LMFDB - Embedded newform 8280.2.p.a.1241.34

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1241,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, 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("8280.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.p (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:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.34
Character	$\chi$	$=$	8280.1241
Dual form			8280.2.p.a.1241.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +2.09276i q^{7} +O(q^{10})$	$q-1.00000 q^{5} +2.09276i q^{7} -3.07645 q^{11} +3.07361 q^{13} +2.79768 q^{17} +7.09530i q^{19} +(-2.07759 - 4.32246i) q^{23} +1.00000 q^{25} +7.02820i q^{29} +2.54854 q^{31} -2.09276i q^{35} +11.8247i q^{37} +1.34987i q^{41} -7.71826i q^{43} +2.64576i q^{47} +2.62034 q^{49} -5.21522 q^{53} +3.07645 q^{55} -9.91273i q^{59} -7.34076i q^{61} -3.07361 q^{65} +1.63371i q^{67} -13.2280i q^{71} -10.0650 q^{73} -6.43829i q^{77} -0.0188146i q^{79} -1.02131 q^{83} -2.79768 q^{85} +8.49568 q^{89} +6.43233i q^{91} -7.09530i q^{95} +12.5278i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{5}+O(q^{10}) $ 48 * q - 48 * q^5	$ 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) $ 48 * q - 48 * q^5 - 8 * q^11 + 4 * q^23 + 48 * q^25 + 8 * q^31 - 32 * q^49 + 8 * q^55 - 16 * q^73 - 32 * q^83 - 8 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8280\mathbb{Z}\right)^\times$.

$n$	$1657$	$2071$	$3961$	$4141$	$4601$
$\chi(n)$	$1$	$1$	$-1$	$1$	$-1$

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
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			2.09276i			0.790991i	0.918468	+	0.395495i	$0.129427\pi$
$7$			2.09276i			0.790991i	−0.918468	+	0.395495i	$0.870573\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.07645			−0.927585			−0.463793	−	0.885944i	$-0.653512\pi$
$11$	−3.07645			−0.927585			−0.463793	+	0.885944i	$0.653512\pi$
$12$		0			0
$13$	3.07361			0.852465			0.426232	−	0.904614i	$-0.359841\pi$
$13$	3.07361			0.852465			0.426232	+	0.904614i	$0.359841\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.79768			0.678537			0.339268	−	0.940690i	$-0.389821\pi$
$17$	2.79768			0.678537			0.339268	+	0.940690i	$0.389821\pi$
$18$		0			0
$19$			7.09530i			1.62777i	0.581024	+	0.813886i	$0.302652\pi$
$19$			7.09530i			1.62777i	−0.581024	+	0.813886i	$0.697348\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.07759	−	4.32246i	−0.433207	−	0.901294i
$23$	−2.07759	−	4.32246i	−0.433207	−	0.901294i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			7.02820i			1.30510i	0.757744	+	0.652552i	$0.226302\pi$
$29$			7.02820i			1.30510i	−0.757744	+	0.652552i	$0.773698\pi$
$30$		0			0
$31$	2.54854			0.457731			0.228865	−	0.973458i	$-0.426498\pi$
$31$	2.54854			0.457731			0.228865	+	0.973458i	$0.426498\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	2.09276i		−	0.353742i
$36$		0			0
$37$			11.8247i			1.94397i	0.235049	+	0.971984i	$0.424475\pi$
$37$			11.8247i			1.94397i	−0.235049	+	0.971984i	$0.575525\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.34987i			0.210814i	0.994429	+	0.105407i	$0.0336145\pi$
$41$			1.34987i			0.210814i	−0.994429	+	0.105407i	$0.966386\pi$
$42$		0			0
$43$		−	7.71826i		−	1.17702i	−0.808489	−	0.588512i	$-0.799714\pi$
$43$		−	7.71826i		−	1.17702i	0.808489	−	0.588512i	$-0.200286\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			2.64576i			0.385923i	0.981206	+	0.192962i	$0.0618093\pi$
$47$			2.64576i			0.385923i	−0.981206	+	0.192962i	$0.938191\pi$
$48$		0			0
$49$	2.62034			0.374334
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.21522			−0.716365			−0.358182	−	0.933652i	$-0.616603\pi$
$53$	−5.21522			−0.716365			−0.358182	+	0.933652i	$0.616603\pi$
$54$		0			0
$55$	3.07645			0.414829
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	9.91273i		−	1.29053i	−0.763960	−	0.645264i	$-0.776748\pi$
$59$		−	9.91273i		−	1.29053i	0.763960	−	0.645264i	$-0.223252\pi$
$60$		0			0
$61$		−	7.34076i		−	0.939888i	−0.882696	−	0.469944i	$-0.844274\pi$
$61$		−	7.34076i		−	0.939888i	0.882696	−	0.469944i	$-0.155726\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.07361			−0.381234
$66$		0			0
$67$			1.63371i			0.199589i	0.995008	+	0.0997947i	$0.0318186\pi$
$67$			1.63371i			0.199589i	−0.995008	+	0.0997947i	$0.968181\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	13.2280i		−	1.56987i	−0.619579	−	0.784934i	$-0.712697\pi$
$71$		−	13.2280i		−	1.56987i	0.619579	−	0.784934i	$-0.287303\pi$
$72$		0			0
$73$	−10.0650			−1.17802			−0.589008	−	0.808127i	$-0.700481\pi$
$73$	−10.0650			−1.17802			−0.589008	+	0.808127i	$0.700481\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	6.43829i		−	0.733711i
$78$		0			0
$79$		−	0.0188146i		−	0.00211681i	−0.999999	−	0.00105840i	$-0.999663\pi$
$79$		−	0.0188146i		−	0.00211681i	0.999999	−	0.00105840i	$-0.000336901\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.02131			−0.112103			−0.0560515	−	0.998428i	$-0.517851\pi$
$83$	−1.02131			−0.112103			−0.0560515	+	0.998428i	$0.517851\pi$
$84$		0			0
$85$	−2.79768			−0.303451
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.49568			0.900540			0.450270	−	0.892892i	$-0.351328\pi$
$89$	8.49568			0.900540			0.450270	+	0.892892i	$0.351328\pi$
$90$		0			0
$91$			6.43233i			0.674292i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	7.09530i		−	0.727962i
$96$		0			0
$97$			12.5278i			1.27201i	0.771685	+	0.636004i	$0.219414\pi$
$97$			12.5278i			1.27201i	−0.771685	+	0.636004i	$0.780586\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			16.5341i			1.64520i	0.568619	+	0.822601i	$0.307478\pi$
$101$			16.5341i			1.64520i	−0.568619	+	0.822601i	$0.692522\pi$
$102$		0			0
$103$		−	3.79697i		−	0.374126i	−0.982348	−	0.187063i	$-0.940103\pi$
$103$		−	3.79697i		−	0.374126i	0.982348	−	0.187063i	$-0.0598969\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.62880			0.640831			0.320415	−	0.947277i	$-0.396178\pi$
$107$	6.62880			0.640831			0.320415	+	0.947277i	$0.396178\pi$
$108$		0			0
$109$			10.7152i			1.02633i	0.858291	+	0.513163i	$0.171526\pi$
$109$			10.7152i			1.02633i	−0.858291	+	0.513163i	$0.828474\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.32932			0.313196			0.156598	−	0.987662i	$-0.449947\pi$
$113$	3.32932			0.313196			0.156598	+	0.987662i	$0.449947\pi$
$114$		0			0
$115$	2.07759	+	4.32246i	0.193736	+	0.403071i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			5.85488i			0.536716i
$120$		0			0
$121$	−1.53544			−0.139586
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−6.87892			−0.610405			−0.305203	−	0.952287i	$-0.598724\pi$
$127$	−6.87892			−0.610405			−0.305203	+	0.952287i	$0.598724\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			8.90544i			0.778072i	0.921223	+	0.389036i	$0.127192\pi$
$131$			8.90544i			0.778072i	−0.921223	+	0.389036i	$0.872808\pi$
$132$		0			0
$133$	−14.8488			−1.28755
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.0328			−0.857160			−0.428580	−	0.903504i	$-0.640986\pi$
$137$	−10.0328			−0.857160			−0.428580	+	0.903504i	$0.640986\pi$
$138$		0			0
$139$	13.7064			1.16256			0.581280	−	0.813703i	$-0.302552\pi$
$139$	13.7064			1.16256			0.581280	+	0.813703i	$0.302552\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−9.45580			−0.790734
$144$		0			0
$145$		−	7.02820i		−	0.583660i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.27123			−0.759528			−0.379764	−	0.925083i	$-0.623995\pi$
$149$	−9.27123			−0.759528			−0.379764	+	0.925083i	$0.623995\pi$
$150$		0			0
$151$	1.24400			0.101235			0.0506177	−	0.998718i	$-0.483881\pi$
$151$	1.24400			0.101235			0.0506177	+	0.998718i	$0.483881\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.54854			−0.204703
$156$		0			0
$157$			4.74224i			0.378472i	0.981932	+	0.189236i	$0.0606011\pi$
$157$			4.74224i			0.378472i	−0.981932	+	0.189236i	$0.939399\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	9.04588	−	4.34790i	0.712916	−	0.342663i
$162$		0			0
$163$	9.12442			0.714680			0.357340	−	0.933974i	$-0.383684\pi$
$163$	9.12442			0.714680			0.357340	+	0.933974i	$0.383684\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	3.76985i		−	0.291719i	−0.989305	−	0.145860i	$-0.953405\pi$
$167$		−	3.76985i		−	0.291719i	0.989305	−	0.145860i	$-0.0465948\pi$
$168$		0			0
$169$	−3.55295			−0.273304
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	4.32929i		−	0.329150i	−0.986365	−	0.164575i	$-0.947375\pi$
$173$		−	4.32929i		−	0.329150i	0.986365	−	0.164575i	$-0.0526253\pi$
$174$		0			0
$175$			2.09276i			0.158198i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			10.7196i			0.801221i	0.916248	+	0.400610i	$0.131202\pi$
$179$			10.7196i			0.801221i	−0.916248	+	0.400610i	$0.868798\pi$
$180$		0			0
$181$			3.63105i			0.269894i	0.990853	+	0.134947i	$0.0430864\pi$
$181$			3.63105i			0.269894i	−0.990853	+	0.134947i	$0.956914\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	11.8247i		−	0.869369i
$186$		0			0
$187$	−8.60692			−0.629400
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10.9287			−0.790776			−0.395388	−	0.918514i	$-0.629390\pi$
$191$	−10.9287			−0.790776			−0.395388	+	0.918514i	$0.629390\pi$
$192$		0			0
$193$	−11.6237			−0.836691			−0.418346	−	0.908288i	$-0.637390\pi$
$193$	−11.6237			−0.836691			−0.418346	+	0.908288i	$0.637390\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	10.7721i		−	0.767478i	−0.923442	−	0.383739i	$-0.874636\pi$
$197$		−	10.7721i		−	0.767478i	0.923442	−	0.383739i	$-0.125364\pi$
$198$		0			0
$199$		−	12.9289i		−	0.916503i	−0.888823	−	0.458252i	$-0.848476\pi$
$199$		−	12.9289i		−	0.916503i	0.888823	−	0.458252i	$-0.151524\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−14.7084			−1.03233
$204$		0			0
$205$		−	1.34987i		−	0.0942788i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	21.8283i		−	1.50990i
$210$		0			0
$211$	−12.6022			−0.867571			−0.433785	−	0.901016i	$-0.642822\pi$
$211$	−12.6022			−0.867571			−0.433785	+	0.901016i	$0.642822\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			7.71826i			0.526381i
$216$		0			0
$217$			5.33349i			0.362061i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	8.59896			0.578429
$222$		0			0
$223$	0.837321			0.0560711			0.0280356	−	0.999607i	$-0.491075\pi$
$223$	0.837321			0.0560711			0.0280356	+	0.999607i	$0.491075\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−15.6146			−1.03638			−0.518189	−	0.855266i	$-0.673393\pi$
$227$	−15.6146			−1.03638			−0.518189	+	0.855266i	$0.673393\pi$
$228$		0			0
$229$			4.41086i			0.291478i	0.989323	+	0.145739i	$0.0465560\pi$
$229$			4.41086i			0.291478i	−0.989323	+	0.145739i	$0.953444\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			23.8541i			1.56274i	0.624070	+	0.781368i	$0.285478\pi$
$233$			23.8541i			1.56274i	−0.624070	+	0.781368i	$0.714522\pi$
$234$		0			0
$235$		−	2.64576i		−	0.172590i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			20.1853i			1.30568i	0.757497	+	0.652839i	$0.226422\pi$
$239$			20.1853i			1.30568i	−0.757497	+	0.652839i	$0.773578\pi$
$240$		0			0
$241$			13.0408i			0.840034i	0.907516	+	0.420017i	$0.137976\pi$
$241$			13.0408i			0.840034i	−0.907516	+	0.420017i	$0.862024\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.62034			−0.167407
$246$		0			0
$247$			21.8081i			1.38762i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−27.2149			−1.71779			−0.858896	−	0.512150i	$-0.828849\pi$
$251$	−27.2149			−1.71779			−0.858896	+	0.512150i	$0.828849\pi$
$252$		0			0
$253$	6.39160	+	13.2978i	0.401836	+	0.836027i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	17.3463i		−	1.08203i	−0.841012	−	0.541017i	$-0.818039\pi$
$257$		−	17.3463i		−	1.08203i	0.841012	−	0.541017i	$-0.181961\pi$
$258$		0			0
$259$	−24.7463			−1.53766
$260$		0			0
$261$		0			0
$262$		0			0
$263$	8.47133			0.522364			0.261182	−	0.965290i	$-0.415888\pi$
$263$	8.47133			0.522364			0.261182	+	0.965290i	$0.415888\pi$
$264$		0			0
$265$	5.21522			0.320368
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	8.35318i		−	0.509302i	−0.967033	−	0.254651i	$-0.918039\pi$
$269$		−	8.35318i		−	0.509302i	0.967033	−	0.254651i	$-0.0819606\pi$
$270$		0			0
$271$	−11.7954			−0.716519			−0.358259	−	0.933622i	$-0.616630\pi$
$271$	−11.7954			−0.716519			−0.358259	+	0.933622i	$0.616630\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.07645			−0.185517
$276$		0			0
$277$	−24.9189			−1.49723			−0.748617	−	0.663003i	$-0.769282\pi$
$277$	−24.9189			−1.49723			−0.748617	+	0.663003i	$0.769282\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.94998			0.354946			0.177473	−	0.984126i	$-0.443208\pi$
$281$	5.94998			0.354946			0.177473	+	0.984126i	$0.443208\pi$
$282$		0			0
$283$			16.1981i			0.962879i	0.876479	+	0.481440i	$0.159886\pi$
$283$			16.1981i			0.962879i	−0.876479	+	0.481440i	$0.840114\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.82495			−0.166752
$288$		0			0
$289$	−9.17300			−0.539588
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.80011			−0.163584			−0.0817922	−	0.996649i	$-0.526064\pi$
$293$	−2.80011			−0.163584			−0.0817922	+	0.996649i	$0.526064\pi$
$294$		0			0
$295$			9.91273i			0.577141i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.38569	−	13.2855i	−0.369294	−	0.768322i
$300$		0			0
$301$	16.1525			0.931015
$302$		0			0
$303$		0			0
$304$		0			0
$305$			7.34076i			0.420331i
$306$		0			0
$307$	15.4933			0.884248			0.442124	−	0.896954i	$-0.354225\pi$
$307$	15.4933			0.884248			0.442124	+	0.896954i	$0.354225\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	25.3465i		−	1.43727i	−0.695389	−	0.718633i	$-0.744768\pi$
$311$		−	25.3465i		−	1.43727i	0.695389	−	0.718633i	$-0.255232\pi$
$312$		0			0
$313$		−	19.3753i		−	1.09516i	−0.836754	−	0.547579i	$-0.815549\pi$
$313$		−	19.3753i		−	1.09516i	0.836754	−	0.547579i	$-0.184451\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			17.9788i			1.00979i	0.863181	+	0.504895i	$0.168469\pi$
$317$			17.9788i			1.00979i	−0.863181	+	0.504895i	$0.831531\pi$
$318$		0			0
$319$		−	21.6219i		−	1.21059i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			19.8504i			1.10450i
$324$		0			0
$325$	3.07361			0.170493
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−5.53695			−0.305262
$330$		0			0
$331$	−3.38098			−0.185835			−0.0929176	−	0.995674i	$-0.529619\pi$
$331$	−3.38098			−0.185835			−0.0929176	+	0.995674i	$0.529619\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	1.63371i		−	0.0892591i
$336$		0			0
$337$		−	2.27395i		−	0.123870i	−0.998080	−	0.0619351i	$-0.980273\pi$
$337$		−	2.27395i		−	0.123870i	0.998080	−	0.0619351i	$-0.0197272\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−7.84045			−0.424584
$342$		0			0
$343$			20.1331i			1.08709i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	14.0479i		−	0.754133i	−0.926186	−	0.377066i	$-0.876933\pi$
$347$		−	14.0479i		−	0.754133i	0.926186	−	0.377066i	$-0.123067\pi$
$348$		0			0
$349$	−14.1740			−0.758714			−0.379357	−	0.925250i	$-0.623855\pi$
$349$	−14.1740			−0.758714			−0.379357	+	0.925250i	$0.623855\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	7.13650i		−	0.379838i	−0.981800	−	0.189919i	$-0.939178\pi$
$353$		−	7.13650i		−	0.379838i	0.981800	−	0.189919i	$-0.0608225\pi$
$354$		0			0
$355$			13.2280i			0.702067i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.04176			−0.107760			−0.0538799	−	0.998547i	$-0.517159\pi$
$359$	−2.04176			−0.107760			−0.0538799	+	0.998547i	$0.517159\pi$
$360$		0			0
$361$	−31.3433			−1.64964
$362$		0			0
$363$		0			0
$364$		0			0
$365$	10.0650			0.526825
$366$		0			0
$367$			13.4289i			0.700985i	0.936566	+	0.350493i	$0.113986\pi$
$367$			13.4289i			0.700985i	−0.936566	+	0.350493i	$0.886014\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	10.9142i		−	0.566638i
$372$		0			0
$373$			27.2918i			1.41312i	0.707655	+	0.706558i	$0.249753\pi$
$373$			27.2918i			1.41312i	−0.707655	+	0.706558i	$0.750247\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			21.6019i			1.11256i
$378$		0			0
$379$			24.0862i			1.23723i	0.785695	+	0.618614i	$0.212305\pi$
$379$			24.0862i			1.23723i	−0.785695	+	0.618614i	$0.787695\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−11.5208			−0.588685			−0.294343	−	0.955700i	$-0.595101\pi$
$383$	−11.5208			−0.588685			−0.294343	+	0.955700i	$0.595101\pi$
$384$		0			0
$385$			6.43829i			0.328126i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10.8756			−0.551417			−0.275708	−	0.961241i	$-0.588912\pi$
$389$	−10.8756			−0.551417			−0.275708	+	0.961241i	$0.588912\pi$
$390$		0			0
$391$	−5.81242	−	12.0928i	−0.293947	−	0.611561i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			0.0188146i			0.000946666i
$396$		0			0
$397$	20.2022			1.01392			0.506960	−	0.861970i	$-0.330769\pi$
$397$	20.2022			1.01392			0.506960	+	0.861970i	$0.330769\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−27.3337			−1.36498			−0.682489	−	0.730896i	$-0.739103\pi$
$401$	−27.3337			−1.36498			−0.682489	+	0.730896i	$0.739103\pi$
$402$		0			0
$403$	7.83320			0.390199
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	36.3781i		−	1.80319i
$408$		0			0
$409$	−38.2315			−1.89043			−0.945213	−	0.326456i	$-0.894146\pi$
$409$	−38.2315			−1.89043			−0.945213	+	0.326456i	$0.894146\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	20.7450			1.02080
$414$		0			0
$415$	1.02131			0.0501340
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.11365			−0.103259			−0.0516294	−	0.998666i	$-0.516441\pi$
$419$	−2.11365			−0.103259			−0.0516294	+	0.998666i	$0.516441\pi$
$420$		0			0
$421$		−	7.36189i		−	0.358797i	−0.983777	−	0.179398i	$-0.942585\pi$
$421$		−	7.36189i		−	0.358797i	0.983777	−	0.179398i	$-0.0574151\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.79768			0.135707
$426$		0			0
$427$	15.3625			0.743442
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−11.8122			−0.568972			−0.284486	−	0.958680i	$-0.591823\pi$
$431$	−11.8122			−0.568972			−0.284486	+	0.958680i	$0.591823\pi$
$432$		0			0
$433$			21.0382i			1.01103i	0.862817	+	0.505516i	$0.168698\pi$
$433$			21.0382i			1.01103i	−0.862817	+	0.505516i	$0.831302\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	30.6691	−	14.7411i	1.46710	−	0.705163i
$438$		0			0
$439$	−2.22130			−0.106017			−0.0530085	−	0.998594i	$-0.516881\pi$
$439$	−2.22130			−0.106017			−0.0530085	+	0.998594i	$0.516881\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	26.9357i		−	1.27975i	−0.768478	−	0.639876i	$-0.778986\pi$
$443$		−	26.9357i		−	1.27975i	0.768478	−	0.639876i	$-0.221014\pi$
$444$		0			0
$445$	−8.49568			−0.402734
$446$		0			0
$447$		0			0
$448$		0			0
$449$			5.74044i			0.270908i	0.990784	+	0.135454i	$0.0432493\pi$
$449$			5.74044i			0.270908i	−0.990784	+	0.135454i	$0.956751\pi$
$450$		0			0
$451$		−	4.15280i		−	0.195548i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	6.43233i		−	0.301552i
$456$		0			0
$457$			12.1356i			0.567682i	0.958871	+	0.283841i	$0.0916087\pi$
$457$			12.1356i			0.567682i	−0.958871	+	0.283841i	$0.908391\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	25.2556i		−	1.17627i	−0.808763	−	0.588135i	$-0.799863\pi$
$461$		−	25.2556i		−	1.17627i	0.808763	−	0.588135i	$-0.200137\pi$
$462$		0			0
$463$	−14.6393			−0.680344			−0.340172	−	0.940363i	$-0.610485\pi$
$463$	−14.6393			−0.680344			−0.340172	+	0.940363i	$0.610485\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.1415			−0.469293			−0.234647	−	0.972081i	$-0.575393\pi$
$467$	−10.1415			−0.469293			−0.234647	+	0.972081i	$0.575393\pi$
$468$		0			0
$469$	−3.41897			−0.157873
$470$		0			0
$471$		0			0
$472$		0			0
$473$			23.7449i			1.09179i
$474$		0			0
$475$			7.09530i			0.325555i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	15.4928			0.707884			0.353942	−	0.935267i	$-0.384841\pi$
$479$	15.4928			0.707884			0.353942	+	0.935267i	$0.384841\pi$
$480$		0			0
$481$			36.3444i			1.65716i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	12.5278i		−	0.568860i
$486$		0			0
$487$	−21.5590			−0.976933			−0.488467	−	0.872583i	$-0.662444\pi$
$487$	−21.5590			−0.976933			−0.488467	+	0.872583i	$0.662444\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	0.416288i		−	0.0187868i	−0.999956	−	0.00939340i	$-0.997010\pi$
$491$		−	0.416288i		−	0.0187868i	0.999956	−	0.00939340i	$-0.00299006\pi$
$492$		0			0
$493$			19.6626i			0.885561i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	27.6830			1.24175
$498$		0			0
$499$	34.6425			1.55081			0.775405	−	0.631464i	$-0.217546\pi$
$499$	34.6425			1.55081			0.775405	+	0.631464i	$0.217546\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−30.5155			−1.36062			−0.680310	−	0.732925i	$-0.738155\pi$
$503$	−30.5155			−1.36062			−0.680310	+	0.732925i	$0.738155\pi$
$504$		0			0
$505$		−	16.5341i		−	0.735757i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			30.4217i			1.34842i	0.738541	+	0.674208i	$0.235515\pi$
$509$			30.4217i			1.34842i	−0.738541	+	0.674208i	$0.764485\pi$
$510$		0			0
$511$		−	21.0636i		−	0.931799i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			3.79697i			0.167314i
$516$		0			0
$517$		−	8.13955i		−	0.357977i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−10.4148			−0.456282			−0.228141	−	0.973628i	$-0.573265\pi$
$521$	−10.4148			−0.456282			−0.228141	+	0.973628i	$0.573265\pi$
$522$		0			0
$523$		−	25.5943i		−	1.11916i	−0.828776	−	0.559580i	$-0.810962\pi$
$523$		−	25.5943i		−	1.11916i	0.828776	−	0.559580i	$-0.189038\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7.12998			0.310587
$528$		0			0
$529$	−14.3673	+	17.9606i	−0.624663	+	0.780894i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			4.14896i			0.179711i
$534$		0			0
$535$	−6.62880			−0.286588
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8.06134			−0.347226
$540$		0			0
$541$	36.2724			1.55947			0.779737	−	0.626108i	$-0.215353\pi$
$541$	36.2724			1.55947			0.779737	+	0.626108i	$0.215353\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	10.7152i		−	0.458987i
$546$		0			0
$547$	−28.7474			−1.22915			−0.614575	−	0.788859i	$-0.710672\pi$
$547$	−28.7474			−1.22915			−0.614575	+	0.788859i	$0.710672\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−49.8672			−2.12441
$552$		0			0
$553$	0.0393746			0.00167438
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.42657			−0.187560			−0.0937799	−	0.995593i	$-0.529895\pi$
$557$	−4.42657			−0.187560			−0.0937799	+	0.995593i	$0.529895\pi$
$558$		0			0
$559$		−	23.7229i		−	1.00337i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	22.0078			0.927520			0.463760	−	0.885961i	$-0.346500\pi$
$563$	22.0078			0.927520			0.463760	+	0.885961i	$0.346500\pi$
$564$		0			0
$565$	−3.32932			−0.140066
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.339615			−0.0142374			−0.00711869	−	0.999975i	$-0.502266\pi$
$569$	−0.339615			−0.0142374			−0.00711869	+	0.999975i	$0.502266\pi$
$570$		0			0
$571$		−	7.06642i		−	0.295720i	−0.989008	−	0.147860i	$-0.952761\pi$
$571$		−	7.06642i		−	0.295720i	0.989008	−	0.147860i	$-0.0472386\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.07759	−	4.32246i	−0.0866414	−	0.180259i
$576$		0			0
$577$	37.3067			1.55310			0.776550	−	0.630056i	$-0.216968\pi$
$577$	37.3067			1.55310			0.776550	+	0.630056i	$0.216968\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	2.13735i		−	0.0886724i
$582$		0			0
$583$	16.0444			0.664489
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	26.1344i		−	1.07868i	−0.842088	−	0.539340i	$-0.818674\pi$
$587$		−	26.1344i		−	1.07868i	0.842088	−	0.539340i	$-0.181326\pi$
$588$		0			0
$589$			18.0826i			0.745082i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	22.2360i		−	0.913123i	−0.889692	−	0.456561i	$-0.849081\pi$
$593$		−	22.2360i		−	0.913123i	0.889692	−	0.456561i	$-0.150919\pi$
$594$		0			0
$595$		−	5.85488i		−	0.240027i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			40.5867i			1.65833i	0.559004	+	0.829165i	$0.311183\pi$
$599$			40.5867i			1.65833i	−0.559004	+	0.829165i	$0.688817\pi$
$600$		0			0
$601$	3.25425			0.132744			0.0663718	−	0.997795i	$-0.478858\pi$
$601$	3.25425			0.132744			0.0663718	+	0.997795i	$0.478858\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.53544			0.0624247
$606$		0			0
$607$	30.7935			1.24987			0.624935	−	0.780677i	$-0.285126\pi$
$607$	30.7935			1.24987			0.624935	+	0.780677i	$0.285126\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			8.13202i			0.328986i
$612$		0			0
$613$			38.9495i			1.57315i	0.617492	+	0.786577i	$0.288149\pi$
$613$			38.9495i			1.57315i	−0.617492	+	0.786577i	$0.711851\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−11.4150			−0.459551			−0.229775	−	0.973244i	$-0.573799\pi$
$617$	−11.4150			−0.459551			−0.229775	+	0.973244i	$0.573799\pi$
$618$		0			0
$619$			17.9167i			0.720135i	0.932926	+	0.360067i	$0.117246\pi$
$619$			17.9167i			0.720135i	−0.932926	+	0.360067i	$0.882754\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			17.7795i			0.712319i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			33.0817i			1.31905i
$630$		0			0
$631$		−	15.9609i		−	0.635392i	−0.948193	−	0.317696i	$-0.897091\pi$
$631$		−	15.9609i		−	0.635392i	0.948193	−	0.317696i	$-0.102909\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6.87892			0.272982
$636$		0			0
$637$	8.05388			0.319106
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−36.5256			−1.44267			−0.721337	−	0.692585i	$-0.756472\pi$
$641$	−36.5256			−1.44267			−0.721337	+	0.692585i	$0.756472\pi$
$642$		0			0
$643$		−	46.5350i		−	1.83516i	−0.397548	−	0.917581i	$-0.630139\pi$
$643$		−	46.5350i		−	1.83516i	0.397548	−	0.917581i	$-0.369861\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			9.05104i			0.355833i	0.984046	+	0.177917i	$0.0569357\pi$
$647$			9.05104i			0.355833i	−0.984046	+	0.177917i	$0.943064\pi$
$648$		0			0
$649$			30.4960i			1.19707i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			28.8899i			1.13055i	0.824903	+	0.565274i	$0.191230\pi$
$653$			28.8899i			1.13055i	−0.824903	+	0.565274i	$0.808770\pi$
$654$		0			0
$655$		−	8.90544i		−	0.347964i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−27.9777			−1.08986			−0.544929	−	0.838482i	$-0.683443\pi$
$659$	−27.9777			−1.08986			−0.544929	+	0.838482i	$0.683443\pi$
$660$		0			0
$661$			19.5857i			0.761795i	0.924617	+	0.380898i	$0.124385\pi$
$661$			19.5857i			0.761795i	−0.924617	+	0.380898i	$0.875615\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	14.8488			0.575811
$666$		0			0
$667$	30.3791	−	14.6017i	1.17628	−	0.565380i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			22.5835i			0.871826i
$672$		0			0
$673$	18.2582			0.703802			0.351901	−	0.936037i	$-0.385535\pi$
$673$	18.2582			0.703802			0.351901	+	0.936037i	$0.385535\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	31.3125			1.20344			0.601718	−	0.798709i	$-0.294483\pi$
$677$	31.3125			1.20344			0.601718	+	0.798709i	$0.294483\pi$
$678$		0			0
$679$	−26.2178			−1.00615
$680$		0			0
$681$		0			0
$682$		0			0
$683$			2.45873i			0.0940807i	0.998893	+	0.0470404i	$0.0149789\pi$
$683$			2.45873i			0.0940807i	−0.998893	+	0.0470404i	$0.985021\pi$
$684$		0			0
$685$	10.0328			0.383333
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−16.0295			−0.610676
$690$		0			0
$691$	47.9601			1.82449			0.912244	−	0.409647i	$-0.134348\pi$
$691$	47.9601			1.82449			0.912244	+	0.409647i	$0.134348\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−13.7064			−0.519913
$696$		0			0
$697$			3.77649i			0.143045i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.31302			0.162901			0.0814503	−	0.996677i	$-0.474045\pi$
$701$	4.31302			0.162901			0.0814503	+	0.996677i	$0.474045\pi$
$702$		0			0
$703$	−83.8997			−3.16434
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−34.6019			−1.30134
$708$		0			0
$709$		−	30.4757i		−	1.14454i	−0.820065	−	0.572270i	$-0.806063\pi$
$709$		−	30.4757i		−	1.14454i	0.820065	−	0.572270i	$-0.193937\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−5.29481	−	11.0159i	−0.198292	−	0.412550i
$714$		0			0
$715$	9.45580			0.353627
$716$		0			0
$717$		0			0
$718$		0			0
$719$			42.7482i			1.59424i	0.603821	+	0.797120i	$0.293644\pi$
$719$			42.7482i			1.59424i	−0.603821	+	0.797120i	$0.706356\pi$
$720$		0			0
$721$	7.94616			0.295930
$722$		0			0
$723$		0			0
$724$		0			0
$725$			7.02820i			0.261021i
$726$		0			0
$727$			31.1931i			1.15689i	0.815722	+	0.578444i	$0.196340\pi$
$727$			31.1931i			1.15689i	−0.815722	+	0.578444i	$0.803660\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	21.5932i		−	0.798654i
$732$		0			0
$733$			24.5956i			0.908459i	0.890885	+	0.454229i	$0.150085\pi$
$733$			24.5956i			0.908459i	−0.890885	+	0.454229i	$0.849915\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	5.02603i		−	0.185136i
$738$		0			0
$739$	14.8144			0.544955			0.272478	−	0.962162i	$-0.412157\pi$
$739$	14.8144			0.544955			0.272478	+	0.962162i	$0.412157\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−48.7808			−1.78959			−0.894797	−	0.446472i	$-0.852680\pi$
$743$	−48.7808			−1.78959			−0.894797	+	0.446472i	$0.852680\pi$
$744$		0			0
$745$	9.27123			0.339671
$746$		0			0
$747$		0			0
$748$		0			0
$749$			13.8725i			0.506891i
$750$		0			0
$751$			24.6795i			0.900568i	0.892885	+	0.450284i	$0.148677\pi$
$751$			24.6795i			0.900568i	−0.892885	+	0.450284i	$0.851323\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.24400			−0.0452738
$756$		0			0
$757$		−	16.4648i		−	0.598423i	−0.954187	−	0.299212i	$-0.903276\pi$
$757$		−	16.4648i		−	0.598423i	0.954187	−	0.299212i	$-0.0967237\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			24.2547i			0.879231i	0.898186	+	0.439615i	$0.144885\pi$
$761$			24.2547i			0.879231i	−0.898186	+	0.439615i	$0.855115\pi$
$762$		0			0
$763$	−22.4243			−0.811814
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	30.4678i		−	1.10013i
$768$		0			0
$769$		−	10.0716i		−	0.363192i	−0.983373	−	0.181596i	$-0.941874\pi$
$769$		−	10.0716i		−	0.363192i	0.983373	−	0.181596i	$-0.0581263\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	43.6499			1.56998			0.784990	−	0.619509i	$-0.212668\pi$
$773$	43.6499			1.56998			0.784990	+	0.619509i	$0.212668\pi$
$774$		0			0
$775$	2.54854			0.0915461
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−9.57771			−0.343157
$780$		0			0
$781$			40.6952i			1.45619i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	4.74224i		−	0.169258i
$786$		0			0
$787$			17.2423i			0.614623i	0.951609	+	0.307312i	$0.0994294\pi$
$787$			17.2423i			0.614623i	−0.951609	+	0.307312i	$0.900571\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			6.96748i			0.247735i
$792$		0			0
$793$		−	22.5626i		−	0.801221i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	47.7562			1.69161			0.845805	−	0.533492i	$-0.179121\pi$
$797$	47.7562			1.69161			0.845805	+	0.533492i	$0.179121\pi$
$798$		0			0
$799$			7.40198i			0.261863i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	30.9644			1.09271
$804$		0			0
$805$	−9.04588	+	4.34790i	−0.318826	+	0.153243i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	23.9844i		−	0.843248i	−0.906771	−	0.421624i	$-0.861460\pi$
$809$		−	23.9844i		−	0.843248i	0.906771	−	0.421624i	$-0.138540\pi$
$810$		0			0
$811$	−19.8949			−0.698604			−0.349302	−	0.937010i	$-0.613581\pi$
$811$	−19.8949			−0.698604			−0.349302	+	0.937010i	$0.613581\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9.12442			−0.319615
$816$		0			0
$817$	54.7634			1.91593
$818$		0			0
$819$		0			0
$820$		0			0
$821$			36.0417i			1.25786i	0.777460	+	0.628932i	$0.216508\pi$
$821$			36.0417i			1.25786i	−0.777460	+	0.628932i	$0.783492\pi$
$822$		0			0
$823$	−15.1097			−0.526691			−0.263345	−	0.964702i	$-0.584826\pi$
$823$	−15.1097			−0.526691			−0.263345	+	0.964702i	$0.584826\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20.0925			0.698686			0.349343	−	0.936995i	$-0.386405\pi$
$827$	20.0925			0.698686			0.349343	+	0.936995i	$0.386405\pi$
$828$		0			0
$829$	−49.2214			−1.70953			−0.854765	−	0.519015i	$-0.826299\pi$
$829$	−49.2214			−1.70953			−0.854765	+	0.519015i	$0.826299\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	7.33085			0.253999
$834$		0			0
$835$			3.76985i			0.130461i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−47.7474			−1.64842			−0.824211	−	0.566283i	$-0.808381\pi$
$839$	−47.7474			−1.64842			−0.824211	+	0.566283i	$0.808381\pi$
$840$		0			0
$841$	−20.3956			−0.703296
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.55295			0.122225
$846$		0			0
$847$		−	3.21332i		−	0.110411i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	51.1117	−	24.5668i	1.75209	−	0.842140i
$852$		0			0
$853$	−30.8767			−1.05720			−0.528599	−	0.848872i	$-0.677283\pi$
$853$	−30.8767			−1.05720			−0.528599	+	0.848872i	$0.677283\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1.77669i		−	0.0606906i	−0.999539	−	0.0303453i	$-0.990339\pi$
$857$		−	1.77669i		−	0.0606906i	0.999539	−	0.0303453i	$-0.00966069\pi$
$858$		0			0
$859$	−44.2431			−1.50955			−0.754777	−	0.655981i	$-0.772255\pi$
$859$	−44.2431			−1.50955			−0.754777	+	0.655981i	$0.772255\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	44.2242i		−	1.50541i	−0.658359	−	0.752704i	$-0.728749\pi$
$863$		−	44.2242i		−	1.50541i	0.658359	−	0.752704i	$-0.271251\pi$
$864$		0			0
$865$			4.32929i			0.147200i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			0.0578823i			0.00196352i
$870$		0			0
$871$			5.02138i			0.170143i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	2.09276i		−	0.0707484i
$876$		0			0
$877$	1.60742			0.0542786			0.0271393	−	0.999632i	$-0.491360\pi$
$877$	1.60742			0.0542786			0.0271393	+	0.999632i	$0.491360\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	30.6692			1.03327			0.516636	−	0.856205i	$-0.327184\pi$
$881$	30.6692			1.03327			0.516636	+	0.856205i	$0.327184\pi$
$882$		0			0
$883$	33.6829			1.13352			0.566760	−	0.823883i	$-0.308197\pi$
$883$	33.6829			1.13352			0.566760	+	0.823883i	$0.308197\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			20.7339i			0.696177i	0.937462	+	0.348088i	$0.113169\pi$
$887$			20.7339i			0.696177i	−0.937462	+	0.348088i	$0.886831\pi$
$888$		0			0
$889$		−	14.3960i		−	0.482825i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−18.7724			−0.628196
$894$		0			0
$895$		−	10.7196i		−	0.358317i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			17.9116i			0.597386i
$900$		0			0
$901$	−14.5905			−0.486080
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	3.63105i		−	0.120700i
$906$		0			0
$907$		−	31.2817i		−	1.03869i	−0.854565	−	0.519345i	$-0.826176\pi$
$907$		−	31.2817i		−	1.03869i	0.854565	−	0.519345i	$-0.173824\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−3.81204			−0.126299			−0.0631493	−	0.998004i	$-0.520114\pi$
$911$	−3.81204			−0.126299			−0.0631493	+	0.998004i	$0.520114\pi$
$912$		0			0
$913$	3.14200			0.103985
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−18.6370			−0.615448
$918$		0			0
$919$		−	15.1045i		−	0.498253i	−0.968471	−	0.249126i	$-0.919857\pi$
$919$		−	15.1045i		−	0.498253i	0.968471	−	0.249126i	$-0.0801434\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	40.6575i		−	1.33826i
$924$		0			0
$925$			11.8247i			0.388793i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	19.6525i		−	0.644778i	−0.946607	−	0.322389i	$-0.895514\pi$
$929$		−	19.6525i		−	0.644778i	0.946607	−	0.322389i	$-0.104486\pi$
$930$		0			0
$931$			18.5921i			0.609330i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	8.60692			0.281476
$936$		0			0
$937$		−	11.4391i		−	0.373700i	−0.982388	−	0.186850i	$-0.940172\pi$
$937$		−	11.4391i		−	0.373700i	0.982388	−	0.186850i	$-0.0598278\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	42.9122			1.39890			0.699448	−	0.714683i	$-0.253429\pi$
$941$	42.9122			1.39890			0.699448	+	0.714683i	$0.253429\pi$
$942$		0			0
$943$	5.83474	−	2.80447i	0.190005	−	0.0913260i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	46.9906i		−	1.52699i	−0.645814	−	0.763495i	$-0.723482\pi$
$947$		−	46.9906i		−	1.52699i	0.645814	−	0.763495i	$-0.276518\pi$
$948$		0			0
$949$	−30.9357			−1.00422
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.321315			−0.0104084			−0.00520421	−	0.999986i	$-0.501657\pi$
$953$	−0.321315			−0.0104084			−0.00520421	+	0.999986i	$0.501657\pi$
$954$		0			0
$955$	10.9287			0.353646
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	20.9963i		−	0.678005i
$960$		0			0
$961$	−24.5050			−0.790483
$962$		0			0
$963$		0			0
$964$		0			0
$965$	11.6237			0.374180
$966$		0			0
$967$	3.74943			0.120574			0.0602868	−	0.998181i	$-0.480798\pi$
$967$	3.74943			0.120574			0.0602868	+	0.998181i	$0.480798\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5.56625			0.178629			0.0893147	−	0.996003i	$-0.471532\pi$
$971$	5.56625			0.178629			0.0893147	+	0.996003i	$0.471532\pi$
$972$		0			0
$973$			28.6843i			0.919575i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	27.5570			0.881628			0.440814	−	0.897598i	$-0.354690\pi$
$977$	27.5570			0.881628			0.440814	+	0.897598i	$0.354690\pi$
$978$		0			0
$979$	−26.1365			−0.835327
$980$		0			0
$981$		0			0
$982$		0			0
$983$	24.8774			0.793466			0.396733	−	0.917934i	$-0.370144\pi$
$983$	24.8774			0.793466			0.396733	+	0.917934i	$0.370144\pi$
$984$		0			0
$985$			10.7721i			0.343226i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−33.3618	+	16.0354i	−1.06084	+	0.509895i
$990$		0			0
$991$	−35.2309			−1.11915			−0.559573	−	0.828781i	$-0.689035\pi$
$991$	−35.2309			−1.11915			−0.559573	+	0.828781i	$0.689035\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			12.9289i			0.409873i
$996$		0			0
$997$	11.4647			0.363092			0.181546	−	0.983382i	$-0.441890\pi$
$997$	11.4647			0.363092			0.181546	+	0.983382i	$0.441890\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.p.a.1241.34	yes	48
3.2	odd	2		8280.2.p.b.1241.34	yes	48
23.22	odd	2		8280.2.p.b.1241.15	yes	48
69.68	even	2	inner	8280.2.p.a.1241.15	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.p.a.1241.15	✓	48	69.68	even	2	inner
8280.2.p.a.1241.34	yes	48	1.1	even	1	trivial
8280.2.p.b.1241.15	yes	48	23.22	odd	2
8280.2.p.b.1241.34	yes	48	3.2	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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +2.09276i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +2.09276i q^{7} -3.07645 q^{11} +3.07361 q^{13} +2.79768 q^{17} +7.09530i q^{19} +(-2.07759 - 4.32246i) q^{23} +1.00000 q^{25} +7.02820i q^{29} +2.54854 q^{31} -2.09276i q^{35} +11.8247i q^{37} +1.34987i q^{41} -7.71826i q^{43} +2.64576i q^{47} +2.62034 q^{49} -5.21522 q^{53} +3.07645 q^{55} -9.91273i q^{59} -7.34076i q^{61} -3.07361 q^{65} +1.63371i q^{67} -13.2280i q^{71} -10.0650 q^{73} -6.43829i q^{77} -0.0188146i q^{79} -1.02131 q^{83} -2.79768 q^{85} +8.49568 q^{89} +6.43233i q^{91} -7.09530i q^{95} +12.5278i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{5}+O(q^{10}) \) 48 * q - 48 * q^5	\( 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) \) 48 * q - 48 * q^5 - 8 * q^11 + 4 * q^23 + 48 * q^25 + 8 * q^31 - 32 * q^49 + 8 * q^55 - 16 * q^73 - 32 * q^83 - 8 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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