LMFDB - Embedded newform 2400.3.g.b.751.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2400,3,Mod(751,2400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2400.751"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 0])) N = Newforms(chi, 3, names="a")

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

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,48,0,-64,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,-192] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(43)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$65.3952634465$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + x^{14} + 24 x^{13} - 44 x^{12} - 32 x^{11} + 180 x^{10} - 64 x^{9} - 352 x^{8} + \cdots + 65536 $ x^16 - 4x^15 + x^14 + 24x^13 - 44x^12 - 32x^11 + 180x^10 - 64x^9 - 352x^8 - 256x^7 + 2880x^6 - 2048x^5 - 11264x^4 + 24576x^3 + 4096x^2 - 65536x + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{34}\cdot 3 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			751.7
Root			$1.56126 + 1.24999i$ of defining polynomial
Character	$\chi$	$=$	2400.751
Dual form			2400.3.g.b.751.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +7.58970i q^{7} +3.00000 q^{9} -16.7055 q^{11} +10.7512i q^{13} -8.61276 q^{17} -35.6616 q^{19} -13.1457i q^{21} -11.2913i q^{23} -5.19615 q^{27} +0.589229i q^{29} -45.0411i q^{31} +28.9347 q^{33} -36.6203i q^{37} -18.6216i q^{39} -2.44967 q^{41} -7.91524 q^{43} +44.3755i q^{47} -8.60348 q^{49} +14.9177 q^{51} +73.3176i q^{53} +61.7677 q^{57} +89.7651 q^{59} -98.7987i q^{61} +22.7691i q^{63} +89.5013 q^{67} +19.5571i q^{69} +90.0121i q^{71} -68.0795 q^{73} -126.790i q^{77} -36.2647i q^{79} +9.00000 q^{81} -95.5904 q^{83} -1.02058i q^{87} -7.89970 q^{89} -81.5982 q^{91} +78.0135i q^{93} +137.133 q^{97} -50.1164 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 48 q^{9} - 64 q^{11} + 32 q^{19} - 192 q^{43} - 80 q^{49} - 96 q^{51} - 96 q^{57} - 128 q^{59} - 64 q^{67} + 160 q^{73} + 144 q^{81} + 192 q^{91} + 224 q^{97} - 192 q^{99}+O(q^{100}) $ 16 * q + 48 * q^9 - 64 * q^11 + 32 * q^19 - 192 * q^43 - 80 * q^49 - 96 * q^51 - 96 * q^57 - 128 * q^59 - 64 * q^67 + 160 * q^73 + 144 * q^81 + 192 * q^91 + 224 * q^97 - 192 * q^99

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
$3$	−1.73205	−0.577350
$3$	−1.73205	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.58970i	1.08424i	0.840300	+	0.542121i	$0.182379\pi$
$7$	7.58970i	1.08424i	−0.840300	+	0.542121i	$0.817621\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	−16.7055	−1.51868	−0.759340	−	0.650694i	$-0.774478\pi$
$11$	−16.7055	−1.51868	−0.759340	+	0.650694i	$0.774478\pi$
$12$	0	0
$13$	10.7512i	0.827014i	0.910501	+	0.413507i	$0.135696\pi$
$13$	10.7512i	0.827014i	−0.910501	+	0.413507i	$0.864304\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−8.61276	−0.506633	−0.253316	−	0.967383i	$-0.581521\pi$
$17$	−8.61276	−0.506633	−0.253316	+	0.967383i	$0.581521\pi$
$18$	0	0
$19$	−35.6616	−1.87693	−0.938464	−	0.345378i	$-0.887751\pi$
$19$	−35.6616	−1.87693	−0.938464	+	0.345378i	$0.887751\pi$
$20$	0	0
$21$	− 13.1457i	− 0.625988i
$22$	0	0
$23$	− 11.2913i	− 0.490925i	−0.969406	−	0.245462i	$-0.921060\pi$
$23$	− 11.2913i	− 0.490925i	0.969406	−	0.245462i	$-0.0789398\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.19615	−0.192450
$28$	0	0
$29$	0.589229i	0.0203183i	0.999948	+	0.0101591i	$0.00323381\pi$
$29$	0.589229i	0.0203183i	−0.999948	+	0.0101591i	$0.996766\pi$
$30$	0	0
$31$	− 45.0411i	− 1.45294i	−0.687198	−	0.726470i	$-0.741160\pi$
$31$	− 45.0411i	− 1.45294i	0.687198	−	0.726470i	$-0.258840\pi$
$32$	0	0
$33$	28.9347	0.876810
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 36.6203i	− 0.989738i	−0.868968	−	0.494869i	$-0.835216\pi$
$37$	− 36.6203i	− 0.989738i	0.868968	−	0.494869i	$-0.164784\pi$
$38$	0	0
$39$	− 18.6216i	− 0.477477i
$40$	0	0
$41$	−2.44967	−0.0597481	−0.0298740	−	0.999554i	$-0.509511\pi$
$41$	−2.44967	−0.0597481	−0.0298740	+	0.999554i	$0.509511\pi$
$42$	0	0
$43$	−7.91524	−0.184075	−0.0920377	−	0.995756i	$-0.529338\pi$
$43$	−7.91524	−0.184075	−0.0920377	+	0.995756i	$0.529338\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	44.3755i	0.944160i	0.881556	+	0.472080i	$0.156497\pi$
$47$	44.3755i	0.944160i	−0.881556	+	0.472080i	$0.843503\pi$
$48$	0	0
$49$	−8.60348	−0.175581
$50$	0	0
$51$	14.9177	0.292505
$52$	0	0
$53$	73.3176i	1.38335i	0.722209	+	0.691675i	$0.243127\pi$
$53$	73.3176i	1.38335i	−0.722209	+	0.691675i	$0.756873\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	61.7677	1.08364
$58$	0	0
$59$	89.7651	1.52144	0.760721	−	0.649079i	$-0.224845\pi$
$59$	89.7651	1.52144	0.760721	+	0.649079i	$0.224845\pi$
$60$	0	0
$61$	− 98.7987i	− 1.61965i	−0.586671	−	0.809826i	$-0.699562\pi$
$61$	− 98.7987i	− 1.61965i	0.586671	−	0.809826i	$-0.300438\pi$
$62$	0	0
$63$	22.7691i	0.361414i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	89.5013	1.33584	0.667920	−	0.744233i	$-0.267185\pi$
$67$	89.5013	1.33584	0.667920	+	0.744233i	$0.267185\pi$
$68$	0	0
$69$	19.5571i	0.283436i
$70$	0	0
$71$	90.0121i	1.26778i	0.773425	+	0.633888i	$0.218542\pi$
$71$	90.0121i	1.26778i	−0.773425	+	0.633888i	$0.781458\pi$
$72$	0	0
$73$	−68.0795	−0.932596	−0.466298	−	0.884628i	$-0.654412\pi$
$73$	−68.0795	−0.932596	−0.466298	+	0.884628i	$0.654412\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 126.790i	− 1.64662i
$78$	0	0
$79$	− 36.2647i	− 0.459046i	−0.973303	−	0.229523i	$-0.926283\pi$
$79$	− 36.2647i	− 0.459046i	0.973303	−	0.229523i	$-0.0737167\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	−95.5904	−1.15169	−0.575846	−	0.817558i	$-0.695327\pi$
$83$	−95.5904	−1.15169	−0.575846	+	0.817558i	$0.695327\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 1.02058i	− 0.0117307i
$88$	0	0
$89$	−7.89970	−0.0887607	−0.0443803	−	0.999015i	$-0.514131\pi$
$89$	−7.89970	−0.0887607	−0.0443803	+	0.999015i	$0.514131\pi$
$90$	0	0
$91$	−81.5982	−0.896683
$92$	0	0
$93$	78.0135i	0.838855i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	137.133	1.41375	0.706873	−	0.707341i	$-0.250105\pi$
$97$	137.133	1.41375	0.706873	+	0.707341i	$0.250105\pi$
$98$	0	0
$99$	−50.1164	−0.506227
$100$	0	0
$101$	92.4295i	0.915144i	0.889173	+	0.457572i	$0.151281\pi$
$101$	92.4295i	0.915144i	−0.889173	+	0.457572i	$0.848719\pi$
$102$	0	0
$103$	123.314i	1.19722i	0.801040	+	0.598610i	$0.204280\pi$
$103$	123.314i	1.19722i	−0.801040	+	0.598610i	$0.795720\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.240104	−0.00224396	−0.00112198	−	0.999999i	$-0.500357\pi$
$107$	−0.240104	−0.00224396	−0.00112198	+	0.999999i	$0.500357\pi$
$108$	0	0
$109$	− 117.597i	− 1.07887i	−0.842028	−	0.539435i	$-0.818638\pi$
$109$	− 117.597i	− 1.07887i	0.842028	−	0.539435i	$-0.181362\pi$
$110$	0	0
$111$	63.4282i	0.571425i
$112$	0	0
$113$	−41.1728	−0.364361	−0.182180	−	0.983265i	$-0.558315\pi$
$113$	−41.1728	−0.364361	−0.182180	+	0.983265i	$0.558315\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	32.2535i	0.275671i
$118$	0	0
$119$	− 65.3682i	− 0.549313i
$120$	0	0
$121$	158.073	1.30639
$122$	0	0
$123$	4.24295	0.0344956
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.15264i	0.00907591i	0.999990	+	0.00453795i	$0.00144448\pi$
$127$	1.15264i	0.00907591i	−0.999990	+	0.00453795i	$0.998556\pi$
$128$	0	0
$129$	13.7096	0.106276
$130$	0	0
$131$	191.878	1.46472	0.732359	−	0.680919i	$-0.238419\pi$
$131$	191.878	1.46472	0.732359	+	0.680919i	$0.238419\pi$
$132$	0	0
$133$	− 270.661i	− 2.03504i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−17.0127	−0.124180	−0.0620901	−	0.998071i	$-0.519777\pi$
$137$	−17.0127	−0.124180	−0.0620901	+	0.998071i	$0.519777\pi$
$138$	0	0
$139$	176.378	1.26890	0.634452	−	0.772963i	$-0.281226\pi$
$139$	176.378	1.26890	0.634452	+	0.772963i	$0.281226\pi$
$140$	0	0
$141$	− 76.8607i	− 0.545111i
$142$	0	0
$143$	− 179.604i	− 1.25597i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	14.9017	0.101372
$148$	0	0
$149$	212.046i	1.42313i	0.702620	+	0.711565i	$0.252013\pi$
$149$	212.046i	1.42313i	−0.702620	+	0.711565i	$0.747987\pi$
$150$	0	0
$151$	143.361i	0.949408i	0.880146	+	0.474704i	$0.157445\pi$
$151$	143.361i	0.949408i	−0.880146	+	0.474704i	$0.842555\pi$
$152$	0	0
$153$	−25.8383	−0.168878
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 82.2710i	− 0.524019i	−0.965065	−	0.262009i	$-0.915615\pi$
$157$	− 82.2710i	− 0.524019i	0.965065	−	0.262009i	$-0.0843852\pi$
$158$	0	0
$159$	− 126.990i	− 0.798678i
$160$	0	0
$161$	85.6973	0.532282
$162$	0	0
$163$	27.8115	0.170623	0.0853113	−	0.996354i	$-0.472812\pi$
$163$	27.8115	0.170623	0.0853113	+	0.996354i	$0.472812\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 207.819i	− 1.24443i	−0.782848	−	0.622214i	$-0.786234\pi$
$167$	− 207.819i	− 1.24443i	0.782848	−	0.622214i	$-0.213766\pi$
$168$	0	0
$169$	53.4122	0.316048
$170$	0	0
$171$	−106.985	−0.625642
$172$	0	0
$173$	45.7625i	0.264523i	0.991215	+	0.132262i	$0.0422239\pi$
$173$	45.7625i	0.264523i	−0.991215	+	0.132262i	$0.957776\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−155.478	−0.878405
$178$	0	0
$179$	33.0983	0.184907	0.0924533	−	0.995717i	$-0.470529\pi$
$179$	33.0983	0.184907	0.0924533	+	0.995717i	$0.470529\pi$
$180$	0	0
$181$	− 177.147i	− 0.978711i	−0.872084	−	0.489356i	$-0.837232\pi$
$181$	− 177.147i	− 0.978711i	0.872084	−	0.489356i	$-0.162768\pi$
$182$	0	0
$183$	171.124i	0.935106i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	143.880	0.769413
$188$	0	0
$189$	− 39.4372i	− 0.208663i
$190$	0	0
$191$	28.8000i	0.150785i	0.997154	+	0.0753927i	$0.0240210\pi$
$191$	28.8000i	0.150785i	−0.997154	+	0.0753927i	$0.975979\pi$
$192$	0	0
$193$	−87.7961	−0.454902	−0.227451	−	0.973789i	$-0.573039\pi$
$193$	−87.7961	−0.454902	−0.227451	+	0.973789i	$0.573039\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 347.625i	− 1.76459i	−0.470696	−	0.882296i	$-0.655997\pi$
$197$	− 347.625i	− 1.76459i	0.470696	−	0.882296i	$-0.344003\pi$
$198$	0	0
$199$	129.347i	0.649985i	0.945717	+	0.324992i	$0.105362\pi$
$199$	129.347i	0.649985i	−0.945717	+	0.324992i	$0.894638\pi$
$200$	0	0
$201$	−155.021	−0.771247
$202$	0	0
$203$	−4.47207	−0.0220299
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 33.8738i	− 0.163642i
$208$	0	0
$209$	595.744	2.85045
$210$	0	0
$211$	190.507	0.902877	0.451439	−	0.892302i	$-0.350911\pi$
$211$	190.507	0.902877	0.451439	+	0.892302i	$0.350911\pi$
$212$	0	0
$213$	− 155.905i	− 0.731951i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	341.848	1.57534
$218$	0	0
$219$	117.917	0.538434
$220$	0	0
$221$	− 92.5973i	− 0.418992i
$222$	0	0
$223$	211.437i	0.948149i	0.880485	+	0.474075i	$0.157217\pi$
$223$	211.437i	0.948149i	−0.880485	+	0.474075i	$0.842783\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	88.8880	0.391577	0.195788	−	0.980646i	$-0.437273\pi$
$227$	88.8880	0.391577	0.195788	+	0.980646i	$0.437273\pi$
$228$	0	0
$229$	− 152.872i	− 0.667565i	−0.942650	−	0.333782i	$-0.891675\pi$
$229$	− 152.872i	− 0.667565i	0.942650	−	0.333782i	$-0.108325\pi$
$230$	0	0
$231$	219.606i	0.950675i
$232$	0	0
$233$	−188.417	−0.808656	−0.404328	−	0.914614i	$-0.632495\pi$
$233$	−188.417	−0.808656	−0.404328	+	0.914614i	$0.632495\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	62.8122i	0.265030i
$238$	0	0
$239$	− 227.979i	− 0.953888i	−0.878934	−	0.476944i	$-0.841745\pi$
$239$	− 227.979i	− 0.953888i	0.878934	−	0.476944i	$-0.158255\pi$
$240$	0	0
$241$	−265.794	−1.10288	−0.551439	−	0.834215i	$-0.685921\pi$
$241$	−265.794	−1.10288	−0.551439	+	0.834215i	$0.685921\pi$
$242$	0	0
$243$	−15.5885	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 383.404i	− 1.55224i
$248$	0	0
$249$	165.567	0.664929
$250$	0	0
$251$	−154.894	−0.617106	−0.308553	−	0.951207i	$-0.599845\pi$
$251$	−154.894	−0.617106	−0.308553	+	0.951207i	$0.599845\pi$
$252$	0	0
$253$	188.626i	0.745558i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	422.025	1.64212	0.821060	−	0.570842i	$-0.193383\pi$
$257$	422.025	1.64212	0.821060	+	0.570842i	$0.193383\pi$
$258$	0	0
$259$	277.937	1.07312
$260$	0	0
$261$	1.76769i	0.00677275i
$262$	0	0
$263$	− 118.083i	− 0.448983i	−0.974476	−	0.224492i	$-0.927928\pi$
$263$	− 118.083i	− 0.448983i	0.974476	−	0.224492i	$-0.0720721\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	13.6827	0.0512460
$268$	0	0
$269$	− 493.643i	− 1.83510i	−0.397617	−	0.917551i	$-0.630163\pi$
$269$	− 493.643i	− 1.83510i	0.397617	−	0.917551i	$-0.369837\pi$
$270$	0	0
$271$	− 362.955i	− 1.33932i	−0.742669	−	0.669659i	$-0.766440\pi$
$271$	− 362.955i	− 1.33932i	0.742669	−	0.669659i	$-0.233560\pi$
$272$	0	0
$273$	141.332	0.517700
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 321.011i	− 1.15888i	−0.815013	−	0.579442i	$-0.803270\pi$
$277$	− 321.011i	− 1.15888i	0.815013	−	0.579442i	$-0.196730\pi$
$278$	0	0
$279$	− 135.123i	− 0.484313i
$280$	0	0
$281$	−185.799	−0.661206	−0.330603	−	0.943770i	$-0.607252\pi$
$281$	−185.799	−0.661206	−0.330603	+	0.943770i	$0.607252\pi$
$282$	0	0
$283$	−208.628	−0.737202	−0.368601	−	0.929588i	$-0.620163\pi$
$283$	−208.628	−0.737202	−0.368601	+	0.929588i	$0.620163\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 18.5923i	− 0.0647814i
$288$	0	0
$289$	−214.820	−0.743323
$290$	0	0
$291$	−237.522	−0.816226
$292$	0	0
$293$	306.925i	1.04752i	0.851864	+	0.523762i	$0.175472\pi$
$293$	306.925i	1.04752i	−0.851864	+	0.523762i	$0.824528\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	86.8042	0.292270
$298$	0	0
$299$	121.394	0.406002
$300$	0	0
$301$	− 60.0743i	− 0.199582i
$302$	0	0
$303$	− 160.093i	− 0.528358i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	283.517	0.923510	0.461755	−	0.887008i	$-0.347220\pi$
$307$	283.517	0.923510	0.461755	+	0.887008i	$0.347220\pi$
$308$	0	0
$309$	− 213.586i	− 0.691216i
$310$	0	0
$311$	− 614.016i	− 1.97433i	−0.159709	−	0.987164i	$-0.551056\pi$
$311$	− 614.016i	− 1.97433i	0.159709	−	0.987164i	$-0.448944\pi$
$312$	0	0
$313$	511.920	1.63553	0.817763	−	0.575555i	$-0.195214\pi$
$313$	511.920	1.63553	0.817763	+	0.575555i	$0.195214\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	429.605i	1.35522i	0.735421	+	0.677611i	$0.236985\pi$
$317$	429.605i	1.35522i	−0.735421	+	0.677611i	$0.763015\pi$
$318$	0	0
$319$	− 9.84336i	− 0.0308569i
$320$	0	0
$321$	0.415872	0.00129555
$322$	0	0
$323$	307.145	0.950913
$324$	0	0
$325$	0	0
$326$	0	0
$327$	203.684i	0.622885i
$328$	0	0
$329$	−336.797	−1.02370
$330$	0	0
$331$	537.298	1.62326	0.811628	−	0.584175i	$-0.198582\pi$
$331$	537.298	1.62326	0.811628	+	0.584175i	$0.198582\pi$
$332$	0	0
$333$	− 109.861i	− 0.329913i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−158.578	−0.470559	−0.235279	−	0.971928i	$-0.575600\pi$
$337$	−158.578	−0.470559	−0.235279	+	0.971928i	$0.575600\pi$
$338$	0	0
$339$	71.3133	0.210364
$340$	0	0
$341$	752.434i	2.20655i
$342$	0	0
$343$	306.597i	0.893870i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	169.697	0.489041	0.244520	−	0.969644i	$-0.421370\pi$
$347$	169.697	0.489041	0.244520	+	0.969644i	$0.421370\pi$
$348$	0	0
$349$	− 296.370i	− 0.849196i	−0.905382	−	0.424598i	$-0.860415\pi$
$349$	− 296.370i	− 0.849196i	0.905382	−	0.424598i	$-0.139585\pi$
$350$	0	0
$351$	− 55.8648i	− 0.159159i
$352$	0	0
$353$	−133.075	−0.376982	−0.188491	−	0.982075i	$-0.560360\pi$
$353$	−133.075	−0.376982	−0.188491	+	0.982075i	$0.560360\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	113.221i	0.317146i
$358$	0	0
$359$	− 238.095i	− 0.663218i	−0.943417	−	0.331609i	$-0.892409\pi$
$359$	− 238.095i	− 0.663218i	0.943417	−	0.331609i	$-0.107591\pi$
$360$	0	0
$361$	910.751	2.52286
$362$	0	0
$363$	−273.791	−0.754244
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 381.665i	− 1.03996i	−0.854179	−	0.519979i	$-0.825940\pi$
$367$	− 381.665i	− 1.03996i	0.854179	−	0.519979i	$-0.174060\pi$
$368$	0	0
$369$	−7.34901	−0.0199160
$370$	0	0
$371$	−556.458	−1.49989
$372$	0	0
$373$	− 42.2729i	− 0.113332i	−0.998393	−	0.0566661i	$-0.981953\pi$
$373$	− 42.2729i	− 0.113332i	0.998393	−	0.0566661i	$-0.0180470\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.33491	−0.0168035
$378$	0	0
$379$	−412.302	−1.08787	−0.543934	−	0.839128i	$-0.683066\pi$
$379$	−412.302	−1.08787	−0.543934	+	0.839128i	$0.683066\pi$
$380$	0	0
$381$	− 1.99643i	− 0.00523998i
$382$	0	0
$383$	− 403.711i	− 1.05408i	−0.849842	−	0.527038i	$-0.823303\pi$
$383$	− 403.711i	− 1.05408i	0.849842	−	0.527038i	$-0.176697\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−23.7457	−0.0613585
$388$	0	0
$389$	78.5053i	0.201813i	0.994896	+	0.100907i	$0.0321743\pi$
$389$	78.5053i	0.201813i	−0.994896	+	0.100907i	$0.967826\pi$
$390$	0	0
$391$	97.2490i	0.248719i
$392$	0	0
$393$	−332.342	−0.845655
$394$	0	0
$395$	0	0
$396$	0	0
$397$	323.080i	0.813803i	0.913472	+	0.406902i	$0.133391\pi$
$397$	323.080i	0.813803i	−0.913472	+	0.406902i	$0.866609\pi$
$398$	0	0
$399$	468.798i	1.17493i
$400$	0	0
$401$	−621.389	−1.54960	−0.774799	−	0.632207i	$-0.782149\pi$
$401$	−621.389	−1.54960	−0.774799	+	0.632207i	$0.782149\pi$
$402$	0	0
$403$	484.245	1.20160
$404$	0	0
$405$	0	0
$406$	0	0
$407$	611.760i	1.50310i
$408$	0	0
$409$	322.888	0.789458	0.394729	−	0.918798i	$-0.370839\pi$
$409$	322.888	0.789458	0.394729	+	0.918798i	$0.370839\pi$
$410$	0	0
$411$	29.4668	0.0716954
$412$	0	0
$413$	681.290i	1.64961i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−305.495	−0.732602
$418$	0	0
$419$	−19.2316	−0.0458988	−0.0229494	−	0.999737i	$-0.507306\pi$
$419$	−19.2316	−0.0458988	−0.0229494	+	0.999737i	$0.507306\pi$
$420$	0	0
$421$	312.791i	0.742970i	0.928439	+	0.371485i	$0.121151\pi$
$421$	312.791i	0.742970i	−0.928439	+	0.371485i	$0.878849\pi$
$422$	0	0
$423$	133.127i	0.314720i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	749.852	1.75609
$428$	0	0
$429$	311.083i	0.725134i
$430$	0	0
$431$	− 417.415i	− 0.968479i	−0.874935	−	0.484240i	$-0.839096\pi$
$431$	− 417.415i	− 0.968479i	0.874935	−	0.484240i	$-0.160904\pi$
$432$	0	0
$433$	−279.474	−0.645436	−0.322718	−	0.946495i	$-0.604596\pi$
$433$	−279.474	−0.645436	−0.322718	+	0.946495i	$0.604596\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	402.665i	0.921430i
$438$	0	0
$439$	30.1336i	0.0686415i	0.999411	+	0.0343207i	$0.0109268\pi$
$439$	30.1336i	0.0686415i	−0.999411	+	0.0343207i	$0.989073\pi$
$440$	0	0
$441$	−25.8104	−0.0585271
$442$	0	0
$443$	−56.2858	−0.127056	−0.0635279	−	0.997980i	$-0.520235\pi$
$443$	−56.2858	−0.127056	−0.0635279	+	0.997980i	$0.520235\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 367.275i	− 0.821644i
$448$	0	0
$449$	−283.497	−0.631396	−0.315698	−	0.948860i	$-0.602239\pi$
$449$	−283.497	−0.631396	−0.315698	+	0.948860i	$0.602239\pi$
$450$	0	0
$451$	40.9229	0.0907382
$452$	0	0
$453$	− 248.308i	− 0.548141i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	265.204	0.580315	0.290158	−	0.956979i	$-0.406292\pi$
$457$	265.204	0.580315	0.290158	+	0.956979i	$0.406292\pi$
$458$	0	0
$459$	44.7532	0.0975015
$460$	0	0
$461$	709.643i	1.53936i	0.638432	+	0.769678i	$0.279583\pi$
$461$	709.643i	1.53936i	−0.638432	+	0.769678i	$0.720417\pi$
$462$	0	0
$463$	− 417.392i	− 0.901495i	−0.892652	−	0.450747i	$-0.851157\pi$
$463$	− 417.392i	− 0.901495i	0.892652	−	0.450747i	$-0.148843\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−679.287	−1.45458	−0.727288	−	0.686332i	$-0.759220\pi$
$467$	−679.287	−1.45458	−0.727288	+	0.686332i	$0.759220\pi$
$468$	0	0
$469$	679.287i	1.44837i
$470$	0	0
$471$	142.498i	0.302543i
$472$	0	0
$473$	132.228	0.279552
$474$	0	0
$475$	0	0
$476$	0	0
$477$	219.953i	0.461117i
$478$	0	0
$479$	157.578i	0.328974i	0.986379	+	0.164487i	$0.0525968\pi$
$479$	157.578i	0.328974i	−0.986379	+	0.164487i	$0.947403\pi$
$480$	0	0
$481$	393.711	0.818527
$482$	0	0
$483$	−148.432	−0.307313
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 736.646i	− 1.51262i	−0.654213	−	0.756311i	$-0.727000\pi$
$487$	− 736.646i	− 1.51262i	0.654213	−	0.756311i	$-0.273000\pi$
$488$	0	0
$489$	−48.1709	−0.0985090
$490$	0	0
$491$	362.408	0.738101	0.369050	−	0.929409i	$-0.379683\pi$
$491$	362.408	0.738101	0.369050	+	0.929409i	$0.379683\pi$
$492$	0	0
$493$	− 5.07489i	− 0.0102939i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−683.164	−1.37458
$498$	0	0
$499$	−648.812	−1.30022	−0.650112	−	0.759838i	$-0.725278\pi$
$499$	−648.812	−1.30022	−0.650112	+	0.759838i	$0.725278\pi$
$500$	0	0
$501$	359.954i	0.718470i
$502$	0	0
$503$	402.612i	0.800421i	0.916423	+	0.400210i	$0.131063\pi$
$503$	402.612i	0.800421i	−0.916423	+	0.400210i	$0.868937\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−92.5126	−0.182471
$508$	0	0
$509$	− 470.175i	− 0.923724i	−0.886952	−	0.461862i	$-0.847182\pi$
$509$	− 470.175i	− 0.923724i	0.886952	−	0.461862i	$-0.152818\pi$
$510$	0	0
$511$	− 516.702i	− 1.01116i
$512$	0	0
$513$	185.303	0.361215
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 741.315i	− 1.43388i
$518$	0	0
$519$	− 79.2630i	− 0.152723i
$520$	0	0
$521$	−155.978	−0.299382	−0.149691	−	0.988733i	$-0.547828\pi$
$521$	−155.978	−0.299382	−0.149691	+	0.988733i	$0.547828\pi$
$522$	0	0
$523$	−313.578	−0.599576	−0.299788	−	0.954006i	$-0.596916\pi$
$523$	−313.578	−0.599576	−0.299788	+	0.954006i	$0.596916\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	387.928i	0.736107i
$528$	0	0
$529$	401.507	0.758993
$530$	0	0
$531$	269.295	0.507147
$532$	0	0
$533$	− 26.3369i	− 0.0494125i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−57.3279	−0.106756
$538$	0	0
$539$	143.725	0.266652
$540$	0	0
$541$	− 892.335i	− 1.64942i	−0.565557	−	0.824709i	$-0.691339\pi$
$541$	− 892.335i	− 1.64942i	0.565557	−	0.824709i	$-0.308661\pi$
$542$	0	0
$543$	306.827i	0.565059i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−671.435	−1.22749	−0.613743	−	0.789506i	$-0.710337\pi$
$547$	−671.435	−1.22749	−0.613743	+	0.789506i	$0.710337\pi$
$548$	0	0
$549$	− 296.396i	− 0.539884i
$550$	0	0
$551$	− 21.0129i	− 0.0381359i
$552$	0	0
$553$	275.238	0.497717
$554$	0	0
$555$	0	0
$556$	0	0
$557$	296.095i	0.531590i	0.964030	+	0.265795i	$0.0856344\pi$
$557$	296.095i	0.531590i	−0.964030	+	0.265795i	$0.914366\pi$
$558$	0	0
$559$	− 85.0982i	− 0.152233i
$560$	0	0
$561$	−249.208	−0.444221
$562$	0	0
$563$	−707.222	−1.25617	−0.628084	−	0.778146i	$-0.716161\pi$
$563$	−707.222	−1.25617	−0.628084	+	0.778146i	$0.716161\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	68.3073i	0.120471i
$568$	0	0
$569$	230.263	0.404680	0.202340	−	0.979315i	$-0.435145\pi$
$569$	230.263	0.404680	0.202340	+	0.979315i	$0.435145\pi$
$570$	0	0
$571$	439.428	0.769575	0.384788	−	0.923005i	$-0.374275\pi$
$571$	439.428	0.769575	0.384788	+	0.923005i	$0.374275\pi$
$572$	0	0
$573$	− 49.8831i	− 0.0870560i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	184.670	0.320053	0.160026	−	0.987113i	$-0.448842\pi$
$577$	184.670	0.320053	0.160026	+	0.987113i	$0.448842\pi$
$578$	0	0
$579$	152.067	0.262638
$580$	0	0
$581$	− 725.502i	− 1.24871i
$582$	0	0
$583$	− 1224.81i	− 2.10087i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	297.341	0.506543	0.253271	−	0.967395i	$-0.418493\pi$
$587$	297.341	0.506543	0.253271	+	0.967395i	$0.418493\pi$
$588$	0	0
$589$	1606.24i	2.72706i
$590$	0	0
$591$	602.103i	1.01879i
$592$	0	0
$593$	607.065	1.02372	0.511859	−	0.859070i	$-0.328957\pi$
$593$	607.065	1.02372	0.511859	+	0.859070i	$0.328957\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 224.036i	− 0.375269i
$598$	0	0
$599$	448.633i	0.748970i	0.927233	+	0.374485i	$0.122180\pi$
$599$	448.633i	0.748970i	−0.927233	+	0.374485i	$0.877820\pi$
$600$	0	0
$601$	580.584	0.966030	0.483015	−	0.875612i	$-0.339542\pi$
$601$	580.584	0.966030	0.483015	+	0.875612i	$0.339542\pi$
$602$	0	0
$603$	268.504	0.445280
$604$	0	0
$605$	0	0
$606$	0	0
$607$	132.576i	0.218413i	0.994019	+	0.109206i	$0.0348309\pi$
$607$	132.576i	0.218413i	−0.994019	+	0.109206i	$0.965169\pi$
$608$	0	0
$609$	7.74585	0.0127190
$610$	0	0
$611$	−477.089	−0.780834
$612$	0	0
$613$	− 175.738i	− 0.286685i	−0.989673	−	0.143343i	$-0.954215\pi$
$613$	− 175.738i	− 0.286685i	0.989673	−	0.143343i	$-0.0457851\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−283.254	−0.459082	−0.229541	−	0.973299i	$-0.573723\pi$
$617$	−283.254	−0.459082	−0.229541	+	0.973299i	$0.573723\pi$
$618$	0	0
$619$	80.2105	0.129581	0.0647904	−	0.997899i	$-0.479362\pi$
$619$	80.2105	0.129581	0.0647904	+	0.997899i	$0.479362\pi$
$620$	0	0
$621$	58.6712i	0.0944785i
$622$	0	0
$623$	− 59.9563i	− 0.0962381i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1031.86	−1.64571
$628$	0	0
$629$	315.402i	0.501434i
$630$	0	0
$631$	− 178.580i	− 0.283011i	−0.989937	−	0.141505i	$-0.954806\pi$
$631$	− 178.580i	− 0.283011i	0.989937	−	0.141505i	$-0.0451942\pi$
$632$	0	0
$633$	−329.968	−0.521277
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 92.4976i	− 0.145208i
$638$	0	0
$639$	270.036i	0.422592i
$640$	0	0
$641$	−379.689	−0.592339	−0.296169	−	0.955135i	$-0.595709\pi$
$641$	−379.689	−0.592339	−0.296169	+	0.955135i	$0.595709\pi$
$642$	0	0
$643$	−847.921	−1.31869	−0.659347	−	0.751839i	$-0.729167\pi$
$643$	−847.921	−1.31869	−0.659347	+	0.751839i	$0.729167\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 372.369i	− 0.575532i	−0.957701	−	0.287766i	$-0.907087\pi$
$647$	− 372.369i	− 0.575532i	0.957701	−	0.287766i	$-0.0929126\pi$
$648$	0	0
$649$	−1499.57	−2.31058
$650$	0	0
$651$	−592.099	−0.909522
$652$	0	0
$653$	− 650.726i	− 0.996517i	−0.867028	−	0.498259i	$-0.833973\pi$
$653$	− 650.726i	− 0.996517i	0.867028	−	0.498259i	$-0.166027\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−204.238	−0.310865
$658$	0	0
$659$	−589.211	−0.894099	−0.447049	−	0.894509i	$-0.647525\pi$
$659$	−589.211	−0.894099	−0.447049	+	0.894509i	$0.647525\pi$
$660$	0	0
$661$	− 402.340i	− 0.608684i	−0.952563	−	0.304342i	$-0.901563\pi$
$661$	− 402.340i	− 0.608684i	0.952563	−	0.304342i	$-0.0984366\pi$
$662$	0	0
$663$	160.383i	0.241905i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.65315	0.00997473
$668$	0	0
$669$	− 366.220i	− 0.547414i
$670$	0	0
$671$	1650.48i	2.45973i
$672$	0	0
$673$	−1237.67	−1.83903	−0.919515	−	0.393056i	$-0.871418\pi$
$673$	−1237.67	−1.83903	−0.919515	+	0.393056i	$0.871418\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 163.462i	− 0.241451i	−0.992686	−	0.120725i	$-0.961478\pi$
$677$	− 163.462i	− 0.241451i	0.992686	−	0.120725i	$-0.0385221\pi$
$678$	0	0
$679$	1040.80i	1.53284i
$680$	0	0
$681$	−153.958	−0.226077
$682$	0	0
$683$	992.033	1.45246	0.726232	−	0.687449i	$-0.241270\pi$
$683$	992.033	1.45246	0.726232	+	0.687449i	$0.241270\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	264.783i	0.385419i
$688$	0	0
$689$	−788.250	−1.14405
$690$	0	0
$691$	465.365	0.673466	0.336733	−	0.941600i	$-0.390678\pi$
$691$	465.365	0.673466	0.336733	+	0.941600i	$0.390678\pi$
$692$	0	0
$693$	− 380.369i	− 0.548872i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	21.0984	0.0302703
$698$	0	0
$699$	326.347	0.466878
$700$	0	0
$701$	261.352i	0.372827i	0.982471	+	0.186414i	$0.0596864\pi$
$701$	261.352i	0.372827i	−0.982471	+	0.186414i	$0.940314\pi$
$702$	0	0
$703$	1305.94i	1.85767i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−701.512	−0.992238
$708$	0	0
$709$	− 186.604i	− 0.263194i	−0.991303	−	0.131597i	$-0.957990\pi$
$709$	− 186.604i	− 0.263194i	0.991303	−	0.131597i	$-0.0420104\pi$
$710$	0	0
$711$	− 108.794i	− 0.153015i
$712$	0	0
$713$	−508.572	−0.713284
$714$	0	0
$715$	0	0
$716$	0	0
$717$	394.871i	0.550727i
$718$	0	0
$719$	448.275i	0.623470i	0.950169	+	0.311735i	$0.100910\pi$
$719$	448.275i	0.623470i	−0.950169	+	0.311735i	$0.899090\pi$
$720$	0	0
$721$	−935.914	−1.29808
$722$	0	0
$723$	460.368	0.636747
$724$	0	0
$725$	0	0
$726$	0	0
$727$	956.205i	1.31527i	0.753335	+	0.657637i	$0.228444\pi$
$727$	956.205i	1.31527i	−0.753335	+	0.657637i	$0.771556\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	68.1721	0.0932586
$732$	0	0
$733$	− 398.986i	− 0.544320i	−0.962252	−	0.272160i	$-0.912262\pi$
$733$	− 398.986i	− 0.544320i	0.962252	−	0.272160i	$-0.0877379\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1495.16	−2.02871
$738$	0	0
$739$	−1011.87	−1.36924	−0.684622	−	0.728899i	$-0.740033\pi$
$739$	−1011.87	−1.36924	−0.684622	+	0.728899i	$0.740033\pi$
$740$	0	0
$741$	664.076i	0.896189i
$742$	0	0
$743$	346.145i	0.465875i	0.972492	+	0.232938i	$0.0748338\pi$
$743$	346.145i	0.465875i	−0.972492	+	0.232938i	$0.925166\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−286.771	−0.383897
$748$	0	0
$749$	− 1.82231i	− 0.00243300i
$750$	0	0
$751$	71.2058i	0.0948146i	0.998876	+	0.0474073i	$0.0150959\pi$
$751$	71.2058i	0.0948146i	−0.998876	+	0.0474073i	$0.984904\pi$
$752$	0	0
$753$	268.283	0.356286
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1452.44i	1.91867i	0.282264	+	0.959337i	$0.408915\pi$
$757$	1452.44i	1.91867i	−0.282264	+	0.959337i	$0.591085\pi$
$758$	0	0
$759$	− 326.710i	− 0.430448i
$760$	0	0
$761$	414.416	0.544567	0.272284	−	0.962217i	$-0.412221\pi$
$761$	414.416	0.544567	0.272284	+	0.962217i	$0.412221\pi$
$762$	0	0
$763$	892.523	1.16976
$764$	0	0
$765$	0	0
$766$	0	0
$767$	965.081i	1.25825i
$768$	0	0
$769$	−132.780	−0.172665	−0.0863326	−	0.996266i	$-0.527515\pi$
$769$	−132.780	−0.172665	−0.0863326	+	0.996266i	$0.527515\pi$
$770$	0	0
$771$	−730.968	−0.948078
$772$	0	0
$773$	135.449i	0.175225i	0.996155	+	0.0876126i	$0.0279238\pi$
$773$	135.449i	0.175225i	−0.996155	+	0.0876126i	$0.972076\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−481.401	−0.619564
$778$	0	0
$779$	87.3592	0.112143
$780$	0	0
$781$	− 1503.70i	− 1.92535i
$782$	0	0
$783$	− 3.06173i	− 0.00391025i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−370.900	−0.471283	−0.235642	−	0.971840i	$-0.575719\pi$
$787$	−370.900	−0.471283	−0.235642	+	0.971840i	$0.575719\pi$
$788$	0	0
$789$	204.525i	0.259220i
$790$	0	0
$791$	− 312.489i	− 0.395055i
$792$	0	0
$793$	1062.20	1.33947
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 980.881i	− 1.23072i	−0.788247	−	0.615358i	$-0.789011\pi$
$797$	− 980.881i	− 1.23072i	0.788247	−	0.615358i	$-0.210989\pi$
$798$	0	0
$799$	− 382.196i	− 0.478343i
$800$	0	0
$801$	−23.6991	−0.0295869
$802$	0	0
$803$	1137.30	1.41631
$804$	0	0
$805$	0	0
$806$	0	0
$807$	855.014i	1.05950i
$808$	0	0
$809$	−464.769	−0.574498	−0.287249	−	0.957856i	$-0.592741\pi$
$809$	−464.769	−0.574498	−0.287249	+	0.957856i	$0.592741\pi$
$810$	0	0
$811$	49.1816	0.0606431	0.0303216	−	0.999540i	$-0.490347\pi$
$811$	49.1816	0.0606431	0.0303216	+	0.999540i	$0.490347\pi$
$812$	0	0
$813$	628.657i	0.773255i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	282.270	0.345496
$818$	0	0
$819$	−244.795	−0.298894
$820$	0	0
$821$	− 911.027i	− 1.10965i	−0.831965	−	0.554827i	$-0.812784\pi$
$821$	− 911.027i	− 1.10965i	0.831965	−	0.554827i	$-0.187216\pi$
$822$	0	0
$823$	− 1278.02i	− 1.55287i	−0.630195	−	0.776437i	$-0.717025\pi$
$823$	− 1278.02i	− 1.55287i	0.630195	−	0.776437i	$-0.282975\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−671.016	−0.811386	−0.405693	−	0.914009i	$-0.632970\pi$
$827$	−671.016	−0.811386	−0.405693	+	0.914009i	$0.632970\pi$
$828$	0	0
$829$	31.8486i	0.0384181i	0.999815	+	0.0192091i	$0.00611481\pi$
$829$	31.8486i	0.0384181i	−0.999815	+	0.0192091i	$0.993885\pi$
$830$	0	0
$831$	556.007i	0.669082i
$832$	0	0
$833$	74.0997	0.0889552
$834$	0	0
$835$	0	0
$836$	0	0
$837$	234.041i	0.279618i
$838$	0	0
$839$	− 112.707i	− 0.134335i	−0.997742	−	0.0671674i	$-0.978604\pi$
$839$	− 112.707i	− 0.134335i	0.997742	−	0.0671674i	$-0.0213961\pi$
$840$	0	0
$841$	840.653	0.999587
$842$	0	0
$843$	321.813	0.381748
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1199.73i	1.41644i
$848$	0	0
$849$	361.354	0.425623
$850$	0	0
$851$	−413.490	−0.485887
$852$	0	0
$853$	− 512.714i	− 0.601072i	−0.953770	−	0.300536i	$-0.902834\pi$
$853$	− 512.714i	− 0.601072i	0.953770	−	0.300536i	$-0.0971655\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1620.81	1.89126	0.945631	−	0.325243i	$-0.105446\pi$
$857$	1620.81	1.89126	0.945631	+	0.325243i	$0.105446\pi$
$858$	0	0
$859$	−225.172	−0.262132	−0.131066	−	0.991374i	$-0.541840\pi$
$859$	−225.172	−0.262132	−0.131066	+	0.991374i	$0.541840\pi$
$860$	0	0
$861$	32.2027i	0.0374016i
$862$	0	0
$863$	1029.21i	1.19260i	0.802763	+	0.596299i	$0.203363\pi$
$863$	1029.21i	1.19260i	−0.802763	+	0.596299i	$0.796637\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	372.080	0.429158
$868$	0	0
$869$	605.818i	0.697144i
$870$	0	0
$871$	962.244i	1.10476i
$872$	0	0
$873$	411.400	0.471248
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 570.056i	− 0.650007i	−0.945713	−	0.325003i	$-0.894635\pi$
$877$	− 570.056i	− 0.650007i	0.945713	−	0.325003i	$-0.105365\pi$
$878$	0	0
$879$	− 531.609i	− 0.604789i
$880$	0	0
$881$	596.788	0.677398	0.338699	−	0.940895i	$-0.390013\pi$
$881$	596.788	0.677398	0.338699	+	0.940895i	$0.390013\pi$
$882$	0	0
$883$	856.738	0.970258	0.485129	−	0.874442i	$-0.338773\pi$
$883$	856.738	0.970258	0.485129	+	0.874442i	$0.338773\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1648.76i	1.85880i	0.369068	+	0.929402i	$0.379677\pi$
$887$	1648.76i	1.85880i	−0.369068	+	0.929402i	$0.620323\pi$
$888$	0	0
$889$	−8.74819	−0.00984048
$890$	0	0
$891$	−150.349	−0.168742
$892$	0	0
$893$	− 1582.50i	− 1.77212i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−210.261	−0.234405
$898$	0	0
$899$	26.5395	0.0295212
$900$	0	0
$901$	− 631.467i	− 0.700851i
$902$	0	0
$903$	104.052i	0.115229i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	844.509	0.931101	0.465551	−	0.885021i	$-0.345856\pi$
$907$	844.509	0.931101	0.465551	+	0.885021i	$0.345856\pi$
$908$	0	0
$909$	277.289i	0.305048i
$910$	0	0
$911$	1074.78i	1.17978i	0.807483	+	0.589891i	$0.200829\pi$
$911$	1074.78i	1.17978i	−0.807483	+	0.589891i	$0.799171\pi$
$912$	0	0
$913$	1596.88	1.74905
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1456.30i	1.58811i
$918$	0	0
$919$	− 1142.19i	− 1.24286i	−0.783471	−	0.621428i	$-0.786553\pi$
$919$	− 1142.19i	− 1.24286i	0.783471	−	0.621428i	$-0.213447\pi$
$920$	0	0
$921$	−491.067	−0.533188
$922$	0	0
$923$	−967.736	−1.04847
$924$	0	0
$925$	0	0
$926$	0	0
$927$	369.941i	0.399074i
$928$	0	0
$929$	415.953	0.447742	0.223871	−	0.974619i	$-0.428131\pi$
$929$	415.953	0.447742	0.223871	+	0.974619i	$0.428131\pi$
$930$	0	0
$931$	306.814	0.329553
$932$	0	0
$933$	1063.51i	1.13988i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	846.979	0.903927	0.451963	−	0.892037i	$-0.350724\pi$
$937$	846.979	0.903927	0.451963	+	0.892037i	$0.350724\pi$
$938$	0	0
$939$	−886.671	−0.944271
$940$	0	0
$941$	149.279i	0.158638i	0.996849	+	0.0793192i	$0.0252746\pi$
$941$	149.279i	0.158638i	−0.996849	+	0.0793192i	$0.974725\pi$
$942$	0	0
$943$	27.6599i	0.0293318i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	543.005	0.573395	0.286697	−	0.958021i	$-0.407443\pi$
$947$	543.005	0.573395	0.286697	+	0.958021i	$0.407443\pi$
$948$	0	0
$949$	− 731.935i	− 0.771269i
$950$	0	0
$951$	− 744.098i	− 0.782438i
$952$	0	0
$953$	344.292	0.361272	0.180636	−	0.983550i	$-0.442184\pi$
$953$	344.292	0.361272	0.180636	+	0.983550i	$0.442184\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	17.0492i	0.0178153i
$958$	0	0
$959$	− 129.121i	− 0.134641i
$960$	0	0
$961$	−1067.70	−1.11103
$962$	0	0
$963$	−0.720311	−0.000747987 0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1044.78i	− 1.08044i	−0.841524	−	0.540219i	$-0.818341\pi$
$967$	− 1044.78i	− 1.08044i	0.841524	−	0.540219i	$-0.181659\pi$
$968$	0	0
$969$	−531.991	−0.549010
$970$	0	0
$971$	−306.656	−0.315814	−0.157907	−	0.987454i	$-0.550475\pi$
$971$	−306.656	−0.315814	−0.157907	+	0.987454i	$0.550475\pi$
$972$	0	0
$973$	1338.65i	1.37580i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1753.81	−1.79509	−0.897547	−	0.440919i	$-0.854653\pi$
$977$	−1753.81	−1.79509	−0.897547	+	0.440919i	$0.854653\pi$
$978$	0	0
$979$	131.968	0.134799
$980$	0	0
$981$	− 352.790i	− 0.359623i
$982$	0	0
$983$	− 951.541i	− 0.967997i	−0.875069	−	0.483999i	$-0.839184\pi$
$983$	− 951.541i	− 0.967997i	0.875069	−	0.483999i	$-0.160816\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	583.349	0.591033
$988$	0	0
$989$	89.3731i	0.0903672i
$990$	0	0
$991$	− 1786.06i	− 1.80229i	−0.433523	−	0.901143i	$-0.642730\pi$
$991$	− 1786.06i	− 1.80229i	0.433523	−	0.901143i	$-0.357270\pi$
$992$	0	0
$993$	−930.627	−0.937187
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1375.80i	− 1.37994i	−0.723839	−	0.689969i	$-0.757624\pi$
$997$	− 1375.80i	− 1.37994i	0.723839	−	0.689969i	$-0.242376\pi$
$998$	0	0
$999$	190.285i	0.190475i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.3.g.b.751.7		16
4.3	odd	2		600.3.g.d.451.12		16
5.2	odd	4		2400.3.p.b.1999.14		32
5.3	odd	4		2400.3.p.b.1999.31		32
5.4	even	2		480.3.g.a.271.9		16
8.3	odd	2	inner	2400.3.g.b.751.2		16
8.5	even	2		600.3.g.d.451.11		16
15.14	odd	2		1440.3.g.c.271.10		16
20.3	even	4		600.3.p.b.499.25		32
20.7	even	4		600.3.p.b.499.8		32
20.19	odd	2		120.3.g.a.91.5	✓	16
40.3	even	4		2400.3.p.b.1999.13		32
40.13	odd	4		600.3.p.b.499.7		32
40.19	odd	2		480.3.g.a.271.16		16
40.27	even	4		2400.3.p.b.1999.32		32
40.29	even	2		120.3.g.a.91.6	yes	16
40.37	odd	4		600.3.p.b.499.26		32
60.59	even	2		360.3.g.c.91.12		16
120.29	odd	2		360.3.g.c.91.11		16
120.59	even	2		1440.3.g.c.271.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.g.a.91.5	✓	16	20.19	odd	2
120.3.g.a.91.6	yes	16	40.29	even	2
360.3.g.c.91.11		16	120.29	odd	2
360.3.g.c.91.12		16	60.59	even	2
480.3.g.a.271.9		16	5.4	even	2
480.3.g.a.271.16		16	40.19	odd	2
600.3.g.d.451.11		16	8.5	even	2
600.3.g.d.451.12		16	4.3	odd	2
600.3.p.b.499.7		32	40.13	odd	4
600.3.p.b.499.8		32	20.7	even	4
600.3.p.b.499.25		32	20.3	even	4
600.3.p.b.499.26		32	40.37	odd	4
1440.3.g.c.271.7		16	120.59	even	2
1440.3.g.c.271.10		16	15.14	odd	2
2400.3.g.b.751.2		16	8.3	odd	2	inner
2400.3.g.b.751.7		16	1.1	even	1	trivial
2400.3.p.b.1999.13		32	40.3	even	4
2400.3.p.b.1999.14		32	5.2	odd	4
2400.3.p.b.1999.31		32	5.3	odd	4
2400.3.p.b.1999.32		32	40.27	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +7.58970i q^{7} +3.00000 q^{9} -16.7055 q^{11} +10.7512i q^{13} -8.61276 q^{17} -35.6616 q^{19} -13.1457i q^{21} -11.2913i q^{23} -5.19615 q^{27} +0.589229i q^{29} -45.0411i q^{31} +28.9347 q^{33} -36.6203i q^{37} -18.6216i q^{39} -2.44967 q^{41} -7.91524 q^{43} +44.3755i q^{47} -8.60348 q^{49} +14.9177 q^{51} +73.3176i q^{53} +61.7677 q^{57} +89.7651 q^{59} -98.7987i q^{61} +22.7691i q^{63} +89.5013 q^{67} +19.5571i q^{69} +90.0121i q^{71} -68.0795 q^{73} -126.790i q^{77} -36.2647i q^{79} +9.00000 q^{81} -95.5904 q^{83} -1.02058i q^{87} -7.89970 q^{89} -81.5982 q^{91} +78.0135i q^{93} +137.133 q^{97} -50.1164 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 48 q^{9} - 64 q^{11} + 32 q^{19} - 192 q^{43} - 80 q^{49} - 96 q^{51} - 96 q^{57} - 128 q^{59} - 64 q^{67} + 160 q^{73} + 144 q^{81} + 192 q^{91} + 224 q^{97} - 192 q^{99}+O(q^{100}) \) 16 * q + 48 * q^9 - 64 * q^11 + 32 * q^19 - 192 * q^43 - 80 * q^49 - 96 * q^51 - 96 * q^57 - 128 * q^59 - 64 * q^67 + 160 * q^73 + 144 * q^81 + 192 * q^91 + 224 * q^97 - 192 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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