LMFDB - Embedded newform 2400.2.k.e.1201.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(1201,2400)]

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, 0]))

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

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

chi := DirichletCharacter("2400.1201");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.k (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:	$\Z[a_1, \ldots, a_{11}]$
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			1201.4
Root			$-0.565036 + 1.29643i$ of defining polynomial
Character	$\chi$	$=$	2400.1201
Dual form			2400.2.k.e.1201.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.72294	1.78510	0.892551	−	0.450947i	$-0.148914\pi$
$7$	4.72294	1.78510	0.892551	+	0.450947i	$0.148914\pi$
$8$	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
$13$	− 3.46733i	− 0.961665i	−0.876812	−	0.480833i	$-0.840334\pi$
$13$	− 3.46733i	− 0.961665i	0.876812	−	0.480833i	$-0.159666\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.51575	−0.852694	−0.426347	−	0.904560i	$-0.640200\pi$
$17$	−3.51575	−0.852694	−0.426347	+	0.904560i	$0.640200\pi$
$18$	0	0
$19$	− 5.44133i	− 1.24833i	−0.781294	−	0.624163i	$-0.785440\pi$
$19$	− 5.44133i	− 1.24833i	0.781294	−	0.624163i	$-0.214560\pi$
$20$	0	0
$21$	− 4.72294i	− 1.03063i
$22$	0	0
$23$	−7.11585	−1.48376	−0.741878	−	0.670534i	$-0.766065\pi$
$23$	−7.11585	−1.48376	−0.741878	+	0.670534i	$0.766065\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	3.66998i	0.681498i	0.940154	+	0.340749i	$0.110681\pi$
$29$	3.66998i	0.681498i	−0.940154	+	0.340749i	$0.889319\pi$
$30$	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
$33$	−3.93012	−0.684147
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 0.414376i	− 0.0681231i	−0.999420	−	0.0340615i	$-0.989156\pi$
$37$	− 0.414376i	− 0.0681231i	0.999420	−	0.0340615i	$-0.0108442\pi$
$38$	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
$43$	− 5.34450i	− 0.815029i	−0.913199	−	0.407514i	$-0.866396\pi$
$43$	− 5.34450i	− 0.815029i	0.913199	−	0.407514i	$-0.133604\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.925579	−0.135010	−0.0675048	−	0.997719i	$-0.521504\pi$
$47$	−0.925579	−0.135010	−0.0675048	+	0.997719i	$0.521504\pi$
$48$	0	0
$49$	15.3061	2.18659
$50$	0	0
$51$	3.51575i	0.492303i
$52$	0	0
$53$	− 0.233196i	− 0.0320320i	−0.999872	−	0.0160160i	$-0.994902\pi$
$53$	− 0.233196i	− 0.0320320i	0.999872	−	0.0160160i	$-0.00509826\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.44133	−0.720721
$58$	0	0
$59$	14.3805i	1.87219i	0.351752	+	0.936093i	$0.385586\pi$
$59$	14.3805i	1.87219i	−0.351752	+	0.936093i	$0.614414\pi$
$60$	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
$63$	−4.72294	−0.595034
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 13.4504i	− 1.64323i	−0.570043	−	0.821615i	$-0.693073\pi$
$67$	− 13.4504i	− 1.64323i	0.570043	−	0.821615i	$-0.306927\pi$
$68$	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
$73$	0.563219	0.0659197	0.0329599	−	0.999457i	$-0.489507\pi$
$73$	0.563219	0.0659197	0.0329599	+	0.999457i	$0.489507\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 18.5617i	− 2.11530i
$78$	0	0
$79$	10.2746	1.15599	0.577993	−	0.816042i	$-0.303836\pi$
$79$	10.2746	1.15599	0.577993	+	0.816042i	$0.303836\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 11.3490i	− 1.24572i	−0.782334	−	0.622860i	$-0.785971\pi$
$83$	− 11.3490i	− 1.24572i	0.782334	−	0.622860i	$-0.214029\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.66998	0.393463
$88$	0	0
$89$	8.88265	0.941559	0.470780	−	0.882251i	$-0.343973\pi$
$89$	8.88265	0.941559	0.470780	+	0.882251i	$0.343973\pi$
$90$	0	0
$91$	− 16.3760i	− 1.71667i
$92$	0	0
$93$	5.23414i	0.542755i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.27462	−0.738626	−0.369313	−	0.929305i	$-0.620407\pi$
$97$	−7.27462	−0.738626	−0.369313	+	0.929305i	$0.620407\pi$
$98$	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
$103$	0.0429270	0.00422972	0.00211486	−	0.999998i	$-0.499327\pi$
$103$	0.0429270	0.00422972	0.00211486	+	0.999998i	$0.499327\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.4728i	1.49581i	0.663804	+	0.747907i	$0.268941\pi$
$107$	15.4728i	1.49581i	−0.663804	+	0.747907i	$0.731059\pi$
$108$	0	0
$109$	12.9561i	1.24097i	0.784217	+	0.620486i	$0.213065\pi$
$109$	12.9561i	1.24097i	−0.784217	+	0.620486i	$0.786935\pi$
$110$	0	0
$111$	−0.414376	−0.0393309
$112$	0	0
$113$	−3.86025	−0.363141	−0.181571	−	0.983378i	$-0.558118\pi$
$113$	−3.86025	−0.363141	−0.181571	+	0.983378i	$0.558118\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.46733i	0.320555i
$118$	0	0
$119$	−16.6046	−1.52215
$120$	0	0
$121$	−4.44587	−0.404170
$122$	0	0
$123$	− 3.00454i	− 0.270911i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−18.3805	−1.63101	−0.815505	−	0.578751i	$-0.803540\pi$
$127$	−18.3805	−1.63101	−0.815505	+	0.578751i	$0.803540\pi$
$128$	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
$133$	− 25.6990i	− 2.22839i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.3714	1.39871	0.699354	−	0.714776i	$-0.253471\pi$
$137$	16.3714	1.39871	0.699354	+	0.714776i	$0.253471\pi$
$138$	0	0
$139$	− 1.95707i	− 0.165997i	−0.996550	−	0.0829984i	$-0.973550\pi$
$139$	− 1.95707i	− 0.165997i	0.996550	−	0.0829984i	$-0.0264496\pi$
$140$	0	0
$141$	0.925579i	0.0779478i
$142$	0	0
$143$	−13.6271	−1.13955
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 15.3061i	− 1.26243i
$148$	0	0
$149$	− 12.0968i	− 0.991011i	−0.868605	−	0.495505i	$-0.834983\pi$
$149$	− 12.0968i	− 0.991011i	0.868605	−	0.495505i	$-0.165017\pi$
$150$	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
$153$	3.51575	0.284231
$154$	0	0
$155$	0	0
$156$	0	0
$157$	21.6561i	1.72835i	0.503195	+	0.864173i	$0.332158\pi$
$157$	21.6561i	1.72835i	−0.503195	+	0.864173i	$0.667842\pi$
$158$	0	0
$159$	−0.233196	−0.0184937
$160$	0	0
$161$	−33.6077	−2.64866
$162$	0	0
$163$	− 16.2362i	− 1.27172i	−0.771804	−	0.635860i	$-0.780645\pi$
$163$	− 16.2362i	− 1.27172i	0.771804	−	0.635860i	$-0.219355\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.69238	−0.517872	−0.258936	−	0.965894i	$-0.583372\pi$
$167$	−6.69238	−0.517872	−0.258936	+	0.965894i	$0.583372\pi$
$168$	0	0
$169$	0.977595	0.0751996
$170$	0	0
$171$	5.44133i	0.416109i
$172$	0	0
$173$	− 22.4220i	− 1.70471i	−0.522963	−	0.852355i	$-0.675173\pi$
$173$	− 22.4220i	− 1.70471i	0.522963	−	0.852355i	$-0.324827\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	14.3805	1.08091
$178$	0	0
$179$	− 0.148842i	− 0.0111250i	−0.999985	−	0.00556249i	$-0.998229\pi$
$179$	− 0.148842i	− 0.0111250i	0.999985	−	0.00556249i	$-0.00177060\pi$
$180$	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
$183$	0.118290	0.00874422
$184$	0	0
$185$	0	0
$186$	0	0
$187$	13.8173i	1.01042i
$188$	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
$193$	0.391971	0.0282147	0.0141074	−	0.999900i	$-0.495509\pi$
$193$	0.391971	0.0282147	0.0141074	+	0.999900i	$0.495509\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.96616i	− 0.425071i	−0.977153	−	0.212536i	$-0.931828\pi$
$197$	− 5.96616i	− 0.425071i	0.977153	−	0.212536i	$-0.0681722\pi$
$198$	0	0
$199$	17.9322	1.27118	0.635591	−	0.772026i	$-0.280757\pi$
$199$	17.9322	1.27118	0.635591	+	0.772026i	$0.280757\pi$
$200$	0	0
$201$	−13.4504	−0.948719
$202$	0	0
$203$	17.3331i	1.21654i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.11585	0.494586
$208$	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
$213$	2.19027i	0.150075i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−24.7205	−1.67814
$218$	0	0
$219$	− 0.563219i	− 0.0380588i
$220$	0	0
$221$	12.1903i	0.820006i
$222$	0	0
$223$	3.14640	0.210699	0.105349	−	0.994435i	$-0.466404\pi$
$223$	3.14640	0.210699	0.105349	+	0.994435i	$0.466404\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3.92103i	− 0.260248i	−0.991498	−	0.130124i	$-0.958462\pi$
$227$	− 3.92103i	− 0.260248i	0.991498	−	0.130124i	$-0.0415376\pi$
$228$	0	0
$229$	− 25.6899i	− 1.69764i	−0.528683	−	0.848820i	$-0.677314\pi$
$229$	− 25.6899i	− 1.69764i	0.528683	−	0.848820i	$-0.322686\pi$
$230$	0	0
$231$	−18.5617	−1.22127
$232$	0	0
$233$	−25.7565	−1.68737	−0.843683	−	0.536841i	$-0.819617\pi$
$233$	−25.7565	−1.68737	−0.843683	+	0.536841i	$0.819617\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 10.2746i	− 0.667409i
$238$	0	0
$239$	15.8727	1.02672	0.513360	−	0.858173i	$-0.328400\pi$
$239$	15.8727	1.02672	0.513360	+	0.858173i	$0.328400\pi$
$240$	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
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−18.8669	−1.20047
$248$	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
$253$	27.9662i	1.75822i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.33996	0.208341	0.104170	−	0.994559i	$-0.466781\pi$
$257$	3.33996	0.208341	0.104170	+	0.994559i	$0.466781\pi$
$258$	0	0
$259$	− 1.95707i	− 0.121607i
$260$	0	0
$261$	− 3.66998i	− 0.227166i
$262$	0	0
$263$	27.5932	1.70147	0.850735	−	0.525595i	$-0.176157\pi$
$263$	27.5932	1.70147	0.850735	+	0.525595i	$0.176157\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 8.88265i	− 0.543609i
$268$	0	0
$269$	21.8727i	1.33360i	0.745235	+	0.666802i	$0.232337\pi$
$269$	21.8727i	1.33360i	−0.745235	+	0.666802i	$0.767663\pi$
$270$	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
$273$	−16.3760	−0.991120
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.3838i	0.623904i	0.950098	+	0.311952i	$0.100983\pi$
$277$	10.3838i	0.623904i	−0.950098	+	0.311952i	$0.899017\pi$
$278$	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
$283$	− 3.54724i	− 0.210862i	−0.994427	−	0.105431i	$-0.966378\pi$
$283$	− 3.54724i	− 0.210862i	0.994427	−	0.105431i	$-0.0336222\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14.1903	0.837625
$288$	0	0
$289$	−4.63952	−0.272913
$290$	0	0
$291$	7.27462i	0.426446i
$292$	0	0
$293$	− 5.72538i	− 0.334480i	−0.985916	−	0.167240i	$-0.946515\pi$
$293$	− 5.72538i	− 0.334480i	0.985916	−	0.167240i	$-0.0534855\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.93012	0.228049
$298$	0	0
$299$	24.6730i	1.42688i
$300$	0	0
$301$	− 25.2417i	− 1.45491i
$302$	0	0
$303$	−4.23320	−0.243191
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.3760i	0.706335i	0.935560	+	0.353168i	$0.114895\pi$
$307$	12.3760i	0.706335i	−0.935560	+	0.353168i	$0.885105\pi$
$308$	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
$313$	12.7114	0.718491	0.359246	−	0.933243i	$-0.383034\pi$
$313$	12.7114	0.718491	0.359246	+	0.933243i	$0.383034\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 15.8602i	− 0.890800i	−0.895332	−	0.445400i	$-0.853061\pi$
$317$	− 15.8602i	− 0.890800i	0.895332	−	0.445400i	$-0.146939\pi$
$318$	0	0
$319$	14.4235	0.807559
$320$	0	0
$321$	15.4728	0.863609
$322$	0	0
$323$	19.1303i	1.06444i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	12.9561	0.716476
$328$	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
$333$	0.414376i	0.0227077i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0.860247	0.0468606	0.0234303	−	0.999725i	$-0.492541\pi$
$337$	0.860247	0.0468606	0.0234303	+	0.999725i	$0.492541\pi$
$338$	0	0
$339$	3.86025i	0.209660i
$340$	0	0
$341$	20.5708i	1.11397i
$342$	0	0
$343$	39.2293	2.11818
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 32.2856i	− 1.73318i	−0.499019	−	0.866591i	$-0.666306\pi$
$347$	− 32.2856i	− 1.73318i	0.499019	−	0.866591i	$-0.333694\pi$
$348$	0	0
$349$	− 0.742899i	− 0.0397665i	−0.999802	−	0.0198832i	$-0.993671\pi$
$349$	− 0.742899i	− 0.0397665i	0.999802	−	0.0198832i	$-0.00632945\pi$
$350$	0	0
$351$	3.46733	0.185073
$352$	0	0
$353$	32.6392	1.73721	0.868604	−	0.495506i	$-0.165018\pi$
$353$	32.6392	1.73721	0.868604	+	0.495506i	$0.165018\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	16.6046i	0.878811i
$358$	0	0
$359$	2.71056	0.143058	0.0715290	−	0.997439i	$-0.477212\pi$
$359$	2.71056	0.143058	0.0715290	+	0.997439i	$0.477212\pi$
$360$	0	0
$361$	−10.6080	−0.558317
$362$	0	0
$363$	4.44587i	0.233348i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0.680008	0.0354961	0.0177481	−	0.999842i	$-0.494350\pi$
$367$	0.680008	0.0354961	0.0177481	+	0.999842i	$0.494350\pi$
$368$	0	0
$369$	−3.00454	−0.156410
$370$	0	0
$371$	− 1.10137i	− 0.0571803i
$372$	0	0
$373$	3.47642i	0.180002i	0.995942	+	0.0900012i	$0.0286871\pi$
$373$	3.47642i	0.180002i	−0.995942	+	0.0900012i	$0.971313\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.7250	0.655373
$378$	0	0
$379$	− 23.0650i	− 1.18477i	−0.805655	−	0.592385i	$-0.798187\pi$
$379$	− 23.0650i	− 1.18477i	0.805655	−	0.592385i	$-0.201813\pi$
$380$	0	0
$381$	18.3805i	0.941664i
$382$	0	0
$383$	−9.81544	−0.501545	−0.250773	−	0.968046i	$-0.580685\pi$
$383$	−9.81544	−0.501545	−0.250773	+	0.968046i	$0.580685\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.34450i	0.271676i
$388$	0	0
$389$	− 9.86175i	− 0.500010i	−0.968244	−	0.250005i	$-0.919568\pi$
$389$	− 9.86175i	− 0.500010i	0.968244	−	0.250005i	$-0.0804323\pi$
$390$	0	0
$391$	25.0175	1.26519
$392$	0	0
$393$	−3.41892	−0.172462
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 12.7783i	− 0.641326i	−0.947193	−	0.320663i	$-0.896094\pi$
$397$	− 12.7783i	− 0.641326i	0.947193	−	0.320663i	$-0.103906\pi$
$398$	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
$403$	18.1485i	0.904041i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.62855	−0.0807243
$408$	0	0
$409$	12.1125	0.598923	0.299461	−	0.954108i	$-0.403193\pi$
$409$	12.1125	0.598923	0.299461	+	0.954108i	$0.403193\pi$
$410$	0	0
$411$	− 16.3714i	− 0.807544i
$412$	0	0
$413$	67.9184i	3.34204i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.95707	−0.0958383
$418$	0	0
$419$	− 4.24767i	− 0.207512i	−0.994603	−	0.103756i	$-0.966914\pi$
$419$	− 4.24767i	− 0.207512i	0.994603	−	0.103756i	$-0.0330862\pi$
$420$	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
$423$	0.925579	0.0450032
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.558674i	0.0270361i
$428$	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
$433$	2.03149	0.0976274	0.0488137	−	0.998808i	$-0.484456\pi$
$433$	2.03149	0.0976274	0.0488137	+	0.998808i	$0.484456\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	38.7196i	1.85221i
$438$	0	0
$439$	22.4864	1.07322	0.536608	−	0.843832i	$-0.319706\pi$
$439$	22.4864	1.07322	0.536608	+	0.843832i	$0.319706\pi$
$440$	0	0
$441$	−15.3061	−0.728863
$442$	0	0
$443$	− 3.39385i	− 0.161247i	−0.996745	−	0.0806234i	$-0.974309\pi$
$443$	− 3.39385i	− 0.161247i	0.996745	−	0.0806234i	$-0.0256911\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−12.0968	−0.572160
$448$	0	0
$449$	−2.17780	−0.102777	−0.0513883	−	0.998679i	$-0.516365\pi$
$449$	−2.17780	−0.102777	−0.0513883	+	0.998679i	$0.516365\pi$
$450$	0	0
$451$	− 11.8082i	− 0.556028i
$452$	0	0
$453$	− 4.87178i	− 0.228896i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.57653	0.167303	0.0836516	−	0.996495i	$-0.473342\pi$
$457$	3.57653	0.167303	0.0836516	+	0.996495i	$0.473342\pi$
$458$	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
$463$	−3.21417	−0.149375	−0.0746877	−	0.997207i	$-0.523796\pi$
$463$	−3.21417	−0.149375	−0.0746877	+	0.997207i	$0.523796\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.3016i	1.40219i	0.713068	+	0.701095i	$0.247305\pi$
$467$	30.3016i	1.40219i	−0.713068	+	0.701095i	$0.752695\pi$
$468$	0	0
$469$	− 63.5254i	− 2.93333i
$470$	0	0
$471$	21.6561	0.997861
$472$	0	0
$473$	−21.0045	−0.965790
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.233196i	0.0106773i
$478$	0	0
$479$	−35.0896	−1.60328	−0.801642	−	0.597804i	$-0.796040\pi$
$479$	−35.0896	−1.60328	−0.801642	+	0.597804i	$0.796040\pi$
$480$	0	0
$481$	−1.43678	−0.0655116
$482$	0	0
$483$	33.6077i	1.52920i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	36.9117	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$487$	36.9117	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$488$	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
$493$	− 12.9027i	− 0.581109i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10.3445	−0.464014
$498$	0	0
$499$	− 15.9906i	− 0.715836i	−0.933753	−	0.357918i	$-0.883487\pi$
$499$	− 15.9906i	− 0.715836i	0.933753	−	0.357918i	$-0.116513\pi$
$500$	0	0
$501$	6.69238i	0.298994i
$502$	0	0
$503$	−37.6023	−1.67660	−0.838302	−	0.545206i	$-0.816451\pi$
$503$	−37.6023	−1.67660	−0.838302	+	0.545206i	$0.816451\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 0.977595i	− 0.0434165i
$508$	0	0
$509$	− 12.8579i	− 0.569917i	−0.958540	−	0.284958i	$-0.908020\pi$
$509$	− 12.8579i	− 0.569917i	0.958540	−	0.284958i	$-0.0919797\pi$
$510$	0	0
$511$	2.66004	0.117673
$512$	0	0
$513$	5.44133	0.240240
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.63764i	0.159983i
$518$	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
$523$	− 17.4952i	− 0.765013i	−0.923953	−	0.382506i	$-0.875061\pi$
$523$	− 17.4952i	− 0.765013i	0.923953	−	0.382506i	$-0.124939\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18.4019	0.801600
$528$	0	0
$529$	27.6353	1.20153
$530$	0	0
$531$	− 14.3805i	− 0.624062i
$532$	0	0
$533$	− 10.4178i	− 0.451243i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−0.148842	−0.00642301
$538$	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
$543$	−10.3929	−0.446003
$544$	0	0
$545$	0	0
$546$	0	0
$547$	35.7406i	1.52816i	0.645124	+	0.764078i	$0.276806\pi$
$547$	35.7406i	1.52816i	−0.645124	+	0.764078i	$0.723194\pi$
$548$	0	0
$549$	− 0.118290i	− 0.00504848i
$550$	0	0
$551$	19.9695	0.850731
$552$	0	0
$553$	48.5264	2.06355
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.65516i	0.112503i	0.998417	+	0.0562514i	$0.0179148\pi$
$557$	2.65516i	0.112503i	−0.998417	+	0.0562514i	$0.982085\pi$
$558$	0	0
$559$	−18.5312	−0.783785
$560$	0	0
$561$	13.8173	0.583368
$562$	0	0
$563$	20.3107i	0.855992i	0.903781	+	0.427996i	$0.140780\pi$
$563$	20.3107i	0.855992i	−0.903781	+	0.427996i	$0.859220\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.72294	0.198345
$568$	0	0
$569$	−28.4274	−1.19174	−0.595868	−	0.803082i	$-0.703192\pi$
$569$	−28.4274	−1.19174	−0.595868	+	0.803082i	$0.703192\pi$
$570$	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
$573$	6.23320i	0.260396i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	30.3600	1.26390	0.631952	−	0.775008i	$-0.282254\pi$
$577$	30.3600	1.26390	0.631952	+	0.775008i	$0.282254\pi$
$578$	0	0
$579$	− 0.391971i	− 0.0162898i
$580$	0	0
$581$	− 53.6008i	− 2.22374i
$582$	0	0
$583$	−0.916490	−0.0379571
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.74070i	0.113121i	0.998399	+	0.0565604i	$0.0180133\pi$
$587$	2.74070i	0.113121i	−0.998399	+	0.0565604i	$0.981987\pi$
$588$	0	0
$589$	28.4807i	1.17352i
$590$	0	0
$591$	−5.96616	−0.245415
$592$	0	0
$593$	22.2586	0.914053	0.457027	−	0.889453i	$-0.348914\pi$
$593$	22.2586	0.914053	0.457027	+	0.889453i	$0.348914\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 17.9322i	− 0.733917i
$598$	0	0
$599$	−48.4526	−1.97972	−0.989860	−	0.142046i	$-0.954632\pi$
$599$	−48.4526	−1.97972	−0.989860	+	0.142046i	$0.954632\pi$
$600$	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
$603$	13.4504i	0.547743i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−7.38288	−0.299662	−0.149831	−	0.988712i	$-0.547873\pi$
$607$	−7.38288	−0.299662	−0.149831	+	0.988712i	$0.547873\pi$
$608$	0	0
$609$	17.3331	0.702371
$610$	0	0
$611$	3.20929i	0.129834i
$612$	0	0
$613$	2.64607i	0.106874i	0.998571	+	0.0534369i	$0.0170176\pi$
$613$	2.64607i	0.106874i	−0.998571	+	0.0534369i	$0.982982\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.0136	0.845977	0.422989	−	0.906135i	$-0.360981\pi$
$617$	21.0136	0.845977	0.422989	+	0.906135i	$0.360981\pi$
$618$	0	0
$619$	− 24.0874i	− 0.968154i	−0.875025	−	0.484077i	$-0.839155\pi$
$619$	− 24.0874i	− 0.968154i	0.875025	−	0.484077i	$-0.160845\pi$
$620$	0	0
$621$	− 7.11585i	− 0.285549i
$622$	0	0
$623$	41.9522	1.68078
$624$	0	0
$625$	0	0
$626$	0	0
$627$	21.3851i	0.854038i
$628$	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
$633$	6.51575	0.258978
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 53.0714i	− 2.10277i
$638$	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
$643$	− 0.636984i	− 0.0251202i	−0.999921	−	0.0125601i	$-0.996002\pi$
$643$	− 0.636984i	− 0.0251202i	0.999921	−	0.0125601i	$-0.00399811\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−32.0182	−1.25876	−0.629382	−	0.777096i	$-0.716692\pi$
$647$	−32.0182	−1.25876	−0.629382	+	0.777096i	$0.716692\pi$
$648$	0	0
$649$	56.5173	2.21850
$650$	0	0
$651$	24.7205i	0.968873i
$652$	0	0
$653$	25.9769i	1.01656i	0.861193	+	0.508278i	$0.169718\pi$
$653$	25.9769i	1.01656i	−0.861193	+	0.508278i	$0.830282\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−0.563219	−0.0219732
$658$	0	0
$659$	21.3422i	0.831372i	0.909508	+	0.415686i	$0.136459\pi$
$659$	21.3422i	0.831372i	−0.909508	+	0.415686i	$0.863541\pi$
$660$	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
$663$	12.1903	0.473431
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 26.1150i	− 1.01118i
$668$	0	0
$669$	− 3.14640i	− 0.121647i
$670$	0	0
$671$	0.464893	0.0179470
$672$	0	0
$673$	18.1167	0.698347	0.349174	−	0.937058i	$-0.386462\pi$
$673$	18.1167	0.698347	0.349174	+	0.937058i	$0.386462\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.5617i	1.17458i	0.809376	+	0.587291i	$0.199806\pi$
$677$	30.5617i	1.17458i	−0.809376	+	0.587291i	$0.800194\pi$
$678$	0	0
$679$	−34.3576	−1.31852
$680$	0	0
$681$	−3.92103	−0.150254
$682$	0	0
$683$	− 22.2027i	− 0.849564i	−0.905296	−	0.424782i	$-0.860351\pi$
$683$	− 22.2027i	− 0.849564i	0.905296	−	0.424782i	$-0.139649\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−25.6899	−0.980132
$688$	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
$693$	18.5617i	0.705101i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−10.5632	−0.400110
$698$	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
$703$	−2.25476	−0.0850398
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 19.9931i	− 0.751918i
$708$	0	0
$709$	5.65610i	0.212419i	0.994344	+	0.106210i	$0.0338715\pi$
$709$	5.65610i	0.212419i	−0.994344	+	0.106210i	$0.966129\pi$
$710$	0	0
$711$	−10.2746	−0.385328
$712$	0	0
$713$	37.2453	1.39485
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 15.8727i	− 0.592778i
$718$	0	0
$719$	20.0844	0.749020	0.374510	−	0.927223i	$-0.377811\pi$
$719$	20.0844	0.749020	0.374510	+	0.927223i	$0.377811\pi$
$720$	0	0
$721$	0.202741	0.00755048
$722$	0	0
$723$	− 28.1664i	− 1.04752i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−13.0424	−0.483715	−0.241857	−	0.970312i	$-0.577757\pi$
$727$	−13.0424	−0.483715	−0.241857	+	0.970312i	$0.577757\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	18.7899i	0.694970i
$732$	0	0
$733$	34.8917i	1.28876i	0.764707	+	0.644378i	$0.222884\pi$
$733$	34.8917i	1.28876i	−0.764707	+	0.644378i	$0.777116\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−52.8618	−1.94719
$738$	0	0
$739$	41.5040i	1.52675i	0.645957	+	0.763374i	$0.276459\pi$
$739$	41.5040i	1.52675i	−0.645957	+	0.763374i	$0.723541\pi$
$740$	0	0
$741$	18.8669i	0.693093i
$742$	0	0
$743$	−7.63764	−0.280198	−0.140099	−	0.990138i	$-0.544742\pi$
$743$	−7.63764	−0.280198	−0.140099	+	0.990138i	$0.544742\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	11.3490i	0.415240i
$748$	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
$753$	−4.66004	−0.169821
$754$	0	0
$755$	0	0
$756$	0	0
$757$	43.7959i	1.59179i	0.605436	+	0.795894i	$0.292999\pi$
$757$	43.7959i	1.59179i	−0.605436	+	0.795894i	$0.707001\pi$
$758$	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
$763$	61.1910i	2.21526i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	49.8621	1.80042
$768$	0	0
$769$	−17.9008	−0.645520	−0.322760	−	0.946481i	$-0.604611\pi$
$769$	−17.9008	−0.645520	−0.322760	+	0.946481i	$0.604611\pi$
$770$	0	0
$771$	− 3.33996i	− 0.120286i
$772$	0	0
$773$	− 20.8182i	− 0.748777i	−0.927272	−	0.374389i	$-0.877853\pi$
$773$	− 20.8182i	− 0.748777i	0.927272	−	0.374389i	$-0.122147\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.95707	−0.0702096
$778$	0	0
$779$	− 16.3487i	− 0.585753i
$780$	0	0
$781$	8.60803i	0.308019i
$782$	0	0
$783$	−3.66998	−0.131154
$784$	0	0
$785$	0	0
$786$	0	0
$787$	27.2047i	0.969744i	0.874585	+	0.484872i	$0.161134\pi$
$787$	27.2047i	0.969744i	−0.874585	+	0.484872i	$0.838866\pi$
$788$	0	0
$789$	− 27.5932i	− 0.982344i
$790$	0	0
$791$	−18.2317	−0.648244
$792$	0	0
$793$	0.410149	0.0145648
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.4344i	0.582138i	0.956702	+	0.291069i	$0.0940109\pi$
$797$	16.4344i	0.582138i	−0.956702	+	0.291069i	$0.905989\pi$
$798$	0	0
$799$	3.25410	0.115122
$800$	0	0
$801$	−8.88265	−0.313853
$802$	0	0
$803$	− 2.21352i	− 0.0781134i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	21.8727	0.769956
$808$	0	0
$809$	28.7698	1.01149	0.505747	−	0.862682i	$-0.331217\pi$
$809$	28.7698	1.01149	0.505747	+	0.862682i	$0.331217\pi$
$810$	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
$813$	− 8.78583i	− 0.308132i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−29.0812	−1.01742
$818$	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
$823$	20.3625	0.709791	0.354895	−	0.934906i	$-0.384517\pi$
$823$	20.3625	0.709791	0.354895	+	0.934906i	$0.384517\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 36.0039i	− 1.25198i	−0.779832	−	0.625989i	$-0.784696\pi$
$827$	− 36.0039i	− 1.25198i	0.779832	−	0.625989i	$-0.215304\pi$
$828$	0	0
$829$	− 7.68666i	− 0.266969i	−0.991051	−	0.133484i	$-0.957383\pi$
$829$	− 7.68666i	− 0.266969i	0.991051	−	0.133484i	$-0.0426166\pi$
$830$	0	0
$831$	10.3838	0.360211
$832$	0	0
$833$	−53.8124	−1.86449
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 5.23414i	− 0.180918i
$838$	0	0
$839$	25.3725	0.875956	0.437978	−	0.898986i	$-0.355695\pi$
$839$	25.3725	0.875956	0.437978	+	0.898986i	$0.355695\pi$
$840$	0	0
$841$	15.5313	0.535561
$842$	0	0
$843$	− 17.0584i	− 0.587524i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−20.9976	−0.721485
$848$	0	0
$849$	−3.54724	−0.121741
$850$	0	0
$851$	2.94864i	0.101078i
$852$	0	0
$853$	− 20.1365i	− 0.689460i	−0.938702	−	0.344730i	$-0.887971\pi$
$853$	− 20.1365i	− 0.689460i	0.938702	−	0.344730i	$-0.112029\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−42.0380	−1.43599	−0.717996	−	0.696047i	$-0.754940\pi$
$857$	−42.0380	−1.43599	−0.717996	+	0.696047i	$0.754940\pi$
$858$	0	0
$859$	12.4095i	0.423406i	0.977334	+	0.211703i	$0.0679010\pi$
$859$	12.4095i	0.423406i	−0.977334	+	0.211703i	$0.932099\pi$
$860$	0	0
$861$	− 14.1903i	− 0.483603i
$862$	0	0
$863$	21.8154	0.742606	0.371303	−	0.928512i	$-0.378911\pi$
$863$	21.8154	0.742606	0.371303	+	0.928512i	$0.378911\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	4.63952i	0.157566i
$868$	0	0
$869$	− 40.3805i	− 1.36982i
$870$	0	0
$871$	−46.6371	−1.58024
$872$	0	0
$873$	7.27462	0.246209
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 5.83690i	− 0.197098i	−0.995132	−	0.0985491i	$-0.968580\pi$
$877$	− 5.83690i	− 0.197098i	0.995132	−	0.0985491i	$-0.0314201\pi$
$878$	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
$883$	− 3.74810i	− 0.126134i	−0.998009	−	0.0630668i	$-0.979912\pi$
$883$	− 3.74810i	− 0.126134i	0.998009	−	0.0630668i	$-0.0200881\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.8101	1.43742	0.718711	−	0.695309i	$-0.244732\pi$
$887$	42.8101	1.43742	0.718711	+	0.695309i	$0.244732\pi$
$888$	0	0
$889$	−86.8101	−2.91152
$890$	0	0
$891$	− 3.93012i	− 0.131664i
$892$	0	0
$893$	5.03638i	0.168536i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	24.6730	0.823808
$898$	0	0
$899$	− 19.2092i	− 0.640662i
$900$	0	0
$901$	0.819859i	0.0273135i
$902$	0	0
$903$	−25.2417	−0.839992
$904$	0	0
$905$	0	0
$906$	0	0
$907$	6.97759i	0.231687i	0.993267	+	0.115844i	$0.0369571\pi$
$907$	6.97759i	0.231687i	−0.993267	+	0.115844i	$0.963043\pi$
$908$	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
$913$	−44.6031	−1.47615
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 16.1473i	− 0.533232i
$918$	0	0
$919$	1.25720	0.0414712	0.0207356	−	0.999785i	$-0.493399\pi$
$919$	1.25720	0.0414712	0.0207356	+	0.999785i	$0.493399\pi$
$920$	0	0
$921$	12.3760	0.407803
$922$	0	0
$923$	7.59440i	0.249973i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−0.0429270	−0.00140991
$928$	0	0
$929$	10.4933	0.344275	0.172138	−	0.985073i	$-0.444933\pi$
$929$	10.4933	0.344275	0.172138	+	0.985073i	$0.444933\pi$
$930$	0	0
$931$	− 83.2856i	− 2.72957i
$932$	0	0
$933$	− 18.2746i	− 0.598284i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−12.9680	−0.423648	−0.211824	−	0.977308i	$-0.567940\pi$
$937$	−12.9680	−0.423648	−0.211824	+	0.977308i	$0.567940\pi$
$938$	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
$943$	−21.3799	−0.696225
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 30.6282i	− 0.995283i	−0.867383	−	0.497642i	$-0.834199\pi$
$947$	− 30.6282i	− 0.995283i	0.867383	−	0.497642i	$-0.165801\pi$
$948$	0	0
$949$	− 1.95287i	− 0.0633927i
$950$	0	0
$951$	−15.8602	−0.514304
$952$	0	0
$953$	−17.5717	−0.569202	−0.284601	−	0.958646i	$-0.591861\pi$
$953$	−17.5717	−0.569202	−0.284601	+	0.958646i	$0.591861\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 14.4235i	− 0.466244i
$958$	0	0
$959$	77.3213	2.49683
$960$	0	0
$961$	−3.60380	−0.116252
$962$	0	0
$963$	− 15.4728i	− 0.498605i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.50932	0.145010	0.0725050	−	0.997368i	$-0.476901\pi$
$967$	4.50932	0.145010	0.0725050	+	0.997368i	$0.476901\pi$
$968$	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
$973$	− 9.24313i	− 0.296321i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	43.9704	1.40674	0.703368	−	0.710826i	$-0.251678\pi$
$977$	43.9704	1.40674	0.703368	+	0.710826i	$0.251678\pi$
$978$	0	0
$979$	− 34.9099i	− 1.11573i
$980$	0	0
$981$	− 12.9561i	− 0.413657i
$982$	0	0
$983$	−4.64187	−0.148053	−0.0740263	−	0.997256i	$-0.523585\pi$
$983$	−4.64187	−0.148053	−0.0740263	+	0.997256i	$0.523585\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.37145i	0.139145i
$988$	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
$993$	23.0315	0.730882
$994$	0	0
$995$	0	0
$996$	0	0
$997$	39.4228i	1.24853i	0.781211	+	0.624266i	$0.214602\pi$
$997$	39.4228i	1.24853i	−0.781211	+	0.624266i	$0.785398\pi$
$998$	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.k.e.1201.4		8
3.2	odd	2		7200.2.k.s.3601.8		8
4.3	odd	2		600.2.k.e.301.6	yes	8
5.2	odd	4		2400.2.d.g.49.8		8
5.3	odd	4		2400.2.d.h.49.1		8
5.4	even	2		2400.2.k.d.1201.5		8
8.3	odd	2		600.2.k.e.301.5	yes	8
8.5	even	2	inner	2400.2.k.e.1201.8		8
12.11	even	2		1800.2.k.q.901.3		8
15.2	even	4		7200.2.d.s.2449.8		8
15.8	even	4		7200.2.d.t.2449.1		8
15.14	odd	2		7200.2.k.r.3601.2		8
20.3	even	4		600.2.d.g.349.7		8
20.7	even	4		600.2.d.h.349.2		8
20.19	odd	2		600.2.k.d.301.3	✓	8
24.5	odd	2		7200.2.k.s.3601.7		8
24.11	even	2		1800.2.k.q.901.4		8
40.3	even	4		600.2.d.h.349.1		8
40.13	odd	4		2400.2.d.g.49.1		8
40.19	odd	2		600.2.k.d.301.4	yes	8
40.27	even	4		600.2.d.g.349.8		8
40.29	even	2		2400.2.k.d.1201.1		8
40.37	odd	4		2400.2.d.h.49.8		8
60.23	odd	4		1800.2.d.t.1549.2		8
60.47	odd	4		1800.2.d.s.1549.7		8
60.59	even	2		1800.2.k.t.901.6		8
120.29	odd	2		7200.2.k.r.3601.1		8
120.53	even	4		7200.2.d.s.2449.1		8
120.59	even	2		1800.2.k.t.901.5		8
120.77	even	4		7200.2.d.t.2449.8		8
120.83	odd	4		1800.2.d.s.1549.8		8
120.107	odd	4		1800.2.d.t.1549.1		8

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.k (of order \(2\), degree \(1\), not minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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