LMFDB - Embedded newform 4620.2.h.e.1849.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4620,2,Mod(1849,4620)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4620.1849"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4620, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4620.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [14,0,0,0,-4,0,0,0,-14,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$36.8908857338$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 4 x^{13} + 12 x^{12} - 32 x^{11} + 86 x^{10} - 220 x^{9} + 585 x^{8} - 1536 x^{7} + \cdots + 78125 $ x^14 - 4x^13 + 12x^12 - 32x^11 + 86x^10 - 220x^9 + 585x^8 - 1536x^7 + 2925x^6 - 5500x^5 + 10750x^4 - 20000x^3 + 37500x^2 - 62500*x + 78125
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.5
Root			$-0.217835 - 2.22543i$ of defining polynomial
Character	$\chi$	$=$	4620.1849
Dual form			4620.2.h.e.1849.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(0.217835 - 2.22543i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +1.00000 q^{11} +5.67621i q^{13} +(-2.22543 - 0.217835i) q^{15} +0.433388i q^{17} +5.87997 q^{19} -1.00000 q^{21} -7.78581i q^{23} +(-4.90510 - 0.969556i) q^{25} +1.00000i q^{27} +4.26921 q^{29} +6.65690 q^{31} -1.00000i q^{33} +(-2.22543 - 0.217835i) q^{35} +3.69831i q^{37} +5.67621 q^{39} +6.88552 q^{41} +6.40776i q^{43} +(-0.217835 + 2.22543i) q^{45} -3.94415i q^{47} -1.00000 q^{49} +0.433388 q^{51} -8.69042i q^{53} +(0.217835 - 2.22543i) q^{55} -5.87997i q^{57} -3.82395 q^{59} -11.2978 q^{61} +1.00000i q^{63} +(12.6320 + 1.23648i) q^{65} -10.6922i q^{67} -7.78581 q^{69} +10.7234 q^{71} -9.32221i q^{73} +(-0.969556 + 4.90510i) q^{75} -1.00000i q^{77} +17.1007 q^{79} +1.00000 q^{81} -0.808171i q^{83} +(0.964477 + 0.0944073i) q^{85} -4.26921i q^{87} -3.85329 q^{89} +5.67621 q^{91} -6.65690i q^{93} +(1.28086 - 13.0855i) q^{95} -1.84873i q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 4 q^{5} - 14 q^{9} + 14 q^{11} + 6 q^{19} - 14 q^{21} - 8 q^{25} + 38 q^{29} + 4 q^{31} + 8 q^{39} - 12 q^{41} + 4 q^{45} - 14 q^{49} + 14 q^{51} - 4 q^{55} + 18 q^{59} - 54 q^{61} - 2 q^{65} + 26 q^{69}+ \cdots - 14 q^{99}+O(q^{100}) $ 14 * q - 4 * q^5 - 14 * q^9 + 14 * q^11 + 6 * q^19 - 14 * q^21 - 8 * q^25 + 38 * q^29 + 4 * q^31 + 8 * q^39 - 12 * q^41 + 4 * q^45 - 14 * q^49 + 14 * q^51 - 4 * q^55 + 18 * q^59 - 54 * q^61 - 2 * q^65 + 26 * q^69 - 24 * q^71 + 48 * q^79 + 14 * q^81 + 12 * q^85 + 46 * q^89 + 8 * q^91 - 72 * q^95 - 14 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4620\mathbb{Z}\right)^\times$.

$n$	$661$	$1541$	$2311$	$2521$	$3697$
$\chi(n)$	$1$	$1$	$1$	$1$	$-1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	0.217835	−	2.22543i	0.0974190	−	0.995243i
$5$	0.217835	−	2.22543i	0.0974190	−	0.995243i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		0			0
$13$			5.67621i			1.57430i	0.616763	+	0.787149i	$0.288444\pi$
$13$			5.67621i			1.57430i	−0.616763	+	0.787149i	$0.711556\pi$
$14$		0			0
$15$	−2.22543	−	0.217835i	−0.574604	−	0.0562449i
$16$		0			0
$17$			0.433388i			0.105112i	0.998618	+	0.0525561i	$0.0167368\pi$
$17$			0.433388i			0.105112i	−0.998618	+	0.0525561i	$0.983263\pi$
$18$		0			0
$19$	5.87997			1.34896			0.674478	−	0.738295i	$-0.264369\pi$
$19$	5.87997			1.34896			0.674478	+	0.738295i	$0.264369\pi$
$20$		0			0
$21$	−1.00000			−0.218218
$22$		0			0
$23$		−	7.78581i		−	1.62345i	−0.584037	−	0.811727i	$-0.698528\pi$
$23$		−	7.78581i		−	1.62345i	0.584037	−	0.811727i	$-0.301472\pi$
$24$		0			0
$25$	−4.90510	−	0.969556i	−0.981019	−	0.193911i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	4.26921			0.792772			0.396386	−	0.918084i	$-0.370264\pi$
$29$	4.26921			0.792772			0.396386	+	0.918084i	$0.370264\pi$
$30$		0			0
$31$	6.65690			1.19561			0.597807	−	0.801640i	$-0.296039\pi$
$31$	6.65690			1.19561			0.597807	+	0.801640i	$0.296039\pi$
$32$		0			0
$33$		−	1.00000i		−	0.174078i
$34$		0			0
$35$	−2.22543	−	0.217835i	−0.376167	−	0.0368209i
$36$		0			0
$37$			3.69831i			0.607998i	0.952672	+	0.303999i	$0.0983220\pi$
$37$			3.69831i			0.607998i	−0.952672	+	0.303999i	$0.901678\pi$
$38$		0			0
$39$	5.67621			0.908921
$40$		0			0
$41$	6.88552			1.07534			0.537669	−	0.843156i	$-0.319305\pi$
$41$	6.88552			1.07534			0.537669	+	0.843156i	$0.319305\pi$
$42$		0			0
$43$			6.40776i			0.977174i	0.872515	+	0.488587i	$0.162488\pi$
$43$			6.40776i			0.977174i	−0.872515	+	0.488587i	$0.837512\pi$
$44$		0			0
$45$	−0.217835	+	2.22543i	−0.0324730	+	0.331748i
$46$		0			0
$47$		−	3.94415i		−	0.575314i	−0.957733	−	0.287657i	$-0.907124\pi$
$47$		−	3.94415i		−	0.575314i	0.957733	−	0.287657i	$-0.0928763\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	0.433388			0.0606865
$52$		0			0
$53$		−	8.69042i		−	1.19372i	−0.802345	−	0.596860i	$-0.796415\pi$
$53$		−	8.69042i		−	1.19372i	0.802345	−	0.596860i	$-0.203585\pi$
$54$		0			0
$55$	0.217835	−	2.22543i	0.0293729	−	0.300077i
$56$		0			0
$57$		−	5.87997i		−	0.778820i
$58$		0			0
$59$	−3.82395			−0.497836			−0.248918	−	0.968525i	$-0.580075\pi$
$59$	−3.82395			−0.497836			−0.248918	+	0.968525i	$0.580075\pi$
$60$		0			0
$61$	−11.2978			−1.44654			−0.723269	−	0.690566i	$-0.757361\pi$
$61$	−11.2978			−1.44654			−0.723269	+	0.690566i	$0.757361\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	12.6320	+	1.23648i	1.56681	+	0.153366i
$66$		0			0
$67$		−	10.6922i		−	1.30626i	−0.757248	−	0.653128i	$-0.773456\pi$
$67$		−	10.6922i		−	1.30626i	0.757248	−	0.653128i	$-0.226544\pi$
$68$		0			0
$69$	−7.78581			−0.937302
$70$		0			0
$71$	10.7234			1.27263			0.636317	−	0.771428i	$-0.280457\pi$
$71$	10.7234			1.27263			0.636317	+	0.771428i	$0.280457\pi$
$72$		0			0
$73$		−	9.32221i		−	1.09108i	−0.838084	−	0.545541i	$-0.816324\pi$
$73$		−	9.32221i		−	1.09108i	0.838084	−	0.545541i	$-0.183676\pi$
$74$		0			0
$75$	−0.969556	+	4.90510i	−0.111955	+	0.566392i
$76$		0			0
$77$		−	1.00000i		−	0.113961i
$78$		0			0
$79$	17.1007			1.92398			0.961989	−	0.273088i	$-0.0880449\pi$
$79$	17.1007			1.92398			0.961989	+	0.273088i	$0.0880449\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	0.808171i		−	0.0887083i	−0.999016	−	0.0443541i	$-0.985877\pi$
$83$		−	0.808171i		−	0.0887083i	0.999016	−	0.0443541i	$-0.0141230\pi$
$84$		0			0
$85$	0.964477	+	0.0944073i	0.104612	+	0.0102399i
$86$		0			0
$87$		−	4.26921i		−	0.457707i
$88$		0			0
$89$	−3.85329			−0.408448			−0.204224	−	0.978924i	$-0.565467\pi$
$89$	−3.85329			−0.408448			−0.204224	+	0.978924i	$0.565467\pi$
$90$		0			0
$91$	5.67621			0.595029
$92$		0			0
$93$		−	6.65690i		−	0.690288i
$94$		0			0
$95$	1.28086	−	13.0855i	0.131414	−	1.34254i
$96$		0			0
$97$		−	1.84873i		−	0.187710i	−0.995586	−	0.0938550i	$-0.970081\pi$
$97$		−	1.84873i		−	0.187710i	0.995586	−	0.0938550i	$-0.0299190\pi$
$98$		0			0
$99$	−1.00000			−0.100504
$100$		0			0
$101$	6.58942			0.655672			0.327836	−	0.944735i	$-0.393681\pi$
$101$	6.58942			0.655672			0.327836	+	0.944735i	$0.393681\pi$
$102$		0			0
$103$		−	1.81144i		−	0.178486i	−0.996010	−	0.0892432i	$-0.971555\pi$
$103$		−	1.81144i		−	0.178486i	0.996010	−	0.0892432i	$-0.0284448\pi$
$104$		0			0
$105$	−0.217835	+	2.22543i	−0.0212586	+	0.217180i
$106$		0			0
$107$			5.76329i			0.557159i	0.960413	+	0.278579i	$0.0898635\pi$
$107$			5.76329i			0.557159i	−0.960413	+	0.278579i	$0.910136\pi$
$108$		0			0
$109$	11.6042			1.11148			0.555738	−	0.831357i	$-0.312436\pi$
$109$	11.6042			1.11148			0.555738	+	0.831357i	$0.312436\pi$
$110$		0			0
$111$	3.69831			0.351028
$112$		0			0
$113$			6.04973i			0.569111i	0.958660	+	0.284555i	$0.0918460\pi$
$113$			6.04973i			0.569111i	−0.958660	+	0.284555i	$0.908154\pi$
$114$		0			0
$115$	−17.3268	−	1.69603i	−1.61573	−	0.158155i
$116$		0			0
$117$		−	5.67621i		−	0.524766i
$118$		0			0
$119$	0.433388			0.0397287
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		−	6.88552i		−	0.620846i
$124$		0			0
$125$	−3.22618	+	10.7048i	−0.288559	+	0.957462i
$126$		0			0
$127$			18.4582i			1.63790i	0.573864	+	0.818950i	$0.305444\pi$
$127$			18.4582i			1.63790i	−0.573864	+	0.818950i	$0.694556\pi$
$128$		0			0
$129$	6.40776			0.564172
$130$		0			0
$131$	−9.49331			−0.829434			−0.414717	−	0.909950i	$-0.636119\pi$
$131$	−9.49331			−0.829434			−0.414717	+	0.909950i	$0.636119\pi$
$132$		0			0
$133$		−	5.87997i		−	0.509858i
$134$		0			0
$135$	2.22543	+	0.217835i	0.191535	+	0.0187483i
$136$		0			0
$137$		−	12.7048i		−	1.08545i	−0.839911	−	0.542724i	$-0.817393\pi$
$137$		−	12.7048i		−	1.08545i	0.839911	−	0.542724i	$-0.182607\pi$
$138$		0			0
$139$	13.0820			1.10960			0.554798	−	0.831985i	$-0.312795\pi$
$139$	13.0820			1.10960			0.554798	+	0.831985i	$0.312795\pi$
$140$		0			0
$141$	−3.94415			−0.332158
$142$		0			0
$143$			5.67621i			0.474669i
$144$		0			0
$145$	0.929984	−	9.50083i	0.0772310	−	0.789001i
$146$		0			0
$147$			1.00000i			0.0824786i
$148$		0			0
$149$	−7.49955			−0.614387			−0.307194	−	0.951647i	$-0.599390\pi$
$149$	−7.49955			−0.614387			−0.307194	+	0.951647i	$0.599390\pi$
$150$		0			0
$151$	−20.4872			−1.66722			−0.833611	−	0.552352i	$-0.813730\pi$
$151$	−20.4872			−1.66722			−0.833611	+	0.552352i	$0.813730\pi$
$152$		0			0
$153$		−	0.433388i		−	0.0350374i
$154$		0			0
$155$	1.45011	−	14.8145i	0.116476	−	1.18993i
$156$		0			0
$157$		−	3.01973i		−	0.241001i	−0.992713	−	0.120501i	$-0.961550\pi$
$157$		−	3.01973i		−	0.241001i	0.992713	−	0.120501i	$-0.0384500\pi$
$158$		0			0
$159$	−8.69042			−0.689195
$160$		0			0
$161$	−7.78581			−0.613608
$162$		0			0
$163$		−	12.5558i		−	0.983450i	−0.870751	−	0.491725i	$-0.836367\pi$
$163$		−	12.5558i		−	0.983450i	0.870751	−	0.491725i	$-0.163633\pi$
$164$		0			0
$165$	−2.22543	−	0.217835i	−0.173250	−	0.0169585i
$166$		0			0
$167$		−	5.39426i		−	0.417421i	−0.977978	−	0.208710i	$-0.933073\pi$
$167$		−	5.39426i		−	0.417421i	0.977978	−	0.208710i	$-0.0669266\pi$
$168$		0			0
$169$	−19.2194			−1.47841
$170$		0			0
$171$	−5.87997			−0.449652
$172$		0			0
$173$		−	20.5911i		−	1.56551i	−0.622328	−	0.782756i	$-0.713813\pi$
$173$		−	20.5911i		−	1.56551i	0.622328	−	0.782756i	$-0.286187\pi$
$174$		0			0
$175$	−0.969556	+	4.90510i	−0.0732915	+	0.370790i
$176$		0			0
$177$			3.82395i			0.287426i
$178$		0			0
$179$	−17.9388			−1.34081			−0.670403	−	0.741997i	$-0.733879\pi$
$179$	−17.9388			−1.34081			−0.670403	+	0.741997i	$0.733879\pi$
$180$		0			0
$181$	−20.3115			−1.50974			−0.754870	−	0.655874i	$-0.772300\pi$
$181$	−20.3115			−1.50974			−0.754870	+	0.655874i	$0.772300\pi$
$182$		0			0
$183$			11.2978i			0.835159i
$184$		0			0
$185$	8.23033	+	0.805622i	0.605106	+	0.0592305i
$186$		0			0
$187$			0.433388i			0.0316925i
$188$		0			0
$189$	1.00000			0.0727393
$190$		0			0
$191$	−19.2708			−1.39438			−0.697192	−	0.716884i	$-0.745568\pi$
$191$	−19.2708			−1.39438			−0.697192	+	0.716884i	$0.745568\pi$
$192$		0			0
$193$			3.13322i			0.225534i	0.993621	+	0.112767i	$0.0359714\pi$
$193$			3.13322i			0.225534i	−0.993621	+	0.112767i	$0.964029\pi$
$194$		0			0
$195$	1.23648	−	12.6320i	0.0885462	−	0.904598i
$196$		0			0
$197$		−	3.76683i		−	0.268376i	−0.990956	−	0.134188i	$-0.957157\pi$
$197$		−	3.76683i		−	0.268376i	0.990956	−	0.134188i	$-0.0428426\pi$
$198$		0			0
$199$	−11.0146			−0.780806			−0.390403	−	0.920644i	$-0.627664\pi$
$199$	−11.0146			−0.780806			−0.390403	+	0.920644i	$0.627664\pi$
$200$		0			0
$201$	−10.6922			−0.754167
$202$		0			0
$203$		−	4.26921i		−	0.299640i
$204$		0			0
$205$	1.49991	−	15.3233i	0.104758	−	1.07022i
$206$		0			0
$207$			7.78581i			0.541151i
$208$		0			0
$209$	5.87997			0.406726
$210$		0			0
$211$	−23.8421			−1.64136			−0.820678	−	0.571391i	$-0.806404\pi$
$211$	−23.8421			−1.64136			−0.820678	+	0.571391i	$0.806404\pi$
$212$		0			0
$213$		−	10.7234i		−	0.734755i
$214$		0			0
$215$	14.2600	+	1.39584i	0.972526	+	0.0951953i
$216$		0			0
$217$		−	6.65690i		−	0.451900i
$218$		0			0
$219$	−9.32221			−0.629936
$220$		0			0
$221$	−2.46000			−0.165478
$222$		0			0
$223$			6.60520i			0.442317i	0.975238	+	0.221158i	$0.0709838\pi$
$223$			6.60520i			0.442317i	−0.975238	+	0.221158i	$0.929016\pi$
$224$		0			0
$225$	4.90510	+	0.969556i	0.327006	+	0.0646370i
$226$		0			0
$227$			24.9423i			1.65548i	0.561111	+	0.827741i	$0.310374\pi$
$227$			24.9423i			1.65548i	−0.561111	+	0.827741i	$0.689626\pi$
$228$		0			0
$229$	17.3402			1.14588			0.572938	−	0.819599i	$-0.305804\pi$
$229$	17.3402			1.14588			0.572938	+	0.819599i	$0.305804\pi$
$230$		0			0
$231$	−1.00000			−0.0657952
$232$		0			0
$233$			8.80675i			0.576950i	0.957487	+	0.288475i	$0.0931482\pi$
$233$			8.80675i			0.576950i	−0.957487	+	0.288475i	$0.906852\pi$
$234$		0			0
$235$	−8.77745	−	0.859176i	−0.572578	−	0.0560465i
$236$		0			0
$237$		−	17.1007i		−	1.11081i
$238$		0			0
$239$	7.99183			0.516949			0.258474	−	0.966018i	$-0.416780\pi$
$239$	7.99183			0.516949			0.258474	+	0.966018i	$0.416780\pi$
$240$		0			0
$241$	19.8225			1.27688			0.638440	−	0.769672i	$-0.279580\pi$
$241$	19.8225			1.27688			0.638440	+	0.769672i	$0.279580\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−0.217835	+	2.22543i	−0.0139170	+	0.142178i
$246$		0			0
$247$			33.3759i			2.12366i
$248$		0			0
$249$	−0.808171			−0.0512157
$250$		0			0
$251$	30.5388			1.92759			0.963796	−	0.266640i	$-0.0859135\pi$
$251$	30.5388			1.92759			0.963796	+	0.266640i	$0.0859135\pi$
$252$		0			0
$253$		−	7.78581i		−	0.489490i
$254$		0			0
$255$	0.0944073	−	0.964477i	0.00591202	−	0.0603979i
$256$		0			0
$257$		−	29.1537i		−	1.81856i	−0.416190	−	0.909278i	$-0.636635\pi$
$257$		−	29.1537i		−	1.81856i	0.416190	−	0.909278i	$-0.363365\pi$
$258$		0			0
$259$	3.69831			0.229802
$260$		0			0
$261$	−4.26921			−0.264257
$262$		0			0
$263$		−	0.585861i		−	0.0361257i	−0.999837	−	0.0180629i	$-0.994250\pi$
$263$		−	0.585861i		−	0.0361257i	0.999837	−	0.0180629i	$-0.00574990\pi$
$264$		0			0
$265$	−19.3399	−	1.89308i	−1.18804	−	0.116291i
$266$		0			0
$267$			3.85329i			0.235818i
$268$		0			0
$269$	6.49648			0.396097			0.198049	−	0.980192i	$-0.436540\pi$
$269$	6.49648			0.396097			0.198049	+	0.980192i	$0.436540\pi$
$270$		0			0
$271$	−13.7204			−0.833452			−0.416726	−	0.909032i	$-0.636823\pi$
$271$	−13.7204			−0.833452			−0.416726	+	0.909032i	$0.636823\pi$
$272$		0			0
$273$		−	5.67621i		−	0.343540i
$274$		0			0
$275$	−4.90510	−	0.969556i	−0.295788	−	0.0584664i
$276$		0			0
$277$		−	6.45187i		−	0.387656i	−0.981036	−	0.193828i	$-0.937910\pi$
$277$		−	6.45187i		−	0.387656i	0.981036	−	0.193828i	$-0.0620903\pi$
$278$		0			0
$279$	−6.65690			−0.398538
$280$		0			0
$281$	−11.0156			−0.657138			−0.328569	−	0.944480i	$-0.606566\pi$
$281$	−11.0156			−0.657138			−0.328569	+	0.944480i	$0.606566\pi$
$282$		0			0
$283$		−	7.45618i		−	0.443224i	−0.975135	−	0.221612i	$-0.928868\pi$
$283$		−	7.45618i		−	0.443224i	0.975135	−	0.221612i	$-0.0711319\pi$
$284$		0			0
$285$	−13.0855	−	1.28086i	−0.775116	−	0.0758719i
$286$		0			0
$287$		−	6.88552i		−	0.406439i
$288$		0			0
$289$	16.8122			0.988951
$290$		0			0
$291$	−1.84873			−0.108374
$292$		0			0
$293$			26.7720i			1.56404i	0.623254	+	0.782020i	$0.285810\pi$
$293$			26.7720i			1.56404i	−0.623254	+	0.782020i	$0.714190\pi$
$294$		0			0
$295$	−0.832992	+	8.50994i	−0.0484987	+	0.495468i
$296$		0			0
$297$			1.00000i			0.0580259i
$298$		0			0
$299$	44.1939			2.55580
$300$		0			0
$301$	6.40776			0.369337
$302$		0			0
$303$		−	6.58942i		−	0.378552i
$304$		0			0
$305$	−2.46107	+	25.1425i	−0.140920	+	1.43966i
$306$		0			0
$307$		−	28.1359i		−	1.60580i	−0.596115	−	0.802899i	$-0.703290\pi$
$307$		−	28.1359i		−	1.60580i	0.596115	−	0.802899i	$-0.296710\pi$
$308$		0			0
$309$	−1.81144			−0.103049
$310$		0			0
$311$	13.3097			0.754726			0.377363	−	0.926065i	$-0.376831\pi$
$311$	13.3097			0.754726			0.377363	+	0.926065i	$0.376831\pi$
$312$		0			0
$313$			6.56503i			0.371077i	0.982637	+	0.185539i	$0.0594030\pi$
$313$			6.56503i			0.371077i	−0.982637	+	0.185539i	$0.940597\pi$
$314$		0			0
$315$	2.22543	+	0.217835i	0.125389	+	0.0122736i
$316$		0			0
$317$		−	13.5458i		−	0.760807i	−0.924821	−	0.380404i	$-0.875785\pi$
$317$		−	13.5458i		−	0.760807i	0.924821	−	0.380404i	$-0.124215\pi$
$318$		0			0
$319$	4.26921			0.239030
$320$		0			0
$321$	5.76329			0.321676
$322$		0			0
$323$			2.54831i			0.141792i
$324$		0			0
$325$	5.50340	−	27.8424i	0.305274	−	1.54442i
$326$		0			0
$327$		−	11.6042i		−	0.641711i
$328$		0			0
$329$	−3.94415			−0.217448
$330$		0			0
$331$	−22.9859			−1.26342			−0.631709	−	0.775206i	$-0.717646\pi$
$331$	−22.9859			−1.26342			−0.631709	+	0.775206i	$0.717646\pi$
$332$		0			0
$333$		−	3.69831i		−	0.202666i
$334$		0			0
$335$	−23.7947	−	2.32913i	−1.30004	−	0.127254i
$336$		0			0
$337$		−	23.7866i		−	1.29574i	−0.761752	−	0.647868i	$-0.775661\pi$
$337$		−	23.7866i		−	1.29574i	0.761752	−	0.647868i	$-0.224339\pi$
$338$		0			0
$339$	6.04973			0.328576
$340$		0			0
$341$	6.65690			0.360491
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	−1.69603	+	17.3268i	−0.0913110	+	0.932843i
$346$		0			0
$347$		−	12.1023i		−	0.649685i	−0.945768	−	0.324843i	$-0.894689\pi$
$347$		−	12.1023i		−	0.649685i	0.945768	−	0.324843i	$-0.105311\pi$
$348$		0			0
$349$	−4.78911			−0.256355			−0.128177	−	0.991751i	$-0.540913\pi$
$349$	−4.78911			−0.256355			−0.128177	+	0.991751i	$0.540913\pi$
$350$		0			0
$351$	−5.67621			−0.302974
$352$		0			0
$353$		−	14.3031i		−	0.761276i	−0.924724	−	0.380638i	$-0.875704\pi$
$353$		−	14.3031i		−	0.761276i	0.924724	−	0.380638i	$-0.124296\pi$
$354$		0			0
$355$	2.33594	−	23.8642i	0.123979	−	1.26658i
$356$		0			0
$357$		−	0.433388i		−	0.0229373i
$358$		0			0
$359$	19.9799			1.05450			0.527250	−	0.849710i	$-0.323223\pi$
$359$	19.9799			1.05450			0.527250	+	0.849710i	$0.323223\pi$
$360$		0			0
$361$	15.5740			0.819684
$362$		0			0
$363$		−	1.00000i		−	0.0524864i
$364$		0			0
$365$	−20.7459	−	2.03071i	−1.08589	−	0.106292i
$366$		0			0
$367$			32.3246i			1.68733i	0.536870	+	0.843665i	$0.319607\pi$
$367$			32.3246i			1.68733i	−0.536870	+	0.843665i	$0.680393\pi$
$368$		0			0
$369$	−6.88552			−0.358446
$370$		0			0
$371$	−8.69042			−0.451184
$372$		0			0
$373$			37.0793i			1.91990i	0.280179	+	0.959948i	$0.409606\pi$
$373$			37.0793i			1.91990i	−0.280179	+	0.959948i	$0.590394\pi$
$374$		0			0
$375$	10.7048	+	3.22618i	0.552791	+	0.166599i
$376$		0			0
$377$			24.2329i			1.24806i
$378$		0			0
$379$	2.02571			0.104054			0.0520270	−	0.998646i	$-0.483432\pi$
$379$	2.02571			0.104054			0.0520270	+	0.998646i	$0.483432\pi$
$380$		0			0
$381$	18.4582			0.945642
$382$		0			0
$383$		−	24.5770i		−	1.25582i	−0.778284	−	0.627912i	$-0.783910\pi$
$383$		−	24.5770i		−	1.25582i	0.778284	−	0.627912i	$-0.216090\pi$
$384$		0			0
$385$	−2.22543	−	0.217835i	−0.113419	−	0.0111019i
$386$		0			0
$387$		−	6.40776i		−	0.325725i
$388$		0			0
$389$	32.8334			1.66472			0.832361	−	0.554234i	$-0.186989\pi$
$389$	32.8334			1.66472			0.832361	+	0.554234i	$0.186989\pi$
$390$		0			0
$391$	3.37428			0.170645
$392$		0			0
$393$			9.49331i			0.478874i
$394$		0			0
$395$	3.72514	−	38.0564i	0.187432	−	1.91483i
$396$		0			0
$397$		−	22.3516i		−	1.12179i	−0.827886	−	0.560896i	$-0.810457\pi$
$397$		−	22.3516i		−	1.12179i	0.827886	−	0.560896i	$-0.189543\pi$
$398$		0			0
$399$	−5.87997			−0.294366
$400$		0			0
$401$	10.6097			0.529822			0.264911	−	0.964273i	$-0.414657\pi$
$401$	10.6097			0.529822			0.264911	+	0.964273i	$0.414657\pi$
$402$		0			0
$403$			37.7860i			1.88225i
$404$		0			0
$405$	0.217835	−	2.22543i	0.0108243	−	0.110583i
$406$		0			0
$407$			3.69831i			0.183318i
$408$		0			0
$409$	−0.943231			−0.0466398			−0.0233199	−	0.999728i	$-0.507424\pi$
$409$	−0.943231			−0.0466398			−0.0233199	+	0.999728i	$0.507424\pi$
$410$		0			0
$411$	−12.7048			−0.626683
$412$		0			0
$413$			3.82395i			0.188164i
$414$		0			0
$415$	−1.79853	−	0.176048i	−0.0882863	−	0.00864187i
$416$		0			0
$417$		−	13.0820i		−	0.640626i
$418$		0			0
$419$	0.279835			0.0136708			0.00683542	−	0.999977i	$-0.497824\pi$
$419$	0.279835			0.0136708			0.00683542	+	0.999977i	$0.497824\pi$
$420$		0			0
$421$	−35.3133			−1.72106			−0.860532	−	0.509396i	$-0.829869\pi$
$421$	−35.3133			−1.72106			−0.860532	+	0.509396i	$0.829869\pi$
$422$		0			0
$423$			3.94415i			0.191771i
$424$		0			0
$425$	0.420194	−	2.12581i	0.0203824	−	0.103117i
$426$		0			0
$427$			11.2978i			0.546740i
$428$		0			0
$429$	5.67621			0.274050
$430$		0			0
$431$	1.69963			0.0818682			0.0409341	−	0.999162i	$-0.486967\pi$
$431$	1.69963			0.0818682			0.0409341	+	0.999162i	$0.486967\pi$
$432$		0			0
$433$		−	20.7461i		−	0.996996i	−0.866891	−	0.498498i	$-0.833885\pi$
$433$		−	20.7461i		−	0.996996i	0.866891	−	0.498498i	$-0.166115\pi$
$434$		0			0
$435$	−9.50083	−	0.929984i	−0.455530	−	0.0445893i
$436$		0			0
$437$		−	45.7803i		−	2.18997i
$438$		0			0
$439$	−4.44320			−0.212062			−0.106031	−	0.994363i	$-0.533814\pi$
$439$	−4.44320			−0.212062			−0.106031	+	0.994363i	$0.533814\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$			23.9390i			1.13737i	0.822554	+	0.568687i	$0.192549\pi$
$443$			23.9390i			1.13737i	−0.822554	+	0.568687i	$0.807451\pi$
$444$		0			0
$445$	−0.839384	+	8.57524i	−0.0397906	+	0.406506i
$446$		0			0
$447$			7.49955i			0.354717i
$448$		0			0
$449$	2.77064			0.130755			0.0653774	−	0.997861i	$-0.479175\pi$
$449$	2.77064			0.130755			0.0653774	+	0.997861i	$0.479175\pi$
$450$		0			0
$451$	6.88552			0.324226
$452$		0			0
$453$			20.4872i			0.962571i
$454$		0			0
$455$	1.23648	−	12.6320i	0.0579671	−	0.592198i
$456$		0			0
$457$		−	27.1298i		−	1.26908i	−0.772891	−	0.634539i	$-0.781190\pi$
$457$		−	27.1298i		−	1.26908i	0.772891	−	0.634539i	$-0.218810\pi$
$458$		0			0
$459$	−0.433388			−0.0202288
$460$		0			0
$461$	18.8843			0.879529			0.439764	−	0.898113i	$-0.355062\pi$
$461$	18.8843			0.879529			0.439764	+	0.898113i	$0.355062\pi$
$462$		0			0
$463$			5.13620i			0.238700i	0.992852	+	0.119350i	$0.0380810\pi$
$463$			5.13620i			0.238700i	−0.992852	+	0.119350i	$0.961919\pi$
$464$		0			0
$465$	−14.8145	−	1.45011i	−0.687005	−	0.0672472i
$466$		0			0
$467$			1.76667i			0.0817515i	0.999164	+	0.0408758i	$0.0130148\pi$
$467$			1.76667i			0.0817515i	−0.999164	+	0.0408758i	$0.986985\pi$
$468$		0			0
$469$	−10.6922			−0.493718
$470$		0			0
$471$	−3.01973			−0.139142
$472$		0			0
$473$			6.40776i			0.294629i
$474$		0			0
$475$	−28.8418	−	5.70095i	−1.32335	−	0.261578i
$476$		0			0
$477$			8.69042i			0.397907i
$478$		0			0
$479$	36.1681			1.65256			0.826281	−	0.563258i	$-0.190452\pi$
$479$	36.1681			1.65256			0.826281	+	0.563258i	$0.190452\pi$
$480$		0			0
$481$	−20.9924			−0.957170
$482$		0			0
$483$			7.78581i			0.354267i
$484$		0			0
$485$	−4.11422	−	0.402719i	−0.186817	−	0.0182865i
$486$		0			0
$487$		−	1.65393i		−	0.0749468i	−0.999298	−	0.0374734i	$-0.988069\pi$
$487$		−	1.65393i		−	0.0749468i	0.999298	−	0.0374734i	$-0.0119310\pi$
$488$		0			0
$489$	−12.5558			−0.567795
$490$		0			0
$491$	35.7826			1.61485			0.807424	−	0.589972i	$-0.200861\pi$
$491$	35.7826			1.61485			0.807424	+	0.589972i	$0.200861\pi$
$492$		0			0
$493$			1.85022i			0.0833299i
$494$		0			0
$495$	−0.217835	+	2.22543i	−0.00979097	+	0.100026i
$496$		0			0
$497$		−	10.7234i		−	0.481010i
$498$		0			0
$499$	36.6187			1.63928			0.819640	−	0.572879i	$-0.194174\pi$
$499$	36.6187			1.63928			0.819640	+	0.572879i	$0.194174\pi$
$500$		0			0
$501$	−5.39426			−0.240998
$502$		0			0
$503$		−	27.9780i		−	1.24748i	−0.781632	−	0.623739i	$-0.785613\pi$
$503$		−	27.9780i		−	1.24748i	0.781632	−	0.623739i	$-0.214387\pi$
$504$		0			0
$505$	1.43541	−	14.6643i	0.0638748	−	0.652553i
$506$		0			0
$507$			19.2194i			0.853563i
$508$		0			0
$509$	−11.2545			−0.498846			−0.249423	−	0.968395i	$-0.580241\pi$
$509$	−11.2545			−0.498846			−0.249423	+	0.968395i	$0.580241\pi$
$510$		0			0
$511$	−9.32221			−0.412390
$512$		0			0
$513$			5.87997i			0.259607i
$514$		0			0
$515$	−4.03124	−	0.394596i	−0.177638	−	0.0173880i
$516$		0			0
$517$		−	3.94415i		−	0.173464i
$518$		0			0
$519$	−20.5911			−0.903849
$520$		0			0
$521$	10.7655			0.471644			0.235822	−	0.971796i	$-0.424222\pi$
$521$	10.7655			0.471644			0.235822	+	0.971796i	$0.424222\pi$
$522$		0			0
$523$			41.8359i			1.82935i	0.404185	+	0.914677i	$0.367555\pi$
$523$			41.8359i			1.82935i	−0.404185	+	0.914677i	$0.632445\pi$
$524$		0			0
$525$	4.90510	+	0.969556i	0.214076	+	0.0423149i
$526$		0			0
$527$			2.88502i			0.125674i
$528$		0			0
$529$	−37.6189			−1.63560
$530$		0			0
$531$	3.82395			0.165945
$532$		0			0
$533$			39.0837i			1.69290i
$534$		0			0
$535$	12.8258	+	1.25545i	0.554509	+	0.0542778i
$536$		0			0
$537$			17.9388i			0.774115i
$538$		0			0
$539$	−1.00000			−0.0430730
$540$		0			0
$541$	4.96464			0.213447			0.106723	−	0.994289i	$-0.465964\pi$
$541$	4.96464			0.213447			0.106723	+	0.994289i	$0.465964\pi$
$542$		0			0
$543$			20.3115i			0.871649i
$544$		0			0
$545$	2.52780	−	25.8243i	0.108279	−	1.10619i
$546$		0			0
$547$			17.6539i			0.754826i	0.926045	+	0.377413i	$0.123186\pi$
$547$			17.6539i			0.754826i	−0.926045	+	0.377413i	$0.876814\pi$
$548$		0			0
$549$	11.2978			0.482180
$550$		0			0
$551$	25.1028			1.06941
$552$		0			0
$553$		−	17.1007i		−	0.727195i
$554$		0			0
$555$	0.805622	−	8.23033i	0.0341968	−	0.349358i
$556$		0			0
$557$			23.2282i			0.984212i	0.870535	+	0.492106i	$0.163773\pi$
$557$			23.2282i			0.984212i	−0.870535	+	0.492106i	$0.836227\pi$
$558$		0			0
$559$	−36.3718			−1.53836
$560$		0			0
$561$	0.433388			0.0182977
$562$		0			0
$563$		−	4.29920i		−	0.181190i	−0.995888	−	0.0905949i	$-0.971123\pi$
$563$		−	4.29920i		−	0.181190i	0.995888	−	0.0905949i	$-0.0288768\pi$
$564$		0			0
$565$	13.4633	+	1.31785i	0.566404	+	0.0554422i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	−24.4765			−1.02611			−0.513054	−	0.858356i	$-0.671486\pi$
$569$	−24.4765			−1.02611			−0.513054	+	0.858356i	$0.671486\pi$
$570$		0			0
$571$	−16.1177			−0.674503			−0.337252	−	0.941415i	$-0.609497\pi$
$571$	−16.1177			−0.674503			−0.337252	+	0.941415i	$0.609497\pi$
$572$		0			0
$573$			19.2708i			0.805049i
$574$		0			0
$575$	−7.54878	+	38.1902i	−0.314806	+	1.59264i
$576$		0			0
$577$		−	16.7954i		−	0.699202i	−0.936899	−	0.349601i	$-0.886317\pi$
$577$		−	16.7954i		−	0.699202i	0.936899	−	0.349601i	$-0.113683\pi$
$578$		0			0
$579$	3.13322			0.130212
$580$		0			0
$581$	−0.808171			−0.0335286
$582$		0			0
$583$		−	8.69042i		−	0.359920i
$584$		0			0
$585$	−12.6320	−	1.23648i	−0.522270	−	0.0511222i
$586$		0			0
$587$			12.2404i			0.505217i	0.967569	+	0.252608i	$0.0812884\pi$
$587$			12.2404i			0.505217i	−0.967569	+	0.252608i	$0.918712\pi$
$588$		0			0
$589$	39.1423			1.61283
$590$		0			0
$591$	−3.76683			−0.154947
$592$		0			0
$593$		−	17.1341i		−	0.703611i	−0.936073	−	0.351806i	$-0.885568\pi$
$593$		−	17.1341i		−	0.703611i	0.936073	−	0.351806i	$-0.114432\pi$
$594$		0			0
$595$	0.0944073	−	0.964477i	0.00387032	−	0.0395397i
$596$		0			0
$597$			11.0146i			0.450799i
$598$		0			0
$599$	−5.19551			−0.212283			−0.106141	−	0.994351i	$-0.533850\pi$
$599$	−5.19551			−0.212283			−0.106141	+	0.994351i	$0.533850\pi$
$600$		0			0
$601$	−16.0347			−0.654069			−0.327035	−	0.945012i	$-0.606049\pi$
$601$	−16.0347			−0.654069			−0.327035	+	0.945012i	$0.606049\pi$
$602$		0			0
$603$			10.6922i			0.435418i
$604$		0			0
$605$	0.217835	−	2.22543i	0.00885627	−	0.0904767i
$606$		0			0
$607$		−	15.7127i		−	0.637757i	−0.947796	−	0.318879i	$-0.896694\pi$
$607$		−	15.7127i		−	0.637757i	0.947796	−	0.318879i	$-0.103306\pi$
$608$		0			0
$609$	−4.26921			−0.172997
$610$		0			0
$611$	22.3879			0.905716
$612$		0			0
$613$		−	31.9917i		−	1.29213i	−0.763281	−	0.646067i	$-0.776413\pi$
$613$		−	31.9917i		−	1.29213i	0.763281	−	0.646067i	$-0.223587\pi$
$614$		0			0
$615$	−15.3233	−	1.49991i	−0.617893	−	0.0604822i
$616$		0			0
$617$		−	27.0459i		−	1.08883i	−0.838818	−	0.544413i	$-0.816753\pi$
$617$		−	27.0459i		−	1.08883i	0.838818	−	0.544413i	$-0.183247\pi$
$618$		0			0
$619$	41.5774			1.67114			0.835568	−	0.549386i	$-0.185139\pi$
$619$	41.5774			1.67114			0.835568	+	0.549386i	$0.185139\pi$
$620$		0			0
$621$	7.78581			0.312434
$622$		0			0
$623$			3.85329i			0.154379i
$624$		0			0
$625$	23.1199	+	9.51153i	0.924797	+	0.380461i
$626$		0			0
$627$		−	5.87997i		−	0.234823i
$628$		0			0
$629$	−1.60280			−0.0639080
$630$		0			0
$631$	18.4082			0.732819			0.366410	−	0.930454i	$-0.380587\pi$
$631$	18.4082			0.732819			0.366410	+	0.930454i	$0.380587\pi$
$632$		0			0
$633$			23.8421i			0.947638i
$634$		0			0
$635$	41.0775	+	4.02085i	1.63011	+	0.159563i
$636$		0			0
$637$		−	5.67621i		−	0.224900i
$638$		0			0
$639$	−10.7234			−0.424211
$640$		0			0
$641$	−35.3955			−1.39804			−0.699020	−	0.715102i	$-0.746380\pi$
$641$	−35.3955			−1.39804			−0.699020	+	0.715102i	$0.746380\pi$
$642$		0			0
$643$		−	10.7233i		−	0.422884i	−0.977391	−	0.211442i	$-0.932184\pi$
$643$		−	10.7233i		−	0.422884i	0.977391	−	0.211442i	$-0.0678159\pi$
$644$		0			0
$645$	1.39584	−	14.2600i	0.0549610	−	0.561488i
$646$		0			0
$647$			22.6713i			0.891299i	0.895207	+	0.445650i	$0.147027\pi$
$647$			22.6713i			0.891299i	−0.895207	+	0.445650i	$0.852973\pi$
$648$		0			0
$649$	−3.82395			−0.150103
$650$		0			0
$651$	−6.65690			−0.260904
$652$		0			0
$653$			38.2867i			1.49827i	0.662415	+	0.749137i	$0.269532\pi$
$653$			38.2867i			1.49827i	−0.662415	+	0.749137i	$0.730468\pi$
$654$		0			0
$655$	−2.06798	+	21.1267i	−0.0808026	+	0.825489i
$656$		0			0
$657$			9.32221i			0.363694i
$658$		0			0
$659$	22.6296			0.881526			0.440763	−	0.897624i	$-0.354708\pi$
$659$	22.6296			0.881526			0.440763	+	0.897624i	$0.354708\pi$
$660$		0			0
$661$	13.9801			0.543763			0.271881	−	0.962331i	$-0.412354\pi$
$661$	13.9801			0.543763			0.271881	+	0.962331i	$0.412354\pi$
$662$		0			0
$663$			2.46000i			0.0955387i
$664$		0			0
$665$	−13.0855	−	1.28086i	−0.507433	−	0.0496698i
$666$		0			0
$667$		−	33.2392i		−	1.28703i
$668$		0			0
$669$	6.60520			0.255372
$670$		0			0
$671$	−11.2978			−0.436148
$672$		0			0
$673$		−	29.5356i		−	1.13851i	−0.822160	−	0.569257i	$-0.807231\pi$
$673$		−	29.5356i		−	1.13851i	0.822160	−	0.569257i	$-0.192769\pi$
$674$		0			0
$675$	0.969556	−	4.90510i	0.0373182	−	0.188797i
$676$		0			0
$677$			36.1349i			1.38878i	0.719600	+	0.694388i	$0.244325\pi$
$677$			36.1349i			1.38878i	−0.719600	+	0.694388i	$0.755675\pi$
$678$		0			0
$679$	−1.84873			−0.0709477
$680$		0			0
$681$	24.9423			0.955793
$682$		0			0
$683$			38.9588i			1.49072i	0.666663	+	0.745359i	$0.267722\pi$
$683$			38.9588i			1.49072i	−0.666663	+	0.745359i	$0.732278\pi$
$684$		0			0
$685$	−28.2737	−	2.76756i	−1.08028	−	0.105743i
$686$		0			0
$687$		−	17.3402i		−	0.661571i
$688$		0			0
$689$	49.3287			1.87927
$690$		0			0
$691$	20.5528			0.781866			0.390933	−	0.920419i	$-0.372152\pi$
$691$	20.5528			0.781866			0.390933	+	0.920419i	$0.372152\pi$
$692$		0			0
$693$			1.00000i			0.0379869i
$694$		0			0
$695$	2.84971	−	29.1130i	0.108096	−	1.10432i
$696$		0			0
$697$			2.98410i			0.113031i
$698$		0			0
$699$	8.80675			0.333102
$700$		0			0
$701$	−41.2759			−1.55897			−0.779484	−	0.626422i	$-0.784519\pi$
$701$	−41.2759			−1.55897			−0.779484	+	0.626422i	$0.784519\pi$
$702$		0			0
$703$			21.7459i			0.820163i
$704$		0			0
$705$	−0.859176	+	8.77745i	−0.0323585	+	0.330578i
$706$		0			0
$707$		−	6.58942i		−	0.247821i
$708$		0			0
$709$	20.2336			0.759890			0.379945	−	0.925009i	$-0.375943\pi$
$709$	20.2336			0.759890			0.379945	+	0.925009i	$0.375943\pi$
$710$		0			0
$711$	−17.1007			−0.641326
$712$		0			0
$713$		−	51.8294i		−	1.94103i
$714$		0			0
$715$	12.6320	+	1.23648i	0.472411	+	0.0462417i
$716$		0			0
$717$		−	7.99183i		−	0.298460i
$718$		0			0
$719$	43.4826			1.62163			0.810814	−	0.585304i	$-0.199025\pi$
$719$	43.4826			1.62163			0.810814	+	0.585304i	$0.199025\pi$
$720$		0			0
$721$	−1.81144			−0.0674615
$722$		0			0
$723$		−	19.8225i		−	0.737207i
$724$		0			0
$725$	−20.9409	−	4.13923i	−0.777724	−	0.153727i
$726$		0			0
$727$			37.2464i			1.38139i	0.723145	+	0.690696i	$0.242696\pi$
$727$			37.2464i			1.38139i	−0.723145	+	0.690696i	$0.757304\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−2.77705			−0.102713
$732$		0			0
$733$			2.22955i			0.0823502i	0.999152	+	0.0411751i	$0.0131101\pi$
$733$			2.22955i			0.0823502i	−0.999152	+	0.0411751i	$0.986890\pi$
$734$		0			0
$735$	2.22543	+	0.217835i	0.0820863	+	0.00803498i
$736$		0			0
$737$		−	10.6922i		−	0.393851i
$738$		0			0
$739$	−48.2592			−1.77524			−0.887621	−	0.460575i	$-0.847643\pi$
$739$	−48.2592			−1.77524			−0.887621	+	0.460575i	$0.847643\pi$
$740$		0			0
$741$	33.3759			1.22610
$742$		0			0
$743$			16.6812i			0.611975i	0.952036	+	0.305987i	$0.0989865\pi$
$743$			16.6812i			0.611975i	−0.952036	+	0.305987i	$0.901013\pi$
$744$		0			0
$745$	−1.63367	+	16.6897i	−0.0598530	+	0.611465i
$746$		0			0
$747$			0.808171i			0.0295694i
$748$		0			0
$749$	5.76329			0.210586
$750$		0			0
$751$	−8.83634			−0.322443			−0.161221	−	0.986918i	$-0.551543\pi$
$751$	−8.83634			−0.322443			−0.161221	+	0.986918i	$0.551543\pi$
$752$		0			0
$753$		−	30.5388i		−	1.11290i
$754$		0			0
$755$	−4.46283	+	45.5928i	−0.162419	+	1.65929i
$756$		0			0
$757$			51.5202i			1.87253i	0.351290	+	0.936267i	$0.385743\pi$
$757$			51.5202i			1.87253i	−0.351290	+	0.936267i	$0.614257\pi$
$758$		0			0
$759$	−7.78581			−0.282607
$760$		0			0
$761$	−7.37450			−0.267325			−0.133663	−	0.991027i	$-0.542674\pi$
$761$	−7.37450			−0.267325			−0.133663	+	0.991027i	$0.542674\pi$
$762$		0			0
$763$		−	11.6042i		−	0.420099i
$764$		0			0
$765$	−0.964477	−	0.0944073i	−0.0348707	−	0.00341330i
$766$		0			0
$767$		−	21.7056i		−	0.783742i
$768$		0			0
$769$	−22.9247			−0.826685			−0.413342	−	0.910576i	$-0.635639\pi$
$769$	−22.9247			−0.826685			−0.413342	+	0.910576i	$0.635639\pi$
$770$		0			0
$771$	−29.1537			−1.04994
$772$		0			0
$773$		−	23.0412i		−	0.828733i	−0.910110	−	0.414366i	$-0.864003\pi$
$773$		−	23.0412i		−	0.828733i	0.910110	−	0.414366i	$-0.135997\pi$
$774$		0			0
$775$	−32.6527	−	6.45424i	−1.17292	−	0.231843i
$776$		0			0
$777$		−	3.69831i		−	0.132676i
$778$		0			0
$779$	40.4866			1.45058
$780$		0			0
$781$	10.7234			0.383713
$782$		0			0
$783$			4.26921i			0.152569i
$784$		0			0
$785$	−6.72021	−	0.657805i	−0.239855	−	0.0234781i
$786$		0			0
$787$			2.28921i			0.0816014i	0.999167	+	0.0408007i	$0.0129909\pi$
$787$			2.28921i			0.0816014i	−0.999167	+	0.0408007i	$0.987009\pi$
$788$		0			0
$789$	−0.585861			−0.0208572
$790$		0			0
$791$	6.04973			0.215104
$792$		0			0
$793$		−	64.1289i		−	2.27728i
$794$		0			0
$795$	−1.89308	+	19.3399i	−0.0671407	+	0.685917i
$796$		0			0
$797$		−	28.8513i		−	1.02197i	−0.859591	−	0.510983i	$-0.829282\pi$
$797$		−	28.8513i		−	1.02197i	0.859591	−	0.510983i	$-0.170718\pi$
$798$		0			0
$799$	1.70935			0.0604725
$800$		0			0
$801$	3.85329			0.136149
$802$		0			0
$803$		−	9.32221i		−	0.328973i
$804$		0			0
$805$	−1.69603	+	17.3268i	−0.0597770	+	0.610689i
$806$		0			0
$807$		−	6.49648i		−	0.228687i
$808$		0			0
$809$	−27.6656			−0.972669			−0.486335	−	0.873773i	$-0.661666\pi$
$809$	−27.6656			−0.972669			−0.486335	+	0.873773i	$0.661666\pi$
$810$		0			0
$811$	−0.216516			−0.00760290			−0.00380145	−	0.999993i	$-0.501210\pi$
$811$	−0.216516			−0.00760290			−0.00380145	+	0.999993i	$0.501210\pi$
$812$		0			0
$813$			13.7204i			0.481194i
$814$		0			0
$815$	−27.9422	−	2.73511i	−0.978772	−	0.0958066i
$816$		0			0
$817$			37.6774i			1.31817i
$818$		0			0
$819$	−5.67621			−0.198343
$820$		0			0
$821$	−3.47329			−0.121218			−0.0606092	−	0.998162i	$-0.519304\pi$
$821$	−3.47329			−0.121218			−0.0606092	+	0.998162i	$0.519304\pi$
$822$		0			0
$823$			38.7760i			1.35165i	0.737063	+	0.675824i	$0.236212\pi$
$823$			38.7760i			1.35165i	−0.737063	+	0.675824i	$0.763788\pi$
$824$		0			0
$825$	−0.969556	+	4.90510i	−0.0337556	+	0.170774i
$826$		0			0
$827$			28.7546i			0.999896i	0.866056	+	0.499948i	$0.166647\pi$
$827$			28.7546i			0.999896i	−0.866056	+	0.499948i	$0.833353\pi$
$828$		0			0
$829$	−42.4781			−1.47532			−0.737662	−	0.675170i	$-0.764070\pi$
$829$	−42.4781			−1.47532			−0.737662	+	0.675170i	$0.764070\pi$
$830$		0			0
$831$	−6.45187			−0.223813
$832$		0			0
$833$		−	0.433388i		−	0.0150160i
$834$		0			0
$835$	−12.0046	−	1.17506i	−0.415435	−	0.0406647i
$836$		0			0
$837$			6.65690i			0.230096i
$838$		0			0
$839$	25.2866			0.872992			0.436496	−	0.899706i	$-0.356219\pi$
$839$	25.2866			0.872992			0.436496	+	0.899706i	$0.356219\pi$
$840$		0			0
$841$	−10.7739			−0.371513
$842$		0			0
$843$			11.0156i			0.379399i
$844$		0			0
$845$	−4.18666	+	42.7714i	−0.144026	+	1.47138i
$846$		0			0
$847$		−	1.00000i		−	0.0343604i
$848$		0			0
$849$	−7.45618			−0.255896
$850$		0			0
$851$	28.7943			0.987057
$852$		0			0
$853$		−	10.7215i		−	0.367098i	−0.983011	−	0.183549i	$-0.941241\pi$
$853$		−	10.7215i		−	0.367098i	0.983011	−	0.183549i	$-0.0587586\pi$
$854$		0			0
$855$	−1.28086	+	13.0855i	−0.0438046	+	0.447513i
$856$		0			0
$857$			8.78941i			0.300240i	0.988668	+	0.150120i	$0.0479661\pi$
$857$			8.78941i			0.300240i	−0.988668	+	0.150120i	$0.952034\pi$
$858$		0			0
$859$	37.6204			1.28359			0.641796	−	0.766875i	$-0.278189\pi$
$859$	37.6204			1.28359			0.641796	+	0.766875i	$0.278189\pi$
$860$		0			0
$861$	−6.88552			−0.234658
$862$		0			0
$863$			26.9014i			0.915734i	0.889021	+	0.457867i	$0.151386\pi$
$863$			26.9014i			0.915734i	−0.889021	+	0.457867i	$0.848614\pi$
$864$		0			0
$865$	−45.8241	−	4.48547i	−1.55807	−	0.152511i
$866$		0			0
$867$		−	16.8122i		−	0.570971i
$868$		0			0
$869$	17.1007			0.580101
$870$		0			0
$871$	60.6910			2.05644
$872$		0			0
$873$			1.84873i			0.0625700i
$874$		0			0
$875$	10.7048	+	3.22618i	0.361887	+	0.109065i
$876$		0			0
$877$			28.8722i			0.974944i	0.873139	+	0.487472i	$0.162081\pi$
$877$			28.8722i			0.974944i	−0.873139	+	0.487472i	$0.837919\pi$
$878$		0			0
$879$	26.7720			0.902999
$880$		0			0
$881$	55.8405			1.88132			0.940658	−	0.339356i	$-0.110209\pi$
$881$	55.8405			1.88132			0.940658	+	0.339356i	$0.110209\pi$
$882$		0			0
$883$			6.67689i			0.224695i	0.993669	+	0.112348i	$0.0358370\pi$
$883$			6.67689i			0.224695i	−0.993669	+	0.112348i	$0.964163\pi$
$884$		0			0
$885$	8.50994	+	0.832992i	0.286059	+	0.0280007i
$886$		0			0
$887$			17.7562i			0.596196i	0.954535	+	0.298098i	$0.0963523\pi$
$887$			17.7562i			0.596196i	−0.954535	+	0.298098i	$0.903648\pi$
$888$		0			0
$889$	18.4582			0.619068
$890$		0			0
$891$	1.00000			0.0335013
$892$		0			0
$893$		−	23.1915i		−	0.776074i
$894$		0			0
$895$	−3.90770	+	39.9215i	−0.130620	+	1.33443i
$896$		0			0
$897$		−	44.1939i		−	1.47559i
$898$		0			0
$899$	28.4197			0.947849
$900$		0			0
$901$	3.76633			0.125475
$902$		0			0
$903$		−	6.40776i		−	0.213237i
$904$		0			0
$905$	−4.42456	+	45.2018i	−0.147077	+	1.50256i
$906$		0			0
$907$			21.6291i			0.718183i	0.933302	+	0.359091i	$0.116913\pi$
$907$			21.6291i			0.718183i	−0.933302	+	0.359091i	$0.883087\pi$
$908$		0			0
$909$	−6.58942			−0.218557
$910$		0			0
$911$	−56.2734			−1.86442			−0.932211	−	0.361916i	$-0.882123\pi$
$911$	−56.2734			−1.86442			−0.932211	+	0.361916i	$0.882123\pi$
$912$		0			0
$913$		−	0.808171i		−	0.0267465i
$914$		0			0
$915$	25.1425	+	2.46107i	0.831187	+	0.0813604i
$916$		0			0
$917$			9.49331i			0.313497i
$918$		0			0
$919$	30.8891			1.01894			0.509469	−	0.860489i	$-0.329842\pi$
$919$	30.8891			1.01894			0.509469	+	0.860489i	$0.329842\pi$
$920$		0			0
$921$	−28.1359			−0.927108
$922$		0			0
$923$			60.8683i			2.00350i
$924$		0			0
$925$	3.58572	−	18.1406i	0.117898	−	0.596458i
$926$		0			0
$927$			1.81144i			0.0594955i
$928$		0			0
$929$	−1.84255			−0.0604520			−0.0302260	−	0.999543i	$-0.509623\pi$
$929$	−1.84255			−0.0604520			−0.0302260	+	0.999543i	$0.509623\pi$
$930$		0			0
$931$	−5.87997			−0.192708
$932$		0			0
$933$		−	13.3097i		−	0.435741i
$934$		0			0
$935$	0.964477	+	0.0944073i	0.0315418	+	0.00308745i
$936$		0			0
$937$			53.2459i			1.73947i	0.493522	+	0.869733i	$0.335709\pi$
$937$			53.2459i			1.73947i	−0.493522	+	0.869733i	$0.664291\pi$
$938$		0			0
$939$	6.56503			0.214242
$940$		0			0
$941$	−4.10568			−0.133841			−0.0669207	−	0.997758i	$-0.521317\pi$
$941$	−4.10568			−0.133841			−0.0669207	+	0.997758i	$0.521317\pi$
$942$		0			0
$943$		−	53.6093i		−	1.74576i
$944$		0			0
$945$	0.217835	−	2.22543i	0.00708619	−	0.0723933i
$946$		0			0
$947$			47.8953i			1.55639i	0.628025	+	0.778193i	$0.283864\pi$
$947$			47.8953i			1.55639i	−0.628025	+	0.778193i	$0.716136\pi$
$948$		0			0
$949$	52.9148			1.71769
$950$		0			0
$951$	−13.5458			−0.439252
$952$		0			0
$953$			29.5318i			0.956628i	0.878189	+	0.478314i	$0.158752\pi$
$953$			29.5318i			0.956628i	−0.878189	+	0.478314i	$0.841248\pi$
$954$		0			0
$955$	−4.19786	+	42.8858i	−0.135840	+	1.38775i
$956$		0			0
$957$		−	4.26921i		−	0.138004i
$958$		0			0
$959$	−12.7048			−0.410260
$960$		0			0
$961$	13.3143			0.429494
$962$		0			0
$963$		−	5.76329i		−	0.185720i
$964$		0			0
$965$	6.97278	+	0.682527i	0.224462	+	0.0219713i
$966$		0			0
$967$		−	14.7651i		−	0.474813i	−0.971410	−	0.237406i	$-0.923703\pi$
$967$		−	14.7651i		−	0.474813i	0.971410	−	0.237406i	$-0.0762973\pi$
$968$		0			0
$969$	2.54831			0.0818635
$970$		0			0
$971$	−57.0117			−1.82959			−0.914796	−	0.403916i	$-0.867649\pi$
$971$	−57.0117			−1.82959			−0.914796	+	0.403916i	$0.867649\pi$
$972$		0			0
$973$		−	13.0820i		−	0.419388i
$974$		0			0
$975$	−27.8424	−	5.50340i	−0.891669	−	0.176250i
$976$		0			0
$977$			22.0910i			0.706753i	0.935481	+	0.353377i	$0.114967\pi$
$977$			22.0910i			0.706753i	−0.935481	+	0.353377i	$0.885033\pi$
$978$		0			0
$979$	−3.85329			−0.123152
$980$		0			0
$981$	−11.6042			−0.370492
$982$		0			0
$983$			19.4362i			0.619917i	0.950750	+	0.309959i	$0.100315\pi$
$983$			19.4362i			0.619917i	−0.950750	+	0.309959i	$0.899685\pi$
$984$		0			0
$985$	−8.38283	−	0.820550i	−0.267099	−	0.0261449i
$986$		0			0
$987$			3.94415i			0.125544i
$988$		0			0
$989$	49.8896			1.58640
$990$		0			0
$991$	−15.0776			−0.478957			−0.239478	−	0.970902i	$-0.576976\pi$
$991$	−15.0776			−0.478957			−0.239478	+	0.970902i	$0.576976\pi$
$992$		0			0
$993$			22.9859i			0.729435i
$994$		0			0
$995$	−2.39938	+	24.5123i	−0.0760653	+	0.777092i
$996$		0			0
$997$			34.4387i			1.09068i	0.838214	+	0.545342i	$0.183600\pi$
$997$			34.4387i			1.09068i	−0.838214	+	0.545342i	$0.816400\pi$
$998$		0			0
$999$	−3.69831			−0.117009

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.h.e.1849.5	✓	14
5.4	even	2	inner	4620.2.h.e.1849.12	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.h.e.1849.5	✓	14	1.1	even	1	trivial
4620.2.h.e.1849.12	yes	14	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4620.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(0.217835 - 2.22543i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +1.00000 q^{11} +5.67621i q^{13} +(-2.22543 - 0.217835i) q^{15} +0.433388i q^{17} +5.87997 q^{19} -1.00000 q^{21} -7.78581i q^{23} +(-4.90510 - 0.969556i) q^{25} +1.00000i q^{27} +4.26921 q^{29} +6.65690 q^{31} -1.00000i q^{33} +(-2.22543 - 0.217835i) q^{35} +3.69831i q^{37} +5.67621 q^{39} +6.88552 q^{41} +6.40776i q^{43} +(-0.217835 + 2.22543i) q^{45} -3.94415i q^{47} -1.00000 q^{49} +0.433388 q^{51} -8.69042i q^{53} +(0.217835 - 2.22543i) q^{55} -5.87997i q^{57} -3.82395 q^{59} -11.2978 q^{61} +1.00000i q^{63} +(12.6320 + 1.23648i) q^{65} -10.6922i q^{67} -7.78581 q^{69} +10.7234 q^{71} -9.32221i q^{73} +(-0.969556 + 4.90510i) q^{75} -1.00000i q^{77} +17.1007 q^{79} +1.00000 q^{81} -0.808171i q^{83} +(0.964477 + 0.0944073i) q^{85} -4.26921i q^{87} -3.85329 q^{89} +5.67621 q^{91} -6.65690i q^{93} +(1.28086 - 13.0855i) q^{95} -1.84873i q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 4 q^{5} - 14 q^{9} + 14 q^{11} + 6 q^{19} - 14 q^{21} - 8 q^{25} + 38 q^{29} + 4 q^{31} + 8 q^{39} - 12 q^{41} + 4 q^{45} - 14 q^{49} + 14 q^{51} - 4 q^{55} + 18 q^{59} - 54 q^{61} - 2 q^{65} + 26 q^{69}+ \cdots - 14 q^{99}+O(q^{100}) \) 14 * q - 4 * q^5 - 14 * q^9 + 14 * q^11 + 6 * q^19 - 14 * q^21 - 8 * q^25 + 38 * q^29 + 4 * q^31 + 8 * q^39 - 12 * q^41 + 4 * q^45 - 14 * q^49 + 14 * q^51 - 4 * q^55 + 18 * q^59 - 54 * q^61 - 2 * q^65 + 26 * q^69 - 24 * q^71 + 48 * q^79 + 14 * q^81 + 12 * q^85 + 46 * q^89 + 8 * q^91 - 72 * q^95 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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