LMFDB - Embedded newform 3060.2.e.a.1801.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3060,2,Mod(1801,3060)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3060, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3060.1801"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3060.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,-12,0,0,0,8,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$24.4342230185$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1020)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1801.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3060.1801
Dual form			3060.2.e.a.1801.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{5} -2.00000i q^{11} -6.00000 q^{13} +(4.00000 + 1.00000i) q^{17} -4.00000 q^{19} +4.00000i q^{23} -1.00000 q^{25} +6.00000i q^{29} +4.00000i q^{31} +2.00000i q^{37} +2.00000i q^{41} -6.00000 q^{43} +8.00000 q^{47} +7.00000 q^{49} -2.00000 q^{55} +12.0000 q^{59} +4.00000i q^{61} +6.00000i q^{65} +2.00000 q^{67} +2.00000i q^{71} -2.00000i q^{73} +16.0000i q^{79} +(1.00000 - 4.00000i) q^{85} -2.00000 q^{89} +4.00000i q^{95} +14.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{13} + 8 q^{17} - 8 q^{19} - 2 q^{25} - 12 q^{43} + 16 q^{47} + 14 q^{49} - 4 q^{55} + 24 q^{59} + 4 q^{67} + 2 q^{85} - 4 q^{89}+O(q^{100}) $ 2 * q - 12 * q^13 + 8 * q^17 - 8 * q^19 - 2 * q^25 - 12 * q^43 + 16 * q^47 + 14 * q^49 - 4 * q^55 + 24 * q^59 + 4 * q^67 + 2 * q^85 - 4 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3060\mathbb{Z}\right)^\times$.

$n$	$1261$	$1361$	$1531$	$1837$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	2.00000i		−	0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$		−	2.00000i		−	0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$		0			0
$13$	−6.00000			−1.66410			−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000			−1.66410			−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.00000	+	1.00000i	0.970143	+	0.242536i
$17$	4.00000	+	1.00000i	0.970143	+	0.242536i
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$			4.00000i			0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$			4.00000i			0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			2.00000i			0.312348i	0.987730	+	0.156174i	$0.0499160\pi$
$41$			2.00000i			0.312348i	−0.987730	+	0.156174i	$0.950084\pi$
$42$		0			0
$43$	−6.00000			−0.914991			−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000			−0.914991			−0.457496	+	0.889212i	$0.651253\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−2.00000			−0.269680
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$		0			0
$61$			4.00000i			0.512148i	0.966657	+	0.256074i	$0.0824290\pi$
$61$			4.00000i			0.512148i	−0.966657	+	0.256074i	$0.917571\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			6.00000i			0.744208i
$66$		0			0
$67$	2.00000			0.244339			0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000			0.244339			0.122169	+	0.992509i	$0.461015\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2.00000i			0.237356i	0.992933	+	0.118678i	$0.0378657\pi$
$71$			2.00000i			0.237356i	−0.992933	+	0.118678i	$0.962134\pi$
$72$		0			0
$73$		−	2.00000i		−	0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$		−	2.00000i		−	0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$			16.0000i			1.80014i	0.435745	+	0.900070i	$0.356485\pi$
$79$			16.0000i			1.80014i	−0.435745	+	0.900070i	$0.643515\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	1.00000	−	4.00000i	0.108465	−	0.433861i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.00000			−0.212000			−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000			−0.212000			−0.106000	+	0.994366i	$0.533804\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$			4.00000i			0.410391i
$96$		0			0
$97$			14.0000i			1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$			14.0000i			1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.00000			−0.199007			−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000			−0.199007			−0.0995037	+	0.995037i	$0.531726\pi$
$102$		0			0
$103$	−6.00000			−0.591198			−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000			−0.591198			−0.295599	+	0.955312i	$0.595519\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$		0			0
$109$		−	12.0000i		−	1.14939i	−0.818367	−	0.574696i	$-0.805120\pi$
$109$		−	12.0000i		−	1.14939i	0.818367	−	0.574696i	$-0.194880\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			14.0000i			1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$			14.0000i			1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$		0			0
$115$	4.00000			0.373002
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	7.00000			0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	−2.00000			−0.177471			−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000			−0.177471			−0.0887357	+	0.996055i	$0.528283\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	6.00000i		−	0.524222i	−0.965038	−	0.262111i	$-0.915581\pi$
$131$		−	6.00000i		−	0.524222i	0.965038	−	0.262111i	$-0.0844187\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−12.0000			−1.02523			−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000			−1.02523			−0.512615	+	0.858619i	$0.671323\pi$
$138$		0			0
$139$			8.00000i			0.678551i	0.940687	+	0.339276i	$0.110182\pi$
$139$			8.00000i			0.678551i	−0.940687	+	0.339276i	$0.889818\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			12.0000i			1.00349i
$144$		0			0
$145$	6.00000			0.498273
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.00000			0.321288
$156$		0			0
$157$	−14.0000			−1.11732			−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000			−1.11732			−0.558661	+	0.829396i	$0.688685\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	20.0000i		−	1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$		−	20.0000i		−	1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			16.0000i			1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$			16.0000i			1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$		0			0
$169$	23.0000			1.76923
$170$		0			0
$171$		0			0
$172$		0			0
$173$			18.0000i			1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$			18.0000i			1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−20.0000			−1.49487			−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000			−1.49487			−0.747435	+	0.664335i	$0.768715\pi$
$180$		0			0
$181$			20.0000i			1.48659i	0.668965	+	0.743294i	$0.266738\pi$
$181$			20.0000i			1.48659i	−0.668965	+	0.743294i	$0.733262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.00000			0.147043
$186$		0			0
$187$	2.00000	−	8.00000i	0.146254	−	0.585018i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$		−	2.00000i		−	0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$		−	2.00000i		−	0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$		0			0
$199$		−	16.0000i		−	1.13421i	−0.823646	−	0.567105i	$-0.808063\pi$
$199$		−	16.0000i		−	1.13421i	0.823646	−	0.567105i	$-0.191937\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	2.00000			0.139686
$206$		0			0
$207$		0			0
$208$		0			0
$209$			8.00000i			0.553372i
$210$		0			0
$211$			12.0000i			0.826114i	0.910705	+	0.413057i	$0.135539\pi$
$211$			12.0000i			0.826114i	−0.910705	+	0.413057i	$0.864461\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			6.00000i			0.409197i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−24.0000	−	6.00000i	−1.61441	−	0.403604i
$222$		0			0
$223$	−2.00000			−0.133930			−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000			−0.133930			−0.0669650	+	0.997755i	$0.521332\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			18.0000i			1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$			18.0000i			1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$		0			0
$235$		−	8.00000i		−	0.521862i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$		0			0
$241$			8.00000i			0.515325i	0.966235	+	0.257663i	$0.0829523\pi$
$241$			8.00000i			0.515325i	−0.966235	+	0.257663i	$0.917048\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	7.00000i		−	0.447214i
$246$		0			0
$247$	24.0000			1.52708
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$	8.00000			0.502956
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−28.0000			−1.74659			−0.873296	−	0.487190i	$-0.838022\pi$
$257$	−28.0000			−1.74659			−0.873296	+	0.487190i	$0.838022\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	8.00000			0.493301			0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000			0.493301			0.246651	+	0.969104i	$0.420670\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			6.00000i			0.365826i	0.983129	+	0.182913i	$0.0585527\pi$
$269$			6.00000i			0.365826i	−0.983129	+	0.182913i	$0.941447\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			2.00000i			0.120605i
$276$		0			0
$277$			22.0000i			1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$			22.0000i			1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$			28.0000i			1.66443i	0.554455	+	0.832214i	$0.312927\pi$
$283$			28.0000i			1.66443i	−0.554455	+	0.832214i	$0.687073\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	15.0000	+	8.00000i	0.882353	+	0.470588i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	12.0000			0.701047			0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000			0.701047			0.350524	+	0.936554i	$0.386004\pi$
$294$		0			0
$295$		−	12.0000i		−	0.698667i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	24.0000i		−	1.38796i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.00000			0.229039
$306$		0			0
$307$	−10.0000			−0.570730			−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000			−0.570730			−0.285365	+	0.958419i	$0.592115\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	18.0000i		−	1.02069i	−0.859971	−	0.510343i	$-0.829518\pi$
$311$		−	18.0000i		−	1.02069i	0.859971	−	0.510343i	$-0.170482\pi$
$312$		0			0
$313$		−	34.0000i		−	1.92179i	−0.276907	−	0.960897i	$-0.589309\pi$
$313$		−	34.0000i		−	1.92179i	0.276907	−	0.960897i	$-0.410691\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			18.0000i			1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$			18.0000i			1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$		0			0
$319$	12.0000			0.671871
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−16.0000	−	4.00000i	−0.890264	−	0.222566i
$324$		0			0
$325$	6.00000			0.332820
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−32.0000			−1.75888			−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000			−1.75888			−0.879440	+	0.476011i	$0.842082\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	2.00000i		−	0.109272i
$336$		0			0
$337$			22.0000i			1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$			22.0000i			1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	8.00000			0.433224
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	16.0000i		−	0.858925i	−0.903085	−	0.429463i	$-0.858703\pi$
$347$		−	16.0000i		−	0.858925i	0.903085	−	0.429463i	$-0.141297\pi$
$348$		0			0
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.00000			0.425797			0.212899	−	0.977074i	$-0.431710\pi$
$353$	8.00000			0.425797			0.212899	+	0.977074i	$0.431710\pi$
$354$		0			0
$355$	2.00000			0.106149
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.00000			−0.104685
$366$		0			0
$367$		−	8.00000i		−	0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$		−	8.00000i		−	0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−10.0000			−0.517780			−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000			−0.517780			−0.258890	+	0.965907i	$0.583357\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	36.0000i		−	1.85409i
$378$		0			0
$379$		−	32.0000i		−	1.64373i	−0.569683	−	0.821865i	$-0.692934\pi$
$379$		−	32.0000i		−	1.64373i	0.569683	−	0.821865i	$-0.307066\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−36.0000			−1.83951			−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000			−1.83951			−0.919757	+	0.392488i	$0.871614\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10.0000			−0.507020			−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000			−0.507020			−0.253510	+	0.967333i	$0.581585\pi$
$390$		0			0
$391$	−4.00000	+	16.0000i	−0.202289	+	0.809155i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	16.0000			0.805047
$396$		0			0
$397$		−	2.00000i		−	0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$		−	2.00000i		−	0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			18.0000i			0.898877i	0.893311	+	0.449439i	$0.148376\pi$
$401$			18.0000i			0.898877i	−0.893311	+	0.449439i	$0.851624\pi$
$402$		0			0
$403$		−	24.0000i		−	1.19553i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.00000			0.198273
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	26.0000i		−	1.27018i	−0.772437	−	0.635092i	$-0.780962\pi$
$419$		−	26.0000i		−	1.27018i	0.772437	−	0.635092i	$-0.219038\pi$
$420$		0			0
$421$	−22.0000			−1.07221			−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000			−1.07221			−0.536107	+	0.844150i	$0.680106\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4.00000	−	1.00000i	−0.194029	−	0.0485071i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$			6.00000i			0.289010i	0.989504	+	0.144505i	$0.0461589\pi$
$431$			6.00000i			0.289010i	−0.989504	+	0.144505i	$0.953841\pi$
$432$		0			0
$433$	−34.0000			−1.63394			−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000			−1.63394			−0.816968	+	0.576683i	$0.804347\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	16.0000i		−	0.765384i
$438$		0			0
$439$			4.00000i			0.190910i	0.995434	+	0.0954548i	$0.0304305\pi$
$439$			4.00000i			0.190910i	−0.995434	+	0.0954548i	$0.969569\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.00000			0.190046			0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000			0.190046			0.0950229	+	0.995475i	$0.469708\pi$
$444$		0			0
$445$			2.00000i			0.0948091i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			18.0000i			0.849473i	0.905317	+	0.424736i	$0.139633\pi$
$449$			18.0000i			0.849473i	−0.905317	+	0.424736i	$0.860367\pi$
$450$		0			0
$451$	4.00000			0.188353
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2.00000			0.0935561			0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000			0.0935561			0.0467780	+	0.998905i	$0.485105\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	42.0000			1.95614			0.978068	−	0.208288i	$-0.0667892\pi$
$461$	42.0000			1.95614			0.978068	+	0.208288i	$0.0667892\pi$
$462$		0			0
$463$	−26.0000			−1.20832			−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000			−1.20832			−0.604161	+	0.796862i	$0.706492\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.0000			0.925490			0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000			0.925490			0.462745	+	0.886492i	$0.346865\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$			12.0000i			0.551761i
$474$		0			0
$475$	4.00000			0.183533
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	6.00000i		−	0.274147i	−0.990561	−	0.137073i	$-0.956230\pi$
$479$		−	6.00000i		−	0.274147i	0.990561	−	0.137073i	$-0.0437697\pi$
$480$		0			0
$481$		−	12.0000i		−	0.547153i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	14.0000			0.635707
$486$		0			0
$487$		−	24.0000i		−	1.08754i	−0.839233	−	0.543772i	$-0.816996\pi$
$487$		−	24.0000i		−	1.08754i	0.839233	−	0.543772i	$-0.183004\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4.00000			−0.180517			−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000			−0.180517			−0.0902587	+	0.995918i	$0.528769\pi$
$492$		0			0
$493$	−6.00000	+	24.0000i	−0.270226	+	1.08091i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			32.0000i			1.43252i	0.697835	+	0.716258i	$0.254147\pi$
$499$			32.0000i			1.43252i	−0.697835	+	0.716258i	$0.745853\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$			2.00000i			0.0889988i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	22.0000			0.975133			0.487566	−	0.873086i	$-0.337885\pi$
$509$	22.0000			0.975133			0.487566	+	0.873086i	$0.337885\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$			6.00000i			0.264392i
$516$		0			0
$517$		−	16.0000i		−	0.703679i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	22.0000i		−	0.963837i	−0.876216	−	0.481919i	$-0.839940\pi$
$521$		−	22.0000i		−	0.963837i	0.876216	−	0.481919i	$-0.160060\pi$
$522$		0			0
$523$	−34.0000			−1.48672			−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000			−1.48672			−0.743358	+	0.668894i	$0.766768\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.00000	+	16.0000i	−0.174243	+	0.696971i
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	12.0000i		−	0.519778i
$534$		0			0
$535$	−12.0000			−0.518805
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	14.0000i		−	0.603023i
$540$		0			0
$541$		−	12.0000i		−	0.515920i	−0.966156	−	0.257960i	$-0.916950\pi$
$541$		−	12.0000i		−	0.515920i	0.966156	−	0.257960i	$-0.0830503\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.0000			−0.514024
$546$		0			0
$547$		−	12.0000i		−	0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$		−	12.0000i		−	0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	24.0000i		−	1.02243i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	32.0000			1.35588			0.677942	−	0.735116i	$-0.262872\pi$
$557$	32.0000			1.35588			0.677942	+	0.735116i	$0.262872\pi$
$558$		0			0
$559$	36.0000			1.52264
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4.00000			0.168580			0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000			0.168580			0.0842900	+	0.996441i	$0.473138\pi$
$564$		0			0
$565$	14.0000			0.588984
$566$		0			0
$567$		0			0
$568$		0			0
$569$	22.0000			0.922288			0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000			0.922288			0.461144	+	0.887325i	$0.347439\pi$
$570$		0			0
$571$			36.0000i			1.50655i	0.657704	+	0.753277i	$0.271528\pi$
$571$			36.0000i			1.50655i	−0.657704	+	0.753277i	$0.728472\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	4.00000i		−	0.166812i
$576$		0			0
$577$	38.0000			1.58196			0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000			1.58196			0.790980	+	0.611842i	$0.209571\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−8.00000			−0.330195			−0.165098	−	0.986277i	$-0.552794\pi$
$587$	−8.00000			−0.330195			−0.165098	+	0.986277i	$0.552794\pi$
$588$		0			0
$589$		−	16.0000i		−	0.659269i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−36.0000			−1.47834			−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000			−1.47834			−0.739171	+	0.673517i	$0.764783\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	8.00000			0.326871			0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000			0.326871			0.163436	+	0.986554i	$0.447742\pi$
$600$		0			0
$601$		−	28.0000i		−	1.14214i	−0.820900	−	0.571072i	$-0.806528\pi$
$601$		−	28.0000i		−	1.14214i	0.820900	−	0.571072i	$-0.193472\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	7.00000i		−	0.284590i
$606$		0			0
$607$		−	8.00000i		−	0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$		−	8.00000i		−	0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−48.0000			−1.94187
$612$		0			0
$613$	−30.0000			−1.21169			−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000			−1.21169			−0.605844	+	0.795583i	$0.707165\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			2.00000i			0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$			2.00000i			0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$		0			0
$619$		−	20.0000i		−	0.803868i	−0.915669	−	0.401934i	$-0.868338\pi$
$619$		−	20.0000i		−	0.803868i	0.915669	−	0.401934i	$-0.131662\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−2.00000	+	8.00000i	−0.0797452	+	0.318981i
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			2.00000i			0.0793676i
$636$		0			0
$637$	−42.0000			−1.66410
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	30.0000i		−	1.18493i	−0.805597	−	0.592464i	$-0.798155\pi$
$641$		−	30.0000i		−	1.18493i	0.805597	−	0.592464i	$-0.201845\pi$
$642$		0			0
$643$			12.0000i			0.473234i	0.971603	+	0.236617i	$0.0760386\pi$
$643$			12.0000i			0.473234i	−0.971603	+	0.236617i	$0.923961\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	20.0000			0.786281			0.393141	−	0.919478i	$-0.371389\pi$
$647$	20.0000			0.786281			0.393141	+	0.919478i	$0.371389\pi$
$648$		0			0
$649$		−	24.0000i		−	0.942082i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	18.0000i		−	0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$		−	18.0000i		−	0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$		0			0
$655$	−6.00000			−0.234439
$656$		0			0
$657$		0			0
$658$		0			0
$659$	20.0000			0.779089			0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000			0.779089			0.389545	+	0.921008i	$0.372632\pi$
$660$		0			0
$661$	42.0000			1.63361			0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000			1.63361			0.816805	+	0.576913i	$0.195743\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−24.0000			−0.929284
$668$		0			0
$669$		0			0
$670$		0			0
$671$	8.00000			0.308837
$672$		0			0
$673$			46.0000i			1.77317i	0.462566	+	0.886585i	$0.346929\pi$
$673$			46.0000i			1.77317i	−0.462566	+	0.886585i	$0.653071\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	38.0000i		−	1.46046i	−0.683202	−	0.730229i	$-0.739413\pi$
$677$		−	38.0000i		−	1.46046i	0.683202	−	0.730229i	$-0.260587\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	40.0000i		−	1.53056i	−0.643699	−	0.765279i	$-0.722601\pi$
$683$		−	40.0000i		−	1.53056i	0.643699	−	0.765279i	$-0.277399\pi$
$684$		0			0
$685$			12.0000i			0.458496i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	8.00000i		−	0.304334i	−0.988355	−	0.152167i	$-0.951375\pi$
$691$		−	8.00000i		−	0.304334i	0.988355	−	0.152167i	$-0.0486252\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.00000			0.303457
$696$		0			0
$697$	−2.00000	+	8.00000i	−0.0757554	+	0.303022i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	50.0000			1.88847			0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000			1.88847			0.944237	+	0.329267i	$0.106802\pi$
$702$		0			0
$703$		−	8.00000i		−	0.301726i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	16.0000i		−	0.600893i	−0.953799	−	0.300446i	$-0.902864\pi$
$709$		−	16.0000i		−	0.600893i	0.953799	−	0.300446i	$-0.0971356\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−16.0000			−0.599205
$714$		0			0
$715$	12.0000			0.448775
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	50.0000i		−	1.86469i	−0.361576	−	0.932343i	$-0.617761\pi$
$719$		−	50.0000i		−	1.86469i	0.361576	−	0.932343i	$-0.382239\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	6.00000i		−	0.222834i
$726$		0			0
$727$	−2.00000			−0.0741759			−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000			−0.0741759			−0.0370879	+	0.999312i	$0.511808\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−24.0000	−	6.00000i	−0.887672	−	0.221918i
$732$		0			0
$733$	−34.0000			−1.25582			−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000			−1.25582			−0.627909	+	0.778287i	$0.716089\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	4.00000i		−	0.147342i
$738$		0			0
$739$	40.0000			1.47142			0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000			1.47142			0.735712	+	0.677295i	$0.236848\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	36.0000i		−	1.32071i	−0.750953	−	0.660356i	$-0.770405\pi$
$743$		−	36.0000i		−	1.32071i	0.750953	−	0.660356i	$-0.229595\pi$
$744$		0			0
$745$		−	10.0000i		−	0.366372i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	32.0000i		−	1.16770i	−0.811863	−	0.583848i	$-0.801546\pi$
$751$		−	32.0000i		−	1.16770i	0.811863	−	0.583848i	$-0.198454\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	8.00000i		−	0.291150i
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−30.0000			−1.08750			−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000			−1.08750			−0.543750	+	0.839248i	$0.682996\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−72.0000			−2.59977
$768$		0			0
$769$	−18.0000			−0.649097			−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000			−0.649097			−0.324548	+	0.945869i	$0.605212\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		−	4.00000i		−	0.143684i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	8.00000i		−	0.286630i
$780$		0			0
$781$	4.00000			0.143131
$782$		0			0
$783$		0			0
$784$		0			0
$785$			14.0000i			0.499681i
$786$		0			0
$787$		−	20.0000i		−	0.712923i	−0.934310	−	0.356462i	$-0.883983\pi$
$787$		−	20.0000i		−	0.712923i	0.934310	−	0.356462i	$-0.116017\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	24.0000i		−	0.852265i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−28.0000			−0.991811			−0.495905	−	0.868377i	$-0.665164\pi$
$797$	−28.0000			−0.991811			−0.495905	+	0.868377i	$0.665164\pi$
$798$		0			0
$799$	32.0000	+	8.00000i	1.13208	+	0.283020i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−4.00000			−0.141157
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			34.0000i			1.19538i	0.801729	+	0.597688i	$0.203914\pi$
$809$			34.0000i			1.19538i	−0.801729	+	0.597688i	$0.796086\pi$
$810$		0			0
$811$		−	56.0000i		−	1.96643i	−0.182462	−	0.983213i	$-0.558407\pi$
$811$		−	56.0000i		−	1.96643i	0.182462	−	0.983213i	$-0.441593\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−20.0000			−0.700569
$816$		0			0
$817$	24.0000			0.839654
$818$		0			0
$819$		0			0
$820$		0			0
$821$			42.0000i			1.46581i	0.680331	+	0.732905i	$0.261836\pi$
$821$			42.0000i			1.46581i	−0.680331	+	0.732905i	$0.738164\pi$
$822$		0			0
$823$			24.0000i			0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$			24.0000i			0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			12.0000i			0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$			12.0000i			0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$		0			0
$829$	−38.0000			−1.31979			−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000			−1.31979			−0.659897	+	0.751356i	$0.729400\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	28.0000	+	7.00000i	0.970143	+	0.242536i
$834$		0			0
$835$	16.0000			0.553703
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	10.0000i		−	0.345238i	−0.984989	−	0.172619i	$-0.944777\pi$
$839$		−	10.0000i		−	0.345238i	0.984989	−	0.172619i	$-0.0552230\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	23.0000i		−	0.791224i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8.00000			−0.274236
$852$		0			0
$853$			14.0000i			0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$			14.0000i			0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			2.00000i			0.0683187i	0.999416	+	0.0341593i	$0.0108754\pi$
$857$			2.00000i			0.0683187i	−0.999416	+	0.0341593i	$0.989125\pi$
$858$		0			0
$859$	52.0000			1.77422			0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000			1.77422			0.887109	+	0.461561i	$0.152710\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−28.0000			−0.953131			−0.476566	−	0.879139i	$-0.658119\pi$
$863$	−28.0000			−0.953131			−0.476566	+	0.879139i	$0.658119\pi$
$864$		0			0
$865$	18.0000			0.612018
$866$		0			0
$867$		0			0
$868$		0			0
$869$	32.0000			1.08553
$870$		0			0
$871$	−12.0000			−0.406604
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$			54.0000i			1.82345i	0.410801	+	0.911725i	$0.365249\pi$
$877$			54.0000i			1.82345i	−0.410801	+	0.911725i	$0.634751\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			14.0000i			0.471672i	0.971793	+	0.235836i	$0.0757828\pi$
$881$			14.0000i			0.471672i	−0.971793	+	0.235836i	$0.924217\pi$
$882$		0			0
$883$	14.0000			0.471138			0.235569	−	0.971858i	$-0.424305\pi$
$883$	14.0000			0.471138			0.235569	+	0.971858i	$0.424305\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			4.00000i			0.134307i	0.997743	+	0.0671534i	$0.0213917\pi$
$887$			4.00000i			0.134307i	−0.997743	+	0.0671534i	$0.978608\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−32.0000			−1.07084
$894$		0			0
$895$			20.0000i			0.668526i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−24.0000			−0.800445
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	20.0000			0.664822
$906$		0			0
$907$			28.0000i			0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$			28.0000i			0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	6.00000i		−	0.198789i	−0.995048	−	0.0993944i	$-0.968309\pi$
$911$		−	6.00000i		−	0.198789i	0.995048	−	0.0993944i	$-0.0316906\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−4.00000			−0.131948			−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000			−0.131948			−0.0659739	+	0.997821i	$0.521015\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	12.0000i		−	0.394985i
$924$		0			0
$925$		−	2.00000i		−	0.0657596i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			46.0000i			1.50921i	0.656179	+	0.754606i	$0.272172\pi$
$929$			46.0000i			1.50921i	−0.656179	+	0.754606i	$0.727828\pi$
$930$		0			0
$931$	−28.0000			−0.917663
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−8.00000	−	2.00000i	−0.261628	−	0.0654070i
$936$		0			0
$937$	−30.0000			−0.980057			−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000			−0.980057			−0.490029	+	0.871706i	$0.663014\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			58.0000i			1.89075i	0.325991	+	0.945373i	$0.394302\pi$
$941$			58.0000i			1.89075i	−0.325991	+	0.945373i	$0.605698\pi$
$942$		0			0
$943$	−8.00000			−0.260516
$944$		0			0
$945$		0			0
$946$		0			0
$947$			32.0000i			1.03986i	0.854209	+	0.519930i	$0.174042\pi$
$947$			32.0000i			1.03986i	−0.854209	+	0.519930i	$0.825958\pi$
$948$		0			0
$949$			12.0000i			0.389536i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−20.0000			−0.647864			−0.323932	−	0.946080i	$-0.605005\pi$
$953$	−20.0000			−0.647864			−0.323932	+	0.946080i	$0.605005\pi$
$954$		0			0
$955$		−	8.00000i		−	0.258874i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.0000			0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.00000			−0.0643823
$966$		0			0
$967$	−14.0000			−0.450210			−0.225105	−	0.974335i	$-0.572272\pi$
$967$	−14.0000			−0.450210			−0.225105	+	0.974335i	$0.572272\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−28.0000			−0.898563			−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000			−0.898563			−0.449281	+	0.893390i	$0.648320\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	48.0000			1.53566			0.767828	−	0.640656i	$-0.221338\pi$
$977$	48.0000			1.53566			0.767828	+	0.640656i	$0.221338\pi$
$978$		0			0
$979$			4.00000i			0.127841i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	36.0000i		−	1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$		−	36.0000i		−	1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$		0			0
$985$	18.0000			0.573528
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	24.0000i		−	0.763156i
$990$		0			0
$991$		−	40.0000i		−	1.27064i	−0.772248	−	0.635321i	$-0.780868\pi$
$991$		−	40.0000i		−	1.27064i	0.772248	−	0.635321i	$-0.219132\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−16.0000			−0.507234
$996$		0			0
$997$			18.0000i			0.570066i	0.958518	+	0.285033i	$0.0920045\pi$
$997$			18.0000i			0.570066i	−0.958518	+	0.285033i	$0.907995\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3060.2.e.a.1801.1		2
3.2	odd	2		1020.2.e.a.781.1	✓	2
12.11	even	2		4080.2.h.a.3841.2		2
15.2	even	4		5100.2.k.b.4249.2		2
15.8	even	4		5100.2.k.e.4249.2		2
15.14	odd	2		5100.2.e.e.1801.2		2
17.16	even	2	inner	3060.2.e.a.1801.2		2
51.50	odd	2		1020.2.e.a.781.2	yes	2
204.203	even	2		4080.2.h.a.3841.1		2
255.152	even	4		5100.2.k.e.4249.1		2
255.203	even	4		5100.2.k.b.4249.1		2
255.254	odd	2		5100.2.e.e.1801.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1020.2.e.a.781.1	✓	2	3.2	odd	2
1020.2.e.a.781.2	yes	2	51.50	odd	2
3060.2.e.a.1801.1		2	1.1	even	1	trivial
3060.2.e.a.1801.2		2	17.16	even	2	inner
4080.2.h.a.3841.1		2	204.203	even	2
4080.2.h.a.3841.2		2	12.11	even	2
5100.2.e.e.1801.1		2	255.254	odd	2
5100.2.e.e.1801.2		2	15.14	odd	2
5100.2.k.b.4249.1		2	255.203	even	4
5100.2.k.b.4249.2		2	15.2	even	4
5100.2.k.e.4249.1		2	255.152	even	4
5100.2.k.e.4249.2		2	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3060.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{5} -2.00000i q^{11} -6.00000 q^{13} +(4.00000 + 1.00000i) q^{17} -4.00000 q^{19} +4.00000i q^{23} -1.00000 q^{25} +6.00000i q^{29} +4.00000i q^{31} +2.00000i q^{37} +2.00000i q^{41} -6.00000 q^{43} +8.00000 q^{47} +7.00000 q^{49} -2.00000 q^{55} +12.0000 q^{59} +4.00000i q^{61} +6.00000i q^{65} +2.00000 q^{67} +2.00000i q^{71} -2.00000i q^{73} +16.0000i q^{79} +(1.00000 - 4.00000i) q^{85} -2.00000 q^{89} +4.00000i q^{95} +14.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{13} + 8 q^{17} - 8 q^{19} - 2 q^{25} - 12 q^{43} + 16 q^{47} + 14 q^{49} - 4 q^{55} + 24 q^{59} + 4 q^{67} + 2 q^{85} - 4 q^{89}+O(q^{100}) \) 2 * q - 12 * q^13 + 8 * q^17 - 8 * q^19 - 2 * q^25 - 12 * q^43 + 16 * q^47 + 14 * q^49 - 4 * q^55 + 24 * q^59 + 4 * q^67 + 2 * q^85 - 4 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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