LMFDB - Embedded newform 2340.2.bi.c.161.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,2,Mod(161,2340)] 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, 0, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.161"); 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.bi (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 10 x^{10} - 16 x^{9} + 50 x^{8} - 32 x^{7} - 110 x^{6} + 40 x^{5} + 417 x^{4} + 712 x^{3} + \cdots + 32 $ x^12 + 10x^10 - 16x^9 + 50x^8 - 32x^7 - 110x^6 + 40x^5 + 417x^4 + 712x^3 + 600x^2 + 224x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			161.4
Root			$-1.29438 - 3.12492i$ of defining polynomial
Character	$\chi$	$=$	2340.161
Dual form			2340.2.bi.c.1061.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} +(-2.78746 - 2.78746i) q^{7} +(-0.479675 + 0.479675i) q^{11} +(-1.10909 + 3.43073i) q^{13} +3.43757 q^{17} -0.154281 q^{23} +1.00000i q^{25} -0.454864i q^{29} +(2.21819 - 2.21819i) q^{31} -3.94206i q^{35} +(5.32728 + 5.32728i) q^{37} +(5.48574 + 5.48574i) q^{41} +5.35673i q^{43} +(5.81114 - 5.81114i) q^{47} +8.53982i q^{49} -11.6303i q^{53} -0.678363 q^{55} +(-1.91870 + 1.91870i) q^{59} +9.97620 q^{61} +(-3.21014 + 1.64165i) q^{65} +(7.57491 - 7.57491i) q^{67} +(4.26745 + 4.26745i) q^{71} +(3.18310 + 3.18310i) q^{73} +2.67415 q^{77} +13.3329 q^{79} +(-12.7359 - 12.7359i) q^{83} +(2.43073 + 2.43073i) q^{85} +(10.6877 - 10.6877i) q^{89} +(12.6546 - 6.47146i) q^{91} +(-8.68401 + 8.68401i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{7} + 12 q^{13} - 24 q^{31} - 12 q^{37} - 8 q^{55} - 32 q^{61} + 40 q^{67} - 12 q^{73} + 8 q^{79} + 4 q^{85} - 28 q^{97}+O(q^{100}) $ 12 * q - 8 * q^7 + 12 * q^13 - 24 * q^31 - 12 * q^37 - 8 * q^55 - 32 * q^61 + 40 * q^67 - 12 * q^73 + 8 * q^79 + 4 * q^85 - 28 * q^97

Character values

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

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

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
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$6$		0			0
$7$	−2.78746	−	2.78746i	−1.05356	−	1.05356i	−0.998482	−	0.0550773i	$-0.982459\pi$
$7$	−2.78746	−	2.78746i	−1.05356	−	1.05356i	−0.0550773	−	0.998482i	$-0.517541\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.479675	+	0.479675i	−0.144627	+	0.144627i	−0.775713	−	0.631086i	$-0.782610\pi$
$11$	−0.479675	+	0.479675i	−0.144627	+	0.144627i	0.631086	+	0.775713i	$0.282610\pi$
$12$		0			0
$13$	−1.10909	+	3.43073i	−0.307607	+	0.951513i
$13$	−1.10909	+	3.43073i	−0.307607	+	0.951513i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.43757			0.833734			0.416867	−	0.908968i	$-0.363128\pi$
$17$	3.43757			0.833734			0.416867	+	0.908968i	$0.363128\pi$
$18$		0			0
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.154281			−0.0321699			−0.0160849	−	0.999871i	$-0.505120\pi$
$23$	−0.154281			−0.0321699			−0.0160849	+	0.999871i	$0.505120\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	0.454864i		−	0.0844661i	−0.999108	−	0.0422330i	$-0.986553\pi$
$29$		−	0.454864i		−	0.0844661i	0.999108	−	0.0422330i	$-0.0134472\pi$
$30$		0			0
$31$	2.21819	−	2.21819i	0.398398	−	0.398398i	−0.479270	−	0.877668i	$-0.659098\pi$
$31$	2.21819	−	2.21819i	0.398398	−	0.398398i	0.877668	+	0.479270i	$0.159098\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	3.94206i		−	0.666329i
$36$		0			0
$37$	5.32728	+	5.32728i	0.875799	+	0.875799i	0.993097	−	0.117297i	$-0.0374231\pi$
$37$	5.32728	+	5.32728i	0.875799	+	0.875799i	−0.117297	+	0.993097i	$0.537423\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.48574	+	5.48574i	0.856729	+	0.856729i	0.990951	−	0.134222i	$-0.0428536\pi$
$41$	5.48574	+	5.48574i	0.856729	+	0.856729i	−0.134222	+	0.990951i	$0.542854\pi$
$42$		0			0
$43$			5.35673i			0.816893i	0.912782	+	0.408447i	$0.133929\pi$
$43$			5.35673i			0.816893i	−0.912782	+	0.408447i	$0.866071\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.81114	−	5.81114i	0.847641	−	0.847641i	−0.142197	−	0.989838i	$-0.545417\pi$
$47$	5.81114	−	5.81114i	0.847641	−	0.847641i	0.989838	+	0.142197i	$0.0454167\pi$
$48$		0			0
$49$			8.53982i			1.21997i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	11.6303i		−	1.59754i	−0.601638	−	0.798769i	$-0.705485\pi$
$53$		−	11.6303i		−	1.59754i	0.601638	−	0.798769i	$-0.294515\pi$
$54$		0			0
$55$	−0.678363			−0.0914704
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.91870	+	1.91870i	−0.249793	+	0.249793i	−0.820886	−	0.571092i	$-0.806520\pi$
$59$	−1.91870	+	1.91870i	−0.249793	+	0.249793i	0.571092	+	0.820886i	$0.306520\pi$
$60$		0			0
$61$	9.97620			1.27732			0.638661	−	0.769489i	$-0.279489\pi$
$61$	9.97620			1.27732			0.638661	+	0.769489i	$0.279489\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.21014	+	1.64165i	−0.398169	+	0.203621i
$66$		0			0
$67$	7.57491	−	7.57491i	0.925423	−	0.925423i	−0.0719830	−	0.997406i	$-0.522933\pi$
$67$	7.57491	−	7.57491i	0.925423	−	0.925423i	0.997406	+	0.0719830i	$0.0229327\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.26745	+	4.26745i	0.506453	+	0.506453i	0.913436	−	0.406983i	$-0.133419\pi$
$71$	4.26745	+	4.26745i	0.506453	+	0.506453i	−0.406983	+	0.913436i	$0.633419\pi$
$72$		0			0
$73$	3.18310	+	3.18310i	0.372553	+	0.372553i	0.868406	−	0.495853i	$-0.165145\pi$
$73$	3.18310	+	3.18310i	0.372553	+	0.372553i	−0.495853	+	0.868406i	$0.665145\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.67415			0.304747
$78$		0			0
$79$	13.3329			1.50007			0.750036	−	0.661398i	$-0.230036\pi$
$79$	13.3329			1.50007			0.750036	+	0.661398i	$0.230036\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−12.7359	−	12.7359i	−1.39795	−	1.39795i	−0.805913	−	0.592034i	$-0.798325\pi$
$83$	−12.7359	−	12.7359i	−1.39795	−	1.39795i	−0.592034	−	0.805913i	$-0.701675\pi$
$84$		0			0
$85$	2.43073	+	2.43073i	0.263650	+	0.263650i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.6877	−	10.6877i	1.13290	−	1.13290i	0.143204	−	0.989693i	$-0.454259\pi$
$89$	10.6877	−	10.6877i	1.13290	−	1.13290i	0.989693	−	0.143204i	$-0.0457406\pi$
$90$		0			0
$91$	12.6546	−	6.47146i	1.32656	−	0.678394i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−8.68401	+	8.68401i	−0.881727	+	0.881727i	−0.993710	−	0.111983i	$-0.964280\pi$
$97$	−8.68401	+	8.68401i	−0.881727	+	0.881727i	0.111983	+	0.993710i	$0.464280\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.68819			0.765004			0.382502	−	0.923955i	$-0.375063\pi$
$101$	7.68819			0.765004			0.382502	+	0.923955i	$0.375063\pi$
$102$		0			0
$103$			11.2978i			1.11321i	0.830778	+	0.556604i	$0.187896\pi$
$103$			11.2978i			1.11321i	−0.830778	+	0.556604i	$0.812104\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	10.7676i		−	1.04094i	−0.853880	−	0.520471i	$-0.825757\pi$
$107$		−	10.7676i		−	1.04094i	0.853880	−	0.520471i	$-0.174243\pi$
$108$		0			0
$109$	1.03509	−	1.03509i	0.0991435	−	0.0991435i	−0.655795	−	0.754939i	$-0.727667\pi$
$109$	1.03509	−	1.03509i	0.0991435	−	0.0991435i	0.754939	+	0.655795i	$0.227667\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			8.33898i			0.784465i	0.919866	+	0.392233i	$0.128297\pi$
$113$			8.33898i			0.784465i	−0.919866	+	0.392233i	$0.871703\pi$
$114$		0			0
$115$	−0.109093	−	0.109093i	−0.0101730	−	0.0101730i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−9.58208	−	9.58208i	−0.878388	−	0.878388i
$120$		0			0
$121$			10.5398i			0.958166i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$			7.78181i			0.690524i	0.938506	+	0.345262i	$0.112210\pi$
$127$			7.78181i			0.690524i	−0.938506	+	0.345262i	$0.887790\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			7.18371i			0.627643i	0.949482	+	0.313822i	$0.101609\pi$
$131$			7.18371i			0.627643i	−0.949482	+	0.313822i	$0.898391\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.86076	+	5.86076i	−0.500718	+	0.500718i	−0.911661	−	0.410943i	$-0.865200\pi$
$137$	−5.86076	+	5.86076i	−0.500718	+	0.500718i	0.410943	+	0.911661i	$0.365200\pi$
$138$		0			0
$139$	−8.96673			−0.760548			−0.380274	−	0.924874i	$-0.624170\pi$
$139$	−8.96673			−0.760548			−0.380274	+	0.924874i	$0.624170\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.11363	−	2.17764i	−0.0931265	−	0.182103i
$144$		0			0
$145$	0.321637	−	0.321637i	0.0267105	−	0.0267105i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.48574	+	5.48574i	0.449409	+	0.449409i	0.895158	−	0.445749i	$-0.147062\pi$
$149$	5.48574	+	5.48574i	0.449409	+	0.449409i	−0.445749	+	0.895158i	$0.647062\pi$
$150$		0			0
$151$	9.32164	+	9.32164i	0.758584	+	0.758584i	0.976065	−	0.217481i	$-0.0697839\pi$
$151$	9.32164	+	9.32164i	0.758584	+	0.758584i	−0.217481	+	0.976065i	$0.569784\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.13699			0.251969
$156$		0			0
$157$	−0.850175			−0.0678514			−0.0339257	−	0.999424i	$-0.510801\pi$
$157$	−0.850175			−0.0678514			−0.0339257	+	0.999424i	$0.510801\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.430052	+	0.430052i	0.0338929	+	0.0338929i
$162$		0			0
$163$	−10.3624	−	10.3624i	−0.811643	−	0.811643i	0.173237	−	0.984880i	$-0.444577\pi$
$163$	−10.3624	−	10.3624i	−0.811643	−	0.811643i	−0.984880	+	0.173237i	$0.944577\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	8.63956	−	8.63956i	0.668549	−	0.668549i	−0.288831	−	0.957380i	$-0.593266\pi$
$167$	8.63956	−	8.63956i	0.668549	−	0.668549i	0.957380	+	0.288831i	$0.0932665\pi$
$168$		0			0
$169$	−10.5398	−	7.61000i	−0.810756	−	0.585385i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	16.2231			1.23342			0.616710	−	0.787191i	$-0.288465\pi$
$173$	16.2231			1.23342			0.616710	+	0.787191i	$0.288465\pi$
$174$		0			0
$175$	2.78746	−	2.78746i	0.210712	−	0.210712i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	13.1828			0.985328			0.492664	−	0.870220i	$-0.336023\pi$
$179$	13.1828			0.985328			0.492664	+	0.870220i	$0.336023\pi$
$180$		0			0
$181$			10.2533i			0.762120i	0.924550	+	0.381060i	$0.124441\pi$
$181$			10.2533i			0.762120i	−0.924550	+	0.381060i	$0.875559\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			7.53391i			0.553904i
$186$		0			0
$187$	−1.64892	+	1.64892i	−0.120581	+	0.120581i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			14.7096i			1.06435i	0.846634	+	0.532176i	$0.178626\pi$
$191$			14.7096i			1.06435i	−0.846634	+	0.532176i	$0.821374\pi$
$192$		0			0
$193$	−1.10909	−	1.10909i	−0.0798343	−	0.0798343i	0.666062	−	0.745896i	$-0.267979\pi$
$193$	−1.10909	−	1.10909i	−0.0798343	−	0.0798343i	−0.745896	+	0.666062i	$0.767979\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−7.53391	−	7.53391i	−0.536769	−	0.536769i	0.385809	−	0.922578i	$-0.373922\pi$
$197$	−7.53391	−	7.53391i	−0.536769	−	0.536769i	−0.922578	+	0.385809i	$0.873922\pi$
$198$		0			0
$199$			20.9429i			1.48460i	0.670065	+	0.742302i	$0.266266\pi$
$199$			20.9429i			1.48460i	−0.670065	+	0.742302i	$0.733734\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.26791	+	1.26791i	−0.0889900	+	0.0889900i
$204$		0			0
$205$			7.75801i			0.541843i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	7.92982			0.545912			0.272956	−	0.962027i	$-0.411999\pi$
$211$	7.92982			0.545912			0.272956	+	0.962027i	$0.411999\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.78778	+	3.78778i	−0.258324	+	0.258324i
$216$		0			0
$217$	−12.3662			−0.839472
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.81259	+	11.7934i	−0.256462	+	0.793309i
$222$		0			0
$223$	18.0113	−	18.0113i	1.20612	−	1.20612i	0.233853	−	0.972272i	$-0.424867\pi$
$223$	18.0113	−	18.0113i	1.20612	−	1.20612i	0.972272	−	0.233853i	$-0.0751334\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.87514	−	6.87514i	−0.456319	−	0.456319i	0.441126	−	0.897445i	$-0.354579\pi$
$227$	−6.87514	−	6.87514i	−0.456319	−	0.456319i	−0.897445	+	0.441126i	$0.854579\pi$
$228$		0			0
$229$	0.781813	+	0.781813i	0.0516637	+	0.0516637i	0.732467	−	0.680803i	$-0.238369\pi$
$229$	0.781813	+	0.781813i	0.0516637	+	0.0516637i	−0.680803	+	0.732467i	$0.738369\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.2224			0.735207			0.367603	−	0.929983i	$-0.380178\pi$
$233$	11.2224			0.735207			0.367603	+	0.929983i	$0.380178\pi$
$234$		0			0
$235$	8.21819			0.536095
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.70694	+	2.70694i	0.175097	+	0.175097i	0.789215	−	0.614117i	$-0.210488\pi$
$239$	2.70694	+	2.70694i	0.175097	+	0.175097i	−0.614117	+	0.789215i	$0.710488\pi$
$240$		0			0
$241$	−16.8282	−	16.8282i	−1.08400	−	1.08400i	−0.996132	−	0.0878666i	$-0.971995\pi$
$241$	−16.8282	−	16.8282i	−1.08400	−	1.08400i	−0.0878666	−	0.996132i	$-0.528005\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.03857	+	6.03857i	−0.385790	+	0.385790i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.2890			0.775676			0.387838	−	0.921728i	$-0.373222\pi$
$251$	12.2890			0.775676			0.387838	+	0.921728i	$0.373222\pi$
$252$		0			0
$253$	0.0740049	−	0.0740049i	0.00465265	−	0.00465265i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.99069			−0.186554			−0.0932770	−	0.995640i	$-0.529734\pi$
$257$	−2.99069			−0.186554			−0.0932770	+	0.995640i	$0.529734\pi$
$258$		0			0
$259$		−	29.6991i		−	1.84541i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	2.21130i		−	0.136355i	−0.997673	−	0.0681774i	$-0.978282\pi$
$263$		−	2.21130i		−	0.136355i	0.997673	−	0.0681774i	$-0.0217184\pi$
$264$		0			0
$265$	8.22383	−	8.22383i	0.505186	−	0.505186i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	31.2413i		−	1.90482i	−0.304827	−	0.952408i	$-0.598599\pi$
$269$		−	31.2413i		−	1.90482i	0.304827	−	0.952408i	$-0.401401\pi$
$270$		0			0
$271$	8.40128	+	8.40128i	0.510342	+	0.510342i	0.914631	−	0.404289i	$-0.132481\pi$
$271$	8.40128	+	8.40128i	0.510342	+	0.510342i	−0.404289	+	0.914631i	$0.632481\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.479675	−	0.479675i	−0.0289255	−	0.0289255i
$276$		0			0
$277$			10.6546i			0.640170i	0.947389	+	0.320085i	$0.103712\pi$
$277$			10.6546i			0.640170i	−0.947389	+	0.320085i	$0.896288\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5.60723	+	5.60723i	−0.334499	+	0.334499i	−0.854292	−	0.519793i	$-0.826009\pi$
$281$	−5.60723	+	5.60723i	−0.334499	+	0.334499i	0.519793	+	0.854292i	$0.326009\pi$
$282$		0			0
$283$			9.64509i			0.573341i	0.958029	+	0.286671i	$0.0925485\pi$
$283$			9.64509i			0.573341i	−0.958029	+	0.286671i	$0.907452\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	30.5825i		−	1.80523i
$288$		0			0
$289$	−5.18310			−0.304888
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4.38894	+	4.38894i	−0.256405	+	0.256405i	−0.823590	−	0.567185i	$-0.808032\pi$
$293$	−4.38894	+	4.38894i	−0.256405	+	0.256405i	0.567185	+	0.823590i	$0.308032\pi$
$294$		0			0
$295$	−2.71345			−0.157983
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.171112	−	0.529298i	0.00989568	−	0.0306101i
$300$		0			0
$301$	14.9316	−	14.9316i	0.860645	−	0.860645i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	7.05424	+	7.05424i	0.403924	+	0.403924i
$306$		0			0
$307$	−6.07400	−	6.07400i	−0.346662	−	0.346662i	0.512203	−	0.858865i	$-0.328830\pi$
$307$	−6.07400	−	6.07400i	−0.346662	−	0.346662i	−0.858865	+	0.512203i	$0.828830\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−17.2791			−0.979809			−0.489905	−	0.871776i	$-0.662968\pi$
$311$	−17.2791			−0.979809			−0.489905	+	0.871776i	$0.662968\pi$
$312$		0			0
$313$	4.14035			0.234027			0.117013	−	0.993130i	$-0.462668\pi$
$313$	4.14035			0.234027			0.117013	+	0.993130i	$0.462668\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.29833	−	9.29833i	−0.522246	−	0.522246i	0.396003	−	0.918249i	$-0.370397\pi$
$317$	−9.29833	−	9.29833i	−0.522246	−	0.522246i	−0.918249	+	0.396003i	$0.870397\pi$
$318$		0			0
$319$	0.218187	+	0.218187i	0.0122161	+	0.0122161i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−3.43073	−	1.10909i	−0.190303	−	0.0615214i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−32.3966			−1.78608
$330$		0			0
$331$	7.92982	−	7.92982i	0.435862	−	0.435862i	−0.454754	−	0.890617i	$-0.650273\pi$
$331$	7.92982	−	7.92982i	0.435862	−	0.435862i	0.890617	+	0.454754i	$0.150273\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	10.7125			0.585289
$336$		0			0
$337$		−	2.20690i		−	0.120218i	−0.998192	−	0.0601088i	$-0.980855\pi$
$337$		−	2.20690i		−	0.120218i	0.998192	−	0.0601088i	$-0.0191448\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			2.12802i			0.115239i
$342$		0			0
$343$	4.29219	−	4.29219i	0.231757	−	0.231757i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	5.11073i		−	0.274358i	−0.990546	−	0.137179i	$-0.956196\pi$
$347$		−	5.11073i		−	0.274358i	0.990546	−	0.137179i	$-0.0438036\pi$
$348$		0			0
$349$	3.28836	+	3.28836i	0.176022	+	0.176022i	0.789619	−	0.613597i	$-0.210278\pi$
$349$	3.28836	+	3.28836i	0.176022	+	0.176022i	−0.613597	+	0.789619i	$0.710278\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	24.7164	+	24.7164i	1.31552	+	1.31552i	0.917283	+	0.398236i	$0.130378\pi$
$353$	24.7164	+	24.7164i	1.31552	+	1.31552i	0.398236	+	0.917283i	$0.369622\pi$
$354$		0			0
$355$			6.03509i			0.320309i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.87514	−	6.87514i	0.362856	−	0.362856i	−0.502007	−	0.864863i	$-0.667405\pi$
$359$	6.87514	−	6.87514i	0.362856	−	0.362856i	0.864863	+	0.502007i	$0.167405\pi$
$360$		0			0
$361$			19.0000i			1.00000i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			4.50158i			0.235624i
$366$		0			0
$367$	−19.8044			−1.03378			−0.516890	−	0.856052i	$-0.672910\pi$
$367$	−19.8044			−1.03378			−0.516890	+	0.856052i	$0.672910\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−32.4188	+	32.4188i	−1.68310	+	1.68310i
$372$		0			0
$373$	−17.4346			−0.902727			−0.451364	−	0.892340i	$-0.649062\pi$
$373$	−17.4346			−0.902727			−0.451364	+	0.892340i	$0.649062\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.56051	+	0.504486i	0.0803706	+	0.0259824i
$378$		0			0
$379$	3.34544	−	3.34544i	0.171844	−	0.171844i	−0.615945	−	0.787789i	$-0.711226\pi$
$379$	3.34544	−	3.34544i	0.171844	−	0.171844i	0.787789	+	0.615945i	$0.211226\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−11.4184	−	11.4184i	−0.583451	−	0.583451i	0.352399	−	0.935850i	$-0.385366\pi$
$383$	−11.4184	−	11.4184i	−0.583451	−	0.583451i	−0.935850	+	0.352399i	$0.885366\pi$
$384$		0			0
$385$	1.89091	+	1.89091i	0.0963695	+	0.0963695i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	18.2410			0.924858			0.462429	−	0.886656i	$-0.346978\pi$
$389$	18.2410			0.924858			0.462429	+	0.886656i	$0.346978\pi$
$390$		0			0
$391$	−0.530353			−0.0268211
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9.42780	+	9.42780i	0.474364	+	0.474364i
$396$		0			0
$397$	−17.9706	−	17.9706i	−0.901916	−	0.901916i	0.0936856	−	0.995602i	$-0.470135\pi$
$397$	−17.9706	−	17.9706i	−0.901916	−	0.901916i	−0.995602	+	0.0936856i	$0.970135\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−0.355618	+	0.355618i	−0.0177587	+	0.0177587i	−0.715930	−	0.698172i	$-0.753997\pi$
$401$	−0.355618	+	0.355618i	−0.0177587	+	0.0177587i	0.698172	+	0.715930i	$0.253997\pi$
$402$		0			0
$403$	5.14982	+	10.0702i	0.256531	+	0.501631i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−5.11073			−0.253329
$408$		0			0
$409$	−20.6991	+	20.6991i	−1.02351	+	1.02351i	−0.0237886	+	0.999717i	$0.507573\pi$
$409$	−20.6991	+	20.6991i	−1.02351	+	1.02351i	−0.999717	+	0.0237886i	$0.992427\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	10.6966			0.526344
$414$		0			0
$415$		−	18.0113i		−	0.884139i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			1.31754i			0.0643658i	0.999482	+	0.0321829i	$0.0102459\pi$
$419$			1.31754i			0.0643658i	−0.999482	+	0.0321829i	$0.989754\pi$
$420$		0			0
$421$	7.86146	−	7.86146i	0.383144	−	0.383144i	−0.489089	−	0.872234i	$-0.662671\pi$
$421$	7.86146	−	7.86146i	0.383144	−	0.383144i	0.872234	+	0.489089i	$0.162671\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			3.43757i			0.166747i
$426$		0			0
$427$	−27.8082	−	27.8082i	−1.34573	−	1.34573i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−14.5003	−	14.5003i	−0.698456	−	0.698456i	0.265622	−	0.964077i	$-0.414423\pi$
$431$	−14.5003	−	14.5003i	−0.698456	−	0.698456i	−0.964077	+	0.265622i	$0.914423\pi$
$432$		0			0
$433$		−	27.3680i		−	1.31522i	−0.753357	−	0.657611i	$-0.771567\pi$
$433$		−	27.3680i		−	1.31522i	0.753357	−	0.657611i	$-0.228433\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	2.68965i		−	0.128370i	−0.997938	−	0.0641850i	$-0.979555\pi$
$439$		−	2.68965i		−	0.128370i	0.997938	−	0.0641850i	$-0.0204448\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			30.5825i			1.45302i	0.687156	+	0.726510i	$0.258859\pi$
$443$			30.5825i			1.45302i	−0.687156	+	0.726510i	$0.741141\pi$
$444$		0			0
$445$	15.1147			0.716507
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−24.5869	+	24.5869i	−1.16033	+	1.16033i	−0.175923	+	0.984404i	$0.556291\pi$
$449$	−24.5869	+	24.5869i	−1.16033	+	1.16033i	−0.984404	+	0.175923i	$0.943709\pi$
$450$		0			0
$451$	−5.26275			−0.247813
$452$		0			0
$453$		0			0
$454$		0			0
$455$	13.5241	+	4.37211i	0.634021	+	0.204968i
$456$		0			0
$457$	−19.9931	+	19.9931i	−0.935239	+	0.935239i	−0.998027	−	0.0627879i	$-0.980001\pi$
$457$	−19.9931	+	19.9931i	−0.935239	+	0.935239i	0.0627879	+	0.998027i	$0.480001\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	11.2552	+	11.2552i	0.524209	+	0.524209i	0.918840	−	0.394631i	$-0.129128\pi$
$461$	11.2552	+	11.2552i	0.524209	+	0.524209i	−0.394631	+	0.918840i	$0.629128\pi$
$462$		0			0
$463$	1.21254	+	1.21254i	0.0563517	+	0.0563517i	0.734721	−	0.678369i	$-0.237313\pi$
$463$	1.21254	+	1.21254i	0.0563517	+	0.0563517i	−0.678369	+	0.734721i	$0.737313\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.8668			−0.502857			−0.251428	−	0.967876i	$-0.580900\pi$
$467$	−10.8668			−0.502857			−0.251428	+	0.967876i	$0.580900\pi$
$468$		0			0
$469$	−42.2295			−1.94998
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.56949	−	2.56949i	−0.118145	−	0.118145i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−28.1787	+	28.1787i	−1.28752	+	1.28752i	−0.351231	+	0.936289i	$0.614237\pi$
$479$	−28.1787	+	28.1787i	−1.28752	+	1.28752i	−0.936289	+	0.351231i	$0.885763\pi$
$480$		0			0
$481$	−24.1849	+	12.3680i	−1.10274	+	0.563933i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−12.2810			−0.557653
$486$		0			0
$487$	14.7875	−	14.7875i	0.670084	−	0.670084i	−0.287652	−	0.957735i	$-0.592874\pi$
$487$	14.7875	−	14.7875i	0.670084	−	0.670084i	0.957735	+	0.287652i	$0.0928745\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−11.2641			−0.508341			−0.254171	−	0.967159i	$-0.581802\pi$
$491$	−11.2641			−0.508341			−0.254171	+	0.967159i	$0.581802\pi$
$492$		0			0
$493$		−	1.56363i		−	0.0704222i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	23.7907i		−	1.06716i
$498$		0			0
$499$	15.5398	−	15.5398i	0.695658	−	0.695658i	−0.267813	−	0.963471i	$-0.586301\pi$
$499$	15.5398	−	15.5398i	0.695658	−	0.695658i	0.963471	+	0.267813i	$0.0863009\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			24.9699i			1.11335i	0.830730	+	0.556676i	$0.187924\pi$
$503$			24.9699i			1.11335i	−0.830730	+	0.556676i	$0.812076\pi$
$504$		0			0
$505$	5.43637	+	5.43637i	0.241915	+	0.241915i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−27.7079	−	27.7079i	−1.22813	−	1.22813i	−0.964670	−	0.263462i	$-0.915136\pi$
$509$	−27.7079	−	27.7079i	−1.22813	−	1.22813i	−0.263462	−	0.964670i	$-0.584864\pi$
$510$		0			0
$511$		−	17.7455i		−	0.785014i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−7.98878	+	7.98878i	−0.352028	+	0.352028i
$516$		0			0
$517$			5.57491i			0.245184i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			25.2599i			1.10666i	0.832963	+	0.553329i	$0.186643\pi$
$521$			25.2599i			1.10666i	−0.832963	+	0.553329i	$0.813357\pi$
$522$		0			0
$523$	−9.09093			−0.397519			−0.198759	−	0.980048i	$-0.563691\pi$
$523$	−9.09093			−0.397519			−0.198759	+	0.980048i	$0.563691\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7.62518	−	7.62518i	0.332158	−	0.332158i
$528$		0			0
$529$	−22.9762			−0.998965
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−24.9043	+	12.7359i	−1.07873	+	0.551653i
$534$		0			0
$535$	7.61383	−	7.61383i	0.329175	−	0.329175i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.09634	−	4.09634i	−0.176442	−	0.176442i
$540$		0			0
$541$	12.1035	+	12.1035i	0.520368	+	0.520368i	0.917683	−	0.397315i	$-0.130058\pi$
$541$	12.1035	+	12.1035i	0.520368	+	0.520368i	−0.397315	+	0.917683i	$0.630058\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.46384			0.0627039
$546$		0			0
$547$	−39.1022			−1.67189			−0.835945	−	0.548813i	$-0.815080\pi$
$547$	−39.1022			−1.67189			−0.835945	+	0.548813i	$0.815080\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−37.1649	−	37.1649i	−1.58041	−	1.58041i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	20.0579	−	20.0579i	0.849882	−	0.849882i	−0.140236	−	0.990118i	$-0.544786\pi$
$557$	20.0579	−	20.0579i	0.849882	−	0.849882i	0.990118	+	0.140236i	$0.0447862\pi$
$558$		0			0
$559$	−18.3775	−	5.94111i	−0.777285	−	0.251282i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−0.964763			−0.0406599			−0.0203300	−	0.999793i	$-0.506472\pi$
$563$	−0.964763			−0.0406599			−0.0203300	+	0.999793i	$0.506472\pi$
$564$		0			0
$565$	−5.89655	+	5.89655i	−0.248070	+	0.248070i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	23.5060			0.985425			0.492712	−	0.870192i	$-0.336005\pi$
$569$	23.5060			0.985425			0.492712	+	0.870192i	$0.336005\pi$
$570$		0			0
$571$		−	19.9762i		−	0.835978i	−0.908452	−	0.417989i	$-0.862735\pi$
$571$		−	19.9762i		−	0.835978i	0.908452	−	0.417989i	$-0.137265\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	0.154281i		−	0.00643397i
$576$		0			0
$577$	−2.83202	+	2.83202i	−0.117898	+	0.117898i	−0.763594	−	0.645696i	$-0.776567\pi$
$577$	−2.83202	+	2.83202i	−0.117898	+	0.117898i	0.645696	+	0.763594i	$0.276567\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			71.0015i			2.94564i
$582$		0			0
$583$	5.57874	+	5.57874i	0.231048	+	0.231048i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11.1098	−	11.1098i	−0.458551	−	0.458551i	0.439629	−	0.898180i	$-0.355110\pi$
$587$	−11.1098	−	11.1098i	−0.458551	−	0.458551i	−0.898180	+	0.439629i	$0.855110\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	25.8770	−	25.8770i	1.06264	−	1.06264i	0.0647408	−	0.997902i	$-0.479378\pi$
$593$	25.8770	−	25.8770i	1.06264	−	1.06264i	0.997902	−	0.0647408i	$-0.0206221\pi$
$594$		0			0
$595$		−	13.5511i		−	0.555541i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	28.5592i		−	1.16690i	−0.812150	−	0.583448i	$-0.801703\pi$
$599$		−	28.5592i		−	1.16690i	0.812150	−	0.583448i	$-0.198297\pi$
$600$		0			0
$601$	18.2533			0.744567			0.372283	−	0.928119i	$-0.378575\pi$
$601$	18.2533			0.744567			0.372283	+	0.928119i	$0.378575\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.45278	+	7.45278i	−0.302999	+	0.302999i
$606$		0			0
$607$	−3.12725			−0.126931			−0.0634656	−	0.997984i	$-0.520215\pi$
$607$	−3.12725			−0.126931			−0.0634656	+	0.997984i	$0.520215\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	13.4913	+	26.3815i	0.545802	+	1.06728i
$612$		0			0
$613$	5.74108	−	5.74108i	0.231880	−	0.231880i	−0.581597	−	0.813477i	$-0.697572\pi$
$613$	5.74108	−	5.74108i	0.231880	−	0.231880i	0.813477	+	0.581597i	$0.197572\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−5.31463	−	5.31463i	−0.213959	−	0.213959i	0.591988	−	0.805947i	$-0.298343\pi$
$617$	−5.31463	−	5.31463i	−0.213959	−	0.213959i	−0.805947	+	0.591988i	$0.798343\pi$
$618$		0			0
$619$	−0.538009	−	0.538009i	−0.0216244	−	0.0216244i	0.696212	−	0.717836i	$-0.254867\pi$
$619$	−0.538009	−	0.538009i	−0.0216244	−	0.0216244i	−0.717836	+	0.696212i	$0.754867\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−59.5832			−2.38715
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	18.3129	+	18.3129i	0.730184	+	0.730184i
$630$		0			0
$631$	17.8044	+	17.8044i	0.708781	+	0.708781i	0.966279	−	0.257498i	$-0.0828979\pi$
$631$	17.8044	+	17.8044i	0.708781	+	0.708781i	−0.257498	+	0.966279i	$0.582898\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.50257	+	5.50257i	−0.218363	+	0.218363i
$636$		0			0
$637$	−29.2978	−	9.47146i	−1.16082	−	0.375273i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−14.9056			−0.588734			−0.294367	−	0.955692i	$-0.595109\pi$
$641$	−14.9056			−0.588734			−0.294367	+	0.955692i	$0.595109\pi$
$642$		0			0
$643$	15.5009	−	15.5009i	0.611296	−	0.611296i	−0.331988	−	0.943284i	$-0.607719\pi$
$643$	15.5009	−	15.5009i	0.611296	−	0.611296i	0.943284	+	0.331988i	$0.107719\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	22.9910			0.903870			0.451935	−	0.892051i	$-0.350734\pi$
$647$	22.9910			0.903870			0.451935	+	0.892051i	$0.350734\pi$
$648$		0			0
$649$		−	1.84070i		−	0.0722540i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	40.1840i		−	1.57252i	−0.617895	−	0.786261i	$-0.712014\pi$
$653$		−	40.1840i		−	1.57252i	0.617895	−	0.786261i	$-0.287986\pi$
$654$		0			0
$655$	−5.07965	+	5.07965i	−0.198478	+	0.198478i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	38.5109i		−	1.50017i	−0.661341	−	0.750085i	$-0.730013\pi$
$659$		−	38.5109i		−	1.50017i	0.661341	−	0.750085i	$-0.269987\pi$
$660$		0			0
$661$	−6.28655	−	6.28655i	−0.244518	−	0.244518i	0.574198	−	0.818716i	$-0.305314\pi$
$661$	−6.28655	−	6.28655i	−0.244518	−	0.244518i	−0.818716	+	0.574198i	$0.805314\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			0.0701770i			0.00271726i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.78533	+	4.78533i	−0.184736	+	0.184736i
$672$		0			0
$673$			11.2087i			0.432064i	0.976386	+	0.216032i	$0.0693116\pi$
$673$			11.2087i			0.432064i	−0.976386	+	0.216032i	$0.930688\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			40.6309i			1.56157i	0.624799	+	0.780786i	$0.285181\pi$
$677$			40.6309i			1.56157i	−0.624799	+	0.780786i	$0.714819\pi$
$678$		0			0
$679$	48.4126			1.85790
$680$		0			0
$681$		0			0
$682$		0			0
$683$	15.9721	−	15.9721i	0.611157	−	0.611157i	−0.332091	−	0.943247i	$-0.607754\pi$
$683$	15.9721	−	15.9721i	0.611157	−	0.611157i	0.943247	+	0.332091i	$0.107754\pi$
$684$		0			0
$685$	−8.28836			−0.316682
$686$		0			0
$687$		0			0
$688$		0			0
$689$	39.9003	+	12.8990i	1.52008	+	0.491414i
$690$		0			0
$691$	25.9875	−	25.9875i	0.988610	−	0.988610i	−0.0113254	−	0.999936i	$-0.503605\pi$
$691$	25.9875	−	25.9875i	0.988610	−	0.988610i	0.999936	+	0.0113254i	$0.00360507\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.34043	−	6.34043i	−0.240506	−	0.240506i
$696$		0			0
$697$	18.8576	+	18.8576i	0.714284	+	0.714284i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	28.4625			1.07501			0.537507	−	0.843259i	$-0.319366\pi$
$701$	28.4625			1.07501			0.537507	+	0.843259i	$0.319366\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−21.4305	−	21.4305i	−0.805977	−	0.805977i
$708$		0			0
$709$	7.48275	+	7.48275i	0.281020	+	0.281020i	0.833516	−	0.552495i	$-0.186324\pi$
$709$	7.48275	+	7.48275i	0.281020	+	0.281020i	−0.552495	+	0.833516i	$0.686324\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.342225	+	0.342225i	−0.0128164	+	0.0128164i
$714$		0			0
$715$	0.752368	−	2.32728i	0.0281370	−	0.0870353i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	18.0400			0.672778			0.336389	−	0.941723i	$-0.390794\pi$
$719$	18.0400			0.672778			0.336389	+	0.941723i	$0.390794\pi$
$720$		0			0
$721$	31.4922	−	31.4922i	1.17283	−	1.17283i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.454864			0.0168932
$726$		0			0
$727$		−	0.920352i		−	0.0341340i	−0.999854	−	0.0170670i	$-0.994567\pi$
$727$		−	0.920352i		−	0.0341340i	0.999854	−	0.0170670i	$-0.00543285\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			18.4141i			0.681071i
$732$		0			0
$733$	−37.1430	+	37.1430i	−1.37191	+	1.37191i	−0.514290	+	0.857617i	$0.671944\pi$
$733$	−37.1430	+	37.1430i	−1.37191	+	1.37191i	−0.857617	+	0.514290i	$0.828056\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			7.26699i			0.267683i
$738$		0			0
$739$	−33.5273	−	33.5273i	−1.23332	−	1.23332i	−0.962680	−	0.270642i	$-0.912764\pi$
$739$	−33.5273	−	33.5273i	−1.23332	−	1.23332i	−0.270642	−	0.962680i	$-0.587236\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−7.18371	−	7.18371i	−0.263545	−	0.263545i	0.562948	−	0.826492i	$-0.309667\pi$
$743$	−7.18371	−	7.18371i	−0.263545	−	0.263545i	−0.826492	+	0.562948i	$0.809667\pi$
$744$		0			0
$745$			7.75801i			0.284232i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−30.0142	+	30.0142i	−1.09669	+	1.09669i
$750$		0			0
$751$			11.5173i			0.420271i	0.977672	+	0.210135i	$0.0673904\pi$
$751$			11.5173i			0.420271i	−0.977672	+	0.210135i	$0.932610\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			13.1828i			0.479771i
$756$		0			0
$757$	34.5880			1.25712			0.628561	−	0.777760i	$-0.283644\pi$
$757$	34.5880			1.25712			0.628561	+	0.777760i	$0.283644\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	20.4817	−	20.4817i	0.742461	−	0.742461i	−0.230590	−	0.973051i	$-0.574066\pi$
$761$	20.4817	−	20.4817i	0.742461	−	0.742461i	0.973051	+	0.230590i	$0.0740656\pi$
$762$		0			0
$763$	−5.77053			−0.208907
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.45452	−	8.71056i	−0.160844	−	0.314520i
$768$		0			0
$769$	−5.14035	+	5.14035i	−0.185366	+	0.185366i	−0.793689	−	0.608323i	$-0.791842\pi$
$769$	−5.14035	+	5.14035i	−0.185366	+	0.185366i	0.608323	+	0.793689i	$0.291842\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−19.7103	−	19.7103i	−0.708930	−	0.708930i	0.257380	−	0.966310i	$-0.417141\pi$
$773$	−19.7103	−	19.7103i	−0.708930	−	0.708930i	−0.966310	+	0.257380i	$0.917141\pi$
$774$		0			0
$775$	2.21819	+	2.21819i	0.0796796	+	0.0796796i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−4.09398			−0.146494
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−0.601165	−	0.601165i	−0.0214565	−	0.0214565i
$786$		0			0
$787$	−16.5731	−	16.5731i	−0.590767	−	0.590767i	0.347072	−	0.937839i	$-0.387176\pi$
$787$	−16.5731	−	16.5731i	−0.590767	−	0.590767i	−0.937839	+	0.347072i	$0.887176\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	23.2445	−	23.2445i	0.826481	−	0.826481i
$792$		0			0
$793$	−11.0645	+	34.2256i	−0.392913	+	1.21539i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.0912650			−0.00323277			−0.00161639	−	0.999999i	$-0.500515\pi$
$797$	−0.0912650			−0.00323277			−0.00161639	+	0.999999i	$0.500515\pi$
$798$		0			0
$799$	19.9762	−	19.9762i	0.706707	−	0.706707i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.05370			−0.107763
$804$		0			0
$805$			0.608186i			0.0214357i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	38.4427i		−	1.35157i	−0.737097	−	0.675787i	$-0.763804\pi$
$809$		−	38.4427i		−	1.35157i	0.737097	−	0.675787i	$-0.236196\pi$
$810$		0			0
$811$	−5.64509	+	5.64509i	−0.198226	+	0.198226i	−0.799239	−	0.601013i	$-0.794764\pi$
$811$	−5.64509	+	5.64509i	−0.198226	+	0.198226i	0.601013	+	0.799239i	$0.294764\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	14.6546i		−	0.513328i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	33.1511	+	33.1511i	1.15698	+	1.15698i	0.985121	+	0.171862i	$0.0549783\pi$
$821$	33.1511	+	33.1511i	1.15698	+	1.15698i	0.171862	+	0.985121i	$0.445022\pi$
$822$		0			0
$823$			7.22766i			0.251940i	0.992034	+	0.125970i	$0.0402044\pi$
$823$			7.22766i			0.251940i	−0.992034	+	0.125970i	$0.959796\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	14.1972	−	14.1972i	0.493684	−	0.493684i	−0.415781	−	0.909465i	$-0.636492\pi$
$827$	14.1972	−	14.1972i	0.493684	−	0.493684i	0.909465	+	0.415781i	$0.136492\pi$
$828$		0			0
$829$			32.0891i			1.11450i	0.830344	+	0.557251i	$0.188144\pi$
$829$			32.0891i			1.11450i	−0.830344	+	0.557251i	$0.811856\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			29.3563i			1.01713i
$834$		0			0
$835$	12.2182			0.422828
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−2.74060	+	2.74060i	−0.0946160	+	0.0946160i	−0.752830	−	0.658214i	$-0.771312\pi$
$839$	−2.74060	+	2.74060i	−0.0946160	+	0.0946160i	0.658214	+	0.752830i	$0.271312\pi$
$840$		0			0
$841$	28.7931			0.992865
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.07170	−	12.8339i	−0.0712686	−	0.441498i
$846$		0			0
$847$	29.3793	−	29.3793i	1.00948	−	1.00948i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.821900	−	0.821900i	−0.0281744	−	0.0281744i
$852$		0			0
$853$	−9.19056	−	9.19056i	−0.314679	−	0.314679i	0.532040	−	0.846719i	$-0.321425\pi$
$853$	−9.19056	−	9.19056i	−0.314679	−	0.314679i	−0.846719	+	0.532040i	$0.821425\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−11.6143			−0.396737			−0.198368	−	0.980128i	$-0.563564\pi$
$857$	−11.6143			−0.396737			−0.198368	+	0.980128i	$0.563564\pi$
$858$		0			0
$859$	−25.3555			−0.865118			−0.432559	−	0.901606i	$-0.642389\pi$
$859$	−25.3555			−0.865118			−0.432559	+	0.901606i	$0.642389\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−24.5125	−	24.5125i	−0.834414	−	0.834414i	0.153703	−	0.988117i	$-0.450880\pi$
$863$	−24.5125	−	24.5125i	−0.834414	−	0.834414i	−0.988117	+	0.153703i	$0.950880\pi$
$864$		0			0
$865$	11.4715	+	11.4715i	0.390041	+	0.390041i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−6.39547	+	6.39547i	−0.216951	+	0.216951i
$870$		0			0
$871$	17.5862	+	34.3888i	0.595886	+	1.16522i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.94206			0.133266
$876$		0			0
$877$	4.61000	−	4.61000i	0.155669	−	0.155669i	−0.624976	−	0.780644i	$-0.714891\pi$
$877$	4.61000	−	4.61000i	0.155669	−	0.155669i	0.780644	+	0.624976i	$0.214891\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−30.5301			−1.02858			−0.514292	−	0.857615i	$-0.671945\pi$
$881$	−30.5301			−1.02858			−0.514292	+	0.857615i	$0.671945\pi$
$882$		0			0
$883$		−	34.4477i		−	1.15926i	−0.814881	−	0.579628i	$-0.803198\pi$
$883$		−	34.4477i		−	1.15926i	0.814881	−	0.579628i	$-0.196802\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1.66517i		−	0.0559110i	−0.999609	−	0.0279555i	$-0.991100\pi$
$887$		−	1.66517i		−	0.0559110i	0.999609	−	0.0279555i	$-0.00889968\pi$
$888$		0			0
$889$	21.6915	−	21.6915i	0.727508	−	0.727508i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	9.32164	+	9.32164i	0.311588	+	0.311588i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.00897	−	1.00897i	−0.0336511	−	0.0336511i
$900$		0			0
$901$		−	39.9798i		−	1.33192i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7.25016	+	7.25016i	−0.241003	+	0.241003i
$906$		0			0
$907$			52.1706i			1.73230i	0.499788	+	0.866148i	$0.333411\pi$
$907$			52.1706i			1.73230i	−0.499788	+	0.866148i	$0.666589\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			17.1354i			0.567721i	0.958866	+	0.283860i	$0.0916152\pi$
$911$			17.1354i			0.567721i	−0.958866	+	0.283860i	$0.908385\pi$
$912$		0			0
$913$	12.2182			0.404363
$914$		0			0
$915$		0			0
$916$		0			0
$917$	20.0243	−	20.0243i	0.661260	−	0.661260i
$918$		0			0
$919$	−51.2664			−1.69112			−0.845561	−	0.533879i	$-0.820734\pi$
$919$	−51.2664			−1.69112			−0.845561	+	0.533879i	$0.820734\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−19.3735	+	9.90748i	−0.637686	+	0.326109i
$924$		0			0
$925$	−5.32728	+	5.32728i	−0.175160	+	0.175160i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−30.9575	−	30.9575i	−1.01568	−	1.01568i	−0.999875	−	0.0158091i	$-0.994968\pi$
$929$	−30.9575	−	30.9575i	−1.01568	−	1.01568i	−0.0158091	−	0.999875i	$-0.505032\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.33192			−0.0762620
$936$		0			0
$937$	−24.5654			−0.802518			−0.401259	−	0.915965i	$-0.631427\pi$
$937$	−24.5654			−0.802518			−0.401259	+	0.915965i	$0.631427\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	17.4166	+	17.4166i	0.567764	+	0.567764i	0.931502	−	0.363737i	$-0.118499\pi$
$941$	17.4166	+	17.4166i	0.567764	+	0.567764i	−0.363737	+	0.931502i	$0.618499\pi$
$942$		0			0
$943$	−0.846347	−	0.846347i	−0.0275609	−	0.0275609i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−31.9001	+	31.9001i	−1.03661	+	1.03661i	−0.0373091	+	0.999304i	$0.511879\pi$
$947$	−31.9001	+	31.9001i	−1.03661	+	1.03661i	−0.999304	+	0.0373091i	$0.988121\pi$
$948$		0			0
$949$	−14.4507	+	7.39000i	−0.469090	+	0.239890i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	39.5067			1.27975			0.639874	−	0.768480i	$-0.278987\pi$
$953$	39.5067			1.27975			0.639874	+	0.768480i	$0.278987\pi$
$954$		0			0
$955$	−10.4013	+	10.4013i	−0.336578	+	0.336578i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	32.6732			1.05507
$960$		0			0
$961$			21.1593i			0.682558i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1.56849i		−	0.0504916i
$966$		0			0
$967$	−3.85199	+	3.85199i	−0.123872	+	0.123872i	−0.766325	−	0.642453i	$-0.777917\pi$
$967$	−3.85199	+	3.85199i	−0.123872	+	0.123872i	0.642453	+	0.766325i	$0.277917\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			59.4626i			1.90824i	0.299418	+	0.954122i	$0.403208\pi$
$971$			59.4626i			1.90824i	−0.299418	+	0.954122i	$0.596792\pi$
$972$		0			0
$973$	24.9944	+	24.9944i	0.801282	+	0.801282i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−31.1207	−	31.1207i	−0.995639	−	0.995639i	0.00435181	−	0.999991i	$-0.498615\pi$
$977$	−31.1207	−	31.1207i	−0.995639	−	0.995639i	−0.999991	+	0.00435181i	$0.998615\pi$
$978$		0			0
$979$			10.2533i			0.327696i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	36.4929	−	36.4929i	1.16394	−	1.16394i	0.180338	−	0.983605i	$-0.442281\pi$
$983$	36.4929	−	36.4929i	1.16394	−	1.16394i	0.983605	−	0.180338i	$-0.0577193\pi$
$984$		0			0
$985$		−	10.6546i		−	0.339483i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	0.826443i		−	0.0262793i
$990$		0			0
$991$	18.1129			0.575376			0.287688	−	0.957724i	$-0.407113\pi$
$991$	18.1129			0.575376			0.287688	+	0.957724i	$0.407113\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−14.8089	+	14.8089i	−0.469473	+	0.469473i
$996$		0			0
$997$	−23.5273			−0.745117			−0.372559	−	0.928009i	$-0.621519\pi$
$997$	−23.5273			−0.745117			−0.372559	+	0.928009i	$0.621519\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.bi.c.161.4	yes	12
3.2	odd	2	inner	2340.2.bi.c.161.1	✓	12
13.8	odd	4	inner	2340.2.bi.c.1061.1	yes	12
39.8	even	4	inner	2340.2.bi.c.1061.4	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.bi.c.161.1	✓	12	3.2	odd	2	inner
2340.2.bi.c.161.4	yes	12	1.1	even	1	trivial
2340.2.bi.c.1061.1	yes	12	13.8	odd	4	inner
2340.2.bi.c.1061.4	yes	12	39.8	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.bi (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} +(-2.78746 - 2.78746i) q^{7} +(-0.479675 + 0.479675i) q^{11} +(-1.10909 + 3.43073i) q^{13} +3.43757 q^{17} -0.154281 q^{23} +1.00000i q^{25} -0.454864i q^{29} +(2.21819 - 2.21819i) q^{31} -3.94206i q^{35} +(5.32728 + 5.32728i) q^{37} +(5.48574 + 5.48574i) q^{41} +5.35673i q^{43} +(5.81114 - 5.81114i) q^{47} +8.53982i q^{49} -11.6303i q^{53} -0.678363 q^{55} +(-1.91870 + 1.91870i) q^{59} +9.97620 q^{61} +(-3.21014 + 1.64165i) q^{65} +(7.57491 - 7.57491i) q^{67} +(4.26745 + 4.26745i) q^{71} +(3.18310 + 3.18310i) q^{73} +2.67415 q^{77} +13.3329 q^{79} +(-12.7359 - 12.7359i) q^{83} +(2.43073 + 2.43073i) q^{85} +(10.6877 - 10.6877i) q^{89} +(12.6546 - 6.47146i) q^{91} +(-8.68401 + 8.68401i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{7} + 12 q^{13} - 24 q^{31} - 12 q^{37} - 8 q^{55} - 32 q^{61} + 40 q^{67} - 12 q^{73} + 8 q^{79} + 4 q^{85} - 28 q^{97}+O(q^{100}) \) 12 * q - 8 * q^7 + 12 * q^13 - 24 * q^31 - 12 * q^37 - 8 * q^55 - 32 * q^61 + 40 * q^67 - 12 * q^73 + 8 * q^79 + 4 * q^85 - 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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