LMFDB - Embedded newform 900.4.j.c.557.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,4,Mod(557,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 1]))

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

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

chi := DirichletCharacter("900.557");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	900.j (of order $4$, degree $2$, 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:	$53.1017190052$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2737x^{12} + 5811553x^{8} - 4597108992x^{4} + 2821109907456 $ x^16 - 2737x^12 + 5811553x^8 - 4597108992*x^4 + 2821109907456
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			557.4
Root			$-6.29861 - 1.68771i$ of defining polynomial
Character	$\chi$	$=$	900.557
Dual form			900.4.j.c.593.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-12.2984 - 12.2984i) q^{7} +O(q^{10})$	$q+(-12.2984 - 12.2984i) q^{7} +34.1176i q^{11} +(25.8215 - 25.8215i) q^{13} +(15.5885 - 15.5885i) q^{17} +32.1248i q^{19} +(72.9623 + 72.9623i) q^{23} -106.419 q^{29} +10.3744 q^{31} +(-2.04195 - 2.04195i) q^{37} -365.573i q^{41} +(-10.2564 + 10.2564i) q^{43} +(260.024 - 260.024i) q^{47} -40.4991i q^{49} +(-42.4337 - 42.4337i) q^{53} +375.294 q^{59} -770.747 q^{61} +(-603.848 - 603.848i) q^{67} -695.440i q^{71} +(262.503 - 262.503i) q^{73} +(419.592 - 419.592i) q^{77} -32.2496i q^{79} +(519.399 + 519.399i) q^{83} -1080.45 q^{89} -635.127 q^{91} +(-416.212 - 416.212i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 1568 q^{31} - 4240 q^{61} - 14208 q^{91}+O(q^{100}) $ 16 * q - 1568 * q^31 - 4240 * q^61 - 14208 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−12.2984	−	12.2984i	−0.664051	−	0.664051i	0.292282	−	0.956332i	$-0.405586\pi$
$7$	−12.2984	−	12.2984i	−0.664051	−	0.664051i	−0.956332	+	0.292282i	$0.905586\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			34.1176i			0.935167i	0.883949	+	0.467584i	$0.154875\pi$
$11$			34.1176i			0.935167i	−0.883949	+	0.467584i	$0.845125\pi$
$12$		0			0
$13$	25.8215	−	25.8215i	0.550893	−	0.550893i	−0.375806	−	0.926698i	$-0.622634\pi$
$13$	25.8215	−	25.8215i	0.550893	−	0.550893i	0.926698	+	0.375806i	$0.122634\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	15.5885	−	15.5885i	0.222397	−	0.222397i	−0.587110	−	0.809507i	$-0.699734\pi$
$17$	15.5885	−	15.5885i	0.222397	−	0.222397i	0.809507	+	0.587110i	$0.199734\pi$
$18$		0			0
$19$			32.1248i			0.387891i	0.981012	+	0.193946i	$0.0621285\pi$
$19$			32.1248i			0.387891i	−0.981012	+	0.193946i	$0.937871\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	72.9623	+	72.9623i	0.661465	+	0.661465i	0.955725	−	0.294261i	$-0.0950734\pi$
$23$	72.9623	+	72.9623i	0.661465	+	0.661465i	−0.294261	+	0.955725i	$0.595073\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−106.419			−0.681431			−0.340716	−	0.940166i	$-0.610669\pi$
$29$	−106.419			−0.681431			−0.340716	+	0.940166i	$0.610669\pi$
$30$		0			0
$31$	10.3744			0.0601061			0.0300530	−	0.999548i	$-0.490432\pi$
$31$	10.3744			0.0601061			0.0300530	+	0.999548i	$0.490432\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.04195	−	2.04195i	−0.00907281	−	0.00907281i	0.702556	−	0.711629i	$-0.252042\pi$
$37$	−2.04195	−	2.04195i	−0.00907281	−	0.00907281i	−0.711629	+	0.702556i	$0.752042\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	365.573i		−	1.39251i	−0.717795	−	0.696255i	$-0.754848\pi$
$41$		−	365.573i		−	1.39251i	0.717795	−	0.696255i	$-0.245152\pi$
$42$		0			0
$43$	−10.2564	+	10.2564i	−0.0363743	+	0.0363743i	−0.725060	−	0.688686i	$-0.758188\pi$
$43$	−10.2564	+	10.2564i	−0.0363743	+	0.0363743i	0.688686	+	0.725060i	$0.258188\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	260.024	−	260.024i	0.806986	−	0.806986i	−0.177190	−	0.984177i	$-0.556701\pi$
$47$	260.024	−	260.024i	0.806986	−	0.806986i	0.984177	+	0.177190i	$0.0567008\pi$
$48$		0			0
$49$		−	40.4991i		−	0.118073i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−42.4337	−	42.4337i	−0.109976	−	0.109976i	0.649978	−	0.759953i	$-0.274778\pi$
$53$	−42.4337	−	42.4337i	−0.109976	−	0.109976i	−0.759953	+	0.649978i	$0.774778\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	375.294			0.828120			0.414060	−	0.910250i	$-0.364110\pi$
$59$	375.294			0.828120			0.414060	+	0.910250i	$0.364110\pi$
$60$		0			0
$61$	−770.747			−1.61777			−0.808886	−	0.587966i	$-0.799929\pi$
$61$	−770.747			−1.61777			−0.808886	+	0.587966i	$0.799929\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−603.848	−	603.848i	−1.10107	−	1.10107i	−0.994282	−	0.106790i	$-0.965943\pi$
$67$	−603.848	−	603.848i	−1.10107	−	1.10107i	−0.106790	−	0.994282i	$-0.534057\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	695.440i		−	1.16244i	−0.813745	−	0.581222i	$-0.802575\pi$
$71$		−	695.440i		−	1.16244i	0.813745	−	0.581222i	$-0.197425\pi$
$72$		0			0
$73$	262.503	−	262.503i	0.420872	−	0.420872i	−0.464632	−	0.885504i	$-0.653813\pi$
$73$	262.503	−	262.503i	0.420872	−	0.420872i	0.885504	+	0.464632i	$0.153813\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	419.592	−	419.592i	0.620999	−	0.620999i
$78$		0			0
$79$		−	32.2496i		−	0.0459286i	−0.999736	−	0.0229643i	$-0.992690\pi$
$79$		−	32.2496i		−	0.0459286i	0.999736	−	0.0229643i	$-0.00731041\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	519.399	+	519.399i	0.686885	+	0.686885i	0.961542	−	0.274657i	$-0.0885644\pi$
$83$	519.399	+	519.399i	0.686885	+	0.686885i	−0.274657	+	0.961542i	$0.588564\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1080.45			−1.28683			−0.643415	−	0.765517i	$-0.722483\pi$
$89$	−1080.45			−1.28683			−0.643415	+	0.765517i	$0.722483\pi$
$90$		0			0
$91$	−635.127			−0.731641
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−416.212	−	416.212i	−0.435669	−	0.435669i	0.454883	−	0.890551i	$-0.349681\pi$
$97$	−416.212	−	416.212i	−0.435669	−	0.435669i	−0.890551	+	0.454883i	$0.849681\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	1161.78i		−	1.14457i	−0.820056	−	0.572283i	$-0.806058\pi$
$101$		−	1161.78i		−	1.14457i	0.820056	−	0.572283i	$-0.193942\pi$
$102$		0			0
$103$	774.548	−	774.548i	0.740956	−	0.740956i	−0.231806	−	0.972762i	$-0.574463\pi$
$103$	774.548	−	774.548i	0.740956	−	0.740956i	0.972762	+	0.231806i	$0.0744634\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1477.87	−	1477.87i	1.33524	−	1.33524i	0.434637	−	0.900606i	$-0.356877\pi$
$107$	1477.87	−	1477.87i	1.33524	−	1.33524i	0.900606	−	0.434637i	$-0.143123\pi$
$108$		0			0
$109$			1480.74i			1.30119i	0.759426	+	0.650594i	$0.225480\pi$
$109$			1480.74i			1.30119i	−0.759426	+	0.650594i	$0.774520\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1490.42	+	1490.42i	1.24077	+	1.24077i	0.959682	+	0.281089i	$0.0906958\pi$
$113$	1490.42	+	1490.42i	1.24077	+	1.24077i	0.281089	+	0.959682i	$0.409304\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−383.426			−0.295366
$120$		0			0
$121$	166.990			0.125462
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−571.347	−	571.347i	−0.399203	−	0.399203i	0.478749	−	0.877952i	$-0.341091\pi$
$127$	−571.347	−	571.347i	−0.399203	−	0.399203i	−0.877952	+	0.478749i	$0.841091\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	1559.88i		−	1.04036i	−0.854056	−	0.520181i	$-0.825865\pi$
$131$		−	1559.88i		−	1.04036i	0.854056	−	0.520181i	$-0.174135\pi$
$132$		0			0
$133$	395.083	−	395.083i	0.257579	−	0.257579i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1897.46	−	1897.46i	1.18329	−	1.18329i	0.204405	−	0.978886i	$-0.434474\pi$
$137$	1897.46	−	1897.46i	1.18329	−	1.18329i	0.978886	−	0.204405i	$-0.0655261\pi$
$138$		0			0
$139$		−	1485.74i		−	0.906610i	−0.891355	−	0.453305i	$-0.850245\pi$
$139$		−	1485.74i		−	0.906610i	0.891355	−	0.453305i	$-0.149755\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	880.968	+	880.968i	0.515177	+	0.515177i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.43380			−0.00408725			−0.00204363	−	0.999998i	$-0.500651\pi$
$149$	−7.43380			−0.00408725			−0.00204363	+	0.999998i	$0.500651\pi$
$150$		0			0
$151$	−2646.62			−1.42635			−0.713174	−	0.700987i	$-0.752743\pi$
$151$	−2646.62			−1.42635			−0.713174	+	0.700987i	$0.752743\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−2653.09	−	2653.09i	−1.34866	−	1.34866i	−0.887117	−	0.461544i	$-0.847296\pi$
$157$	−2653.09	−	2653.09i	−1.34866	−	1.34866i	−0.461544	−	0.887117i	$-0.652704\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	1794.64i		−	0.878492i
$162$		0			0
$163$	912.484	−	912.484i	0.438474	−	0.438474i	−0.453024	−	0.891498i	$-0.649655\pi$
$163$	912.484	−	912.484i	0.438474	−	0.438474i	0.891498	+	0.453024i	$0.149655\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	245.525	−	245.525i	0.113768	−	0.113768i	−0.647931	−	0.761699i	$-0.724365\pi$
$167$	245.525	−	245.525i	0.113768	−	0.113768i	0.761699	+	0.647931i	$0.224365\pi$
$168$		0			0
$169$			863.497i			0.393035i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1164.36	−	1164.36i	−0.511704	−	0.511704i	0.403344	−	0.915048i	$-0.367848\pi$
$173$	−1164.36	−	1164.36i	−0.511704	−	0.511704i	−0.915048	+	0.403344i	$0.867848\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	593.087			0.247650			0.123825	−	0.992304i	$-0.460484\pi$
$179$	593.087			0.247650			0.123825	+	0.992304i	$0.460484\pi$
$180$		0			0
$181$	−3358.23			−1.37909			−0.689545	−	0.724243i	$-0.742189\pi$
$181$	−3358.23			−1.37909			−0.689545	+	0.724243i	$0.742189\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	531.841	+	531.841i	0.207979	+	0.207979i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	3766.21i		−	1.42677i	−0.700771	−	0.713387i	$-0.747160\pi$
$191$		−	3766.21i		−	1.42677i	0.700771	−	0.713387i	$-0.252840\pi$
$192$		0			0
$193$	−928.758	+	928.758i	−0.346391	+	0.346391i	−0.858763	−	0.512372i	$-0.828767\pi$
$193$	−928.758	+	928.758i	−0.346391	+	0.346391i	0.512372	+	0.858763i	$0.328767\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1323.72	−	1323.72i	0.478738	−	0.478738i	−0.425990	−	0.904728i	$-0.640074\pi$
$197$	1323.72	−	1323.72i	0.478738	−	0.478738i	0.904728	+	0.425990i	$0.140074\pi$
$198$		0			0
$199$			1744.36i			0.621380i	0.950511	+	0.310690i	$0.100560\pi$
$199$			1744.36i			0.621380i	−0.950511	+	0.310690i	$0.899440\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1308.78	+	1308.78i	0.452505	+	0.452505i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1096.02			−0.362743
$210$		0			0
$211$	−96.6308			−0.0315277			−0.0157638	−	0.999876i	$-0.505018\pi$
$211$	−96.6308			−0.0315277			−0.0157638	+	0.999876i	$0.505018\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−127.588	−	127.588i	−0.0399135	−	0.0399135i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	805.036i		−	0.245034i
$222$		0			0
$223$	1928.21	−	1928.21i	0.579024	−	0.579024i	−0.355611	−	0.934634i	$-0.615727\pi$
$223$	1928.21	−	1928.21i	0.579024	−	0.579024i	0.934634	+	0.355611i	$0.115727\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−997.661	+	997.661i	−0.291705	+	0.291705i	−0.837754	−	0.546048i	$-0.816131\pi$
$227$	−997.661	+	997.661i	−0.291705	+	0.291705i	0.546048	+	0.837754i	$0.316131\pi$
$228$		0			0
$229$		−	2842.72i		−	0.820316i	−0.912014	−	0.410158i	$-0.865474\pi$
$229$		−	2842.72i		−	0.820316i	0.912014	−	0.410158i	$-0.134526\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−337.759	−	337.759i	−0.0949670	−	0.0949670i	0.658027	−	0.752994i	$-0.271391\pi$
$233$	−337.759	−	337.759i	−0.0949670	−	0.0949670i	−0.752994	+	0.658027i	$0.771391\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−519.705			−0.140657			−0.0703283	−	0.997524i	$-0.522405\pi$
$239$	−519.705			−0.140657			−0.0703283	+	0.997524i	$0.522405\pi$
$240$		0			0
$241$	4246.97			1.13515			0.567575	−	0.823321i	$-0.307882\pi$
$241$	4246.97			1.13515			0.567575	+	0.823321i	$0.307882\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	829.511	+	829.511i	0.213686	+	0.213686i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			2398.14i			0.603066i	0.953456	+	0.301533i	$0.0974983\pi$
$251$			2398.14i			0.603066i	−0.953456	+	0.301533i	$0.902502\pi$
$252$		0			0
$253$	−2489.30	+	2489.30i	−0.618580	+	0.618580i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3527.30	−	3527.30i	0.856135	−	0.856135i	−0.134745	−	0.990880i	$-0.543022\pi$
$257$	3527.30	−	3527.30i	0.856135	−	0.856135i	0.990880	+	0.134745i	$0.0430216\pi$
$258$		0			0
$259$			50.2253i			0.0120496i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4523.66	+	4523.66i	1.06061	+	1.06061i	0.998041	+	0.0625711i	$0.0199300\pi$
$263$	4523.66	+	4523.66i	1.06061	+	1.06061i	0.0625711	+	0.998041i	$0.480070\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−345.433			−0.0782953			−0.0391477	−	0.999233i	$-0.512464\pi$
$269$	−345.433			−0.0782953			−0.0391477	+	0.999233i	$0.512464\pi$
$270$		0			0
$271$	2124.74			0.476269			0.238134	−	0.971232i	$-0.423464\pi$
$271$	2124.74			0.476269			0.238134	+	0.971232i	$0.423464\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−47.4509	−	47.4509i	−0.0102926	−	0.0102926i	0.701942	−	0.712234i	$-0.252317\pi$
$277$	−47.4509	−	47.4509i	−0.0102926	−	0.0102926i	−0.712234	+	0.701942i	$0.752317\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			8408.88i			1.78517i	0.450883	+	0.892583i	$0.351109\pi$
$281$			8408.88i			1.78517i	−0.450883	+	0.892583i	$0.648891\pi$
$282$		0			0
$283$	1288.89	−	1288.89i	0.270730	−	0.270730i	−0.558664	−	0.829394i	$-0.688686\pi$
$283$	1288.89	−	1288.89i	0.270730	−	0.270730i	0.829394	+	0.558664i	$0.188686\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4495.96	+	4495.96i	−0.924697	+	0.924697i
$288$		0			0
$289$			4427.00i			0.901079i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6951.13	−	6951.13i	−1.38597	−	1.38597i	−0.833597	−	0.552373i	$-0.813722\pi$
$293$	−6951.13	−	6951.13i	−1.38597	−	1.38597i	−0.552373	−	0.833597i	$-0.686278\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3767.99			0.728792
$300$		0			0
$301$	252.276			0.0483087
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	839.729	+	839.729i	0.156110	+	0.156110i	0.780841	−	0.624730i	$-0.214791\pi$
$307$	839.729	+	839.729i	0.156110	+	0.156110i	−0.624730	+	0.780841i	$0.714791\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			3067.60i			0.559317i	0.960100	+	0.279658i	$0.0902212\pi$
$311$			3067.60i			0.559317i	−0.960100	+	0.279658i	$0.909779\pi$
$312$		0			0
$313$	1687.26	−	1687.26i	0.304696	−	0.304696i	−0.538152	−	0.842848i	$-0.680877\pi$
$313$	1687.26	−	1687.26i	0.304696	−	0.304696i	0.842848	+	0.538152i	$0.180877\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4304.13	+	4304.13i	−0.762599	+	0.762599i	−0.976791	−	0.214193i	$-0.931288\pi$
$317$	−4304.13	+	4304.13i	−0.762599	+	0.762599i	0.214193	+	0.976791i	$0.431288\pi$
$318$		0			0
$319$		−	3630.76i		−	0.637252i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	500.776	+	500.776i	0.0862660	+	0.0862660i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6395.75			−1.07176
$330$		0			0
$331$	377.276			0.0626494			0.0313247	−	0.999509i	$-0.490027\pi$
$331$	377.276			0.0626494			0.0313247	+	0.999509i	$0.490027\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	2820.79	+	2820.79i	0.455959	+	0.455959i	0.897326	−	0.441368i	$-0.145507\pi$
$337$	2820.79	+	2820.79i	0.455959	+	0.455959i	−0.441368	+	0.897326i	$0.645507\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			353.948i			0.0562093i
$342$		0			0
$343$	−4716.42	+	4716.42i	−0.742457	+	0.742457i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8401.09	+	8401.09i	−1.29969	+	1.29969i	−0.371103	+	0.928592i	$0.621020\pi$
$347$	−8401.09	+	8401.09i	−1.29969	+	1.29969i	−0.928592	+	0.371103i	$0.878980\pi$
$348$		0			0
$349$		−	799.494i		−	0.122624i	−0.998119	−	0.0613122i	$-0.980471\pi$
$349$		−	799.494i		−	0.122624i	0.998119	−	0.0613122i	$-0.0195285\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5656.02	−	5656.02i	−0.852803	−	0.852803i	0.137675	−	0.990477i	$-0.456037\pi$
$353$	−5656.02	−	5656.02i	−0.852803	−	0.852803i	−0.990477	+	0.137675i	$0.956037\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10749.4			1.58031			0.790155	−	0.612907i	$-0.210000\pi$
$359$	10749.4			1.58031			0.790155	+	0.612907i	$0.210000\pi$
$360$		0			0
$361$	5827.00			0.849540
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	7245.02	+	7245.02i	1.03048	+	1.03048i	0.999521	+	0.0309613i	$0.00985685\pi$
$367$	7245.02	+	7245.02i	1.03048	+	1.03048i	0.0309613	+	0.999521i	$0.490143\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			1043.73i			0.146059i
$372$		0			0
$373$	−6870.69	+	6870.69i	−0.953756	+	0.953756i	−0.998977	−	0.0452209i	$-0.985601\pi$
$373$	−6870.69	+	6870.69i	−0.953756	+	0.953756i	0.0452209	+	0.998977i	$0.485601\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2747.90	+	2747.90i	−0.375395	+	0.375395i
$378$		0			0
$379$		−	6570.58i		−	0.890522i	−0.895401	−	0.445261i	$-0.853111\pi$
$379$		−	6570.58i		−	0.890522i	0.895401	−	0.445261i	$-0.146889\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3805.71	−	3805.71i	−0.507735	−	0.507735i	0.406095	−	0.913831i	$-0.366890\pi$
$383$	−3805.71	−	3805.71i	−0.507735	−	0.507735i	−0.913831	+	0.406095i	$0.866890\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1875.26			−0.244420			−0.122210	−	0.992504i	$-0.538998\pi$
$389$	−1875.26			−0.244420			−0.122210	+	0.992504i	$0.538998\pi$
$390$		0			0
$391$	2274.74			0.294216
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1776.78	+	1776.78i	0.224619	+	0.224619i	0.810440	−	0.585821i	$-0.199228\pi$
$397$	1776.78	+	1776.78i	0.224619	+	0.224619i	−0.585821	+	0.810440i	$0.699228\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			9060.16i			1.12829i	0.825677	+	0.564143i	$0.190793\pi$
$401$			9060.16i			1.12829i	−0.825677	+	0.564143i	$0.809207\pi$
$402$		0			0
$403$	267.882	−	267.882i	0.0331120	−	0.0331120i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	69.6663	−	69.6663i	0.00848460	−	0.00848460i
$408$		0			0
$409$			7155.45i			0.865072i	0.901617	+	0.432536i	$0.142381\pi$
$409$			7155.45i			0.865072i	−0.901617	+	0.432536i	$0.857619\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4615.51	−	4615.51i	−0.549913	−	0.549913i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	11104.0			1.29467			0.647336	−	0.762204i	$-0.275883\pi$
$419$	11104.0			1.29467			0.647336	+	0.762204i	$0.275883\pi$
$420$		0			0
$421$	7180.48			0.831247			0.415624	−	0.909537i	$-0.363563\pi$
$421$	7180.48			0.831247			0.415624	+	0.909537i	$0.363563\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	9478.95	+	9478.95i	1.07428	+	1.07428i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			8309.00i			0.928609i	0.885676	+	0.464305i	$0.153696\pi$
$431$			8309.00i			0.928609i	−0.885676	+	0.464305i	$0.846304\pi$
$432$		0			0
$433$	−10540.8	+	10540.8i	−1.16988	+	1.16988i	−0.187639	+	0.982238i	$0.560084\pi$
$433$	−10540.8	+	10540.8i	−1.16988	+	1.16988i	−0.982238	+	0.187639i	$0.939916\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2343.90	+	2343.90i	−0.256576	+	0.256576i
$438$		0			0
$439$		−	17125.6i		−	1.86186i	−0.365192	−	0.930932i	$-0.618997\pi$
$439$		−	17125.6i		−	1.86186i	0.365192	−	0.930932i	$-0.381003\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1463.14	−	1463.14i	−0.156920	−	0.156920i	0.624280	−	0.781200i	$-0.285392\pi$
$443$	−1463.14	−	1463.14i	−0.156920	−	0.156920i	−0.781200	+	0.624280i	$0.785392\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−11506.7			−1.20943			−0.604717	−	0.796441i	$-0.706714\pi$
$449$	−11506.7			−1.20943			−0.604717	+	0.796441i	$0.706714\pi$
$450$		0			0
$451$	12472.5			1.30223
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−2582.94	−	2582.94i	−0.264387	−	0.264387i	0.562446	−	0.826834i	$-0.309860\pi$
$457$	−2582.94	−	2582.94i	−0.264387	−	0.264387i	−0.826834	+	0.562446i	$0.809860\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			7353.90i			0.742962i	0.928441	+	0.371481i	$0.121150\pi$
$461$			7353.90i			0.742962i	−0.928441	+	0.371481i	$0.878850\pi$
$462$		0			0
$463$	13239.8	−	13239.8i	1.32896	−	1.32896i	0.422677	−	0.906280i	$-0.361090\pi$
$463$	13239.8	−	13239.8i	1.32896	−	1.32896i	0.906280	−	0.422677i	$-0.138910\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10856.7	+	10856.7i	−1.07578	+	1.07578i	−0.0788945	+	0.996883i	$0.525139\pi$
$467$	−10856.7	+	10856.7i	−1.07578	+	1.07578i	−0.996883	+	0.0788945i	$0.974861\pi$
$468$		0			0
$469$			14852.7i			1.46233i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−349.925	−	349.925i	−0.0340160	−	0.0340160i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7058.53			0.673304			0.336652	−	0.941629i	$-0.390705\pi$
$479$	7058.53			0.673304			0.336652	+	0.941629i	$0.390705\pi$
$480$		0			0
$481$	−105.452			−0.00999629
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−8019.32	−	8019.32i	−0.746181	−	0.746181i	0.227579	−	0.973760i	$-0.426919\pi$
$487$	−8019.32	−	8019.32i	−0.746181	−	0.746181i	−0.973760	+	0.227579i	$0.926919\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			11763.0i			1.08118i	0.841288	+	0.540588i	$0.181798\pi$
$491$			11763.0i			1.08118i	−0.841288	+	0.540588i	$0.818202\pi$
$492$		0			0
$493$	−1658.91	+	1658.91i	−0.151549	+	0.151549i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8552.79	+	8552.79i	−0.771922	+	0.771922i
$498$		0			0
$499$			2807.05i			0.251825i	0.992041	+	0.125913i	$0.0401859\pi$
$499$			2807.05i			0.251825i	−0.992041	+	0.125913i	$0.959814\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	6923.87	+	6923.87i	0.613758	+	0.613758i	0.943923	−	0.330166i	$-0.107105\pi$
$503$	6923.87	+	6923.87i	0.613758	+	0.613758i	−0.330166	+	0.943923i	$0.607105\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9795.18			−0.852974			−0.426487	−	0.904494i	$-0.640249\pi$
$509$	−9795.18			−0.852974			−0.426487	+	0.904494i	$0.640249\pi$
$510$		0			0
$511$	−6456.73			−0.558961
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	8871.39	+	8871.39i	0.754667	+	0.754667i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			7208.22i			0.606138i	0.952969	+	0.303069i	$0.0980112\pi$
$521$			7208.22i			0.606138i	−0.952969	+	0.303069i	$0.901989\pi$
$522$		0			0
$523$	−759.129	+	759.129i	−0.0634692	+	0.0634692i	−0.738129	−	0.674660i	$-0.764290\pi$
$523$	−759.129	+	759.129i	−0.0634692	+	0.0634692i	0.674660	+	0.738129i	$0.264290\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	161.720	−	161.720i	0.0133674	−	0.0133674i
$528$		0			0
$529$		−	1520.02i		−	0.124929i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−9439.65	−	9439.65i	−0.767123	−	0.767123i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1381.73			0.110418
$540$		0			0
$541$	5208.22			0.413898			0.206949	−	0.978352i	$-0.433647\pi$
$541$	5208.22			0.413898			0.206949	+	0.978352i	$0.433647\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	10838.3	+	10838.3i	0.847185	+	0.847185i	0.989781	−	0.142596i	$-0.0455448\pi$
$547$	10838.3	+	10838.3i	0.847185	+	0.847185i	−0.142596	+	0.989781i	$0.545545\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	3418.69i		−	0.264321i
$552$		0			0
$553$	−396.618	+	396.618i	−0.0304989	+	0.0304989i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	11311.5	−	11311.5i	0.860476	−	0.860476i	−0.130917	−	0.991393i	$-0.541792\pi$
$557$	11311.5	−	11311.5i	0.860476	−	0.860476i	0.991393	+	0.130917i	$0.0417922\pi$
$558$		0			0
$559$			529.674i			0.0400766i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10980.9	+	10980.9i	0.822010	+	0.822010i	0.986396	−	0.164386i	$-0.0525644\pi$
$563$	10980.9	+	10980.9i	0.822010	+	0.822010i	−0.164386	+	0.986396i	$0.552564\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2015.99			−0.148532			−0.0742660	−	0.997238i	$-0.523661\pi$
$569$	−2015.99			−0.148532			−0.0742660	+	0.997238i	$0.523661\pi$
$570$		0			0
$571$	−786.891			−0.0576714			−0.0288357	−	0.999584i	$-0.509180\pi$
$571$	−786.891			−0.0576714			−0.0288357	+	0.999584i	$0.509180\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−15028.6	−	15028.6i	−1.08431	−	1.08431i	−0.996102	−	0.0882125i	$-0.971885\pi$
$577$	−15028.6	−	15028.6i	−1.08431	−	1.08431i	−0.0882125	−	0.996102i	$-0.528115\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	12775.5i		−	0.912253i
$582$		0			0
$583$	1447.74	−	1447.74i	0.102846	−	0.102846i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	190.484	−	190.484i	0.0133937	−	0.0133937i	−0.700378	−	0.713772i	$-0.746985\pi$
$587$	190.484	−	190.484i	0.0133937	−	0.0133937i	0.713772	+	0.700378i	$0.246985\pi$
$588$		0			0
$589$			333.274i			0.0233146i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−16660.5	−	16660.5i	−1.15374	−	1.15374i	−0.985798	−	0.167937i	$-0.946289\pi$
$593$	−16660.5	−	16660.5i	−1.15374	−	1.15374i	−0.167937	−	0.985798i	$-0.553711\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	13527.2			0.922716			0.461358	−	0.887214i	$-0.347362\pi$
$599$	13527.2			0.922716			0.461358	+	0.887214i	$0.347362\pi$
$600$		0			0
$601$	−2751.50			−0.186749			−0.0933745	−	0.995631i	$-0.529765\pi$
$601$	−2751.50			−0.186749			−0.0933745	+	0.995631i	$0.529765\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	1583.36	+	1583.36i	0.105875	+	0.105875i	0.758060	−	0.652185i	$-0.226147\pi$
$607$	1583.36	+	1583.36i	0.105875	+	0.105875i	−0.652185	+	0.758060i	$0.726147\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	13428.4i		−	0.889126i
$612$		0			0
$613$	−17215.4	+	17215.4i	−1.13429	+	1.13429i	−0.144840	+	0.989455i	$0.546267\pi$
$613$	−17215.4	+	17215.4i	−1.13429	+	1.13429i	−0.989455	+	0.144840i	$0.953733\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6984.51	−	6984.51i	0.455731	−	0.455731i	−0.441520	−	0.897251i	$-0.645561\pi$
$617$	6984.51	−	6984.51i	0.455731	−	0.455731i	0.897251	+	0.441520i	$0.145561\pi$
$618$		0			0
$619$		−	24718.3i		−	1.60503i	−0.596632	−	0.802515i	$-0.703495\pi$
$619$		−	24718.3i		−	1.60503i	0.596632	−	0.802515i	$-0.296505\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	13287.8	+	13287.8i	0.854521	+	0.854521i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−63.6616			−0.00403554
$630$		0			0
$631$	153.092			0.00965847			0.00482923	−	0.999988i	$-0.498463\pi$
$631$	153.092			0.00965847			0.00482923	+	0.999988i	$0.498463\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1045.75	−	1045.75i	−0.0650457	−	0.0650457i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	13681.1i		−	0.843012i	−0.906826	−	0.421506i	$-0.861502\pi$
$641$		−	13681.1i		−	0.843012i	0.906826	−	0.421506i	$-0.138498\pi$
$642$		0			0
$643$	7036.19	−	7036.19i	0.431540	−	0.431540i	−0.457612	−	0.889152i	$-0.651295\pi$
$643$	7036.19	−	7036.19i	0.431540	−	0.431540i	0.889152	+	0.457612i	$0.151295\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3073.13	−	3073.13i	0.186735	−	0.186735i	−0.607548	−	0.794283i	$-0.707847\pi$
$647$	3073.13	−	3073.13i	0.186735	−	0.186735i	0.794283	+	0.607548i	$0.207847\pi$
$648$		0			0
$649$			12804.1i			0.774431i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−358.120	−	358.120i	−0.0214615	−	0.0214615i	0.696295	−	0.717756i	$-0.254831\pi$
$653$	−358.120	−	358.120i	−0.0214615	−	0.0214615i	−0.717756	+	0.696295i	$0.754831\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−376.499			−0.0222554			−0.0111277	−	0.999938i	$-0.503542\pi$
$659$	−376.499			−0.0222554			−0.0111277	+	0.999938i	$0.503542\pi$
$660$		0			0
$661$	−11926.5			−0.701794			−0.350897	−	0.936414i	$-0.614123\pi$
$661$	−11926.5			−0.701794			−0.350897	+	0.936414i	$0.614123\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−7764.57	−	7764.57i	−0.450743	−	0.450743i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	26296.0i		−	1.51289i
$672$		0			0
$673$	16792.8	−	16792.8i	0.961836	−	0.961836i	−0.0374623	−	0.999298i	$-0.511927\pi$
$673$	16792.8	−	16792.8i	0.961836	−	0.961836i	0.999298	+	0.0374623i	$0.0119274\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	19698.1	−	19698.1i	1.11826	−	1.11826i	0.126261	−	0.991997i	$-0.459702\pi$
$677$	19698.1	−	19698.1i	1.11826	−	1.11826i	0.991997	−	0.126261i	$-0.0402976\pi$
$678$		0			0
$679$			10237.5i			0.578613i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	3238.98	+	3238.98i	0.181458	+	0.181458i	0.791991	−	0.610533i	$-0.209045\pi$
$683$	3238.98	+	3238.98i	0.181458	+	0.181458i	−0.610533	+	0.791991i	$0.709045\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2191.41			−0.121170
$690$		0			0
$691$	−25894.9			−1.42560			−0.712799	−	0.701368i	$-0.752573\pi$
$691$	−25894.9			−1.42560			−0.712799	+	0.701368i	$0.752573\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−5698.72	−	5698.72i	−0.309691	−	0.309691i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			7282.04i			0.392352i	0.980569	+	0.196176i	$0.0628524\pi$
$701$			7282.04i			0.392352i	−0.980569	+	0.196176i	$0.937148\pi$
$702$		0			0
$703$	65.5971	−	65.5971i	0.00351926	−	0.00351926i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−14288.0	+	14288.0i	−0.760050	+	0.760050i
$708$		0			0
$709$		−	4010.75i		−	0.212450i	−0.994342	−	0.106225i	$-0.966124\pi$
$709$		−	4010.75i		−	0.212450i	0.994342	−	0.106225i	$-0.0338763\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	756.936	+	756.936i	0.0397580	+	0.0397580i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−7758.80			−0.402440			−0.201220	−	0.979546i	$-0.564491\pi$
$719$	−7758.80			−0.402440			−0.201220	+	0.979546i	$0.564491\pi$
$720$		0			0
$721$	−19051.4			−0.984065
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−7046.07	−	7046.07i	−0.359456	−	0.359456i	0.504157	−	0.863612i	$-0.331803\pi$
$727$	−7046.07	−	7046.07i	−0.359456	−	0.359456i	−0.863612	+	0.504157i	$0.831803\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			319.764i			0.0161791i
$732$		0			0
$733$	9924.06	−	9924.06i	0.500073	−	0.500073i	−0.411387	−	0.911461i	$-0.634956\pi$
$733$	9924.06	−	9924.06i	0.500073	−	0.500073i	0.911461	+	0.411387i	$0.134956\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	20601.8	−	20601.8i	1.02969	−	1.02969i
$738$		0			0
$739$			36353.6i			1.80959i	0.425844	+	0.904796i	$0.359977\pi$
$739$			36353.6i			1.80959i	−0.425844	+	0.904796i	$0.640023\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	20580.2	+	20580.2i	1.01617	+	1.01617i	0.999867	+	0.0163022i	$0.00518938\pi$
$743$	20580.2	+	20580.2i	1.01617	+	1.01617i	0.0163022	+	0.999867i	$0.494811\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−36350.8			−1.77334
$750$		0			0
$751$	−13694.3			−0.665395			−0.332697	−	0.943034i	$-0.607959\pi$
$751$	−13694.3			−0.665395			−0.332697	+	0.943034i	$0.607959\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−27376.6	−	27376.6i	−1.31442	−	1.31442i	−0.918119	−	0.396305i	$-0.870292\pi$
$757$	−27376.6	−	27376.6i	−1.31442	−	1.31442i	−0.396305	−	0.918119i	$-0.629708\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	6407.89i		−	0.305237i	−0.988285	−	0.152619i	$-0.951229\pi$
$761$		−	6407.89i		−	0.305237i	0.988285	−	0.152619i	$-0.0487707\pi$
$762$		0			0
$763$	18210.8	−	18210.8i	0.864055	−	0.864055i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	9690.65	−	9690.65i	0.456205	−	0.456205i
$768$		0			0
$769$		−	20338.9i		−	0.953758i	−0.878969	−	0.476879i	$-0.841768\pi$
$769$		−	20338.9i		−	0.953758i	0.878969	−	0.476879i	$-0.158232\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	18015.0	+	18015.0i	0.838232	+	0.838232i	0.988626	−	0.150394i	$-0.0480543\pi$
$773$	18015.0	+	18015.0i	0.838232	+	0.838232i	−0.150394	+	0.988626i	$0.548054\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	11744.0			0.540142
$780$		0			0
$781$	23726.7			1.08708
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−2474.01	−	2474.01i	−0.112057	−	0.112057i	0.648855	−	0.760912i	$-0.275248\pi$
$787$	−2474.01	−	2474.01i	−0.112057	−	0.112057i	−0.760912	+	0.648855i	$0.775248\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	36659.6i		−	1.64787i
$792$		0			0
$793$	−19901.9	+	19901.9i	−0.891218	+	0.891218i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	21457.4	−	21457.4i	0.953653	−	0.953653i	−0.0453193	−	0.998973i	$-0.514431\pi$
$797$	21457.4	−	21457.4i	0.953653	−	0.953653i	0.998973	+	0.0453193i	$0.0144305\pi$
$798$		0			0
$799$		−	8106.74i		−	0.358943i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	8955.97	+	8955.97i	0.393586	+	0.393586i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1927.61			−0.0837717			−0.0418859	−	0.999122i	$-0.513337\pi$
$809$	−1927.61			−0.0837717			−0.0418859	+	0.999122i	$0.513337\pi$
$810$		0			0
$811$	9971.05			0.431727			0.215864	−	0.976424i	$-0.430743\pi$
$811$	9971.05			0.431727			0.215864	+	0.976424i	$0.430743\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−329.486	−	329.486i	−0.0141092	−	0.0141092i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			2696.43i			0.114624i	0.998356	+	0.0573118i	$0.0182529\pi$
$821$			2696.43i			0.114624i	−0.998356	+	0.0573118i	$0.981747\pi$
$822$		0			0
$823$	19441.7	−	19441.7i	0.823443	−	0.823443i	−0.163157	−	0.986600i	$-0.552168\pi$
$823$	19441.7	−	19441.7i	0.823443	−	0.823443i	0.986600	+	0.163157i	$0.0521676\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	14089.3	−	14089.3i	0.592420	−	0.592420i	−0.345864	−	0.938285i	$-0.612414\pi$
$827$	14089.3	−	14089.3i	0.592420	−	0.592420i	0.938285	+	0.345864i	$0.112414\pi$
$828$		0			0
$829$			32154.0i			1.34711i	0.739138	+	0.673554i	$0.235233\pi$
$829$			32154.0i			1.34711i	−0.739138	+	0.673554i	$0.764767\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−631.319	−	631.319i	−0.0262592	−	0.0262592i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	4243.48			0.174614			0.0873070	−	0.996181i	$-0.472174\pi$
$839$	4243.48			0.174614			0.0873070	+	0.996181i	$0.472174\pi$
$840$		0			0
$841$	−13064.0			−0.535652
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−2053.70	−	2053.70i	−0.0833130	−	0.0833130i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	297.970i		−	0.0120027i
$852$		0			0
$853$	27081.0	−	27081.0i	1.08703	−	1.08703i	0.0911961	−	0.995833i	$-0.470931\pi$
$853$	27081.0	−	27081.0i	1.08703	−	1.08703i	0.995833	−	0.0911961i	$-0.0290690\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	13880.6	−	13880.6i	0.553271	−	0.553271i	−0.374112	−	0.927383i	$-0.622053\pi$
$857$	13880.6	−	13880.6i	0.553271	−	0.553271i	0.927383	+	0.374112i	$0.122053\pi$
$858$		0			0
$859$		−	41873.7i		−	1.66323i	−0.555354	−	0.831614i	$-0.687417\pi$
$859$		−	41873.7i		−	1.66323i	0.555354	−	0.831614i	$-0.312583\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	9614.63	+	9614.63i	0.379242	+	0.379242i	0.870829	−	0.491587i	$-0.163583\pi$
$863$	9614.63	+	9614.63i	0.379242	+	0.379242i	−0.491587	+	0.870829i	$0.663583\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1100.28			0.0429509
$870$		0			0
$871$	−31184.6			−1.21314
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−10205.0	−	10205.0i	−0.392929	−	0.392929i	0.482801	−	0.875730i	$-0.339619\pi$
$877$	−10205.0	−	10205.0i	−0.392929	−	0.392929i	−0.875730	+	0.482801i	$0.839619\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	27928.4i		−	1.06803i	−0.845476	−	0.534013i	$-0.820683\pi$
$881$		−	27928.4i		−	1.06803i	0.845476	−	0.534013i	$-0.179317\pi$
$882$		0			0
$883$	−29194.0	+	29194.0i	−1.11263	+	1.11263i	−0.119840	+	0.992793i	$0.538238\pi$
$883$	−29194.0	+	29194.0i	−1.11263	+	1.11263i	−0.992793	+	0.119840i	$0.961762\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	25382.5	−	25382.5i	0.960835	−	0.960835i	−0.0384264	−	0.999261i	$-0.512235\pi$
$887$	25382.5	−	25382.5i	0.960835	−	0.960835i	0.999261	+	0.0384264i	$0.0122345\pi$
$888$		0			0
$889$			14053.3i			0.530182i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	8353.21	+	8353.21i	0.313023	+	0.313023i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1104.03			−0.0409582
$900$		0			0
$901$	−1322.95			−0.0489167
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	17048.2	+	17048.2i	0.624120	+	0.624120i	0.946582	−	0.322462i	$-0.104510\pi$
$907$	17048.2	+	17048.2i	0.624120	+	0.624120i	−0.322462	+	0.946582i	$0.604510\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	40281.8i		−	1.46498i	−0.680778	−	0.732490i	$-0.738358\pi$
$911$		−	40281.8i		−	1.46498i	0.680778	−	0.732490i	$-0.261642\pi$
$912$		0			0
$913$	−17720.6	+	17720.6i	−0.642353	+	0.642353i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−19184.0	+	19184.0i	−0.690853	+	0.690853i
$918$		0			0
$919$			28470.1i			1.02192i	0.859605	+	0.510959i	$0.170710\pi$
$919$			28470.1i			1.02192i	−0.859605	+	0.510959i	$0.829290\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−17957.3	−	17957.3i	−0.640382	−	0.640382i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	23799.7			0.840518			0.420259	−	0.907404i	$-0.361939\pi$
$929$	23799.7			0.840518			0.420259	+	0.907404i	$0.361939\pi$
$930$		0			0
$931$	1301.03			0.0457996
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−8925.58	−	8925.58i	−0.311191	−	0.311191i	0.534180	−	0.845371i	$-0.320620\pi$
$937$	−8925.58	−	8925.58i	−0.311191	−	0.311191i	−0.845371	+	0.534180i	$0.820620\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	9979.82i		−	0.345731i	−0.984945	−	0.172865i	$-0.944697\pi$
$941$		−	9979.82i		−	0.345731i	0.984945	−	0.172865i	$-0.0553026\pi$
$942$		0			0
$943$	26673.0	−	26673.0i	0.921096	−	0.921096i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−14180.9	+	14180.9i	−0.486608	+	0.486608i	−0.907234	−	0.420626i	$-0.861810\pi$
$947$	−14180.9	+	14180.9i	−0.486608	+	0.486608i	0.420626	+	0.907234i	$0.361810\pi$
$948$		0			0
$949$		−	13556.5i		−	0.463710i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−7943.99	−	7943.99i	−0.270022	−	0.270022i	0.559087	−	0.829109i	$-0.311152\pi$
$953$	−7943.99	−	7943.99i	−0.270022	−	0.270022i	−0.829109	+	0.559087i	$0.811152\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−46671.4			−1.57153
$960$		0			0
$961$	−29683.4			−0.996387
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−28769.3	−	28769.3i	−0.956731	−	0.956731i	0.0423711	−	0.999102i	$-0.486509\pi$
$967$	−28769.3	−	28769.3i	−0.956731	−	0.956731i	−0.999102	+	0.0423711i	$0.986509\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			29221.3i			0.965762i	0.875686	+	0.482881i	$0.160410\pi$
$971$			29221.3i			0.965762i	−0.875686	+	0.482881i	$0.839590\pi$
$972$		0			0
$973$	−18272.2	+	18272.2i	−0.602035	+	0.602035i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−12510.5	+	12510.5i	−0.409670	+	0.409670i	−0.881623	−	0.471954i	$-0.843549\pi$
$977$	−12510.5	+	12510.5i	−0.409670	+	0.409670i	0.471954	+	0.881623i	$0.343549\pi$
$978$		0			0
$979$		−	36862.5i		−	1.20340i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1612.48	−	1612.48i	−0.0523196	−	0.0523196i	0.680463	−	0.732783i	$-0.261779\pi$
$983$	−1612.48	−	1612.48i	−0.0523196	−	0.0523196i	−0.732783	+	0.680463i	$0.761779\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1496.67			−0.0481206
$990$		0			0
$991$	−35794.4			−1.14737			−0.573686	−	0.819075i	$-0.694487\pi$
$991$	−35794.4			−1.14737			−0.573686	+	0.819075i	$0.694487\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	7344.86	+	7344.86i	0.233314	+	0.233314i	0.814075	−	0.580760i	$-0.197245\pi$
$997$	7344.86	+	7344.86i	0.233314	+	0.233314i	−0.580760	+	0.814075i	$0.697245\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.4.j.c.557.4	yes	16
3.2	odd	2	inner	900.4.j.c.557.3	✓	16
5.2	odd	4	inner	900.4.j.c.593.6	yes	16
5.3	odd	4	inner	900.4.j.c.593.4	yes	16
5.4	even	2	inner	900.4.j.c.557.6	yes	16
15.2	even	4	inner	900.4.j.c.593.5	yes	16
15.8	even	4	inner	900.4.j.c.593.3	yes	16
15.14	odd	2	inner	900.4.j.c.557.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.4.j.c.557.3	✓	16	3.2	odd	2	inner
900.4.j.c.557.4	yes	16	1.1	even	1	trivial
900.4.j.c.557.5	yes	16	15.14	odd	2	inner
900.4.j.c.557.6	yes	16	5.4	even	2	inner
900.4.j.c.593.3	yes	16	15.8	even	4	inner
900.4.j.c.593.4	yes	16	5.3	odd	4	inner
900.4.j.c.593.5	yes	16	15.2	even	4	inner
900.4.j.c.593.6	yes	16	5.2	odd	4	inner

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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	900.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-12.2984 - 12.2984i) q^{7} +O(q^{10})\)	\(q+(-12.2984 - 12.2984i) q^{7} +34.1176i q^{11} +(25.8215 - 25.8215i) q^{13} +(15.5885 - 15.5885i) q^{17} +32.1248i q^{19} +(72.9623 + 72.9623i) q^{23} -106.419 q^{29} +10.3744 q^{31} +(-2.04195 - 2.04195i) q^{37} -365.573i q^{41} +(-10.2564 + 10.2564i) q^{43} +(260.024 - 260.024i) q^{47} -40.4991i q^{49} +(-42.4337 - 42.4337i) q^{53} +375.294 q^{59} -770.747 q^{61} +(-603.848 - 603.848i) q^{67} -695.440i q^{71} +(262.503 - 262.503i) q^{73} +(419.592 - 419.592i) q^{77} -32.2496i q^{79} +(519.399 + 519.399i) q^{83} -1080.45 q^{89} -635.127 q^{91} +(-416.212 - 416.212i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 1568 q^{31} - 4240 q^{61} - 14208 q^{91}+O(q^{100}) \) 16 * q - 1568 * q^31 - 4240 * q^61 - 14208 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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