LMFDB - Embedded newform 126.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [126,4,Mod(1,126)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(126, 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("126.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	126.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:	$7.43424066072$
Analytic rank:	$1$
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	$\chi$	$=$	126.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} -22.0000 q^{5} -7.00000 q^{7} +8.00000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} +4.00000 q^{4} -22.0000 q^{5} -7.00000 q^{7} +8.00000 q^{8} -44.0000 q^{10} -26.0000 q^{11} -54.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} -74.0000 q^{17} +116.000 q^{19} -88.0000 q^{20} -52.0000 q^{22} +58.0000 q^{23} +359.000 q^{25} -108.000 q^{26} -28.0000 q^{28} -208.000 q^{29} -252.000 q^{31} +32.0000 q^{32} -148.000 q^{34} +154.000 q^{35} +50.0000 q^{37} +232.000 q^{38} -176.000 q^{40} -126.000 q^{41} +164.000 q^{43} -104.000 q^{44} +116.000 q^{46} +444.000 q^{47} +49.0000 q^{49} +718.000 q^{50} -216.000 q^{52} -12.0000 q^{53} +572.000 q^{55} -56.0000 q^{56} -416.000 q^{58} -124.000 q^{59} -162.000 q^{61} -504.000 q^{62} +64.0000 q^{64} +1188.00 q^{65} -860.000 q^{67} -296.000 q^{68} +308.000 q^{70} +238.000 q^{71} -146.000 q^{73} +100.000 q^{74} +464.000 q^{76} +182.000 q^{77} -984.000 q^{79} -352.000 q^{80} -252.000 q^{82} -656.000 q^{83} +1628.00 q^{85} +328.000 q^{86} -208.000 q^{88} +954.000 q^{89} +378.000 q^{91} +232.000 q^{92} +888.000 q^{94} -2552.00 q^{95} +526.000 q^{97} +98.0000 q^{98} +O(q^{100})\)

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−22.0000	−1.96774	−0.983870	−	0.178885i	$-0.942751\pi$
$5$	−22.0000	−1.96774	−0.983870	+	0.178885i	$0.942751\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	0	0
$10$	−44.0000	−1.39140
$11$	−26.0000	−0.712663	−0.356332	−	0.934360i	$-0.615973\pi$
$11$	−26.0000	−0.712663	−0.356332	+	0.934360i	$0.615973\pi$
$12$	0	0
$13$	−54.0000	−1.15207	−0.576035	−	0.817425i	$-0.695401\pi$
$13$	−54.0000	−1.15207	−0.576035	+	0.817425i	$0.695401\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−74.0000	−1.05574	−0.527872	−	0.849324i	$-0.677010\pi$
$17$	−74.0000	−1.05574	−0.527872	+	0.849324i	$0.677010\pi$
$18$	0	0
$19$	116.000	1.40064	0.700322	−	0.713827i	$-0.253040\pi$
$19$	116.000	1.40064	0.700322	+	0.713827i	$0.253040\pi$
$20$	−88.0000	−0.983870
$21$	0	0
$22$	−52.0000	−0.503929
$23$	58.0000	0.525819	0.262909	−	0.964821i	$-0.415318\pi$
$23$	58.0000	0.525819	0.262909	+	0.964821i	$0.415318\pi$
$24$	0	0
$25$	359.000	2.87200
$26$	−108.000	−0.814636
$27$	0	0
$28$	−28.0000	−0.188982
$29$	−208.000	−1.33188	−0.665942	−	0.746004i	$-0.731970\pi$
$29$	−208.000	−1.33188	−0.665942	+	0.746004i	$0.731970\pi$
$30$	0	0
$31$	−252.000	−1.46002	−0.730009	−	0.683438i	$-0.760484\pi$
$31$	−252.000	−1.46002	−0.730009	+	0.683438i	$0.760484\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−148.000	−0.746523
$35$	154.000	0.743736
$36$	0	0
$37$	50.0000	0.222161	0.111080	−	0.993811i	$-0.464569\pi$
$37$	50.0000	0.222161	0.111080	+	0.993811i	$0.464569\pi$
$38$	232.000	0.990404
$39$	0	0
$40$	−176.000	−0.695701
$41$	−126.000	−0.479949	−0.239974	−	0.970779i	$-0.577139\pi$
$41$	−126.000	−0.479949	−0.239974	+	0.970779i	$0.577139\pi$
$42$	0	0
$43$	164.000	0.581622	0.290811	−	0.956780i	$-0.406075\pi$
$43$	164.000	0.581622	0.290811	+	0.956780i	$0.406075\pi$
$44$	−104.000	−0.356332
$45$	0	0
$46$	116.000	0.371810
$47$	444.000	1.37796	0.688979	−	0.724781i	$-0.258059\pi$
$47$	444.000	1.37796	0.688979	+	0.724781i	$0.258059\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	718.000	2.03081
$51$	0	0
$52$	−216.000	−0.576035
$53$	−12.0000	−0.0311005	−0.0155503	−	0.999879i	$-0.504950\pi$
$53$	−12.0000	−0.0311005	−0.0155503	+	0.999879i	$0.504950\pi$
$54$	0	0
$55$	572.000	1.40234
$56$	−56.0000	−0.133631
$57$	0	0
$58$	−416.000	−0.941784
$59$	−124.000	−0.273617	−0.136809	−	0.990597i	$-0.543685\pi$
$59$	−124.000	−0.273617	−0.136809	+	0.990597i	$0.543685\pi$
$60$	0	0
$61$	−162.000	−0.340032	−0.170016	−	0.985441i	$-0.554382\pi$
$61$	−162.000	−0.340032	−0.170016	+	0.985441i	$0.554382\pi$
$62$	−504.000	−1.03239
$63$	0	0
$64$	64.0000	0.125000
$65$	1188.00	2.26697
$66$	0	0
$67$	−860.000	−1.56815	−0.784073	−	0.620669i	$-0.786861\pi$
$67$	−860.000	−1.56815	−0.784073	+	0.620669i	$0.786861\pi$
$68$	−296.000	−0.527872
$69$	0	0
$70$	308.000	0.525901
$71$	238.000	0.397823	0.198911	−	0.980017i	$-0.436259\pi$
$71$	238.000	0.397823	0.198911	+	0.980017i	$0.436259\pi$
$72$	0	0
$73$	−146.000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−146.000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	100.000	0.157091
$75$	0	0
$76$	464.000	0.700322
$77$	182.000	0.269361
$78$	0	0
$79$	−984.000	−1.40138	−0.700688	−	0.713468i	$-0.747123\pi$
$79$	−984.000	−1.40138	−0.700688	+	0.713468i	$0.747123\pi$
$80$	−352.000	−0.491935
$81$	0	0
$82$	−252.000	−0.339375
$83$	−656.000	−0.867534	−0.433767	−	0.901025i	$-0.642816\pi$
$83$	−656.000	−0.867534	−0.433767	+	0.901025i	$0.642816\pi$
$84$	0	0
$85$	1628.00	2.07743
$86$	328.000	0.411269
$87$	0	0
$88$	−208.000	−0.251964
$89$	954.000	1.13622	0.568111	−	0.822952i	$-0.307674\pi$
$89$	954.000	1.13622	0.568111	+	0.822952i	$0.307674\pi$
$90$	0	0
$91$	378.000	0.435441
$92$	232.000	0.262909
$93$	0	0
$94$	888.000	0.974364
$95$	−2552.00	−2.75610
$96$	0	0
$97$	526.000	0.550590	0.275295	−	0.961360i	$-0.411225\pi$
$97$	526.000	0.550590	0.275295	+	0.961360i	$0.411225\pi$
$98$	98.0000	0.101015
$99$	0	0
$100$	1436.00	1.43600
$101$	−1306.00	−1.28665	−0.643326	−	0.765592i	$-0.722446\pi$
$101$	−1306.00	−1.28665	−0.643326	+	0.765592i	$0.722446\pi$
$102$	0	0
$103$	−508.000	−0.485968	−0.242984	−	0.970030i	$-0.578126\pi$
$103$	−508.000	−0.485968	−0.242984	+	0.970030i	$0.578126\pi$
$104$	−432.000	−0.407318
$105$	0	0
$106$	−24.0000	−0.0219914
$107$	−498.000	−0.449939	−0.224970	−	0.974366i	$-0.572228\pi$
$107$	−498.000	−0.449939	−0.224970	+	0.974366i	$0.572228\pi$
$108$	0	0
$109$	−614.000	−0.539546	−0.269773	−	0.962924i	$-0.586949\pi$
$109$	−614.000	−0.539546	−0.269773	+	0.962924i	$0.586949\pi$
$110$	1144.00	0.991601
$111$	0	0
$112$	−112.000	−0.0944911
$113$	1232.00	1.02564	0.512818	−	0.858498i	$-0.328602\pi$
$113$	1232.00	1.02564	0.512818	+	0.858498i	$0.328602\pi$
$114$	0	0
$115$	−1276.00	−1.03467
$116$	−832.000	−0.665942
$117$	0	0
$118$	−248.000	−0.193477
$119$	518.000	0.399033
$120$	0	0
$121$	−655.000	−0.492111
$122$	−324.000	−0.240439
$123$	0	0
$124$	−1008.00	−0.730009
$125$	−5148.00	−3.68361
$126$	0	0
$127$	−2808.00	−1.96197	−0.980983	−	0.194093i	$-0.937824\pi$
$127$	−2808.00	−1.96197	−0.980983	+	0.194093i	$0.937824\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	2376.00	1.60299
$131$	−520.000	−0.346814	−0.173407	−	0.984850i	$-0.555478\pi$
$131$	−520.000	−0.346814	−0.173407	+	0.984850i	$0.555478\pi$
$132$	0	0
$133$	−812.000	−0.529393
$134$	−1720.00	−1.10885
$135$	0	0
$136$	−592.000	−0.373262
$137$	2516.00	1.56902	0.784512	−	0.620113i	$-0.212913\pi$
$137$	2516.00	1.56902	0.784512	+	0.620113i	$0.212913\pi$
$138$	0	0
$139$	−2672.00	−1.63048	−0.815238	−	0.579126i	$-0.803394\pi$
$139$	−2672.00	−1.63048	−0.815238	+	0.579126i	$0.803394\pi$
$140$	616.000	0.371868
$141$	0	0
$142$	476.000	0.281303
$143$	1404.00	0.821038
$144$	0	0
$145$	4576.00	2.62080
$146$	−292.000	−0.165521
$147$	0	0
$148$	200.000	0.111080
$149$	1164.00	0.639991	0.319995	−	0.947419i	$-0.396319\pi$
$149$	1164.00	0.639991	0.319995	+	0.947419i	$0.396319\pi$
$150$	0	0
$151$	1672.00	0.901096	0.450548	−	0.892752i	$-0.351229\pi$
$151$	1672.00	0.901096	0.450548	+	0.892752i	$0.351229\pi$
$152$	928.000	0.495202
$153$	0	0
$154$	364.000	0.190467
$155$	5544.00	2.87293
$156$	0	0
$157$	446.000	0.226718	0.113359	−	0.993554i	$-0.463839\pi$
$157$	446.000	0.226718	0.113359	+	0.993554i	$0.463839\pi$
$158$	−1968.00	−0.990922
$159$	0	0
$160$	−704.000	−0.347851
$161$	−406.000	−0.198741
$162$	0	0
$163$	428.000	0.205666	0.102833	−	0.994699i	$-0.467209\pi$
$163$	428.000	0.205666	0.102833	+	0.994699i	$0.467209\pi$
$164$	−504.000	−0.239974
$165$	0	0
$166$	−1312.00	−0.613439
$167$	−4.00000	−0.00185347	−0.000926734	−	1.00000i	$-0.500295\pi$
$167$	−4.00000	−0.00185347	−0.000926734		1.00000i	$0.500295\pi$
$168$	0	0
$169$	719.000	0.327264
$170$	3256.00	1.46896
$171$	0	0
$172$	656.000	0.290811
$173$	590.000	0.259288	0.129644	−	0.991561i	$-0.458617\pi$
$173$	590.000	0.259288	0.129644	+	0.991561i	$0.458617\pi$
$174$	0	0
$175$	−2513.00	−1.08551
$176$	−416.000	−0.178166
$177$	0	0
$178$	1908.00	0.803431
$179$	−3534.00	−1.47566	−0.737831	−	0.674985i	$-0.764150\pi$
$179$	−3534.00	−1.47566	−0.737831	+	0.674985i	$0.764150\pi$
$180$	0	0
$181$	1098.00	0.450904	0.225452	−	0.974254i	$-0.427614\pi$
$181$	1098.00	0.450904	0.225452	+	0.974254i	$0.427614\pi$
$182$	756.000	0.307904
$183$	0	0
$184$	464.000	0.185905
$185$	−1100.00	−0.437155
$186$	0	0
$187$	1924.00	0.752389
$188$	1776.00	0.688979
$189$	0	0
$190$	−5104.00	−1.94886
$191$	4854.00	1.83886	0.919432	−	0.393248i	$-0.128649\pi$
$191$	4854.00	1.83886	0.919432	+	0.393248i	$0.128649\pi$
$192$	0	0
$193$	−1498.00	−0.558696	−0.279348	−	0.960190i	$-0.590118\pi$
$193$	−1498.00	−0.558696	−0.279348	+	0.960190i	$0.590118\pi$
$194$	1052.00	0.389326
$195$	0	0
$196$	196.000	0.0714286
$197$	−620.000	−0.224229	−0.112115	−	0.993695i	$-0.535762\pi$
$197$	−620.000	−0.224229	−0.112115	+	0.993695i	$0.535762\pi$
$198$	0	0
$199$	32.0000	0.0113991	0.00569955	−	0.999984i	$-0.498186\pi$
$199$	32.0000	0.0113991	0.00569955	+	0.999984i	$0.498186\pi$
$200$	2872.00	1.01541
$201$	0	0
$202$	−2612.00	−0.909800
$203$	1456.00	0.503405
$204$	0	0
$205$	2772.00	0.944414
$206$	−1016.00	−0.343631
$207$	0	0
$208$	−864.000	−0.288017
$209$	−3016.00	−0.998187
$210$	0	0
$211$	4268.00	1.39252	0.696259	−	0.717791i	$-0.254847\pi$
$211$	4268.00	1.39252	0.696259	+	0.717791i	$0.254847\pi$
$212$	−48.0000	−0.0155503
$213$	0	0
$214$	−996.000	−0.318155
$215$	−3608.00	−1.14448
$216$	0	0
$217$	1764.00	0.551835
$218$	−1228.00	−0.381517
$219$	0	0
$220$	2288.00	0.701168
$221$	3996.00	1.21629
$222$	0	0
$223$	3464.00	1.04021	0.520104	−	0.854103i	$-0.325893\pi$
$223$	3464.00	1.04021	0.520104	+	0.854103i	$0.325893\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	2464.00	0.725234
$227$	3252.00	0.950849	0.475425	−	0.879756i	$-0.342294\pi$
$227$	3252.00	0.950849	0.475425	+	0.879756i	$0.342294\pi$
$228$	0	0
$229$	418.000	0.120621	0.0603105	−	0.998180i	$-0.480791\pi$
$229$	418.000	0.120621	0.0603105	+	0.998180i	$0.480791\pi$
$230$	−2552.00	−0.731626
$231$	0	0
$232$	−1664.00	−0.470892
$233$	−2084.00	−0.585954	−0.292977	−	0.956119i	$-0.594646\pi$
$233$	−2084.00	−0.585954	−0.292977	+	0.956119i	$0.594646\pi$
$234$	0	0
$235$	−9768.00	−2.71146
$236$	−496.000	−0.136809
$237$	0	0
$238$	1036.00	0.282159
$239$	−1662.00	−0.449815	−0.224908	−	0.974380i	$-0.572208\pi$
$239$	−1662.00	−0.449815	−0.224908	+	0.974380i	$0.572208\pi$
$240$	0	0
$241$	6182.00	1.65236	0.826178	−	0.563410i	$-0.190511\pi$
$241$	6182.00	1.65236	0.826178	+	0.563410i	$0.190511\pi$
$242$	−1310.00	−0.347975
$243$	0	0
$244$	−648.000	−0.170016
$245$	−1078.00	−0.281106
$246$	0	0
$247$	−6264.00	−1.61364
$248$	−2016.00	−0.516194
$249$	0	0
$250$	−10296.0	−2.60470
$251$	996.000	0.250466	0.125233	−	0.992127i	$-0.460032\pi$
$251$	996.000	0.250466	0.125233	+	0.992127i	$0.460032\pi$
$252$	0	0
$253$	−1508.00	−0.374732
$254$	−5616.00	−1.38732
$255$	0	0
$256$	256.000	0.0625000
$257$	5994.00	1.45485	0.727423	−	0.686189i	$-0.240718\pi$
$257$	5994.00	1.45485	0.727423	+	0.686189i	$0.240718\pi$
$258$	0	0
$259$	−350.000	−0.0839689
$260$	4752.00	1.13349
$261$	0	0
$262$	−1040.00	−0.245234
$263$	−6414.00	−1.50382	−0.751909	−	0.659267i	$-0.770867\pi$
$263$	−6414.00	−1.50382	−0.751909	+	0.659267i	$0.770867\pi$
$264$	0	0
$265$	264.000	0.0611977
$266$	−1624.00	−0.374338
$267$	0	0
$268$	−3440.00	−0.784073
$269$	−2686.00	−0.608804	−0.304402	−	0.952544i	$-0.598457\pi$
$269$	−2686.00	−0.608804	−0.304402	+	0.952544i	$0.598457\pi$
$270$	0	0
$271$	5100.00	1.14318	0.571592	−	0.820538i	$-0.306326\pi$
$271$	5100.00	1.14318	0.571592	+	0.820538i	$0.306326\pi$
$272$	−1184.00	−0.263936
$273$	0	0
$274$	5032.00	1.10947
$275$	−9334.00	−2.04677
$276$	0	0
$277$	−4426.00	−0.960045	−0.480023	−	0.877256i	$-0.659372\pi$
$277$	−4426.00	−0.960045	−0.480023	+	0.877256i	$0.659372\pi$
$278$	−5344.00	−1.15292
$279$	0	0
$280$	1232.00	0.262950
$281$	−7508.00	−1.59391	−0.796957	−	0.604036i	$-0.793558\pi$
$281$	−7508.00	−1.59391	−0.796957	+	0.604036i	$0.793558\pi$
$282$	0	0
$283$	3412.00	0.716687	0.358343	−	0.933590i	$-0.383342\pi$
$283$	3412.00	0.716687	0.358343	+	0.933590i	$0.383342\pi$
$284$	952.000	0.198911
$285$	0	0
$286$	2808.00	0.580561
$287$	882.000	0.181404
$288$	0	0
$289$	563.000	0.114594
$290$	9152.00	1.85319
$291$	0	0
$292$	−584.000	−0.117041
$293$	4734.00	0.943902	0.471951	−	0.881625i	$-0.343550\pi$
$293$	4734.00	0.943902	0.471951	+	0.881625i	$0.343550\pi$
$294$	0	0
$295$	2728.00	0.538408
$296$	400.000	0.0785457
$297$	0	0
$298$	2328.00	0.452542
$299$	−3132.00	−0.605780
$300$	0	0
$301$	−1148.00	−0.219833
$302$	3344.00	0.637171
$303$	0	0
$304$	1856.00	0.350161
$305$	3564.00	0.669095
$306$	0	0
$307$	5836.00	1.08494	0.542472	−	0.840074i	$-0.317488\pi$
$307$	5836.00	1.08494	0.542472	+	0.840074i	$0.317488\pi$
$308$	728.000	0.134681
$309$	0	0
$310$	11088.0	2.03147
$311$	−5620.00	−1.02470	−0.512349	−	0.858777i	$-0.671225\pi$
$311$	−5620.00	−1.02470	−0.512349	+	0.858777i	$0.671225\pi$
$312$	0	0
$313$	6082.00	1.09832	0.549161	−	0.835716i	$-0.314947\pi$
$313$	6082.00	1.09832	0.549161	+	0.835716i	$0.314947\pi$
$314$	892.000	0.160314
$315$	0	0
$316$	−3936.00	−0.700688
$317$	7308.00	1.29482	0.647410	−	0.762142i	$-0.275852\pi$
$317$	7308.00	1.29482	0.647410	+	0.762142i	$0.275852\pi$
$318$	0	0
$319$	5408.00	0.949185
$320$	−1408.00	−0.245967
$321$	0	0
$322$	−812.000	−0.140531
$323$	−8584.00	−1.47872
$324$	0	0
$325$	−19386.0	−3.30874
$326$	856.000	0.145428
$327$	0	0
$328$	−1008.00	−0.169687
$329$	−3108.00	−0.520819
$330$	0	0
$331$	−8020.00	−1.33178	−0.665890	−	0.746050i	$-0.731948\pi$
$331$	−8020.00	−1.33178	−0.665890	+	0.746050i	$0.731948\pi$
$332$	−2624.00	−0.433767
$333$	0	0
$334$	−8.00000	−0.00131060
$335$	18920.0	3.08570
$336$	0	0
$337$	4590.00	0.741938	0.370969	−	0.928645i	$-0.379026\pi$
$337$	4590.00	0.741938	0.370969	+	0.928645i	$0.379026\pi$
$338$	1438.00	0.231411
$339$	0	0
$340$	6512.00	1.03871
$341$	6552.00	1.04050
$342$	0	0
$343$	−343.000	−0.0539949
$344$	1312.00	0.205635
$345$	0	0
$346$	1180.00	0.183344
$347$	6546.00	1.01270	0.506351	−	0.862327i	$-0.330994\pi$
$347$	6546.00	1.01270	0.506351	+	0.862327i	$0.330994\pi$
$348$	0	0
$349$	−7994.00	−1.22610	−0.613050	−	0.790044i	$-0.710058\pi$
$349$	−7994.00	−1.22610	−0.613050	+	0.790044i	$0.710058\pi$
$350$	−5026.00	−0.767574
$351$	0	0
$352$	−832.000	−0.125982
$353$	−4650.00	−0.701118	−0.350559	−	0.936541i	$-0.614008\pi$
$353$	−4650.00	−0.701118	−0.350559	+	0.936541i	$0.614008\pi$
$354$	0	0
$355$	−5236.00	−0.782811
$356$	3816.00	0.568111
$357$	0	0
$358$	−7068.00	−1.04345
$359$	−346.000	−0.0508668	−0.0254334	−	0.999677i	$-0.508097\pi$
$359$	−346.000	−0.0508668	−0.0254334	+	0.999677i	$0.508097\pi$
$360$	0	0
$361$	6597.00	0.961802
$362$	2196.00	0.318838
$363$	0	0
$364$	1512.00	0.217721
$365$	3212.00	0.460613
$366$	0	0
$367$	−6784.00	−0.964910	−0.482455	−	0.875921i	$-0.660255\pi$
$367$	−6784.00	−0.964910	−0.482455	+	0.875921i	$0.660255\pi$
$368$	928.000	0.131455
$369$	0	0
$370$	−2200.00	−0.309115
$371$	84.0000	0.0117549
$372$	0	0
$373$	−6098.00	−0.846495	−0.423247	−	0.906014i	$-0.639110\pi$
$373$	−6098.00	−0.846495	−0.423247	+	0.906014i	$0.639110\pi$
$374$	3848.00	0.532020
$375$	0	0
$376$	3552.00	0.487182
$377$	11232.0	1.53442
$378$	0	0
$379$	−2660.00	−0.360515	−0.180257	−	0.983619i	$-0.557693\pi$
$379$	−2660.00	−0.360515	−0.180257	+	0.983619i	$0.557693\pi$
$380$	−10208.0	−1.37805
$381$	0	0
$382$	9708.00	1.30027
$383$	760.000	0.101395	0.0506974	−	0.998714i	$-0.483856\pi$
$383$	760.000	0.101395	0.0506974	+	0.998714i	$0.483856\pi$
$384$	0	0
$385$	−4004.00	−0.530033
$386$	−2996.00	−0.395058
$387$	0	0
$388$	2104.00	0.275295
$389$	104.000	0.0135553	0.00677765	−	0.999977i	$-0.497843\pi$
$389$	104.000	0.0135553	0.00677765	+	0.999977i	$0.497843\pi$
$390$	0	0
$391$	−4292.00	−0.555130
$392$	392.000	0.0505076
$393$	0	0
$394$	−1240.00	−0.158554
$395$	21648.0	2.75754
$396$	0	0
$397$	4398.00	0.555993	0.277997	−	0.960582i	$-0.410330\pi$
$397$	4398.00	0.555993	0.277997	+	0.960582i	$0.410330\pi$
$398$	64.0000	0.00806038
$399$	0	0
$400$	5744.00	0.718000
$401$	−13236.0	−1.64831	−0.824157	−	0.566361i	$-0.808351\pi$
$401$	−13236.0	−1.64831	−0.824157	+	0.566361i	$0.808351\pi$
$402$	0	0
$403$	13608.0	1.68204
$404$	−5224.00	−0.643326
$405$	0	0
$406$	2912.00	0.355961
$407$	−1300.00	−0.158326
$408$	0	0
$409$	−9490.00	−1.14731	−0.573656	−	0.819097i	$-0.694475\pi$
$409$	−9490.00	−1.14731	−0.573656	+	0.819097i	$0.694475\pi$
$410$	5544.00	0.667802
$411$	0	0
$412$	−2032.00	−0.242984
$413$	868.000	0.103418
$414$	0	0
$415$	14432.0	1.70708
$416$	−1728.00	−0.203659
$417$	0	0
$418$	−6032.00	−0.705825
$419$	−4236.00	−0.493895	−0.246948	−	0.969029i	$-0.579428\pi$
$419$	−4236.00	−0.493895	−0.246948	+	0.969029i	$0.579428\pi$
$420$	0	0
$421$	918.000	0.106272	0.0531361	−	0.998587i	$-0.483078\pi$
$421$	918.000	0.106272	0.0531361	+	0.998587i	$0.483078\pi$
$422$	8536.00	0.984659
$423$	0	0
$424$	−96.0000	−0.0109957
$425$	−26566.0	−3.03209
$426$	0	0
$427$	1134.00	0.128520
$428$	−1992.00	−0.224970
$429$	0	0
$430$	−7216.00	−0.809271
$431$	11814.0	1.32033	0.660163	−	0.751123i	$-0.270487\pi$
$431$	11814.0	1.32033	0.660163	+	0.751123i	$0.270487\pi$
$432$	0	0
$433$	−8374.00	−0.929397	−0.464698	−	0.885469i	$-0.653837\pi$
$433$	−8374.00	−0.929397	−0.464698	+	0.885469i	$0.653837\pi$
$434$	3528.00	0.390206
$435$	0	0
$436$	−2456.00	−0.269773
$437$	6728.00	0.736485
$438$	0	0
$439$	3840.00	0.417479	0.208739	−	0.977971i	$-0.433064\pi$
$439$	3840.00	0.417479	0.208739	+	0.977971i	$0.433064\pi$
$440$	4576.00	0.495801
$441$	0	0
$442$	7992.00	0.860047
$443$	−10166.0	−1.09030	−0.545148	−	0.838340i	$-0.683527\pi$
$443$	−10166.0	−1.09030	−0.545148	+	0.838340i	$0.683527\pi$
$444$	0	0
$445$	−20988.0	−2.23579
$446$	6928.00	0.735539
$447$	0	0
$448$	−448.000	−0.0472456
$449$	−8200.00	−0.861875	−0.430938	−	0.902382i	$-0.641817\pi$
$449$	−8200.00	−0.861875	−0.430938	+	0.902382i	$0.641817\pi$
$450$	0	0
$451$	3276.00	0.342042
$452$	4928.00	0.512818
$453$	0	0
$454$	6504.00	0.672352
$455$	−8316.00	−0.856835
$456$	0	0
$457$	−6074.00	−0.621728	−0.310864	−	0.950454i	$-0.600618\pi$
$457$	−6074.00	−0.621728	−0.310864	+	0.950454i	$0.600618\pi$
$458$	836.000	0.0852920
$459$	0	0
$460$	−5104.00	−0.517337
$461$	2006.00	0.202665	0.101333	−	0.994853i	$-0.467689\pi$
$461$	2006.00	0.202665	0.101333	+	0.994853i	$0.467689\pi$
$462$	0	0
$463$	−3728.00	−0.374201	−0.187100	−	0.982341i	$-0.559909\pi$
$463$	−3728.00	−0.374201	−0.187100	+	0.982341i	$0.559909\pi$
$464$	−3328.00	−0.332971
$465$	0	0
$466$	−4168.00	−0.414332
$467$	6380.00	0.632187	0.316093	−	0.948728i	$-0.397629\pi$
$467$	6380.00	0.632187	0.316093	+	0.948728i	$0.397629\pi$
$468$	0	0
$469$	6020.00	0.592703
$470$	−19536.0	−1.91729
$471$	0	0
$472$	−992.000	−0.0967383
$473$	−4264.00	−0.414501
$474$	0	0
$475$	41644.0	4.02265
$476$	2072.00	0.199517
$477$	0	0
$478$	−3324.00	−0.318067
$479$	−17180.0	−1.63878	−0.819389	−	0.573239i	$-0.805687\pi$
$479$	−17180.0	−1.63878	−0.819389	+	0.573239i	$0.805687\pi$
$480$	0	0
$481$	−2700.00	−0.255945
$482$	12364.0	1.16839
$483$	0	0
$484$	−2620.00	−0.246056
$485$	−11572.0	−1.08342
$486$	0	0
$487$	−2728.00	−0.253835	−0.126917	−	0.991913i	$-0.540508\pi$
$487$	−2728.00	−0.253835	−0.126917	+	0.991913i	$0.540508\pi$
$488$	−1296.00	−0.120220
$489$	0	0
$490$	−2156.00	−0.198772
$491$	−2574.00	−0.236585	−0.118292	−	0.992979i	$-0.537742\pi$
$491$	−2574.00	−0.236585	−0.118292	+	0.992979i	$0.537742\pi$
$492$	0	0
$493$	15392.0	1.40613
$494$	−12528.0	−1.14101
$495$	0	0
$496$	−4032.00	−0.365004
$497$	−1666.00	−0.150363
$498$	0	0
$499$	−7484.00	−0.671403	−0.335701	−	0.941969i	$-0.608973\pi$
$499$	−7484.00	−0.671403	−0.335701	+	0.941969i	$0.608973\pi$
$500$	−20592.0	−1.84180
$501$	0	0
$502$	1992.00	0.177106
$503$	−7920.00	−0.702058	−0.351029	−	0.936365i	$-0.614168\pi$
$503$	−7920.00	−0.702058	−0.351029	+	0.936365i	$0.614168\pi$
$504$	0	0
$505$	28732.0	2.53180
$506$	−3016.00	−0.264975
$507$	0	0
$508$	−11232.0	−0.980983
$509$	−7254.00	−0.631685	−0.315843	−	0.948812i	$-0.602287\pi$
$509$	−7254.00	−0.631685	−0.315843	+	0.948812i	$0.602287\pi$
$510$	0	0
$511$	1022.00	0.0884748
$512$	512.000	0.0441942
$513$	0	0
$514$	11988.0	1.02873
$515$	11176.0	0.956259
$516$	0	0
$517$	−11544.0	−0.982020
$518$	−700.000	−0.0593750
$519$	0	0
$520$	9504.00	0.801496
$521$	17862.0	1.50201	0.751006	−	0.660295i	$-0.229569\pi$
$521$	17862.0	1.50201	0.751006	+	0.660295i	$0.229569\pi$
$522$	0	0
$523$	−592.000	−0.0494959	−0.0247479	−	0.999694i	$-0.507878\pi$
$523$	−592.000	−0.0494959	−0.0247479	+	0.999694i	$0.507878\pi$
$524$	−2080.00	−0.173407
$525$	0	0
$526$	−12828.0	−1.06336
$527$	18648.0	1.54140
$528$	0	0
$529$	−8803.00	−0.723514
$530$	528.000	0.0432733
$531$	0	0
$532$	−3248.00	−0.264697
$533$	6804.00	0.552934
$534$	0	0
$535$	10956.0	0.885363
$536$	−6880.00	−0.554423
$537$	0	0
$538$	−5372.00	−0.430490
$539$	−1274.00	−0.101809
$540$	0	0
$541$	−6402.00	−0.508768	−0.254384	−	0.967103i	$-0.581873\pi$
$541$	−6402.00	−0.508768	−0.254384	+	0.967103i	$0.581873\pi$
$542$	10200.0	0.808353
$543$	0	0
$544$	−2368.00	−0.186631
$545$	13508.0	1.06169
$546$	0	0
$547$	−8988.00	−0.702558	−0.351279	−	0.936271i	$-0.614253\pi$
$547$	−8988.00	−0.702558	−0.351279	+	0.936271i	$0.614253\pi$
$548$	10064.0	0.784512
$549$	0	0
$550$	−18668.0	−1.44728
$551$	−24128.0	−1.86549
$552$	0	0
$553$	6888.00	0.529670
$554$	−8852.00	−0.678855
$555$	0	0
$556$	−10688.0	−0.815238
$557$	−3244.00	−0.246773	−0.123387	−	0.992359i	$-0.539376\pi$
$557$	−3244.00	−0.246773	−0.123387	+	0.992359i	$0.539376\pi$
$558$	0	0
$559$	−8856.00	−0.670070
$560$	2464.00	0.185934
$561$	0	0
$562$	−15016.0	−1.12707
$563$	−9812.00	−0.734505	−0.367253	−	0.930121i	$-0.619702\pi$
$563$	−9812.00	−0.734505	−0.367253	+	0.930121i	$0.619702\pi$
$564$	0	0
$565$	−27104.0	−2.01818
$566$	6824.00	0.506774
$567$	0	0
$568$	1904.00	0.140652
$569$	12156.0	0.895617	0.447808	−	0.894130i	$-0.352205\pi$
$569$	12156.0	0.895617	0.447808	+	0.894130i	$0.352205\pi$
$570$	0	0
$571$	6876.00	0.503943	0.251972	−	0.967735i	$-0.418921\pi$
$571$	6876.00	0.503943	0.251972	+	0.967735i	$0.418921\pi$
$572$	5616.00	0.410519
$573$	0	0
$574$	1764.00	0.128272
$575$	20822.0	1.51015
$576$	0	0
$577$	20002.0	1.44314	0.721572	−	0.692339i	$-0.243420\pi$
$577$	20002.0	1.44314	0.721572	+	0.692339i	$0.243420\pi$
$578$	1126.00	0.0810301
$579$	0	0
$580$	18304.0	1.31040
$581$	4592.00	0.327897
$582$	0	0
$583$	312.000	0.0221642
$584$	−1168.00	−0.0827606
$585$	0	0
$586$	9468.00	0.667439
$587$	−18404.0	−1.29406	−0.647031	−	0.762464i	$-0.723990\pi$
$587$	−18404.0	−1.29406	−0.647031	+	0.762464i	$0.723990\pi$
$588$	0	0
$589$	−29232.0	−2.04496
$590$	5456.00	0.380712
$591$	0	0
$592$	800.000	0.0555402
$593$	9846.00	0.681833	0.340916	−	0.940094i	$-0.389263\pi$
$593$	9846.00	0.681833	0.340916	+	0.940094i	$0.389263\pi$
$594$	0	0
$595$	−11396.0	−0.785194
$596$	4656.00	0.319995
$597$	0	0
$598$	−6264.00	−0.428351
$599$	−9234.00	−0.629868	−0.314934	−	0.949114i	$-0.601982\pi$
$599$	−9234.00	−0.629868	−0.314934	+	0.949114i	$0.601982\pi$
$600$	0	0
$601$	−1510.00	−0.102486	−0.0512431	−	0.998686i	$-0.516318\pi$
$601$	−1510.00	−0.102486	−0.0512431	+	0.998686i	$0.516318\pi$
$602$	−2296.00	−0.155445
$603$	0	0
$604$	6688.00	0.450548
$605$	14410.0	0.968347
$606$	0	0
$607$	17544.0	1.17313	0.586564	−	0.809903i	$-0.300480\pi$
$607$	17544.0	1.17313	0.586564	+	0.809903i	$0.300480\pi$
$608$	3712.00	0.247601
$609$	0	0
$610$	7128.00	0.473122
$611$	−23976.0	−1.58750
$612$	0	0
$613$	9246.00	0.609205	0.304602	−	0.952480i	$-0.401476\pi$
$613$	9246.00	0.609205	0.304602	+	0.952480i	$0.401476\pi$
$614$	11672.0	0.767172
$615$	0	0
$616$	1456.00	0.0952336
$617$	29212.0	1.90605	0.953023	−	0.302897i	$-0.0979537\pi$
$617$	29212.0	1.90605	0.953023	+	0.302897i	$0.0979537\pi$
$618$	0	0
$619$	7096.00	0.460763	0.230382	−	0.973100i	$-0.426003\pi$
$619$	7096.00	0.460763	0.230382	+	0.973100i	$0.426003\pi$
$620$	22176.0	1.43647
$621$	0	0
$622$	−11240.0	−0.724571
$623$	−6678.00	−0.429452
$624$	0	0
$625$	68381.0	4.37638
$626$	12164.0	0.776631
$627$	0	0
$628$	1784.00	0.113359
$629$	−3700.00	−0.234545
$630$	0	0
$631$	488.000	0.0307876	0.0153938	−	0.999882i	$-0.495100\pi$
$631$	488.000	0.0307876	0.0153938	+	0.999882i	$0.495100\pi$
$632$	−7872.00	−0.495461
$633$	0	0
$634$	14616.0	0.915577
$635$	61776.0	3.86064
$636$	0	0
$637$	−2646.00	−0.164581
$638$	10816.0	0.671175
$639$	0	0
$640$	−2816.00	−0.173925
$641$	−8756.00	−0.539534	−0.269767	−	0.962926i	$-0.586947\pi$
$641$	−8756.00	−0.539534	−0.269767	+	0.962926i	$0.586947\pi$
$642$	0	0
$643$	−3364.00	−0.206319	−0.103160	−	0.994665i	$-0.532895\pi$
$643$	−3364.00	−0.206319	−0.103160	+	0.994665i	$0.532895\pi$
$644$	−1624.00	−0.0993704
$645$	0	0
$646$	−17168.0	−1.04561
$647$	−21804.0	−1.32489	−0.662445	−	0.749111i	$-0.730481\pi$
$647$	−21804.0	−1.32489	−0.662445	+	0.749111i	$0.730481\pi$
$648$	0	0
$649$	3224.00	0.194997
$650$	−38772.0	−2.33964
$651$	0	0
$652$	1712.00	0.102833
$653$	−13488.0	−0.808310	−0.404155	−	0.914691i	$-0.632434\pi$
$653$	−13488.0	−0.808310	−0.404155	+	0.914691i	$0.632434\pi$
$654$	0	0
$655$	11440.0	0.682439
$656$	−2016.00	−0.119987
$657$	0	0
$658$	−6216.00	−0.368275
$659$	−28946.0	−1.71104	−0.855521	−	0.517769i	$-0.826763\pi$
$659$	−28946.0	−1.71104	−0.855521	+	0.517769i	$0.826763\pi$
$660$	0	0
$661$	−20642.0	−1.21465	−0.607323	−	0.794455i	$-0.707757\pi$
$661$	−20642.0	−1.21465	−0.607323	+	0.794455i	$0.707757\pi$
$662$	−16040.0	−0.941710
$663$	0	0
$664$	−5248.00	−0.306720
$665$	17864.0	1.04171
$666$	0	0
$667$	−12064.0	−0.700330
$668$	−16.0000	−0.000926734 0
$669$	0	0
$670$	37840.0	2.18192
$671$	4212.00	0.242329
$672$	0	0
$673$	−17602.0	−1.00818	−0.504092	−	0.863650i	$-0.668173\pi$
$673$	−17602.0	−1.00818	−0.504092	+	0.863650i	$0.668173\pi$
$674$	9180.00	0.524630
$675$	0	0
$676$	2876.00	0.163632
$677$	−4266.00	−0.242180	−0.121090	−	0.992642i	$-0.538639\pi$
$677$	−4266.00	−0.242180	−0.121090	+	0.992642i	$0.538639\pi$
$678$	0	0
$679$	−3682.00	−0.208103
$680$	13024.0	0.734482
$681$	0	0
$682$	13104.0	0.735745
$683$	26874.0	1.50557	0.752786	−	0.658266i	$-0.228710\pi$
$683$	26874.0	1.50557	0.752786	+	0.658266i	$0.228710\pi$
$684$	0	0
$685$	−55352.0	−3.08743
$686$	−686.000	−0.0381802
$687$	0	0
$688$	2624.00	0.145406
$689$	648.000	0.0358299
$690$	0	0
$691$	−17128.0	−0.942952	−0.471476	−	0.881879i	$-0.656279\pi$
$691$	−17128.0	−0.942952	−0.471476	+	0.881879i	$0.656279\pi$
$692$	2360.00	0.129644
$693$	0	0
$694$	13092.0	0.716089
$695$	58784.0	3.20835
$696$	0	0
$697$	9324.00	0.506703
$698$	−15988.0	−0.866984
$699$	0	0
$700$	−10052.0	−0.542757
$701$	−11968.0	−0.644829	−0.322414	−	0.946599i	$-0.604494\pi$
$701$	−11968.0	−0.644829	−0.322414	+	0.946599i	$0.604494\pi$
$702$	0	0
$703$	5800.00	0.311168
$704$	−1664.00	−0.0890829
$705$	0	0
$706$	−9300.00	−0.495765
$707$	9142.00	0.486309
$708$	0	0
$709$	−5278.00	−0.279576	−0.139788	−	0.990181i	$-0.544642\pi$
$709$	−5278.00	−0.279576	−0.139788	+	0.990181i	$0.544642\pi$
$710$	−10472.0	−0.553531
$711$	0	0
$712$	7632.00	0.401715
$713$	−14616.0	−0.767705
$714$	0	0
$715$	−30888.0	−1.61559
$716$	−14136.0	−0.737831
$717$	0	0
$718$	−692.000	−0.0359683
$719$	−6720.00	−0.348559	−0.174279	−	0.984696i	$-0.555760\pi$
$719$	−6720.00	−0.348559	−0.174279	+	0.984696i	$0.555760\pi$
$720$	0	0
$721$	3556.00	0.183679
$722$	13194.0	0.680097
$723$	0	0
$724$	4392.00	0.225452
$725$	−74672.0	−3.82517
$726$	0	0
$727$	16804.0	0.857257	0.428629	−	0.903481i	$-0.358997\pi$
$727$	16804.0	0.857257	0.428629	+	0.903481i	$0.358997\pi$
$728$	3024.00	0.153952
$729$	0	0
$730$	6424.00	0.325703
$731$	−12136.0	−0.614044
$732$	0	0
$733$	27522.0	1.38683	0.693416	−	0.720537i	$-0.256105\pi$
$733$	27522.0	1.38683	0.693416	+	0.720537i	$0.256105\pi$
$734$	−13568.0	−0.682294
$735$	0	0
$736$	1856.00	0.0929525
$737$	22360.0	1.11756
$738$	0	0
$739$	21132.0	1.05190	0.525949	−	0.850516i	$-0.323710\pi$
$739$	21132.0	1.05190	0.525949	+	0.850516i	$0.323710\pi$
$740$	−4400.00	−0.218577
$741$	0	0
$742$	168.000	0.00831196
$743$	30.0000	0.00148128	0.000740641	−	1.00000i	$-0.499764\pi$
$743$	30.0000	0.00148128	0.000740641		1.00000i	$0.499764\pi$
$744$	0	0
$745$	−25608.0	−1.25933
$746$	−12196.0	−0.598562
$747$	0	0
$748$	7696.00	0.376195
$749$	3486.00	0.170061
$750$	0	0
$751$	−15480.0	−0.752161	−0.376081	−	0.926587i	$-0.622728\pi$
$751$	−15480.0	−0.752161	−0.376081	+	0.926587i	$0.622728\pi$
$752$	7104.00	0.344490
$753$	0	0
$754$	22464.0	1.08500
$755$	−36784.0	−1.77312
$756$	0	0
$757$	28770.0	1.38133	0.690663	−	0.723177i	$-0.257319\pi$
$757$	28770.0	1.38133	0.690663	+	0.723177i	$0.257319\pi$
$758$	−5320.00	−0.254922
$759$	0	0
$760$	−20416.0	−0.974429
$761$	12418.0	0.591527	0.295764	−	0.955261i	$-0.404426\pi$
$761$	12418.0	0.591527	0.295764	+	0.955261i	$0.404426\pi$
$762$	0	0
$763$	4298.00	0.203929
$764$	19416.0	0.919432
$765$	0	0
$766$	1520.00	0.0716969
$767$	6696.00	0.315226
$768$	0	0
$769$	12346.0	0.578944	0.289472	−	0.957186i	$-0.406520\pi$
$769$	12346.0	0.578944	0.289472	+	0.957186i	$0.406520\pi$
$770$	−8008.00	−0.374790
$771$	0	0
$772$	−5992.00	−0.279348
$773$	−38098.0	−1.77269	−0.886345	−	0.463025i	$-0.846764\pi$
$773$	−38098.0	−1.77269	−0.886345	+	0.463025i	$0.846764\pi$
$774$	0	0
$775$	−90468.0	−4.19317
$776$	4208.00	0.194663
$777$	0	0
$778$	208.000	0.00958504
$779$	−14616.0	−0.672237
$780$	0	0
$781$	−6188.00	−0.283514
$782$	−8584.00	−0.392536
$783$	0	0
$784$	784.000	0.0357143
$785$	−9812.00	−0.446121
$786$	0	0
$787$	13824.0	0.626140	0.313070	−	0.949730i	$-0.398643\pi$
$787$	13824.0	0.626140	0.313070	+	0.949730i	$0.398643\pi$
$788$	−2480.00	−0.112115
$789$	0	0
$790$	43296.0	1.94988
$791$	−8624.00	−0.387654
$792$	0	0
$793$	8748.00	0.391741
$794$	8796.00	0.393147
$795$	0	0
$796$	128.000	0.00569955
$797$	−22170.0	−0.985322	−0.492661	−	0.870221i	$-0.663976\pi$
$797$	−22170.0	−0.985322	−0.492661	+	0.870221i	$0.663976\pi$
$798$	0	0
$799$	−32856.0	−1.45477
$800$	11488.0	0.507703
$801$	0	0
$802$	−26472.0	−1.16553
$803$	3796.00	0.166822
$804$	0	0
$805$	8932.00	0.391070
$806$	27216.0	1.18938
$807$	0	0
$808$	−10448.0	−0.454900
$809$	−16288.0	−0.707856	−0.353928	−	0.935273i	$-0.615154\pi$
$809$	−16288.0	−0.707856	−0.353928	+	0.935273i	$0.615154\pi$
$810$	0	0
$811$	−8720.00	−0.377559	−0.188780	−	0.982019i	$-0.560453\pi$
$811$	−8720.00	−0.377559	−0.188780	+	0.982019i	$0.560453\pi$
$812$	5824.00	0.251702
$813$	0	0
$814$	−2600.00	−0.111953
$815$	−9416.00	−0.404697
$816$	0	0
$817$	19024.0	0.814646
$818$	−18980.0	−0.811272
$819$	0	0
$820$	11088.0	0.472207
$821$	31372.0	1.33361	0.666803	−	0.745234i	$-0.267662\pi$
$821$	31372.0	1.33361	0.666803	+	0.745234i	$0.267662\pi$
$822$	0	0
$823$	−17648.0	−0.747473	−0.373737	−	0.927535i	$-0.621924\pi$
$823$	−17648.0	−0.747473	−0.373737	+	0.927535i	$0.621924\pi$
$824$	−4064.00	−0.171816
$825$	0	0
$826$	1736.00	0.0731273
$827$	2382.00	0.100158	0.0500788	−	0.998745i	$-0.484053\pi$
$827$	2382.00	0.100158	0.0500788	+	0.998745i	$0.484053\pi$
$828$	0	0
$829$	26650.0	1.11652	0.558259	−	0.829667i	$-0.311470\pi$
$829$	26650.0	1.11652	0.558259	+	0.829667i	$0.311470\pi$
$830$	28864.0	1.20709
$831$	0	0
$832$	−3456.00	−0.144009
$833$	−3626.00	−0.150820
$834$	0	0
$835$	88.0000	0.00364714
$836$	−12064.0	−0.499093
$837$	0	0
$838$	−8472.00	−0.349237
$839$	−24092.0	−0.991357	−0.495678	−	0.868506i	$-0.665080\pi$
$839$	−24092.0	−0.991357	−0.495678	+	0.868506i	$0.665080\pi$
$840$	0	0
$841$	18875.0	0.773914
$842$	1836.00	0.0751458
$843$	0	0
$844$	17072.0	0.696259
$845$	−15818.0	−0.643971
$846$	0	0
$847$	4585.00	0.186001
$848$	−192.000	−0.00777513
$849$	0	0
$850$	−53132.0	−2.14401
$851$	2900.00	0.116816
$852$	0	0
$853$	−8194.00	−0.328906	−0.164453	−	0.986385i	$-0.552586\pi$
$853$	−8194.00	−0.328906	−0.164453	+	0.986385i	$0.552586\pi$
$854$	2268.00	0.0908775
$855$	0	0
$856$	−3984.00	−0.159077
$857$	16962.0	0.676092	0.338046	−	0.941130i	$-0.390234\pi$
$857$	16962.0	0.676092	0.338046	+	0.941130i	$0.390234\pi$
$858$	0	0
$859$	−48556.0	−1.92865	−0.964324	−	0.264723i	$-0.914719\pi$
$859$	−48556.0	−1.92865	−0.964324	+	0.264723i	$0.914719\pi$
$860$	−14432.0	−0.572241
$861$	0	0
$862$	23628.0	0.933611
$863$	34274.0	1.35191	0.675956	−	0.736942i	$-0.263731\pi$
$863$	34274.0	1.35191	0.675956	+	0.736942i	$0.263731\pi$
$864$	0	0
$865$	−12980.0	−0.510212
$866$	−16748.0	−0.657183
$867$	0	0
$868$	7056.00	0.275917
$869$	25584.0	0.998709
$870$	0	0
$871$	46440.0	1.80661
$872$	−4912.00	−0.190758
$873$	0	0
$874$	13456.0	0.520773
$875$	36036.0	1.39227
$876$	0	0
$877$	7126.00	0.274376	0.137188	−	0.990545i	$-0.456194\pi$
$877$	7126.00	0.274376	0.137188	+	0.990545i	$0.456194\pi$
$878$	7680.00	0.295202
$879$	0	0
$880$	9152.00	0.350584
$881$	9222.00	0.352664	0.176332	−	0.984331i	$-0.443577\pi$
$881$	9222.00	0.352664	0.176332	+	0.984331i	$0.443577\pi$
$882$	0	0
$883$	37652.0	1.43498	0.717492	−	0.696567i	$-0.245290\pi$
$883$	37652.0	1.43498	0.717492	+	0.696567i	$0.245290\pi$
$884$	15984.0	0.608145
$885$	0	0
$886$	−20332.0	−0.770956
$887$	21996.0	0.832642	0.416321	−	0.909218i	$-0.363319\pi$
$887$	21996.0	0.832642	0.416321	+	0.909218i	$0.363319\pi$
$888$	0	0
$889$	19656.0	0.741554
$890$	−41976.0	−1.58094
$891$	0	0
$892$	13856.0	0.520104
$893$	51504.0	1.93003
$894$	0	0
$895$	77748.0	2.90372
$896$	−896.000	−0.0334077
$897$	0	0
$898$	−16400.0	−0.609438
$899$	52416.0	1.94457
$900$	0	0
$901$	888.000	0.0328342
$902$	6552.00	0.241860
$903$	0	0
$904$	9856.00	0.362617
$905$	−24156.0	−0.887263
$906$	0	0
$907$	14844.0	0.543426	0.271713	−	0.962378i	$-0.412410\pi$
$907$	14844.0	0.543426	0.271713	+	0.962378i	$0.412410\pi$
$908$	13008.0	0.475425
$909$	0	0
$910$	−16632.0	−0.605874
$911$	19446.0	0.707217	0.353609	−	0.935394i	$-0.384955\pi$
$911$	19446.0	0.707217	0.353609	+	0.935394i	$0.384955\pi$
$912$	0	0
$913$	17056.0	0.618260
$914$	−12148.0	−0.439628
$915$	0	0
$916$	1672.00	0.0603105
$917$	3640.00	0.131083
$918$	0	0
$919$	−39200.0	−1.40706	−0.703530	−	0.710665i	$-0.748394\pi$
$919$	−39200.0	−1.40706	−0.703530	+	0.710665i	$0.748394\pi$
$920$	−10208.0	−0.365813
$921$	0	0
$922$	4012.00	0.143306
$923$	−12852.0	−0.458319
$924$	0	0
$925$	17950.0	0.638046
$926$	−7456.00	−0.264600
$927$	0	0
$928$	−6656.00	−0.235446
$929$	15954.0	0.563438	0.281719	−	0.959497i	$-0.409095\pi$
$929$	15954.0	0.563438	0.281719	+	0.959497i	$0.409095\pi$
$930$	0	0
$931$	5684.00	0.200092
$932$	−8336.00	−0.292977
$933$	0	0
$934$	12760.0	0.447024
$935$	−42328.0	−1.48051
$936$	0	0
$937$	2546.00	0.0887665	0.0443832	−	0.999015i	$-0.485868\pi$
$937$	2546.00	0.0887665	0.0443832	+	0.999015i	$0.485868\pi$
$938$	12040.0	0.419104
$939$	0	0
$940$	−39072.0	−1.35573
$941$	−430.000	−0.0148965	−0.00744825	−	0.999972i	$-0.502371\pi$
$941$	−430.000	−0.0148965	−0.00744825	+	0.999972i	$0.502371\pi$
$942$	0	0
$943$	−7308.00	−0.252366
$944$	−1984.00	−0.0684043
$945$	0	0
$946$	−8528.00	−0.293096
$947$	−38266.0	−1.31307	−0.656535	−	0.754295i	$-0.727979\pi$
$947$	−38266.0	−1.31307	−0.656535	+	0.754295i	$0.727979\pi$
$948$	0	0
$949$	7884.00	0.269679
$950$	83288.0	2.84444
$951$	0	0
$952$	4144.00	0.141080
$953$	28216.0	0.959083	0.479541	−	0.877519i	$-0.340803\pi$
$953$	28216.0	0.959083	0.479541	+	0.877519i	$0.340803\pi$
$954$	0	0
$955$	−106788.	−3.61841
$956$	−6648.00	−0.224908
$957$	0	0
$958$	−34360.0	−1.15879
$959$	−17612.0	−0.593036
$960$	0	0
$961$	33713.0	1.13165
$962$	−5400.00	−0.180980
$963$	0	0
$964$	24728.0	0.826178
$965$	32956.0	1.09937
$966$	0	0
$967$	−27712.0	−0.921570	−0.460785	−	0.887512i	$-0.652432\pi$
$967$	−27712.0	−0.921570	−0.460785	+	0.887512i	$0.652432\pi$
$968$	−5240.00	−0.173988
$969$	0	0
$970$	−23144.0	−0.766092
$971$	−32976.0	−1.08986	−0.544928	−	0.838483i	$-0.683443\pi$
$971$	−32976.0	−1.08986	−0.544928	+	0.838483i	$0.683443\pi$
$972$	0	0
$973$	18704.0	0.616262
$974$	−5456.00	−0.179488
$975$	0	0
$976$	−2592.00	−0.0850081
$977$	8940.00	0.292749	0.146375	−	0.989229i	$-0.453240\pi$
$977$	8940.00	0.292749	0.146375	+	0.989229i	$0.453240\pi$
$978$	0	0
$979$	−24804.0	−0.809744
$980$	−4312.00	−0.140553
$981$	0	0
$982$	−5148.00	−0.167291
$983$	3288.00	0.106685	0.0533423	−	0.998576i	$-0.483013\pi$
$983$	3288.00	0.106685	0.0533423	+	0.998576i	$0.483013\pi$
$984$	0	0
$985$	13640.0	0.441225
$986$	30784.0	0.994282
$987$	0	0
$988$	−25056.0	−0.806819
$989$	9512.00	0.305828
$990$	0	0
$991$	33944.0	1.08806	0.544030	−	0.839066i	$-0.316898\pi$
$991$	33944.0	1.08806	0.544030	+	0.839066i	$0.316898\pi$
$992$	−8064.00	−0.258097
$993$	0	0
$994$	−3332.00	−0.106323
$995$	−704.000	−0.0224305
$996$	0	0
$997$	−54562.0	−1.73320	−0.866598	−	0.499007i	$-0.833698\pi$
$997$	−54562.0	−1.73320	−0.866598	+	0.499007i	$0.833698\pi$
$998$	−14968.0	−0.474753
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.4.a.f.1.1	yes	1
3.2	odd	2		126.4.a.e.1.1	✓	1
4.3	odd	2		1008.4.a.a.1.1		1
7.2	even	3		882.4.g.m.361.1		2
7.3	odd	6		882.4.g.a.667.1		2
7.4	even	3		882.4.g.m.667.1		2
7.5	odd	6		882.4.g.a.361.1		2
7.6	odd	2		882.4.a.s.1.1		1
12.11	even	2		1008.4.a.w.1.1		1
21.2	odd	6		882.4.g.n.361.1		2
21.5	even	6		882.4.g.x.361.1		2
21.11	odd	6		882.4.g.n.667.1		2
21.17	even	6		882.4.g.x.667.1		2
21.20	even	2		882.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.4.a.e.1.1	✓	1	3.2	odd	2
126.4.a.f.1.1	yes	1	1.1	even	1	trivial
882.4.a.a.1.1		1	21.20	even	2
882.4.a.s.1.1		1	7.6	odd	2
882.4.g.a.361.1		2	7.5	odd	6
882.4.g.a.667.1		2	7.3	odd	6
882.4.g.m.361.1		2	7.2	even	3
882.4.g.m.667.1		2	7.4	even	3
882.4.g.n.361.1		2	21.2	odd	6
882.4.g.n.667.1		2	21.11	odd	6
882.4.g.x.361.1		2	21.5	even	6
882.4.g.x.667.1		2	21.17	even	6
1008.4.a.a.1.1		1	4.3	odd	2
1008.4.a.w.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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