LMFDB - Embedded newform 3060.2.e.j.1801.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,16] 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:	$6$
Coefficient field:	6.0.37161216.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 15x^{4} + 51x^{2} + 1 $ x^6 + 15x^4 + 51x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 340)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1801.6
Root			$2.27307i$ of defining polynomial
Character	$\chi$	$=$	3060.1801
Dual form			3060.2.e.j.1801.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} +1.27307i q^{7} -5.43993i q^{11} -0.166860 q^{13} +(-2.27307 + 3.43993i) q^{17} -2.54615 q^{19} +3.27307i q^{23} -1.00000 q^{25} -6.54615i q^{29} +5.98608i q^{31} -1.27307 q^{35} +4.00000i q^{37} +8.54615i q^{41} +2.71301 q^{43} -10.7130 q^{47} +5.37929 q^{49} -12.8799 q^{53} +5.43993 q^{55} +6.21243 q^{59} +7.42601i q^{61} -0.166860i q^{65} -6.37929 q^{67} -2.89379i q^{71} +12.5461i q^{73} +6.92543 q^{77} -3.77365i q^{79} -13.5929 q^{83} +(-3.43993 - 2.27307i) q^{85} +5.83314 q^{89} -0.212425i q^{91} -2.54615i q^{95} +1.45385i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{17} + 16 q^{19} - 6 q^{25} + 8 q^{35} - 16 q^{43} - 32 q^{47} + 2 q^{49} - 44 q^{53} + 16 q^{55} + 8 q^{59} - 8 q^{67} - 20 q^{77} - 16 q^{83} - 4 q^{85} + 36 q^{89}+O(q^{100}) $ 6 * q + 2 * q^17 + 16 * q^19 - 6 * q^25 + 8 * q^35 - 16 * q^43 - 32 * q^47 + 2 * q^49 - 44 * q^53 + 16 * q^55 + 8 * q^59 - 8 * q^67 - 20 * q^77 - 16 * q^83 - 4 * q^85 + 36 * q^89

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$			1.27307i			0.481176i	0.970627	+	0.240588i	$0.0773403\pi$
$7$			1.27307i			0.481176i	−0.970627	+	0.240588i	$0.922660\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	5.43993i		−	1.64020i	−0.572219	−	0.820101i	$-0.693917\pi$
$11$		−	5.43993i		−	1.64020i	0.572219	−	0.820101i	$-0.306083\pi$
$12$		0			0
$13$	−0.166860			−0.0462787			−0.0231394	−	0.999732i	$-0.507366\pi$
$13$	−0.166860			−0.0462787			−0.0231394	+	0.999732i	$0.507366\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.27307	+	3.43993i	−0.551301	+	0.834306i
$17$	−2.27307	+	3.43993i	−0.551301	+	0.834306i
$18$		0			0
$19$	−2.54615			−0.584126			−0.292063	−	0.956399i	$-0.594342\pi$
$19$	−2.54615			−0.584126			−0.292063	+	0.956399i	$0.594342\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.27307i			0.682483i	0.939976	+	0.341241i	$0.110847\pi$
$23$			3.27307i			0.682483i	−0.939976	+	0.341241i	$0.889153\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	6.54615i		−	1.21559i	−0.794094	−	0.607794i	$-0.792054\pi$
$29$		−	6.54615i		−	1.21559i	0.794094	−	0.607794i	$-0.207946\pi$
$30$		0			0
$31$			5.98608i			1.07513i	0.843222	+	0.537566i	$0.180656\pi$
$31$			5.98608i			1.07513i	−0.843222	+	0.537566i	$0.819344\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.27307			−0.215189
$36$		0			0
$37$			4.00000i			0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$			4.00000i			0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			8.54615i			1.33468i	0.744752	+	0.667342i	$0.232568\pi$
$41$			8.54615i			1.33468i	−0.744752	+	0.667342i	$0.767432\pi$
$42$		0			0
$43$	2.71301			0.413730			0.206865	−	0.978370i	$-0.433674\pi$
$43$	2.71301			0.413730			0.206865	+	0.978370i	$0.433674\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−10.7130			−1.56265			−0.781326	−	0.624123i	$-0.785456\pi$
$47$	−10.7130			−1.56265			−0.781326	+	0.624123i	$0.785456\pi$
$48$		0			0
$49$	5.37929			0.768469
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−12.8799			−1.76919			−0.884593	−	0.466364i	$-0.845564\pi$
$53$	−12.8799			−1.76919			−0.884593	+	0.466364i	$0.845564\pi$
$54$		0			0
$55$	5.43993			0.733520
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.21243			0.808789			0.404394	−	0.914585i	$-0.367482\pi$
$59$	6.21243			0.808789			0.404394	+	0.914585i	$0.367482\pi$
$60$		0			0
$61$			7.42601i			0.950803i	0.879769	+	0.475402i	$0.157697\pi$
$61$			7.42601i			0.950803i	−0.879769	+	0.475402i	$0.842303\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	0.166860i		−	0.0206965i
$66$		0			0
$67$	−6.37929			−0.779354			−0.389677	−	0.920952i	$-0.627413\pi$
$67$	−6.37929			−0.779354			−0.389677	+	0.920952i	$0.627413\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	2.89379i		−	0.343429i	−0.985147	−	0.171715i	$-0.945069\pi$
$71$		−	2.89379i		−	0.343429i	0.985147	−	0.171715i	$-0.0549307\pi$
$72$		0			0
$73$			12.5461i			1.46842i	0.678925	+	0.734208i	$0.262446\pi$
$73$			12.5461i			1.46842i	−0.678925	+	0.734208i	$0.737554\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.92543			0.789226
$78$		0			0
$79$		−	3.77365i		−	0.424569i	−0.977208	−	0.212285i	$-0.931910\pi$
$79$		−	3.77365i		−	0.424569i	0.977208	−	0.212285i	$-0.0680904\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−13.5929			−1.49201			−0.746006	−	0.665939i	$-0.768031\pi$
$83$	−13.5929			−1.49201			−0.746006	+	0.665939i	$0.768031\pi$
$84$		0			0
$85$	−3.43993	−	2.27307i	−0.373113	−	0.246549i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.83314			0.618312			0.309156	−	0.951011i	$-0.399954\pi$
$89$	5.83314			0.618312			0.309156	+	0.951011i	$0.399954\pi$
$90$		0			0
$91$		−	0.212425i		−	0.0222682i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	2.54615i		−	0.261229i
$96$		0			0
$97$			1.45385i			0.147617i	0.997272	+	0.0738083i	$0.0235153\pi$
$97$			1.45385i			0.147617i	−0.997272	+	0.0738083i	$0.976485\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−11.5929			−1.15353			−0.576767	−	0.816909i	$-0.695686\pi$
$101$	−11.5929			−1.15353			−0.576767	+	0.816909i	$0.695686\pi$
$102$		0			0
$103$	8.92543			0.879449			0.439724	−	0.898133i	$-0.355076\pi$
$103$	8.92543			0.879449			0.439724	+	0.898133i	$0.355076\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1.84706i		−	0.178562i	−0.996006	−	0.0892811i	$-0.971543\pi$
$107$		−	1.84706i		−	0.178562i	0.996006	−	0.0892811i	$-0.0284569\pi$
$108$		0			0
$109$		−	10.0000i		−	0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$		−	10.0000i		−	0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			17.9722i			1.69068i	0.534230	+	0.845339i	$0.320602\pi$
$113$			17.9722i			1.69068i	−0.534230	+	0.845339i	$0.679398\pi$
$114$		0			0
$115$	−3.27307			−0.305216
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.37929	−	2.89379i	−0.401448	−	0.265273i
$120$		0			0
$121$	−18.5929			−1.69026
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	−11.8053			−1.04755			−0.523775	−	0.851856i	$-0.675477\pi$
$127$	−11.8053			−1.04755			−0.523775	+	0.851856i	$0.675477\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			4.56007i			0.398415i	0.979957	+	0.199207i	$0.0638368\pi$
$131$			4.56007i			0.398415i	−0.979957	+	0.199207i	$0.936163\pi$
$132$		0			0
$133$		−	3.24143i		−	0.281068i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.16686			0.697742			0.348871	−	0.937171i	$-0.386565\pi$
$137$	8.16686			0.697742			0.348871	+	0.937171i	$0.386565\pi$
$138$		0			0
$139$		−	8.31980i		−	0.705676i	−0.935684	−	0.352838i	$-0.885217\pi$
$139$		−	8.31980i		−	0.705676i	0.935684	−	0.352838i	$-0.114783\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			0.907709i			0.0759064i
$144$		0			0
$145$	6.54615			0.543628
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−21.9722			−1.80003			−0.900015	−	0.435860i	$-0.856444\pi$
$149$	−21.9722			−1.80003			−0.900015	+	0.435860i	$0.856444\pi$
$150$		0			0
$151$	−10.8799			−0.885391			−0.442695	−	0.896672i	$-0.645978\pi$
$151$	−10.8799			−0.885391			−0.442695	+	0.896672i	$0.645978\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.98608			−0.480813
$156$		0			0
$157$	−13.9722			−1.11510			−0.557550	−	0.830144i	$-0.688258\pi$
$157$	−13.9722			−1.11510			−0.557550	+	0.830144i	$0.688258\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.16686			−0.328395
$162$		0			0
$163$		−	8.93935i		−	0.700184i	−0.936715	−	0.350092i	$-0.886150\pi$
$163$		−	8.93935i		−	0.700184i	0.936715	−	0.350092i	$-0.113850\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2.39321i			0.185192i	0.995704	+	0.0925959i	$0.0295165\pi$
$167$			2.39321i			0.185192i	−0.995704	+	0.0925959i	$0.970484\pi$
$168$		0			0
$169$	−12.9722			−0.997858
$170$		0			0
$171$		0			0
$172$		0			0
$173$			1.42601i			0.108418i	0.998530	+	0.0542088i	$0.0172637\pi$
$173$			1.42601i			0.108418i	−0.998530	+	0.0542088i	$0.982736\pi$
$174$		0			0
$175$		−	1.27307i		−	0.0962353i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−13.0923			−0.978564			−0.489282	−	0.872126i	$-0.662741\pi$
$179$	−13.0923			−0.978564			−0.489282	+	0.872126i	$0.662741\pi$
$180$		0			0
$181$		−	19.9722i		−	1.48452i	−0.670113	−	0.742259i	$-0.733754\pi$
$181$		−	19.9722i		−	1.48452i	0.670113	−	0.742259i	$-0.266246\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.00000			−0.294086
$186$		0			0
$187$	18.7130	+	12.3654i	1.36843	+	0.904245i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	20.3059			1.46928			0.734641	−	0.678456i	$-0.237351\pi$
$191$	20.3059			1.46928			0.734641	+	0.678456i	$0.237351\pi$
$192$		0			0
$193$		−	12.2124i		−	0.879070i	−0.898226	−	0.439535i	$-0.855143\pi$
$193$		−	12.2124i		−	0.879070i	0.898226	−	0.439535i	$-0.144857\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$			13.3186i			0.944133i	0.881563	+	0.472067i	$0.156492\pi$
$199$			13.3186i			0.944133i	−0.881563	+	0.472067i	$0.843508\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.33372			0.584913
$204$		0			0
$205$	−8.54615			−0.596889
$206$		0			0
$207$		0			0
$208$		0			0
$209$			13.8509i			0.958084i
$210$		0			0
$211$			20.8659i			1.43647i	0.695800	+	0.718235i	$0.255050\pi$
$211$			20.8659i			1.43647i	−0.695800	+	0.718235i	$0.744950\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			2.71301i			0.185025i
$216$		0			0
$217$	−7.62071			−0.517328
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.379285	−	0.573988i	0.0255135	−	0.0386106i
$222$		0			0
$223$	13.2592			0.887898			0.443949	−	0.896052i	$-0.353577\pi$
$223$	13.2592			0.887898			0.443949	+	0.896052i	$0.353577\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			9.27307i			0.615475i	0.951471	+	0.307738i	$0.0995719\pi$
$227$			9.27307i			0.615475i	−0.951471	+	0.307738i	$0.900428\pi$
$228$		0			0
$229$	−14.1669			−0.936172			−0.468086	−	0.883683i	$-0.655056\pi$
$229$	−14.1669			−0.936172			−0.468086	+	0.883683i	$0.655056\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			29.0644i			1.90408i	0.305981	+	0.952038i	$0.401016\pi$
$233$			29.0644i			1.90408i	−0.305981	+	0.952038i	$0.598984\pi$
$234$		0			0
$235$		−	10.7130i		−	0.698839i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5.42601			0.350980			0.175490	−	0.984481i	$-0.443849\pi$
$239$	5.42601			0.350980			0.175490	+	0.984481i	$0.443849\pi$
$240$		0			0
$241$			20.2124i			1.30200i	0.759079	+	0.650998i	$0.225650\pi$
$241$			20.2124i			1.30200i	−0.759079	+	0.650998i	$0.774350\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			5.37929i			0.343670i
$246$		0			0
$247$	0.424850			0.0270326
$248$		0			0
$249$		0			0
$250$		0			0
$251$	19.3047			1.21850			0.609251	−	0.792977i	$-0.291470\pi$
$251$	19.3047			1.21850			0.609251	+	0.792977i	$0.291470\pi$
$252$		0			0
$253$	17.8053			1.11941
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.59287			−0.0993606			−0.0496803	−	0.998765i	$-0.515820\pi$
$257$	−1.59287			−0.0993606			−0.0496803	+	0.998765i	$0.515820\pi$
$258$		0			0
$259$	−5.09229			−0.316420
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.37929			0.146713			0.0733565	−	0.997306i	$-0.476629\pi$
$263$	2.37929			0.146713			0.0733565	+	0.997306i	$0.476629\pi$
$264$		0			0
$265$		−	12.8799i		−	0.791204i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			18.4249i			1.12338i	0.827347	+	0.561691i	$0.189849\pi$
$269$			18.4249i			1.12338i	−0.827347	+	0.561691i	$0.810151\pi$
$270$		0			0
$271$	25.0923			1.52425			0.762124	−	0.647431i	$-0.224157\pi$
$271$	25.0923			1.52425			0.762124	+	0.647431i	$0.224157\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			5.43993i			0.328040i
$276$		0			0
$277$		−	12.1846i		−	0.732101i	−0.930595	−	0.366050i	$-0.880710\pi$
$277$		−	12.1846i		−	0.732101i	0.930595	−	0.366050i	$-0.119290\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.21243			−0.251292			−0.125646	−	0.992075i	$-0.540100\pi$
$281$	−4.21243			−0.251292			−0.125646	+	0.992075i	$0.540100\pi$
$282$		0			0
$283$			16.4867i			0.980030i	0.871714	+	0.490015i	$0.163009\pi$
$283$			16.4867i			0.980030i	−0.871714	+	0.490015i	$0.836991\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.8799			−0.642218
$288$		0			0
$289$	−6.66628	−	15.6384i	−0.392134	−	0.919908i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.57399			0.267215			0.133608	−	0.991034i	$-0.457344\pi$
$293$	4.57399			0.267215			0.133608	+	0.991034i	$0.457344\pi$
$294$		0			0
$295$			6.21243i			0.361701i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	0.546146i		−	0.0315844i
$300$		0			0
$301$			3.45385i			0.199077i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−7.42601			−0.425212
$306$		0			0
$307$	15.0189			0.857173			0.428586	−	0.903501i	$-0.359012\pi$
$307$	15.0189			0.857173			0.428586	+	0.903501i	$0.359012\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	6.22635i		−	0.353064i	−0.984295	−	0.176532i	$-0.943512\pi$
$311$		−	6.22635i		−	0.353064i	0.984295	−	0.176532i	$-0.0564879\pi$
$312$		0			0
$313$			4.54615i			0.256963i	0.991712	+	0.128482i	$0.0410103\pi$
$313$			4.54615i			0.256963i	−0.991712	+	0.128482i	$0.958990\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	11.4260i		−	0.641749i	−0.947122	−	0.320874i	$-0.896023\pi$
$317$		−	11.4260i		−	0.641749i	0.947122	−	0.320874i	$-0.103977\pi$
$318$		0			0
$319$	−35.6106			−1.99381
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5.78757	−	8.75857i	0.322029	−	0.487340i
$324$		0			0
$325$	0.166860			0.00925574
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	13.6384i		−	0.751911i
$330$		0			0
$331$	3.63844			0.199987			0.0999933	−	0.994988i	$-0.468118\pi$
$331$	3.63844			0.199987			0.0999933	+	0.994988i	$0.468118\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	6.37929i		−	0.348538i
$336$		0			0
$337$			7.97216i			0.434271i	0.976141	+	0.217136i	$0.0696714\pi$
$337$			7.97216i			0.434271i	−0.976141	+	0.217136i	$0.930329\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	32.5639			1.76343
$342$		0			0
$343$			15.7597i			0.850946i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	0.152939i		−	0.00821020i	−0.999992	−	0.00410510i	$-0.998693\pi$
$347$		−	0.152939i		−	0.00821020i	0.999992	−	0.00410510i	$-0.00130670\pi$
$348$		0			0
$349$	−4.21243			−0.225486			−0.112743	−	0.993624i	$-0.535964\pi$
$349$	−4.21243			−0.225486			−0.112743	+	0.993624i	$0.535964\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.66744			−0.354872			−0.177436	−	0.984132i	$-0.556780\pi$
$353$	−6.66744			−0.354872			−0.177436	+	0.984132i	$0.556780\pi$
$354$		0			0
$355$	2.89379			0.153586
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.51830			0.132911			0.0664555	−	0.997789i	$-0.478831\pi$
$359$	2.51830			0.132911			0.0664555	+	0.997789i	$0.478831\pi$
$360$		0			0
$361$	−12.5171			−0.658797
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−12.5461			−0.656695
$366$		0			0
$367$		−	19.1518i		−	0.999715i	−0.866108	−	0.499857i	$-0.833386\pi$
$367$		−	19.1518i		−	0.999715i	0.866108	−	0.499857i	$-0.166614\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	16.3970i		−	0.851290i
$372$		0			0
$373$	24.9254			1.29059			0.645295	−	0.763934i	$-0.276734\pi$
$373$	24.9254			1.29059			0.645295	+	0.763934i	$0.276734\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			1.09229i			0.0562559i
$378$		0			0
$379$		−	7.41209i		−	0.380734i	−0.981713	−	0.190367i	$-0.939032\pi$
$379$		−	7.41209i		−	0.380734i	0.981713	−	0.190367i	$-0.0609677\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−30.0177			−1.53383			−0.766917	−	0.641746i	$-0.778210\pi$
$383$	−30.0177			−1.53383			−0.766917	+	0.641746i	$0.778210\pi$
$384$		0			0
$385$			6.92543i			0.352953i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−24.6573			−1.25018			−0.625088	−	0.780554i	$-0.714937\pi$
$389$	−24.6573			−1.25018			−0.625088	+	0.780554i	$0.714937\pi$
$390$		0			0
$391$	−11.2592	−	7.43993i	−0.569400	−	0.376254i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.77365			0.189873
$396$		0			0
$397$			16.8520i			0.845779i	0.906181	+	0.422889i	$0.138984\pi$
$397$			16.8520i			0.845779i	−0.906181	+	0.422889i	$0.861016\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			22.5183i			1.12451i	0.826964	+	0.562255i	$0.190066\pi$
$401$			22.5183i			1.12451i	−0.826964	+	0.562255i	$0.809934\pi$
$402$		0			0
$403$		−	0.998839i		−	0.0497557i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	21.7597			1.07859
$408$		0			0
$409$	14.0000			0.692255			0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000			0.692255			0.346128	+	0.938187i	$0.387496\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			7.90887i			0.389170i
$414$		0			0
$415$		−	13.5929i		−	0.667248i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	35.4399i		−	1.73135i	−0.500603	−	0.865677i	$-0.666888\pi$
$419$		−	35.4399i		−	1.73135i	0.500603	−	0.865677i	$-0.333112\pi$
$420$		0			0
$421$	−27.2313			−1.32717			−0.663586	−	0.748100i	$-0.730966\pi$
$421$	−27.2313			−1.32717			−0.663586	+	0.748100i	$0.730966\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.27307	−	3.43993i	0.110260	−	0.166861i
$426$		0			0
$427$	−9.45385			−0.457504
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	8.56007i		−	0.412324i	−0.978518	−	0.206162i	$-0.933903\pi$
$431$		−	8.56007i		−	0.412324i	0.978518	−	0.206162i	$-0.0660974\pi$
$432$		0			0
$433$	6.68516			0.321268			0.160634	−	0.987014i	$-0.448646\pi$
$433$	6.68516			0.321268			0.160634	+	0.987014i	$0.448646\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	8.33372i		−	0.398656i
$438$		0			0
$439$		−	35.2908i		−	1.68434i	−0.539214	−	0.842169i	$-0.681279\pi$
$439$		−	35.2908i		−	1.68434i	0.539214	−	0.842169i	$-0.318721\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	27.7775			1.31975			0.659873	−	0.751377i	$-0.270610\pi$
$443$	27.7775			1.31975			0.659873	+	0.751377i	$0.270610\pi$
$444$		0			0
$445$			5.83314i			0.276517i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	6.30588i		−	0.297593i	−0.988868	−	0.148796i	$-0.952460\pi$
$449$		−	6.30588i		−	0.297593i	0.988868	−	0.148796i	$-0.0475399\pi$
$450$		0			0
$451$	46.4905			2.18915
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.212425			0.00995865
$456$		0			0
$457$	15.1657			0.709421			0.354711	−	0.934976i	$-0.384579\pi$
$457$	15.1657			0.709421			0.354711	+	0.934976i	$0.384579\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−29.3047			−1.36486			−0.682428	−	0.730952i	$-0.739076\pi$
$461$	−29.3047			−1.36486			−0.682428	+	0.730952i	$0.739076\pi$
$462$		0			0
$463$	−17.2592			−0.802101			−0.401050	−	0.916056i	$-0.631355\pi$
$463$	−17.2592			−0.802101			−0.401050	+	0.916056i	$0.631355\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	16.8065			0.777710			0.388855	−	0.921299i	$-0.372871\pi$
$467$	16.8065			0.777710			0.388855	+	0.921299i	$0.372871\pi$
$468$		0			0
$469$		−	8.12130i		−	0.375007i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	14.7586i		−	0.678600i
$474$		0			0
$475$	2.54615			0.116825
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	12.5044i		−	0.571340i	−0.958328	−	0.285670i	$-0.907784\pi$
$479$		−	12.5044i		−	0.571340i	0.958328	−	0.285670i	$-0.0922161\pi$
$480$		0			0
$481$		−	0.667441i		−	0.0304327i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.45385			−0.0660161
$486$		0			0
$487$		−	24.4867i		−	1.10960i	−0.831985	−	0.554798i	$-0.812795\pi$
$487$		−	24.4867i		−	1.10960i	0.831985	−	0.554798i	$-0.187205\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	14.0935			0.636029			0.318014	−	0.948086i	$-0.396984\pi$
$491$	14.0935			0.636029			0.318014	+	0.948086i	$0.396984\pi$
$492$		0			0
$493$	22.5183	+	14.8799i	1.01417	+	0.670155i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3.68400			0.165250
$498$		0			0
$499$		−	5.89495i		−	0.263894i	−0.991257	−	0.131947i	$-0.957877\pi$
$499$		−	5.89495i		−	0.263894i	0.991257	−	0.131947i	$-0.0421229\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	15.0328i		−	0.670280i	−0.942168	−	0.335140i	$-0.891216\pi$
$503$		−	15.0328i		−	0.670280i	0.942168	−	0.335140i	$-0.108784\pi$
$504$		0			0
$505$		−	11.5929i		−	0.515876i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−14.4249			−0.639370			−0.319685	−	0.947524i	$-0.603577\pi$
$509$	−14.4249			−0.639370			−0.319685	+	0.947524i	$0.603577\pi$
$510$		0			0
$511$	−15.9722			−0.706567
$512$		0			0
$513$		0			0
$514$		0			0
$515$			8.92543i			0.393301i
$516$		0			0
$517$			58.2780i			2.56307i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	9.42601i		−	0.412961i	−0.978451	−	0.206481i	$-0.933799\pi$
$521$		−	9.42601i		−	0.412961i	0.978451	−	0.206481i	$-0.0662010\pi$
$522$		0			0
$523$	19.7142			0.862040			0.431020	−	0.902342i	$-0.358154\pi$
$523$	19.7142			0.862040			0.431020	+	0.902342i	$0.358154\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−20.5917	−	13.6068i	−0.896989	−	0.592721i
$528$		0			0
$529$	12.2870			0.534217
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	1.42601i		−	0.0617674i
$534$		0			0
$535$	1.84706			0.0798554
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	29.2630i		−	1.26044i
$540$		0			0
$541$			15.2136i			0.654083i	0.945010	+	0.327042i	$0.106052\pi$
$541$			15.2136i			0.654083i	−0.945010	+	0.327042i	$0.893948\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10.0000			0.428353
$546$		0			0
$547$		−	27.5790i		−	1.17919i	−0.807699	−	0.589595i	$-0.799287\pi$
$547$		−	27.5790i		−	1.17919i	0.807699	−	0.589595i	$-0.200713\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			16.6674i			0.710057i
$552$		0			0
$553$	4.80414			0.204293
$554$		0			0
$555$		0			0
$556$		0			0
$557$	36.1112			1.53008			0.765040	−	0.643983i	$-0.222719\pi$
$557$	36.1112			1.53008			0.765040	+	0.643983i	$0.222719\pi$
$558$		0			0
$559$	−0.452693			−0.0191469
$560$		0			0
$561$		0			0
$562$		0			0
$563$	24.1112			1.01616			0.508082	−	0.861308i	$-0.330355\pi$
$563$	24.1112			1.01616			0.508082	+	0.861308i	$0.330355\pi$
$564$		0			0
$565$	−17.9722			−0.756094
$566$		0			0
$567$		0			0
$568$		0			0
$569$	23.7597			0.996060			0.498030	−	0.867160i	$-0.334057\pi$
$569$	23.7597			0.996060			0.498030	+	0.867160i	$0.334057\pi$
$570$		0			0
$571$			30.2920i			1.26768i	0.773465	+	0.633839i	$0.218522\pi$
$571$			30.2920i			1.26768i	−0.773465	+	0.633839i	$0.781478\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	3.27307i		−	0.136497i
$576$		0			0
$577$	10.6852			0.444829			0.222415	−	0.974952i	$-0.428606\pi$
$577$	10.6852			0.444829			0.222415	+	0.974952i	$0.428606\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	17.3047i		−	0.717921i
$582$		0			0
$583$			70.0656i			2.90182i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.64856			−0.0680432			−0.0340216	−	0.999421i	$-0.510832\pi$
$587$	−1.64856			−0.0680432			−0.0340216	+	0.999421i	$0.510832\pi$
$588$		0			0
$589$		−	15.2414i		−	0.628012i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.21359			0.0498360			0.0249180	−	0.999689i	$-0.492068\pi$
$593$	1.21359			0.0498360			0.0249180	+	0.999689i	$0.492068\pi$
$594$		0			0
$595$	2.89379	−	4.37929i	0.118634	−	0.179533i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	8.36156			0.341644			0.170822	−	0.985302i	$-0.445358\pi$
$599$	8.36156			0.341644			0.170822	+	0.985302i	$0.445358\pi$
$600$		0			0
$601$			7.06445i			0.288165i	0.989566	+	0.144082i	$0.0460231\pi$
$601$			7.06445i			0.288165i	−0.989566	+	0.144082i	$0.953977\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	18.5929i		−	0.755908i
$606$		0			0
$607$			22.9115i			0.929950i	0.885324	+	0.464975i	$0.153937\pi$
$607$			22.9115i			0.929950i	−0.885324	+	0.464975i	$0.846063\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.78757			0.0723175
$612$		0			0
$613$	−0.573988			−0.0231832			−0.0115916	−	0.999933i	$-0.503690\pi$
$613$	−0.573988			−0.0231832			−0.0115916	+	0.999933i	$0.503690\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	11.7319i		−	0.472308i	−0.971716	−	0.236154i	$-0.924113\pi$
$617$		−	11.7319i		−	0.472308i	0.971716	−	0.236154i	$-0.0758870\pi$
$618$		0			0
$619$		−	2.50438i		−	0.100660i	−0.998733	−	0.0503298i	$-0.983973\pi$
$619$		−	2.50438i		−	0.100660i	0.998733	−	0.0503298i	$-0.0160272\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			7.42601i			0.297517i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−13.7597	−	9.09229i	−0.548636	−	0.362533i
$630$		0			0
$631$	38.0656			1.51537			0.757684	−	0.652622i	$-0.226331\pi$
$631$	38.0656			1.51537			0.757684	+	0.652622i	$0.226331\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	11.8053i		−	0.468479i
$636$		0			0
$637$	−0.897589			−0.0355638
$638$		0			0
$639$		0			0
$640$		0			0
$641$			18.7307i			0.739819i	0.929068	+	0.369910i	$0.120611\pi$
$641$			18.7307i			0.739819i	−0.929068	+	0.369910i	$0.879389\pi$
$642$		0			0
$643$		−	8.72693i		−	0.344156i	−0.985083	−	0.172078i	$-0.944952\pi$
$643$		−	8.72693i		−	0.344156i	0.985083	−	0.172078i	$-0.0550482\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−14.6573			−0.576239			−0.288119	−	0.957594i	$-0.593030\pi$
$647$	−14.6573			−0.576239			−0.288119	+	0.957594i	$0.593030\pi$
$648$		0			0
$649$		−	33.7952i		−	1.32658i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	17.4260i		−	0.681933i	−0.940076	−	0.340966i	$-0.889246\pi$
$653$		−	17.4260i		−	0.681933i	0.940076	−	0.340966i	$-0.110754\pi$
$654$		0			0
$655$	−4.56007			−0.178177
$656$		0			0
$657$		0			0
$658$		0			0
$659$	39.8532			1.55246			0.776230	−	0.630450i	$-0.217130\pi$
$659$	39.8532			1.55246			0.776230	+	0.630450i	$0.217130\pi$
$660$		0			0
$661$	40.8242			1.58788			0.793938	−	0.607998i	$-0.208027\pi$
$661$	40.8242			1.58788			0.793938	+	0.607998i	$0.208027\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.24143			0.125697
$666$		0			0
$667$	21.4260			0.829618
$668$		0			0
$669$		0			0
$670$		0			0
$671$	40.3970			1.55951
$672$		0			0
$673$		−	25.5473i		−	0.984776i	−0.870376	−	0.492388i	$-0.836124\pi$
$673$		−	25.5473i		−	0.984776i	0.870376	−	0.492388i	$-0.163876\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			3.33488i			0.128170i	0.997944	+	0.0640850i	$0.0204129\pi$
$677$			3.33488i			0.128170i	−0.997944	+	0.0640850i	$0.979587\pi$
$678$		0			0
$679$	−1.85086			−0.0710296
$680$		0			0
$681$		0			0
$682$		0			0
$683$			46.0973i			1.76386i	0.471378	+	0.881931i	$0.343757\pi$
$683$			46.0973i			1.76386i	−0.471378	+	0.881931i	$0.656243\pi$
$684$		0			0
$685$			8.16686i			0.312040i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.14914			0.0818756
$690$		0			0
$691$		−	18.9849i		−	0.722220i	−0.932523	−	0.361110i	$-0.882398\pi$
$691$		−	18.9849i		−	0.722220i	0.932523	−	0.361110i	$-0.117602\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.31980			0.315588
$696$		0			0
$697$	−29.3982	−	19.4260i	−1.11354	−	0.735813i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−42.8976			−1.62022			−0.810110	−	0.586278i	$-0.800593\pi$
$701$	−42.8976			−1.62022			−0.810110	+	0.586278i	$0.800593\pi$
$702$		0			0
$703$		−	10.1846i		−	0.384119i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	14.7586i		−	0.555053i
$708$		0			0
$709$			25.5450i			0.959362i	0.877443	+	0.479681i	$0.159248\pi$
$709$			25.5450i			0.959362i	−0.877443	+	0.479681i	$0.840752\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−19.5929			−0.733759
$714$		0			0
$715$	−0.907709			−0.0339464
$716$		0			0
$717$		0			0
$718$		0			0
$719$			5.10621i			0.190430i	0.995457	+	0.0952148i	$0.0303538\pi$
$719$			5.10621i			0.190430i	−0.995457	+	0.0952148i	$0.969646\pi$
$720$		0			0
$721$			11.3627i			0.423170i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			6.54615i			0.243118i
$726$		0			0
$727$	23.3526			0.866100			0.433050	−	0.901370i	$-0.357437\pi$
$727$	23.3526			0.866100			0.433050	+	0.901370i	$0.357437\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−6.16686	+	9.33256i	−0.228090	+	0.345177i
$732$		0			0
$733$	−35.3703			−1.30643			−0.653216	−	0.757171i	$-0.726581\pi$
$733$	−35.3703			−1.30643			−0.653216	+	0.757171i	$0.726581\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			34.7029i			1.27830i
$738$		0			0
$739$	−22.3036			−0.820450			−0.410225	−	0.911984i	$-0.634550\pi$
$739$	−22.3036			−0.820450			−0.410225	+	0.911984i	$0.634550\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			10.9394i			0.401326i	0.979660	+	0.200663i	$0.0643096\pi$
$743$			10.9394i			0.401326i	−0.979660	+	0.200663i	$0.935690\pi$
$744$		0			0
$745$		−	21.9722i		−	0.804998i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.35144			0.0859199
$750$		0			0
$751$		−	14.9216i		−	0.544498i	−0.962227	−	0.272249i	$-0.912233\pi$
$751$		−	14.9216i		−	0.544498i	0.962227	−	0.272249i	$-0.0877674\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	10.8799i		−	0.395959i
$756$		0			0
$757$	22.2603			0.809065			0.404532	−	0.914524i	$-0.367434\pi$
$757$	22.2603			0.809065			0.404532	+	0.914524i	$0.367434\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	19.5929			0.710241			0.355121	−	0.934821i	$-0.384440\pi$
$761$	19.5929			0.710241			0.355121	+	0.934821i	$0.384440\pi$
$762$		0			0
$763$	12.7307			0.460883
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.03661			−0.0374297
$768$		0			0
$769$	−37.1100			−1.33822			−0.669111	−	0.743163i	$-0.733325\pi$
$769$	−37.1100			−1.33822			−0.669111	+	0.743163i	$0.733325\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	38.7763			1.39469			0.697343	−	0.716737i	$-0.254365\pi$
$773$	38.7763			1.39469			0.697343	+	0.716737i	$0.254365\pi$
$774$		0			0
$775$		−	5.98608i		−	0.215026i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	21.7597i		−	0.779623i
$780$		0			0
$781$	−15.7420			−0.563293
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	13.9722i		−	0.498688i
$786$		0			0
$787$		−	10.6080i		−	0.378133i	−0.981964	−	0.189066i	$-0.939454\pi$
$787$		−	10.6080i		−	0.378133i	0.981964	−	0.189066i	$-0.0605461\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−22.8799			−0.813514
$792$		0			0
$793$		−	1.23911i		−	0.0440019i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−9.66628			−0.342397			−0.171199	−	0.985237i	$-0.554764\pi$
$797$	−9.66628			−0.342397			−0.171199	+	0.985237i	$0.554764\pi$
$798$		0			0
$799$	24.3514	−	36.8520i	0.861492	−	1.30373i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	68.2502			2.40850
$804$		0			0
$805$		−	4.16686i		−	0.146863i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	42.5183i		−	1.49486i	−0.664338	−	0.747432i	$-0.731287\pi$
$809$		−	42.5183i		−	1.49486i	0.664338	−	0.747432i	$-0.268713\pi$
$810$		0			0
$811$		−	6.65120i		−	0.233555i	−0.993158	−	0.116778i	$-0.962744\pi$
$811$		−	6.65120i		−	0.233555i	0.993158	−	0.116778i	$-0.0372565\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8.93935			0.313132
$816$		0			0
$817$	−6.90771			−0.241670
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	44.7963i		−	1.56340i	−0.623653	−	0.781702i	$-0.714352\pi$
$821$		−	44.7963i		−	1.56340i	0.623653	−	0.781702i	$-0.285648\pi$
$822$		0			0
$823$		−	9.36653i		−	0.326497i	−0.986585	−	0.163248i	$-0.947803\pi$
$823$		−	9.36653i		−	0.326497i	0.986585	−	0.163248i	$-0.0521972\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	44.7902i		−	1.55751i	−0.627329	−	0.778754i	$-0.715852\pi$
$827$		−	44.7902i		−	1.55751i	0.627329	−	0.778754i	$-0.284148\pi$
$828$		0			0
$829$	36.8520			1.27992			0.639962	−	0.768407i	$-0.278950\pi$
$829$	36.8520			1.27992			0.639962	+	0.768407i	$0.278950\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−12.2275	+	18.5044i	−0.423658	+	0.641139i
$834$		0			0
$835$	−2.39321			−0.0828203
$836$		0			0
$837$		0			0
$838$		0			0
$839$			11.6245i			0.401323i	0.979661	+	0.200661i	$0.0643091\pi$
$839$			11.6245i			0.401323i	−0.979661	+	0.200661i	$0.935691\pi$
$840$		0			0
$841$	−13.8520			−0.477656
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	12.9722i		−	0.446256i
$846$		0			0
$847$		−	23.6701i		−	0.813314i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−13.0923			−0.448798
$852$		0			0
$853$			49.2792i			1.68729i	0.536903	+	0.843644i	$0.319594\pi$
$853$			49.2792i			1.68729i	−0.536903	+	0.843644i	$0.680406\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			23.6408i			0.807553i	0.914858	+	0.403776i	$0.132303\pi$
$857$			23.6408i			0.807553i	−0.914858	+	0.403776i	$0.867697\pi$
$858$		0			0
$859$	−14.0935			−0.480862			−0.240431	−	0.970666i	$-0.577289\pi$
$859$	−14.0935			−0.480862			−0.240431	+	0.970666i	$0.577289\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−4.28583			−0.145891			−0.0729457	−	0.997336i	$-0.523240\pi$
$863$	−4.28583			−0.145891			−0.0729457	+	0.997336i	$0.523240\pi$
$864$		0			0
$865$	−1.42601			−0.0484859
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−20.5284			−0.696379
$870$		0			0
$871$	1.06445			0.0360675
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.27307			0.0430377
$876$		0			0
$877$			7.85086i			0.265105i	0.991176	+	0.132552i	$0.0423173\pi$
$877$			7.85086i			0.265105i	−0.991176	+	0.132552i	$0.957683\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			5.30472i			0.178720i	0.995999	+	0.0893602i	$0.0284822\pi$
$881$			5.30472i			0.178720i	−0.995999	+	0.0893602i	$0.971518\pi$
$882$		0			0
$883$	3.74201			0.125929			0.0629643	−	0.998016i	$-0.479945\pi$
$883$	3.74201			0.125929			0.0629643	+	0.998016i	$0.479945\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			4.75477i			0.159650i	0.996809	+	0.0798248i	$0.0254361\pi$
$887$			4.75477i			0.159650i	−0.996809	+	0.0798248i	$0.974564\pi$
$888$		0			0
$889$		−	15.0290i		−	0.504057i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	27.2769			0.912786
$894$		0			0
$895$		−	13.0923i		−	0.437627i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	39.1857			1.30692
$900$		0			0
$901$	29.2769	−	44.3059i	0.975354	−	1.47604i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	19.9722			0.663897
$906$		0			0
$907$		−	34.9771i		−	1.16140i	−0.814119	−	0.580698i	$-0.802780\pi$
$907$		−	34.9771i		−	1.16140i	0.814119	−	0.580698i	$-0.197220\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			20.2920i			0.672303i	0.941808	+	0.336151i	$0.109125\pi$
$911$			20.2920i			0.672303i	−0.941808	+	0.336151i	$0.890875\pi$
$912$		0			0
$913$			73.9443i			2.44720i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−5.80530			−0.191708
$918$		0			0
$919$	−26.8242			−0.884848			−0.442424	−	0.896806i	$-0.645881\pi$
$919$	−26.8242			−0.884848			−0.442424	+	0.896806i	$0.645881\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			0.482858i			0.0158935i
$924$		0			0
$925$		−	4.00000i		−	0.131519i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	0.546146i		−	0.0179185i	−0.999960	−	0.00895923i	$-0.997148\pi$
$929$		−	0.546146i		−	0.0179185i	0.999960	−	0.00895923i	$-0.00285185\pi$
$930$		0			0
$931$	−13.6964			−0.448883
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−12.3654	+	18.7130i	−0.404391	+	0.611981i
$936$		0			0
$937$	−22.7864			−0.744400			−0.372200	−	0.928153i	$-0.621396\pi$
$937$	−22.7864			−0.744400			−0.372200	+	0.928153i	$0.621396\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			8.69528i			0.283458i	0.989905	+	0.141729i	$0.0452662\pi$
$941$			8.69528i			0.283458i	−0.989905	+	0.141729i	$0.954734\pi$
$942$		0			0
$943$	−27.9722			−0.910899
$944$		0			0
$945$		0			0
$946$		0			0
$947$			55.3665i			1.79917i	0.436745	+	0.899585i	$0.356131\pi$
$947$			55.3665i			1.79917i	−0.436745	+	0.899585i	$0.643869\pi$
$948$		0			0
$949$		−	2.09345i		−	0.0679564i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−50.4426			−1.63400			−0.816998	−	0.576641i	$-0.804363\pi$
$953$	−50.4426			−1.63400			−0.816998	+	0.576641i	$0.804363\pi$
$954$		0			0
$955$			20.3059i			0.657083i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			10.3970i			0.335737i
$960$		0			0
$961$	−4.83314			−0.155908
$962$		0			0
$963$		0			0
$964$		0			0
$965$	12.2124			0.393132
$966$		0			0
$967$	51.1379			1.64448			0.822241	−	0.569139i	$-0.192723\pi$
$967$	51.1379			1.64448			0.822241	+	0.569139i	$0.192723\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	36.1869			1.16129			0.580647	−	0.814156i	$-0.302800\pi$
$971$	36.1869			1.16129			0.580647	+	0.814156i	$0.302800\pi$
$972$		0			0
$973$	10.5917			0.339555
$974$		0			0
$975$		0			0
$976$		0			0
$977$	25.5828			0.818465			0.409232	−	0.912430i	$-0.365797\pi$
$977$	25.5828			0.818465			0.409232	+	0.912430i	$0.365797\pi$
$978$		0			0
$979$		−	31.7319i		−	1.01416i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	18.7548i		−	0.598184i	−0.954224	−	0.299092i	$-0.903316\pi$
$983$		−	18.7548i		−	0.598184i	0.954224	−	0.299092i	$-0.0966838\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			8.87987i			0.282363i
$990$		0			0
$991$		−	20.7168i		−	0.658091i	−0.944314	−	0.329046i	$-0.893273\pi$
$991$		−	20.7168i		−	0.658091i	0.944314	−	0.329046i	$-0.106727\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−13.3186			−0.422229
$996$		0			0
$997$			20.9455i			0.663350i	0.943394	+	0.331675i	$0.107614\pi$
$997$			20.9455i			0.663350i	−0.943394	+	0.331675i	$0.892386\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3060.2.e.j.1801.6		6
3.2	odd	2		340.2.c.a.101.6	yes	6
12.11	even	2		1360.2.c.e.1121.1		6
15.2	even	4		1700.2.g.c.849.6		6
15.8	even	4		1700.2.g.b.849.2		6
15.14	odd	2		1700.2.c.b.101.1		6
17.16	even	2	inner	3060.2.e.j.1801.1		6
51.38	odd	4		5780.2.a.k.1.3		3
51.47	odd	4		5780.2.a.i.1.1		3
51.50	odd	2		340.2.c.a.101.1	✓	6
204.203	even	2		1360.2.c.e.1121.6		6
255.152	even	4		1700.2.g.b.849.1		6
255.203	even	4		1700.2.g.c.849.5		6
255.254	odd	2		1700.2.c.b.101.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.c.a.101.1	✓	6	51.50	odd	2
340.2.c.a.101.6	yes	6	3.2	odd	2
1360.2.c.e.1121.1		6	12.11	even	2
1360.2.c.e.1121.6		6	204.203	even	2
1700.2.c.b.101.1		6	15.14	odd	2
1700.2.c.b.101.6		6	255.254	odd	2
1700.2.g.b.849.1		6	255.152	even	4
1700.2.g.b.849.2		6	15.8	even	4
1700.2.g.c.849.5		6	255.203	even	4
1700.2.g.c.849.6		6	15.2	even	4
3060.2.e.j.1801.1		6	17.16	even	2	inner
3060.2.e.j.1801.6		6	1.1	even	1	trivial
5780.2.a.i.1.1		3	51.47	odd	4
5780.2.a.k.1.3		3	51.38	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} +1.27307i q^{7} -5.43993i q^{11} -0.166860 q^{13} +(-2.27307 + 3.43993i) q^{17} -2.54615 q^{19} +3.27307i q^{23} -1.00000 q^{25} -6.54615i q^{29} +5.98608i q^{31} -1.27307 q^{35} +4.00000i q^{37} +8.54615i q^{41} +2.71301 q^{43} -10.7130 q^{47} +5.37929 q^{49} -12.8799 q^{53} +5.43993 q^{55} +6.21243 q^{59} +7.42601i q^{61} -0.166860i q^{65} -6.37929 q^{67} -2.89379i q^{71} +12.5461i q^{73} +6.92543 q^{77} -3.77365i q^{79} -13.5929 q^{83} +(-3.43993 - 2.27307i) q^{85} +5.83314 q^{89} -0.212425i q^{91} -2.54615i q^{95} +1.45385i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{17} + 16 q^{19} - 6 q^{25} + 8 q^{35} - 16 q^{43} - 32 q^{47} + 2 q^{49} - 44 q^{53} + 16 q^{55} + 8 q^{59} - 8 q^{67} - 20 q^{77} - 16 q^{83} - 4 q^{85} + 36 q^{89}+O(q^{100}) \) 6 * q + 2 * q^17 + 16 * q^19 - 6 * q^25 + 8 * q^35 - 16 * q^43 - 32 * q^47 + 2 * q^49 - 44 * q^53 + 16 * q^55 + 8 * q^59 - 8 * q^67 - 20 * q^77 - 16 * q^83 - 4 * q^85 + 36 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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