LMFDB - Embedded newform 1620.4.i.r.541.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(541,1620)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1620.541");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - x^{2} - 2x + 4 $ x^4 - x^3 - x^2 - 2*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			541.1
Root			$-0.895644 + 1.09445i$ of defining polynomial
Character	$\chi$	$=$	1620.541
Dual form			1620.4.i.r.1081.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 + 4.33013i) q^{5} +(-6.37386 + 11.0399i) q^{7} +O(q^{10})$	$q+(2.50000 + 4.33013i) q^{5} +(-6.37386 + 11.0399i) q^{7} +(-17.3739 + 30.0924i) q^{11} +(37.8693 + 65.5916i) q^{13} -84.2341 q^{17} -83.7477 q^{19} +(-86.4909 - 149.807i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(-44.1125 + 76.4051i) q^{29} +(-102.856 - 178.151i) q^{31} -63.7386 q^{35} -286.000 q^{37} +(160.851 + 278.602i) q^{41} +(84.3830 - 146.156i) q^{43} +(195.234 - 338.155i) q^{47} +(90.2477 + 156.314i) q^{49} +91.6932 q^{53} -173.739 q^{55} +(-61.3830 - 106.318i) q^{59} +(439.202 - 760.721i) q^{61} +(-189.347 + 327.958i) q^{65} +(518.702 + 898.419i) q^{67} +605.216 q^{71} -13.8114 q^{73} +(-221.477 - 383.610i) q^{77} +(286.649 - 496.490i) q^{79} +(313.968 - 543.809i) q^{83} +(-210.585 - 364.744i) q^{85} -1013.11 q^{89} -965.495 q^{91} +(-209.369 - 362.638i) q^{95} +(-577.909 + 1000.97i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{5} + 2 q^{7}+O(q^{10}) $ 4 * q + 10 * q^5 + 2 * q^7	$ 4 q + 10 q^{5} + 2 q^{7} - 42 q^{11} + 14 q^{13} + 48 q^{17} - 280 q^{19} - 126 q^{23} - 50 q^{25} + 126 q^{29} + 56 q^{31} + 20 q^{35} - 1144 q^{37} + 66 q^{41} + 530 q^{43} + 396 q^{47} + 306 q^{49} - 1008 q^{53} - 420 q^{55} - 438 q^{59} + 602 q^{61} - 70 q^{65} + 920 q^{67} + 1596 q^{71} - 1540 q^{73} - 336 q^{77} + 1724 q^{79} + 486 q^{83} + 120 q^{85} - 588 q^{89} - 3752 q^{91} - 700 q^{95} - 112 q^{97}+O(q^{100}) $ 4 * q + 10 * q^5 + 2 * q^7 - 42 * q^11 + 14 * q^13 + 48 * q^17 - 280 * q^19 - 126 * q^23 - 50 * q^25 + 126 * q^29 + 56 * q^31 + 20 * q^35 - 1144 * q^37 + 66 * q^41 + 530 * q^43 + 396 * q^47 + 306 * q^49 - 1008 * q^53 - 420 * q^55 - 438 * q^59 + 602 * q^61 - 70 * q^65 + 920 * q^67 + 1596 * q^71 - 1540 * q^73 - 336 * q^77 + 1724 * q^79 + 486 * q^83 + 120 * q^85 - 588 * q^89 - 3752 * q^91 - 700 * q^95 - 112 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$6$		0			0
$7$	−6.37386	+	11.0399i	−0.344156	+	0.596096i	−0.985200	−	0.171408i	$-0.945168\pi$
$7$	−6.37386	+	11.0399i	−0.344156	+	0.596096i	0.641044	+	0.767504i	$0.278502\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−17.3739	+	30.0924i	−0.476220	+	0.824837i	−0.999629	−	0.0272448i	$-0.991327\pi$
$11$	−17.3739	+	30.0924i	−0.476220	+	0.824837i	0.523409	+	0.852082i	$0.324660\pi$
$12$		0			0
$13$	37.8693	+	65.5916i	0.807928	+	1.39937i	0.914297	+	0.405045i	$0.132744\pi$
$13$	37.8693	+	65.5916i	0.807928	+	1.39937i	−0.106369	+	0.994327i	$0.533923\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−84.2341			−1.20175			−0.600876	−	0.799343i	$-0.705181\pi$
$17$	−84.2341			−1.20175			−0.600876	+	0.799343i	$0.705181\pi$
$18$		0			0
$19$	−83.7477			−1.01121			−0.505606	−	0.862764i	$-0.668731\pi$
$19$	−83.7477			−1.01121			−0.505606	+	0.862764i	$0.668731\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−86.4909	−	149.807i	−0.784113	−	1.35812i	−0.929527	−	0.368753i	$-0.879785\pi$
$23$	−86.4909	−	149.807i	−0.784113	−	1.35812i	0.145414	−	0.989371i	$-0.453548\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−44.1125	+	76.4051i	−0.282465	+	0.489244i	−0.971991	−	0.235017i	$-0.924485\pi$
$29$	−44.1125	+	76.4051i	−0.282465	+	0.489244i	0.689526	+	0.724261i	$0.257819\pi$
$30$		0			0
$31$	−102.856	−	178.151i	−0.595917	−	1.03216i	−0.993417	−	0.114556i	$-0.963455\pi$
$31$	−102.856	−	178.151i	−0.595917	−	1.03216i	0.397500	−	0.917602i	$-0.369878\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−63.7386			−0.307823
$36$		0			0
$37$	−286.000			−1.27076			−0.635380	−	0.772200i	$-0.719156\pi$
$37$	−286.000			−1.27076			−0.635380	+	0.772200i	$0.719156\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	160.851	+	278.602i	0.612701	+	1.06123i	0.990783	+	0.135457i	$0.0432502\pi$
$41$	160.851	+	278.602i	0.612701	+	1.06123i	−0.378083	+	0.925772i	$0.623416\pi$
$42$		0			0
$43$	84.3830	−	146.156i	0.299262	−	0.518338i	−0.676705	−	0.736254i	$-0.736593\pi$
$43$	84.3830	−	146.156i	0.299262	−	0.518338i	0.975967	+	0.217917i	$0.0699261\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	195.234	−	338.155i	0.605911	−	1.04947i	−0.385996	−	0.922501i	$-0.626142\pi$
$47$	195.234	−	338.155i	0.605911	−	1.04947i	0.991907	−	0.126968i	$-0.0405246\pi$
$48$		0			0
$49$	90.2477	+	156.314i	0.263113	+	0.455725i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	91.6932			0.237642			0.118821	−	0.992916i	$-0.462089\pi$
$53$	91.6932			0.237642			0.118821	+	0.992916i	$0.462089\pi$
$54$		0			0
$55$	−173.739			−0.425944
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−61.3830	−	106.318i	−0.135447	−	0.234601i	0.790321	−	0.612693i	$-0.209914\pi$
$59$	−61.3830	−	106.318i	−0.135447	−	0.234601i	−0.925768	+	0.378092i	$0.876580\pi$
$60$		0			0
$61$	439.202	−	760.721i	0.921870	−	1.59673i	0.125352	−	0.992112i	$-0.459994\pi$
$61$	439.202	−	760.721i	0.921870	−	1.59673i	0.796519	−	0.604614i	$-0.206673\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−189.347	+	327.958i	−0.361316	+	0.625818i
$66$		0			0
$67$	518.702	+	898.419i	0.945814	+	1.63820i	0.754112	+	0.656746i	$0.228067\pi$
$67$	518.702	+	898.419i	0.945814	+	1.63820i	0.191702	+	0.981453i	$0.438599\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	605.216			1.01163			0.505816	−	0.862641i	$-0.331191\pi$
$71$	605.216			1.01163			0.505816	+	0.862641i	$0.331191\pi$
$72$		0			0
$73$	−13.8114			−0.0221438			−0.0110719	−	0.999939i	$-0.503524\pi$
$73$	−13.8114			−0.0221438			−0.0110719	+	0.999939i	$0.503524\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−221.477	−	383.610i	−0.327788	−	0.567746i
$78$		0			0
$79$	286.649	−	496.490i	0.408234	−	0.707083i	−0.586458	−	0.809980i	$-0.699478\pi$
$79$	286.649	−	496.490i	0.408234	−	0.707083i	0.994692	+	0.102897i	$0.0328113\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	313.968	−	543.809i	0.415211	−	0.719166i	−0.580240	−	0.814446i	$-0.697041\pi$
$83$	313.968	−	543.809i	0.415211	−	0.719166i	0.995451	+	0.0952797i	$0.0303745\pi$
$84$		0			0
$85$	−210.585	−	364.744i	−0.268720	−	0.465436i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1013.11			−1.20662			−0.603310	−	0.797507i	$-0.706152\pi$
$89$	−1013.11			−1.20662			−0.603310	+	0.797507i	$0.706152\pi$
$90$		0			0
$91$	−965.495			−1.11221
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−209.369	−	362.638i	−0.226114	−	0.391641i
$96$		0			0
$97$	−577.909	+	1000.97i	−0.604926	+	1.04776i	0.387138	+	0.922022i	$0.373464\pi$
$97$	−577.909	+	1000.97i	−0.604926	+	1.04776i	−0.992063	+	0.125740i	$0.959870\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	426.936	−	739.475i	0.420611	−	0.728520i	−0.575388	−	0.817881i	$-0.695149\pi$
$101$	426.936	−	739.475i	0.420611	−	0.728520i	0.995999	+	0.0893602i	$0.0284822\pi$
$102$		0			0
$103$	514.973	+	891.959i	0.492639	+	0.853275i	0.999964	−	0.00847952i	$-0.00269915\pi$
$103$	514.973	+	891.959i	0.492639	+	0.853275i	−0.507326	+	0.861755i	$0.669366\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	514.845			0.465159			0.232579	−	0.972577i	$-0.425283\pi$
$107$	514.845			0.465159			0.232579	+	0.972577i	$0.425283\pi$
$108$		0			0
$109$	−656.873			−0.577220			−0.288610	−	0.957447i	$-0.593193\pi$
$109$	−656.873			−0.577220			−0.288610	+	0.957447i	$0.593193\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−850.720	−	1473.49i	−0.708222	−	1.22668i	−0.965516	−	0.260343i	$-0.916164\pi$
$113$	−850.720	−	1473.49i	−0.708222	−	1.22668i	0.257295	−	0.966333i	$-0.417169\pi$
$114$		0			0
$115$	432.455	−	749.033i	0.350666	−	0.607371i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	536.897	−	929.932i	0.413590	−	0.716359i
$120$		0			0
$121$	61.7977	+	107.037i	0.0464296	+	0.0804183i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$	345.405			0.241336			0.120668	−	0.992693i	$-0.461496\pi$
$127$	345.405			0.241336			0.120668	+	0.992693i	$0.461496\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1413.94	−	2449.01i	−0.943024	−	1.63337i	−0.759659	−	0.650321i	$-0.774634\pi$
$131$	−1413.94	−	2449.01i	−0.943024	−	1.63337i	−0.183365	−	0.983045i	$-0.558699\pi$
$132$		0			0
$133$	533.797	−	924.563i	0.348015	−	0.602780i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	613.919	−	1063.34i	0.382852	−	0.663118i	−0.608617	−	0.793464i	$-0.708275\pi$
$137$	613.919	−	1063.34i	0.382852	−	0.663118i	0.991469	+	0.130346i	$0.0416087\pi$
$138$		0			0
$139$	20.0545	+	34.7355i	0.0122374	+	0.0211959i	0.872079	−	0.489365i	$-0.162771\pi$
$139$	20.0545	+	34.7355i	0.0122374	+	0.0211959i	−0.859842	+	0.510560i	$0.829438\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2631.75			−1.53900
$144$		0			0
$145$	−441.125			−0.252644
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−240.365	−	416.324i	−0.132157	−	0.228903i	0.792351	−	0.610066i	$-0.208857\pi$
$149$	−240.365	−	416.324i	−0.132157	−	0.228903i	−0.924508	+	0.381163i	$0.875524\pi$
$150$		0			0
$151$	38.0000	−	65.8179i	0.0204794	−	0.0354714i	−0.855604	−	0.517631i	$-0.826814\pi$
$151$	38.0000	−	65.8179i	0.0204794	−	0.0354714i	0.876083	+	0.482159i	$0.160147\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	514.278	−	890.756i	0.266502	−	0.461595i
$156$		0			0
$157$	219.301	+	379.841i	0.111479	+	0.193087i	0.916367	−	0.400340i	$-0.131108\pi$
$157$	219.301	+	379.841i	0.111479	+	0.193087i	−0.804888	+	0.593427i	$0.797775\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2205.12			1.07943
$162$		0			0
$163$	1755.87			0.843746			0.421873	−	0.906655i	$-0.361373\pi$
$163$	1755.87			0.843746			0.421873	+	0.906655i	$0.361373\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1393.78	+	2414.09i	0.645831	+	1.11861i	0.984109	+	0.177565i	$0.0568221\pi$
$167$	1393.78	+	2414.09i	0.645831	+	1.11861i	−0.338278	+	0.941046i	$0.609845\pi$
$168$		0			0
$169$	−1769.67	+	3065.16i	−0.805494	+	1.39516i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	608.181	−	1053.40i	0.267278	−	0.462939i	−0.700880	−	0.713279i	$-0.747209\pi$
$173$	608.181	−	1053.40i	0.267278	−	0.462939i	0.968158	+	0.250340i	$0.0805424\pi$
$174$		0			0
$175$	−159.347	−	275.996i	−0.0688313	−	0.119219i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2531.98			1.05726			0.528628	−	0.848854i	$-0.322707\pi$
$179$	2531.98			1.05726			0.528628	+	0.848854i	$0.322707\pi$
$180$		0			0
$181$	3444.82			1.41465			0.707324	−	0.706889i	$-0.249902\pi$
$181$	3444.82			1.41465			0.707324	+	0.706889i	$0.249902\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−715.000	−	1238.42i	−0.284151	−	0.492163i
$186$		0			0
$187$	1463.47	−	2534.81i	0.572298	−	0.991248i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−851.286	+	1474.47i	−0.322497	+	0.558581i	−0.981003	−	0.193995i	$-0.937856\pi$
$191$	−851.286	+	1474.47i	−0.322497	+	0.558581i	0.658506	+	0.752576i	$0.271189\pi$
$192$		0			0
$193$	−342.744	−	593.651i	−0.127830	−	0.221409i	0.795005	−	0.606602i	$-0.207468\pi$
$193$	−342.744	−	593.651i	−0.127830	−	0.221409i	−0.922836	+	0.385194i	$0.874135\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4362.95			−1.57791			−0.788953	−	0.614454i	$-0.789377\pi$
$197$	−4362.95			−1.57791			−0.788953	+	0.614454i	$0.789377\pi$
$198$		0			0
$199$	−2920.20			−1.04024			−0.520119	−	0.854094i	$-0.674112\pi$
$199$	−2920.20			−1.04024			−0.520119	+	0.854094i	$0.674112\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−562.334	−	973.991i	−0.194424	−	0.336753i
$204$		0			0
$205$	−804.256	+	1393.01i	−0.274008	+	0.474596i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1455.02	−	2520.17i	0.481560	−	0.834086i
$210$		0			0
$211$	106.692	+	184.796i	0.0348103	+	0.0602933i	0.882906	−	0.469550i	$-0.155584\pi$
$211$	106.692	+	184.796i	0.0348103	+	0.0602933i	−0.848095	+	0.529844i	$0.822251\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	843.830			0.267668
$216$		0			0
$217$	2622.35			0.820354
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3189.89	−	5525.05i	−0.970928	−	1.68170i
$222$		0			0
$223$	−1045.77	+	1811.33i	−0.314036	+	0.543927i	−0.979232	−	0.202742i	$-0.935015\pi$
$223$	−1045.77	+	1811.33i	−0.314036	+	0.543927i	0.665196	+	0.746669i	$0.268348\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2241.80	−	3882.91i	0.655477	−	1.13532i	−0.326297	−	0.945267i	$-0.605801\pi$
$227$	2241.80	−	3882.91i	0.655477	−	1.13532i	0.981774	−	0.190053i	$-0.0608658\pi$
$228$		0			0
$229$	−2465.54	−	4270.45i	−0.711475	−	1.23231i	−0.964304	−	0.264799i	$-0.914694\pi$
$229$	−2465.54	−	4270.45i	−0.711475	−	1.23231i	0.252829	−	0.967511i	$-0.418639\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−773.782			−0.217563			−0.108781	−	0.994066i	$-0.534695\pi$
$233$	−773.782			−0.217563			−0.108781	+	0.994066i	$0.534695\pi$
$234$		0			0
$235$	1952.34			0.541943
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2206.20	−	3821.25i	−0.597101	−	1.03421i	−0.993247	−	0.116021i	$-0.962986\pi$
$239$	−2206.20	−	3821.25i	−0.597101	−	1.03421i	0.396146	−	0.918188i	$-0.370347\pi$
$240$		0			0
$241$	−3071.71	+	5320.35i	−0.821021	+	1.42205i	0.0839019	+	0.996474i	$0.473262\pi$
$241$	−3071.71	+	5320.35i	−0.821021	+	1.42205i	−0.904923	+	0.425576i	$0.860072\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−451.239	+	781.568i	−0.117668	+	0.203806i
$246$		0			0
$247$	−3171.47	−	5493.15i	−0.816987	−	1.41506i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−746.895			−0.187823			−0.0939116	−	0.995581i	$-0.529937\pi$
$251$	−746.895			−0.187823			−0.0939116	+	0.995581i	$0.529937\pi$
$252$		0			0
$253$	6010.72			1.49364
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3773.31	−	6535.56i	−0.915847	−	1.58629i	−0.805657	−	0.592383i	$-0.798187\pi$
$257$	−3773.31	−	6535.56i	−0.915847	−	1.58629i	−0.110190	−	0.993911i	$-0.535146\pi$
$258$		0			0
$259$	1822.92	−	3157.40i	0.437340	−	0.757495i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	691.993	−	1198.57i	0.162244	−	0.281015i	−0.773429	−	0.633883i	$-0.781460\pi$
$263$	691.993	−	1198.57i	0.162244	−	0.281015i	0.935673	+	0.352868i	$0.114794\pi$
$264$		0			0
$265$	229.233	+	397.043i	0.0531384	+	0.0920384i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6987.28			−1.58372			−0.791862	−	0.610700i	$-0.790888\pi$
$269$	−6987.28			−1.58372			−0.791862	+	0.610700i	$0.790888\pi$
$270$		0			0
$271$	−3600.27			−0.807014			−0.403507	−	0.914977i	$-0.632209\pi$
$271$	−3600.27			−0.807014			−0.403507	+	0.914977i	$0.632209\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−434.347	−	752.310i	−0.0952439	−	0.164967i
$276$		0			0
$277$	−901.872	+	1562.09i	−0.195625	+	0.338833i	−0.947105	−	0.320923i	$-0.896007\pi$
$277$	−901.872	+	1562.09i	−0.195625	+	0.338833i	0.751480	+	0.659756i	$0.229340\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1856.48	+	3215.52i	−0.394123	+	0.682641i	−0.992989	−	0.118209i	$-0.962285\pi$
$281$	−1856.48	+	3215.52i	−0.394123	+	0.682641i	0.598866	+	0.800849i	$0.295618\pi$
$282$		0			0
$283$	414.481	+	717.902i	0.0870612	+	0.150794i	0.906268	−	0.422704i	$-0.138919\pi$
$283$	414.481	+	717.902i	0.0870612	+	0.150794i	−0.819206	+	0.573499i	$0.805586\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4100.97			−0.843459
$288$		0			0
$289$	2182.38			0.444206
$290$		0			0
$291$		0			0
$292$		0			0
$293$	843.160	+	1460.40i	0.168116	+	0.291185i	0.937757	−	0.347291i	$-0.112898\pi$
$293$	843.160	+	1460.40i	0.168116	+	0.291185i	−0.769642	+	0.638476i	$0.779565\pi$
$294$		0			0
$295$	306.915	−	531.592i	0.0605738	−	0.104917i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6550.70	−	11346.2i	1.26701	−	2.19453i
$300$		0			0
$301$	1075.69	+	1863.15i	0.205986	+	0.356778i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4392.02			0.824546
$306$		0			0
$307$	−5760.81			−1.07097			−0.535483	−	0.844546i	$-0.679871\pi$
$307$	−5760.81			−1.07097			−0.535483	+	0.844546i	$0.679871\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3443.98	−	5965.15i	−0.627942	−	1.08763i	−0.987964	−	0.154684i	$-0.950564\pi$
$311$	−3443.98	−	5965.15i	−0.627942	−	1.08763i	0.360022	−	0.932944i	$-0.382769\pi$
$312$		0			0
$313$	1528.33	−	2647.14i	0.275995	−	0.478037i	−0.694391	−	0.719598i	$-0.744326\pi$
$313$	1528.33	−	2647.14i	0.275995	−	0.478037i	0.970386	+	0.241561i	$0.0776595\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4828.23	−	8362.74i	0.855459	−	1.48170i	−0.0207591	−	0.999785i	$-0.506608\pi$
$317$	4828.23	−	8362.74i	0.855459	−	1.48170i	0.876218	−	0.481914i	$-0.160058\pi$
$318$		0			0
$319$	−1532.81	−	2654.90i	−0.269031	−	0.465975i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	7054.41			1.21523
$324$		0			0
$325$	−1893.47			−0.323171
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2488.79	+	4310.71i	0.417056	+	0.722362i
$330$		0			0
$331$	−4439.53	+	7689.50i	−0.737217	+	1.27690i	0.216527	+	0.976277i	$0.430527\pi$
$331$	−4439.53	+	7689.50i	−0.737217	+	1.27690i	−0.953744	+	0.300621i	$0.902806\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2593.51	+	4492.09i	−0.422981	+	0.732625i
$336$		0			0
$337$	−1305.09	−	2260.47i	−0.210957	−	0.365388i	0.741057	−	0.671442i	$-0.234325\pi$
$337$	−1305.09	−	2260.47i	−0.210957	−	0.365388i	−0.952014	+	0.306054i	$0.900991\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	7148.00			1.13515
$342$		0			0
$343$	−6673.38			−1.05052
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1903.24	+	3296.51i	0.294442	+	0.509988i	0.974855	−	0.222841i	$-0.0715329\pi$
$347$	1903.24	+	3296.51i	0.294442	+	0.509988i	−0.680413	+	0.732829i	$0.738200\pi$
$348$		0			0
$349$	−2040.50	+	3534.24i	−0.312966	+	0.542073i	−0.979003	−	0.203845i	$-0.934656\pi$
$349$	−2040.50	+	3534.24i	−0.312966	+	0.542073i	0.666037	+	0.745919i	$0.267989\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2508.32	−	4344.54i	0.378200	−	0.655061i	−0.612601	−	0.790392i	$-0.709877\pi$
$353$	2508.32	−	4344.54i	0.378200	−	0.655061i	0.990800	+	0.135332i	$0.0432100\pi$
$354$		0			0
$355$	1513.04	+	2620.66i	0.226208	+	0.391804i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	515.859			0.0758384			0.0379192	−	0.999281i	$-0.487927\pi$
$359$	515.859			0.0758384			0.0379192	+	0.999281i	$0.487927\pi$
$360$		0			0
$361$	154.682			0.0225517
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−34.5284	−	59.8050i	−0.00495151	−	0.00857626i
$366$		0			0
$367$	−2281.15	+	3951.06i	−0.324455	+	0.561972i	−0.981402	−	0.191965i	$-0.938514\pi$
$367$	−2281.15	+	3951.06i	−0.324455	+	0.561972i	0.656947	+	0.753937i	$0.271847\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−584.440	+	1012.28i	−0.0817860	+	0.141657i
$372$		0			0
$373$	−5219.07	−	9039.69i	−0.724485	−	1.25485i	−0.959186	−	0.282777i	$-0.908744\pi$
$373$	−5219.07	−	9039.69i	−0.724485	−	1.25485i	0.234700	−	0.972068i	$-0.424589\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6682.04			−0.912845
$378$		0			0
$379$	7375.64			0.999634			0.499817	−	0.866131i	$-0.333401\pi$
$379$	7375.64			0.999634			0.499817	+	0.866131i	$0.333401\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3638.57	−	6302.18i	−0.485436	−	0.840800i	0.514424	−	0.857536i	$-0.328006\pi$
$383$	−3638.57	−	6302.18i	−0.485436	−	0.840800i	−0.999860	+	0.0167361i	$0.994672\pi$
$384$		0			0
$385$	1107.39	−	1918.05i	0.146591	−	0.253904i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5549.70	+	9612.36i	−0.723344	+	1.25287i	0.236308	+	0.971678i	$0.424063\pi$
$389$	−5549.70	+	9612.36i	−0.723344	+	1.25287i	−0.959652	+	0.281191i	$0.909271\pi$
$390$		0			0
$391$	7285.48	+	12618.8i	0.942309	+	1.63213i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2866.49			0.365136
$396$		0			0
$397$	5.73405			0.000724896			0.000362448	−	1.00000i	$-0.499885\pi$
$397$	5.73405			0.000724896			0.000362448		1.00000i	$0.499885\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3039.33	−	5264.27i	−0.378496	−	0.655574i	0.612348	−	0.790588i	$-0.290225\pi$
$401$	−3039.33	−	5264.27i	−0.378496	−	0.655574i	−0.990844	+	0.135015i	$0.956892\pi$
$402$		0			0
$403$	7790.15	−	13492.9i	0.962916	−	1.66782i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4968.92	−	8606.43i	0.605161	−	1.04817i
$408$		0			0
$409$	5577.26	+	9660.10i	0.674273	+	1.16788i	0.976681	+	0.214697i	$0.0688763\pi$
$409$	5577.26	+	9660.10i	0.674273	+	1.16788i	−0.302408	+	0.953179i	$0.597790\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1564.99			0.186460
$414$		0			0
$415$	3139.68			0.371376
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3359.41	−	5818.66i	−0.391689	−	0.678425i	0.600983	−	0.799261i	$-0.294776\pi$
$419$	−3359.41	−	5818.66i	−0.391689	−	0.678425i	−0.992672	+	0.120836i	$0.961442\pi$
$420$		0			0
$421$	4649.74	−	8053.58i	0.538276	−	0.932322i	−0.460721	−	0.887545i	$-0.652409\pi$
$421$	4649.74	−	8053.58i	0.538276	−	0.932322i	0.998997	−	0.0447766i	$-0.0142576\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1052.93	−	1823.72i	0.120175	−	0.208149i
$426$		0			0
$427$	5598.83	+	9697.46i	0.634535	+	1.09905i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2338.96			0.261400			0.130700	−	0.991422i	$-0.458277\pi$
$431$	2338.96			0.261400			0.130700	+	0.991422i	$0.458277\pi$
$432$		0			0
$433$	11685.4			1.29691			0.648457	−	0.761251i	$-0.275414\pi$
$433$	11685.4			1.29691			0.648457	+	0.761251i	$0.275414\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7243.42	+	12546.0i	0.792905	+	1.37335i
$438$		0			0
$439$	8051.88	−	13946.3i	0.875388	−	1.51622i	0.0190395	−	0.999819i	$-0.493939\pi$
$439$	8051.88	−	13946.3i	0.875388	−	1.51622i	0.856349	−	0.516398i	$-0.172727\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5631.94	−	9754.80i	0.604021	−	1.04620i	−0.388184	−	0.921582i	$-0.626897\pi$
$443$	5631.94	−	9754.80i	0.604021	−	1.04620i	0.992205	−	0.124614i	$-0.0397692\pi$
$444$		0			0
$445$	−2532.77	−	4386.88i	−0.269808	−	0.467322i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−10612.8			−1.11548			−0.557739	−	0.830016i	$-0.688331\pi$
$449$	−10612.8			−1.11548			−0.557739	+	0.830016i	$0.688331\pi$
$450$		0			0
$451$	−11178.4			−1.16712
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2413.74	−	4180.72i	−0.248698	−	0.430758i
$456$		0			0
$457$	−1246.51	+	2159.02i	−0.127592	+	0.220995i	−0.922743	−	0.385416i	$-0.874058\pi$
$457$	−1246.51	+	2159.02i	−0.127592	+	0.220995i	0.795151	+	0.606411i	$0.207391\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7705.72	+	13346.7i	−0.778506	+	1.34841i	0.154296	+	0.988025i	$0.450689\pi$
$461$	−7705.72	+	13346.7i	−0.778506	+	1.34841i	−0.932803	+	0.360388i	$0.882644\pi$
$462$		0			0
$463$	4415.89	+	7648.55i	0.443248	+	0.767729i	0.997928	−	0.0643352i	$-0.0204927\pi$
$463$	4415.89	+	7648.55i	0.443248	+	0.767729i	−0.554680	+	0.832064i	$0.687159\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8430.83			−0.835401			−0.417701	−	0.908585i	$-0.637164\pi$
$467$	−8430.83			−0.835401			−0.417701	+	0.908585i	$0.637164\pi$
$468$		0			0
$469$	−13224.5			−1.30203
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2932.12	+	5078.57i	0.285029	+	0.493685i
$474$		0			0
$475$	1046.85	−	1813.19i	0.101121	−	0.175147i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4313.02	+	7470.38i	−0.411414	+	0.712589i	−0.995045	−	0.0994298i	$-0.968298\pi$
$479$	−4313.02	+	7470.38i	−0.411414	+	0.712589i	0.583631	+	0.812019i	$0.301631\pi$
$480$		0			0
$481$	−10830.6	−	18759.2i	−1.02668	−	1.77827i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5779.09			−0.541062
$486$		0			0
$487$	−12182.8			−1.13358			−0.566792	−	0.823861i	$-0.691816\pi$
$487$	−12182.8			−1.13358			−0.566792	+	0.823861i	$0.691816\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−10510.5	−	18204.6i	−0.966049	−	1.67325i	−0.706769	−	0.707445i	$-0.749848\pi$
$491$	−10510.5	−	18204.6i	−0.966049	−	1.67325i	−0.259281	−	0.965802i	$-0.583485\pi$
$492$		0			0
$493$	3715.78	−	6435.91i	0.339453	−	0.587949i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3857.56	+	6681.50i	−0.348160	+	0.603030i
$498$		0			0
$499$	4445.35	+	7699.57i	0.398800	+	0.690742i	0.993578	−	0.113148i	$-0.0360934\pi$
$499$	4445.35	+	7699.57i	0.398800	+	0.690742i	−0.594778	+	0.803890i	$0.702760\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2433.17			−0.215686			−0.107843	−	0.994168i	$-0.534394\pi$
$503$	−2433.17			−0.215686			−0.107843	+	0.994168i	$0.534394\pi$
$504$		0			0
$505$	4269.36			0.376206
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7628.81	+	13213.5i	0.664324	+	1.15064i	0.979468	+	0.201599i	$0.0646139\pi$
$509$	7628.81	+	13213.5i	0.664324	+	1.15064i	−0.315144	+	0.949044i	$0.602053\pi$
$510$		0			0
$511$	88.0318	−	152.476i	0.00762093	−	0.0131998i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2574.86	+	4459.79i	−0.220315	+	0.381596i
$516$		0			0
$517$	6783.94	+	11750.1i	0.577094	+	0.999555i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12991.2			1.09242			0.546212	−	0.837647i	$-0.316069\pi$
$521$	12991.2			1.09242			0.546212	+	0.837647i	$0.316069\pi$
$522$		0			0
$523$	1501.60			0.125545			0.0627727	−	0.998028i	$-0.480006\pi$
$523$	1501.60			0.125545			0.0627727	+	0.998028i	$0.480006\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8663.95	+	15006.4i	0.716144	+	1.24040i
$528$		0			0
$529$	−8877.85	+	15376.9i	−0.729667	+	1.26382i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−12182.6	+	21101.0i	−0.990036	+	1.71479i
$534$		0			0
$535$	1287.11	+	2229.35i	0.104013	+	0.180155i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6271.81			−0.501198
$540$		0			0
$541$	−18683.1			−1.48475			−0.742375	−	0.669985i	$-0.766301\pi$
$541$	−18683.1			−1.48475			−0.742375	+	0.669985i	$0.766301\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1642.18	−	2844.34i	−0.129070	−	0.223556i
$546$		0			0
$547$	1415.45	−	2451.63i	0.110640	−	0.191634i	−0.805388	−	0.592747i	$-0.798043\pi$
$547$	1415.45	−	2451.63i	0.110640	−	0.191634i	0.916029	+	0.401113i	$0.131377\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3694.32	−	6398.75i	0.285632	−	0.494730i
$552$		0			0
$553$	3654.12	+	6329.12i	0.280993	+	0.486694i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2065.19			−0.157100			−0.0785500	−	0.996910i	$-0.525029\pi$
$557$	−2065.19			−0.157100			−0.0785500	+	0.996910i	$0.525029\pi$
$558$		0			0
$559$	12782.1			0.967129
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10264.0	+	17777.8i	0.768341	+	1.33081i	0.938462	+	0.345383i	$0.112251\pi$
$563$	10264.0	+	17777.8i	0.768341	+	1.33081i	−0.170120	+	0.985423i	$0.554416\pi$
$564$		0			0
$565$	4253.60	−	7367.46i	0.316726	−	0.548586i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8879.24	−	15379.3i	0.654195	−	1.13310i	−0.327900	−	0.944712i	$-0.606341\pi$
$569$	8879.24	−	15379.3i	0.654195	−	1.13310i	0.982095	−	0.188386i	$-0.0603257\pi$
$570$		0			0
$571$	−6304.57	−	10919.8i	−0.462063	−	0.800316i	0.537001	−	0.843582i	$-0.319557\pi$
$571$	−6304.57	−	10919.8i	−0.462063	−	0.800316i	−0.999064	+	0.0432654i	$0.986224\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4324.55			0.313645
$576$		0			0
$577$	1266.62			0.0913863			0.0456932	−	0.998956i	$-0.485450\pi$
$577$	1266.62			0.0913863			0.0456932	+	0.998956i	$0.485450\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4002.38	+	6932.33i	0.285795	+	0.495011i
$582$		0			0
$583$	−1593.06	+	2759.27i	−0.113170	+	0.196016i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4471.76	+	7745.31i	−0.314428	+	0.544605i	−0.979316	−	0.202338i	$-0.935146\pi$
$587$	−4471.76	+	7745.31i	−0.314428	+	0.544605i	0.664888	+	0.746943i	$0.268479\pi$
$588$		0			0
$589$	8613.93	+	14919.8i	0.602599	+	1.04373i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−16200.0			−1.12185			−0.560924	−	0.827868i	$-0.689554\pi$
$593$	−16200.0			−1.12185			−0.560924	+	0.827868i	$0.689554\pi$
$594$		0			0
$595$	5368.97			0.369926
$596$		0			0
$597$		0			0
$598$		0			0
$599$	9815.03	+	17000.1i	0.669501	+	1.15961i	0.978044	+	0.208399i	$0.0668254\pi$
$599$	9815.03	+	17000.1i	0.669501	+	1.15961i	−0.308543	+	0.951211i	$0.599841\pi$
$600$		0			0
$601$	−5768.73	+	9991.73i	−0.391533	+	0.678155i	−0.992652	−	0.121004i	$-0.961388\pi$
$601$	−5768.73	+	9991.73i	−0.391533	+	0.678155i	0.601119	+	0.799160i	$0.294722\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−308.989	+	535.184i	−0.0207639	+	0.0359642i
$606$		0			0
$607$	3173.17	+	5496.10i	0.212183	+	0.367512i	0.952397	−	0.304859i	$-0.0986093\pi$
$607$	3173.17	+	5496.10i	0.212183	+	0.367512i	−0.740214	+	0.672371i	$0.765276\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	29573.5			1.95813
$612$		0			0
$613$	−12007.3			−0.791144			−0.395572	−	0.918435i	$-0.629454\pi$
$613$	−12007.3			−0.791144			−0.395572	+	0.918435i	$0.629454\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4325.42	−	7491.85i	−0.282229	−	0.488834i	0.689705	−	0.724091i	$-0.257740\pi$
$617$	−4325.42	−	7491.85i	−0.282229	−	0.488834i	−0.971933	+	0.235257i	$0.924407\pi$
$618$		0			0
$619$	−5764.40	+	9984.24i	−0.374299	+	0.648304i	−0.990222	−	0.139502i	$-0.955450\pi$
$619$	−5764.40	+	9984.24i	−0.374299	+	0.648304i	0.615923	+	0.787806i	$0.288783\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6457.40	−	11184.6i	0.415266	−	0.719261i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	24090.9			1.52714
$630$		0			0
$631$	−4739.76			−0.299028			−0.149514	−	0.988760i	$-0.547771\pi$
$631$	−4739.76			−0.299028			−0.149514	+	0.988760i	$0.547771\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	863.511	+	1495.65i	0.0539644	+	0.0934691i
$636$		0			0
$637$	−6835.24	+	11839.0i	−0.425152	+	0.736385i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−14606.9	+	25299.9i	−0.900060	+	1.55895i	−0.0726460	+	0.997358i	$0.523144\pi$
$641$	−14606.9	+	25299.9i	−0.900060	+	1.55895i	−0.827414	+	0.561592i	$0.810189\pi$
$642$		0			0
$643$	−11722.2	−	20303.5i	−0.718943	−	1.24525i	−0.961419	−	0.275088i	$-0.911293\pi$
$643$	−11722.2	−	20303.5i	−0.718943	−	1.24525i	0.242476	−	0.970157i	$-0.422040\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	27783.6			1.68823			0.844116	−	0.536160i	$-0.180126\pi$
$647$	27783.6			1.68823			0.844116	+	0.536160i	$0.180126\pi$
$648$		0			0
$649$	4265.84			0.258010
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−4266.52	−	7389.84i	−0.255684	−	0.442859i	0.709397	−	0.704809i	$-0.248967\pi$
$653$	−4266.52	−	7389.84i	−0.255684	−	0.442859i	−0.965081	+	0.261951i	$0.915634\pi$
$654$		0			0
$655$	7069.68	−	12245.0i	0.421733	−	0.730464i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−542.638	+	939.876i	−0.0320761	+	0.0555575i	−0.881618	−	0.471964i	$-0.843545\pi$
$659$	−542.638	+	939.876i	−0.0320761	+	0.0555575i	0.849542	+	0.527521i	$0.176879\pi$
$660$		0			0
$661$	201.868	+	349.646i	0.0118786	+	0.0205744i	0.871904	−	0.489678i	$-0.162885\pi$
$661$	201.868	+	349.646i	0.0118786	+	0.0205744i	−0.860025	+	0.510252i	$0.829552\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5337.97			0.311274
$666$		0			0
$667$	15261.3			0.885938
$668$		0			0
$669$		0			0
$670$		0			0
$671$	15261.3	+	26433.3i	0.878026	+	1.52079i
$672$		0			0
$673$	12562.5	−	21758.9i	0.719537	−	1.24627i	−0.241646	−	0.970364i	$-0.577687\pi$
$673$	12562.5	−	21758.9i	0.719537	−	1.24627i	0.961183	−	0.275910i	$-0.0889793\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2180.83	−	3777.32i	0.123805	−	0.214437i	−0.797460	−	0.603372i	$-0.793823\pi$
$677$	2180.83	−	3777.32i	0.123805	−	0.214437i	0.921265	+	0.388935i	$0.127157\pi$
$678$		0			0
$679$	−7367.03	−	12760.1i	−0.416378	−	0.721188i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−14945.1			−0.837275			−0.418637	−	0.908153i	$-0.637492\pi$
$683$	−14945.1			−0.837275			−0.418637	+	0.908153i	$0.637492\pi$
$684$		0			0
$685$	6139.19			0.342433
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3472.36	+	6014.30i	0.191998	+	0.332549i
$690$		0			0
$691$	−10749.5	+	18618.7i	−0.591797	+	1.02502i	0.402193	+	0.915555i	$0.368248\pi$
$691$	−10749.5	+	18618.7i	−0.591797	+	1.02502i	−0.993990	+	0.109468i	$0.965085\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−100.273	+	173.677i	−0.00547275	+	0.00947908i
$696$		0			0
$697$	−13549.1	−	23467.8i	−0.736314	−	1.27533i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−28878.4			−1.55595			−0.777977	−	0.628293i	$-0.783754\pi$
$701$	−28878.4			−1.55595			−0.777977	+	0.628293i	$0.783754\pi$
$702$		0			0
$703$	23951.8			1.28501
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5442.47	+	9426.63i	0.289512	+	0.501450i
$708$		0			0
$709$	9972.75	−	17273.3i	0.528257	−	0.914969i	−0.471200	−	0.882026i	$-0.656179\pi$
$709$	9972.75	−	17273.3i	0.528257	−	0.914969i	0.999457	−	0.0329422i	$-0.0104877\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−17792.2	+	30816.9i	−0.934533	+	1.61866i
$714$		0			0
$715$	−6579.36	−	11395.8i	−0.344132	−	0.596054i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	18280.0			0.948160			0.474080	−	0.880482i	$-0.342781\pi$
$719$	18280.0			0.948160			0.474080	+	0.880482i	$0.342781\pi$
$720$		0			0
$721$	−13129.5			−0.678179
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1102.81	−	1910.13i	−0.0564930	−	0.0978488i
$726$		0			0
$727$	−8999.55	+	15587.7i	−0.459112	+	0.795206i	−0.998914	−	0.0465860i	$-0.985166\pi$
$727$	−8999.55	+	15587.7i	−0.459112	+	0.795206i	0.539802	+	0.841792i	$0.318499\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7107.92	+	12311.3i	−0.359639	+	0.622913i
$732$		0			0
$733$	−4949.94	−	8573.54i	−0.249427	−	0.432020i	0.713940	−	0.700207i	$-0.246909\pi$
$733$	−4949.94	−	8573.54i	−0.249427	−	0.432020i	−0.963367	+	0.268187i	$0.913576\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−36047.4			−1.80166
$738$		0			0
$739$	−16358.7			−0.814294			−0.407147	−	0.913363i	$-0.633476\pi$
$739$	−16358.7			−0.814294			−0.407147	+	0.913363i	$0.633476\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	378.014	+	654.739i	0.0186648	+	0.0323285i	0.875207	−	0.483749i	$-0.160725\pi$
$743$	378.014	+	654.739i	0.0186648	+	0.0323285i	−0.856542	+	0.516077i	$0.827392\pi$
$744$		0			0
$745$	1201.82	−	2081.62i	0.0591026	−	0.102369i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3281.55	+	5683.82i	−0.160087	+	0.277279i
$750$		0			0
$751$	15684.2	+	27165.8i	0.762083	+	1.31997i	0.941775	+	0.336243i	$0.109156\pi$
$751$	15684.2	+	27165.8i	0.762083	+	1.31997i	−0.179692	+	0.983723i	$0.557510\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	380.000			0.0183174
$756$		0			0
$757$	−17559.4			−0.843072			−0.421536	−	0.906812i	$-0.638509\pi$
$757$	−17559.4			−0.843072			−0.421536	+	0.906812i	$0.638509\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	11751.2	+	20353.7i	0.559765	+	0.969541i	0.997516	+	0.0704449i	$0.0224419\pi$
$761$	11751.2	+	20353.7i	0.559765	+	0.969541i	−0.437751	+	0.899096i	$0.644225\pi$
$762$		0			0
$763$	4186.82	−	7251.78i	0.198654	−	0.344079i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4649.06	−	8052.41i	0.218863	−	0.379082i
$768$		0			0
$769$	−16138.0	−	27951.8i	−0.756762	−	1.31075i	−0.944493	−	0.328531i	$-0.893447\pi$
$769$	−16138.0	−	27951.8i	−0.756762	−	1.31075i	0.187731	−	0.982221i	$-0.439887\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	8693.38			0.404501			0.202251	−	0.979334i	$-0.435174\pi$
$773$	8693.38			0.404501			0.202251	+	0.979334i	$0.435174\pi$
$774$		0			0
$775$	5142.78			0.238367
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−13470.9	−	23332.3i	−0.619571	−	1.07313i
$780$		0			0
$781$	−10514.9	+	18212.4i	−0.481759	+	0.834432i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1096.51	+	1899.20i	−0.0498547	+	0.0863509i
$786$		0			0
$787$	−10408.5	−	18028.1i	−0.471440	−	0.816558i	0.528026	−	0.849228i	$-0.322932\pi$
$787$	−10408.5	−	18028.1i	−0.471440	−	0.816558i	−0.999466	+	0.0326698i	$0.989599\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	21689.5			0.974956
$792$		0			0
$793$	66529.2			2.97922
$794$		0			0
$795$		0			0
$796$		0			0
$797$	21233.0	+	36776.6i	0.943678	+	1.63450i	0.758377	+	0.651816i	$0.225993\pi$
$797$	21233.0	+	36776.6i	0.943678	+	1.63450i	0.185301	+	0.982682i	$0.440674\pi$
$798$		0			0
$799$	−16445.4	+	28484.2i	−0.728154	+	1.26120i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	239.957	−	415.617i	0.0105453	−	0.0182650i
$804$		0			0
$805$	5512.81	+	9548.47i	0.241368	+	0.418061i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−32955.4			−1.43220			−0.716101	−	0.697997i	$-0.754075\pi$
$809$	−32955.4			−1.43220			−0.716101	+	0.697997i	$0.754075\pi$
$810$		0			0
$811$	−3396.82			−0.147076			−0.0735379	−	0.997292i	$-0.523429\pi$
$811$	−3396.82			−0.147076			−0.0735379	+	0.997292i	$0.523429\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4389.68	+	7603.15i	0.188667	+	0.326781i
$816$		0			0
$817$	−7066.88	+	12240.2i	−0.302618	+	0.524150i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−6569.47	+	11378.7i	−0.279265	+	0.483700i	−0.971202	−	0.238257i	$-0.923424\pi$
$821$	−6569.47	+	11378.7i	−0.279265	+	0.483700i	0.691938	+	0.721957i	$0.256757\pi$
$822$		0			0
$823$	−12322.6	−	21343.4i	−0.521918	−	0.903989i	−0.999675	−	0.0254966i	$-0.991883\pi$
$823$	−12322.6	−	21343.4i	−0.521918	−	0.903989i	0.477757	−	0.878492i	$-0.341450\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	14508.9			0.610064			0.305032	−	0.952342i	$-0.401333\pi$
$827$	14508.9			0.610064			0.305032	+	0.952342i	$0.401333\pi$
$828$		0			0
$829$	28408.6			1.19019			0.595097	−	0.803654i	$-0.297113\pi$
$829$	28408.6			1.19019			0.595097	+	0.803654i	$0.297113\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−7601.94	−	13166.9i	−0.316196	−	0.547668i
$834$		0			0
$835$	−6968.89	+	12070.5i	−0.288824	+	0.500258i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−7474.98	+	12947.1i	−0.307587	+	0.532756i	−0.977834	−	0.209382i	$-0.932855\pi$
$839$	−7474.98	+	12947.1i	−0.307587	+	0.532756i	0.670247	+	0.742138i	$0.266188\pi$
$840$		0			0
$841$	8302.67	+	14380.7i	0.340427	+	0.589637i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−17696.7			−0.720456
$846$		0			0
$847$	−1575.56			−0.0639161
$848$		0			0
$849$		0			0
$850$		0			0
$851$	24736.4	+	42844.7i	0.996419	+	1.72585i
$852$		0			0
$853$	23927.7	−	41444.0i	0.960456	−	1.66356i	0.239099	−	0.970995i	$-0.423148\pi$
$853$	23927.7	−	41444.0i	0.960456	−	1.66356i	0.721357	−	0.692563i	$-0.243519\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−12969.1	+	22463.2i	−0.516939	+	0.895364i	0.482868	+	0.875693i	$0.339595\pi$
$857$	−12969.1	+	22463.2i	−0.516939	+	0.895364i	−0.999807	+	0.0196710i	$0.993738\pi$
$858$		0			0
$859$	3797.40	+	6577.29i	0.150833	+	0.261250i	0.931534	−	0.363654i	$-0.118471\pi$
$859$	3797.40	+	6577.29i	0.150833	+	0.261250i	−0.780701	+	0.624905i	$0.785138\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	20686.0			0.815943			0.407972	−	0.912995i	$-0.366236\pi$
$863$	20686.0			0.815943			0.407972	+	0.912995i	$0.366236\pi$
$864$		0			0
$865$	6081.81			0.239061
$866$		0			0
$867$		0			0
$868$		0			0
$869$	9960.40	+	17251.9i	0.388819	+	0.673453i
$870$		0			0
$871$	−39285.8	+	68045.0i	−1.52830	+	2.64709i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	796.733	−	1379.98i	0.0307823	−	0.0533165i
$876$		0			0
$877$	20011.5	+	34661.0i	0.770515	+	1.33457i	0.937281	+	0.348574i	$0.113334\pi$
$877$	20011.5	+	34661.0i	0.770515	+	1.33457i	−0.166766	+	0.985996i	$0.553333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−21863.6			−0.836100			−0.418050	−	0.908424i	$-0.637286\pi$
$881$	−21863.6			−0.836100			−0.418050	+	0.908424i	$0.637286\pi$
$882$		0			0
$883$	25239.1			0.961906			0.480953	−	0.876746i	$-0.340291\pi$
$883$	25239.1			0.961906			0.480953	+	0.876746i	$0.340291\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−21080.2	−	36511.9i	−0.797974	−	1.38213i	−0.920933	−	0.389720i	$-0.872572\pi$
$887$	−21080.2	−	36511.9i	−0.797974	−	1.38213i	0.122959	−	0.992412i	$-0.460762\pi$
$888$		0			0
$889$	−2201.56	+	3813.22i	−0.0830574	+	0.143860i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−16350.4	+	28319.7i	−0.612705	+	1.06124i
$894$		0			0
$895$	6329.94	+	10963.8i	0.236410	+	0.409474i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	18148.9			0.673303
$900$		0			0
$901$	−7723.69			−0.285587
$902$		0			0
$903$		0			0
$904$		0			0
$905$	8612.05	+	14916.5i	0.316325	+	0.547891i
$906$		0			0
$907$	4645.50	−	8046.23i	0.170067	−	0.294565i	−0.768376	−	0.639999i	$-0.778935\pi$
$907$	4645.50	−	8046.23i	0.170067	−	0.294565i	0.938443	+	0.345434i	$0.112268\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	24224.4	−	41957.9i	0.881000	−	1.52594i	0.0307692	−	0.999527i	$-0.490204\pi$
$911$	24224.4	−	41957.9i	0.881000	−	1.52594i	0.850231	−	0.526410i	$-0.176462\pi$
$912$		0			0
$913$	10909.7	+	18896.1i	0.395463	+	0.684962i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	36048.9			1.29819
$918$		0			0
$919$	−39905.6			−1.43239			−0.716193	−	0.697902i	$-0.754117\pi$
$919$	−39905.6			−1.43239			−0.716193	+	0.697902i	$0.754117\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	22919.1	+	39697.1i	0.817326	+	1.41565i
$924$		0			0
$925$	3575.00	−	6192.08i	0.127076	−	0.220102i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−17244.2	+	29867.8i	−0.609002	+	1.05482i	0.382403	+	0.923996i	$0.375097\pi$
$929$	−17244.2	+	29867.8i	−0.609002	+	1.05482i	−0.991405	+	0.130827i	$0.958237\pi$
$930$		0			0
$931$	−7558.04	−	13090.9i	−0.266063	−	0.460835i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	14634.7			0.511878
$936$		0			0
$937$	−57056.3			−1.98927			−0.994635	−	0.103443i	$-0.967014\pi$
$937$	−57056.3			−1.98927			−0.994635	+	0.103443i	$0.967014\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−26236.7	−	45443.3i	−0.908919	−	1.57429i	−0.815569	−	0.578660i	$-0.803576\pi$
$941$	−26236.7	−	45443.3i	−0.908919	−	1.57429i	−0.0933496	−	0.995633i	$-0.529757\pi$
$942$		0			0
$943$	27824.3	−	48193.1i	0.960853	−	1.66425i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9531.11	+	16508.4i	−0.327053	+	0.566473i	−0.981926	−	0.189266i	$-0.939389\pi$
$947$	−9531.11	+	16508.4i	−0.327053	+	0.566473i	0.654872	+	0.755739i	$0.272722\pi$
$948$		0			0
$949$	−523.027	−	905.910i	−0.0178906	−	0.0309874i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	36107.0			1.22730			0.613652	−	0.789577i	$-0.289700\pi$
$953$	36107.0			1.22730			0.613652	+	0.789577i	$0.289700\pi$
$954$		0			0
$955$	−8512.86			−0.288450
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7826.08	+	13555.2i	0.263522	+	0.456433i
$960$		0			0
$961$	−6263.08	+	10848.0i	−0.210234	+	0.364136i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1713.72	−	2968.25i	0.0571675	−	0.0990170i
$966$		0			0
$967$	−21374.8	−	37022.2i	−0.710824	−	1.23118i	−0.964548	−	0.263907i	$-0.914989\pi$
$967$	−21374.8	−	37022.2i	−0.710824	−	1.23118i	0.253724	−	0.967277i	$-0.418345\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−42208.9			−1.39500			−0.697501	−	0.716583i	$-0.745705\pi$
$971$	−42208.9			−1.39500			−0.697501	+	0.716583i	$0.745705\pi$
$972$		0			0
$973$	−511.300			−0.0168464
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3108.76	+	5384.52i	0.101799	+	0.176322i	0.912426	−	0.409242i	$-0.134207\pi$
$977$	3108.76	+	5384.52i	0.101799	+	0.176322i	−0.810627	+	0.585563i	$0.800873\pi$
$978$		0			0
$979$	17601.6	−	30486.8i	0.574616	−	0.995264i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	20466.0	−	35448.1i	0.664053	−	1.15017i	−0.315488	−	0.948929i	$-0.602168\pi$
$983$	20466.0	−	35448.1i	0.664053	−	1.15017i	0.979541	−	0.201244i	$-0.0644984\pi$
$984$		0			0
$985$	−10907.4	−	18892.1i	−0.352831	−	0.611120i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−29193.4			−0.938622
$990$		0			0
$991$	13878.5			0.444870			0.222435	−	0.974947i	$-0.428599\pi$
$991$	13878.5			0.444870			0.222435	+	0.974947i	$0.428599\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−7300.49	−	12644.8i	−0.232604	−	0.402882i
$996$		0			0
$997$	−3366.90	+	5831.64i	−0.106952	+	0.185246i	−0.914534	−	0.404509i	$-0.867442\pi$
$997$	−3366.90	+	5831.64i	−0.106952	+	0.185246i	0.807582	+	0.589755i	$0.200776\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.i.r.541.1		4
3.2	odd	2		1620.4.i.o.541.1		4
9.2	odd	6		540.4.a.h.1.2	yes	2
9.4	even	3	inner	1620.4.i.r.1081.1		4
9.5	odd	6		1620.4.i.o.1081.1		4
9.7	even	3		540.4.a.e.1.2	✓	2
36.7	odd	6		2160.4.a.x.1.1		2
36.11	even	6		2160.4.a.bc.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.4.a.e.1.2	✓	2	9.7	even	3
540.4.a.h.1.2	yes	2	9.2	odd	6
1620.4.i.o.541.1		4	3.2	odd	2
1620.4.i.o.1081.1		4	9.5	odd	6
1620.4.i.r.541.1		4	1.1	even	1	trivial
1620.4.i.r.1081.1		4	9.4	even	3	inner
2160.4.a.x.1.1		2	36.7	odd	6
2160.4.a.bc.1.1		2	36.11	even	6

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.50000 + 4.33013i) q^{5} +(-6.37386 + 11.0399i) q^{7} +O(q^{10})\)	\(q+(2.50000 + 4.33013i) q^{5} +(-6.37386 + 11.0399i) q^{7} +(-17.3739 + 30.0924i) q^{11} +(37.8693 + 65.5916i) q^{13} -84.2341 q^{17} -83.7477 q^{19} +(-86.4909 - 149.807i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(-44.1125 + 76.4051i) q^{29} +(-102.856 - 178.151i) q^{31} -63.7386 q^{35} -286.000 q^{37} +(160.851 + 278.602i) q^{41} +(84.3830 - 146.156i) q^{43} +(195.234 - 338.155i) q^{47} +(90.2477 + 156.314i) q^{49} +91.6932 q^{53} -173.739 q^{55} +(-61.3830 - 106.318i) q^{59} +(439.202 - 760.721i) q^{61} +(-189.347 + 327.958i) q^{65} +(518.702 + 898.419i) q^{67} +605.216 q^{71} -13.8114 q^{73} +(-221.477 - 383.610i) q^{77} +(286.649 - 496.490i) q^{79} +(313.968 - 543.809i) q^{83} +(-210.585 - 364.744i) q^{85} -1013.11 q^{89} -965.495 q^{91} +(-209.369 - 362.638i) q^{95} +(-577.909 + 1000.97i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{5} + 2 q^{7}+O(q^{10}) \) 4 * q + 10 * q^5 + 2 * q^7	\( 4 q + 10 q^{5} + 2 q^{7} - 42 q^{11} + 14 q^{13} + 48 q^{17} - 280 q^{19} - 126 q^{23} - 50 q^{25} + 126 q^{29} + 56 q^{31} + 20 q^{35} - 1144 q^{37} + 66 q^{41} + 530 q^{43} + 396 q^{47} + 306 q^{49} - 1008 q^{53} - 420 q^{55} - 438 q^{59} + 602 q^{61} - 70 q^{65} + 920 q^{67} + 1596 q^{71} - 1540 q^{73} - 336 q^{77} + 1724 q^{79} + 486 q^{83} + 120 q^{85} - 588 q^{89} - 3752 q^{91} - 700 q^{95} - 112 q^{97}+O(q^{100}) \) 4 * q + 10 * q^5 + 2 * q^7 - 42 * q^11 + 14 * q^13 + 48 * q^17 - 280 * q^19 - 126 * q^23 - 50 * q^25 + 126 * q^29 + 56 * q^31 + 20 * q^35 - 1144 * q^37 + 66 * q^41 + 530 * q^43 + 396 * q^47 + 306 * q^49 - 1008 * q^53 - 420 * q^55 - 438 * q^59 + 602 * q^61 - 70 * q^65 + 920 * q^67 + 1596 * q^71 - 1540 * q^73 - 336 * q^77 + 1724 * q^79 + 486 * q^83 + 120 * q^85 - 588 * q^89 - 3752 * q^91 - 700 * q^95 - 112 * q^97

Introduction

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