LMFDB - Embedded newform 2340.2.y.b.53.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,2,Mod(53,2340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2340, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.53"); 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.y (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			53.2
Character	$\chi$	$=$	2340.53
Dual form			2340.2.y.b.1457.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.02935 - 0.939016i) q^{5} +(-0.932729 + 0.932729i) q^{7} -2.46212i q^{11} +(0.707107 + 0.707107i) q^{13} +(1.21893 + 1.21893i) q^{17} -7.52960i q^{19} +(-4.70605 + 4.70605i) q^{23} +(3.23650 + 3.81118i) q^{25} -6.52718 q^{29} +7.11591 q^{31} +(2.76868 - 1.01698i) q^{35} +(-5.27885 + 5.27885i) q^{37} -4.71432i q^{41} +(7.56537 + 7.56537i) q^{43} +(-1.63641 - 1.63641i) q^{47} +5.26003i q^{49} +(3.59649 - 3.59649i) q^{53} +(-2.31197 + 4.99650i) q^{55} -5.39616 q^{59} -12.1396 q^{61} +(-0.770981 - 2.09895i) q^{65} +(-2.18639 + 2.18639i) q^{67} +5.08457i q^{71} +(3.03059 + 3.03059i) q^{73} +(2.29649 + 2.29649i) q^{77} +16.0016i q^{79} +(-7.93799 + 7.93799i) q^{83} +(-1.32903 - 3.61821i) q^{85} -0.935268 q^{89} -1.31908 q^{91} +(-7.07041 + 15.2802i) q^{95} +(-4.95103 + 4.95103i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{5} - 8 q^{7} + 8 q^{17} + 8 q^{23} + 16 q^{25} - 32 q^{29} - 8 q^{35} + 16 q^{37} - 8 q^{43} + 40 q^{47} + 8 q^{53} + 8 q^{55} - 56 q^{59} + 8 q^{61} + 4 q^{65} - 16 q^{67} + 72 q^{77} + 32 q^{83}+ \cdots + 8 q^{97}+O(q^{100}) $ 24 * q + 4 * q^5 - 8 * q^7 + 8 * q^17 + 8 * q^23 + 16 * q^25 - 32 * q^29 - 8 * q^35 + 16 * q^37 - 8 * q^43 + 40 * q^47 + 8 * q^53 + 8 * q^55 - 56 * q^59 + 8 * q^61 + 4 * q^65 - 16 * q^67 + 72 * q^77 + 32 * q^83 - 32 * q^85 - 128 * q^89 - 8 * q^91 - 16 * q^95 + 8 * q^97

Character values

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

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$	$-1$

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
$5$	−2.02935	−	0.939016i	−0.907552	−	0.419941i
$5$	−2.02935	−	0.939016i	−0.907552	−	0.419941i
$6$		0			0
$7$	−0.932729	+	0.932729i	−0.352538	+	0.352538i	−0.861053	−	0.508515i	$-0.830195\pi$
$7$	−0.932729	+	0.932729i	−0.352538	+	0.352538i	0.508515	+	0.861053i	$0.330195\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	2.46212i		−	0.742357i	−0.928561	−	0.371179i	$-0.878954\pi$
$11$		−	2.46212i		−	0.742357i	0.928561	−	0.371179i	$-0.121046\pi$
$12$		0			0
$13$	0.707107	+	0.707107i	0.196116	+	0.196116i
$13$	0.707107	+	0.707107i	0.196116	+	0.196116i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.21893	+	1.21893i	0.295633	+	0.295633i	0.839301	−	0.543668i	$-0.182965\pi$
$17$	1.21893	+	1.21893i	0.295633	+	0.295633i	−0.543668	+	0.839301i	$0.682965\pi$
$18$		0			0
$19$		−	7.52960i		−	1.72741i	−0.503999	−	0.863704i	$-0.668139\pi$
$19$		−	7.52960i		−	1.72741i	0.503999	−	0.863704i	$-0.331861\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.70605	+	4.70605i	−0.981279	+	0.981279i	−0.999828	−	0.0185491i	$-0.994095\pi$
$23$	−4.70605	+	4.70605i	−0.981279	+	0.981279i	0.0185491	+	0.999828i	$0.494095\pi$
$24$		0			0
$25$	3.23650	+	3.81118i	0.647300	+	0.762236i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.52718			−1.21207			−0.606033	−	0.795440i	$-0.707240\pi$
$29$	−6.52718			−1.21207			−0.606033	+	0.795440i	$0.707240\pi$
$30$		0			0
$31$	7.11591			1.27806			0.639028	−	0.769184i	$-0.279337\pi$
$31$	7.11591			1.27806			0.639028	+	0.769184i	$0.279337\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.76868	−	1.01698i	0.467992	−	0.171902i
$36$		0			0
$37$	−5.27885	+	5.27885i	−0.867838	+	0.867838i	−0.992233	−	0.124395i	$-0.960301\pi$
$37$	−5.27885	+	5.27885i	−0.867838	+	0.867838i	0.124395	+	0.992233i	$0.460301\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	4.71432i		−	0.736253i	−0.929776	−	0.368126i	$-0.879999\pi$
$41$		−	4.71432i		−	0.736253i	0.929776	−	0.368126i	$-0.120001\pi$
$42$		0			0
$43$	7.56537	+	7.56537i	1.15371	+	1.15371i	0.985803	+	0.167904i	$0.0537000\pi$
$43$	7.56537	+	7.56537i	1.15371	+	1.15371i	0.167904	+	0.985803i	$0.446300\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.63641	−	1.63641i	−0.238695	−	0.238695i	0.577615	−	0.816310i	$-0.303984\pi$
$47$	−1.63641	−	1.63641i	−0.238695	−	0.238695i	−0.816310	+	0.577615i	$0.803984\pi$
$48$		0			0
$49$			5.26003i			0.751433i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.59649	−	3.59649i	0.494016	−	0.494016i	−0.415553	−	0.909569i	$-0.636412\pi$
$53$	3.59649	−	3.59649i	0.494016	−	0.494016i	0.909569	+	0.415553i	$0.136412\pi$
$54$		0			0
$55$	−2.31197	+	4.99650i	−0.311746	+	0.673728i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.39616			−0.702520			−0.351260	−	0.936278i	$-0.614247\pi$
$59$	−5.39616			−0.702520			−0.351260	+	0.936278i	$0.614247\pi$
$60$		0			0
$61$	−12.1396			−1.55432			−0.777159	−	0.629305i	$-0.783340\pi$
$61$	−12.1396			−1.55432			−0.777159	+	0.629305i	$0.783340\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.770981	−	2.09895i	−0.0956284	−	0.260343i
$66$		0			0
$67$	−2.18639	+	2.18639i	−0.267110	+	0.267110i	−0.827935	−	0.560825i	$-0.810484\pi$
$67$	−2.18639	+	2.18639i	−0.267110	+	0.267110i	0.560825	+	0.827935i	$0.310484\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			5.08457i			0.603427i	0.953399	+	0.301714i	$0.0975587\pi$
$71$			5.08457i			0.603427i	−0.953399	+	0.301714i	$0.902441\pi$
$72$		0			0
$73$	3.03059	+	3.03059i	0.354704	+	0.354704i	0.861856	−	0.507153i	$-0.169302\pi$
$73$	3.03059	+	3.03059i	0.354704	+	0.354704i	−0.507153	+	0.861856i	$0.669302\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.29649	+	2.29649i	0.261710	+	0.261710i
$78$		0			0
$79$			16.0016i			1.80032i	0.435559	+	0.900160i	$0.356551\pi$
$79$			16.0016i			1.80032i	−0.435559	+	0.900160i	$0.643449\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.93799	+	7.93799i	−0.871308	+	0.871308i	−0.992615	−	0.121307i	$-0.961291\pi$
$83$	−7.93799	+	7.93799i	−0.871308	+	0.871308i	0.121307	+	0.992615i	$0.461291\pi$
$84$		0			0
$85$	−1.32903	−	3.61821i	−0.144154	−	0.392450i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.935268			−0.0991382			−0.0495691	−	0.998771i	$-0.515785\pi$
$89$	−0.935268			−0.0991382			−0.0495691	+	0.998771i	$0.515785\pi$
$90$		0			0
$91$	−1.31908			−0.138277
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−7.07041	+	15.2802i	−0.725409	+	1.56771i
$96$		0			0
$97$	−4.95103	+	4.95103i	−0.502701	+	0.502701i	−0.912276	−	0.409575i	$-0.865677\pi$
$97$	−4.95103	+	4.95103i	−0.502701	+	0.502701i	0.409575	+	0.912276i	$0.365677\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			12.1040i			1.20439i	0.798348	+	0.602196i	$0.205707\pi$
$101$			12.1040i			1.20439i	−0.798348	+	0.602196i	$0.794293\pi$
$102$		0			0
$103$	−1.32538	−	1.32538i	−0.130593	−	0.130593i	0.638789	−	0.769382i	$-0.279436\pi$
$103$	−1.32538	−	1.32538i	−0.130593	−	0.130593i	−0.769382	+	0.638789i	$0.779436\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9.57371	−	9.57371i	−0.925525	−	0.925525i	0.0718874	−	0.997413i	$-0.477098\pi$
$107$	−9.57371	−	9.57371i	−0.925525	−	0.925525i	−0.997413	+	0.0718874i	$0.977098\pi$
$108$		0			0
$109$			6.16462i			0.590463i	0.955426	+	0.295232i	$0.0953968\pi$
$109$			6.16462i			0.590463i	−0.955426	+	0.295232i	$0.904603\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.31966	−	4.31966i	0.406360	−	0.406360i	−0.474107	−	0.880467i	$-0.657229\pi$
$113$	4.31966	−	4.31966i	0.406360	−	0.406360i	0.880467	+	0.474107i	$0.157229\pi$
$114$		0			0
$115$	13.9693	−	5.13115i	1.30264	−	0.478482i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.27385			−0.208444
$120$		0			0
$121$	4.93796			0.448905
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.98922	−	10.7733i	−0.267364	−	0.963596i
$126$		0			0
$127$	−7.81705	+	7.81705i	−0.693651	+	0.693651i	−0.963033	−	0.269383i	$-0.913180\pi$
$127$	−7.81705	+	7.81705i	−0.693651	+	0.693651i	0.269383	+	0.963033i	$0.413180\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			10.1135i			0.883624i	0.897108	+	0.441812i	$0.145664\pi$
$131$			10.1135i			0.883624i	−0.897108	+	0.441812i	$0.854336\pi$
$132$		0			0
$133$	7.02307	+	7.02307i	0.608978	+	0.608978i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	12.4342	+	12.4342i	1.06233	+	1.06233i	0.997924	+	0.0644032i	$0.0205144\pi$
$137$	12.4342	+	12.4342i	1.06233	+	1.06233i	0.0644032	+	0.997924i	$0.479486\pi$
$138$		0			0
$139$			2.79014i			0.236657i	0.992975	+	0.118328i	$0.0377536\pi$
$139$			2.79014i			0.236657i	−0.992975	+	0.118328i	$0.962246\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.74098	−	1.74098i	0.145588	−	0.145588i
$144$		0			0
$145$	13.2459	+	6.12912i	1.10001	+	0.508996i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−23.0719			−1.89013			−0.945063	−	0.326889i	$-0.894000\pi$
$149$	−23.0719			−1.89013			−0.945063	+	0.326889i	$0.894000\pi$
$150$		0			0
$151$	0.562608			0.0457844			0.0228922	−	0.999738i	$-0.492713\pi$
$151$	0.562608			0.0457844			0.0228922	+	0.999738i	$0.492713\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−14.4406	−	6.68195i	−1.15990	−	0.536707i
$156$		0			0
$157$	−0.533485	+	0.533485i	−0.0425767	+	0.0425767i	−0.728075	−	0.685498i	$-0.759585\pi$
$157$	−0.533485	+	0.533485i	−0.0425767	+	0.0425767i	0.685498	+	0.728075i	$0.259585\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	8.77894i		−	0.691877i
$162$		0			0
$163$	3.32364	+	3.32364i	0.260327	+	0.260327i	0.825187	−	0.564860i	$-0.191070\pi$
$163$	3.32364	+	3.32364i	0.260327	+	0.260327i	−0.564860	+	0.825187i	$0.691070\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	14.9029	+	14.9029i	1.15322	+	1.15322i	0.985902	+	0.167322i	$0.0535121\pi$
$167$	14.9029	+	14.9029i	1.15322	+	1.15322i	0.167322	+	0.985902i	$0.446488\pi$
$168$		0			0
$169$			1.00000i			0.0769231i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	15.7519	−	15.7519i	1.19759	−	1.19759i	0.222710	−	0.974885i	$-0.428510\pi$
$173$	15.7519	−	15.7519i	1.19759	−	1.19759i	0.974885	−	0.222710i	$-0.0714903\pi$
$174$		0			0
$175$	−6.57357	−	0.536020i	−0.496915	−	0.0405193i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	0.771165			0.0576396			0.0288198	−	0.999585i	$-0.490825\pi$
$179$	0.771165			0.0576396			0.0288198	+	0.999585i	$0.490825\pi$
$180$		0			0
$181$	−22.6881			−1.68640			−0.843198	−	0.537603i	$-0.819330\pi$
$181$	−22.6881			−1.68640			−0.843198	+	0.537603i	$0.819330\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	15.6695	−	5.75570i	1.15205	−	0.423167i
$186$		0			0
$187$	3.00114	−	3.00114i	0.219465	−	0.219465i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			18.2796i			1.32267i	0.750093	+	0.661333i	$0.230009\pi$
$191$			18.2796i			1.32267i	−0.750093	+	0.661333i	$0.769991\pi$
$192$		0			0
$193$	−4.26791	−	4.26791i	−0.307211	−	0.307211i	0.536616	−	0.843827i	$-0.319702\pi$
$193$	−4.26791	−	4.26791i	−0.307211	−	0.307211i	−0.843827	+	0.536616i	$0.819702\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−13.7146	−	13.7146i	−0.977123	−	0.977123i	0.0226212	−	0.999744i	$-0.492799\pi$
$197$	−13.7146	−	13.7146i	−0.977123	−	0.977123i	−0.999744	+	0.0226212i	$0.992799\pi$
$198$		0			0
$199$		−	22.5559i		−	1.59895i	−0.600700	−	0.799474i	$-0.705111\pi$
$199$		−	22.5559i		−	1.59895i	0.600700	−	0.799474i	$-0.294889\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.08809	−	6.08809i	0.427300	−	0.427300i
$204$		0			0
$205$	−4.42682	+	9.56699i	−0.309183	+	0.668188i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−18.5388			−1.28235
$210$		0			0
$211$	3.79202			0.261053			0.130527	−	0.991445i	$-0.458333\pi$
$211$	3.79202			0.261053			0.130527	+	0.991445i	$0.458333\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.24876	−	22.4568i	−0.562561	−	1.53154i
$216$		0			0
$217$	−6.63722	+	6.63722i	−0.450564	+	0.450564i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			1.72382i			0.115957i
$222$		0			0
$223$	14.5849	+	14.5849i	0.976674	+	0.976674i	0.999734	−	0.0230597i	$-0.00734077\pi$
$223$	14.5849	+	14.5849i	0.976674	+	0.976674i	−0.0230597	+	0.999734i	$0.507341\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	16.1903	+	16.1903i	1.07459	+	1.07459i	0.996984	+	0.0776050i	$0.0247273\pi$
$227$	16.1903	+	16.1903i	1.07459	+	1.07459i	0.0776050	+	0.996984i	$0.475273\pi$
$228$		0			0
$229$		−	6.52126i		−	0.430937i	−0.976511	−	0.215469i	$-0.930872\pi$
$229$		−	6.52126i		−	0.430937i	0.976511	−	0.215469i	$-0.0691279\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−18.7189	+	18.7189i	−1.22632	+	1.22632i	−0.260971	+	0.965347i	$0.584043\pi$
$233$	−18.7189	+	18.7189i	−1.22632	+	1.22632i	−0.965347	+	0.260971i	$0.915957\pi$
$234$		0			0
$235$	1.78423	+	4.85746i	0.116390	+	0.316866i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.88154			0.251076			0.125538	−	0.992089i	$-0.459934\pi$
$239$	3.88154			0.251076			0.125538	+	0.992089i	$0.459934\pi$
$240$		0			0
$241$	14.5300			0.935963			0.467981	−	0.883738i	$-0.344981\pi$
$241$	14.5300			0.935963			0.467981	+	0.883738i	$0.344981\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.93925	−	10.6744i	0.315557	−	0.681964i
$246$		0			0
$247$	5.32423	−	5.32423i	0.338773	−	0.338773i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			9.87535i			0.623327i	0.950193	+	0.311663i	$0.100886\pi$
$251$			9.87535i			0.623327i	−0.950193	+	0.311663i	$0.899114\pi$
$252$		0			0
$253$	11.5869	+	11.5869i	0.728460	+	0.728460i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−15.8597	−	15.8597i	−0.989298	−	0.989298i	0.0106454	−	0.999943i	$-0.496611\pi$
$257$	−15.8597	−	15.8597i	−0.989298	−	0.989298i	−0.999943	+	0.0106454i	$0.996611\pi$
$258$		0			0
$259$		−	9.84748i		−	0.611892i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	10.6873	−	10.6873i	0.659006	−	0.659006i	−0.296139	−	0.955145i	$-0.595699\pi$
$263$	10.6873	−	10.6873i	0.659006	−	0.659006i	0.955145	+	0.296139i	$0.0956992\pi$
$264$		0			0
$265$	−10.6757	+	3.92137i	−0.655802	+	0.240888i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−27.2478			−1.66133			−0.830663	−	0.556776i	$-0.812038\pi$
$269$	−27.2478			−1.66133			−0.830663	+	0.556776i	$0.812038\pi$
$270$		0			0
$271$	24.7612			1.50414			0.752068	−	0.659085i	$-0.229056\pi$
$271$	24.7612			1.50414			0.752068	+	0.659085i	$0.229056\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	9.38358	−	7.96865i	0.565851	−	0.480528i
$276$		0			0
$277$	11.2845	−	11.2845i	0.678018	−	0.678018i	−0.281534	−	0.959551i	$-0.590843\pi$
$277$	11.2845	−	11.2845i	0.678018	−	0.678018i	0.959551	+	0.281534i	$0.0908431\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	8.98033i		−	0.535722i	−0.963458	−	0.267861i	$-0.913683\pi$
$281$		−	8.98033i		−	0.535722i	0.963458	−	0.267861i	$-0.0863167\pi$
$282$		0			0
$283$	−9.00334	−	9.00334i	−0.535193	−	0.535193i	0.386920	−	0.922113i	$-0.373539\pi$
$283$	−9.00334	−	9.00334i	−0.535193	−	0.535193i	−0.922113	+	0.386920i	$0.873539\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.39718	+	4.39718i	0.259557	+	0.259557i
$288$		0			0
$289$		−	14.0284i		−	0.825202i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.94386	+	9.94386i	−0.580926	+	0.580926i	−0.935158	−	0.354231i	$-0.884743\pi$
$293$	−9.94386	+	9.94386i	−0.580926	+	0.580926i	0.354231	+	0.935158i	$0.384743\pi$
$294$		0			0
$295$	10.9507	+	5.06708i	0.637574	+	0.295017i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.65536			−0.384889
$300$		0			0
$301$	−14.1129			−0.813453
$302$		0			0
$303$		0			0
$304$		0			0
$305$	24.6355	+	11.3993i	1.41062	+	0.652721i
$306$		0			0
$307$	18.9943	−	18.9943i	1.08406	−	1.08406i	0.0879339	−	0.996126i	$-0.471974\pi$
$307$	18.9943	−	18.9943i	1.08406	−	1.08406i	0.996126	−	0.0879339i	$-0.0280264\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			1.78543i			0.101243i	0.998718	+	0.0506213i	$0.0161201\pi$
$311$			1.78543i			0.101243i	−0.998718	+	0.0506213i	$0.983880\pi$
$312$		0			0
$313$	−23.4660	−	23.4660i	−1.32638	−	1.32638i	−0.908505	−	0.417873i	$-0.862776\pi$
$313$	−23.4660	−	23.4660i	−1.32638	−	1.32638i	−0.417873	−	0.908505i	$-0.637224\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.75170	+	7.75170i	0.435379	+	0.435379i	0.890453	−	0.455075i	$-0.150387\pi$
$317$	7.75170	+	7.75170i	0.435379	+	0.435379i	−0.455075	+	0.890453i	$0.650387\pi$
$318$		0			0
$319$			16.0707i			0.899786i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	9.17802	−	9.17802i	0.510678	−	0.510678i
$324$		0			0
$325$	−0.406359	+	4.98346i	−0.0225408	+	0.276433i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.05266			0.168298
$330$		0			0
$331$	−7.25581			−0.398815			−0.199408	−	0.979917i	$-0.563902\pi$
$331$	−7.25581			−0.398815			−0.199408	+	0.979917i	$0.563902\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.48999	−	2.38389i	0.354586	−	0.130246i
$336$		0			0
$337$	−12.4606	+	12.4606i	−0.678770	+	0.678770i	−0.959722	−	0.280952i	$-0.909350\pi$
$337$	−12.4606	+	12.4606i	−0.678770	+	0.678770i	0.280952	+	0.959722i	$0.409350\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	17.5202i		−	0.948774i
$342$		0			0
$343$	−11.4353	−	11.4353i	−0.617448	−	0.617448i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.947599	−	0.947599i	−0.0508698	−	0.0508698i	0.681214	−	0.732084i	$-0.261452\pi$
$347$	−0.947599	−	0.947599i	−0.0508698	−	0.0508698i	−0.732084	+	0.681214i	$0.761452\pi$
$348$		0			0
$349$		−	11.7708i		−	0.630074i	−0.949079	−	0.315037i	$-0.897983\pi$
$349$		−	11.7708i		−	0.630074i	0.949079	−	0.315037i	$-0.102017\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.06174	+	3.06174i	−0.162960	+	0.162960i	−0.783877	−	0.620917i	$-0.786761\pi$
$353$	−3.06174	+	3.06174i	−0.162960	+	0.162960i	0.620917	+	0.783877i	$0.286761\pi$
$354$		0			0
$355$	4.77449	−	10.3184i	0.253404	−	0.547641i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.6325			−0.561162			−0.280581	−	0.959830i	$-0.590527\pi$
$359$	−10.6325			−0.561162			−0.280581	+	0.959830i	$0.590527\pi$
$360$		0			0
$361$	−37.6948			−1.98394
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.30435	−	8.99589i	−0.172957	−	0.470866i
$366$		0			0
$367$	−23.8669	+	23.8669i	−1.24584	+	1.24584i	−0.288305	+	0.957539i	$0.593092\pi$
$367$	−23.8669	+	23.8669i	−1.24584	+	1.24584i	−0.957539	+	0.288305i	$0.906908\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			6.70910i			0.348319i
$372$		0			0
$373$	7.35733	+	7.35733i	0.380948	+	0.380948i	0.871444	−	0.490496i	$-0.163184\pi$
$373$	7.35733	+	7.35733i	0.380948	+	0.380948i	−0.490496	+	0.871444i	$0.663184\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.61541	−	4.61541i	−0.237706	−	0.237706i
$378$		0			0
$379$		−	12.8168i		−	0.658357i	−0.944268	−	0.329179i	$-0.893228\pi$
$379$		−	12.8168i		−	0.658357i	0.944268	−	0.329179i	$-0.106772\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.95089	+	9.95089i	−0.508467	+	0.508467i	−0.914056	−	0.405589i	$-0.867066\pi$
$383$	−9.95089	+	9.95089i	−0.508467	+	0.508467i	0.405589	+	0.914056i	$0.367066\pi$
$384$		0			0
$385$	−2.50394	−	6.81682i	−0.127612	−	0.347417i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	25.8520			1.31075			0.655375	−	0.755303i	$-0.272511\pi$
$389$	25.8520			1.31075			0.655375	+	0.755303i	$0.272511\pi$
$390$		0			0
$391$	−11.4726			−0.580196
$392$		0			0
$393$		0			0
$394$		0			0
$395$	15.0258	−	32.4728i	0.756028	−	1.63388i
$396$		0			0
$397$	−7.36720	+	7.36720i	−0.369749	+	0.369749i	−0.867386	−	0.497637i	$-0.834201\pi$
$397$	−7.36720	+	7.36720i	−0.369749	+	0.369749i	0.497637	+	0.867386i	$0.334201\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			12.0303i			0.600763i	0.953819	+	0.300382i	$0.0971140\pi$
$401$			12.0303i			0.600763i	−0.953819	+	0.300382i	$0.902886\pi$
$402$		0			0
$403$	5.03171	+	5.03171i	0.250647	+	0.250647i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.9972	+	12.9972i	0.644246	+	0.644246i
$408$		0			0
$409$		−	29.6258i		−	1.46490i	−0.680820	−	0.732451i	$-0.738376\pi$
$409$		−	29.6258i		−	1.46490i	0.680820	−	0.732451i	$-0.261624\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.03316	−	5.03316i	0.247665	−	0.247665i
$414$		0			0
$415$	23.5628	−	8.65504i	1.15665	−	0.424859i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.07031			−0.101141			−0.0505706	−	0.998720i	$-0.516104\pi$
$419$	−2.07031			−0.101141			−0.0505706	+	0.998720i	$0.516104\pi$
$420$		0			0
$421$	9.96717			0.485770			0.242885	−	0.970055i	$-0.421906\pi$
$421$	9.96717			0.485770			0.242885	+	0.970055i	$0.421906\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.700491	+	8.59059i	−0.0339788	+	0.416705i
$426$		0			0
$427$	11.3230	−	11.3230i	0.547957	−	0.547957i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	7.70549i		−	0.371160i	−0.982629	−	0.185580i	$-0.940584\pi$
$431$		−	7.70549i		−	0.371160i	0.982629	−	0.185580i	$-0.0594164\pi$
$432$		0			0
$433$	−9.74531	−	9.74531i	−0.468330	−	0.468330i	0.433043	−	0.901373i	$-0.357440\pi$
$433$	−9.74531	−	9.74531i	−0.468330	−	0.468330i	−0.901373	+	0.433043i	$0.857440\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	35.4346	+	35.4346i	1.69507	+	1.69507i
$438$		0			0
$439$		−	1.08773i		−	0.0519143i	−0.999663	−	0.0259572i	$-0.991737\pi$
$439$		−	1.08773i		−	0.0519143i	0.999663	−	0.0259572i	$-0.00826335\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.96876	−	4.96876i	0.236073	−	0.236073i	−0.579149	−	0.815222i	$-0.696615\pi$
$443$	4.96876	−	4.96876i	0.236073	−	0.236073i	0.815222	+	0.579149i	$0.196615\pi$
$444$		0			0
$445$	1.89798	+	0.878231i	0.0899730	+	0.0416322i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	11.2250			0.529742			0.264871	−	0.964284i	$-0.414671\pi$
$449$	11.2250			0.529742			0.264871	+	0.964284i	$0.414671\pi$
$450$		0			0
$451$	−11.6072			−0.546563
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.67687	+	1.23864i	0.125493	+	0.0580681i
$456$		0			0
$457$	−20.6809	+	20.6809i	−0.967413	+	0.967413i	−0.999486	−	0.0320721i	$-0.989789\pi$
$457$	−20.6809	+	20.6809i	−0.967413	+	0.967413i	0.0320721	+	0.999486i	$0.489789\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			36.7705i			1.71257i	0.516501	+	0.856286i	$0.327234\pi$
$461$			36.7705i			1.71257i	−0.516501	+	0.856286i	$0.672766\pi$
$462$		0			0
$463$	0.627674	+	0.627674i	0.0291705	+	0.0291705i	0.721542	−	0.692371i	$-0.243434\pi$
$463$	0.627674	+	0.627674i	0.0291705	+	0.0291705i	−0.692371	+	0.721542i	$0.743434\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	21.9883	+	21.9883i	1.01750	+	1.01750i	0.999844	+	0.0176551i	$0.00562008\pi$
$467$	21.9883	+	21.9883i	1.01750	+	1.01750i	0.0176551	+	0.999844i	$0.494380\pi$
$468$		0			0
$469$		−	4.07862i		−	0.188333i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	18.6269	−	18.6269i	0.856463	−	0.856463i
$474$		0			0
$475$	28.6966	−	24.3695i	1.31669	−	1.11815i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.33254			0.426414			0.213207	−	0.977007i	$-0.431609\pi$
$479$	9.33254			0.426414			0.213207	+	0.977007i	$0.431609\pi$
$480$		0			0
$481$	−7.46542			−0.340394
$482$		0			0
$483$		0			0
$484$		0			0
$485$	14.6965	−	5.39826i	0.667331	−	0.245122i
$486$		0			0
$487$	18.0280	−	18.0280i	0.816929	−	0.816929i	−0.168733	−	0.985662i	$-0.553968\pi$
$487$	18.0280	−	18.0280i	0.816929	−	0.816929i	0.985662	+	0.168733i	$0.0539675\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			24.9098i			1.12416i	0.827082	+	0.562081i	$0.189999\pi$
$491$			24.9098i			1.12416i	−0.827082	+	0.562081i	$0.810001\pi$
$492$		0			0
$493$	−7.95614	−	7.95614i	−0.358326	−	0.358326i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.74253	−	4.74253i	−0.212731	−	0.212731i
$498$		0			0
$499$		−	24.1119i		−	1.07940i	−0.841858	−	0.539699i	$-0.818538\pi$
$499$		−	24.1119i		−	1.07940i	0.841858	−	0.539699i	$-0.181462\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	18.5211	−	18.5211i	0.825816	−	0.825816i	−0.161119	−	0.986935i	$-0.551510\pi$
$503$	18.5211	−	18.5211i	0.825816	−	0.825816i	0.986935	+	0.161119i	$0.0515104\pi$
$504$		0			0
$505$	11.3658	−	24.5632i	0.505773	−	1.09305i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.898160			−0.0398103			−0.0199051	−	0.999802i	$-0.506336\pi$
$509$	−0.898160			−0.0398103			−0.0199051	+	0.999802i	$0.506336\pi$
$510$		0			0
$511$	−5.65344			−0.250093
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.44510	+	3.93421i	0.0636788	+	0.173362i
$516$		0			0
$517$	−4.02904	+	4.02904i	−0.177197	+	0.177197i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	32.4911i		−	1.42346i	−0.702452	−	0.711731i	$-0.747911\pi$
$521$		−	32.4911i		−	1.42346i	0.702452	−	0.711731i	$-0.252089\pi$
$522$		0			0
$523$	17.8580	+	17.8580i	0.780877	+	0.780877i	0.979979	−	0.199102i	$-0.0638024\pi$
$523$	17.8580	+	17.8580i	0.780877	+	0.780877i	−0.199102	+	0.979979i	$0.563802\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.67376	+	8.67376i	0.377835	+	0.377835i
$528$		0			0
$529$		−	21.2938i		−	0.925816i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.33353	−	3.33353i	0.144391	−	0.144391i
$534$		0			0
$535$	10.4385	+	28.4182i	0.451296	+	1.22863i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	12.9508			0.557832
$540$		0			0
$541$	2.39106			0.102800			0.0513998	−	0.998678i	$-0.483632\pi$
$541$	2.39106			0.102800			0.0513998	+	0.998678i	$0.483632\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5.78867	−	12.5101i	0.247960	−	0.535876i
$546$		0			0
$547$	−9.12276	+	9.12276i	−0.390061	+	0.390061i	−0.874709	−	0.484648i	$-0.838948\pi$
$547$	−9.12276	+	9.12276i	−0.390061	+	0.390061i	0.484648	+	0.874709i	$0.338948\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			49.1470i			2.09373i
$552$		0			0
$553$	−14.9252	−	14.9252i	−0.634682	−	0.634682i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−13.2341	−	13.2341i	−0.560748	−	0.560748i	0.368772	−	0.929520i	$-0.379778\pi$
$557$	−13.2341	−	13.2341i	−0.560748	−	0.560748i	−0.929520	+	0.368772i	$0.879778\pi$
$558$		0			0
$559$			10.6990i			0.452521i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	19.0737	−	19.0737i	0.803861	−	0.803861i	−0.179835	−	0.983697i	$-0.557556\pi$
$563$	19.0737	−	19.0737i	0.803861	−	0.803861i	0.983697	+	0.179835i	$0.0575565\pi$
$564$		0			0
$565$	−12.8223	+	4.70986i	−0.539439	+	0.198145i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−38.8443			−1.62844			−0.814218	−	0.580559i	$-0.802834\pi$
$569$	−38.8443			−1.62844			−0.814218	+	0.580559i	$0.802834\pi$
$570$		0			0
$571$	7.02706			0.294073			0.147037	−	0.989131i	$-0.453026\pi$
$571$	7.02706			0.294073			0.147037	+	0.989131i	$0.453026\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−33.1667	−	2.70447i	−1.38315	−	0.112784i
$576$		0			0
$577$	−11.4785	+	11.4785i	−0.477856	+	0.477856i	−0.904446	−	0.426589i	$-0.859715\pi$
$577$	−11.4785	+	11.4785i	−0.477856	+	0.477856i	0.426589	+	0.904446i	$0.359715\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	14.8080i		−	0.614339i
$582$		0			0
$583$	−8.85499	−	8.85499i	−0.366736	−	0.366736i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10.9683	−	10.9683i	−0.452711	−	0.452711i	0.443542	−	0.896253i	$-0.353722\pi$
$587$	−10.9683	−	10.9683i	−0.452711	−	0.452711i	−0.896253	+	0.443542i	$0.853722\pi$
$588$		0			0
$589$		−	53.5799i		−	2.20772i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	24.5934	−	24.5934i	1.00993	−	1.00993i	0.00997888	−	0.999950i	$-0.496824\pi$
$593$	24.5934	−	24.5934i	1.00993	−	1.00993i	0.999950	−	0.00997888i	$-0.00317643\pi$
$594$		0			0
$595$	4.61444	+	2.13518i	0.189174	+	0.0875340i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−24.1117			−0.985177			−0.492589	−	0.870262i	$-0.663949\pi$
$599$	−24.1117			−0.985177			−0.492589	+	0.870262i	$0.663949\pi$
$600$		0			0
$601$	−7.91998			−0.323063			−0.161531	−	0.986868i	$-0.551643\pi$
$601$	−7.91998			−0.323063			−0.161531	+	0.986868i	$0.551643\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−10.0208	−	4.63682i	−0.407405	−	0.188514i
$606$		0			0
$607$	17.4916	−	17.4916i	0.709961	−	0.709961i	−0.256566	−	0.966527i	$-0.582591\pi$
$607$	17.4916	−	17.4916i	0.709961	−	0.709961i	0.966527	+	0.256566i	$0.0825911\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	2.31423i		−	0.0936239i
$612$		0			0
$613$	−20.0407	−	20.0407i	−0.809434	−	0.809434i	0.175114	−	0.984548i	$-0.443971\pi$
$613$	−20.0407	−	20.0407i	−0.809434	−	0.809434i	−0.984548	+	0.175114i	$0.943971\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.21684	−	6.21684i	−0.250281	−	0.250281i	0.570805	−	0.821086i	$-0.306631\pi$
$617$	−6.21684	−	6.21684i	−0.250281	−	0.250281i	−0.821086	+	0.570805i	$0.806631\pi$
$618$		0			0
$619$		−	29.0915i		−	1.16929i	−0.811290	−	0.584643i	$-0.801234\pi$
$619$		−	29.0915i		−	1.16929i	0.811290	−	0.584643i	$-0.198766\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.872352	−	0.872352i	0.0349500	−	0.0349500i
$624$		0			0
$625$	−4.05015	+	24.6697i	−0.162006	+	0.986790i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.8691			−0.513123
$630$		0			0
$631$	−35.3792			−1.40842			−0.704211	−	0.709991i	$-0.748699\pi$
$631$	−35.3792			−1.40842			−0.704211	+	0.709991i	$0.748699\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	23.2038	−	8.52317i	0.920816	−	0.338232i
$636$		0			0
$637$	−3.71940	+	3.71940i	−0.147368	+	0.147368i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			20.1152i			0.794501i	0.917710	+	0.397251i	$0.130036\pi$
$641$			20.1152i			0.794501i	−0.917710	+	0.397251i	$0.869964\pi$
$642$		0			0
$643$	−10.1144	−	10.1144i	−0.398871	−	0.398871i	0.478964	−	0.877835i	$-0.341013\pi$
$643$	−10.1144	−	10.1144i	−0.398871	−	0.398871i	−0.877835	+	0.478964i	$0.841013\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3.36761	+	3.36761i	0.132394	+	0.132394i	0.770199	−	0.637804i	$-0.220157\pi$
$647$	3.36761	+	3.36761i	0.132394	+	0.132394i	−0.637804	+	0.770199i	$0.720157\pi$
$648$		0			0
$649$			13.2860i			0.521521i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.413873	−	0.413873i	0.0161961	−	0.0161961i	−0.698962	−	0.715158i	$-0.746355\pi$
$653$	0.413873	−	0.413873i	0.0161961	−	0.0161961i	0.715158	+	0.698962i	$0.246355\pi$
$654$		0			0
$655$	9.49678	−	20.5239i	0.371070	−	0.801935i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	22.3354			0.870064			0.435032	−	0.900415i	$-0.356737\pi$
$659$	22.3354			0.870064			0.435032	+	0.900415i	$0.356737\pi$
$660$		0			0
$661$	1.64547			0.0640014			0.0320007	−	0.999488i	$-0.489812\pi$
$661$	1.64547			0.0640014			0.0320007	+	0.999488i	$0.489812\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−7.65748	−	20.8470i	−0.296944	−	0.808413i
$666$		0			0
$667$	30.7172	−	30.7172i	1.18937	−	1.18937i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			29.8892i			1.15386i
$672$		0			0
$673$	10.5570	+	10.5570i	0.406944	+	0.406944i	0.880672	−	0.473727i	$-0.157092\pi$
$673$	10.5570	+	10.5570i	0.406944	+	0.406944i	−0.473727	+	0.880672i	$0.657092\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	11.9259	+	11.9259i	0.458350	+	0.458350i	0.898114	−	0.439764i	$-0.144938\pi$
$677$	11.9259	+	11.9259i	0.458350	+	0.458350i	−0.439764	+	0.898114i	$0.644938\pi$
$678$		0			0
$679$		−	9.23594i		−	0.354443i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	15.8560	−	15.8560i	0.606712	−	0.606712i	−0.335374	−	0.942085i	$-0.608863\pi$
$683$	15.8560	−	15.8560i	0.606712	−	0.606712i	0.942085	+	0.335374i	$0.108863\pi$
$684$		0			0
$685$	−13.5574	−	36.9093i	−0.518002	−	1.41023i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	5.08621			0.193769
$690$		0			0
$691$	−22.1692			−0.843357			−0.421678	−	0.906746i	$-0.638559\pi$
$691$	−22.1692			−0.843357			−0.421678	+	0.906746i	$0.638559\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.61999	−	5.66217i	0.0993818	−	0.214778i
$696$		0			0
$697$	5.74640	−	5.74640i	0.217661	−	0.217661i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	31.1856i		−	1.17787i	−0.808182	−	0.588933i	$-0.799548\pi$
$701$		−	31.1856i		−	1.17787i	0.808182	−	0.588933i	$-0.200452\pi$
$702$		0			0
$703$	39.7476	+	39.7476i	1.49911	+	1.49911i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−11.2897	−	11.2897i	−0.424594	−	0.424594i
$708$		0			0
$709$		−	18.2641i		−	0.685921i	−0.939350	−	0.342961i	$-0.888570\pi$
$709$		−	18.2641i		−	0.685921i	0.939350	−	0.342961i	$-0.111430\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−33.4878	+	33.4878i	−1.25413	+	1.25413i
$714$		0			0
$715$	−5.16787	+	1.89825i	−0.193267	+	0.0709904i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	5.09190			0.189896			0.0949479	−	0.995482i	$-0.469732\pi$
$719$	5.09190			0.189896			0.0949479	+	0.995482i	$0.469732\pi$
$720$		0			0
$721$	2.47244			0.0920784
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−21.1252	−	24.8762i	−0.784570	−	0.923880i
$726$		0			0
$727$	−20.8594	+	20.8594i	−0.773632	+	0.773632i	−0.978739	−	0.205108i	$-0.934246\pi$
$727$	−20.8594	+	20.8594i	−0.773632	+	0.773632i	0.205108	+	0.978739i	$0.434246\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			18.4432i			0.682148i
$732$		0			0
$733$	26.9173	+	26.9173i	0.994212	+	0.994212i	0.999983	−	0.00577109i	$-0.00183701\pi$
$733$	26.9173	+	26.9173i	0.994212	+	0.994212i	−0.00577109	+	0.999983i	$0.501837\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.38315	+	5.38315i	0.198291	+	0.198291i
$738$		0			0
$739$		−	18.5886i		−	0.683794i	−0.939737	−	0.341897i	$-0.888931\pi$
$739$		−	18.5886i		−	0.683794i	0.939737	−	0.341897i	$-0.111069\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−10.3229	+	10.3229i	−0.378710	+	0.378710i	−0.870637	−	0.491926i	$-0.836293\pi$
$743$	−10.3229	+	10.3229i	−0.378710	+	0.378710i	0.491926	+	0.870637i	$0.336293\pi$
$744$		0			0
$745$	46.8209	+	21.6649i	1.71539	+	0.793741i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	17.8594			0.652567
$750$		0			0
$751$	−2.89363			−0.105590			−0.0527950	−	0.998605i	$-0.516813\pi$
$751$	−2.89363			−0.105590			−0.0527950	+	0.998605i	$0.516813\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.14173	−	0.528298i	−0.0415517	−	0.0192267i
$756$		0			0
$757$	−15.3615	+	15.3615i	−0.558324	+	0.558324i	−0.928830	−	0.370506i	$-0.879184\pi$
$757$	−15.3615	+	15.3615i	−0.558324	+	0.558324i	0.370506	+	0.928830i	$0.379184\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	19.7719i		−	0.716730i	−0.933582	−	0.358365i	$-0.883334\pi$
$761$		−	19.7719i		−	0.716730i	0.933582	−	0.358365i	$-0.116666\pi$
$762$		0			0
$763$	−5.74992	−	5.74992i	−0.208161	−	0.208161i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.81566	−	3.81566i	−0.137776	−	0.137776i
$768$		0			0
$769$			29.3463i			1.05825i	0.848543	+	0.529127i	$0.177481\pi$
$769$			29.3463i			1.05825i	−0.848543	+	0.529127i	$0.822519\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	21.4655	−	21.4655i	0.772059	−	0.772059i	−0.206407	−	0.978466i	$-0.566177\pi$
$773$	21.4655	−	21.4655i	0.772059	−	0.772059i	0.978466	+	0.206407i	$0.0661772\pi$
$774$		0			0
$775$	23.0306	+	27.1200i	0.827285	+	0.974179i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−35.4969			−1.27181
$780$		0			0
$781$	12.5188			0.447959
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.58358	−	0.581675i	0.0565202	−	0.0207609i
$786$		0			0
$787$	−9.96255	+	9.96255i	−0.355126	+	0.355126i	−0.862013	−	0.506886i	$-0.830796\pi$
$787$	−9.96255	+	9.96255i	−0.355126	+	0.355126i	0.506886	+	0.862013i	$0.330796\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			8.05815i			0.286515i
$792$		0			0
$793$	−8.58400	−	8.58400i	−0.304827	−	0.304827i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	29.9101	+	29.9101i	1.05947	+	1.05947i	0.998116	+	0.0613542i	$0.0195419\pi$
$797$	29.9101	+	29.9101i	1.05947	+	1.05947i	0.0613542	+	0.998116i	$0.480458\pi$
$798$		0			0
$799$		−	3.98933i		−	0.141132i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7.46168	−	7.46168i	0.263317	−	0.263317i
$804$		0			0
$805$	−8.24356	+	17.8155i	−0.290547	+	0.627914i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−46.5232			−1.63567			−0.817834	−	0.575454i	$-0.804825\pi$
$809$	−46.5232			−1.63567			−0.817834	+	0.575454i	$0.804825\pi$
$810$		0			0
$811$	−6.18421			−0.217157			−0.108578	−	0.994088i	$-0.534630\pi$
$811$	−6.18421			−0.217157			−0.108578	+	0.994088i	$0.534630\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.62387	−	9.86576i	−0.126939	−	0.345583i
$816$		0			0
$817$	56.9642	−	56.9642i	1.99292	−	1.99292i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	29.6630i		−	1.03525i	−0.855608	−	0.517624i	$-0.826817\pi$
$821$		−	29.6630i		−	1.03525i	0.855608	−	0.517624i	$-0.173183\pi$
$822$		0			0
$823$	14.5754	+	14.5754i	0.508065	+	0.508065i	0.913932	−	0.405867i	$-0.133030\pi$
$823$	14.5754	+	14.5754i	0.508065	+	0.508065i	−0.405867	+	0.913932i	$0.633030\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	9.65283	+	9.65283i	0.335662	+	0.335662i	0.854732	−	0.519070i	$-0.173722\pi$
$827$	9.65283	+	9.65283i	0.335662	+	0.335662i	−0.519070	+	0.854732i	$0.673722\pi$
$828$		0			0
$829$			43.5282i			1.51180i	0.654688	+	0.755899i	$0.272800\pi$
$829$			43.5282i			1.51180i	−0.654688	+	0.755899i	$0.727200\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−6.41159	+	6.41159i	−0.222148	+	0.222148i
$834$		0			0
$835$	−16.2491	−	44.2374i	−0.562325	−	1.53090i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−2.74571			−0.0947926			−0.0473963	−	0.998876i	$-0.515092\pi$
$839$	−2.74571			−0.0947926			−0.0473963	+	0.998876i	$0.515092\pi$
$840$		0			0
$841$	13.6040			0.469104
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.939016	−	2.02935i	0.0323031	−	0.0698117i
$846$		0			0
$847$	−4.60578	+	4.60578i	−0.158256	+	0.158256i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	49.6851i		−	1.70318i
$852$		0			0
$853$	−26.9180	−	26.9180i	−0.921655	−	0.921655i	0.0754913	−	0.997146i	$-0.475947\pi$
$853$	−26.9180	−	26.9180i	−0.921655	−	0.921655i	−0.997146	+	0.0754913i	$0.975947\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.01644	+	9.01644i	0.307996	+	0.307996i	0.844132	−	0.536136i	$-0.180117\pi$
$857$	9.01644	+	9.01644i	0.307996	+	0.307996i	−0.536136	+	0.844132i	$0.680117\pi$
$858$		0			0
$859$		−	25.4335i		−	0.867779i	−0.900966	−	0.433889i	$-0.857141\pi$
$859$		−	25.4335i		−	0.867779i	0.900966	−	0.433889i	$-0.142859\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−21.9159	+	21.9159i	−0.746027	+	0.746027i	−0.973731	−	0.227703i	$-0.926878\pi$
$863$	−21.9159	+	21.9159i	−0.746027	+	0.746027i	0.227703	+	0.973731i	$0.426878\pi$
$864$		0			0
$865$	−46.7574	+	17.1748i	−1.58980	+	0.583960i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	39.3979			1.33648
$870$		0			0
$871$	−3.09202			−0.104769
$872$		0			0
$873$		0			0
$874$		0			0
$875$	12.8367	+	7.26046i	0.433961	+	0.245448i
$876$		0			0
$877$	−19.5319	+	19.5319i	−0.659547	+	0.659547i	−0.955273	−	0.295726i	$-0.904439\pi$
$877$	−19.5319	+	19.5319i	−0.659547	+	0.659547i	0.295726	+	0.955273i	$0.404439\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	27.9504i		−	0.941671i	−0.882221	−	0.470836i	$-0.843952\pi$
$881$		−	27.9504i		−	0.941671i	0.882221	−	0.470836i	$-0.156048\pi$
$882$		0			0
$883$	−5.71709	−	5.71709i	−0.192396	−	0.192396i	0.604335	−	0.796730i	$-0.293439\pi$
$883$	−5.71709	−	5.71709i	−0.192396	−	0.192396i	−0.796730	+	0.604335i	$0.793439\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	11.4029	+	11.4029i	0.382871	+	0.382871i	0.872135	−	0.489265i	$-0.162735\pi$
$887$	11.4029	+	11.4029i	0.382871	+	0.382871i	−0.489265	+	0.872135i	$0.662735\pi$
$888$		0			0
$889$		−	14.5824i		−	0.489077i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−12.3215	+	12.3215i	−0.412324	+	0.412324i
$894$		0			0
$895$	−1.56496	−	0.724136i	−0.0523109	−	0.0242052i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−46.4468			−1.54909
$900$		0			0
$901$	8.76771			0.292095
$902$		0			0
$903$		0			0
$904$		0			0
$905$	46.0421	+	21.3045i	1.53049	+	0.708186i
$906$		0			0
$907$	−29.2239	+	29.2239i	−0.970364	+	0.970364i	−0.999573	−	0.0292097i	$-0.990701\pi$
$907$	−29.2239	+	29.2239i	−0.970364	+	0.970364i	0.0292097	+	0.999573i	$0.490701\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	29.3299i		−	0.971744i	−0.874030	−	0.485872i	$-0.838502\pi$
$911$		−	29.3299i		−	0.971744i	0.874030	−	0.485872i	$-0.161498\pi$
$912$		0			0
$913$	19.5443	+	19.5443i	0.646822	+	0.646822i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9.43320	−	9.43320i	−0.311512	−	0.311512i
$918$		0			0
$919$			25.3332i			0.835667i	0.908524	+	0.417833i	$0.137210\pi$
$919$			25.3332i			0.835667i	−0.908524	+	0.417833i	$0.862790\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−3.59533	+	3.59533i	−0.118342	+	0.118342i
$924$		0			0
$925$	−37.2036	−	3.03365i	−1.22325	−	0.0997456i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	9.53879			0.312958			0.156479	−	0.987681i	$-0.449986\pi$
$929$	9.53879			0.312958			0.156479	+	0.987681i	$0.449986\pi$
$930$		0			0
$931$	39.6059			1.29803
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−8.90848	+	3.27224i	−0.291338	+	0.107014i
$936$		0			0
$937$	−31.8235	+	31.8235i	−1.03963	+	1.03963i	−0.0404469	+	0.999182i	$0.512878\pi$
$937$	−31.8235	+	31.8235i	−1.03963	+	1.03963i	−0.999182	+	0.0404469i	$0.987122\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			43.6984i			1.42453i	0.701912	+	0.712264i	$0.252330\pi$
$941$			43.6984i			1.42453i	−0.701912	+	0.712264i	$0.747670\pi$
$942$		0			0
$943$	22.1858	+	22.1858i	0.722469	+	0.722469i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	27.6085	+	27.6085i	0.897154	+	0.897154i	0.995184	−	0.0980291i	$-0.0312538\pi$
$947$	27.6085	+	27.6085i	0.897154	+	0.897154i	−0.0980291	+	0.995184i	$0.531254\pi$
$948$		0			0
$949$			4.28590i			0.139126i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−6.99634	+	6.99634i	−0.226634	+	0.226634i	−0.811285	−	0.584651i	$-0.801231\pi$
$953$	−6.99634	+	6.99634i	−0.226634	+	0.226634i	0.584651	+	0.811285i	$0.301231\pi$
$954$		0			0
$955$	17.1648	−	37.0956i	0.555441	−	1.20039i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−23.1955			−0.749022
$960$		0			0
$961$	19.6362			0.633425
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.65344	+	12.6687i	0.149799	+	0.407820i
$966$		0			0
$967$	−18.1830	+	18.1830i	−0.584726	+	0.584726i	−0.936198	−	0.351472i	$-0.885681\pi$
$967$	−18.1830	+	18.1830i	−0.584726	+	0.584726i	0.351472	+	0.936198i	$0.385681\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			25.3915i			0.814851i	0.913238	+	0.407426i	$0.133573\pi$
$971$			25.3915i			0.814851i	−0.913238	+	0.407426i	$0.866427\pi$
$972$		0			0
$973$	−2.60245	−	2.60245i	−0.0834306	−	0.0834306i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	30.4477	+	30.4477i	0.974108	+	0.974108i	0.999673	−	0.0255653i	$-0.00813856\pi$
$977$	30.4477	+	30.4477i	0.974108	+	0.974108i	−0.0255653	+	0.999673i	$0.508139\pi$
$978$		0			0
$979$			2.30274i			0.0735960i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−29.2000	+	29.2000i	−0.931335	+	0.931335i	−0.997789	−	0.0664547i	$-0.978831\pi$
$983$	−29.2000	+	29.2000i	−0.931335	+	0.931335i	0.0664547	+	0.997789i	$0.478831\pi$
$984$		0			0
$985$	14.9534	+	40.7098i	0.476456	+	1.29712i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−71.2060			−2.26422
$990$		0			0
$991$	−8.19522			−0.260330			−0.130165	−	0.991492i	$-0.541551\pi$
$991$	−8.19522			−0.260330			−0.130165	+	0.991492i	$0.541551\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−21.1804	+	45.7738i	−0.671464	+	1.45113i
$996$		0			0
$997$	−13.2182	+	13.2182i	−0.418625	+	0.418625i	−0.884730	−	0.466104i	$-0.845657\pi$
$997$	−13.2182	+	13.2182i	−0.418625	+	0.418625i	0.466104	+	0.884730i	$0.345657\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.y.b.53.2	yes	24
3.2	odd	2		2340.2.y.a.53.11	✓	24
5.2	odd	4		2340.2.y.a.1457.11	yes	24
15.2	even	4	inner	2340.2.y.b.1457.2	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.y.a.53.11	✓	24	3.2	odd	2
2340.2.y.a.1457.11	yes	24	5.2	odd	4
2340.2.y.b.53.2	yes	24	1.1	even	1	trivial
2340.2.y.b.1457.2	yes	24	15.2	even	4	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.y (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.02935 - 0.939016i) q^{5} +(-0.932729 + 0.932729i) q^{7} -2.46212i q^{11} +(0.707107 + 0.707107i) q^{13} +(1.21893 + 1.21893i) q^{17} -7.52960i q^{19} +(-4.70605 + 4.70605i) q^{23} +(3.23650 + 3.81118i) q^{25} -6.52718 q^{29} +7.11591 q^{31} +(2.76868 - 1.01698i) q^{35} +(-5.27885 + 5.27885i) q^{37} -4.71432i q^{41} +(7.56537 + 7.56537i) q^{43} +(-1.63641 - 1.63641i) q^{47} +5.26003i q^{49} +(3.59649 - 3.59649i) q^{53} +(-2.31197 + 4.99650i) q^{55} -5.39616 q^{59} -12.1396 q^{61} +(-0.770981 - 2.09895i) q^{65} +(-2.18639 + 2.18639i) q^{67} +5.08457i q^{71} +(3.03059 + 3.03059i) q^{73} +(2.29649 + 2.29649i) q^{77} +16.0016i q^{79} +(-7.93799 + 7.93799i) q^{83} +(-1.32903 - 3.61821i) q^{85} -0.935268 q^{89} -1.31908 q^{91} +(-7.07041 + 15.2802i) q^{95} +(-4.95103 + 4.95103i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{5} - 8 q^{7} + 8 q^{17} + 8 q^{23} + 16 q^{25} - 32 q^{29} - 8 q^{35} + 16 q^{37} - 8 q^{43} + 40 q^{47} + 8 q^{53} + 8 q^{55} - 56 q^{59} + 8 q^{61} + 4 q^{65} - 16 q^{67} + 72 q^{77} + 32 q^{83}+ \cdots + 8 q^{97}+O(q^{100}) \) 24 * q + 4 * q^5 - 8 * q^7 + 8 * q^17 + 8 * q^23 + 16 * q^25 - 32 * q^29 - 8 * q^35 + 16 * q^37 - 8 * q^43 + 40 * q^47 + 8 * q^53 + 8 * q^55 - 56 * q^59 + 8 * q^61 + 4 * q^65 - 16 * q^67 + 72 * q^77 + 32 * q^83 - 32 * q^85 - 128 * q^89 - 8 * q^91 - 16 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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