LMFDB - Embedded newform 2400.2.d.e.49.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(49,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2400.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.d (of order $2$, degree $1$, not 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:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.4
Root			$0.264658 - 1.38923i$ of defining polynomial
Character	$\chi$	$=$	2400.49
Dual form			2400.2.d.e.49.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.941367i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.941367i q^{7} +1.00000 q^{9} +4.49828i q^{11} -5.55691 q^{13} -7.55691i q^{17} -1.05863i q^{19} -0.941367i q^{21} +1.05863i q^{23} -1.00000 q^{27} -2.00000i q^{29} -3.55691 q^{31} -4.49828i q^{33} +7.43965 q^{37} +5.55691 q^{39} -3.88273 q^{41} +1.88273 q^{43} -10.0552i q^{47} +6.11383 q^{49} +7.55691i q^{51} +2.00000 q^{53} +1.05863i q^{57} +8.49828i q^{59} -8.99656i q^{61} +0.941367i q^{63} -4.00000 q^{67} -1.05863i q^{69} +12.9966 q^{71} -6.00000i q^{73} -4.23453 q^{77} +11.5569 q^{79} +1.00000 q^{81} -5.88273 q^{83} +2.00000i q^{87} +4.11727 q^{89} -5.23109i q^{91} +3.55691 q^{93} -17.1138i q^{97} +4.49828i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^9	$ 6 q - 6 q^{3} + 6 q^{9} - 6 q^{27} + 12 q^{31} + 8 q^{37} - 20 q^{41} + 8 q^{43} - 30 q^{49} + 12 q^{53} - 24 q^{67} + 8 q^{71} - 32 q^{77} + 36 q^{79} + 6 q^{81} - 32 q^{83} + 28 q^{89} - 12 q^{93}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^9 - 6 * q^27 + 12 * q^31 + 8 * q^37 - 20 * q^41 + 8 * q^43 - 30 * q^49 + 12 * q^53 - 24 * q^67 + 8 * q^71 - 32 * q^77 + 36 * q^79 + 6 * q^81 - 32 * q^83 + 28 * q^89 - 12 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.941367i	0.355803i	0.984048	+	0.177902i	$0.0569309\pi$
$7$	0.941367i	0.355803i	−0.984048	+	0.177902i	$0.943069\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.49828i	1.35628i	0.734931	+	0.678141i	$0.237214\pi$
$11$	4.49828i	1.35628i	−0.734931	+	0.678141i	$0.762786\pi$
$12$	0	0
$13$	−5.55691	−1.54121	−0.770605	−	0.637313i	$-0.780046\pi$
$13$	−5.55691	−1.54121	−0.770605	+	0.637313i	$0.780046\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.55691i	− 1.83282i	−0.400240	−	0.916410i	$-0.631073\pi$
$17$	− 7.55691i	− 1.83282i	0.400240	−	0.916410i	$-0.368927\pi$
$18$	0	0
$19$	− 1.05863i	− 0.242867i	−0.992600	−	0.121434i	$-0.961251\pi$
$19$	− 1.05863i	− 0.242867i	0.992600	−	0.121434i	$-0.0387491\pi$
$20$	0	0
$21$	− 0.941367i	− 0.205423i
$22$	0	0
$23$	1.05863i	0.220740i	0.993891	+	0.110370i	$0.0352036\pi$
$23$	1.05863i	0.220740i	−0.993891	+	0.110370i	$0.964796\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	− 2.00000i	− 0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	0	0
$31$	−3.55691	−0.638841	−0.319420	−	0.947613i	$-0.603488\pi$
$31$	−3.55691	−0.638841	−0.319420	+	0.947613i	$0.603488\pi$
$32$	0	0
$33$	− 4.49828i	− 0.783050i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.43965	1.22307	0.611535	−	0.791217i	$-0.290552\pi$
$37$	7.43965	1.22307	0.611535	+	0.791217i	$0.290552\pi$
$38$	0	0
$39$	5.55691	0.889818
$40$	0	0
$41$	−3.88273	−0.606381	−0.303191	−	0.952930i	$-0.598052\pi$
$41$	−3.88273	−0.606381	−0.303191	+	0.952930i	$0.598052\pi$
$42$	0	0
$43$	1.88273	0.287114	0.143557	−	0.989642i	$-0.454146\pi$
$43$	1.88273	0.287114	0.143557	+	0.989642i	$0.454146\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 10.0552i	− 1.46670i	−0.679851	−	0.733350i	$-0.737955\pi$
$47$	− 10.0552i	− 1.46670i	0.679851	−	0.733350i	$-0.262045\pi$
$48$	0	0
$49$	6.11383	0.873404
$50$	0	0
$51$	7.55691i	1.05818i
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.05863i	0.140219i
$58$	0	0
$59$	8.49828i	1.10638i	0.833054	+	0.553191i	$0.186590\pi$
$59$	8.49828i	1.10638i	−0.833054	+	0.553191i	$0.813410\pi$
$60$	0	0
$61$	− 8.99656i	− 1.15189i	−0.817488	−	0.575946i	$-0.804634\pi$
$61$	− 8.99656i	− 1.15189i	0.817488	−	0.575946i	$-0.195366\pi$
$62$	0	0
$63$	0.941367i	0.118601i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	− 1.05863i	− 0.127444i
$70$	0	0
$71$	12.9966	1.54241	0.771204	−	0.636588i	$-0.219655\pi$
$71$	12.9966	1.54241	0.771204	+	0.636588i	$0.219655\pi$
$72$	0	0
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.23453	−0.482570
$78$	0	0
$79$	11.5569	1.30025	0.650127	−	0.759825i	$-0.274716\pi$
$79$	11.5569	1.30025	0.650127	+	0.759825i	$0.274716\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.88273	−0.645714	−0.322857	−	0.946448i	$-0.604643\pi$
$83$	−5.88273	−0.645714	−0.322857	+	0.946448i	$0.604643\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.00000i	0.214423i
$88$	0	0
$89$	4.11727	0.436429	0.218215	−	0.975901i	$-0.429977\pi$
$89$	4.11727	0.436429	0.218215	+	0.975901i	$0.429977\pi$
$90$	0	0
$91$	− 5.23109i	− 0.548368i
$92$	0	0
$93$	3.55691	0.368835
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 17.1138i	− 1.73765i	−0.495123	−	0.868823i	$-0.664877\pi$
$97$	− 17.1138i	− 1.73765i	0.495123	−	0.868823i	$-0.335123\pi$
$98$	0	0
$99$	4.49828i	0.452094i
$100$	0	0
$101$	− 2.00000i	− 0.199007i	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$	− 2.00000i	− 0.199007i	0.995037	−	0.0995037i	$-0.0317255\pi$
$102$	0	0
$103$	− 10.1725i	− 1.00232i	−0.865354	−	0.501161i	$-0.832906\pi$
$103$	− 10.1725i	− 1.00232i	0.865354	−	0.501161i	$-0.167094\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.2311	−1.66579	−0.832896	−	0.553429i	$-0.813319\pi$
$107$	−17.2311	−1.66579	−0.832896	+	0.553429i	$0.813319\pi$
$108$	0	0
$109$	1.88273i	0.180333i	0.995927	+	0.0901666i	$0.0287399\pi$
$109$	1.88273i	0.180333i	−0.995927	+	0.0901666i	$0.971260\pi$
$110$	0	0
$111$	−7.43965	−0.706140
$112$	0	0
$113$	15.3224i	1.44141i	0.693243	+	0.720704i	$0.256181\pi$
$113$	15.3224i	1.44141i	−0.693243	+	0.720704i	$0.743819\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5.55691	−0.513737
$118$	0	0
$119$	7.11383	0.652124
$120$	0	0
$121$	−9.23453	−0.839503
$122$	0	0
$123$	3.88273	0.350094
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 18.1725i	− 1.61255i	−0.591544	−	0.806273i	$-0.701481\pi$
$127$	− 18.1725i	− 1.61255i	0.591544	−	0.806273i	$-0.298519\pi$
$128$	0	0
$129$	−1.88273	−0.165765
$130$	0	0
$131$	− 6.38101i	− 0.557512i	−0.960362	−	0.278756i	$-0.910078\pi$
$131$	− 6.38101i	− 0.557512i	0.960362	−	0.278756i	$-0.0899220\pi$
$132$	0	0
$133$	0.996562	0.0864129
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.44309i	0.379598i	0.981823	+	0.189799i	$0.0607837\pi$
$137$	4.44309i	0.379598i	−0.981823	+	0.189799i	$0.939216\pi$
$138$	0	0
$139$	− 20.1725i	− 1.71101i	−0.517798	−	0.855503i	$-0.673248\pi$
$139$	− 20.1725i	− 1.71101i	0.517798	−	0.855503i	$-0.326752\pi$
$140$	0	0
$141$	10.0552i	0.846800i
$142$	0	0
$143$	− 24.9966i	− 2.09032i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−6.11383	−0.504260
$148$	0	0
$149$	− 2.00000i	− 0.163846i	−0.996639	−	0.0819232i	$-0.973894\pi$
$149$	− 2.00000i	− 0.163846i	0.996639	−	0.0819232i	$-0.0261062\pi$
$150$	0	0
$151$	−9.67418	−0.787274	−0.393637	−	0.919266i	$-0.628783\pi$
$151$	−9.67418	−0.787274	−0.393637	+	0.919266i	$0.628783\pi$
$152$	0	0
$153$	− 7.55691i	− 0.610940i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.32582	−0.345238	−0.172619	−	0.984989i	$-0.555223\pi$
$157$	−4.32582	−0.345238	−0.172619	+	0.984989i	$0.555223\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	−0.996562	−0.0785401
$162$	0	0
$163$	6.11727	0.479141	0.239571	−	0.970879i	$-0.422993\pi$
$163$	6.11727	0.479141	0.239571	+	0.970879i	$0.422993\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 6.05520i	− 0.468565i	−0.972169	−	0.234283i	$-0.924726\pi$
$167$	− 6.05520i	− 0.468565i	0.972169	−	0.234283i	$-0.0752741\pi$
$168$	0	0
$169$	17.8793	1.37533
$170$	0	0
$171$	− 1.05863i	− 0.0809557i
$172$	0	0
$173$	16.8793	1.28331	0.641655	−	0.766994i	$-0.278248\pi$
$173$	16.8793	1.28331	0.641655	+	0.766994i	$0.278248\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 8.49828i	− 0.638770i
$178$	0	0
$179$	− 10.6155i	− 0.793443i	−0.917939	−	0.396722i	$-0.870148\pi$
$179$	− 10.6155i	− 0.793443i	0.917939	−	0.396722i	$-0.129852\pi$
$180$	0	0
$181$	− 14.1173i	− 1.04933i	−0.851309	−	0.524664i	$-0.824191\pi$
$181$	− 14.1173i	− 1.04933i	0.851309	−	0.524664i	$-0.175809\pi$
$182$	0	0
$183$	8.99656i	0.665045i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	33.9931	2.48582
$188$	0	0
$189$	− 0.941367i	− 0.0684744i
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	− 4.87930i	− 0.351219i	−0.984460	−	0.175610i	$-0.943810\pi$
$193$	− 4.87930i	− 0.351219i	0.984460	−	0.175610i	$-0.0561897\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.88617	0.205631	0.102816	−	0.994700i	$-0.467215\pi$
$197$	2.88617	0.205631	0.102816	+	0.994700i	$0.467215\pi$
$198$	0	0
$199$	−17.6742	−1.25289	−0.626445	−	0.779466i	$-0.715491\pi$
$199$	−17.6742	−1.25289	−0.626445	+	0.779466i	$0.715491\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	1.88273	0.132142
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.05863i	0.0735801i
$208$	0	0
$209$	4.76203	0.329396
$210$	0	0
$211$	− 23.9379i	− 1.64795i	−0.566623	−	0.823977i	$-0.691750\pi$
$211$	− 23.9379i	− 1.64795i	0.566623	−	0.823977i	$-0.308250\pi$
$212$	0	0
$213$	−12.9966	−0.890510
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 3.34836i	− 0.227302i
$218$	0	0
$219$	6.00000i	0.405442i
$220$	0	0
$221$	41.9931i	2.82476i
$222$	0	0
$223$	24.0552i	1.61086i	0.592694	+	0.805428i	$0.298064\pi$
$223$	24.0552i	1.61086i	−0.592694	+	0.805428i	$0.701936\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.1138	0.737651	0.368825	−	0.929499i	$-0.379760\pi$
$227$	11.1138	0.737651	0.368825	+	0.929499i	$0.379760\pi$
$228$	0	0
$229$	− 17.2311i	− 1.13866i	−0.822108	−	0.569331i	$-0.807202\pi$
$229$	− 17.2311i	− 1.13866i	0.822108	−	0.569331i	$-0.192798\pi$
$230$	0	0
$231$	4.23453	0.278612
$232$	0	0
$233$	− 8.44309i	− 0.553125i	−0.960996	−	0.276562i	$-0.910805\pi$
$233$	− 8.44309i	− 0.553125i	0.960996	−	0.276562i	$-0.0891953\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−11.5569	−0.750702
$238$	0	0
$239$	10.1173	0.654432	0.327216	−	0.944950i	$-0.393890\pi$
$239$	10.1173	0.654432	0.327216	+	0.944950i	$0.393890\pi$
$240$	0	0
$241$	16.8793	1.08729	0.543646	−	0.839315i	$-0.317044\pi$
$241$	16.8793	1.08729	0.543646	+	0.839315i	$0.317044\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.88273i	0.374309i
$248$	0	0
$249$	5.88273	0.372803
$250$	0	0
$251$	11.8466i	0.747753i	0.927478	+	0.373877i	$0.121972\pi$
$251$	11.8466i	0.747753i	−0.927478	+	0.373877i	$0.878028\pi$
$252$	0	0
$253$	−4.76203	−0.299386
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.6707i	0.665623i	0.942993	+	0.332811i	$0.107997\pi$
$257$	10.6707i	0.665623i	−0.942993	+	0.332811i	$0.892003\pi$
$258$	0	0
$259$	7.00344i	0.435172i
$260$	0	0
$261$	− 2.00000i	− 0.123797i
$262$	0	0
$263$	1.94480i	0.119922i	0.998201	+	0.0599609i	$0.0190976\pi$
$263$	1.94480i	0.119922i	−0.998201	+	0.0599609i	$0.980902\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4.11727	−0.251973
$268$	0	0
$269$	9.76547i	0.595411i	0.954658	+	0.297706i	$0.0962214\pi$
$269$	9.76547i	0.595411i	−0.954658	+	0.297706i	$0.903779\pi$
$270$	0	0
$271$	−3.44652	−0.209361	−0.104681	−	0.994506i	$-0.533382\pi$
$271$	−3.44652	−0.209361	−0.104681	+	0.994506i	$0.533382\pi$
$272$	0	0
$273$	5.23109i	0.316600i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.7880	1.12886	0.564431	−	0.825480i	$-0.309096\pi$
$277$	18.7880	1.12886	0.564431	+	0.825480i	$0.309096\pi$
$278$	0	0
$279$	−3.55691	−0.212947
$280$	0	0
$281$	−16.8793	−1.00693	−0.503467	−	0.864014i	$-0.667943\pi$
$281$	−16.8793	−1.00693	−0.503467	+	0.864014i	$0.667943\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 3.65508i	− 0.215752i
$288$	0	0
$289$	−40.1070	−2.35923
$290$	0	0
$291$	17.1138i	1.00323i
$292$	0	0
$293$	−20.2277	−1.18171	−0.590856	−	0.806777i	$-0.701210\pi$
$293$	−20.2277	−1.18171	−0.590856	+	0.806777i	$0.701210\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.49828i	− 0.261017i
$298$	0	0
$299$	− 5.88273i	− 0.340207i
$300$	0	0
$301$	1.77234i	0.102156i
$302$	0	0
$303$	2.00000i	0.114897i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.11039	0.462884	0.231442	−	0.972849i	$-0.425656\pi$
$307$	8.11039	0.462884	0.231442	+	0.972849i	$0.425656\pi$
$308$	0	0
$309$	10.1725i	0.578691i
$310$	0	0
$311$	−31.8759	−1.80751	−0.903757	−	0.428046i	$-0.859202\pi$
$311$	−31.8759	−1.80751	−0.903757	+	0.428046i	$0.859202\pi$
$312$	0	0
$313$	− 5.11383i	− 0.289051i	−0.989501	−	0.144525i	$-0.953834\pi$
$313$	− 5.11383i	− 0.289051i	0.989501	−	0.144525i	$-0.0461655\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.6448	−1.38419	−0.692094	−	0.721807i	$-0.743312\pi$
$317$	−24.6448	−1.38419	−0.692094	+	0.721807i	$0.743312\pi$
$318$	0	0
$319$	8.99656	0.503711
$320$	0	0
$321$	17.2311	0.961746
$322$	0	0
$323$	−8.00000	−0.445132
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1.88273i	− 0.104115i
$328$	0	0
$329$	9.46563	0.521857
$330$	0	0
$331$	− 11.0518i	− 0.607460i	−0.952758	−	0.303730i	$-0.901768\pi$
$331$	− 11.0518i	− 0.607460i	0.952758	−	0.303730i	$-0.0982320\pi$
$332$	0	0
$333$	7.43965	0.407690
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.9931i	1.08909i	0.838730	+	0.544547i	$0.183299\pi$
$337$	19.9931i	1.08909i	−0.838730	+	0.544547i	$0.816701\pi$
$338$	0	0
$339$	− 15.3224i	− 0.832198i
$340$	0	0
$341$	− 16.0000i	− 0.866449i
$342$	0	0
$343$	12.3449i	0.666563i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.87930	−0.369300	−0.184650	−	0.982804i	$-0.559115\pi$
$347$	−6.87930	−0.369300	−0.184650	+	0.982804i	$0.559115\pi$
$348$	0	0
$349$	− 4.76203i	− 0.254906i	−0.991845	−	0.127453i	$-0.959320\pi$
$349$	− 4.76203i	− 0.254906i	0.991845	−	0.127453i	$-0.0406801\pi$
$350$	0	0
$351$	5.55691	0.296606
$352$	0	0
$353$	− 3.79145i	− 0.201798i	−0.994897	−	0.100899i	$-0.967828\pi$
$353$	− 3.79145i	− 0.201798i	0.994897	−	0.100899i	$-0.0321720\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−7.11383	−0.376504
$358$	0	0
$359$	−12.9966	−0.685932	−0.342966	−	0.939348i	$-0.611432\pi$
$359$	−12.9966	−0.685932	−0.342966	+	0.939348i	$0.611432\pi$
$360$	0	0
$361$	17.8793	0.941016
$362$	0	0
$363$	9.23453	0.484687
$364$	0	0
$365$	0	0
$366$	0	0
$367$	22.9345i	1.19717i	0.801059	+	0.598585i	$0.204270\pi$
$367$	22.9345i	1.19717i	−0.801059	+	0.598585i	$0.795730\pi$
$368$	0	0
$369$	−3.88273	−0.202127
$370$	0	0
$371$	1.88273i	0.0977467i
$372$	0	0
$373$	15.4396	0.799435	0.399717	−	0.916638i	$-0.369108\pi$
$373$	15.4396	0.799435	0.399717	+	0.916638i	$0.369108\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11.1138i	0.572391i
$378$	0	0
$379$	6.28973i	0.323082i	0.986866	+	0.161541i	$0.0516463\pi$
$379$	6.28973i	0.323082i	−0.986866	+	0.161541i	$0.948354\pi$
$380$	0	0
$381$	18.1725i	0.931003i
$382$	0	0
$383$	− 2.94137i	− 0.150297i	−0.997172	−	0.0751484i	$-0.976057\pi$
$383$	− 2.94137i	− 0.150297i	0.997172	−	0.0751484i	$-0.0239431\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.88273	0.0957047
$388$	0	0
$389$	12.2277i	0.619967i	0.950742	+	0.309983i	$0.100324\pi$
$389$	12.2277i	0.619967i	−0.950742	+	0.309983i	$0.899676\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	6.38101i	0.321880i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	5.32238	0.267123	0.133561	−	0.991041i	$-0.457359\pi$
$397$	5.32238	0.267123	0.133561	+	0.991041i	$0.457359\pi$
$398$	0	0
$399$	−0.996562	−0.0498905
$400$	0	0
$401$	6.99656	0.349392	0.174696	−	0.984622i	$-0.444106\pi$
$401$	6.99656	0.349392	0.174696	+	0.984622i	$0.444106\pi$
$402$	0	0
$403$	19.7655	0.984588
$404$	0	0
$405$	0	0
$406$	0	0
$407$	33.4656i	1.65883i
$408$	0	0
$409$	−16.2277	−0.802406	−0.401203	−	0.915989i	$-0.631408\pi$
$409$	−16.2277	−0.802406	−0.401203	+	0.915989i	$0.631408\pi$
$410$	0	0
$411$	− 4.44309i	− 0.219161i
$412$	0	0
$413$	−8.00000	−0.393654
$414$	0	0
$415$	0	0
$416$	0	0
$417$	20.1725i	0.987850i
$418$	0	0
$419$	15.6121i	0.762701i	0.924430	+	0.381351i	$0.124541\pi$
$419$	15.6121i	0.762701i	−0.924430	+	0.381351i	$0.875459\pi$
$420$	0	0
$421$	− 33.2311i	− 1.61958i	−0.586717	−	0.809792i	$-0.699580\pi$
$421$	− 33.2311i	− 1.61958i	0.586717	−	0.809792i	$-0.300420\pi$
$422$	0	0
$423$	− 10.0552i	− 0.488900i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.46907	0.409847
$428$	0	0
$429$	24.9966i	1.20685i
$430$	0	0
$431$	12.9966	0.626022	0.313011	−	0.949749i	$-0.398662\pi$
$431$	12.9966	0.626022	0.313011	+	0.949749i	$0.398662\pi$
$432$	0	0
$433$	20.2277i	0.972079i	0.873937	+	0.486040i	$0.161559\pi$
$433$	20.2277i	0.972079i	−0.873937	+	0.486040i	$0.838441\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.12070	0.0536106
$438$	0	0
$439$	−5.43965	−0.259620	−0.129810	−	0.991539i	$-0.541437\pi$
$439$	−5.43965	−0.259620	−0.129810	+	0.991539i	$0.541437\pi$
$440$	0	0
$441$	6.11383	0.291135
$442$	0	0
$443$	15.3484	0.729223	0.364611	−	0.931160i	$-0.381202\pi$
$443$	15.3484	0.729223	0.364611	+	0.931160i	$0.381202\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.00000i	0.0945968i
$448$	0	0
$449$	−4.22766	−0.199515	−0.0997577	−	0.995012i	$-0.531807\pi$
$449$	−4.22766	−0.199515	−0.0997577	+	0.995012i	$0.531807\pi$
$450$	0	0
$451$	− 17.4656i	− 0.822424i
$452$	0	0
$453$	9.67418	0.454533
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2.65164i	− 0.124038i	−0.998075	−	0.0620192i	$-0.980246\pi$
$457$	− 2.65164i	− 0.124038i	0.998075	−	0.0620192i	$-0.0197540\pi$
$458$	0	0
$459$	7.55691i	0.352727i
$460$	0	0
$461$	10.2345i	0.476670i	0.971183	+	0.238335i	$0.0766016\pi$
$461$	10.2345i	0.476670i	−0.971183	+	0.238335i	$0.923398\pi$
$462$	0	0
$463$	− 19.0586i	− 0.885730i	−0.896588	−	0.442865i	$-0.853962\pi$
$463$	− 19.0586i	− 0.885730i	0.896588	−	0.442865i	$-0.146038\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.11039	−0.190206	−0.0951031	−	0.995467i	$-0.530318\pi$
$467$	−4.11039	−0.190206	−0.0951031	+	0.995467i	$0.530318\pi$
$468$	0	0
$469$	− 3.76547i	− 0.173873i
$470$	0	0
$471$	4.32582	0.199323
$472$	0	0
$473$	8.46907i	0.389408i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	−25.2311	−1.15284	−0.576419	−	0.817154i	$-0.695550\pi$
$479$	−25.2311	−1.15284	−0.576419	+	0.817154i	$0.695550\pi$
$480$	0	0
$481$	−41.3415	−1.88501
$482$	0	0
$483$	0.996562	0.0453451
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 21.9379i	− 0.994102i	−0.867721	−	0.497051i	$-0.834416\pi$
$487$	− 21.9379i	− 0.994102i	0.867721	−	0.497051i	$-0.165584\pi$
$488$	0	0
$489$	−6.11727	−0.276632
$490$	0	0
$491$	− 7.50172i	− 0.338548i	−0.985569	−	0.169274i	$-0.945858\pi$
$491$	− 7.50172i	− 0.338548i	0.985569	−	0.169274i	$-0.0541423\pi$
$492$	0	0
$493$	−15.1138	−0.680693
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.2345i	0.548794i
$498$	0	0
$499$	29.1690i	1.30578i	0.757451	+	0.652892i	$0.226445\pi$
$499$	29.1690i	1.30578i	−0.757451	+	0.652892i	$0.773555\pi$
$500$	0	0
$501$	6.05520i	0.270526i
$502$	0	0
$503$	23.9379i	1.06734i	0.845693	+	0.533670i	$0.179187\pi$
$503$	23.9379i	1.06734i	−0.845693	+	0.533670i	$0.820813\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−17.8793	−0.794047
$508$	0	0
$509$	28.6967i	1.27196i	0.771706	+	0.635980i	$0.219404\pi$
$509$	28.6967i	1.27196i	−0.771706	+	0.635980i	$0.780596\pi$
$510$	0	0
$511$	5.64820	0.249862
$512$	0	0
$513$	1.05863i	0.0467398i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	45.2311	1.98926
$518$	0	0
$519$	−16.8793	−0.740919
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	−25.7586	−1.12634	−0.563172	−	0.826340i	$-0.690419\pi$
$523$	−25.7586	−1.12634	−0.563172	+	0.826340i	$0.690419\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	26.8793i	1.17088i
$528$	0	0
$529$	21.8793	0.951274
$530$	0	0
$531$	8.49828i	0.368794i
$532$	0	0
$533$	21.5760	0.934561
$534$	0	0
$535$	0	0
$536$	0	0
$537$	10.6155i	0.458095i
$538$	0	0
$539$	27.5017i	1.18458i
$540$	0	0
$541$	− 12.3449i	− 0.530750i	−0.964145	−	0.265375i	$-0.914504\pi$
$541$	− 12.3449i	− 0.530750i	0.964145	−	0.265375i	$-0.0854957\pi$
$542$	0	0
$543$	14.1173i	0.605830i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−19.8759	−0.849830	−0.424915	−	0.905233i	$-0.639696\pi$
$547$	−19.8759	−0.849830	−0.424915	+	0.905233i	$0.639696\pi$
$548$	0	0
$549$	− 8.99656i	− 0.383964i
$550$	0	0
$551$	−2.11727	−0.0901986
$552$	0	0
$553$	10.8793i	0.462635i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.12070	−0.132228	−0.0661142	−	0.997812i	$-0.521060\pi$
$557$	−3.12070	−0.132228	−0.0661142	+	0.997812i	$0.521060\pi$
$558$	0	0
$559$	−10.4622	−0.442503
$560$	0	0
$561$	−33.9931	−1.43519
$562$	0	0
$563$	0.651639	0.0274633	0.0137317	−	0.999906i	$-0.495629\pi$
$563$	0.651639	0.0274633	0.0137317	+	0.999906i	$0.495629\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.941367i	0.0395337i
$568$	0	0
$569$	−26.9966	−1.13175	−0.565877	−	0.824489i	$-0.691462\pi$
$569$	−26.9966	−1.13175	−0.565877	+	0.824489i	$0.691462\pi$
$570$	0	0
$571$	14.9414i	0.625277i	0.949872	+	0.312638i	$0.101213\pi$
$571$	14.9414i	0.625277i	−0.949872	+	0.312638i	$0.898787\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8.87930i	0.369650i	0.982771	+	0.184825i	$0.0591718\pi$
$577$	8.87930i	0.369650i	−0.982771	+	0.184825i	$0.940828\pi$
$578$	0	0
$579$	4.87930i	0.202777i
$580$	0	0
$581$	− 5.53781i	− 0.229747i
$582$	0	0
$583$	8.99656i	0.372600i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.23109	0.0508127	0.0254064	−	0.999677i	$-0.491912\pi$
$587$	1.23109	0.0508127	0.0254064	+	0.999677i	$0.491912\pi$
$588$	0	0
$589$	3.76547i	0.155153i
$590$	0	0
$591$	−2.88617	−0.118721
$592$	0	0
$593$	3.55691i	0.146065i	0.997330	+	0.0730325i	$0.0232677\pi$
$593$	3.55691i	0.146065i	−0.997330	+	0.0730325i	$0.976732\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	17.6742	0.723356
$598$	0	0
$599$	−19.2242	−0.785480	−0.392740	−	0.919649i	$-0.628473\pi$
$599$	−19.2242	−0.785480	−0.392740	+	0.919649i	$0.628473\pi$
$600$	0	0
$601$	−27.7586	−1.13230	−0.566148	−	0.824303i	$-0.691567\pi$
$601$	−27.7586	−1.13230	−0.566148	+	0.824303i	$0.691567\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7.16902i	0.290982i	0.989360	+	0.145491i	$0.0464761\pi$
$607$	7.16902i	0.290982i	−0.989360	+	0.145491i	$0.953524\pi$
$608$	0	0
$609$	−1.88273	−0.0762922
$610$	0	0
$611$	55.8759i	2.26050i
$612$	0	0
$613$	9.55691	0.386000	0.193000	−	0.981199i	$-0.438178\pi$
$613$	9.55691	0.386000	0.193000	+	0.981199i	$0.438178\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.32926i	− 0.0535139i	−0.999642	−	0.0267569i	$-0.991482\pi$
$617$	− 1.32926i	− 0.0535139i	0.999642	−	0.0267569i	$-0.00851802\pi$
$618$	0	0
$619$	− 28.1725i	− 1.13235i	−0.824286	−	0.566173i	$-0.808423\pi$
$619$	− 28.1725i	− 1.13235i	0.824286	−	0.566173i	$-0.191577\pi$
$620$	0	0
$621$	− 1.05863i	− 0.0424815i
$622$	0	0
$623$	3.87586i	0.155283i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.76203	−0.190177
$628$	0	0
$629$	− 56.2208i	− 2.24167i
$630$	0	0
$631$	23.3224	0.928449	0.464225	−	0.885717i	$-0.346333\pi$
$631$	23.3224	0.928449	0.464225	+	0.885717i	$0.346333\pi$
$632$	0	0
$633$	23.9379i	0.951447i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−33.9740	−1.34610
$638$	0	0
$639$	12.9966	0.514136
$640$	0	0
$641$	27.1070	1.07066	0.535330	−	0.844643i	$-0.320187\pi$
$641$	27.1070	1.07066	0.535330	+	0.844643i	$0.320187\pi$
$642$	0	0
$643$	−20.3449	−0.802325	−0.401163	−	0.916007i	$-0.631394\pi$
$643$	−20.3449	−0.802325	−0.401163	+	0.916007i	$0.631394\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	37.6965i	1.48200i	0.671503	+	0.741002i	$0.265649\pi$
$647$	37.6965i	1.48200i	−0.671503	+	0.741002i	$0.734351\pi$
$648$	0	0
$649$	−38.2277	−1.50057
$650$	0	0
$651$	3.34836i	0.131233i
$652$	0	0
$653$	−8.64476	−0.338296	−0.169148	−	0.985591i	$-0.554102\pi$
$653$	−8.64476	−0.338296	−0.169148	+	0.985591i	$0.554102\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 6.00000i	− 0.234082i
$658$	0	0
$659$	− 29.2603i	− 1.13982i	−0.821707	−	0.569910i	$-0.806978\pi$
$659$	− 29.2603i	− 1.13982i	0.821707	−	0.569910i	$-0.193022\pi$
$660$	0	0
$661$	− 28.7620i	− 1.11871i	−0.828927	−	0.559357i	$-0.811048\pi$
$661$	− 28.7620i	− 1.11871i	0.828927	−	0.559357i	$-0.188952\pi$
$662$	0	0
$663$	− 41.9931i	− 1.63088i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.11727	0.0819809
$668$	0	0
$669$	− 24.0552i	− 0.930028i
$670$	0	0
$671$	40.4691	1.56229
$672$	0	0
$673$	− 18.0000i	− 0.693849i	−0.937893	−	0.346925i	$-0.887226\pi$
$673$	− 18.0000i	− 0.693849i	0.937893	−	0.346925i	$-0.112774\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.8724	1.64772	0.823860	−	0.566793i	$-0.191816\pi$
$677$	42.8724	1.64772	0.823860	+	0.566793i	$0.191816\pi$
$678$	0	0
$679$	16.1104	0.618260
$680$	0	0
$681$	−11.1138	−0.425883
$682$	0	0
$683$	26.1173	0.999349	0.499675	−	0.866213i	$-0.333453\pi$
$683$	26.1173	0.999349	0.499675	+	0.866213i	$0.333453\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	17.2311i	0.657407i
$688$	0	0
$689$	−11.1138	−0.423403
$690$	0	0
$691$	− 5.29317i	− 0.201362i	−0.994919	−	0.100681i	$-0.967898\pi$
$691$	− 5.29317i	− 0.201362i	0.994919	−	0.100681i	$-0.0321021\pi$
$692$	0	0
$693$	−4.23453	−0.160857
$694$	0	0
$695$	0	0
$696$	0	0
$697$	29.3415i	1.11139i
$698$	0	0
$699$	8.44309i	0.319347i
$700$	0	0
$701$	− 7.99312i	− 0.301896i	−0.988542	−	0.150948i	$-0.951767\pi$
$701$	− 7.99312i	− 0.301896i	0.988542	−	0.150948i	$-0.0482326\pi$
$702$	0	0
$703$	− 7.87586i	− 0.297044i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.88273	0.0708075
$708$	0	0
$709$	− 28.9966i	− 1.08899i	−0.838764	−	0.544494i	$-0.816722\pi$
$709$	− 28.9966i	− 1.08899i	0.838764	−	0.544494i	$-0.183278\pi$
$710$	0	0
$711$	11.5569	0.433418
$712$	0	0
$713$	− 3.76547i	− 0.141018i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−10.1173	−0.377836
$718$	0	0
$719$	26.8793	1.00243	0.501214	−	0.865323i	$-0.332887\pi$
$719$	26.8793	1.00243	0.501214	+	0.865323i	$0.332887\pi$
$720$	0	0
$721$	9.57602	0.356630
$722$	0	0
$723$	−16.8793	−0.627748
$724$	0	0
$725$	0	0
$726$	0	0
$727$	41.8138i	1.55079i	0.631478	+	0.775394i	$0.282449\pi$
$727$	41.8138i	1.55079i	−0.631478	+	0.775394i	$0.717551\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	− 14.2277i	− 0.526229i
$732$	0	0
$733$	30.0844	1.11119	0.555597	−	0.831452i	$-0.312490\pi$
$733$	30.0844	1.11119	0.555597	+	0.831452i	$0.312490\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 17.9931i	− 0.662785i
$738$	0	0
$739$	− 29.0449i	− 1.06843i	−0.845348	−	0.534217i	$-0.820607\pi$
$739$	− 29.0449i	− 1.06843i	0.845348	−	0.534217i	$-0.179393\pi$
$740$	0	0
$741$	− 5.88273i	− 0.216108i
$742$	0	0
$743$	43.2863i	1.58802i	0.607905	+	0.794010i	$0.292010\pi$
$743$	43.2863i	1.58802i	−0.607905	+	0.794010i	$0.707990\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−5.88273	−0.215238
$748$	0	0
$749$	− 16.2208i	− 0.592694i
$750$	0	0
$751$	−41.7846	−1.52474	−0.762370	−	0.647141i	$-0.775964\pi$
$751$	−41.7846	−1.52474	−0.762370	+	0.647141i	$0.775964\pi$
$752$	0	0
$753$	− 11.8466i	− 0.431716i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	16.3258	0.593372	0.296686	−	0.954975i	$-0.404119\pi$
$757$	16.3258	0.593372	0.296686	+	0.954975i	$0.404119\pi$
$758$	0	0
$759$	4.76203	0.172851
$760$	0	0
$761$	50.2208	1.82050	0.910251	−	0.414057i	$-0.135889\pi$
$761$	50.2208	1.82050	0.910251	+	0.414057i	$0.135889\pi$
$762$	0	0
$763$	−1.77234	−0.0641631
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 47.2242i	− 1.70517i
$768$	0	0
$769$	31.3415	1.13020	0.565101	−	0.825021i	$-0.308837\pi$
$769$	31.3415	1.13020	0.565101	+	0.825021i	$0.308837\pi$
$770$	0	0
$771$	− 10.6707i	− 0.384297i
$772$	0	0
$773$	−9.11383	−0.327802	−0.163901	−	0.986477i	$-0.552408\pi$
$773$	−9.11383	−0.327802	−0.163901	+	0.986477i	$0.552408\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 7.00344i	− 0.251247i
$778$	0	0
$779$	4.11039i	0.147270i
$780$	0	0
$781$	58.4622i	2.09194i
$782$	0	0
$783$	2.00000i	0.0714742i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	36.2208	1.29113	0.645566	−	0.763705i	$-0.276622\pi$
$787$	36.2208	1.29113	0.645566	+	0.763705i	$0.276622\pi$
$788$	0	0
$789$	− 1.94480i	− 0.0692369i
$790$	0	0
$791$	−14.4240	−0.512858
$792$	0	0
$793$	49.9931i	1.77531i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10.0000	−0.354218	−0.177109	−	0.984191i	$-0.556675\pi$
$797$	−10.0000	−0.354218	−0.177109	+	0.984191i	$0.556675\pi$
$798$	0	0
$799$	−75.9862	−2.68820
$800$	0	0
$801$	4.11727	0.145476
$802$	0	0
$803$	26.9897	0.952445
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 9.76547i	− 0.343761i
$808$	0	0
$809$	−47.5760	−1.67268	−0.836342	−	0.548208i	$-0.815310\pi$
$809$	−47.5760	−1.67268	−0.836342	+	0.548208i	$0.815310\pi$
$810$	0	0
$811$	20.5174i	0.720463i	0.932863	+	0.360231i	$0.117302\pi$
$811$	20.5174i	0.720463i	−0.932863	+	0.360231i	$0.882698\pi$
$812$	0	0
$813$	3.44652	0.120875
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.99312i	− 0.0697306i
$818$	0	0
$819$	− 5.23109i	− 0.182789i
$820$	0	0
$821$	− 44.4622i	− 1.55174i	−0.630892	−	0.775871i	$-0.717311\pi$
$821$	− 44.4622i	− 1.55174i	0.630892	−	0.775871i	$-0.282689\pi$
$822$	0	0
$823$	− 32.1656i	− 1.12122i	−0.828079	−	0.560611i	$-0.810566\pi$
$823$	− 32.1656i	− 1.12122i	0.828079	−	0.560611i	$-0.189434\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20.0000	−0.695468	−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000	−0.695468	−0.347734	+	0.937593i	$0.613049\pi$
$828$	0	0
$829$	33.8827i	1.17680i	0.808571	+	0.588398i	$0.200241\pi$
$829$	33.8827i	1.17680i	−0.808571	+	0.588398i	$0.799759\pi$
$830$	0	0
$831$	−18.7880	−0.651749
$832$	0	0
$833$	− 46.2017i	− 1.60079i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.55691	0.122945
$838$	0	0
$839$	4.52750	0.156307	0.0781533	−	0.996941i	$-0.475098\pi$
$839$	4.52750	0.156307	0.0781533	+	0.996941i	$0.475098\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	16.8793	0.581354
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 8.69308i	− 0.298698i
$848$	0	0
$849$	20.0000	0.686398
$850$	0	0
$851$	7.87586i	0.269981i
$852$	0	0
$853$	−50.4293	−1.72667	−0.863334	−	0.504633i	$-0.831628\pi$
$853$	−50.4293	−1.72667	−0.863334	+	0.504633i	$0.831628\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 26.4362i	− 0.903044i	−0.892260	−	0.451522i	$-0.850881\pi$
$857$	− 26.4362i	− 0.903044i	0.892260	−	0.451522i	$-0.149119\pi$
$858$	0	0
$859$	− 0.406994i	− 0.0138865i	−0.999976	−	0.00694323i	$-0.997790\pi$
$859$	− 0.406994i	− 0.0138865i	0.999976	−	0.00694323i	$-0.00221012\pi$
$860$	0	0
$861$	3.65508i	0.124565i
$862$	0	0
$863$	29.9311i	1.01886i	0.860511	+	0.509432i	$0.170145\pi$
$863$	29.9311i	1.01886i	−0.860511	+	0.509432i	$0.829855\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	40.1070	1.36210
$868$	0	0
$869$	51.9862i	1.76351i
$870$	0	0
$871$	22.2277	0.753155
$872$	0	0
$873$	− 17.1138i	− 0.579215i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	11.2051	0.378370	0.189185	−	0.981941i	$-0.439415\pi$
$877$	11.2051	0.378370	0.189185	+	0.981941i	$0.439415\pi$
$878$	0	0
$879$	20.2277	0.682262
$880$	0	0
$881$	−48.3380	−1.62855	−0.814275	−	0.580479i	$-0.802865\pi$
$881$	−48.3380	−1.62855	−0.814275	+	0.580479i	$0.802865\pi$
$882$	0	0
$883$	50.5726	1.70190	0.850951	−	0.525244i	$-0.176026\pi$
$883$	50.5726	1.70190	0.850951	+	0.525244i	$0.176026\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 48.0483i	− 1.61330i	−0.591026	−	0.806652i	$-0.701277\pi$
$887$	− 48.0483i	− 1.61330i	0.591026	−	0.806652i	$-0.298723\pi$
$888$	0	0
$889$	17.1070	0.573749
$890$	0	0
$891$	4.49828i	0.150698i
$892$	0	0
$893$	−10.6448	−0.356213
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.88273i	0.196419i
$898$	0	0
$899$	7.11383i	0.237259i
$900$	0	0
$901$	− 15.1138i	− 0.503515i
$902$	0	0
$903$	− 1.77234i	− 0.0589799i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−6.46219	−0.214573	−0.107287	−	0.994228i	$-0.534216\pi$
$907$	−6.46219	−0.214573	−0.107287	+	0.994228i	$0.534216\pi$
$908$	0	0
$909$	− 2.00000i	− 0.0663358i
$910$	0	0
$911$	−50.3380	−1.66777	−0.833887	−	0.551935i	$-0.813890\pi$
$911$	−50.3380	−1.66777	−0.833887	+	0.551935i	$0.813890\pi$
$912$	0	0
$913$	− 26.4622i	− 0.875771i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.00688	0.198365
$918$	0	0
$919$	46.4362	1.53179	0.765895	−	0.642966i	$-0.222296\pi$
$919$	46.4362	1.53179	0.765895	+	0.642966i	$0.222296\pi$
$920$	0	0
$921$	−8.11039	−0.267246
$922$	0	0
$923$	−72.2208	−2.37718
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 10.1725i	− 0.334107i
$928$	0	0
$929$	35.9931	1.18090	0.590448	−	0.807076i	$-0.298951\pi$
$929$	35.9931	1.18090	0.590448	+	0.807076i	$0.298951\pi$
$930$	0	0
$931$	− 6.47230i	− 0.212121i
$932$	0	0
$933$	31.8759	1.04357
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 2.70360i	− 0.0883227i	−0.999024	−	0.0441613i	$-0.985938\pi$
$937$	− 2.70360i	− 0.0883227i	0.999024	−	0.0441613i	$-0.0140616\pi$
$938$	0	0
$939$	5.11383i	0.166883i
$940$	0	0
$941$	17.7655i	0.579138i	0.957157	+	0.289569i	$0.0935119\pi$
$941$	17.7655i	0.579138i	−0.957157	+	0.289569i	$0.906488\pi$
$942$	0	0
$943$	− 4.11039i	− 0.133853i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26.2277	−0.852284	−0.426142	−	0.904656i	$-0.640128\pi$
$947$	−26.2277	−0.852284	−0.426142	+	0.904656i	$0.640128\pi$
$948$	0	0
$949$	33.3415i	1.08231i
$950$	0	0
$951$	24.6448	0.799161
$952$	0	0
$953$	9.09472i	0.294607i	0.989091	+	0.147304i	$0.0470594\pi$
$953$	9.09472i	0.294607i	−0.989091	+	0.147304i	$0.952941\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−8.99656	−0.290818
$958$	0	0
$959$	−4.18257	−0.135062
$960$	0	0
$961$	−18.3484	−0.591883
$962$	0	0
$963$	−17.2311	−0.555264
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 7.47574i	− 0.240404i	−0.992749	−	0.120202i	$-0.961646\pi$
$967$	− 7.47574i	− 0.240404i	0.992749	−	0.120202i	$-0.0383542\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	− 41.0777i	− 1.31825i	−0.752035	−	0.659124i	$-0.770927\pi$
$971$	− 41.0777i	− 1.31825i	0.752035	−	0.659124i	$-0.229073\pi$
$972$	0	0
$973$	18.9897	0.608781
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.20855i	0.134644i	0.997731	+	0.0673218i	$0.0214454\pi$
$977$	4.20855i	0.134644i	−0.997731	+	0.0673218i	$0.978555\pi$
$978$	0	0
$979$	18.5206i	0.591922i
$980$	0	0
$981$	1.88273i	0.0601111i
$982$	0	0
$983$	8.35504i	0.266484i	0.991084	+	0.133242i	$0.0425388\pi$
$983$	8.35504i	0.266484i	−0.991084	+	0.133242i	$0.957461\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−9.46563	−0.301294
$988$	0	0
$989$	1.99312i	0.0633777i
$990$	0	0
$991$	13.9087	0.441825	0.220912	−	0.975294i	$-0.429097\pi$
$991$	13.9087	0.441825	0.220912	+	0.975294i	$0.429097\pi$
$992$	0	0
$993$	11.0518i	0.350717i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−34.8984	−1.10524	−0.552622	−	0.833432i	$-0.686373\pi$
$997$	−34.8984	−1.10524	−0.552622	+	0.833432i	$0.686373\pi$
$998$	0	0
$999$	−7.43965	−0.235380

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.d.e.49.4		6
3.2	odd	2		7200.2.d.r.2449.4		6
4.3	odd	2		600.2.d.f.349.1		6
5.2	odd	4		480.2.k.b.241.5		6
5.3	odd	4		2400.2.k.c.1201.2		6
5.4	even	2		2400.2.d.f.49.3		6
8.3	odd	2		600.2.d.e.349.5		6
8.5	even	2		2400.2.d.f.49.4		6
12.11	even	2		1800.2.d.r.1549.6		6
15.2	even	4		1440.2.k.f.721.5		6
15.8	even	4		7200.2.k.p.3601.3		6
15.14	odd	2		7200.2.d.q.2449.3		6
20.3	even	4		600.2.k.c.301.4		6
20.7	even	4		120.2.k.b.61.3	✓	6
20.19	odd	2		600.2.d.e.349.6		6
24.5	odd	2		7200.2.d.q.2449.4		6
24.11	even	2		1800.2.d.q.1549.2		6
40.3	even	4		600.2.k.c.301.3		6
40.13	odd	4		2400.2.k.c.1201.5		6
40.19	odd	2		600.2.d.f.349.2		6
40.27	even	4		120.2.k.b.61.4	yes	6
40.29	even	2	inner	2400.2.d.e.49.3		6
40.37	odd	4		480.2.k.b.241.2		6
60.23	odd	4		1800.2.k.p.901.3		6
60.47	odd	4		360.2.k.f.181.4		6
60.59	even	2		1800.2.d.q.1549.1		6
80.27	even	4		3840.2.a.bp.1.2		3
80.37	odd	4		3840.2.a.br.1.2		3
80.67	even	4		3840.2.a.bq.1.2		3
80.77	odd	4		3840.2.a.bo.1.2		3
120.29	odd	2		7200.2.d.r.2449.3		6
120.53	even	4		7200.2.k.p.3601.4		6
120.59	even	2		1800.2.d.r.1549.5		6
120.77	even	4		1440.2.k.f.721.2		6
120.83	odd	4		1800.2.k.p.901.4		6
120.107	odd	4		360.2.k.f.181.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.k.b.61.3	✓	6	20.7	even	4
120.2.k.b.61.4	yes	6	40.27	even	4
360.2.k.f.181.3		6	120.107	odd	4
360.2.k.f.181.4		6	60.47	odd	4
480.2.k.b.241.2		6	40.37	odd	4
480.2.k.b.241.5		6	5.2	odd	4
600.2.d.e.349.5		6	8.3	odd	2
600.2.d.e.349.6		6	20.19	odd	2
600.2.d.f.349.1		6	4.3	odd	2
600.2.d.f.349.2		6	40.19	odd	2
600.2.k.c.301.3		6	40.3	even	4
600.2.k.c.301.4		6	20.3	even	4
1440.2.k.f.721.2		6	120.77	even	4
1440.2.k.f.721.5		6	15.2	even	4
1800.2.d.q.1549.1		6	60.59	even	2
1800.2.d.q.1549.2		6	24.11	even	2
1800.2.d.r.1549.5		6	120.59	even	2
1800.2.d.r.1549.6		6	12.11	even	2
1800.2.k.p.901.3		6	60.23	odd	4
1800.2.k.p.901.4		6	120.83	odd	4
2400.2.d.e.49.3		6	40.29	even	2	inner
2400.2.d.e.49.4		6	1.1	even	1	trivial
2400.2.d.f.49.3		6	5.4	even	2
2400.2.d.f.49.4		6	8.5	even	2
2400.2.k.c.1201.2		6	5.3	odd	4
2400.2.k.c.1201.5		6	40.13	odd	4
3840.2.a.bo.1.2		3	80.77	odd	4
3840.2.a.bp.1.2		3	80.27	even	4
3840.2.a.bq.1.2		3	80.67	even	4
3840.2.a.br.1.2		3	80.37	odd	4
7200.2.d.q.2449.3		6	15.14	odd	2
7200.2.d.q.2449.4		6	24.5	odd	2
7200.2.d.r.2449.3		6	120.29	odd	2
7200.2.d.r.2449.4		6	3.2	odd	2
7200.2.k.p.3601.3		6	15.8	even	4
7200.2.k.p.3601.4		6	120.53	even	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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.941367i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.941367i q^{7} +1.00000 q^{9} +4.49828i q^{11} -5.55691 q^{13} -7.55691i q^{17} -1.05863i q^{19} -0.941367i q^{21} +1.05863i q^{23} -1.00000 q^{27} -2.00000i q^{29} -3.55691 q^{31} -4.49828i q^{33} +7.43965 q^{37} +5.55691 q^{39} -3.88273 q^{41} +1.88273 q^{43} -10.0552i q^{47} +6.11383 q^{49} +7.55691i q^{51} +2.00000 q^{53} +1.05863i q^{57} +8.49828i q^{59} -8.99656i q^{61} +0.941367i q^{63} -4.00000 q^{67} -1.05863i q^{69} +12.9966 q^{71} -6.00000i q^{73} -4.23453 q^{77} +11.5569 q^{79} +1.00000 q^{81} -5.88273 q^{83} +2.00000i q^{87} +4.11727 q^{89} -5.23109i q^{91} +3.55691 q^{93} -17.1138i q^{97} +4.49828i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^9	\( 6 q - 6 q^{3} + 6 q^{9} - 6 q^{27} + 12 q^{31} + 8 q^{37} - 20 q^{41} + 8 q^{43} - 30 q^{49} + 12 q^{53} - 24 q^{67} + 8 q^{71} - 32 q^{77} + 36 q^{79} + 6 q^{81} - 32 q^{83} + 28 q^{89} - 12 q^{93}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^9 - 6 * q^27 + 12 * q^31 + 8 * q^37 - 20 * q^41 + 8 * q^43 - 30 * q^49 + 12 * q^53 - 24 * q^67 + 8 * q^71 - 32 * q^77 + 36 * q^79 + 6 * q^81 - 32 * q^83 + 28 * q^89 - 12 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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