LMFDB - Embedded newform 2400.2.k.f.1201.3

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:	$12$
Coefficient field:	12.0.180227832610816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 $ x^12 + x^10 - 8x^6 + 16x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1201.3
Root			$0.806504 + 1.16170i$ of defining polynomial
Character	$\chi$	$=$	2400.1201
Dual form			2400.2.k.f.1201.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -0.746175 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -0.746175 q^{7} -1.00000 q^{9} +5.36068i q^{11} -2.92520i q^{13} +2.13466 q^{17} +1.73367i q^{19} +0.746175i q^{21} -7.49534 q^{23} +1.00000i q^{27} -6.74916i q^{29} -2.64681 q^{31} +5.36068 q^{33} +1.07480i q^{37} -2.92520 q^{39} -11.2936 q^{41} -7.44322i q^{43} -1.73367 q^{47} -6.44322 q^{49} -2.13466i q^{51} +7.72161i q^{53} +1.73367 q^{57} +6.85302i q^{59} +6.45203i q^{61} +0.746175 q^{63} -7.44322i q^{67} +7.49534i q^{69} -13.2936 q^{71} -0.690358 q^{73} -4.00000i q^{77} -2.64681 q^{79} +1.00000 q^{81} +5.85039i q^{83} -6.74916 q^{87} +7.59283 q^{89} +2.18271i q^{91} +2.64681i q^{93} -14.1887 q^{97} -5.36068i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^9	$ 12 q - 12 q^{9} + 32 q^{31} - 16 q^{39} - 8 q^{41} + 12 q^{49} - 32 q^{71} + 32 q^{79} + 12 q^{81} + 40 q^{89}+O(q^{100}) $ 12 * q - 12 * q^9 + 32 * q^31 - 16 * q^39 - 8 * q^41 + 12 * q^49 - 32 * q^71 + 32 * q^79 + 12 * q^81 + 40 * q^89

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
$7$	−0.746175	−0.282028	−0.141014	−	0.990008i	$-0.545036\pi$
$7$	−0.746175	−0.282028	−0.141014	+	0.990008i	$0.545036\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.36068i	1.61630i	0.588974	+	0.808152i	$0.299532\pi$
$11$	5.36068i	1.61630i	−0.588974	+	0.808152i	$0.700468\pi$
$12$	0	0
$13$	− 2.92520i	− 0.811304i	−0.914028	−	0.405652i	$-0.867045\pi$
$13$	− 2.92520i	− 0.811304i	0.914028	−	0.405652i	$-0.132955\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.13466	0.517731	0.258866	−	0.965913i	$-0.416651\pi$
$17$	2.13466	0.517731	0.258866	+	0.965913i	$0.416651\pi$
$18$	0	0
$19$	1.73367i	0.397730i	0.980027	+	0.198865i	$0.0637255\pi$
$19$	1.73367i	0.397730i	−0.980027	+	0.198865i	$0.936274\pi$
$20$	0	0
$21$	0.746175i	0.162829i
$22$	0	0
$23$	−7.49534	−1.56289	−0.781443	−	0.623977i	$-0.785516\pi$
$23$	−7.49534	−1.56289	−0.781443	+	0.623977i	$0.785516\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 6.74916i	− 1.25329i	−0.779306	−	0.626644i	$-0.784428\pi$
$29$	− 6.74916i	− 1.25329i	0.779306	−	0.626644i	$-0.215572\pi$
$30$	0	0
$31$	−2.64681	−0.475381	−0.237690	−	0.971341i	$-0.576390\pi$
$31$	−2.64681	−0.475381	−0.237690	+	0.971341i	$0.576390\pi$
$32$	0	0
$33$	5.36068	0.933174
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.07480i	0.176697i	0.996090	+	0.0883483i	$0.0281588\pi$
$37$	1.07480i	0.176697i	−0.996090	+	0.0883483i	$0.971841\pi$
$38$	0	0
$39$	−2.92520	−0.468406
$40$	0	0
$41$	−11.2936	−1.76377	−0.881883	−	0.471468i	$-0.843724\pi$
$41$	−11.2936	−1.76377	−0.881883	+	0.471468i	$0.843724\pi$
$42$	0	0
$43$	− 7.44322i	− 1.13508i	−0.823346	−	0.567540i	$-0.807895\pi$
$43$	− 7.44322i	− 1.13508i	0.823346	−	0.567540i	$-0.192105\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.73367	−0.252881	−0.126441	−	0.991974i	$-0.540355\pi$
$47$	−1.73367	−0.252881	−0.126441	+	0.991974i	$0.540355\pi$
$48$	0	0
$49$	−6.44322	−0.920460
$50$	0	0
$51$	− 2.13466i	− 0.298912i
$52$	0	0
$53$	7.72161i	1.06064i	0.847796	+	0.530322i	$0.177929\pi$
$53$	7.72161i	1.06064i	−0.847796	+	0.530322i	$0.822071\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.73367	0.229630
$58$	0	0
$59$	6.85302i	0.892188i	0.894986	+	0.446094i	$0.147185\pi$
$59$	6.85302i	0.892188i	−0.894986	+	0.446094i	$0.852815\pi$
$60$	0	0
$61$	6.45203i	0.826098i	0.910709	+	0.413049i	$0.135536\pi$
$61$	6.45203i	0.826098i	−0.910709	+	0.413049i	$0.864464\pi$
$62$	0	0
$63$	0.746175	0.0940092
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 7.44322i	− 0.909334i	−0.890661	−	0.454667i	$-0.849758\pi$
$67$	− 7.44322i	− 0.909334i	0.890661	−	0.454667i	$-0.150242\pi$
$68$	0	0
$69$	7.49534i	0.902332i
$70$	0	0
$71$	−13.2936	−1.57766	−0.788831	−	0.614610i	$-0.789313\pi$
$71$	−13.2936	−1.57766	−0.788831	+	0.614610i	$0.789313\pi$
$72$	0	0
$73$	−0.690358	−0.0808003	−0.0404002	−	0.999184i	$-0.512863\pi$
$73$	−0.690358	−0.0808003	−0.0404002	+	0.999184i	$0.512863\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	0	0
$79$	−2.64681	−0.297789	−0.148895	−	0.988853i	$-0.547572\pi$
$79$	−2.64681	−0.297789	−0.148895	+	0.988853i	$0.547572\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.85039i	0.642164i	0.947051	+	0.321082i	$0.104047\pi$
$83$	5.85039i	0.642164i	−0.947051	+	0.321082i	$0.895953\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.74916	−0.723586
$88$	0	0
$89$	7.59283	0.804838	0.402419	−	0.915456i	$-0.368169\pi$
$89$	7.59283	0.804838	0.402419	+	0.915456i	$0.368169\pi$
$90$	0	0
$91$	2.18271i	0.228810i
$92$	0	0
$93$	2.64681i	0.274461i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.1887	−1.44064	−0.720321	−	0.693641i	$-0.756006\pi$
$97$	−14.1887	−1.44064	−0.720321	+	0.693641i	$0.756006\pi$
$98$	0	0
$99$	− 5.36068i	− 0.538768i
$100$	0	0
$101$	7.43952i	0.740260i	0.928980	+	0.370130i	$0.120687\pi$
$101$	7.43952i	0.740260i	−0.928980	+	0.370130i	$0.879313\pi$
$102$	0	0
$103$	−7.19820	−0.709260	−0.354630	−	0.935007i	$-0.615393\pi$
$103$	−7.19820	−0.709260	−0.354630	+	0.935007i	$0.615393\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	19.9504i	1.91090i	0.295158	+	0.955449i	$0.404628\pi$
$109$	19.9504i	1.91090i	−0.295158	+	0.955449i	$0.595372\pi$
$110$	0	0
$111$	1.07480	0.102016
$112$	0	0
$113$	12.0540	1.13395	0.566973	−	0.823736i	$-0.308114\pi$
$113$	12.0540	1.13395	0.566973	+	0.823736i	$0.308114\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.92520i	0.270435i
$118$	0	0
$119$	−1.59283	−0.146014
$120$	0	0
$121$	−17.7368	−1.61244
$122$	0	0
$123$	11.2936i	1.01831i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.21351	0.373888	0.186944	−	0.982371i	$-0.440142\pi$
$127$	4.21351	0.373888	0.186944	+	0.982371i	$0.440142\pi$
$128$	0	0
$129$	−7.44322	−0.655339
$130$	0	0
$131$	10.3204i	0.901694i	0.892601	+	0.450847i	$0.148878\pi$
$131$	10.3204i	0.901694i	−0.892601	+	0.450847i	$0.851122\pi$
$132$	0	0
$133$	− 1.29362i	− 0.112171i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.0387	1.28484	0.642422	−	0.766351i	$-0.277930\pi$
$137$	15.0387	1.28484	0.642422	+	0.766351i	$0.277930\pi$
$138$	0	0
$139$	− 9.47032i	− 0.803262i	−0.915802	−	0.401631i	$-0.868443\pi$
$139$	− 9.47032i	− 0.803262i	0.915802	−	0.401631i	$-0.131557\pi$
$140$	0	0
$141$	1.73367i	0.146001i
$142$	0	0
$143$	15.6810	1.31131
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.44322i	0.531428i
$148$	0	0
$149$	− 1.78948i	− 0.146600i	−0.997310	−	0.0733000i	$-0.976647\pi$
$149$	− 1.78948i	− 0.146600i	0.997310	−	0.0733000i	$-0.0233531\pi$
$150$	0	0
$151$	−10.6468	−0.866425	−0.433212	−	0.901292i	$-0.642620\pi$
$151$	−10.6468	−0.866425	−0.433212	+	0.901292i	$0.642620\pi$
$152$	0	0
$153$	−2.13466	−0.172577
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 6.92520i	− 0.552691i	−0.961058	−	0.276345i	$-0.910877\pi$
$157$	− 6.92520i	− 0.552691i	0.961058	−	0.276345i	$-0.0891234\pi$
$158$	0	0
$159$	7.72161	0.612364
$160$	0	0
$161$	5.59283	0.440777
$162$	0	0
$163$	− 7.70079i	− 0.603172i	−0.953439	−	0.301586i	$-0.902484\pi$
$163$	− 7.70079i	− 0.603172i	0.953439	−	0.301586i	$-0.0975161\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.22601	0.249637	0.124818	−	0.992180i	$-0.460165\pi$
$167$	3.22601	0.249637	0.124818	+	0.992180i	$0.460165\pi$
$168$	0	0
$169$	4.44322	0.341786
$170$	0	0
$171$	− 1.73367i	− 0.132577i
$172$	0	0
$173$	− 6.42799i	− 0.488711i	−0.969686	−	0.244356i	$-0.921424\pi$
$173$	− 6.42799i	− 0.488711i	0.969686	−	0.244356i	$-0.0785764\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.85302	0.515105
$178$	0	0
$179$	8.13765i	0.608236i	0.952634	+	0.304118i	$0.0983618\pi$
$179$	8.13765i	0.608236i	−0.952634	+	0.304118i	$0.901638\pi$
$180$	0	0
$181$	1.49235i	0.110925i	0.998461	+	0.0554627i	$0.0176634\pi$
$181$	1.49235i	0.110925i	−0.998461	+	0.0554627i	$0.982337\pi$
$182$	0	0
$183$	6.45203	0.476948
$184$	0	0
$185$	0	0
$186$	0	0
$187$	11.4432i	0.836811i
$188$	0	0
$189$	− 0.746175i	− 0.0542762i
$190$	0	0
$191$	−6.88645	−0.498286	−0.249143	−	0.968467i	$-0.580149\pi$
$191$	−6.88645	−0.498286	−0.249143	+	0.968467i	$0.580149\pi$
$192$	0	0
$193$	−16.4830	−1.18647	−0.593237	−	0.805028i	$-0.702150\pi$
$193$	−16.4830	−1.18647	−0.593237	+	0.805028i	$0.702150\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.5720i	0.966965i	0.875354	+	0.483483i	$0.160628\pi$
$197$	13.5720i	0.966965i	−0.875354	+	0.483483i	$0.839372\pi$
$198$	0	0
$199$	−9.05398	−0.641820	−0.320910	−	0.947110i	$-0.603989\pi$
$199$	−9.05398	−0.641820	−0.320910	+	0.947110i	$0.603989\pi$
$200$	0	0
$201$	−7.44322	−0.525005
$202$	0	0
$203$	5.03605i	0.353462i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.49534	0.520962
$208$	0	0
$209$	−9.29362	−0.642853
$210$	0	0
$211$	2.53566i	0.174562i	0.996184	+	0.0872809i	$0.0278178\pi$
$211$	2.53566i	0.174562i	−0.996184	+	0.0872809i	$0.972182\pi$
$212$	0	0
$213$	13.2936i	0.910864i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.97498	0.134070
$218$	0	0
$219$	0.690358i	0.0466501i
$220$	0	0
$221$	− 6.24430i	− 0.420037i
$222$	0	0
$223$	−12.1579	−0.814152	−0.407076	−	0.913394i	$-0.633452\pi$
$223$	−12.1579	−0.814152	−0.407076	+	0.913394i	$0.633452\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.7368i	1.37635i	0.725544	+	0.688176i	$0.241588\pi$
$227$	20.7368i	1.37635i	−0.725544	+	0.688176i	$0.758412\pi$
$228$	0	0
$229$	− 19.9504i	− 1.31836i	−0.751987	−	0.659178i	$-0.770904\pi$
$229$	− 19.9504i	− 1.31836i	0.751987	−	0.659178i	$-0.229096\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	−13.3386	−0.873844	−0.436922	−	0.899499i	$-0.643931\pi$
$233$	−13.3386	−0.873844	−0.436922	+	0.899499i	$0.643931\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.64681i	0.171929i
$238$	0	0
$239$	−22.8864	−1.48040	−0.740201	−	0.672386i	$-0.765269\pi$
$239$	−22.8864	−1.48040	−0.740201	+	0.672386i	$0.765269\pi$
$240$	0	0
$241$	3.59283	0.231435	0.115717	−	0.993282i	$-0.463083\pi$
$241$	3.59283	0.231435	0.115717	+	0.993282i	$0.463083\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.07131	0.322680
$248$	0	0
$249$	5.85039	0.370754
$250$	0	0
$251$	− 8.82801i	− 0.557219i	−0.960404	−	0.278609i	$-0.910127\pi$
$251$	− 8.82801i	− 0.557219i	0.960404	−	0.278609i	$-0.0898735\pi$
$252$	0	0
$253$	− 40.1801i	− 2.52610i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−22.2927	−1.39058	−0.695291	−	0.718728i	$-0.744725\pi$
$257$	−22.2927	−1.39058	−0.695291	+	0.718728i	$0.744725\pi$
$258$	0	0
$259$	− 0.801991i	− 0.0498333i
$260$	0	0
$261$	6.74916i	0.417763i
$262$	0	0
$263$	21.2014	1.30733	0.653667	−	0.756783i	$-0.273230\pi$
$263$	21.2014	1.30733	0.653667	+	0.756783i	$0.273230\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 7.59283i	− 0.464674i
$268$	0	0
$269$	− 14.6935i	− 0.895881i	−0.894063	−	0.447940i	$-0.852158\pi$
$269$	− 14.6935i	− 0.895881i	0.894063	−	0.447940i	$-0.147842\pi$
$270$	0	0
$271$	20.2396	1.22947	0.614735	−	0.788734i	$-0.289263\pi$
$271$	20.2396	1.22947	0.614735	+	0.788734i	$0.289263\pi$
$272$	0	0
$273$	2.18271	0.132103
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 0.518027i	− 0.0311252i	−0.999879	−	0.0155626i	$-0.995046\pi$
$277$	− 0.518027i	− 0.0311252i	0.999879	−	0.0155626i	$-0.00495393\pi$
$278$	0	0
$279$	2.64681	0.158460
$280$	0	0
$281$	13.7008	0.817320	0.408660	−	0.912687i	$-0.365996\pi$
$281$	13.7008	0.817320	0.408660	+	0.912687i	$0.365996\pi$
$282$	0	0
$283$	− 18.0305i	− 1.07180i	−0.844282	−	0.535900i	$-0.819973\pi$
$283$	− 18.0305i	− 1.07180i	0.844282	−	0.535900i	$-0.180027\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.42701	0.497431
$288$	0	0
$289$	−12.4432	−0.731954
$290$	0	0
$291$	14.1887i	0.831755i
$292$	0	0
$293$	15.9792i	0.933513i	0.884386	+	0.466757i	$0.154578\pi$
$293$	15.9792i	0.933513i	−0.884386	+	0.466757i	$0.845422\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.36068	−0.311058
$298$	0	0
$299$	21.9253i	1.26797i
$300$	0	0
$301$	5.55394i	0.320124i
$302$	0	0
$303$	7.43952	0.427389
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 22.5872i	− 1.28912i	−0.764553	−	0.644561i	$-0.777040\pi$
$307$	− 22.5872i	− 1.28912i	0.764553	−	0.644561i	$-0.222960\pi$
$308$	0	0
$309$	7.19820i	0.409492i
$310$	0	0
$311$	18.5872	1.05399	0.526993	−	0.849870i	$-0.323320\pi$
$311$	18.5872	1.05399	0.526993	+	0.849870i	$0.323320\pi$
$312$	0	0
$313$	−29.3871	−1.66106	−0.830528	−	0.556977i	$-0.811961\pi$
$313$	−29.3871	−1.66106	−0.830528	+	0.556977i	$0.811961\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.57201i	0.312955i	0.987682	+	0.156478i	$0.0500139\pi$
$317$	5.57201i	0.312955i	−0.987682	+	0.156478i	$0.949986\pi$
$318$	0	0
$319$	36.1801	2.02569
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	3.70079i	0.205917i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	19.9504	1.10326
$328$	0	0
$329$	1.29362	0.0713194
$330$	0	0
$331$	13.7396i	0.755199i	0.925969	+	0.377599i	$0.123250\pi$
$331$	13.7396i	0.755199i	−0.925969	+	0.377599i	$0.876750\pi$
$332$	0	0
$333$	− 1.07480i	− 0.0588989i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	20.7523	1.13045	0.565226	−	0.824936i	$-0.308789\pi$
$337$	20.7523	1.13045	0.565226	+	0.824936i	$0.308789\pi$
$338$	0	0
$339$	− 12.0540i	− 0.654685i
$340$	0	0
$341$	− 14.1887i	− 0.768360i
$342$	0	0
$343$	10.0310	0.541623
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4.73684i	− 0.254287i	−0.991884	−	0.127143i	$-0.959419\pi$
$347$	− 4.73684i	− 0.254287i	0.991884	−	0.127143i	$-0.0405809\pi$
$348$	0	0
$349$	− 0.482632i	− 0.0258347i	−0.999917	−	0.0129174i	$-0.995888\pi$
$349$	− 0.482632i	− 0.0258347i	0.999917	−	0.0129174i	$-0.00411184\pi$
$350$	0	0
$351$	2.92520	0.156135
$352$	0	0
$353$	−2.13466	−0.113617	−0.0568083	−	0.998385i	$-0.518092\pi$
$353$	−2.13466	−0.113617	−0.0568083	+	0.998385i	$0.518092\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.59283i	0.0843015i
$358$	0	0
$359$	9.59283	0.506290	0.253145	−	0.967428i	$-0.418535\pi$
$359$	9.59283	0.506290	0.253145	+	0.967428i	$0.418535\pi$
$360$	0	0
$361$	15.9944	0.841811
$362$	0	0
$363$	17.7368i	0.930943i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−34.0832	−1.77913	−0.889565	−	0.456809i	$-0.848992\pi$
$367$	−34.0832	−1.77913	−0.889565	+	0.456809i	$0.848992\pi$
$368$	0	0
$369$	11.2936	0.587922
$370$	0	0
$371$	− 5.76167i	− 0.299131i
$372$	0	0
$373$	4.33796i	0.224611i	0.993674	+	0.112306i	$0.0358236\pi$
$373$	4.33796i	0.224611i	−0.993674	+	0.112306i	$0.964176\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−19.7426	−1.01680
$378$	0	0
$379$	− 6.90107i	− 0.354484i	−0.984167	−	0.177242i	$-0.943282\pi$
$379$	− 6.90107i	− 0.354484i	0.984167	−	0.177242i	$-0.0567176\pi$
$380$	0	0
$381$	− 4.21351i	− 0.215864i
$382$	0	0
$383$	−22.3744	−1.14328	−0.571639	−	0.820506i	$-0.693692\pi$
$383$	−22.3744	−1.14328	−0.571639	+	0.820506i	$0.693692\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.44322i	0.378360i
$388$	0	0
$389$	11.0185i	0.558659i	0.960195	+	0.279330i	$0.0901122\pi$
$389$	11.0185i	0.558659i	−0.960195	+	0.279330i	$0.909888\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	0	0
$393$	10.3204	0.520593
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 25.2549i	− 1.26751i	−0.773536	−	0.633753i	$-0.781514\pi$
$397$	− 25.2549i	− 1.26751i	0.773536	−	0.633753i	$-0.218486\pi$
$398$	0	0
$399$	−1.29362	−0.0647619
$400$	0	0
$401$	7.29362	0.364226	0.182113	−	0.983278i	$-0.441706\pi$
$401$	7.29362	0.364226	0.182113	+	0.983278i	$0.441706\pi$
$402$	0	0
$403$	7.74244i	0.385678i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.76167	−0.285595
$408$	0	0
$409$	15.8504	0.783752	0.391876	−	0.920018i	$-0.371826\pi$
$409$	15.8504	0.783752	0.391876	+	0.920018i	$0.371826\pi$
$410$	0	0
$411$	− 15.0387i	− 0.741805i
$412$	0	0
$413$	− 5.11355i	− 0.251622i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−9.47032	−0.463763
$418$	0	0
$419$	− 8.02602i	− 0.392097i	−0.980594	−	0.196048i	$-0.937189\pi$
$419$	− 8.02602i	− 0.392097i	0.980594	−	0.196048i	$-0.0628109\pi$
$420$	0	0
$421$	− 22.9351i	− 1.11779i	−0.829240	−	0.558893i	$-0.811226\pi$
$421$	− 22.9351i	− 1.11779i	0.829240	−	0.558893i	$-0.188774\pi$
$422$	0	0
$423$	1.73367	0.0842937
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 4.81434i	− 0.232982i
$428$	0	0
$429$	− 15.6810i	− 0.757087i
$430$	0	0
$431$	35.0665	1.68909	0.844547	−	0.535481i	$-0.179870\pi$
$431$	35.0665	1.68909	0.844547	+	0.535481i	$0.179870\pi$
$432$	0	0
$433$	17.0773	0.820682	0.410341	−	0.911932i	$-0.365410\pi$
$433$	17.0773	0.820682	0.410341	+	0.911932i	$0.365410\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 12.9944i	− 0.621607i
$438$	0	0
$439$	−8.53885	−0.407537	−0.203769	−	0.979019i	$-0.565319\pi$
$439$	−8.53885	−0.407537	−0.203769	+	0.979019i	$0.565319\pi$
$440$	0	0
$441$	6.44322	0.306820
$442$	0	0
$443$	20.7368i	0.985237i	0.870245	+	0.492619i	$0.163960\pi$
$443$	20.7368i	0.985237i	−0.870245	+	0.492619i	$0.836040\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1.78948	−0.0846396
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	− 60.5414i	− 2.85078i
$452$	0	0
$453$	10.6468i	0.500231i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.28462	0.0600921	0.0300461	−	0.999549i	$-0.490435\pi$
$457$	1.28462	0.0600921	0.0300461	+	0.999549i	$0.490435\pi$
$458$	0	0
$459$	2.13466i	0.0996374i
$460$	0	0
$461$	− 15.7033i	− 0.731374i	−0.930738	−	0.365687i	$-0.880834\pi$
$461$	− 15.7033i	− 0.731374i	0.930738	−	0.365687i	$-0.119166\pi$
$462$	0	0
$463$	−18.7215	−0.870064	−0.435032	−	0.900415i	$-0.643263\pi$
$463$	−18.7215	−0.870064	−0.435032	+	0.900415i	$0.643263\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 2.14961i	− 0.0994719i	−0.998762	−	0.0497360i	$-0.984162\pi$
$467$	− 2.14961i	− 0.0994719i	0.998762	−	0.0497360i	$-0.0158380\pi$
$468$	0	0
$469$	5.55394i	0.256457i
$470$	0	0
$471$	−6.92520	−0.319096
$472$	0	0
$473$	39.9007	1.83464
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 7.72161i	− 0.353548i
$478$	0	0
$479$	12.1801	0.556521	0.278261	−	0.960506i	$-0.410242\pi$
$479$	12.1801	0.556521	0.278261	+	0.960506i	$0.410242\pi$
$480$	0	0
$481$	3.14401	0.143355
$482$	0	0
$483$	− 5.59283i	− 0.254483i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−25.7678	−1.16765	−0.583826	−	0.811879i	$-0.698445\pi$
$487$	−25.7678	−1.16765	−0.583826	+	0.811879i	$0.698445\pi$
$488$	0	0
$489$	−7.70079	−0.348242
$490$	0	0
$491$	− 16.7724i	− 0.756927i	−0.925616	−	0.378464i	$-0.876453\pi$
$491$	− 16.7724i	− 0.756927i	0.925616	−	0.378464i	$-0.123547\pi$
$492$	0	0
$493$	− 14.4072i	− 0.648866i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.91936	0.444944
$498$	0	0
$499$	17.6224i	0.788888i	0.918920	+	0.394444i	$0.129063\pi$
$499$	17.6224i	0.788888i	−0.918920	+	0.394444i	$0.870937\pi$
$500$	0	0
$501$	− 3.22601i	− 0.144128i
$502$	0	0
$503$	27.1263	1.20950	0.604752	−	0.796414i	$-0.293272\pi$
$503$	27.1263	1.20950	0.604752	+	0.796414i	$0.293272\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 4.44322i	− 0.197330i
$508$	0	0
$509$	15.9782i	0.708220i	0.935204	+	0.354110i	$0.115216\pi$
$509$	15.9782i	0.708220i	−0.935204	+	0.354110i	$0.884784\pi$
$510$	0	0
$511$	0.515128	0.0227879
$512$	0	0
$513$	−1.73367	−0.0765432
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 9.29362i	− 0.408733i
$518$	0	0
$519$	−6.42799	−0.282158
$520$	0	0
$521$	0.886447	0.0388359	0.0194180	−	0.999811i	$-0.493819\pi$
$521$	0.886447	0.0388359	0.0194180	+	0.999811i	$0.493819\pi$
$522$	0	0
$523$	− 41.7729i	− 1.82660i	−0.407286	−	0.913301i	$-0.633525\pi$
$523$	− 41.7729i	− 1.82660i	0.407286	−	0.913301i	$-0.366475\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.65004	−0.246120
$528$	0	0
$529$	33.1801	1.44261
$530$	0	0
$531$	− 6.85302i	− 0.297396i
$532$	0	0
$533$	33.0361i	1.43095i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.13765	0.351165
$538$	0	0
$539$	− 34.5400i	− 1.48774i
$540$	0	0
$541$	4.47705i	0.192483i	0.995358	+	0.0962417i	$0.0306822\pi$
$541$	4.47705i	0.192483i	−0.995358	+	0.0962417i	$0.969318\pi$
$542$	0	0
$543$	1.49235	0.0640428
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 14.3297i	− 0.612692i	−0.951920	−	0.306346i	$-0.900893\pi$
$547$	− 14.3297i	− 0.612692i	0.951920	−	0.306346i	$-0.0991065\pi$
$548$	0	0
$549$	− 6.45203i	− 0.275366i
$550$	0	0
$551$	11.7008	0.498470
$552$	0	0
$553$	1.97498	0.0839848
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.68556i	− 0.113791i	−0.998380	−	0.0568954i	$-0.981880\pi$
$557$	− 2.68556i	− 0.113791i	0.998380	−	0.0568954i	$-0.0181201\pi$
$558$	0	0
$559$	−21.7729	−0.920895
$560$	0	0
$561$	11.4432	0.483133
$562$	0	0
$563$	− 20.7368i	− 0.873954i	−0.899473	−	0.436977i	$-0.856049\pi$
$563$	− 20.7368i	− 0.873954i	0.899473	−	0.436977i	$-0.143951\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−0.746175	−0.0313364
$568$	0	0
$569$	4.40717	0.184758	0.0923791	−	0.995724i	$-0.470553\pi$
$569$	4.40717	0.184758	0.0923791	+	0.995724i	$0.470553\pi$
$570$	0	0
$571$	23.6590i	0.990098i	0.868865	+	0.495049i	$0.164850\pi$
$571$	23.6590i	0.990098i	−0.868865	+	0.495049i	$0.835150\pi$
$572$	0	0
$573$	6.88645i	0.287685i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−6.56366	−0.273249	−0.136624	−	0.990623i	$-0.543625\pi$
$577$	−6.56366	−0.273249	−0.136624	+	0.990623i	$0.543625\pi$
$578$	0	0
$579$	16.4830i	0.685011i
$580$	0	0
$581$	− 4.36542i	− 0.181108i
$582$	0	0
$583$	−41.3931	−1.71433
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 16.2992i	− 0.672741i	−0.941730	−	0.336370i	$-0.890801\pi$
$587$	− 16.2992i	− 0.672741i	0.941730	−	0.336370i	$-0.109199\pi$
$588$	0	0
$589$	− 4.58868i	− 0.189073i
$590$	0	0
$591$	13.5720	0.558278
$592$	0	0
$593$	−16.3233	−0.670319	−0.335160	−	0.942161i	$-0.608790\pi$
$593$	−16.3233	−0.670319	−0.335160	+	0.942161i	$0.608790\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.05398i	0.370555i
$598$	0	0
$599$	25.5928	1.04569	0.522847	−	0.852426i	$-0.324870\pi$
$599$	25.5928	1.04569	0.522847	+	0.852426i	$0.324870\pi$
$600$	0	0
$601$	29.9225	1.22056	0.610282	−	0.792184i	$-0.291056\pi$
$601$	29.9225	1.22056	0.610282	+	0.792184i	$0.291056\pi$
$602$	0	0
$603$	7.44322i	0.303111i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	20.6965	0.840046	0.420023	−	0.907513i	$-0.362022\pi$
$607$	20.6965	0.840046	0.420023	+	0.907513i	$0.362022\pi$
$608$	0	0
$609$	5.03605	0.204071
$610$	0	0
$611$	5.07131i	0.205163i
$612$	0	0
$613$	22.6676i	0.915537i	0.889071	+	0.457769i	$0.151351\pi$
$613$	22.6676i	0.915537i	−0.889071	+	0.457769i	$0.848649\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.1966	0.893603	0.446802	−	0.894633i	$-0.352563\pi$
$617$	22.1966	0.893603	0.446802	+	0.894633i	$0.352563\pi$
$618$	0	0
$619$	− 16.8204i	− 0.676070i	−0.941133	−	0.338035i	$-0.890238\pi$
$619$	− 16.8204i	− 0.676070i	0.941133	−	0.338035i	$-0.109762\pi$
$620$	0	0
$621$	− 7.49534i	− 0.300777i
$622$	0	0
$623$	−5.66558	−0.226987
$624$	0	0
$625$	0	0
$626$	0	0
$627$	9.29362i	0.371151i
$628$	0	0
$629$	2.29434i	0.0914813i
$630$	0	0
$631$	44.1205	1.75641	0.878204	−	0.478285i	$-0.158742\pi$
$631$	44.1205	1.75641	0.878204	+	0.478285i	$0.158742\pi$
$632$	0	0
$633$	2.53566	0.100783
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18.8477i	0.746773i
$638$	0	0
$639$	13.2936	0.525887
$640$	0	0
$641$	−1.18566	−0.0468307	−0.0234154	−	0.999726i	$-0.507454\pi$
$641$	−1.18566	−0.0468307	−0.0234154	+	0.999726i	$0.507454\pi$
$642$	0	0
$643$	22.5872i	0.890754i	0.895343	+	0.445377i	$0.146930\pi$
$643$	22.5872i	0.890754i	−0.895343	+	0.445377i	$0.853070\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.7090	0.774842	0.387421	−	0.921903i	$-0.373366\pi$
$647$	19.7090	0.774842	0.387421	+	0.921903i	$0.373366\pi$
$648$	0	0
$649$	−36.7368	−1.44205
$650$	0	0
$651$	− 1.97498i	− 0.0774056i
$652$	0	0
$653$	− 44.4585i	− 1.73979i	−0.493234	−	0.869897i	$-0.664185\pi$
$653$	− 44.4585i	− 1.73979i	0.493234	−	0.869897i	$-0.335815\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.690358	0.0269334
$658$	0	0
$659$	− 41.5863i	− 1.61997i	−0.586448	−	0.809987i	$-0.699474\pi$
$659$	− 41.5863i	− 1.61997i	0.586448	−	0.809987i	$-0.300526\pi$
$660$	0	0
$661$	12.0060i	0.466978i	0.972359	+	0.233489i	$0.0750143\pi$
$661$	12.0060i	0.466978i	−0.972359	+	0.233489i	$0.924986\pi$
$662$	0	0
$663$	−6.24430	−0.242509
$664$	0	0
$665$	0	0
$666$	0	0
$667$	50.5872i	1.95875i
$668$	0	0
$669$	12.1579i	0.470051i
$670$	0	0
$671$	−34.5872	−1.33523
$672$	0	0
$673$	14.5080	0.559244	0.279622	−	0.960110i	$-0.409791\pi$
$673$	14.5080	0.559244	0.279622	+	0.960110i	$0.409791\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	43.8600i	1.68568i	0.538166	+	0.842839i	$0.319117\pi$
$677$	43.8600i	1.68568i	−0.538166	+	0.842839i	$0.680883\pi$
$678$	0	0
$679$	10.5872	0.406301
$680$	0	0
$681$	20.7368	0.794637
$682$	0	0
$683$	− 5.33527i	− 0.204148i	−0.994777	−	0.102074i	$-0.967452\pi$
$683$	− 5.33527i	− 0.204148i	0.994777	−	0.102074i	$-0.0325479\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−19.9504	−0.761153
$688$	0	0
$689$	22.5872	0.860505
$690$	0	0
$691$	39.7710i	1.51296i	0.654016	+	0.756480i	$0.273083\pi$
$691$	39.7710i	1.51296i	−0.654016	+	0.756480i	$0.726917\pi$
$692$	0	0
$693$	4.00000i	0.151947i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−24.1080	−0.913157
$698$	0	0
$699$	13.3386i	0.504514i
$700$	0	0
$701$	27.5015i	1.03872i	0.854556	+	0.519359i	$0.173829\pi$
$701$	27.5015i	1.03872i	−0.854556	+	0.519359i	$0.826171\pi$
$702$	0	0
$703$	−1.86335	−0.0702775
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 5.55118i	− 0.208774i
$708$	0	0
$709$	− 0.111632i	− 0.00419244i	−0.999998	−	0.00209622i	$-0.999333\pi$
$709$	− 0.111632i	− 0.00419244i	0.999998	−	0.00209622i	$-0.000667249\pi$
$710$	0	0
$711$	2.64681	0.0992631
$712$	0	0
$713$	19.8387	0.742966
$714$	0	0
$715$	0	0
$716$	0	0
$717$	22.8864i	0.854710i
$718$	0	0
$719$	−10.7064	−0.399281	−0.199640	−	0.979869i	$-0.563977\pi$
$719$	−10.7064	−0.399281	−0.199640	+	0.979869i	$0.563977\pi$
$720$	0	0
$721$	5.37112	0.200031
$722$	0	0
$723$	− 3.59283i	− 0.133619i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	25.6562	0.951536	0.475768	−	0.879571i	$-0.342170\pi$
$727$	25.6562	0.951536	0.475768	+	0.879571i	$0.342170\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 15.8888i	− 0.587667i
$732$	0	0
$733$	− 30.3684i	− 1.12168i	−0.827923	−	0.560842i	$-0.810478\pi$
$733$	− 30.3684i	− 1.12168i	0.827923	−	0.560842i	$-0.189522\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	39.9007	1.46976
$738$	0	0
$739$	20.1917i	0.742763i	0.928480	+	0.371381i	$0.121116\pi$
$739$	20.1917i	0.742763i	−0.928480	+	0.371381i	$0.878884\pi$
$740$	0	0
$741$	− 5.07131i	− 0.186299i
$742$	0	0
$743$	−46.3863	−1.70175	−0.850875	−	0.525369i	$-0.823927\pi$
$743$	−46.3863	−1.70175	−0.850875	+	0.525369i	$0.823927\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 5.85039i	− 0.214055i
$748$	0	0
$749$	2.98470i	0.109059i
$750$	0	0
$751$	27.1261	0.989845	0.494922	−	0.868937i	$-0.335196\pi$
$751$	27.1261	0.989845	0.494922	+	0.868937i	$0.335196\pi$
$752$	0	0
$753$	−8.82801	−0.321710
$754$	0	0
$755$	0	0
$756$	0	0
$757$	45.2549i	1.64482i	0.568898	+	0.822408i	$0.307370\pi$
$757$	45.2549i	1.64482i	−0.568898	+	0.822408i	$0.692630\pi$
$758$	0	0
$759$	−40.1801	−1.45844
$760$	0	0
$761$	16.8864	0.612133	0.306067	−	0.952010i	$-0.400987\pi$
$761$	16.8864	0.612133	0.306067	+	0.952010i	$0.400987\pi$
$762$	0	0
$763$	− 14.8864i	− 0.538926i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	20.0464	0.723835
$768$	0	0
$769$	16.3297	0.588863	0.294431	−	0.955673i	$-0.404870\pi$
$769$	16.3297	0.588863	0.294431	+	0.955673i	$0.404870\pi$
$770$	0	0
$771$	22.2927i	0.802853i
$772$	0	0
$773$	41.3144i	1.48598i	0.669304	+	0.742989i	$0.266592\pi$
$773$	41.3144i	1.48598i	−0.669304	+	0.742989i	$0.733408\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.801991	−0.0287713
$778$	0	0
$779$	− 19.5794i	− 0.701503i
$780$	0	0
$781$	− 71.2628i	− 2.54998i
$782$	0	0
$783$	6.74916	0.241195
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 11.4849i	− 0.409391i	−0.978826	−	0.204696i	$-0.934380\pi$
$787$	− 11.4849i	− 0.409391i	0.978826	−	0.204696i	$-0.0656205\pi$
$788$	0	0
$789$	− 21.2014i	− 0.754789i
$790$	0	0
$791$	−8.99440	−0.319804
$792$	0	0
$793$	18.8735	0.670216
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 45.4945i	− 1.61150i	−0.592257	−	0.805749i	$-0.701763\pi$
$797$	− 45.4945i	− 1.61150i	0.592257	−	0.805749i	$-0.298237\pi$
$798$	0	0
$799$	−3.70079	−0.130924
$800$	0	0
$801$	−7.59283	−0.268279
$802$	0	0
$803$	− 3.70079i	− 0.130598i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−14.6935	−0.517237
$808$	0	0
$809$	−36.0721	−1.26823	−0.634114	−	0.773240i	$-0.718635\pi$
$809$	−36.0721	−1.26823	−0.634114	+	0.773240i	$0.718635\pi$
$810$	0	0
$811$	− 44.5230i	− 1.56341i	−0.623646	−	0.781707i	$-0.714349\pi$
$811$	− 44.5230i	− 1.56341i	0.623646	−	0.781707i	$-0.285651\pi$
$812$	0	0
$813$	− 20.2396i	− 0.709835i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	12.9041	0.451456
$818$	0	0
$819$	− 2.18271i	− 0.0762700i
$820$	0	0
$821$	34.1613i	1.19224i	0.802897	+	0.596118i	$0.203291\pi$
$821$	34.1613i	1.19224i	−0.802897	+	0.596118i	$0.796709\pi$
$822$	0	0
$823$	2.12689	0.0741388	0.0370694	−	0.999313i	$-0.488198\pi$
$823$	2.12689	0.0741388	0.0370694	+	0.999313i	$0.488198\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38.5872i	1.34181i	0.741543	+	0.670905i	$0.234094\pi$
$827$	38.5872i	1.34181i	−0.741543	+	0.670905i	$0.765906\pi$
$828$	0	0
$829$	− 34.2351i	− 1.18904i	−0.804083	−	0.594518i	$-0.797343\pi$
$829$	− 34.2351i	− 1.18904i	0.804083	−	0.594518i	$-0.202657\pi$
$830$	0	0
$831$	−0.518027	−0.0179701
$832$	0	0
$833$	−13.7541	−0.476551
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2.64681i	− 0.0914871i
$838$	0	0
$839$	41.5928	1.43594	0.717972	−	0.696072i	$-0.245071\pi$
$839$	41.5928	1.43594	0.717972	+	0.696072i	$0.245071\pi$
$840$	0	0
$841$	−16.5512	−0.570730
$842$	0	0
$843$	− 13.7008i	− 0.471880i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	13.2348	0.454752
$848$	0	0
$849$	−18.0305	−0.618804
$850$	0	0
$851$	− 8.05601i	− 0.276156i
$852$	0	0
$853$	23.1828i	0.793763i	0.917870	+	0.396881i	$0.129908\pi$
$853$	23.1828i	0.793763i	−0.917870	+	0.396881i	$0.870092\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.38868	0.320711	0.160356	−	0.987059i	$-0.448736\pi$
$857$	9.38868	0.320711	0.160356	+	0.987059i	$0.448736\pi$
$858$	0	0
$859$	8.98769i	0.306656i	0.988175	+	0.153328i	$0.0489991\pi$
$859$	8.98769i	0.306656i	−0.988175	+	0.153328i	$0.951001\pi$
$860$	0	0
$861$	− 8.42701i	− 0.287192i
$862$	0	0
$863$	12.2473	0.416903	0.208451	−	0.978033i	$-0.433158\pi$
$863$	12.2473	0.416903	0.208451	+	0.978033i	$0.433158\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	12.4432i	0.422594i
$868$	0	0
$869$	− 14.1887i	− 0.481318i
$870$	0	0
$871$	−21.7729	−0.737746
$872$	0	0
$873$	14.1887	0.480214
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 26.1109i	− 0.881701i	−0.897581	−	0.440850i	$-0.854677\pi$
$877$	− 26.1109i	− 0.881701i	0.897581	−	0.440850i	$-0.145323\pi$
$878$	0	0
$879$	15.9792	0.538964
$880$	0	0
$881$	−38.4793	−1.29640	−0.648200	−	0.761470i	$-0.724478\pi$
$881$	−38.4793	−1.29640	−0.648200	+	0.761470i	$0.724478\pi$
$882$	0	0
$883$	6.58723i	0.221678i	0.993838	+	0.110839i	$0.0353538\pi$
$883$	6.58723i	0.221678i	−0.993838	+	0.110839i	$0.964646\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−50.9595	−1.71105	−0.855526	−	0.517760i	$-0.826766\pi$
$887$	−50.9595	−1.71105	−0.855526	+	0.517760i	$0.826766\pi$
$888$	0	0
$889$	−3.14401	−0.105447
$890$	0	0
$891$	5.36068i	0.179589i
$892$	0	0
$893$	− 3.00560i	− 0.100578i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	21.9253	0.732066
$898$	0	0
$899$	17.8637i	0.595789i
$900$	0	0
$901$	16.4830i	0.549129i
$902$	0	0
$903$	5.55394	0.184824
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 24.5568i	− 0.815394i	−0.913117	−	0.407697i	$-0.866332\pi$
$907$	− 24.5568i	− 0.815394i	0.913117	−	0.407697i	$-0.133668\pi$
$908$	0	0
$909$	− 7.43952i	− 0.246753i
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	0	0
$913$	−31.3621	−1.03793
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 7.70079i	− 0.254302i
$918$	0	0
$919$	−28.7548	−0.948532	−0.474266	−	0.880382i	$-0.657287\pi$
$919$	−28.7548	−0.948532	−0.474266	+	0.880382i	$0.657287\pi$
$920$	0	0
$921$	−22.5872	−0.744275
$922$	0	0
$923$	38.8864i	1.27996i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	7.19820	0.236420
$928$	0	0
$929$	28.2880	0.928100	0.464050	−	0.885809i	$-0.346396\pi$
$929$	28.2880	0.928100	0.464050	+	0.885809i	$0.346396\pi$
$930$	0	0
$931$	− 11.1704i	− 0.366095i
$932$	0	0
$933$	− 18.5872i	− 0.608519i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−33.9313	−1.10849	−0.554244	−	0.832354i	$-0.686992\pi$
$937$	−33.9313	−1.10849	−0.554244	+	0.832354i	$0.686992\pi$
$938$	0	0
$939$	29.3871i	0.959011i
$940$	0	0
$941$	38.8016i	1.26490i	0.774603	+	0.632448i	$0.217950\pi$
$941$	38.8016i	1.26490i	−0.774603	+	0.632448i	$0.782050\pi$
$942$	0	0
$943$	84.6495	2.75657
$944$	0	0
$945$	0	0
$946$	0	0
$947$	17.7729i	0.577541i	0.957398	+	0.288771i	$0.0932465\pi$
$947$	17.7729i	0.577541i	−0.957398	+	0.288771i	$0.906753\pi$
$948$	0	0
$949$	2.01943i	0.0655536i
$950$	0	0
$951$	5.57201	0.180685
$952$	0	0
$953$	−46.3047	−1.49996	−0.749978	−	0.661463i	$-0.769936\pi$
$953$	−46.3047	−1.49996	−0.749978	+	0.661463i	$0.769936\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 36.1801i	− 1.16954i
$958$	0	0
$959$	−11.2215	−0.362361
$960$	0	0
$961$	−23.9944	−0.774013
$962$	0	0
$963$	4.00000i	0.128898i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−9.28482	−0.298580	−0.149290	−	0.988793i	$-0.547699\pi$
$967$	−9.28482	−0.298580	−0.149290	+	0.988793i	$0.547699\pi$
$968$	0	0
$969$	3.70079	0.118886
$970$	0	0
$971$	20.9301i	0.671678i	0.941919	+	0.335839i	$0.109020\pi$
$971$	20.9301i	0.671678i	−0.941919	+	0.335839i	$0.890980\pi$
$972$	0	0
$973$	7.06651i	0.226542i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30.8314	−0.986383	−0.493192	−	0.869921i	$-0.664170\pi$
$977$	−30.8314	−0.986383	−0.493192	+	0.869921i	$0.664170\pi$
$978$	0	0
$979$	40.7027i	1.30086i
$980$	0	0
$981$	− 19.9504i	− 0.636966i
$982$	0	0
$983$	−31.3285	−0.999223	−0.499612	−	0.866250i	$-0.666524\pi$
$983$	−31.3285	−0.999223	−0.499612	+	0.866250i	$0.666524\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 1.29362i	− 0.0411763i
$988$	0	0
$989$	55.7895i	1.77400i
$990$	0	0
$991$	−31.3420	−0.995611	−0.497806	−	0.867289i	$-0.665861\pi$
$991$	−31.3420	−0.995611	−0.497806	+	0.867289i	$0.665861\pi$
$992$	0	0
$993$	13.7396	0.436014
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 20.9557i	− 0.663672i	−0.943337	−	0.331836i	$-0.892332\pi$
$997$	− 20.9557i	− 0.663672i	0.943337	−	0.331836i	$-0.107668\pi$
$998$	0	0
$999$	−1.07480	−0.0340053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.k.f.1201.3		12
3.2	odd	2		7200.2.k.u.3601.5		12
4.3	odd	2		600.2.k.f.301.9		12
5.2	odd	4		480.2.d.a.49.1		6
5.3	odd	4		480.2.d.b.49.5		6
5.4	even	2	inner	2400.2.k.f.1201.10		12
8.3	odd	2		600.2.k.f.301.10		12
8.5	even	2	inner	2400.2.k.f.1201.9		12
12.11	even	2		1800.2.k.u.901.4		12
15.2	even	4		1440.2.d.e.1009.6		6
15.8	even	4		1440.2.d.f.1009.2		6
15.14	odd	2		7200.2.k.u.3601.7		12
20.3	even	4		120.2.d.b.109.1	yes	6
20.7	even	4		120.2.d.a.109.6	yes	6
20.19	odd	2		600.2.k.f.301.4		12
24.5	odd	2		7200.2.k.u.3601.6		12
24.11	even	2		1800.2.k.u.901.3		12
40.3	even	4		120.2.d.a.109.5	✓	6
40.13	odd	4		480.2.d.a.49.2		6
40.19	odd	2		600.2.k.f.301.3		12
40.27	even	4		120.2.d.b.109.2	yes	6
40.29	even	2	inner	2400.2.k.f.1201.4		12
40.37	odd	4		480.2.d.b.49.6		6
60.23	odd	4		360.2.d.e.109.6		6
60.47	odd	4		360.2.d.f.109.1		6
60.59	even	2		1800.2.k.u.901.9		12
80.3	even	4		3840.2.f.l.769.2		12
80.13	odd	4		3840.2.f.m.769.8		12
80.27	even	4		3840.2.f.l.769.5		12
80.37	odd	4		3840.2.f.m.769.11		12
80.43	even	4		3840.2.f.l.769.11		12
80.53	odd	4		3840.2.f.m.769.5		12
80.67	even	4		3840.2.f.l.769.8		12
80.77	odd	4		3840.2.f.m.769.2		12
120.29	odd	2		7200.2.k.u.3601.8		12
120.53	even	4		1440.2.d.e.1009.5		6
120.59	even	2		1800.2.k.u.901.10		12
120.77	even	4		1440.2.d.f.1009.1		6
120.83	odd	4		360.2.d.f.109.2		6
120.107	odd	4		360.2.d.e.109.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.d.a.109.5	✓	6	40.3	even	4
120.2.d.a.109.6	yes	6	20.7	even	4
120.2.d.b.109.1	yes	6	20.3	even	4
120.2.d.b.109.2	yes	6	40.27	even	4
360.2.d.e.109.5		6	120.107	odd	4
360.2.d.e.109.6		6	60.23	odd	4
360.2.d.f.109.1		6	60.47	odd	4
360.2.d.f.109.2		6	120.83	odd	4
480.2.d.a.49.1		6	5.2	odd	4
480.2.d.a.49.2		6	40.13	odd	4
480.2.d.b.49.5		6	5.3	odd	4
480.2.d.b.49.6		6	40.37	odd	4
600.2.k.f.301.3		12	40.19	odd	2
600.2.k.f.301.4		12	20.19	odd	2
600.2.k.f.301.9		12	4.3	odd	2
600.2.k.f.301.10		12	8.3	odd	2
1440.2.d.e.1009.5		6	120.53	even	4
1440.2.d.e.1009.6		6	15.2	even	4
1440.2.d.f.1009.1		6	120.77	even	4
1440.2.d.f.1009.2		6	15.8	even	4
1800.2.k.u.901.3		12	24.11	even	2
1800.2.k.u.901.4		12	12.11	even	2
1800.2.k.u.901.9		12	60.59	even	2
1800.2.k.u.901.10		12	120.59	even	2
2400.2.k.f.1201.3		12	1.1	even	1	trivial
2400.2.k.f.1201.4		12	40.29	even	2	inner
2400.2.k.f.1201.9		12	8.5	even	2	inner
2400.2.k.f.1201.10		12	5.4	even	2	inner
3840.2.f.l.769.2		12	80.3	even	4
3840.2.f.l.769.5		12	80.27	even	4
3840.2.f.l.769.8		12	80.67	even	4
3840.2.f.l.769.11		12	80.43	even	4
3840.2.f.m.769.2		12	80.77	odd	4
3840.2.f.m.769.5		12	80.53	odd	4
3840.2.f.m.769.8		12	80.13	odd	4
3840.2.f.m.769.11		12	80.37	odd	4
7200.2.k.u.3601.5		12	3.2	odd	2
7200.2.k.u.3601.6		12	24.5	odd	2
7200.2.k.u.3601.7		12	15.14	odd	2
7200.2.k.u.3601.8		12	120.29	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -0.746175 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -0.746175 q^{7} -1.00000 q^{9} +5.36068i q^{11} -2.92520i q^{13} +2.13466 q^{17} +1.73367i q^{19} +0.746175i q^{21} -7.49534 q^{23} +1.00000i q^{27} -6.74916i q^{29} -2.64681 q^{31} +5.36068 q^{33} +1.07480i q^{37} -2.92520 q^{39} -11.2936 q^{41} -7.44322i q^{43} -1.73367 q^{47} -6.44322 q^{49} -2.13466i q^{51} +7.72161i q^{53} +1.73367 q^{57} +6.85302i q^{59} +6.45203i q^{61} +0.746175 q^{63} -7.44322i q^{67} +7.49534i q^{69} -13.2936 q^{71} -0.690358 q^{73} -4.00000i q^{77} -2.64681 q^{79} +1.00000 q^{81} +5.85039i q^{83} -6.74916 q^{87} +7.59283 q^{89} +2.18271i q^{91} +2.64681i q^{93} -14.1887 q^{97} -5.36068i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^9	\( 12 q - 12 q^{9} + 32 q^{31} - 16 q^{39} - 8 q^{41} + 12 q^{49} - 32 q^{71} + 32 q^{79} + 12 q^{81} + 40 q^{89}+O(q^{100}) \) 12 * q - 12 * q^9 + 32 * q^31 - 16 * q^39 - 8 * q^41 + 12 * q^49 - 32 * q^71 + 32 * q^79 + 12 * q^81 + 40 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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