LMFDB - Embedded newform 810.3.b.c.809.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,3,Mod(809,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("810.809");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	810.b (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:	$22.0709014132$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			809.7
Character	$\chi$	$=$	810.809
Dual form			810.3.b.c.809.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +(-0.174155 - 4.99697i) q^{5} -9.47560i q^{7} -2.82843 q^{8} +O(q^{10})$	\(q-1.41421 q^{2} +2.00000 q^{4} +(-0.174155 - 4.99697i) q^{5} -9.47560i q^{7} -2.82843 q^{8} +(0.246293 + 7.06678i) q^{10} +20.3859i q^{11} -6.10986i q^{13} +13.4005i q^{14} +4.00000 q^{16} +24.0002 q^{17} +16.0752 q^{19} +(-0.348311 - 9.99393i) q^{20} -28.8301i q^{22} +43.2779 q^{23} +(-24.9393 + 1.74050i) q^{25} +8.64064i q^{26} -18.9512i q^{28} -19.6215i q^{29} +30.5072 q^{31} -5.65685 q^{32} -33.9414 q^{34} +(-47.3492 + 1.65023i) q^{35} -44.8335i q^{37} -22.7337 q^{38} +(0.492585 + 14.1336i) q^{40} +48.0908i q^{41} -48.1045i q^{43} +40.7719i q^{44} -61.2042 q^{46} -68.0770 q^{47} -40.7869 q^{49} +(35.2696 - 2.46143i) q^{50} -12.2197i q^{52} -61.7484 q^{53} +(101.868 - 3.55032i) q^{55} +26.8010i q^{56} +27.7489i q^{58} -50.4411i q^{59} -48.9975 q^{61} -43.1438 q^{62} +8.00000 q^{64} +(-30.5308 + 1.06406i) q^{65} -55.0907i q^{67} +48.0004 q^{68} +(66.9619 - 2.33377i) q^{70} -23.5982i q^{71} -40.4303i q^{73} +63.4041i q^{74} +32.1503 q^{76} +193.169 q^{77} +33.2338 q^{79} +(-0.696621 - 19.9879i) q^{80} -68.0107i q^{82} -22.7144 q^{83} +(-4.17976 - 119.928i) q^{85} +68.0300i q^{86} -57.6601i q^{88} -79.5266i q^{89} -57.8946 q^{91} +86.5558 q^{92} +96.2754 q^{94} +(-2.79958 - 80.3271i) q^{95} +119.886i q^{97} +57.6815 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 48 q^{4}+O(q^{10}) $ 24 * q + 48 * q^4	$ 24 q + 48 q^{4} - 12 q^{10} + 96 q^{16} - 48 q^{25} - 120 q^{34} - 24 q^{40} + 72 q^{49} + 216 q^{55} + 120 q^{61} + 192 q^{64} + 192 q^{70} + 480 q^{79} + 444 q^{85} + 48 q^{91} + 96 q^{94}+O(q^{100}) $ 24 * q + 48 * q^4 - 12 * q^10 + 96 * q^16 - 48 * q^25 - 120 * q^34 - 24 * q^40 + 72 * q^49 + 216 * q^55 + 120 * q^61 + 192 * q^64 + 192 * q^70 + 480 * q^79 + 444 * q^85 + 48 * q^91 + 96 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$	−0.174155	−	4.99697i	−0.0348311	−	0.999393i
$5$	−0.174155	−	4.99697i	−0.0348311	−	0.999393i
$6$		0			0
$7$		−	9.47560i		−	1.35366i	−0.736141	−	0.676828i	$-0.763354\pi$
$7$		−	9.47560i		−	1.35366i	0.736141	−	0.676828i	$-0.236646\pi$
$8$	−2.82843			−0.353553
$9$		0			0
$10$	0.246293	+	7.06678i	0.0246293	+	0.706678i
$11$			20.3859i			1.85327i	0.375966	+	0.926633i	$0.377311\pi$
$11$			20.3859i			1.85327i	−0.375966	+	0.926633i	$0.622689\pi$
$12$		0			0
$13$		−	6.10986i		−	0.469989i	−0.971997	−	0.234995i	$-0.924493\pi$
$13$		−	6.10986i		−	0.469989i	0.971997	−	0.234995i	$-0.0755072\pi$
$14$			13.4005i			0.957180i
$15$		0			0
$16$	4.00000			0.250000
$17$	24.0002			1.41178			0.705888	−	0.708323i	$-0.250548\pi$
$17$	24.0002			1.41178			0.705888	+	0.708323i	$0.250548\pi$
$18$		0			0
$19$	16.0752			0.846062			0.423031	−	0.906115i	$-0.360966\pi$
$19$	16.0752			0.846062			0.423031	+	0.906115i	$0.360966\pi$
$20$	−0.348311	−	9.99393i	−0.0174155	−	0.499697i
$21$		0			0
$22$		−	28.8301i		−	1.31046i
$23$	43.2779			1.88165			0.940824	−	0.338896i	$-0.110054\pi$
$23$	43.2779			1.88165			0.940824	+	0.338896i	$0.110054\pi$
$24$		0			0
$25$	−24.9393	+	1.74050i	−0.997574	+	0.0696198i
$26$			8.64064i			0.332332i
$27$		0			0
$28$		−	18.9512i		−	0.676828i
$29$		−	19.6215i		−	0.676602i	−0.941038	−	0.338301i	$-0.890148\pi$
$29$		−	19.6215i		−	0.676602i	0.941038	−	0.338301i	$-0.109852\pi$
$30$		0			0
$31$	30.5072			0.984105			0.492052	−	0.870566i	$-0.336247\pi$
$31$	30.5072			0.984105			0.492052	+	0.870566i	$0.336247\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$	−33.9414			−0.998277
$35$	−47.3492	+	1.65023i	−1.35284	+	0.0471493i
$36$		0			0
$37$		−	44.8335i		−	1.21172i	−0.795573	−	0.605858i	$-0.792830\pi$
$37$		−	44.8335i		−	1.21172i	0.795573	−	0.605858i	$-0.207170\pi$
$38$	−22.7337			−0.598256
$39$		0			0
$40$	0.492585	+	14.1336i	0.0123146	+	0.353339i
$41$			48.0908i			1.17295i	0.809968	+	0.586473i	$0.199484\pi$
$41$			48.0908i			1.17295i	−0.809968	+	0.586473i	$0.800516\pi$
$42$		0			0
$43$		−	48.1045i		−	1.11871i	−0.828929	−	0.559354i	$-0.811049\pi$
$43$		−	48.1045i		−	1.11871i	0.828929	−	0.559354i	$-0.188951\pi$
$44$			40.7719i			0.926633i
$45$		0			0
$46$	−61.2042			−1.33053
$47$	−68.0770			−1.44845			−0.724223	−	0.689565i	$-0.757802\pi$
$47$	−68.0770			−1.44845			−0.724223	+	0.689565i	$0.757802\pi$
$48$		0			0
$49$	−40.7869			−0.832387
$50$	35.2696	−	2.46143i	0.705391	−	0.0492287i
$51$		0			0
$52$		−	12.2197i		−	0.234995i
$53$	−61.7484			−1.16506			−0.582532	−	0.812808i	$-0.697938\pi$
$53$	−61.7484			−1.16506			−0.582532	+	0.812808i	$0.697938\pi$
$54$		0			0
$55$	101.868	−	3.55032i	1.85214	−	0.0645512i
$56$			26.8010i			0.478590i
$57$		0			0
$58$			27.7489i			0.478430i
$59$		−	50.4411i		−	0.854934i	−0.904031	−	0.427467i	$-0.859406\pi$
$59$		−	50.4411i		−	0.854934i	0.904031	−	0.427467i	$-0.140594\pi$
$60$		0			0
$61$	−48.9975			−0.803237			−0.401619	−	0.915807i	$-0.631552\pi$
$61$	−48.9975			−0.803237			−0.401619	+	0.915807i	$0.631552\pi$
$62$	−43.1438			−0.695867
$63$		0			0
$64$	8.00000			0.125000
$65$	−30.5308	+	1.06406i	−0.469704	+	0.0163702i
$66$		0			0
$67$		−	55.0907i		−	0.822250i	−0.911579	−	0.411125i	$-0.865136\pi$
$67$		−	55.0907i		−	0.822250i	0.911579	−	0.411125i	$-0.134864\pi$
$68$	48.0004			0.705888
$69$		0			0
$70$	66.9619	−	2.33377i	0.956599	−	0.0333396i
$71$		−	23.5982i		−	0.332369i	−0.986095	−	0.166184i	$-0.946855\pi$
$71$		−	23.5982i		−	0.332369i	0.986095	−	0.166184i	$-0.0531447\pi$
$72$		0			0
$73$		−	40.4303i		−	0.553840i	−0.960893	−	0.276920i	$-0.910686\pi$
$73$		−	40.4303i		−	0.553840i	0.960893	−	0.276920i	$-0.0893137\pi$
$74$			63.4041i			0.856812i
$75$		0			0
$76$	32.1503			0.423031
$77$	193.169			2.50869
$78$		0			0
$79$	33.2338			0.420681			0.210340	−	0.977628i	$-0.432543\pi$
$79$	33.2338			0.420681			0.210340	+	0.977628i	$0.432543\pi$
$80$	−0.696621	−	19.9879i	−0.00870776	−	0.249848i
$81$		0			0
$82$		−	68.0107i		−	0.829399i
$83$	−22.7144			−0.273668			−0.136834	−	0.990594i	$-0.543693\pi$
$83$	−22.7144			−0.273668			−0.136834	+	0.990594i	$0.543693\pi$
$84$		0			0
$85$	−4.17976	−	119.928i	−0.0491737	−	1.41092i
$86$			68.0300i			0.791046i
$87$		0			0
$88$		−	57.6601i		−	0.655229i
$89$		−	79.5266i		−	0.893557i	−0.894644	−	0.446779i	$-0.852571\pi$
$89$		−	79.5266i		−	0.893557i	0.894644	−	0.446779i	$-0.147429\pi$
$90$		0			0
$91$	−57.8946			−0.636204
$92$	86.5558			0.940824
$93$		0			0
$94$	96.2754			1.02421
$95$	−2.79958	−	80.3271i	−0.0294692	−	0.845548i
$96$		0			0
$97$			119.886i			1.23594i	0.786202	+	0.617970i	$0.212045\pi$
$97$			119.886i			1.23594i	−0.786202	+	0.617970i	$0.787955\pi$
$98$	57.6815			0.588586
$99$		0			0
$100$	−49.8787	+	3.48099i	−0.498787	+	0.0348099i
$101$		−	22.0774i		−	0.218588i	−0.994009	−	0.109294i	$-0.965141\pi$
$101$		−	22.0774i		−	0.218588i	0.994009	−	0.109294i	$-0.0348591\pi$
$102$		0			0
$103$			18.3872i			0.178516i	0.996009	+	0.0892581i	$0.0284496\pi$
$103$			18.3872i			0.178516i	−0.996009	+	0.0892581i	$0.971550\pi$
$104$			17.2813i			0.166166i
$105$		0			0
$106$	87.3254			0.823825
$107$	−60.8601			−0.568786			−0.284393	−	0.958708i	$-0.591792\pi$
$107$	−60.8601			−0.568786			−0.284393	+	0.958708i	$0.591792\pi$
$108$		0			0
$109$	80.4892			0.738433			0.369216	−	0.929343i	$-0.379626\pi$
$109$	80.4892			0.738433			0.369216	+	0.929343i	$0.379626\pi$
$110$	−144.063	+	5.02091i	−1.30966	+	0.0456446i
$111$		0			0
$112$		−	37.9024i		−	0.338414i
$113$	22.7556			0.201377			0.100689	−	0.994918i	$-0.467895\pi$
$113$	22.7556			0.201377			0.100689	+	0.994918i	$0.467895\pi$
$114$		0			0
$115$	−7.53707	−	216.258i	−0.0655398	−	1.88051i
$116$		−	39.2429i		−	0.338301i
$117$		0			0
$118$			71.3345i			0.604530i
$119$		−	227.416i		−	1.91106i
$120$		0			0
$121$	−294.586			−2.43460
$122$	69.2929			0.567974
$123$		0			0
$124$	61.0145			0.492052
$125$	13.0405	+	124.318i	0.104324	+	0.994543i
$126$		0			0
$127$			73.3562i			0.577608i	0.957388	+	0.288804i	$0.0932576\pi$
$127$			73.3562i			0.577608i	−0.957388	+	0.288804i	$0.906742\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$	43.1770	−	1.50481i	0.332131	−	0.0115755i
$131$		−	91.0508i		−	0.695045i	−0.937672	−	0.347522i	$-0.887023\pi$
$131$		−	91.0508i		−	0.695045i	0.937672	−	0.347522i	$-0.112977\pi$
$132$		0			0
$133$		−	152.322i		−	1.14528i
$134$			77.9101i			0.581419i
$135$		0			0
$136$	−67.8828			−0.499138
$137$	27.8837			0.203531			0.101765	−	0.994808i	$-0.467551\pi$
$137$	27.8837			0.203531			0.101765	+	0.994808i	$0.467551\pi$
$138$		0			0
$139$	34.7379			0.249913			0.124957	−	0.992162i	$-0.460121\pi$
$139$	34.7379			0.249913			0.124957	+	0.992162i	$0.460121\pi$
$140$	−94.6985	+	3.30045i	−0.676418	+	0.0235746i
$141$		0			0
$142$			33.3729i			0.235020i
$143$	124.555			0.871015
$144$		0			0
$145$	−98.0477	+	3.41718i	−0.676191	+	0.0235668i
$146$			57.1771i			0.391624i
$147$		0			0
$148$		−	89.6669i		−	0.605858i
$149$			0.317069i			0.00212798i	0.999999	+	0.00106399i	$0.000338678\pi$
$149$			0.317069i			0.00212798i	−0.999999	+	0.00106399i	$0.999661\pi$
$150$		0			0
$151$	−105.328			−0.697534			−0.348767	−	0.937209i	$-0.613400\pi$
$151$	−105.328			−0.697534			−0.348767	+	0.937209i	$0.613400\pi$
$152$	−45.4674			−0.299128
$153$		0			0
$154$	−273.182			−1.77391
$155$	−5.31300	−	152.444i	−0.0342774	−	0.983507i
$156$		0			0
$157$		−	45.6023i		−	0.290460i	−0.989398	−	0.145230i	$-0.953608\pi$
$157$		−	45.6023i		−	0.290460i	0.989398	−	0.145230i	$-0.0463923\pi$
$158$	−46.9997			−0.297466
$159$		0			0
$160$	0.985171	+	28.2671i	0.00615732	+	0.176669i
$161$		−	410.084i		−	2.54711i
$162$		0			0
$163$		−	152.794i		−	0.937384i	−0.883362	−	0.468692i	$-0.844725\pi$
$163$		−	152.794i		−	0.937384i	0.883362	−	0.468692i	$-0.155275\pi$
$164$			96.1816i			0.586473i
$165$		0			0
$166$	32.1230			0.193512
$167$	13.7284			0.0822060			0.0411030	−	0.999155i	$-0.486913\pi$
$167$	13.7284			0.0822060			0.0411030	+	0.999155i	$0.486913\pi$
$168$		0			0
$169$	131.670			0.779110
$170$	5.91107	+	169.604i	0.0347710	+	0.997671i
$171$		0			0
$172$		−	96.2089i		−	0.559354i
$173$	10.4307			0.0602932			0.0301466	−	0.999545i	$-0.490403\pi$
$173$	10.4307			0.0602932			0.0301466	+	0.999545i	$0.490403\pi$
$174$		0			0
$175$	16.4922	+	236.315i	0.0942414	+	1.35037i
$176$			81.5437i			0.463317i
$177$		0			0
$178$			112.468i			0.631841i
$179$			62.4650i			0.348967i	0.984660	+	0.174483i	$0.0558255\pi$
$179$			62.4650i			0.348967i	−0.984660	+	0.174483i	$0.944174\pi$
$180$		0			0
$181$	343.029			1.89519			0.947594	−	0.319477i	$-0.103507\pi$
$181$	343.029			1.89519			0.947594	+	0.319477i	$0.103507\pi$
$182$	81.8753			0.449864
$183$		0			0
$184$	−122.408			−0.665263
$185$	−224.031	+	7.80799i	−1.21098	+	0.0422053i
$186$		0			0
$187$			489.266i			2.61640i
$188$	−136.154			−0.724223
$189$		0			0
$190$	3.95920	+	113.600i	0.0208379	+	0.597893i
$191$			12.6376i			0.0661657i	0.999453	+	0.0330829i	$0.0105325\pi$
$191$			12.6376i			0.0661657i	−0.999453	+	0.0330829i	$0.989467\pi$
$192$		0			0
$193$			168.124i			0.871111i	0.900162	+	0.435556i	$0.143448\pi$
$193$			168.124i			0.871111i	−0.900162	+	0.435556i	$0.856552\pi$
$194$		−	169.545i		−	0.873942i
$195$		0			0
$196$	−81.5739			−0.416193
$197$	372.044			1.88855			0.944275	−	0.329157i	$-0.106765\pi$
$197$	372.044			1.88855			0.944275	+	0.329157i	$0.106765\pi$
$198$		0			0
$199$	231.821			1.16493			0.582466	−	0.812855i	$-0.302088\pi$
$199$	231.821			1.16493			0.582466	+	0.812855i	$0.302088\pi$
$200$	70.5391	−	4.92287i	0.352696	−	0.0246143i
$201$		0			0
$202$			31.2222i			0.154565i
$203$	−185.925			−0.915887
$204$		0			0
$205$	240.308	−	8.37527i	1.17224	−	0.0408550i
$206$		−	26.0034i		−	0.126230i
$207$		0			0
$208$		−	24.4394i		−	0.117497i
$209$			327.707i			1.56798i
$210$		0			0
$211$	20.9012			0.0990579			0.0495289	−	0.998773i	$-0.484228\pi$
$211$	20.9012			0.0990579			0.0495289	+	0.998773i	$0.484228\pi$
$212$	−123.497			−0.582532
$213$		0			0
$214$	86.0692			0.402192
$215$	−240.376	+	8.37765i	−1.11803	+	0.0389658i
$216$		0			0
$217$		−	289.074i		−	1.33214i
$218$	−113.829			−0.522151
$219$		0			0
$220$	203.736	−	7.10063i	0.926071	−	0.0322756i
$221$		−	146.638i		−	0.663520i
$222$		0			0
$223$		−	15.7730i		−	0.0707311i	−0.999374	−	0.0353655i	$-0.988740\pi$
$223$		−	15.7730i		−	0.0707311i	0.999374	−	0.0353655i	$-0.0112595\pi$
$224$			53.6021i			0.239295i
$225$		0			0
$226$	−32.1813			−0.142395
$227$	341.346			1.50373			0.751863	−	0.659319i	$-0.229155\pi$
$227$	341.346			1.50373			0.751863	+	0.659319i	$0.229155\pi$
$228$		0			0
$229$	120.453			0.525994			0.262997	−	0.964797i	$-0.415289\pi$
$229$	120.453			0.525994			0.262997	+	0.964797i	$0.415289\pi$
$230$	10.6590	+	305.835i	0.0463436	+	1.32972i
$231$		0			0
$232$			55.4979i			0.239215i
$233$	−307.400			−1.31931			−0.659657	−	0.751567i	$-0.729298\pi$
$233$	−307.400			−1.31931			−0.659657	+	0.751567i	$0.729298\pi$
$234$		0			0
$235$	11.8560	+	340.178i	0.0504509	+	1.44757i
$236$		−	100.882i		−	0.427467i
$237$		0			0
$238$			321.615i			1.35132i
$239$			119.083i			0.498257i	0.968470	+	0.249128i	$0.0801441\pi$
$239$			119.083i			0.498257i	−0.968470	+	0.249128i	$0.919856\pi$
$240$		0			0
$241$	−314.600			−1.30539			−0.652697	−	0.757619i	$-0.726362\pi$
$241$	−314.600			−1.30539			−0.652697	+	0.757619i	$0.726362\pi$
$242$	416.608			1.72152
$243$		0			0
$244$	−97.9949			−0.401619
$245$	7.10326	+	203.811i	0.0289929	+	0.831882i
$246$		0			0
$247$		−	98.2170i		−	0.397640i
$248$	−86.2875			−0.347933
$249$		0			0
$250$	−18.4421	−	175.812i	−0.0737683	−	0.703248i
$251$		−	148.193i		−	0.590411i	−0.955434	−	0.295206i	$-0.904612\pi$
$251$		−	148.193i		−	0.590411i	0.955434	−	0.295206i	$-0.0953882\pi$
$252$		0			0
$253$			882.260i			3.48719i
$254$		−	103.741i		−	0.408431i
$255$		0			0
$256$	16.0000			0.0625000
$257$	−294.316			−1.14520			−0.572598	−	0.819836i	$-0.694065\pi$
$257$	−294.316			−1.14520			−0.572598	+	0.819836i	$0.694065\pi$
$258$		0			0
$259$	−424.824			−1.64025
$260$	−61.0615	+	2.12813i	−0.234852	+	0.00818511i
$261$		0			0
$262$			128.765i			0.491471i
$263$	69.4645			0.264123			0.132062	−	0.991241i	$-0.457840\pi$
$263$	69.4645			0.264123			0.132062	+	0.991241i	$0.457840\pi$
$264$		0			0
$265$	10.7538	+	308.555i	0.0405804	+	1.16436i
$266$			215.416i			0.809833i
$267$		0			0
$268$		−	110.181i		−	0.411125i
$269$			262.460i			0.975687i	0.872931	+	0.487844i	$0.162216\pi$
$269$			262.460i			0.975687i	−0.872931	+	0.487844i	$0.837784\pi$
$270$		0			0
$271$	−375.110			−1.38417			−0.692084	−	0.721817i	$-0.743307\pi$
$271$	−375.110			−1.38417			−0.692084	+	0.721817i	$0.743307\pi$
$272$	96.0008			0.352944
$273$		0			0
$274$	−39.4335			−0.143918
$275$	−35.4816	−	508.412i	−0.129024	−	1.84877i
$276$		0			0
$277$			146.370i			0.528412i	0.964466	+	0.264206i	$0.0851100\pi$
$277$			146.370i			0.528412i	−0.964466	+	0.264206i	$0.914890\pi$
$278$	−49.1268			−0.176715
$279$		0			0
$280$	133.924	−	4.66754i	0.478300	−	0.0166698i
$281$		−	79.1692i		−	0.281741i	−0.990028	−	0.140870i	$-0.955010\pi$
$281$		−	79.1692i		−	0.281741i	0.990028	−	0.140870i	$-0.0449901\pi$
$282$		0			0
$283$		−	216.326i		−	0.764401i	−0.924079	−	0.382201i	$-0.875166\pi$
$283$		−	216.326i		−	0.764401i	0.924079	−	0.382201i	$-0.124834\pi$
$284$		−	47.1963i		−	0.166184i
$285$		0			0
$286$	−176.148			−0.615901
$287$	455.689			1.58777
$288$		0			0
$289$	287.010			0.993113
$290$	138.660	−	4.83262i	0.478140	−	0.0166642i
$291$		0			0
$292$		−	80.8606i		−	0.276920i
$293$	−416.332			−1.42093			−0.710464	−	0.703734i	$-0.751515\pi$
$293$	−416.332			−1.42093			−0.710464	+	0.703734i	$0.751515\pi$
$294$		0			0
$295$	−252.053	+	8.78458i	−0.854415	+	0.0297783i
$296$			126.808i			0.428406i
$297$		0			0
$298$		−	0.448403i		−	0.00150471i
$299$		−	264.422i		−	0.884354i
$300$		0			0
$301$	−455.819			−1.51435
$302$	148.956			0.493231
$303$		0			0
$304$	64.3007			0.211515
$305$	8.53317	+	244.839i	0.0279776	+	0.802750i
$306$		0			0
$307$		−	331.571i		−	1.08004i	−0.841653	−	0.540018i	$-0.818417\pi$
$307$		−	331.571i		−	1.08004i	0.841653	−	0.540018i	$-0.181583\pi$
$308$	386.338			1.25434
$309$		0			0
$310$	7.51371	+	215.588i	0.0242378	+	0.695445i
$311$			393.568i			1.26549i	0.774360	+	0.632745i	$0.218072\pi$
$311$			393.568i			1.26549i	−0.774360	+	0.632745i	$0.781928\pi$
$312$		0			0
$313$			254.130i			0.811917i	0.913891	+	0.405959i	$0.133062\pi$
$313$			254.130i			0.811917i	−0.913891	+	0.405959i	$0.866938\pi$
$314$			64.4914i			0.205387i
$315$		0			0
$316$	66.4675			0.210340
$317$	431.983			1.36272			0.681361	−	0.731947i	$-0.261388\pi$
$317$	431.983			1.36272			0.681361	+	0.731947i	$0.261388\pi$
$318$		0			0
$319$	400.002			1.25392
$320$	−1.39324	−	39.9757i	−0.00435388	−	0.124924i
$321$		0			0
$322$			579.946i			1.80108i
$323$	385.807			1.19445
$324$		0			0
$325$	10.6342	+	152.376i	0.0327206	+	0.468849i
$326$			216.083i			0.662831i
$327$		0			0
$328$		−	136.021i		−	0.414699i
$329$			645.070i			1.96070i
$330$		0			0
$331$	191.732			0.579249			0.289625	−	0.957140i	$-0.406469\pi$
$331$	191.732			0.579249			0.289625	+	0.957140i	$0.406469\pi$
$332$	−45.4288			−0.136834
$333$		0			0
$334$	−19.4149			−0.0581284
$335$	−275.287	+	9.59434i	−0.821751	+	0.0286398i
$336$		0			0
$337$		−	254.110i		−	0.754035i	−0.926206	−	0.377017i	$-0.876950\pi$
$337$		−	254.110i		−	0.754035i	0.926206	−	0.377017i	$-0.123050\pi$
$338$	−186.209			−0.550914
$339$		0			0
$340$	−8.35952	−	239.856i	−0.0245868	−	0.705460i
$341$			621.919i			1.82381i
$342$		0			0
$343$		−	77.8236i		−	0.226891i
$344$			136.060i			0.395523i
$345$		0			0
$346$	−14.7513			−0.0426337
$347$	320.795			0.924481			0.462241	−	0.886755i	$-0.347046\pi$
$347$	320.795			0.924481			0.462241	+	0.886755i	$0.347046\pi$
$348$		0			0
$349$	304.524			0.872563			0.436281	−	0.899810i	$-0.356295\pi$
$349$	304.524			0.872563			0.436281	+	0.899810i	$0.356295\pi$
$350$	−23.3235	−	334.200i	−0.0666387	−	0.954857i
$351$		0			0
$352$		−	115.320i		−	0.327614i
$353$	−215.758			−0.611213			−0.305607	−	0.952158i	$-0.598859\pi$
$353$	−215.758			−0.611213			−0.305607	+	0.952158i	$0.598859\pi$
$354$		0			0
$355$	−117.919	+	4.10975i	−0.332167	+	0.0115767i
$356$		−	159.053i		−	0.446779i
$357$		0			0
$358$		−	88.3389i		−	0.246757i
$359$		−	638.110i		−	1.77746i	−0.458427	−	0.888732i	$-0.651587\pi$
$359$		−	638.110i		−	1.77746i	0.458427	−	0.888732i	$-0.348413\pi$
$360$		0			0
$361$	−102.589			−0.284180
$362$	−485.116			−1.34010
$363$		0			0
$364$	−115.789			−0.318102
$365$	−202.029	+	7.04115i	−0.553504	+	0.0192908i
$366$		0			0
$367$			370.600i			1.00981i	0.863176	+	0.504904i	$0.168472\pi$
$367$			370.600i			1.00981i	−0.863176	+	0.504904i	$0.831528\pi$
$368$	173.112			0.470412
$369$		0			0
$370$	316.828	−	11.0422i	0.856292	−	0.0298437i
$371$			585.103i			1.57710i
$372$		0			0
$373$			514.661i			1.37979i	0.723911	+	0.689894i	$0.242343\pi$
$373$			514.661i			1.37979i	−0.723911	+	0.689894i	$0.757657\pi$
$374$		−	691.927i		−	1.85007i
$375$		0			0
$376$	192.551			0.512103
$377$	−119.884			−0.317996
$378$		0			0
$379$	−159.583			−0.421063			−0.210532	−	0.977587i	$-0.567519\pi$
$379$	−159.583			−0.421063			−0.210532	+	0.977587i	$0.567519\pi$
$380$	−5.59915	−	160.654i	−0.0147346	−	0.422774i
$381$		0			0
$382$		−	17.8723i		−	0.0467862i
$383$	−448.043			−1.16983			−0.584913	−	0.811096i	$-0.698871\pi$
$383$	−448.043			−1.16983			−0.584913	+	0.811096i	$0.698871\pi$
$384$		0			0
$385$	−33.6414	−	965.258i	−0.0873802	−	2.50716i
$386$		−	237.764i		−	0.615969i
$387$		0			0
$388$			239.772i			0.617970i
$389$		−	558.830i		−	1.43658i	−0.695743	−	0.718291i	$-0.744925\pi$
$389$		−	558.830i		−	1.43658i	0.695743	−	0.718291i	$-0.255075\pi$
$390$		0			0
$391$	1038.68			2.65647
$392$	115.363			0.294293
$393$		0			0
$394$	−526.150			−1.33541
$395$	−5.78784	−	166.068i	−0.0146527	−	0.420425i
$396$		0			0
$397$		−	266.143i		−	0.670385i	−0.942150	−	0.335192i	$-0.891199\pi$
$397$		−	266.143i		−	0.670385i	0.942150	−	0.335192i	$-0.108801\pi$
$398$	−327.845			−0.823731
$399$		0			0
$400$	−99.7574	+	6.96198i	−0.249393	+	0.0174050i
$401$		−	17.4204i		−	0.0434423i	−0.999764	−	0.0217212i	$-0.993085\pi$
$401$		−	17.4204i		−	0.0434423i	0.999764	−	0.0217212i	$-0.00691460\pi$
$402$		0			0
$403$		−	186.395i		−	0.462518i
$404$		−	44.1549i		−	0.109294i
$405$		0			0
$406$	262.938			0.647630
$407$	913.972			2.24563
$408$		0			0
$409$	−445.763			−1.08989			−0.544943	−	0.838473i	$-0.683449\pi$
$409$	−445.763			−1.08989			−0.544943	+	0.838473i	$0.683449\pi$
$410$	−339.847	+	11.8444i	−0.828895	+	0.0288888i
$411$		0			0
$412$			36.7743i			0.0892581i
$413$	−477.960			−1.15729
$414$		0			0
$415$	3.95583	+	113.503i	0.00953213	+	0.273502i
$416$			34.5626i			0.0830831i
$417$		0			0
$418$		−	463.448i		−	1.10873i
$419$		−	357.460i		−	0.853127i	−0.904458	−	0.426563i	$-0.859724\pi$
$419$		−	357.460i		−	0.853127i	0.904458	−	0.426563i	$-0.140276\pi$
$420$		0			0
$421$	575.027			1.36586			0.682930	−	0.730484i	$-0.260705\pi$
$421$	575.027			1.36586			0.682930	+	0.730484i	$0.260705\pi$
$422$	−29.5588			−0.0700445
$423$		0			0
$424$	174.651			0.411912
$425$	−598.549	+	41.7722i	−1.40835	+	0.0982876i
$426$		0			0
$427$			464.280i			1.08731i
$428$	−121.720			−0.284393
$429$		0			0
$430$	339.944	−	11.8478i	0.790566	−	0.0275530i
$431$		−	181.318i		−	0.420691i	−0.977627	−	0.210346i	$-0.932541\pi$
$431$		−	181.318i		−	0.420691i	0.977627	−	0.210346i	$-0.0674589\pi$
$432$		0			0
$433$		−	705.627i		−	1.62962i	−0.579726	−	0.814811i	$-0.696840\pi$
$433$		−	705.627i		−	1.62962i	0.579726	−	0.814811i	$-0.303160\pi$
$434$			408.813i			0.941965i
$435$		0			0
$436$	160.978			0.369216
$437$	695.700			1.59199
$438$		0			0
$439$	285.553			0.650463			0.325231	−	0.945634i	$-0.394558\pi$
$439$	285.553			0.650463			0.325231	+	0.945634i	$0.394558\pi$
$440$	−288.126	+	10.0418i	−0.654831	+	0.0228223i
$441$		0			0
$442$			207.377i			0.469179i
$443$	−706.104			−1.59391			−0.796957	−	0.604036i	$-0.793558\pi$
$443$	−706.104			−1.59391			−0.796957	+	0.604036i	$0.793558\pi$
$444$		0			0
$445$	−397.392	+	13.8500i	−0.893015	+	0.0311235i
$446$			22.3064i			0.0500144i
$447$		0			0
$448$		−	75.8048i		−	0.169207i
$449$			505.807i			1.12652i	0.826280	+	0.563260i	$0.190453\pi$
$449$			505.807i			1.12652i	−0.826280	+	0.563260i	$0.809547\pi$
$450$		0			0
$451$	−980.376			−2.17378
$452$	45.5113			0.100689
$453$		0			0
$454$	−482.736			−1.06330
$455$	10.0826	+	289.297i	0.0221597	+	0.635818i
$456$		0			0
$457$			710.680i			1.55510i	0.628822	+	0.777550i	$0.283538\pi$
$457$			710.680i			1.55510i	−0.628822	+	0.777550i	$0.716462\pi$
$458$	−170.346			−0.371934
$459$		0			0
$460$	−15.0741	−	432.516i	−0.0327699	−	0.940253i
$461$			137.615i			0.298513i	0.988799	+	0.149257i	$0.0476880\pi$
$461$			137.615i			0.298513i	−0.988799	+	0.149257i	$0.952312\pi$
$462$		0			0
$463$		−	344.162i		−	0.743330i	−0.928367	−	0.371665i	$-0.878787\pi$
$463$		−	344.162i		−	0.743330i	0.928367	−	0.371665i	$-0.121213\pi$
$464$		−	78.4858i		−	0.169150i
$465$		0			0
$466$	434.730			0.932896
$467$	−322.715			−0.691038			−0.345519	−	0.938412i	$-0.612297\pi$
$467$	−322.715			−0.691038			−0.345519	+	0.938412i	$0.612297\pi$
$468$		0			0
$469$	−522.018			−1.11304
$470$	−16.7669	−	481.085i	−0.0356742	−	1.02359i
$471$		0			0
$472$			142.669i			0.302265i
$473$	980.654			2.07327
$474$		0			0
$475$	−400.904	+	27.9788i	−0.844009	+	0.0589027i
$476$		−	454.832i		−	0.955530i
$477$		0			0
$478$		−	168.409i		−	0.352321i
$479$		−	85.1304i		−	0.177725i	−0.996044	−	0.0888626i	$-0.971677\pi$
$479$		−	85.1304i		−	0.177725i	0.996044	−	0.0888626i	$-0.0283232\pi$
$480$		0			0
$481$	−273.926			−0.569493
$482$	444.911			0.923053
$483$		0			0
$484$	−589.173			−1.21730
$485$	599.067	−	20.8788i	1.23519	−	0.0430491i
$486$		0			0
$487$		−	437.344i		−	0.898037i	−0.893522	−	0.449018i	$-0.851774\pi$
$487$		−	437.344i		−	0.898037i	0.893522	−	0.449018i	$-0.148226\pi$
$488$	138.586			0.283987
$489$		0			0
$490$	−10.0455	−	288.232i	−0.0205011	−	0.588229i
$491$		−	679.475i		−	1.38386i	−0.721965	−	0.691930i	$-0.756761\pi$
$491$		−	679.475i		−	1.38386i	0.721965	−	0.691930i	$-0.243239\pi$
$492$		0			0
$493$		−	470.919i		−	0.955211i
$494$			138.900i			0.281174i
$495$		0			0
$496$	122.029			0.246026
$497$	−223.607			−0.449913
$498$		0			0
$499$	300.560			0.602325			0.301162	−	0.953573i	$-0.402625\pi$
$499$	300.560			0.602325			0.301162	+	0.953573i	$0.402625\pi$
$500$	26.0810	+	248.636i	0.0521621	+	0.497272i
$501$		0			0
$502$			209.577i			0.417484i
$503$	−371.067			−0.737707			−0.368853	−	0.929488i	$-0.620250\pi$
$503$	−371.067			−0.737707			−0.368853	+	0.929488i	$0.620250\pi$
$504$		0			0
$505$	−110.320	+	3.84490i	−0.218456	+	0.00761367i
$506$		−	1247.70i		−	2.46582i
$507$		0			0
$508$			146.712i			0.288804i
$509$			581.653i			1.14274i	0.820693	+	0.571369i	$0.193587\pi$
$509$			581.653i			1.14274i	−0.820693	+	0.571369i	$0.806413\pi$
$510$		0			0
$511$	−383.101			−0.749709
$512$	−22.6274			−0.0441942
$513$		0			0
$514$	416.225			0.809777
$515$	91.8800	−	3.20222i	0.178408	−	0.00621791i
$516$		0			0
$517$		−	1387.81i		−	2.68436i
$518$	600.792			1.15983
$519$		0			0
$520$	86.3540	−	3.00963i	0.166065	−	0.00578775i
$521$			1006.68i			1.93222i	0.258138	+	0.966108i	$0.416891\pi$
$521$			1006.68i			1.93222i	−0.258138	+	0.966108i	$0.583109\pi$
$522$		0			0
$523$			674.227i			1.28915i	0.764540	+	0.644576i	$0.222966\pi$
$523$			674.227i			1.28915i	−0.764540	+	0.644576i	$0.777034\pi$
$524$		−	182.102i		−	0.347522i
$525$		0			0
$526$	−98.2376			−0.186763
$527$	732.180			1.38934
$528$		0			0
$529$	1343.98			2.54060
$530$	−15.2082	−	436.362i	−0.0286947	−	0.823325i
$531$		0			0
$532$		−	304.644i		−	0.572639i
$533$	293.828			0.551272
$534$		0			0
$535$	10.5991	+	304.116i	0.0198114	+	0.568441i
$536$			155.820i			0.290709i
$537$		0			0
$538$		−	371.174i		−	0.689915i
$539$		−	831.480i		−	1.54263i
$540$		0			0
$541$	178.075			0.329159			0.164579	−	0.986364i	$-0.447373\pi$
$541$	178.075			0.329159			0.164579	+	0.986364i	$0.447373\pi$
$542$	530.485			0.978755
$543$		0			0
$544$	−135.766			−0.249569
$545$	−14.0176	−	402.202i	−0.0257204	−	0.737985i
$546$		0			0
$547$		−	206.149i		−	0.376873i	−0.982085	−	0.188436i	$-0.939658\pi$
$547$		−	206.149i		−	0.376873i	0.982085	−	0.188436i	$-0.0603419\pi$
$548$	55.7674			0.101765
$549$		0			0
$550$	50.1786	+	719.003i	0.0912338	+	1.30728i
$551$		−	315.418i		−	0.572447i
$552$		0			0
$553$		−	314.910i		−	0.569457i
$554$		−	206.999i		−	0.373644i
$555$		0			0
$556$	69.4759			0.124957
$557$	−37.8724			−0.0679935			−0.0339968	−	0.999422i	$-0.510824\pi$
$557$	−37.8724			−0.0679935			−0.0339968	+	0.999422i	$0.510824\pi$
$558$		0			0
$559$	−293.911			−0.525781
$560$	−189.397	+	6.60090i	−0.338209	+	0.0117873i
$561$		0			0
$562$			111.962i			0.199221i
$563$	317.255			0.563508			0.281754	−	0.959487i	$-0.409084\pi$
$563$	317.255			0.563508			0.281754	+	0.959487i	$0.409084\pi$
$564$		0			0
$565$	−3.96301	−	113.709i	−0.00701418	−	0.201255i
$566$			305.931i			0.540513i
$567$		0			0
$568$			66.7457i			0.117510i
$569$			403.753i			0.709584i	0.934945	+	0.354792i	$0.115448\pi$
$569$			403.753i			0.709584i	−0.934945	+	0.354792i	$0.884552\pi$
$570$		0			0
$571$	284.074			0.497502			0.248751	−	0.968567i	$-0.419980\pi$
$571$	284.074			0.497502			0.248751	+	0.968567i	$0.419980\pi$
$572$	249.110			0.435508
$573$		0			0
$574$	−644.442			−1.12272
$575$	−1079.32	+	75.3250i	−1.87708	+	0.131000i
$576$		0			0
$577$			105.236i			0.182386i	0.995833	+	0.0911928i	$0.0290679\pi$
$577$			105.236i			0.182386i	−0.995833	+	0.0911928i	$0.970932\pi$
$578$	−405.893			−0.702237
$579$		0			0
$580$	−196.095	+	6.83436i	−0.338096	+	0.0117834i
$581$			215.233i			0.370452i
$582$		0			0
$583$		−	1258.80i		−	2.15917i
$584$			114.354i			0.195812i
$585$		0			0
$586$	588.782			1.00475
$587$	−877.574			−1.49502			−0.747508	−	0.664253i	$-0.768750\pi$
$587$	−877.574			−1.49502			−0.747508	+	0.664253i	$0.768750\pi$
$588$		0			0
$589$	490.409			0.832613
$590$	356.456	−	12.4233i	0.604163	−	0.0210564i
$591$		0			0
$592$		−	179.334i		−	0.302929i
$593$	−47.7971			−0.0806022			−0.0403011	−	0.999188i	$-0.512832\pi$
$593$	−47.7971			−0.0806022			−0.0403011	+	0.999188i	$0.512832\pi$
$594$		0			0
$595$	−1136.39	+	39.6057i	−1.90990	+	0.0665643i
$596$			0.634137i			0.00106399i
$597$		0			0
$598$			373.949i			0.625333i
$599$			1137.50i			1.89900i	0.313761	+	0.949502i	$0.398411\pi$
$599$			1137.50i			1.89900i	−0.313761	+	0.949502i	$0.601589\pi$
$600$		0			0
$601$	190.726			0.317348			0.158674	−	0.987331i	$-0.449278\pi$
$601$	190.726			0.317348			0.158674	+	0.987331i	$0.449278\pi$
$602$	644.625			1.07081
$603$		0			0
$604$	−210.655			−0.348767
$605$	51.3037	+	1472.04i	0.0847996	+	2.43312i
$606$		0			0
$607$		−	988.920i		−	1.62919i	−0.580028	−	0.814597i	$-0.696958\pi$
$607$		−	988.920i		−	1.62919i	0.580028	−	0.814597i	$-0.303042\pi$
$608$	−90.9349			−0.149564
$609$		0			0
$610$	−12.0677	−	346.254i	−0.0197831	−	0.567630i
$611$			415.941i			0.680754i
$612$		0			0
$613$		−	131.155i		−	0.213956i	−0.994261	−	0.106978i	$-0.965882\pi$
$613$		−	131.155i		−	0.213956i	0.994261	−	0.106978i	$-0.0341175\pi$
$614$			468.912i			0.763701i
$615$		0			0
$616$	−546.364			−0.886955
$617$	−433.317			−0.702296			−0.351148	−	0.936320i	$-0.614209\pi$
$617$	−433.317			−0.702296			−0.351148	+	0.936320i	$0.614209\pi$
$618$		0			0
$619$	579.893			0.936823			0.468411	−	0.883511i	$-0.344827\pi$
$619$	579.893			0.936823			0.468411	+	0.883511i	$0.344827\pi$
$620$	−10.6260	−	304.887i	−0.0171387	−	0.491754i
$621$		0			0
$622$		−	556.589i		−	0.894837i
$623$	−753.562			−1.20957
$624$		0			0
$625$	618.941	−	86.8136i	0.990306	−	0.138902i
$626$		−	359.394i		−	0.574112i
$627$		0			0
$628$		−	91.2046i		−	0.145230i
$629$		−	1076.01i		−	1.71067i
$630$		0			0
$631$	1026.45			1.62671			0.813354	−	0.581768i	$-0.197639\pi$
$631$	1026.45			1.62671			0.813354	+	0.581768i	$0.197639\pi$
$632$	−93.9993			−0.148733
$633$		0			0
$634$	−610.916			−0.963591
$635$	366.559	−	12.7754i	0.577258	−	0.0201187i
$636$		0			0
$637$			249.202i			0.391213i
$638$	−565.688			−0.886658
$639$		0			0
$640$	1.97034	+	56.5342i	0.00307866	+	0.0883347i
$641$			326.380i			0.509173i	0.967050	+	0.254586i	$0.0819394\pi$
$641$			326.380i			0.509173i	−0.967050	+	0.254586i	$0.918061\pi$
$642$		0			0
$643$			558.564i			0.868685i	0.900748	+	0.434342i	$0.143019\pi$
$643$			558.564i			0.868685i	−0.900748	+	0.434342i	$0.856981\pi$
$644$		−	820.168i		−	1.27355i
$645$		0			0
$646$	−545.614			−0.844604
$647$	404.040			0.624483			0.312241	−	0.950003i	$-0.398920\pi$
$647$	404.040			0.624483			0.312241	+	0.950003i	$0.398920\pi$
$648$		0			0
$649$	1028.29			1.58442
$650$	−15.0390	−	215.492i	−0.0231369	−	0.331526i
$651$		0			0
$652$		−	305.587i		−	0.468692i
$653$	714.002			1.09342			0.546709	−	0.837323i	$-0.315880\pi$
$653$	714.002			1.09342			0.546709	+	0.837323i	$0.315880\pi$
$654$		0			0
$655$	−454.978	+	15.8570i	−0.694623	+	0.0242091i
$656$			192.363i			0.293237i
$657$		0			0
$658$		−	912.267i		−	1.38642i
$659$		−	127.690i		−	0.193763i	−0.995296	−	0.0968817i	$-0.969113\pi$
$659$		−	127.690i		−	0.193763i	0.995296	−	0.0968817i	$-0.0308868\pi$
$660$		0			0
$661$	−404.692			−0.612242			−0.306121	−	0.951993i	$-0.599031\pi$
$661$	−404.692			−0.612242			−0.306121	+	0.951993i	$0.599031\pi$
$662$	−271.149			−0.409591
$663$		0			0
$664$	64.2461			0.0967561
$665$	−761.147	+	26.5277i	−1.14458	+	0.0398912i
$666$		0			0
$667$		−	849.175i		−	1.27313i
$668$	27.4568			0.0411030
$669$		0			0
$670$	389.314	−	13.5685i	0.581066	−	0.0202514i
$671$		−	998.859i		−	1.48861i
$672$		0			0
$673$			1239.08i			1.84112i	0.390595	+	0.920562i	$0.372269\pi$
$673$			1239.08i			1.84112i	−0.390595	+	0.920562i	$0.627731\pi$
$674$			359.365i			0.533183i
$675$		0			0
$676$	263.339			0.389555
$677$	−679.087			−1.00308			−0.501541	−	0.865134i	$-0.667234\pi$
$677$	−679.087			−1.00308			−0.501541	+	0.865134i	$0.667234\pi$
$678$		0			0
$679$	1135.99			1.67304
$680$	11.8221	+	339.208i	0.0173855	+	0.498835i
$681$		0			0
$682$		−	879.526i		−	1.28963i
$683$	439.600			0.643632			0.321816	−	0.946802i	$-0.395707\pi$
$683$	439.600			0.643632			0.321816	+	0.946802i	$0.395707\pi$
$684$		0			0
$685$	−4.85610	−	139.334i	−0.00708919	−	0.203407i
$686$			110.059i			0.160436i
$687$		0			0
$688$		−	192.418i		−	0.279677i
$689$			377.274i			0.547567i
$690$		0			0
$691$	−324.689			−0.469883			−0.234941	−	0.972010i	$-0.575490\pi$
$691$	−324.689			−0.469883			−0.234941	+	0.972010i	$0.575490\pi$
$692$	20.8614			0.0301466
$693$		0			0
$694$	−453.673			−0.653707
$695$	−6.04979	−	173.584i	−0.00870474	−	0.249761i
$696$		0			0
$697$			1154.19i			1.65594i
$698$	−430.662			−0.616995
$699$		0			0
$700$	32.9845	+	472.630i	0.0471207	+	0.675186i
$701$			319.512i			0.455795i	0.973685	+	0.227898i	$0.0731851\pi$
$701$			319.512i			0.455795i	−0.973685	+	0.227898i	$0.926815\pi$
$702$		0			0
$703$		−	720.706i		−	1.02519i
$704$			163.087i			0.231658i
$705$		0			0
$706$	305.128			0.432193
$707$	−209.197			−0.295894
$708$		0			0
$709$	−1106.62			−1.56082			−0.780411	−	0.625267i	$-0.784990\pi$
$709$	−1106.62			−1.56082			−0.780411	+	0.625267i	$0.784990\pi$
$710$	166.763	−	5.81206i	0.234877	−	0.00818600i
$711$		0			0
$712$			224.935i			0.315920i
$713$	1320.29			1.85174
$714$		0			0
$715$	−21.6919	−	622.398i	−0.0303384	−	0.870487i
$716$			124.930i			0.174483i
$717$		0			0
$718$			902.424i			1.25686i
$719$			992.085i			1.37981i	0.723899	+	0.689906i	$0.242348\pi$
$719$			992.085i			1.37981i	−0.723899	+	0.689906i	$0.757652\pi$
$720$		0			0
$721$	174.229			0.241650
$722$	145.083			0.200945
$723$		0			0
$724$	686.058			0.947594
$725$	34.1511	+	489.346i	0.0471049	+	0.674960i
$726$		0			0
$727$		−	6.95114i		−	0.00956140i	−0.999989	−	0.00478070i	$-0.998478\pi$
$727$		−	6.95114i		−	0.00956140i	0.999989	−	0.00478070i	$-0.00152175\pi$
$728$	163.751			0.224932
$729$		0			0
$730$	285.712	−	9.95769i	0.391386	−	0.0136407i
$731$		−	1154.52i		−	1.57937i
$732$		0			0
$733$		−	249.539i		−	0.340436i	−0.985406	−	0.170218i	$-0.945553\pi$
$733$		−	249.539i		−	0.340436i	0.985406	−	0.170218i	$-0.0544471\pi$
$734$		−	524.107i		−	0.714042i
$735$		0			0
$736$	−244.817			−0.332631
$737$	1123.08			1.52385
$738$		0			0
$739$	−961.297			−1.30081			−0.650404	−	0.759588i	$-0.725400\pi$
$739$	−961.297			−1.30081			−0.650404	+	0.759588i	$0.725400\pi$
$740$	−448.063	+	15.6160i	−0.605490	+	0.0211027i
$741$		0			0
$742$		−	827.460i		−	1.11518i
$743$	−796.752			−1.07234			−0.536172	−	0.844108i	$-0.680130\pi$
$743$	−796.752			−1.07234			−0.536172	+	0.844108i	$0.680130\pi$
$744$		0			0
$745$	1.58438	−	0.0552192i	0.00212669	−	7.41197e-5i
$746$		−	727.840i		−	0.975657i
$747$		0			0
$748$			978.533i			1.30820i
$749$			576.686i			0.769941i
$750$		0			0
$751$	306.137			0.407639			0.203820	−	0.979008i	$-0.434664\pi$
$751$	306.137			0.407639			0.203820	+	0.979008i	$0.434664\pi$
$752$	−272.308			−0.362112
$753$		0			0
$754$	169.542			0.224857
$755$	18.3434	+	526.319i	0.0242958	+	0.697111i
$756$		0			0
$757$		−	411.616i		−	0.543746i	−0.962333	−	0.271873i	$-0.912357\pi$
$757$		−	411.616i		−	0.543746i	0.962333	−	0.271873i	$-0.0876431\pi$
$758$	225.684			0.297737
$759$		0			0
$760$	7.91840	+	227.199i	0.0104189	+	0.298946i
$761$			535.619i			0.703835i	0.936031	+	0.351918i	$0.114470\pi$
$761$			535.619i			0.703835i	−0.936031	+	0.351918i	$0.885530\pi$
$762$		0			0
$763$		−	762.683i		−	0.999584i
$764$			25.2753i			0.0330829i
$765$		0			0
$766$	633.629			0.827192
$767$	−308.188			−0.401810
$768$		0			0
$769$	−1081.83			−1.40680			−0.703402	−	0.710792i	$-0.748337\pi$
$769$	−1081.83			−1.40680			−0.703402	+	0.710792i	$0.748337\pi$
$770$	47.5761	+	1365.08i	0.0617871	+	1.77283i
$771$		0			0
$772$			336.249i			0.435556i
$773$	166.623			0.215554			0.107777	−	0.994175i	$-0.465627\pi$
$773$	166.623			0.215554			0.107777	+	0.994175i	$0.465627\pi$
$774$		0			0
$775$	−760.830	+	53.0977i	−0.981717	+	0.0685132i
$776$		−	339.089i		−	0.436971i
$777$		0			0
$778$			790.306i			1.01582i
$779$			773.068i			0.992385i
$780$		0			0
$781$	481.071			0.615968
$782$	−1468.91			−1.87841
$783$		0			0
$784$	−163.148			−0.208097
$785$	−227.873	+	7.94188i	−0.290284	+	0.0101170i
$786$		0			0
$787$			406.762i			0.516852i	0.966031	+	0.258426i	$0.0832038\pi$
$787$			406.762i			0.516852i	−0.966031	+	0.258426i	$0.916796\pi$
$788$	744.089			0.944275
$789$		0			0
$790$	8.18524	+	234.856i	0.0103611	+	0.297286i
$791$		−	215.623i		−	0.272596i
$792$		0			0
$793$			299.368i			0.377513i
$794$			376.383i			0.474034i
$795$		0			0
$796$	463.643			0.582466
$797$	103.236			0.129530			0.0647652	−	0.997901i	$-0.479370\pi$
$797$	103.236			0.129530			0.0647652	+	0.997901i	$0.479370\pi$
$798$		0			0
$799$	−1633.86			−2.04488
$800$	141.078	−	9.84573i	0.176348	−	0.0123072i
$801$		0			0
$802$			24.6361i			0.0307184i
$803$	824.210			1.02641
$804$		0			0
$805$	−2049.18	+	71.4183i	−2.54556	+	0.0887183i
$806$			263.602i			0.327050i
$807$		0			0
$808$			62.4444i			0.0772827i
$809$			833.545i			1.03034i	0.857088	+	0.515170i	$0.172271\pi$
$809$			833.545i			1.03034i	−0.857088	+	0.515170i	$0.827729\pi$
$810$		0			0
$811$	814.193			1.00394			0.501968	−	0.864886i	$-0.332609\pi$
$811$	814.193			1.00394			0.501968	+	0.864886i	$0.332609\pi$
$812$	−371.850			−0.457943
$813$		0			0
$814$	−1292.55			−1.58790
$815$	−763.504	+	26.6098i	−0.936815	+	0.0326501i
$816$		0			0
$817$		−	773.287i		−	0.946496i
$818$	630.405			0.770666
$819$		0			0
$820$	480.616	−	16.7505i	0.586118	−	0.0204275i
$821$			154.644i			0.188361i	0.995555	+	0.0941803i	$0.0300230\pi$
$821$			154.644i			0.188361i	−0.995555	+	0.0941803i	$0.969977\pi$
$822$		0			0
$823$			1103.86i			1.34127i	0.741788	+	0.670635i	$0.233978\pi$
$823$			1103.86i			1.34127i	−0.741788	+	0.670635i	$0.766022\pi$
$824$		−	52.0068i		−	0.0631150i
$825$		0			0
$826$	675.937			0.818326
$827$	−423.629			−0.512248			−0.256124	−	0.966644i	$-0.582446\pi$
$827$	−423.629			−0.512248			−0.256124	+	0.966644i	$0.582446\pi$
$828$		0			0
$829$	325.734			0.392924			0.196462	−	0.980511i	$-0.437055\pi$
$829$	325.734			0.392924			0.196462	+	0.980511i	$0.437055\pi$
$830$	−5.59439	−	160.518i	−0.00674023	−	0.193395i
$831$		0			0
$832$		−	48.8789i		−	0.0587486i
$833$	−978.895			−1.17514
$834$		0			0
$835$	−2.39087	−	68.6003i	−0.00286332	−	0.0821561i
$836$			655.415i			0.783989i
$837$		0			0
$838$			505.525i			0.603252i
$839$		−	1213.40i		−	1.44624i	−0.690721	−	0.723121i	$-0.742707\pi$
$839$		−	1213.40i		−	1.44624i	0.690721	−	0.723121i	$-0.257293\pi$
$840$		0			0
$841$	455.998			0.542210
$842$	−813.211			−0.965809
$843$		0			0
$844$	41.8024			0.0495289
$845$	−22.9310	−	657.949i	−0.0271372	−	0.778637i
$846$		0			0
$847$			2791.38i			3.29561i
$848$	−246.994			−0.291266
$849$		0			0
$850$	846.476	−	59.0749i	0.995855	−	0.0694999i
$851$		−	1940.30i		−	2.28002i
$852$		0			0
$853$			35.0492i			0.0410893i	0.999789	+	0.0205446i	$0.00654002\pi$
$853$			35.0492i			0.0410893i	−0.999789	+	0.0205446i	$0.993460\pi$
$854$		−	656.591i		−	0.768842i
$855$		0			0
$856$	172.138			0.201096
$857$	1097.41			1.28053			0.640265	−	0.768154i	$-0.278825\pi$
$857$	1097.41			1.28053			0.640265	+	0.768154i	$0.278825\pi$
$858$		0			0
$859$	904.289			1.05272			0.526362	−	0.850261i	$-0.323556\pi$
$859$	904.289			1.05272			0.526362	+	0.850261i	$0.323556\pi$
$860$	−480.753	+	16.7553i	−0.559015	+	0.0194829i
$861$		0			0
$862$			256.422i			0.297473i
$863$	−179.328			−0.207796			−0.103898	−	0.994588i	$-0.533132\pi$
$863$	−179.328			−0.207796			−0.103898	+	0.994588i	$0.533132\pi$
$864$		0			0
$865$	−1.81656	−	52.1219i	−0.00210007	−	0.0602566i
$866$			997.907i			1.15232i
$867$		0			0
$868$		−	578.149i		−	0.666070i
$869$			677.501i			0.779633i
$870$		0			0
$871$	−336.597			−0.386449
$872$	−227.658			−0.261075
$873$		0			0
$874$	−983.868			−1.12571
$875$	1177.99	−	123.567i	1.34627	−	0.141219i
$876$		0			0
$877$			1207.53i			1.37689i	0.725290	+	0.688443i	$0.241706\pi$
$877$			1207.53i			1.37689i	−0.725290	+	0.688443i	$0.758294\pi$
$878$	−403.833			−0.459947
$879$		0			0
$880$	407.471	−	14.2013i	0.463036	−	0.0161378i
$881$			407.569i			0.462621i	0.972880	+	0.231311i	$0.0743013\pi$
$881$			407.569i			0.462621i	−0.972880	+	0.231311i	$0.925699\pi$
$882$		0			0
$883$			85.9086i			0.0972918i	0.998816	+	0.0486459i	$0.0154906\pi$
$883$			85.9086i			0.0972918i	−0.998816	+	0.0486459i	$0.984509\pi$
$884$		−	293.276i		−	0.331760i
$885$		0			0
$886$	998.582			1.12707
$887$	398.565			0.449340			0.224670	−	0.974435i	$-0.427870\pi$
$887$	398.565			0.449340			0.224670	+	0.974435i	$0.427870\pi$
$888$		0			0
$889$	695.094			0.781883
$890$	561.997	−	19.5868i	0.631457	−	0.0220077i
$891$		0			0
$892$		−	31.5461i		−	0.0353655i
$893$	−1094.35			−1.22548
$894$		0			0
$895$	312.136	−	10.8786i	0.348755	−	0.0121549i
$896$			107.204i			0.119647i
$897$		0			0
$898$		−	715.320i		−	0.796570i
$899$		−	598.596i		−	0.665847i
$900$		0			0
$901$	−1481.97			−1.64481
$902$	1386.46			1.53710
$903$		0			0
$904$	−64.3626			−0.0711976
$905$	−59.7403	−	1714.10i	−0.0660114	−	1.89404i
$906$		0			0
$907$			1194.39i			1.31686i	0.752643	+	0.658429i	$0.228779\pi$
$907$			1194.39i			1.31686i	−0.752643	+	0.658429i	$0.771221\pi$
$908$	682.692			0.751863
$909$		0			0
$910$	−14.2590	−	409.128i	−0.0156692	−	0.449591i
$911$			1348.96i			1.48074i	0.672199	+	0.740371i	$0.265350\pi$
$911$			1348.96i			1.48074i	−0.672199	+	0.740371i	$0.734650\pi$
$912$		0			0
$913$		−	463.054i		−	0.507179i
$914$		−	1005.05i		−	1.09962i
$915$		0			0
$916$	240.905			0.262997
$917$	−862.761			−0.940852
$918$		0			0
$919$	1571.32			1.70982			0.854908	−	0.518780i	$-0.173614\pi$
$919$	1571.32			1.70982			0.854908	+	0.518780i	$0.173614\pi$
$920$	21.3181	+	611.670i	0.0231718	+	0.664859i
$921$		0			0
$922$		−	194.616i		−	0.211081i
$923$	−144.181			−0.156210
$924$		0			0
$925$	78.0325	+	1118.12i	0.0843594	+	1.20878i
$926$			486.718i			0.525614i
$927$		0			0
$928$			110.996i			0.119607i
$929$		−	251.609i		−	0.270839i	−0.990788	−	0.135420i	$-0.956762\pi$
$929$		−	251.609i		−	0.270839i	0.990788	−	0.135420i	$-0.0432382\pi$
$930$		0			0
$931$	−655.657			−0.704250
$932$	−614.800			−0.659657
$933$		0			0
$934$	456.388			0.488638
$935$	2444.85	−	85.2083i	2.61481	−	0.0911319i
$936$		0			0
$937$			1592.01i			1.69905i	0.527546	+	0.849526i	$0.323112\pi$
$937$			1592.01i			1.69905i	−0.527546	+	0.849526i	$0.676888\pi$
$938$	738.245			0.787041
$939$		0			0
$940$	23.7119	+	680.357i	0.0252255	+	0.723784i
$941$			1020.39i			1.08437i	0.840260	+	0.542184i	$0.182402\pi$
$941$			1020.39i			1.08437i	−0.840260	+	0.542184i	$0.817598\pi$
$942$		0			0
$943$			2081.27i			2.20707i
$944$		−	201.764i		−	0.213734i
$945$		0			0
$946$	−1386.85			−1.46602
$947$	674.066			0.711791			0.355895	−	0.934526i	$-0.384176\pi$
$947$	674.066			0.711791			0.355895	+	0.934526i	$0.384176\pi$
$948$		0			0
$949$	−247.024			−0.260299
$950$	566.964	−	39.5680i	0.596804	−	0.0416505i
$951$		0			0
$952$			643.230i			0.675662i
$953$	836.522			0.877778			0.438889	−	0.898541i	$-0.355372\pi$
$953$	836.522			0.877778			0.438889	+	0.898541i	$0.355372\pi$
$954$		0			0
$955$	63.1499	−	2.20091i	0.0661256	−	0.00230462i
$956$			238.167i			0.249128i
$957$		0			0
$958$			120.393i			0.125671i
$959$		−	264.215i		−	0.275511i
$960$		0			0
$961$	−30.3083			−0.0315383
$962$	387.390			0.402692
$963$		0			0
$964$	−629.200			−0.652697
$965$	840.112	−	29.2798i	0.870583	−	0.0303417i
$966$		0			0
$967$			1018.19i			1.05294i	0.850194	+	0.526470i	$0.176485\pi$
$967$			1018.19i			1.05294i	−0.850194	+	0.526470i	$0.823515\pi$
$968$	833.216			0.860760
$969$		0			0
$970$	−847.209	+	29.5271i	−0.873412	+	0.0304403i
$971$		−	997.549i		−	1.02734i	−0.857987	−	0.513671i	$-0.828285\pi$
$971$		−	997.549i		−	1.02734i	0.857987	−	0.513671i	$-0.171715\pi$
$972$		0			0
$973$		−	329.163i		−	0.338297i
$974$			618.498i			0.635008i
$975$		0			0
$976$	−195.990			−0.200809
$977$	−1104.02			−1.13001			−0.565003	−	0.825089i	$-0.691125\pi$
$977$	−1104.02			−1.13001			−0.565003	+	0.825089i	$0.691125\pi$
$978$		0			0
$979$	1621.22			1.65600
$980$	14.2065	+	407.622i	0.0144965	+	0.415941i
$981$		0			0
$982$			960.923i			0.978537i
$983$	1216.60			1.23764			0.618822	−	0.785531i	$-0.287610\pi$
$983$	1216.60			1.23764			0.618822	+	0.785531i	$0.287610\pi$
$984$		0			0
$985$	−64.7935	−	1859.09i	−0.0657802	−	1.88740i
$986$			665.980i			0.675436i
$987$		0			0
$988$		−	196.434i		−	0.198820i
$989$		−	2081.86i		−	2.10502i
$990$		0			0
$991$	736.595			0.743285			0.371642	−	0.928376i	$-0.378795\pi$
$991$	736.595			0.743285			0.371642	+	0.928376i	$0.378795\pi$
$992$	−172.575			−0.173967
$993$		0			0
$994$	316.228			0.318137
$995$	−40.3729	−	1158.40i	−0.0405758	−	1.16422i
$996$		0			0
$997$		−	751.131i		−	0.753391i	−0.926337	−	0.376696i	$-0.877060\pi$
$997$		−	751.131i		−	0.753391i	0.926337	−	0.376696i	$-0.122940\pi$
$998$	−425.056			−0.425908
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.3.b.c.809.7	✓	24
3.2	odd	2	inner	810.3.b.c.809.18	yes	24
5.4	even	2	inner	810.3.b.c.809.17	yes	24
9.2	odd	6		810.3.j.g.539.6		24
9.4	even	3		810.3.j.h.269.12		24
9.5	odd	6		810.3.j.h.269.6		24
9.7	even	3		810.3.j.g.539.12		24
15.14	odd	2	inner	810.3.b.c.809.8	yes	24
45.4	even	6		810.3.j.g.269.6		24
45.14	odd	6		810.3.j.g.269.12		24
45.29	odd	6		810.3.j.h.539.12		24
45.34	even	6		810.3.j.h.539.6		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
810.3.b.c.809.7	✓	24	1.1	even	1	trivial
810.3.b.c.809.8	yes	24	15.14	odd	2	inner
810.3.b.c.809.17	yes	24	5.4	even	2	inner
810.3.b.c.809.18	yes	24	3.2	odd	2	inner
810.3.j.g.269.6		24	45.4	even	6
810.3.j.g.269.12		24	45.14	odd	6
810.3.j.g.539.6		24	9.2	odd	6
810.3.j.g.539.12		24	9.7	even	3
810.3.j.h.269.6		24	9.5	odd	6
810.3.j.h.269.12		24	9.4	even	3
810.3.j.h.539.6		24	45.34	even	6
810.3.j.h.539.12		24	45.29	odd	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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	810.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +(-0.174155 - 4.99697i) q^{5} -9.47560i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} +(-0.174155 - 4.99697i) q^{5} -9.47560i q^{7} -2.82843 q^{8} +(0.246293 + 7.06678i) q^{10} +20.3859i q^{11} -6.10986i q^{13} +13.4005i q^{14} +4.00000 q^{16} +24.0002 q^{17} +16.0752 q^{19} +(-0.348311 - 9.99393i) q^{20} -28.8301i q^{22} +43.2779 q^{23} +(-24.9393 + 1.74050i) q^{25} +8.64064i q^{26} -18.9512i q^{28} -19.6215i q^{29} +30.5072 q^{31} -5.65685 q^{32} -33.9414 q^{34} +(-47.3492 + 1.65023i) q^{35} -44.8335i q^{37} -22.7337 q^{38} +(0.492585 + 14.1336i) q^{40} +48.0908i q^{41} -48.1045i q^{43} +40.7719i q^{44} -61.2042 q^{46} -68.0770 q^{47} -40.7869 q^{49} +(35.2696 - 2.46143i) q^{50} -12.2197i q^{52} -61.7484 q^{53} +(101.868 - 3.55032i) q^{55} +26.8010i q^{56} +27.7489i q^{58} -50.4411i q^{59} -48.9975 q^{61} -43.1438 q^{62} +8.00000 q^{64} +(-30.5308 + 1.06406i) q^{65} -55.0907i q^{67} +48.0004 q^{68} +(66.9619 - 2.33377i) q^{70} -23.5982i q^{71} -40.4303i q^{73} +63.4041i q^{74} +32.1503 q^{76} +193.169 q^{77} +33.2338 q^{79} +(-0.696621 - 19.9879i) q^{80} -68.0107i q^{82} -22.7144 q^{83} +(-4.17976 - 119.928i) q^{85} +68.0300i q^{86} -57.6601i q^{88} -79.5266i q^{89} -57.8946 q^{91} +86.5558 q^{92} +96.2754 q^{94} +(-2.79958 - 80.3271i) q^{95} +119.886i q^{97} +57.6815 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 48 q^{4}+O(q^{10}) \) 24 * q + 48 * q^4	\( 24 q + 48 q^{4} - 12 q^{10} + 96 q^{16} - 48 q^{25} - 120 q^{34} - 24 q^{40} + 72 q^{49} + 216 q^{55} + 120 q^{61} + 192 q^{64} + 192 q^{70} + 480 q^{79} + 444 q^{85} + 48 q^{91} + 96 q^{94}+O(q^{100}) \) 24 * q + 48 * q^4 - 12 * q^10 + 96 * q^16 - 48 * q^25 - 120 * q^34 - 24 * q^40 + 72 * q^49 + 216 * q^55 + 120 * q^61 + 192 * q^64 + 192 * q^70 + 480 * q^79 + 444 * q^85 + 48 * q^91 + 96 * q^94

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