LMFDB - Embedded newform 1800.2.s.e.593.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,2,Mod(593,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 2, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1800.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1800.s (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:	$14.3730723638$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.2
Root			$0.437016 + 0.437016i$ of defining polynomial
Character	$\chi$	$=$	1800.593
Dual form			1800.2.s.e.1457.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.23607 + 1.23607i) q^{7} +O(q^{10})$	$q+(-1.23607 + 1.23607i) q^{7} +1.74806i q^{11} +(-0.236068 - 0.236068i) q^{13} +(-4.57649 - 4.57649i) q^{17} -6.47214i q^{19} +(2.82843 - 2.82843i) q^{23} +0.333851 q^{29} +10.4721 q^{31} +(2.23607 - 2.23607i) q^{37} -7.07107i q^{41} +(6.47214 + 6.47214i) q^{43} +(4.57649 + 4.57649i) q^{47} +3.94427i q^{49} -7.40492 q^{59} +1.52786 q^{61} +(10.4721 - 10.4721i) q^{67} -12.6491i q^{71} +(9.47214 + 9.47214i) q^{73} +(-2.16073 - 2.16073i) q^{77} -5.52786i q^{79} +(7.40492 - 7.40492i) q^{83} -13.3956 q^{89} +0.583592 q^{91} +(-1.00000 + 1.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7}+O(q^{10}) $ 8 * q + 8 * q^7	$ 8 q + 8 q^{7} + 16 q^{13} + 48 q^{31} + 16 q^{43} + 48 q^{61} + 48 q^{67} + 40 q^{73} + 112 q^{91} - 8 q^{97}+O(q^{100}) $ 8 * q + 8 * q^7 + 16 * q^13 + 48 * q^31 + 16 * q^43 + 48 * q^61 + 48 * q^67 + 40 * q^73 + 112 * q^91 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.23607	+	1.23607i	−0.467190	+	0.467190i	−0.901003	−	0.433813i	$-0.857168\pi$
$7$	−1.23607	+	1.23607i	−0.467190	+	0.467190i	0.433813	+	0.901003i	$0.357168\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			1.74806i			0.527061i	0.964651	+	0.263531i	$0.0848870\pi$
$11$			1.74806i			0.527061i	−0.964651	+	0.263531i	$0.915113\pi$
$12$		0			0
$13$	−0.236068	−	0.236068i	−0.0654735	−	0.0654735i	0.673612	−	0.739085i	$-0.264742\pi$
$13$	−0.236068	−	0.236068i	−0.0654735	−	0.0654735i	−0.739085	+	0.673612i	$0.764742\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.57649	−	4.57649i	−1.10996	−	1.10996i	−0.993155	−	0.116808i	$-0.962734\pi$
$17$	−4.57649	−	4.57649i	−1.10996	−	1.10996i	−0.116808	−	0.993155i	$-0.537266\pi$
$18$		0			0
$19$		−	6.47214i		−	1.48481i	−0.669951	−	0.742405i	$-0.733685\pi$
$19$		−	6.47214i		−	1.48481i	0.669951	−	0.742405i	$-0.266315\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.82843	−	2.82843i	0.589768	−	0.589768i	−0.347801	−	0.937568i	$-0.613071\pi$
$23$	2.82843	−	2.82843i	0.589768	−	0.589768i	0.937568	+	0.347801i	$0.113071\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.333851			0.0619945			0.0309972	−	0.999519i	$-0.490132\pi$
$29$	0.333851			0.0619945			0.0309972	+	0.999519i	$0.490132\pi$
$30$		0			0
$31$	10.4721			1.88085			0.940426	−	0.340000i	$-0.110427\pi$
$31$	10.4721			1.88085			0.940426	+	0.340000i	$0.110427\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.23607	−	2.23607i	0.367607	−	0.367607i	−0.498997	−	0.866604i	$-0.666298\pi$
$37$	2.23607	−	2.23607i	0.367607	−	0.367607i	0.866604	+	0.498997i	$0.166298\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	7.07107i		−	1.10432i	−0.833740	−	0.552158i	$-0.813805\pi$
$41$		−	7.07107i		−	1.10432i	0.833740	−	0.552158i	$-0.186195\pi$
$42$		0			0
$43$	6.47214	+	6.47214i	0.986991	+	0.986991i	0.999916	−	0.0129250i	$-0.00411427\pi$
$43$	6.47214	+	6.47214i	0.986991	+	0.986991i	−0.0129250	+	0.999916i	$0.504114\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.57649	+	4.57649i	0.667550	+	0.667550i	0.957148	−	0.289598i	$-0.0935217\pi$
$47$	4.57649	+	4.57649i	0.667550	+	0.667550i	−0.289598	+	0.957148i	$0.593522\pi$
$48$		0			0
$49$			3.94427i			0.563467i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.40492			−0.964038			−0.482019	−	0.876161i	$-0.660096\pi$
$59$	−7.40492			−0.964038			−0.482019	+	0.876161i	$0.660096\pi$
$60$		0			0
$61$	1.52786			0.195623			0.0978115	−	0.995205i	$-0.468816\pi$
$61$	1.52786			0.195623			0.0978115	+	0.995205i	$0.468816\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	10.4721	−	10.4721i	1.27938	−	1.27938i	0.338357	−	0.941018i	$-0.390129\pi$
$67$	10.4721	−	10.4721i	1.27938	−	1.27938i	0.941018	−	0.338357i	$-0.109871\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	12.6491i		−	1.50117i	−0.660772	−	0.750587i	$-0.729771\pi$
$71$		−	12.6491i		−	1.50117i	0.660772	−	0.750587i	$-0.270229\pi$
$72$		0			0
$73$	9.47214	+	9.47214i	1.10863	+	1.10863i	0.993331	+	0.115299i	$0.0367826\pi$
$73$	9.47214	+	9.47214i	1.10863	+	1.10863i	0.115299	+	0.993331i	$0.463217\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.16073	−	2.16073i	−0.246238	−	0.246238i
$78$		0			0
$79$		−	5.52786i		−	0.621933i	−0.950421	−	0.310967i	$-0.899347\pi$
$79$		−	5.52786i		−	0.621933i	0.950421	−	0.310967i	$-0.100653\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7.40492	−	7.40492i	0.812795	−	0.812795i	−0.172257	−	0.985052i	$-0.555106\pi$
$83$	7.40492	−	7.40492i	0.812795	−	0.812795i	0.985052	+	0.172257i	$0.0551059\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−13.3956			−1.41993			−0.709967	−	0.704235i	$-0.751290\pi$
$89$	−13.3956			−1.41993			−0.709967	+	0.704235i	$0.751290\pi$
$90$		0			0
$91$	0.583592			0.0611771
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.00000	+	1.00000i	−0.101535	+	0.101535i	−0.756049	−	0.654515i	$-0.772873\pi$
$97$	−1.00000	+	1.00000i	−0.101535	+	0.101535i	0.654515	+	0.756049i	$0.272873\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	17.3044i		−	1.72185i	−0.508729	−	0.860927i	$-0.669884\pi$
$101$		−	17.3044i		−	1.72185i	0.508729	−	0.860927i	$-0.330116\pi$
$102$		0			0
$103$	−10.1803	−	10.1803i	−1.00310	−	1.00310i	−0.999995	−	0.00310351i	$-0.999012\pi$
$103$	−10.1803	−	10.1803i	−1.00310	−	1.00310i	−0.00310351	−	0.999995i	$-0.500988\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.90879	+	3.90879i	0.377877	+	0.377877i	0.870336	−	0.492459i	$-0.163902\pi$
$107$	3.90879	+	3.90879i	0.377877	+	0.377877i	−0.492459	+	0.870336i	$0.663902\pi$
$108$		0			0
$109$		−	11.4164i		−	1.09349i	−0.837298	−	0.546747i	$-0.815866\pi$
$109$		−	11.4164i		−	1.09349i	0.837298	−	0.546747i	$-0.184134\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.16228	+	3.16228i	−0.297482	+	0.297482i	−0.840027	−	0.542545i	$-0.817461\pi$
$113$	−3.16228	+	3.16228i	−0.297482	+	0.297482i	0.542545	+	0.840027i	$0.317461\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	11.3137			1.03713
$120$		0			0
$121$	7.94427			0.722207
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	2.76393	−	2.76393i	0.245259	−	0.245259i	−0.573762	−	0.819022i	$-0.694517\pi$
$127$	2.76393	−	2.76393i	0.245259	−	0.245259i	0.819022	+	0.573762i	$0.194517\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			16.5579i			1.44667i	0.690497	+	0.723335i	$0.257392\pi$
$131$			16.5579i			1.44667i	−0.690497	+	0.723335i	$0.742608\pi$
$132$		0			0
$133$	8.00000	+	8.00000i	0.693688	+	0.693688i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.49458	−	2.49458i	−0.213126	−	0.213126i	0.592468	−	0.805594i	$-0.298154\pi$
$137$	−2.49458	−	2.49458i	−0.213126	−	0.213126i	−0.805594	+	0.592468i	$0.798154\pi$
$138$		0			0
$139$			8.94427i			0.758643i	0.925265	+	0.379322i	$0.123843\pi$
$139$			8.94427i			0.758643i	−0.925265	+	0.379322i	$0.876157\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.412662	−	0.412662i	0.0345085	−	0.0345085i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.15143			−0.667791			−0.333896	−	0.942610i	$-0.608363\pi$
$149$	−8.15143			−0.667791			−0.333896	+	0.942610i	$0.608363\pi$
$150$		0			0
$151$	12.9443			1.05339			0.526695	−	0.850054i	$-0.323431\pi$
$151$	12.9443			1.05339			0.526695	+	0.850054i	$0.323431\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−12.7082	+	12.7082i	−1.01423	+	1.01423i	−0.0143277	+	0.999897i	$0.504561\pi$
$157$	−12.7082	+	12.7082i	−1.01423	+	1.01423i	−0.999897	+	0.0143277i	$0.995439\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			6.99226i			0.551067i
$162$		0			0
$163$	6.47214	+	6.47214i	0.506937	+	0.506937i	0.913585	−	0.406648i	$-0.133302\pi$
$163$	6.47214	+	6.47214i	0.506937	+	0.506937i	−0.406648	+	0.913585i	$0.633302\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.32456	−	6.32456i	−0.489409	−	0.489409i	0.418711	−	0.908120i	$-0.362482\pi$
$167$	−6.32456	−	6.32456i	−0.489409	−	0.489409i	−0.908120	+	0.418711i	$0.862482\pi$
$168$		0			0
$169$		−	12.8885i		−	0.991426i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.333851	−	0.333851i	0.0253822	−	0.0253822i	−0.694302	−	0.719684i	$-0.744287\pi$
$173$	0.333851	−	0.333851i	0.0253822	−	0.0253822i	0.719684	+	0.694302i	$0.244287\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.24419			0.391969			0.195985	−	0.980607i	$-0.437210\pi$
$179$	5.24419			0.391969			0.195985	+	0.980607i	$0.437210\pi$
$180$		0			0
$181$	−1.52786			−0.113565			−0.0567826	−	0.998387i	$-0.518084\pi$
$181$	−1.52786			−0.113565			−0.0567826	+	0.998387i	$0.518084\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	8.00000	−	8.00000i	0.585018	−	0.585018i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			6.99226i			0.505942i	0.967474	+	0.252971i	$0.0814077\pi$
$191$			6.99226i			0.505942i	−0.967474	+	0.252971i	$0.918592\pi$
$192$		0			0
$193$	−7.47214	−	7.47214i	−0.537856	−	0.537856i	0.385043	−	0.922899i	$-0.374187\pi$
$193$	−7.47214	−	7.47214i	−0.537856	−	0.537856i	−0.922899	+	0.385043i	$0.874187\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.1437	−	15.1437i	−1.07894	−	1.07894i	−0.996604	−	0.0823386i	$-0.973761\pi$
$197$	−15.1437	−	15.1437i	−1.07894	−	1.07894i	−0.0823386	−	0.996604i	$-0.526239\pi$
$198$		0			0
$199$		−	8.00000i		−	0.567105i	−0.958957	−	0.283552i	$-0.908487\pi$
$199$		−	8.00000i		−	0.567105i	0.958957	−	0.283552i	$-0.0915130\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.412662	+	0.412662i	−0.0289632	+	0.0289632i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	11.3137			0.782586
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−12.9443	+	12.9443i	−0.878714	+	0.878714i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			2.16073i			0.145346i
$222$		0			0
$223$	−14.7639	−	14.7639i	−0.988666	−	0.988666i	0.0112705	−	0.999936i	$-0.496412\pi$
$223$	−14.7639	−	14.7639i	−0.988666	−	0.988666i	−0.999936	+	0.0112705i	$0.996412\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.81758	+	7.81758i	0.518871	+	0.518871i	0.917230	−	0.398359i	$-0.130420\pi$
$227$	7.81758	+	7.81758i	0.518871	+	0.518871i	−0.398359	+	0.917230i	$0.630420\pi$
$228$		0			0
$229$			15.8885i			1.04994i	0.851119	+	0.524972i	$0.175924\pi$
$229$			15.8885i			1.04994i	−0.851119	+	0.524972i	$0.824076\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.41577	−	2.41577i	0.158262	−	0.158262i	−0.623534	−	0.781796i	$-0.714304\pi$
$233$	2.41577	−	2.41577i	0.158262	−	0.158262i	0.781796	+	0.623534i	$0.214304\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.6491			0.818203			0.409101	−	0.912489i	$-0.365842\pi$
$239$	12.6491			0.818203			0.409101	+	0.912489i	$0.365842\pi$
$240$		0			0
$241$	−14.9443			−0.962645			−0.481323	−	0.876544i	$-0.659843\pi$
$241$	−14.9443			−0.962645			−0.481323	+	0.876544i	$0.659843\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−1.52786	+	1.52786i	−0.0972157	+	0.0972157i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			11.7264i			0.740162i	0.929000	+	0.370081i	$0.120670\pi$
$251$			11.7264i			0.740162i	−0.929000	+	0.370081i	$0.879330\pi$
$252$		0			0
$253$	4.94427	+	4.94427i	0.310844	+	0.310844i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.16228	−	3.16228i	−0.197257	−	0.197257i	0.601566	−	0.798823i	$-0.294544\pi$
$257$	−3.16228	−	3.16228i	−0.197257	−	0.197257i	−0.798823	+	0.601566i	$0.794544\pi$
$258$		0			0
$259$			5.52786i			0.343485i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	14.1421	−	14.1421i	0.872041	−	0.872041i	−0.120653	−	0.992695i	$-0.538499\pi$
$263$	14.1421	−	14.1421i	0.872041	−	0.872041i	0.992695	+	0.120653i	$0.0384989\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.65841			−0.405970			−0.202985	−	0.979182i	$-0.565064\pi$
$269$	−6.65841			−0.405970			−0.202985	+	0.979182i	$0.565064\pi$
$270$		0			0
$271$	−11.0557			−0.671588			−0.335794	−	0.941936i	$-0.609005\pi$
$271$	−11.0557			−0.671588			−0.335794	+	0.941936i	$0.609005\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	7.18034	−	7.18034i	0.431425	−	0.431425i	−0.457688	−	0.889113i	$-0.651322\pi$
$277$	7.18034	−	7.18034i	0.431425	−	0.431425i	0.889113	+	0.457688i	$0.151322\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	20.3879i		−	1.21624i	−0.793846	−	0.608119i	$-0.791924\pi$
$281$		−	20.3879i		−	1.21624i	0.793846	−	0.608119i	$-0.208076\pi$
$282$		0			0
$283$	8.00000	+	8.00000i	0.475551	+	0.475551i	0.903705	−	0.428155i	$-0.140836\pi$
$283$	8.00000	+	8.00000i	0.475551	+	0.475551i	−0.428155	+	0.903705i	$0.640836\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.74032	+	8.74032i	0.515925	+	0.515925i
$288$		0			0
$289$			24.8885i			1.46403i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−14.4760	+	14.4760i	−0.845696	+	0.845696i	−0.989593	−	0.143897i	$-0.954037\pi$
$293$	−14.4760	+	14.4760i	−0.845696	+	0.845696i	0.143897	+	0.989593i	$0.454037\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.33540			−0.0772283
$300$		0			0
$301$	−16.0000			−0.922225
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−1.52786	+	1.52786i	−0.0871998	+	0.0871998i	−0.749361	−	0.662161i	$-0.769639\pi$
$307$	−1.52786	+	1.52786i	−0.0871998	+	0.0871998i	0.662161	+	0.749361i	$0.269639\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			2.16073i			0.122524i	0.998122	+	0.0612618i	$0.0195124\pi$
$311$			2.16073i			0.122524i	−0.998122	+	0.0612618i	$0.980488\pi$
$312$		0			0
$313$	−1.47214	−	1.47214i	−0.0832100	−	0.0832100i	0.664277	−	0.747487i	$-0.268740\pi$
$313$	−1.47214	−	1.47214i	−0.0832100	−	0.0832100i	−0.747487	+	0.664277i	$0.768740\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	20.4667	+	20.4667i	1.14952	+	1.14952i	0.986646	+	0.162878i	$0.0520776\pi$
$317$	20.4667	+	20.4667i	1.14952	+	1.14952i	0.162878	+	0.986646i	$0.447922\pi$
$318$		0			0
$319$			0.583592i			0.0326749i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−29.6197	+	29.6197i	−1.64808	+	1.64808i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−11.3137			−0.623745
$330$		0			0
$331$	−14.4721			−0.795461			−0.397730	−	0.917502i	$-0.630202\pi$
$331$	−14.4721			−0.795461			−0.397730	+	0.917502i	$0.630202\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	12.4164	−	12.4164i	0.676365	−	0.676365i	−0.282811	−	0.959176i	$-0.591267\pi$
$337$	12.4164	−	12.4164i	0.676365	−	0.676365i	0.959176	+	0.282811i	$0.0912669\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			18.3060i			0.991324i
$342$		0			0
$343$	−13.5279	−	13.5279i	−0.730436	−	0.730436i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	18.3060	+	18.3060i	0.982716	+	0.982716i	0.999853	−	0.0171375i	$-0.00545529\pi$
$347$	18.3060	+	18.3060i	0.982716	+	0.982716i	−0.0171375	+	0.999853i	$0.505455\pi$
$348$		0			0
$349$			14.4721i			0.774676i	0.921938	+	0.387338i	$0.126605\pi$
$349$			14.4721i			0.774676i	−0.921938	+	0.387338i	$0.873395\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−11.5687	+	11.5687i	−0.615742	+	0.615742i	−0.944436	−	0.328694i	$-0.893391\pi$
$353$	−11.5687	+	11.5687i	−0.615742	+	0.615742i	0.328694	+	0.944436i	$0.393391\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4.83153			−0.254998			−0.127499	−	0.991839i	$-0.540695\pi$
$359$	−4.83153			−0.254998			−0.127499	+	0.991839i	$0.540695\pi$
$360$		0			0
$361$	−22.8885			−1.20466
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−15.7082	+	15.7082i	−0.819962	+	0.819962i	−0.986102	−	0.166141i	$-0.946869\pi$
$367$	−15.7082	+	15.7082i	−0.819962	+	0.819962i	0.166141	+	0.986102i	$0.446869\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−24.7082	−	24.7082i	−1.27934	−	1.27934i	−0.941036	−	0.338306i	$-0.890146\pi$
$373$	−24.7082	−	24.7082i	−1.27934	−	1.27934i	−0.338306	−	0.941036i	$-0.609854\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.0788114	−	0.0788114i	−0.00405899	−	0.00405899i
$378$		0			0
$379$		−	0.944272i		−	0.0485040i	−0.999706	−	0.0242520i	$-0.992280\pi$
$379$		−	0.944272i		−	0.0485040i	0.999706	−	0.0242520i	$-0.00772041\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	10.2333	−	10.2333i	0.522900	−	0.522900i	−0.395546	−	0.918446i	$-0.629445\pi$
$383$	10.2333	−	10.2333i	0.522900	−	0.522900i	0.918446	+	0.395546i	$0.129445\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	37.7711			1.91507			0.957536	−	0.288315i	$-0.0930952\pi$
$389$	37.7711			1.91507			0.957536	+	0.288315i	$0.0930952\pi$
$390$		0			0
$391$	−25.8885			−1.30924
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.76393	−	1.76393i	0.0885292	−	0.0885292i	−0.661455	−	0.749985i	$-0.730061\pi$
$397$	1.76393	−	1.76393i	0.0885292	−	0.0885292i	0.749985	+	0.661455i	$0.230061\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			14.8886i			0.743504i	0.928332	+	0.371752i	$0.121243\pi$
$401$			14.8886i			0.743504i	−0.928332	+	0.371752i	$0.878757\pi$
$402$		0			0
$403$	−2.47214	−	2.47214i	−0.123146	−	0.123146i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.90879	+	3.90879i	0.193752	+	0.193752i
$408$		0			0
$409$		−	9.88854i		−	0.488957i	−0.969655	−	0.244479i	$-0.921383\pi$
$409$		−	9.88854i		−	0.488957i	0.969655	−	0.244479i	$-0.0786168\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	9.15298	−	9.15298i	0.450389	−	0.450389i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−37.0246			−1.80877			−0.904385	−	0.426718i	$-0.859670\pi$
$419$	−37.0246			−1.80877			−0.904385	+	0.426718i	$0.859670\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.88854	+	1.88854i	−0.0913930	+	0.0913930i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	5.65685i		−	0.272481i	−0.990676	−	0.136241i	$-0.956498\pi$
$431$		−	5.65685i		−	0.272481i	0.990676	−	0.136241i	$-0.0435020\pi$
$432$		0			0
$433$	−10.4164	−	10.4164i	−0.500581	−	0.500581i	0.411038	−	0.911618i	$-0.365166\pi$
$433$	−10.4164	−	10.4164i	−0.500581	−	0.500581i	−0.911618	+	0.411038i	$0.865166\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−18.3060	−	18.3060i	−0.875693	−	0.875693i
$438$		0			0
$439$		−	3.05573i		−	0.145842i	−0.997338	−	0.0729210i	$-0.976768\pi$
$439$		−	3.05573i		−	0.145842i	0.997338	−	0.0729210i	$-0.0232321\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7.81758	+	7.81758i	−0.371424	+	0.371424i	−0.867996	−	0.496571i	$-0.834592\pi$
$443$	−7.81758	+	7.81758i	−0.371424	+	0.371424i	0.496571	+	0.867996i	$0.334592\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	24.5517			1.15867			0.579333	−	0.815091i	$-0.303313\pi$
$449$	24.5517			1.15867			0.579333	+	0.815091i	$0.303313\pi$
$450$		0			0
$451$	12.3607			0.582042
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−13.0000	+	13.0000i	−0.608114	+	0.608114i	−0.942453	−	0.334339i	$-0.891487\pi$
$457$	−13.0000	+	13.0000i	−0.608114	+	0.608114i	0.334339	+	0.942453i	$0.391487\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			9.48683i			0.441846i	0.975291	+	0.220923i	$0.0709069\pi$
$461$			9.48683i			0.441846i	−0.975291	+	0.220923i	$0.929093\pi$
$462$		0			0
$463$	20.6525	+	20.6525i	0.959802	+	0.959802i	0.999223	−	0.0394208i	$-0.0125513\pi$
$463$	20.6525	+	20.6525i	0.959802	+	0.959802i	−0.0394208	+	0.999223i	$0.512551\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.74806	+	1.74806i	0.0808908	+	0.0808908i	0.746395	−	0.665504i	$-0.231783\pi$
$467$	1.74806	+	1.74806i	0.0808908	+	0.0808908i	−0.665504	+	0.746395i	$0.731783\pi$
$468$		0			0
$469$			25.8885i			1.19542i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−11.3137	+	11.3137i	−0.520205	+	0.520205i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	39.5980			1.80928			0.904639	−	0.426179i	$-0.140141\pi$
$479$	39.5980			1.80928			0.904639	+	0.426179i	$0.140141\pi$
$480$		0			0
$481$	−1.05573			−0.0481371
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−13.2361	+	13.2361i	−0.599783	+	0.599783i	−0.940255	−	0.340471i	$-0.889413\pi$
$487$	−13.2361	+	13.2361i	−0.599783	+	0.599783i	0.340471	+	0.940255i	$0.389413\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			11.7264i			0.529204i	0.964358	+	0.264602i	$0.0852405\pi$
$491$			11.7264i			0.529204i	−0.964358	+	0.264602i	$0.914759\pi$
$492$		0			0
$493$	−1.52786	−	1.52786i	−0.0688115	−	0.0688115i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	15.6352	+	15.6352i	0.701333	+	0.701333i
$498$		0			0
$499$			17.5279i			0.784655i	0.919826	+	0.392327i	$0.128330\pi$
$499$			17.5279i			0.784655i	−0.919826	+	0.392327i	$0.871670\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	15.0649	−	15.0649i	0.671710	−	0.671710i	−0.286400	−	0.958110i	$-0.592459\pi$
$503$	15.0649	−	15.0649i	0.671710	−	0.671710i	0.958110	+	0.286400i	$0.0924588\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−18.6398			−0.826195			−0.413098	−	0.910687i	$-0.635553\pi$
$509$	−18.6398			−0.826195			−0.413098	+	0.910687i	$0.635553\pi$
$510$		0			0
$511$	−23.4164			−1.03588
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−8.00000	+	8.00000i	−0.351840	+	0.351840i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			31.7016i			1.38887i	0.719555	+	0.694436i	$0.244346\pi$
$521$			31.7016i			1.38887i	−0.719555	+	0.694436i	$0.755654\pi$
$522$		0			0
$523$	−25.8885	−	25.8885i	−1.13203	−	1.13203i	−0.989840	−	0.142187i	$-0.954587\pi$
$523$	−25.8885	−	25.8885i	−1.13203	−	1.13203i	−0.142187	−	0.989840i	$-0.545413\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−47.9256	−	47.9256i	−2.08767	−	2.08767i
$528$		0			0
$529$			7.00000i			0.304348i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.66925	+	1.66925i	−0.0723034	+	0.0723034i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6.89484			−0.296982
$540$		0			0
$541$	37.3050			1.60387			0.801933	−	0.597415i	$-0.203805\pi$
$541$	37.3050			1.60387			0.801933	+	0.597415i	$0.203805\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−12.0000	+	12.0000i	−0.513083	+	0.513083i	−0.915470	−	0.402387i	$-0.868181\pi$
$547$	−12.0000	+	12.0000i	−0.513083	+	0.513083i	0.402387	+	0.915470i	$0.368181\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	2.16073i		−	0.0920500i
$552$		0			0
$553$	6.83282	+	6.83282i	0.290561	+	0.290561i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	13.4744	+	13.4744i	0.570930	+	0.570930i	0.932388	−	0.361458i	$-0.117721\pi$
$557$	13.4744	+	13.4744i	0.570930	+	0.570930i	−0.361458	+	0.932388i	$0.617721\pi$
$558$		0			0
$559$		−	3.05573i		−	0.129244i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−23.9628	+	23.9628i	−1.00991	+	1.00991i	−0.00996204	+	0.999950i	$0.503171\pi$
$563$	−23.9628	+	23.9628i	−1.00991	+	1.00991i	−0.999950	+	0.00996204i	$0.996829\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	17.7171			0.742738			0.371369	−	0.928485i	$-0.378888\pi$
$569$	17.7171			0.742738			0.371369	+	0.928485i	$0.378888\pi$
$570$		0			0
$571$	19.4164			0.812551			0.406276	−	0.913751i	$-0.366827\pi$
$571$	19.4164			0.812551			0.406276	+	0.913751i	$0.366827\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	29.9443	−	29.9443i	1.24660	−	1.24660i	0.289383	−	0.957213i	$-0.406550\pi$
$577$	29.9443	−	29.9443i	1.24660	−	1.24660i	0.957213	−	0.289383i	$-0.0934501\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			18.3060i			0.759459i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−8.74032	−	8.74032i	−0.360752	−	0.360752i	0.503338	−	0.864090i	$-0.332105\pi$
$587$	−8.74032	−	8.74032i	−0.360752	−	0.360752i	−0.864090	+	0.503338i	$0.832105\pi$
$588$		0			0
$589$		−	67.7771i		−	2.79271i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	22.1359	−	22.1359i	0.909014	−	0.909014i	−0.0871785	−	0.996193i	$-0.527785\pi$
$593$	22.1359	−	22.1359i	0.909014	−	0.909014i	0.996193	+	0.0871785i	$0.0277850\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.6491			0.516829			0.258414	−	0.966034i	$-0.416800\pi$
$599$	12.6491			0.516829			0.258414	+	0.966034i	$0.416800\pi$
$600$		0			0
$601$	22.8328			0.931370			0.465685	−	0.884951i	$-0.345808\pi$
$601$	22.8328			0.931370			0.465685	+	0.884951i	$0.345808\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	0.652476	−	0.652476i	0.0264832	−	0.0264832i	−0.693741	−	0.720224i	$-0.744039\pi$
$607$	0.652476	−	0.652476i	0.0264832	−	0.0264832i	0.720224	+	0.693741i	$0.244039\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	2.16073i		−	0.0874136i
$612$		0			0
$613$	13.2918	+	13.2918i	0.536851	+	0.536851i	0.922603	−	0.385752i	$-0.126058\pi$
$613$	13.2918	+	13.2918i	0.536851	+	0.536851i	−0.385752	+	0.922603i	$0.626058\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.41577	+	2.41577i	0.0972550	+	0.0972550i	0.754060	−	0.656805i	$-0.228093\pi$
$617$	2.41577	+	2.41577i	0.0972550	+	0.0972550i	−0.656805	+	0.754060i	$0.728093\pi$
$618$		0			0
$619$			29.8885i			1.20132i	0.799504	+	0.600661i	$0.205096\pi$
$619$			29.8885i			1.20132i	−0.799504	+	0.600661i	$0.794904\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	16.5579	−	16.5579i	0.663378	−	0.663378i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−20.4667			−0.816060
$630$		0			0
$631$	−36.3607			−1.44750			−0.723748	−	0.690064i	$-0.757582\pi$
$631$	−36.3607			−1.44750			−0.723748	+	0.690064i	$0.757582\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.931116	−	0.931116i	0.0368922	−	0.0368922i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	19.7202i		−	0.778900i	−0.921048	−	0.389450i	$-0.872665\pi$
$641$		−	19.7202i		−	0.778900i	0.921048	−	0.389450i	$-0.127335\pi$
$642$		0			0
$643$	20.0000	+	20.0000i	0.788723	+	0.788723i	0.981285	−	0.192562i	$-0.0616796\pi$
$643$	20.0000	+	20.0000i	0.788723	+	0.788723i	−0.192562	+	0.981285i	$0.561680\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−18.0509	−	18.0509i	−0.709655	−	0.709655i	0.256807	−	0.966463i	$-0.417329\pi$
$647$	−18.0509	−	18.0509i	−0.709655	−	0.709655i	−0.966463	+	0.256807i	$0.917329\pi$
$648$		0			0
$649$		−	12.9443i		−	0.508107i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	11.3137	−	11.3137i	0.442740	−	0.442740i	−0.450192	−	0.892932i	$-0.648644\pi$
$653$	11.3137	−	11.3137i	0.442740	−	0.442740i	0.892932	+	0.450192i	$0.148644\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	21.3894			0.833214			0.416607	−	0.909087i	$-0.363219\pi$
$659$	21.3894			0.833214			0.416607	+	0.909087i	$0.363219\pi$
$660$		0			0
$661$	−21.3050			−0.828667			−0.414333	−	0.910125i	$-0.635985\pi$
$661$	−21.3050			−0.828667			−0.414333	+	0.910125i	$0.635985\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	0.944272	−	0.944272i	0.0365624	−	0.0365624i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			2.67080i			0.103105i
$672$		0			0
$673$	−10.5279	−	10.5279i	−0.405819	−	0.405819i	0.474459	−	0.880278i	$-0.342644\pi$
$673$	−10.5279	−	10.5279i	−0.405819	−	0.405819i	−0.880278	+	0.474459i	$0.842644\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6.99226	+	6.99226i	0.268734	+	0.268734i	0.828590	−	0.559856i	$-0.189143\pi$
$677$	6.99226	+	6.99226i	0.268734	+	0.268734i	−0.559856	+	0.828590i	$0.689143\pi$
$678$		0			0
$679$		−	2.47214i		−	0.0948719i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−29.2070	+	29.2070i	−1.11758	+	1.11758i	−0.125479	+	0.992096i	$0.540047\pi$
$683$	−29.2070	+	29.2070i	−1.11758	+	1.11758i	−0.992096	+	0.125479i	$0.959953\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	24.3607			0.926724			0.463362	−	0.886169i	$-0.346643\pi$
$691$	24.3607			0.926724			0.463362	+	0.886169i	$0.346643\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−32.3607	+	32.3607i	−1.22575	+	1.22575i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	13.6507i		−	0.515578i	−0.966201	−	0.257789i	$-0.917006\pi$
$701$		−	13.6507i		−	0.515578i	0.966201	−	0.257789i	$-0.0829940\pi$
$702$		0			0
$703$	−14.4721	−	14.4721i	−0.545827	−	0.545827i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	21.3894	+	21.3894i	0.804432	+	0.804432i
$708$		0			0
$709$			19.8885i			0.746930i	0.927644	+	0.373465i	$0.121830\pi$
$709$			19.8885i			0.746930i	−0.927644	+	0.373465i	$0.878170\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	29.6197	−	29.6197i	1.10927	−	1.10927i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−29.6197			−1.10463			−0.552314	−	0.833636i	$-0.686255\pi$
$719$	−29.6197			−1.10463			−0.552314	+	0.833636i	$0.686255\pi$
$720$		0			0
$721$	25.1672			0.937275
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	31.1246	−	31.1246i	1.15435	−	1.15435i	0.168676	−	0.985671i	$-0.446051\pi$
$727$	31.1246	−	31.1246i	1.15435	−	1.15435i	0.985671	−	0.168676i	$-0.0539493\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	59.2393i		−	2.19105i
$732$		0			0
$733$	11.1803	+	11.1803i	0.412955	+	0.412955i	0.882767	−	0.469811i	$-0.155678\pi$
$733$	11.1803	+	11.1803i	0.412955	+	0.412955i	−0.469811	+	0.882767i	$0.655678\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	18.3060	+	18.3060i	0.674309	+	0.674309i
$738$		0			0
$739$			32.3607i			1.19041i	0.803575	+	0.595203i	$0.202929\pi$
$739$			32.3607i			1.19041i	−0.803575	+	0.595203i	$0.797071\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−15.0649	+	15.0649i	−0.552677	+	0.552677i	−0.927212	−	0.374536i	$-0.877802\pi$
$743$	−15.0649	+	15.0649i	−0.552677	+	0.552677i	0.374536	+	0.927212i	$0.377802\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−9.66306			−0.353081
$750$		0			0
$751$	−12.3607			−0.451048			−0.225524	−	0.974238i	$-0.572409\pi$
$751$	−12.3607			−0.451048			−0.225524	+	0.974238i	$0.572409\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	2.70820	−	2.70820i	0.0984313	−	0.0984313i	−0.656176	−	0.754608i	$-0.727827\pi$
$757$	2.70820	−	2.70820i	0.0984313	−	0.0984313i	0.754608	+	0.656176i	$0.227827\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	11.9026i		−	0.431469i	−0.976452	−	0.215734i	$-0.930785\pi$
$761$		−	11.9026i		−	0.431469i	0.976452	−	0.215734i	$-0.0692145\pi$
$762$		0			0
$763$	14.1115	+	14.1115i	0.510869	+	0.510869i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.74806	+	1.74806i	0.0631189	+	0.0631189i
$768$		0			0
$769$		−	32.0000i		−	1.15395i	−0.816762	−	0.576975i	$-0.804233\pi$
$769$		−	32.0000i		−	1.15395i	0.816762	−	0.576975i	$-0.195767\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−20.4667	+	20.4667i	−0.736136	+	0.736136i	−0.971828	−	0.235692i	$-0.924264\pi$
$773$	−20.4667	+	20.4667i	−0.736136	+	0.736136i	0.235692	+	0.971828i	$0.424264\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−45.7649			−1.63970
$780$		0			0
$781$	22.1115			0.791210
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	21.5279	−	21.5279i	0.767385	−	0.767385i	−0.210260	−	0.977645i	$-0.567431\pi$
$787$	21.5279	−	21.5279i	0.767385	−	0.767385i	0.977645	+	0.210260i	$0.0674311\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	7.81758i		−	0.277961i
$792$		0			0
$793$	−0.360680	−	0.360680i	−0.0128081	−	0.0128081i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	29.9535	+	29.9535i	1.06101	+	1.06101i	0.998014	+	0.0629944i	$0.0200650\pi$
$797$	29.9535	+	29.9535i	1.06101	+	1.06101i	0.0629944	+	0.998014i	$0.479935\pi$
$798$		0			0
$799$		−	41.8885i		−	1.48191i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−16.5579	+	16.5579i	−0.584316	+	0.584316i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−32.3693			−1.13804			−0.569022	−	0.822322i	$-0.692678\pi$
$809$	−32.3693			−1.13804			−0.569022	+	0.822322i	$0.692678\pi$
$810$		0			0
$811$	21.8885			0.768611			0.384305	−	0.923206i	$-0.374441\pi$
$811$	21.8885			0.768611			0.384305	+	0.923206i	$0.374441\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	41.8885	−	41.8885i	1.46549	−	1.46549i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	24.2967i		−	0.847960i	−0.905672	−	0.423980i	$-0.860633\pi$
$821$		−	24.2967i		−	0.847960i	0.905672	−	0.423980i	$-0.139367\pi$
$822$		0			0
$823$	27.1246	+	27.1246i	0.945505	+	0.945505i	0.998590	−	0.0530854i	$-0.0169056\pi$
$823$	27.1246	+	27.1246i	0.945505	+	0.945505i	−0.0530854	+	0.998590i	$0.516906\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−20.4667	−	20.4667i	−0.711697	−	0.711697i	0.255193	−	0.966890i	$-0.417861\pi$
$827$	−20.4667	−	20.4667i	−0.711697	−	0.711697i	−0.966890	+	0.255193i	$0.917861\pi$
$828$		0			0
$829$		−	8.36068i		−	0.290378i	−0.989404	−	0.145189i	$-0.953621\pi$
$829$		−	8.36068i		−	0.290378i	0.989404	−	0.145189i	$-0.0463791\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	18.0509	−	18.0509i	0.625428	−	0.625428i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−16.1452			−0.557396			−0.278698	−	0.960379i	$-0.589903\pi$
$839$	−16.1452			−0.557396			−0.278698	+	0.960379i	$0.589903\pi$
$840$		0			0
$841$	−28.8885			−0.996157
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−9.81966	+	9.81966i	−0.337408	+	0.337408i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	12.6491i		−	0.433606i
$852$		0			0
$853$	−12.1246	−	12.1246i	−0.415139	−	0.415139i	0.468385	−	0.883524i	$-0.344836\pi$
$853$	−12.1246	−	12.1246i	−0.415139	−	0.415139i	−0.883524	+	0.468385i	$0.844836\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18.0509	+	18.0509i	0.616608	+	0.616608i	0.944660	−	0.328052i	$-0.106392\pi$
$857$	18.0509	+	18.0509i	0.616608	+	0.616608i	−0.328052	+	0.944660i	$0.606392\pi$
$858$		0			0
$859$		−	4.00000i		−	0.136478i	−0.997669	−	0.0682391i	$-0.978262\pi$
$859$		−	4.00000i		−	0.136478i	0.997669	−	0.0682391i	$-0.0217381\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	34.6088	−	34.6088i	1.17810	−	1.17810i	0.197870	−	0.980228i	$-0.436598\pi$
$863$	34.6088	−	34.6088i	1.17810	−	1.17810i	0.980228	−	0.197870i	$-0.0634023\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	9.66306			0.327797
$870$		0			0
$871$	−4.94427			−0.167530
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	20.7082	−	20.7082i	0.699266	−	0.699266i	−0.264986	−	0.964252i	$-0.585367\pi$
$877$	20.7082	−	20.7082i	0.699266	−	0.699266i	0.964252	+	0.264986i	$0.0853673\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			14.8886i			0.501611i	0.968037	+	0.250806i	$0.0806955\pi$
$881$			14.8886i			0.501611i	−0.968037	+	0.250806i	$0.919304\pi$
$882$		0			0
$883$	22.4721	+	22.4721i	0.756248	+	0.756248i	0.975637	−	0.219390i	$-0.0704066\pi$
$883$	22.4721	+	22.4721i	0.756248	+	0.756248i	−0.219390	+	0.975637i	$0.570407\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−40.2657	−	40.2657i	−1.35199	−	1.35199i	−0.883434	−	0.468555i	$-0.844775\pi$
$887$	−40.2657	−	40.2657i	−1.35199	−	1.35199i	−0.468555	−	0.883434i	$-0.655225\pi$
$888$		0			0
$889$			6.83282i			0.229165i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	29.6197	−	29.6197i	0.991185	−	0.991185i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.49613			0.116602
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−34.4721	+	34.4721i	−1.14463	+	1.14463i	−0.157036	+	0.987593i	$0.550194\pi$
$907$	−34.4721	+	34.4721i	−1.14463	+	1.14463i	−0.987593	+	0.157036i	$0.949806\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	43.0941i		−	1.42777i	−0.700262	−	0.713886i	$-0.746934\pi$
$911$		−	43.0941i		−	1.42777i	0.700262	−	0.713886i	$-0.253066\pi$
$912$		0			0
$913$	12.9443	+	12.9443i	0.428393	+	0.428393i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−20.4667	−	20.4667i	−0.675870	−	0.675870i
$918$		0			0
$919$		−	13.5279i		−	0.446243i	−0.974791	−	0.223122i	$-0.928375\pi$
$919$		−	13.5279i		−	0.446243i	0.974791	−	0.223122i	$-0.0716247\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2.98605	+	2.98605i	−0.0982870	+	0.0982870i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	10.5672			0.346698			0.173349	−	0.984860i	$-0.444541\pi$
$929$	10.5672			0.346698			0.173349	+	0.984860i	$0.444541\pi$
$930$		0			0
$931$	25.5279			0.836642
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	19.0000	−	19.0000i	0.620703	−	0.620703i	−0.325008	−	0.945711i	$-0.605367\pi$
$937$	19.0000	−	19.0000i	0.620703	−	0.620703i	0.945711	+	0.325008i	$0.105367\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			8.81913i			0.287495i	0.989614	+	0.143748i	$0.0459154\pi$
$941$			8.81913i			0.287495i	−0.989614	+	0.143748i	$0.954085\pi$
$942$		0			0
$943$	−20.0000	−	20.0000i	−0.651290	−	0.651290i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	14.8098	+	14.8098i	0.481255	+	0.481255i	0.905532	−	0.424277i	$-0.139472\pi$
$947$	14.8098	+	14.8098i	0.481255	+	0.481255i	−0.424277	+	0.905532i	$0.639472\pi$
$948$		0			0
$949$		−	4.47214i		−	0.145172i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−34.7851	+	34.7851i	−1.12680	+	1.12680i	−0.136104	+	0.990695i	$0.543458\pi$
$953$	−34.7851	+	34.7851i	−1.12680	+	1.12680i	−0.990695	+	0.136104i	$0.956542\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	6.16693			0.199141
$960$		0			0
$961$	78.6656			2.53760
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−24.0689	+	24.0689i	−0.774003	+	0.774003i	−0.978804	−	0.204801i	$-0.934345\pi$
$967$	−24.0689	+	24.0689i	−0.774003	+	0.774003i	0.204801	+	0.978804i	$0.434345\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			3.08347i			0.0989531i	0.998775	+	0.0494766i	$0.0157553\pi$
$971$			3.08347i			0.0989531i	−0.998775	+	0.0494766i	$0.984245\pi$
$972$		0			0
$973$	−11.0557	−	11.0557i	−0.354430	−	0.354430i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−29.8747	−	29.8747i	−0.955777	−	0.955777i	0.0432860	−	0.999063i	$-0.486217\pi$
$977$	−29.8747	−	29.8747i	−0.955777	−	0.955777i	−0.999063	+	0.0432860i	$0.986217\pi$
$978$		0			0
$979$		−	23.4164i		−	0.748392i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	30.7000	−	30.7000i	0.979179	−	0.979179i	−0.0206085	−	0.999788i	$-0.506560\pi$
$983$	30.7000	−	30.7000i	0.979179	−	0.979179i	0.999788	+	0.0206085i	$0.00656036\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	36.6119			1.16419
$990$		0			0
$991$	30.8328			0.979437			0.489718	−	0.871881i	$-0.337100\pi$
$991$	30.8328			0.979437			0.489718	+	0.871881i	$0.337100\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	5.65248	−	5.65248i	0.179016	−	0.179016i	−0.611911	−	0.790927i	$-0.709599\pi$
$997$	5.65248	−	5.65248i	0.179016	−	0.179016i	0.790927	+	0.611911i	$0.209599\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.2.s.e.593.2		8
3.2	odd	2	inner	1800.2.s.e.593.1		8
4.3	odd	2		3600.2.w.j.593.3		8
5.2	odd	4	inner	1800.2.s.e.1457.2		8
5.3	odd	4		360.2.s.b.17.2	✓	8
5.4	even	2		360.2.s.b.233.4	yes	8
12.11	even	2		3600.2.w.j.593.4		8
15.2	even	4	inner	1800.2.s.e.1457.1		8
15.8	even	4		360.2.s.b.17.4	yes	8
15.14	odd	2		360.2.s.b.233.2	yes	8
20.3	even	4		720.2.w.e.17.1		8
20.7	even	4		3600.2.w.j.1457.3		8
20.19	odd	2		720.2.w.e.593.3		8
40.3	even	4		2880.2.w.o.2177.3		8
40.13	odd	4		2880.2.w.m.2177.4		8
40.19	odd	2		2880.2.w.o.2753.1		8
40.29	even	2		2880.2.w.m.2753.2		8
60.23	odd	4		720.2.w.e.17.3		8
60.47	odd	4		3600.2.w.j.1457.4		8
60.59	even	2		720.2.w.e.593.1		8
120.29	odd	2		2880.2.w.m.2753.4		8
120.53	even	4		2880.2.w.m.2177.2		8
120.59	even	2		2880.2.w.o.2753.3		8
120.83	odd	4		2880.2.w.o.2177.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.s.b.17.2	✓	8	5.3	odd	4
360.2.s.b.17.4	yes	8	15.8	even	4
360.2.s.b.233.2	yes	8	15.14	odd	2
360.2.s.b.233.4	yes	8	5.4	even	2
720.2.w.e.17.1		8	20.3	even	4
720.2.w.e.17.3		8	60.23	odd	4
720.2.w.e.593.1		8	60.59	even	2
720.2.w.e.593.3		8	20.19	odd	2
1800.2.s.e.593.1		8	3.2	odd	2	inner
1800.2.s.e.593.2		8	1.1	even	1	trivial
1800.2.s.e.1457.1		8	15.2	even	4	inner
1800.2.s.e.1457.2		8	5.2	odd	4	inner
2880.2.w.m.2177.2		8	120.53	even	4
2880.2.w.m.2177.4		8	40.13	odd	4
2880.2.w.m.2753.2		8	40.29	even	2
2880.2.w.m.2753.4		8	120.29	odd	2
2880.2.w.o.2177.1		8	120.83	odd	4
2880.2.w.o.2177.3		8	40.3	even	4
2880.2.w.o.2753.1		8	40.19	odd	2
2880.2.w.o.2753.3		8	120.59	even	2
3600.2.w.j.593.3		8	4.3	odd	2
3600.2.w.j.593.4		8	12.11	even	2
3600.2.w.j.1457.3		8	20.7	even	4
3600.2.w.j.1457.4		8	60.47	odd	4

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:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1800.s (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.23607 + 1.23607i) q^{7} +O(q^{10})\)	\(q+(-1.23607 + 1.23607i) q^{7} +1.74806i q^{11} +(-0.236068 - 0.236068i) q^{13} +(-4.57649 - 4.57649i) q^{17} -6.47214i q^{19} +(2.82843 - 2.82843i) q^{23} +0.333851 q^{29} +10.4721 q^{31} +(2.23607 - 2.23607i) q^{37} -7.07107i q^{41} +(6.47214 + 6.47214i) q^{43} +(4.57649 + 4.57649i) q^{47} +3.94427i q^{49} -7.40492 q^{59} +1.52786 q^{61} +(10.4721 - 10.4721i) q^{67} -12.6491i q^{71} +(9.47214 + 9.47214i) q^{73} +(-2.16073 - 2.16073i) q^{77} -5.52786i q^{79} +(7.40492 - 7.40492i) q^{83} -13.3956 q^{89} +0.583592 q^{91} +(-1.00000 + 1.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7}+O(q^{10}) \) 8 * q + 8 * q^7	\( 8 q + 8 q^{7} + 16 q^{13} + 48 q^{31} + 16 q^{43} + 48 q^{61} + 48 q^{67} + 40 q^{73} + 112 q^{91} - 8 q^{97}+O(q^{100}) \) 8 * q + 8 * q^7 + 16 * q^13 + 48 * q^31 + 16 * q^43 + 48 * q^61 + 48 * q^67 + 40 * q^73 + 112 * q^91 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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