LMFDB - Embedded newform 2808.2.r.f.2089.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2808,2,Mod(289,2808)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2808, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2808.289"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2808 = 2^{3} \cdot 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2808.r (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [40,0,0,0,-1,0,7] 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:	$22.4219928876$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 936)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2089.1
Character	$\chi$	$=$	2808.2089
Dual form			2808.2.r.f.289.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.01584 + 3.49154i) q^{5} +(1.55709 - 2.69696i) q^{7} +2.67730 q^{11} +(3.02511 + 1.96181i) q^{13} +(-3.16121 - 5.47537i) q^{17} +(-2.62008 - 4.53812i) q^{19} +(1.70309 + 2.94984i) q^{23} +(-5.62723 - 9.74665i) q^{25} -8.61066 q^{29} +(2.18835 - 3.79033i) q^{31} +(6.27770 + 10.8733i) q^{35} +(3.81043 - 6.59986i) q^{37} +(-3.05166 - 5.28564i) q^{41} +(2.42580 - 4.20161i) q^{43} +(1.83976 + 3.18655i) q^{47} +(-1.34906 - 2.33665i) q^{49} +9.70053 q^{53} +(-5.39701 + 9.34790i) q^{55} -2.34157 q^{59} +(-3.97927 + 6.89230i) q^{61} +(-12.9479 + 6.60759i) q^{65} +(5.04356 + 8.73571i) q^{67} +(-4.55512 - 7.88971i) q^{71} -7.32900 q^{73} +(4.16880 - 7.22057i) q^{77} +(0.0416773 + 0.0721871i) q^{79} +(-2.29262 - 3.97093i) q^{83} +25.4900 q^{85} +(3.92221 - 6.79347i) q^{89} +(10.0013 - 5.10388i) q^{91} +21.1267 q^{95} +(0.879736 - 1.52375i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - q^{5} + 7 q^{7} + q^{17} + 2 q^{19} + q^{23} - 23 q^{25} + 24 q^{29} + 8 q^{31} + 12 q^{35} + 18 q^{37} + 3 q^{41} + 8 q^{43} - 4 q^{47} - 23 q^{49} + 40 q^{53} - 14 q^{55} - 8 q^{59} - 3 q^{61}+ \cdots + 35 q^{97}+O(q^{100}) $ 40 * q - q^5 + 7 * q^7 + q^17 + 2 * q^19 + q^23 - 23 * q^25 + 24 * q^29 + 8 * q^31 + 12 * q^35 + 18 * q^37 + 3 * q^41 + 8 * q^43 - 4 * q^47 - 23 * q^49 + 40 * q^53 - 14 * q^55 - 8 * q^59 - 3 * q^61 + 32 * q^65 + 27 * q^67 - q^71 - 12 * q^73 + 4 * q^77 + 13 * q^79 + 15 * q^83 - 10 * q^85 - 12 * q^89 - 57 * q^91 + 2 * q^95 + 35 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2808\mathbb{Z}\right)^\times$.

$n$	$703$	$1081$	$1405$	$2081$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−2.01584	+	3.49154i	−0.901512	+	1.56146i	−0.0759791	+	0.997109i	$0.524208\pi$
$5$	−2.01584	+	3.49154i	−0.901512	+	1.56146i	−0.825533	+	0.564355i	$0.809125\pi$
$6$		0			0
$7$	1.55709	−	2.69696i	0.588525	−	1.01936i	−0.405901	−	0.913917i	$-0.633042\pi$
$7$	1.55709	−	2.69696i	0.588525	−	1.01936i	0.994426	−	0.105438i	$-0.0336245\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.67730			0.807236			0.403618	−	0.914928i	$-0.367752\pi$
$11$	2.67730			0.807236			0.403618	+	0.914928i	$0.367752\pi$
$12$		0			0
$13$	3.02511	+	1.96181i	0.839015	+	0.544109i
$13$	3.02511	+	1.96181i	0.839015	+	0.544109i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.16121	−	5.47537i	−0.766705	−	1.32797i	−0.939340	−	0.342987i	$-0.888561\pi$
$17$	−3.16121	−	5.47537i	−0.766705	−	1.32797i	0.172635	−	0.984986i	$-0.444772\pi$
$18$		0			0
$19$	−2.62008	−	4.53812i	−0.601088	−	1.04112i	−0.992657	−	0.120966i	$-0.961401\pi$
$19$	−2.62008	−	4.53812i	−0.601088	−	1.04112i	0.391568	−	0.920149i	$-0.371933\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.70309	+	2.94984i	0.355119	+	0.615084i	0.987138	−	0.159869i	$-0.0511070\pi$
$23$	1.70309	+	2.94984i	0.355119	+	0.615084i	−0.632019	+	0.774953i	$0.717774\pi$
$24$		0			0
$25$	−5.62723	−	9.74665i	−1.12545	−	1.94933i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.61066			−1.59896			−0.799480	−	0.600693i	$-0.794892\pi$
$29$	−8.61066			−1.59896			−0.799480	+	0.600693i	$0.794892\pi$
$30$		0			0
$31$	2.18835	−	3.79033i	0.393039	−	0.680764i	−0.599810	−	0.800143i	$-0.704757\pi$
$31$	2.18835	−	3.79033i	0.393039	−	0.680764i	0.992849	+	0.119379i	$0.0380904\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.27770	+	10.8733i	1.06112	+	1.83792i
$36$		0			0
$37$	3.81043	−	6.59986i	0.626431	−	1.08501i	−0.361831	−	0.932244i	$-0.617848\pi$
$37$	3.81043	−	6.59986i	0.626431	−	1.08501i	0.988262	−	0.152767i	$-0.0488185\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.05166	−	5.28564i	−0.476590	−	0.825478i	0.523050	−	0.852302i	$-0.324794\pi$
$41$	−3.05166	−	5.28564i	−0.476590	−	0.825478i	−0.999640	+	0.0268240i	$0.991461\pi$
$42$		0			0
$43$	2.42580	−	4.20161i	0.369932	−	0.640740i	−0.619623	−	0.784900i	$-0.712714\pi$
$43$	2.42580	−	4.20161i	0.369932	−	0.640740i	0.989554	+	0.144159i	$0.0460478\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.83976	+	3.18655i	0.268356	+	0.464806i	0.968437	−	0.249257i	$-0.0801864\pi$
$47$	1.83976	+	3.18655i	0.268356	+	0.464806i	−0.700082	+	0.714063i	$0.746853\pi$
$48$		0			0
$49$	−1.34906	−	2.33665i	−0.192724	−	0.333807i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.70053			1.33247			0.666235	−	0.745742i	$-0.267905\pi$
$53$	9.70053			1.33247			0.666235	+	0.745742i	$0.267905\pi$
$54$		0			0
$55$	−5.39701	+	9.34790i	−0.727733	+	1.26047i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.34157			−0.304846			−0.152423	−	0.988315i	$-0.548708\pi$
$59$	−2.34157			−0.304846			−0.152423	+	0.988315i	$0.548708\pi$
$60$		0			0
$61$	−3.97927	+	6.89230i	−0.509493	+	0.882468i	0.490446	+	0.871471i	$0.336834\pi$
$61$	−3.97927	+	6.89230i	−0.509493	+	0.882468i	−0.999940	+	0.0109967i	$0.996500\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−12.9479	+	6.60759i	−1.60599	+	0.819571i
$66$		0			0
$67$	5.04356	+	8.73571i	0.616169	+	1.06724i	0.990178	+	0.139812i	$0.0446497\pi$
$67$	5.04356	+	8.73571i	0.616169	+	1.06724i	−0.374009	+	0.927425i	$0.622017\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.55512	−	7.88971i	−0.540594	−	0.936336i	−0.998870	−	0.0475260i	$-0.984866\pi$
$71$	−4.55512	−	7.88971i	−0.540594	−	0.936336i	0.458276	−	0.888810i	$-0.348467\pi$
$72$		0			0
$73$	−7.32900			−0.857795			−0.428897	−	0.903353i	$-0.641098\pi$
$73$	−7.32900			−0.857795			−0.428897	+	0.903353i	$0.641098\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.16880	−	7.22057i	0.475079	−	0.822861i
$78$		0			0
$79$	0.0416773	+	0.0721871i	0.00468906	+	0.00812169i	0.868360	−	0.495934i	$-0.165174\pi$
$79$	0.0416773	+	0.0721871i	0.00468906	+	0.00812169i	−0.863671	+	0.504055i	$0.831841\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.29262	−	3.97093i	−0.251647	−	0.435866i	0.712332	−	0.701842i	$-0.247639\pi$
$83$	−2.29262	−	3.97093i	−0.251647	−	0.435866i	−0.963979	+	0.265977i	$0.914306\pi$
$84$		0			0
$85$	25.4900			2.76478
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.92221	−	6.79347i	0.415754	−	0.720106i	−0.579754	−	0.814792i	$-0.696851\pi$
$89$	3.92221	−	6.79347i	0.415754	−	0.720106i	0.995507	+	0.0946855i	$0.0301846\pi$
$90$		0			0
$91$	10.0013	−	5.10388i	1.04842	−	0.535032i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	21.1267			2.16755
$96$		0			0
$97$	0.879736	−	1.52375i	0.0893236	−	0.154713i	−0.817902	−	0.575358i	$-0.804863\pi$
$97$	0.879736	−	1.52375i	0.0893236	−	0.154713i	0.907225	+	0.420645i	$0.138196\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.64723			0.462416			0.231208	−	0.972904i	$-0.425732\pi$
$101$	4.64723			0.462416			0.231208	+	0.972904i	$0.425732\pi$
$102$		0			0
$103$	4.49995	−	7.79414i	0.443393	−	0.767979i	−0.554546	−	0.832153i	$-0.687108\pi$
$103$	4.49995	−	7.79414i	0.443393	−	0.767979i	0.997939	+	0.0641740i	$0.0204413\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.24718	−	12.5525i	0.700611	−	1.21349i	−0.267641	−	0.963519i	$-0.586244\pi$
$107$	7.24718	−	12.5525i	0.700611	−	1.21349i	0.968252	−	0.249975i	$-0.0804225\pi$
$108$		0			0
$109$	1.60985			0.154196			0.0770980	−	0.997024i	$-0.475435\pi$
$109$	1.60985			0.154196			0.0770980	+	0.997024i	$0.475435\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.28737			0.497394			0.248697	−	0.968581i	$-0.419998\pi$
$113$	5.28737			0.497394			0.248697	+	0.968581i	$0.419998\pi$
$114$		0			0
$115$	−13.7326			−1.28058
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−19.6891			−1.80490
$120$		0			0
$121$	−3.83207			−0.348370
$122$		0			0
$123$		0			0
$124$		0			0
$125$	25.2160			2.25539
$126$		0			0
$127$	−1.94770	+	3.37351i	−0.172830	+	0.299350i	−0.939408	−	0.342801i	$-0.888624\pi$
$127$	−1.94770	+	3.37351i	−0.172830	+	0.299350i	0.766578	+	0.642151i	$0.221958\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.67425	−	16.7563i	0.845244	−	1.46400i	−0.0401660	−	0.999193i	$-0.512789\pi$
$131$	9.67425	−	16.7563i	0.845244	−	1.46400i	0.885410	−	0.464812i	$-0.153878\pi$
$132$		0			0
$133$	−16.3188			−1.41502
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.59439	−	11.4218i	0.563396	−	0.975831i	−0.433800	−	0.901009i	$-0.642828\pi$
$137$	6.59439	−	11.4218i	0.563396	−	0.975831i	0.997197	−	0.0748223i	$-0.0238389\pi$
$138$		0			0
$139$	−5.94787			−0.504492			−0.252246	−	0.967663i	$-0.581169\pi$
$139$	−5.94787			−0.504492			−0.252246	+	0.967663i	$0.581169\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.09913	+	5.25236i	0.677283	+	0.439224i
$144$		0			0
$145$	17.3577	−	30.0645i	1.44148	−	2.49672i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.50109			−0.204897			−0.102448	−	0.994738i	$-0.532668\pi$
$149$	−2.50109			−0.204897			−0.102448	+	0.994738i	$0.532668\pi$
$150$		0			0
$151$	−2.26475	−	3.92266i	−0.184303	−	0.319222i	0.759039	−	0.651046i	$-0.225669\pi$
$151$	−2.26475	−	3.92266i	−0.184303	−	0.319222i	−0.943341	+	0.331824i	$0.892336\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.82273	+	15.2814i	0.708659	+	1.22743i
$156$		0			0
$157$	8.74024	−	15.1385i	0.697547	−	1.20819i	−0.271767	−	0.962363i	$-0.587608\pi$
$157$	8.74024	−	15.1385i	0.697547	−	1.20819i	0.969314	−	0.245824i	$-0.0790586\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	10.6075			0.835986
$162$		0			0
$163$	6.86395	+	11.8887i	0.537626	+	0.931196i	0.999031	+	0.0440063i	$0.0140122\pi$
$163$	6.86395	+	11.8887i	0.537626	+	0.931196i	−0.461405	+	0.887190i	$0.652654\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.969025	+	1.67840i	0.0749854	+	0.129879i	0.901080	−	0.433653i	$-0.142776\pi$
$167$	0.969025	+	1.67840i	0.0749854	+	0.129879i	−0.826095	+	0.563532i	$0.809442\pi$
$168$		0			0
$169$	5.30259	+	11.8694i	0.407891	+	0.913030i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.36002	−	9.28383i	0.407515	−	0.705837i	−0.587096	−	0.809518i	$-0.699729\pi$
$173$	5.36002	−	9.28383i	0.407515	−	0.705837i	0.994611	+	0.103681i	$0.0330621\pi$
$174$		0			0
$175$	−35.0485			−2.64941
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.31874	+	16.1405i	−0.696516	+	1.20640i	0.273151	+	0.961971i	$0.411934\pi$
$179$	−9.31874	+	16.1405i	−0.696516	+	1.20640i	−0.969667	+	0.244429i	$0.921399\pi$
$180$		0			0
$181$	22.1248			1.64453			0.822263	−	0.569107i	$-0.192711\pi$
$181$	22.1248			1.64453			0.822263	+	0.569107i	$0.192711\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	15.3625	+	26.6086i	1.12947	+	1.95630i
$186$		0			0
$187$	−8.46350	−	14.6592i	−0.618912	−	1.07199i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	12.3606	−	21.4092i	0.894384	−	1.54912i	0.0598191	−	0.998209i	$-0.480948\pi$
$191$	12.3606	−	21.4092i	0.894384	−	1.54912i	0.834565	−	0.550909i	$-0.185719\pi$
$192$		0			0
$193$	8.63957	+	14.9642i	0.621890	+	1.07714i	0.989134	+	0.147020i	$0.0469682\pi$
$193$	8.63957	+	14.9642i	0.621890	+	1.07714i	−0.367244	+	0.930125i	$0.619698\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.51491	+	4.35596i	−0.179180	+	0.310349i	−0.941600	−	0.336734i	$-0.890678\pi$
$197$	−2.51491	+	4.35596i	−0.179180	+	0.310349i	0.762420	+	0.647083i	$0.224011\pi$
$198$		0			0
$199$	−1.13337	−	1.96306i	−0.0803427	−	0.139158i	0.823054	−	0.567962i	$-0.192268\pi$
$199$	−1.13337	−	1.96306i	−0.0803427	−	0.139158i	−0.903397	+	0.428805i	$0.858935\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−13.4076	+	23.2226i	−0.941028	+	1.62991i
$204$		0			0
$205$	24.6067			1.71861
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7.01475	−	12.1499i	−0.485220	−	0.840426i
$210$		0			0
$211$	10.3004	+	17.8409i	0.709112	+	1.22822i	0.965187	+	0.261560i	$0.0842371\pi$
$211$	10.3004	+	17.8409i	0.709112	+	1.22822i	−0.256076	+	0.966657i	$0.582430\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	9.78007	+	16.9396i	0.666995	+	1.15527i
$216$		0			0
$217$	−6.81492	−	11.8038i	−0.462627	−	0.801293i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.17865	−	22.7653i	0.0792844	−	1.53136i
$222$		0			0
$223$	−25.4540			−1.70452			−0.852262	−	0.523115i	$-0.824770\pi$
$223$	−25.4540			−1.70452			−0.852262	+	0.523115i	$0.824770\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.63041	−	2.82395i	0.108214	−	0.187432i	−0.806833	−	0.590780i	$-0.798820\pi$
$227$	1.63041	−	2.82395i	0.108214	−	0.187432i	0.915047	+	0.403348i	$0.132154\pi$
$228$		0			0
$229$	7.97842	−	13.8190i	0.527229	−	0.913187i	−0.472268	−	0.881455i	$-0.656564\pi$
$229$	7.97842	−	13.8190i	0.527229	−	0.913187i	0.999496	−	0.0317320i	$-0.0101023\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−15.5070			−1.01590			−0.507950	−	0.861387i	$-0.669597\pi$
$233$	−15.5070			−1.01590			−0.507950	+	0.861387i	$0.669597\pi$
$234$		0			0
$235$	−14.8346			−0.967704
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.76503	+	3.05712i	−0.114170	+	0.197748i	−0.917448	−	0.397856i	$-0.869754\pi$
$239$	−1.76503	+	3.05712i	−0.114170	+	0.197748i	0.803278	+	0.595605i	$0.203088\pi$
$240$		0			0
$241$	−1.40310	+	2.43024i	−0.0903817	+	0.156546i	−0.907672	−	0.419681i	$-0.862142\pi$
$241$	−1.40310	+	2.43024i	−0.0903817	+	0.156546i	0.817290	+	0.576226i	$0.195475\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	10.8780			0.694970
$246$		0			0
$247$	0.976891	−	18.8684i	0.0621581	−	1.20057i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	0.126564	+	0.219215i	0.00798865	+	0.0138367i	0.869992	−	0.493066i	$-0.164124\pi$
$251$	0.126564	+	0.219215i	0.00798865	+	0.0138367i	−0.862003	+	0.506902i	$0.830790\pi$
$252$		0			0
$253$	4.55968	+	7.89760i	0.286665	+	0.496518i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.00646	−	6.93939i	−0.249916	−	0.432867i	0.713586	−	0.700567i	$-0.247070\pi$
$257$	−4.00646	−	6.93939i	−0.249916	−	0.432867i	−0.963502	+	0.267700i	$0.913736\pi$
$258$		0			0
$259$	−11.8664	−	20.5532i	−0.737341	−	1.27711i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	21.0928			1.30064			0.650319	−	0.759662i	$-0.274635\pi$
$263$	21.0928			1.30064			0.650319	+	0.759662i	$0.274635\pi$
$264$		0			0
$265$	−19.5547	+	33.8698i	−1.20124	+	2.08060i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.24959	−	3.89641i	−0.137160	−	0.237568i	0.789260	−	0.614059i	$-0.210464\pi$
$269$	−2.24959	−	3.89641i	−0.137160	−	0.237568i	−0.926421	+	0.376490i	$0.877131\pi$
$270$		0			0
$271$	−12.3274	+	21.3517i	−0.748836	+	1.29702i	0.199544	+	0.979889i	$0.436054\pi$
$271$	−12.3274	+	21.3517i	−0.748836	+	1.29702i	−0.948381	+	0.317134i	$0.897280\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−15.0658	−	26.0947i	−0.908501	−	1.57357i
$276$		0			0
$277$	1.98477	−	3.43772i	0.119253	−	0.206553i	−0.800219	−	0.599708i	$-0.795283\pi$
$277$	1.98477	−	3.43772i	0.119253	−	0.206553i	0.919472	+	0.393156i	$0.128617\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	12.4046	+	21.4854i	0.739995	+	1.28171i	0.952497	+	0.304548i	$0.0985055\pi$
$281$	12.4046	+	21.4854i	0.739995	+	1.28171i	−0.212502	+	0.977161i	$0.568161\pi$
$282$		0			0
$283$	7.21597	+	12.4984i	0.428945	+	0.742955i	0.996780	−	0.0801880i	$-0.0255521\pi$
$283$	7.21597	+	12.4984i	0.428945	+	0.742955i	−0.567835	+	0.823143i	$0.692219\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−19.0069			−1.12194
$288$		0			0
$289$	−11.4865	+	19.8951i	−0.675674	+	1.17030i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−10.3017			−0.601833			−0.300916	−	0.953651i	$-0.597293\pi$
$293$	−10.3017			−0.601833			−0.300916	+	0.953651i	$0.597293\pi$
$294$		0			0
$295$	4.72023	−	8.17568i	0.274823	−	0.476007i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.634993	+	12.2647i	−0.0367226	+	0.709288i
$300$		0			0
$301$	−7.55439	−	13.0846i	−0.435428	−	0.754183i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−16.0431	−	27.7875i	−0.918628	−	1.59111i
$306$		0			0
$307$	11.8921			0.678721			0.339360	−	0.940656i	$-0.389789\pi$
$307$	11.8921			0.678721			0.339360	+	0.940656i	$0.389789\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.13967	−	5.43806i	0.178034	−	0.308364i	−0.763173	−	0.646194i	$-0.776360\pi$
$311$	3.13967	−	5.43806i	0.178034	−	0.308364i	0.941207	+	0.337830i	$0.109693\pi$
$312$		0			0
$313$	−13.9544	−	24.1697i	−0.788749	−	1.36615i	−0.926734	−	0.375718i	$-0.877396\pi$
$313$	−13.9544	−	24.1697i	−0.788749	−	1.36615i	0.137985	−	0.990434i	$-0.455937\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.7361	+	23.7916i	0.771495	+	1.33627i	0.936744	+	0.350016i	$0.113824\pi$
$317$	13.7361	+	23.7916i	0.771495	+	1.33627i	−0.165249	+	0.986252i	$0.552843\pi$
$318$		0			0
$319$	−23.0533			−1.29074
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−16.5653	+	28.6919i	−0.921715	+	1.59646i
$324$		0			0
$325$	2.09810	−	40.5243i	0.116382	−	2.24788i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	11.4587			0.631737
$330$		0			0
$331$	2.01407	−	3.48847i	0.110703	−	0.191744i	−0.805351	−	0.592799i	$-0.798023\pi$
$331$	2.01407	−	3.48847i	0.110703	−	0.191744i	0.916054	+	0.401055i	$0.131356\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−40.6681			−2.22194
$336$		0			0
$337$	−11.8750	+	20.5680i	−0.646870	+	1.12041i	0.336996	+	0.941506i	$0.390589\pi$
$337$	−11.8750	+	20.5680i	−0.646870	+	1.12041i	−0.983866	+	0.178906i	$0.942744\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	5.85887	−	10.1479i	0.317275	−	0.549537i
$342$		0			0
$343$	13.3968			0.723360
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−36.8976			−1.98077			−0.990384	−	0.138344i	$-0.955822\pi$
$347$	−36.8976			−1.98077			−0.990384	+	0.138344i	$0.955822\pi$
$348$		0			0
$349$	3.85911			0.206574			0.103287	−	0.994652i	$-0.467064\pi$
$349$	3.85911			0.206574			0.103287	+	0.994652i	$0.467064\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	16.2963			0.867364			0.433682	−	0.901066i	$-0.357214\pi$
$353$	16.2963			0.867364			0.433682	+	0.901066i	$0.357214\pi$
$354$		0			0
$355$	36.7296			1.94941
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−11.0745			−0.584492			−0.292246	−	0.956343i	$-0.594403\pi$
$359$	−11.0745			−0.584492			−0.292246	+	0.956343i	$0.594403\pi$
$360$		0			0
$361$	−4.22967	+	7.32600i	−0.222614	+	0.385579i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	14.7741	−	25.5895i	0.773312	−	1.33942i
$366$		0			0
$367$	−17.9995			−0.939566			−0.469783	−	0.882782i	$-0.655668\pi$
$367$	−17.9995			−0.939566			−0.469783	+	0.882782i	$0.655668\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	15.1046	−	26.1619i	0.784192	−	1.35826i
$372$		0			0
$373$	−21.5883			−1.11780			−0.558901	−	0.829234i	$-0.688777\pi$
$373$	−21.5883			−1.11780			−0.558901	+	0.829234i	$0.688777\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−26.0482	−	16.8925i	−1.34155	−	0.870008i
$378$		0			0
$379$	1.10605	−	1.91573i	0.0568139	−	0.0984046i	−0.836220	−	0.548395i	$-0.815239\pi$
$379$	1.10605	−	1.91573i	0.0568139	−	0.0984046i	0.893034	+	0.449990i	$0.148572\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.24127			−0.472207			−0.236104	−	0.971728i	$-0.575870\pi$
$383$	−9.24127			−0.472207			−0.236104	+	0.971728i	$0.575870\pi$
$384$		0			0
$385$	16.8073	+	29.1111i	0.856578	+	1.48364i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8.88767	−	15.3939i	−0.450623	−	0.780501i	0.547802	−	0.836608i	$-0.315465\pi$
$389$	−8.88767	−	15.3939i	−0.450623	−	0.780501i	−0.998425	+	0.0561066i	$0.982131\pi$
$390$		0			0
$391$	10.7676	−	18.6501i	0.544543	−	0.943177i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.336059			−0.0169090
$396$		0			0
$397$	4.13275	+	7.15813i	0.207417	+	0.359256i	0.950900	−	0.309498i	$-0.100161\pi$
$397$	4.13275	+	7.15813i	0.207417	+	0.359256i	−0.743483	+	0.668754i	$0.766828\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.14850	−	15.8457i	−0.456854	−	0.791295i	0.541938	−	0.840418i	$-0.317691\pi$
$401$	−9.14850	−	15.8457i	−0.456854	−	0.791295i	−0.998793	+	0.0491231i	$0.984357\pi$
$402$		0			0
$403$	14.0559	−	7.17304i	0.700175	−	0.357315i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	10.2017	−	17.6698i	0.505678	−	0.875860i
$408$		0			0
$409$	−6.51546			−0.322169			−0.161084	−	0.986941i	$-0.551499\pi$
$409$	−6.51546			−0.322169			−0.161084	+	0.986941i	$0.551499\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.64604	+	6.31512i	−0.179410	+	0.310747i
$414$		0			0
$415$	18.4862			0.907452
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.4406	+	21.5477i	0.607761	+	1.05267i	0.991609	+	0.129276i	$0.0412654\pi$
$419$	12.4406	+	21.5477i	0.607761	+	1.05267i	−0.383848	+	0.923396i	$0.625401\pi$
$420$		0			0
$421$	−19.1169	−	33.1115i	−0.931703	−	1.61376i	−0.780411	−	0.625267i	$-0.784990\pi$
$421$	−19.1169	−	33.1115i	−0.931703	−	1.61376i	−0.151292	−	0.988489i	$-0.548343\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−35.5777	+	61.6224i	−1.72577	+	2.98912i
$426$		0			0
$427$	12.3922	+	21.4639i	0.599699	+	1.03871i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.42332	+	14.5896i	−0.405737	+	0.702757i	−0.994407	−	0.105616i	$-0.966318\pi$
$431$	−8.42332	+	14.5896i	−0.405737	+	0.702757i	0.588670	+	0.808373i	$0.299652\pi$
$432$		0			0
$433$	−17.7816	−	30.7987i	−0.854531	−	1.48009i	−0.877079	−	0.480346i	$-0.840511\pi$
$433$	−17.7816	−	30.7987i	−0.854531	−	1.48009i	0.0225475	−	0.999746i	$-0.492822\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.92448	−	15.4576i	0.426916	−	0.739440i
$438$		0			0
$439$	4.52799			0.216109			0.108054	−	0.994145i	$-0.465538\pi$
$439$	4.52799			0.216109			0.108054	+	0.994145i	$0.465538\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−14.0231	−	24.2888i	−0.666259	−	1.15399i	−0.978942	−	0.204137i	$-0.934561\pi$
$443$	−14.0231	−	24.2888i	−0.666259	−	1.15399i	0.312683	−	0.949857i	$-0.398772\pi$
$444$		0			0
$445$	15.8131	+	27.3891i	0.749613	+	1.29837i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	14.7395	+	25.5295i	0.695599	+	1.20481i	0.969978	+	0.243191i	$0.0781941\pi$
$449$	14.7395	+	25.5295i	0.695599	+	1.20481i	−0.274380	+	0.961621i	$0.588473\pi$
$450$		0			0
$451$	−8.17022	−	14.1512i	−0.384721	−	0.666356i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.34062	+	45.2086i	−0.109730	+	2.11941i
$456$		0			0
$457$	−12.2938			−0.575078			−0.287539	−	0.957769i	$-0.592837\pi$
$457$	−12.2938			−0.575078			−0.287539	+	0.957769i	$0.592837\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12.3623	+	21.4121i	−0.575768	+	0.997260i	0.420189	+	0.907436i	$0.361964\pi$
$461$	−12.3623	+	21.4121i	−0.575768	+	0.997260i	−0.995958	+	0.0898237i	$0.971370\pi$
$462$		0			0
$463$	−16.7335	+	28.9832i	−0.777670	+	1.34696i	0.155611	+	0.987818i	$0.450265\pi$
$463$	−16.7335	+	28.9832i	−0.777670	+	1.34696i	−0.933281	+	0.359146i	$0.883068\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−0.115149			−0.00532846			−0.00266423	−	0.999996i	$-0.500848\pi$
$467$	−0.115149			−0.00532846			−0.00266423	+	0.999996i	$0.500848\pi$
$468$		0			0
$469$	31.4132			1.45052
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.49460	−	11.2490i	0.298622	−	0.517229i
$474$		0			0
$475$	−29.4876	+	51.0741i	−1.35299	+	2.34344i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	42.8386			1.95734			0.978672	−	0.205427i	$-0.0658582\pi$
$479$	42.8386			1.95734			0.978672	+	0.205427i	$0.0658582\pi$
$480$		0			0
$481$	24.4747	−	12.4900i	1.11595	−	0.569493i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.54681	+	6.14326i	0.161053	+	0.278951i
$486$		0			0
$487$	−1.61647	−	2.79981i	−0.0732493	−	0.126872i	0.827074	−	0.562093i	$-0.190004\pi$
$487$	−1.61647	−	2.79981i	−0.0732493	−	0.126872i	−0.900324	+	0.435221i	$0.856670\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	10.8148	+	18.7318i	0.488065	+	0.845353i	0.999906	−	0.0137269i	$-0.00436955\pi$
$491$	10.8148	+	18.7318i	0.488065	+	0.845353i	−0.511841	+	0.859080i	$0.671036\pi$
$492$		0			0
$493$	27.2201	+	47.1466i	1.22593	+	2.12338i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−28.3710			−1.27261
$498$		0			0
$499$	−5.03855	+	8.72702i	−0.225556	+	0.390675i	−0.956486	−	0.291777i	$-0.905753\pi$
$499$	−5.03855	+	8.72702i	−0.225556	+	0.390675i	0.730930	+	0.682453i	$0.239087\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	7.24603	+	12.5505i	0.323084	+	0.559598i	0.981123	−	0.193386i	$-0.0619469\pi$
$503$	7.24603	+	12.5505i	0.323084	+	0.559598i	−0.658038	+	0.752984i	$0.728614\pi$
$504$		0			0
$505$	−9.36807	+	16.2260i	−0.416874	+	0.722047i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.14025	−	3.70703i	−0.0948650	−	0.164311i	0.814687	−	0.579901i	$-0.196909\pi$
$509$	−2.14025	−	3.70703i	−0.0948650	−	0.164311i	−0.909552	+	0.415590i	$0.863575\pi$
$510$		0			0
$511$	−11.4119	+	19.7660i	−0.504834	+	0.874398i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	18.1424	+	31.4235i	0.799448	+	1.38468i
$516$		0			0
$517$	4.92558	+	8.53135i	0.216627	+	0.375208i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3.36236			−0.147307			−0.0736537	−	0.997284i	$-0.523466\pi$
$521$	−3.36236			−0.147307			−0.0736537	+	0.997284i	$0.523466\pi$
$522$		0			0
$523$	6.02690	−	10.4389i	0.263538	−	0.456461i	−0.703641	−	0.710555i	$-0.748444\pi$
$523$	6.02690	−	10.4389i	0.263538	−	0.456461i	0.967180	+	0.254094i	$0.0817772\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−27.6713			−1.20538
$528$		0			0
$529$	5.69896	−	9.87090i	0.247781	−	0.429169i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.13781	−	21.9764i	0.0492838	−	0.951905i
$534$		0			0
$535$	29.2183	+	50.6076i	1.26322	+	2.18796i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.61185	−	6.25591i	−0.155573	−	0.269461i
$540$		0			0
$541$	2.77390			0.119259			0.0596296	−	0.998221i	$-0.481008\pi$
$541$	2.77390			0.119259			0.0596296	+	0.998221i	$0.481008\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.24521	+	5.62087i	−0.139009	+	0.240771i
$546$		0			0
$547$	5.30305	+	9.18515i	0.226742	+	0.392729i	0.956841	−	0.290613i	$-0.0938593\pi$
$547$	5.30305	+	9.18515i	0.226742	+	0.392729i	−0.730099	+	0.683342i	$0.760526\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	22.5607	+	39.0762i	0.961116	+	1.66470i
$552$		0			0
$553$	0.259581			0.0110385
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3.46387	−	5.99960i	0.146769	−	0.254211i	−0.783263	−	0.621691i	$-0.786446\pi$
$557$	3.46387	−	5.99960i	0.146769	−	0.254211i	0.930031	+	0.367480i	$0.119779\pi$
$558$		0			0
$559$	15.5811	−	7.95138i	0.659010	−	0.336307i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−30.9047			−1.30248			−0.651239	−	0.758872i	$-0.725751\pi$
$563$	−30.9047			−1.30248			−0.651239	+	0.758872i	$0.725751\pi$
$564$		0			0
$565$	−10.6585	+	18.4610i	−0.448406	+	0.776662i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.537887			0.0225494			0.0112747	−	0.999936i	$-0.496411\pi$
$569$	0.537887			0.0225494			0.0112747	+	0.999936i	$0.496411\pi$
$570$		0			0
$571$	23.3527	−	40.4481i	0.977281	−	1.69270i	0.305089	−	0.952324i	$-0.401314\pi$
$571$	23.3527	−	40.4481i	0.977281	−	1.69270i	0.672192	−	0.740377i	$-0.265353\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	19.1674	−	33.1989i	0.799335	−	1.38449i
$576$		0			0
$577$	16.0641			0.668757			0.334378	−	0.942439i	$-0.391474\pi$
$577$	16.0641			0.668757			0.334378	+	0.942439i	$0.391474\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−14.2792			−0.592403
$582$		0			0
$583$	25.9712			1.07562
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10.4814			−0.432615			−0.216307	−	0.976325i	$-0.569401\pi$
$587$	−10.4814			−0.432615			−0.216307	+	0.976325i	$0.569401\pi$
$588$		0			0
$589$	−22.9346			−0.945005
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−8.47081			−0.347855			−0.173927	−	0.984759i	$-0.555646\pi$
$593$	−8.47081			−0.347855			−0.173927	+	0.984759i	$0.555646\pi$
$594$		0			0
$595$	39.6902	−	68.7454i	1.62714	−	2.81829i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	15.4513	−	26.7624i	0.631322	−	1.09348i	−0.355959	−	0.934501i	$-0.615846\pi$
$599$	15.4513	−	26.7624i	0.631322	−	1.09348i	0.987282	−	0.158981i	$-0.0508208\pi$
$600$		0			0
$601$	−15.0003			−0.611876			−0.305938	−	0.952051i	$-0.598970\pi$
$601$	−15.0003			−0.611876			−0.305938	+	0.952051i	$0.598970\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.72484	−	13.3798i	0.314059	−	0.543967i
$606$		0			0
$607$	21.1383			0.857978			0.428989	−	0.903310i	$-0.358870\pi$
$607$	21.1383			0.857978			0.428989	+	0.903310i	$0.358870\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.685948	+	13.2489i	−0.0277505	+	0.535994i
$612$		0			0
$613$	3.93979	−	6.82392i	0.159127	−	0.275615i	−0.775427	−	0.631437i	$-0.782466\pi$
$613$	3.93979	−	6.82392i	0.159127	−	0.275615i	0.934554	+	0.355821i	$0.115799\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−21.2504			−0.855510			−0.427755	−	0.903895i	$-0.640695\pi$
$617$	−21.2504			−0.855510			−0.427755	+	0.903895i	$0.640695\pi$
$618$		0			0
$619$	−12.1361	−	21.0204i	−0.487792	−	0.844881i	0.512109	−	0.858920i	$-0.328864\pi$
$619$	−12.1361	−	21.0204i	−0.487792	−	0.844881i	−0.999901	+	0.0140395i	$0.995531\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−12.2145	−	21.1561i	−0.489363	−	0.847601i
$624$		0			0
$625$	−22.6953	+	39.3095i	−0.907813	+	1.57238i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−48.1823			−1.92115
$630$		0			0
$631$	7.65676	+	13.2619i	0.304811	+	0.527948i	0.977219	−	0.212232i	$-0.0680734\pi$
$631$	7.65676	+	13.2619i	0.304811	+	0.527948i	−0.672408	+	0.740180i	$0.734740\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.85249	−	13.6009i	−0.311617	−	0.539736i
$636$		0			0
$637$	0.502995	−	9.71523i	0.0199294	−	0.384932i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.266316	−	0.461273i	0.0105189	−	0.0182192i	−0.860718	−	0.509082i	$-0.829985\pi$
$641$	0.266316	−	0.461273i	0.0105189	−	0.0182192i	0.871237	+	0.490863i	$0.163318\pi$
$642$		0			0
$643$	−19.7650			−0.779455			−0.389727	−	0.920930i	$-0.627431\pi$
$643$	−19.7650			−0.779455			−0.389727	+	0.920930i	$0.627431\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.96131	+	15.5214i	−0.352305	+	0.610211i	−0.986653	−	0.162837i	$-0.947935\pi$
$647$	−8.96131	+	15.5214i	−0.352305	+	0.610211i	0.634348	+	0.773048i	$0.281269\pi$
$648$		0			0
$649$	−6.26908			−0.246083
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.11946	−	1.93896i	−0.0438077	−	0.0758772i	0.843290	−	0.537459i	$-0.180616\pi$
$653$	−1.11946	−	1.93896i	−0.0438077	−	0.0758772i	−0.887098	+	0.461581i	$0.847282\pi$
$654$		0			0
$655$	39.0035	+	67.5561i	1.52399	+	2.63963i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.55709	+	4.42902i	−0.0996103	+	0.172530i	−0.911523	−	0.411248i	$-0.865093\pi$
$659$	−2.55709	+	4.42902i	−0.0996103	+	0.172530i	0.811913	+	0.583778i	$0.198426\pi$
$660$		0			0
$661$	−1.08959	−	1.88722i	−0.0423801	−	0.0734045i	0.844057	−	0.536253i	$-0.180161\pi$
$661$	−1.08959	−	1.88722i	−0.0423801	−	0.0734045i	−0.886437	+	0.462849i	$0.846827\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	32.8962	−	56.9778i	1.27566	−	2.20951i
$666$		0			0
$667$	−14.6647	−	25.4001i	−0.567821	−	0.983495i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−10.6537	+	18.4527i	−0.411281	+	0.712360i
$672$		0			0
$673$	2.38389			0.0918924			0.0459462	−	0.998944i	$-0.485370\pi$
$673$	2.38389			0.0918924			0.0459462	+	0.998944i	$0.485370\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−12.4674	−	21.5942i	−0.479162	−	0.829933i	0.520552	−	0.853830i	$-0.325726\pi$
$677$	−12.4674	−	21.5942i	−0.479162	−	0.829933i	−0.999714	+	0.0238965i	$0.992393\pi$
$678$		0			0
$679$	−2.73966	−	4.74522i	−0.105138	−	0.182105i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.09921	−	1.90389i	−0.0420601	−	0.0728503i	0.844229	−	0.535983i	$-0.180059\pi$
$683$	−1.09921	−	1.90389i	−0.0420601	−	0.0728503i	−0.886289	+	0.463132i	$0.846725\pi$
$684$		0			0
$685$	26.5865	+	46.0491i	1.01582	+	1.75945i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	29.3452	+	19.0306i	1.11796	+	0.725009i
$690$		0			0
$691$	−26.0324			−0.990321			−0.495160	−	0.868802i	$-0.664891\pi$
$691$	−26.0324			−0.990321			−0.495160	+	0.868802i	$0.664891\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	11.9900	−	20.7672i	0.454805	−	0.787746i
$696$		0			0
$697$	−19.2939	+	33.4180i	−0.730808	+	1.26580i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−25.1052			−0.948212			−0.474106	−	0.880468i	$-0.657229\pi$
$701$	−25.1052			−0.948212			−0.474106	+	0.880468i	$0.657229\pi$
$702$		0			0
$703$	−39.9346			−1.50616
$704$		0			0
$705$		0			0
$706$		0			0
$707$	7.23616	−	12.5334i	0.272144	−	0.471367i
$708$		0			0
$709$	7.96118	−	13.7892i	0.298988	−	0.517863i	−0.676916	−	0.736060i	$-0.736684\pi$
$709$	7.96118	−	13.7892i	0.298988	−	0.517863i	0.975905	+	0.218197i	$0.0700175\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	14.9078			0.558302
$714$		0			0
$715$	−34.6654	+	17.6905i	−1.29641	+	0.661587i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−7.81560	−	13.5370i	−0.291473	−	0.504845i	0.682686	−	0.730712i	$-0.260812\pi$
$719$	−7.81560	−	13.5370i	−0.291473	−	0.504845i	−0.974158	+	0.225867i	$0.927479\pi$
$720$		0			0
$721$	−14.0137	−	24.2724i	−0.521896	−	0.903950i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	48.4542	+	83.9251i	1.79954	+	3.11690i
$726$		0			0
$727$	13.9681	+	24.1935i	0.518049	+	0.897288i	0.999780	+	0.0209685i	$0.00667497\pi$
$727$	13.9681	+	24.1935i	0.518049	+	0.897288i	−0.481731	+	0.876319i	$0.659992\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−30.6739			−1.13451
$732$		0			0
$733$	13.5016	−	23.3855i	0.498693	−	0.863762i	−0.501306	−	0.865270i	$-0.667147\pi$
$733$	13.5016	−	23.3855i	0.498693	−	0.863762i	0.999999	+	0.00150851i	$0.000480173\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	13.5031	+	23.3881i	0.497394	+	0.861512i
$738$		0			0
$739$	−4.92983	+	8.53872i	−0.181347	+	0.314102i	−0.942339	−	0.334659i	$-0.891379\pi$
$739$	−4.92983	+	8.53872i	−0.181347	+	0.314102i	0.760993	+	0.648760i	$0.224712\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	18.3352	+	31.7574i	0.672652	+	1.16507i	0.977149	+	0.212554i	$0.0681781\pi$
$743$	18.3352	+	31.7574i	0.672652	+	1.16507i	−0.304498	+	0.952513i	$0.598489\pi$
$744$		0			0
$745$	5.04179	−	8.73264i	0.184717	−	0.319939i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−22.5690	−	39.0907i	−0.824654	−	1.42834i
$750$		0			0
$751$	−5.63561	−	9.76117i	−0.205647	−	0.356190i	0.744692	−	0.667408i	$-0.232596\pi$
$751$	−5.63561	−	9.76117i	−0.205647	−	0.356190i	−0.950339	+	0.311218i	$0.899263\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	18.2615			0.664604
$756$		0			0
$757$	−20.5518	+	35.5968i	−0.746970	+	1.29379i	0.202299	+	0.979324i	$0.435159\pi$
$757$	−20.5518	+	35.5968i	−0.746970	+	1.29379i	−0.949269	+	0.314466i	$0.898175\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−34.2836			−1.24278			−0.621390	−	0.783501i	$-0.713432\pi$
$761$	−34.2836			−1.24278			−0.621390	+	0.783501i	$0.713432\pi$
$762$		0			0
$763$	2.50669	−	4.34171i	0.0907482	−	0.157180i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−7.08351	−	4.59372i	−0.255771	−	0.165870i
$768$		0			0
$769$	−22.8320	−	39.5462i	−0.823344	−	1.42607i	−0.903178	−	0.429265i	$-0.858773\pi$
$769$	−22.8320	−	39.5462i	−0.823344	−	1.42607i	0.0798346	−	0.996808i	$-0.474561\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	8.02739	+	13.9038i	0.288725	+	0.500087i	0.973506	−	0.228663i	$-0.0734353\pi$
$773$	8.02739	+	13.9038i	0.288725	+	0.500087i	−0.684781	+	0.728749i	$0.740102\pi$
$774$		0			0
$775$	−49.2574			−1.76938
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−15.9912	+	27.6976i	−0.572945	+	0.992370i
$780$		0			0
$781$	−12.1954	−	21.1231i	−0.436387	−	0.755844i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	35.2379	+	61.0338i	1.25769	+	2.17839i
$786$		0			0
$787$	1.80768			0.0644367			0.0322183	−	0.999481i	$-0.489743\pi$
$787$	1.80768			0.0644367			0.0322183	+	0.999481i	$0.489743\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	8.23291	−	14.2598i	0.292729	−	0.507021i
$792$		0			0
$793$	−25.5591	+	13.0434i	−0.907631	+	0.463184i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	21.8242			0.773051			0.386526	−	0.922279i	$-0.373675\pi$
$797$	21.8242			0.773051			0.386526	+	0.922279i	$0.373675\pi$
$798$		0			0
$799$	11.6317	−	20.1467i	0.411500	−	0.712739i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−19.6219			−0.692443
$804$		0			0
$805$	−21.3830	+	37.0364i	−0.753651	+	1.30536i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	9.25389	−	16.0282i	0.325349	−	0.563522i	−0.656234	−	0.754558i	$-0.727851\pi$
$809$	9.25389	−	16.0282i	0.325349	−	0.563522i	0.981583	+	0.191036i	$0.0611848\pi$
$810$		0			0
$811$	−8.50964			−0.298814			−0.149407	−	0.988776i	$-0.547736\pi$
$811$	−8.50964			−0.298814			−0.149407	+	0.988776i	$0.547736\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−55.3466			−1.93871
$816$		0			0
$817$	−25.4232			−0.889446
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−46.7924			−1.63307			−0.816534	−	0.577298i	$-0.804107\pi$
$821$	−46.7924			−1.63307			−0.816534	+	0.577298i	$0.804107\pi$
$822$		0			0
$823$	42.6709			1.48741			0.743707	−	0.668506i	$-0.233066\pi$
$823$	42.6709			1.48741			0.743707	+	0.668506i	$0.233066\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−38.6497			−1.34398			−0.671991	−	0.740560i	$-0.734560\pi$
$827$	−38.6497			−1.34398			−0.671991	+	0.740560i	$0.734560\pi$
$828$		0			0
$829$	3.50070	−	6.06339i	0.121584	−	0.210590i	−0.798808	−	0.601586i	$-0.794536\pi$
$829$	3.50070	−	6.06339i	0.121584	−	0.210590i	0.920393	+	0.390995i	$0.127869\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−8.52935	+	14.7733i	−0.295524	+	0.511863i
$834$		0			0
$835$	−7.81360			−0.270401
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10.0334	−	17.3784i	0.346393	−	0.599970i	−0.639213	−	0.769030i	$-0.720740\pi$
$839$	10.0334	−	17.3784i	0.346393	−	0.599970i	0.985606	+	0.169059i	$0.0540730\pi$
$840$		0			0
$841$	45.1435			1.55667
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−52.1316	−	5.41262i	−1.79338	−	0.186200i
$846$		0			0
$847$	−5.96688	+	10.3349i	−0.205024	+	0.355113i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	25.9581			0.889831
$852$		0			0
$853$	0.00947654	+	0.0164138i	0.000324471	+	0.000561999i	0.866188	−	0.499719i	$-0.166563\pi$
$853$	0.00947654	+	0.0164138i	0.000324471	+	0.000561999i	−0.865863	+	0.500281i	$0.833230\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−7.52291	−	13.0301i	−0.256978	−	0.445099i	0.708453	−	0.705758i	$-0.249393\pi$
$857$	−7.52291	−	13.0301i	−0.256978	−	0.445099i	−0.965431	+	0.260659i	$0.916060\pi$
$858$		0			0
$859$	−23.5351	+	40.7639i	−0.803006	+	1.39085i	0.114624	+	0.993409i	$0.463434\pi$
$859$	−23.5351	+	40.7639i	−0.803006	+	1.39085i	−0.917629	+	0.397438i	$0.869900\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	14.8794			0.506501			0.253251	−	0.967401i	$-0.418500\pi$
$863$	14.8794			0.506501			0.253251	+	0.967401i	$0.418500\pi$
$864$		0			0
$865$	21.6099	+	37.4295i	0.734759	+	1.27264i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.111583	+	0.193267i	0.00378518	+	0.00655612i
$870$		0			0
$871$	−1.88048	+	36.3210i	−0.0637176	+	1.23069i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	39.2636	−	68.0066i	1.32735	−	2.29904i
$876$		0			0
$877$	27.9803			0.944828			0.472414	−	0.881377i	$-0.343383\pi$
$877$	27.9803			0.944828			0.472414	+	0.881377i	$0.343383\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	16.9488	−	29.3563i	0.571021	−	0.989038i	−0.425440	−	0.904987i	$-0.639881\pi$
$881$	16.9488	−	29.3563i	0.571021	−	0.989038i	0.996461	−	0.0840512i	$-0.0267859\pi$
$882$		0			0
$883$	6.94722			0.233793			0.116896	−	0.993144i	$-0.462705\pi$
$883$	6.94722			0.233793			0.116896	+	0.993144i	$0.462705\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−18.5972	−	32.2114i	−0.624434	−	1.08155i	−0.988650	−	0.150238i	$-0.951996\pi$
$887$	−18.5972	−	32.2114i	−0.624434	−	1.08155i	0.364215	−	0.931315i	$-0.381337\pi$
$888$		0			0
$889$	6.06548	+	10.5057i	0.203430	+	0.352350i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	9.64062	−	16.6980i	0.322611	−	0.558779i
$894$		0			0
$895$	−37.5702	−	65.0735i	−1.25583	−	2.17517i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−18.8431	+	32.6373i	−0.628454	+	1.08851i
$900$		0			0
$901$	−30.6654	−	53.1140i	−1.02161	−	1.76948i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−44.6002	+	77.2498i	−1.48256	+	2.56787i
$906$		0			0
$907$	−27.1399			−0.901167			−0.450583	−	0.892734i	$-0.648784\pi$
$907$	−27.1399			−0.901167			−0.450583	+	0.892734i	$0.648784\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	22.6069	+	39.1563i	0.748999	+	1.29730i	0.948303	+	0.317367i	$0.102799\pi$
$911$	22.6069	+	39.1563i	0.748999	+	1.29730i	−0.199304	+	0.979938i	$0.563868\pi$
$912$		0			0
$913$	−6.13802	−	10.6314i	−0.203139	−	0.351847i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−30.1274	−	52.1822i	−0.994894	−	1.72321i
$918$		0			0
$919$	−22.1780	−	38.4134i	−0.731585	−	1.26714i	−0.956206	−	0.292696i	$-0.905448\pi$
$919$	−22.1780	−	38.4134i	−0.731585	−	1.26714i	0.224621	−	0.974446i	$-0.427886\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.69837	−	32.8035i	0.0559024	−	1.07974i
$924$		0			0
$925$	−85.7688			−2.82006
$926$		0			0
$927$		0			0
$928$		0			0
$929$	2.52003	−	4.36482i	0.0826796	−	0.143205i	−0.821721	−	0.569891i	$-0.806986\pi$
$929$	2.52003	−	4.36482i	0.0826796	−	0.143205i	0.904400	+	0.426685i	$0.140319\pi$
$930$		0			0
$931$	−7.06932	+	12.2444i	−0.231688	+	0.401295i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	68.2443			2.23183
$936$		0			0
$937$	12.0649			0.394144			0.197072	−	0.980389i	$-0.436857\pi$
$937$	12.0649			0.394144			0.197072	+	0.980389i	$0.436857\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−17.6379	+	30.5497i	−0.574979	+	0.995892i	0.421065	+	0.907030i	$0.361656\pi$
$941$	−17.6379	+	30.5497i	−0.574979	+	0.995892i	−0.996044	+	0.0888620i	$0.971677\pi$
$942$		0			0
$943$	10.3945	−	18.0038i	0.338492	−	0.586286i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−20.0420			−0.651279			−0.325639	−	0.945494i	$-0.605580\pi$
$947$	−20.0420			−0.651279			−0.325639	+	0.945494i	$0.605580\pi$
$948$		0			0
$949$	−22.1710	−	14.3781i	−0.719703	−	0.466734i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	2.65975	+	4.60682i	0.0861577	+	0.149230i	0.905884	−	0.423526i	$-0.139208\pi$
$953$	2.65975	+	4.60682i	0.0861577	+	0.149230i	−0.819726	+	0.572756i	$0.805874\pi$
$954$		0			0
$955$	49.8342	+	86.3153i	1.61260	+	2.79310i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−20.5361	−	35.5696i	−0.663146	−	1.14860i
$960$		0			0
$961$	5.92226	+	10.2577i	0.191041	+	0.330892i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−69.6640			−2.24256
$966$		0			0
$967$	1.03787	−	1.79765i	0.0333757	−	0.0578084i	−0.848855	−	0.528626i	$-0.822708\pi$
$967$	1.03787	−	1.79765i	0.0333757	−	0.0578084i	0.882231	+	0.470817i	$0.156041\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	19.8093	+	34.3106i	0.635709	+	1.10108i	0.986364	+	0.164576i	$0.0526256\pi$
$971$	19.8093	+	34.3106i	0.635709	+	1.10108i	−0.350655	+	0.936505i	$0.614041\pi$
$972$		0			0
$973$	−9.26138	+	16.0412i	−0.296906	+	0.514257i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	16.6176	+	28.7825i	0.531643	+	0.920833i	0.999318	+	0.0369323i	$0.0117586\pi$
$977$	16.6176	+	28.7825i	0.531643	+	0.920833i	−0.467675	+	0.883901i	$0.654908\pi$
$978$		0			0
$979$	10.5009	−	18.1882i	0.335611	−	0.581296i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−24.5472	−	42.5171i	−0.782936	−	1.35608i	−0.930225	−	0.366991i	$-0.880388\pi$
$983$	−24.5472	−	42.5171i	−0.782936	−	1.35608i	0.147289	−	0.989094i	$-0.452945\pi$
$984$		0			0
$985$	−10.1393	−	17.5618i	−0.323066	−	0.559566i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	16.5255			0.525479
$990$		0			0
$991$	13.4804	−	23.3487i	0.428219	−	0.741697i	−0.568496	−	0.822686i	$-0.692475\pi$
$991$	13.4804	−	23.3487i	0.428219	−	0.741697i	0.996715	+	0.0809893i	$0.0258080\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	9.13881			0.289720
$996$		0			0
$997$	−0.274519	+	0.475481i	−0.00869410	+	0.0150586i	−0.870340	−	0.492452i	$-0.836101\pi$
$997$	−0.274519	+	0.475481i	−0.00869410	+	0.0150586i	0.861646	+	0.507510i	$0.169434\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2808.2.r.f.2089.1		40
3.2	odd	2		936.2.r.f.841.13	yes	40
9.2	odd	6		936.2.s.f.529.1	yes	40
9.7	even	3		2808.2.s.f.1153.1		40
13.3	even	3		2808.2.s.f.1225.1		40
39.29	odd	6		936.2.s.f.913.1	yes	40
117.16	even	3	inner	2808.2.r.f.289.1		40
117.29	odd	6		936.2.r.f.601.13	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.r.f.601.13	✓	40	117.29	odd	6
936.2.r.f.841.13	yes	40	3.2	odd	2
936.2.s.f.529.1	yes	40	9.2	odd	6
936.2.s.f.913.1	yes	40	39.29	odd	6
2808.2.r.f.289.1		40	117.16	even	3	inner
2808.2.r.f.2089.1		40	1.1	even	1	trivial
2808.2.s.f.1153.1		40	9.7	even	3
2808.2.s.f.1225.1		40	13.3	even	3

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:	\( N \)	\(=\)	\( 2808 = 2^{3} \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2808.r (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.01584 + 3.49154i) q^{5} +(1.55709 - 2.69696i) q^{7} +2.67730 q^{11} +(3.02511 + 1.96181i) q^{13} +(-3.16121 - 5.47537i) q^{17} +(-2.62008 - 4.53812i) q^{19} +(1.70309 + 2.94984i) q^{23} +(-5.62723 - 9.74665i) q^{25} -8.61066 q^{29} +(2.18835 - 3.79033i) q^{31} +(6.27770 + 10.8733i) q^{35} +(3.81043 - 6.59986i) q^{37} +(-3.05166 - 5.28564i) q^{41} +(2.42580 - 4.20161i) q^{43} +(1.83976 + 3.18655i) q^{47} +(-1.34906 - 2.33665i) q^{49} +9.70053 q^{53} +(-5.39701 + 9.34790i) q^{55} -2.34157 q^{59} +(-3.97927 + 6.89230i) q^{61} +(-12.9479 + 6.60759i) q^{65} +(5.04356 + 8.73571i) q^{67} +(-4.55512 - 7.88971i) q^{71} -7.32900 q^{73} +(4.16880 - 7.22057i) q^{77} +(0.0416773 + 0.0721871i) q^{79} +(-2.29262 - 3.97093i) q^{83} +25.4900 q^{85} +(3.92221 - 6.79347i) q^{89} +(10.0013 - 5.10388i) q^{91} +21.1267 q^{95} +(0.879736 - 1.52375i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - q^{5} + 7 q^{7} + q^{17} + 2 q^{19} + q^{23} - 23 q^{25} + 24 q^{29} + 8 q^{31} + 12 q^{35} + 18 q^{37} + 3 q^{41} + 8 q^{43} - 4 q^{47} - 23 q^{49} + 40 q^{53} - 14 q^{55} - 8 q^{59} - 3 q^{61}+ \cdots + 35 q^{97}+O(q^{100}) \) 40 * q - q^5 + 7 * q^7 + q^17 + 2 * q^19 + q^23 - 23 * q^25 + 24 * q^29 + 8 * q^31 + 12 * q^35 + 18 * q^37 + 3 * q^41 + 8 * q^43 - 4 * q^47 - 23 * q^49 + 40 * q^53 - 14 * q^55 - 8 * q^59 - 3 * q^61 + 32 * q^65 + 27 * q^67 - q^71 - 12 * q^73 + 4 * q^77 + 13 * q^79 + 15 * q^83 - 10 * q^85 - 12 * q^89 - 57 * q^91 + 2 * q^95 + 35 * q^97

Introduction

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