LMFDB - Embedded newform 28.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [28,4,Mod(1,28)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(28, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("28.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 28 = 2^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	28.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:	$1.65205348016$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	28.1

$q$-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	$-0.413529\pi$
$5$	6.00000	0.536656	0.268328	+	0.963328i	$0.413529\pi$
$6$	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	$-0.838954\pi$
$13$	−82.0000	−1.74944	−0.874720	+	0.484629i	$0.838954\pi$
$14$	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	$-0.0651750\pi$
$23$	216.000	1.95822	0.979111	+	0.203326i	$0.0651750\pi$
$24$	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	$-0.211321\pi$
$29$	246.000	1.57521	0.787604	+	0.616181i	$0.211321\pi$
$30$	0	0
$31$	−112.000	−0.648897	−0.324448	−	0.945903i	$-0.605179\pi$
$31$	−112.000	−0.648897	−0.324448	+	0.945903i	$0.605179\pi$
$32$	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	$-0.421417\pi$
$37$	110.000	0.488754	0.244377	+	0.969680i	$0.421417\pi$
$38$	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	$-0.403701\pi$
$47$	192.000	0.595874	0.297937	+	0.954586i	$0.403701\pi$
$48$	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	$-0.242723\pi$
$53$	558.000	1.44617	0.723087	+	0.690757i	$0.242723\pi$
$54$	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	$-0.463171\pi$
$61$	110.000	0.230886	0.115443	+	0.993314i	$0.463171\pi$
$62$	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	$-0.747727\pi$
$71$	−840.000	−1.40408	−0.702040	+	0.712138i	$0.747727\pi$
$72$	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	$-0.547320\pi$
$79$	−208.000	−0.296226	−0.148113	+	0.988970i	$0.547320\pi$
$80$	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	$-0.709554\pi$
$101$	−1242.00	−1.22360	−0.611800	+	0.791012i	$0.709554\pi$
$102$	0	0
$103$	680.000	0.650509	0.325254	−	0.945627i	$-0.394550\pi$
$103$	680.000	0.650509	0.325254	+	0.945627i	$0.394550\pi$
$104$	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	$-0.292289\pi$
$109$	1382.00	1.21442	0.607209	+	0.794542i	$0.292289\pi$
$110$	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	$-0.480416\pi$
$127$	176.000	0.122972	0.0614861	+	0.998108i	$0.480416\pi$
$128$	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	$-0.425285\pi$
$149$	846.000	0.465148	0.232574	+	0.972579i	$0.425285\pi$
$150$	0	0
$151$	−2536.00	−1.36673	−0.683367	−	0.730075i	$-0.739485\pi$
$151$	−2536.00	−1.36673	−0.683367	+	0.730075i	$0.739485\pi$
$152$	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	$-0.597468\pi$
$157$	−1186.00	−0.602886	−0.301443	+	0.953484i	$0.597468\pi$
$158$	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	$-0.648716\pi$
$167$	−1944.00	−0.900786	−0.450393	+	0.892830i	$0.648716\pi$
$168$	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	$-0.596747\pi$
$173$	−1362.00	−0.598560	−0.299280	+	0.954165i	$0.596747\pi$
$174$	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	$-0.612802\pi$
$181$	−1690.00	−0.694015	−0.347007	+	0.937862i	$0.612802\pi$
$182$	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	$-0.265087\pi$
$191$	3552.00	1.34562	0.672811	+	0.739815i	$0.265087\pi$
$192$	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	$-0.582066\pi$
$197$	−1410.00	−0.509941	−0.254970	+	0.966949i	$0.582066\pi$
$198$	0	0
$199$	−2968.00	−1.05727	−0.528633	−	0.848850i	$-0.677295\pi$
$199$	−2968.00	−1.05727	−0.528633	+	0.848850i	$0.677295\pi$
$200$	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	$-0.302519\pi$
$223$	3872.00	1.16273	0.581364	+	0.813644i	$0.302519\pi$
$224$	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	$-0.540247\pi$
$229$	−874.000	−0.252208	−0.126104	+	0.992017i	$0.540247\pi$
$230$	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	$-0.416336\pi$
$239$	1920.00	0.519642	0.259821	+	0.965657i	$0.416336\pi$
$240$	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	$-0.354368\pi$
$263$	3768.00	0.883440	0.441720	+	0.897153i	$0.354368\pi$
$264$	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	$-0.353551\pi$
$269$	3918.00	0.888047	0.444024	+	0.896015i	$0.353551\pi$
$270$	0	0
$271$	4880.00	1.09387	0.546935	−	0.837175i	$-0.315794\pi$
$271$	4880.00	1.09387	0.546935	+	0.837175i	$0.315794\pi$
$272$	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	$-0.625355\pi$
$277$	−3538.00	−0.767429	−0.383714	+	0.923452i	$0.625355\pi$
$278$	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	$-0.457029\pi$
$293$	1350.00	0.269174	0.134587	+	0.990902i	$0.457029\pi$
$294$	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	$-0.641849\pi$
$311$	−4728.00	−0.862059	−0.431029	+	0.902338i	$0.641849\pi$
$312$	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	$-0.279607\pi$
$317$	7206.00	1.27675	0.638374	+	0.769726i	$0.279607\pi$
$318$	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	$-0.422740\pi$
$349$	3134.00	0.480685	0.240343	+	0.970688i	$0.422740\pi$
$350$	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	$-0.763127\pi$
$359$	−10008.0	−1.47131	−0.735657	+	0.677354i	$0.763127\pi$
$360$	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	$-0.524291\pi$
$367$	−1072.00	−0.152474	−0.0762370	+	0.997090i	$0.524291\pi$
$368$	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	$-0.506054\pi$
$373$	−274.000	−0.0380353	−0.0190177	+	0.999819i	$0.506054\pi$
$374$	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	$-0.453975\pi$
$383$	2160.00	0.288175	0.144087	+	0.989565i	$0.453975\pi$
$384$	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	$-0.522297\pi$
$389$	−1074.00	−0.139984	−0.0699922	+	0.997548i	$0.522297\pi$
$390$	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	$-0.355761\pi$
$397$	6926.00	0.875582	0.437791	+	0.899077i	$0.355761\pi$
$398$	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	$-0.436531\pi$
$421$	3422.00	0.396147	0.198074	+	0.980187i	$0.436531\pi$
$422$	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	$-0.460626\pi$
$431$	2208.00	0.246765	0.123382	+	0.992359i	$0.460626\pi$
$432$	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	$-0.260214\pi$
$439$	12584.0	1.36811	0.684056	+	0.729429i	$0.260214\pi$
$440$	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	$-0.444437\pi$
$461$	3438.00	0.347340	0.173670	+	0.984804i	$0.444437\pi$
$462$	0	0
$463$	9392.00	0.942728	0.471364	−	0.881939i	$-0.343762\pi$
$463$	9392.00	0.942728	0.471364	+	0.881939i	$0.343762\pi$
$464$	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	$-0.939719\pi$
$479$	−20592.0	−1.96424	−0.982122	+	0.188248i	$0.939719\pi$
$480$	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	$-0.714864\pi$
$487$	−13432.0	−1.24982	−0.624910	+	0.780697i	$0.714864\pi$
$488$	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	$-0.229064\pi$
$503$	16968.0	1.50411	0.752053	+	0.659102i	$0.229064\pi$
$504$	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	$-0.427102\pi$
$509$	5214.00	0.454040	0.227020	+	0.973890i	$0.427102\pi$
$510$	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	$-0.771769\pi$
$541$	−18970.0	−1.50755	−0.753774	+	0.657133i	$0.771769\pi$
$542$	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	$-0.397635\pi$
$557$	8310.00	0.632147	0.316074	+	0.948735i	$0.397635\pi$
$558$	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	$-0.500261\pi$
$599$	−24.0000	−0.00163708	−0.000818542		1.00000i	$0.500261\pi$
$600$	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	$-0.408498\pi$
$607$	8480.00	0.567039	0.283519	+	0.958966i	$0.408498\pi$
$608$	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	$-0.520000\pi$
$613$	−1906.00	−0.125583	−0.0627917	+	0.998027i	$0.520000\pi$
$614$	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	$-0.454460\pi$
$631$	4520.00	0.285164	0.142582	+	0.989783i	$0.454460\pi$
$632$	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	$-0.168834\pi$
$647$	28392.0	1.72520	0.862600	+	0.505886i	$0.168834\pi$
$648$	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	$-0.676394\pi$
$653$	−17562.0	−1.05246	−0.526228	+	0.850343i	$0.676394\pi$
$654$	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	$-0.733576\pi$
$661$	−22762.0	−1.33939	−0.669697	+	0.742635i	$0.733576\pi$
$662$	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	$-0.290397\pi$
$677$	21558.0	1.22384	0.611921	+	0.790919i	$0.290397\pi$
$678$	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	$-0.411126\pi$
$701$	10230.0	0.551187	0.275593	+	0.961274i	$0.411126\pi$
$702$	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	$-0.413015\pi$
$709$	10190.0	0.539765	0.269883	+	0.962893i	$0.413015\pi$
$710$	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	$-0.421543\pi$
$719$	9408.00	0.487982	0.243991	+	0.969777i	$0.421543\pi$
$720$	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	$-0.819438\pi$
$727$	−33064.0	−1.68676	−0.843381	+	0.537316i	$0.819438\pi$
$728$	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	$-0.550918\pi$
$733$	−6322.00	−0.318565	−0.159283	+	0.987233i	$0.550918\pi$
$734$	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	$-0.209558\pi$
$743$	32040.0	1.58201	0.791005	+	0.611810i	$0.209558\pi$
$744$	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	$-0.600915\pi$
$751$	−12832.0	−0.623497	−0.311749	+	0.950165i	$0.600915\pi$
$752$	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	$-0.658591\pi$
$757$	−19906.0	−0.955741	−0.477870	+	0.878430i	$0.658591\pi$
$758$	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	$-0.771165\pi$
$773$	−32346.0	−1.50505	−0.752526	+	0.658563i	$0.771165\pi$
$774$	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	$-0.546811\pi$
$797$	−6594.00	−0.293063	−0.146532	+	0.989206i	$0.546811\pi$
$798$	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	$-0.234683\pi$
$821$	34830.0	1.48060	0.740302	+	0.672275i	$0.234683\pi$
$822$	0	0
$823$	31016.0	1.31367	0.656835	−	0.754035i	$-0.271895\pi$
$823$	31016.0	1.31367	0.656835	+	0.754035i	$0.271895\pi$
$824$	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	$-0.521046\pi$
$829$	−3154.00	−0.132139	−0.0660693	+	0.997815i	$0.521046\pi$
$830$	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	$-0.225230\pi$
$839$	36936.0	1.51987	0.759936	+	0.649998i	$0.225230\pi$
$840$	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	$-0.438037\pi$
$853$	9638.00	0.386869	0.193434	+	0.981113i	$0.438037\pi$
$854$	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	$-0.501205\pi$
$863$	−192.000	−0.00757330	−0.00378665	+	0.999993i	$0.501205\pi$
$864$	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	$-0.374787\pi$
$877$	19910.0	0.766605	0.383303	+	0.923623i	$0.374787\pi$
$878$	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	$-0.491179\pi$
$887$	1464.00	0.0554186	0.0277093	+	0.999616i	$0.491179\pi$
$888$	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	$-0.450907\pi$
$911$	8448.00	0.307239	0.153619	+	0.988130i	$0.450907\pi$
$912$	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	$-0.415608\pi$
$919$	14600.0	0.524058	0.262029	+	0.965060i	$0.415608\pi$
$920$	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	$-0.360432\pi$
$941$	24510.0	0.849100	0.424550	+	0.905404i	$0.360432\pi$
$942$	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	$-0.554592\pi$
$967$	−10264.0	−0.341332	−0.170666	+	0.985329i	$0.554592\pi$
$968$	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	$-0.358977\pi$
$983$	26424.0	0.857370	0.428685	+	0.903454i	$0.358977\pi$
$984$	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	$-0.281871\pi$
$991$	39488.0	1.26577	0.632885	+	0.774246i	$0.281871\pi$
$992$	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	$-0.336979\pi$
$997$	30854.0	0.980096	0.490048	+	0.871695i	$0.336979\pi$
$998$	0	0
$999$	−16720.0	−0.529527

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.4.a.b.1.1	✓	1
3.2	odd	2		252.4.a.c.1.1		1
4.3	odd	2		112.4.a.c.1.1		1
5.2	odd	4		700.4.e.f.449.1		2
5.3	odd	4		700.4.e.f.449.2		2
5.4	even	2		700.4.a.e.1.1		1
7.2	even	3		196.4.e.c.165.1		2
7.3	odd	6		196.4.e.d.177.1		2
7.4	even	3		196.4.e.c.177.1		2
7.5	odd	6		196.4.e.d.165.1		2
7.6	odd	2		196.4.a.b.1.1		1
8.3	odd	2		448.4.a.m.1.1		1
8.5	even	2		448.4.a.d.1.1		1
12.11	even	2		1008.4.a.f.1.1		1
21.2	odd	6		1764.4.k.k.361.1		2
21.5	even	6		1764.4.k.e.361.1		2
21.11	odd	6		1764.4.k.k.1549.1		2
21.17	even	6		1764.4.k.e.1549.1		2
21.20	even	2		1764.4.a.k.1.1		1
28.27	even	2		784.4.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.4.a.b.1.1	✓	1	1.1	even	1	trivial
112.4.a.c.1.1		1	4.3	odd	2
196.4.a.b.1.1		1	7.6	odd	2
196.4.e.c.165.1		2	7.2	even	3
196.4.e.c.177.1		2	7.4	even	3
196.4.e.d.165.1		2	7.5	odd	6
196.4.e.d.177.1		2	7.3	odd	6
252.4.a.c.1.1		1	3.2	odd	2
448.4.a.d.1.1		1	8.5	even	2
448.4.a.m.1.1		1	8.3	odd	2
700.4.a.e.1.1		1	5.4	even	2
700.4.e.f.449.1		2	5.2	odd	4
700.4.e.f.449.2		2	5.3	odd	4
784.4.a.n.1.1		1	28.27	even	2
1008.4.a.f.1.1		1	12.11	even	2
1764.4.a.k.1.1		1	21.20	even	2
1764.4.k.e.361.1		2	21.5	even	6
1764.4.k.e.1549.1		2	21.17	even	6
1764.4.k.k.361.1		2	21.2	odd	6
1764.4.k.k.1549.1		2	21.11	odd	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:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	28.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more