LMFDB - Embedded newform 90.6.c.c.19.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [90,6,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, 6, names="a")

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

chi := DirichletCharacter("90.19");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
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:	$14.4345437832$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{1249})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 625x^{2} + 97344 $ x^4 + 625*x^2 + 97344
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 5 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.3
Root			$-17.1706i$ of defining polynomial
Character	$\chi$	$=$	90.19
Dual form			90.6.c.c.19.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -16.0000 q^{4} +(-19.1706 + 52.5118i) q^{5} -233.706i q^{7} -64.0000i q^{8} +O(q^{10})$	\(q+4.00000i q^{2} -16.0000 q^{4} +(-19.1706 + 52.5118i) q^{5} -233.706i q^{7} -64.0000i q^{8} +(-210.047 - 76.6824i) q^{10} +89.7060 q^{11} -209.118i q^{13} +934.824 q^{14} +256.000 q^{16} +226.882i q^{17} +2180.47 q^{19} +(306.730 - 840.189i) q^{20} +358.824i q^{22} -4296.47i q^{23} +(-2389.98 - 2013.36i) q^{25} +836.472 q^{26} +3739.30i q^{28} +3556.06 q^{29} +6902.24 q^{31} +1024.00i q^{32} -907.528 q^{34} +(12272.3 + 4480.28i) q^{35} +5435.59i q^{37} +8721.89i q^{38} +(3360.75 + 1226.92i) q^{40} -6484.00 q^{41} -21979.5i q^{43} -1435.30 q^{44} +17185.9 q^{46} -7718.24i q^{47} -37811.5 q^{49} +(8053.46 - 9559.91i) q^{50} +3345.89i q^{52} -12765.4i q^{53} +(-1719.72 + 4710.62i) q^{55} -14957.2 q^{56} +14224.2i q^{58} -44555.6 q^{59} +11598.2 q^{61} +27608.9i q^{62} -4096.00 q^{64} +(10981.2 + 4008.92i) q^{65} +3961.29i q^{67} -3630.11i q^{68} +(-17921.1 + 49089.3i) q^{70} -46413.6 q^{71} -61591.2i q^{73} -21742.3 q^{74} -34887.5 q^{76} -20964.8i q^{77} +27877.8 q^{79} +(-4907.67 + 13443.0i) q^{80} -25936.0i q^{82} +50941.0i q^{83} +(-11914.0 - 4349.47i) q^{85} +87918.1 q^{86} -5741.18i q^{88} +5131.19 q^{89} -48872.1 q^{91} +68743.5i q^{92} +30872.9 q^{94} +(-41800.9 + 114500. i) q^{95} +88435.3i q^{97} -151246. i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 64 q^{4} - 6 q^{5}+O(q^{10}) $ 4 * q - 64 * q^4 - 6 * q^5	$ 4 q - 64 q^{4} - 6 q^{5} + 8 q^{10} - 348 q^{11} + 912 q^{14} + 1024 q^{16} + 240 q^{19} + 96 q^{20} - 9984 q^{25} - 5136 q^{26} - 4860 q^{29} + 23368 q^{31} - 12112 q^{34} + 37356 q^{35} - 128 q^{40} - 49968 q^{41} + 5568 q^{44} + 34816 q^{46} - 70668 q^{49} + 29952 q^{50} - 11968 q^{55} - 14592 q^{56} - 47460 q^{59} + 114248 q^{61} - 16384 q^{64} + 113052 q^{65} - 51328 q^{70} + 22152 q^{71} - 140688 q^{74} - 3840 q^{76} - 28440 q^{79} - 1536 q^{80} + 113924 q^{85} + 193344 q^{86} - 120840 q^{89} - 301512 q^{91} + 106528 q^{94} - 150240 q^{95}+O(q^{100}) $ 4 * q - 64 * q^4 - 6 * q^5 + 8 * q^10 - 348 * q^11 + 912 * q^14 + 1024 * q^16 + 240 * q^19 + 96 * q^20 - 9984 * q^25 - 5136 * q^26 - 4860 * q^29 + 23368 * q^31 - 12112 * q^34 + 37356 * q^35 - 128 * q^40 - 49968 * q^41 + 5568 * q^44 + 34816 * q^46 - 70668 * q^49 + 29952 * q^50 - 11968 * q^55 - 14592 * q^56 - 47460 * q^59 + 114248 * q^61 - 16384 * q^64 + 113052 * q^65 - 51328 * q^70 + 22152 * q^71 - 140688 * q^74 - 3840 * q^76 - 28440 * q^79 - 1536 * q^80 + 113924 * q^85 + 193344 * q^86 - 120840 * q^89 - 301512 * q^91 + 106528 * q^94 - 150240 * 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 $

$2$			4.00000i			0.707107i
$2$			4.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−16.0000			−0.500000
$5$	−19.1706	+	52.5118i	−0.342934	+	0.939359i
$5$	−19.1706	+	52.5118i	−0.342934	+	0.939359i
$6$		0			0
$7$		−	233.706i		−	1.80271i	−0.433086	−	0.901353i	$-0.642575\pi$
$7$		−	233.706i		−	1.80271i	0.433086	−	0.901353i	$-0.357425\pi$
$8$		−	64.0000i		−	0.353553i
$9$		0			0
$10$	−210.047	−	76.6824i	−0.664227	−	0.242491i
$11$	89.7060			0.223532			0.111766	−	0.993735i	$-0.464349\pi$
$11$	89.7060			0.223532			0.111766	+	0.993735i	$0.464349\pi$
$12$		0			0
$13$		−	209.118i		−	0.343189i	−0.985168	−	0.171594i	$-0.945108\pi$
$13$		−	209.118i		−	0.343189i	0.985168	−	0.171594i	$-0.0548918\pi$
$14$	934.824			1.27471
$15$		0			0
$16$	256.000			0.250000
$17$			226.882i			0.190405i	0.995458	+	0.0952024i	$0.0303498\pi$
$17$			226.882i			0.190405i	−0.995458	+	0.0952024i	$0.969650\pi$
$18$		0			0
$19$	2180.47			1.38569			0.692846	−	0.721086i	$-0.256357\pi$
$19$	2180.47			1.38569			0.692846	+	0.721086i	$0.256357\pi$
$20$	306.730	−	840.189i	0.171467	−	0.469680i
$21$		0			0
$22$			358.824i			0.158061i
$23$		−	4296.47i		−	1.69353i	−0.531969	−	0.846764i	$-0.678548\pi$
$23$		−	4296.47i		−	1.69353i	0.531969	−	0.846764i	$-0.321452\pi$
$24$		0			0
$25$	−2389.98	−	2013.36i	−0.764792	−	0.644277i
$26$	836.472			0.242671
$27$		0			0
$28$			3739.30i			0.901353i
$29$	3556.06			0.785189			0.392595	−	0.919712i	$-0.371578\pi$
$29$	3556.06			0.785189			0.392595	+	0.919712i	$0.371578\pi$
$30$		0			0
$31$	6902.24			1.28999			0.644994	−	0.764188i	$-0.276860\pi$
$31$	6902.24			1.28999			0.644994	+	0.764188i	$0.276860\pi$
$32$			1024.00i			0.176777i
$33$		0			0
$34$	−907.528			−0.134637
$35$	12272.3	+	4480.28i	1.69339	+	0.618209i
$36$		0			0
$37$			5435.59i			0.652743i	0.945242	+	0.326371i	$0.105826\pi$
$37$			5435.59i			0.652743i	−0.945242	+	0.326371i	$0.894174\pi$
$38$			8721.89i			0.979832i
$39$		0			0
$40$	3360.75	+	1226.92i	0.332114	+	0.121246i
$41$	−6484.00			−0.602398			−0.301199	−	0.953561i	$-0.597387\pi$
$41$	−6484.00			−0.602398			−0.301199	+	0.953561i	$0.597387\pi$
$42$		0			0
$43$		−	21979.5i		−	1.81279i	−0.422432	−	0.906395i	$-0.638823\pi$
$43$		−	21979.5i		−	1.81279i	0.422432	−	0.906395i	$-0.361177\pi$
$44$	−1435.30			−0.111766
$45$		0			0
$46$	17185.9			1.19751
$47$		−	7718.24i		−	0.509652i	−0.966987	−	0.254826i	$-0.917982\pi$
$47$		−	7718.24i		−	0.509652i	0.966987	−	0.254826i	$-0.0820181\pi$
$48$		0			0
$49$	−37811.5			−2.24975
$50$	8053.46	−	9559.91i	0.455572	−	0.540790i
$51$		0			0
$52$			3345.89i			0.171594i
$53$		−	12765.4i		−	0.624228i	−0.950045	−	0.312114i	$-0.898963\pi$
$53$		−	12765.4i		−	0.624228i	0.950045	−	0.312114i	$-0.101037\pi$
$54$		0			0
$55$	−1719.72	+	4710.62i	−0.0766567	+	0.209977i
$56$	−14957.2			−0.637353
$57$		0			0
$58$			14224.2i			0.555212i
$59$	−44555.6			−1.66637			−0.833187	−	0.552992i	$-0.813486\pi$
$59$	−44555.6			−1.66637			−0.833187	+	0.552992i	$0.813486\pi$
$60$		0			0
$61$	11598.2			0.399086			0.199543	−	0.979889i	$-0.436054\pi$
$61$	11598.2			0.399086			0.199543	+	0.979889i	$0.436054\pi$
$62$			27608.9i			0.912159i
$63$		0			0
$64$	−4096.00			−0.125000
$65$	10981.2	+	4008.92i	0.322377	+	0.117691i
$66$		0			0
$67$			3961.29i			0.107808i	0.998546	+	0.0539038i	$0.0171664\pi$
$67$			3961.29i			0.107808i	−0.998546	+	0.0539038i	$0.982834\pi$
$68$		−	3630.11i		−	0.0952024i
$69$		0			0
$70$	−17921.1	+	49089.3i	−0.437140	+	1.19741i
$71$	−46413.6			−1.09270			−0.546348	−	0.837559i	$-0.683982\pi$
$71$	−46413.6			−1.09270			−0.546348	+	0.837559i	$0.683982\pi$
$72$		0			0
$73$		−	61591.2i		−	1.35273i	−0.736566	−	0.676365i	$-0.763554\pi$
$73$		−	61591.2i		−	1.35273i	0.736566	−	0.676365i	$-0.236446\pi$
$74$	−21742.3			−0.461559
$75$		0			0
$76$	−34887.5			−0.692846
$77$		−	20964.8i		−	0.402962i
$78$		0			0
$79$	27877.8			0.502563			0.251281	−	0.967914i	$-0.419148\pi$
$79$	27877.8			0.502563			0.251281	+	0.967914i	$0.419148\pi$
$80$	−4907.67	+	13443.0i	−0.0857335	+	0.234840i
$81$		0			0
$82$		−	25936.0i		−	0.425959i
$83$			50941.0i			0.811656i	0.913949	+	0.405828i	$0.133017\pi$
$83$			50941.0i			0.811656i	−0.913949	+	0.405828i	$0.866983\pi$
$84$		0			0
$85$	−11914.0	−	4349.47i	−0.178859	−	0.0652963i
$86$	87918.1			1.28184
$87$		0			0
$88$		−	5741.18i		−	0.0790305i
$89$	5131.19			0.0686663			0.0343331	−	0.999410i	$-0.489069\pi$
$89$	5131.19			0.0686663			0.0343331	+	0.999410i	$0.489069\pi$
$90$		0			0
$91$	−48872.1			−0.618668
$92$			68743.5i			0.846764i
$93$		0			0
$94$	30872.9			0.360378
$95$	−41800.9	+	114500.i	−0.475201	+	1.30166i
$96$		0			0
$97$			88435.3i			0.954325i	0.878815	+	0.477162i	$0.158335\pi$
$97$			88435.3i			0.954325i	−0.878815	+	0.477162i	$0.841665\pi$
$98$		−	151246.i		−	1.59081i
$99$		0			0
$100$	38239.6	+	32213.8i	0.382396	+	0.322138i
$101$	116471.			1.13609			0.568047	−	0.822996i	$-0.307699\pi$
$101$	116471.			1.13609			0.568047	+	0.822996i	$0.307699\pi$
$102$		0			0
$103$			22369.1i			0.207756i	0.994590	+	0.103878i	$0.0331252\pi$
$103$			22369.1i			0.207756i	−0.994590	+	0.103878i	$0.966875\pi$
$104$	−13383.5			−0.121335
$105$		0			0
$106$	51061.5			0.441396
$107$		−	130626.i		−	1.10299i	−0.834178	−	0.551495i	$-0.814058\pi$
$107$		−	130626.i		−	1.10299i	0.834178	−	0.551495i	$-0.185942\pi$
$108$		0			0
$109$	51443.5			0.414729			0.207365	−	0.978264i	$-0.433511\pi$
$109$	51443.5			0.414729			0.207365	+	0.978264i	$0.433511\pi$
$110$	−18842.5	−	6878.87i	−0.148476	−	0.0542045i
$111$		0			0
$112$		−	59828.7i		−	0.450676i
$113$		−	138259.i		−	1.01858i	−0.860594	−	0.509291i	$-0.829908\pi$
$113$		−	138259.i		−	1.01858i	0.860594	−	0.509291i	$-0.170092\pi$
$114$		0			0
$115$	225615.	+	82365.9i	1.59083	+	0.580768i
$116$	−56897.0			−0.392595
$117$		0			0
$118$		−	178222.i		−	1.17830i
$119$	53023.7			0.343244
$120$		0			0
$121$	−153004.			−0.950033
$122$			46392.9i			0.282197i
$123$		0			0
$124$	−110436.			−0.644994
$125$	151543.	−	86904.5i	0.867481	−	0.497471i
$126$		0			0
$127$			199662.i			1.09847i	0.835669	+	0.549233i	$0.185080\pi$
$127$			199662.i			1.09847i	−0.835669	+	0.549233i	$0.814920\pi$
$128$		−	16384.0i		−	0.0883883i
$129$		0			0
$130$	−16035.7	+	43924.6i	−0.0832201	+	0.227955i
$131$	314223.			1.59978			0.799889	−	0.600148i	$-0.204891\pi$
$131$	314223.			1.59978			0.799889	+	0.600148i	$0.204891\pi$
$132$		0			0
$133$		−	509589.i		−	2.49799i
$134$	−15845.1			−0.0762315
$135$		0			0
$136$	14520.5			0.0673183
$137$			256841.i			1.16913i	0.811348	+	0.584564i	$0.198734\pi$
$137$			256841.i			1.16913i	−0.811348	+	0.584564i	$0.801266\pi$
$138$		0			0
$139$	10777.2			0.0473118			0.0236559	−	0.999720i	$-0.492469\pi$
$139$	10777.2			0.0473118			0.0236559	+	0.999720i	$0.492469\pi$
$140$	−196357.	−	71684.5i	−0.846694	−	0.309105i
$141$		0			0
$142$		−	185654.i		−	0.772652i
$143$		−	18759.1i		−	0.0767136i
$144$		0			0
$145$	−68171.8	+	186735.i	−0.269268	+	0.737575i
$146$	246365.			0.956525
$147$		0			0
$148$		−	86969.4i		−	0.326371i
$149$	4103.27			0.0151413			0.00757067	−	0.999971i	$-0.497590\pi$
$149$	4103.27			0.0151413			0.00757067	+	0.999971i	$0.497590\pi$
$150$		0			0
$151$	107449.			0.383495			0.191748	−	0.981444i	$-0.438585\pi$
$151$	107449.			0.383495			0.191748	+	0.981444i	$0.438585\pi$
$152$		−	139550.i		−	0.489916i
$153$		0			0
$154$	83859.3			0.284937
$155$	−132320.	+	362449.i	−0.442381	+	1.21176i
$156$		0			0
$157$			345824.i			1.11971i	0.828591	+	0.559855i	$0.189143\pi$
$157$			345824.i			1.11971i	−0.828591	+	0.559855i	$0.810857\pi$
$158$			111511.i			0.355366i
$159$		0			0
$160$	−53772.1	−	19630.7i	−0.166057	−	0.0606228i
$161$	−1.00411e6			−3.05293
$162$		0			0
$163$			58776.3i			0.173274i	0.996240	+	0.0866370i	$0.0276120\pi$
$163$			58776.3i			0.173274i	−0.996240	+	0.0866370i	$0.972388\pi$
$164$	103744.			0.301199
$165$		0			0
$166$	−203764.			−0.573928
$167$			154654.i			0.429111i	0.976712	+	0.214556i	$0.0688303\pi$
$167$			154654.i			0.429111i	−0.976712	+	0.214556i	$0.931170\pi$
$168$		0			0
$169$	327563.			0.882222
$170$	17397.9	−	47655.9i	0.0461715	−	0.126472i
$171$		0			0
$172$			351673.i			0.906395i
$173$		−	171600.i		−	0.435916i	−0.975958	−	0.217958i	$-0.930060\pi$
$173$		−	171600.i		−	0.435916i	0.975958	−	0.217958i	$-0.0699395\pi$
$174$		0			0
$175$	−470535.	+	558552.i	−1.16144	+	1.37870i
$176$	22964.7			0.0558830
$177$		0			0
$178$			20524.8i			0.0485544i
$179$	595571.			1.38932			0.694658	−	0.719340i	$-0.255556\pi$
$179$	595571.			1.38932			0.694658	+	0.719340i	$0.255556\pi$
$180$		0			0
$181$	144518.			0.327889			0.163944	−	0.986470i	$-0.447578\pi$
$181$	144518.			0.327889			0.163944	+	0.986470i	$0.447578\pi$
$182$		−	195488.i		−	0.437464i
$183$		0			0
$184$	−274974.			−0.598753
$185$	−285432.	−	104203.i	−0.613160	−	0.223848i
$186$		0			0
$187$			20352.7i			0.0425616i
$188$			123492.i			0.254826i
$189$		0			0
$190$	−458002.	−	167204.i	−0.920414	−	0.336018i
$191$	−815918.			−1.61831			−0.809157	−	0.587592i	$-0.800076\pi$
$191$	−815918.			−1.61831			−0.809157	+	0.587592i	$0.800076\pi$
$192$		0			0
$193$		−	665030.i		−	1.28513i	−0.766230	−	0.642566i	$-0.777870\pi$
$193$		−	665030.i		−	1.28513i	0.766230	−	0.642566i	$-0.222130\pi$
$194$	−353741.			−0.674810
$195$		0			0
$196$	604984.			1.12487
$197$			332972.i			0.611283i	0.952147	+	0.305642i	$0.0988709\pi$
$197$			332972.i			0.611283i	−0.952147	+	0.305642i	$0.901129\pi$
$198$		0			0
$199$	19565.3			0.0350231			0.0175115	−	0.999847i	$-0.494426\pi$
$199$	19565.3			0.0350231			0.0175115	+	0.999847i	$0.494426\pi$
$200$	−128855.	+	152958.i	−0.227786	+	0.270395i
$201$		0			0
$202$			465884.i			0.803339i
$203$		−	831073.i		−	1.41546i
$204$		0			0
$205$	124302.	−	340486.i	0.206583	−	0.565868i
$206$	−89476.2			−0.146906
$207$		0			0
$208$		−	53534.2i		−	0.0857971i
$209$	195601.			0.309746
$210$		0			0
$211$	−669474.			−1.03521			−0.517604	−	0.855620i	$-0.673176\pi$
$211$	−669474.			−1.03521			−0.517604	+	0.855620i	$0.673176\pi$
$212$			204246.i			0.312114i
$213$		0			0
$214$	522506.			0.779932
$215$	1.15418e6	+	421361.i	1.70286	+	0.621667i
$216$		0			0
$217$		−	1.61309e6i		−	2.32547i
$218$			205774.i			0.293258i
$219$		0			0
$220$	27515.5	−	75369.9i	0.0383284	−	0.104988i
$221$	47445.1			0.0653448
$222$		0			0
$223$			841915.i			1.13372i	0.823814	+	0.566860i	$0.191842\pi$
$223$			841915.i			1.13372i	−0.823814	+	0.566860i	$0.808158\pi$
$224$	239315.			0.318676
$225$		0			0
$226$	553034.			0.720246
$227$		−	106296.i		−	0.136916i	−0.997654	−	0.0684580i	$-0.978192\pi$
$227$		−	106296.i		−	0.136916i	0.997654	−	0.0684580i	$-0.0218079\pi$
$228$		0			0
$229$	−1.52931e6			−1.92711			−0.963553	−	0.267518i	$-0.913797\pi$
$229$	−1.52931e6			−1.92711			−0.963553	+	0.267518i	$0.913797\pi$
$230$	−329464.	+	902462.i	−0.410665	+	1.12489i
$231$		0			0
$232$		−	227588.i		−	0.277606i
$233$			362144.i			0.437011i	0.975836	+	0.218505i	$0.0701181\pi$
$233$			362144.i			0.437011i	−0.975836	+	0.218505i	$0.929882\pi$
$234$		0			0
$235$	405298.	+	147963.i	0.478746	+	0.174777i
$236$	712890.			0.833187
$237$		0			0
$238$			212095.i			0.242710i
$239$	38495.6			0.0435929			0.0217965	−	0.999762i	$-0.493061\pi$
$239$	38495.6			0.0435929			0.0217965	+	0.999762i	$0.493061\pi$
$240$		0			0
$241$	567433.			0.629320			0.314660	−	0.949204i	$-0.398109\pi$
$241$	567433.			0.629320			0.314660	+	0.949204i	$0.398109\pi$
$242$		−	612015.i		−	0.671775i
$243$		0			0
$244$	−185572.			−0.199543
$245$	724869.	−	1.98555e6i	0.771515	−	2.11332i
$246$		0			0
$247$		−	455976.i		−	0.475553i
$248$		−	441743.i		−	0.456080i
$249$		0			0
$250$	347618.	+	606171.i	0.351765	+	0.613402i
$251$	−19067.0			−0.0191028			−0.00955141	−	0.999954i	$-0.503040\pi$
$251$	−19067.0			−0.0191028			−0.00955141	+	0.999954i	$0.503040\pi$
$252$		0			0
$253$		−	385419.i		−	0.378558i
$254$	−798649.			−0.776733
$255$		0			0
$256$	65536.0			0.0625000
$257$		−	854054.i		−	0.806589i	−0.915070	−	0.403295i	$-0.867865\pi$
$257$		−	854054.i		−	0.806589i	0.915070	−	0.403295i	$-0.132135\pi$
$258$		0			0
$259$	1.27033e6			1.17670
$260$	−175698.	−	64142.6i	−0.161189	−	0.0588455i
$261$		0			0
$262$			1.25689e6i			1.13121i
$263$			755220.i			0.673262i	0.941637	+	0.336631i	$0.109288\pi$
$263$			755220.i			0.673262i	−0.941637	+	0.336631i	$0.890712\pi$
$264$		0			0
$265$	670332.	+	244720.i	0.586375	+	0.214069i
$266$	2.03836e6			1.76635
$267$		0			0
$268$		−	63380.6i		−	0.0539038i
$269$	−256615.			−0.216222			−0.108111	−	0.994139i	$-0.534480\pi$
$269$	−256615.			−0.216222			−0.108111	+	0.994139i	$0.534480\pi$
$270$		0			0
$271$	598518.			0.495056			0.247528	−	0.968881i	$-0.420382\pi$
$271$	598518.			0.495056			0.247528	+	0.968881i	$0.420382\pi$
$272$			58081.8i			0.0476012i
$273$		0			0
$274$	−1.02736e6			−0.826698
$275$	−214395.	−	180611.i	−0.170956	−	0.144016i
$276$		0			0
$277$			425045.i			0.332840i	0.986055	+	0.166420i	$0.0532208\pi$
$277$			425045.i			0.332840i	−0.986055	+	0.166420i	$0.946779\pi$
$278$			43108.9i			0.0334545i
$279$		0			0
$280$	286738.	−	785428.i	0.218570	−	0.598703i
$281$	949963.			0.717696			0.358848	−	0.933396i	$-0.383170\pi$
$281$	949963.			0.717696			0.358848	+	0.933396i	$0.383170\pi$
$282$		0			0
$283$			2.12422e6i			1.57664i	0.615264	+	0.788321i	$0.289050\pi$
$283$			2.12422e6i			1.57664i	−0.615264	+	0.788321i	$0.710950\pi$
$284$	742617.			0.546348
$285$		0			0
$286$	75036.5			0.0542447
$287$			1.51535e6i			1.08595i
$288$		0			0
$289$	1.36838e6			0.963746
$290$	−746941.	−	272687.i	−0.521544	−	0.190401i
$291$		0			0
$292$			985459.i			0.676365i
$293$			1.17556e6i			0.799977i	0.916520	+	0.399988i	$0.130986\pi$
$293$			1.17556e6i			0.799977i	−0.916520	+	0.399988i	$0.869014\pi$
$294$		0			0
$295$	854158.	−	2.33969e6i	0.571456	−	1.56532i
$296$	347878.			0.230779
$297$		0			0
$298$			16413.1i			0.0107065i
$299$	−898469.			−0.581199
$300$		0			0
$301$	−5.13675e6			−3.26792
$302$			429796.i			0.271172i
$303$		0			0
$304$	558201.			0.346423
$305$	−222345.	+	609044.i	−0.136860	+	0.374886i
$306$		0			0
$307$		−	588520.i		−	0.356382i	−0.983996	−	0.178191i	$-0.942976\pi$
$307$		−	588520.i		−	0.356382i	0.983996	−	0.178191i	$-0.0570244\pi$
$308$			335437.i			0.201481i
$309$		0			0
$310$	−1.44980e6	−	529280.i	−0.856845	−	0.312810i
$311$	−1.61863e6			−0.948958			−0.474479	−	0.880267i	$-0.657363\pi$
$311$	−1.61863e6			−0.948958			−0.474479	+	0.880267i	$0.657363\pi$
$312$		0			0
$313$		−	1.58901e6i		−	0.916783i	−0.888750	−	0.458392i	$-0.848426\pi$
$313$		−	1.58901e6i		−	0.916783i	0.888750	−	0.458392i	$-0.151574\pi$
$314$	−1.38329e6			−0.791754
$315$		0			0
$316$	−446045.			−0.251281
$317$		−	2.35137e6i		−	1.31423i	−0.753789	−	0.657117i	$-0.771776\pi$
$317$		−	2.35137e6i		−	1.31423i	0.753789	−	0.657117i	$-0.228224\pi$
$318$		0			0
$319$	319000.			0.175515
$320$	78522.8	−	215088.i	0.0428668	−	0.117420i
$321$		0			0
$322$		−	4.01644e6i		−	2.15875i
$323$			494710.i			0.263842i
$324$		0			0
$325$	−421031.	+	499787.i	−0.221108	+	0.262468i
$326$	−235105.			−0.122523
$327$		0			0
$328$			414976.i			0.212980i
$329$	−1.80380e6			−0.918752
$330$		0			0
$331$	480922.			0.241271			0.120635	−	0.992697i	$-0.461507\pi$
$331$	480922.			0.241271			0.120635	+	0.992697i	$0.461507\pi$
$332$		−	815056.i		−	0.405828i
$333$		0			0
$334$	−618616.			−0.303427
$335$	−208014.	−	75940.2i	−0.101270	−	0.0369709i
$336$		0			0
$337$			981449.i			0.470753i	0.971904	+	0.235376i	$0.0756323\pi$
$337$			981449.i			0.470753i	−0.971904	+	0.235376i	$0.924368\pi$
$338$			1.31025e6i			0.623825i
$339$		0			0
$340$	190624.	+	69591.4i	0.0894293	+	0.0326481i
$341$	619172.			0.288353
$342$		0			0
$343$			4.90887e6i			2.25292i
$344$	−1.40669e6			−0.640918
$345$		0			0
$346$	686400.			0.308239
$347$			1.91647e6i			0.854432i	0.904150	+	0.427216i	$0.140506\pi$
$347$			1.91647e6i			0.854432i	−0.904150	+	0.427216i	$0.859494\pi$
$348$		0			0
$349$	−1.22795e6			−0.539658			−0.269829	−	0.962908i	$-0.586967\pi$
$349$	−1.22795e6			−0.539658			−0.269829	+	0.962908i	$0.586967\pi$
$350$	−2.23421e6	−	1.88214e6i	−0.974885	−	0.821263i
$351$		0			0
$352$			91858.9i			0.0395152i
$353$			547823.i			0.233993i	0.993132	+	0.116997i	$0.0373267\pi$
$353$			547823.i			0.233993i	−0.993132	+	0.116997i	$0.962673\pi$
$354$		0			0
$355$	889776.	−	2.43726e6i	0.374722	−	1.02643i
$356$	−82099.1			−0.0343331
$357$		0			0
$358$			2.38228e6i			0.982395i
$359$	1.02314e6			0.418984			0.209492	−	0.977810i	$-0.432819\pi$
$359$	1.02314e6			0.418984			0.209492	+	0.977810i	$0.432819\pi$
$360$		0			0
$361$	2.27836e6			0.920140
$362$			578073.i			0.231852i
$363$		0			0
$364$	781954.			0.309334
$365$	3.23426e6	+	1.18074e6i	1.27070	+	0.463898i
$366$		0			0
$367$			3.05305e6i			1.18323i	0.806221	+	0.591614i	$0.201509\pi$
$367$			3.05305e6i			1.18323i	−0.806221	+	0.591614i	$0.798491\pi$
$368$		−	1.09990e6i		−	0.423382i
$369$		0			0
$370$	416814.	−	1.14173e6i	0.158284	−	0.433570i
$371$	−2.98334e6			−1.12530
$372$		0			0
$373$			3.69411e6i			1.37479i	0.726282	+	0.687397i	$0.241247\pi$
$373$			3.69411e6i			1.37479i	−0.726282	+	0.687397i	$0.758753\pi$
$374$	−81410.7			−0.0300956
$375$		0			0
$376$	−493967.			−0.180189
$377$		−	743636.i		−	0.269468i
$378$		0			0
$379$	−4.48524e6			−1.60394			−0.801969	−	0.597365i	$-0.796214\pi$
$379$	−4.48524e6			−1.60394			−0.801969	+	0.597365i	$0.796214\pi$
$380$	668815.	−	1.83201e6i	0.237600	−	0.650831i
$381$		0			0
$382$		−	3.26367e6i		−	1.14432i
$383$		−	4.53062e6i		−	1.57819i	−0.614269	−	0.789097i	$-0.710549\pi$
$383$		−	4.53062e6i		−	1.57819i	0.614269	−	0.789097i	$-0.289451\pi$
$384$		0			0
$385$	1.10090e6	+	401908.i	0.378526	+	0.138189i
$386$	2.66012e6			0.908726
$387$		0			0
$388$		−	1.41496e6i		−	0.477162i
$389$	2.73557e6			0.916588			0.458294	−	0.888801i	$-0.348461\pi$
$389$	2.73557e6			0.916588			0.458294	+	0.888801i	$0.348461\pi$
$390$		0			0
$391$	974792.			0.322456
$392$			2.41993e6i			0.795405i
$393$		0			0
$394$	−1.33189e6			−0.432242
$395$	−534434.	+	1.46391e6i	−0.172346	+	0.472087i
$396$		0			0
$397$		−	1.54948e6i		−	0.493413i	−0.969090	−	0.246707i	$-0.920652\pi$
$397$		−	1.54948e6i		−	0.493413i	0.969090	−	0.246707i	$-0.0793484\pi$
$398$			78261.3i			0.0247651i
$399$		0			0
$400$	−611834.	−	515421.i	−0.191198	−	0.161069i
$401$	−4.46136e6			−1.38550			−0.692749	−	0.721179i	$-0.743601\pi$
$401$	−4.46136e6			−1.38550			−0.692749	+	0.721179i	$0.743601\pi$
$402$		0			0
$403$		−	1.44338e6i		−	0.442709i
$404$	−1.86353e6			−0.568047
$405$		0			0
$406$	3.32429e6			1.00088
$407$			487605.i			0.145909i
$408$		0			0
$409$	2.36815e6			0.700003			0.350002	−	0.936749i	$-0.386181\pi$
$409$	2.36815e6			0.700003			0.350002	+	0.936749i	$0.386181\pi$
$410$	1.36195e6	+	497208.i	0.400129	+	0.146076i
$411$		0			0
$412$		−	357905.i		−	0.103878i
$413$			1.04129e7i			3.00398i
$414$		0			0
$415$	−2.67500e6	−	976569.i	−0.762437	−	0.278345i
$416$	214137.			0.0606677
$417$		0			0
$418$			782405.i			0.219024i
$419$	5.76363e6			1.60384			0.801920	−	0.597432i	$-0.203812\pi$
$419$	5.76363e6			1.60384			0.801920	+	0.597432i	$0.203812\pi$
$420$		0			0
$421$	3.26572e6			0.897995			0.448998	−	0.893533i	$-0.351781\pi$
$421$	3.26572e6			0.897995			0.448998	+	0.893533i	$0.351781\pi$
$422$		−	2.67789e6i		−	0.732002i
$423$		0			0
$424$	−816983.			−0.220698
$425$	456796.	−	542243.i	0.122673	−	0.145620i
$426$		0			0
$427$		−	2.71057e6i		−	0.719435i
$428$			2.09002e6i			0.551495i
$429$		0			0
$430$	−1.68544e6	+	4.61674e6i	−0.439585	+	1.20410i
$431$	5.62110e6			1.45756			0.728782	−	0.684745i	$-0.240087\pi$
$431$	5.62110e6			1.45756			0.728782	+	0.684745i	$0.240087\pi$
$432$		0			0
$433$		−	3.18656e6i		−	0.816774i	−0.912809	−	0.408387i	$-0.866091\pi$
$433$		−	3.18656e6i		−	0.816774i	0.912809	−	0.408387i	$-0.133909\pi$
$434$	6.45237e6			1.64435
$435$		0			0
$436$	−823097.			−0.207365
$437$		−	9.36833e6i		−	2.34671i
$438$		0			0
$439$	7.41102e6			1.83534			0.917671	−	0.397342i	$-0.130067\pi$
$439$	7.41102e6			1.83534			0.917671	+	0.397342i	$0.130067\pi$
$440$	301480.	+	110062.i	0.0742380	+	0.0271022i
$441$		0			0
$442$			189780.i			0.0462057i
$443$		−	2.62862e6i		−	0.636382i	−0.948027	−	0.318191i	$-0.896925\pi$
$443$		−	2.62862e6i		−	0.636382i	0.948027	−	0.318191i	$-0.103075\pi$
$444$		0			0
$445$	−98368.1	+	269448.i	−0.0235480	+	0.0645023i
$446$	−3.36766e6			−0.801662
$447$		0			0
$448$			957260.i			0.225338i
$449$	71764.4			0.0167994			0.00839969	−	0.999965i	$-0.497326\pi$
$449$	71764.4			0.0167994			0.00839969	+	0.999965i	$0.497326\pi$
$450$		0			0
$451$	−581653.			−0.134655
$452$			2.21214e6i			0.509291i
$453$		0			0
$454$	425186.			0.0968142
$455$	936907.	−	2.56636e6i	0.212162	−	0.581152i
$456$		0			0
$457$			3.51471e6i			0.787225i	0.919276	+	0.393613i	$0.128775\pi$
$457$			3.51471e6i			0.787225i	−0.919276	+	0.393613i	$0.871225\pi$
$458$		−	6.11722e6i		−	1.36267i
$459$		0			0
$460$	−3.60985e6	−	1.31785e6i	−0.795416	−	0.290384i
$461$	−3.35569e6			−0.735410			−0.367705	−	0.929943i	$-0.619856\pi$
$461$	−3.35569e6			−0.735410			−0.367705	+	0.929943i	$0.619856\pi$
$462$		0			0
$463$		−	3.47269e6i		−	0.752860i	−0.926445	−	0.376430i	$-0.877152\pi$
$463$		−	3.47269e6i		−	0.752860i	0.926445	−	0.376430i	$-0.122848\pi$
$464$	910352.			0.196297
$465$		0			0
$466$	−1.44858e6			−0.309013
$467$		−	2.35495e6i		−	0.499678i	−0.968287	−	0.249839i	$-0.919622\pi$
$467$		−	2.35495e6i		−	0.499678i	0.968287	−	0.249839i	$-0.0803777\pi$
$468$		0			0
$469$	925776.			0.194345
$470$	−591853.	+	1.62119e6i	−0.123586	+	0.338525i
$471$		0			0
$472$			2.85156e6i			0.589152i
$473$		−	1.97170e6i		−	0.405216i
$474$		0			0
$475$	−5.21128e6	−	4.39008e6i	−1.05977	−	0.892769i
$476$	−848379.			−0.171622
$477$		0			0
$478$			153982.i			0.0308249i
$479$	−187771.			−0.0373929			−0.0186964	−	0.999825i	$-0.505952\pi$
$479$	−187771.			−0.0373929			−0.0186964	+	0.999825i	$0.505952\pi$
$480$		0			0
$481$	1.13668e6			0.224014
$482$			2.26973e6i			0.444997i
$483$		0			0
$484$	2.44806e6			0.475017
$485$	−4.64390e6	−	1.69536e6i	−0.896454	−	0.327271i
$486$		0			0
$487$		−	2.01505e6i		−	0.385002i	−0.981297	−	0.192501i	$-0.938340\pi$
$487$		−	2.01505e6i		−	0.385002i	0.981297	−	0.192501i	$-0.0616598\pi$
$488$		−	742287.i		−	0.141098i
$489$		0			0
$490$	7.94219e6	+	2.89947e6i	1.49434	+	0.545543i
$491$	−5.80814e6			−1.08726			−0.543630	−	0.839325i	$-0.682951\pi$
$491$	−5.80814e6			−1.08726			−0.543630	+	0.839325i	$0.682951\pi$
$492$		0			0
$493$			806807.i			0.149504i
$494$	1.82390e6			0.336267
$495$		0			0
$496$	1.76697e6			0.322497
$497$			1.08471e7i			1.96981i
$498$		0			0
$499$	−580611.			−0.104384			−0.0521920	−	0.998637i	$-0.516621\pi$
$499$	−580611.			−0.104384			−0.0521920	+	0.998637i	$0.516621\pi$
$500$	−2.42468e6	+	1.39047e6i	−0.433740	+	0.248735i
$501$		0			0
$502$		−	76267.9i		−	0.0135077i
$503$		−	557751.i		−	0.0982926i	−0.998792	−	0.0491463i	$-0.984350\pi$
$503$		−	557751.i		−	0.0982926i	0.998792	−	0.0491463i	$-0.0156501\pi$
$504$		0			0
$505$	−2.23282e6	+	6.11610e6i	−0.389605	+	1.06720i
$506$	1.54168e6			0.267681
$507$		0			0
$508$		−	3.19460e6i		−	0.549233i
$509$	−2.63318e6			−0.450491			−0.225246	−	0.974302i	$-0.572318\pi$
$509$	−2.63318e6			−0.450491			−0.225246	+	0.974302i	$0.572318\pi$
$510$		0			0
$511$	−1.43942e7			−2.43858
$512$			262144.i			0.0441942i
$513$		0			0
$514$	3.41621e6			0.570345
$515$	−1.17464e6	−	428828.i	−0.195158	−	0.0712468i
$516$		0			0
$517$		−	692372.i		−	0.113923i
$518$			5.08132e6i			0.832055i
$519$		0			0
$520$	256571.	−	702794.i	0.0416101	−	0.113978i
$521$	6.78129e6			1.09451			0.547253	−	0.836967i	$-0.315674\pi$
$521$	6.78129e6			1.09451			0.547253	+	0.836967i	$0.315674\pi$
$522$		0			0
$523$		−	1.55863e6i		−	0.249166i	−0.992209	−	0.124583i	$-0.960241\pi$
$523$		−	1.55863e6i		−	0.249166i	0.992209	−	0.124583i	$-0.0397592\pi$
$524$	−5.02757e6			−0.799889
$525$		0			0
$526$	−3.02088e6			−0.476068
$527$			1.56599e6i			0.245620i
$528$		0			0
$529$	−1.20233e7			−1.86804
$530$	−978878.	+	2.68133e6i	−0.151370	+	0.414630i
$531$		0			0
$532$			8.15343e6i			1.24900i
$533$			1.35592e6i			0.206736i
$534$		0			0
$535$	6.85943e6	+	2.50419e6i	1.03610	+	0.378253i
$536$	253522.			0.0381157
$537$		0			0
$538$		−	1.02646e6i		−	0.152892i
$539$	−3.39192e6			−0.502890
$540$		0			0
$541$	7.29465e6			1.07155			0.535774	−	0.844362i	$-0.320020\pi$
$541$	7.29465e6			1.07155			0.535774	+	0.844362i	$0.320020\pi$
$542$			2.39407e6i			0.350057i
$543$		0			0
$544$	−232327.			−0.0336591
$545$	−986203.	+	2.70139e6i	−0.142225	+	0.389580i
$546$		0			0
$547$		−	5.15944e6i		−	0.737283i	−0.929572	−	0.368642i	$-0.879823\pi$
$547$		−	5.15944e6i		−	0.737283i	0.929572	−	0.368642i	$-0.120177\pi$
$548$		−	4.10945e6i		−	0.584564i
$549$		0			0
$550$	722443.	−	857581.i	0.101835	−	0.120884i
$551$	7.75389e6			1.08803
$552$		0			0
$553$		−	6.51520e6i		−	0.905973i
$554$	−1.70018e6			−0.235353
$555$		0			0
$556$	−172436.			−0.0236559
$557$			2.18378e6i			0.298244i	0.988819	+	0.149122i	$0.0476447\pi$
$557$			2.18378e6i			0.298244i	−0.988819	+	0.149122i	$0.952355\pi$
$558$		0			0
$559$	−4.59631e6			−0.622129
$560$	3.14171e6	+	1.14695e6i	0.423347	+	0.154552i
$561$		0			0
$562$			3.79985e6i			0.507488i
$563$		−	1.13003e7i		−	1.50252i	−0.660007	−	0.751259i	$-0.729447\pi$
$563$		−	1.13003e7i		−	1.50252i	0.660007	−	0.751259i	$-0.270553\pi$
$564$		0			0
$565$	7.26021e6	+	2.65050e6i	0.956815	+	0.349306i
$566$	−8.49688e6			−1.11485
$567$		0			0
$568$			2.97047e6i			0.386326i
$569$	−1.01686e7			−1.31669			−0.658343	−	0.752718i	$-0.728742\pi$
$569$	−1.01686e7			−1.31669			−0.658343	+	0.752718i	$0.728742\pi$
$570$		0			0
$571$	−284871.			−0.0365644			−0.0182822	−	0.999833i	$-0.505820\pi$
$571$	−284871.			−0.0365644			−0.0182822	+	0.999833i	$0.505820\pi$
$572$			300146.i			0.0383568i
$573$		0			0
$574$	−6.06140e6			−0.767879
$575$	−8.65036e6	+	1.02685e7i	−1.09110	+	1.29520i
$576$		0			0
$577$			9.14119e6i			1.14304i	0.820587	+	0.571522i	$0.193647\pi$
$577$			9.14119e6i			1.14304i	−0.820587	+	0.571522i	$0.806353\pi$
$578$			5.47353e6i			0.681471i
$579$		0			0
$580$	1.09075e6	−	2.98776e6i	0.134634	−	0.368787i
$581$	1.19052e7			1.46318
$582$		0			0
$583$		−	1.14513e6i		−	0.139535i
$584$	−3.94184e6			−0.478263
$585$		0			0
$586$	−4.70226e6			−0.565669
$587$			1.85347e6i			0.222019i	0.993819	+	0.111010i	$0.0354084\pi$
$587$			1.85347e6i			0.222019i	−0.993819	+	0.111010i	$0.964592\pi$
$588$		0			0
$589$	1.50501e7			1.78752
$590$	9.35878e6	+	3.41663e6i	1.10685	+	0.404081i
$591$		0			0
$592$			1.39151e6i			0.163186i
$593$		−	1.30967e7i		−	1.52941i	−0.644380	−	0.764705i	$-0.722884\pi$
$593$		−	1.30967e7i		−	1.52941i	0.644380	−	0.764705i	$-0.277116\pi$
$594$		0			0
$595$	−1.01650e6	+	2.78437e6i	−0.117710	+	0.322429i
$596$	−65652.3			−0.00757067
$597$		0			0
$598$		−	3.59388e6i		−	0.410970i
$599$	−7.13874e6			−0.812933			−0.406466	−	0.913666i	$-0.633239\pi$
$599$	−7.13874e6			−0.812933			−0.406466	+	0.913666i	$0.633239\pi$
$600$		0			0
$601$	−7.27856e6			−0.821976			−0.410988	−	0.911641i	$-0.634816\pi$
$601$	−7.27856e6			−0.821976			−0.410988	+	0.911641i	$0.634816\pi$
$602$		−	2.05470e7i		−	2.31077i
$603$		0			0
$604$	−1.71918e6			−0.191748
$605$	2.93317e6	−	8.03451e6i	0.325799	−	0.892423i
$606$		0			0
$607$		−	466696.i		−	0.0514117i	−0.999670	−	0.0257058i	$-0.991817\pi$
$607$		−	466696.i		−	0.0514117i	0.999670	−	0.0257058i	$-0.00818333\pi$
$608$			2.23280e6i			0.244958i
$609$		0			0
$610$	−2.43617e6	−	889380.i	−0.265084	−	0.0967749i
$611$	−1.61402e6			−0.174907
$612$		0			0
$613$		−	9.21342e6i		−	0.990307i	−0.868806	−	0.495153i	$-0.835112\pi$
$613$		−	9.21342e6i		−	0.990307i	0.868806	−	0.495153i	$-0.164888\pi$
$614$	2.35408e6			0.252000
$615$		0			0
$616$	−1.34175e6			−0.142469
$617$			8.53638e6i			0.902736i	0.892338	+	0.451368i	$0.149064\pi$
$617$			8.53638e6i			0.902736i	−0.892338	+	0.451368i	$0.850936\pi$
$618$		0			0
$619$	−7.72476e6			−0.810323			−0.405162	−	0.914245i	$-0.632785\pi$
$619$	−7.72476e6			−0.810323			−0.405162	+	0.914245i	$0.632785\pi$
$620$	2.11712e6	−	5.79918e6i	0.221190	−	0.605881i
$621$		0			0
$622$		−	6.47452e6i		−	0.671014i
$623$		−	1.19919e6i		−	0.123785i
$624$		0			0
$625$	1.65835e6	+	9.62379e6i	0.169815	+	0.985476i
$626$	6.35605e6			0.648264
$627$		0			0
$628$		−	5.53318e6i		−	0.559855i
$629$	−1.23324e6			−0.124285
$630$		0			0
$631$	1.50119e7			1.50094			0.750470	−	0.660904i	$-0.229827\pi$
$631$	1.50119e7			1.50094			0.750470	+	0.660904i	$0.229827\pi$
$632$		−	1.78418e6i		−	0.177683i
$633$		0			0
$634$	9.40547e6			0.929303
$635$	−1.04846e7	−	3.82765e6i	−1.03185	−	0.376702i
$636$		0			0
$637$			7.90706e6i			0.772087i
$638$			1.27600e6i			0.124108i
$639$		0			0
$640$	860353.	+	314091.i	0.0830284	+	0.0303114i
$641$	8.60718e6			0.827401			0.413700	−	0.910413i	$-0.364236\pi$
$641$	8.60718e6			0.827401			0.413700	+	0.910413i	$0.364236\pi$
$642$		0			0
$643$		−	1.97196e7i		−	1.88092i	−0.339904	−	0.940460i	$-0.610395\pi$
$643$		−	1.97196e7i		−	1.88092i	0.339904	−	0.940460i	$-0.389605\pi$
$644$	1.60658e7			1.52647
$645$		0			0
$646$	−1.97884e6			−0.186565
$647$		−	541311.i		−	0.0508378i	−0.999677	−	0.0254189i	$-0.991908\pi$
$647$		−	541311.i		−	0.0508378i	0.999677	−	0.0254189i	$-0.00809195\pi$
$648$		0			0
$649$	−3.99690e6			−0.372488
$650$	−1.99915e6	−	1.68412e6i	−0.185593	−	0.156347i
$651$		0			0
$652$		−	940421.i		−	0.0866370i
$653$			9.28231e6i			0.851870i	0.904754	+	0.425935i	$0.140055\pi$
$653$			9.28231e6i			0.851870i	−0.904754	+	0.425935i	$0.859945\pi$
$654$		0			0
$655$	−6.02384e6	+	1.65004e7i	−0.548619	+	1.50277i
$656$	−1.65990e6			−0.150599
$657$		0			0
$658$		−	7.21519e6i		−	0.649655i
$659$	1.79964e7			1.61425			0.807127	−	0.590378i	$-0.201021\pi$
$659$	1.79964e7			1.61425			0.807127	+	0.590378i	$0.201021\pi$
$660$		0			0
$661$	7.78607e6			0.693130			0.346565	−	0.938026i	$-0.387348\pi$
$661$	7.78607e6			0.693130			0.346565	+	0.938026i	$0.387348\pi$
$662$			1.92369e6i			0.170604i
$663$		0			0
$664$	3.26022e6			0.286964
$665$	2.67594e7	+	9.76913e6i	2.34651	+	0.856647i
$666$		0			0
$667$		−	1.52785e7i		−	1.32974i
$668$		−	2.47446e6i		−	0.214556i
$669$		0			0
$670$	303761.	−	832057.i	0.0261424	−	0.0716087i
$671$	1.04043e6			0.0892086
$672$		0			0
$673$			1.35540e7i			1.15353i	0.816910	+	0.576766i	$0.195685\pi$
$673$			1.35540e7i			1.15353i	−0.816910	+	0.576766i	$0.804315\pi$
$674$	−3.92579e6			−0.332872
$675$		0			0
$676$	−5.24100e6			−0.441111
$677$			1.46008e7i			1.22435i	0.790724	+	0.612173i	$0.209704\pi$
$677$			1.46008e7i			1.22435i	−0.790724	+	0.612173i	$0.790296\pi$
$678$		0			0
$679$	2.06679e7			1.72037
$680$	−278366.	+	762495.i	−0.0230857	+	0.0632361i
$681$		0			0
$682$			2.47669e6i			0.203897i
$683$			9.71085e6i			0.796536i	0.917269	+	0.398268i	$0.130389\pi$
$683$			9.71085e6i			0.796536i	−0.917269	+	0.398268i	$0.869611\pi$
$684$		0			0
$685$	−1.34872e7	−	4.92379e6i	−1.09823	−	0.400934i
$686$	−1.96355e7			−1.59306
$687$		0			0
$688$		−	5.62676e6i		−	0.453197i
$689$	−2.66947e6			−0.214228
$690$		0			0
$691$	1.30382e7			1.03878			0.519389	−	0.854538i	$-0.326159\pi$
$691$	1.30382e7			1.03878			0.519389	+	0.854538i	$0.326159\pi$
$692$			2.74560e6i			0.217958i
$693$		0			0
$694$	−7.66587e6			−0.604175
$695$	−206606.	+	565931.i	−0.0162248	+	0.0444428i
$696$		0			0
$697$		−	1.47110e6i		−	0.114699i
$698$		−	4.91182e6i		−	0.381596i
$699$		0			0
$700$	7.52857e6	−	8.93683e6i	0.580720	−	0.689348i
$701$	−4.65952e6			−0.358134			−0.179067	−	0.983837i	$-0.557308\pi$
$701$	−4.65952e6			−0.358134			−0.179067	+	0.983837i	$0.557308\pi$
$702$		0			0
$703$			1.18521e7i			0.904500i
$704$	−367436.			−0.0279415
$705$		0			0
$706$	−2.19129e6			−0.165458
$707$		−	2.72199e7i		−	2.04804i
$708$		0			0
$709$	−1.94059e7			−1.44983			−0.724916	−	0.688837i	$-0.758122\pi$
$709$	−1.94059e7			−1.44983			−0.724916	+	0.688837i	$0.758122\pi$
$710$	9.74904e6	+	3.55910e6i	0.725798	+	0.264969i
$711$		0			0
$712$		−	328396.i		−	0.0242772i
$713$		−	2.96553e7i		−	2.18463i
$714$		0			0
$715$	985075.	+	359624.i	0.0720617	+	0.0263077i
$716$	−9.52914e6			−0.694658
$717$		0			0
$718$			4.09254e6i			0.296266i
$719$	1.20100e7			0.866405			0.433202	−	0.901297i	$-0.357384\pi$
$719$	1.20100e7			0.866405			0.433202	+	0.901297i	$0.357384\pi$
$720$		0			0
$721$	5.22778e6			0.374524
$722$			9.11343e6i			0.650637i
$723$		0			0
$724$	−2.31229e6			−0.163944
$725$	−8.49890e6	−	7.15965e6i	−0.600507	−	0.505879i
$726$		0			0
$727$			1.14993e7i			0.806927i	0.914996	+	0.403463i	$0.132194\pi$
$727$			1.14993e7i			0.806927i	−0.914996	+	0.403463i	$0.867806\pi$
$728$			3.12781e6i			0.218732i
$729$		0			0
$730$	−4.72296e6	+	1.29371e7i	−0.328025	+	0.898521i
$731$	4.98676e6			0.345164
$732$		0			0
$733$			2.04473e7i			1.40565i	0.711364	+	0.702823i	$0.248078\pi$
$733$			2.04473e7i			1.40565i	−0.711364	+	0.702823i	$0.751922\pi$
$734$	−1.22122e7			−0.836668
$735$		0			0
$736$	4.39959e6			0.299376
$737$			355351.i			0.0240984i
$738$		0			0
$739$	−220621.			−0.0148606			−0.00743030	−	0.999972i	$-0.502365\pi$
$739$	−220621.			−0.0148606			−0.00743030	+	0.999972i	$0.502365\pi$
$740$	4.56692e6	+	1.66726e6i	0.306580	+	0.111924i
$741$		0			0
$742$		−	1.19334e7i		−	0.795707i
$743$			2.51435e7i			1.67091i	0.549558	+	0.835456i	$0.314796\pi$
$743$			2.51435e7i			1.67091i	−0.549558	+	0.835456i	$0.685204\pi$
$744$		0			0
$745$	−78662.1	+	215470.i	−0.00519248	+	0.0142232i
$746$	−1.47764e7			−0.972126
$747$		0			0
$748$		−	325643.i		−	0.0212808i
$749$	−3.05282e7			−1.98837
$750$		0			0
$751$	−667547.			−0.0431899			−0.0215950	−	0.999767i	$-0.506874\pi$
$751$	−667547.			−0.0431899			−0.0215950	+	0.999767i	$0.506874\pi$
$752$		−	1.97587e6i		−	0.127413i
$753$		0			0
$754$	2.97454e6			0.190543
$755$	−2.05986e6	+	5.64234e6i	−0.131514	+	0.360240i
$756$		0			0
$757$			1.62490e7i			1.03059i	0.857012	+	0.515297i	$0.172318\pi$
$757$			1.62490e7i			1.03059i	−0.857012	+	0.515297i	$0.827682\pi$
$758$		−	1.79410e7i		−	1.13416i
$759$		0			0
$760$	7.32803e6	+	2.67526e6i	0.460207	+	0.168009i
$761$	1.43662e7			0.899253			0.449626	−	0.893217i	$-0.351557\pi$
$761$	1.43662e7			0.899253			0.449626	+	0.893217i	$0.351557\pi$
$762$		0			0
$763$		−	1.20227e7i		−	0.747634i
$764$	1.30547e7			0.809157
$765$		0			0
$766$	1.81225e7			1.11595
$767$			9.31737e6i			0.571880i
$768$		0			0
$769$	−1.92705e7			−1.17511			−0.587553	−	0.809185i	$-0.699909\pi$
$769$	−1.92705e7			−1.17511			−0.587553	+	0.809185i	$0.699909\pi$
$770$	−1.60763e6	+	4.40360e6i	−0.0977147	+	0.267659i
$771$		0			0
$772$			1.06405e7i			0.642566i
$773$			2.40693e7i			1.44882i	0.689370	+	0.724410i	$0.257888\pi$
$773$			2.40693e7i			1.44882i	−0.689370	+	0.724410i	$0.742112\pi$
$774$		0			0
$775$	−1.64962e7	−	1.38967e7i	−0.986573	−	0.831109i
$776$	5.65986e6			0.337405
$777$		0			0
$778$			1.09423e7i			0.648125i
$779$	−1.41382e7			−0.834737
$780$		0			0
$781$	−4.16357e6			−0.244252
$782$			3.89917e6i			0.228011i
$783$		0			0
$784$	−9.67974e6			−0.562436
$785$	−1.81598e7	−	6.62964e6i	−1.05181	−	0.383987i
$786$		0			0
$787$			3.76223e6i			0.216525i	0.994122	+	0.108263i	$0.0345288\pi$
$787$			3.76223e6i			0.216525i	−0.994122	+	0.108263i	$0.965471\pi$
$788$		−	5.32756e6i		−	0.305642i
$789$		0			0
$790$	−5.85565e6	−	2.13773e6i	−0.333816	−	0.121867i
$791$	−3.23119e7			−1.83620
$792$		0			0
$793$		−	2.42540e6i		−	0.136962i
$794$	6.19794e6			0.348896
$795$		0			0
$796$	−313045.			−0.0175115
$797$		−	9.10423e6i		−	0.507689i	−0.967245	−	0.253844i	$-0.918305\pi$
$797$		−	9.10423e6i		−	0.507689i	0.967245	−	0.253844i	$-0.0816951\pi$
$798$		0			0
$799$	1.75113e6			0.0970401
$800$	2.06169e6	−	2.44734e6i	0.113893	−	0.135197i
$801$		0			0
$802$		−	1.78454e7i		−	0.979695i
$803$		−	5.52510e6i		−	0.302379i
$804$		0			0
$805$	1.92494e7	−	5.27277e7i	1.04695	−	2.86780i
$806$	5.77352e6			0.313043
$807$		0			0
$808$		−	7.45414e6i		−	0.401670i
$809$	1.20232e7			0.645877			0.322938	−	0.946420i	$-0.395329\pi$
$809$	1.20232e7			0.645877			0.322938	+	0.946420i	$0.395329\pi$
$810$		0			0
$811$	−5.50728e6			−0.294026			−0.147013	−	0.989135i	$-0.546966\pi$
$811$	−5.50728e6			−0.294026			−0.147013	+	0.989135i	$0.546966\pi$
$812$			1.32972e7i			0.707732i
$813$		0			0
$814$	−1.95042e6			−0.103173
$815$	−3.08645e6	−	1.12678e6i	−0.162766	−	0.0594215i
$816$		0			0
$817$		−	4.79258e7i		−	2.51197i
$818$			9.47258e6i			0.494977i
$819$		0			0
$820$	−1.98883e6	+	5.44778e6i	−0.103291	+	0.282934i
$821$	−1.33971e7			−0.693672			−0.346836	−	0.937926i	$-0.612744\pi$
$821$	−1.33971e7			−0.693672			−0.346836	+	0.937926i	$0.612744\pi$
$822$		0			0
$823$			1.41005e7i			0.725664i	0.931855	+	0.362832i	$0.118190\pi$
$823$			1.41005e7i			0.725664i	−0.931855	+	0.362832i	$0.881810\pi$
$824$	1.43162e6			0.0734530
$825$		0			0
$826$	−4.16516e7			−2.12413
$827$		−	1.28958e7i		−	0.655669i	−0.944735	−	0.327834i	$-0.893681\pi$
$827$		−	1.28958e7i		−	0.655669i	0.944735	−	0.327834i	$-0.106319\pi$
$828$		0			0
$829$	372726.			0.0188366			0.00941831	−	0.999956i	$-0.497002\pi$
$829$	372726.			0.0188366			0.00941831	+	0.999956i	$0.497002\pi$
$830$	3.90628e6	−	1.07000e7i	0.196819	−	0.539124i
$831$		0			0
$832$			856547.i			0.0428986i
$833$		−	8.57875e6i		−	0.428362i
$834$		0			0
$835$	−8.12115e6	−	2.96481e6i	−0.403089	−	0.147157i
$836$	−3.12962e6			−0.154873
$837$		0			0
$838$			2.30545e7i			1.13409i
$839$	1.57724e7			0.773560			0.386780	−	0.922172i	$-0.373587\pi$
$839$	1.57724e7			0.773560			0.386780	+	0.922172i	$0.373587\pi$
$840$		0			0
$841$	−7.86558e6			−0.383478
$842$			1.30629e7i			0.634978i
$843$		0			0
$844$	1.07116e7			0.517604
$845$	−6.27957e6	+	1.72009e7i	−0.302544	+	0.828723i
$846$		0			0
$847$			3.57579e7i			1.71263i
$848$		−	3.26793e6i		−	0.156057i
$849$		0			0
$850$	2.16897e6	+	1.82719e6i	0.102969	+	0.0867432i
$851$	2.33538e7			1.10544
$852$		0			0
$853$		−	3.39555e6i		−	0.159786i	−0.996803	−	0.0798928i	$-0.974542\pi$
$853$		−	3.39555e6i		−	0.159786i	0.996803	−	0.0798928i	$-0.0254578\pi$
$854$	1.08423e7			0.508718
$855$		0			0
$856$	−8.36009e6			−0.389966
$857$			4.62981e6i			0.215333i	0.994187	+	0.107667i	$0.0343379\pi$
$857$			4.62981e6i			0.215333i	−0.994187	+	0.107667i	$0.965662\pi$
$858$		0			0
$859$	1.57703e7			0.729218			0.364609	−	0.931161i	$-0.381203\pi$
$859$	1.57703e7			0.729218			0.364609	+	0.931161i	$0.381203\pi$
$860$	−1.84670e7	−	6.74177e6i	−0.851430	−	0.310834i
$861$		0			0
$862$			2.24844e7i			1.03065i
$863$			3.26217e7i			1.49101i	0.666502	+	0.745503i	$0.267791\pi$
$863$			3.26217e7i			1.49101i	−0.666502	+	0.745503i	$0.732209\pi$
$864$		0			0
$865$	9.01103e6	+	3.28968e6i	0.409481	+	0.149490i
$866$	1.27462e7			0.577546
$867$		0			0
$868$			2.58095e7i			1.16273i
$869$	2.50080e6			0.112339
$870$		0			0
$871$	828376.			0.0369983
$872$		−	3.29239e6i		−	0.146629i
$873$		0			0
$874$	3.74733e7			1.65937
$875$	−2.03101e7	−	3.54164e7i	−0.896793	−	1.56381i
$876$		0			0
$877$		−	3.88139e6i		−	0.170407i	−0.996364	−	0.0852037i	$-0.972846\pi$
$877$		−	3.88139e6i		−	0.170407i	0.996364	−	0.0852037i	$-0.0271541\pi$
$878$			2.96441e7i			1.29778i
$879$		0			0
$880$	−440248.	+	1.20592e6i	−0.0191642	+	0.0524942i
$881$	2.11909e7			0.919834			0.459917	−	0.887962i	$-0.347879\pi$
$881$	2.11909e7			0.919834			0.459917	+	0.887962i	$0.347879\pi$
$882$		0			0
$883$			257162.i			0.0110995i	0.999985	+	0.00554976i	$0.00176655\pi$
$883$			257162.i			0.0110995i	−0.999985	+	0.00554976i	$0.998233\pi$
$884$	−759122.			−0.0326724
$885$		0			0
$886$	1.05145e7			0.449990
$887$		−	1.02364e7i		−	0.436856i	−0.975853	−	0.218428i	$-0.929907\pi$
$887$		−	1.02364e7i		−	0.436856i	0.975853	−	0.218428i	$-0.0700929\pi$
$888$		0			0
$889$	4.66623e7			1.98021
$890$	−1.07779e6	−	393472.i	−0.0456100	−	0.0166510i
$891$		0			0
$892$		−	1.34706e7i		−	0.566860i
$893$		−	1.68294e7i		−	0.706220i
$894$		0			0
$895$	−1.14175e7	+	3.12745e7i	−0.476444	+	1.30507i
$896$	−3.82904e6			−0.159338
$897$		0			0
$898$			287058.i			0.0118790i
$899$	2.45448e7			1.01288
$900$		0			0
$901$	2.89623e6			0.118856
$902$		−	2.32661e6i		−	0.0952155i
$903$		0			0
$904$	−8.84855e6			−0.360123
$905$	−2.77050e6	+	7.58891e6i	−0.112444	+	0.308005i
$906$		0			0
$907$		−	1.30652e7i		−	0.527350i	−0.964612	−	0.263675i	$-0.915065\pi$
$907$		−	1.30652e7i		−	0.527350i	0.964612	−	0.263675i	$-0.0849346\pi$
$908$			1.70074e6i			0.0684580i
$909$		0			0
$910$	1.02654e7	+	3.74763e6i	0.410936	+	0.150021i
$911$	9.81608e6			0.391870			0.195935	−	0.980617i	$-0.437226\pi$
$911$	9.81608e6			0.391870			0.195935	+	0.980617i	$0.437226\pi$
$912$		0			0
$913$			4.56971e6i			0.181431i
$914$	−1.40588e7			−0.556652
$915$		0			0
$916$	2.44689e7			0.963553
$917$		−	7.34358e7i		−	2.88393i
$918$		0			0
$919$	−2.71414e6			−0.106009			−0.0530045	−	0.998594i	$-0.516880\pi$
$919$	−2.71414e6			−0.106009			−0.0530045	+	0.998594i	$0.516880\pi$
$920$	5.27142e6	−	1.44394e7i	0.205333	−	0.562444i
$921$		0			0
$922$		−	1.34228e7i		−	0.520013i
$923$			9.70591e6i			0.375000i
$924$		0			0
$925$	1.09438e7	−	1.29909e7i	0.420547	−	0.499213i
$926$	1.38908e7			0.532352
$927$		0			0
$928$			3.64141e6i			0.138803i
$929$	−4.31963e6			−0.164213			−0.0821064	−	0.996624i	$-0.526165\pi$
$929$	−4.31963e6			−0.164213			−0.0821064	+	0.996624i	$0.526165\pi$
$930$		0			0
$931$	−8.24469e7			−3.11745
$932$		−	5.79431e6i		−	0.218505i
$933$		0			0
$934$	9.41981e6			0.353326
$935$	−1.06876e6	−	390173.i	−0.0399806	−	0.0145958i
$936$		0			0
$937$		−	2.17380e7i		−	0.808854i	−0.914570	−	0.404427i	$-0.867471\pi$
$937$		−	2.17380e7i		−	0.808854i	0.914570	−	0.404427i	$-0.132529\pi$
$938$			3.70311e6i			0.137423i
$939$		0			0
$940$	−6.48477e6	−	2.36741e6i	−0.239373	−	0.0873884i
$941$	2.17794e7			0.801811			0.400905	−	0.916119i	$-0.368696\pi$
$941$	2.17794e7			0.801811			0.400905	+	0.916119i	$0.368696\pi$
$942$		0			0
$943$			2.78583e7i			1.02018i
$944$	−1.14062e7			−0.416593
$945$		0			0
$946$	7.88678e6			0.286531
$947$			4.93262e7i			1.78732i	0.448745	+	0.893660i	$0.351871\pi$
$947$			4.93262e7i			1.78732i	−0.448745	+	0.893660i	$0.648129\pi$
$948$		0			0
$949$	−1.28798e7			−0.464242
$950$	1.75603e7	−	2.08451e7i	0.631283	−	0.749368i
$951$		0			0
$952$		−	3.39352e6i		−	0.121355i
$953$		−	3.37247e7i		−	1.20286i	−0.798924	−	0.601432i	$-0.794597\pi$
$953$		−	3.37247e7i		−	1.20286i	0.798924	−	0.601432i	$-0.205403\pi$
$954$		0			0
$955$	1.56416e7	−	4.28453e7i	0.554975	−	1.52018i
$956$	−615929.			−0.0217965
$957$		0			0
$958$		−	751082.i		−	0.0264407i
$959$	6.00252e7			2.10759
$960$		0			0
$961$	1.90117e7			0.664068
$962$			4.54671e6i			0.158402i
$963$		0			0
$964$	−9.07892e6			−0.314660
$965$	3.49219e7	+	1.27490e7i	1.20720	+	0.440716i
$966$		0			0
$967$			2.77310e7i			0.953673i	0.878992	+	0.476836i	$0.158217\pi$
$967$			2.77310e7i			0.953673i	−0.878992	+	0.476836i	$0.841783\pi$
$968$			9.79225e6i			0.335888i
$969$		0			0
$970$	6.78143e6	−	1.85756e7i	0.231415	−	0.633889i
$971$	6.67841e6			0.227314			0.113657	−	0.993520i	$-0.463744\pi$
$971$	6.67841e6			0.227314			0.113657	+	0.993520i	$0.463744\pi$
$972$		0			0
$973$		−	2.51870e6i		−	0.0852893i
$974$	8.06018e6			0.272237
$975$		0			0
$976$	2.96915e6			0.0997716
$977$			4.34946e7i			1.45780i	0.684618	+	0.728902i	$0.259969\pi$
$977$			4.34946e7i			1.45780i	−0.684618	+	0.728902i	$0.740031\pi$
$978$		0			0
$979$	460299.			0.0153491
$980$	−1.15979e7	+	3.17688e7i	−0.385757	+	1.05666i
$981$		0			0
$982$		−	2.32326e7i		−	0.768809i
$983$			1.24213e7i			0.410000i	0.978762	+	0.205000i	$0.0657194\pi$
$983$			1.24213e7i			0.410000i	−0.978762	+	0.205000i	$0.934281\pi$
$984$		0			0
$985$	−1.74850e7	−	6.38328e6i	−0.574215	−	0.209630i
$986$	−3.22723e6			−0.105715
$987$		0			0
$988$			7.29561e6i			0.237777i
$989$	−9.44344e7			−3.07001
$990$		0			0
$991$	5.93479e7			1.91965			0.959824	−	0.280604i	$-0.0905348\pi$
$991$	5.93479e7			1.91965			0.959824	+	0.280604i	$0.0905348\pi$
$992$			7.06789e6i			0.228040i
$993$		0			0
$994$	−4.33885e7			−1.39286
$995$	−375079.	+	1.02741e6i	−0.0120106	+	0.0328993i
$996$		0			0
$997$		−	2.44456e7i		−	0.778867i	−0.921055	−	0.389434i	$-0.872671\pi$
$997$		−	2.44456e7i		−	0.778867i	0.921055	−	0.389434i	$-0.127329\pi$
$998$		−	2.32244e6i		−	0.0738107i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	90.6.c.c.19.3		4
3.2	odd	2		30.6.c.b.19.2	✓	4
4.3	odd	2		720.6.f.i.289.2		4
5.2	odd	4		450.6.a.bb.1.2		2
5.3	odd	4		450.6.a.bc.1.1		2
5.4	even	2	inner	90.6.c.c.19.1		4
12.11	even	2		240.6.f.b.49.2		4
15.2	even	4		150.6.a.o.1.2		2
15.8	even	4		150.6.a.n.1.1		2
15.14	odd	2		30.6.c.b.19.4	yes	4
20.19	odd	2		720.6.f.i.289.1		4
60.59	even	2		240.6.f.b.49.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.6.c.b.19.2	✓	4	3.2	odd	2
30.6.c.b.19.4	yes	4	15.14	odd	2
90.6.c.c.19.1		4	5.4	even	2	inner
90.6.c.c.19.3		4	1.1	even	1	trivial
150.6.a.n.1.1		2	15.8	even	4
150.6.a.o.1.2		2	15.2	even	4
240.6.f.b.49.2		4	12.11	even	2
240.6.f.b.49.4		4	60.59	even	2
450.6.a.bb.1.2		2	5.2	odd	4
450.6.a.bc.1.1		2	5.3	odd	4
720.6.f.i.289.1		4	20.19	odd	2
720.6.f.i.289.2		4	4.3	odd	2

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	90.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -16.0000 q^{4} +(-19.1706 + 52.5118i) q^{5} -233.706i q^{7} -64.0000i q^{8} +O(q^{10})\)	\(q+4.00000i q^{2} -16.0000 q^{4} +(-19.1706 + 52.5118i) q^{5} -233.706i q^{7} -64.0000i q^{8} +(-210.047 - 76.6824i) q^{10} +89.7060 q^{11} -209.118i q^{13} +934.824 q^{14} +256.000 q^{16} +226.882i q^{17} +2180.47 q^{19} +(306.730 - 840.189i) q^{20} +358.824i q^{22} -4296.47i q^{23} +(-2389.98 - 2013.36i) q^{25} +836.472 q^{26} +3739.30i q^{28} +3556.06 q^{29} +6902.24 q^{31} +1024.00i q^{32} -907.528 q^{34} +(12272.3 + 4480.28i) q^{35} +5435.59i q^{37} +8721.89i q^{38} +(3360.75 + 1226.92i) q^{40} -6484.00 q^{41} -21979.5i q^{43} -1435.30 q^{44} +17185.9 q^{46} -7718.24i q^{47} -37811.5 q^{49} +(8053.46 - 9559.91i) q^{50} +3345.89i q^{52} -12765.4i q^{53} +(-1719.72 + 4710.62i) q^{55} -14957.2 q^{56} +14224.2i q^{58} -44555.6 q^{59} +11598.2 q^{61} +27608.9i q^{62} -4096.00 q^{64} +(10981.2 + 4008.92i) q^{65} +3961.29i q^{67} -3630.11i q^{68} +(-17921.1 + 49089.3i) q^{70} -46413.6 q^{71} -61591.2i q^{73} -21742.3 q^{74} -34887.5 q^{76} -20964.8i q^{77} +27877.8 q^{79} +(-4907.67 + 13443.0i) q^{80} -25936.0i q^{82} +50941.0i q^{83} +(-11914.0 - 4349.47i) q^{85} +87918.1 q^{86} -5741.18i q^{88} +5131.19 q^{89} -48872.1 q^{91} +68743.5i q^{92} +30872.9 q^{94} +(-41800.9 + 114500. i) q^{95} +88435.3i q^{97} -151246. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 64 q^{4} - 6 q^{5}+O(q^{10}) \) 4 * q - 64 * q^4 - 6 * q^5	\( 4 q - 64 q^{4} - 6 q^{5} + 8 q^{10} - 348 q^{11} + 912 q^{14} + 1024 q^{16} + 240 q^{19} + 96 q^{20} - 9984 q^{25} - 5136 q^{26} - 4860 q^{29} + 23368 q^{31} - 12112 q^{34} + 37356 q^{35} - 128 q^{40} - 49968 q^{41} + 5568 q^{44} + 34816 q^{46} - 70668 q^{49} + 29952 q^{50} - 11968 q^{55} - 14592 q^{56} - 47460 q^{59} + 114248 q^{61} - 16384 q^{64} + 113052 q^{65} - 51328 q^{70} + 22152 q^{71} - 140688 q^{74} - 3840 q^{76} - 28440 q^{79} - 1536 q^{80} + 113924 q^{85} + 193344 q^{86} - 120840 q^{89} - 301512 q^{91} + 106528 q^{94} - 150240 q^{95}+O(q^{100}) \) 4 * q - 64 * q^4 - 6 * q^5 + 8 * q^10 - 348 * q^11 + 912 * q^14 + 1024 * q^16 + 240 * q^19 + 96 * q^20 - 9984 * q^25 - 5136 * q^26 - 4860 * q^29 + 23368 * q^31 - 12112 * q^34 + 37356 * q^35 - 128 * q^40 - 49968 * q^41 + 5568 * q^44 + 34816 * q^46 - 70668 * q^49 + 29952 * q^50 - 11968 * q^55 - 14592 * q^56 - 47460 * q^59 + 114248 * q^61 - 16384 * q^64 + 113052 * q^65 - 51328 * q^70 + 22152 * q^71 - 140688 * q^74 - 3840 * q^76 - 28440 * q^79 - 1536 * q^80 + 113924 * q^85 + 193344 * q^86 - 120840 * q^89 - 301512 * q^91 + 106528 * q^94 - 150240 * q^95

Introduction

L-functions

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