LMFDB - Embedded newform 3060.2.e.i.1801.4

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)

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);

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")

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 = [4,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,28] 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:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1801.4
Root			$1.30278i$ of defining polynomial
Character	$\chi$	$=$	3060.1801
Dual form			3060.2.e.i.1801.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} +3.60555i q^{7} -1.00000i q^{11} +4.00000 q^{13} +(3.60555 + 2.00000i) q^{17} +7.00000 q^{19} -1.00000 q^{25} -9.00000i q^{29} -7.21110i q^{31} -3.60555 q^{35} +3.60555i q^{37} +5.00000i q^{41} -2.00000 q^{43} +10.8167 q^{47} -6.00000 q^{49} -3.60555 q^{53} +1.00000 q^{55} -14.4222 q^{59} +14.4222i q^{61} +4.00000i q^{65} +8.00000 q^{67} -8.00000i q^{71} +10.8167i q^{73} +3.60555 q^{77} +(-2.00000 + 3.60555i) q^{85} +14.4222 q^{89} +14.4222i q^{91} +7.00000i q^{95} +7.21110i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{13} + 28 q^{19} - 4 q^{25} - 8 q^{43} - 24 q^{49} + 4 q^{55} + 32 q^{67} - 8 q^{85}+O(q^{100}) $ 4 * q + 16 * q^13 + 28 * q^19 - 4 * q^25 - 8 * q^43 - 24 * q^49 + 4 * q^55 + 32 * q^67 - 8 * q^85

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
$7$			3.60555i			1.36277i	0.731925	+	0.681385i	$0.238622\pi$
$7$			3.60555i			1.36277i	−0.731925	+	0.681385i	$0.761378\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.00000i		−	0.301511i	−0.988571	−	0.150756i	$-0.951829\pi$
$11$		−	1.00000i		−	0.301511i	0.988571	−	0.150756i	$-0.0481707\pi$
$12$		0			0
$13$	4.00000			1.10940			0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000			1.10940			0.554700	+	0.832050i	$0.312833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.60555	+	2.00000i	0.874475	+	0.485071i
$17$	3.60555	+	2.00000i	0.874475	+	0.485071i
$18$		0			0
$19$	7.00000			1.60591			0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000			1.60591			0.802955	+	0.596040i	$0.203260\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	9.00000i		−	1.67126i	−0.549294	−	0.835629i	$-0.685103\pi$
$29$		−	9.00000i		−	1.67126i	0.549294	−	0.835629i	$-0.314897\pi$
$30$		0			0
$31$		−	7.21110i		−	1.29515i	−0.762001	−	0.647576i	$-0.775783\pi$
$31$		−	7.21110i		−	1.29515i	0.762001	−	0.647576i	$-0.224217\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.60555			−0.609449
$36$		0			0
$37$			3.60555i			0.592749i	0.955072	+	0.296374i	$0.0957776\pi$
$37$			3.60555i			0.592749i	−0.955072	+	0.296374i	$0.904222\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			5.00000i			0.780869i	0.920631	+	0.390434i	$0.127675\pi$
$41$			5.00000i			0.780869i	−0.920631	+	0.390434i	$0.872325\pi$
$42$		0			0
$43$	−2.00000			−0.304997			−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000			−0.304997			−0.152499	+	0.988304i	$0.548732\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	10.8167			1.57777			0.788886	−	0.614540i	$-0.210658\pi$
$47$	10.8167			1.57777			0.788886	+	0.614540i	$0.210658\pi$
$48$		0			0
$49$	−6.00000			−0.857143
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.60555			−0.495261			−0.247630	−	0.968855i	$-0.579652\pi$
$53$	−3.60555			−0.495261			−0.247630	+	0.968855i	$0.579652\pi$
$54$		0			0
$55$	1.00000			0.134840
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−14.4222			−1.87761			−0.938806	−	0.344447i	$-0.888066\pi$
$59$	−14.4222			−1.87761			−0.938806	+	0.344447i	$0.888066\pi$
$60$		0			0
$61$			14.4222i			1.84657i	0.384111	+	0.923287i	$0.374508\pi$
$61$			14.4222i			1.84657i	−0.384111	+	0.923287i	$0.625492\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			4.00000i			0.496139i
$66$		0			0
$67$	8.00000			0.977356			0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000			0.977356			0.488678	+	0.872464i	$0.337479\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8.00000i		−	0.949425i	−0.880141	−	0.474713i	$-0.842552\pi$
$71$		−	8.00000i		−	0.949425i	0.880141	−	0.474713i	$-0.157448\pi$
$72$		0			0
$73$			10.8167i			1.26599i	0.774154	+	0.632997i	$0.218175\pi$
$73$			10.8167i			1.26599i	−0.774154	+	0.632997i	$0.781825\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.60555			0.410891
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		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
$85$	−2.00000	+	3.60555i	−0.216930	+	0.391077i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	14.4222			1.52875			0.764375	−	0.644772i	$-0.223047\pi$
$89$	14.4222			1.52875			0.764375	+	0.644772i	$0.223047\pi$
$90$		0			0
$91$			14.4222i			1.51186i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			7.00000i			0.718185i
$96$		0			0
$97$			7.21110i			0.732177i	0.930580	+	0.366088i	$0.119303\pi$
$97$			7.21110i			0.732177i	−0.930580	+	0.366088i	$0.880697\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.21110			0.717532			0.358766	−	0.933428i	$-0.383198\pi$
$101$	7.21110			0.717532			0.358766	+	0.933428i	$0.383198\pi$
$102$		0			0
$103$	6.00000			0.591198			0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000			0.591198			0.295599	+	0.955312i	$0.404481\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			16.0000i			1.54678i	0.633932	+	0.773389i	$0.281440\pi$
$107$			16.0000i			1.54678i	−0.633932	+	0.773389i	$0.718560\pi$
$108$		0			0
$109$			7.21110i			0.690698i	0.938474	+	0.345349i	$0.112240\pi$
$109$			7.21110i			0.690698i	−0.938474	+	0.345349i	$0.887760\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	16.0000i		−	1.50515i	−0.658505	−	0.752577i	$-0.728811\pi$
$113$		−	16.0000i		−	1.50515i	0.658505	−	0.752577i	$-0.271189\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−7.21110	+	13.0000i	−0.661041	+	1.19171i
$120$		0			0
$121$	10.0000			0.909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	−12.0000			−1.06483			−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000			−1.06483			−0.532414	+	0.846484i	$0.678715\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	12.0000i		−	1.04844i	−0.851581	−	0.524222i	$-0.824356\pi$
$131$		−	12.0000i		−	1.04844i	0.851581	−	0.524222i	$-0.175644\pi$
$132$		0			0
$133$			25.2389i			2.18849i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−18.0278			−1.54022			−0.770108	−	0.637914i	$-0.779798\pi$
$137$	−18.0278			−1.54022			−0.770108	+	0.637914i	$0.779798\pi$
$138$		0			0
$139$			7.21110i			0.611638i	0.952090	+	0.305819i	$0.0989302\pi$
$139$			7.21110i			0.611638i	−0.952090	+	0.305819i	$0.901070\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	4.00000i		−	0.334497i
$144$		0			0
$145$	9.00000			0.747409
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−21.6333			−1.77227			−0.886135	−	0.463428i	$-0.846619\pi$
$149$	−21.6333			−1.77227			−0.886135	+	0.463428i	$0.846619\pi$
$150$		0			0
$151$	7.00000			0.569652			0.284826	−	0.958579i	$-0.408064\pi$
$151$	7.00000			0.569652			0.284826	+	0.958579i	$0.408064\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	7.21110			0.579210
$156$		0			0
$157$	−4.00000			−0.319235			−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000			−0.319235			−0.159617	+	0.987179i	$0.551026\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	18.0278i		−	1.41204i	−0.708190	−	0.706021i	$-0.750488\pi$
$163$		−	18.0278i		−	1.41204i	0.708190	−	0.706021i	$-0.249512\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2.00000i			0.154765i	0.997001	+	0.0773823i	$0.0246562\pi$
$167$			2.00000i			0.154765i	−0.997001	+	0.0773823i	$0.975344\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$		0			0
$172$		0			0
$173$			4.00000i			0.304114i	0.988372	+	0.152057i	$0.0485898\pi$
$173$			4.00000i			0.304114i	−0.988372	+	0.152057i	$0.951410\pi$
$174$		0			0
$175$		−	3.60555i		−	0.272554i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−21.6333			−1.61695			−0.808475	−	0.588531i	$-0.799706\pi$
$179$	−21.6333			−1.61695			−0.808475	+	0.588531i	$0.799706\pi$
$180$		0			0
$181$			7.21110i			0.535997i	0.963419	+	0.267999i	$0.0863622\pi$
$181$			7.21110i			0.535997i	−0.963419	+	0.267999i	$0.913638\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.60555			−0.265085
$186$		0			0
$187$	2.00000	−	3.60555i	0.146254	−	0.263664i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.21110			0.521777			0.260889	−	0.965369i	$-0.415984\pi$
$191$	7.21110			0.521777			0.260889	+	0.965369i	$0.415984\pi$
$192$		0			0
$193$		−	7.21110i		−	0.519067i	−0.965734	−	0.259533i	$-0.916431\pi$
$193$		−	7.21110i		−	0.519067i	0.965734	−	0.259533i	$-0.0835687\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			24.0000i			1.70993i	0.518686	+	0.854965i	$0.326421\pi$
$197$			24.0000i			1.70993i	−0.518686	+	0.854965i	$0.673579\pi$
$198$		0			0
$199$			14.4222i			1.02236i	0.859473	+	0.511182i	$0.170792\pi$
$199$			14.4222i			1.02236i	−0.859473	+	0.511182i	$0.829208\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	32.4500			2.27754
$204$		0			0
$205$	−5.00000			−0.349215
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	7.00000i		−	0.484200i
$210$		0			0
$211$		−	7.21110i		−	0.496433i	−0.968705	−	0.248216i	$-0.920156\pi$
$211$		−	7.21110i		−	0.496433i	0.968705	−	0.248216i	$-0.0798444\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	2.00000i		−	0.136399i
$216$		0			0
$217$	26.0000			1.76500
$218$		0			0
$219$		0			0
$220$		0			0
$221$	14.4222	+	8.00000i	0.970143	+	0.538138i
$222$		0			0
$223$	8.00000			0.535720			0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000			0.535720			0.267860	+	0.963458i	$0.413684\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	18.0000i		−	1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$		−	18.0000i		−	1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$		0			0
$229$	29.0000			1.91637			0.958187	−	0.286143i	$-0.0923732\pi$
$229$	29.0000			1.91637			0.958187	+	0.286143i	$0.0923732\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	18.0000i		−	1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$		−	18.0000i		−	1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$		0			0
$235$			10.8167i			0.705601i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−14.4222			−0.932895			−0.466447	−	0.884549i	$-0.654466\pi$
$239$	−14.4222			−0.932895			−0.466447	+	0.884549i	$0.654466\pi$
$240$		0			0
$241$		−	7.21110i		−	0.464508i	−0.972655	−	0.232254i	$-0.925390\pi$
$241$		−	7.21110i		−	0.464508i	0.972655	−	0.232254i	$-0.0746101\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	6.00000i		−	0.383326i
$246$		0			0
$247$	28.0000			1.78160
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.21110			0.455161			0.227580	−	0.973759i	$-0.426919\pi$
$251$	7.21110			0.455161			0.227580	+	0.973759i	$0.426919\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	21.6333			1.34945			0.674724	−	0.738070i	$-0.264263\pi$
$257$	21.6333			1.34945			0.674724	+	0.738070i	$0.264263\pi$
$258$		0			0
$259$	−13.0000			−0.807781
$260$		0			0
$261$		0			0
$262$		0			0
$263$	10.8167			0.666983			0.333492	−	0.942753i	$-0.391773\pi$
$263$	10.8167			0.666983			0.333492	+	0.942753i	$0.391773\pi$
$264$		0			0
$265$		−	3.60555i		−	0.221487i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			1.00000i			0.0609711i	0.999535	+	0.0304855i	$0.00970535\pi$
$269$			1.00000i			0.0609711i	−0.999535	+	0.0304855i	$0.990295\pi$
$270$		0			0
$271$	−24.0000			−1.45790			−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000			−1.45790			−0.728948	+	0.684569i	$0.759990\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			1.00000i			0.0603023i
$276$		0			0
$277$			7.21110i			0.433273i	0.976252	+	0.216637i	$0.0695087\pi$
$277$			7.21110i			0.433273i	−0.976252	+	0.216637i	$0.930491\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−21.6333			−1.29053			−0.645267	−	0.763957i	$-0.723254\pi$
$281$	−21.6333			−1.29053			−0.645267	+	0.763957i	$0.723254\pi$
$282$		0			0
$283$			18.0278i			1.07164i	0.844333	+	0.535819i	$0.179997\pi$
$283$			18.0278i			1.07164i	−0.844333	+	0.535819i	$0.820003\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−18.0278			−1.06414
$288$		0			0
$289$	9.00000	+	14.4222i	0.529412	+	0.848365i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	18.0278			1.05319			0.526596	−	0.850115i	$-0.323468\pi$
$293$	18.0278			1.05319			0.526596	+	0.850115i	$0.323468\pi$
$294$		0			0
$295$		−	14.4222i		−	0.839693i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		−	7.21110i		−	0.415641i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−14.4222			−0.825813
$306$		0			0
$307$	−20.0000			−1.14146			−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000			−1.14146			−0.570730	+	0.821138i	$0.693340\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			5.00000i			0.283524i	0.989901	+	0.141762i	$0.0452768\pi$
$311$			5.00000i			0.283524i	−0.989901	+	0.141762i	$0.954723\pi$
$312$		0			0
$313$			3.60555i			0.203798i	0.994795	+	0.101899i	$0.0324918\pi$
$313$			3.60555i			0.203798i	−0.994795	+	0.101899i	$0.967508\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			4.00000i			0.224662i	0.993671	+	0.112331i	$0.0358318\pi$
$317$			4.00000i			0.224662i	−0.993671	+	0.112331i	$0.964168\pi$
$318$		0			0
$319$	−9.00000			−0.503903
$320$		0			0
$321$		0			0
$322$		0			0
$323$	25.2389	+	14.0000i	1.40433	+	0.778981i
$324$		0			0
$325$	−4.00000			−0.221880
$326$		0			0
$327$		0			0
$328$		0			0
$329$			39.0000i			2.15014i
$330$		0			0
$331$	−1.00000			−0.0549650			−0.0274825	−	0.999622i	$-0.508749\pi$
$331$	−1.00000			−0.0549650			−0.0274825	+	0.999622i	$0.508749\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			8.00000i			0.437087i
$336$		0			0
$337$		−	18.0278i		−	0.982034i	−0.871150	−	0.491017i	$-0.836625\pi$
$337$		−	18.0278i		−	0.982034i	0.871150	−	0.491017i	$-0.163375\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−7.21110			−0.390503
$342$		0			0
$343$			3.60555i			0.194681i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	36.0000i		−	1.93258i	−0.257454	−	0.966291i	$-0.582883\pi$
$347$		−	36.0000i		−	1.93258i	0.257454	−	0.966291i	$-0.417117\pi$
$348$		0			0
$349$	−27.0000			−1.44528			−0.722638	−	0.691226i	$-0.757071\pi$
$349$	−27.0000			−1.44528			−0.722638	+	0.691226i	$0.757071\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	25.2389			1.34333			0.671664	−	0.740855i	$-0.265580\pi$
$353$	25.2389			1.34333			0.671664	+	0.740855i	$0.265580\pi$
$354$		0			0
$355$	8.00000			0.424596
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7.21110			−0.380587			−0.190294	−	0.981727i	$-0.560944\pi$
$359$	−7.21110			−0.380587			−0.190294	+	0.981727i	$0.560944\pi$
$360$		0			0
$361$	30.0000			1.57895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−10.8167			−0.566170
$366$		0			0
$367$			14.4222i			0.752833i	0.926451	+	0.376416i	$0.122844\pi$
$367$			14.4222i			0.752833i	−0.926451	+	0.376416i	$0.877156\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	13.0000i		−	0.674926i
$372$		0			0
$373$	−26.0000			−1.34623			−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000			−1.34623			−0.673114	+	0.739538i	$0.735044\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	36.0000i		−	1.85409i
$378$		0			0
$379$		−	28.8444i		−	1.48164i	−0.671705	−	0.740819i	$-0.734438\pi$
$379$		−	28.8444i		−	1.48164i	0.671705	−	0.740819i	$-0.265562\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−18.0278			−0.921175			−0.460588	−	0.887614i	$-0.652361\pi$
$383$	−18.0278			−0.921175			−0.460588	+	0.887614i	$0.652361\pi$
$384$		0			0
$385$			3.60555i			0.183756i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$			25.2389i			1.26670i	0.773865	+	0.633351i	$0.218321\pi$
$397$			25.2389i			1.26670i	−0.773865	+	0.633351i	$0.781679\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			3.00000i			0.149813i	0.997191	+	0.0749064i	$0.0238658\pi$
$401$			3.00000i			0.149813i	−0.997191	+	0.0749064i	$0.976134\pi$
$402$		0			0
$403$		−	28.8444i		−	1.43684i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.60555			0.178721
$408$		0			0
$409$	5.00000			0.247234			0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000			0.247234			0.123617	+	0.992330i	$0.460551\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	52.0000i		−	2.55875i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			29.0000i			1.41674i	0.705840	+	0.708371i	$0.250570\pi$
$419$			29.0000i			1.41674i	−0.705840	+	0.708371i	$0.749430\pi$
$420$		0			0
$421$	−15.0000			−0.731055			−0.365528	−	0.930800i	$-0.619111\pi$
$421$	−15.0000			−0.731055			−0.365528	+	0.930800i	$0.619111\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.60555	−	2.00000i	−0.174895	−	0.0970143i
$426$		0			0
$427$	−52.0000			−2.51646
$428$		0			0
$429$		0			0
$430$		0			0
$431$			5.00000i			0.240842i	0.992723	+	0.120421i	$0.0384244\pi$
$431$			5.00000i			0.240842i	−0.992723	+	0.120421i	$0.961576\pi$
$432$		0			0
$433$	28.0000			1.34559			0.672797	−	0.739827i	$-0.265093\pi$
$433$	28.0000			1.34559			0.672797	+	0.739827i	$0.265093\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	14.4222i		−	0.688334i	−0.938908	−	0.344167i	$-0.888161\pi$
$439$		−	14.4222i		−	0.688334i	0.938908	−	0.344167i	$-0.111839\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$			14.4222i			0.683678i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			30.0000i			1.41579i	0.706319	+	0.707894i	$0.250354\pi$
$449$			30.0000i			1.41579i	−0.706319	+	0.707894i	$0.749646\pi$
$450$		0			0
$451$	5.00000			0.235441
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−14.4222			−0.676123
$456$		0			0
$457$	−18.0000			−0.842004			−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000			−0.842004			−0.421002	+	0.907060i	$0.638322\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	21.6333			1.00756			0.503782	−	0.863831i	$-0.331942\pi$
$461$	21.6333			1.00756			0.503782	+	0.863831i	$0.331942\pi$
$462$		0			0
$463$	−30.0000			−1.39422			−0.697109	−	0.716965i	$-0.745531\pi$
$463$	−30.0000			−1.39422			−0.697109	+	0.716965i	$0.745531\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−18.0278			−0.834225			−0.417113	−	0.908855i	$-0.636958\pi$
$467$	−18.0278			−0.834225			−0.417113	+	0.908855i	$0.636958\pi$
$468$		0			0
$469$			28.8444i			1.33191i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			2.00000i			0.0919601i
$474$		0			0
$475$	−7.00000			−0.321182
$476$		0			0
$477$		0			0
$478$		0			0
$479$			24.0000i			1.09659i	0.836286	+	0.548294i	$0.184723\pi$
$479$			24.0000i			1.09659i	−0.836286	+	0.548294i	$0.815277\pi$
$480$		0			0
$481$			14.4222i			0.657596i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.21110			−0.327439
$486$		0			0
$487$		−	14.4222i		−	0.653532i	−0.945105	−	0.326766i	$-0.894041\pi$
$487$		−	14.4222i		−	0.653532i	0.945105	−	0.326766i	$-0.105959\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$	18.0000	−	32.4500i	0.810679	−	1.46147i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	28.8444			1.29385
$498$		0			0
$499$		−	36.0555i		−	1.61407i	−0.590506	−	0.807033i	$-0.701072\pi$
$499$		−	36.0555i		−	1.61407i	0.590506	−	0.807033i	$-0.298928\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			6.00000i			0.267527i	0.991013	+	0.133763i	$0.0427062\pi$
$503$			6.00000i			0.267527i	−0.991013	+	0.133763i	$0.957294\pi$
$504$		0			0
$505$			7.21110i			0.320890i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−21.6333			−0.958880			−0.479440	−	0.877575i	$-0.659160\pi$
$509$	−21.6333			−0.958880			−0.479440	+	0.877575i	$0.659160\pi$
$510$		0			0
$511$	−39.0000			−1.72526
$512$		0			0
$513$		0			0
$514$		0			0
$515$			6.00000i			0.264392i
$516$		0			0
$517$		−	10.8167i		−	0.475716i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			39.0000i			1.70862i	0.519763	+	0.854311i	$0.326020\pi$
$521$			39.0000i			1.70862i	−0.519763	+	0.854311i	$0.673980\pi$
$522$		0			0
$523$	−16.0000			−0.699631			−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000			−0.699631			−0.349816	+	0.936819i	$0.613756\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	14.4222	−	26.0000i	0.628241	−	1.13258i
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$			20.0000i			0.866296i
$534$		0			0
$535$	−16.0000			−0.691740
$536$		0			0
$537$		0			0
$538$		0			0
$539$			6.00000i			0.258438i
$540$		0			0
$541$		−	7.21110i		−	0.310030i	−0.987912	−	0.155015i	$-0.950457\pi$
$541$		−	7.21110i		−	0.310030i	0.987912	−	0.155015i	$-0.0495425\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−7.21110			−0.308890
$546$		0			0
$547$		−	32.4500i		−	1.38746i	−0.720235	−	0.693730i	$-0.755966\pi$
$547$		−	32.4500i		−	1.38746i	0.720235	−	0.693730i	$-0.244034\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	63.0000i		−	2.68389i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−21.6333			−0.916633			−0.458316	−	0.888789i	$-0.651547\pi$
$557$	−21.6333			−0.916633			−0.458316	+	0.888789i	$0.651547\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	32.4500			1.36760			0.683801	−	0.729668i	$-0.260325\pi$
$563$	32.4500			1.36760			0.683801	+	0.729668i	$0.260325\pi$
$564$		0			0
$565$	16.0000			0.673125
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−21.6333			−0.906915			−0.453458	−	0.891278i	$-0.649810\pi$
$569$	−21.6333			−0.906915			−0.453458	+	0.891278i	$0.649810\pi$
$570$		0			0
$571$		−	28.8444i		−	1.20710i	−0.797325	−	0.603550i	$-0.793752\pi$
$571$		−	28.8444i		−	1.20710i	0.797325	−	0.603550i	$-0.206248\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	32.0000			1.33218			0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000			1.33218			0.666089	+	0.745873i	$0.267967\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$			3.60555i			0.149327i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−18.0278			−0.744085			−0.372043	−	0.928216i	$-0.621342\pi$
$587$	−18.0278			−0.744085			−0.372043	+	0.928216i	$0.621342\pi$
$588$		0			0
$589$		−	50.4777i		−	2.07990i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	18.0278			0.740311			0.370156	−	0.928970i	$-0.379304\pi$
$593$	18.0278			0.740311			0.370156	+	0.928970i	$0.379304\pi$
$594$		0			0
$595$	−13.0000	−	7.21110i	−0.532948	−	0.295626i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−28.8444			−1.17855			−0.589275	−	0.807932i	$-0.700587\pi$
$599$	−28.8444			−1.17855			−0.589275	+	0.807932i	$0.700587\pi$
$600$		0			0
$601$			7.21110i			0.294147i	0.989126	+	0.147074i	$0.0469854\pi$
$601$			7.21110i			0.294147i	−0.989126	+	0.147074i	$0.953015\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			10.0000i			0.406558i
$606$		0			0
$607$		−	32.4500i		−	1.31710i	−0.752536	−	0.658552i	$-0.771169\pi$
$607$		−	32.4500i		−	1.31710i	0.752536	−	0.658552i	$-0.228831\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	43.2666			1.75038
$612$		0			0
$613$	2.00000			0.0807792			0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000			0.0807792			0.0403896	+	0.999184i	$0.487140\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	20.0000i		−	0.805170i	−0.915383	−	0.402585i	$-0.868112\pi$
$617$		−	20.0000i		−	0.805170i	0.915383	−	0.402585i	$-0.131888\pi$
$618$		0			0
$619$		−	28.8444i		−	1.15935i	−0.814846	−	0.579677i	$-0.803179\pi$
$619$		−	28.8444i		−	1.15935i	0.814846	−	0.579677i	$-0.196821\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			52.0000i			2.08334i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−7.21110	+	13.0000i	−0.287525	+	0.518344i
$630$		0			0
$631$	35.0000			1.39333			0.696664	−	0.717398i	$-0.254667\pi$
$631$	35.0000			1.39333			0.696664	+	0.717398i	$0.254667\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	12.0000i		−	0.476205i
$636$		0			0
$637$	−24.0000			−0.950915
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	11.0000i		−	0.434474i	−0.976119	−	0.217237i	$-0.930296\pi$
$641$		−	11.0000i		−	0.434474i	0.976119	−	0.217237i	$-0.0697044\pi$
$642$		0			0
$643$		−	32.4500i		−	1.27970i	−0.768499	−	0.639851i	$-0.778996\pi$
$643$		−	32.4500i		−	1.27970i	0.768499	−	0.639851i	$-0.221004\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	14.4222			0.566995			0.283498	−	0.958973i	$-0.408505\pi$
$647$	14.4222			0.566995			0.283498	+	0.958973i	$0.408505\pi$
$648$		0			0
$649$			14.4222i			0.566121i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	26.0000i		−	1.01746i	−0.860927	−	0.508729i	$-0.830115\pi$
$653$		−	26.0000i		−	1.01746i	0.860927	−	0.508729i	$-0.169885\pi$
$654$		0			0
$655$	12.0000			0.468879
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7.21110			0.280905			0.140452	−	0.990087i	$-0.455144\pi$
$659$	7.21110			0.280905			0.140452	+	0.990087i	$0.455144\pi$
$660$		0			0
$661$	−3.00000			−0.116686			−0.0583432	−	0.998297i	$-0.518582\pi$
$661$	−3.00000			−0.116686			−0.0583432	+	0.998297i	$0.518582\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−25.2389			−0.978721
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	14.4222			0.556763
$672$		0			0
$673$		−	21.6333i		−	0.833903i	−0.908929	−	0.416951i	$-0.863099\pi$
$673$		−	21.6333i		−	0.833903i	0.908929	−	0.416951i	$-0.136901\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			12.0000i			0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$			12.0000i			0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$		0			0
$679$	−26.0000			−0.997788
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	34.0000i		−	1.30097i	−0.759517	−	0.650487i	$-0.774565\pi$
$683$		−	34.0000i		−	1.30097i	0.759517	−	0.650487i	$-0.225435\pi$
$684$		0			0
$685$		−	18.0278i		−	0.688805i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−14.4222			−0.549442
$690$		0			0
$691$		−	43.2666i		−	1.64594i	−0.568085	−	0.822970i	$-0.692316\pi$
$691$		−	43.2666i		−	1.64594i	0.568085	−	0.822970i	$-0.307684\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−7.21110			−0.273533
$696$		0			0
$697$	−10.0000	+	18.0278i	−0.378777	+	0.682850i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	21.6333			0.817079			0.408539	−	0.912741i	$-0.366038\pi$
$701$	21.6333			0.817079			0.408539	+	0.912741i	$0.366038\pi$
$702$		0			0
$703$			25.2389i			0.951902i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			26.0000i			0.977831i
$708$		0			0
$709$			28.8444i			1.08327i	0.840612	+	0.541637i	$0.182195\pi$
$709$			28.8444i			1.08327i	−0.840612	+	0.541637i	$0.817805\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	4.00000			0.149592
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	21.0000i		−	0.783168i	−0.920142	−	0.391584i	$-0.871927\pi$
$719$		−	21.0000i		−	0.783168i	0.920142	−	0.391584i	$-0.128073\pi$
$720$		0			0
$721$			21.6333i			0.805666i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			9.00000i			0.334252i
$726$		0			0
$727$	52.0000			1.92857			0.964287	−	0.264861i	$-0.0853260\pi$
$727$	52.0000			1.92857			0.964287	+	0.264861i	$0.0853260\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7.21110	−	4.00000i	−0.266712	−	0.147945i
$732$		0			0
$733$	6.00000			0.221615			0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000			0.221615			0.110808	+	0.993842i	$0.464656\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	8.00000i		−	0.294684i
$738$		0			0
$739$	−12.0000			−0.441427			−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000			−0.441427			−0.220714	+	0.975339i	$0.570839\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	8.00000i		−	0.293492i	−0.989174	−	0.146746i	$-0.953120\pi$
$743$		−	8.00000i		−	0.293492i	0.989174	−	0.146746i	$-0.0468799\pi$
$744$		0			0
$745$		−	21.6333i		−	0.792583i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−57.6888			−2.10790
$750$		0			0
$751$		−	43.2666i		−	1.57882i	−0.613866	−	0.789411i	$-0.710386\pi$
$751$		−	43.2666i		−	1.57882i	0.613866	−	0.789411i	$-0.289614\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			7.00000i			0.254756i
$756$		0			0
$757$	16.0000			0.581530			0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000			0.581530			0.290765	+	0.956795i	$0.406090\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$	−26.0000			−0.941263
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−57.6888			−2.08302
$768$		0			0
$769$	−41.0000			−1.47850			−0.739249	−	0.673432i	$-0.764819\pi$
$769$	−41.0000			−1.47850			−0.739249	+	0.673432i	$0.764819\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−10.8167			−0.389048			−0.194524	−	0.980898i	$-0.562316\pi$
$773$	−10.8167			−0.389048			−0.194524	+	0.980898i	$0.562316\pi$
$774$		0			0
$775$			7.21110i			0.259030i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			35.0000i			1.25401i
$780$		0			0
$781$	−8.00000			−0.286263
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	4.00000i		−	0.142766i
$786$		0			0
$787$		−	3.60555i		−	0.128524i	−0.997933	−	0.0642620i	$-0.979531\pi$
$787$		−	3.60555i		−	0.128524i	0.997933	−	0.0642620i	$-0.0204693\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	57.6888			2.05118
$792$		0			0
$793$			57.6888i			2.04859i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	32.4500			1.14944			0.574718	−	0.818351i	$-0.305112\pi$
$797$	32.4500			1.14944			0.574718	+	0.818351i	$0.305112\pi$
$798$		0			0
$799$	39.0000	+	21.6333i	1.37972	+	0.765331i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	10.8167			0.381711
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			10.0000i			0.351581i	0.984428	+	0.175791i	$0.0562482\pi$
$809$			10.0000i			0.351581i	−0.984428	+	0.175791i	$0.943752\pi$
$810$		0			0
$811$		−	7.21110i		−	0.253216i	−0.991953	−	0.126608i	$-0.959591\pi$
$811$		−	7.21110i		−	0.253216i	0.991953	−	0.126608i	$-0.0404090\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	18.0278			0.631485
$816$		0			0
$817$	−14.0000			−0.489798
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	42.0000i		−	1.46581i	−0.680331	−	0.732905i	$-0.738164\pi$
$821$		−	42.0000i		−	1.46581i	0.680331	−	0.732905i	$-0.261836\pi$
$822$		0			0
$823$		−	3.60555i		−	0.125682i	−0.998024	−	0.0628408i	$-0.979984\pi$
$823$		−	3.60555i		−	0.125682i	0.998024	−	0.0628408i	$-0.0200160\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			46.0000i			1.59958i	0.600282	+	0.799788i	$0.295055\pi$
$827$			46.0000i			1.59958i	−0.600282	+	0.799788i	$0.704945\pi$
$828$		0			0
$829$	−15.0000			−0.520972			−0.260486	−	0.965478i	$-0.583883\pi$
$829$	−15.0000			−0.520972			−0.260486	+	0.965478i	$0.583883\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−21.6333	−	12.0000i	−0.749550	−	0.415775i
$834$		0			0
$835$	−2.00000			−0.0692129
$836$		0			0
$837$		0			0
$838$		0			0
$839$			5.00000i			0.172619i	0.996268	+	0.0863096i	$0.0275074\pi$
$839$			5.00000i			0.172619i	−0.996268	+	0.0863096i	$0.972493\pi$
$840$		0			0
$841$	−52.0000			−1.79310
$842$		0			0
$843$		0			0
$844$		0			0
$845$			3.00000i			0.103203i
$846$		0			0
$847$			36.0555i			1.23888i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	36.0555i		−	1.23452i	−0.786760	−	0.617259i	$-0.788243\pi$
$853$		−	36.0555i		−	1.23452i	0.786760	−	0.617259i	$-0.211757\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	46.0000i		−	1.57133i	−0.618652	−	0.785665i	$-0.712321\pi$
$857$		−	46.0000i		−	1.57133i	0.618652	−	0.785665i	$-0.287679\pi$
$858$		0			0
$859$	−27.0000			−0.921228			−0.460614	−	0.887601i	$-0.652371\pi$
$859$	−27.0000			−0.921228			−0.460614	+	0.887601i	$0.652371\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	18.0278			0.613672			0.306836	−	0.951762i	$-0.400730\pi$
$863$	18.0278			0.613672			0.306836	+	0.951762i	$0.400730\pi$
$864$		0			0
$865$	−4.00000			−0.136004
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	32.0000			1.08428
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.60555			0.121890
$876$		0			0
$877$		−	46.8722i		−	1.58276i	−0.611324	−	0.791380i	$-0.709363\pi$
$877$		−	46.8722i		−	1.58276i	0.611324	−	0.791380i	$-0.290637\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			15.0000i			0.505363i	0.967550	+	0.252681i	$0.0813125\pi$
$881$			15.0000i			0.505363i	−0.967550	+	0.252681i	$0.918688\pi$
$882$		0			0
$883$	26.0000			0.874970			0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000			0.874970			0.437485	+	0.899226i	$0.355869\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			42.0000i			1.41022i	0.709097	+	0.705111i	$0.249103\pi$
$887$			42.0000i			1.41022i	−0.709097	+	0.705111i	$0.750897\pi$
$888$		0			0
$889$		−	43.2666i		−	1.45112i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	75.7166			2.53376
$894$		0			0
$895$		−	21.6333i		−	0.723122i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−64.8999			−2.16453
$900$		0			0
$901$	−13.0000	−	7.21110i	−0.433093	−	0.240237i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7.21110			−0.239705
$906$		0			0
$907$		−	54.0833i		−	1.79581i	−0.440194	−	0.897903i	$-0.645090\pi$
$907$		−	54.0833i		−	1.79581i	0.440194	−	0.897903i	$-0.354910\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	9.00000i		−	0.298183i	−0.988823	−	0.149092i	$-0.952365\pi$
$911$		−	9.00000i		−	0.298183i	0.988823	−	0.149092i	$-0.0476349\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	43.2666			1.42879
$918$		0			0
$919$	29.0000			0.956622			0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000			0.956622			0.478311	+	0.878191i	$0.341249\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	32.0000i		−	1.05329i
$924$		0			0
$925$		−	3.60555i		−	0.118550i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			45.0000i			1.47640i	0.674581	+	0.738201i	$0.264324\pi$
$929$			45.0000i			1.47640i	−0.674581	+	0.738201i	$0.735676\pi$
$930$		0			0
$931$	−42.0000			−1.37649
$932$		0			0
$933$		0			0
$934$		0			0
$935$	3.60555	+	2.00000i	0.117914	+	0.0654070i
$936$		0			0
$937$	−22.0000			−0.718709			−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000			−0.718709			−0.359354	+	0.933201i	$0.617003\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			2.00000i			0.0651981i	0.999469	+	0.0325991i	$0.0103784\pi$
$941$			2.00000i			0.0651981i	−0.999469	+	0.0325991i	$0.989622\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	10.0000i		−	0.324956i	−0.986712	−	0.162478i	$-0.948051\pi$
$947$		−	10.0000i		−	0.324956i	0.986712	−	0.162478i	$-0.0519487\pi$
$948$		0			0
$949$			43.2666i			1.40449i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	25.2389			0.817567			0.408783	−	0.912631i	$-0.365953\pi$
$953$	25.2389			0.817567			0.408783	+	0.912631i	$0.365953\pi$
$954$		0			0
$955$			7.21110i			0.233346i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	65.0000i		−	2.09896i
$960$		0			0
$961$	−21.0000			−0.677419
$962$		0			0
$963$		0			0
$964$		0			0
$965$	7.21110			0.232134
$966$		0			0
$967$	−58.0000			−1.86515			−0.932577	−	0.360971i	$-0.882445\pi$
$967$	−58.0000			−1.86515			−0.932577	+	0.360971i	$0.882445\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−14.4222			−0.462831			−0.231415	−	0.972855i	$-0.574336\pi$
$971$	−14.4222			−0.462831			−0.231415	+	0.972855i	$0.574336\pi$
$972$		0			0
$973$	−26.0000			−0.833522
$974$		0			0
$975$		0			0
$976$		0			0
$977$	36.0555			1.15352			0.576759	−	0.816914i	$-0.304317\pi$
$977$	36.0555			1.15352			0.576759	+	0.816914i	$0.304317\pi$
$978$		0			0
$979$		−	14.4222i		−	0.460936i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			10.0000i			0.318950i	0.987202	+	0.159475i	$0.0509802\pi$
$983$			10.0000i			0.318950i	−0.987202	+	0.159475i	$0.949020\pi$
$984$		0			0
$985$	−24.0000			−0.764704
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$			50.4777i			1.60348i	0.597675	+	0.801739i	$0.296091\pi$
$991$			50.4777i			1.60348i	−0.597675	+	0.801739i	$0.703909\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−14.4222			−0.457215
$996$		0			0
$997$			32.4500i			1.02770i	0.857880	+	0.513850i	$0.171781\pi$
$997$			32.4500i			1.02770i	−0.857880	+	0.513850i	$0.828219\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3060.2.e.i.1801.4	yes	4
3.2	odd	2	inner	3060.2.e.i.1801.2	yes	4
17.16	even	2	inner	3060.2.e.i.1801.1	✓	4
51.50	odd	2	inner	3060.2.e.i.1801.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3060.2.e.i.1801.1	✓	4	17.16	even	2	inner
3060.2.e.i.1801.2	yes	4	3.2	odd	2	inner
3060.2.e.i.1801.3	yes	4	51.50	odd	2	inner
3060.2.e.i.1801.4	yes	4	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} +3.60555i q^{7} -1.00000i q^{11} +4.00000 q^{13} +(3.60555 + 2.00000i) q^{17} +7.00000 q^{19} -1.00000 q^{25} -9.00000i q^{29} -7.21110i q^{31} -3.60555 q^{35} +3.60555i q^{37} +5.00000i q^{41} -2.00000 q^{43} +10.8167 q^{47} -6.00000 q^{49} -3.60555 q^{53} +1.00000 q^{55} -14.4222 q^{59} +14.4222i q^{61} +4.00000i q^{65} +8.00000 q^{67} -8.00000i q^{71} +10.8167i q^{73} +3.60555 q^{77} +(-2.00000 + 3.60555i) q^{85} +14.4222 q^{89} +14.4222i q^{91} +7.00000i q^{95} +7.21110i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{13} + 28 q^{19} - 4 q^{25} - 8 q^{43} - 24 q^{49} + 4 q^{55} + 32 q^{67} - 8 q^{85}+O(q^{100}) \) 4 * q + 16 * q^13 + 28 * q^19 - 4 * q^25 - 8 * q^43 - 24 * q^49 + 4 * q^55 + 32 * q^67 - 8 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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