LMFDB - Embedded newform 2400.2.d.h.49.1

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:	$8$
Coefficient field:	8.0.214798336.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 $ x^8 - 2x^7 - 2x^5 + 9x^4 - 4x^3 - 16*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-0.565036 - 1.29643i$ of defining polynomial
Character	$\chi$	$=$	2400.49
Dual form			2400.2.d.h.49.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -4.72294i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.72294i q^{7} +1.00000 q^{9} -3.93012i q^{11} +3.46733 q^{13} +3.51575i q^{17} +5.44133i q^{19} -4.72294i q^{21} -7.11585i q^{23} +1.00000 q^{27} -3.66998i q^{29} -5.23414 q^{31} -3.93012i q^{33} -0.414376 q^{37} +3.46733 q^{39} +3.00454 q^{41} +5.34450 q^{43} +0.925579i q^{47} -15.3061 q^{49} +3.51575i q^{51} +0.233196 q^{53} +5.44133i q^{57} -14.3805i q^{59} +0.118290i q^{61} -4.72294i q^{63} -13.4504 q^{67} -7.11585i q^{69} -2.19027 q^{71} +0.563219i q^{73} -18.5617 q^{77} -10.2746 q^{79} +1.00000 q^{81} +11.3490 q^{83} -3.66998i q^{87} -8.88265 q^{89} -16.3760i q^{91} -5.23414 q^{93} +7.27462i q^{97} -3.93012i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 + 8 * q^9	$ 8 q + 8 q^{3} + 8 q^{9} + 8 q^{27} - 8 q^{31} + 8 q^{43} + 8 q^{53} - 24 q^{67} + 40 q^{71} - 24 q^{77} - 16 q^{79} + 8 q^{81} + 32 q^{83} - 8 q^{93}+O(q^{100}) $ 8 * q + 8 * q^3 + 8 * q^9 + 8 * q^27 - 8 * q^31 + 8 * q^43 + 8 * q^53 - 24 * q^67 + 40 * q^71 - 24 * q^77 - 16 * q^79 + 8 * q^81 + 32 * q^83 - 8 * 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
$7$	− 4.72294i	− 1.78510i	−0.450947	−	0.892551i	$-0.648914\pi$
$7$	− 4.72294i	− 1.78510i	0.450947	−	0.892551i	$-0.351086\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 3.93012i	− 1.18498i	−0.805579	−	0.592488i	$-0.798146\pi$
$11$	− 3.93012i	− 1.18498i	0.805579	−	0.592488i	$-0.201854\pi$
$12$	0	0
$13$	3.46733	0.961665	0.480833	−	0.876812i	$-0.340334\pi$
$13$	3.46733	0.961665	0.480833	+	0.876812i	$0.340334\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.51575i	0.852694i	0.904560	+	0.426347i	$0.140200\pi$
$17$	3.51575i	0.852694i	−0.904560	+	0.426347i	$0.859800\pi$
$18$	0	0
$19$	5.44133i	1.24833i	0.781294	+	0.624163i	$0.214560\pi$
$19$	5.44133i	1.24833i	−0.781294	+	0.624163i	$0.785440\pi$
$20$	0	0
$21$	− 4.72294i	− 1.03063i
$22$	0	0
$23$	− 7.11585i	− 1.48376i	−0.670534	−	0.741878i	$-0.733935\pi$
$23$	− 7.11585i	− 1.48376i	0.670534	−	0.741878i	$-0.266065\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	− 3.66998i	− 0.681498i	−0.940154	−	0.340749i	$-0.889319\pi$
$29$	− 3.66998i	− 0.681498i	0.940154	−	0.340749i	$-0.110681\pi$
$30$	0	0
$31$	−5.23414	−0.940079	−0.470039	−	0.882645i	$-0.655760\pi$
$31$	−5.23414	−0.940079	−0.470039	+	0.882645i	$0.655760\pi$
$32$	0	0
$33$	− 3.93012i	− 0.684147i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.414376	−0.0681231	−0.0340615	−	0.999420i	$-0.510844\pi$
$37$	−0.414376	−0.0681231	−0.0340615	+	0.999420i	$0.510844\pi$
$38$	0	0
$39$	3.46733	0.555218
$40$	0	0
$41$	3.00454	0.469231	0.234616	−	0.972088i	$-0.424617\pi$
$41$	3.00454	0.469231	0.234616	+	0.972088i	$0.424617\pi$
$42$	0	0
$43$	5.34450	0.815029	0.407514	−	0.913199i	$-0.366396\pi$
$43$	5.34450	0.815029	0.407514	+	0.913199i	$0.366396\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.925579i	0.135010i	0.997719	+	0.0675048i	$0.0215038\pi$
$47$	0.925579i	0.135010i	−0.997719	+	0.0675048i	$0.978496\pi$
$48$	0	0
$49$	−15.3061	−2.18659
$50$	0	0
$51$	3.51575i	0.492303i
$52$	0	0
$53$	0.233196	0.0320320	0.0160160	−	0.999872i	$-0.494902\pi$
$53$	0.233196	0.0320320	0.0160160	+	0.999872i	$0.494902\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.44133i	0.720721i
$58$	0	0
$59$	− 14.3805i	− 1.87219i	−0.351752	−	0.936093i	$-0.614414\pi$
$59$	− 14.3805i	− 1.87219i	0.351752	−	0.936093i	$-0.385586\pi$
$60$	0	0
$61$	0.118290i	0.0151454i	0.999971	+	0.00757271i	$0.00241049\pi$
$61$	0.118290i	0.0151454i	−0.999971	+	0.00757271i	$0.997590\pi$
$62$	0	0
$63$	− 4.72294i	− 0.595034i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−13.4504	−1.64323	−0.821615	−	0.570043i	$-0.806927\pi$
$67$	−13.4504	−1.64323	−0.821615	+	0.570043i	$0.806927\pi$
$68$	0	0
$69$	− 7.11585i	− 0.856647i
$70$	0	0
$71$	−2.19027	−0.259937	−0.129969	−	0.991518i	$-0.541488\pi$
$71$	−2.19027	−0.259937	−0.129969	+	0.991518i	$0.541488\pi$
$72$	0	0
$73$	0.563219i	0.0659197i	0.999457	+	0.0329599i	$0.0104934\pi$
$73$	0.563219i	0.0659197i	−0.999457	+	0.0329599i	$0.989507\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−18.5617	−2.11530
$78$	0	0
$79$	−10.2746	−1.15599	−0.577993	−	0.816042i	$-0.696164\pi$
$79$	−10.2746	−1.15599	−0.577993	+	0.816042i	$0.696164\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.3490	1.24572	0.622860	−	0.782334i	$-0.285971\pi$
$83$	11.3490	1.24572	0.622860	+	0.782334i	$0.285971\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.66998i	− 0.393463i
$88$	0	0
$89$	−8.88265	−0.941559	−0.470780	−	0.882251i	$-0.656027\pi$
$89$	−8.88265	−0.941559	−0.470780	+	0.882251i	$0.656027\pi$
$90$	0	0
$91$	− 16.3760i	− 1.71667i
$92$	0	0
$93$	−5.23414	−0.542755
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.27462i	0.738626i	0.929305	+	0.369313i	$0.120407\pi$
$97$	7.27462i	0.738626i	−0.929305	+	0.369313i	$0.879593\pi$
$98$	0	0
$99$	− 3.93012i	− 0.394992i
$100$	0	0
$101$	− 4.23320i	− 0.421219i	−0.977570	−	0.210609i	$-0.932455\pi$
$101$	− 4.23320i	− 0.421219i	0.977570	−	0.210609i	$-0.0675448\pi$
$102$	0	0
$103$	0.0429270i	0.00422972i	0.999998	+	0.00211486i	$0.000673181\pi$
$103$	0.0429270i	0.00422972i	−0.999998	+	0.00211486i	$0.999327\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.4728	1.49581	0.747907	−	0.663804i	$-0.231059\pi$
$107$	15.4728	1.49581	0.747907	+	0.663804i	$0.231059\pi$
$108$	0	0
$109$	− 12.9561i	− 1.24097i	−0.784217	−	0.620486i	$-0.786935\pi$
$109$	− 12.9561i	− 1.24097i	0.784217	−	0.620486i	$-0.213065\pi$
$110$	0	0
$111$	−0.414376	−0.0393309
$112$	0	0
$113$	− 3.86025i	− 0.363141i	−0.983378	−	0.181571i	$-0.941882\pi$
$113$	− 3.86025i	− 0.363141i	0.983378	−	0.181571i	$-0.0581181\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.46733	0.320555
$118$	0	0
$119$	16.6046	1.52215
$120$	0	0
$121$	−4.44587	−0.404170
$122$	0	0
$123$	3.00454	0.270911
$124$	0	0
$125$	0	0
$126$	0	0
$127$	18.3805i	1.63101i	0.578751	+	0.815505i	$0.303540\pi$
$127$	18.3805i	1.63101i	−0.578751	+	0.815505i	$0.696460\pi$
$128$	0	0
$129$	5.34450	0.470557
$130$	0	0
$131$	− 3.41892i	− 0.298713i	−0.988783	−	0.149356i	$-0.952280\pi$
$131$	− 3.41892i	− 0.298713i	0.988783	−	0.149356i	$-0.0477201\pi$
$132$	0	0
$133$	25.6990	2.22839
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 16.3714i	− 1.39871i	−0.714776	−	0.699354i	$-0.753471\pi$
$137$	− 16.3714i	− 1.39871i	0.714776	−	0.699354i	$-0.246529\pi$
$138$	0	0
$139$	1.95707i	0.165997i	0.996550	+	0.0829984i	$0.0264496\pi$
$139$	1.95707i	0.165997i	−0.996550	+	0.0829984i	$0.973550\pi$
$140$	0	0
$141$	0.925579i	0.0779478i
$142$	0	0
$143$	− 13.6271i	− 1.13955i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−15.3061	−1.26243
$148$	0	0
$149$	12.0968i	0.991011i	0.868605	+	0.495505i	$0.165017\pi$
$149$	12.0968i	0.991011i	−0.868605	+	0.495505i	$0.834983\pi$
$150$	0	0
$151$	4.87178	0.396460	0.198230	−	0.980156i	$-0.436481\pi$
$151$	4.87178	0.396460	0.198230	+	0.980156i	$0.436481\pi$
$152$	0	0
$153$	3.51575i	0.284231i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	21.6561	1.72835	0.864173	−	0.503195i	$-0.167842\pi$
$157$	21.6561	1.72835	0.864173	+	0.503195i	$0.167842\pi$
$158$	0	0
$159$	0.233196	0.0184937
$160$	0	0
$161$	−33.6077	−2.64866
$162$	0	0
$163$	16.2362	1.27172	0.635860	−	0.771804i	$-0.280645\pi$
$163$	16.2362	1.27172	0.635860	+	0.771804i	$0.280645\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.69238i	0.517872i	0.965894	+	0.258936i	$0.0833719\pi$
$167$	6.69238i	0.517872i	−0.965894	+	0.258936i	$0.916628\pi$
$168$	0	0
$169$	−0.977595	−0.0751996
$170$	0	0
$171$	5.44133i	0.416109i
$172$	0	0
$173$	22.4220	1.70471	0.852355	−	0.522963i	$-0.175173\pi$
$173$	22.4220	1.70471	0.852355	+	0.522963i	$0.175173\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 14.3805i	− 1.08091i
$178$	0	0
$179$	0.148842i	0.0111250i	0.999985	+	0.00556249i	$0.00177060\pi$
$179$	0.148842i	0.0111250i	−0.999985	+	0.00556249i	$0.998229\pi$
$180$	0	0
$181$	− 10.3929i	− 0.772499i	−0.922394	−	0.386250i	$-0.873770\pi$
$181$	− 10.3929i	− 0.772499i	0.922394	−	0.386250i	$-0.126230\pi$
$182$	0	0
$183$	0.118290i	0.00874422i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	13.8173	1.01042
$188$	0	0
$189$	− 4.72294i	− 0.343543i
$190$	0	0
$191$	−6.23320	−0.451018	−0.225509	−	0.974241i	$-0.572405\pi$
$191$	−6.23320	−0.451018	−0.225509	+	0.974241i	$0.572405\pi$
$192$	0	0
$193$	0.391971i	0.0282147i	0.999900	+	0.0141074i	$0.00449066\pi$
$193$	0.391971i	0.0282147i	−0.999900	+	0.0141074i	$0.995509\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.96616	−0.425071	−0.212536	−	0.977153i	$-0.568172\pi$
$197$	−5.96616	−0.425071	−0.212536	+	0.977153i	$0.568172\pi$
$198$	0	0
$199$	−17.9322	−1.27118	−0.635591	−	0.772026i	$-0.719243\pi$
$199$	−17.9322	−1.27118	−0.635591	+	0.772026i	$0.719243\pi$
$200$	0	0
$201$	−13.4504	−0.948719
$202$	0	0
$203$	−17.3331	−1.21654
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 7.11585i	− 0.494586i
$208$	0	0
$209$	21.3851	1.47924
$210$	0	0
$211$	6.51575i	0.448563i	0.974524	+	0.224281i	$0.0720034\pi$
$211$	6.51575i	0.448563i	−0.974524	+	0.224281i	$0.927997\pi$
$212$	0	0
$213$	−2.19027	−0.150075
$214$	0	0
$215$	0	0
$216$	0	0
$217$	24.7205i	1.67814i
$218$	0	0
$219$	0.563219i	0.0380588i
$220$	0	0
$221$	12.1903i	0.820006i
$222$	0	0
$223$	3.14640i	0.210699i	0.994435	+	0.105349i	$0.0335961\pi$
$223$	3.14640i	0.210699i	−0.994435	+	0.105349i	$0.966404\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.92103	−0.260248	−0.130124	−	0.991498i	$-0.541538\pi$
$227$	−3.92103	−0.260248	−0.130124	+	0.991498i	$0.541538\pi$
$228$	0	0
$229$	25.6899i	1.69764i	0.528683	+	0.848820i	$0.322686\pi$
$229$	25.6899i	1.69764i	−0.528683	+	0.848820i	$0.677314\pi$
$230$	0	0
$231$	−18.5617	−1.22127
$232$	0	0
$233$	− 25.7565i	− 1.68737i	−0.536841	−	0.843683i	$-0.680383\pi$
$233$	− 25.7565i	− 1.68737i	0.536841	−	0.843683i	$-0.319617\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−10.2746	−0.667409
$238$	0	0
$239$	−15.8727	−1.02672	−0.513360	−	0.858173i	$-0.671600\pi$
$239$	−15.8727	−1.02672	−0.513360	+	0.858173i	$0.671600\pi$
$240$	0	0
$241$	28.1664	1.81436	0.907178	−	0.420748i	$-0.138232\pi$
$241$	28.1664	1.81436	0.907178	+	0.420748i	$0.138232\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	18.8669i	1.20047i
$248$	0	0
$249$	11.3490	0.719216
$250$	0	0
$251$	− 4.66004i	− 0.294139i	−0.989126	−	0.147070i	$-0.953016\pi$
$251$	− 4.66004i	− 0.294139i	0.989126	−	0.147070i	$-0.0469842\pi$
$252$	0	0
$253$	−27.9662	−1.75822
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3.33996i	− 0.208341i	−0.994559	−	0.104170i	$-0.966781\pi$
$257$	− 3.33996i	− 0.208341i	0.994559	−	0.104170i	$-0.0332187\pi$
$258$	0	0
$259$	1.95707i	0.121607i
$260$	0	0
$261$	− 3.66998i	− 0.227166i
$262$	0	0
$263$	27.5932i	1.70147i	0.525595	+	0.850735i	$0.323843\pi$
$263$	27.5932i	1.70147i	−0.525595	+	0.850735i	$0.676157\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−8.88265	−0.543609
$268$	0	0
$269$	− 21.8727i	− 1.33360i	−0.745235	−	0.666802i	$-0.767663\pi$
$269$	− 21.8727i	− 1.33360i	0.745235	−	0.666802i	$-0.232337\pi$
$270$	0	0
$271$	8.78583	0.533701	0.266850	−	0.963738i	$-0.414017\pi$
$271$	8.78583	0.533701	0.266850	+	0.963738i	$0.414017\pi$
$272$	0	0
$273$	− 16.3760i	− 0.991120i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.3838	0.623904	0.311952	−	0.950098i	$-0.399017\pi$
$277$	10.3838	0.623904	0.311952	+	0.950098i	$0.399017\pi$
$278$	0	0
$279$	−5.23414	−0.313360
$280$	0	0
$281$	17.0584	1.01762	0.508811	−	0.860878i	$-0.330085\pi$
$281$	17.0584	1.01762	0.508811	+	0.860878i	$0.330085\pi$
$282$	0	0
$283$	3.54724	0.210862	0.105431	−	0.994427i	$-0.466378\pi$
$283$	3.54724	0.210862	0.105431	+	0.994427i	$0.466378\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 14.1903i	− 0.837625i
$288$	0	0
$289$	4.63952	0.272913
$290$	0	0
$291$	7.27462i	0.426446i
$292$	0	0
$293$	5.72538	0.334480	0.167240	−	0.985916i	$-0.446515\pi$
$293$	5.72538	0.334480	0.167240	+	0.985916i	$0.446515\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 3.93012i	− 0.228049i
$298$	0	0
$299$	− 24.6730i	− 1.42688i
$300$	0	0
$301$	− 25.2417i	− 1.45491i
$302$	0	0
$303$	− 4.23320i	− 0.243191i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.3760	0.706335	0.353168	−	0.935560i	$-0.385105\pi$
$307$	12.3760	0.706335	0.353168	+	0.935560i	$0.385105\pi$
$308$	0	0
$309$	0.0429270i	0.00244203i
$310$	0	0
$311$	18.2746	1.03626	0.518129	−	0.855302i	$-0.326629\pi$
$311$	18.2746	1.03626	0.518129	+	0.855302i	$0.326629\pi$
$312$	0	0
$313$	12.7114i	0.718491i	0.933243	+	0.359246i	$0.116966\pi$
$313$	12.7114i	0.718491i	−0.933243	+	0.359246i	$0.883034\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.8602	−0.890800	−0.445400	−	0.895332i	$-0.646939\pi$
$317$	−15.8602	−0.890800	−0.445400	+	0.895332i	$0.646939\pi$
$318$	0	0
$319$	−14.4235	−0.807559
$320$	0	0
$321$	15.4728	0.863609
$322$	0	0
$323$	−19.1303	−1.06444
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 12.9561i	− 0.716476i
$328$	0	0
$329$	4.37145	0.241006
$330$	0	0
$331$	23.0315i	1.26593i	0.774182	+	0.632963i	$0.218161\pi$
$331$	23.0315i	1.26593i	−0.774182	+	0.632963i	$0.781839\pi$
$332$	0	0
$333$	−0.414376	−0.0227077
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 0.860247i	− 0.0468606i	−0.999725	−	0.0234303i	$-0.992541\pi$
$337$	− 0.860247i	− 0.0468606i	0.999725	−	0.0234303i	$-0.00745878\pi$
$338$	0	0
$339$	− 3.86025i	− 0.209660i
$340$	0	0
$341$	20.5708i	1.11397i
$342$	0	0
$343$	39.2293i	2.11818i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−32.2856	−1.73318	−0.866591	−	0.499019i	$-0.833694\pi$
$347$	−32.2856	−1.73318	−0.866591	+	0.499019i	$0.833694\pi$
$348$	0	0
$349$	0.742899i	0.0397665i	0.999802	+	0.0198832i	$0.00632945\pi$
$349$	0.742899i	0.0397665i	−0.999802	+	0.0198832i	$0.993671\pi$
$350$	0	0
$351$	3.46733	0.185073
$352$	0	0
$353$	32.6392i	1.73721i	0.495506	+	0.868604i	$0.334982\pi$
$353$	32.6392i	1.73721i	−0.495506	+	0.868604i	$0.665018\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	16.6046	0.878811
$358$	0	0
$359$	−2.71056	−0.143058	−0.0715290	−	0.997439i	$-0.522788\pi$
$359$	−2.71056	−0.143058	−0.0715290	+	0.997439i	$0.522788\pi$
$360$	0	0
$361$	−10.6080	−0.558317
$362$	0	0
$363$	−4.44587	−0.233348
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 0.680008i	− 0.0354961i	−0.999842	−	0.0177481i	$-0.994350\pi$
$367$	− 0.680008i	− 0.0354961i	0.999842	−	0.0177481i	$-0.00564968\pi$
$368$	0	0
$369$	3.00454	0.156410
$370$	0	0
$371$	− 1.10137i	− 0.0571803i
$372$	0	0
$373$	−3.47642	−0.180002	−0.0900012	−	0.995942i	$-0.528687\pi$
$373$	−3.47642	−0.180002	−0.0900012	+	0.995942i	$0.528687\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.7250i	− 0.655373i
$378$	0	0
$379$	23.0650i	1.18477i	0.805655	+	0.592385i	$0.201813\pi$
$379$	23.0650i	1.18477i	−0.805655	+	0.592385i	$0.798187\pi$
$380$	0	0
$381$	18.3805i	0.941664i
$382$	0	0
$383$	− 9.81544i	− 0.501545i	−0.968046	−	0.250773i	$-0.919315\pi$
$383$	− 9.81544i	− 0.501545i	0.968046	−	0.250773i	$-0.0806847\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.34450	0.271676
$388$	0	0
$389$	9.86175i	0.500010i	0.968244	+	0.250005i	$0.0804323\pi$
$389$	9.86175i	0.500010i	−0.968244	+	0.250005i	$0.919568\pi$
$390$	0	0
$391$	25.0175	1.26519
$392$	0	0
$393$	− 3.41892i	− 0.172462i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−12.7783	−0.641326	−0.320663	−	0.947193i	$-0.603906\pi$
$397$	−12.7783	−0.641326	−0.320663	+	0.947193i	$0.603906\pi$
$398$	0	0
$399$	25.6990	1.28656
$400$	0	0
$401$	3.17325	0.158465	0.0792323	−	0.996856i	$-0.474753\pi$
$401$	3.17325	0.158465	0.0792323	+	0.996856i	$0.474753\pi$
$402$	0	0
$403$	−18.1485	−0.904041
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.62855i	0.0807243i
$408$	0	0
$409$	−12.1125	−0.598923	−0.299461	−	0.954108i	$-0.596807\pi$
$409$	−12.1125	−0.598923	−0.299461	+	0.954108i	$0.596807\pi$
$410$	0	0
$411$	− 16.3714i	− 0.807544i
$412$	0	0
$413$	−67.9184	−3.34204
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.95707i	0.0958383i
$418$	0	0
$419$	4.24767i	0.207512i	0.994603	+	0.103756i	$0.0330862\pi$
$419$	4.24767i	0.207512i	−0.994603	+	0.103756i	$0.966914\pi$
$420$	0	0
$421$	− 3.77928i	− 0.184191i	−0.995750	−	0.0920953i	$-0.970644\pi$
$421$	− 3.77928i	− 0.184191i	0.995750	−	0.0920953i	$-0.0293564\pi$
$422$	0	0
$423$	0.925579i	0.0450032i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.558674	0.0270361
$428$	0	0
$429$	− 13.6271i	− 0.657920i
$430$	0	0
$431$	−25.6271	−1.23441	−0.617206	−	0.786802i	$-0.711735\pi$
$431$	−25.6271	−1.23441	−0.617206	+	0.786802i	$0.711735\pi$
$432$	0	0
$433$	2.03149i	0.0976274i	0.998808	+	0.0488137i	$0.0155441\pi$
$433$	2.03149i	0.0976274i	−0.998808	+	0.0488137i	$0.984456\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	38.7196	1.85221
$438$	0	0
$439$	−22.4864	−1.07322	−0.536608	−	0.843832i	$-0.680294\pi$
$439$	−22.4864	−1.07322	−0.536608	+	0.843832i	$0.680294\pi$
$440$	0	0
$441$	−15.3061	−0.728863
$442$	0	0
$443$	3.39385	0.161247	0.0806234	−	0.996745i	$-0.474309\pi$
$443$	3.39385	0.161247	0.0806234	+	0.996745i	$0.474309\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	12.0968i	0.572160i
$448$	0	0
$449$	2.17780	0.102777	0.0513883	−	0.998679i	$-0.483635\pi$
$449$	2.17780	0.102777	0.0513883	+	0.998679i	$0.483635\pi$
$450$	0	0
$451$	− 11.8082i	− 0.556028i
$452$	0	0
$453$	4.87178	0.228896
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 3.57653i	− 0.167303i	−0.996495	−	0.0836516i	$-0.973342\pi$
$457$	− 3.57653i	− 0.167303i	0.996495	−	0.0836516i	$-0.0266583\pi$
$458$	0	0
$459$	3.51575i	0.164101i
$460$	0	0
$461$	13.1158i	0.610866i	0.952214	+	0.305433i	$0.0988012\pi$
$461$	13.1158i	0.610866i	−0.952214	+	0.305433i	$0.901199\pi$
$462$	0	0
$463$	− 3.21417i	− 0.149375i	−0.997207	−	0.0746877i	$-0.976204\pi$
$463$	− 3.21417i	− 0.149375i	0.997207	−	0.0746877i	$-0.0237960\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.3016	1.40219	0.701095	−	0.713068i	$-0.252695\pi$
$467$	30.3016	1.40219	0.701095	+	0.713068i	$0.252695\pi$
$468$	0	0
$469$	63.5254i	2.93333i
$470$	0	0
$471$	21.6561	0.997861
$472$	0	0
$473$	− 21.0045i	− 0.965790i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.233196	0.0106773
$478$	0	0
$479$	35.0896	1.60328	0.801642	−	0.597804i	$-0.203960\pi$
$479$	35.0896	1.60328	0.801642	+	0.597804i	$0.203960\pi$
$480$	0	0
$481$	−1.43678	−0.0655116
$482$	0	0
$483$	−33.6077	−1.52920
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 36.9117i	− 1.67263i	−0.548250	−	0.836315i	$-0.684706\pi$
$487$	− 36.9117i	− 1.67263i	0.548250	−	0.836315i	$-0.315294\pi$
$488$	0	0
$489$	16.2362	0.734228
$490$	0	0
$491$	20.9867i	0.947116i	0.880763	+	0.473558i	$0.157031\pi$
$491$	20.9867i	0.947116i	−0.880763	+	0.473558i	$0.842969\pi$
$492$	0	0
$493$	12.9027	0.581109
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.3445i	0.464014i
$498$	0	0
$499$	15.9906i	0.715836i	0.933753	+	0.357918i	$0.116513\pi$
$499$	15.9906i	0.715836i	−0.933753	+	0.357918i	$0.883487\pi$
$500$	0	0
$501$	6.69238i	0.298994i
$502$	0	0
$503$	− 37.6023i	− 1.67660i	−0.545206	−	0.838302i	$-0.683549\pi$
$503$	− 37.6023i	− 1.67660i	0.545206	−	0.838302i	$-0.316451\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−0.977595	−0.0434165
$508$	0	0
$509$	12.8579i	0.569917i	0.958540	+	0.284958i	$0.0919797\pi$
$509$	12.8579i	0.569917i	−0.958540	+	0.284958i	$0.908020\pi$
$510$	0	0
$511$	2.66004	0.117673
$512$	0	0
$513$	5.44133i	0.240240i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.63764	0.159983
$518$	0	0
$519$	22.4220	0.984215
$520$	0	0
$521$	37.0015	1.62107	0.810534	−	0.585692i	$-0.199177\pi$
$521$	37.0015	1.62107	0.810534	+	0.585692i	$0.199177\pi$
$522$	0	0
$523$	17.4952	0.765013	0.382506	−	0.923953i	$-0.375061\pi$
$523$	17.4952	0.765013	0.382506	+	0.923953i	$0.375061\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 18.4019i	− 0.801600i
$528$	0	0
$529$	−27.6353	−1.20153
$530$	0	0
$531$	− 14.3805i	− 0.624062i
$532$	0	0
$533$	10.4178	0.451243
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.148842i	0.00642301i
$538$	0	0
$539$	60.1549i	2.59106i
$540$	0	0
$541$	38.1225i	1.63901i	0.573069	+	0.819507i	$0.305753\pi$
$541$	38.1225i	1.63901i	−0.573069	+	0.819507i	$0.694247\pi$
$542$	0	0
$543$	− 10.3929i	− 0.446003i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	35.7406	1.52816	0.764078	−	0.645124i	$-0.223194\pi$
$547$	35.7406	1.52816	0.764078	+	0.645124i	$0.223194\pi$
$548$	0	0
$549$	0.118290i	0.00504848i
$550$	0	0
$551$	19.9695	0.850731
$552$	0	0
$553$	48.5264i	2.06355i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.65516	0.112503	0.0562514	−	0.998417i	$-0.482085\pi$
$557$	2.65516	0.112503	0.0562514	+	0.998417i	$0.482085\pi$
$558$	0	0
$559$	18.5312	0.783785
$560$	0	0
$561$	13.8173	0.583368
$562$	0	0
$563$	−20.3107	−0.855992	−0.427996	−	0.903781i	$-0.640780\pi$
$563$	−20.3107	−0.855992	−0.427996	+	0.903781i	$0.640780\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 4.72294i	− 0.198345i
$568$	0	0
$569$	28.4274	1.19174	0.595868	−	0.803082i	$-0.296808\pi$
$569$	28.4274	1.19174	0.595868	+	0.803082i	$0.296808\pi$
$570$	0	0
$571$	16.1485i	0.675794i	0.941183	+	0.337897i	$0.109716\pi$
$571$	16.1485i	0.675794i	−0.941183	+	0.337897i	$0.890284\pi$
$572$	0	0
$573$	−6.23320	−0.260396
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 30.3600i	− 1.26390i	−0.775008	−	0.631952i	$-0.782254\pi$
$577$	− 30.3600i	− 1.26390i	0.775008	−	0.631952i	$-0.217746\pi$
$578$	0	0
$579$	0.391971i	0.0162898i
$580$	0	0
$581$	− 53.6008i	− 2.22374i
$582$	0	0
$583$	− 0.916490i	− 0.0379571i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.74070	0.113121	0.0565604	−	0.998399i	$-0.481987\pi$
$587$	2.74070	0.113121	0.0565604	+	0.998399i	$0.481987\pi$
$588$	0	0
$589$	− 28.4807i	− 1.17352i
$590$	0	0
$591$	−5.96616	−0.245415
$592$	0	0
$593$	22.2586i	0.914053i	0.889453	+	0.457027i	$0.151086\pi$
$593$	22.2586i	0.914053i	−0.889453	+	0.457027i	$0.848914\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−17.9322	−0.733917
$598$	0	0
$599$	48.4526	1.97972	0.989860	−	0.142046i	$-0.0453680\pi$
$599$	48.4526	1.97972	0.989860	+	0.142046i	$0.0453680\pi$
$600$	0	0
$601$	5.26553	0.214786	0.107393	−	0.994217i	$-0.465750\pi$
$601$	5.26553	0.214786	0.107393	+	0.994217i	$0.465750\pi$
$602$	0	0
$603$	−13.4504	−0.547743
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7.38288i	0.299662i	0.988712	+	0.149831i	$0.0478730\pi$
$607$	7.38288i	0.299662i	−0.988712	+	0.149831i	$0.952127\pi$
$608$	0	0
$609$	−17.3331	−0.702371
$610$	0	0
$611$	3.20929i	0.129834i
$612$	0	0
$613$	−2.64607	−0.106874	−0.0534369	−	0.998571i	$-0.517018\pi$
$613$	−2.64607	−0.106874	−0.0534369	+	0.998571i	$0.517018\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 21.0136i	− 0.845977i	−0.906135	−	0.422989i	$-0.860981\pi$
$617$	− 21.0136i	− 0.845977i	0.906135	−	0.422989i	$-0.139019\pi$
$618$	0	0
$619$	24.0874i	0.968154i	0.875025	+	0.484077i	$0.160845\pi$
$619$	24.0874i	0.968154i	−0.875025	+	0.484077i	$0.839155\pi$
$620$	0	0
$621$	− 7.11585i	− 0.285549i
$622$	0	0
$623$	41.9522i	1.68078i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	21.3851	0.854038
$628$	0	0
$629$	− 1.45684i	− 0.0580881i
$630$	0	0
$631$	−25.2094	−1.00357	−0.501785	−	0.864992i	$-0.667323\pi$
$631$	−25.2094	−1.00357	−0.501785	+	0.864992i	$0.667323\pi$
$632$	0	0
$633$	6.51575i	0.258978i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−53.0714	−2.10277
$638$	0	0
$639$	−2.19027	−0.0866457
$640$	0	0
$641$	−18.4755	−0.729738	−0.364869	−	0.931059i	$-0.618886\pi$
$641$	−18.4755	−0.729738	−0.364869	+	0.931059i	$0.618886\pi$
$642$	0	0
$643$	0.636984	0.0251202	0.0125601	−	0.999921i	$-0.496002\pi$
$643$	0.636984	0.0251202	0.0125601	+	0.999921i	$0.496002\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.0182i	1.25876i	0.777096	+	0.629382i	$0.216692\pi$
$647$	32.0182i	1.25876i	−0.777096	+	0.629382i	$0.783308\pi$
$648$	0	0
$649$	−56.5173	−2.21850
$650$	0	0
$651$	24.7205i	0.968873i
$652$	0	0
$653$	−25.9769	−1.01656	−0.508278	−	0.861193i	$-0.669718\pi$
$653$	−25.9769	−1.01656	−0.508278	+	0.861193i	$0.669718\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.563219i	0.0219732i
$658$	0	0
$659$	− 21.3422i	− 0.831372i	−0.909508	−	0.415686i	$-0.863541\pi$
$659$	− 21.3422i	− 0.831372i	0.909508	−	0.415686i	$-0.136459\pi$
$660$	0	0
$661$	14.8397i	0.577198i	0.957450	+	0.288599i	$0.0931895\pi$
$661$	14.8397i	0.577198i	−0.957450	+	0.288599i	$0.906810\pi$
$662$	0	0
$663$	12.1903i	0.473431i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−26.1150	−1.01118
$668$	0	0
$669$	3.14640i	0.121647i
$670$	0	0
$671$	0.464893	0.0179470
$672$	0	0
$673$	18.1167i	0.698347i	0.937058	+	0.349174i	$0.113538\pi$
$673$	18.1167i	0.698347i	−0.937058	+	0.349174i	$0.886462\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.5617	1.17458	0.587291	−	0.809376i	$-0.300194\pi$
$677$	30.5617	1.17458	0.587291	+	0.809376i	$0.300194\pi$
$678$	0	0
$679$	34.3576	1.31852
$680$	0	0
$681$	−3.92103	−0.150254
$682$	0	0
$683$	22.2027	0.849564	0.424782	−	0.905296i	$-0.360351\pi$
$683$	22.2027	0.849564	0.424782	+	0.905296i	$0.360351\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	25.6899i	0.980132i
$688$	0	0
$689$	0.808569	0.0308040
$690$	0	0
$691$	12.6890i	0.482712i	0.970437	+	0.241356i	$0.0775922\pi$
$691$	12.6890i	0.482712i	−0.970437	+	0.241356i	$0.922408\pi$
$692$	0	0
$693$	−18.5617	−0.705101
$694$	0	0
$695$	0	0
$696$	0	0
$697$	10.5632i	0.400110i
$698$	0	0
$699$	− 25.7565i	− 0.974202i
$700$	0	0
$701$	34.9241i	1.31906i	0.751676	+	0.659532i	$0.229246\pi$
$701$	34.9241i	1.31906i	−0.751676	+	0.659532i	$0.770754\pi$
$702$	0	0
$703$	− 2.25476i	− 0.0850398i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−19.9931	−0.751918
$708$	0	0
$709$	− 5.65610i	− 0.212419i	−0.994344	−	0.106210i	$-0.966129\pi$
$709$	− 5.65610i	− 0.212419i	0.994344	−	0.106210i	$-0.0338715\pi$
$710$	0	0
$711$	−10.2746	−0.385328
$712$	0	0
$713$	37.2453i	1.39485i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−15.8727	−0.592778
$718$	0	0
$719$	−20.0844	−0.749020	−0.374510	−	0.927223i	$-0.622189\pi$
$719$	−20.0844	−0.749020	−0.374510	+	0.927223i	$0.622189\pi$
$720$	0	0
$721$	0.202741	0.00755048
$722$	0	0
$723$	28.1664	1.04752
$724$	0	0
$725$	0	0
$726$	0	0
$727$	13.0424i	0.483715i	0.970312	+	0.241857i	$0.0777566\pi$
$727$	13.0424i	0.483715i	−0.970312	+	0.241857i	$0.922243\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	18.7899i	0.694970i
$732$	0	0
$733$	−34.8917	−1.28876	−0.644378	−	0.764707i	$-0.722884\pi$
$733$	−34.8917	−1.28876	−0.644378	+	0.764707i	$0.722884\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	52.8618i	1.94719i
$738$	0	0
$739$	− 41.5040i	− 1.52675i	−0.645957	−	0.763374i	$-0.723541\pi$
$739$	− 41.5040i	− 1.52675i	0.645957	−	0.763374i	$-0.276459\pi$
$740$	0	0
$741$	18.8669i	0.693093i
$742$	0	0
$743$	− 7.63764i	− 0.280198i	−0.990138	−	0.140099i	$-0.955258\pi$
$743$	− 7.63764i	− 0.280198i	0.990138	−	0.140099i	$-0.0447421\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	11.3490	0.415240
$748$	0	0
$749$	− 73.0771i	− 2.67018i
$750$	0	0
$751$	19.4029	0.708023	0.354012	−	0.935241i	$-0.384817\pi$
$751$	19.4029	0.708023	0.354012	+	0.935241i	$0.384817\pi$
$752$	0	0
$753$	− 4.66004i	− 0.169821i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	43.7959	1.59179	0.795894	−	0.605436i	$-0.207001\pi$
$757$	43.7959	1.59179	0.795894	+	0.605436i	$0.207001\pi$
$758$	0	0
$759$	−27.9662	−1.01511
$760$	0	0
$761$	9.95519	0.360875	0.180438	−	0.983586i	$-0.442249\pi$
$761$	9.95519	0.360875	0.180438	+	0.983586i	$0.442249\pi$
$762$	0	0
$763$	−61.1910	−2.21526
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 49.8621i	− 1.80042i
$768$	0	0
$769$	17.9008	0.645520	0.322760	−	0.946481i	$-0.395389\pi$
$769$	17.9008	0.645520	0.322760	+	0.946481i	$0.395389\pi$
$770$	0	0
$771$	− 3.33996i	− 0.120286i
$772$	0	0
$773$	20.8182	0.748777	0.374389	−	0.927272i	$-0.377853\pi$
$773$	20.8182	0.748777	0.374389	+	0.927272i	$0.377853\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.95707i	0.0702096i
$778$	0	0
$779$	16.3487i	0.585753i
$780$	0	0
$781$	8.60803i	0.308019i
$782$	0	0
$783$	− 3.66998i	− 0.131154i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	27.2047	0.969744	0.484872	−	0.874585i	$-0.338866\pi$
$787$	27.2047	0.969744	0.484872	+	0.874585i	$0.338866\pi$
$788$	0	0
$789$	27.5932i	0.982344i
$790$	0	0
$791$	−18.2317	−0.648244
$792$	0	0
$793$	0.410149i	0.0145648i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.4344	0.582138	0.291069	−	0.956702i	$-0.405989\pi$
$797$	16.4344	0.582138	0.291069	+	0.956702i	$0.405989\pi$
$798$	0	0
$799$	−3.25410	−0.115122
$800$	0	0
$801$	−8.88265	−0.313853
$802$	0	0
$803$	2.21352	0.0781134
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 21.8727i	− 0.769956i
$808$	0	0
$809$	−28.7698	−1.01149	−0.505747	−	0.862682i	$-0.668783\pi$
$809$	−28.7698	−1.01149	−0.505747	+	0.862682i	$0.668783\pi$
$810$	0	0
$811$	− 14.3512i	− 0.503940i	−0.967735	−	0.251970i	$-0.918922\pi$
$811$	− 14.3512i	− 0.503940i	0.967735	−	0.251970i	$-0.0810785\pi$
$812$	0	0
$813$	8.78583	0.308132
$814$	0	0
$815$	0	0
$816$	0	0
$817$	29.0812i	1.01742i
$818$	0	0
$819$	− 16.3760i	− 0.572224i
$820$	0	0
$821$	4.97271i	0.173549i	0.996228	+	0.0867744i	$0.0276559\pi$
$821$	4.97271i	0.173549i	−0.996228	+	0.0867744i	$0.972344\pi$
$822$	0	0
$823$	20.3625i	0.709791i	0.934906	+	0.354895i	$0.115483\pi$
$823$	20.3625i	0.709791i	−0.934906	+	0.354895i	$0.884517\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−36.0039	−1.25198	−0.625989	−	0.779832i	$-0.715304\pi$
$827$	−36.0039	−1.25198	−0.625989	+	0.779832i	$0.715304\pi$
$828$	0	0
$829$	7.68666i	0.266969i	0.991051	+	0.133484i	$0.0426166\pi$
$829$	7.68666i	0.266969i	−0.991051	+	0.133484i	$0.957383\pi$
$830$	0	0
$831$	10.3838	0.360211
$832$	0	0
$833$	− 53.8124i	− 1.86449i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−5.23414	−0.180918
$838$	0	0
$839$	−25.3725	−0.875956	−0.437978	−	0.898986i	$-0.644305\pi$
$839$	−25.3725	−0.875956	−0.437978	+	0.898986i	$0.644305\pi$
$840$	0	0
$841$	15.5313	0.535561
$842$	0	0
$843$	17.0584	0.587524
$844$	0	0
$845$	0	0
$846$	0	0
$847$	20.9976i	0.721485i
$848$	0	0
$849$	3.54724	0.121741
$850$	0	0
$851$	2.94864i	0.101078i
$852$	0	0
$853$	20.1365	0.689460	0.344730	−	0.938702i	$-0.387971\pi$
$853$	20.1365	0.689460	0.344730	+	0.938702i	$0.387971\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	42.0380i	1.43599i	0.696047	+	0.717996i	$0.254940\pi$
$857$	42.0380i	1.43599i	−0.696047	+	0.717996i	$0.745060\pi$
$858$	0	0
$859$	− 12.4095i	− 0.423406i	−0.977334	−	0.211703i	$-0.932099\pi$
$859$	− 12.4095i	− 0.423406i	0.977334	−	0.211703i	$-0.0679010\pi$
$860$	0	0
$861$	− 14.1903i	− 0.483603i
$862$	0	0
$863$	21.8154i	0.742606i	0.928512	+	0.371303i	$0.121089\pi$
$863$	21.8154i	0.742606i	−0.928512	+	0.371303i	$0.878911\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	4.63952	0.157566
$868$	0	0
$869$	40.3805i	1.36982i
$870$	0	0
$871$	−46.6371	−1.58024
$872$	0	0
$873$	7.27462i	0.246209i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−5.83690	−0.197098	−0.0985491	−	0.995132i	$-0.531420\pi$
$877$	−5.83690	−0.197098	−0.0985491	+	0.995132i	$0.531420\pi$
$878$	0	0
$879$	5.72538	0.193112
$880$	0	0
$881$	−11.9613	−0.402986	−0.201493	−	0.979490i	$-0.564579\pi$
$881$	−11.9613	−0.402986	−0.201493	+	0.979490i	$0.564579\pi$
$882$	0	0
$883$	3.74810	0.126134	0.0630668	−	0.998009i	$-0.479912\pi$
$883$	3.74810	0.126134	0.0630668	+	0.998009i	$0.479912\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 42.8101i	− 1.43742i	−0.695309	−	0.718711i	$-0.744732\pi$
$887$	− 42.8101i	− 1.43742i	0.695309	−	0.718711i	$-0.255268\pi$
$888$	0	0
$889$	86.8101	2.91152
$890$	0	0
$891$	− 3.93012i	− 0.131664i
$892$	0	0
$893$	−5.03638	−0.168536
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 24.6730i	− 0.823808i
$898$	0	0
$899$	19.2092i	0.640662i
$900$	0	0
$901$	0.819859i	0.0273135i
$902$	0	0
$903$	− 25.2417i	− 0.839992i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	6.97759	0.231687	0.115844	−	0.993267i	$-0.463043\pi$
$907$	6.97759	0.231687	0.115844	+	0.993267i	$0.463043\pi$
$908$	0	0
$909$	− 4.23320i	− 0.140406i
$910$	0	0
$911$	−24.2837	−0.804555	−0.402278	−	0.915518i	$-0.631781\pi$
$911$	−24.2837	−0.804555	−0.402278	+	0.915518i	$0.631781\pi$
$912$	0	0
$913$	− 44.6031i	− 1.47615i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−16.1473	−0.533232
$918$	0	0
$919$	−1.25720	−0.0414712	−0.0207356	−	0.999785i	$-0.506601\pi$
$919$	−1.25720	−0.0414712	−0.0207356	+	0.999785i	$0.506601\pi$
$920$	0	0
$921$	12.3760	0.407803
$922$	0	0
$923$	−7.59440	−0.249973
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.0429270i	0.00140991i
$928$	0	0
$929$	−10.4933	−0.344275	−0.172138	−	0.985073i	$-0.555067\pi$
$929$	−10.4933	−0.344275	−0.172138	+	0.985073i	$0.555067\pi$
$930$	0	0
$931$	− 83.2856i	− 2.72957i
$932$	0	0
$933$	18.2746	0.598284
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12.9680i	0.423648i	0.977308	+	0.211824i	$0.0679403\pi$
$937$	12.9680i	0.423648i	−0.977308	+	0.211824i	$0.932060\pi$
$938$	0	0
$939$	12.7114i	0.414821i
$940$	0	0
$941$	− 44.9947i	− 1.46678i	−0.679806	−	0.733392i	$-0.737936\pi$
$941$	− 44.9947i	− 1.46678i	0.679806	−	0.733392i	$-0.262064\pi$
$942$	0	0
$943$	− 21.3799i	− 0.696225i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.6282	−0.995283	−0.497642	−	0.867383i	$-0.665801\pi$
$947$	−30.6282	−0.995283	−0.497642	+	0.867383i	$0.665801\pi$
$948$	0	0
$949$	1.95287i	0.0633927i
$950$	0	0
$951$	−15.8602	−0.514304
$952$	0	0
$953$	− 17.5717i	− 0.569202i	−0.958646	−	0.284601i	$-0.908139\pi$
$953$	− 17.5717i	− 0.569202i	0.958646	−	0.284601i	$-0.0918610\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−14.4235	−0.466244
$958$	0	0
$959$	−77.3213	−2.49683
$960$	0	0
$961$	−3.60380	−0.116252
$962$	0	0
$963$	15.4728	0.498605
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 4.50932i	− 0.145010i	−0.997368	−	0.0725050i	$-0.976901\pi$
$967$	− 4.50932i	− 0.145010i	0.997368	−	0.0725050i	$-0.0230993\pi$
$968$	0	0
$969$	−19.1303	−0.614555
$970$	0	0
$971$	35.8512i	1.15052i	0.817971	+	0.575259i	$0.195099\pi$
$971$	35.8512i	1.15052i	−0.817971	+	0.575259i	$0.804901\pi$
$972$	0	0
$973$	9.24313	0.296321
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 43.9704i	− 1.40674i	−0.710826	−	0.703368i	$-0.751678\pi$
$977$	− 43.9704i	− 1.40674i	0.710826	−	0.703368i	$-0.248322\pi$
$978$	0	0
$979$	34.9099i	1.11573i
$980$	0	0
$981$	− 12.9561i	− 0.413657i
$982$	0	0
$983$	− 4.64187i	− 0.148053i	−0.997256	−	0.0740263i	$-0.976415\pi$
$983$	− 4.64187i	− 0.148053i	0.997256	−	0.0740263i	$-0.0235849\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.37145	0.139145
$988$	0	0
$989$	− 38.0307i	− 1.20930i
$990$	0	0
$991$	52.9117	1.68080	0.840398	−	0.541970i	$-0.182321\pi$
$991$	52.9117	1.68080	0.840398	+	0.541970i	$0.182321\pi$
$992$	0	0
$993$	23.0315i	0.730882i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	39.4228	1.24853	0.624266	−	0.781211i	$-0.285398\pi$
$997$	39.4228	1.24853	0.624266	+	0.781211i	$0.285398\pi$
$998$	0	0
$999$	−0.414376	−0.0131103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.d.h.49.1		8
3.2	odd	2		7200.2.d.t.2449.1		8
4.3	odd	2		600.2.d.g.349.7		8
5.2	odd	4		2400.2.k.e.1201.4		8
5.3	odd	4		2400.2.k.d.1201.5		8
5.4	even	2		2400.2.d.g.49.8		8
8.3	odd	2		600.2.d.h.349.1		8
8.5	even	2		2400.2.d.g.49.1		8
12.11	even	2		1800.2.d.t.1549.2		8
15.2	even	4		7200.2.k.s.3601.8		8
15.8	even	4		7200.2.k.r.3601.2		8
15.14	odd	2		7200.2.d.s.2449.8		8
20.3	even	4		600.2.k.d.301.3	✓	8
20.7	even	4		600.2.k.e.301.6	yes	8
20.19	odd	2		600.2.d.h.349.2		8
24.5	odd	2		7200.2.d.s.2449.1		8
24.11	even	2		1800.2.d.s.1549.8		8
40.3	even	4		600.2.k.d.301.4	yes	8
40.13	odd	4		2400.2.k.d.1201.1		8
40.19	odd	2		600.2.d.g.349.8		8
40.27	even	4		600.2.k.e.301.5	yes	8
40.29	even	2	inner	2400.2.d.h.49.8		8
40.37	odd	4		2400.2.k.e.1201.8		8
60.23	odd	4		1800.2.k.t.901.6		8
60.47	odd	4		1800.2.k.q.901.3		8
60.59	even	2		1800.2.d.s.1549.7		8
120.29	odd	2		7200.2.d.t.2449.8		8
120.53	even	4		7200.2.k.r.3601.1		8
120.59	even	2		1800.2.d.t.1549.1		8
120.77	even	4		7200.2.k.s.3601.7		8
120.83	odd	4		1800.2.k.t.901.5		8
120.107	odd	4		1800.2.k.q.901.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.d.g.349.7		8	4.3	odd	2
600.2.d.g.349.8		8	40.19	odd	2
600.2.d.h.349.1		8	8.3	odd	2
600.2.d.h.349.2		8	20.19	odd	2
600.2.k.d.301.3	✓	8	20.3	even	4
600.2.k.d.301.4	yes	8	40.3	even	4
600.2.k.e.301.5	yes	8	40.27	even	4
600.2.k.e.301.6	yes	8	20.7	even	4
1800.2.d.s.1549.7		8	60.59	even	2
1800.2.d.s.1549.8		8	24.11	even	2
1800.2.d.t.1549.1		8	120.59	even	2
1800.2.d.t.1549.2		8	12.11	even	2
1800.2.k.q.901.3		8	60.47	odd	4
1800.2.k.q.901.4		8	120.107	odd	4
1800.2.k.t.901.5		8	120.83	odd	4
1800.2.k.t.901.6		8	60.23	odd	4
2400.2.d.g.49.1		8	8.5	even	2
2400.2.d.g.49.8		8	5.4	even	2
2400.2.d.h.49.1		8	1.1	even	1	trivial
2400.2.d.h.49.8		8	40.29	even	2	inner
2400.2.k.d.1201.1		8	40.13	odd	4
2400.2.k.d.1201.5		8	5.3	odd	4
2400.2.k.e.1201.4		8	5.2	odd	4
2400.2.k.e.1201.8		8	40.37	odd	4
7200.2.d.s.2449.1		8	24.5	odd	2
7200.2.d.s.2449.8		8	15.14	odd	2
7200.2.d.t.2449.1		8	3.2	odd	2
7200.2.d.t.2449.8		8	120.29	odd	2
7200.2.k.r.3601.1		8	120.53	even	4
7200.2.k.r.3601.2		8	15.8	even	4
7200.2.k.s.3601.7		8	120.77	even	4
7200.2.k.s.3601.8		8	15.2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.72294i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.72294i q^{7} +1.00000 q^{9} -3.93012i q^{11} +3.46733 q^{13} +3.51575i q^{17} +5.44133i q^{19} -4.72294i q^{21} -7.11585i q^{23} +1.00000 q^{27} -3.66998i q^{29} -5.23414 q^{31} -3.93012i q^{33} -0.414376 q^{37} +3.46733 q^{39} +3.00454 q^{41} +5.34450 q^{43} +0.925579i q^{47} -15.3061 q^{49} +3.51575i q^{51} +0.233196 q^{53} +5.44133i q^{57} -14.3805i q^{59} +0.118290i q^{61} -4.72294i q^{63} -13.4504 q^{67} -7.11585i q^{69} -2.19027 q^{71} +0.563219i q^{73} -18.5617 q^{77} -10.2746 q^{79} +1.00000 q^{81} +11.3490 q^{83} -3.66998i q^{87} -8.88265 q^{89} -16.3760i q^{91} -5.23414 q^{93} +7.27462i q^{97} -3.93012i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 + 8 * q^9	\( 8 q + 8 q^{3} + 8 q^{9} + 8 q^{27} - 8 q^{31} + 8 q^{43} + 8 q^{53} - 24 q^{67} + 40 q^{71} - 24 q^{77} - 16 q^{79} + 8 q^{81} + 32 q^{83} - 8 q^{93}+O(q^{100}) \) 8 * q + 8 * q^3 + 8 * q^9 + 8 * q^27 - 8 * q^31 + 8 * q^43 + 8 * q^53 - 24 * q^67 + 40 * q^71 - 24 * q^77 - 16 * q^79 + 8 * q^81 + 32 * q^83 - 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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