LMFDB - Embedded newform 3060.2.e.j.1801.3

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			1801.3
Root			$3.13264i$ of defining polynomial
Character	$\chi$	$=$	3060.1801
Dual form			3060.2.e.j.1801.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{5} +4.13264i q^{7} +4.68078i q^{11} -4.81342 q^{13} +(3.13264 - 2.68078i) q^{17} +8.26527 q^{19} +2.13264i q^{23} -1.00000 q^{25} -4.26527i q^{29} +5.58449i q^{31} +4.13264 q^{35} -4.00000i q^{37} +2.26527i q^{41} -3.45186 q^{43} -4.54814 q^{47} -10.0787 q^{49} -11.3616 q^{53} +4.68078 q^{55} -13.8921 q^{59} +4.90371i q^{61} +4.81342i q^{65} +9.07869 q^{67} +12.9461i q^{71} -1.73473i q^{73} -19.3440 q^{77} +12.3076i q^{79} -5.90970 q^{83} +(-2.68078 - 3.13264i) q^{85} +1.18658 q^{89} -19.8921i q^{91} -8.26527i q^{95} -12.2653i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$			4.13264i			1.56199i	0.624537	+	0.780995i	$0.285288\pi$
$7$			4.13264i			1.56199i	−0.624537	+	0.780995i	$0.714712\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			4.68078i			1.41131i	0.708556	+	0.705654i	$0.249347\pi$
$11$			4.68078i			1.41131i	−0.708556	+	0.705654i	$0.750653\pi$
$12$		0			0
$13$	−4.81342			−1.33500			−0.667501	−	0.744609i	$-0.732636\pi$
$13$	−4.81342			−1.33500			−0.667501	+	0.744609i	$0.732636\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.13264	−	2.68078i	0.759776	−	0.650185i
$17$	3.13264	−	2.68078i	0.759776	−	0.650185i
$18$		0			0
$19$	8.26527			1.89618			0.948092	−	0.317996i	$-0.103010\pi$
$19$	8.26527			1.89618			0.948092	+	0.317996i	$0.103010\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			2.13264i			0.444686i	0.974969	+	0.222343i	$0.0713704\pi$
$23$			2.13264i			0.444686i	−0.974969	+	0.222343i	$0.928630\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.26527i		−	0.792042i	−0.918242	−	0.396021i	$-0.870391\pi$
$29$		−	4.26527i		−	0.792042i	0.918242	−	0.396021i	$-0.129609\pi$
$30$		0			0
$31$			5.58449i			1.00300i	0.865156	+	0.501502i	$0.167219\pi$
$31$			5.58449i			1.00300i	−0.865156	+	0.501502i	$0.832781\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.13264			0.698543
$36$		0			0
$37$		−	4.00000i		−	0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$		−	4.00000i		−	0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			2.26527i			0.353777i	0.984231	+	0.176888i	$0.0566031\pi$
$41$			2.26527i			0.353777i	−0.984231	+	0.176888i	$0.943397\pi$
$42$		0			0
$43$	−3.45186			−0.526403			−0.263202	−	0.964741i	$-0.584778\pi$
$43$	−3.45186			−0.526403			−0.263202	+	0.964741i	$0.584778\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.54814			−0.663415			−0.331707	−	0.943382i	$-0.607625\pi$
$47$	−4.54814			−0.663415			−0.331707	+	0.943382i	$0.607625\pi$
$48$		0			0
$49$	−10.0787			−1.43981
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−11.3616			−1.56063			−0.780315	−	0.625386i	$-0.784941\pi$
$53$	−11.3616			−1.56063			−0.780315	+	0.625386i	$0.784941\pi$
$54$		0			0
$55$	4.68078			0.631156
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−13.8921			−1.80860			−0.904299	−	0.426899i	$-0.859606\pi$
$59$	−13.8921			−1.80860			−0.904299	+	0.426899i	$0.859606\pi$
$60$		0			0
$61$			4.90371i			0.627856i	0.949447	+	0.313928i	$0.101645\pi$
$61$			4.90371i			0.627856i	−0.949447	+	0.313928i	$0.898355\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			4.81342i			0.597031i
$66$		0			0
$67$	9.07869			1.10914			0.554569	−	0.832137i	$-0.312883\pi$
$67$	9.07869			1.10914			0.554569	+	0.832137i	$0.312883\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			12.9461i			1.53641i	0.640201	+	0.768207i	$0.278851\pi$
$71$			12.9461i			1.53641i	−0.640201	+	0.768207i	$0.721149\pi$
$72$		0			0
$73$		−	1.73473i		−	0.203034i	−0.994834	−	0.101517i	$-0.967630\pi$
$73$		−	1.73473i		−	0.203034i	0.994834	−	0.101517i	$-0.0323697\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−19.3440			−2.20445
$78$		0			0
$79$			12.3076i			1.38471i	0.721555	+	0.692357i	$0.243428\pi$
$79$			12.3076i			1.38471i	−0.721555	+	0.692357i	$0.756572\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.90970			−0.648674			−0.324337	−	0.945942i	$-0.605141\pi$
$83$	−5.90970			−0.648674			−0.324337	+	0.945942i	$0.605141\pi$
$84$		0			0
$85$	−2.68078	−	3.13264i	−0.290771	−	0.339782i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.18658			0.125777			0.0628887	−	0.998021i	$-0.479969\pi$
$89$	1.18658			0.125777			0.0628887	+	0.998021i	$0.479969\pi$
$90$		0			0
$91$		−	19.8921i		−	2.08526i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	8.26527i		−	0.847999i
$96$		0			0
$97$		−	12.2653i		−	1.24535i	−0.782481	−	0.622675i	$-0.786046\pi$
$97$		−	12.2653i		−	1.24535i	0.782481	−	0.622675i	$-0.213954\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.90970			−0.389030			−0.194515	−	0.980900i	$-0.562313\pi$
$101$	−3.90970			−0.389030			−0.194515	+	0.980900i	$0.562313\pi$
$102$		0			0
$103$	−17.3440			−1.70895			−0.854476	−	0.519491i	$-0.826122\pi$
$103$	−17.3440			−1.70895			−0.854476	+	0.519491i	$0.826122\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			8.77108i			0.847932i	0.905678	+	0.423966i	$0.139362\pi$
$107$			8.77108i			0.847932i	−0.905678	+	0.423966i	$0.860638\pi$
$108$		0			0
$109$			10.0000i			0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$			10.0000i			0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			5.16899i			0.486258i	0.969994	+	0.243129i	$0.0781738\pi$
$113$			5.16899i			0.486258i	−0.969994	+	0.243129i	$0.921826\pi$
$114$		0			0
$115$	2.13264			0.198869
$116$		0			0
$117$		0			0
$118$		0			0
$119$	11.0787	+	12.9461i	1.01558	+	1.18676i
$120$		0			0
$121$	−10.9097			−0.991791
$122$		0			0
$123$		0			0
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	15.9824			1.41821			0.709105	−	0.705103i	$-0.249099\pi$
$127$	15.9824			1.41821			0.709105	+	0.705103i	$0.249099\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	5.31922i		−	0.464742i	−0.972627	−	0.232371i	$-0.925352\pi$
$131$		−	5.31922i		−	0.464742i	0.972627	−	0.232371i	$-0.0746484\pi$
$132$		0			0
$133$			34.1574i			2.96182i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	12.8134			1.09472			0.547362	−	0.836896i	$-0.315632\pi$
$137$	12.8134			1.09472			0.547362	+	0.836896i	$0.315632\pi$
$138$		0			0
$139$			6.04234i			0.512505i	0.966610	+	0.256252i	$0.0824878\pi$
$139$			6.04234i			0.512505i	−0.966610	+	0.256252i	$0.917512\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	22.5305i		−	1.88410i
$144$		0			0
$145$	−4.26527			−0.354212
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.16899			0.0957674			0.0478837	−	0.998853i	$-0.484752\pi$
$149$	1.16899			0.0957674			0.0478837	+	0.998853i	$0.484752\pi$
$150$		0			0
$151$	−9.36156			−0.761833			−0.380916	−	0.924609i	$-0.624391\pi$
$151$	−9.36156			−0.761833			−0.380916	+	0.924609i	$0.624391\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	5.58449			0.448557
$156$		0			0
$157$	9.16899			0.731765			0.365883	−	0.930661i	$-0.380767\pi$
$157$	9.16899			0.731765			0.365883	+	0.930661i	$0.380767\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−8.81342			−0.694595
$162$		0			0
$163$		−	5.75947i		−	0.451117i	−0.974230	−	0.225558i	$-0.927579\pi$
$163$		−	5.75947i		−	0.451117i	0.974230	−	0.225558i	$-0.0724206\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			1.49420i			0.115625i	0.998327	+	0.0578123i	$0.0184125\pi$
$167$			1.49420i			0.115625i	−0.998327	+	0.0578123i	$0.981588\pi$
$168$		0			0
$169$	10.1690			0.782230
$170$		0			0
$171$		0			0
$172$		0			0
$173$			10.9037i			0.828994i	0.910051	+	0.414497i	$0.136042\pi$
$173$			10.9037i			0.828994i	−0.910051	+	0.414497i	$0.863958\pi$
$174$		0			0
$175$		−	4.13264i		−	0.312398i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.53055			0.637603			0.318802	−	0.947821i	$-0.396720\pi$
$179$	8.53055			0.637603			0.318802	+	0.947821i	$0.396720\pi$
$180$		0			0
$181$		−	3.16899i		−	0.235549i	−0.993040	−	0.117775i	$-0.962424\pi$
$181$		−	3.16899i		−	0.235549i	0.993040	−	0.117775i	$-0.0375760\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.00000			−0.294086
$186$		0			0
$187$	12.5481	+	14.6632i	0.917611	+	1.07228i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.45785			0.467273			0.233637	−	0.972324i	$-0.424937\pi$
$191$	6.45785			0.467273			0.233637	+	0.972324i	$0.424937\pi$
$192$		0			0
$193$		−	7.89211i		−	0.568087i	−0.958811	−	0.284043i	$-0.908324\pi$
$193$		−	7.89211i		−	0.568087i	0.958811	−	0.284043i	$-0.0916759\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$			16.8382i			1.19363i	0.802380	+	0.596813i	$0.203567\pi$
$199$			16.8382i			1.19363i	−0.802380	+	0.596813i	$0.796433\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	17.6268			1.23716
$204$		0			0
$205$	2.26527			0.158214
$206$		0			0
$207$		0			0
$208$		0			0
$209$			38.6879i			2.67610i
$210$		0			0
$211$		−	7.77707i		−	0.535395i	−0.963503	−	0.267698i	$-0.913737\pi$
$211$		−	7.77707i		−	0.535395i	0.963503	−	0.267698i	$-0.0862628\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			3.45186i			0.235415i
$216$		0			0
$217$	−23.0787			−1.56668
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−15.0787	+	12.9037i	−1.01430	+	0.867998i
$222$		0			0
$223$	−3.71713			−0.248918			−0.124459	−	0.992225i	$-0.539719\pi$
$223$	−3.71713			−0.248918			−0.124459	+	0.992225i	$0.539719\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	3.86736i		−	0.256686i	−0.991730	−	0.128343i	$-0.959034\pi$
$227$		−	3.86736i		−	0.256686i	0.991730	−	0.128343i	$-0.0409658\pi$
$228$		0			0
$229$	−18.8134			−1.24323			−0.621613	−	0.783325i	$-0.713522\pi$
$229$	−18.8134			−1.24323			−0.621613	+	0.783325i	$0.713522\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			15.6995i			1.02851i	0.857637	+	0.514256i	$0.171932\pi$
$233$			15.6995i			1.02851i	−0.857637	+	0.514256i	$0.828068\pi$
$234$		0			0
$235$			4.54814i			0.296688i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.90371			−0.446564			−0.223282	−	0.974754i	$-0.571677\pi$
$239$	−6.90371			−0.446564			−0.223282	+	0.974754i	$0.571677\pi$
$240$		0			0
$241$		−	0.107890i		−	0.00694979i	−0.999994	−	0.00347489i	$-0.998894\pi$
$241$		−	0.107890i		−	0.00694979i	0.999994	−	0.00347489i	$-0.00110609\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			10.0787i			0.643904i
$246$		0			0
$247$	−39.7842			−2.53141
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−22.4227			−1.41531			−0.707653	−	0.706560i	$-0.750246\pi$
$251$	−22.4227			−1.41531			−0.707653	+	0.706560i	$0.750246\pi$
$252$		0			0
$253$	−9.98241			−0.627589
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.09030			0.379902			0.189951	−	0.981794i	$-0.439167\pi$
$257$	6.09030			0.379902			0.189951	+	0.981794i	$0.439167\pi$
$258$		0			0
$259$	16.5305			1.02716
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−13.0787			−0.806467			−0.403233	−	0.915097i	$-0.632114\pi$
$263$	−13.0787			−0.806467			−0.403233	+	0.915097i	$0.632114\pi$
$264$		0			0
$265$			11.3616i			0.697935i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			21.7842i			1.32821i	0.747640	+	0.664104i	$0.231187\pi$
$269$			21.7842i			1.32821i	−0.747640	+	0.664104i	$0.768813\pi$
$270$		0			0
$271$	3.46945			0.210754			0.105377	−	0.994432i	$-0.466395\pi$
$271$	3.46945			0.210754			0.105377	+	0.994432i	$0.466395\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	4.68078i		−	0.282262i
$276$		0			0
$277$		−	31.0611i		−	1.86628i	−0.359512	−	0.933140i	$-0.617057\pi$
$277$		−	31.0611i		−	1.86628i	0.359512	−	0.933140i	$-0.382943\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	15.8921			0.948044			0.474022	−	0.880513i	$-0.342802\pi$
$281$	15.8921			0.948044			0.474022	+	0.880513i	$0.342802\pi$
$282$		0			0
$283$		−	18.8558i		−	1.12086i	−0.828202	−	0.560429i	$-0.810636\pi$
$283$		−	18.8558i		−	1.12086i	0.828202	−	0.560429i	$-0.189364\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.36156			−0.552595
$288$		0			0
$289$	2.62684	−	16.7958i	0.154520	−	0.987990i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	16.9037			0.987526			0.493763	−	0.869597i	$-0.335621\pi$
$293$	16.9037			0.987526			0.493763	+	0.869597i	$0.335621\pi$
$294$		0			0
$295$			13.8921i			0.808830i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	10.2653i		−	0.593656i
$300$		0			0
$301$		−	14.2653i		−	0.822237i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.90371			0.280786
$306$		0			0
$307$	−4.99401			−0.285023			−0.142512	−	0.989793i	$-0.545518\pi$
$307$	−4.99401			−0.285023			−0.142512	+	0.989793i	$0.545518\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	2.30762i		−	0.130853i	−0.997857	−	0.0654264i	$-0.979159\pi$
$311$		−	2.30762i		−	0.130853i	0.997857	−	0.0654264i	$-0.0208408\pi$
$312$		0			0
$313$			6.26527i			0.354134i	0.984199	+	0.177067i	$0.0566610\pi$
$313$			6.26527i			0.354134i	−0.984199	+	0.177067i	$0.943339\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	0.903715i		−	0.0507577i	−0.999678	−	0.0253788i	$-0.991921\pi$
$317$		−	0.903715i		−	0.0507577i	0.999678	−	0.0253788i	$-0.00807920\pi$
$318$		0			0
$319$	19.9648			1.11782
$320$		0			0
$321$		0			0
$322$		0			0
$323$	25.8921	−	22.1574i	1.44068	−	1.23287i
$324$		0			0
$325$	4.81342			0.267000
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	18.7958i		−	1.03625i
$330$		0			0
$331$	−28.7958			−1.58276			−0.791381	−	0.611323i	$-0.790637\pi$
$331$	−28.7958			−1.58276			−0.791381	+	0.611323i	$0.790637\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	9.07869i		−	0.496022i
$336$		0			0
$337$			15.1690i			0.826308i	0.910661	+	0.413154i	$0.135573\pi$
$337$			15.1690i			0.826308i	−0.910661	+	0.413154i	$0.864427\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−26.1398			−1.41555
$342$		0			0
$343$		−	12.7231i		−	0.686984i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	6.77108i		−	0.363490i	−0.983346	−	0.181745i	$-0.941825\pi$
$347$		−	6.77108i		−	0.363490i	0.983346	−	0.181745i	$-0.0581746\pi$
$348$		0			0
$349$	15.8921			0.850685			0.425343	−	0.905032i	$-0.360154\pi$
$349$	15.8921			0.850685			0.425343	+	0.905032i	$0.360154\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−25.2537			−1.34412			−0.672059	−	0.740498i	$-0.734590\pi$
$353$	−25.2537			−1.34412			−0.672059	+	0.740498i	$0.734590\pi$
$354$		0			0
$355$	12.9461			0.687105
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−31.4343			−1.65904			−0.829519	−	0.558479i	$-0.811385\pi$
$359$	−31.4343			−1.65904			−0.829519	+	0.558479i	$0.811385\pi$
$360$		0			0
$361$	49.3148			2.59551
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.73473			−0.0907997
$366$		0			0
$367$		−	15.6516i		−	0.817006i	−0.912757	−	0.408503i	$-0.866051\pi$
$367$		−	15.6516i		−	0.817006i	0.912757	−	0.408503i	$-0.133949\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	46.9532i		−	2.43769i
$372$		0			0
$373$	−1.34397			−0.0695880			−0.0347940	−	0.999395i	$-0.511078\pi$
$373$	−1.34397			−0.0695880			−0.0347940	+	0.999395i	$0.511078\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			20.5305i			1.05738i
$378$		0			0
$379$		−	16.4882i		−	0.846942i	−0.905910	−	0.423471i	$-0.860811\pi$
$379$		−	16.4882i		−	0.846942i	0.905910	−	0.423471i	$-0.139189\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	17.8745			0.913345			0.456673	−	0.889635i	$-0.349041\pi$
$383$	17.8745			0.913345			0.456673	+	0.889635i	$0.349041\pi$
$384$		0			0
$385$			19.3440i			0.985860i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	27.7898			1.40900			0.704500	−	0.709704i	$-0.251171\pi$
$389$	27.7898			1.40900			0.704500	+	0.709704i	$0.251171\pi$
$390$		0			0
$391$	5.71713	+	6.68078i	0.289128	+	0.337862i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	12.3076			0.619263
$396$		0			0
$397$			7.80743i			0.391844i	0.980620	+	0.195922i	$0.0627699\pi$
$397$			7.80743i			0.391844i	−0.980620	+	0.195922i	$0.937230\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			11.4343i			0.571000i	0.958379	+	0.285500i	$0.0921596\pi$
$401$			11.4343i			0.571000i	−0.958379	+	0.285500i	$0.907840\pi$
$402$		0			0
$403$		−	26.8805i		−	1.33901i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	18.7231			0.928071
$408$		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
$413$		−	57.4111i		−	2.82501i
$414$		0			0
$415$			5.90970i			0.290096i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			34.6808i			1.69427i	0.531380	+	0.847133i	$0.321674\pi$
$419$			34.6808i			1.69427i	−0.531380	+	0.847133i	$0.678326\pi$
$420$		0			0
$421$	12.8861			0.628031			0.314016	−	0.949418i	$-0.398326\pi$
$421$	12.8861			0.628031			0.314016	+	0.949418i	$0.398326\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.13264	+	2.68078i	−0.151955	+	0.130037i
$426$		0			0
$427$	−20.2653			−0.980705
$428$		0			0
$429$		0			0
$430$		0			0
$431$			9.31922i			0.448891i	0.974487	+	0.224446i	$0.0720571\pi$
$431$			9.31922i			0.448891i	−0.974487	+	0.224446i	$0.927943\pi$
$432$		0			0
$433$	−22.6208			−1.08709			−0.543544	−	0.839381i	$-0.682918\pi$
$433$	−22.6208			−1.08709			−0.543544	+	0.839381i	$0.682918\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			17.6268i			0.843206i
$438$		0			0
$439$		−	18.0072i		−	0.859435i	−0.902963	−	0.429717i	$-0.858613\pi$
$439$		−	18.0072i		−	0.859435i	0.902963	−	0.429717i	$-0.141387\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−23.1514			−1.09996			−0.549978	−	0.835179i	$-0.685364\pi$
$443$	−23.1514			−1.09996			−0.549978	+	0.835179i	$0.685364\pi$
$444$		0			0
$445$		−	1.18658i		−	0.0562494i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	7.54215i		−	0.355936i	−0.984036	−	0.177968i	$-0.943048\pi$
$449$		−	7.54215i		−	0.355936i	0.984036	−	0.177968i	$-0.0569524\pi$
$450$		0			0
$451$	−10.6033			−0.499288
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−19.8921			−0.932557
$456$		0			0
$457$	−8.06709			−0.377362			−0.188681	−	0.982038i	$-0.560421\pi$
$457$	−8.06709			−0.377362			−0.188681	+	0.982038i	$0.560421\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	12.4227			0.578581			0.289290	−	0.957241i	$-0.406581\pi$
$461$	12.4227			0.578581			0.289290	+	0.957241i	$0.406581\pi$
$462$		0			0
$463$	−0.282868			−0.0131460			−0.00657299	−	0.999978i	$-0.502092\pi$
$463$	−0.282868			−0.0131460			−0.00657299	+	0.999978i	$0.502092\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	16.8981			0.781951			0.390975	−	0.920401i	$-0.372138\pi$
$467$	16.8981			0.781951			0.390975	+	0.920401i	$0.372138\pi$
$468$		0			0
$469$			37.5189i			1.73246i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	16.1574i		−	0.742917i
$474$		0			0
$475$	−8.26527			−0.379237
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	33.0188i		−	1.50867i	−0.656492	−	0.754333i	$-0.727961\pi$
$479$		−	33.0188i		−	1.50867i	0.656492	−	0.754333i	$-0.272039\pi$
$480$		0			0
$481$			19.2537i			0.877892i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−12.2653			−0.556937
$486$		0			0
$487$			26.8558i			1.21695i	0.793573	+	0.608475i	$0.208219\pi$
$487$			26.8558i			1.21695i	−0.793573	+	0.608475i	$0.791781\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	20.3500			0.918381			0.459190	−	0.888338i	$-0.348140\pi$
$491$	20.3500			0.918381			0.459190	+	0.888338i	$0.348140\pi$
$492$		0			0
$493$	−11.4343	−	13.3616i	−0.514973	−	0.601774i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−53.5014			−2.39986
$498$		0			0
$499$			43.8266i			1.96195i	0.194143	+	0.980973i	$0.437807\pi$
$499$			43.8266i			1.96195i	−0.194143	+	0.980973i	$0.562193\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			6.59048i			0.293855i	0.989147	+	0.146928i	$0.0469384\pi$
$503$			6.59048i			0.293855i	−0.989147	+	0.146928i	$0.953062\pi$
$504$		0			0
$505$			3.90970i			0.173980i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	25.7842			1.14287			0.571433	−	0.820649i	$-0.306388\pi$
$509$	25.7842			1.14287			0.571433	+	0.820649i	$0.306388\pi$
$510$		0			0
$511$	7.16899			0.317137
$512$		0			0
$513$		0			0
$514$		0			0
$515$			17.3440i			0.764267i
$516$		0			0
$517$		−	21.2889i		−	0.936283i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	2.90371i		−	0.127214i	−0.997975	−	0.0636070i	$-0.979740\pi$
$521$		−	2.90371i		−	0.127214i	0.997975	−	0.0636070i	$-0.0202604\pi$
$522$		0			0
$523$	41.4286			1.81155			0.905774	−	0.423761i	$-0.139290\pi$
$523$	41.4286			1.81155			0.905774	+	0.423761i	$0.139290\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	14.9708	+	17.4942i	0.652138	+	0.762059i
$528$		0			0
$529$	18.4519			0.802255
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	10.9037i		−	0.472292i
$534$		0			0
$535$	8.77108			0.379207
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	47.1761i		−	2.03202i
$540$		0			0
$541$		−	22.9884i		−	0.988348i	−0.869363	−	0.494174i	$-0.835471\pi$
$541$		−	22.9884i		−	0.988348i	0.869363	−	0.494174i	$-0.164529\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10.0000			0.428353
$546$		0			0
$547$			8.32521i			0.355960i	0.984034	+	0.177980i	$0.0569563\pi$
$547$			8.32521i			0.355960i	−0.984034	+	0.177980i	$0.943044\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	35.2537i		−	1.50186i
$552$		0			0
$553$	−50.8629			−2.16291
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.52456			−0.234083			−0.117042	−	0.993127i	$-0.537341\pi$
$557$	−5.52456			−0.234083			−0.117042	+	0.993127i	$0.537341\pi$
$558$		0			0
$559$	16.6152			0.702749
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−17.5246			−0.738572			−0.369286	−	0.929316i	$-0.620398\pi$
$563$	−17.5246			−0.738572			−0.369286	+	0.929316i	$0.620398\pi$
$564$		0			0
$565$	5.16899			0.217461
$566$		0			0
$567$		0			0
$568$		0			0
$569$	20.7231			0.868758			0.434379	−	0.900730i	$-0.356968\pi$
$569$	20.7231			0.868758			0.434379	+	0.900730i	$0.356968\pi$
$570$		0			0
$571$		−	4.87335i		−	0.203943i	−0.994787	−	0.101972i	$-0.967485\pi$
$571$		−	4.87335i		−	0.203943i	0.994787	−	0.101972i	$-0.0325151\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	2.13264i		−	0.0889371i
$576$		0			0
$577$	−18.6208			−0.775196			−0.387598	−	0.921829i	$-0.626695\pi$
$577$	−18.6208			−0.775196			−0.387598	+	0.921829i	$0.626695\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	24.4227i		−	1.01322i
$582$		0			0
$583$		−	53.1810i		−	2.20253i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−40.2477			−1.66120			−0.830600	−	0.556870i	$-0.812002\pi$
$587$	−40.2477			−1.66120			−0.830600	+	0.556870i	$0.812002\pi$
$588$		0			0
$589$			46.1574i			1.90188i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8.98840			0.369109			0.184555	−	0.982822i	$-0.440916\pi$
$593$	8.98840			0.369109			0.184555	+	0.982822i	$0.440916\pi$
$594$		0			0
$595$	12.9461	−	11.0787i	0.530737	−	0.454182i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	40.7958			1.66687			0.833436	−	0.552616i	$-0.186370\pi$
$599$	40.7958			1.66687			0.833436	+	0.552616i	$0.186370\pi$
$600$		0			0
$601$			37.6995i			1.53780i	0.639371	+	0.768898i	$0.279195\pi$
$601$			37.6995i			1.53780i	−0.639371	+	0.768898i	$0.720805\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			10.9097i			0.443543i
$606$		0			0
$607$			14.9285i			0.605928i	0.953002	+	0.302964i	$0.0979761\pi$
$607$			14.9285i			0.605928i	−0.953002	+	0.302964i	$0.902024\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	21.8921			0.885660
$612$		0			0
$613$	−12.9037			−0.521176			−0.260588	−	0.965450i	$-0.583916\pi$
$613$	−12.9037			−0.521176			−0.260588	+	0.965450i	$0.583916\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	14.4459i		−	0.581569i	−0.956789	−	0.290784i	$-0.906084\pi$
$617$		−	14.4459i		−	0.581569i	0.956789	−	0.290784i	$-0.0939162\pi$
$618$		0			0
$619$		−	43.0188i		−	1.72907i	−0.502573	−	0.864535i	$-0.667613\pi$
$619$		−	43.0188i		−	1.72907i	0.502573	−	0.864535i	$-0.332387\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			4.90371i			0.196463i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−10.7231	−	12.5305i	−0.427559	−	0.499626i
$630$		0			0
$631$	21.1810			0.843201			0.421600	−	0.906782i	$-0.361468\pi$
$631$	21.1810			0.843201			0.421600	+	0.906782i	$0.361468\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	15.9824i		−	0.634242i
$636$		0			0
$637$	48.5130			1.92215
$638$		0			0
$639$		0			0
$640$		0			0
$641$			35.3264i			1.39531i	0.716435	+	0.697654i	$0.245773\pi$
$641$			35.3264i			1.39531i	−0.716435	+	0.697654i	$0.754227\pi$
$642$		0			0
$643$			14.1326i			0.557337i	0.960387	+	0.278668i	$0.0898930\pi$
$643$			14.1326i			0.557337i	−0.960387	+	0.278668i	$0.910107\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	37.7898			1.48567			0.742836	−	0.669474i	$-0.233480\pi$
$647$	37.7898			1.48567			0.742836	+	0.669474i	$0.233480\pi$
$648$		0			0
$649$		−	65.0259i		−	2.55249i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			5.09629i			0.199433i	0.995016	+	0.0997165i	$0.0317936\pi$
$653$			5.09629i			0.199433i	−0.995016	+	0.0997165i	$0.968206\pi$
$654$		0			0
$655$	−5.31922			−0.207839
$656$		0			0
$657$		0			0
$658$		0			0
$659$	43.0731			1.67789			0.838944	−	0.544217i	$-0.183173\pi$
$659$	43.0731			1.67789			0.838944	+	0.544217i	$0.183173\pi$
$660$		0			0
$661$	−6.97642			−0.271351			−0.135676	−	0.990753i	$-0.543320\pi$
$661$	−6.97642			−0.271351			−0.135676	+	0.990753i	$0.543320\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	34.1574			1.32457
$666$		0			0
$667$	9.09629			0.352210
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−22.9532			−0.886099
$672$		0			0
$673$			42.6152i			1.64270i	0.570427	+	0.821348i	$0.306778\pi$
$673$			42.6152i			1.64270i	−0.570427	+	0.821348i	$0.693222\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	40.5073i		−	1.55682i	−0.627754	−	0.778412i	$-0.716026\pi$
$677$		−	40.5073i		−	1.55682i	0.627754	−	0.778412i	$-0.283974\pi$
$678$		0			0
$679$	50.6879			1.94522
$680$		0			0
$681$		0			0
$682$		0			0
$683$			7.10906i			0.272020i	0.990707	+	0.136010i	$0.0434280\pi$
$683$			7.10906i			0.272020i	−0.990707	+	0.136010i	$0.956572\pi$
$684$		0			0
$685$		−	12.8134i		−	0.489576i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	54.6879			2.08344
$690$		0			0
$691$		−	20.4650i		−	0.778525i	−0.921127	−	0.389263i	$-0.872730\pi$
$691$		−	20.4650i		−	0.778525i	0.921127	−	0.389263i	$-0.127270\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.04234			0.229199
$696$		0			0
$697$	6.07270	+	7.09629i	0.230020	+	0.268791i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	6.51296			0.245991			0.122996	−	0.992407i	$-0.460750\pi$
$701$	6.51296			0.245991			0.122996	+	0.992407i	$0.460750\pi$
$702$		0			0
$703$		−	33.0611i		−	1.24692i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	16.1574i		−	0.607661i
$708$		0			0
$709$			13.1458i			0.493700i	0.969054	+	0.246850i	$0.0793955\pi$
$709$			13.1458i			0.493700i	−0.969054	+	0.246850i	$0.920604\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−11.9097			−0.446022
$714$		0			0
$715$	−22.5305			−0.842595
$716$		0			0
$717$		0			0
$718$		0			0
$719$			4.94606i			0.184457i	0.995738	+	0.0922284i	$0.0293990\pi$
$719$			4.94606i			0.184457i	−0.995738	+	0.0922284i	$0.970601\pi$
$720$		0			0
$721$		−	71.6763i		−	2.66937i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			4.26527i			0.158408i
$726$		0			0
$727$	12.6328			0.468525			0.234263	−	0.972173i	$-0.424732\pi$
$727$	12.6328			0.468525			0.234263	+	0.972173i	$0.424732\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−10.8134	+	9.25367i	−0.399949	+	0.342259i
$732$		0			0
$733$	23.2417			0.858452			0.429226	−	0.903197i	$-0.358786\pi$
$733$	23.2417			0.858452			0.429226	+	0.903197i	$0.358786\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			42.4954i			1.56534i
$738$		0			0
$739$	47.3032			1.74008			0.870038	−	0.492985i	$-0.164094\pi$
$739$	47.3032			1.74008			0.870038	+	0.492985i	$0.164094\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			3.75947i			0.137922i	0.997619	+	0.0689608i	$0.0219684\pi$
$743$			3.75947i			0.137922i	−0.997619	+	0.0689608i	$0.978032\pi$
$744$		0			0
$745$		−	1.16899i		−	0.0428285i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−36.2477			−1.32446
$750$		0			0
$751$			48.1150i			1.75574i	0.478896	+	0.877871i	$0.341037\pi$
$751$			48.1150i			1.75574i	−0.478896	+	0.877871i	$0.658963\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			9.36156i			0.340702i
$756$		0			0
$757$	33.1634			1.20534			0.602672	−	0.797989i	$-0.294103\pi$
$757$	33.1634			1.20534			0.602672	+	0.797989i	$0.294103\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	11.9097			0.431726			0.215863	−	0.976424i	$-0.430743\pi$
$761$	11.9097			0.431726			0.215863	+	0.976424i	$0.430743\pi$
$762$		0			0
$763$	−41.3264			−1.49612
$764$		0			0
$765$		0			0
$766$		0			0
$767$	66.8685			2.41448
$768$		0			0
$769$	32.4051			1.16856			0.584278	−	0.811553i	$-0.301378\pi$
$769$	32.4051			1.16856			0.584278	+	0.811553i	$0.301378\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−40.0319			−1.43985			−0.719924	−	0.694053i	$-0.755823\pi$
$773$	−40.0319			−1.43985			−0.719924	+	0.694053i	$0.755823\pi$
$774$		0			0
$775$		−	5.58449i		−	0.200601i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			18.7231i			0.670825i
$780$		0			0
$781$	−60.5976			−2.16835
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	9.16899i		−	0.327255i
$786$		0			0
$787$			42.3747i			1.51050i	0.655440	+	0.755248i	$0.272483\pi$
$787$			42.3747i			1.51050i	−0.655440	+	0.755248i	$0.727517\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−21.3616			−0.759530
$792$		0			0
$793$		−	23.6036i		−	0.838189i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.373165			−0.0132182			−0.00660909	−	0.999978i	$-0.502104\pi$
$797$	−0.373165			−0.0132182			−0.00660909	+	0.999978i	$0.502104\pi$
$798$		0			0
$799$	−14.2477	+	12.1926i	−0.504047	+	0.431342i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	8.11987			0.286544
$804$		0			0
$805$			8.81342i			0.310632i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			8.56574i			0.301155i	0.988598	+	0.150578i	$0.0481133\pi$
$809$			8.56574i			0.301155i	−0.988598	+	0.150578i	$0.951887\pi$
$810$		0			0
$811$		−	42.0918i		−	1.47804i	−0.673681	−	0.739022i	$-0.735288\pi$
$811$		−	42.0918i		−	1.47804i	0.673681	−	0.739022i	$-0.264712\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−5.75947			−0.201746
$816$		0			0
$817$	−28.5305			−0.998158
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	26.1454i		−	0.912481i	−0.889856	−	0.456241i	$-0.849196\pi$
$821$		−	26.1454i		−	0.912481i	0.889856	−	0.456241i	$-0.150804\pi$
$822$		0			0
$823$			10.2173i			0.356153i	0.984017	+	0.178077i	$0.0569875\pi$
$823$			10.2173i			0.356153i	−0.984017	+	0.178077i	$0.943012\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	22.4474i		−	0.780573i	−0.920693	−	0.390286i	$-0.872376\pi$
$827$		−	22.4474i		−	0.780573i	0.920693	−	0.390286i	$-0.127624\pi$
$828$		0			0
$829$	12.1926			0.423465			0.211733	−	0.977328i	$-0.432089\pi$
$829$	12.1926			0.423465			0.211733	+	0.977328i	$0.432089\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−31.5729	+	27.0188i	−1.09394	+	0.936145i
$834$		0			0
$835$	1.49420			0.0517088
$836$		0			0
$837$		0			0
$838$		0			0
$839$			32.3803i			1.11789i	0.829204	+	0.558946i	$0.188794\pi$
$839$			32.3803i			1.11789i	−0.829204	+	0.558946i	$0.811206\pi$
$840$		0			0
$841$	10.8074			0.372670
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	10.1690i		−	0.349824i
$846$		0			0
$847$		−	45.0858i		−	1.54917i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	8.53055			0.292423
$852$		0			0
$853$		−	40.1694i		−	1.37537i	−0.726008	−	0.687686i	$-0.758626\pi$
$853$		−	40.1694i		−	1.37537i	0.726008	−	0.687686i	$-0.241374\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	46.9652i		−	1.60430i	−0.597122	−	0.802150i	$-0.703689\pi$
$857$		−	46.9652i		−	1.60430i	0.597122	−	0.802150i	$-0.296311\pi$
$858$		0			0
$859$	−20.3500			−0.694332			−0.347166	−	0.937804i	$-0.612856\pi$
$859$	−20.3500			−0.694332			−0.347166	+	0.937804i	$0.612856\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	17.4286			0.593278			0.296639	−	0.954990i	$-0.404134\pi$
$863$	17.4286			0.593278			0.296639	+	0.954990i	$0.404134\pi$
$864$		0			0
$865$	10.9037			0.370737
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−57.6092			−1.95426
$870$		0			0
$871$	−43.6995			−1.48070
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4.13264			−0.139709
$876$		0			0
$877$			44.6879i			1.50900i	0.656298	+	0.754502i	$0.272122\pi$
$877$			44.6879i			1.50900i	−0.656298	+	0.754502i	$0.727878\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			36.4227i			1.22711i	0.789652	+	0.613555i	$0.210261\pi$
$881$			36.4227i			1.22711i	−0.789652	+	0.613555i	$0.789739\pi$
$882$		0			0
$883$	48.5976			1.63544			0.817720	−	0.575616i	$-0.195238\pi$
$883$	48.5976			1.63544			0.817720	+	0.575616i	$0.195238\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	33.3016i		−	1.11816i	−0.829114	−	0.559080i	$-0.811155\pi$
$887$		−	33.3016i		−	1.11816i	0.829114	−	0.559080i	$-0.188845\pi$
$888$		0			0
$889$			66.0495i			2.21523i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−37.5916			−1.25796
$894$		0			0
$895$		−	8.53055i		−	0.285145i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	23.8194			0.794422
$900$		0			0
$901$	−35.5916	+	30.4578i	−1.18573	+	1.01470i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3.16899			−0.105341
$906$		0			0
$907$		−	19.7475i		−	0.655705i	−0.944729	−	0.327852i	$-0.893675\pi$
$907$		−	19.7475i		−	0.655705i	0.944729	−	0.327852i	$-0.106325\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			5.12665i			0.169853i	0.996387	+	0.0849267i	$0.0270656\pi$
$911$			5.12665i			0.169853i	−0.996387	+	0.0849267i	$0.972934\pi$
$912$		0			0
$913$		−	27.6620i		−	0.915479i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	21.9824			0.725923
$918$		0			0
$919$	20.9764			0.691948			0.345974	−	0.938244i	$-0.387548\pi$
$919$	20.9764			0.691948			0.345974	+	0.938244i	$0.387548\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	62.3148i		−	2.05112i
$924$		0			0
$925$			4.00000i			0.131519i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	10.2653i		−	0.336793i	−0.985719	−	0.168396i	$-0.946141\pi$
$929$		−	10.2653i		−	0.336793i	0.985719	−	0.168396i	$-0.0538589\pi$
$930$		0			0
$931$	−83.3032			−2.73015
$932$		0			0
$933$		0			0
$934$		0			0
$935$	14.6632	−	12.5481i	0.479538	−	0.410368i
$936$		0			0
$937$	−15.0116			−0.490408			−0.245204	−	0.969472i	$-0.578855\pi$
$937$	−15.0116			−0.490408			−0.245204	+	0.969472i	$0.578855\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	50.4227i		−	1.64373i	−0.569681	−	0.821866i	$-0.692933\pi$
$941$		−	50.4227i		−	1.64373i	0.569681	−	0.821866i	$-0.307067\pi$
$942$		0			0
$943$	−4.83101			−0.157319
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	56.2173i		−	1.82682i	−0.407044	−	0.913409i	$-0.633440\pi$
$947$		−	56.2173i		−	1.82682i	0.407044	−	0.913409i	$-0.366560\pi$
$948$		0			0
$949$			8.34996i			0.271051i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	37.6587			1.21989			0.609943	−	0.792445i	$-0.291192\pi$
$953$	37.6587			1.21989			0.609943	+	0.792445i	$0.291192\pi$
$954$		0			0
$955$		−	6.45785i		−	0.208971i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			52.9532i			1.70995i
$960$		0			0
$961$	−0.186582			−0.00601878
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.89211			−0.254056
$966$		0			0
$967$	4.76392			0.153197			0.0765987	−	0.997062i	$-0.475594\pi$
$967$	4.76392			0.153197			0.0765987	+	0.997062i	$0.475594\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	48.6999			1.56285			0.781427	−	0.623996i	$-0.214492\pi$
$971$	48.6999			1.56285			0.781427	+	0.623996i	$0.214492\pi$
$972$		0			0
$973$	−24.9708			−0.800527
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−53.1338			−1.69990			−0.849950	−	0.526863i	$-0.823368\pi$
$977$	−53.1338			−1.69990			−0.849950	+	0.526863i	$0.823368\pi$
$978$		0			0
$979$			5.55413i			0.177511i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			47.3016i			1.50869i	0.656480	+	0.754344i	$0.272045\pi$
$983$			47.3016i			1.50869i	−0.656480	+	0.754344i	$0.727955\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	7.36156i		−	0.234084i
$990$		0			0
$991$		−	44.9109i		−	1.42664i	−0.700838	−	0.713320i	$-0.747190\pi$
$991$		−	44.9109i		−	1.42664i	0.700838	−	0.713320i	$-0.252810\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	16.8382			0.533806
$996$		0			0
$997$		−	2.54253i		−	0.0805226i	−0.999189	−	0.0402613i	$-0.987181\pi$
$997$		−	2.54253i		−	0.0805226i	0.999189	−	0.0402613i	$-0.0128190\pi$
$998$		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.1801.3		6
3.2	odd	2		340.2.c.a.101.5	yes	6
12.11	even	2		1360.2.c.e.1121.2		6
15.2	even	4		1700.2.g.b.849.5		6
15.8	even	4		1700.2.g.c.849.1		6
15.14	odd	2		1700.2.c.b.101.2		6
17.16	even	2	inner	3060.2.e.j.1801.4		6
51.38	odd	4		5780.2.a.i.1.3		3
51.47	odd	4		5780.2.a.k.1.1		3
51.50	odd	2		340.2.c.a.101.2	✓	6
204.203	even	2		1360.2.c.e.1121.5		6
255.152	even	4		1700.2.g.c.849.2		6
255.203	even	4		1700.2.g.b.849.6		6
255.254	odd	2		1700.2.c.b.101.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.c.a.101.2	✓	6	51.50	odd	2
340.2.c.a.101.5	yes	6	3.2	odd	2
1360.2.c.e.1121.2		6	12.11	even	2
1360.2.c.e.1121.5		6	204.203	even	2
1700.2.c.b.101.2		6	15.14	odd	2
1700.2.c.b.101.5		6	255.254	odd	2
1700.2.g.b.849.5		6	15.2	even	4
1700.2.g.b.849.6		6	255.203	even	4
1700.2.g.c.849.1		6	15.8	even	4
1700.2.g.c.849.2		6	255.152	even	4
3060.2.e.j.1801.3		6	1.1	even	1	trivial
3060.2.e.j.1801.4		6	17.16	even	2	inner
5780.2.a.i.1.3		3	51.38	odd	4
5780.2.a.k.1.1		3	51.47	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{5} +4.13264i q^{7} +4.68078i q^{11} -4.81342 q^{13} +(3.13264 - 2.68078i) q^{17} +8.26527 q^{19} +2.13264i q^{23} -1.00000 q^{25} -4.26527i q^{29} +5.58449i q^{31} +4.13264 q^{35} -4.00000i q^{37} +2.26527i q^{41} -3.45186 q^{43} -4.54814 q^{47} -10.0787 q^{49} -11.3616 q^{53} +4.68078 q^{55} -13.8921 q^{59} +4.90371i q^{61} +4.81342i q^{65} +9.07869 q^{67} +12.9461i q^{71} -1.73473i q^{73} -19.3440 q^{77} +12.3076i q^{79} -5.90970 q^{83} +(-2.68078 - 3.13264i) q^{85} +1.18658 q^{89} -19.8921i q^{91} -8.26527i q^{95} -12.2653i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{17} + 16 q^{19} - 6 q^{25} + 8 q^{35} - 16 q^{43} - 32 q^{47} + 2 q^{49} - 44 q^{53} + 16 q^{55} + 8 q^{59} - 8 q^{67} - 20 q^{77} - 16 q^{83} - 4 q^{85} + 36 q^{89}+O(q^{100}) \) 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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