LMFDB - Embedded newform 2340.2.c.e.181.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,2,Mod(181,2340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2340, 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("2340.181"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2340.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2732361984.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 13x^{6} + 36x^{4} + 13x^{2} + 1 $ x^8 + 13x^6 + 36x^4 + 13*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.5
Root			$0.328543i$ of defining polynomial
Character	$\chi$	$=$	2340.181
Dual form			2340.2.c.e.181.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} -2.71519i q^{7} -1.37228i q^{11} +(-2.37228 + 2.71519i) q^{13} +3.72601 q^{17} +1.01082i q^{19} +1.70438 q^{23} -1.00000 q^{25} +5.43039 q^{29} -1.01082i q^{31} +2.71519 q^{35} -3.72601i q^{37} -1.37228i q^{41} -4.00000 q^{43} -8.74456i q^{47} -0.372281 q^{49} +9.15640 q^{53} +1.37228 q^{55} -8.74456i q^{59} +8.11684 q^{61} +(-2.71519 - 2.37228i) q^{65} -6.44121i q^{67} -4.11684i q^{71} +7.45202i q^{73} -3.72601 q^{77} -3.37228 q^{79} -11.4891i q^{83} +3.72601i q^{85} -1.37228i q^{89} +(7.37228 + 6.44121i) q^{91} -1.01082 q^{95} -1.70438i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{13} - 8 q^{25} - 32 q^{43} + 20 q^{49} - 12 q^{55} - 4 q^{61} - 4 q^{79} + 36 q^{91}+O(q^{100}) $ 8 * q + 4 * q^13 - 8 * q^25 - 32 * q^43 + 20 * q^49 - 12 * q^55 - 4 * q^61 - 4 * q^79 + 36 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$7$		−	2.71519i		−	1.02625i	−0.858315	−	0.513124i	$-0.828488\pi$
$7$		−	2.71519i		−	1.02625i	0.858315	−	0.513124i	$-0.171512\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.37228i		−	0.413758i	−0.978366	−	0.206879i	$-0.933669\pi$
$11$		−	1.37228i		−	0.413758i	0.978366	−	0.206879i	$-0.0663307\pi$
$12$		0			0
$13$	−2.37228	+	2.71519i	−0.657952	+	0.753059i
$13$	−2.37228	+	2.71519i	−0.657952	+	0.753059i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.72601			0.903690			0.451845	−	0.892096i	$-0.350766\pi$
$17$	3.72601			0.903690			0.451845	+	0.892096i	$0.350766\pi$
$18$		0			0
$19$			1.01082i			0.231897i	0.993255	+	0.115949i	$0.0369908\pi$
$19$			1.01082i			0.231897i	−0.993255	+	0.115949i	$0.963009\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.70438			0.355387			0.177694	−	0.984086i	$-0.443136\pi$
$23$	1.70438			0.355387			0.177694	+	0.984086i	$0.443136\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.43039			1.00840			0.504199	−	0.863588i	$-0.331788\pi$
$29$	5.43039			1.00840			0.504199	+	0.863588i	$0.331788\pi$
$30$		0			0
$31$		−	1.01082i		−	0.181548i	−0.995872	−	0.0907740i	$-0.971066\pi$
$31$		−	1.01082i		−	0.181548i	0.995872	−	0.0907740i	$-0.0289341\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.71519			0.458952
$36$		0			0
$37$		−	3.72601i		−	0.612552i	−0.951943	−	0.306276i	$-0.900917\pi$
$37$		−	3.72601i		−	0.612552i	0.951943	−	0.306276i	$-0.0990831\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1.37228i		−	0.214314i	−0.994242	−	0.107157i	$-0.965825\pi$
$41$		−	1.37228i		−	0.214314i	0.994242	−	0.107157i	$-0.0341748\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	8.74456i		−	1.27553i	−0.770233	−	0.637763i	$-0.779860\pi$
$47$		−	8.74456i		−	1.27553i	0.770233	−	0.637763i	$-0.220140\pi$
$48$		0			0
$49$	−0.372281			−0.0531830
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.15640			1.25773			0.628864	−	0.777515i	$-0.283520\pi$
$53$	9.15640			1.25773			0.628864	+	0.777515i	$0.283520\pi$
$54$		0			0
$55$	1.37228			0.185038
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	8.74456i		−	1.13845i	−0.822183	−	0.569223i	$-0.807244\pi$
$59$		−	8.74456i		−	1.13845i	0.822183	−	0.569223i	$-0.192756\pi$
$60$		0			0
$61$	8.11684			1.03926			0.519628	−	0.854393i	$-0.326071\pi$
$61$	8.11684			1.03926			0.519628	+	0.854393i	$0.326071\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.71519	−	2.37228i	−0.336778	−	0.294245i
$66$		0			0
$67$		−	6.44121i		−	0.786918i	−0.919342	−	0.393459i	$-0.871278\pi$
$67$		−	6.44121i		−	0.786918i	0.919342	−	0.393459i	$-0.128722\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.11684i		−	0.488579i	−0.969702	−	0.244290i	$-0.921445\pi$
$71$		−	4.11684i		−	0.488579i	0.969702	−	0.244290i	$-0.0785548\pi$
$72$		0			0
$73$			7.45202i			0.872193i	0.899900	+	0.436097i	$0.143639\pi$
$73$			7.45202i			0.872193i	−0.899900	+	0.436097i	$0.856361\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.72601			−0.424618
$78$		0			0
$79$	−3.37228			−0.379411			−0.189706	−	0.981841i	$-0.560753\pi$
$79$	−3.37228			−0.379411			−0.189706	+	0.981841i	$0.560753\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	11.4891i		−	1.26110i	−0.776151	−	0.630548i	$-0.782830\pi$
$83$		−	11.4891i		−	1.26110i	0.776151	−	0.630548i	$-0.217170\pi$
$84$		0			0
$85$			3.72601i			0.404143i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	1.37228i		−	0.145462i	−0.997352	−	0.0727308i	$-0.976829\pi$
$89$		−	1.37228i		−	0.145462i	0.997352	−	0.0727308i	$-0.0231714\pi$
$90$		0			0
$91$	7.37228	+	6.44121i	0.772825	+	0.675222i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.01082			−0.103708
$96$		0			0
$97$		−	1.70438i		−	0.173053i	−0.996250	−	0.0865267i	$-0.972423\pi$
$97$		−	1.70438i		−	0.173053i	0.996250	−	0.0865267i	$-0.0275768\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.40876			0.339184			0.169592	−	0.985514i	$-0.445755\pi$
$101$	3.40876			0.339184			0.169592	+	0.985514i	$0.445755\pi$
$102$		0			0
$103$	−6.74456			−0.664562			−0.332281	−	0.943181i	$-0.607818\pi$
$103$	−6.74456			−0.664562			−0.332281	+	0.943181i	$0.607818\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	17.9955			1.73970			0.869848	−	0.493321i	$-0.164217\pi$
$107$	17.9955			1.73970			0.869848	+	0.493321i	$0.164217\pi$
$108$		0			0
$109$		−	2.02163i		−	0.193637i	−0.995302	−	0.0968186i	$-0.969133\pi$
$109$		−	2.02163i		−	0.193637i	0.995302	−	0.0968186i	$-0.0308667\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.43039			−0.510848			−0.255424	−	0.966829i	$-0.582215\pi$
$113$	−5.43039			−0.510848			−0.255424	+	0.966829i	$0.582215\pi$
$114$		0			0
$115$			1.70438i			0.158934i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	10.1168i		−	0.927410i
$120$		0			0
$121$	9.11684			0.828804
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	−1.48913			−0.132139			−0.0660693	−	0.997815i	$-0.521046\pi$
$127$	−1.48913			−0.132139			−0.0660693	+	0.997815i	$0.521046\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	18.3128			1.60000			0.799998	−	0.600002i	$-0.204834\pi$
$131$	18.3128			1.60000			0.799998	+	0.600002i	$0.204834\pi$
$132$		0			0
$133$	2.74456			0.237984
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	8.74456i		−	0.747098i	−0.927610	−	0.373549i	$-0.878141\pi$
$137$		−	8.74456i		−	0.747098i	0.927610	−	0.373549i	$-0.121859\pi$
$138$		0			0
$139$	3.37228			0.286033			0.143017	−	0.989720i	$-0.454320\pi$
$139$	3.37228			0.286033			0.143017	+	0.989720i	$0.454320\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.72601	+	3.25544i	0.311585	+	0.272233i
$144$		0			0
$145$			5.43039i			0.450969i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			13.3723i			1.09550i	0.836642	+	0.547750i	$0.184515\pi$
$149$			13.3723i			1.09550i	−0.836642	+	0.547750i	$0.815485\pi$
$150$		0			0
$151$			9.84996i			0.801579i	0.916170	+	0.400789i	$0.131264\pi$
$151$			9.84996i			0.801579i	−0.916170	+	0.400789i	$0.868736\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.01082			0.0811907
$156$		0			0
$157$	−15.4891			−1.23617			−0.618083	−	0.786113i	$-0.712091\pi$
$157$	−15.4891			−1.23617			−0.618083	+	0.786113i	$0.712091\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	4.62772i		−	0.364715i
$162$		0			0
$163$		−	2.71519i		−	0.212670i	−0.994330	−	0.106335i	$-0.966088\pi$
$163$		−	2.71519i		−	0.212670i	0.994330	−	0.106335i	$-0.0339117\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			6.00000i			0.464294i	0.972681	+	0.232147i	$0.0745750\pi$
$167$			6.00000i			0.464294i	−0.972681	+	0.232147i	$0.925425\pi$
$168$		0			0
$169$	−1.74456	−	12.8824i	−0.134197	−	0.990955i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.45202			0.566567			0.283283	−	0.959036i	$-0.408576\pi$
$173$	7.45202			0.566567			0.283283	+	0.959036i	$0.408576\pi$
$174$		0			0
$175$			2.71519i			0.205249i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	25.7648			1.92575			0.962877	−	0.269942i	$-0.0870045\pi$
$179$	25.7648			1.92575			0.962877	+	0.269942i	$0.0870045\pi$
$180$		0			0
$181$	2.86141			0.212687			0.106343	−	0.994329i	$-0.466086\pi$
$181$	2.86141			0.212687			0.106343	+	0.994329i	$0.466086\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.72601			0.273942
$186$		0			0
$187$		−	5.11313i		−	0.373909i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	5.43039			0.392929			0.196465	−	0.980511i	$-0.437054\pi$
$191$	5.43039			0.392929			0.196465	+	0.980511i	$0.437054\pi$
$192$		0			0
$193$		−	1.70438i		−	0.122684i	−0.998117	−	0.0613419i	$-0.980462\pi$
$193$		−	1.70438i		−	0.122684i	0.998117	−	0.0613419i	$-0.0195380\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			6.00000i			0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$			6.00000i			0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$		0			0
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	14.7446i		−	1.03487i
$204$		0			0
$205$	1.37228			0.0958443
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.38712			0.0959494
$210$		0			0
$211$	9.48913			0.653258			0.326629	−	0.945153i	$-0.394087\pi$
$211$	9.48913			0.653258			0.326629	+	0.945153i	$0.394087\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	4.00000i		−	0.272798i
$216$		0			0
$217$	−2.74456			−0.186313
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.83915	+	10.1168i	−0.594585	+	0.680533i
$222$		0			0
$223$		−	17.3020i		−	1.15863i	−0.815105	−	0.579313i	$-0.803321\pi$
$223$		−	17.3020i		−	1.15863i	0.815105	−	0.579313i	$-0.196679\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			0.510875i			0.0339079i	0.999856	+	0.0169540i	$0.00539688\pi$
$227$			0.510875i			0.0339079i	−0.999856	+	0.0169540i	$0.994603\pi$
$228$		0			0
$229$		−	19.6999i		−	1.30181i	−0.759160	−	0.650904i	$-0.774390\pi$
$229$		−	19.6999i		−	1.30181i	0.759160	−	0.650904i	$-0.225610\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.6084			−1.08805			−0.544027	−	0.839068i	$-0.683101\pi$
$233$	−16.6084			−1.08805			−0.544027	+	0.839068i	$0.683101\pi$
$234$		0			0
$235$	8.74456			0.570432
$236$		0			0
$237$		0			0
$238$		0			0
$239$			1.37228i			0.0887655i	0.999015	+	0.0443827i	$0.0141321\pi$
$239$			1.37228i			0.0887655i	−0.999015	+	0.0443827i	$0.985868\pi$
$240$		0			0
$241$			14.2695i			0.919182i	0.888131	+	0.459591i	$0.152004\pi$
$241$			14.2695i			0.919182i	−0.888131	+	0.459591i	$0.847996\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	0.372281i		−	0.0237842i
$246$		0			0
$247$	−2.74456	−	2.39794i	−0.174632	−	0.152577i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.38712			0.0875545			0.0437773	−	0.999041i	$-0.486061\pi$
$251$	1.38712			0.0875545			0.0437773	+	0.999041i	$0.486061\pi$
$252$		0			0
$253$		−	2.33889i		−	0.147045i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−14.2695			−0.890109			−0.445055	−	0.895503i	$-0.646816\pi$
$257$	−14.2695			−0.890109			−0.445055	+	0.895503i	$0.646816\pi$
$258$		0			0
$259$	−10.1168			−0.628630
$260$		0			0
$261$		0			0
$262$		0			0
$263$	8.83915			0.545045			0.272523	−	0.962149i	$-0.412142\pi$
$263$	8.83915			0.545045			0.272523	+	0.962149i	$0.412142\pi$
$264$		0			0
$265$			9.15640i			0.562473i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−27.1519			−1.65548			−0.827742	−	0.561109i	$-0.810375\pi$
$269$	−27.1519			−1.65548			−0.827742	+	0.561109i	$0.810375\pi$
$270$		0			0
$271$			19.3236i			1.17383i	0.809650	+	0.586913i	$0.199657\pi$
$271$			19.3236i			1.17383i	−0.809650	+	0.586913i	$0.800343\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			1.37228i			0.0827517i
$276$		0			0
$277$	−18.2337			−1.09556			−0.547778	−	0.836624i	$-0.684526\pi$
$277$	−18.2337			−1.09556			−0.547778	+	0.836624i	$0.684526\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	18.0000i		−	1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$		−	18.0000i		−	1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$		0			0
$283$	1.25544			0.0746280			0.0373140	−	0.999304i	$-0.488120\pi$
$283$	1.25544			0.0746280			0.0373140	+	0.999304i	$0.488120\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.72601			−0.219939
$288$		0			0
$289$	−3.11684			−0.183344
$290$		0			0
$291$		0			0
$292$		0			0
$293$			2.23369i			0.130493i	0.997869	+	0.0652467i	$0.0207834\pi$
$293$			2.23369i			0.130493i	−0.997869	+	0.0652467i	$0.979217\pi$
$294$		0			0
$295$	8.74456			0.509128
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.04326	+	4.62772i	−0.233828	+	0.267628i
$300$		0			0
$301$			10.8608i			0.626005i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			8.11684i			0.464769i
$306$		0			0
$307$		−	19.6409i		−	1.12096i	−0.828167	−	0.560482i	$-0.810616\pi$
$307$		−	19.6409i		−	1.12096i	0.828167	−	0.560482i	$-0.189384\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−19.6999			−1.11708			−0.558540	−	0.829478i	$-0.688638\pi$
$311$	−19.6999			−1.11708			−0.558540	+	0.829478i	$0.688638\pi$
$312$		0			0
$313$	10.0000			0.565233			0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000			0.565233			0.282617	+	0.959233i	$0.408798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	3.25544i		−	0.182844i	−0.995812	−	0.0914218i	$-0.970859\pi$
$317$		−	3.25544i		−	0.182844i	0.995812	−	0.0914218i	$-0.0291411\pi$
$318$		0			0
$319$		−	7.45202i		−	0.417233i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			3.76631i			0.209563i
$324$		0			0
$325$	2.37228	−	2.71519i	0.131590	−	0.150612i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−23.7432			−1.30900
$330$		0			0
$331$			28.1628i			1.54797i	0.633207	+	0.773983i	$0.281738\pi$
$331$			28.1628i			1.54797i	−0.633207	+	0.773983i	$0.718262\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.44121			0.351921
$336$		0			0
$337$	−22.2337			−1.21115			−0.605573	−	0.795790i	$-0.707056\pi$
$337$	−22.2337			−1.21115			−0.605573	+	0.795790i	$0.707056\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.38712			−0.0751170
$342$		0			0
$343$		−	17.9955i		−	0.971668i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	9.15640			0.491541			0.245771	−	0.969328i	$-0.420959\pi$
$347$	9.15640			0.491541			0.245771	+	0.969328i	$0.420959\pi$
$348$		0			0
$349$		−	1.38712i		−	0.0742511i	−0.999311	−	0.0371255i	$-0.988180\pi$
$349$		−	1.38712i		−	0.0742511i	0.999311	−	0.0371255i	$-0.0118201\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			14.2337i			0.757583i	0.925482	+	0.378791i	$0.123660\pi$
$353$			14.2337i			0.757583i	−0.925482	+	0.378791i	$0.876340\pi$
$354$		0			0
$355$	4.11684			0.218499
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	15.2554i		−	0.805151i	−0.915387	−	0.402576i	$-0.868115\pi$
$359$		−	15.2554i		−	0.805151i	0.915387	−	0.402576i	$-0.131885\pi$
$360$		0			0
$361$	17.9783			0.946224
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7.45202			−0.390057
$366$		0			0
$367$	−12.2337			−0.638593			−0.319297	−	0.947655i	$-0.603447\pi$
$367$	−12.2337			−0.638593			−0.319297	+	0.947655i	$0.603447\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	24.8614i		−	1.29074i
$372$		0			0
$373$	22.0000			1.13912			0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000			1.13912			0.569558	+	0.821951i	$0.307114\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.8824	+	14.7446i	−0.663478	+	0.759384i
$378$		0			0
$379$			37.6364i			1.93325i	0.256190	+	0.966626i	$0.417533\pi$
$379$			37.6364i			1.93325i	−0.256190	+	0.966626i	$0.582467\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			31.7228i			1.62096i	0.585766	+	0.810480i	$0.300794\pi$
$383$			31.7228i			1.62096i	−0.585766	+	0.810480i	$0.699206\pi$
$384$		0			0
$385$		−	3.72601i		−	0.189895i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	16.9257			0.858166			0.429083	−	0.903265i	$-0.358837\pi$
$389$	16.9257			0.858166			0.429083	+	0.903265i	$0.358837\pi$
$390$		0			0
$391$	6.35053			0.321160
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	3.37228i		−	0.169678i
$396$		0			0
$397$		−	10.5435i		−	0.529164i	−0.964363	−	0.264582i	$-0.914766\pi$
$397$		−	10.5435i		−	0.529164i	0.964363	−	0.264582i	$-0.0852340\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			34.4674i			1.72122i	0.509266	+	0.860609i	$0.329917\pi$
$401$			34.4674i			1.72122i	−0.509266	+	0.860609i	$0.670083\pi$
$402$		0			0
$403$	2.74456	+	2.39794i	0.136716	+	0.119450i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−5.11313			−0.253449
$408$		0			0
$409$		−	29.1736i		−	1.44254i	−0.692654	−	0.721270i	$-0.743559\pi$
$409$		−	29.1736i		−	1.44254i	0.692654	−	0.721270i	$-0.256441\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−23.7432			−1.16833
$414$		0			0
$415$	11.4891			0.563979
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3.40876			−0.166529			−0.0832643	−	0.996527i	$-0.526535\pi$
$419$	−3.40876			−0.166529			−0.0832643	+	0.996527i	$0.526535\pi$
$420$		0			0
$421$			31.1952i			1.52036i	0.649712	+	0.760181i	$0.274890\pi$
$421$			31.1952i			1.52036i	−0.649712	+	0.760181i	$0.725110\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.72601			−0.180738
$426$		0			0
$427$		−	22.0388i		−	1.06653i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			3.25544i			0.156809i	0.996922	+	0.0784044i	$0.0249826\pi$
$431$			3.25544i			0.156809i	−0.996922	+	0.0784044i	$0.975017\pi$
$432$		0			0
$433$	−14.0000			−0.672797			−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000			−0.672797			−0.336399	+	0.941720i	$0.609209\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			1.72281i			0.0824133i
$438$		0			0
$439$	0.627719			0.0299594			0.0149797	−	0.999888i	$-0.495232\pi$
$439$	0.627719			0.0299594			0.0149797	+	0.999888i	$0.495232\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11.1780			0.531084			0.265542	−	0.964099i	$-0.414449\pi$
$443$	11.1780			0.531084			0.265542	+	0.964099i	$0.414449\pi$
$444$		0			0
$445$	1.37228			0.0650524
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	5.13859i		−	0.242505i	−0.992622	−	0.121253i	$-0.961309\pi$
$449$		−	5.13859i		−	0.242505i	0.992622	−	0.121253i	$-0.0386911\pi$
$450$		0			0
$451$	−1.88316			−0.0886744
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.44121	+	7.37228i	−0.301968	+	0.345618i
$456$		0			0
$457$		−	12.5652i		−	0.587773i	−0.955840	−	0.293887i	$-0.905051\pi$
$457$		−	12.5652i		−	0.587773i	0.955840	−	0.293887i	$-0.0949488\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			34.6277i			1.61277i	0.591388	+	0.806387i	$0.298580\pi$
$461$			34.6277i			1.61277i	−0.591388	+	0.806387i	$0.701420\pi$
$462$		0			0
$463$			8.14558i			0.378557i	0.981923	+	0.189279i	$0.0606150\pi$
$463$			8.14558i			0.378557i	−0.981923	+	0.189279i	$0.939385\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9.15640			−0.423708			−0.211854	−	0.977301i	$-0.567950\pi$
$467$	−9.15640			−0.423708			−0.211854	+	0.977301i	$0.567950\pi$
$468$		0			0
$469$	−17.4891			−0.807573
$470$		0			0
$471$		0			0
$472$		0			0
$473$			5.48913i			0.252390i
$474$		0			0
$475$		−	1.01082i		−	0.0463794i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			30.8614i			1.41009i	0.709161	+	0.705047i	$0.249074\pi$
$479$			30.8614i			1.41009i	−0.709161	+	0.705047i	$0.750926\pi$
$480$		0			0
$481$	10.1168	+	8.83915i	0.461288	+	0.403030i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.70438			0.0773918
$486$		0			0
$487$		−	26.4584i		−	1.19894i	−0.800396	−	0.599472i	$-0.795377\pi$
$487$		−	26.4584i		−	1.19894i	0.800396	−	0.599472i	$-0.204623\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−9.47365			−0.427540			−0.213770	−	0.976884i	$-0.568574\pi$
$491$	−9.47365			−0.427540			−0.213770	+	0.976884i	$0.568574\pi$
$492$		0			0
$493$	20.2337			0.911279
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−11.1780			−0.501403
$498$		0			0
$499$			24.1195i			1.07974i	0.841749	+	0.539868i	$0.181526\pi$
$499$			24.1195i			1.07974i	−0.841749	+	0.539868i	$0.818474\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	2.02163			0.0901401			0.0450701	−	0.998984i	$-0.485649\pi$
$503$	2.02163			0.0901401			0.0450701	+	0.998984i	$0.485649\pi$
$504$		0			0
$505$			3.40876i			0.151688i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			36.3505i			1.61121i	0.592454	+	0.805604i	$0.298159\pi$
$509$			36.3505i			1.61121i	−0.592454	+	0.805604i	$0.701841\pi$
$510$		0			0
$511$	20.2337			0.895086
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	6.74456i		−	0.297201i
$516$		0			0
$517$	−12.0000			−0.527759
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−34.6040			−1.51603			−0.758014	−	0.652239i	$-0.773830\pi$
$521$	−34.6040			−1.51603			−0.758014	+	0.652239i	$0.773830\pi$
$522$		0			0
$523$	−29.7228			−1.29969			−0.649844	−	0.760068i	$-0.725166\pi$
$523$	−29.7228			−1.29969			−0.649844	+	0.760068i	$0.725166\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	3.76631i		−	0.164063i
$528$		0			0
$529$	−20.0951			−0.873700
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.72601	+	3.25544i	0.161391	+	0.141009i
$534$		0			0
$535$			17.9955i			0.778015i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			0.510875i			0.0220049i
$540$		0			0
$541$			38.0127i			1.63429i	0.576429	+	0.817147i	$0.304446\pi$
$541$			38.0127i			1.63429i	−0.576429	+	0.817147i	$0.695554\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.02163			0.0865972
$546$		0			0
$547$	−4.00000			−0.171028			−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000			−0.171028			−0.0855138	+	0.996337i	$0.527253\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			5.48913i			0.233845i
$552$		0			0
$553$			9.15640i			0.389370i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			16.9783i			0.719392i	0.933070	+	0.359696i	$0.117120\pi$
$557$			16.9783i			0.719392i	−0.933070	+	0.359696i	$0.882880\pi$
$558$		0			0
$559$	9.48913	−	10.8608i	0.401347	−	0.459362i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−17.9955			−0.758422			−0.379211	−	0.925310i	$-0.623805\pi$
$563$	−17.9955			−0.758422			−0.379211	+	0.925310i	$0.623805\pi$
$564$		0			0
$565$		−	5.43039i		−	0.228458i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6.81751			−0.285805			−0.142903	−	0.989737i	$-0.545644\pi$
$569$	−6.81751			−0.285805			−0.142903	+	0.989737i	$0.545644\pi$
$570$		0			0
$571$	−0.627719			−0.0262692			−0.0131346	−	0.999914i	$-0.504181\pi$
$571$	−0.627719			−0.0262692			−0.0131346	+	0.999914i	$0.504181\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.70438			−0.0710775
$576$		0			0
$577$		−	23.4259i		−	0.975234i	−0.873058	−	0.487617i	$-0.837866\pi$
$577$		−	23.4259i		−	0.975234i	0.873058	−	0.487617i	$-0.162134\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−31.1952			−1.29420
$582$		0			0
$583$		−	12.5652i		−	0.520396i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	26.2337i		−	1.08278i	−0.840772	−	0.541390i	$-0.817898\pi$
$587$		−	26.2337i		−	1.08278i	0.840772	−	0.541390i	$-0.182102\pi$
$588$		0			0
$589$	1.02175			0.0421005
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	15.2554i		−	0.626466i	−0.949676	−	0.313233i	$-0.898588\pi$
$593$		−	15.2554i		−	0.626466i	0.949676	−	0.313233i	$-0.101412\pi$
$594$		0			0
$595$	10.1168			0.414750
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−29.1736			−1.19200			−0.596000	−	0.802984i	$-0.703244\pi$
$599$	−29.1736			−1.19200			−0.596000	+	0.802984i	$0.703244\pi$
$600$		0			0
$601$	40.3505			1.64593			0.822966	−	0.568090i	$-0.192318\pi$
$601$	40.3505			1.64593			0.822966	+	0.568090i	$0.192318\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			9.11684i			0.370652i
$606$		0			0
$607$	−32.4674			−1.31781			−0.658905	−	0.752226i	$-0.728980\pi$
$607$	−32.4674			−1.31781			−0.658905	+	0.752226i	$0.728980\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	23.7432	+	20.7446i	0.960547	+	0.839235i
$612$		0			0
$613$		−	21.4043i		−	0.864512i	−0.901751	−	0.432256i	$-0.857718\pi$
$613$		−	21.4043i		−	0.864512i	0.901751	−	0.432256i	$-0.142282\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			47.4891i			1.91184i	0.293627	+	0.955920i	$0.405138\pi$
$617$			47.4891i			1.91184i	−0.293627	+	0.955920i	$0.594862\pi$
$618$		0			0
$619$		−	26.1411i		−	1.05070i	−0.850886	−	0.525350i	$-0.823934\pi$
$619$		−	26.1411i		−	1.05070i	0.850886	−	0.525350i	$-0.176066\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.72601			−0.149279
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	13.8832i		−	0.553558i
$630$		0			0
$631$			6.44121i			0.256420i	0.991747	+	0.128210i	$0.0409232\pi$
$631$			6.44121i			0.256420i	−0.991747	+	0.128210i	$0.959077\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	1.48913i		−	0.0590941i
$636$		0			0
$637$	0.883156	−	1.01082i	0.0349919	−	0.0400500i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	27.7865			1.09750			0.548749	−	0.835987i	$-0.315104\pi$
$641$	27.7865			1.09750			0.548749	+	0.835987i	$0.315104\pi$
$642$		0			0
$643$			21.6625i			0.854286i	0.904184	+	0.427143i	$0.140480\pi$
$643$			21.6625i			0.854286i	−0.904184	+	0.427143i	$0.859520\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3.72601			−0.146485			−0.0732423	−	0.997314i	$-0.523335\pi$
$647$	−3.72601			−0.146485			−0.0732423	+	0.997314i	$0.523335\pi$
$648$		0			0
$649$	−12.0000			−0.471041
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−48.8735			−1.91257			−0.956284	−	0.292440i	$-0.905533\pi$
$653$	−48.8735			−1.91257			−0.956284	+	0.292440i	$0.905533\pi$
$654$		0			0
$655$			18.3128i			0.715540i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	30.5607			1.19048			0.595238	−	0.803549i	$-0.297058\pi$
$659$	30.5607			1.19048			0.595238	+	0.803549i	$0.297058\pi$
$660$		0			0
$661$			1.38712i			0.0539529i	0.999636	+	0.0269764i	$0.00858791\pi$
$661$			1.38712i			0.0539529i	−0.999636	+	0.0269764i	$0.991412\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			2.74456i			0.106430i
$666$		0			0
$667$	9.25544			0.358372
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	11.1386i		−	0.430001i
$672$		0			0
$673$	−1.76631			−0.0680863			−0.0340432	−	0.999420i	$-0.510838\pi$
$673$	−1.76631			−0.0680863			−0.0340432	+	0.999420i	$0.510838\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	13.1997			0.507304			0.253652	−	0.967295i	$-0.418368\pi$
$677$	13.1997			0.507304			0.253652	+	0.967295i	$0.418368\pi$
$678$		0			0
$679$	−4.62772			−0.177596
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	47.4891i		−	1.81712i	−0.417753	−	0.908560i	$-0.637182\pi$
$683$		−	47.4891i		−	1.81712i	0.417753	−	0.908560i	$-0.362818\pi$
$684$		0			0
$685$	8.74456			0.334113
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−21.7216	+	24.8614i	−0.827525	+	0.947144i
$690$		0			0
$691$		−	4.41957i		−	0.168128i	−0.996460	−	0.0840642i	$-0.973210\pi$
$691$		−	4.41957i		−	0.168128i	0.996460	−	0.0840642i	$-0.0267901\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			3.37228i			0.127918i
$696$		0			0
$697$		−	5.11313i		−	0.193674i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.79588			0.181138			0.0905690	−	0.995890i	$-0.471131\pi$
$701$	4.79588			0.181138			0.0905690	+	0.995890i	$0.471131\pi$
$702$		0			0
$703$	3.76631			0.142049
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	9.25544i		−	0.348087i
$708$		0			0
$709$		−	34.6040i		−	1.29958i	−0.760114	−	0.649790i	$-0.774857\pi$
$709$		−	34.6040i		−	1.29958i	0.760114	−	0.649790i	$-0.225143\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1.72281i		−	0.0645199i
$714$		0			0
$715$	−3.25544	+	3.72601i	−0.121746	+	0.139345i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	5.43039			0.202519			0.101260	−	0.994860i	$-0.467713\pi$
$719$	5.43039			0.202519			0.101260	+	0.994860i	$0.467713\pi$
$720$		0			0
$721$			18.3128i			0.682004i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.43039			−0.201680
$726$		0			0
$727$	−16.2337			−0.602074			−0.301037	−	0.953612i	$-0.597333\pi$
$727$	−16.2337			−0.602074			−0.301037	+	0.953612i	$0.597333\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−14.9040			−0.551246
$732$		0			0
$733$		−	26.0821i		−	0.963363i	−0.876346	−	0.481682i	$-0.840026\pi$
$733$		−	26.0821i		−	0.963363i	0.876346	−	0.481682i	$-0.159974\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.83915			−0.325594
$738$		0			0
$739$			20.7107i			0.761857i	0.924605	+	0.380928i	$0.124396\pi$
$739$			20.7107i			0.761857i	−0.924605	+	0.380928i	$0.875604\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			4.97825i			0.182634i	0.995822	+	0.0913172i	$0.0291077\pi$
$743$			4.97825i			0.182634i	−0.995822	+	0.0913172i	$0.970892\pi$
$744$		0			0
$745$	−13.3723			−0.489922
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	48.8614i		−	1.78536i
$750$		0			0
$751$	35.6060			1.29928			0.649640	−	0.760242i	$-0.274920\pi$
$751$	35.6060			1.29928			0.649640	+	0.760242i	$0.274920\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−9.84996			−0.358477
$756$		0			0
$757$	−0.978251			−0.0355551			−0.0177776	−	0.999842i	$-0.505659\pi$
$757$	−0.978251			−0.0355551			−0.0177776	+	0.999842i	$0.505659\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	12.5109i		−	0.453519i	−0.973951	−	0.226759i	$-0.927187\pi$
$761$		−	12.5109i		−	0.453519i	0.973951	−	0.226759i	$-0.0728131\pi$
$762$		0			0
$763$	−5.48913			−0.198720
$764$		0			0
$765$		0			0
$766$		0			0
$767$	23.7432	+	20.7446i	0.857317	+	0.749043i
$768$		0			0
$769$		−	0.634508i		−	0.0228810i	−0.999935	−	0.0114405i	$-0.996358\pi$
$769$		−	0.634508i		−	0.0228810i	0.999935	−	0.0114405i	$-0.00364170\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			52.9783i			1.90550i	0.303763	+	0.952748i	$0.401757\pi$
$773$			52.9783i			1.90550i	−0.303763	+	0.952748i	$0.598243\pi$
$774$		0			0
$775$			1.01082i			0.0363096i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.38712			0.0496989
$780$		0			0
$781$	−5.64947			−0.202154
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	15.4891i		−	0.552831i
$786$		0			0
$787$		−	2.39794i		−	0.0854773i	−0.999086	−	0.0427387i	$-0.986392\pi$
$787$		−	2.39794i		−	0.0854773i	0.999086	−	0.0427387i	$-0.0136083\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			14.7446i			0.524256i
$792$		0			0
$793$	−19.2554	+	22.0388i	−0.683781	+	0.782621i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−29.4908			−1.04462			−0.522309	−	0.852756i	$-0.674929\pi$
$797$	−29.4908			−1.04462			−0.522309	+	0.852756i	$0.674929\pi$
$798$		0			0
$799$		−	32.5823i		−	1.15268i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	10.2263			0.360877
$804$		0			0
$805$	4.62772			0.163106
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.38712			0.0487687			0.0243843	−	0.999703i	$-0.492237\pi$
$809$	1.38712			0.0487687			0.0243843	+	0.999703i	$0.492237\pi$
$810$		0			0
$811$		−	36.2493i		−	1.27288i	−0.771324	−	0.636442i	$-0.780405\pi$
$811$		−	36.2493i		−	1.27288i	0.771324	−	0.636442i	$-0.219595\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.71519			0.0951091
$816$		0			0
$817$		−	4.04326i		−	0.141456i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	12.3505i		−	0.431037i	−0.976500	−	0.215518i	$-0.930856\pi$
$821$		−	12.3505i		−	0.431037i	0.976500	−	0.215518i	$-0.0691441\pi$
$822$		0			0
$823$	−36.2337			−1.26303			−0.631513	−	0.775365i	$-0.717566\pi$
$823$	−36.2337			−1.26303			−0.631513	+	0.775365i	$0.717566\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	43.7228i		−	1.52039i	−0.649694	−	0.760196i	$-0.725103\pi$
$827$		−	43.7228i		−	1.52039i	0.649694	−	0.760196i	$-0.274897\pi$
$828$		0			0
$829$	−19.4891			−0.676885			−0.338443	−	0.940987i	$-0.609900\pi$
$829$	−19.4891			−0.676885			−0.338443	+	0.940987i	$0.609900\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.38712			−0.0480610
$834$		0			0
$835$	−6.00000			−0.207639
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	13.3723i		−	0.461662i	−0.972994	−	0.230831i	$-0.925855\pi$
$839$		−	13.3723i		−	0.461662i	0.972994	−	0.230831i	$-0.0741445\pi$
$840$		0			0
$841$	0.489125			0.0168664
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.8824	−	1.74456i	0.443168	−	0.0600148i
$846$		0			0
$847$		−	24.7540i		−	0.850558i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	6.35053i		−	0.217693i
$852$		0			0
$853$			43.1259i			1.47660i	0.674472	+	0.738301i	$0.264371\pi$
$853$			43.1259i			1.47660i	−0.674472	+	0.738301i	$0.735629\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	41.7387			1.42577			0.712884	−	0.701282i	$-0.247389\pi$
$857$	41.7387			1.42577			0.712884	+	0.701282i	$0.247389\pi$
$858$		0			0
$859$	34.3505			1.17203			0.586013	−	0.810302i	$-0.300697\pi$
$859$	34.3505			1.17203			0.586013	+	0.810302i	$0.300697\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			0.510875i			0.0173904i	0.999962	+	0.00869519i	$0.00276780\pi$
$863$			0.510875i			0.0173904i	−0.999962	+	0.00869519i	$0.997232\pi$
$864$		0			0
$865$			7.45202i			0.253376i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			4.62772i			0.156985i
$870$		0			0
$871$	17.4891	+	15.2804i	0.592596	+	0.517755i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.71519			−0.0917903
$876$		0			0
$877$			20.3344i			0.686645i	0.939218	+	0.343322i	$0.111552\pi$
$877$			20.3344i			0.686645i	−0.939218	+	0.343322i	$0.888448\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−6.81751			−0.229688			−0.114844	−	0.993384i	$-0.536637\pi$
$881$	−6.81751			−0.229688			−0.114844	+	0.993384i	$0.536637\pi$
$882$		0			0
$883$	−4.23369			−0.142475			−0.0712375	−	0.997459i	$-0.522695\pi$
$883$	−4.23369			−0.142475			−0.0712375	+	0.997459i	$0.522695\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−37.6955			−1.26569			−0.632845	−	0.774279i	$-0.718113\pi$
$887$	−37.6955			−1.26569			−0.632845	+	0.774279i	$0.718113\pi$
$888$		0			0
$889$			4.04326i			0.135607i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	8.83915			0.295791
$894$		0			0
$895$			25.7648i			0.861223i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	5.48913i		−	0.183073i
$900$		0			0
$901$	34.1168			1.13660
$902$		0			0
$903$		0			0
$904$		0			0
$905$			2.86141i			0.0951164i
$906$		0			0
$907$	−13.4891			−0.447899			−0.223950	−	0.974601i	$-0.571895\pi$
$907$	−13.4891			−0.447899			−0.223950	+	0.974601i	$0.571895\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	9.47365			0.313876			0.156938	−	0.987608i	$-0.449838\pi$
$911$	9.47365			0.313876			0.156938	+	0.987608i	$0.449838\pi$
$912$		0			0
$913$	−15.7663			−0.521789
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	49.7228i		−	1.64199i
$918$		0			0
$919$	7.60597			0.250898			0.125449	−	0.992100i	$-0.459963\pi$
$919$	7.60597			0.250898			0.125449	+	0.992100i	$0.459963\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	11.1780	+	9.76631i	0.367929	+	0.321462i
$924$		0			0
$925$			3.72601i			0.122510i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			31.8832i			1.04605i	0.852317	+	0.523026i	$0.175197\pi$
$929$			31.8832i			1.04605i	−0.852317	+	0.523026i	$0.824803\pi$
$930$		0			0
$931$		−	0.376308i		−	0.0123330i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5.11313			0.167217
$936$		0			0
$937$	−10.2337			−0.334320			−0.167160	−	0.985930i	$-0.553460\pi$
$937$	−10.2337			−0.334320			−0.167160	+	0.985930i	$0.553460\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			0.350532i			0.0114270i	0.999984	+	0.00571351i	$0.00181868\pi$
$941$			0.350532i			0.0114270i	−0.999984	+	0.00571351i	$0.998181\pi$
$942$		0			0
$943$		−	2.33889i		−	0.0761646i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			36.5109i			1.18644i	0.805039	+	0.593222i	$0.202144\pi$
$947$			36.5109i			1.18644i	−0.805039	+	0.593222i	$0.797856\pi$
$948$		0			0
$949$	−20.2337	−	17.6783i	−0.656813	−	0.573862i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	45.1475			1.46247			0.731235	−	0.682125i	$-0.238944\pi$
$953$	45.1475			1.46247			0.731235	+	0.682125i	$0.238944\pi$
$954$		0			0
$955$			5.43039i			0.175723i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−23.7432			−0.766708
$960$		0			0
$961$	29.9783			0.967040
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.70438			0.0548659
$966$		0			0
$967$			4.41957i			0.142124i	0.997472	+	0.0710619i	$0.0226388\pi$
$967$			4.41957i			0.142124i	−0.997472	+	0.0710619i	$0.977361\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	46.0993			1.47940			0.739698	−	0.672939i	$-0.234968\pi$
$971$	46.0993			1.47940			0.739698	+	0.672939i	$0.234968\pi$
$972$		0			0
$973$		−	9.15640i		−	0.293541i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			30.0000i			0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$			30.0000i			0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$		0			0
$979$	−1.88316			−0.0601859
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	39.2554i		−	1.25205i	−0.779801	−	0.626027i	$-0.784680\pi$
$983$		−	39.2554i		−	1.25205i	0.779801	−	0.626027i	$-0.215320\pi$
$984$		0			0
$985$	−6.00000			−0.191176
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−6.81751			−0.216784
$990$		0			0
$991$	9.88316			0.313949			0.156974	−	0.987603i	$-0.449826\pi$
$991$	9.88316			0.313949			0.156974	+	0.987603i	$0.449826\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	8.00000i		−	0.253617i
$996$		0			0
$997$	−4.51087			−0.142861			−0.0714304	−	0.997446i	$-0.522756\pi$
$997$	−4.51087			−0.142861			−0.0714304	+	0.997446i	$0.522756\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.c.e.181.5	yes	8
3.2	odd	2	inner	2340.2.c.e.181.1	✓	8
13.12	even	2	inner	2340.2.c.e.181.4	yes	8
39.38	odd	2	inner	2340.2.c.e.181.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.c.e.181.1	✓	8	3.2	odd	2	inner
2340.2.c.e.181.4	yes	8	13.12	even	2	inner
2340.2.c.e.181.5	yes	8	1.1	even	1	trivial
2340.2.c.e.181.8	yes	8	39.38	odd	2	inner

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:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2340.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} -2.71519i q^{7} -1.37228i q^{11} +(-2.37228 + 2.71519i) q^{13} +3.72601 q^{17} +1.01082i q^{19} +1.70438 q^{23} -1.00000 q^{25} +5.43039 q^{29} -1.01082i q^{31} +2.71519 q^{35} -3.72601i q^{37} -1.37228i q^{41} -4.00000 q^{43} -8.74456i q^{47} -0.372281 q^{49} +9.15640 q^{53} +1.37228 q^{55} -8.74456i q^{59} +8.11684 q^{61} +(-2.71519 - 2.37228i) q^{65} -6.44121i q^{67} -4.11684i q^{71} +7.45202i q^{73} -3.72601 q^{77} -3.37228 q^{79} -11.4891i q^{83} +3.72601i q^{85} -1.37228i q^{89} +(7.37228 + 6.44121i) q^{91} -1.01082 q^{95} -1.70438i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{13} - 8 q^{25} - 32 q^{43} + 20 q^{49} - 12 q^{55} - 4 q^{61} - 4 q^{79} + 36 q^{91}+O(q^{100}) \) 8 * q + 4 * q^13 - 8 * q^25 - 32 * q^43 + 20 * q^49 - 12 * q^55 - 4 * q^61 - 4 * q^79 + 36 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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