LMFDB - Embedded newform 4140.2.i.a.1241.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1241,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4140.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.i (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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 62x^{14} + 1303x^{12} + 12842x^{10} + 65359x^{8} + 170834x^{6} + 207293x^{4} + 91366x^{2} + 9604 $ x^16 + 62x^14 + 1303x^12 + 12842x^10 + 65359x^8 + 170834x^6 + 207293x^4 + 91366*x^2 + 9604
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.8
Root			$-0.773378i$ of defining polynomial
Character	$\chi$	$=$	4140.1241
Dual form			4140.2.i.a.1241.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -0.420015i q^{7} +O(q^{10})$	$q-1.00000 q^{5} -0.420015i q^{7} +4.79041 q^{11} +0.120193 q^{13} -6.14914 q^{17} -0.691886i q^{19} +(-2.21391 - 4.25424i) q^{23} +1.00000 q^{25} +2.25003i q^{29} -2.99102 q^{31} +0.420015i q^{35} -1.72813i q^{37} -0.604833i q^{41} -7.79229i q^{43} -2.63522i q^{47} +6.82359 q^{49} -5.37185 q^{53} -4.79041 q^{55} -14.4099i q^{59} +10.5601i q^{61} -0.120193 q^{65} -4.65866i q^{67} +15.5728i q^{71} +10.5895 q^{73} -2.01204i q^{77} +1.04381i q^{79} -15.0579 q^{83} +6.14914 q^{85} -10.7709 q^{89} -0.0504831i q^{91} +0.691886i q^{95} -11.6961i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{5}+O(q^{10}) $ 16 * q - 16 * q^5	$ 16 q - 16 q^{5} - 8 q^{23} + 16 q^{25} - 8 q^{31} - 40 q^{49} + 4 q^{53} + 8 q^{73} + 20 q^{83} + 32 q^{89}+O(q^{100}) $ 16 * q - 16 * q^5 - 8 * q^23 + 16 * q^25 - 8 * q^31 - 40 * q^49 + 4 * q^53 + 8 * q^73 + 20 * q^83 + 32 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	0.420015i		−	0.158751i	−0.996845	−	0.0793754i	$-0.974707\pi$
$7$		−	0.420015i		−	0.158751i	0.996845	−	0.0793754i	$-0.0252926\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.79041			1.44436			0.722181	−	0.691704i	$-0.243140\pi$
$11$	4.79041			1.44436			0.722181	+	0.691704i	$0.243140\pi$
$12$		0			0
$13$	0.120193			0.0333357			0.0166678	−	0.999861i	$-0.494694\pi$
$13$	0.120193			0.0333357			0.0166678	+	0.999861i	$0.494694\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.14914			−1.49139			−0.745693	−	0.666289i	$-0.767882\pi$
$17$	−6.14914			−1.49139			−0.745693	+	0.666289i	$0.767882\pi$
$18$		0			0
$19$		−	0.691886i		−	0.158730i	−0.996846	−	0.0793648i	$-0.974711\pi$
$19$		−	0.691886i		−	0.158730i	0.996846	−	0.0793648i	$-0.0252892\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.21391	−	4.25424i	−0.461633	−	0.887071i
$23$	−2.21391	−	4.25424i	−0.461633	−	0.887071i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			2.25003i			0.417821i	0.977935	+	0.208911i	$0.0669917\pi$
$29$			2.25003i			0.417821i	−0.977935	+	0.208911i	$0.933008\pi$
$30$		0			0
$31$	−2.99102			−0.537203			−0.268601	−	0.963251i	$-0.586561\pi$
$31$	−2.99102			−0.537203			−0.268601	+	0.963251i	$0.586561\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			0.420015i			0.0709955i
$36$		0			0
$37$		−	1.72813i		−	0.284103i	−0.989859	−	0.142051i	$-0.954630\pi$
$37$		−	1.72813i		−	0.284103i	0.989859	−	0.142051i	$-0.0453698\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	0.604833i		−	0.0944591i	−0.998884	−	0.0472296i	$-0.984961\pi$
$41$		−	0.604833i		−	0.0944591i	0.998884	−	0.0472296i	$-0.0150392\pi$
$42$		0			0
$43$		−	7.79229i		−	1.18831i	−0.804349	−	0.594157i	$-0.797486\pi$
$43$		−	7.79229i		−	1.18831i	0.804349	−	0.594157i	$-0.202514\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	2.63522i		−	0.384386i	−0.981357	−	0.192193i	$-0.938440\pi$
$47$		−	2.63522i		−	0.384386i	0.981357	−	0.192193i	$-0.0615600\pi$
$48$		0			0
$49$	6.82359			0.974798
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.37185			−0.737881			−0.368941	−	0.929453i	$-0.620279\pi$
$53$	−5.37185			−0.737881			−0.368941	+	0.929453i	$0.620279\pi$
$54$		0			0
$55$	−4.79041			−0.645938
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	14.4099i		−	1.87601i	−0.346626	−	0.938004i	$-0.612673\pi$
$59$		−	14.4099i		−	1.87601i	0.346626	−	0.938004i	$-0.387327\pi$
$60$		0			0
$61$			10.5601i			1.35208i	0.736867	+	0.676038i	$0.236305\pi$
$61$			10.5601i			1.35208i	−0.736867	+	0.676038i	$0.763695\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.120193			−0.0149082
$66$		0			0
$67$		−	4.65866i		−	0.569145i	−0.958654	−	0.284573i	$-0.908148\pi$
$67$		−	4.65866i		−	0.569145i	0.958654	−	0.284573i	$-0.0918517\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			15.5728i			1.84815i	0.382216	+	0.924073i	$0.375161\pi$
$71$			15.5728i			1.84815i	−0.382216	+	0.924073i	$0.624839\pi$
$72$		0			0
$73$	10.5895			1.23941			0.619705	−	0.784835i	$-0.287252\pi$
$73$	10.5895			1.23941			0.619705	+	0.784835i	$0.287252\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	2.01204i		−	0.229294i
$78$		0			0
$79$			1.04381i			0.117438i	0.998275	+	0.0587191i	$0.0187016\pi$
$79$			1.04381i			0.117438i	−0.998275	+	0.0587191i	$0.981298\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−15.0579			−1.65282			−0.826412	−	0.563066i	$-0.809621\pi$
$83$	−15.0579			−1.65282			−0.826412	+	0.563066i	$0.809621\pi$
$84$		0			0
$85$	6.14914			0.666968
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.7709			−1.14171			−0.570857	−	0.821050i	$-0.693389\pi$
$89$	−10.7709			−1.14171			−0.570857	+	0.821050i	$0.693389\pi$
$90$		0			0
$91$		−	0.0504831i		−	0.00529206i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			0.691886i			0.0709860i
$96$		0			0
$97$		−	11.6961i		−	1.18755i	−0.804630	−	0.593777i	$-0.797636\pi$
$97$		−	11.6961i		−	1.18755i	0.804630	−	0.593777i	$-0.202364\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			7.80766i			0.776891i	0.921472	+	0.388446i	$0.126988\pi$
$101$			7.80766i			0.776891i	−0.921472	+	0.388446i	$0.873012\pi$
$102$		0			0
$103$		−	1.18762i		−	0.117019i	−0.998287	−	0.0585096i	$-0.981365\pi$
$103$		−	1.18762i		−	0.117019i	0.998287	−	0.0585096i	$-0.0186348\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.210810			0.0203798			0.0101899	−	0.999948i	$-0.496756\pi$
$107$	0.210810			0.0203798			0.0101899	+	0.999948i	$0.496756\pi$
$108$		0			0
$109$		−	11.6888i		−	1.11959i	−0.828632	−	0.559794i	$-0.810880\pi$
$109$		−	11.6888i		−	1.11959i	0.828632	−	0.559794i	$-0.189120\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.78684			0.450308			0.225154	−	0.974323i	$-0.427712\pi$
$113$	4.78684			0.450308			0.225154	+	0.974323i	$0.427712\pi$
$114$		0			0
$115$	2.21391	+	4.25424i	0.206449	+	0.396710i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			2.58273i			0.236759i
$120$		0			0
$121$	11.9480			1.08618
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	11.0600			0.981418			0.490709	−	0.871323i	$-0.336738\pi$
$127$	11.0600			0.981418			0.490709	+	0.871323i	$0.336738\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	18.7895i		−	1.64165i	−0.571181	−	0.820824i	$-0.693515\pi$
$131$		−	18.7895i		−	1.64165i	0.571181	−	0.820824i	$-0.306485\pi$
$132$		0			0
$133$	−0.290603			−0.0251984
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−19.4411			−1.66097			−0.830483	−	0.557043i	$-0.811936\pi$
$137$	−19.4411			−1.66097			−0.830483	+	0.557043i	$0.811936\pi$
$138$		0			0
$139$	−8.11323			−0.688155			−0.344078	−	0.938941i	$-0.611808\pi$
$139$	−8.11323			−0.688155			−0.344078	+	0.938941i	$0.611808\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.575775			0.0481487
$144$		0			0
$145$		−	2.25003i		−	0.186855i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.8663			−1.05405			−0.527023	−	0.849851i	$-0.676692\pi$
$149$	−12.8663			−1.05405			−0.527023	+	0.849851i	$0.676692\pi$
$150$		0			0
$151$	12.4771			1.01537			0.507687	−	0.861542i	$-0.330501\pi$
$151$	12.4771			1.01537			0.507687	+	0.861542i	$0.330501\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.99102			0.240244
$156$		0			0
$157$		−	9.25137i		−	0.738339i	−0.929362	−	0.369170i	$-0.879642\pi$
$157$		−	9.25137i		−	0.738339i	0.929362	−	0.369170i	$-0.120358\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.78685	+	0.929878i	−0.140823	+	0.0732846i
$162$		0			0
$163$	−10.6408			−0.833453			−0.416726	−	0.909032i	$-0.636823\pi$
$163$	−10.6408			−0.833453			−0.416726	+	0.909032i	$0.636823\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	16.0260i		−	1.24013i	−0.784552	−	0.620063i	$-0.787107\pi$
$167$		−	16.0260i		−	1.24013i	0.784552	−	0.620063i	$-0.212893\pi$
$168$		0			0
$169$	−12.9856			−0.998889
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	21.6296i		−	1.64447i	−0.569148	−	0.822235i	$-0.692727\pi$
$173$		−	21.6296i		−	1.64447i	0.569148	−	0.822235i	$-0.307273\pi$
$174$		0			0
$175$		−	0.420015i		−	0.0317502i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			0.916993i			0.0685393i	0.999413	+	0.0342696i	$0.0109105\pi$
$179$			0.916993i			0.0685393i	−0.999413	+	0.0342696i	$0.989089\pi$
$180$		0			0
$181$		−	5.07502i		−	0.377223i	−0.982052	−	0.188612i	$-0.939601\pi$
$181$		−	5.07502i		−	0.377223i	0.982052	−	0.188612i	$-0.0603987\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			1.72813i			0.127055i
$186$		0			0
$187$	−29.4569			−2.15410
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.88191			−0.425600			−0.212800	−	0.977096i	$-0.568258\pi$
$191$	−5.88191			−0.425600			−0.212800	+	0.977096i	$0.568258\pi$
$192$		0			0
$193$	−16.6809			−1.20072			−0.600359	−	0.799730i	$-0.704976\pi$
$193$	−16.6809			−1.20072			−0.600359	+	0.799730i	$0.704976\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	14.3370i		−	1.02147i	−0.859739	−	0.510734i	$-0.829374\pi$
$197$		−	14.3370i		−	1.02147i	0.859739	−	0.510734i	$-0.170626\pi$
$198$		0			0
$199$		−	21.5450i		−	1.52729i	−0.645638	−	0.763644i	$-0.723408\pi$
$199$		−	21.5450i		−	1.52729i	0.645638	−	0.763644i	$-0.276592\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.945049			0.0663294
$204$		0			0
$205$			0.604833i			0.0422434i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	3.31442i		−	0.229263i
$210$		0			0
$211$	−2.14658			−0.147777			−0.0738884	−	0.997267i	$-0.523541\pi$
$211$	−2.14658			−0.147777			−0.0738884	+	0.997267i	$0.523541\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			7.79229i			0.531430i
$216$		0			0
$217$			1.25627i			0.0852813i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.739087			−0.0497163
$222$		0			0
$223$	21.6059			1.44684			0.723418	−	0.690411i	$-0.242570\pi$
$223$	21.6059			1.44684			0.723418	+	0.690411i	$0.242570\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	16.9240			1.12328			0.561642	−	0.827381i	$-0.310170\pi$
$227$	16.9240			1.12328			0.561642	+	0.827381i	$0.310170\pi$
$228$		0			0
$229$		−	4.96089i		−	0.327825i	−0.986475	−	0.163913i	$-0.947589\pi$
$229$		−	4.96089i		−	0.327825i	0.986475	−	0.163913i	$-0.0524115\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			9.49267i			0.621886i	0.950429	+	0.310943i	$0.100645\pi$
$233$			9.49267i			0.621886i	−0.950429	+	0.310943i	$0.899355\pi$
$234$		0			0
$235$			2.63522i			0.171903i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			10.6659i			0.689918i	0.938618	+	0.344959i	$0.112107\pi$
$239$			10.6659i			0.689918i	−0.938618	+	0.344959i	$0.887893\pi$
$240$		0			0
$241$		−	22.7497i		−	1.46544i	−0.680532	−	0.732718i	$-0.738251\pi$
$241$		−	22.7497i		−	1.46544i	0.680532	−	0.732718i	$-0.261749\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.82359			−0.435943
$246$		0			0
$247$		−	0.0831601i		−	0.00529135i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−29.8041			−1.88122			−0.940609	−	0.339492i	$-0.889745\pi$
$251$	−29.8041			−1.88122			−0.940609	+	0.339492i	$0.889745\pi$
$252$		0			0
$253$	−10.6056	−	20.3796i	−0.666765	−	1.28125i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	30.9596i		−	1.93120i	−0.260023	−	0.965602i	$-0.583730\pi$
$257$		−	30.9596i		−	1.93120i	0.260023	−	0.965602i	$-0.416270\pi$
$258$		0			0
$259$	−0.725840			−0.0451015
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.99845			−0.308218			−0.154109	−	0.988054i	$-0.549251\pi$
$263$	−4.99845			−0.308218			−0.154109	+	0.988054i	$0.549251\pi$
$264$		0			0
$265$	5.37185			0.329990
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	22.9913i		−	1.40181i	−0.713256	−	0.700903i	$-0.752781\pi$
$269$		−	22.9913i		−	1.40181i	0.713256	−	0.700903i	$-0.247219\pi$
$270$		0			0
$271$	13.4349			0.816112			0.408056	−	0.912957i	$-0.366207\pi$
$271$	13.4349			0.816112			0.408056	+	0.912957i	$0.366207\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.79041			0.288872
$276$		0			0
$277$	0.0580060			0.00348524			0.00174262	−	0.999998i	$-0.499445\pi$
$277$	0.0580060			0.00348524			0.00174262	+	0.999998i	$0.499445\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−20.4170			−1.21797			−0.608987	−	0.793180i	$-0.708424\pi$
$281$	−20.4170			−1.21797			−0.608987	+	0.793180i	$0.708424\pi$
$282$		0			0
$283$			10.2646i			0.610166i	0.952326	+	0.305083i	$0.0986842\pi$
$283$			10.2646i			0.610166i	−0.952326	+	0.305083i	$0.901316\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.254039			−0.0149955
$288$		0			0
$289$	20.8120			1.22423
$290$		0			0
$291$		0			0
$292$		0			0
$293$	16.9225			0.988624			0.494312	−	0.869285i	$-0.335420\pi$
$293$	16.9225			0.988624			0.494312	+	0.869285i	$0.335420\pi$
$294$		0			0
$295$			14.4099i			0.838976i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.266098	−	0.511332i	−0.0153888	−	0.0295711i
$300$		0			0
$301$	−3.27288			−0.188646
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	10.5601i		−	0.604667i
$306$		0			0
$307$	2.25469			0.128682			0.0643410	−	0.997928i	$-0.479505\pi$
$307$	2.25469			0.128682			0.0643410	+	0.997928i	$0.479505\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			6.46410i			0.366546i	0.983062	+	0.183273i	$0.0586692\pi$
$311$			6.46410i			0.366546i	−0.983062	+	0.183273i	$0.941331\pi$
$312$		0			0
$313$			7.42383i			0.419620i	0.977742	+	0.209810i	$0.0672845\pi$
$313$			7.42383i			0.419620i	−0.977742	+	0.209810i	$0.932716\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			12.3930i			0.696059i	0.937484	+	0.348029i	$0.113149\pi$
$317$			12.3930i			0.696059i	−0.937484	+	0.348029i	$0.886851\pi$
$318$		0			0
$319$			10.7786i			0.603485i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			4.25451i			0.236727i
$324$		0			0
$325$	0.120193			0.00666713
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.10683			−0.0610216
$330$		0			0
$331$	31.1247			1.71077			0.855384	−	0.517994i	$-0.173321\pi$
$331$	31.1247			1.71077			0.855384	+	0.517994i	$0.173321\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			4.65866i			0.254530i
$336$		0			0
$337$		−	21.4914i		−	1.17071i	−0.810777	−	0.585355i	$-0.800955\pi$
$337$		−	21.4914i		−	1.17071i	0.810777	−	0.585355i	$-0.199045\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−14.3282			−0.775915
$342$		0			0
$343$		−	5.80612i		−	0.313501i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			33.1297i			1.77849i	0.457428	+	0.889247i	$0.348771\pi$
$347$			33.1297i			1.77849i	−0.457428	+	0.889247i	$0.651229\pi$
$348$		0			0
$349$	13.5930			0.727617			0.363808	−	0.931474i	$-0.381476\pi$
$349$	13.5930			0.727617			0.363808	+	0.931474i	$0.381476\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			26.8678i			1.43003i	0.699110	+	0.715014i	$0.253580\pi$
$353$			26.8678i			1.43003i	−0.699110	+	0.715014i	$0.746420\pi$
$354$		0			0
$355$		−	15.5728i		−	0.826516i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	14.0841			0.743328			0.371664	−	0.928367i	$-0.378787\pi$
$359$	14.0841			0.743328			0.371664	+	0.928367i	$0.378787\pi$
$360$		0			0
$361$	18.5213			0.974805
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−10.5895			−0.554281
$366$		0			0
$367$			14.1953i			0.740988i	0.928835	+	0.370494i	$0.120812\pi$
$367$			14.1953i			0.740988i	−0.928835	+	0.370494i	$0.879188\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			2.25626i			0.117139i
$372$		0			0
$373$			29.8562i			1.54590i	0.634470	+	0.772948i	$0.281219\pi$
$373$			29.8562i			1.54590i	−0.634470	+	0.772948i	$0.718781\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			0.270439i			0.0139283i
$378$		0			0
$379$			5.65465i			0.290460i	0.989398	+	0.145230i	$0.0463922\pi$
$379$			5.65465i			0.290460i	−0.989398	+	0.145230i	$0.953608\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.96045			−0.508955			−0.254478	−	0.967079i	$-0.581904\pi$
$383$	−9.96045			−0.508955			−0.254478	+	0.967079i	$0.581904\pi$
$384$		0			0
$385$			2.01204i			0.102543i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.61635			−0.234058			−0.117029	−	0.993128i	$-0.537337\pi$
$389$	−4.61635			−0.234058			−0.117029	+	0.993128i	$0.537337\pi$
$390$		0			0
$391$	13.6137	+	26.1600i	0.688473	+	1.32297i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	1.04381i		−	0.0525200i
$396$		0			0
$397$	−18.0224			−0.904521			−0.452260	−	0.891886i	$-0.649382\pi$
$397$	−18.0224			−0.904521			−0.452260	+	0.891886i	$0.649382\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−29.8052			−1.48840			−0.744200	−	0.667957i	$-0.767169\pi$
$401$	−29.8052			−1.48840			−0.744200	+	0.667957i	$0.767169\pi$
$402$		0			0
$403$	−0.359501			−0.0179080
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	8.27844i		−	0.410347i
$408$		0			0
$409$	−8.05200			−0.398146			−0.199073	−	0.979985i	$-0.563793\pi$
$409$	−8.05200			−0.398146			−0.199073	+	0.979985i	$0.563793\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.05237			−0.297818
$414$		0			0
$415$	15.0579			0.739165
$416$		0			0
$417$		0			0
$418$		0			0
$419$	21.6088			1.05566			0.527830	−	0.849350i	$-0.323006\pi$
$419$	21.6088			1.05566			0.527830	+	0.849350i	$0.323006\pi$
$420$		0			0
$421$		−	1.79396i		−	0.0874324i	−0.999044	−	0.0437162i	$-0.986080\pi$
$421$		−	1.79396i		−	0.0874324i	0.999044	−	0.0437162i	$-0.0139197\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6.14914			−0.298277
$426$		0			0
$427$	4.43538			0.214643
$428$		0			0
$429$		0			0
$430$		0			0
$431$	5.62143			0.270775			0.135387	−	0.990793i	$-0.456772\pi$
$431$	5.62143			0.270775			0.135387	+	0.990793i	$0.456772\pi$
$432$		0			0
$433$			3.33200i			0.160126i	0.996790	+	0.0800630i	$0.0255121\pi$
$433$			3.33200i			0.160126i	−0.996790	+	0.0800630i	$0.974488\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.94345	+	1.53178i	−0.140804	+	0.0732748i
$438$		0			0
$439$	−35.5935			−1.69878			−0.849392	−	0.527762i	$-0.823031\pi$
$439$	−35.5935			−1.69878			−0.849392	+	0.527762i	$0.823031\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	0.738622i		−	0.0350930i	−0.999846	−	0.0175465i	$-0.994414\pi$
$443$		−	0.738622i		−	0.0350930i	0.999846	−	0.0175465i	$-0.00558551\pi$
$444$		0			0
$445$	10.7709			0.510590
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	27.4708i		−	1.29643i	−0.761459	−	0.648213i	$-0.775517\pi$
$449$		−	27.4708i		−	1.29643i	0.761459	−	0.648213i	$-0.224483\pi$
$450$		0			0
$451$		−	2.89740i		−	0.136433i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			0.0504831i			0.00236668i
$456$		0			0
$457$		−	0.00157376i		−	7.36174e-5i	−1.00000		3.68087e-5i	$-0.999988\pi$
$457$		−	0.00157376i		−	7.36174e-5i	1.00000		3.68087e-5i	$-1.17166e-5\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			4.74825i			0.221148i	0.993868	+	0.110574i	$0.0352689\pi$
$461$			4.74825i			0.221148i	−0.993868	+	0.110574i	$0.964731\pi$
$462$		0			0
$463$	19.6536			0.913382			0.456691	−	0.889625i	$-0.349034\pi$
$463$	19.6536			0.913382			0.456691	+	0.889625i	$0.349034\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	16.7766			0.776329			0.388164	−	0.921590i	$-0.373109\pi$
$467$	16.7766			0.776329			0.388164	+	0.921590i	$0.373109\pi$
$468$		0			0
$469$	−1.95671			−0.0903523
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	37.3283i		−	1.71635i
$474$		0			0
$475$		−	0.691886i		−	0.0317459i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.6762			−0.579192			−0.289596	−	0.957149i	$-0.593521\pi$
$479$	−12.6762			−0.579192			−0.289596	+	0.957149i	$0.593521\pi$
$480$		0			0
$481$		−	0.207710i		−	0.00947074i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			11.6961i			0.531090i
$486$		0			0
$487$	−32.4291			−1.46950			−0.734752	−	0.678336i	$-0.762701\pi$
$487$	−32.4291			−1.46950			−0.734752	+	0.678336i	$0.762701\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			20.8626i			0.941517i	0.882262	+	0.470759i	$0.156020\pi$
$491$			20.8626i			0.941517i	−0.882262	+	0.470759i	$0.843980\pi$
$492$		0			0
$493$		−	13.8358i		−	0.623133i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	6.54079			0.293395
$498$		0			0
$499$	−13.6075			−0.609156			−0.304578	−	0.952487i	$-0.598515\pi$
$499$	−13.6075			−0.609156			−0.304578	+	0.952487i	$0.598515\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12.6316			0.563213			0.281607	−	0.959530i	$-0.409133\pi$
$503$	12.6316			0.563213			0.281607	+	0.959530i	$0.409133\pi$
$504$		0			0
$505$		−	7.80766i		−	0.347436i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			25.0568i			1.11062i	0.831642	+	0.555312i	$0.187401\pi$
$509$			25.0568i			1.11062i	−0.831642	+	0.555312i	$0.812599\pi$
$510$		0			0
$511$		−	4.44776i		−	0.196757i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			1.18762i			0.0523326i
$516$		0			0
$517$		−	12.6238i		−	0.555193i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	18.2486			0.799486			0.399743	−	0.916627i	$-0.369099\pi$
$521$	18.2486			0.799486			0.399743	+	0.916627i	$0.369099\pi$
$522$		0			0
$523$		−	38.2290i		−	1.67164i	−0.549005	−	0.835819i	$-0.684993\pi$
$523$		−	38.2290i		−	1.67164i	0.549005	−	0.835819i	$-0.315007\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	18.3922			0.801177
$528$		0			0
$529$	−13.1972	+	18.8371i	−0.573790	+	0.819003i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	0.0726970i		−	0.00314886i
$534$		0			0
$535$	−0.210810			−0.00911413
$536$		0			0
$537$		0			0
$538$		0			0
$539$	32.6878			1.40796
$540$		0			0
$541$	39.4184			1.69473			0.847364	−	0.531013i	$-0.178188\pi$
$541$	39.4184			1.69473			0.847364	+	0.531013i	$0.178188\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			11.6888i			0.500695i
$546$		0			0
$547$	−27.7704			−1.18738			−0.593688	−	0.804695i	$-0.702329\pi$
$547$	−27.7704			−1.18738			−0.593688	+	0.804695i	$0.702329\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.55677			0.0663205
$552$		0			0
$553$	0.438418			0.0186434
$554$		0			0
$555$		0			0
$556$		0			0
$557$	23.1447			0.980671			0.490336	−	0.871534i	$-0.336874\pi$
$557$	23.1447			0.980671			0.490336	+	0.871534i	$0.336874\pi$
$558$		0			0
$559$		−	0.936582i		−	0.0396132i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	9.99039			0.421045			0.210522	−	0.977589i	$-0.432484\pi$
$563$	9.99039			0.421045			0.210522	+	0.977589i	$0.432484\pi$
$564$		0			0
$565$	−4.78684			−0.201384
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−17.5957			−0.737649			−0.368824	−	0.929499i	$-0.620240\pi$
$569$	−17.5957			−0.737649			−0.368824	+	0.929499i	$0.620240\pi$
$570$		0			0
$571$			9.10580i			0.381066i	0.981681	+	0.190533i	$0.0610216\pi$
$571$			9.10580i			0.381066i	−0.981681	+	0.190533i	$0.938978\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.21391	−	4.25424i	−0.0923266	−	0.177414i
$576$		0			0
$577$	−29.3630			−1.22240			−0.611199	−	0.791477i	$-0.709312\pi$
$577$	−29.3630			−1.22240			−0.611199	+	0.791477i	$0.709312\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			6.32456i			0.262387i
$582$		0			0
$583$	−25.7334			−1.06577
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	27.5701i		−	1.13794i	−0.822359	−	0.568969i	$-0.807342\pi$
$587$		−	27.5701i		−	1.13794i	0.822359	−	0.568969i	$-0.192658\pi$
$588$		0			0
$589$			2.06944i			0.0852699i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	28.6623i		−	1.17702i	−0.808490	−	0.588509i	$-0.799715\pi$
$593$		−	28.6623i		−	1.17702i	0.808490	−	0.588509i	$-0.200285\pi$
$594$		0			0
$595$		−	2.58273i		−	0.105882i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	6.44581i		−	0.263369i	−0.991292	−	0.131684i	$-0.957961\pi$
$599$		−	6.44581i		−	0.263369i	0.991292	−	0.131684i	$-0.0420385\pi$
$600$		0			0
$601$	4.57211			0.186500			0.0932500	−	0.995643i	$-0.470274\pi$
$601$	4.57211			0.186500			0.0932500	+	0.995643i	$0.470274\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11.9480			−0.485755
$606$		0			0
$607$	−28.0490			−1.13847			−0.569237	−	0.822174i	$-0.692761\pi$
$607$	−28.0490			−1.13847			−0.569237	+	0.822174i	$0.692761\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	0.316736i		−	0.0128138i
$612$		0			0
$613$		−	12.6684i		−	0.511670i	−0.966720	−	0.255835i	$-0.917650\pi$
$613$		−	12.6684i		−	0.511670i	0.966720	−	0.255835i	$-0.0823504\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−5.24511			−0.211160			−0.105580	−	0.994411i	$-0.533670\pi$
$617$	−5.24511			−0.211160			−0.105580	+	0.994411i	$0.533670\pi$
$618$		0			0
$619$			32.0621i			1.28868i	0.764737	+	0.644342i	$0.222869\pi$
$619$			32.0621i			1.28868i	−0.764737	+	0.644342i	$0.777131\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			4.52394i			0.181248i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			10.6265i			0.423707i
$630$		0			0
$631$			39.4691i			1.57124i	0.618709	+	0.785620i	$0.287656\pi$
$631$			39.4691i			1.57124i	−0.618709	+	0.785620i	$0.712344\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−11.0600			−0.438904
$636$		0			0
$637$	0.820150			0.0324955
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.26556			−0.0499865			−0.0249933	−	0.999688i	$-0.507956\pi$
$641$	−1.26556			−0.0499865			−0.0249933	+	0.999688i	$0.507956\pi$
$642$		0			0
$643$			37.3164i			1.47162i	0.677190	+	0.735808i	$0.263198\pi$
$643$			37.3164i			1.47162i	−0.677190	+	0.735808i	$0.736802\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	17.8201i		−	0.700582i	−0.936641	−	0.350291i	$-0.886083\pi$
$647$		−	17.8201i		−	0.700582i	0.936641	−	0.350291i	$-0.113917\pi$
$648$		0			0
$649$		−	69.0292i		−	2.70963i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			20.7057i			0.810277i	0.914255	+	0.405139i	$0.132777\pi$
$653$			20.7057i			0.810277i	−0.914255	+	0.405139i	$0.867223\pi$
$654$		0			0
$655$			18.7895i			0.734168i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	37.6023			1.46478			0.732389	−	0.680887i	$-0.238406\pi$
$659$	37.6023			1.46478			0.732389	+	0.680887i	$0.238406\pi$
$660$		0			0
$661$			27.3245i			1.06280i	0.847121	+	0.531399i	$0.178334\pi$
$661$			27.3245i			1.06280i	−0.847121	+	0.531399i	$0.821666\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.290603			0.0112691
$666$		0			0
$667$	9.57220	−	4.98139i	0.370637	−	0.192880i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			50.5870i			1.95289i
$672$		0			0
$673$	−0.975177			−0.0375903			−0.0187952	−	0.999823i	$-0.505983\pi$
$673$	−0.975177			−0.0375903			−0.0187952	+	0.999823i	$0.505983\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	22.7727			0.875227			0.437614	−	0.899163i	$-0.355824\pi$
$677$	22.7727			0.875227			0.437614	+	0.899163i	$0.355824\pi$
$678$		0			0
$679$	−4.91252			−0.188525
$680$		0			0
$681$		0			0
$682$		0			0
$683$			0.546387i			0.0209069i	0.999945	+	0.0104535i	$0.00332750\pi$
$683$			0.546387i			0.0209069i	−0.999945	+	0.0104535i	$0.996673\pi$
$684$		0			0
$685$	19.4411			0.742807
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.645662			−0.0245977
$690$		0			0
$691$	−11.7056			−0.445301			−0.222651	−	0.974898i	$-0.571471\pi$
$691$	−11.7056			−0.445301			−0.222651	+	0.974898i	$0.571471\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.11323			0.307752
$696$		0			0
$697$			3.71921i			0.140875i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	38.8682			1.46803			0.734016	−	0.679133i	$-0.237644\pi$
$701$	38.8682			1.46803			0.734016	+	0.679133i	$0.237644\pi$
$702$		0			0
$703$	−1.19567			−0.0450955
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.27934			0.123332
$708$		0			0
$709$			36.1899i			1.35914i	0.733611	+	0.679569i	$0.237833\pi$
$709$			36.1899i			1.35914i	−0.733611	+	0.679569i	$0.762167\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.62186	+	12.7245i	0.247990	+	0.476537i
$714$		0			0
$715$	−0.575775			−0.0215328
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	5.55677i		−	0.207233i	−0.994617	−	0.103616i	$-0.966959\pi$
$719$		−	5.55677i		−	0.207233i	0.994617	−	0.103616i	$-0.0330414\pi$
$720$		0			0
$721$	−0.498816			−0.0185769
$722$		0			0
$723$		0			0
$724$		0			0
$725$			2.25003i			0.0835642i
$726$		0			0
$727$			5.22231i			0.193685i	0.995300	+	0.0968423i	$0.0308743\pi$
$727$			5.22231i			0.193685i	−0.995300	+	0.0968423i	$0.969126\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			47.9159i			1.77223i
$732$		0			0
$733$		−	25.5233i		−	0.942724i	−0.881940	−	0.471362i	$-0.843763\pi$
$733$		−	25.5233i		−	0.942724i	0.881940	−	0.471362i	$-0.156237\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	22.3169i		−	0.822052i
$738$		0			0
$739$	6.16953			0.226950			0.113475	−	0.993541i	$-0.463802\pi$
$739$	6.16953			0.226950			0.113475	+	0.993541i	$0.463802\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	4.71275			0.172894			0.0864470	−	0.996256i	$-0.472449\pi$
$743$	4.71275			0.172894			0.0864470	+	0.996256i	$0.472449\pi$
$744$		0			0
$745$	12.8663			0.471384
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	0.0885436i		−	0.00323531i
$750$		0			0
$751$			14.7663i			0.538830i	0.963024	+	0.269415i	$0.0868304\pi$
$751$			14.7663i			0.538830i	−0.963024	+	0.269415i	$0.913170\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−12.4771			−0.454089
$756$		0			0
$757$			21.5575i			0.783522i	0.920067	+	0.391761i	$0.128134\pi$
$757$			21.5575i			0.783522i	−0.920067	+	0.391761i	$0.871866\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	4.26088i		−	0.154457i	−0.997013	−	0.0772284i	$-0.975393\pi$
$761$		−	4.26088i		−	0.154457i	0.997013	−	0.0772284i	$-0.0246071\pi$
$762$		0			0
$763$	−4.90949			−0.177735
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	1.73197i		−	0.0625379i
$768$		0			0
$769$		−	27.2866i		−	0.983982i	−0.870600	−	0.491991i	$-0.836269\pi$
$769$		−	27.2866i		−	0.983982i	0.870600	−	0.491991i	$-0.163731\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−31.2736			−1.12483			−0.562416	−	0.826855i	$-0.690128\pi$
$773$	−31.2736			−1.12483			−0.562416	+	0.826855i	$0.690128\pi$
$774$		0			0
$775$	−2.99102			−0.107441
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.418476			−0.0149934
$780$		0			0
$781$			74.5998i			2.66939i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			9.25137i			0.330195i
$786$		0			0
$787$		−	6.12584i		−	0.218363i	−0.994022	−	0.109181i	$-0.965177\pi$
$787$		−	6.12584i		−	0.218363i	0.994022	−	0.109181i	$-0.0348229\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	2.01054i		−	0.0714867i
$792$		0			0
$793$			1.26925i			0.0450724i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12.2780			0.434910			0.217455	−	0.976070i	$-0.430224\pi$
$797$	12.2780			0.434910			0.217455	+	0.976070i	$0.430224\pi$
$798$		0			0
$799$			16.2043i			0.573268i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	50.7281			1.79016
$804$		0			0
$805$	1.78685	−	0.929878i	0.0629781	−	0.0327739i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			3.23023i			0.113569i	0.998386	+	0.0567845i	$0.0180848\pi$
$809$			3.23023i			0.113569i	−0.998386	+	0.0567845i	$0.981915\pi$
$810$		0			0
$811$	19.7411			0.693202			0.346601	−	0.938013i	$-0.387336\pi$
$811$	19.7411			0.693202			0.346601	+	0.938013i	$0.387336\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10.6408			0.372731
$816$		0			0
$817$	−5.39138			−0.188620
$818$		0			0
$819$		0			0
$820$		0			0
$821$			33.5478i			1.17083i	0.810735	+	0.585414i	$0.199068\pi$
$821$			33.5478i			1.17083i	−0.810735	+	0.585414i	$0.800932\pi$
$822$		0			0
$823$	−9.59829			−0.334575			−0.167288	−	0.985908i	$-0.553501\pi$
$823$	−9.59829			−0.334575			−0.167288	+	0.985908i	$0.553501\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	8.29733			0.288526			0.144263	−	0.989539i	$-0.453919\pi$
$827$	8.29733			0.288526			0.144263	+	0.989539i	$0.453919\pi$
$828$		0			0
$829$	−7.43313			−0.258163			−0.129082	−	0.991634i	$-0.541203\pi$
$829$	−7.43313			−0.258163			−0.129082	+	0.991634i	$0.541203\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−41.9592			−1.45380
$834$		0			0
$835$			16.0260i			0.554601i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	28.9046			0.997898			0.498949	−	0.866631i	$-0.333719\pi$
$839$	28.9046			0.997898			0.498949	+	0.866631i	$0.333719\pi$
$840$		0			0
$841$	23.9373			0.825426
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.9856			0.446717
$846$		0			0
$847$		−	5.01834i		−	0.172432i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.35188	+	3.82593i	−0.252019	+	0.131151i
$852$		0			0
$853$	30.5600			1.04635			0.523176	−	0.852224i	$-0.324747\pi$
$853$	30.5600			1.04635			0.523176	+	0.852224i	$0.324747\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	17.6063i		−	0.601421i	−0.953715	−	0.300711i	$-0.902776\pi$
$857$		−	17.6063i		−	0.601421i	0.953715	−	0.300711i	$-0.0972238\pi$
$858$		0			0
$859$	−26.2413			−0.895340			−0.447670	−	0.894199i	$-0.647746\pi$
$859$	−26.2413			−0.895340			−0.447670	+	0.894199i	$0.647746\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			30.4686i			1.03716i	0.855028	+	0.518582i	$0.173540\pi$
$863$			30.4686i			1.03716i	−0.855028	+	0.518582i	$0.826460\pi$
$864$		0			0
$865$			21.6296i			0.735429i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			5.00029i			0.169623i
$870$		0			0
$871$		−	0.559940i		−	0.0189728i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			0.420015i			0.0141991i
$876$		0			0
$877$	−24.0190			−0.811064			−0.405532	−	0.914081i	$-0.632914\pi$
$877$	−24.0190			−0.811064			−0.405532	+	0.914081i	$0.632914\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−10.6343			−0.358280			−0.179140	−	0.983824i	$-0.557332\pi$
$881$	−10.6343			−0.358280			−0.179140	+	0.983824i	$0.557332\pi$
$882$		0			0
$883$	28.5194			0.959754			0.479877	−	0.877336i	$-0.340681\pi$
$883$	28.5194			0.959754			0.479877	+	0.877336i	$0.340681\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			9.13646i			0.306772i	0.988166	+	0.153386i	$0.0490178\pi$
$887$			9.13646i			0.306772i	−0.988166	+	0.153386i	$0.950982\pi$
$888$		0			0
$889$		−	4.64538i		−	0.155801i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.82327			−0.0610134
$894$		0			0
$895$		−	0.916993i		−	0.0306517i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	6.72989i		−	0.224455i
$900$		0			0
$901$	33.0323			1.10047
$902$		0			0
$903$		0			0
$904$		0			0
$905$			5.07502i			0.168699i
$906$		0			0
$907$			12.0830i			0.401208i	0.979672	+	0.200604i	$0.0642905\pi$
$907$			12.0830i			0.401208i	−0.979672	+	0.200604i	$0.935709\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	34.0926			1.12954			0.564770	−	0.825249i	$-0.308965\pi$
$911$	34.0926			1.12954			0.564770	+	0.825249i	$0.308965\pi$
$912$		0			0
$913$	−72.1336			−2.38727
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−7.89189			−0.260613
$918$		0			0
$919$		−	51.3844i		−	1.69501i	−0.530784	−	0.847507i	$-0.678102\pi$
$919$		−	51.3844i		−	1.69501i	0.530784	−	0.847507i	$-0.321898\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			1.87174i			0.0616091i
$924$		0			0
$925$		−	1.72813i		−	0.0568205i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	42.8495i		−	1.40585i	−0.711266	−	0.702923i	$-0.751878\pi$
$929$		−	42.8495i		−	1.40585i	0.711266	−	0.702923i	$-0.248122\pi$
$930$		0			0
$931$		−	4.72114i		−	0.154729i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	29.4569			0.963344
$936$		0			0
$937$		−	50.7164i		−	1.65683i	−0.560113	−	0.828416i	$-0.689242\pi$
$937$		−	50.7164i		−	1.65683i	0.560113	−	0.828416i	$-0.310758\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9.53254			0.310752			0.155376	−	0.987855i	$-0.450341\pi$
$941$	9.53254			0.310752			0.155376	+	0.987855i	$0.450341\pi$
$942$		0			0
$943$	−2.57311	+	1.33905i	−0.0837919	+	0.0436054i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			39.8748i			1.29576i	0.761744	+	0.647878i	$0.224343\pi$
$947$			39.8748i			1.29576i	−0.761744	+	0.647878i	$0.775657\pi$
$948$		0			0
$949$	1.27279			0.0413165
$950$		0			0
$951$		0			0
$952$		0			0
$953$	29.2075			0.946122			0.473061	−	0.881030i	$-0.343149\pi$
$953$	29.2075			0.946122			0.473061	+	0.881030i	$0.343149\pi$
$954$		0			0
$955$	5.88191			0.190334
$956$		0			0
$957$		0			0
$958$		0			0
$959$			8.16556i			0.263680i
$960$		0			0
$961$	−22.0538			−0.711413
$962$		0			0
$963$		0			0
$964$		0			0
$965$	16.6809			0.536978
$966$		0			0
$967$	−6.97120			−0.224179			−0.112089	−	0.993698i	$-0.535754\pi$
$967$	−6.97120			−0.224179			−0.112089	+	0.993698i	$0.535754\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	28.7401			0.922313			0.461156	−	0.887319i	$-0.347435\pi$
$971$	28.7401			0.922313			0.461156	+	0.887319i	$0.347435\pi$
$972$		0			0
$973$			3.40768i			0.109245i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	36.1685			1.15713			0.578566	−	0.815636i	$-0.303613\pi$
$977$	36.1685			1.15713			0.578566	+	0.815636i	$0.303613\pi$
$978$		0			0
$979$	−51.5970			−1.64905
$980$		0			0
$981$		0			0
$982$		0			0
$983$	46.4464			1.48141			0.740705	−	0.671830i	$-0.234492\pi$
$983$	46.4464			1.48141			0.740705	+	0.671830i	$0.234492\pi$
$984$		0			0
$985$			14.3370i			0.456815i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−33.1503	+	17.2515i	−1.05412	+	0.548565i
$990$		0			0
$991$	22.0715			0.701124			0.350562	−	0.936540i	$-0.385991\pi$
$991$	22.0715			0.701124			0.350562	+	0.936540i	$0.385991\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			21.5450i			0.683024i
$996$		0			0
$997$	−27.5479			−0.872450			−0.436225	−	0.899838i	$-0.643685\pi$
$997$	−27.5479			−0.872450			−0.436225	+	0.899838i	$0.643685\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.i.a.1241.8	✓	16
3.2	odd	2		4140.2.i.b.1241.8	yes	16
23.22	odd	2		4140.2.i.b.1241.9	yes	16
69.68	even	2	inner	4140.2.i.a.1241.9	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.i.a.1241.8	✓	16	1.1	even	1	trivial
4140.2.i.a.1241.9	yes	16	69.68	even	2	inner
4140.2.i.b.1241.8	yes	16	3.2	odd	2
4140.2.i.b.1241.9	yes	16	23.22	odd	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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -0.420015i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -0.420015i q^{7} +4.79041 q^{11} +0.120193 q^{13} -6.14914 q^{17} -0.691886i q^{19} +(-2.21391 - 4.25424i) q^{23} +1.00000 q^{25} +2.25003i q^{29} -2.99102 q^{31} +0.420015i q^{35} -1.72813i q^{37} -0.604833i q^{41} -7.79229i q^{43} -2.63522i q^{47} +6.82359 q^{49} -5.37185 q^{53} -4.79041 q^{55} -14.4099i q^{59} +10.5601i q^{61} -0.120193 q^{65} -4.65866i q^{67} +15.5728i q^{71} +10.5895 q^{73} -2.01204i q^{77} +1.04381i q^{79} -15.0579 q^{83} +6.14914 q^{85} -10.7709 q^{89} -0.0504831i q^{91} +0.691886i q^{95} -11.6961i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5}+O(q^{10}) \) 16 * q - 16 * q^5	\( 16 q - 16 q^{5} - 8 q^{23} + 16 q^{25} - 8 q^{31} - 40 q^{49} + 4 q^{53} + 8 q^{73} + 20 q^{83} + 32 q^{89}+O(q^{100}) \) 16 * q - 16 * q^5 - 8 * q^23 + 16 * q^25 - 8 * q^31 - 40 * q^49 + 4 * q^53 + 8 * q^73 + 20 * q^83 + 32 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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