LMFDB - Embedded newform 2496.4.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,4,Mod(1,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2496.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2496.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:	$147.268767374$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2496.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +2.00000 q^{5} +32.0000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} +2.00000 q^{5} +32.0000 q^{7} +9.00000 q^{9} -68.0000 q^{11} -13.0000 q^{13} +6.00000 q^{15} -14.0000 q^{17} +4.00000 q^{19} +96.0000 q^{21} -72.0000 q^{23} -121.000 q^{25} +27.0000 q^{27} -102.000 q^{29} +136.000 q^{31} -204.000 q^{33} +64.0000 q^{35} +386.000 q^{37} -39.0000 q^{39} +250.000 q^{41} -140.000 q^{43} +18.0000 q^{45} +296.000 q^{47} +681.000 q^{49} -42.0000 q^{51} -526.000 q^{53} -136.000 q^{55} +12.0000 q^{57} +332.000 q^{59} +410.000 q^{61} +288.000 q^{63} -26.0000 q^{65} +596.000 q^{67} -216.000 q^{69} +880.000 q^{71} +506.000 q^{73} -363.000 q^{75} -2176.00 q^{77} +640.000 q^{79} +81.0000 q^{81} +1380.00 q^{83} -28.0000 q^{85} -306.000 q^{87} +1450.00 q^{89} -416.000 q^{91} +408.000 q^{93} +8.00000 q^{95} -446.000 q^{97} -612.000 q^{99} +O(q^{100})$

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	2.00000	0.178885	0.0894427	−	0.995992i	$-0.471491\pi$
$5$	2.00000	0.178885	0.0894427	+	0.995992i	$0.471491\pi$
$6$	0	0
$7$	32.0000	1.72784	0.863919	−	0.503631i	$-0.168003\pi$
$7$	32.0000	1.72784	0.863919	+	0.503631i	$0.168003\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−68.0000	−1.86389	−0.931944	−	0.362602i	$-0.881889\pi$
$11$	−68.0000	−1.86389	−0.931944	+	0.362602i	$0.881889\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	6.00000	0.103280
$16$	0	0
$17$	−14.0000	−0.199735	−0.0998676	−	0.995001i	$-0.531842\pi$
$17$	−14.0000	−0.199735	−0.0998676	+	0.995001i	$0.531842\pi$
$18$	0	0
$19$	4.00000	0.0482980	0.0241490	−	0.999708i	$-0.492312\pi$
$19$	4.00000	0.0482980	0.0241490	+	0.999708i	$0.492312\pi$
$20$	0	0
$21$	96.0000	0.997567
$22$	0	0
$23$	−72.0000	−0.652741	−0.326370	−	0.945242i	$-0.605826\pi$
$23$	−72.0000	−0.652741	−0.326370	+	0.945242i	$0.605826\pi$
$24$	0	0
$25$	−121.000	−0.968000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−102.000	−0.653135	−0.326568	−	0.945174i	$-0.605892\pi$
$29$	−102.000	−0.653135	−0.326568	+	0.945174i	$0.605892\pi$
$30$	0	0
$31$	136.000	0.787946	0.393973	−	0.919122i	$-0.371100\pi$
$31$	136.000	0.787946	0.393973	+	0.919122i	$0.371100\pi$
$32$	0	0
$33$	−204.000	−1.07612
$34$	0	0
$35$	64.0000	0.309085
$36$	0	0
$37$	386.000	1.71508	0.857541	−	0.514416i	$-0.171991\pi$
$37$	386.000	1.71508	0.857541	+	0.514416i	$0.171991\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	250.000	0.952279	0.476140	−	0.879370i	$-0.342036\pi$
$41$	250.000	0.952279	0.476140	+	0.879370i	$0.342036\pi$
$42$	0	0
$43$	−140.000	−0.496507	−0.248253	−	0.968695i	$-0.579857\pi$
$43$	−140.000	−0.496507	−0.248253	+	0.968695i	$0.579857\pi$
$44$	0	0
$45$	18.0000	0.0596285
$46$	0	0
$47$	296.000	0.918639	0.459320	−	0.888271i	$-0.348093\pi$
$47$	296.000	0.918639	0.459320	+	0.888271i	$0.348093\pi$
$48$	0	0
$49$	681.000	1.98542
$50$	0	0
$51$	−42.0000	−0.115317
$52$	0	0
$53$	−526.000	−1.36324	−0.681619	−	0.731707i	$-0.738724\pi$
$53$	−526.000	−1.36324	−0.681619	+	0.731707i	$0.738724\pi$
$54$	0	0
$55$	−136.000	−0.333422
$56$	0	0
$57$	12.0000	0.0278849
$58$	0	0
$59$	332.000	0.732588	0.366294	−	0.930499i	$-0.380626\pi$
$59$	332.000	0.732588	0.366294	+	0.930499i	$0.380626\pi$
$60$	0	0
$61$	410.000	0.860576	0.430288	−	0.902692i	$-0.358412\pi$
$61$	410.000	0.860576	0.430288	+	0.902692i	$0.358412\pi$
$62$	0	0
$63$	288.000	0.575946
$64$	0	0
$65$	−26.0000	−0.0496139
$66$	0	0
$67$	596.000	1.08676	0.543381	−	0.839487i	$-0.317144\pi$
$67$	596.000	1.08676	0.543381	+	0.839487i	$0.317144\pi$
$68$	0	0
$69$	−216.000	−0.376860
$70$	0	0
$71$	880.000	1.47094	0.735470	−	0.677557i	$-0.236961\pi$
$71$	880.000	1.47094	0.735470	+	0.677557i	$0.236961\pi$
$72$	0	0
$73$	506.000	0.811272	0.405636	−	0.914035i	$-0.367050\pi$
$73$	506.000	0.811272	0.405636	+	0.914035i	$0.367050\pi$
$74$	0	0
$75$	−363.000	−0.558875
$76$	0	0
$77$	−2176.00	−3.22050
$78$	0	0
$79$	640.000	0.911464	0.455732	−	0.890117i	$-0.349378\pi$
$79$	640.000	0.911464	0.455732	+	0.890117i	$0.349378\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1380.00	1.82500	0.912498	−	0.409081i	$-0.134151\pi$
$83$	1380.00	1.82500	0.912498	+	0.409081i	$0.134151\pi$
$84$	0	0
$85$	−28.0000	−0.0357297
$86$	0	0
$87$	−306.000	−0.377088
$88$	0	0
$89$	1450.00	1.72696	0.863481	−	0.504381i	$-0.168279\pi$
$89$	1450.00	1.72696	0.863481	+	0.504381i	$0.168279\pi$
$90$	0	0
$91$	−416.000	−0.479216
$92$	0	0
$93$	408.000	0.454921
$94$	0	0
$95$	8.00000	0.00863982
$96$	0	0
$97$	−446.000	−0.466850	−0.233425	−	0.972375i	$-0.574993\pi$
$97$	−446.000	−0.466850	−0.233425	+	0.972375i	$0.574993\pi$
$98$	0	0
$99$	−612.000	−0.621296
$100$	0	0
$101$	610.000	0.600963	0.300482	−	0.953788i	$-0.402853\pi$
$101$	610.000	0.600963	0.300482	+	0.953788i	$0.402853\pi$
$102$	0	0
$103$	1352.00	1.29336	0.646682	−	0.762760i	$-0.276156\pi$
$103$	1352.00	1.29336	0.646682	+	0.762760i	$0.276156\pi$
$104$	0	0
$105$	192.000	0.178450
$106$	0	0
$107$	−732.000	−0.661356	−0.330678	−	0.943744i	$-0.607277\pi$
$107$	−732.000	−0.661356	−0.330678	+	0.943744i	$0.607277\pi$
$108$	0	0
$109$	1514.00	1.33041	0.665206	−	0.746660i	$-0.268344\pi$
$109$	1514.00	1.33041	0.665206	+	0.746660i	$0.268344\pi$
$110$	0	0
$111$	1158.00	0.990203
$112$	0	0
$113$	−1518.00	−1.26373	−0.631865	−	0.775079i	$-0.717710\pi$
$113$	−1518.00	−1.26373	−0.631865	+	0.775079i	$0.717710\pi$
$114$	0	0
$115$	−144.000	−0.116766
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	−448.000	−0.345110
$120$	0	0
$121$	3293.00	2.47408
$122$	0	0
$123$	750.000	0.549799
$124$	0	0
$125$	−492.000	−0.352047
$126$	0	0
$127$	96.0000	0.0670758	0.0335379	−	0.999437i	$-0.489323\pi$
$127$	96.0000	0.0670758	0.0335379	+	0.999437i	$0.489323\pi$
$128$	0	0
$129$	−420.000	−0.286658
$130$	0	0
$131$	−2548.00	−1.69939	−0.849694	−	0.527276i	$-0.823213\pi$
$131$	−2548.00	−1.69939	−0.849694	+	0.527276i	$0.823213\pi$
$132$	0	0
$133$	128.000	0.0834512
$134$	0	0
$135$	54.0000	0.0344265
$136$	0	0
$137$	−230.000	−0.143432	−0.0717162	−	0.997425i	$-0.522848\pi$
$137$	−230.000	−0.143432	−0.0717162	+	0.997425i	$0.522848\pi$
$138$	0	0
$139$	516.000	0.314867	0.157434	−	0.987530i	$-0.449678\pi$
$139$	516.000	0.314867	0.157434	+	0.987530i	$0.449678\pi$
$140$	0	0
$141$	888.000	0.530377
$142$	0	0
$143$	884.000	0.516950
$144$	0	0
$145$	−204.000	−0.116836
$146$	0	0
$147$	2043.00	1.14628
$148$	0	0
$149$	1842.00	1.01277	0.506384	−	0.862308i	$-0.330982\pi$
$149$	1842.00	1.01277	0.506384	+	0.862308i	$0.330982\pi$
$150$	0	0
$151$	528.000	0.284556	0.142278	−	0.989827i	$-0.454557\pi$
$151$	528.000	0.284556	0.142278	+	0.989827i	$0.454557\pi$
$152$	0	0
$153$	−126.000	−0.0665784
$154$	0	0
$155$	272.000	0.140952
$156$	0	0
$157$	1306.00	0.663886	0.331943	−	0.943299i	$-0.392296\pi$
$157$	1306.00	0.663886	0.331943	+	0.943299i	$0.392296\pi$
$158$	0	0
$159$	−1578.00	−0.787066
$160$	0	0
$161$	−2304.00	−1.12783
$162$	0	0
$163$	−3772.00	−1.81255	−0.906276	−	0.422687i	$-0.861087\pi$
$163$	−3772.00	−1.81255	−0.906276	+	0.422687i	$0.861087\pi$
$164$	0	0
$165$	−408.000	−0.192502
$166$	0	0
$167$	−384.000	−0.177933	−0.0889665	−	0.996035i	$-0.528356\pi$
$167$	−384.000	−0.177933	−0.0889665	+	0.996035i	$0.528356\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	36.0000	0.0160993
$172$	0	0
$173$	−1462.00	−0.642508	−0.321254	−	0.946993i	$-0.604104\pi$
$173$	−1462.00	−0.642508	−0.321254	+	0.946993i	$0.604104\pi$
$174$	0	0
$175$	−3872.00	−1.67255
$176$	0	0
$177$	996.000	0.422960
$178$	0	0
$179$	−1332.00	−0.556192	−0.278096	−	0.960553i	$-0.589703\pi$
$179$	−1332.00	−0.556192	−0.278096	+	0.960553i	$0.589703\pi$
$180$	0	0
$181$	−2030.00	−0.833639	−0.416820	−	0.908989i	$-0.636855\pi$
$181$	−2030.00	−0.833639	−0.416820	+	0.908989i	$0.636855\pi$
$182$	0	0
$183$	1230.00	0.496854
$184$	0	0
$185$	772.000	0.306803
$186$	0	0
$187$	952.000	0.372284
$188$	0	0
$189$	864.000	0.332522
$190$	0	0
$191$	16.0000	0.00606136	0.00303068	−	0.999995i	$-0.499035\pi$
$191$	16.0000	0.00606136	0.00303068	+	0.999995i	$0.499035\pi$
$192$	0	0
$193$	−2078.00	−0.775014	−0.387507	−	0.921867i	$-0.626664\pi$
$193$	−2078.00	−0.775014	−0.387507	+	0.921867i	$0.626664\pi$
$194$	0	0
$195$	−78.0000	−0.0286446
$196$	0	0
$197$	−3486.00	−1.26075	−0.630374	−	0.776292i	$-0.717098\pi$
$197$	−3486.00	−1.26075	−0.630374	+	0.776292i	$0.717098\pi$
$198$	0	0
$199$	−568.000	−0.202334	−0.101167	−	0.994869i	$-0.532258\pi$
$199$	−568.000	−0.202334	−0.101167	+	0.994869i	$0.532258\pi$
$200$	0	0
$201$	1788.00	0.627442
$202$	0	0
$203$	−3264.00	−1.12851
$204$	0	0
$205$	500.000	0.170349
$206$	0	0
$207$	−648.000	−0.217580
$208$	0	0
$209$	−272.000	−0.0900222
$210$	0	0
$211$	3804.00	1.24113	0.620564	−	0.784156i	$-0.286904\pi$
$211$	3804.00	1.24113	0.620564	+	0.784156i	$0.286904\pi$
$212$	0	0
$213$	2640.00	0.849248
$214$	0	0
$215$	−280.000	−0.0888179
$216$	0	0
$217$	4352.00	1.36144
$218$	0	0
$219$	1518.00	0.468388
$220$	0	0
$221$	182.000	0.0553966
$222$	0	0
$223$	−5912.00	−1.77532	−0.887661	−	0.460498i	$-0.847671\pi$
$223$	−5912.00	−1.77532	−0.887661	+	0.460498i	$0.847671\pi$
$224$	0	0
$225$	−1089.00	−0.322667
$226$	0	0
$227$	−3308.00	−0.967223	−0.483612	−	0.875283i	$-0.660675\pi$
$227$	−3308.00	−0.967223	−0.483612	+	0.875283i	$0.660675\pi$
$228$	0	0
$229$	6050.00	1.74583	0.872915	−	0.487872i	$-0.162227\pi$
$229$	6050.00	1.74583	0.872915	+	0.487872i	$0.162227\pi$
$230$	0	0
$231$	−6528.00	−1.85935
$232$	0	0
$233$	4794.00	1.34792	0.673960	−	0.738768i	$-0.264592\pi$
$233$	4794.00	1.34792	0.673960	+	0.738768i	$0.264592\pi$
$234$	0	0
$235$	592.000	0.164331
$236$	0	0
$237$	1920.00	0.526234
$238$	0	0
$239$	4440.00	1.20167	0.600836	−	0.799372i	$-0.294834\pi$
$239$	4440.00	1.20167	0.600836	+	0.799372i	$0.294834\pi$
$240$	0	0
$241$	1330.00	0.355489	0.177744	−	0.984077i	$-0.443120\pi$
$241$	1330.00	0.355489	0.177744	+	0.984077i	$0.443120\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	1362.00	0.355163
$246$	0	0
$247$	−52.0000	−0.0133955
$248$	0	0
$249$	4140.00	1.05366
$250$	0	0
$251$	−5116.00	−1.28653	−0.643265	−	0.765644i	$-0.722421\pi$
$251$	−5116.00	−1.28653	−0.643265	+	0.765644i	$0.722421\pi$
$252$	0	0
$253$	4896.00	1.21664
$254$	0	0
$255$	−84.0000	−0.0206286
$256$	0	0
$257$	642.000	0.155824	0.0779122	−	0.996960i	$-0.475175\pi$
$257$	642.000	0.155824	0.0779122	+	0.996960i	$0.475175\pi$
$258$	0	0
$259$	12352.0	2.96338
$260$	0	0
$261$	−918.000	−0.217712
$262$	0	0
$263$	−4264.00	−0.999732	−0.499866	−	0.866103i	$-0.666617\pi$
$263$	−4264.00	−0.999732	−0.499866	+	0.866103i	$0.666617\pi$
$264$	0	0
$265$	−1052.00	−0.243864
$266$	0	0
$267$	4350.00	0.997062
$268$	0	0
$269$	3850.00	0.872634	0.436317	−	0.899793i	$-0.356283\pi$
$269$	3850.00	0.872634	0.436317	+	0.899793i	$0.356283\pi$
$270$	0	0
$271$	−2936.00	−0.658115	−0.329058	−	0.944310i	$-0.606731\pi$
$271$	−2936.00	−0.658115	−0.329058	+	0.944310i	$0.606731\pi$
$272$	0	0
$273$	−1248.00	−0.276675
$274$	0	0
$275$	8228.00	1.80424
$276$	0	0
$277$	2066.00	0.448137	0.224068	−	0.974573i	$-0.428066\pi$
$277$	2066.00	0.448137	0.224068	+	0.974573i	$0.428066\pi$
$278$	0	0
$279$	1224.00	0.262649
$280$	0	0
$281$	−214.000	−0.0454312	−0.0227156	−	0.999742i	$-0.507231\pi$
$281$	−214.000	−0.0454312	−0.0227156	+	0.999742i	$0.507231\pi$
$282$	0	0
$283$	−2620.00	−0.550328	−0.275164	−	0.961397i	$-0.588732\pi$
$283$	−2620.00	−0.550328	−0.275164	+	0.961397i	$0.588732\pi$
$284$	0	0
$285$	24.0000	0.00498820
$286$	0	0
$287$	8000.00	1.64538
$288$	0	0
$289$	−4717.00	−0.960106
$290$	0	0
$291$	−1338.00	−0.269536
$292$	0	0
$293$	1154.00	0.230094	0.115047	−	0.993360i	$-0.463298\pi$
$293$	1154.00	0.230094	0.115047	+	0.993360i	$0.463298\pi$
$294$	0	0
$295$	664.000	0.131049
$296$	0	0
$297$	−1836.00	−0.358705
$298$	0	0
$299$	936.000	0.181038
$300$	0	0
$301$	−4480.00	−0.857883
$302$	0	0
$303$	1830.00	0.346966
$304$	0	0
$305$	820.000	0.153944
$306$	0	0
$307$	−4076.00	−0.757751	−0.378876	−	0.925448i	$-0.623689\pi$
$307$	−4076.00	−0.757751	−0.378876	+	0.925448i	$0.623689\pi$
$308$	0	0
$309$	4056.00	0.746724
$310$	0	0
$311$	6456.00	1.17713	0.588563	−	0.808451i	$-0.299694\pi$
$311$	6456.00	1.17713	0.588563	+	0.808451i	$0.299694\pi$
$312$	0	0
$313$	−5526.00	−0.997917	−0.498958	−	0.866626i	$-0.666284\pi$
$313$	−5526.00	−0.997917	−0.498958	+	0.866626i	$0.666284\pi$
$314$	0	0
$315$	576.000	0.103028
$316$	0	0
$317$	10458.0	1.85293	0.926467	−	0.376377i	$-0.122830\pi$
$317$	10458.0	1.85293	0.926467	+	0.376377i	$0.122830\pi$
$318$	0	0
$319$	6936.00	1.21737
$320$	0	0
$321$	−2196.00	−0.381834
$322$	0	0
$323$	−56.0000	−0.00964682
$324$	0	0
$325$	1573.00	0.268475
$326$	0	0
$327$	4542.00	0.768114
$328$	0	0
$329$	9472.00	1.58726
$330$	0	0
$331$	2348.00	0.389903	0.194951	−	0.980813i	$-0.437545\pi$
$331$	2348.00	0.389903	0.194951	+	0.980813i	$0.437545\pi$
$332$	0	0
$333$	3474.00	0.571694
$334$	0	0
$335$	1192.00	0.194406
$336$	0	0
$337$	5298.00	0.856381	0.428191	−	0.903688i	$-0.359151\pi$
$337$	5298.00	0.856381	0.428191	+	0.903688i	$0.359151\pi$
$338$	0	0
$339$	−4554.00	−0.729615
$340$	0	0
$341$	−9248.00	−1.46864
$342$	0	0
$343$	10816.0	1.70265
$344$	0	0
$345$	−432.000	−0.0674148
$346$	0	0
$347$	9876.00	1.52787	0.763936	−	0.645292i	$-0.223264\pi$
$347$	9876.00	1.52787	0.763936	+	0.645292i	$0.223264\pi$
$348$	0	0
$349$	5370.00	0.823638	0.411819	−	0.911266i	$-0.364894\pi$
$349$	5370.00	0.823638	0.411819	+	0.911266i	$0.364894\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	7330.00	1.10520	0.552601	−	0.833446i	$-0.313635\pi$
$353$	7330.00	1.10520	0.552601	+	0.833446i	$0.313635\pi$
$354$	0	0
$355$	1760.00	0.263130
$356$	0	0
$357$	−1344.00	−0.199249
$358$	0	0
$359$	−7488.00	−1.10084	−0.550420	−	0.834888i	$-0.685532\pi$
$359$	−7488.00	−1.10084	−0.550420	+	0.834888i	$0.685532\pi$
$360$	0	0
$361$	−6843.00	−0.997667
$362$	0	0
$363$	9879.00	1.42841
$364$	0	0
$365$	1012.00	0.145125
$366$	0	0
$367$	−1504.00	−0.213919	−0.106959	−	0.994263i	$-0.534111\pi$
$367$	−1504.00	−0.213919	−0.106959	+	0.994263i	$0.534111\pi$
$368$	0	0
$369$	2250.00	0.317426
$370$	0	0
$371$	−16832.0	−2.35546
$372$	0	0
$373$	−6702.00	−0.930339	−0.465169	−	0.885222i	$-0.654007\pi$
$373$	−6702.00	−0.930339	−0.465169	+	0.885222i	$0.654007\pi$
$374$	0	0
$375$	−1476.00	−0.203254
$376$	0	0
$377$	1326.00	0.181147
$378$	0	0
$379$	−5700.00	−0.772531	−0.386266	−	0.922388i	$-0.626235\pi$
$379$	−5700.00	−0.772531	−0.386266	+	0.922388i	$0.626235\pi$
$380$	0	0
$381$	288.000	0.0387262
$382$	0	0
$383$	−2328.00	−0.310588	−0.155294	−	0.987868i	$-0.549633\pi$
$383$	−2328.00	−0.310588	−0.155294	+	0.987868i	$0.549633\pi$
$384$	0	0
$385$	−4352.00	−0.576100
$386$	0	0
$387$	−1260.00	−0.165502
$388$	0	0
$389$	11554.0	1.50594	0.752971	−	0.658054i	$-0.228620\pi$
$389$	11554.0	1.50594	0.752971	+	0.658054i	$0.228620\pi$
$390$	0	0
$391$	1008.00	0.130375
$392$	0	0
$393$	−7644.00	−0.981142
$394$	0	0
$395$	1280.00	0.163048
$396$	0	0
$397$	−6486.00	−0.819957	−0.409979	−	0.912095i	$-0.634464\pi$
$397$	−6486.00	−0.819957	−0.409979	+	0.912095i	$0.634464\pi$
$398$	0	0
$399$	384.000	0.0481806
$400$	0	0
$401$	7698.00	0.958653	0.479326	−	0.877637i	$-0.340881\pi$
$401$	7698.00	0.958653	0.479326	+	0.877637i	$0.340881\pi$
$402$	0	0
$403$	−1768.00	−0.218537
$404$	0	0
$405$	162.000	0.0198762
$406$	0	0
$407$	−26248.0	−3.19672
$408$	0	0
$409$	3338.00	0.403554	0.201777	−	0.979432i	$-0.435328\pi$
$409$	3338.00	0.403554	0.201777	+	0.979432i	$0.435328\pi$
$410$	0	0
$411$	−690.000	−0.0828107
$412$	0	0
$413$	10624.0	1.26579
$414$	0	0
$415$	2760.00	0.326465
$416$	0	0
$417$	1548.00	0.181789
$418$	0	0
$419$	−52.0000	−0.00606293	−0.00303146	−	0.999995i	$-0.500965\pi$
$419$	−52.0000	−0.00606293	−0.00303146	+	0.999995i	$0.500965\pi$
$420$	0	0
$421$	5858.00	0.678151	0.339075	−	0.940759i	$-0.389886\pi$
$421$	5858.00	0.678151	0.339075	+	0.940759i	$0.389886\pi$
$422$	0	0
$423$	2664.00	0.306213
$424$	0	0
$425$	1694.00	0.193344
$426$	0	0
$427$	13120.0	1.48694
$428$	0	0
$429$	2652.00	0.298461
$430$	0	0
$431$	−8840.00	−0.987953	−0.493977	−	0.869475i	$-0.664457\pi$
$431$	−8840.00	−0.987953	−0.493977	+	0.869475i	$0.664457\pi$
$432$	0	0
$433$	11346.0	1.25925	0.629624	−	0.776900i	$-0.283209\pi$
$433$	11346.0	1.25925	0.629624	+	0.776900i	$0.283209\pi$
$434$	0	0
$435$	−612.000	−0.0674555
$436$	0	0
$437$	−288.000	−0.0315261
$438$	0	0
$439$	−16456.0	−1.78907	−0.894535	−	0.446997i	$-0.852493\pi$
$439$	−16456.0	−1.78907	−0.894535	+	0.446997i	$0.852493\pi$
$440$	0	0
$441$	6129.00	0.661808
$442$	0	0
$443$	−3788.00	−0.406260	−0.203130	−	0.979152i	$-0.565111\pi$
$443$	−3788.00	−0.406260	−0.203130	+	0.979152i	$0.565111\pi$
$444$	0	0
$445$	2900.00	0.308929
$446$	0	0
$447$	5526.00	0.584722
$448$	0	0
$449$	546.000	0.0573883	0.0286941	−	0.999588i	$-0.490865\pi$
$449$	546.000	0.0573883	0.0286941	+	0.999588i	$0.490865\pi$
$450$	0	0
$451$	−17000.0	−1.77494
$452$	0	0
$453$	1584.00	0.164289
$454$	0	0
$455$	−832.000	−0.0857248
$456$	0	0
$457$	3546.00	0.362965	0.181482	−	0.983394i	$-0.441910\pi$
$457$	3546.00	0.362965	0.181482	+	0.983394i	$0.441910\pi$
$458$	0	0
$459$	−378.000	−0.0384391
$460$	0	0
$461$	−12918.0	−1.30510	−0.652550	−	0.757746i	$-0.726301\pi$
$461$	−12918.0	−1.30510	−0.652550	+	0.757746i	$0.726301\pi$
$462$	0	0
$463$	−18328.0	−1.83969	−0.919843	−	0.392287i	$-0.871684\pi$
$463$	−18328.0	−1.83969	−0.919843	+	0.392287i	$0.871684\pi$
$464$	0	0
$465$	816.000	0.0813787
$466$	0	0
$467$	11980.0	1.18708	0.593542	−	0.804803i	$-0.297729\pi$
$467$	11980.0	1.18708	0.593542	+	0.804803i	$0.297729\pi$
$468$	0	0
$469$	19072.0	1.87775
$470$	0	0
$471$	3918.00	0.383295
$472$	0	0
$473$	9520.00	0.925434
$474$	0	0
$475$	−484.000	−0.0467525
$476$	0	0
$477$	−4734.00	−0.454413
$478$	0	0
$479$	12344.0	1.17748	0.588739	−	0.808323i	$-0.299625\pi$
$479$	12344.0	1.17748	0.588739	+	0.808323i	$0.299625\pi$
$480$	0	0
$481$	−5018.00	−0.475678
$482$	0	0
$483$	−6912.00	−0.651153
$484$	0	0
$485$	−892.000	−0.0835126
$486$	0	0
$487$	80.0000	0.00744383	0.00372192	−	0.999993i	$-0.498815\pi$
$487$	80.0000	0.00744383	0.00372192	+	0.999993i	$0.498815\pi$
$488$	0	0
$489$	−11316.0	−1.04648
$490$	0	0
$491$	−15660.0	−1.43936	−0.719680	−	0.694306i	$-0.755712\pi$
$491$	−15660.0	−1.43936	−0.719680	+	0.694306i	$0.755712\pi$
$492$	0	0
$493$	1428.00	0.130454
$494$	0	0
$495$	−1224.00	−0.111141
$496$	0	0
$497$	28160.0	2.54155
$498$	0	0
$499$	−60.0000	−0.00538270	−0.00269135	−	0.999996i	$-0.500857\pi$
$499$	−60.0000	−0.00538270	−0.00269135	+	0.999996i	$0.500857\pi$
$500$	0	0
$501$	−1152.00	−0.102730
$502$	0	0
$503$	−12248.0	−1.08571	−0.542854	−	0.839827i	$-0.682656\pi$
$503$	−12248.0	−1.08571	−0.542854	+	0.839827i	$0.682656\pi$
$504$	0	0
$505$	1220.00	0.107504
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	90.0000	0.00783729	0.00391864	−	0.999992i	$-0.498753\pi$
$509$	90.0000	0.00783729	0.00391864	+	0.999992i	$0.498753\pi$
$510$	0	0
$511$	16192.0	1.40175
$512$	0	0
$513$	108.000	0.00929496
$514$	0	0
$515$	2704.00	0.231364
$516$	0	0
$517$	−20128.0	−1.71224
$518$	0	0
$519$	−4386.00	−0.370952
$520$	0	0
$521$	9818.00	0.825594	0.412797	−	0.910823i	$-0.364552\pi$
$521$	9818.00	0.825594	0.412797	+	0.910823i	$0.364552\pi$
$522$	0	0
$523$	−20252.0	−1.69323	−0.846614	−	0.532208i	$-0.821363\pi$
$523$	−20252.0	−1.69323	−0.846614	+	0.532208i	$0.821363\pi$
$524$	0	0
$525$	−11616.0	−0.965645
$526$	0	0
$527$	−1904.00	−0.157381
$528$	0	0
$529$	−6983.00	−0.573929
$530$	0	0
$531$	2988.00	0.244196
$532$	0	0
$533$	−3250.00	−0.264115
$534$	0	0
$535$	−1464.00	−0.118307
$536$	0	0
$537$	−3996.00	−0.321118
$538$	0	0
$539$	−46308.0	−3.70061
$540$	0	0
$541$	12634.0	1.00403	0.502013	−	0.864860i	$-0.332593\pi$
$541$	12634.0	1.00403	0.502013	+	0.864860i	$0.332593\pi$
$542$	0	0
$543$	−6090.00	−0.481302
$544$	0	0
$545$	3028.00	0.237991
$546$	0	0
$547$	11756.0	0.918922	0.459461	−	0.888198i	$-0.348043\pi$
$547$	11756.0	0.918922	0.459461	+	0.888198i	$0.348043\pi$
$548$	0	0
$549$	3690.00	0.286859
$550$	0	0
$551$	−408.000	−0.0315452
$552$	0	0
$553$	20480.0	1.57486
$554$	0	0
$555$	2316.00	0.177133
$556$	0	0
$557$	−17622.0	−1.34052	−0.670259	−	0.742128i	$-0.733817\pi$
$557$	−17622.0	−1.34052	−0.670259	+	0.742128i	$0.733817\pi$
$558$	0	0
$559$	1820.00	0.137706
$560$	0	0
$561$	2856.00	0.214938
$562$	0	0
$563$	−23092.0	−1.72862	−0.864309	−	0.502961i	$-0.832244\pi$
$563$	−23092.0	−1.72862	−0.864309	+	0.502961i	$0.832244\pi$
$564$	0	0
$565$	−3036.00	−0.226063
$566$	0	0
$567$	2592.00	0.191982
$568$	0	0
$569$	−1302.00	−0.0959274	−0.0479637	−	0.998849i	$-0.515273\pi$
$569$	−1302.00	−0.0959274	−0.0479637	+	0.998849i	$0.515273\pi$
$570$	0	0
$571$	24868.0	1.82258	0.911290	−	0.411765i	$-0.135087\pi$
$571$	24868.0	1.82258	0.911290	+	0.411765i	$0.135087\pi$
$572$	0	0
$573$	48.0000	0.00349953
$574$	0	0
$575$	8712.00	0.631853
$576$	0	0
$577$	2562.00	0.184848	0.0924241	−	0.995720i	$-0.470538\pi$
$577$	2562.00	0.184848	0.0924241	+	0.995720i	$0.470538\pi$
$578$	0	0
$579$	−6234.00	−0.447455
$580$	0	0
$581$	44160.0	3.15330
$582$	0	0
$583$	35768.0	2.54092
$584$	0	0
$585$	−234.000	−0.0165380
$586$	0	0
$587$	13484.0	0.948116	0.474058	−	0.880494i	$-0.342789\pi$
$587$	13484.0	0.948116	0.474058	+	0.880494i	$0.342789\pi$
$588$	0	0
$589$	544.000	0.0380562
$590$	0	0
$591$	−10458.0	−0.727893
$592$	0	0
$593$	−16974.0	−1.17544	−0.587722	−	0.809063i	$-0.699975\pi$
$593$	−16974.0	−1.17544	−0.587722	+	0.809063i	$0.699975\pi$
$594$	0	0
$595$	−896.000	−0.0617352
$596$	0	0
$597$	−1704.00	−0.116818
$598$	0	0
$599$	−3864.00	−0.263571	−0.131785	−	0.991278i	$-0.542071\pi$
$599$	−3864.00	−0.263571	−0.131785	+	0.991278i	$0.542071\pi$
$600$	0	0
$601$	17546.0	1.19088	0.595438	−	0.803401i	$-0.296979\pi$
$601$	17546.0	1.19088	0.595438	+	0.803401i	$0.296979\pi$
$602$	0	0
$603$	5364.00	0.362254
$604$	0	0
$605$	6586.00	0.442577
$606$	0	0
$607$	−9296.00	−0.621603	−0.310801	−	0.950475i	$-0.600597\pi$
$607$	−9296.00	−0.621603	−0.310801	+	0.950475i	$0.600597\pi$
$608$	0	0
$609$	−9792.00	−0.651547
$610$	0	0
$611$	−3848.00	−0.254785
$612$	0	0
$613$	6914.00	0.455553	0.227776	−	0.973713i	$-0.426854\pi$
$613$	6914.00	0.455553	0.227776	+	0.973713i	$0.426854\pi$
$614$	0	0
$615$	1500.00	0.0983510
$616$	0	0
$617$	−25446.0	−1.66032	−0.830160	−	0.557525i	$-0.811751\pi$
$617$	−25446.0	−1.66032	−0.830160	+	0.557525i	$0.811751\pi$
$618$	0	0
$619$	−11236.0	−0.729585	−0.364792	−	0.931089i	$-0.618860\pi$
$619$	−11236.0	−0.729585	−0.364792	+	0.931089i	$0.618860\pi$
$620$	0	0
$621$	−1944.00	−0.125620
$622$	0	0
$623$	46400.0	2.98391
$624$	0	0
$625$	14141.0	0.905024
$626$	0	0
$627$	−816.000	−0.0519743
$628$	0	0
$629$	−5404.00	−0.342562
$630$	0	0
$631$	29424.0	1.85634	0.928170	−	0.372156i	$-0.121381\pi$
$631$	29424.0	1.85634	0.928170	+	0.372156i	$0.121381\pi$
$632$	0	0
$633$	11412.0	0.716566
$634$	0	0
$635$	192.000	0.0119989
$636$	0	0
$637$	−8853.00	−0.550657
$638$	0	0
$639$	7920.00	0.490314
$640$	0	0
$641$	−5054.00	−0.311421	−0.155711	−	0.987803i	$-0.549767\pi$
$641$	−5054.00	−0.311421	−0.155711	+	0.987803i	$0.549767\pi$
$642$	0	0
$643$	−1132.00	−0.0694273	−0.0347136	−	0.999397i	$-0.511052\pi$
$643$	−1132.00	−0.0694273	−0.0347136	+	0.999397i	$0.511052\pi$
$644$	0	0
$645$	−840.000	−0.0512790
$646$	0	0
$647$	−1464.00	−0.0889579	−0.0444790	−	0.999010i	$-0.514163\pi$
$647$	−1464.00	−0.0889579	−0.0444790	+	0.999010i	$0.514163\pi$
$648$	0	0
$649$	−22576.0	−1.36546
$650$	0	0
$651$	13056.0	0.786029
$652$	0	0
$653$	−5494.00	−0.329245	−0.164622	−	0.986357i	$-0.552641\pi$
$653$	−5494.00	−0.329245	−0.164622	+	0.986357i	$0.552641\pi$
$654$	0	0
$655$	−5096.00	−0.303996
$656$	0	0
$657$	4554.00	0.270424
$658$	0	0
$659$	11580.0	0.684511	0.342256	−	0.939607i	$-0.388809\pi$
$659$	11580.0	0.684511	0.342256	+	0.939607i	$0.388809\pi$
$660$	0	0
$661$	1298.00	0.0763787	0.0381894	−	0.999271i	$-0.487841\pi$
$661$	1298.00	0.0763787	0.0381894	+	0.999271i	$0.487841\pi$
$662$	0	0
$663$	546.000	0.0319832
$664$	0	0
$665$	256.000	0.0149282
$666$	0	0
$667$	7344.00	0.426328
$668$	0	0
$669$	−17736.0	−1.02498
$670$	0	0
$671$	−27880.0	−1.60402
$672$	0	0
$673$	16162.0	0.925705	0.462852	−	0.886435i	$-0.346826\pi$
$673$	16162.0	0.925705	0.462852	+	0.886435i	$0.346826\pi$
$674$	0	0
$675$	−3267.00	−0.186292
$676$	0	0
$677$	9890.00	0.561453	0.280726	−	0.959788i	$-0.409425\pi$
$677$	9890.00	0.561453	0.280726	+	0.959788i	$0.409425\pi$
$678$	0	0
$679$	−14272.0	−0.806641
$680$	0	0
$681$	−9924.00	−0.558427
$682$	0	0
$683$	27100.0	1.51823	0.759116	−	0.650955i	$-0.225631\pi$
$683$	27100.0	1.51823	0.759116	+	0.650955i	$0.225631\pi$
$684$	0	0
$685$	−460.000	−0.0256580
$686$	0	0
$687$	18150.0	1.00796
$688$	0	0
$689$	6838.00	0.378094
$690$	0	0
$691$	−11132.0	−0.612853	−0.306426	−	0.951894i	$-0.599133\pi$
$691$	−11132.0	−0.612853	−0.306426	+	0.951894i	$0.599133\pi$
$692$	0	0
$693$	−19584.0	−1.07350
$694$	0	0
$695$	1032.00	0.0563252
$696$	0	0
$697$	−3500.00	−0.190204
$698$	0	0
$699$	14382.0	0.778222
$700$	0	0
$701$	−1862.00	−0.100323	−0.0501617	−	0.998741i	$-0.515974\pi$
$701$	−1862.00	−0.100323	−0.0501617	+	0.998741i	$0.515974\pi$
$702$	0	0
$703$	1544.00	0.0828351
$704$	0	0
$705$	1776.00	0.0948766
$706$	0	0
$707$	19520.0	1.03837
$708$	0	0
$709$	7938.00	0.420477	0.210238	−	0.977650i	$-0.432576\pi$
$709$	7938.00	0.420477	0.210238	+	0.977650i	$0.432576\pi$
$710$	0	0
$711$	5760.00	0.303821
$712$	0	0
$713$	−9792.00	−0.514324
$714$	0	0
$715$	1768.00	0.0924748
$716$	0	0
$717$	13320.0	0.693786
$718$	0	0
$719$	24240.0	1.25730	0.628651	−	0.777688i	$-0.283608\pi$
$719$	24240.0	1.25730	0.628651	+	0.777688i	$0.283608\pi$
$720$	0	0
$721$	43264.0	2.23472
$722$	0	0
$723$	3990.00	0.205242
$724$	0	0
$725$	12342.0	0.632235
$726$	0	0
$727$	13720.0	0.699927	0.349963	−	0.936763i	$-0.386194\pi$
$727$	13720.0	0.699927	0.349963	+	0.936763i	$0.386194\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	1960.00	0.0991699
$732$	0	0
$733$	−21958.0	−1.10646	−0.553231	−	0.833028i	$-0.686605\pi$
$733$	−21958.0	−1.10646	−0.553231	+	0.833028i	$0.686605\pi$
$734$	0	0
$735$	4086.00	0.205054
$736$	0	0
$737$	−40528.0	−2.02560
$738$	0	0
$739$	13348.0	0.664430	0.332215	−	0.943204i	$-0.392204\pi$
$739$	13348.0	0.664430	0.332215	+	0.943204i	$0.392204\pi$
$740$	0	0
$741$	−156.000	−0.00773388
$742$	0	0
$743$	−20304.0	−1.00253	−0.501266	−	0.865293i	$-0.667132\pi$
$743$	−20304.0	−1.00253	−0.501266	+	0.865293i	$0.667132\pi$
$744$	0	0
$745$	3684.00	0.181170
$746$	0	0
$747$	12420.0	0.608332
$748$	0	0
$749$	−23424.0	−1.14272
$750$	0	0
$751$	3952.00	0.192025	0.0960123	−	0.995380i	$-0.469391\pi$
$751$	3952.00	0.192025	0.0960123	+	0.995380i	$0.469391\pi$
$752$	0	0
$753$	−15348.0	−0.742779
$754$	0	0
$755$	1056.00	0.0509030
$756$	0	0
$757$	22386.0	1.07481	0.537406	−	0.843324i	$-0.319404\pi$
$757$	22386.0	1.07481	0.537406	+	0.843324i	$0.319404\pi$
$758$	0	0
$759$	14688.0	0.702425
$760$	0	0
$761$	12458.0	0.593433	0.296716	−	0.954966i	$-0.404108\pi$
$761$	12458.0	0.593433	0.296716	+	0.954966i	$0.404108\pi$
$762$	0	0
$763$	48448.0	2.29874
$764$	0	0
$765$	−252.000	−0.0119099
$766$	0	0
$767$	−4316.00	−0.203183
$768$	0	0
$769$	−28126.0	−1.31892	−0.659460	−	0.751740i	$-0.729215\pi$
$769$	−28126.0	−1.31892	−0.659460	+	0.751740i	$0.729215\pi$
$770$	0	0
$771$	1926.00	0.0899652
$772$	0	0
$773$	35778.0	1.66474	0.832371	−	0.554219i	$-0.186983\pi$
$773$	35778.0	1.66474	0.832371	+	0.554219i	$0.186983\pi$
$774$	0	0
$775$	−16456.0	−0.762732
$776$	0	0
$777$	37056.0	1.71091
$778$	0	0
$779$	1000.00	0.0459932
$780$	0	0
$781$	−59840.0	−2.74167
$782$	0	0
$783$	−2754.00	−0.125696
$784$	0	0
$785$	2612.00	0.118760
$786$	0	0
$787$	7252.00	0.328470	0.164235	−	0.986421i	$-0.447484\pi$
$787$	7252.00	0.328470	0.164235	+	0.986421i	$0.447484\pi$
$788$	0	0
$789$	−12792.0	−0.577196
$790$	0	0
$791$	−48576.0	−2.18352
$792$	0	0
$793$	−5330.00	−0.238681
$794$	0	0
$795$	−3156.00	−0.140795
$796$	0	0
$797$	−4454.00	−0.197953	−0.0989766	−	0.995090i	$-0.531557\pi$
$797$	−4454.00	−0.197953	−0.0989766	+	0.995090i	$0.531557\pi$
$798$	0	0
$799$	−4144.00	−0.183485
$800$	0	0
$801$	13050.0	0.575654
$802$	0	0
$803$	−34408.0	−1.51212
$804$	0	0
$805$	−4608.00	−0.201752
$806$	0	0
$807$	11550.0	0.503816
$808$	0	0
$809$	−34118.0	−1.48273	−0.741363	−	0.671104i	$-0.765820\pi$
$809$	−34118.0	−1.48273	−0.741363	+	0.671104i	$0.765820\pi$
$810$	0	0
$811$	16428.0	0.711301	0.355650	−	0.934619i	$-0.384259\pi$
$811$	16428.0	0.711301	0.355650	+	0.934619i	$0.384259\pi$
$812$	0	0
$813$	−8808.00	−0.379963
$814$	0	0
$815$	−7544.00	−0.324239
$816$	0	0
$817$	−560.000	−0.0239803
$818$	0	0
$819$	−3744.00	−0.159739
$820$	0	0
$821$	18738.0	0.796542	0.398271	−	0.917268i	$-0.369610\pi$
$821$	18738.0	0.796542	0.398271	+	0.917268i	$0.369610\pi$
$822$	0	0
$823$	−13928.0	−0.589914	−0.294957	−	0.955510i	$-0.595305\pi$
$823$	−13928.0	−0.589914	−0.294957	+	0.955510i	$0.595305\pi$
$824$	0	0
$825$	24684.0	1.04168
$826$	0	0
$827$	41804.0	1.75776	0.878880	−	0.477043i	$-0.158291\pi$
$827$	41804.0	1.75776	0.878880	+	0.477043i	$0.158291\pi$
$828$	0	0
$829$	43226.0	1.81098	0.905489	−	0.424369i	$-0.139504\pi$
$829$	43226.0	1.81098	0.905489	+	0.424369i	$0.139504\pi$
$830$	0	0
$831$	6198.00	0.258732
$832$	0	0
$833$	−9534.00	−0.396559
$834$	0	0
$835$	−768.000	−0.0318296
$836$	0	0
$837$	3672.00	0.151640
$838$	0	0
$839$	−10240.0	−0.421364	−0.210682	−	0.977555i	$-0.567568\pi$
$839$	−10240.0	−0.421364	−0.210682	+	0.977555i	$0.567568\pi$
$840$	0	0
$841$	−13985.0	−0.573414
$842$	0	0
$843$	−642.000	−0.0262297
$844$	0	0
$845$	338.000	0.0137604
$846$	0	0
$847$	105376.	4.27481
$848$	0	0
$849$	−7860.00	−0.317732
$850$	0	0
$851$	−27792.0	−1.11950
$852$	0	0
$853$	25682.0	1.03087	0.515437	−	0.856928i	$-0.327630\pi$
$853$	25682.0	1.03087	0.515437	+	0.856928i	$0.327630\pi$
$854$	0	0
$855$	72.0000	0.00287994
$856$	0	0
$857$	−21558.0	−0.859285	−0.429643	−	0.902999i	$-0.641360\pi$
$857$	−21558.0	−0.859285	−0.429643	+	0.902999i	$0.641360\pi$
$858$	0	0
$859$	−14060.0	−0.558465	−0.279232	−	0.960224i	$-0.590080\pi$
$859$	−14060.0	−0.558465	−0.279232	+	0.960224i	$0.590080\pi$
$860$	0	0
$861$	24000.0	0.949963
$862$	0	0
$863$	−42008.0	−1.65697	−0.828487	−	0.560008i	$-0.810798\pi$
$863$	−42008.0	−1.65697	−0.828487	+	0.560008i	$0.810798\pi$
$864$	0	0
$865$	−2924.00	−0.114935
$866$	0	0
$867$	−14151.0	−0.554317
$868$	0	0
$869$	−43520.0	−1.69887
$870$	0	0
$871$	−7748.00	−0.301413
$872$	0	0
$873$	−4014.00	−0.155617
$874$	0	0
$875$	−15744.0	−0.608279
$876$	0	0
$877$	−23734.0	−0.913843	−0.456921	−	0.889507i	$-0.651048\pi$
$877$	−23734.0	−0.913843	−0.456921	+	0.889507i	$0.651048\pi$
$878$	0	0
$879$	3462.00	0.132845
$880$	0	0
$881$	−37550.0	−1.43597	−0.717986	−	0.696057i	$-0.754936\pi$
$881$	−37550.0	−1.43597	−0.717986	+	0.696057i	$0.754936\pi$
$882$	0	0
$883$	12556.0	0.478531	0.239266	−	0.970954i	$-0.423093\pi$
$883$	12556.0	0.478531	0.239266	+	0.970954i	$0.423093\pi$
$884$	0	0
$885$	1992.00	0.0756614
$886$	0	0
$887$	−37368.0	−1.41454	−0.707269	−	0.706945i	$-0.750073\pi$
$887$	−37368.0	−1.41454	−0.707269	+	0.706945i	$0.750073\pi$
$888$	0	0
$889$	3072.00	0.115896
$890$	0	0
$891$	−5508.00	−0.207099
$892$	0	0
$893$	1184.00	0.0443685
$894$	0	0
$895$	−2664.00	−0.0994946
$896$	0	0
$897$	2808.00	0.104522
$898$	0	0
$899$	−13872.0	−0.514635
$900$	0	0
$901$	7364.00	0.272287
$902$	0	0
$903$	−13440.0	−0.495299
$904$	0	0
$905$	−4060.00	−0.149126
$906$	0	0
$907$	33364.0	1.22143	0.610713	−	0.791852i	$-0.290883\pi$
$907$	33364.0	1.22143	0.610713	+	0.791852i	$0.290883\pi$
$908$	0	0
$909$	5490.00	0.200321
$910$	0	0
$911$	−50432.0	−1.83412	−0.917062	−	0.398745i	$-0.869446\pi$
$911$	−50432.0	−1.83412	−0.917062	+	0.398745i	$0.869446\pi$
$912$	0	0
$913$	−93840.0	−3.40159
$914$	0	0
$915$	2460.00	0.0888799
$916$	0	0
$917$	−81536.0	−2.93627
$918$	0	0
$919$	−11864.0	−0.425851	−0.212926	−	0.977068i	$-0.568299\pi$
$919$	−11864.0	−0.425851	−0.212926	+	0.977068i	$0.568299\pi$
$920$	0	0
$921$	−12228.0	−0.437488
$922$	0	0
$923$	−11440.0	−0.407966
$924$	0	0
$925$	−46706.0	−1.66020
$926$	0	0
$927$	12168.0	0.431121
$928$	0	0
$929$	−5950.00	−0.210133	−0.105066	−	0.994465i	$-0.533505\pi$
$929$	−5950.00	−0.210133	−0.105066	+	0.994465i	$0.533505\pi$
$930$	0	0
$931$	2724.00	0.0958920
$932$	0	0
$933$	19368.0	0.679614
$934$	0	0
$935$	1904.00	0.0665962
$936$	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
$939$	−16578.0	−0.576148
$940$	0	0
$941$	22346.0	0.774133	0.387066	−	0.922052i	$-0.373488\pi$
$941$	22346.0	0.774133	0.387066	+	0.922052i	$0.373488\pi$
$942$	0	0
$943$	−18000.0	−0.621591
$944$	0	0
$945$	1728.00	0.0594834
$946$	0	0
$947$	31636.0	1.08557	0.542783	−	0.839873i	$-0.317370\pi$
$947$	31636.0	1.08557	0.542783	+	0.839873i	$0.317370\pi$
$948$	0	0
$949$	−6578.00	−0.225006
$950$	0	0
$951$	31374.0	1.06979
$952$	0	0
$953$	−23190.0	−0.788245	−0.394123	−	0.919058i	$-0.628951\pi$
$953$	−23190.0	−0.788245	−0.394123	+	0.919058i	$0.628951\pi$
$954$	0	0
$955$	32.0000	0.00108429
$956$	0	0
$957$	20808.0	0.702850
$958$	0	0
$959$	−7360.00	−0.247828
$960$	0	0
$961$	−11295.0	−0.379141
$962$	0	0
$963$	−6588.00	−0.220452
$964$	0	0
$965$	−4156.00	−0.138639
$966$	0	0
$967$	−304.000	−0.0101096	−0.00505480	−	0.999987i	$-0.501609\pi$
$967$	−304.000	−0.0101096	−0.00505480	+	0.999987i	$0.501609\pi$
$968$	0	0
$969$	−168.000	−0.00556960
$970$	0	0
$971$	−53372.0	−1.76394	−0.881972	−	0.471302i	$-0.843784\pi$
$971$	−53372.0	−1.76394	−0.881972	+	0.471302i	$0.843784\pi$
$972$	0	0
$973$	16512.0	0.544039
$974$	0	0
$975$	4719.00	0.155004
$976$	0	0
$977$	13650.0	0.446983	0.223491	−	0.974706i	$-0.428255\pi$
$977$	13650.0	0.446983	0.223491	+	0.974706i	$0.428255\pi$
$978$	0	0
$979$	−98600.0	−3.21887
$980$	0	0
$981$	13626.0	0.443471
$982$	0	0
$983$	−34992.0	−1.13537	−0.567686	−	0.823245i	$-0.692161\pi$
$983$	−34992.0	−1.13537	−0.567686	+	0.823245i	$0.692161\pi$
$984$	0	0
$985$	−6972.00	−0.225529
$986$	0	0
$987$	28416.0	0.916405
$988$	0	0
$989$	10080.0	0.324090
$990$	0	0
$991$	−18096.0	−0.580059	−0.290029	−	0.957018i	$-0.593665\pi$
$991$	−18096.0	−0.580059	−0.290029	+	0.957018i	$0.593665\pi$
$992$	0	0
$993$	7044.00	0.225110
$994$	0	0
$995$	−1136.00	−0.0361946
$996$	0	0
$997$	18914.0	0.600815	0.300407	−	0.953811i	$-0.402877\pi$
$997$	18914.0	0.600815	0.300407	+	0.953811i	$0.402877\pi$
$998$	0	0
$999$	10422.0	0.330068

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.4.a.m.1.1		1
4.3	odd	2		2496.4.a.d.1.1		1
8.3	odd	2		156.4.a.b.1.1	✓	1
8.5	even	2		624.4.a.b.1.1		1
24.5	odd	2		1872.4.a.i.1.1		1
24.11	even	2		468.4.a.a.1.1		1
104.51	odd	2		2028.4.a.b.1.1		1
104.83	even	4		2028.4.b.d.337.2		2
104.99	even	4		2028.4.b.d.337.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.4.a.b.1.1	✓	1	8.3	odd	2
468.4.a.a.1.1		1	24.11	even	2
624.4.a.b.1.1		1	8.5	even	2
1872.4.a.i.1.1		1	24.5	odd	2
2028.4.a.b.1.1		1	104.51	odd	2
2028.4.b.d.337.1		2	104.99	even	4
2028.4.b.d.337.2		2	104.83	even	4
2496.4.a.d.1.1		1	4.3	odd	2
2496.4.a.m.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2496.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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