LMFDB - Embedded newform 448.4.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("448.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.a (trivial)

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:	yes
Analytic conductor:	$26.4328556826$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{3} -6.00000 q^{5} -7.00000 q^{7} -11.0000 q^{9} +O(q^{10})$

$q+4.00000 q^{3} -6.00000 q^{5} -7.00000 q^{7} -11.0000 q^{9} -12.0000 q^{11} +82.0000 q^{13} -24.0000 q^{15} -30.0000 q^{17} +68.0000 q^{19} -28.0000 q^{21} -216.000 q^{23} -89.0000 q^{25} -152.000 q^{27} -246.000 q^{29} +112.000 q^{31} -48.0000 q^{33} +42.0000 q^{35} -110.000 q^{37} +328.000 q^{39} -246.000 q^{41} -172.000 q^{43} +66.0000 q^{45} -192.000 q^{47} +49.0000 q^{49} -120.000 q^{51} -558.000 q^{53} +72.0000 q^{55} +272.000 q^{57} +540.000 q^{59} -110.000 q^{61} +77.0000 q^{63} -492.000 q^{65} +140.000 q^{67} -864.000 q^{69} +840.000 q^{71} -550.000 q^{73} -356.000 q^{75} +84.0000 q^{77} +208.000 q^{79} -311.000 q^{81} +516.000 q^{83} +180.000 q^{85} -984.000 q^{87} -1398.00 q^{89} -574.000 q^{91} +448.000 q^{93} -408.000 q^{95} +1586.00 q^{97} +132.000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	4.00000	0.769800	0.384900	−	0.922958i	$-0.374236\pi$
$3$	4.00000	0.769800	0.384900	+	0.922958i	$0.374236\pi$
$4$	0	0
$5$	−6.00000	−0.536656	−0.268328	−	0.963328i	$-0.586471\pi$
$5$	−6.00000	−0.536656	−0.268328	+	0.963328i	$0.586471\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−11.0000	−0.407407
$10$	0	0
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	82.0000	1.74944	0.874720	−	0.484629i	$-0.161046\pi$
$13$	82.0000	1.74944	0.874720	+	0.484629i	$0.161046\pi$
$14$	0	0
$15$	−24.0000	−0.413118
$16$	0	0
$17$	−30.0000	−0.428004	−0.214002	−	0.976833i	$-0.568650\pi$
$17$	−30.0000	−0.428004	−0.214002	+	0.976833i	$0.568650\pi$
$18$	0	0
$19$	68.0000	0.821067	0.410533	−	0.911846i	$-0.365343\pi$
$19$	68.0000	0.821067	0.410533	+	0.911846i	$0.365343\pi$
$20$	0	0
$21$	−28.0000	−0.290957
$22$	0	0
$23$	−216.000	−1.95822	−0.979111	−	0.203326i	$-0.934825\pi$
$23$	−216.000	−1.95822	−0.979111	+	0.203326i	$0.934825\pi$
$24$	0	0
$25$	−89.0000	−0.712000
$26$	0	0
$27$	−152.000	−1.08342
$28$	0	0
$29$	−246.000	−1.57521	−0.787604	−	0.616181i	$-0.788679\pi$
$29$	−246.000	−1.57521	−0.787604	+	0.616181i	$0.788679\pi$
$30$	0	0
$31$	112.000	0.648897	0.324448	−	0.945903i	$-0.394821\pi$
$31$	112.000	0.648897	0.324448	+	0.945903i	$0.394821\pi$
$32$	0	0
$33$	−48.0000	−0.253204
$34$	0	0
$35$	42.0000	0.202837
$36$	0	0
$37$	−110.000	−0.488754	−0.244377	−	0.969680i	$-0.578583\pi$
$37$	−110.000	−0.488754	−0.244377	+	0.969680i	$0.578583\pi$
$38$	0	0
$39$	328.000	1.34672
$40$	0	0
$41$	−246.000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−246.000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−172.000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−172.000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	66.0000	0.218638
$46$	0	0
$47$	−192.000	−0.595874	−0.297937	−	0.954586i	$-0.596299\pi$
$47$	−192.000	−0.595874	−0.297937	+	0.954586i	$0.596299\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−120.000	−0.329478
$52$	0	0
$53$	−558.000	−1.44617	−0.723087	−	0.690757i	$-0.757277\pi$
$53$	−558.000	−1.44617	−0.723087	+	0.690757i	$0.757277\pi$
$54$	0	0
$55$	72.0000	0.176518
$56$	0	0
$57$	272.000	0.632058
$58$	0	0
$59$	540.000	1.19156	0.595780	−	0.803148i	$-0.296843\pi$
$59$	540.000	1.19156	0.595780	+	0.803148i	$0.296843\pi$
$60$	0	0
$61$	−110.000	−0.230886	−0.115443	−	0.993314i	$-0.536829\pi$
$61$	−110.000	−0.230886	−0.115443	+	0.993314i	$0.536829\pi$
$62$	0	0
$63$	77.0000	0.153986
$64$	0	0
$65$	−492.000	−0.938848
$66$	0	0
$67$	140.000	0.255279	0.127640	−	0.991821i	$-0.459260\pi$
$67$	140.000	0.255279	0.127640	+	0.991821i	$0.459260\pi$
$68$	0	0
$69$	−864.000	−1.50744
$70$	0	0
$71$	840.000	1.40408	0.702040	−	0.712138i	$-0.252273\pi$
$71$	840.000	1.40408	0.702040	+	0.712138i	$0.252273\pi$
$72$	0	0
$73$	−550.000	−0.881817	−0.440908	−	0.897552i	$-0.645344\pi$
$73$	−550.000	−0.881817	−0.440908	+	0.897552i	$0.645344\pi$
$74$	0	0
$75$	−356.000	−0.548098
$76$	0	0
$77$	84.0000	0.124321
$78$	0	0
$79$	208.000	0.296226	0.148113	−	0.988970i	$-0.452680\pi$
$79$	208.000	0.296226	0.148113	+	0.988970i	$0.452680\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	516.000	0.682390	0.341195	−	0.939993i	$-0.389168\pi$
$83$	516.000	0.682390	0.341195	+	0.939993i	$0.389168\pi$
$84$	0	0
$85$	180.000	0.229691
$86$	0	0
$87$	−984.000	−1.21260
$88$	0	0
$89$	−1398.00	−1.66503	−0.832515	−	0.554002i	$-0.813100\pi$
$89$	−1398.00	−1.66503	−0.832515	+	0.554002i	$0.813100\pi$
$90$	0	0
$91$	−574.000	−0.661226
$92$	0	0
$93$	448.000	0.499521
$94$	0	0
$95$	−408.000	−0.440631
$96$	0	0
$97$	1586.00	1.66014	0.830072	−	0.557657i	$-0.188299\pi$
$97$	1586.00	1.66014	0.830072	+	0.557657i	$0.188299\pi$
$98$	0	0
$99$	132.000	0.134005
$100$	0	0
$101$	1242.00	1.22360	0.611800	−	0.791012i	$-0.290446\pi$
$101$	1242.00	1.22360	0.611800	+	0.791012i	$0.290446\pi$
$102$	0	0
$103$	−680.000	−0.650509	−0.325254	−	0.945627i	$-0.605450\pi$
$103$	−680.000	−0.650509	−0.325254	+	0.945627i	$0.605450\pi$
$104$	0	0
$105$	168.000	0.156144
$106$	0	0
$107$	996.000	0.899878	0.449939	−	0.893059i	$-0.351446\pi$
$107$	996.000	0.899878	0.449939	+	0.893059i	$0.351446\pi$
$108$	0	0
$109$	−1382.00	−1.21442	−0.607209	−	0.794542i	$-0.707711\pi$
$109$	−1382.00	−1.21442	−0.607209	+	0.794542i	$0.707711\pi$
$110$	0	0
$111$	−440.000	−0.376243
$112$	0	0
$113$	−750.000	−0.624372	−0.312186	−	0.950021i	$-0.601061\pi$
$113$	−750.000	−0.624372	−0.312186	+	0.950021i	$0.601061\pi$
$114$	0	0
$115$	1296.00	1.05089
$116$	0	0
$117$	−902.000	−0.712734
$118$	0	0
$119$	210.000	0.161770
$120$	0	0
$121$	−1187.00	−0.891811
$122$	0	0
$123$	−984.000	−0.721336
$124$	0	0
$125$	1284.00	0.918756
$126$	0	0
$127$	−176.000	−0.122972	−0.0614861	−	0.998108i	$-0.519584\pi$
$127$	−176.000	−0.122972	−0.0614861	+	0.998108i	$0.519584\pi$
$128$	0	0
$129$	−688.000	−0.469574
$130$	0	0
$131$	−1548.00	−1.03244	−0.516219	−	0.856457i	$-0.672661\pi$
$131$	−1548.00	−1.03244	−0.516219	+	0.856457i	$0.672661\pi$
$132$	0	0
$133$	−476.000	−0.310334
$134$	0	0
$135$	912.000	0.581426
$136$	0	0
$137$	378.000	0.235728	0.117864	−	0.993030i	$-0.462395\pi$
$137$	378.000	0.235728	0.117864	+	0.993030i	$0.462395\pi$
$138$	0	0
$139$	−2500.00	−1.52552	−0.762760	−	0.646682i	$-0.776156\pi$
$139$	−2500.00	−1.52552	−0.762760	+	0.646682i	$0.776156\pi$
$140$	0	0
$141$	−768.000	−0.458704
$142$	0	0
$143$	−984.000	−0.575428
$144$	0	0
$145$	1476.00	0.845346
$146$	0	0
$147$	196.000	0.109971
$148$	0	0
$149$	−846.000	−0.465148	−0.232574	−	0.972579i	$-0.574715\pi$
$149$	−846.000	−0.465148	−0.232574	+	0.972579i	$0.574715\pi$
$150$	0	0
$151$	2536.00	1.36673	0.683367	−	0.730075i	$-0.260515\pi$
$151$	2536.00	1.36673	0.683367	+	0.730075i	$0.260515\pi$
$152$	0	0
$153$	330.000	0.174372
$154$	0	0
$155$	−672.000	−0.348234
$156$	0	0
$157$	1186.00	0.602886	0.301443	−	0.953484i	$-0.402532\pi$
$157$	1186.00	0.602886	0.301443	+	0.953484i	$0.402532\pi$
$158$	0	0
$159$	−2232.00	−1.11326
$160$	0	0
$161$	1512.00	0.740138
$162$	0	0
$163$	2108.00	1.01295	0.506476	−	0.862254i	$-0.330948\pi$
$163$	2108.00	1.01295	0.506476	+	0.862254i	$0.330948\pi$
$164$	0	0
$165$	288.000	0.135883
$166$	0	0
$167$	1944.00	0.900786	0.450393	−	0.892830i	$-0.351284\pi$
$167$	1944.00	0.900786	0.450393	+	0.892830i	$0.351284\pi$
$168$	0	0
$169$	4527.00	2.06054
$170$	0	0
$171$	−748.000	−0.334509
$172$	0	0
$173$	1362.00	0.598560	0.299280	−	0.954165i	$-0.403253\pi$
$173$	1362.00	0.598560	0.299280	+	0.954165i	$0.403253\pi$
$174$	0	0
$175$	623.000	0.269111
$176$	0	0
$177$	2160.00	0.917263
$178$	0	0
$179$	1596.00	0.666428	0.333214	−	0.942851i	$-0.391867\pi$
$179$	1596.00	0.666428	0.333214	+	0.942851i	$0.391867\pi$
$180$	0	0
$181$	1690.00	0.694015	0.347007	−	0.937862i	$-0.387198\pi$
$181$	1690.00	0.694015	0.347007	+	0.937862i	$0.387198\pi$
$182$	0	0
$183$	−440.000	−0.177736
$184$	0	0
$185$	660.000	0.262293
$186$	0	0
$187$	360.000	0.140780
$188$	0	0
$189$	1064.00	0.409495
$190$	0	0
$191$	−3552.00	−1.34562	−0.672811	−	0.739815i	$-0.734913\pi$
$191$	−3552.00	−1.34562	−0.672811	+	0.739815i	$0.734913\pi$
$192$	0	0
$193$	−2686.00	−1.00177	−0.500887	−	0.865512i	$-0.666993\pi$
$193$	−2686.00	−1.00177	−0.500887	+	0.865512i	$0.666993\pi$
$194$	0	0
$195$	−1968.00	−0.722725
$196$	0	0
$197$	1410.00	0.509941	0.254970	−	0.966949i	$-0.417934\pi$
$197$	1410.00	0.509941	0.254970	+	0.966949i	$0.417934\pi$
$198$	0	0
$199$	2968.00	1.05727	0.528633	−	0.848850i	$-0.322705\pi$
$199$	2968.00	1.05727	0.528633	+	0.848850i	$0.322705\pi$
$200$	0	0
$201$	560.000	0.196514
$202$	0	0
$203$	1722.00	0.595373
$204$	0	0
$205$	1476.00	0.502870
$206$	0	0
$207$	2376.00	0.797794
$208$	0	0
$209$	−816.000	−0.270067
$210$	0	0
$211$	−1348.00	−0.439811	−0.219906	−	0.975521i	$-0.570575\pi$
$211$	−1348.00	−0.439811	−0.219906	+	0.975521i	$0.570575\pi$
$212$	0	0
$213$	3360.00	1.08086
$214$	0	0
$215$	1032.00	0.327357
$216$	0	0
$217$	−784.000	−0.245260
$218$	0	0
$219$	−2200.00	−0.678823
$220$	0	0
$221$	−2460.00	−0.748767
$222$	0	0
$223$	−3872.00	−1.16273	−0.581364	−	0.813644i	$-0.697481\pi$
$223$	−3872.00	−1.16273	−0.581364	+	0.813644i	$0.697481\pi$
$224$	0	0
$225$	979.000	0.290074
$226$	0	0
$227$	5364.00	1.56838	0.784188	−	0.620524i	$-0.213080\pi$
$227$	5364.00	1.56838	0.784188	+	0.620524i	$0.213080\pi$
$228$	0	0
$229$	874.000	0.252208	0.126104	−	0.992017i	$-0.459753\pi$
$229$	874.000	0.252208	0.126104	+	0.992017i	$0.459753\pi$
$230$	0	0
$231$	336.000	0.0957021
$232$	0	0
$233$	378.000	0.106282	0.0531408	−	0.998587i	$-0.483077\pi$
$233$	378.000	0.106282	0.0531408	+	0.998587i	$0.483077\pi$
$234$	0	0
$235$	1152.00	0.319780
$236$	0	0
$237$	832.000	0.228035
$238$	0	0
$239$	−1920.00	−0.519642	−0.259821	−	0.965657i	$-0.583664\pi$
$239$	−1920.00	−0.519642	−0.259821	+	0.965657i	$0.583664\pi$
$240$	0	0
$241$	4322.00	1.15521	0.577603	−	0.816318i	$-0.303988\pi$
$241$	4322.00	1.15521	0.577603	+	0.816318i	$0.303988\pi$
$242$	0	0
$243$	2860.00	0.755017
$244$	0	0
$245$	−294.000	−0.0766652
$246$	0	0
$247$	5576.00	1.43641
$248$	0	0
$249$	2064.00	0.525304
$250$	0	0
$251$	5292.00	1.33079	0.665395	−	0.746492i	$-0.268263\pi$
$251$	5292.00	1.33079	0.665395	+	0.746492i	$0.268263\pi$
$252$	0	0
$253$	2592.00	0.644101
$254$	0	0
$255$	720.000	0.176816
$256$	0	0
$257$	−5118.00	−1.24223	−0.621113	−	0.783721i	$-0.713319\pi$
$257$	−5118.00	−1.24223	−0.621113	+	0.783721i	$0.713319\pi$
$258$	0	0
$259$	770.000	0.184732
$260$	0	0
$261$	2706.00	0.641752
$262$	0	0
$263$	−3768.00	−0.883440	−0.441720	−	0.897153i	$-0.645632\pi$
$263$	−3768.00	−0.883440	−0.441720	+	0.897153i	$0.645632\pi$
$264$	0	0
$265$	3348.00	0.776098
$266$	0	0
$267$	−5592.00	−1.28174
$268$	0	0
$269$	−3918.00	−0.888047	−0.444024	−	0.896015i	$-0.646449\pi$
$269$	−3918.00	−0.888047	−0.444024	+	0.896015i	$0.646449\pi$
$270$	0	0
$271$	−4880.00	−1.09387	−0.546935	−	0.837175i	$-0.684206\pi$
$271$	−4880.00	−1.09387	−0.546935	+	0.837175i	$0.684206\pi$
$272$	0	0
$273$	−2296.00	−0.509012
$274$	0	0
$275$	1068.00	0.234192
$276$	0	0
$277$	3538.00	0.767429	0.383714	−	0.923452i	$-0.374645\pi$
$277$	3538.00	0.767429	0.383714	+	0.923452i	$0.374645\pi$
$278$	0	0
$279$	−1232.00	−0.264365
$280$	0	0
$281$	−5430.00	−1.15276	−0.576382	−	0.817180i	$-0.695536\pi$
$281$	−5430.00	−1.15276	−0.576382	+	0.817180i	$0.695536\pi$
$282$	0	0
$283$	−6436.00	−1.35187	−0.675937	−	0.736959i	$-0.736261\pi$
$283$	−6436.00	−1.35187	−0.675937	+	0.736959i	$0.736261\pi$
$284$	0	0
$285$	−1632.00	−0.339198
$286$	0	0
$287$	1722.00	0.354169
$288$	0	0
$289$	−4013.00	−0.816813
$290$	0	0
$291$	6344.00	1.27798
$292$	0	0
$293$	−1350.00	−0.269174	−0.134587	−	0.990902i	$-0.542971\pi$
$293$	−1350.00	−0.269174	−0.134587	+	0.990902i	$0.542971\pi$
$294$	0	0
$295$	−3240.00	−0.639458
$296$	0	0
$297$	1824.00	0.356361
$298$	0	0
$299$	−17712.0	−3.42579
$300$	0	0
$301$	1204.00	0.230556
$302$	0	0
$303$	4968.00	0.941928
$304$	0	0
$305$	660.000	0.123907
$306$	0	0
$307$	3332.00	0.619437	0.309719	−	0.950828i	$-0.399765\pi$
$307$	3332.00	0.619437	0.309719	+	0.950828i	$0.399765\pi$
$308$	0	0
$309$	−2720.00	−0.500762
$310$	0	0
$311$	4728.00	0.862059	0.431029	−	0.902338i	$-0.358151\pi$
$311$	4728.00	0.862059	0.431029	+	0.902338i	$0.358151\pi$
$312$	0	0
$313$	5114.00	0.923516	0.461758	−	0.887006i	$-0.347219\pi$
$313$	5114.00	0.923516	0.461758	+	0.887006i	$0.347219\pi$
$314$	0	0
$315$	−462.000	−0.0826373
$316$	0	0
$317$	−7206.00	−1.27675	−0.638374	−	0.769726i	$-0.720393\pi$
$317$	−7206.00	−1.27675	−0.638374	+	0.769726i	$0.720393\pi$
$318$	0	0
$319$	2952.00	0.518120
$320$	0	0
$321$	3984.00	0.692726
$322$	0	0
$323$	−2040.00	−0.351420
$324$	0	0
$325$	−7298.00	−1.24560
$326$	0	0
$327$	−5528.00	−0.934860
$328$	0	0
$329$	1344.00	0.225219
$330$	0	0
$331$	6260.00	1.03952	0.519759	−	0.854313i	$-0.326022\pi$
$331$	6260.00	1.03952	0.519759	+	0.854313i	$0.326022\pi$
$332$	0	0
$333$	1210.00	0.199122
$334$	0	0
$335$	−840.000	−0.136997
$336$	0	0
$337$	−5326.00	−0.860907	−0.430454	−	0.902613i	$-0.641646\pi$
$337$	−5326.00	−0.860907	−0.430454	+	0.902613i	$0.641646\pi$
$338$	0	0
$339$	−3000.00	−0.480642
$340$	0	0
$341$	−1344.00	−0.213436
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	5184.00	0.808977
$346$	0	0
$347$	36.0000	0.00556940	0.00278470	−	0.999996i	$-0.499114\pi$
$347$	36.0000	0.00556940	0.00278470	+	0.999996i	$0.499114\pi$
$348$	0	0
$349$	−3134.00	−0.480685	−0.240343	−	0.970688i	$-0.577260\pi$
$349$	−3134.00	−0.480685	−0.240343	+	0.970688i	$0.577260\pi$
$350$	0	0
$351$	−12464.0	−1.89538
$352$	0	0
$353$	1218.00	0.183648	0.0918238	−	0.995775i	$-0.470730\pi$
$353$	1218.00	0.183648	0.0918238	+	0.995775i	$0.470730\pi$
$354$	0	0
$355$	−5040.00	−0.753508
$356$	0	0
$357$	840.000	0.124531
$358$	0	0
$359$	10008.0	1.47131	0.735657	−	0.677354i	$-0.236873\pi$
$359$	10008.0	1.47131	0.735657	+	0.677354i	$0.236873\pi$
$360$	0	0
$361$	−2235.00	−0.325849
$362$	0	0
$363$	−4748.00	−0.686516
$364$	0	0
$365$	3300.00	0.473233
$366$	0	0
$367$	1072.00	0.152474	0.0762370	−	0.997090i	$-0.475709\pi$
$367$	1072.00	0.152474	0.0762370	+	0.997090i	$0.475709\pi$
$368$	0	0
$369$	2706.00	0.381758
$370$	0	0
$371$	3906.00	0.546602
$372$	0	0
$373$	274.000	0.0380353	0.0190177	−	0.999819i	$-0.493946\pi$
$373$	274.000	0.0380353	0.0190177	+	0.999819i	$0.493946\pi$
$374$	0	0
$375$	5136.00	0.707258
$376$	0	0
$377$	−20172.0	−2.75573
$378$	0	0
$379$	7652.00	1.03709	0.518545	−	0.855051i	$-0.326474\pi$
$379$	7652.00	1.03709	0.518545	+	0.855051i	$0.326474\pi$
$380$	0	0
$381$	−704.000	−0.0946641
$382$	0	0
$383$	−2160.00	−0.288175	−0.144087	−	0.989565i	$-0.546025\pi$
$383$	−2160.00	−0.288175	−0.144087	+	0.989565i	$0.546025\pi$
$384$	0	0
$385$	−504.000	−0.0667175
$386$	0	0
$387$	1892.00	0.248516
$388$	0	0
$389$	1074.00	0.139984	0.0699922	−	0.997548i	$-0.477703\pi$
$389$	1074.00	0.139984	0.0699922	+	0.997548i	$0.477703\pi$
$390$	0	0
$391$	6480.00	0.838127
$392$	0	0
$393$	−6192.00	−0.794771
$394$	0	0
$395$	−1248.00	−0.158971
$396$	0	0
$397$	−6926.00	−0.875582	−0.437791	−	0.899077i	$-0.644239\pi$
$397$	−6926.00	−0.875582	−0.437791	+	0.899077i	$0.644239\pi$
$398$	0	0
$399$	−1904.00	−0.238895
$400$	0	0
$401$	1938.00	0.241344	0.120672	−	0.992692i	$-0.461495\pi$
$401$	1938.00	0.241344	0.120672	+	0.992692i	$0.461495\pi$
$402$	0	0
$403$	9184.00	1.13521
$404$	0	0
$405$	1866.00	0.228944
$406$	0	0
$407$	1320.00	0.160762
$408$	0	0
$409$	−9574.00	−1.15747	−0.578733	−	0.815517i	$-0.696453\pi$
$409$	−9574.00	−1.15747	−0.578733	+	0.815517i	$0.696453\pi$
$410$	0	0
$411$	1512.00	0.181463
$412$	0	0
$413$	−3780.00	−0.450367
$414$	0	0
$415$	−3096.00	−0.366209
$416$	0	0
$417$	−10000.0	−1.17435
$418$	0	0
$419$	−5052.00	−0.589037	−0.294518	−	0.955646i	$-0.595159\pi$
$419$	−5052.00	−0.589037	−0.294518	+	0.955646i	$0.595159\pi$
$420$	0	0
$421$	−3422.00	−0.396147	−0.198074	−	0.980187i	$-0.563469\pi$
$421$	−3422.00	−0.396147	−0.198074	+	0.980187i	$0.563469\pi$
$422$	0	0
$423$	2112.00	0.242763
$424$	0	0
$425$	2670.00	0.304739
$426$	0	0
$427$	770.000	0.0872668
$428$	0	0
$429$	−3936.00	−0.442965
$430$	0	0
$431$	−2208.00	−0.246765	−0.123382	−	0.992359i	$-0.539374\pi$
$431$	−2208.00	−0.246765	−0.123382	+	0.992359i	$0.539374\pi$
$432$	0	0
$433$	−6814.00	−0.756259	−0.378129	−	0.925753i	$-0.623433\pi$
$433$	−6814.00	−0.756259	−0.378129	+	0.925753i	$0.623433\pi$
$434$	0	0
$435$	5904.00	0.650747
$436$	0	0
$437$	−14688.0	−1.60783
$438$	0	0
$439$	−12584.0	−1.36811	−0.684056	−	0.729429i	$-0.739786\pi$
$439$	−12584.0	−1.36811	−0.684056	+	0.729429i	$0.739786\pi$
$440$	0	0
$441$	−539.000	−0.0582011
$442$	0	0
$443$	6996.00	0.750316	0.375158	−	0.926961i	$-0.377588\pi$
$443$	6996.00	0.750316	0.375158	+	0.926961i	$0.377588\pi$
$444$	0	0
$445$	8388.00	0.893549
$446$	0	0
$447$	−3384.00	−0.358071
$448$	0	0
$449$	9474.00	0.995781	0.497891	−	0.867240i	$-0.334108\pi$
$449$	9474.00	0.995781	0.497891	+	0.867240i	$0.334108\pi$
$450$	0	0
$451$	2952.00	0.308213
$452$	0	0
$453$	10144.0	1.05211
$454$	0	0
$455$	3444.00	0.354851
$456$	0	0
$457$	5786.00	0.592249	0.296124	−	0.955149i	$-0.404306\pi$
$457$	5786.00	0.592249	0.296124	+	0.955149i	$0.404306\pi$
$458$	0	0
$459$	4560.00	0.463709
$460$	0	0
$461$	−3438.00	−0.347340	−0.173670	−	0.984804i	$-0.555563\pi$
$461$	−3438.00	−0.347340	−0.173670	+	0.984804i	$0.555563\pi$
$462$	0	0
$463$	−9392.00	−0.942728	−0.471364	−	0.881939i	$-0.656238\pi$
$463$	−9392.00	−0.942728	−0.471364	+	0.881939i	$0.656238\pi$
$464$	0	0
$465$	−2688.00	−0.268071
$466$	0	0
$467$	−4956.00	−0.491084	−0.245542	−	0.969386i	$-0.578966\pi$
$467$	−4956.00	−0.491084	−0.245542	+	0.969386i	$0.578966\pi$
$468$	0	0
$469$	−980.000	−0.0964866
$470$	0	0
$471$	4744.00	0.464102
$472$	0	0
$473$	2064.00	0.200640
$474$	0	0
$475$	−6052.00	−0.584600
$476$	0	0
$477$	6138.00	0.589182
$478$	0	0
$479$	20592.0	1.96424	0.982122	−	0.188248i	$-0.0602807\pi$
$479$	20592.0	1.96424	0.982122	+	0.188248i	$0.0602807\pi$
$480$	0	0
$481$	−9020.00	−0.855045
$482$	0	0
$483$	6048.00	0.569759
$484$	0	0
$485$	−9516.00	−0.890926
$486$	0	0
$487$	13432.0	1.24982	0.624910	−	0.780697i	$-0.285136\pi$
$487$	13432.0	1.24982	0.624910	+	0.780697i	$0.285136\pi$
$488$	0	0
$489$	8432.00	0.779771
$490$	0	0
$491$	−14172.0	−1.30259	−0.651297	−	0.758823i	$-0.725775\pi$
$491$	−14172.0	−1.30259	−0.651297	+	0.758823i	$0.725775\pi$
$492$	0	0
$493$	7380.00	0.674196
$494$	0	0
$495$	−792.000	−0.0719147
$496$	0	0
$497$	−5880.00	−0.530692
$498$	0	0
$499$	−5956.00	−0.534323	−0.267162	−	0.963652i	$-0.586086\pi$
$499$	−5956.00	−0.534323	−0.267162	+	0.963652i	$0.586086\pi$
$500$	0	0
$501$	7776.00	0.693425
$502$	0	0
$503$	−16968.0	−1.50411	−0.752053	−	0.659102i	$-0.770936\pi$
$503$	−16968.0	−1.50411	−0.752053	+	0.659102i	$0.770936\pi$
$504$	0	0
$505$	−7452.00	−0.656653
$506$	0	0
$507$	18108.0	1.58620
$508$	0	0
$509$	−5214.00	−0.454040	−0.227020	−	0.973890i	$-0.572898\pi$
$509$	−5214.00	−0.454040	−0.227020	+	0.973890i	$0.572898\pi$
$510$	0	0
$511$	3850.00	0.333295
$512$	0	0
$513$	−10336.0	−0.889562
$514$	0	0
$515$	4080.00	0.349100
$516$	0	0
$517$	2304.00	0.195996
$518$	0	0
$519$	5448.00	0.460772
$520$	0	0
$521$	−1398.00	−0.117558	−0.0587788	−	0.998271i	$-0.518721\pi$
$521$	−1398.00	−0.117558	−0.0587788	+	0.998271i	$0.518721\pi$
$522$	0	0
$523$	−18580.0	−1.55344	−0.776718	−	0.629849i	$-0.783117\pi$
$523$	−18580.0	−1.55344	−0.776718	+	0.629849i	$0.783117\pi$
$524$	0	0
$525$	2492.00	0.207162
$526$	0	0
$527$	−3360.00	−0.277730
$528$	0	0
$529$	34489.0	2.83463
$530$	0	0
$531$	−5940.00	−0.485450
$532$	0	0
$533$	−20172.0	−1.63930
$534$	0	0
$535$	−5976.00	−0.482925
$536$	0	0
$537$	6384.00	0.513017
$538$	0	0
$539$	−588.000	−0.0469888
$540$	0	0
$541$	18970.0	1.50755	0.753774	−	0.657133i	$-0.228231\pi$
$541$	18970.0	1.50755	0.753774	+	0.657133i	$0.228231\pi$
$542$	0	0
$543$	6760.00	0.534253
$544$	0	0
$545$	8292.00	0.651725
$546$	0	0
$547$	−16036.0	−1.25347	−0.626737	−	0.779231i	$-0.715610\pi$
$547$	−16036.0	−1.25347	−0.626737	+	0.779231i	$0.715610\pi$
$548$	0	0
$549$	1210.00	0.0940647
$550$	0	0
$551$	−16728.0	−1.29335
$552$	0	0
$553$	−1456.00	−0.111963
$554$	0	0
$555$	2640.00	0.201913
$556$	0	0
$557$	−8310.00	−0.632147	−0.316074	−	0.948735i	$-0.602365\pi$
$557$	−8310.00	−0.632147	−0.316074	+	0.948735i	$0.602365\pi$
$558$	0	0
$559$	−14104.0	−1.06715
$560$	0	0
$561$	1440.00	0.108372
$562$	0	0
$563$	7092.00	0.530892	0.265446	−	0.964126i	$-0.414481\pi$
$563$	7092.00	0.530892	0.265446	+	0.964126i	$0.414481\pi$
$564$	0	0
$565$	4500.00	0.335073
$566$	0	0
$567$	2177.00	0.161244
$568$	0	0
$569$	−7158.00	−0.527380	−0.263690	−	0.964608i	$-0.584940\pi$
$569$	−7158.00	−0.527380	−0.263690	+	0.964608i	$0.584940\pi$
$570$	0	0
$571$	6500.00	0.476386	0.238193	−	0.971218i	$-0.423445\pi$
$571$	6500.00	0.476386	0.238193	+	0.971218i	$0.423445\pi$
$572$	0	0
$573$	−14208.0	−1.03586
$574$	0	0
$575$	19224.0	1.39425
$576$	0	0
$577$	21794.0	1.57244	0.786218	−	0.617949i	$-0.212036\pi$
$577$	21794.0	1.57244	0.786218	+	0.617949i	$0.212036\pi$
$578$	0	0
$579$	−10744.0	−0.771166
$580$	0	0
$581$	−3612.00	−0.257919
$582$	0	0
$583$	6696.00	0.475678
$584$	0	0
$585$	5412.00	0.382493
$586$	0	0
$587$	9756.00	0.685985	0.342993	−	0.939338i	$-0.388559\pi$
$587$	9756.00	0.685985	0.342993	+	0.939338i	$0.388559\pi$
$588$	0	0
$589$	7616.00	0.532787
$590$	0	0
$591$	5640.00	0.392553
$592$	0	0
$593$	5586.00	0.386829	0.193414	−	0.981117i	$-0.438044\pi$
$593$	5586.00	0.386829	0.193414	+	0.981117i	$0.438044\pi$
$594$	0	0
$595$	−1260.00	−0.0868151
$596$	0	0
$597$	11872.0	0.813884
$598$	0	0
$599$	24.0000	0.00163708	0.000818542	−	1.00000i	$-0.499739\pi$
$599$	24.0000	0.00163708	0.000818542		1.00000i	$0.499739\pi$
$600$	0	0
$601$	4298.00	0.291712	0.145856	−	0.989306i	$-0.453406\pi$
$601$	4298.00	0.291712	0.145856	+	0.989306i	$0.453406\pi$
$602$	0	0
$603$	−1540.00	−0.104003
$604$	0	0
$605$	7122.00	0.478596
$606$	0	0
$607$	−8480.00	−0.567039	−0.283519	−	0.958966i	$-0.591502\pi$
$607$	−8480.00	−0.567039	−0.283519	+	0.958966i	$0.591502\pi$
$608$	0	0
$609$	6888.00	0.458318
$610$	0	0
$611$	−15744.0	−1.04245
$612$	0	0
$613$	1906.00	0.125583	0.0627917	−	0.998027i	$-0.480000\pi$
$613$	1906.00	0.125583	0.0627917	+	0.998027i	$0.480000\pi$
$614$	0	0
$615$	5904.00	0.387109
$616$	0	0
$617$	7482.00	0.488191	0.244096	−	0.969751i	$-0.421509\pi$
$617$	7482.00	0.488191	0.244096	+	0.969751i	$0.421509\pi$
$618$	0	0
$619$	−7348.00	−0.477126	−0.238563	−	0.971127i	$-0.576676\pi$
$619$	−7348.00	−0.477126	−0.238563	+	0.971127i	$0.576676\pi$
$620$	0	0
$621$	32832.0	2.12158
$622$	0	0
$623$	9786.00	0.629322
$624$	0	0
$625$	3421.00	0.218944
$626$	0	0
$627$	−3264.00	−0.207897
$628$	0	0
$629$	3300.00	0.209189
$630$	0	0
$631$	−4520.00	−0.285164	−0.142582	−	0.989783i	$-0.545540\pi$
$631$	−4520.00	−0.285164	−0.142582	+	0.989783i	$0.545540\pi$
$632$	0	0
$633$	−5392.00	−0.338567
$634$	0	0
$635$	1056.00	0.0659938
$636$	0	0
$637$	4018.00	0.249920
$638$	0	0
$639$	−9240.00	−0.572032
$640$	0	0
$641$	−19806.0	−1.22042	−0.610211	−	0.792239i	$-0.708915\pi$
$641$	−19806.0	−1.22042	−0.610211	+	0.792239i	$0.708915\pi$
$642$	0	0
$643$	−5020.00	−0.307884	−0.153942	−	0.988080i	$-0.549197\pi$
$643$	−5020.00	−0.307884	−0.153942	+	0.988080i	$0.549197\pi$
$644$	0	0
$645$	4128.00	0.252000
$646$	0	0
$647$	−28392.0	−1.72520	−0.862600	−	0.505886i	$-0.831166\pi$
$647$	−28392.0	−1.72520	−0.862600	+	0.505886i	$0.831166\pi$
$648$	0	0
$649$	−6480.00	−0.391930
$650$	0	0
$651$	−3136.00	−0.188801
$652$	0	0
$653$	17562.0	1.05246	0.526228	−	0.850343i	$-0.323606\pi$
$653$	17562.0	1.05246	0.526228	+	0.850343i	$0.323606\pi$
$654$	0	0
$655$	9288.00	0.554064
$656$	0	0
$657$	6050.00	0.359259
$658$	0	0
$659$	4716.00	0.278770	0.139385	−	0.990238i	$-0.455487\pi$
$659$	4716.00	0.278770	0.139385	+	0.990238i	$0.455487\pi$
$660$	0	0
$661$	22762.0	1.33939	0.669697	−	0.742635i	$-0.266424\pi$
$661$	22762.0	1.33939	0.669697	+	0.742635i	$0.266424\pi$
$662$	0	0
$663$	−9840.00	−0.576401
$664$	0	0
$665$	2856.00	0.166543
$666$	0	0
$667$	53136.0	3.08461
$668$	0	0
$669$	−15488.0	−0.895068
$670$	0	0
$671$	1320.00	0.0759434
$672$	0	0
$673$	4802.00	0.275042	0.137521	−	0.990499i	$-0.456086\pi$
$673$	4802.00	0.275042	0.137521	+	0.990499i	$0.456086\pi$
$674$	0	0
$675$	13528.0	0.771397
$676$	0	0
$677$	−21558.0	−1.22384	−0.611921	−	0.790919i	$-0.709603\pi$
$677$	−21558.0	−1.22384	−0.611921	+	0.790919i	$0.709603\pi$
$678$	0	0
$679$	−11102.0	−0.627475
$680$	0	0
$681$	21456.0	1.20734
$682$	0	0
$683$	3780.00	0.211768	0.105884	−	0.994378i	$-0.466233\pi$
$683$	3780.00	0.211768	0.105884	+	0.994378i	$0.466233\pi$
$684$	0	0
$685$	−2268.00	−0.126505
$686$	0	0
$687$	3496.00	0.194150
$688$	0	0
$689$	−45756.0	−2.52999
$690$	0	0
$691$	−5500.00	−0.302793	−0.151396	−	0.988473i	$-0.548377\pi$
$691$	−5500.00	−0.302793	−0.151396	+	0.988473i	$0.548377\pi$
$692$	0	0
$693$	−924.000	−0.0506491
$694$	0	0
$695$	15000.0	0.818680
$696$	0	0
$697$	7380.00	0.401058
$698$	0	0
$699$	1512.00	0.0818156
$700$	0	0
$701$	−10230.0	−0.551187	−0.275593	−	0.961274i	$-0.588874\pi$
$701$	−10230.0	−0.551187	−0.275593	+	0.961274i	$0.588874\pi$
$702$	0	0
$703$	−7480.00	−0.401299
$704$	0	0
$705$	4608.00	0.246166
$706$	0	0
$707$	−8694.00	−0.462477
$708$	0	0
$709$	−10190.0	−0.539765	−0.269883	−	0.962893i	$-0.586985\pi$
$709$	−10190.0	−0.539765	−0.269883	+	0.962893i	$0.586985\pi$
$710$	0	0
$711$	−2288.00	−0.120685
$712$	0	0
$713$	−24192.0	−1.27068
$714$	0	0
$715$	5904.00	0.308807
$716$	0	0
$717$	−7680.00	−0.400021
$718$	0	0
$719$	−9408.00	−0.487982	−0.243991	−	0.969777i	$-0.578457\pi$
$719$	−9408.00	−0.487982	−0.243991	+	0.969777i	$0.578457\pi$
$720$	0	0
$721$	4760.00	0.245869
$722$	0	0
$723$	17288.0	0.889278
$724$	0	0
$725$	21894.0	1.12155
$726$	0	0
$727$	33064.0	1.68676	0.843381	−	0.537316i	$-0.180562\pi$
$727$	33064.0	1.68676	0.843381	+	0.537316i	$0.180562\pi$
$728$	0	0
$729$	19837.0	1.00782
$730$	0	0
$731$	5160.00	0.261080
$732$	0	0
$733$	6322.00	0.318565	0.159283	−	0.987233i	$-0.449082\pi$
$733$	6322.00	0.318565	0.159283	+	0.987233i	$0.449082\pi$
$734$	0	0
$735$	−1176.00	−0.0590169
$736$	0	0
$737$	−1680.00	−0.0839669
$738$	0	0
$739$	−20740.0	−1.03239	−0.516193	−	0.856472i	$-0.672651\pi$
$739$	−20740.0	−1.03239	−0.516193	+	0.856472i	$0.672651\pi$
$740$	0	0
$741$	22304.0	1.10575
$742$	0	0
$743$	−32040.0	−1.58201	−0.791005	−	0.611810i	$-0.790442\pi$
$743$	−32040.0	−1.58201	−0.791005	+	0.611810i	$0.790442\pi$
$744$	0	0
$745$	5076.00	0.249624
$746$	0	0
$747$	−5676.00	−0.278011
$748$	0	0
$749$	−6972.00	−0.340122
$750$	0	0
$751$	12832.0	0.623497	0.311749	−	0.950165i	$-0.399085\pi$
$751$	12832.0	0.623497	0.311749	+	0.950165i	$0.399085\pi$
$752$	0	0
$753$	21168.0	1.02444
$754$	0	0
$755$	−15216.0	−0.733466
$756$	0	0
$757$	19906.0	0.955741	0.477870	−	0.878430i	$-0.341409\pi$
$757$	19906.0	0.955741	0.477870	+	0.878430i	$0.341409\pi$
$758$	0	0
$759$	10368.0	0.495829
$760$	0	0
$761$	10842.0	0.516455	0.258227	−	0.966084i	$-0.416862\pi$
$761$	10842.0	0.516455	0.258227	+	0.966084i	$0.416862\pi$
$762$	0	0
$763$	9674.00	0.459007
$764$	0	0
$765$	−1980.00	−0.0935778
$766$	0	0
$767$	44280.0	2.08456
$768$	0	0
$769$	28274.0	1.32586	0.662930	−	0.748681i	$-0.269313\pi$
$769$	28274.0	1.32586	0.662930	+	0.748681i	$0.269313\pi$
$770$	0	0
$771$	−20472.0	−0.956266
$772$	0	0
$773$	32346.0	1.50505	0.752526	−	0.658563i	$-0.228835\pi$
$773$	32346.0	1.50505	0.752526	+	0.658563i	$0.228835\pi$
$774$	0	0
$775$	−9968.00	−0.462014
$776$	0	0
$777$	3080.00	0.142206
$778$	0	0
$779$	−16728.0	−0.769375
$780$	0	0
$781$	−10080.0	−0.461832
$782$	0	0
$783$	37392.0	1.70662
$784$	0	0
$785$	−7116.00	−0.323543
$786$	0	0
$787$	30116.0	1.36407	0.682033	−	0.731322i	$-0.261096\pi$
$787$	30116.0	1.36407	0.682033	+	0.731322i	$0.261096\pi$
$788$	0	0
$789$	−15072.0	−0.680073
$790$	0	0
$791$	5250.00	0.235991
$792$	0	0
$793$	−9020.00	−0.403921
$794$	0	0
$795$	13392.0	0.597441
$796$	0	0
$797$	6594.00	0.293063	0.146532	−	0.989206i	$-0.453189\pi$
$797$	6594.00	0.293063	0.146532	+	0.989206i	$0.453189\pi$
$798$	0	0
$799$	5760.00	0.255036
$800$	0	0
$801$	15378.0	0.678346
$802$	0	0
$803$	6600.00	0.290048
$804$	0	0
$805$	−9072.00	−0.397200
$806$	0	0
$807$	−15672.0	−0.683619
$808$	0	0
$809$	−43014.0	−1.86933	−0.934667	−	0.355524i	$-0.884303\pi$
$809$	−43014.0	−1.86933	−0.934667	+	0.355524i	$0.884303\pi$
$810$	0	0
$811$	−14164.0	−0.613274	−0.306637	−	0.951827i	$-0.599204\pi$
$811$	−14164.0	−0.613274	−0.306637	+	0.951827i	$0.599204\pi$
$812$	0	0
$813$	−19520.0	−0.842062
$814$	0	0
$815$	−12648.0	−0.543608
$816$	0	0
$817$	−11696.0	−0.500846
$818$	0	0
$819$	6314.00	0.269388
$820$	0	0
$821$	−34830.0	−1.48060	−0.740302	−	0.672275i	$-0.765317\pi$
$821$	−34830.0	−1.48060	−0.740302	+	0.672275i	$0.765317\pi$
$822$	0	0
$823$	−31016.0	−1.31367	−0.656835	−	0.754035i	$-0.728105\pi$
$823$	−31016.0	−1.31367	−0.656835	+	0.754035i	$0.728105\pi$
$824$	0	0
$825$	4272.00	0.180281
$826$	0	0
$827$	9876.00	0.415263	0.207631	−	0.978207i	$-0.433425\pi$
$827$	9876.00	0.415263	0.207631	+	0.978207i	$0.433425\pi$
$828$	0	0
$829$	3154.00	0.132139	0.0660693	−	0.997815i	$-0.478954\pi$
$829$	3154.00	0.132139	0.0660693	+	0.997815i	$0.478954\pi$
$830$	0	0
$831$	14152.0	0.590767
$832$	0	0
$833$	−1470.00	−0.0611434
$834$	0	0
$835$	−11664.0	−0.483412
$836$	0	0
$837$	−17024.0	−0.703029
$838$	0	0
$839$	−36936.0	−1.51987	−0.759936	−	0.649998i	$-0.774770\pi$
$839$	−36936.0	−1.51987	−0.759936	+	0.649998i	$0.774770\pi$
$840$	0	0
$841$	36127.0	1.48128
$842$	0	0
$843$	−21720.0	−0.887398
$844$	0	0
$845$	−27162.0	−1.10580
$846$	0	0
$847$	8309.00	0.337073
$848$	0	0
$849$	−25744.0	−1.04067
$850$	0	0
$851$	23760.0	0.957088
$852$	0	0
$853$	−9638.00	−0.386869	−0.193434	−	0.981113i	$-0.561963\pi$
$853$	−9638.00	−0.386869	−0.193434	+	0.981113i	$0.561963\pi$
$854$	0	0
$855$	4488.00	0.179516
$856$	0	0
$857$	10266.0	0.409195	0.204597	−	0.978846i	$-0.434411\pi$
$857$	10266.0	0.409195	0.204597	+	0.978846i	$0.434411\pi$
$858$	0	0
$859$	−4084.00	−0.162217	−0.0811084	−	0.996705i	$-0.525846\pi$
$859$	−4084.00	−0.162217	−0.0811084	+	0.996705i	$0.525846\pi$
$860$	0	0
$861$	6888.00	0.272639
$862$	0	0
$863$	192.000	0.00757330	0.00378665	−	0.999993i	$-0.498795\pi$
$863$	192.000	0.00757330	0.00378665	+	0.999993i	$0.498795\pi$
$864$	0	0
$865$	−8172.00	−0.321221
$866$	0	0
$867$	−16052.0	−0.628783
$868$	0	0
$869$	−2496.00	−0.0974350
$870$	0	0
$871$	11480.0	0.446596
$872$	0	0
$873$	−17446.0	−0.676355
$874$	0	0
$875$	−8988.00	−0.347257
$876$	0	0
$877$	−19910.0	−0.766605	−0.383303	−	0.923623i	$-0.625213\pi$
$877$	−19910.0	−0.766605	−0.383303	+	0.923623i	$0.625213\pi$
$878$	0	0
$879$	−5400.00	−0.207210
$880$	0	0
$881$	14802.0	0.566052	0.283026	−	0.959112i	$-0.408662\pi$
$881$	14802.0	0.566052	0.283026	+	0.959112i	$0.408662\pi$
$882$	0	0
$883$	−32548.0	−1.24046	−0.620231	−	0.784419i	$-0.712961\pi$
$883$	−32548.0	−1.24046	−0.620231	+	0.784419i	$0.712961\pi$
$884$	0	0
$885$	−12960.0	−0.492255
$886$	0	0
$887$	−1464.00	−0.0554186	−0.0277093	−	0.999616i	$-0.508821\pi$
$887$	−1464.00	−0.0554186	−0.0277093	+	0.999616i	$0.508821\pi$
$888$	0	0
$889$	1232.00	0.0464791
$890$	0	0
$891$	3732.00	0.140322
$892$	0	0
$893$	−13056.0	−0.489252
$894$	0	0
$895$	−9576.00	−0.357643
$896$	0	0
$897$	−70848.0	−2.63717
$898$	0	0
$899$	−27552.0	−1.02215
$900$	0	0
$901$	16740.0	0.618968
$902$	0	0
$903$	4816.00	0.177482
$904$	0	0
$905$	−10140.0	−0.372448
$906$	0	0
$907$	−49564.0	−1.81449	−0.907247	−	0.420599i	$-0.861820\pi$
$907$	−49564.0	−1.81449	−0.907247	+	0.420599i	$0.861820\pi$
$908$	0	0
$909$	−13662.0	−0.498504
$910$	0	0
$911$	−8448.00	−0.307239	−0.153619	−	0.988130i	$-0.549093\pi$
$911$	−8448.00	−0.307239	−0.153619	+	0.988130i	$0.549093\pi$
$912$	0	0
$913$	−6192.00	−0.224453
$914$	0	0
$915$	2640.00	0.0953833
$916$	0	0
$917$	10836.0	0.390225
$918$	0	0
$919$	−14600.0	−0.524058	−0.262029	−	0.965060i	$-0.584392\pi$
$919$	−14600.0	−0.524058	−0.262029	+	0.965060i	$0.584392\pi$
$920$	0	0
$921$	13328.0	0.476843
$922$	0	0
$923$	68880.0	2.45635
$924$	0	0
$925$	9790.00	0.347993
$926$	0	0
$927$	7480.00	0.265022
$928$	0	0
$929$	−21102.0	−0.745247	−0.372623	−	0.927983i	$-0.621542\pi$
$929$	−21102.0	−0.745247	−0.372623	+	0.927983i	$0.621542\pi$
$930$	0	0
$931$	3332.00	0.117295
$932$	0	0
$933$	18912.0	0.663613
$934$	0	0
$935$	−2160.00	−0.0755503
$936$	0	0
$937$	−20806.0	−0.725403	−0.362701	−	0.931905i	$-0.618146\pi$
$937$	−20806.0	−0.725403	−0.362701	+	0.931905i	$0.618146\pi$
$938$	0	0
$939$	20456.0	0.710923
$940$	0	0
$941$	−24510.0	−0.849100	−0.424550	−	0.905404i	$-0.639568\pi$
$941$	−24510.0	−0.849100	−0.424550	+	0.905404i	$0.639568\pi$
$942$	0	0
$943$	53136.0	1.83494
$944$	0	0
$945$	−6384.00	−0.219758
$946$	0	0
$947$	−44148.0	−1.51491	−0.757454	−	0.652889i	$-0.773557\pi$
$947$	−44148.0	−1.51491	−0.757454	+	0.652889i	$0.773557\pi$
$948$	0	0
$949$	−45100.0	−1.54268
$950$	0	0
$951$	−28824.0	−0.982841
$952$	0	0
$953$	27114.0	0.921625	0.460812	−	0.887498i	$-0.347558\pi$
$953$	27114.0	0.921625	0.460812	+	0.887498i	$0.347558\pi$
$954$	0	0
$955$	21312.0	0.722136
$956$	0	0
$957$	11808.0	0.398849
$958$	0	0
$959$	−2646.00	−0.0890968
$960$	0	0
$961$	−17247.0	−0.578933
$962$	0	0
$963$	−10956.0	−0.366617
$964$	0	0
$965$	16116.0	0.537609
$966$	0	0
$967$	10264.0	0.341332	0.170666	−	0.985329i	$-0.445408\pi$
$967$	10264.0	0.341332	0.170666	+	0.985329i	$0.445408\pi$
$968$	0	0
$969$	−8160.00	−0.270523
$970$	0	0
$971$	51468.0	1.70102	0.850508	−	0.525962i	$-0.176295\pi$
$971$	51468.0	1.70102	0.850508	+	0.525962i	$0.176295\pi$
$972$	0	0
$973$	17500.0	0.576592
$974$	0	0
$975$	−29192.0	−0.958864
$976$	0	0
$977$	−23790.0	−0.779027	−0.389514	−	0.921021i	$-0.627357\pi$
$977$	−23790.0	−0.779027	−0.389514	+	0.921021i	$0.627357\pi$
$978$	0	0
$979$	16776.0	0.547664
$980$	0	0
$981$	15202.0	0.494763
$982$	0	0
$983$	−26424.0	−0.857370	−0.428685	−	0.903454i	$-0.641023\pi$
$983$	−26424.0	−0.857370	−0.428685	+	0.903454i	$0.641023\pi$
$984$	0	0
$985$	−8460.00	−0.273663
$986$	0	0
$987$	5376.00	0.173374
$988$	0	0
$989$	37152.0	1.19450
$990$	0	0
$991$	−39488.0	−1.26577	−0.632885	−	0.774246i	$-0.718129\pi$
$991$	−39488.0	−1.26577	−0.632885	+	0.774246i	$0.718129\pi$
$992$	0	0
$993$	25040.0	0.800222
$994$	0	0
$995$	−17808.0	−0.567388
$996$	0	0
$997$	−30854.0	−0.980096	−0.490048	−	0.871695i	$-0.663021\pi$
$997$	−30854.0	−0.980096	−0.490048	+	0.871695i	$0.663021\pi$
$998$	0	0
$999$	16720.0	0.529527

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.a.m.1.1		1
4.3	odd	2		448.4.a.d.1.1		1
8.3	odd	2		28.4.a.b.1.1	✓	1
8.5	even	2		112.4.a.c.1.1		1
24.5	odd	2		1008.4.a.f.1.1		1
24.11	even	2		252.4.a.c.1.1		1
40.3	even	4		700.4.e.f.449.2		2
40.19	odd	2		700.4.a.e.1.1		1
40.27	even	4		700.4.e.f.449.1		2
56.3	even	6		196.4.e.d.177.1		2
56.11	odd	6		196.4.e.c.177.1		2
56.13	odd	2		784.4.a.n.1.1		1
56.19	even	6		196.4.e.d.165.1		2
56.27	even	2		196.4.a.b.1.1		1
56.51	odd	6		196.4.e.c.165.1		2
168.11	even	6		1764.4.k.k.1549.1		2
168.59	odd	6		1764.4.k.e.1549.1		2
168.83	odd	2		1764.4.a.k.1.1		1
168.107	even	6		1764.4.k.k.361.1		2
168.131	odd	6		1764.4.k.e.361.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.4.a.b.1.1	✓	1	8.3	odd	2
112.4.a.c.1.1		1	8.5	even	2
196.4.a.b.1.1		1	56.27	even	2
196.4.e.c.165.1		2	56.51	odd	6
196.4.e.c.177.1		2	56.11	odd	6
196.4.e.d.165.1		2	56.19	even	6
196.4.e.d.177.1		2	56.3	even	6
252.4.a.c.1.1		1	24.11	even	2
448.4.a.d.1.1		1	4.3	odd	2
448.4.a.m.1.1		1	1.1	even	1	trivial
700.4.a.e.1.1		1	40.19	odd	2
700.4.e.f.449.1		2	40.27	even	4
700.4.e.f.449.2		2	40.3	even	4
784.4.a.n.1.1		1	56.13	odd	2
1008.4.a.f.1.1		1	24.5	odd	2
1764.4.a.k.1.1		1	168.83	odd	2
1764.4.k.e.361.1		2	168.131	odd	6
1764.4.k.e.1549.1		2	168.59	odd	6
1764.4.k.k.361.1		2	168.107	even	6
1764.4.k.k.1549.1		2	168.11	even	6

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more