LMFDB - Embedded newform 90.4.c.a.19.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [90,4,Mod(19,90)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(90, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("90.19");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	90.c (of order $2$, degree $1$, 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:	$5.31017190052$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	90.19
Dual form			90.4.c.a.19.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -4.00000 q^{4} +(-2.00000 + 11.0000i) q^{5} +2.00000i q^{7} -8.00000i q^{8} +O(q^{10})$	\(q+2.00000i q^{2} -4.00000 q^{4} +(-2.00000 + 11.0000i) q^{5} +2.00000i q^{7} -8.00000i q^{8} +(-22.0000 - 4.00000i) q^{10} -70.0000 q^{11} +54.0000i q^{13} -4.00000 q^{14} +16.0000 q^{16} -22.0000i q^{17} -24.0000 q^{19} +(8.00000 - 44.0000i) q^{20} -140.000i q^{22} +100.000i q^{23} +(-117.000 - 44.0000i) q^{25} -108.000 q^{26} -8.00000i q^{28} +216.000 q^{29} +208.000 q^{31} +32.0000i q^{32} +44.0000 q^{34} +(-22.0000 - 4.00000i) q^{35} +254.000i q^{37} -48.0000i q^{38} +(88.0000 + 16.0000i) q^{40} +206.000 q^{41} +292.000i q^{43} +280.000 q^{44} -200.000 q^{46} -320.000i q^{47} +339.000 q^{49} +(88.0000 - 234.000i) q^{50} -216.000i q^{52} +402.000i q^{53} +(140.000 - 770.000i) q^{55} +16.0000 q^{56} +432.000i q^{58} -370.000 q^{59} -550.000 q^{61} +416.000i q^{62} -64.0000 q^{64} +(-594.000 - 108.000i) q^{65} -728.000i q^{67} +88.0000i q^{68} +(8.00000 - 44.0000i) q^{70} +540.000 q^{71} +604.000i q^{73} -508.000 q^{74} +96.0000 q^{76} -140.000i q^{77} -792.000 q^{79} +(-32.0000 + 176.000i) q^{80} +412.000i q^{82} -404.000i q^{83} +(242.000 + 44.0000i) q^{85} -584.000 q^{86} +560.000i q^{88} -938.000 q^{89} -108.000 q^{91} -400.000i q^{92} +640.000 q^{94} +(48.0000 - 264.000i) q^{95} -56.0000i q^{97} +678.000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 4 q^{5}+O(q^{10}) $ 2 * q - 8 * q^4 - 4 * q^5	$ 2 q - 8 q^{4} - 4 q^{5} - 44 q^{10} - 140 q^{11} - 8 q^{14} + 32 q^{16} - 48 q^{19} + 16 q^{20} - 234 q^{25} - 216 q^{26} + 432 q^{29} + 416 q^{31} + 88 q^{34} - 44 q^{35} + 176 q^{40} + 412 q^{41} + 560 q^{44} - 400 q^{46} + 678 q^{49} + 176 q^{50} + 280 q^{55} + 32 q^{56} - 740 q^{59} - 1100 q^{61} - 128 q^{64} - 1188 q^{65} + 16 q^{70} + 1080 q^{71} - 1016 q^{74} + 192 q^{76} - 1584 q^{79} - 64 q^{80} + 484 q^{85} - 1168 q^{86} - 1876 q^{89} - 216 q^{91} + 1280 q^{94} + 96 q^{95}+O(q^{100}) $ 2 * q - 8 * q^4 - 4 * q^5 - 44 * q^10 - 140 * q^11 - 8 * q^14 + 32 * q^16 - 48 * q^19 + 16 * q^20 - 234 * q^25 - 216 * q^26 + 432 * q^29 + 416 * q^31 + 88 * q^34 - 44 * q^35 + 176 * q^40 + 412 * q^41 + 560 * q^44 - 400 * q^46 + 678 * q^49 + 176 * q^50 + 280 * q^55 + 32 * q^56 - 740 * q^59 - 1100 * q^61 - 128 * q^64 - 1188 * q^65 + 16 * q^70 + 1080 * q^71 - 1016 * q^74 + 192 * q^76 - 1584 * q^79 - 64 * q^80 + 484 * q^85 - 1168 * q^86 - 1876 * q^89 - 216 * q^91 + 1280 * q^94 + 96 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/90\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			2.00000i			0.707107i
$2$			2.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−4.00000			−0.500000
$5$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$5$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$6$		0			0
$7$			2.00000i			0.107990i	0.998541	+	0.0539949i	$0.0171955\pi$
$7$			2.00000i			0.107990i	−0.998541	+	0.0539949i	$0.982805\pi$
$8$		−	8.00000i		−	0.353553i
$9$		0			0
$10$	−22.0000	−	4.00000i	−0.695701	−	0.126491i
$11$	−70.0000			−1.91871			−0.959354	−	0.282204i	$-0.908934\pi$
$11$	−70.0000			−1.91871			−0.959354	+	0.282204i	$0.908934\pi$
$12$		0			0
$13$			54.0000i			1.15207i	0.817425	+	0.576035i	$0.195401\pi$
$13$			54.0000i			1.15207i	−0.817425	+	0.576035i	$0.804599\pi$
$14$	−4.00000			−0.0763604
$15$		0			0
$16$	16.0000			0.250000
$17$		−	22.0000i		−	0.313870i	−0.987609	−	0.156935i	$-0.949839\pi$
$17$		−	22.0000i		−	0.313870i	0.987609	−	0.156935i	$-0.0501613\pi$
$18$		0			0
$19$	−24.0000			−0.289788			−0.144894	−	0.989447i	$-0.546284\pi$
$19$	−24.0000			−0.289788			−0.144894	+	0.989447i	$0.546284\pi$
$20$	8.00000	−	44.0000i	0.0894427	−	0.491935i
$21$		0			0
$22$		−	140.000i		−	1.35673i
$23$			100.000i			0.906584i	0.891362	+	0.453292i	$0.149751\pi$
$23$			100.000i			0.906584i	−0.891362	+	0.453292i	$0.850249\pi$
$24$		0			0
$25$	−117.000	−	44.0000i	−0.936000	−	0.352000i
$26$	−108.000			−0.814636
$27$		0			0
$28$		−	8.00000i		−	0.0539949i
$29$	216.000			1.38311			0.691555	−	0.722324i	$-0.256926\pi$
$29$	216.000			1.38311			0.691555	+	0.722324i	$0.256926\pi$
$30$		0			0
$31$	208.000			1.20509			0.602547	−	0.798084i	$-0.294153\pi$
$31$	208.000			1.20509			0.602547	+	0.798084i	$0.294153\pi$
$32$			32.0000i			0.176777i
$33$		0			0
$34$	44.0000			0.221939
$35$	−22.0000	−	4.00000i	−0.106248	−	0.0193178i
$36$		0			0
$37$			254.000i			1.12858i	0.825578	+	0.564288i	$0.190849\pi$
$37$			254.000i			1.12858i	−0.825578	+	0.564288i	$0.809151\pi$
$38$		−	48.0000i		−	0.204911i
$39$		0			0
$40$	88.0000	+	16.0000i	0.347851	+	0.0632456i
$41$	206.000			0.784678			0.392339	−	0.919821i	$-0.371666\pi$
$41$	206.000			0.784678			0.392339	+	0.919821i	$0.371666\pi$
$42$		0			0
$43$			292.000i			1.03557i	0.855510	+	0.517786i	$0.173244\pi$
$43$			292.000i			1.03557i	−0.855510	+	0.517786i	$0.826756\pi$
$44$	280.000			0.959354
$45$		0			0
$46$	−200.000			−0.641052
$47$		−	320.000i		−	0.993123i	−0.868001	−	0.496562i	$-0.834596\pi$
$47$		−	320.000i		−	0.993123i	0.868001	−	0.496562i	$-0.165404\pi$
$48$		0			0
$49$	339.000			0.988338
$50$	88.0000	−	234.000i	0.248902	−	0.661852i
$51$		0			0
$52$		−	216.000i		−	0.576035i
$53$			402.000i			1.04187i	0.853597	+	0.520933i	$0.174416\pi$
$53$			402.000i			1.04187i	−0.853597	+	0.520933i	$0.825584\pi$
$54$		0			0
$55$	140.000	−	770.000i	0.343229	−	1.88776i
$56$	16.0000			0.0381802
$57$		0			0
$58$			432.000i			0.978007i
$59$	−370.000			−0.816439			−0.408219	−	0.912884i	$-0.633850\pi$
$59$	−370.000			−0.816439			−0.408219	+	0.912884i	$0.633850\pi$
$60$		0			0
$61$	−550.000			−1.15443			−0.577215	−	0.816592i	$-0.695861\pi$
$61$	−550.000			−1.15443			−0.577215	+	0.816592i	$0.695861\pi$
$62$			416.000i			0.852130i
$63$		0			0
$64$	−64.0000			−0.125000
$65$	−594.000	−	108.000i	−1.13349	−	0.206088i
$66$		0			0
$67$		−	728.000i		−	1.32745i	−0.747975	−	0.663727i	$-0.768974\pi$
$67$		−	728.000i		−	1.32745i	0.747975	−	0.663727i	$-0.231026\pi$
$68$			88.0000i			0.156935i
$69$		0			0
$70$	8.00000	−	44.0000i	0.0136598	−	0.0751287i
$71$	540.000			0.902623			0.451311	−	0.892367i	$-0.350956\pi$
$71$	540.000			0.902623			0.451311	+	0.892367i	$0.350956\pi$
$72$		0			0
$73$			604.000i			0.968395i	0.874959	+	0.484198i	$0.160888\pi$
$73$			604.000i			0.968395i	−0.874959	+	0.484198i	$0.839112\pi$
$74$	−508.000			−0.798024
$75$		0			0
$76$	96.0000			0.144894
$77$		−	140.000i		−	0.207201i
$78$		0			0
$79$	−792.000			−1.12794			−0.563968	−	0.825797i	$-0.690726\pi$
$79$	−792.000			−1.12794			−0.563968	+	0.825797i	$0.690726\pi$
$80$	−32.0000	+	176.000i	−0.0447214	+	0.245967i
$81$		0			0
$82$			412.000i			0.554851i
$83$		−	404.000i		−	0.534274i	−0.963659	−	0.267137i	$-0.913922\pi$
$83$		−	404.000i		−	0.534274i	0.963659	−	0.267137i	$-0.0860777\pi$
$84$		0			0
$85$	242.000	+	44.0000i	0.308807	+	0.0561467i
$86$	−584.000			−0.732260
$87$		0			0
$88$			560.000i			0.678366i
$89$	−938.000			−1.11717			−0.558583	−	0.829449i	$-0.688655\pi$
$89$	−938.000			−1.11717			−0.558583	+	0.829449i	$0.688655\pi$
$90$		0			0
$91$	−108.000			−0.124412
$92$		−	400.000i		−	0.453292i
$93$		0			0
$94$	640.000			0.702244
$95$	48.0000	−	264.000i	0.0518389	−	0.285114i
$96$		0			0
$97$		−	56.0000i		−	0.0586179i	−0.999570	−	0.0293090i	$-0.990669\pi$
$97$		−	56.0000i		−	0.0586179i	0.999570	−	0.0293090i	$-0.00933067\pi$
$98$			678.000i			0.698861i
$99$		0			0
$100$	468.000	+	176.000i	0.468000	+	0.176000i
$101$	592.000			0.583230			0.291615	−	0.956536i	$-0.405807\pi$
$101$	592.000			0.583230			0.291615	+	0.956536i	$0.405807\pi$
$102$		0			0
$103$			62.0000i			0.0593111i	0.999560	+	0.0296555i	$0.00944104\pi$
$103$			62.0000i			0.0593111i	−0.999560	+	0.0296555i	$0.990559\pi$
$104$	432.000			0.407318
$105$		0			0
$106$	−804.000			−0.736711
$107$			84.0000i			0.0758933i	0.999280	+	0.0379467i	$0.0120817\pi$
$107$			84.0000i			0.0758933i	−0.999280	+	0.0379467i	$0.987918\pi$
$108$		0			0
$109$	−370.000			−0.325134			−0.162567	−	0.986698i	$-0.551977\pi$
$109$	−370.000			−0.325134			−0.162567	+	0.986698i	$0.551977\pi$
$110$	1540.00	+	280.000i	1.33485	+	0.242700i
$111$		0			0
$112$			32.0000i			0.0269975i
$113$			1746.00i			1.45354i	0.686882	+	0.726769i	$0.258979\pi$
$113$			1746.00i			1.45354i	−0.686882	+	0.726769i	$0.741021\pi$
$114$		0			0
$115$	−1100.00	−	200.000i	−0.891961	−	0.162175i
$116$	−864.000			−0.691555
$117$		0			0
$118$		−	740.000i		−	0.577310i
$119$	44.0000			0.0338947
$120$		0			0
$121$	3569.00			2.68144
$122$		−	1100.00i		−	0.816306i
$123$		0			0
$124$	−832.000			−0.602547
$125$	718.000	−	1199.00i	0.513759	−	0.857935i
$126$		0			0
$127$		−	1630.00i		−	1.13889i	−0.822029	−	0.569445i	$-0.807158\pi$
$127$		−	1630.00i		−	1.13889i	0.822029	−	0.569445i	$-0.192842\pi$
$128$		−	128.000i		−	0.0883883i
$129$		0			0
$130$	216.000	−	1188.00i	0.145727	−	0.801496i
$131$	870.000			0.580246			0.290123	−	0.956989i	$-0.406304\pi$
$131$	870.000			0.580246			0.290123	+	0.956989i	$0.406304\pi$
$132$		0			0
$133$		−	48.0000i		−	0.0312942i
$134$	1456.00			0.938651
$135$		0			0
$136$	−176.000			−0.110970
$137$			918.000i			0.572482i	0.958158	+	0.286241i	$0.0924058\pi$
$137$			918.000i			0.572482i	−0.958158	+	0.286241i	$0.907594\pi$
$138$		0			0
$139$	596.000			0.363684			0.181842	−	0.983328i	$-0.441794\pi$
$139$	596.000			0.363684			0.181842	+	0.983328i	$0.441794\pi$
$140$	88.0000	+	16.0000i	0.0531240	+	0.00965891i
$141$		0			0
$142$			1080.00i			0.638251i
$143$		−	3780.00i		−	2.21049i
$144$		0			0
$145$	−432.000	+	2376.00i	−0.247418	+	1.36080i
$146$	−1208.00			−0.684759
$147$		0			0
$148$		−	1016.00i		−	0.564288i
$149$	1076.00			0.591606			0.295803	−	0.955249i	$-0.404413\pi$
$149$	1076.00			0.591606			0.295803	+	0.955249i	$0.404413\pi$
$150$		0			0
$151$	−32.0000			−0.0172458			−0.00862292	−	0.999963i	$-0.502745\pi$
$151$	−32.0000			−0.0172458			−0.00862292	+	0.999963i	$0.502745\pi$
$152$			192.000i			0.102456i
$153$		0			0
$154$	280.000			0.146513
$155$	−416.000	+	2288.00i	−0.215574	+	1.18566i
$156$		0			0
$157$			2554.00i			1.29829i	0.760665	+	0.649145i	$0.224873\pi$
$157$			2554.00i			1.29829i	−0.760665	+	0.649145i	$0.775127\pi$
$158$		−	1584.00i		−	0.797571i
$159$		0			0
$160$	−352.000	−	64.0000i	−0.173925	−	0.0316228i
$161$	−200.000			−0.0979019
$162$		0			0
$163$			752.000i			0.361357i	0.983542	+	0.180678i	$0.0578293\pi$
$163$			752.000i			0.361357i	−0.983542	+	0.180678i	$0.942171\pi$
$164$	−824.000			−0.392339
$165$		0			0
$166$	808.000			0.377789
$167$			2700.00i			1.25109i	0.780188	+	0.625546i	$0.215124\pi$
$167$			2700.00i			1.25109i	−0.780188	+	0.625546i	$0.784876\pi$
$168$		0			0
$169$	−719.000			−0.327264
$170$	−88.0000	+	484.000i	−0.0397017	+	0.218359i
$171$		0			0
$172$		−	1168.00i		−	0.517786i
$173$			1334.00i			0.586255i	0.956073	+	0.293128i	$0.0946961\pi$
$173$			1334.00i			0.586255i	−0.956073	+	0.293128i	$0.905304\pi$
$174$		0			0
$175$	88.0000	−	234.000i	0.0380124	−	0.101078i
$176$	−1120.00			−0.479677
$177$		0			0
$178$		−	1876.00i		−	0.789956i
$179$	−1714.00			−0.715700			−0.357850	−	0.933779i	$-0.616490\pi$
$179$	−1714.00			−0.715700			−0.357850	+	0.933779i	$0.616490\pi$
$180$		0			0
$181$	−4006.00			−1.64510			−0.822551	−	0.568691i	$-0.807450\pi$
$181$	−4006.00			−1.64510			−0.822551	+	0.568691i	$0.807450\pi$
$182$		−	216.000i		−	0.0879724i
$183$		0			0
$184$	800.000			0.320526
$185$	−2794.00	−	508.000i	−1.11037	−	0.201886i
$186$		0			0
$187$			1540.00i			0.602224i
$188$			1280.00i			0.496562i
$189$		0			0
$190$	528.000	+	96.0000i	0.201606	+	0.0366556i
$191$	684.000			0.259123			0.129562	−	0.991571i	$-0.458643\pi$
$191$	684.000			0.259123			0.129562	+	0.991571i	$0.458643\pi$
$192$		0			0
$193$		−	4484.00i		−	1.67236i	−0.548455	−	0.836180i	$-0.684784\pi$
$193$		−	4484.00i		−	1.67236i	0.548455	−	0.836180i	$-0.315216\pi$
$194$	112.000			0.0414491
$195$		0			0
$196$	−1356.00			−0.494169
$197$		−	1058.00i		−	0.382636i	−0.981528	−	0.191318i	$-0.938724\pi$
$197$		−	1058.00i		−	0.382636i	0.981528	−	0.191318i	$-0.0612762\pi$
$198$		0			0
$199$	1128.00			0.401818			0.200909	−	0.979610i	$-0.435610\pi$
$199$	1128.00			0.401818			0.200909	+	0.979610i	$0.435610\pi$
$200$	−352.000	+	936.000i	−0.124451	+	0.330926i
$201$		0			0
$202$			1184.00i			0.412406i
$203$			432.000i			0.149362i
$204$		0			0
$205$	−412.000	+	2266.00i	−0.140367	+	0.772021i
$206$	−124.000			−0.0419393
$207$		0			0
$208$			864.000i			0.288017i
$209$	1680.00			0.556019
$210$		0			0
$211$	780.000			0.254490			0.127245	−	0.991871i	$-0.459387\pi$
$211$	780.000			0.254490			0.127245	+	0.991871i	$0.459387\pi$
$212$		−	1608.00i		−	0.520933i
$213$		0			0
$214$	−168.000			−0.0536647
$215$	−3212.00	−	584.000i	−1.01887	−	0.185249i
$216$		0			0
$217$			416.000i			0.130138i
$218$		−	740.000i		−	0.229904i
$219$		0			0
$220$	−560.000	+	3080.00i	−0.171615	+	0.943880i
$221$	1188.00			0.361600
$222$		0			0
$223$		−	2570.00i		−	0.771749i	−0.922551	−	0.385874i	$-0.873900\pi$
$223$		−	2570.00i		−	0.771749i	0.922551	−	0.385874i	$-0.126100\pi$
$224$	−64.0000			−0.0190901
$225$		0			0
$226$	−3492.00			−1.02781
$227$		−	2836.00i		−	0.829216i	−0.910000	−	0.414608i	$-0.863919\pi$
$227$		−	2836.00i		−	0.829216i	0.910000	−	0.414608i	$-0.136081\pi$
$228$		0			0
$229$	610.000			0.176026			0.0880130	−	0.996119i	$-0.471948\pi$
$229$	610.000			0.176026			0.0880130	+	0.996119i	$0.471948\pi$
$230$	400.000	−	2200.00i	0.114675	−	0.630712i
$231$		0			0
$232$		−	1728.00i		−	0.489003i
$233$		−	3514.00i		−	0.988025i	−0.869455	−	0.494012i	$-0.835530\pi$
$233$		−	3514.00i		−	0.988025i	0.869455	−	0.494012i	$-0.164470\pi$
$234$		0			0
$235$	3520.00	+	640.000i	0.977104	+	0.177655i
$236$	1480.00			0.408219
$237$		0			0
$238$			88.0000i			0.0239672i
$239$	−1844.00			−0.499073			−0.249536	−	0.968365i	$-0.580278\pi$
$239$	−1844.00			−0.499073			−0.249536	+	0.968365i	$0.580278\pi$
$240$		0			0
$241$	982.000			0.262474			0.131237	−	0.991351i	$-0.458105\pi$
$241$	982.000			0.262474			0.131237	+	0.991351i	$0.458105\pi$
$242$			7138.00i			1.89607i
$243$		0			0
$244$	2200.00			0.577215
$245$	−678.000	+	3729.00i	−0.176799	+	0.972396i
$246$		0			0
$247$		−	1296.00i		−	0.333856i
$248$		−	1664.00i		−	0.426065i
$249$		0			0
$250$	2398.00	+	1436.00i	0.606651	+	0.363282i
$251$	3174.00			0.798172			0.399086	−	0.916914i	$-0.369328\pi$
$251$	3174.00			0.798172			0.399086	+	0.916914i	$0.369328\pi$
$252$		0			0
$253$		−	7000.00i		−	1.73947i
$254$	3260.00			0.805317
$255$		0			0
$256$	256.000			0.0625000
$257$		−	1194.00i		−	0.289804i	−0.989446	−	0.144902i	$-0.953713\pi$
$257$		−	1194.00i		−	0.289804i	0.989446	−	0.144902i	$-0.0462867\pi$
$258$		0			0
$259$	−508.000			−0.121875
$260$	2376.00	+	432.000i	0.566743	+	0.103044i
$261$		0			0
$262$			1740.00i			0.410296i
$263$		−	140.000i		−	0.0328242i	−0.999865	−	0.0164121i	$-0.994776\pi$
$263$		−	140.000i		−	0.0328242i	0.999865	−	0.0164121i	$-0.00522437\pi$
$264$		0			0
$265$	−4422.00	−	804.000i	−1.02506	−	0.186375i
$266$	96.0000			0.0221283
$267$		0			0
$268$			2912.00i			0.663727i
$269$	5256.00			1.19132			0.595658	−	0.803238i	$-0.296891\pi$
$269$	5256.00			1.19132			0.595658	+	0.803238i	$0.296891\pi$
$270$		0			0
$271$	544.000			0.121940			0.0609698	−	0.998140i	$-0.480581\pi$
$271$	544.000			0.121940			0.0609698	+	0.998140i	$0.480581\pi$
$272$		−	352.000i		−	0.0784674i
$273$		0			0
$274$	−1836.00			−0.404806
$275$	8190.00	+	3080.00i	1.79591	+	0.675385i
$276$		0			0
$277$		−	946.000i		−	0.205197i	−0.994723	−	0.102599i	$-0.967284\pi$
$277$		−	946.000i		−	0.205197i	0.994723	−	0.102599i	$-0.0327157\pi$
$278$			1192.00i			0.257163i
$279$		0			0
$280$	−32.0000	+	176.000i	−0.00682988	+	0.0375643i
$281$	−1278.00			−0.271313			−0.135657	−	0.990756i	$-0.543314\pi$
$281$	−1278.00			−0.271313			−0.135657	+	0.990756i	$0.543314\pi$
$282$		0			0
$283$			7424.00i			1.55940i	0.626152	+	0.779701i	$0.284629\pi$
$283$			7424.00i			1.55940i	−0.626152	+	0.779701i	$0.715371\pi$
$284$	−2160.00			−0.451311
$285$		0			0
$286$	7560.00			1.56305
$287$			412.000i			0.0847373i
$288$		0			0
$289$	4429.00			0.901486
$290$	−4752.00	−	864.000i	−0.962231	−	0.174951i
$291$		0			0
$292$		−	2416.00i		−	0.484198i
$293$			1362.00i			0.271566i	0.990739	+	0.135783i	$0.0433550\pi$
$293$			1362.00i			0.271566i	−0.990739	+	0.135783i	$0.956645\pi$
$294$		0			0
$295$	740.000	−	4070.00i	0.146049	−	0.803270i
$296$	2032.00			0.399012
$297$		0			0
$298$			2152.00i			0.418329i
$299$	−5400.00			−1.04445
$300$		0			0
$301$	−584.000			−0.111831
$302$		−	64.0000i		−	0.0121947i
$303$		0			0
$304$	−384.000			−0.0724471
$305$	1100.00	−	6050.00i	0.206511	−	1.13581i
$306$		0			0
$307$			7740.00i			1.43891i	0.694539	+	0.719455i	$0.255608\pi$
$307$			7740.00i			1.43891i	−0.694539	+	0.719455i	$0.744392\pi$
$308$			560.000i			0.103601i
$309$		0			0
$310$	−4576.00	−	832.000i	−0.838385	−	0.152434i
$311$	−4980.00			−0.908006			−0.454003	−	0.891000i	$-0.650004\pi$
$311$	−4980.00			−0.908006			−0.454003	+	0.891000i	$0.650004\pi$
$312$		0			0
$313$			604.000i			0.109074i	0.998512	+	0.0545369i	$0.0173683\pi$
$313$			604.000i			0.109074i	−0.998512	+	0.0545369i	$0.982632\pi$
$314$	−5108.00			−0.918029
$315$		0			0
$316$	3168.00			0.563968
$317$		−	8566.00i		−	1.51771i	−0.651259	−	0.758856i	$-0.725759\pi$
$317$		−	8566.00i		−	1.51771i	0.651259	−	0.758856i	$-0.274241\pi$
$318$		0			0
$319$	−15120.0			−2.65379
$320$	128.000	−	704.000i	0.0223607	−	0.122984i
$321$		0			0
$322$		−	400.000i		−	0.0692271i
$323$			528.000i			0.0909557i
$324$		0			0
$325$	2376.00	−	6318.00i	0.405529	−	1.07834i
$326$	−1504.00			−0.255518
$327$		0			0
$328$		−	1648.00i		−	0.277426i
$329$	640.000			0.107247
$330$		0			0
$331$	3472.00			0.576551			0.288275	−	0.957548i	$-0.406918\pi$
$331$	3472.00			0.576551			0.288275	+	0.957548i	$0.406918\pi$
$332$			1616.00i			0.267137i
$333$		0			0
$334$	−5400.00			−0.884655
$335$	8008.00	+	1456.00i	1.30604	+	0.237462i
$336$		0			0
$337$		−	5668.00i		−	0.916189i	−0.888904	−	0.458094i	$-0.848532\pi$
$337$		−	5668.00i		−	0.916189i	0.888904	−	0.458094i	$-0.151468\pi$
$338$		−	1438.00i		−	0.231411i
$339$		0			0
$340$	−968.000	−	176.000i	−0.154403	−	0.0280734i
$341$	−14560.0			−2.31222
$342$		0			0
$343$			1364.00i			0.214720i
$344$	2336.00			0.366130
$345$		0			0
$346$	−2668.00			−0.414545
$347$			10836.0i			1.67639i	0.545371	+	0.838194i	$0.316389\pi$
$347$			10836.0i			1.67639i	−0.545371	+	0.838194i	$0.683611\pi$
$348$		0			0
$349$	8990.00			1.37886			0.689432	−	0.724350i	$-0.257860\pi$
$349$	8990.00			1.37886			0.689432	+	0.724350i	$0.257860\pi$
$350$	468.000	+	176.000i	0.0714733	+	0.0268788i
$351$		0			0
$352$		−	2240.00i		−	0.339183i
$353$			5078.00i			0.765651i	0.923821	+	0.382825i	$0.125049\pi$
$353$			5078.00i			0.765651i	−0.923821	+	0.382825i	$0.874951\pi$
$354$		0			0
$355$	−1080.00	+	5940.00i	−0.161466	+	0.888063i
$356$	3752.00			0.558583
$357$		0			0
$358$		−	3428.00i		−	0.506077i
$359$	−3696.00			−0.543363			−0.271682	−	0.962387i	$-0.587580\pi$
$359$	−3696.00			−0.543363			−0.271682	+	0.962387i	$0.587580\pi$
$360$		0			0
$361$	−6283.00			−0.916023
$362$		−	8012.00i		−	1.16326i
$363$		0			0
$364$	432.000			0.0622059
$365$	−6644.00	−	1208.00i	−0.952775	−	0.173232i
$366$		0			0
$367$		−	286.000i		−	0.0406787i	−0.999793	−	0.0203393i	$-0.993525\pi$
$367$		−	286.000i		−	0.0406787i	0.999793	−	0.0203393i	$-0.00647466\pi$
$368$			1600.00i			0.226646i
$369$		0			0
$370$	1016.00	−	5588.00i	0.142755	−	0.785152i
$371$	−804.000			−0.112511
$372$		0			0
$373$			8262.00i			1.14689i	0.819244	+	0.573445i	$0.194393\pi$
$373$			8262.00i			1.14689i	−0.819244	+	0.573445i	$0.805607\pi$
$374$	−3080.00			−0.425837
$375$		0			0
$376$	−2560.00			−0.351122
$377$			11664.0i			1.59344i
$378$		0			0
$379$	2956.00			0.400632			0.200316	−	0.979731i	$-0.435803\pi$
$379$	2956.00			0.400632			0.200316	+	0.979731i	$0.435803\pi$
$380$	−192.000	+	1056.00i	−0.0259195	+	0.142557i
$381$		0			0
$382$			1368.00i			0.183228i
$383$			5240.00i			0.699090i	0.936920	+	0.349545i	$0.113664\pi$
$383$			5240.00i			0.699090i	−0.936920	+	0.349545i	$0.886336\pi$
$384$		0			0
$385$	1540.00	+	280.000i	0.203859	+	0.0370653i
$386$	8968.00			1.18254
$387$		0			0
$388$			224.000i			0.0293090i
$389$	−884.000			−0.115220			−0.0576100	−	0.998339i	$-0.518348\pi$
$389$	−884.000			−0.115220			−0.0576100	+	0.998339i	$0.518348\pi$
$390$		0			0
$391$	2200.00			0.284549
$392$		−	2712.00i		−	0.349430i
$393$		0			0
$394$	2116.00			0.270565
$395$	1584.00	−	8712.00i	0.201771	−	1.10974i
$396$		0			0
$397$		−	3394.00i		−	0.429068i	−0.976717	−	0.214534i	$-0.931177\pi$
$397$		−	3394.00i		−	0.429068i	0.976717	−	0.214534i	$-0.0688233\pi$
$398$			2256.00i			0.284128i
$399$		0			0
$400$	−1872.00	−	704.000i	−0.234000	−	0.0880000i
$401$	6826.00			0.850060			0.425030	−	0.905179i	$-0.360263\pi$
$401$	6826.00			0.850060			0.425030	+	0.905179i	$0.360263\pi$
$402$		0			0
$403$			11232.0i			1.38835i
$404$	−2368.00			−0.291615
$405$		0			0
$406$	−864.000			−0.105615
$407$		−	17780.0i		−	2.16541i
$408$		0			0
$409$	−7814.00			−0.944688			−0.472344	−	0.881414i	$-0.656592\pi$
$409$	−7814.00			−0.944688			−0.472344	+	0.881414i	$0.656592\pi$
$410$	−4532.00	−	824.000i	−0.545901	−	0.0992548i
$411$		0			0
$412$		−	248.000i		−	0.0296555i
$413$		−	740.000i		−	0.0881671i
$414$		0			0
$415$	4444.00	+	808.000i	0.525656	+	0.0955739i
$416$	−1728.00			−0.203659
$417$		0			0
$418$			3360.00i			0.393165i
$419$	8290.00			0.966570			0.483285	−	0.875463i	$-0.339443\pi$
$419$	8290.00			0.966570			0.483285	+	0.875463i	$0.339443\pi$
$420$		0			0
$421$	2110.00			0.244264			0.122132	−	0.992514i	$-0.461027\pi$
$421$	2110.00			0.244264			0.122132	+	0.992514i	$0.461027\pi$
$422$			1560.00i			0.179952i
$423$		0			0
$424$	3216.00			0.368356
$425$	−968.000	+	2574.00i	−0.110482	+	0.293782i
$426$		0			0
$427$		−	1100.00i		−	0.124667i
$428$		−	336.000i		−	0.0379467i
$429$		0			0
$430$	1168.00	−	6424.00i	0.130991	−	0.720448i
$431$	12080.0			1.35005			0.675027	−	0.737793i	$-0.264132\pi$
$431$	12080.0			1.35005			0.675027	+	0.737793i	$0.264132\pi$
$432$		0			0
$433$			16492.0i			1.83038i	0.403022	+	0.915190i	$0.367960\pi$
$433$			16492.0i			1.83038i	−0.403022	+	0.915190i	$0.632040\pi$
$434$	−832.000			−0.0920214
$435$		0			0
$436$	1480.00			0.162567
$437$		−	2400.00i		−	0.262718i
$438$		0			0
$439$	15048.0			1.63600			0.817998	−	0.575222i	$-0.195084\pi$
$439$	15048.0			1.63600			0.817998	+	0.575222i	$0.195084\pi$
$440$	−6160.00	−	1120.00i	−0.667424	−	0.121350i
$441$		0			0
$442$			2376.00i			0.255690i
$443$			9876.00i			1.05919i	0.848249	+	0.529597i	$0.177657\pi$
$443$			9876.00i			1.05919i	−0.848249	+	0.529597i	$0.822343\pi$
$444$		0			0
$445$	1876.00	−	10318.0i	0.199845	−	1.09915i
$446$	5140.00			0.545709
$447$		0			0
$448$		−	128.000i		−	0.0134987i
$449$	17166.0			1.80426			0.902131	−	0.431462i	$-0.142002\pi$
$449$	17166.0			1.80426			0.902131	+	0.431462i	$0.142002\pi$
$450$		0			0
$451$	−14420.0			−1.50557
$452$		−	6984.00i		−	0.726769i
$453$		0			0
$454$	5672.00			0.586344
$455$	216.000	−	1188.00i	0.0222555	−	0.122405i
$456$		0			0
$457$		−	14848.0i		−	1.51983i	−0.650025	−	0.759913i	$-0.725242\pi$
$457$		−	14848.0i		−	1.51983i	0.650025	−	0.759913i	$-0.274758\pi$
$458$			1220.00i			0.124469i
$459$		0			0
$460$	4400.00	+	800.000i	0.445981	+	0.0810874i
$461$	1260.00			0.127297			0.0636486	−	0.997972i	$-0.479726\pi$
$461$	1260.00			0.127297			0.0636486	+	0.997972i	$0.479726\pi$
$462$		0			0
$463$		−	11238.0i		−	1.12802i	−0.825767	−	0.564011i	$-0.809258\pi$
$463$		−	11238.0i		−	1.12802i	0.825767	−	0.564011i	$-0.190742\pi$
$464$	3456.00			0.345778
$465$		0			0
$466$	7028.00			0.698639
$467$			14772.0i			1.46374i	0.681444	+	0.731870i	$0.261352\pi$
$467$			14772.0i			1.46374i	−0.681444	+	0.731870i	$0.738648\pi$
$468$		0			0
$469$	1456.00			0.143351
$470$	−1280.00	+	7040.00i	−0.125621	+	0.690917i
$471$		0			0
$472$			2960.00i			0.288655i
$473$		−	20440.0i		−	1.98696i
$474$		0			0
$475$	2808.00	+	1056.00i	0.271242	+	0.102005i
$476$	−176.000			−0.0169474
$477$		0			0
$478$		−	3688.00i		−	0.352898i
$479$	−6116.00			−0.583397			−0.291699	−	0.956510i	$-0.594220\pi$
$479$	−6116.00			−0.583397			−0.291699	+	0.956510i	$0.594220\pi$
$480$		0			0
$481$	−13716.0			−1.30020
$482$			1964.00i			0.185597i
$483$		0			0
$484$	−14276.0			−1.34072
$485$	616.000	+	112.000i	0.0576724	+	0.0104859i
$486$		0			0
$487$		−	15906.0i		−	1.48002i	−0.672596	−	0.740010i	$-0.734821\pi$
$487$		−	15906.0i		−	1.48002i	0.672596	−	0.740010i	$-0.265179\pi$
$488$			4400.00i			0.408153i
$489$		0			0
$490$	−7458.00	−	1356.00i	−0.687588	−	0.125016i
$491$	−18714.0			−1.72006			−0.860032	−	0.510241i	$-0.829556\pi$
$491$	−18714.0			−1.72006			−0.860032	+	0.510241i	$0.829556\pi$
$492$		0			0
$493$		−	4752.00i		−	0.434116i
$494$	2592.00			0.236072
$495$		0			0
$496$	3328.00			0.301273
$497$			1080.00i			0.0974741i
$498$		0			0
$499$	4056.00			0.363871			0.181935	−	0.983310i	$-0.441764\pi$
$499$	4056.00			0.363871			0.181935	+	0.983310i	$0.441764\pi$
$500$	−2872.00	+	4796.00i	−0.256879	+	0.428967i
$501$		0			0
$502$			6348.00i			0.564393i
$503$		−	6288.00i		−	0.557392i	−0.960379	−	0.278696i	$-0.910098\pi$
$503$		−	6288.00i		−	0.557392i	0.960379	−	0.278696i	$-0.0899021\pi$
$504$		0			0
$505$	−1184.00	+	6512.00i	−0.104331	+	0.573822i
$506$	14000.0			1.22999
$507$		0			0
$508$			6520.00i			0.569445i
$509$	2856.00			0.248703			0.124352	−	0.992238i	$-0.460315\pi$
$509$	2856.00			0.248703			0.124352	+	0.992238i	$0.460315\pi$
$510$		0			0
$511$	−1208.00			−0.104577
$512$			512.000i			0.0441942i
$513$		0			0
$514$	2388.00			0.204922
$515$	−682.000	−	124.000i	−0.0583544	−	0.0106099i
$516$		0			0
$517$			22400.0i			1.90551i
$518$		−	1016.00i		−	0.0861785i
$519$		0			0
$520$	−864.000	+	4752.00i	−0.0728633	+	0.400748i
$521$	−17078.0			−1.43609			−0.718043	−	0.695999i	$-0.754962\pi$
$521$	−17078.0			−1.43609			−0.718043	+	0.695999i	$0.754962\pi$
$522$		0			0
$523$		−	8560.00i		−	0.715684i	−0.933782	−	0.357842i	$-0.883513\pi$
$523$		−	8560.00i		−	0.715684i	0.933782	−	0.357842i	$-0.116487\pi$
$524$	−3480.00			−0.290123
$525$		0			0
$526$	280.000			0.0232102
$527$		−	4576.00i		−	0.378242i
$528$		0			0
$529$	2167.00			0.178105
$530$	1608.00	−	8844.00i	0.131787	−	0.724828i
$531$		0			0
$532$			192.000i			0.0156471i
$533$			11124.0i			0.904004i
$534$		0			0
$535$	−924.000	−	168.000i	−0.0746692	−	0.0135762i
$536$	−5824.00			−0.469326
$537$		0			0
$538$			10512.0i			0.842388i
$539$	−23730.0			−1.89633
$540$		0			0
$541$	15970.0			1.26914			0.634569	−	0.772866i	$-0.281178\pi$
$541$	15970.0			1.26914			0.634569	+	0.772866i	$0.281178\pi$
$542$			1088.00i			0.0862244i
$543$		0			0
$544$	704.000			0.0554848
$545$	740.000	−	4070.00i	0.0581617	−	0.319889i
$546$		0			0
$547$			15524.0i			1.21345i	0.794911	+	0.606726i	$0.207518\pi$
$547$			15524.0i			1.21345i	−0.794911	+	0.606726i	$0.792482\pi$
$548$		−	3672.00i		−	0.286241i
$549$		0			0
$550$	−6160.00	+	16380.0i	−0.477570	+	1.26990i
$551$	−5184.00			−0.400809
$552$		0			0
$553$		−	1584.00i		−	0.121806i
$554$	1892.00			0.145096
$555$		0			0
$556$	−2384.00			−0.181842
$557$			6774.00i			0.515303i	0.966238	+	0.257651i	$0.0829486\pi$
$557$			6774.00i			0.515303i	−0.966238	+	0.257651i	$0.917051\pi$
$558$		0			0
$559$	−15768.0			−1.19305
$560$	−352.000	−	64.0000i	−0.0265620	−	0.00482945i
$561$		0			0
$562$		−	2556.00i		−	0.191848i
$563$			10484.0i			0.784810i	0.919793	+	0.392405i	$0.128357\pi$
$563$			10484.0i			0.784810i	−0.919793	+	0.392405i	$0.871643\pi$
$564$		0			0
$565$	−19206.0	−	3492.00i	−1.43009	−	0.260017i
$566$	−14848.0			−1.10266
$567$		0			0
$568$		−	4320.00i		−	0.319125i
$569$	23302.0			1.71682			0.858410	−	0.512964i	$-0.171453\pi$
$569$	23302.0			1.71682			0.858410	+	0.512964i	$0.171453\pi$
$570$		0			0
$571$	21520.0			1.57720			0.788602	−	0.614903i	$-0.210805\pi$
$571$	21520.0			1.57720			0.788602	+	0.614903i	$0.210805\pi$
$572$			15120.0i			1.10524i
$573$		0			0
$574$	−824.000			−0.0599183
$575$	4400.00	−	11700.0i	0.319118	−	0.848563i
$576$		0			0
$577$		−	3856.00i		−	0.278210i	−0.990278	−	0.139105i	$-0.955577\pi$
$577$		−	3856.00i		−	0.278210i	0.990278	−	0.139105i	$-0.0444226\pi$
$578$			8858.00i			0.637447i
$579$		0			0
$580$	1728.00	−	9504.00i	0.123709	−	0.680400i
$581$	808.000			0.0576962
$582$		0			0
$583$		−	28140.0i		−	1.99904i
$584$	4832.00			0.342379
$585$		0			0
$586$	−2724.00			−0.192026
$587$		−	26796.0i		−	1.88414i	−0.335418	−	0.942069i	$-0.608878\pi$
$587$		−	26796.0i		−	1.88414i	0.335418	−	0.942069i	$-0.391122\pi$
$588$		0			0
$589$	−4992.00			−0.349222
$590$	8140.00	+	1480.00i	0.567997	+	0.103272i
$591$		0			0
$592$			4064.00i			0.282144i
$593$		−	9870.00i		−	0.683495i	−0.939792	−	0.341747i	$-0.888981\pi$
$593$		−	9870.00i		−	0.683495i	0.939792	−	0.341747i	$-0.111019\pi$
$594$		0			0
$595$	−88.0000	+	484.000i	−0.00606327	+	0.0333480i
$596$	−4304.00			−0.295803
$597$		0			0
$598$		−	10800.0i		−	0.738537i
$599$	−13296.0			−0.906945			−0.453472	−	0.891270i	$-0.649815\pi$
$599$	−13296.0			−0.906945			−0.453472	+	0.891270i	$0.649815\pi$
$600$		0			0
$601$	−9262.00			−0.628627			−0.314314	−	0.949319i	$-0.601774\pi$
$601$	−9262.00			−0.628627			−0.314314	+	0.949319i	$0.601774\pi$
$602$		−	1168.00i		−	0.0790766i
$603$		0			0
$604$	128.000			0.00862292
$605$	−7138.00	+	39259.0i	−0.479671	+	2.63819i
$606$		0			0
$607$		−	5498.00i		−	0.367639i	−0.982960	−	0.183820i	$-0.941154\pi$
$607$		−	5498.00i		−	0.367639i	0.982960	−	0.183820i	$-0.0588462\pi$
$608$		−	768.000i		−	0.0512278i
$609$		0			0
$610$	12100.0	+	2200.00i	0.803139	+	0.146025i
$611$	17280.0			1.14415
$612$		0			0
$613$			394.000i			0.0259600i	0.999916	+	0.0129800i	$0.00413179\pi$
$613$			394.000i			0.0259600i	−0.999916	+	0.0129800i	$0.995868\pi$
$614$	−15480.0			−1.01746
$615$		0			0
$616$	−1120.00			−0.0732566
$617$			7370.00i			0.480883i	0.970664	+	0.240442i	$0.0772923\pi$
$617$			7370.00i			0.480883i	−0.970664	+	0.240442i	$0.922708\pi$
$618$		0			0
$619$	−25316.0			−1.64384			−0.821919	−	0.569604i	$-0.807097\pi$
$619$	−25316.0			−1.64384			−0.821919	+	0.569604i	$0.807097\pi$
$620$	1664.00	−	9152.00i	0.107787	−	0.592828i
$621$		0			0
$622$		−	9960.00i		−	0.642057i
$623$		−	1876.00i		−	0.120643i
$624$		0			0
$625$	11753.0	+	10296.0i	0.752192	+	0.658944i
$626$	−1208.00			−0.0771268
$627$		0			0
$628$		−	10216.0i		−	0.649145i
$629$	5588.00			0.354226
$630$		0			0
$631$	2552.00			0.161004			0.0805020	−	0.996754i	$-0.474348\pi$
$631$	2552.00			0.161004			0.0805020	+	0.996754i	$0.474348\pi$
$632$			6336.00i			0.398786i
$633$		0			0
$634$	17132.0			1.07318
$635$	17930.0	+	3260.00i	1.12052	+	0.203731i
$636$		0			0
$637$			18306.0i			1.13863i
$638$		−	30240.0i		−	1.87651i
$639$		0			0
$640$	1408.00	+	256.000i	0.0869626	+	0.0158114i
$641$	−8050.00			−0.496031			−0.248016	−	0.968756i	$-0.579778\pi$
$641$	−8050.00			−0.496031			−0.248016	+	0.968756i	$0.579778\pi$
$642$		0			0
$643$		−	19368.0i		−	1.18787i	−0.804514	−	0.593934i	$-0.797574\pi$
$643$		−	19368.0i		−	1.18787i	0.804514	−	0.593934i	$-0.202426\pi$
$644$	800.000			0.0489510
$645$		0			0
$646$	−1056.00			−0.0643154
$647$			9912.00i			0.602289i	0.953579	+	0.301144i	$0.0973686\pi$
$647$			9912.00i			0.602289i	−0.953579	+	0.301144i	$0.902631\pi$
$648$		0			0
$649$	25900.0			1.56651
$650$	12636.0	+	4752.00i	0.762500	+	0.286752i
$651$		0			0
$652$		−	3008.00i		−	0.180678i
$653$		−	27986.0i		−	1.67715i	−0.544789	−	0.838573i	$-0.683390\pi$
$653$		−	27986.0i		−	1.67715i	0.544789	−	0.838573i	$-0.316610\pi$
$654$		0			0
$655$	−1740.00	+	9570.00i	−0.103798	+	0.570887i
$656$	3296.00			0.196169
$657$		0			0
$658$			1280.00i			0.0758353i
$659$	7562.00			0.447001			0.223501	−	0.974704i	$-0.428252\pi$
$659$	7562.00			0.447001			0.223501	+	0.974704i	$0.428252\pi$
$660$		0			0
$661$	20234.0			1.19064			0.595319	−	0.803490i	$-0.297026\pi$
$661$	20234.0			1.19064			0.595319	+	0.803490i	$0.297026\pi$
$662$			6944.00i			0.407683i
$663$		0			0
$664$	−3232.00			−0.188894
$665$	528.000	+	96.0000i	0.0307894	+	0.00559808i
$666$		0			0
$667$			21600.0i			1.25391i
$668$		−	10800.0i		−	0.625546i
$669$		0			0
$670$	−2912.00	+	16016.0i	−0.167911	+	0.923511i
$671$	38500.0			2.21502
$672$		0			0
$673$			25332.0i			1.45093i	0.688258	+	0.725466i	$0.258376\pi$
$673$			25332.0i			1.45093i	−0.688258	+	0.725466i	$0.741624\pi$
$674$	11336.0			0.647843
$675$		0			0
$676$	2876.00			0.163632
$677$		−	18358.0i		−	1.04218i	−0.853502	−	0.521090i	$-0.825526\pi$
$677$		−	18358.0i		−	1.04218i	0.853502	−	0.521090i	$-0.174474\pi$
$678$		0			0
$679$	112.000			0.00633014
$680$	352.000	−	1936.00i	0.0198509	−	0.109180i
$681$		0			0
$682$		−	29120.0i		−	1.63499i
$683$			124.000i			0.00694689i	0.999994	+	0.00347345i	$0.00110563\pi$
$683$			124.000i			0.00694689i	−0.999994	+	0.00347345i	$0.998894\pi$
$684$		0			0
$685$	−10098.0	−	1836.00i	−0.563248	−	0.102409i
$686$	−2728.00			−0.151830
$687$		0			0
$688$			4672.00i			0.258893i
$689$	−21708.0			−1.20030
$690$		0			0
$691$	−17456.0			−0.961009			−0.480505	−	0.876992i	$-0.659547\pi$
$691$	−17456.0			−0.961009			−0.480505	+	0.876992i	$0.659547\pi$
$692$		−	5336.00i		−	0.293128i
$693$		0			0
$694$	−21672.0			−1.18539
$695$	−1192.00	+	6556.00i	−0.0650578	+	0.357818i
$696$		0			0
$697$		−	4532.00i		−	0.246287i
$698$			17980.0i			0.975004i
$699$		0			0
$700$	−352.000	+	936.000i	−0.0190062	+	0.0505392i
$701$	17816.0			0.959916			0.479958	−	0.877291i	$-0.340652\pi$
$701$	17816.0			0.959916			0.479958	+	0.877291i	$0.340652\pi$
$702$		0			0
$703$		−	6096.00i		−	0.327048i
$704$	4480.00			0.239839
$705$		0			0
$706$	−10156.0			−0.541397
$707$			1184.00i			0.0629829i
$708$		0			0
$709$	14298.0			0.757366			0.378683	−	0.925526i	$-0.376377\pi$
$709$	14298.0			0.757366			0.378683	+	0.925526i	$0.376377\pi$
$710$	−11880.0	−	2160.00i	−0.627956	−	0.114174i
$711$		0			0
$712$			7504.00i			0.394978i
$713$			20800.0i			1.09252i
$714$		0			0
$715$	41580.0	+	7560.00i	2.17483	+	0.395424i
$716$	6856.00			0.357850
$717$		0			0
$718$		−	7392.00i		−	0.384216i
$719$	−18440.0			−0.956462			−0.478231	−	0.878234i	$-0.658722\pi$
$719$	−18440.0			−0.956462			−0.478231	+	0.878234i	$0.658722\pi$
$720$		0			0
$721$	−124.000			−0.00640499
$722$		−	12566.0i		−	0.647726i
$723$		0			0
$724$	16024.0			0.822551
$725$	−25272.0	−	9504.00i	−1.29459	−	0.486855i
$726$		0			0
$727$		−	9666.00i		−	0.493112i	−0.969129	−	0.246556i	$-0.920701\pi$
$727$		−	9666.00i		−	0.493112i	0.969129	−	0.246556i	$-0.0792989\pi$
$728$			864.000i			0.0439862i
$729$		0			0
$730$	2416.00	−	13288.0i	0.122493	−	0.673714i
$731$	6424.00			0.325035
$732$		0			0
$733$		−	6094.00i		−	0.307076i	−0.988143	−	0.153538i	$-0.950933\pi$
$733$		−	6094.00i		−	0.307076i	0.988143	−	0.153538i	$-0.0490668\pi$
$734$	572.000			0.0287642
$735$		0			0
$736$	−3200.00			−0.160263
$737$			50960.0i			2.54700i
$738$		0			0
$739$	−9952.00			−0.495386			−0.247693	−	0.968839i	$-0.579672\pi$
$739$	−9952.00			−0.495386			−0.247693	+	0.968839i	$0.579672\pi$
$740$	11176.0	+	2032.00i	0.555186	+	0.100943i
$741$		0			0
$742$		−	1608.00i		−	0.0795573i
$743$			2208.00i			0.109022i	0.998513	+	0.0545112i	$0.0173601\pi$
$743$			2208.00i			0.109022i	−0.998513	+	0.0545112i	$0.982640\pi$
$744$		0			0
$745$	−2152.00	+	11836.0i	−0.105830	+	0.582064i
$746$	−16524.0			−0.810974
$747$		0			0
$748$		−	6160.00i		−	0.301112i
$749$	−168.000			−0.00819571
$750$		0			0
$751$	−9400.00			−0.456739			−0.228369	−	0.973575i	$-0.573339\pi$
$751$	−9400.00			−0.456739			−0.228369	+	0.973575i	$0.573339\pi$
$752$		−	5120.00i		−	0.248281i
$753$		0			0
$754$	−23328.0			−1.12673
$755$	64.0000	−	352.000i	0.00308503	−	0.0169677i
$756$		0			0
$757$			22574.0i			1.08384i	0.840430	+	0.541919i	$0.182302\pi$
$757$			22574.0i			1.08384i	−0.840430	+	0.541919i	$0.817698\pi$
$758$			5912.00i			0.283290i
$759$		0			0
$760$	−2112.00	−	384.000i	−0.100803	−	0.0183278i
$761$	−7278.00			−0.346685			−0.173343	−	0.984862i	$-0.555457\pi$
$761$	−7278.00			−0.346685			−0.173343	+	0.984862i	$0.555457\pi$
$762$		0			0
$763$		−	740.000i		−	0.0351111i
$764$	−2736.00			−0.129562
$765$		0			0
$766$	−10480.0			−0.494331
$767$		−	19980.0i		−	0.940595i
$768$		0			0
$769$	16542.0			0.775708			0.387854	−	0.921721i	$-0.373216\pi$
$769$	16542.0			0.775708			0.387854	+	0.921721i	$0.373216\pi$
$770$	−560.000	+	3080.00i	−0.0262091	+	0.144150i
$771$		0			0
$772$			17936.0i			0.836180i
$773$			28926.0i			1.34592i	0.739679	+	0.672960i	$0.234977\pi$
$773$			28926.0i			1.34592i	−0.739679	+	0.672960i	$0.765023\pi$
$774$		0			0
$775$	−24336.0	−	9152.00i	−1.12797	−	0.424193i
$776$	−448.000			−0.0207246
$777$		0			0
$778$		−	1768.00i		−	0.0814728i
$779$	−4944.00			−0.227390
$780$		0			0
$781$	−37800.0			−1.73187
$782$			4400.00i			0.201207i
$783$		0			0
$784$	5424.00			0.247085
$785$	−28094.0	−	5108.00i	−1.27735	−	0.232245i
$786$		0			0
$787$		−	20608.0i

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:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	90.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -4.00000 q^{4} +(-2.00000 + 11.0000i) q^{5} +2.00000i q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+2.00000i q^{2} -4.00000 q^{4} +(-2.00000 + 11.0000i) q^{5} +2.00000i q^{7} -8.00000i q^{8} +(-22.0000 - 4.00000i) q^{10} -70.0000 q^{11} +54.0000i q^{13} -4.00000 q^{14} +16.0000 q^{16} -22.0000i q^{17} -24.0000 q^{19} +(8.00000 - 44.0000i) q^{20} -140.000i q^{22} +100.000i q^{23} +(-117.000 - 44.0000i) q^{25} -108.000 q^{26} -8.00000i q^{28} +216.000 q^{29} +208.000 q^{31} +32.0000i q^{32} +44.0000 q^{34} +(-22.0000 - 4.00000i) q^{35} +254.000i q^{37} -48.0000i q^{38} +(88.0000 + 16.0000i) q^{40} +206.000 q^{41} +292.000i q^{43} +280.000 q^{44} -200.000 q^{46} -320.000i q^{47} +339.000 q^{49} +(88.0000 - 234.000i) q^{50} -216.000i q^{52} +402.000i q^{53} +(140.000 - 770.000i) q^{55} +16.0000 q^{56} +432.000i q^{58} -370.000 q^{59} -550.000 q^{61} +416.000i q^{62} -64.0000 q^{64} +(-594.000 - 108.000i) q^{65} -728.000i q^{67} +88.0000i q^{68} +(8.00000 - 44.0000i) q^{70} +540.000 q^{71} +604.000i q^{73} -508.000 q^{74} +96.0000 q^{76} -140.000i q^{77} -792.000 q^{79} +(-32.0000 + 176.000i) q^{80} +412.000i q^{82} -404.000i q^{83} +(242.000 + 44.0000i) q^{85} -584.000 q^{86} +560.000i q^{88} -938.000 q^{89} -108.000 q^{91} -400.000i q^{92} +640.000 q^{94} +(48.0000 - 264.000i) q^{95} -56.0000i q^{97} +678.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 4 q^{5}+O(q^{10}) \) 2 * q - 8 * q^4 - 4 * q^5	\( 2 q - 8 q^{4} - 4 q^{5} - 44 q^{10} - 140 q^{11} - 8 q^{14} + 32 q^{16} - 48 q^{19} + 16 q^{20} - 234 q^{25} - 216 q^{26} + 432 q^{29} + 416 q^{31} + 88 q^{34} - 44 q^{35} + 176 q^{40} + 412 q^{41} + 560 q^{44} - 400 q^{46} + 678 q^{49} + 176 q^{50} + 280 q^{55} + 32 q^{56} - 740 q^{59} - 1100 q^{61} - 128 q^{64} - 1188 q^{65} + 16 q^{70} + 1080 q^{71} - 1016 q^{74} + 192 q^{76} - 1584 q^{79} - 64 q^{80} + 484 q^{85} - 1168 q^{86} - 1876 q^{89} - 216 q^{91} + 1280 q^{94} + 96 q^{95}+O(q^{100}) \) 2 * q - 8 * q^4 - 4 * q^5 - 44 * q^10 - 140 * q^11 - 8 * q^14 + 32 * q^16 - 48 * q^19 + 16 * q^20 - 234 * q^25 - 216 * q^26 + 432 * q^29 + 416 * q^31 + 88 * q^34 - 44 * q^35 + 176 * q^40 + 412 * q^41 + 560 * q^44 - 400 * q^46 + 678 * q^49 + 176 * q^50 + 280 * q^55 + 32 * q^56 - 740 * q^59 - 1100 * q^61 - 128 * q^64 - 1188 * q^65 + 16 * q^70 + 1080 * q^71 - 1016 * q^74 + 192 * q^76 - 1584 * q^79 - 64 * q^80 + 484 * q^85 - 1168 * q^86 - 1876 * q^89 - 216 * q^91 + 1280 * q^94 + 96 * q^95

Introduction

L-functions

Modular forms

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