LMFDB - Embedded newform 3120.2.l.j.1249.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3120,2,Mod(1249,3120)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3120.1249"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-4,0,0,0,-4,0,-8,0,0,0,-4,0,0,0,8] 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:	$24.9133254306$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 780)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.4
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	3120.1249
Dual form			3120.2.l.j.1249.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(0.224745 + 2.22474i) q^{5} -2.00000i q^{7} -1.00000 q^{9} +0.449490 q^{11} +1.00000i q^{13} +(-2.22474 + 0.224745i) q^{15} +2.00000i q^{17} +6.89898 q^{19} +2.00000 q^{21} -4.89898i q^{23} +(-4.89898 + 1.00000i) q^{25} -1.00000i q^{27} +2.00000 q^{29} +2.89898 q^{31} +0.449490i q^{33} +(4.44949 - 0.449490i) q^{35} +10.8990i q^{37} -1.00000 q^{39} +3.55051 q^{41} +7.79796i q^{43} +(-0.224745 - 2.22474i) q^{45} -5.34847i q^{47} +3.00000 q^{49} -2.00000 q^{51} +2.89898i q^{53} +(0.101021 + 1.00000i) q^{55} +6.89898i q^{57} -4.44949 q^{59} +4.00000 q^{61} +2.00000i q^{63} +(-2.22474 + 0.224745i) q^{65} +7.79796i q^{67} +4.89898 q^{69} -14.2474 q^{71} +7.79796i q^{73} +(-1.00000 - 4.89898i) q^{75} -0.898979i q^{77} +10.0000 q^{79} +1.00000 q^{81} -8.44949i q^{83} +(-4.44949 + 0.449490i) q^{85} +2.00000i q^{87} -1.34847 q^{89} +2.00000 q^{91} +2.89898i q^{93} +(1.55051 + 15.3485i) q^{95} +6.00000i q^{97} -0.449490 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 4 q^{9} - 8 q^{11} - 4 q^{15} + 8 q^{19} + 8 q^{21} + 8 q^{29} - 8 q^{31} + 8 q^{35} - 4 q^{39} + 24 q^{41} + 4 q^{45} + 12 q^{49} - 8 q^{51} + 20 q^{55} - 8 q^{59} + 16 q^{61} - 4 q^{65}+ \cdots + 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^5 - 4 * q^9 - 8 * q^11 - 4 * q^15 + 8 * q^19 + 8 * q^21 + 8 * q^29 - 8 * q^31 + 8 * q^35 - 4 * q^39 + 24 * q^41 + 4 * q^45 + 12 * q^49 - 8 * q^51 + 20 * q^55 - 8 * q^59 + 16 * q^61 - 4 * q^65 - 8 * q^71 - 4 * q^75 + 40 * q^79 + 4 * q^81 - 8 * q^85 + 24 * q^89 + 8 * q^91 + 16 * q^95 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times$.

$n$	$1951$	$2081$	$2341$	$2497$	$2641$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	0.224745	+	2.22474i	0.100509	+	0.994936i
$5$	0.224745	+	2.22474i	0.100509	+	0.994936i
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	0.449490			0.135526			0.0677631	−	0.997701i	$-0.478414\pi$
$11$	0.449490			0.135526			0.0677631	+	0.997701i	$0.478414\pi$
$12$		0			0
$13$			1.00000i			0.277350i
$13$			1.00000i			0.277350i
$14$		0			0
$15$	−2.22474	+	0.224745i	−0.574427	+	0.0580289i
$16$		0			0
$17$			2.00000i			0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$			2.00000i			0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$		0			0
$19$	6.89898			1.58273			0.791367	−	0.611341i	$-0.209370\pi$
$19$	6.89898			1.58273			0.791367	+	0.611341i	$0.209370\pi$
$20$		0			0
$21$	2.00000			0.436436
$22$		0			0
$23$		−	4.89898i		−	1.02151i	−0.859727	−	0.510754i	$-0.829366\pi$
$23$		−	4.89898i		−	1.02151i	0.859727	−	0.510754i	$-0.170634\pi$
$24$		0			0
$25$	−4.89898	+	1.00000i	−0.979796	+	0.200000i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	2.89898			0.520672			0.260336	−	0.965518i	$-0.416167\pi$
$31$	2.89898			0.520672			0.260336	+	0.965518i	$0.416167\pi$
$32$		0			0
$33$			0.449490i			0.0782461i
$34$		0			0
$35$	4.44949	−	0.449490i	0.752101	−	0.0759776i
$36$		0			0
$37$			10.8990i			1.79178i	0.444275	+	0.895891i	$0.353461\pi$
$37$			10.8990i			1.79178i	−0.444275	+	0.895891i	$0.646539\pi$
$38$		0			0
$39$	−1.00000			−0.160128
$40$		0			0
$41$	3.55051			0.554497			0.277248	−	0.960798i	$-0.410578\pi$
$41$	3.55051			0.554497			0.277248	+	0.960798i	$0.410578\pi$
$42$		0			0
$43$			7.79796i			1.18918i	0.804030	+	0.594589i	$0.202685\pi$
$43$			7.79796i			1.18918i	−0.804030	+	0.594589i	$0.797315\pi$
$44$		0			0
$45$	−0.224745	−	2.22474i	−0.0335030	−	0.331645i
$46$		0			0
$47$		−	5.34847i		−	0.780154i	−0.920782	−	0.390077i	$-0.872448\pi$
$47$		−	5.34847i		−	0.780154i	0.920782	−	0.390077i	$-0.127552\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$			2.89898i			0.398205i	0.979979	+	0.199103i	$0.0638027\pi$
$53$			2.89898i			0.398205i	−0.979979	+	0.199103i	$0.936197\pi$
$54$		0			0
$55$	0.101021	+	1.00000i	0.0136216	+	0.134840i
$56$		0			0
$57$			6.89898i			0.913792i
$58$		0			0
$59$	−4.44949			−0.579274			−0.289637	−	0.957137i	$-0.593535\pi$
$59$	−4.44949			−0.579274			−0.289637	+	0.957137i	$0.593535\pi$
$60$		0			0
$61$	4.00000			0.512148			0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000			0.512148			0.256074	+	0.966657i	$0.417571\pi$
$62$		0			0
$63$			2.00000i			0.251976i
$64$		0			0
$65$	−2.22474	+	0.224745i	−0.275946	+	0.0278762i
$66$		0			0
$67$			7.79796i			0.952672i	0.879263	+	0.476336i	$0.158035\pi$
$67$			7.79796i			0.952672i	−0.879263	+	0.476336i	$0.841965\pi$
$68$		0			0
$69$	4.89898			0.589768
$70$		0			0
$71$	−14.2474			−1.69086			−0.845431	−	0.534085i	$-0.820656\pi$
$71$	−14.2474			−1.69086			−0.845431	+	0.534085i	$0.820656\pi$
$72$		0			0
$73$			7.79796i			0.912682i	0.889805	+	0.456341i	$0.150840\pi$
$73$			7.79796i			0.912682i	−0.889805	+	0.456341i	$0.849160\pi$
$74$		0			0
$75$	−1.00000	−	4.89898i	−0.115470	−	0.565685i
$76$		0			0
$77$		−	0.898979i		−	0.102448i
$78$		0			0
$79$	10.0000			1.12509			0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000			1.12509			0.562544	+	0.826767i	$0.309823\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	8.44949i		−	0.927452i	−0.885979	−	0.463726i	$-0.846512\pi$
$83$		−	8.44949i		−	0.927452i	0.885979	−	0.463726i	$-0.153488\pi$
$84$		0			0
$85$	−4.44949	+	0.449490i	−0.482615	+	0.0487540i
$86$		0			0
$87$			2.00000i			0.214423i
$88$		0			0
$89$	−1.34847			−0.142937			−0.0714687	−	0.997443i	$-0.522769\pi$
$89$	−1.34847			−0.142937			−0.0714687	+	0.997443i	$0.522769\pi$
$90$		0			0
$91$	2.00000			0.209657
$92$		0			0
$93$			2.89898i			0.300610i
$94$		0			0
$95$	1.55051	+	15.3485i	0.159079	+	1.57472i
$96$		0			0
$97$			6.00000i			0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$			6.00000i			0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$		0			0
$99$	−0.449490			−0.0451754
$100$		0			0
$101$	14.8990			1.48250			0.741252	−	0.671227i	$-0.234232\pi$
$101$	14.8990			1.48250			0.741252	+	0.671227i	$0.234232\pi$
$102$		0			0
$103$			3.79796i			0.374224i	0.982339	+	0.187112i	$0.0599127\pi$
$103$			3.79796i			0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$104$		0			0
$105$	0.449490	+	4.44949i	0.0438657	+	0.434226i
$106$		0			0
$107$			13.7980i			1.33390i	0.745103	+	0.666950i	$0.232400\pi$
$107$			13.7980i			1.33390i	−0.745103	+	0.666950i	$0.767600\pi$
$108$		0			0
$109$	−10.8990			−1.04393			−0.521966	−	0.852966i	$-0.674801\pi$
$109$	−10.8990			−1.04393			−0.521966	+	0.852966i	$0.674801\pi$
$110$		0			0
$111$	−10.8990			−1.03449
$112$		0			0
$113$		−	8.69694i		−	0.818139i	−0.912503	−	0.409070i	$-0.865853\pi$
$113$		−	8.69694i		−	0.818139i	0.912503	−	0.409070i	$-0.134147\pi$
$114$		0			0
$115$	10.8990	−	1.10102i	1.01634	−	0.102671i
$116$		0			0
$117$		−	1.00000i		−	0.0924500i
$118$		0			0
$119$	4.00000			0.366679
$120$		0			0
$121$	−10.7980			−0.981633
$122$		0			0
$123$			3.55051i			0.320139i
$124$		0			0
$125$	−3.32577	−	10.6742i	−0.297465	−	0.954733i
$126$		0			0
$127$			12.0000i			1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$			12.0000i			1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$		0			0
$129$	−7.79796			−0.686572
$130$		0			0
$131$	4.89898			0.428026			0.214013	−	0.976831i	$-0.431347\pi$
$131$	4.89898			0.428026			0.214013	+	0.976831i	$0.431347\pi$
$132$		0			0
$133$		−	13.7980i		−	1.19643i
$134$		0			0
$135$	2.22474	−	0.224745i	0.191476	−	0.0193430i
$136$		0			0
$137$			10.2474i			0.875499i	0.899097	+	0.437749i	$0.144224\pi$
$137$			10.2474i			0.875499i	−0.899097	+	0.437749i	$0.855776\pi$
$138$		0			0
$139$	16.0000			1.35710			0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000			1.35710			0.678551	+	0.734553i	$0.262608\pi$
$140$		0			0
$141$	5.34847			0.450422
$142$		0			0
$143$			0.449490i			0.0375882i
$144$		0			0
$145$	0.449490	+	4.44949i	0.0373281	+	0.369510i
$146$		0			0
$147$			3.00000i			0.247436i
$148$		0			0
$149$	6.24745			0.511811			0.255905	−	0.966702i	$-0.417626\pi$
$149$	6.24745			0.511811			0.255905	+	0.966702i	$0.417626\pi$
$150$		0			0
$151$	−18.8990			−1.53798			−0.768989	−	0.639263i	$-0.779240\pi$
$151$	−18.8990			−1.53798			−0.768989	+	0.639263i	$0.779240\pi$
$152$		0			0
$153$		−	2.00000i		−	0.161690i
$154$		0			0
$155$	0.651531	+	6.44949i	0.0523322	+	0.518035i
$156$		0			0
$157$		−	5.79796i		−	0.462728i	−0.972867	−	0.231364i	$-0.925681\pi$
$157$		−	5.79796i		−	0.462728i	0.972867	−	0.231364i	$-0.0743187\pi$
$158$		0			0
$159$	−2.89898			−0.229904
$160$		0			0
$161$	−9.79796			−0.772187
$162$		0			0
$163$		−	6.89898i		−	0.540370i	−0.962808	−	0.270185i	$-0.912915\pi$
$163$		−	6.89898i		−	0.540370i	0.962808	−	0.270185i	$-0.0870848\pi$
$164$		0			0
$165$	−1.00000	+	0.101021i	−0.0778499	+	0.00786444i
$166$		0			0
$167$			3.55051i			0.274747i	0.990519	+	0.137373i	$0.0438660\pi$
$167$			3.55051i			0.274747i	−0.990519	+	0.137373i	$0.956134\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$	−6.89898			−0.527578
$172$		0			0
$173$		−	2.00000i		−	0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$		−	2.00000i		−	0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$		0			0
$175$	2.00000	+	9.79796i	0.151186	+	0.740656i
$176$		0			0
$177$		−	4.44949i		−	0.334444i
$178$		0			0
$179$	−19.5959			−1.46467			−0.732334	−	0.680946i	$-0.761569\pi$
$179$	−19.5959			−1.46467			−0.732334	+	0.680946i	$0.761569\pi$
$180$		0			0
$181$	−3.79796			−0.282300			−0.141150	−	0.989988i	$-0.545080\pi$
$181$	−3.79796			−0.282300			−0.141150	+	0.989988i	$0.545080\pi$
$182$		0			0
$183$			4.00000i			0.295689i
$184$		0			0
$185$	−24.2474	+	2.44949i	−1.78271	+	0.180090i
$186$		0			0
$187$			0.898979i			0.0657399i
$188$		0			0
$189$	−2.00000			−0.145479
$190$		0			0
$191$	−16.8990			−1.22277			−0.611384	−	0.791334i	$-0.709387\pi$
$191$	−16.8990			−1.22277			−0.611384	+	0.791334i	$0.709387\pi$
$192$		0			0
$193$		−	1.10102i		−	0.0792532i	−0.999215	−	0.0396266i	$-0.987383\pi$
$193$		−	1.10102i		−	0.0792532i	0.999215	−	0.0396266i	$-0.0126168\pi$
$194$		0			0
$195$	−0.224745	−	2.22474i	−0.0160943	−	0.159317i
$196$		0			0
$197$		−	11.5505i		−	0.822940i	−0.911423	−	0.411470i	$-0.865016\pi$
$197$		−	11.5505i		−	0.822940i	0.911423	−	0.411470i	$-0.134984\pi$
$198$		0			0
$199$	6.00000			0.425329			0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000			0.425329			0.212664	+	0.977125i	$0.431786\pi$
$200$		0			0
$201$	−7.79796			−0.550026
$202$		0			0
$203$		−	4.00000i		−	0.280745i
$204$		0			0
$205$	0.797959	+	7.89898i	0.0557319	+	0.551689i
$206$		0			0
$207$			4.89898i			0.340503i
$208$		0			0
$209$	3.10102			0.214502
$210$		0			0
$211$	−11.7980			−0.812205			−0.406102	−	0.913828i	$-0.633112\pi$
$211$	−11.7980			−0.812205			−0.406102	+	0.913828i	$0.633112\pi$
$212$		0			0
$213$		−	14.2474i		−	0.976219i
$214$		0			0
$215$	−17.3485	+	1.75255i	−1.18316	+	0.119523i
$216$		0			0
$217$		−	5.79796i		−	0.393591i
$218$		0			0
$219$	−7.79796			−0.526937
$220$		0			0
$221$	−2.00000			−0.134535
$222$		0			0
$223$			14.8990i			0.997709i	0.866686	+	0.498855i	$0.166246\pi$
$223$			14.8990i			0.997709i	−0.866686	+	0.498855i	$0.833754\pi$
$224$		0			0
$225$	4.89898	−	1.00000i	0.326599	−	0.0666667i
$226$		0			0
$227$			20.4495i			1.35728i	0.734471	+	0.678640i	$0.237430\pi$
$227$			20.4495i			1.35728i	−0.734471	+	0.678640i	$0.762570\pi$
$228$		0			0
$229$	24.6969			1.63202			0.816010	−	0.578038i	$-0.196181\pi$
$229$	24.6969			1.63202			0.816010	+	0.578038i	$0.196181\pi$
$230$		0			0
$231$	0.898979			0.0591485
$232$		0			0
$233$			26.0000i			1.70332i	0.524097	+	0.851658i	$0.324403\pi$
$233$			26.0000i			1.70332i	−0.524097	+	0.851658i	$0.675597\pi$
$234$		0			0
$235$	11.8990	−	1.20204i	0.776204	−	0.0784125i
$236$		0			0
$237$			10.0000i			0.649570i
$238$		0			0
$239$	20.4495			1.32277			0.661384	−	0.750048i	$-0.269969\pi$
$239$	20.4495			1.32277			0.661384	+	0.750048i	$0.269969\pi$
$240$		0			0
$241$	10.8990			0.702065			0.351032	−	0.936363i	$-0.385831\pi$
$241$	10.8990			0.702065			0.351032	+	0.936363i	$0.385831\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	0.674235	+	6.67423i	0.0430753	+	0.426401i
$246$		0			0
$247$			6.89898i			0.438972i
$248$		0			0
$249$	8.44949			0.535465
$250$		0			0
$251$	−19.5959			−1.23688			−0.618442	−	0.785831i	$-0.712236\pi$
$251$	−19.5959			−1.23688			−0.618442	+	0.785831i	$0.712236\pi$
$252$		0			0
$253$		−	2.20204i		−	0.138441i
$254$		0			0
$255$	−0.449490	−	4.44949i	−0.0281481	−	0.278638i
$256$		0			0
$257$			3.79796i			0.236910i	0.992959	+	0.118455i	$0.0377942\pi$
$257$			3.79796i			0.236910i	−0.992959	+	0.118455i	$0.962206\pi$
$258$		0			0
$259$	21.7980			1.35446
$260$		0			0
$261$	−2.00000			−0.123797
$262$		0			0
$263$		−	17.7980i		−	1.09747i	−0.835997	−	0.548735i	$-0.815110\pi$
$263$		−	17.7980i		−	1.09747i	0.835997	−	0.548735i	$-0.184890\pi$
$264$		0			0
$265$	−6.44949	+	0.651531i	−0.396189	+	0.0400232i
$266$		0			0
$267$		−	1.34847i		−	0.0825250i
$268$		0			0
$269$	−31.7980			−1.93876			−0.969378	−	0.245574i	$-0.921024\pi$
$269$	−31.7980			−1.93876			−0.969378	+	0.245574i	$0.921024\pi$
$270$		0			0
$271$	−10.0000			−0.607457			−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000			−0.607457			−0.303728	+	0.952759i	$0.598232\pi$
$272$		0			0
$273$			2.00000i			0.121046i
$274$		0			0
$275$	−2.20204	+	0.449490i	−0.132788	+	0.0271053i
$276$		0			0
$277$			13.7980i			0.829039i	0.910040	+	0.414520i	$0.136050\pi$
$277$			13.7980i			0.829039i	−0.910040	+	0.414520i	$0.863950\pi$
$278$		0			0
$279$	−2.89898			−0.173557
$280$		0			0
$281$	30.2474			1.80441			0.902206	−	0.431306i	$-0.141947\pi$
$281$	30.2474			1.80441			0.902206	+	0.431306i	$0.141947\pi$
$282$		0			0
$283$			16.0000i			0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$			16.0000i			0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$		0			0
$285$	−15.3485	+	1.55051i	−0.909165	+	0.0918443i
$286$		0			0
$287$		−	7.10102i		−	0.419160i
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$	−6.00000			−0.351726
$292$		0			0
$293$			7.55051i			0.441106i	0.975375	+	0.220553i	$0.0707861\pi$
$293$			7.55051i			0.441106i	−0.975375	+	0.220553i	$0.929214\pi$
$294$		0			0
$295$	−1.00000	−	9.89898i	−0.0582223	−	0.576341i
$296$		0			0
$297$		−	0.449490i		−	0.0260820i
$298$		0			0
$299$	4.89898			0.283315
$300$		0			0
$301$	15.5959			0.898934
$302$		0			0
$303$			14.8990i			0.855924i
$304$		0			0
$305$	0.898979	+	8.89898i	0.0514754	+	0.509554i
$306$		0			0
$307$		−	9.59592i		−	0.547668i	−0.961777	−	0.273834i	$-0.911708\pi$
$307$		−	9.59592i		−	0.547668i	0.961777	−	0.273834i	$-0.0882919\pi$
$308$		0			0
$309$	−3.79796			−0.216058
$310$		0			0
$311$	23.5959			1.33800			0.669001	−	0.743262i	$-0.266722\pi$
$311$	23.5959			1.33800			0.669001	+	0.743262i	$0.266722\pi$
$312$		0			0
$313$		−	12.2020i		−	0.689700i	−0.938658	−	0.344850i	$-0.887930\pi$
$313$		−	12.2020i		−	0.689700i	0.938658	−	0.344850i	$-0.112070\pi$
$314$		0			0
$315$	−4.44949	+	0.449490i	−0.250700	+	0.0253259i
$316$		0			0
$317$		−	23.1464i		−	1.30003i	−0.759920	−	0.650016i	$-0.774762\pi$
$317$		−	23.1464i		−	1.30003i	0.759920	−	0.650016i	$-0.225238\pi$
$318$		0			0
$319$	0.898979			0.0503332
$320$		0			0
$321$	−13.7980			−0.770127
$322$		0			0
$323$			13.7980i			0.767739i
$324$		0			0
$325$	−1.00000	−	4.89898i	−0.0554700	−	0.271746i
$326$		0			0
$327$		−	10.8990i		−	0.602715i
$328$		0			0
$329$	−10.6969			−0.589741
$330$		0			0
$331$	25.5959			1.40688			0.703439	−	0.710755i	$-0.251647\pi$
$331$	25.5959			1.40688			0.703439	+	0.710755i	$0.251647\pi$
$332$		0			0
$333$		−	10.8990i		−	0.597260i
$334$		0			0
$335$	−17.3485	+	1.75255i	−0.947848	+	0.0957521i
$336$		0			0
$337$			2.00000i			0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$			2.00000i			0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$		0			0
$339$	8.69694			0.472353
$340$		0			0
$341$	1.30306			0.0705647
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$		0			0
$345$	1.10102	+	10.8990i	0.0592770	+	0.586781i
$346$		0			0
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$		0			0
$349$	12.2020			0.653160			0.326580	−	0.945170i	$-0.394104\pi$
$349$	12.2020			0.653160			0.326580	+	0.945170i	$0.394104\pi$
$350$		0			0
$351$	1.00000			0.0533761
$352$		0			0
$353$			36.9444i			1.96635i	0.182663	+	0.983176i	$0.441528\pi$
$353$			36.9444i			1.96635i	−0.182663	+	0.983176i	$0.558472\pi$
$354$		0			0
$355$	−3.20204	−	31.6969i	−0.169947	−	1.68230i
$356$		0			0
$357$			4.00000i			0.211702i
$358$		0			0
$359$	9.34847			0.493393			0.246697	−	0.969093i	$-0.420655\pi$
$359$	9.34847			0.493393			0.246697	+	0.969093i	$0.420655\pi$
$360$		0			0
$361$	28.5959			1.50505
$362$		0			0
$363$		−	10.7980i		−	0.566746i
$364$		0			0
$365$	−17.3485	+	1.75255i	−0.908060	+	0.0917327i
$366$		0			0
$367$		−	11.7980i		−	0.615848i	−0.951411	−	0.307924i	$-0.900366\pi$
$367$		−	11.7980i		−	0.615848i	0.951411	−	0.307924i	$-0.0996343\pi$
$368$		0			0
$369$	−3.55051			−0.184832
$370$		0			0
$371$	5.79796			0.301015
$372$		0			0
$373$		−	23.7980i		−	1.23221i	−0.787663	−	0.616106i	$-0.788709\pi$
$373$		−	23.7980i		−	1.23221i	0.787663	−	0.616106i	$-0.211291\pi$
$374$		0			0
$375$	10.6742	−	3.32577i	0.551215	−	0.171742i
$376$		0			0
$377$			2.00000i			0.103005i
$378$		0			0
$379$	−0.696938			−0.0357993			−0.0178997	−	0.999840i	$-0.505698\pi$
$379$	−0.696938			−0.0357993			−0.0178997	+	0.999840i	$0.505698\pi$
$380$		0			0
$381$	−12.0000			−0.614779
$382$		0			0
$383$		−	27.5505i		−	1.40777i	−0.710316	−	0.703883i	$-0.751448\pi$
$383$		−	27.5505i		−	1.40777i	0.710316	−	0.703883i	$-0.248552\pi$
$384$		0			0
$385$	2.00000	−	0.202041i	0.101929	−	0.0102970i
$386$		0			0
$387$		−	7.79796i		−	0.396393i
$388$		0			0
$389$	6.89898			0.349792			0.174896	−	0.984587i	$-0.444041\pi$
$389$	6.89898			0.349792			0.174896	+	0.984587i	$0.444041\pi$
$390$		0			0
$391$	9.79796			0.495504
$392$		0			0
$393$			4.89898i			0.247121i
$394$		0			0
$395$	2.24745	+	22.2474i	0.113081	+	1.11939i
$396$		0			0
$397$		−	1.10102i		−	0.0552586i	−0.999618	−	0.0276293i	$-0.991204\pi$
$397$		−	1.10102i		−	0.0552586i	0.999618	−	0.0276293i	$-0.00879580\pi$
$398$		0			0
$399$	13.7980			0.690762
$400$		0			0
$401$	−35.1464			−1.75513			−0.877564	−	0.479459i	$-0.840833\pi$
$401$	−35.1464			−1.75513			−0.877564	+	0.479459i	$0.840833\pi$
$402$		0			0
$403$			2.89898i			0.144408i
$404$		0			0
$405$	0.224745	+	2.22474i	0.0111677	+	0.110548i
$406$		0			0
$407$			4.89898i			0.242833i
$408$		0			0
$409$	21.1010			1.04338			0.521689	−	0.853136i	$-0.325302\pi$
$409$	21.1010			1.04338			0.521689	+	0.853136i	$0.325302\pi$
$410$		0			0
$411$	−10.2474			−0.505469
$412$		0			0
$413$			8.89898i			0.437890i
$414$		0			0
$415$	18.7980	−	1.89898i	0.922756	−	0.0932172i
$416$		0			0
$417$			16.0000i			0.783523i
$418$		0			0
$419$	18.6969			0.913405			0.456703	−	0.889619i	$-0.349030\pi$
$419$	18.6969			0.913405			0.456703	+	0.889619i	$0.349030\pi$
$420$		0			0
$421$	0.696938			0.0339667			0.0169834	−	0.999856i	$-0.494594\pi$
$421$	0.696938			0.0339667			0.0169834	+	0.999856i	$0.494594\pi$
$422$		0			0
$423$			5.34847i			0.260051i
$424$		0			0
$425$	−2.00000	−	9.79796i	−0.0970143	−	0.475271i
$426$		0			0
$427$		−	8.00000i		−	0.387147i
$428$		0			0
$429$	−0.449490			−0.0217016
$430$		0			0
$431$	19.1464			0.922251			0.461125	−	0.887335i	$-0.347446\pi$
$431$	19.1464			0.922251			0.461125	+	0.887335i	$0.347446\pi$
$432$		0			0
$433$			37.5959i			1.80674i	0.428857	+	0.903372i	$0.358916\pi$
$433$			37.5959i			1.80674i	−0.428857	+	0.903372i	$0.641084\pi$
$434$		0			0
$435$	−4.44949	+	0.449490i	−0.213337	+	0.0215514i
$436$		0			0
$437$		−	33.7980i		−	1.61678i
$438$		0			0
$439$	−35.5959			−1.69890			−0.849450	−	0.527669i	$-0.823066\pi$
$439$	−35.5959			−1.69890			−0.849450	+	0.527669i	$0.823066\pi$
$440$		0			0
$441$	−3.00000			−0.142857
$442$		0			0
$443$			11.5959i			0.550939i	0.961310	+	0.275469i	$0.0888333\pi$
$443$			11.5959i			0.550939i	−0.961310	+	0.275469i	$0.911167\pi$
$444$		0			0
$445$	−0.303062	−	3.00000i	−0.0143665	−	0.142214i
$446$		0			0
$447$			6.24745i			0.295494i
$448$		0			0
$449$	35.1464			1.65866			0.829331	−	0.558757i	$-0.188722\pi$
$449$	35.1464			1.65866			0.829331	+	0.558757i	$0.188722\pi$
$450$		0			0
$451$	1.59592			0.0751488
$452$		0			0
$453$		−	18.8990i		−	0.887952i
$454$		0			0
$455$	0.449490	+	4.44949i	0.0210724	+	0.208595i
$456$		0			0
$457$			19.3939i			0.907207i	0.891204	+	0.453604i	$0.149862\pi$
$457$			19.3939i			0.907207i	−0.891204	+	0.453604i	$0.850138\pi$
$458$		0			0
$459$	2.00000			0.0933520
$460$		0			0
$461$	19.1464			0.891738			0.445869	−	0.895098i	$-0.352895\pi$
$461$	19.1464			0.891738			0.445869	+	0.895098i	$0.352895\pi$
$462$		0			0
$463$		−	25.5959i		−	1.18954i	−0.803895	−	0.594772i	$-0.797242\pi$
$463$		−	25.5959i		−	1.18954i	0.803895	−	0.594772i	$-0.202758\pi$
$464$		0			0
$465$	−6.44949	+	0.651531i	−0.299088	+	0.0302140i
$466$		0			0
$467$		−	8.89898i		−	0.411796i	−0.978573	−	0.205898i	$-0.933989\pi$
$467$		−	8.89898i		−	0.411796i	0.978573	−	0.205898i	$-0.0660115\pi$
$468$		0			0
$469$	15.5959			0.720153
$470$		0			0
$471$	5.79796			0.267156
$472$		0			0
$473$			3.50510i			0.161165i
$474$		0			0
$475$	−33.7980	+	6.89898i	−1.55076	+	0.316547i
$476$		0			0
$477$		−	2.89898i		−	0.132735i
$478$		0			0
$479$	0.853572			0.0390007			0.0195003	−	0.999810i	$-0.493792\pi$
$479$	0.853572			0.0390007			0.0195003	+	0.999810i	$0.493792\pi$
$480$		0			0
$481$	−10.8990			−0.496951
$482$		0			0
$483$		−	9.79796i		−	0.445823i
$484$		0			0
$485$	−13.3485	+	1.34847i	−0.606123	+	0.0612308i
$486$		0			0
$487$		−	5.10102i		−	0.231149i	−0.993299	−	0.115575i	$-0.963129\pi$
$487$		−	5.10102i		−	0.231149i	0.993299	−	0.115575i	$-0.0368710\pi$
$488$		0			0
$489$	6.89898			0.311983
$490$		0			0
$491$	−37.3939			−1.68756			−0.843781	−	0.536688i	$-0.819675\pi$
$491$	−37.3939			−1.68756			−0.843781	+	0.536688i	$0.819675\pi$
$492$		0			0
$493$			4.00000i			0.180151i
$494$		0			0
$495$	−0.101021	−	1.00000i	−0.00454053	−	0.0449467i
$496$		0			0
$497$			28.4949i			1.27817i
$498$		0			0
$499$	23.3939			1.04725			0.523627	−	0.851948i	$-0.324579\pi$
$499$	23.3939			1.04725			0.523627	+	0.851948i	$0.324579\pi$
$500$		0			0
$501$	−3.55051			−0.158625
$502$		0			0
$503$		−	19.5959i		−	0.873739i	−0.899525	−	0.436869i	$-0.856087\pi$
$503$		−	19.5959i		−	0.873739i	0.899525	−	0.436869i	$-0.143913\pi$
$504$		0			0
$505$	3.34847	+	33.1464i	0.149005	+	1.47500i
$506$		0			0
$507$		−	1.00000i		−	0.0444116i
$508$		0			0
$509$	−9.34847			−0.414364			−0.207182	−	0.978302i	$-0.566429\pi$
$509$	−9.34847			−0.414364			−0.207182	+	0.978302i	$0.566429\pi$
$510$		0			0
$511$	15.5959			0.689923
$512$		0			0
$513$		−	6.89898i		−	0.304597i
$514$		0			0
$515$	−8.44949	+	0.853572i	−0.372329	+	0.0376129i
$516$		0			0
$517$		−	2.40408i		−	0.105731i
$518$		0			0
$519$	2.00000			0.0877903
$520$		0			0
$521$	−10.0000			−0.438108			−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000			−0.438108			−0.219054	+	0.975713i	$0.570297\pi$
$522$		0			0
$523$			0.202041i			0.00883464i	0.999990	+	0.00441732i	$0.00140608\pi$
$523$			0.202041i			0.00883464i	−0.999990	+	0.00441732i	$0.998594\pi$
$524$		0			0
$525$	−9.79796	+	2.00000i	−0.427618	+	0.0872872i
$526$		0			0
$527$			5.79796i			0.252563i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	4.44949			0.193091
$532$		0			0
$533$			3.55051i			0.153790i
$534$		0			0
$535$	−30.6969	+	3.10102i	−1.32714	+	0.134069i
$536$		0			0
$537$		−	19.5959i		−	0.845626i
$538$		0			0
$539$	1.34847			0.0580827
$540$		0			0
$541$	29.1010			1.25115			0.625575	−	0.780164i	$-0.284864\pi$
$541$	29.1010			1.25115			0.625575	+	0.780164i	$0.284864\pi$
$542$		0			0
$543$		−	3.79796i		−	0.162986i
$544$		0			0
$545$	−2.44949	−	24.2474i	−0.104925	−	1.03865i
$546$		0			0
$547$		−	2.40408i		−	0.102791i	−0.998678	−	0.0513956i	$-0.983633\pi$
$547$		−	2.40408i		−	0.102791i	0.998678	−	0.0513956i	$-0.0163669\pi$
$548$		0			0
$549$	−4.00000			−0.170716
$550$		0			0
$551$	13.7980			0.587813
$552$		0			0
$553$		−	20.0000i		−	0.850487i
$554$		0			0
$555$	−2.44949	−	24.2474i	−0.103975	−	1.02925i
$556$		0			0
$557$		−	28.9444i		−	1.22641i	−0.789923	−	0.613207i	$-0.789879\pi$
$557$		−	28.9444i		−	1.22641i	0.789923	−	0.613207i	$-0.210121\pi$
$558$		0			0
$559$	−7.79796			−0.329819
$560$		0			0
$561$	−0.898979			−0.0379549
$562$		0			0
$563$		−	3.10102i		−	0.130692i	−0.997863	−	0.0653462i	$-0.979185\pi$
$563$		−	3.10102i		−	0.130692i	0.997863	−	0.0653462i	$-0.0208152\pi$
$564$		0			0
$565$	19.3485	−	1.95459i	0.813996	−	0.0822303i
$566$		0			0
$567$		−	2.00000i		−	0.0839921i
$568$		0			0
$569$	13.1010			0.549223			0.274612	−	0.961555i	$-0.411451\pi$
$569$	13.1010			0.549223			0.274612	+	0.961555i	$0.411451\pi$
$570$		0			0
$571$	−1.59592			−0.0667871			−0.0333935	−	0.999442i	$-0.510631\pi$
$571$	−1.59592			−0.0667871			−0.0333935	+	0.999442i	$0.510631\pi$
$572$		0			0
$573$		−	16.8990i		−	0.705965i
$574$		0			0
$575$	4.89898	+	24.0000i	0.204302	+	1.00087i
$576$		0			0
$577$		−	28.6969i		−	1.19467i	−0.801992	−	0.597335i	$-0.796226\pi$
$577$		−	28.6969i		−	1.19467i	0.801992	−	0.597335i	$-0.203774\pi$
$578$		0			0
$579$	1.10102			0.0457569
$580$		0			0
$581$	−16.8990			−0.701088
$582$		0			0
$583$			1.30306i			0.0539673i
$584$		0			0
$585$	2.22474	−	0.224745i	0.0919819	−	0.00929206i
$586$		0			0
$587$		−	18.2474i		−	0.753153i	−0.926386	−	0.376576i	$-0.877101\pi$
$587$		−	18.2474i		−	0.753153i	0.926386	−	0.376576i	$-0.122899\pi$
$588$		0			0
$589$	20.0000			0.824086
$590$		0			0
$591$	11.5505			0.475125
$592$		0			0
$593$			21.3485i			0.876677i	0.898810	+	0.438338i	$0.144433\pi$
$593$			21.3485i			0.876677i	−0.898810	+	0.438338i	$0.855567\pi$
$594$		0			0
$595$	0.898979	+	8.89898i	0.0368546	+	0.364823i
$596$		0			0
$597$			6.00000i			0.245564i
$598$		0			0
$599$	22.6969			0.927372			0.463686	−	0.886000i	$-0.346527\pi$
$599$	22.6969			0.927372			0.463686	+	0.886000i	$0.346527\pi$
$600$		0			0
$601$	−6.20204			−0.252987			−0.126493	−	0.991967i	$-0.540372\pi$
$601$	−6.20204			−0.252987			−0.126493	+	0.991967i	$0.540372\pi$
$602$		0			0
$603$		−	7.79796i		−	0.317557i
$604$		0			0
$605$	−2.42679	−	24.0227i	−0.0986629	−	0.976662i
$606$		0			0
$607$		−	25.5959i		−	1.03891i	−0.854499	−	0.519453i	$-0.826136\pi$
$607$		−	25.5959i		−	1.03891i	0.854499	−	0.519453i	$-0.173864\pi$
$608$		0			0
$609$	4.00000			0.162088
$610$		0			0
$611$	5.34847			0.216376
$612$		0			0
$613$		−	46.4949i		−	1.87791i	−0.344038	−	0.938956i	$-0.611795\pi$
$613$		−	46.4949i		−	1.87791i	0.344038	−	0.938956i	$-0.388205\pi$
$614$		0			0
$615$	−7.89898	+	0.797959i	−0.318518	+	0.0321768i
$616$		0			0
$617$		−	16.4495i		−	0.662232i	−0.943590	−	0.331116i	$-0.892575\pi$
$617$		−	16.4495i		−	0.662232i	0.943590	−	0.331116i	$-0.107425\pi$
$618$		0			0
$619$	−1.59592			−0.0641454			−0.0320727	−	0.999486i	$-0.510211\pi$
$619$	−1.59592			−0.0641454			−0.0320727	+	0.999486i	$0.510211\pi$
$620$		0			0
$621$	−4.89898			−0.196589
$622$		0			0
$623$			2.69694i			0.108051i
$624$		0			0
$625$	23.0000	−	9.79796i	0.920000	−	0.391918i
$626$		0			0
$627$			3.10102i			0.123843i
$628$		0			0
$629$	−21.7980			−0.869142
$630$		0			0
$631$	−20.2929			−0.807846			−0.403923	−	0.914793i	$-0.632354\pi$
$631$	−20.2929			−0.807846			−0.403923	+	0.914793i	$0.632354\pi$
$632$		0			0
$633$		−	11.7980i		−	0.468927i
$634$		0			0
$635$	−26.6969	+	2.69694i	−1.05944	+	0.107025i
$636$		0			0
$637$			3.00000i			0.118864i
$638$		0			0
$639$	14.2474			0.563620
$640$		0			0
$641$	−33.1010			−1.30741			−0.653706	−	0.756749i	$-0.726787\pi$
$641$	−33.1010			−1.30741			−0.653706	+	0.756749i	$0.726787\pi$
$642$		0			0
$643$		−	34.0000i		−	1.34083i	−0.741987	−	0.670415i	$-0.766116\pi$
$643$		−	34.0000i		−	1.34083i	0.741987	−	0.670415i	$-0.233884\pi$
$644$		0			0
$645$	−1.75255	−	17.3485i	−0.0690066	−	0.683095i
$646$		0			0
$647$		−	48.4949i		−	1.90653i	−0.302134	−	0.953266i	$-0.597699\pi$
$647$		−	48.4949i		−	1.90653i	0.302134	−	0.953266i	$-0.402301\pi$
$648$		0			0
$649$	−2.00000			−0.0785069
$650$		0			0
$651$	5.79796			0.227240
$652$		0			0
$653$		−	2.49490i		−	0.0976329i	−0.998808	−	0.0488164i	$-0.984455\pi$
$653$		−	2.49490i		−	0.0976329i	0.998808	−	0.0488164i	$-0.0155449\pi$
$654$		0			0
$655$	1.10102	+	10.8990i	0.0430204	+	0.425858i
$656$		0			0
$657$		−	7.79796i		−	0.304227i
$658$		0			0
$659$	−10.6969			−0.416694			−0.208347	−	0.978055i	$-0.566808\pi$
$659$	−10.6969			−0.416694			−0.208347	+	0.978055i	$0.566808\pi$
$660$		0			0
$661$	−15.7980			−0.614469			−0.307235	−	0.951634i	$-0.599404\pi$
$661$	−15.7980			−0.614469			−0.307235	+	0.951634i	$0.599404\pi$
$662$		0			0
$663$		−	2.00000i		−	0.0776736i
$664$		0			0
$665$	30.6969	−	3.10102i	1.19038	−	0.120252i
$666$		0			0
$667$		−	9.79796i		−	0.379378i
$668$		0			0
$669$	−14.8990			−0.576028
$670$		0			0
$671$	1.79796			0.0694094
$672$		0			0
$673$		−	36.0000i		−	1.38770i	−0.720121	−	0.693849i	$-0.755914\pi$
$673$		−	36.0000i		−	1.38770i	0.720121	−	0.693849i	$-0.244086\pi$
$674$		0			0
$675$	1.00000	+	4.89898i	0.0384900	+	0.188562i
$676$		0			0
$677$		−	19.3031i		−	0.741877i	−0.928657	−	0.370938i	$-0.879036\pi$
$677$		−	19.3031i		−	0.741877i	0.928657	−	0.370938i	$-0.120964\pi$
$678$		0			0
$679$	12.0000			0.460518
$680$		0			0
$681$	−20.4495			−0.783626
$682$		0			0
$683$			0.853572i			0.0326610i	0.999867	+	0.0163305i	$0.00519839\pi$
$683$			0.853572i			0.0326610i	−0.999867	+	0.0163305i	$0.994802\pi$
$684$		0			0
$685$	−22.7980	+	2.30306i	−0.871065	+	0.0879955i
$686$		0			0
$687$			24.6969i			0.942247i
$688$		0			0
$689$	−2.89898			−0.110442
$690$		0			0
$691$	10.0000			0.380418			0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000			0.380418			0.190209	+	0.981744i	$0.439083\pi$
$692$		0			0
$693$			0.898979i			0.0341494i
$694$		0			0
$695$	3.59592	+	35.5959i	0.136401	+	1.35023i
$696$		0			0
$697$			7.10102i			0.268970i
$698$		0			0
$699$	−26.0000			−0.983410
$700$		0			0
$701$	−34.8990			−1.31812			−0.659058	−	0.752092i	$-0.729045\pi$
$701$	−34.8990			−1.31812			−0.659058	+	0.752092i	$0.729045\pi$
$702$		0			0
$703$			75.1918i			2.83591i
$704$		0			0
$705$	1.20204	+	11.8990i	0.0452715	+	0.448141i
$706$		0			0
$707$		−	29.7980i		−	1.12067i
$708$		0			0
$709$	14.4949			0.544367			0.272184	−	0.962245i	$-0.412254\pi$
$709$	14.4949			0.544367			0.272184	+	0.962245i	$0.412254\pi$
$710$		0			0
$711$	−10.0000			−0.375029
$712$		0			0
$713$		−	14.2020i		−	0.531871i
$714$		0			0
$715$	−1.00000	+	0.101021i	−0.0373979	+	0.00377795i
$716$		0			0
$717$			20.4495i			0.763700i
$718$		0			0
$719$	−9.79796			−0.365402			−0.182701	−	0.983169i	$-0.558484\pi$
$719$	−9.79796			−0.365402			−0.182701	+	0.983169i	$0.558484\pi$
$720$		0			0
$721$	7.59592			0.282887
$722$		0			0
$723$			10.8990i			0.405337i
$724$		0			0
$725$	−9.79796	+	2.00000i	−0.363887	+	0.0742781i
$726$		0			0
$727$			25.3939i			0.941807i	0.882185	+	0.470903i	$0.156072\pi$
$727$			25.3939i			0.941807i	−0.882185	+	0.470903i	$0.843928\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−15.5959			−0.576836
$732$		0			0
$733$		−	11.7980i		−	0.435768i	−0.975975	−	0.217884i	$-0.930085\pi$
$733$		−	11.7980i		−	0.435768i	0.975975	−	0.217884i	$-0.0699154\pi$
$734$		0			0
$735$	−6.67423	+	0.674235i	−0.246183	+	0.0248695i
$736$		0			0
$737$			3.50510i			0.129112i
$738$		0			0
$739$	15.3939			0.566273			0.283136	−	0.959080i	$-0.408625\pi$
$739$	15.3939			0.566273			0.283136	+	0.959080i	$0.408625\pi$
$740$		0			0
$741$	−6.89898			−0.253440
$742$		0			0
$743$		−	6.65153i		−	0.244021i	−0.992529	−	0.122010i	$-0.961066\pi$
$743$		−	6.65153i		−	0.244021i	0.992529	−	0.122010i	$-0.0389342\pi$
$744$		0			0
$745$	1.40408	+	13.8990i	0.0514416	+	0.509219i
$746$		0			0
$747$			8.44949i			0.309151i
$748$		0			0
$749$	27.5959			1.00833
$750$		0			0
$751$	4.40408			0.160707			0.0803536	−	0.996766i	$-0.474395\pi$
$751$	4.40408			0.160707			0.0803536	+	0.996766i	$0.474395\pi$
$752$		0			0
$753$		−	19.5959i		−	0.714115i
$754$		0			0
$755$	−4.24745	−	42.0454i	−0.154580	−	1.53019i
$756$		0			0
$757$		−	53.1918i		−	1.93329i	−0.256120	−	0.966645i	$-0.582444\pi$
$757$		−	53.1918i		−	1.93329i	0.256120	−	0.966645i	$-0.417556\pi$
$758$		0			0
$759$	2.20204			0.0799290
$760$		0			0
$761$	−43.5505			−1.57870			−0.789352	−	0.613940i	$-0.789584\pi$
$761$	−43.5505			−1.57870			−0.789352	+	0.613940i	$0.789584\pi$
$762$		0			0
$763$			21.7980i			0.789139i
$764$		0			0
$765$	4.44949	−	0.449490i	0.160872	−	0.0162513i
$766$		0			0
$767$		−	4.44949i		−	0.160662i
$768$		0			0
$769$	39.3939			1.42058			0.710290	−	0.703909i	$-0.248564\pi$
$769$	39.3939			1.42058			0.710290	+	0.703909i	$0.248564\pi$
$770$		0			0
$771$	−3.79796			−0.136780
$772$		0			0
$773$			26.2474i			0.944055i	0.881584	+	0.472028i	$0.156478\pi$
$773$			26.2474i			0.944055i	−0.881584	+	0.472028i	$0.843522\pi$
$774$		0			0
$775$	−14.2020	+	2.89898i	−0.510152	+	0.104134i
$776$		0			0
$777$			21.7980i			0.781997i
$778$		0			0
$779$	24.4949			0.877621
$780$		0			0
$781$	−6.40408			−0.229156
$782$		0			0
$783$		−	2.00000i		−	0.0714742i
$784$		0			0
$785$	12.8990	−	1.30306i	0.460384	−	0.0465083i
$786$		0			0
$787$			16.2020i			0.577540i	0.957398	+	0.288770i	$0.0932464\pi$
$787$			16.2020i			0.577540i	−0.957398	+	0.288770i	$0.906754\pi$
$788$		0			0
$789$	17.7980			0.633624
$790$		0			0
$791$	−17.3939			−0.618455
$792$		0			0
$793$			4.00000i			0.142044i
$794$		0			0
$795$	−0.651531	−	6.44949i	−0.0231074	−	0.228740i
$796$		0			0
$797$		−	27.3939i		−	0.970341i	−0.874420	−	0.485170i	$-0.838758\pi$
$797$		−	27.3939i		−	0.970341i	0.874420	−	0.485170i	$-0.161242\pi$
$798$		0			0
$799$	10.6969			0.378430
$800$		0			0
$801$	1.34847			0.0476458
$802$		0			0
$803$			3.50510i			0.123692i
$804$		0			0
$805$	−2.20204	−	21.7980i	−0.0776117	−	0.768277i
$806$		0			0
$807$		−	31.7980i		−	1.11934i
$808$		0			0
$809$	53.5959			1.88433			0.942166	−	0.335146i	$-0.108786\pi$
$809$	53.5959			1.88433			0.942166	+	0.335146i	$0.108786\pi$
$810$		0			0
$811$	16.6969			0.586309			0.293154	−	0.956065i	$-0.405295\pi$
$811$	16.6969			0.586309			0.293154	+	0.956065i	$0.405295\pi$
$812$		0			0
$813$		−	10.0000i		−	0.350715i
$814$		0			0
$815$	15.3485	−	1.55051i	0.537633	−	0.0543120i
$816$		0			0
$817$			53.7980i			1.88215i
$818$		0			0
$819$	−2.00000			−0.0698857
$820$		0			0
$821$	16.0454			0.559989			0.279994	−	0.960002i	$-0.409667\pi$
$821$	16.0454			0.559989			0.279994	+	0.960002i	$0.409667\pi$
$822$		0			0
$823$		−	7.39388i		−	0.257734i	−0.991662	−	0.128867i	$-0.958866\pi$
$823$		−	7.39388i		−	0.257734i	0.991662	−	0.128867i	$-0.0411341\pi$
$824$		0			0
$825$	−0.449490	−	2.20204i	−0.0156492	−	0.0766652i
$826$		0			0
$827$			44.4495i			1.54566i	0.634613	+	0.772830i	$0.281160\pi$
$827$			44.4495i			1.54566i	−0.634613	+	0.772830i	$0.718840\pi$
$828$		0			0
$829$	17.3939			0.604114			0.302057	−	0.953290i	$-0.402327\pi$
$829$	17.3939			0.604114			0.302057	+	0.953290i	$0.402327\pi$
$830$		0			0
$831$	−13.7980			−0.478646
$832$		0			0
$833$			6.00000i			0.207888i
$834$		0			0
$835$	−7.89898	+	0.797959i	−0.273355	+	0.0276145i
$836$		0			0
$837$		−	2.89898i		−	0.100203i
$838$		0			0
$839$	−45.3485			−1.56560			−0.782802	−	0.622271i	$-0.786210\pi$
$839$	−45.3485			−1.56560			−0.782802	+	0.622271i	$0.786210\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$			30.2474i			1.04178i
$844$		0			0
$845$	−0.224745	−	2.22474i	−0.00773146	−	0.0765336i
$846$		0			0
$847$			21.5959i			0.742045i
$848$		0			0
$849$	−16.0000			−0.549119
$850$		0			0
$851$	53.3939			1.83032
$852$		0			0
$853$			44.6969i			1.53039i	0.643796	+	0.765197i	$0.277358\pi$
$853$			44.6969i			1.53039i	−0.643796	+	0.765197i	$0.722642\pi$
$854$		0			0
$855$	−1.55051	−	15.3485i	−0.0530263	−	0.524907i
$856$		0			0
$857$		−	57.5959i		−	1.96744i	−0.179711	−	0.983720i	$-0.557516\pi$
$857$		−	57.5959i		−	1.96744i	0.179711	−	0.983720i	$-0.442484\pi$
$858$		0			0
$859$	47.7980			1.63085			0.815423	−	0.578866i	$-0.196505\pi$
$859$	47.7980			1.63085			0.815423	+	0.578866i	$0.196505\pi$
$860$		0			0
$861$	7.10102			0.242002
$862$		0			0
$863$		−	0.853572i		−	0.0290559i	−0.999894	−	0.0145280i	$-0.995375\pi$
$863$		−	0.853572i		−	0.0290559i	0.999894	−	0.0145280i	$-0.00462456\pi$
$864$		0			0
$865$	4.44949	−	0.449490i	0.151287	−	0.0152831i
$866$		0			0
$867$			13.0000i			0.441503i
$868$		0			0
$869$	4.49490			0.152479
$870$		0			0
$871$	−7.79796			−0.264224
$872$		0			0
$873$		−	6.00000i		−	0.203069i
$874$		0			0
$875$	−21.3485	+	6.65153i	−0.721710	+	0.224863i
$876$		0			0
$877$			38.0908i			1.28624i	0.765767	+	0.643118i	$0.222359\pi$
$877$			38.0908i			1.28624i	−0.765767	+	0.643118i	$0.777641\pi$
$878$		0			0
$879$	−7.55051			−0.254672
$880$		0			0
$881$	−32.2929			−1.08797			−0.543987	−	0.839094i	$-0.683086\pi$
$881$	−32.2929			−1.08797			−0.543987	+	0.839094i	$0.683086\pi$
$882$		0			0
$883$			5.39388i			0.181518i	0.995873	+	0.0907592i	$0.0289294\pi$
$883$			5.39388i			0.181518i	−0.995873	+	0.0907592i	$0.971071\pi$
$884$		0			0
$885$	9.89898	−	1.00000i	0.332751	−	0.0336146i
$886$		0			0
$887$		−	24.0000i		−	0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$		−	24.0000i		−	0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$		0			0
$889$	24.0000			0.804934
$890$		0			0
$891$	0.449490			0.0150585
$892$		0			0
$893$		−	36.8990i		−	1.23478i
$894$		0			0
$895$	−4.40408	−	43.5959i	−0.147212	−	1.45725i
$896$		0			0
$897$			4.89898i			0.163572i
$898$		0			0
$899$	5.79796			0.193373
$900$		0			0
$901$	−5.79796			−0.193158
$902$		0			0
$903$			15.5959i			0.519000i
$904$		0			0
$905$	−0.853572	−	8.44949i	−0.0283737	−	0.280871i
$906$		0			0
$907$			31.1918i			1.03571i	0.855469	+	0.517854i	$0.173269\pi$
$907$			31.1918i			1.03571i	−0.855469	+	0.517854i	$0.826731\pi$
$908$		0			0
$909$	−14.8990			−0.494168
$910$		0			0
$911$	15.1010			0.500319			0.250160	−	0.968205i	$-0.419517\pi$
$911$	15.1010			0.500319			0.250160	+	0.968205i	$0.419517\pi$
$912$		0			0
$913$		−	3.79796i		−	0.125694i
$914$		0			0
$915$	−8.89898	+	0.898979i	−0.294191	+	0.0297193i
$916$		0			0
$917$		−	9.79796i		−	0.323557i
$918$		0			0
$919$	13.5959			0.448488			0.224244	−	0.974533i	$-0.428009\pi$
$919$	13.5959			0.448488			0.224244	+	0.974533i	$0.428009\pi$
$920$		0			0
$921$	9.59592			0.316196
$922$		0			0
$923$		−	14.2474i		−	0.468960i
$924$		0			0
$925$	−10.8990	−	53.3939i	−0.358356	−	1.75558i
$926$		0			0
$927$		−	3.79796i		−	0.124741i
$928$		0			0
$929$	21.3485			0.700421			0.350210	−	0.936671i	$-0.386110\pi$
$929$	21.3485			0.700421			0.350210	+	0.936671i	$0.386110\pi$
$930$		0			0
$931$	20.6969			0.678315
$932$		0			0
$933$			23.5959i			0.772496i
$934$		0			0
$935$	−2.00000	+	0.202041i	−0.0654070	+	0.00660745i
$936$		0			0
$937$			31.3939i			1.02559i	0.858510	+	0.512797i	$0.171391\pi$
$937$			31.3939i			1.02559i	−0.858510	+	0.512797i	$0.828609\pi$
$938$		0			0
$939$	12.2020			0.398199
$940$		0			0
$941$	−4.44949			−0.145049			−0.0725246	−	0.997367i	$-0.523106\pi$
$941$	−4.44949			−0.145049			−0.0725246	+	0.997367i	$0.523106\pi$
$942$		0			0
$943$		−	17.3939i		−	0.566423i
$944$		0			0
$945$	−0.449490	−	4.44949i	−0.0146219	−	0.144742i
$946$		0			0
$947$		−	18.2474i		−	0.592962i	−0.955039	−	0.296481i	$-0.904187\pi$
$947$		−	18.2474i		−	0.592962i	0.955039	−	0.296481i	$-0.0958132\pi$
$948$		0			0
$949$	−7.79796			−0.253132
$950$		0			0
$951$	23.1464			0.750574
$952$		0			0
$953$		−	35.3939i		−	1.14652i	−0.819373	−	0.573260i	$-0.805678\pi$
$953$		−	35.3939i		−	1.14652i	0.819373	−	0.573260i	$-0.194322\pi$
$954$		0			0
$955$	−3.79796	−	37.5959i	−0.122899	−	1.21658i
$956$		0			0
$957$			0.898979i			0.0290599i
$958$		0			0
$959$	20.4949			0.661815
$960$		0			0
$961$	−22.5959			−0.728901
$962$		0			0
$963$		−	13.7980i		−	0.444633i
$964$		0			0
$965$	2.44949	−	0.247449i	0.0788519	−	0.00796566i
$966$		0			0
$967$		−	12.6969i		−	0.408306i	−0.978939	−	0.204153i	$-0.934556\pi$
$967$		−	12.6969i		−	0.408306i	0.978939	−	0.204153i	$-0.0654440\pi$
$968$		0			0
$969$	−13.7980			−0.443254
$970$		0			0
$971$	−27.1010			−0.869713			−0.434857	−	0.900500i	$-0.643201\pi$
$971$	−27.1010			−0.869713			−0.434857	+	0.900500i	$0.643201\pi$
$972$		0			0
$973$		−	32.0000i		−	1.02587i
$974$		0			0
$975$	4.89898	−	1.00000i	0.156893	−	0.0320256i
$976$		0			0
$977$		−	27.5505i		−	0.881419i	−0.897650	−	0.440709i	$-0.854727\pi$
$977$		−	27.5505i		−	0.881419i	0.897650	−	0.440709i	$-0.145273\pi$
$978$		0			0
$979$	−0.606123			−0.0193718
$980$		0			0
$981$	10.8990			0.347978
$982$		0			0
$983$		−	48.4495i		−	1.54530i	−0.634833	−	0.772649i	$-0.718931\pi$
$983$		−	48.4495i		−	1.54530i	0.634833	−	0.772649i	$-0.281069\pi$
$984$		0			0
$985$	25.6969	−	2.59592i	0.818773	−	0.0827128i
$986$		0			0
$987$		−	10.6969i		−	0.340487i
$988$		0			0
$989$	38.2020			1.21475
$990$		0			0
$991$	−0.404082			−0.0128361			−0.00641804	−	0.999979i	$-0.502043\pi$
$991$	−0.404082			−0.0128361			−0.00641804	+	0.999979i	$0.502043\pi$
$992$		0			0
$993$			25.5959i			0.812262i
$994$		0			0
$995$	1.34847	+	13.3485i	0.0427493	+	0.423175i
$996$		0			0
$997$			8.40408i			0.266160i	0.991105	+	0.133080i	$0.0424867\pi$
$997$			8.40408i			0.266160i	−0.991105	+	0.133080i	$0.957513\pi$
$998$		0			0
$999$	10.8990			0.344828

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.l.j.1249.4		4
4.3	odd	2		780.2.h.c.469.2	✓	4
5.4	even	2	inner	3120.2.l.j.1249.2		4
12.11	even	2		2340.2.h.d.469.1		4
20.3	even	4		3900.2.a.t.1.1		2
20.7	even	4		3900.2.a.o.1.1		2
20.19	odd	2		780.2.h.c.469.4	yes	4
60.59	even	2		2340.2.h.d.469.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
780.2.h.c.469.2	✓	4	4.3	odd	2
780.2.h.c.469.4	yes	4	20.19	odd	2
2340.2.h.d.469.1		4	12.11	even	2
2340.2.h.d.469.2		4	60.59	even	2
3120.2.l.j.1249.2		4	5.4	even	2	inner
3120.2.l.j.1249.4		4	1.1	even	1	trivial
3900.2.a.o.1.1		2	20.7	even	4
3900.2.a.t.1.1		2	20.3	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:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(0.224745 + 2.22474i) q^{5} -2.00000i q^{7} -1.00000 q^{9} +0.449490 q^{11} +1.00000i q^{13} +(-2.22474 + 0.224745i) q^{15} +2.00000i q^{17} +6.89898 q^{19} +2.00000 q^{21} -4.89898i q^{23} +(-4.89898 + 1.00000i) q^{25} -1.00000i q^{27} +2.00000 q^{29} +2.89898 q^{31} +0.449490i q^{33} +(4.44949 - 0.449490i) q^{35} +10.8990i q^{37} -1.00000 q^{39} +3.55051 q^{41} +7.79796i q^{43} +(-0.224745 - 2.22474i) q^{45} -5.34847i q^{47} +3.00000 q^{49} -2.00000 q^{51} +2.89898i q^{53} +(0.101021 + 1.00000i) q^{55} +6.89898i q^{57} -4.44949 q^{59} +4.00000 q^{61} +2.00000i q^{63} +(-2.22474 + 0.224745i) q^{65} +7.79796i q^{67} +4.89898 q^{69} -14.2474 q^{71} +7.79796i q^{73} +(-1.00000 - 4.89898i) q^{75} -0.898979i q^{77} +10.0000 q^{79} +1.00000 q^{81} -8.44949i q^{83} +(-4.44949 + 0.449490i) q^{85} +2.00000i q^{87} -1.34847 q^{89} +2.00000 q^{91} +2.89898i q^{93} +(1.55051 + 15.3485i) q^{95} +6.00000i q^{97} -0.449490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 4 q^{9} - 8 q^{11} - 4 q^{15} + 8 q^{19} + 8 q^{21} + 8 q^{29} - 8 q^{31} + 8 q^{35} - 4 q^{39} + 24 q^{41} + 4 q^{45} + 12 q^{49} - 8 q^{51} + 20 q^{55} - 8 q^{59} + 16 q^{61} - 4 q^{65}+ \cdots + 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^5 - 4 * q^9 - 8 * q^11 - 4 * q^15 + 8 * q^19 + 8 * q^21 + 8 * q^29 - 8 * q^31 + 8 * q^35 - 4 * q^39 + 24 * q^41 + 4 * q^45 + 12 * q^49 - 8 * q^51 + 20 * q^55 - 8 * q^59 + 16 * q^61 - 4 * q^65 - 8 * q^71 - 4 * q^75 + 40 * q^79 + 4 * q^81 - 8 * q^85 + 24 * q^89 + 8 * q^91 + 16 * q^95 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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