * q^77 + 56 * q^83 + 88 * q^91 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3960\mathbb{Z}\right)^\times$.

$n$	$991$	$1981$	$2377$	$2521$	$3521$
$\chi(n)$	$1$	$1$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	4.60668i		−	1.74116i	−0.492024	−	0.870581i	$-0.663743\pi$
$7$		−	4.60668i		−	1.74116i	0.492024	−	0.870581i	$-0.336257\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.38159	−	2.30825i	0.718077	−	0.695963i
$11$	2.38159	−	2.30825i	0.718077	−	0.695963i
$12$		0			0
$13$			4.20374i			1.16591i	0.812505	+	0.582954i	$0.198103\pi$
$13$			4.20374i			1.16591i	−0.812505	+	0.582954i	$0.801897\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.65334			−1.61367			−0.806836	−	0.590776i	$-0.798822\pi$
$17$	−6.65334			−1.61367			−0.806836	+	0.590776i	$0.798822\pi$
$18$		0			0
$19$			7.35714i			1.68784i	0.536466	+	0.843922i	$0.319759\pi$
$19$			7.35714i			1.68784i	−0.536466	+	0.843922i	$0.680241\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			2.38494i			0.497294i	0.968594	+	0.248647i	$0.0799860\pi$
$23$			2.38494i			0.497294i	−0.968594	+	0.248647i	$0.920014\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.56985			−0.477208			−0.238604	−	0.971117i	$-0.576690\pi$
$29$	−2.56985			−0.477208			−0.238604	+	0.971117i	$0.576690\pi$
$30$		0			0
$31$	−3.29455			−0.591718			−0.295859	−	0.955232i	$-0.595606\pi$
$31$	−3.29455			−0.591718			−0.295859	+	0.955232i	$0.595606\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.60668			0.778672
$36$		0			0
$37$	5.15225			0.847025			0.423512	−	0.905890i	$-0.360797\pi$
$37$	5.15225			0.847025			0.423512	+	0.905890i	$0.360797\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.69872			−0.577643			−0.288822	−	0.957383i	$-0.593263\pi$
$41$	−3.69872			−0.577643			−0.288822	+	0.957383i	$0.593263\pi$
$42$		0			0
$43$			10.3397i			1.57680i	0.615165	+	0.788398i	$0.289089\pi$
$43$			10.3397i			1.57680i	−0.615165	+	0.788398i	$0.710911\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	6.86252i		−	1.00100i	−0.865736	−	0.500501i	$-0.833149\pi$
$47$		−	6.86252i		−	1.00100i	0.865736	−	0.500501i	$-0.166851\pi$
$48$		0			0
$49$	−14.2215			−2.03165
$50$		0			0
$51$		0			0
$52$		0			0
$53$			9.94953i			1.36667i	0.730104	+	0.683337i	$0.239472\pi$
$53$			9.94953i			1.36667i	−0.730104	+	0.683337i	$0.760528\pi$
$54$		0			0
$55$	2.30825	+	2.38159i	0.311244	+	0.321134i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	8.67992i		−	1.13003i	−0.825081	−	0.565015i	$-0.808870\pi$
$59$		−	8.67992i		−	1.13003i	0.825081	−	0.565015i	$-0.191130\pi$
$60$		0			0
$61$			13.2998i			1.70287i	0.524464	+	0.851433i	$0.324266\pi$
$61$			13.2998i			1.70287i	−0.524464	+	0.851433i	$0.675734\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.20374			−0.521410
$66$		0			0
$67$	13.8347			1.69018			0.845090	−	0.534624i	$-0.179547\pi$
$67$	13.8347			1.69018			0.845090	+	0.534624i	$0.179547\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			13.8088i			1.63880i	0.573220	+	0.819401i	$0.305694\pi$
$71$			13.8088i			1.63880i	−0.573220	+	0.819401i	$0.694306\pi$
$72$		0			0
$73$		−	1.11541i		−	0.130549i	−0.997867	−	0.0652745i	$-0.979208\pi$
$73$		−	1.11541i		−	0.130549i	0.997867	−	0.0652745i	$-0.0207923\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.6334	−	10.9712i	−1.21179	−	1.25029i
$78$		0			0
$79$			4.14669i			0.466539i	0.972412	+	0.233269i	$0.0749424\pi$
$79$			4.14669i			0.466539i	−0.972412	+	0.233269i	$0.925058\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	10.8521			1.19118			0.595588	−	0.803290i	$-0.296919\pi$
$83$	10.8521			1.19118			0.595588	+	0.803290i	$0.296919\pi$
$84$		0			0
$85$		−	6.65334i		−	0.721656i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			0.932418i			0.0988361i	0.998778	+	0.0494180i	$0.0157367\pi$
$89$			0.932418i			0.0988361i	−0.998778	+	0.0494180i	$0.984263\pi$
$90$		0			0
$91$	19.3653			2.03004
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−7.35714			−0.754827
$96$		0			0
$97$	17.2301			1.74945			0.874725	−	0.484620i	$-0.161042\pi$
$97$	17.2301			1.74945			0.874725	+	0.484620i	$0.161042\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.95479			0.493020			0.246510	−	0.969140i	$-0.420716\pi$
$101$	4.95479			0.493020			0.246510	+	0.969140i	$0.420716\pi$
$102$		0			0
$103$	1.07737			0.106156			0.0530780	−	0.998590i	$-0.483097\pi$
$103$	1.07737			0.106156			0.0530780	+	0.998590i	$0.483097\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.33496			0.612424			0.306212	−	0.951963i	$-0.400938\pi$
$107$	6.33496			0.612424			0.306212	+	0.951963i	$0.400938\pi$
$108$		0			0
$109$		−	12.5194i		−	1.19914i	−0.800323	−	0.599569i	$-0.795339\pi$
$109$		−	12.5194i		−	1.19914i	0.800323	−	0.599569i	$-0.204661\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	7.96520i		−	0.749303i	−0.927166	−	0.374652i	$-0.877762\pi$
$113$		−	7.96520i		−	0.749303i	0.927166	−	0.374652i	$-0.122238\pi$
$114$		0			0
$115$	−2.38494			−0.222397
$116$		0			0
$117$		0			0
$118$		0			0
$119$			30.6498i			2.80966i
$120$		0			0
$121$	0.343972	−	10.9946i	0.0312702	−	0.999511i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	18.4275i		−	1.63518i	−0.575802	−	0.817589i	$-0.695310\pi$
$127$		−	18.4275i		−	1.63518i	0.575802	−	0.817589i	$-0.304690\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−15.2949			−1.33633			−0.668163	−	0.744015i	$-0.732919\pi$
$131$	−15.2949			−1.33633			−0.668163	+	0.744015i	$0.732919\pi$
$132$		0			0
$133$	33.8920			2.93881
$134$		0			0
$135$		0			0
$136$		0			0
$137$			16.4569i			1.40601i	0.711186	+	0.703004i	$0.248159\pi$
$137$			16.4569i			1.40601i	−0.711186	+	0.703004i	$0.751841\pi$
$138$		0			0
$139$			20.7670i			1.76143i	0.473646	+	0.880716i	$0.342938\pi$
$139$			20.7670i			1.76143i	−0.473646	+	0.880716i	$0.657062\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	9.70328	+	10.0116i	0.811429	+	0.837212i
$144$		0			0
$145$		−	2.56985i		−	0.213414i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.03175			−0.739910			−0.369955	−	0.929050i	$-0.620627\pi$
$149$	−9.03175			−0.739910			−0.369955	+	0.929050i	$0.620627\pi$
$150$		0			0
$151$			0.476374i			0.0387668i	0.999812	+	0.0193834i	$0.00617031\pi$
$151$			0.476374i			0.0387668i	−0.999812	+	0.0193834i	$0.993830\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	3.29455i		−	0.264625i
$156$		0			0
$157$	12.0281			0.959944			0.479972	−	0.877284i	$-0.340647\pi$
$157$	12.0281			0.959944			0.479972	+	0.877284i	$0.340647\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	10.9867			0.865871
$162$		0			0
$163$	−9.22970			−0.722926			−0.361463	−	0.932386i	$-0.617723\pi$
$163$	−9.22970			−0.722926			−0.361463	+	0.932386i	$0.617723\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−15.5824			−1.20580			−0.602902	−	0.797815i	$-0.705989\pi$
$167$	−15.5824			−1.20580			−0.602902	+	0.797815i	$0.705989\pi$
$168$		0			0
$169$	−4.67145			−0.359342
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−6.55420			−0.498307			−0.249153	−	0.968464i	$-0.580152\pi$
$173$	−6.55420			−0.498307			−0.249153	+	0.968464i	$0.580152\pi$
$174$		0			0
$175$			4.60668i			0.348233i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			26.6716i			1.99353i	0.0803844	+	0.996764i	$0.474385\pi$
$179$			26.6716i			1.99353i	−0.0803844	+	0.996764i	$0.525615\pi$
$180$		0			0
$181$	4.50829			0.335099			0.167549	−	0.985864i	$-0.446415\pi$
$181$	4.50829			0.335099			0.167549	+	0.985864i	$0.446415\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			5.15225i			0.378801i
$186$		0			0
$187$	−15.8455	+	15.3576i	−1.15874	+	1.12306i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	1.02901i		−	0.0744563i	−0.999307	−	0.0372281i	$-0.988147\pi$
$191$		−	1.02901i		−	0.0744563i	0.999307	−	0.0372281i	$-0.0118528\pi$
$192$		0			0
$193$			17.1653i			1.23558i	0.786342	+	0.617791i	$0.211972\pi$
$193$			17.1653i			1.23558i	−0.786342	+	0.617791i	$0.788028\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−9.75931			−0.695322			−0.347661	−	0.937620i	$-0.613024\pi$
$197$	−9.75931			−0.695322			−0.347661	+	0.937620i	$0.613024\pi$
$198$		0			0
$199$	10.9374			0.775332			0.387666	−	0.921800i	$-0.373281\pi$
$199$	10.9374			0.775332			0.387666	+	0.921800i	$0.373281\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			11.8385i			0.830898i
$204$		0			0
$205$		−	3.69872i		−	0.258330i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	16.9821	+	17.5217i	1.17468	+	1.21200i
$210$		0			0
$211$			4.49462i			0.309422i	0.987960	+	0.154711i	$0.0494447\pi$
$211$			4.49462i			0.309422i	−0.987960	+	0.154711i	$0.950555\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−10.3397			−0.705165
$216$		0			0
$217$			15.1769i			1.03028i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	27.9689i		−	1.88139i
$222$		0			0
$223$	−14.0775			−0.942696			−0.471348	−	0.881947i	$-0.656232\pi$
$223$	−14.0775			−0.942696			−0.471348	+	0.881947i	$0.656232\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.20693			−0.411969			−0.205984	−	0.978555i	$-0.566040\pi$
$227$	−6.20693			−0.411969			−0.205984	+	0.978555i	$0.566040\pi$
$228$		0			0
$229$	−19.5312			−1.29066			−0.645328	−	0.763906i	$-0.723279\pi$
$229$	−19.5312			−1.29066			−0.645328	+	0.763906i	$0.723279\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.12229			−0.401085			−0.200542	−	0.979685i	$-0.564270\pi$
$233$	−6.12229			−0.401085			−0.200542	+	0.979685i	$0.564270\pi$
$234$		0			0
$235$	6.86252			0.447662
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.0295797			0.00191335			0.000956675	−	1.00000i	$-0.499695\pi$
$239$	0.0295797			0.00191335			0.000956675		1.00000i	$0.499695\pi$
$240$		0			0
$241$			19.0336i			1.22606i	0.790058	+	0.613032i	$0.210050\pi$
$241$			19.0336i			1.22606i	−0.790058	+	0.613032i	$0.789950\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	14.2215i		−	0.908581i
$246$		0			0
$247$	−30.9275			−1.96787
$248$		0			0
$249$		0			0
$250$		0			0
$251$			5.59583i			0.353206i	0.984282	+	0.176603i	$0.0565108\pi$
$251$			5.59583i			0.353206i	−0.984282	+	0.176603i	$0.943489\pi$
$252$		0			0
$253$	5.50504	+	5.67996i	0.346099	+	0.357096i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	26.9360i		−	1.68022i	−0.542416	−	0.840110i	$-0.682490\pi$
$257$		−	26.9360i		−	1.68022i	0.542416	−	0.840110i	$-0.317510\pi$
$258$		0			0
$259$		−	23.7348i		−	1.47481i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	7.37164			0.454555			0.227277	−	0.973830i	$-0.427018\pi$
$263$	7.37164			0.454555			0.227277	+	0.973830i	$0.427018\pi$
$264$		0			0
$265$	−9.94953			−0.611195
$266$		0			0
$267$		0			0
$268$		0			0
$269$			4.96173i			0.302522i	0.988494	+	0.151261i	$0.0483334\pi$
$269$			4.96173i			0.302522i	−0.988494	+	0.151261i	$0.951667\pi$
$270$		0			0
$271$			11.6128i			0.705428i	0.935731	+	0.352714i	$0.114741\pi$
$271$			11.6128i			0.705428i	−0.935731	+	0.352714i	$0.885259\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.38159	+	2.30825i	−0.143615	+	0.139193i
$276$		0			0
$277$			17.3933i			1.04506i	0.852621	+	0.522530i	$0.175012\pi$
$277$			17.3933i			1.04506i	−0.852621	+	0.522530i	$0.824988\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−30.1187			−1.79673			−0.898365	−	0.439249i	$-0.855244\pi$
$281$	−30.1187			−1.79673			−0.898365	+	0.439249i	$0.855244\pi$
$282$		0			0
$283$			5.73851i			0.341119i	0.985347	+	0.170559i	$0.0545575\pi$
$283$			5.73851i			0.341119i	−0.985347	+	0.170559i	$0.945443\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			17.0388i			1.00577i
$288$		0			0
$289$	27.2669			1.60393
$290$		0			0
$291$		0			0
$292$		0			0
$293$	14.0553			0.821121			0.410561	−	0.911833i	$-0.365333\pi$
$293$	14.0553			0.821121			0.410561	+	0.911833i	$0.365333\pi$
$294$		0			0
$295$	8.67992			0.505365
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−10.0257			−0.579800
$300$		0			0
$301$	47.6319			2.74546
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−13.2998			−0.761544
$306$		0			0
$307$			24.8614i			1.41892i	0.704747	+	0.709459i	$0.251061\pi$
$307$			24.8614i			1.41892i	−0.704747	+	0.709459i	$0.748939\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			3.13136i			0.177563i	0.996051	+	0.0887815i	$0.0282973\pi$
$311$			3.13136i			0.177563i	−0.996051	+	0.0887815i	$0.971703\pi$
$312$		0			0
$313$	20.9214			1.18255			0.591273	−	0.806471i	$-0.298625\pi$
$313$	20.9214			1.18255			0.591273	+	0.806471i	$0.298625\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			7.07729i			0.397500i	0.980050	+	0.198750i	$0.0636882\pi$
$317$			7.07729i			0.397500i	−0.980050	+	0.198750i	$0.936312\pi$
$318$		0			0
$319$	−6.12033	+	5.93184i	−0.342673	+	0.332120i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	48.9495i		−	2.72362i
$324$		0			0
$325$		−	4.20374i		−	0.233182i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−31.6135			−1.74291
$330$		0			0
$331$	−18.5333			−1.01868			−0.509340	−	0.860565i	$-0.670111\pi$
$331$	−18.5333			−1.01868			−0.509340	+	0.860565i	$0.670111\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			13.8347i			0.755872i
$336$		0			0
$337$		−	9.02373i		−	0.491554i	−0.969326	−	0.245777i	$-0.920957\pi$
$337$		−	9.02373i		−	0.491554i	0.969326	−	0.245777i	$-0.0790431\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−7.84628	+	7.60464i	−0.424900	+	0.411814i
$342$		0			0
$343$			33.2673i			1.79627i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	18.7727			1.00777			0.503885	−	0.863771i	$-0.331904\pi$
$347$	18.7727			1.00777			0.503885	+	0.863771i	$0.331904\pi$
$348$		0			0
$349$		−	4.03233i		−	0.215846i	−0.994159	−	0.107923i	$-0.965580\pi$
$349$		−	4.03233i		−	0.215846i	0.994159	−	0.107923i	$-0.0344199\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	19.5460i		−	1.04033i	−0.854066	−	0.520164i	$-0.825871\pi$
$353$		−	19.5460i		−	1.04033i	0.854066	−	0.520164i	$-0.174129\pi$
$354$		0			0
$355$	−13.8088			−0.732895
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−11.6720			−0.616024			−0.308012	−	0.951383i	$-0.599664\pi$
$359$	−11.6720			−0.616024			−0.308012	+	0.951383i	$0.599664\pi$
$360$		0			0
$361$	−35.1275			−1.84882
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.11541			0.0583833
$366$		0			0
$367$	12.8486			0.670690			0.335345	−	0.942095i	$-0.391147\pi$
$367$	12.8486			0.670690			0.335345	+	0.942095i	$0.391147\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	45.8343			2.37960
$372$		0			0
$373$		−	34.6983i		−	1.79661i	−0.439374	−	0.898304i	$-0.644800\pi$
$373$		−	34.6983i		−	1.79661i	0.439374	−	0.898304i	$-0.355200\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	10.8030i		−	0.556381i
$378$		0			0
$379$	15.7073			0.806829			0.403415	−	0.915017i	$-0.367823\pi$
$379$	15.7073			0.806829			0.403415	+	0.915017i	$0.367823\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	26.5507i		−	1.35668i	−0.734750	−	0.678338i	$-0.762701\pi$
$383$		−	26.5507i		−	1.35668i	0.734750	−	0.678338i	$-0.237299\pi$
$384$		0			0
$385$	10.9712	−	10.6334i	0.559147	−	0.541927i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	18.0963i		−	0.917519i	−0.888560	−	0.458760i	$-0.848294\pi$
$389$		−	18.0963i		−	0.917519i	0.888560	−	0.458760i	$-0.151706\pi$
$390$		0			0
$391$		−	15.8678i		−	0.802470i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.14669			−0.208643
$396$		0			0
$397$	−36.6888			−1.84136			−0.920678	−	0.390323i	$-0.872363\pi$
$397$	−36.6888			−1.84136			−0.920678	+	0.390323i	$0.872363\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			6.49923i			0.324556i	0.986745	+	0.162278i	$0.0518841\pi$
$401$			6.49923i			0.324556i	−0.986745	+	0.162278i	$0.948116\pi$
$402$		0			0
$403$		−	13.8494i		−	0.689889i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.2706	−	11.8927i	0.608229	−	0.589498i
$408$		0			0
$409$			17.9456i			0.887352i	0.896187	+	0.443676i	$0.146326\pi$
$409$			17.9456i			0.887352i	−0.896187	+	0.443676i	$0.853674\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−39.9857			−1.96757
$414$		0			0
$415$			10.8521i			0.532710i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			31.4497i			1.53642i	0.640199	+	0.768209i	$0.278852\pi$
$419$			31.4497i			1.53642i	−0.640199	+	0.768209i	$0.721148\pi$
$420$		0			0
$421$	−1.94449			−0.0947688			−0.0473844	−	0.998877i	$-0.515089\pi$
$421$	−1.94449			−0.0947688			−0.0473844	+	0.998877i	$0.515089\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.65334			0.322734
$426$		0			0
$427$	61.2680			2.96497
$428$		0			0
$429$		0			0
$430$		0			0
$431$	26.1904			1.26154			0.630772	−	0.775968i	$-0.282738\pi$
$431$	26.1904			1.26154			0.630772	+	0.775968i	$0.282738\pi$
$432$		0			0
$433$	2.27288			0.109228			0.0546138	−	0.998508i	$-0.482607\pi$
$433$	2.27288			0.109228			0.0546138	+	0.998508i	$0.482607\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−17.5463			−0.839356
$438$		0			0
$439$			2.07907i			0.0992288i	0.998768	+	0.0496144i	$0.0157992\pi$
$439$			2.07907i			0.0992288i	−0.998768	+	0.0496144i	$0.984201\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	25.2254i		−	1.19849i	−0.800564	−	0.599247i	$-0.795467\pi$
$443$		−	25.2254i		−	1.19849i	0.800564	−	0.599247i	$-0.204533\pi$
$444$		0			0
$445$	−0.932418			−0.0442008
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	13.3541i		−	0.630220i	−0.949055	−	0.315110i	$-0.897959\pi$
$449$		−	13.3541i		−	0.630220i	0.949055	−	0.315110i	$-0.102041\pi$
$450$		0			0
$451$	−8.80885	+	8.53757i	−0.414793	+	0.402018i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			19.3653i			0.907860i
$456$		0			0
$457$			0.487396i			0.0227994i	0.999935	+	0.0113997i	$0.00362872\pi$
$457$			0.487396i			0.0227994i	−0.999935	+	0.0113997i	$0.996371\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	12.0118			0.559447			0.279724	−	0.960081i	$-0.409757\pi$
$461$	12.0118			0.559447			0.279724	+	0.960081i	$0.409757\pi$
$462$		0			0
$463$	−1.18798			−0.0552102			−0.0276051	−	0.999619i	$-0.508788\pi$
$463$	−1.18798			−0.0552102			−0.0276051	+	0.999619i	$0.508788\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			35.7662i			1.65506i	0.561419	+	0.827532i	$0.310256\pi$
$467$			35.7662i			1.65506i	−0.561419	+	0.827532i	$0.689744\pi$
$468$		0			0
$469$		−	63.7322i		−	2.94288i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	23.8667	+	24.6251i	1.09739	+	1.13226i
$474$		0			0
$475$		−	7.35714i		−	0.337569i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7.08375			−0.323665			−0.161832	−	0.986818i	$-0.551740\pi$
$479$	−7.08375			−0.323665			−0.161832	+	0.986818i	$0.551740\pi$
$480$		0			0
$481$			21.6587i			0.987553i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			17.2301i			0.782378i
$486$		0			0
$487$	−8.48448			−0.384468			−0.192234	−	0.981349i	$-0.561573\pi$
$487$	−8.48448			−0.384468			−0.192234	+	0.981349i	$0.561573\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5.61003			0.253177			0.126589	−	0.991955i	$-0.459597\pi$
$491$	5.61003			0.253177			0.126589	+	0.991955i	$0.459597\pi$
$492$		0			0
$493$	17.0980			0.770057
$494$		0			0
$495$		0			0
$496$		0			0
$497$	63.6128			2.85342
$498$		0			0
$499$	−10.1321			−0.453577			−0.226788	−	0.973944i	$-0.572823\pi$
$499$	−10.1321			−0.453577			−0.226788	+	0.973944i	$0.572823\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2.47761			−0.110471			−0.0552356	−	0.998473i	$-0.517591\pi$
$503$	−2.47761			−0.110471			−0.0552356	+	0.998473i	$0.517591\pi$
$504$		0			0
$505$			4.95479i			0.220485i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			16.6543i			0.738188i	0.929392	+	0.369094i	$0.120332\pi$
$509$			16.6543i			0.738188i	−0.929392	+	0.369094i	$0.879668\pi$
$510$		0			0
$511$	−5.13835			−0.227307
$512$		0			0
$513$		0			0
$514$		0			0
$515$			1.07737i			0.0474744i
$516$		0			0
$517$	−15.8404	−	16.3437i	−0.696660	−	0.718797i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			28.9552i			1.26855i	0.773107	+	0.634276i	$0.218702\pi$
$521$			28.9552i			1.26855i	−0.773107	+	0.634276i	$0.781298\pi$
$522$		0			0
$523$		−	11.0407i		−	0.482778i	−0.970428	−	0.241389i	$-0.922397\pi$
$523$		−	11.0407i		−	0.482778i	0.970428	−	0.241389i	$-0.0776030\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	21.9197			0.954839
$528$		0			0
$529$	17.3121			0.752698
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	15.5485i		−	0.673479i
$534$		0			0
$535$			6.33496i			0.273884i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−33.8699	+	32.8268i	−1.45888	+	1.41395i
$540$		0			0
$541$			5.12557i			0.220365i	0.993911	+	0.110183i	$0.0351436\pi$
$541$			5.12557i			0.220365i	−0.993911	+	0.110183i	$0.964856\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	12.5194			0.536271
$546$		0			0
$547$			11.5465i			0.493694i	0.969054	+	0.246847i	$0.0793945\pi$
$547$			11.5465i			0.493694i	−0.969054	+	0.246847i	$0.920605\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	18.9067i		−	0.805453i
$552$		0			0
$553$	19.1025			0.812320
$554$		0			0
$555$		0			0
$556$		0			0
$557$	0.334959			0.0141927			0.00709633	−	0.999975i	$-0.497741\pi$
$557$	0.334959			0.0141927			0.00709633	+	0.999975i	$0.497741\pi$
$558$		0			0
$559$	−43.4656			−1.83840
$560$		0			0
$561$		0			0
$562$		0			0
$563$	39.7072			1.67346			0.836729	−	0.547617i	$-0.184465\pi$
$563$	39.7072			1.67346			0.836729	+	0.547617i	$0.184465\pi$
$564$		0			0
$565$	7.96520			0.335099
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11.3575			−0.476131			−0.238066	−	0.971249i	$-0.576513\pi$
$569$	−11.3575			−0.476131			−0.238066	+	0.971249i	$0.576513\pi$
$570$		0			0
$571$		−	28.6115i		−	1.19735i	−0.800991	−	0.598676i	$-0.795694\pi$
$571$		−	28.6115i		−	1.19735i	0.800991	−	0.598676i	$-0.204306\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	2.38494i		−	0.0994589i
$576$		0			0
$577$	−16.5984			−0.691002			−0.345501	−	0.938418i	$-0.612291\pi$
$577$	−16.5984			−0.691002			−0.345501	+	0.938418i	$0.612291\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	49.9923i		−	2.07403i
$582$		0			0
$583$	22.9660	+	23.6957i	0.951154	+	0.981377i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			32.7380i			1.35124i	0.737250	+	0.675620i	$0.236124\pi$
$587$			32.7380i			1.35124i	−0.737250	+	0.675620i	$0.763876\pi$
$588$		0			0
$589$		−	24.2385i		−	0.998728i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	16.4314			0.674756			0.337378	−	0.941369i	$-0.390460\pi$
$593$	16.4314			0.674756			0.337378	+	0.941369i	$0.390460\pi$
$594$		0			0
$595$	−30.6498			−1.25652
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	28.6394i		−	1.17017i	−0.810970	−	0.585087i	$-0.801060\pi$
$599$		−	28.6394i		−	1.17017i	0.810970	−	0.585087i	$-0.198940\pi$
$600$		0			0
$601$		−	5.67181i		−	0.231358i	−0.993287	−	0.115679i	$-0.963096\pi$
$601$		−	5.67181i		−	0.231358i	0.993287	−	0.115679i	$-0.0369044\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	10.9946	+	0.343972i	0.446995	+	0.0139845i
$606$		0			0
$607$		−	7.15149i		−	0.290270i	−0.989412	−	0.145135i	$-0.953638\pi$
$607$		−	7.15149i		−	0.290270i	0.989412	−	0.145135i	$-0.0463617\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	28.8483			1.16708
$612$		0			0
$613$			14.5290i			0.586823i	0.955986	+	0.293411i	$0.0947905\pi$
$613$			14.5290i			0.586823i	−0.955986	+	0.293411i	$0.905209\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			21.8546i			0.879832i	0.898039	+	0.439916i	$0.144992\pi$
$617$			21.8546i			0.879832i	−0.898039	+	0.439916i	$0.855008\pi$
$618$		0			0
$619$	−21.5373			−0.865657			−0.432828	−	0.901476i	$-0.642484\pi$
$619$	−21.5373			−0.865657			−0.432828	+	0.901476i	$0.642484\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	4.29535			0.172090
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−34.2796			−1.36682
$630$		0			0
$631$	46.9484			1.86899			0.934494	−	0.355979i	$-0.115852\pi$
$631$	46.9484			1.86899			0.934494	+	0.355979i	$0.115852\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	18.4275			0.731274
$636$		0			0
$637$		−	59.7837i		−	2.36872i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	26.0602i		−	1.02932i	−0.857396	−	0.514658i	$-0.827919\pi$
$641$		−	26.0602i		−	1.02932i	0.857396	−	0.514658i	$-0.172081\pi$
$642$		0			0
$643$	−14.4794			−0.571013			−0.285506	−	0.958377i	$-0.592162\pi$
$643$	−14.4794			−0.571013			−0.285506	+	0.958377i	$0.592162\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	3.84457i		−	0.151146i	−0.997140	−	0.0755728i	$-0.975921\pi$
$647$		−	3.84457i		−	0.151146i	0.997140	−	0.0755728i	$-0.0240785\pi$
$648$		0			0
$649$	−20.0354	−	20.6720i	−0.786459	−	0.811449i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	20.9946i		−	0.821581i	−0.911730	−	0.410791i	$-0.865253\pi$
$653$		−	20.9946i		−	0.821581i	0.911730	−	0.410791i	$-0.134747\pi$
$654$		0			0
$655$		−	15.2949i		−	0.597623i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−45.3461			−1.76643			−0.883216	−	0.468966i	$-0.844627\pi$
$659$	−45.3461			−1.76643			−0.883216	+	0.468966i	$0.844627\pi$
$660$		0			0
$661$	−31.9578			−1.24301			−0.621507	−	0.783409i	$-0.713479\pi$
$661$	−31.9578			−1.24301			−0.621507	+	0.783409i	$0.713479\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			33.8920i			1.31428i
$666$		0			0
$667$		−	6.12893i		−	0.237313i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	30.6993	+	31.6747i	1.18513	+	1.22279i
$672$		0			0
$673$		−	22.6026i		−	0.871266i	−0.900124	−	0.435633i	$-0.856525\pi$
$673$		−	22.6026i		−	0.871266i	0.900124	−	0.435633i	$-0.143475\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−22.4898			−0.864353			−0.432176	−	0.901789i	$-0.642254\pi$
$677$	−22.4898			−0.864353			−0.432176	+	0.901789i	$0.642254\pi$
$678$		0			0
$679$		−	79.3735i		−	3.04608i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	10.8908i		−	0.416724i	−0.978052	−	0.208362i	$-0.933187\pi$
$683$		−	10.8908i		−	0.416724i	0.978052	−	0.208362i	$-0.0668133\pi$
$684$		0			0
$685$	−16.4569			−0.628786
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−41.8253			−1.59342
$690$		0			0
$691$	−22.8205			−0.868133			−0.434066	−	0.900881i	$-0.642922\pi$
$691$	−22.8205			−0.868133			−0.434066	+	0.900881i	$0.642922\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−20.7670			−0.787736
$696$		0			0
$697$	24.6088			0.932126
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−25.9683			−0.980809			−0.490405	−	0.871495i	$-0.663151\pi$
$701$	−25.9683			−0.980809			−0.490405	+	0.871495i	$0.663151\pi$
$702$		0			0
$703$			37.9058i			1.42965i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	22.8251i		−	0.858428i
$708$		0			0
$709$	7.73679			0.290561			0.145281	−	0.989390i	$-0.453591\pi$
$709$	7.73679			0.290561			0.145281	+	0.989390i	$0.453591\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	7.85730i		−	0.294258i
$714$		0			0
$715$	−10.0116	+	9.70328i	−0.374413	+	0.362882i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	11.1400i		−	0.415451i	−0.978187	−	0.207726i	$-0.933394\pi$
$719$		−	11.1400i		−	0.415451i	0.978187	−	0.207726i	$-0.0666061\pi$
$720$		0			0
$721$		−	4.96308i		−	0.184835i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.56985			0.0954417
$726$		0			0
$727$	−7.40708			−0.274714			−0.137357	−	0.990522i	$-0.543861\pi$
$727$	−7.40708			−0.274714			−0.137357	+	0.990522i	$0.543861\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	68.7938i		−	2.54443i
$732$		0			0
$733$			37.0276i			1.36764i	0.729649	+	0.683822i	$0.239683\pi$
$733$			37.0276i			1.36764i	−0.729649	+	0.683822i	$0.760317\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	32.9487	−	31.9340i	1.21368	−	1.17630i
$738$		0			0
$739$		−	33.2805i		−	1.22424i	−0.790764	−	0.612122i	$-0.790316\pi$
$739$		−	33.2805i		−	1.22424i	0.790764	−	0.612122i	$-0.209684\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.83660			−0.140751			−0.0703756	−	0.997521i	$-0.522420\pi$
$743$	−3.83660			−0.140751			−0.0703756	+	0.997521i	$0.522420\pi$
$744$		0			0
$745$		−	9.03175i		−	0.330898i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	29.1832i		−	1.06633i
$750$		0			0
$751$	31.0330			1.13241			0.566206	−	0.824264i	$-0.308411\pi$
$751$	31.0330			1.13241			0.566206	+	0.824264i	$0.308411\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.476374			−0.0173370
$756$		0			0
$757$	32.4366			1.17893			0.589465	−	0.807794i	$-0.299339\pi$
$757$	32.4366			1.17893			0.589465	+	0.807794i	$0.299339\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−25.6641			−0.930323			−0.465162	−	0.885226i	$-0.654004\pi$
$761$	−25.6641			−0.930323			−0.465162	+	0.885226i	$0.654004\pi$
$762$		0			0
$763$	−57.6728			−2.08790
$764$		0			0
$765$		0			0
$766$		0			0
$767$	36.4881			1.31751
$768$		0			0
$769$			47.2202i			1.70281i	0.524513	+	0.851403i	$0.324248\pi$
$769$			47.2202i			1.70281i	−0.524513	+	0.851403i	$0.675752\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	10.9147i		−	0.392576i	−0.980546	−	0.196288i	$-0.937111\pi$
$773$		−	10.9147i		−	0.392576i	0.980546	−	0.196288i	$-0.0628887\pi$
$774$		0			0
$775$	3.29455			0.118344
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	27.2120i		−	0.974972i
$780$		0			0
$781$	31.8741	+	32.8869i	1.14055	+	1.17679i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			12.0281i			0.429300i
$786$		0			0
$787$			20.1492i			0.718242i	0.933291	+	0.359121i	$0.116923\pi$
$787$			20.1492i			0.718242i	−0.933291	+	0.359121i	$0.883077\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−36.6932			−1.30466
$792$		0			0
$793$	−55.9089			−1.98538
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	24.3186i		−	0.861408i	−0.902493	−	0.430704i	$-0.858265\pi$
$797$		−	24.3186i		−	0.861408i	0.902493	−	0.430704i	$-0.141735\pi$
$798$		0			0
$799$			45.6587i			1.61529i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.57465	−	2.65646i	−0.0908573	−	0.0937443i
$804$		0			0
$805$			10.9867i			0.387229i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−2.85903			−0.100518			−0.0502590	−	0.998736i	$-0.516005\pi$
$809$	−2.85903			−0.100518			−0.0502590	+	0.998736i	$0.516005\pi$
$810$		0			0
$811$			4.01366i			0.140939i	0.997514	+	0.0704693i	$0.0224497\pi$
$811$			4.01366i			0.140939i	−0.997514	+	0.0704693i	$0.977550\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	9.22970i		−	0.323302i
$816$		0			0
$817$	−76.0710			−2.66139
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−28.6774			−1.00085			−0.500425	−	0.865780i	$-0.666823\pi$
$821$	−28.6774			−1.00085			−0.500425	+	0.865780i	$0.666823\pi$
$822$		0			0
$823$	1.95050			0.0679902			0.0339951	−	0.999422i	$-0.489177\pi$
$823$	1.95050			0.0679902			0.0339951	+	0.999422i	$0.489177\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	13.0151			0.452581			0.226290	−	0.974060i	$-0.427340\pi$
$827$	13.0151			0.452581			0.226290	+	0.974060i	$0.427340\pi$
$828$		0			0
$829$	−41.2247			−1.43179			−0.715896	−	0.698207i	$-0.753981\pi$
$829$	−41.2247			−1.43179			−0.715896	+	0.698207i	$0.753981\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	94.6207			3.27841
$834$		0			0
$835$		−	15.5824i		−	0.539252i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			21.1633i			0.730639i	0.930882	+	0.365319i	$0.119040\pi$
$839$			21.1633i			0.730639i	−0.930882	+	0.365319i	$0.880960\pi$
$840$		0			0
$841$	−22.3959			−0.772272
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	4.67145i		−	0.160703i
$846$		0			0
$847$	−50.6487	−	1.58457i	−1.74031	−	0.0544466i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			12.2878i			0.421221i
$852$		0			0
$853$		−	51.4155i		−	1.76043i	−0.474571	−	0.880217i	$-0.657397\pi$
$853$		−	51.4155i		−	1.76043i	0.474571	−	0.880217i	$-0.342603\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−51.8295			−1.77046			−0.885231	−	0.465151i	$-0.846000\pi$
$857$	−51.8295			−1.77046			−0.885231	+	0.465151i	$0.846000\pi$
$858$		0			0
$859$	15.5028			0.528948			0.264474	−	0.964393i	$-0.414802\pi$
$859$	15.5028			0.528948			0.264474	+	0.964393i	$0.414802\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	33.3310i		−	1.13460i	−0.823511	−	0.567300i	$-0.807988\pi$
$863$		−	33.3310i		−	1.13460i	0.823511	−	0.567300i	$-0.192012\pi$
$864$		0			0
$865$		−	6.55420i		−	0.222850i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	9.57159	+	9.87572i	0.324694	+	0.335011i
$870$		0			0
$871$			58.1576i			1.97060i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4.60668			−0.155734
$876$		0			0
$877$			29.3432i			0.990850i	0.868651	+	0.495425i	$0.164988\pi$
$877$			29.3432i			0.990850i	−0.868651	+	0.495425i	$0.835012\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	5.67352i		−	0.191146i	−0.995422	−	0.0955728i	$-0.969532\pi$
$881$		−	5.67352i		−	0.191146i	0.995422	−	0.0955728i	$-0.0304683\pi$
$882$		0			0
$883$	32.4878			1.09330			0.546650	−	0.837361i	$-0.315903\pi$
$883$	32.4878			1.09330			0.546650	+	0.837361i	$0.315903\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	46.1323			1.54897			0.774486	−	0.632591i	$-0.218009\pi$
$887$	46.1323			1.54897			0.774486	+	0.632591i	$0.218009\pi$
$888$		0			0
$889$	−84.8898			−2.84711
$890$		0			0
$891$		0			0
$892$		0			0
$893$	50.4885			1.68953
$894$		0			0
$895$	−26.6716			−0.891533
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.46648			0.282373
$900$		0			0
$901$		−	66.1976i		−	2.20536i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			4.50829i			0.149861i
$906$		0			0
$907$	2.02685			0.0673004			0.0336502	−	0.999434i	$-0.489287\pi$
$907$	2.02685			0.0673004			0.0336502	+	0.999434i	$0.489287\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	12.9410i		−	0.428755i	−0.976751	−	0.214378i	$-0.931228\pi$
$911$		−	12.9410i		−	0.428755i	0.976751	−	0.214378i	$-0.0687723\pi$
$912$		0			0
$913$	25.8453	−	25.0494i	0.855356	−	0.829014i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			70.4590i			2.32676i
$918$		0			0
$919$		−	44.3958i		−	1.46448i	−0.681045	−	0.732242i	$-0.738474\pi$
$919$		−	44.3958i		−	1.46448i	0.681045	−	0.732242i	$-0.261526\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−58.0486			−1.91069
$924$		0			0
$925$	−5.15225			−0.169405
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	10.9250i		−	0.358439i	−0.983809	−	0.179220i	$-0.942643\pi$
$929$		−	10.9250i		−	0.358439i	0.983809	−	0.179220i	$-0.0573572\pi$
$930$		0			0
$931$		−	104.630i		−	3.42910i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−15.3576	−	15.8455i	−0.502246	−	0.518205i
$936$		0			0
$937$			32.9724i			1.07716i	0.842574	+	0.538581i	$0.181039\pi$
$937$			32.9724i			1.07716i	−0.842574	+	0.538581i	$0.818961\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7.64199			−0.249122			−0.124561	−	0.992212i	$-0.539752\pi$
$941$	−7.64199			−0.249122			−0.124561	+	0.992212i	$0.539752\pi$
$942$		0			0
$943$		−	8.82123i		−	0.287259i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			29.0979i			0.945553i	0.881182	+	0.472777i	$0.156748\pi$
$947$			29.0979i			0.945553i	−0.881182	+	0.472777i	$0.843252\pi$
$948$		0			0
$949$	4.68890			0.152208
$950$		0			0
$951$		0			0
$952$		0			0
$953$	45.7997			1.48360			0.741799	−	0.670622i	$-0.233973\pi$
$953$	45.7997			1.48360			0.741799	+	0.670622i	$0.233973\pi$
$954$		0			0
$955$	1.02901			0.0332979
$956$		0			0
$957$		0			0
$958$		0			0
$959$	75.8118			2.44809
$960$		0			0
$961$	−20.1459			−0.649869
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−17.1653			−0.552569
$966$		0			0
$967$		−	10.2436i		−	0.329411i	−0.986343	−	0.164705i	$-0.947333\pi$
$967$		−	10.2436i		−	0.329411i	0.986343	−	0.164705i	$-0.0526674\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			29.5889i			0.949553i	0.880106	+	0.474776i	$0.157471\pi$
$971$			29.5889i			0.949553i	−0.880106	+	0.474776i	$0.842529\pi$
$972$		0			0
$973$	95.6669			3.06694
$974$		0			0
$975$		0			0
$976$		0			0
$977$			30.2757i			0.968605i	0.874901	+	0.484302i	$0.160927\pi$
$977$			30.2757i			0.968605i	−0.874901	+	0.484302i	$0.839073\pi$
$978$		0			0
$979$	2.15225	+	2.22064i	0.0687863	+	0.0709720i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	33.8872i		−	1.08083i	−0.841397	−	0.540417i	$-0.818266\pi$
$983$		−	33.8872i		−	1.08083i	0.841397	−	0.540417i	$-0.181734\pi$
$984$		0			0
$985$		−	9.75931i		−	0.310957i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−24.6597			−0.784132
$990$		0			0
$991$	33.3172			1.05836			0.529179	−	0.848510i	$-0.322500\pi$
$991$	33.3172			1.05836			0.529179	+	0.848510i	$0.322500\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			10.9374i			0.346739i
$996$		0			0
$997$			45.5622i			1.44297i	0.692430	+	0.721485i	$0.256540\pi$
$997$			45.5622i			1.44297i	−0.692430	+	0.721485i	$0.743460\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3960.2.f.d.3761.7	yes	12
3.2	odd	2		3960.2.f.a.3761.1	✓	12
4.3	odd	2		7920.2.f.d.3761.12		12
11.10	odd	2		3960.2.f.a.3761.12	yes	12
12.11	even	2		7920.2.f.e.3761.6		12
33.32	even	2	inner	3960.2.f.d.3761.6	yes	12
44.43	even	2		7920.2.f.e.3761.7		12
132.131	odd	2		7920.2.f.d.3761.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3960.2.f.a.3761.1	✓	12	3.2	odd	2
3960.2.f.a.3761.12	yes	12	11.10	odd	2
3960.2.f.d.3761.6	yes	12	33.32	even	2	inner
3960.2.f.d.3761.7	yes	12	1.1	even	1	trivial
7920.2.f.d.3761.1		12	132.131	odd	2
7920.2.f.d.3761.12		12	4.3	odd	2
7920.2.f.e.3761.6		12	12.11	even	2
7920.2.f.e.3761.7		12	44.43	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:	\( N \)	\(=\)	\( 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3960.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} -4.60668i q^{7} +O(q^{10})\)	\(q+1.00000i q^{5} -4.60668i q^{7} +(2.38159 - 2.30825i) q^{11} +4.20374i q^{13} -6.65334 q^{17} +7.35714i q^{19} +2.38494i q^{23} -1.00000 q^{25} -2.56985 q^{29} -3.29455 q^{31} +4.60668 q^{35} +5.15225 q^{37} -3.69872 q^{41} +10.3397i q^{43} -6.86252i q^{47} -14.2215 q^{49} +9.94953i q^{53} +(2.30825 + 2.38159i) q^{55} -8.67992i q^{59} +13.2998i q^{61} -4.20374 q^{65} +13.8347 q^{67} +13.8088i q^{71} -1.11541i q^{73} +(-10.6334 - 10.9712i) q^{77} +4.14669i q^{79} +10.8521 q^{83} -6.65334i q^{85} +0.932418i q^{89} +19.3653 q^{91} -7.35714 q^{95} +17.2301 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 4 q^{11} + 32 q^{17} - 12 q^{25} - 24 q^{29} + 16 q^{35} + 16 q^{37} - 16 q^{41} - 36 q^{49} - 12 q^{55} - 8 q^{65} + 48 q^{67} + 4 q^{77} + 56 q^{83} + 88 q^{91} + 8 q^{97}+O(q^{100}) \) 12 * q + 4 * q^11 + 32 * q^17 - 12 * q^25 - 24 * q^29 + 16 * q^35 + 16 * q^37 - 16 * q^41 - 36 * q^49 - 12 * q^55 - 8 * q^65 + 48 * q^67 + 4 * q^77 + 56 * q^83 + 88 * q^91 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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