LMFDB - Embedded newform 912.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,4,Mod(1,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, 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("912.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	912.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:	$53.8097419252$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	912.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -11.0000 q^{5} +15.0000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} -11.0000 q^{5} +15.0000 q^{7} +9.00000 q^{9} +29.0000 q^{11} -82.0000 q^{13} -33.0000 q^{15} +27.0000 q^{17} +19.0000 q^{19} +45.0000 q^{21} -100.000 q^{23} -4.00000 q^{25} +27.0000 q^{27} -118.000 q^{29} -70.0000 q^{31} +87.0000 q^{33} -165.000 q^{35} +232.000 q^{37} -246.000 q^{39} +8.00000 q^{41} +287.000 q^{43} -99.0000 q^{45} -385.000 q^{47} -118.000 q^{49} +81.0000 q^{51} +538.000 q^{53} -319.000 q^{55} +57.0000 q^{57} +300.000 q^{59} -901.000 q^{61} +135.000 q^{63} +902.000 q^{65} -132.000 q^{67} -300.000 q^{69} -472.000 q^{71} -1131.00 q^{73} -12.0000 q^{75} +435.000 q^{77} +52.0000 q^{79} +81.0000 q^{81} -276.000 q^{83} -297.000 q^{85} -354.000 q^{87} -1302.00 q^{89} -1230.00 q^{91} -210.000 q^{93} -209.000 q^{95} -1310.00 q^{97} +261.000 q^{99} +O(q^{100})$

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
$5$	−11.0000	−0.983870	−0.491935	−	0.870632i	$-0.663710\pi$
$5$	−11.0000	−0.983870	−0.491935	+	0.870632i	$0.663710\pi$
$6$	0	0
$7$	15.0000	0.809924	0.404962	−	0.914334i	$-0.367285\pi$
$7$	15.0000	0.809924	0.404962	+	0.914334i	$0.367285\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	29.0000	0.794894	0.397447	−	0.917625i	$-0.369896\pi$
$11$	29.0000	0.794894	0.397447	+	0.917625i	$0.369896\pi$
$12$	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
$15$	−33.0000	−0.568038
$16$	0	0
$17$	27.0000	0.385204	0.192602	−	0.981277i	$-0.438307\pi$
$17$	27.0000	0.385204	0.192602	+	0.981277i	$0.438307\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	45.0000	0.467610
$22$	0	0
$23$	−100.000	−0.906584	−0.453292	−	0.891362i	$-0.649751\pi$
$23$	−100.000	−0.906584	−0.453292	+	0.891362i	$0.649751\pi$
$24$	0	0
$25$	−4.00000	−0.0320000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−118.000	−0.755588	−0.377794	−	0.925890i	$-0.623317\pi$
$29$	−118.000	−0.755588	−0.377794	+	0.925890i	$0.623317\pi$
$30$	0	0
$31$	−70.0000	−0.405560	−0.202780	−	0.979224i	$-0.564998\pi$
$31$	−70.0000	−0.405560	−0.202780	+	0.979224i	$0.564998\pi$
$32$	0	0
$33$	87.0000	0.458932
$34$	0	0
$35$	−165.000	−0.796860
$36$	0	0
$37$	232.000	1.03083	0.515413	−	0.856942i	$-0.327639\pi$
$37$	232.000	1.03083	0.515413	+	0.856942i	$0.327639\pi$
$38$	0	0
$39$	−246.000	−1.01004
$40$	0	0
$41$	8.00000	0.0304729	0.0152365	−	0.999884i	$-0.495150\pi$
$41$	8.00000	0.0304729	0.0152365	+	0.999884i	$0.495150\pi$
$42$	0	0
$43$	287.000	1.01784	0.508920	−	0.860814i	$-0.330045\pi$
$43$	287.000	1.01784	0.508920	+	0.860814i	$0.330045\pi$
$44$	0	0
$45$	−99.0000	−0.327957
$46$	0	0
$47$	−385.000	−1.19485	−0.597426	−	0.801924i	$-0.703810\pi$
$47$	−385.000	−1.19485	−0.597426	+	0.801924i	$0.703810\pi$
$48$	0	0
$49$	−118.000	−0.344023
$50$	0	0
$51$	81.0000	0.222397
$52$	0	0
$53$	538.000	1.39434	0.697170	−	0.716906i	$-0.254442\pi$
$53$	538.000	1.39434	0.697170	+	0.716906i	$0.254442\pi$
$54$	0	0
$55$	−319.000	−0.782072
$56$	0	0
$57$	57.0000	0.132453
$58$	0	0
$59$	300.000	0.661978	0.330989	−	0.943635i	$-0.392618\pi$
$59$	300.000	0.661978	0.330989	+	0.943635i	$0.392618\pi$
$60$	0	0
$61$	−901.000	−1.89117	−0.945584	−	0.325379i	$-0.894508\pi$
$61$	−901.000	−1.89117	−0.945584	+	0.325379i	$0.894508\pi$
$62$	0	0
$63$	135.000	0.269975
$64$	0	0
$65$	902.000	1.72122
$66$	0	0
$67$	−132.000	−0.240692	−0.120346	−	0.992732i	$-0.538400\pi$
$67$	−132.000	−0.240692	−0.120346	+	0.992732i	$0.538400\pi$
$68$	0	0
$69$	−300.000	−0.523417
$70$	0	0
$71$	−472.000	−0.788959	−0.394480	−	0.918905i	$-0.629075\pi$
$71$	−472.000	−0.788959	−0.394480	+	0.918905i	$0.629075\pi$
$72$	0	0
$73$	−1131.00	−1.81334	−0.906668	−	0.421845i	$-0.861383\pi$
$73$	−1131.00	−1.81334	−0.906668	+	0.421845i	$0.861383\pi$
$74$	0	0
$75$	−12.0000	−0.0184752
$76$	0	0
$77$	435.000	0.643803
$78$	0	0
$79$	52.0000	0.0740564	0.0370282	−	0.999314i	$-0.488211\pi$
$79$	52.0000	0.0740564	0.0370282	+	0.999314i	$0.488211\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−276.000	−0.364999	−0.182500	−	0.983206i	$-0.558419\pi$
$83$	−276.000	−0.364999	−0.182500	+	0.983206i	$0.558419\pi$
$84$	0	0
$85$	−297.000	−0.378990
$86$	0	0
$87$	−354.000	−0.436239
$88$	0	0
$89$	−1302.00	−1.55069	−0.775347	−	0.631536i	$-0.782425\pi$
$89$	−1302.00	−1.55069	−0.775347	+	0.631536i	$0.782425\pi$
$90$	0	0
$91$	−1230.00	−1.41691
$92$	0	0
$93$	−210.000	−0.234150
$94$	0	0
$95$	−209.000	−0.225715
$96$	0	0
$97$	−1310.00	−1.37124	−0.685620	−	0.727959i	$-0.740469\pi$
$97$	−1310.00	−1.37124	−0.685620	+	0.727959i	$0.740469\pi$
$98$	0	0
$99$	261.000	0.264965
$100$	0	0
$101$	−638.000	−0.628548	−0.314274	−	0.949332i	$-0.601761\pi$
$101$	−638.000	−0.628548	−0.314274	+	0.949332i	$0.601761\pi$
$102$	0	0
$103$	−786.000	−0.751911	−0.375956	−	0.926638i	$-0.622686\pi$
$103$	−786.000	−0.751911	−0.375956	+	0.926638i	$0.622686\pi$
$104$	0	0
$105$	−495.000	−0.460067
$106$	0	0
$107$	1310.00	1.18357	0.591787	−	0.806094i	$-0.298423\pi$
$107$	1310.00	1.18357	0.591787	+	0.806094i	$0.298423\pi$
$108$	0	0
$109$	−1296.00	−1.13885	−0.569423	−	0.822044i	$-0.692833\pi$
$109$	−1296.00	−1.13885	−0.569423	+	0.822044i	$0.692833\pi$
$110$	0	0
$111$	696.000	0.595148
$112$	0	0
$113$	−1130.00	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−1130.00	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	1100.00	0.891961
$116$	0	0
$117$	−738.000	−0.583146
$118$	0	0
$119$	405.000	0.311986
$120$	0	0
$121$	−490.000	−0.368144
$122$	0	0
$123$	24.0000	0.0175936
$124$	0	0
$125$	1419.00	1.01535
$126$	0	0
$127$	−750.000	−0.524029	−0.262015	−	0.965064i	$-0.584387\pi$
$127$	−750.000	−0.524029	−0.262015	+	0.965064i	$0.584387\pi$
$128$	0	0
$129$	861.000	0.587650
$130$	0	0
$131$	1275.00	0.850361	0.425180	−	0.905109i	$-0.360211\pi$
$131$	1275.00	0.850361	0.425180	+	0.905109i	$0.360211\pi$
$132$	0	0
$133$	285.000	0.185809
$134$	0	0
$135$	−297.000	−0.189346
$136$	0	0
$137$	2123.00	1.32394	0.661971	−	0.749529i	$-0.269720\pi$
$137$	2123.00	1.32394	0.661971	+	0.749529i	$0.269720\pi$
$138$	0	0
$139$	−2277.00	−1.38944	−0.694722	−	0.719279i	$-0.744472\pi$
$139$	−2277.00	−1.38944	−0.694722	+	0.719279i	$0.744472\pi$
$140$	0	0
$141$	−1155.00	−0.689848
$142$	0	0
$143$	−2378.00	−1.39062
$144$	0	0
$145$	1298.00	0.743400
$146$	0	0
$147$	−354.000	−0.198622
$148$	0	0
$149$	2235.00	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	2235.00	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	2764.00	1.48961	0.744805	−	0.667282i	$-0.232542\pi$
$151$	2764.00	1.48961	0.744805	+	0.667282i	$0.232542\pi$
$152$	0	0
$153$	243.000	0.128401
$154$	0	0
$155$	770.000	0.399019
$156$	0	0
$157$	1702.00	0.865187	0.432594	−	0.901589i	$-0.357599\pi$
$157$	1702.00	0.865187	0.432594	+	0.901589i	$0.357599\pi$
$158$	0	0
$159$	1614.00	0.805022
$160$	0	0
$161$	−1500.00	−0.734264
$162$	0	0
$163$	−2840.00	−1.36470	−0.682350	−	0.731026i	$-0.739042\pi$
$163$	−2840.00	−1.36470	−0.682350	+	0.731026i	$0.739042\pi$
$164$	0	0
$165$	−957.000	−0.451529
$166$	0	0
$167$	−1194.00	−0.553260	−0.276630	−	0.960976i	$-0.589218\pi$
$167$	−1194.00	−0.553260	−0.276630	+	0.960976i	$0.589218\pi$
$168$	0	0
$169$	4527.00	2.06054
$170$	0	0
$171$	171.000	0.0764719
$172$	0	0
$173$	−3786.00	−1.66384	−0.831920	−	0.554896i	$-0.812758\pi$
$173$	−3786.00	−1.66384	−0.831920	+	0.554896i	$0.812758\pi$
$174$	0	0
$175$	−60.0000	−0.0259176
$176$	0	0
$177$	900.000	0.382193
$178$	0	0
$179$	−2094.00	−0.874374	−0.437187	−	0.899371i	$-0.644025\pi$
$179$	−2094.00	−0.874374	−0.437187	+	0.899371i	$0.644025\pi$
$180$	0	0
$181$	−350.000	−0.143731	−0.0718655	−	0.997414i	$-0.522895\pi$
$181$	−350.000	−0.143731	−0.0718655	+	0.997414i	$0.522895\pi$
$182$	0	0
$183$	−2703.00	−1.09187
$184$	0	0
$185$	−2552.00	−1.01420
$186$	0	0
$187$	783.000	0.306196
$188$	0	0
$189$	405.000	0.155870
$190$	0	0
$191$	1489.00	0.564085	0.282043	−	0.959402i	$-0.408988\pi$
$191$	1489.00	0.564085	0.282043	+	0.959402i	$0.408988\pi$
$192$	0	0
$193$	600.000	0.223777	0.111888	−	0.993721i	$-0.464310\pi$
$193$	600.000	0.223777	0.111888	+	0.993721i	$0.464310\pi$
$194$	0	0
$195$	2706.00	0.993747
$196$	0	0
$197$	−290.000	−0.104881	−0.0524407	−	0.998624i	$-0.516700\pi$
$197$	−290.000	−0.104881	−0.0524407	+	0.998624i	$0.516700\pi$
$198$	0	0
$199$	−4169.00	−1.48509	−0.742544	−	0.669797i	$-0.766381\pi$
$199$	−4169.00	−1.48509	−0.742544	+	0.669797i	$0.766381\pi$
$200$	0	0
$201$	−396.000	−0.138964
$202$	0	0
$203$	−1770.00	−0.611969
$204$	0	0
$205$	−88.0000	−0.0299814
$206$	0	0
$207$	−900.000	−0.302195
$208$	0	0
$209$	551.000	0.182361
$210$	0	0
$211$	1368.00	0.446337	0.223168	−	0.974780i	$-0.428360\pi$
$211$	1368.00	0.446337	0.223168	+	0.974780i	$0.428360\pi$
$212$	0	0
$213$	−1416.00	−0.455506
$214$	0	0
$215$	−3157.00	−1.00142
$216$	0	0
$217$	−1050.00	−0.328473
$218$	0	0
$219$	−3393.00	−1.04693
$220$	0	0
$221$	−2214.00	−0.673890
$222$	0	0
$223$	−2540.00	−0.762740	−0.381370	−	0.924423i	$-0.624548\pi$
$223$	−2540.00	−0.762740	−0.381370	+	0.924423i	$0.624548\pi$
$224$	0	0
$225$	−36.0000	−0.0106667
$226$	0	0
$227$	5974.00	1.74673	0.873366	−	0.487064i	$-0.161932\pi$
$227$	5974.00	1.74673	0.873366	+	0.487064i	$0.161932\pi$
$228$	0	0
$229$	−355.000	−0.102441	−0.0512207	−	0.998687i	$-0.516311\pi$
$229$	−355.000	−0.102441	−0.0512207	+	0.998687i	$0.516311\pi$
$230$	0	0
$231$	1305.00	0.371700
$232$	0	0
$233$	237.000	0.0666369	0.0333184	−	0.999445i	$-0.489392\pi$
$233$	237.000	0.0666369	0.0333184	+	0.999445i	$0.489392\pi$
$234$	0	0
$235$	4235.00	1.17558
$236$	0	0
$237$	156.000	0.0427565
$238$	0	0
$239$	−1635.00	−0.442508	−0.221254	−	0.975216i	$-0.571015\pi$
$239$	−1635.00	−0.442508	−0.221254	+	0.975216i	$0.571015\pi$
$240$	0	0
$241$	−164.000	−0.0438347	−0.0219174	−	0.999760i	$-0.506977\pi$
$241$	−164.000	−0.0438347	−0.0219174	+	0.999760i	$0.506977\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	1298.00	0.338474
$246$	0	0
$247$	−1558.00	−0.401349
$248$	0	0
$249$	−828.000	−0.210732
$250$	0	0
$251$	2099.00	0.527839	0.263920	−	0.964545i	$-0.414985\pi$
$251$	2099.00	0.527839	0.263920	+	0.964545i	$0.414985\pi$
$252$	0	0
$253$	−2900.00	−0.720638
$254$	0	0
$255$	−891.000	−0.218810
$256$	0	0
$257$	5536.00	1.34368	0.671841	−	0.740696i	$-0.265504\pi$
$257$	5536.00	1.34368	0.671841	+	0.740696i	$0.265504\pi$
$258$	0	0
$259$	3480.00	0.834891
$260$	0	0
$261$	−1062.00	−0.251863
$262$	0	0
$263$	−2073.00	−0.486033	−0.243016	−	0.970022i	$-0.578137\pi$
$263$	−2073.00	−0.486033	−0.243016	+	0.970022i	$0.578137\pi$
$264$	0	0
$265$	−5918.00	−1.37185
$266$	0	0
$267$	−3906.00	−0.895293
$268$	0	0
$269$	1482.00	0.335908	0.167954	−	0.985795i	$-0.446284\pi$
$269$	1482.00	0.335908	0.167954	+	0.985795i	$0.446284\pi$
$270$	0	0
$271$	7268.00	1.62915	0.814575	−	0.580058i	$-0.196970\pi$
$271$	7268.00	1.62915	0.814575	+	0.580058i	$0.196970\pi$
$272$	0	0
$273$	−3690.00	−0.818055
$274$	0	0
$275$	−116.000	−0.0254366
$276$	0	0
$277$	−4583.00	−0.994100	−0.497050	−	0.867722i	$-0.665584\pi$
$277$	−4583.00	−0.994100	−0.497050	+	0.867722i	$0.665584\pi$
$278$	0	0
$279$	−630.000	−0.135187
$280$	0	0
$281$	5762.00	1.22325	0.611623	−	0.791149i	$-0.290517\pi$
$281$	5762.00	1.22325	0.611623	+	0.791149i	$0.290517\pi$
$282$	0	0
$283$	−3909.00	−0.821081	−0.410541	−	0.911842i	$-0.634660\pi$
$283$	−3909.00	−0.821081	−0.410541	+	0.911842i	$0.634660\pi$
$284$	0	0
$285$	−627.000	−0.130317
$286$	0	0
$287$	120.000	0.0246808
$288$	0	0
$289$	−4184.00	−0.851618
$290$	0	0
$291$	−3930.00	−0.791686
$292$	0	0
$293$	−8148.00	−1.62461	−0.812306	−	0.583232i	$-0.801788\pi$
$293$	−8148.00	−1.62461	−0.812306	+	0.583232i	$0.801788\pi$
$294$	0	0
$295$	−3300.00	−0.651300
$296$	0	0
$297$	783.000	0.152977
$298$	0	0
$299$	8200.00	1.58601
$300$	0	0
$301$	4305.00	0.824372
$302$	0	0
$303$	−1914.00	−0.362892
$304$	0	0
$305$	9911.00	1.86066
$306$	0	0
$307$	−600.000	−0.111543	−0.0557717	−	0.998444i	$-0.517762\pi$
$307$	−600.000	−0.111543	−0.0557717	+	0.998444i	$0.517762\pi$
$308$	0	0
$309$	−2358.00	−0.434116
$310$	0	0
$311$	4963.00	0.904906	0.452453	−	0.891788i	$-0.350549\pi$
$311$	4963.00	0.904906	0.452453	+	0.891788i	$0.350549\pi$
$312$	0	0
$313$	5462.00	0.986359	0.493180	−	0.869927i	$-0.335835\pi$
$313$	5462.00	0.986359	0.493180	+	0.869927i	$0.335835\pi$
$314$	0	0
$315$	−1485.00	−0.265620
$316$	0	0
$317$	−984.000	−0.174344	−0.0871718	−	0.996193i	$-0.527783\pi$
$317$	−984.000	−0.174344	−0.0871718	+	0.996193i	$0.527783\pi$
$318$	0	0
$319$	−3422.00	−0.600612
$320$	0	0
$321$	3930.00	0.683337
$322$	0	0
$323$	513.000	0.0883718
$324$	0	0
$325$	328.000	0.0559821
$326$	0	0
$327$	−3888.00	−0.657513
$328$	0	0
$329$	−5775.00	−0.967739
$330$	0	0
$331$	2632.00	0.437063	0.218531	−	0.975830i	$-0.429873\pi$
$331$	2632.00	0.437063	0.218531	+	0.975830i	$0.429873\pi$
$332$	0	0
$333$	2088.00	0.343609
$334$	0	0
$335$	1452.00	0.236810
$336$	0	0
$337$	−11234.0	−1.81589	−0.907945	−	0.419089i	$-0.862349\pi$
$337$	−11234.0	−1.81589	−0.907945	+	0.419089i	$0.862349\pi$
$338$	0	0
$339$	−3390.00	−0.543125
$340$	0	0
$341$	−2030.00	−0.322377
$342$	0	0
$343$	−6915.00	−1.08856
$344$	0	0
$345$	3300.00	0.514974
$346$	0	0
$347$	3211.00	0.496759	0.248380	−	0.968663i	$-0.420102\pi$
$347$	3211.00	0.496759	0.248380	+	0.968663i	$0.420102\pi$
$348$	0	0
$349$	2341.00	0.359057	0.179528	−	0.983753i	$-0.442543\pi$
$349$	2341.00	0.359057	0.179528	+	0.983753i	$0.442543\pi$
$350$	0	0
$351$	−2214.00	−0.336680
$352$	0	0
$353$	6366.00	0.959853	0.479926	−	0.877309i	$-0.340663\pi$
$353$	6366.00	0.959853	0.479926	+	0.877309i	$0.340663\pi$
$354$	0	0
$355$	5192.00	0.776233
$356$	0	0
$357$	1215.00	0.180125
$358$	0	0
$359$	7989.00	1.17449	0.587247	−	0.809408i	$-0.300212\pi$
$359$	7989.00	1.17449	0.587247	+	0.809408i	$0.300212\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−1470.00	−0.212548
$364$	0	0
$365$	12441.0	1.78409
$366$	0	0
$367$	10544.0	1.49971	0.749853	−	0.661604i	$-0.230124\pi$
$367$	10544.0	1.49971	0.749853	+	0.661604i	$0.230124\pi$
$368$	0	0
$369$	72.0000	0.0101576
$370$	0	0
$371$	8070.00	1.12931
$372$	0	0
$373$	7616.00	1.05722	0.528608	−	0.848866i	$-0.322714\pi$
$373$	7616.00	1.05722	0.528608	+	0.848866i	$0.322714\pi$
$374$	0	0
$375$	4257.00	0.586215
$376$	0	0
$377$	9676.00	1.32186
$378$	0	0
$379$	−5958.00	−0.807498	−0.403749	−	0.914870i	$-0.632293\pi$
$379$	−5958.00	−0.807498	−0.403749	+	0.914870i	$0.632293\pi$
$380$	0	0
$381$	−2250.00	−0.302549
$382$	0	0
$383$	12382.0	1.65193	0.825967	−	0.563719i	$-0.190630\pi$
$383$	12382.0	1.65193	0.825967	+	0.563719i	$0.190630\pi$
$384$	0	0
$385$	−4785.00	−0.633419
$386$	0	0
$387$	2583.00	0.339280
$388$	0	0
$389$	7989.00	1.04128	0.520641	−	0.853776i	$-0.325693\pi$
$389$	7989.00	1.04128	0.520641	+	0.853776i	$0.325693\pi$
$390$	0	0
$391$	−2700.00	−0.349220
$392$	0	0
$393$	3825.00	0.490956
$394$	0	0
$395$	−572.000	−0.0728619
$396$	0	0
$397$	7061.00	0.892648	0.446324	−	0.894871i	$-0.352733\pi$
$397$	7061.00	0.892648	0.446324	+	0.894871i	$0.352733\pi$
$398$	0	0
$399$	855.000	0.107277
$400$	0	0
$401$	−14512.0	−1.80722	−0.903609	−	0.428357i	$-0.859092\pi$
$401$	−14512.0	−1.80722	−0.903609	+	0.428357i	$0.859092\pi$
$402$	0	0
$403$	5740.00	0.709503
$404$	0	0
$405$	−891.000	−0.109319
$406$	0	0
$407$	6728.00	0.819397
$408$	0	0
$409$	12634.0	1.52741	0.763705	−	0.645565i	$-0.223378\pi$
$409$	12634.0	1.52741	0.763705	+	0.645565i	$0.223378\pi$
$410$	0	0
$411$	6369.00	0.764379
$412$	0	0
$413$	4500.00	0.536151
$414$	0	0
$415$	3036.00	0.359112
$416$	0	0
$417$	−6831.00	−0.802195
$418$	0	0
$419$	8268.00	0.964005	0.482003	−	0.876170i	$-0.339910\pi$
$419$	8268.00	0.964005	0.482003	+	0.876170i	$0.339910\pi$
$420$	0	0
$421$	−1534.00	−0.177583	−0.0887917	−	0.996050i	$-0.528301\pi$
$421$	−1534.00	−0.177583	−0.0887917	+	0.996050i	$0.528301\pi$
$422$	0	0
$423$	−3465.00	−0.398284
$424$	0	0
$425$	−108.000	−0.0123265
$426$	0	0
$427$	−13515.0	−1.53170
$428$	0	0
$429$	−7134.00	−0.802874
$430$	0	0
$431$	−14358.0	−1.60464	−0.802321	−	0.596893i	$-0.796402\pi$
$431$	−14358.0	−1.60464	−0.802321	+	0.596893i	$0.796402\pi$
$432$	0	0
$433$	−4534.00	−0.503210	−0.251605	−	0.967830i	$-0.580958\pi$
$433$	−4534.00	−0.503210	−0.251605	+	0.967830i	$0.580958\pi$
$434$	0	0
$435$	3894.00	0.429202
$436$	0	0
$437$	−1900.00	−0.207985
$438$	0	0
$439$	12766.0	1.38790	0.693950	−	0.720023i	$-0.255869\pi$
$439$	12766.0	1.38790	0.693950	+	0.720023i	$0.255869\pi$
$440$	0	0
$441$	−1062.00	−0.114674
$442$	0	0
$443$	16711.0	1.79224	0.896121	−	0.443809i	$-0.146373\pi$
$443$	16711.0	1.79224	0.896121	+	0.443809i	$0.146373\pi$
$444$	0	0
$445$	14322.0	1.52568
$446$	0	0
$447$	6705.00	0.709476
$448$	0	0
$449$	3932.00	0.413280	0.206640	−	0.978417i	$-0.433747\pi$
$449$	3932.00	0.413280	0.206640	+	0.978417i	$0.433747\pi$
$450$	0	0
$451$	232.000	0.0242227
$452$	0	0
$453$	8292.00	0.860027
$454$	0	0
$455$	13530.0	1.39406
$456$	0	0
$457$	4451.00	0.455600	0.227800	−	0.973708i	$-0.426847\pi$
$457$	4451.00	0.455600	0.227800	+	0.973708i	$0.426847\pi$
$458$	0	0
$459$	729.000	0.0741325
$460$	0	0
$461$	9619.00	0.971804	0.485902	−	0.874013i	$-0.338491\pi$
$461$	9619.00	0.971804	0.485902	+	0.874013i	$0.338491\pi$
$462$	0	0
$463$	−16603.0	−1.66654	−0.833269	−	0.552868i	$-0.813533\pi$
$463$	−16603.0	−1.66654	−0.833269	+	0.552868i	$0.813533\pi$
$464$	0	0
$465$	2310.00	0.230374
$466$	0	0
$467$	−12857.0	−1.27399	−0.636993	−	0.770870i	$-0.719822\pi$
$467$	−12857.0	−1.27399	−0.636993	+	0.770870i	$0.719822\pi$
$468$	0	0
$469$	−1980.00	−0.194942
$470$	0	0
$471$	5106.00	0.499516
$472$	0	0
$473$	8323.00	0.809074
$474$	0	0
$475$	−76.0000	−0.00734130
$476$	0	0
$477$	4842.00	0.464780
$478$	0	0
$479$	−11072.0	−1.05614	−0.528072	−	0.849200i	$-0.677085\pi$
$479$	−11072.0	−1.05614	−0.528072	+	0.849200i	$0.677085\pi$
$480$	0	0
$481$	−19024.0	−1.80337
$482$	0	0
$483$	−4500.00	−0.423928
$484$	0	0
$485$	14410.0	1.34912
$486$	0	0
$487$	−11284.0	−1.04995	−0.524976	−	0.851117i	$-0.675926\pi$
$487$	−11284.0	−1.04995	−0.524976	+	0.851117i	$0.675926\pi$
$488$	0	0
$489$	−8520.00	−0.787909
$490$	0	0
$491$	11984.0	1.10149	0.550744	−	0.834674i	$-0.314344\pi$
$491$	11984.0	1.10149	0.550744	+	0.834674i	$0.314344\pi$
$492$	0	0
$493$	−3186.00	−0.291055
$494$	0	0
$495$	−2871.00	−0.260691
$496$	0	0
$497$	−7080.00	−0.638997
$498$	0	0
$499$	18701.0	1.67770	0.838849	−	0.544364i	$-0.183229\pi$
$499$	18701.0	1.67770	0.838849	+	0.544364i	$0.183229\pi$
$500$	0	0
$501$	−3582.00	−0.319425
$502$	0	0
$503$	−21888.0	−1.94023	−0.970117	−	0.242638i	$-0.921987\pi$
$503$	−21888.0	−1.94023	−0.970117	+	0.242638i	$0.921987\pi$
$504$	0	0
$505$	7018.00	0.618410
$506$	0	0
$507$	13581.0	1.18965
$508$	0	0
$509$	−14238.0	−1.23986	−0.619930	−	0.784657i	$-0.712839\pi$
$509$	−14238.0	−1.23986	−0.619930	+	0.784657i	$0.712839\pi$
$510$	0	0
$511$	−16965.0	−1.46866
$512$	0	0
$513$	513.000	0.0441511
$514$	0	0
$515$	8646.00	0.739783
$516$	0	0
$517$	−11165.0	−0.949780
$518$	0	0
$519$	−11358.0	−0.960618
$520$	0	0
$521$	−12992.0	−1.09249	−0.546247	−	0.837624i	$-0.683944\pi$
$521$	−12992.0	−1.09249	−0.546247	+	0.837624i	$0.683944\pi$
$522$	0	0
$523$	9070.00	0.758324	0.379162	−	0.925330i	$-0.376212\pi$
$523$	9070.00	0.758324	0.379162	+	0.925330i	$0.376212\pi$
$524$	0	0
$525$	−180.000	−0.0149635
$526$	0	0
$527$	−1890.00	−0.156223
$528$	0	0
$529$	−2167.00	−0.178105
$530$	0	0
$531$	2700.00	0.220659
$532$	0	0
$533$	−656.000	−0.0533105
$534$	0	0
$535$	−14410.0	−1.16448
$536$	0	0
$537$	−6282.00	−0.504820
$538$	0	0
$539$	−3422.00	−0.273462
$540$	0	0
$541$	−2473.00	−0.196530	−0.0982649	−	0.995160i	$-0.531329\pi$
$541$	−2473.00	−0.196530	−0.0982649	+	0.995160i	$0.531329\pi$
$542$	0	0
$543$	−1050.00	−0.0829831
$544$	0	0
$545$	14256.0	1.12048
$546$	0	0
$547$	11282.0	0.881871	0.440936	−	0.897539i	$-0.354647\pi$
$547$	11282.0	0.881871	0.440936	+	0.897539i	$0.354647\pi$
$548$	0	0
$549$	−8109.00	−0.630389
$550$	0	0
$551$	−2242.00	−0.173344
$552$	0	0
$553$	780.000	0.0599801
$554$	0	0
$555$	−7656.00	−0.585548
$556$	0	0
$557$	23975.0	1.82379	0.911897	−	0.410419i	$-0.134618\pi$
$557$	23975.0	1.82379	0.911897	+	0.410419i	$0.134618\pi$
$558$	0	0
$559$	−23534.0	−1.78065
$560$	0	0
$561$	2349.00	0.176782
$562$	0	0
$563$	−17892.0	−1.33936	−0.669678	−	0.742651i	$-0.733568\pi$
$563$	−17892.0	−1.33936	−0.669678	+	0.742651i	$0.733568\pi$
$564$	0	0
$565$	12430.0	0.925547
$566$	0	0
$567$	1215.00	0.0899915
$568$	0	0
$569$	10778.0	0.794090	0.397045	−	0.917799i	$-0.370036\pi$
$569$	10778.0	0.794090	0.397045	+	0.917799i	$0.370036\pi$
$570$	0	0
$571$	−8984.00	−0.658439	−0.329220	−	0.944253i	$-0.606786\pi$
$571$	−8984.00	−0.658439	−0.329220	+	0.944253i	$0.606786\pi$
$572$	0	0
$573$	4467.00	0.325675
$574$	0	0
$575$	400.000	0.0290107
$576$	0	0
$577$	9539.00	0.688239	0.344119	−	0.938926i	$-0.388178\pi$
$577$	9539.00	0.688239	0.344119	+	0.938926i	$0.388178\pi$
$578$	0	0
$579$	1800.00	0.129198
$580$	0	0
$581$	−4140.00	−0.295622
$582$	0	0
$583$	15602.0	1.10835
$584$	0	0
$585$	8118.00	0.573740
$586$	0	0
$587$	−21789.0	−1.53208	−0.766038	−	0.642796i	$-0.777774\pi$
$587$	−21789.0	−1.53208	−0.766038	+	0.642796i	$0.777774\pi$
$588$	0	0
$589$	−1330.00	−0.0930419
$590$	0	0
$591$	−870.000	−0.0605533
$592$	0	0
$593$	5474.00	0.379073	0.189536	−	0.981874i	$-0.439301\pi$
$593$	5474.00	0.379073	0.189536	+	0.981874i	$0.439301\pi$
$594$	0	0
$595$	−4455.00	−0.306953
$596$	0	0
$597$	−12507.0	−0.857416
$598$	0	0
$599$	17748.0	1.21062	0.605312	−	0.795988i	$-0.293048\pi$
$599$	17748.0	1.21062	0.605312	+	0.795988i	$0.293048\pi$
$600$	0	0
$601$	−8972.00	−0.608944	−0.304472	−	0.952521i	$-0.598480\pi$
$601$	−8972.00	−0.608944	−0.304472	+	0.952521i	$0.598480\pi$
$602$	0	0
$603$	−1188.00	−0.0802307
$604$	0	0
$605$	5390.00	0.362206
$606$	0	0
$607$	17710.0	1.18423	0.592114	−	0.805854i	$-0.298293\pi$
$607$	17710.0	1.18423	0.592114	+	0.805854i	$0.298293\pi$
$608$	0	0
$609$	−5310.00	−0.353320
$610$	0	0
$611$	31570.0	2.09032
$612$	0	0
$613$	11557.0	0.761473	0.380736	−	0.924684i	$-0.375671\pi$
$613$	11557.0	0.761473	0.380736	+	0.924684i	$0.375671\pi$
$614$	0	0
$615$	−264.000	−0.0173098
$616$	0	0
$617$	−4473.00	−0.291858	−0.145929	−	0.989295i	$-0.546617\pi$
$617$	−4473.00	−0.291858	−0.145929	+	0.989295i	$0.546617\pi$
$618$	0	0
$619$	−1948.00	−0.126489	−0.0632445	−	0.997998i	$-0.520145\pi$
$619$	−1948.00	−0.126489	−0.0632445	+	0.997998i	$0.520145\pi$
$620$	0	0
$621$	−2700.00	−0.174472
$622$	0	0
$623$	−19530.0	−1.25594
$624$	0	0
$625$	−15109.0	−0.966976
$626$	0	0
$627$	1653.00	0.105286
$628$	0	0
$629$	6264.00	0.397078
$630$	0	0
$631$	−12653.0	−0.798269	−0.399135	−	0.916892i	$-0.630689\pi$
$631$	−12653.0	−0.798269	−0.399135	+	0.916892i	$0.630689\pi$
$632$	0	0
$633$	4104.00	0.257693
$634$	0	0
$635$	8250.00	0.515577
$636$	0	0
$637$	9676.00	0.601848
$638$	0	0
$639$	−4248.00	−0.262986
$640$	0	0
$641$	−11874.0	−0.731661	−0.365831	−	0.930681i	$-0.619215\pi$
$641$	−11874.0	−0.731661	−0.365831	+	0.930681i	$0.619215\pi$
$642$	0	0
$643$	26783.0	1.64264	0.821321	−	0.570467i	$-0.193238\pi$
$643$	26783.0	1.64264	0.821321	+	0.570467i	$0.193238\pi$
$644$	0	0
$645$	−9471.00	−0.578171
$646$	0	0
$647$	24593.0	1.49436	0.747180	−	0.664622i	$-0.231407\pi$
$647$	24593.0	1.49436	0.747180	+	0.664622i	$0.231407\pi$
$648$	0	0
$649$	8700.00	0.526202
$650$	0	0
$651$	−3150.00	−0.189644
$652$	0	0
$653$	−3261.00	−0.195425	−0.0977127	−	0.995215i	$-0.531153\pi$
$653$	−3261.00	−0.195425	−0.0977127	+	0.995215i	$0.531153\pi$
$654$	0	0
$655$	−14025.0	−0.836644
$656$	0	0
$657$	−10179.0	−0.604445
$658$	0	0
$659$	20714.0	1.22444	0.612218	−	0.790689i	$-0.290278\pi$
$659$	20714.0	1.22444	0.612218	+	0.790689i	$0.290278\pi$
$660$	0	0
$661$	−30572.0	−1.79896	−0.899480	−	0.436961i	$-0.856055\pi$
$661$	−30572.0	−1.79896	−0.899480	+	0.436961i	$0.856055\pi$
$662$	0	0
$663$	−6642.00	−0.389071
$664$	0	0
$665$	−3135.00	−0.182812
$666$	0	0
$667$	11800.0	0.685004
$668$	0	0
$669$	−7620.00	−0.440368
$670$	0	0
$671$	−26129.0	−1.50328
$672$	0	0
$673$	−9772.00	−0.559707	−0.279854	−	0.960043i	$-0.590286\pi$
$673$	−9772.00	−0.559707	−0.279854	+	0.960043i	$0.590286\pi$
$674$	0	0
$675$	−108.000	−0.00615840
$676$	0	0
$677$	12350.0	0.701106	0.350553	−	0.936543i	$-0.385994\pi$
$677$	12350.0	0.701106	0.350553	+	0.936543i	$0.385994\pi$
$678$	0	0
$679$	−19650.0	−1.11060
$680$	0	0
$681$	17922.0	1.00848
$682$	0	0
$683$	8686.00	0.486619	0.243309	−	0.969949i	$-0.421767\pi$
$683$	8686.00	0.486619	0.243309	+	0.969949i	$0.421767\pi$
$684$	0	0
$685$	−23353.0	−1.30259
$686$	0	0
$687$	−1065.00	−0.0591445
$688$	0	0
$689$	−44116.0	−2.43931
$690$	0	0
$691$	22449.0	1.23589	0.617945	−	0.786221i	$-0.287965\pi$
$691$	22449.0	1.23589	0.617945	+	0.786221i	$0.287965\pi$
$692$	0	0
$693$	3915.00	0.214601
$694$	0	0
$695$	25047.0	1.36703
$696$	0	0
$697$	216.000	0.0117383
$698$	0	0
$699$	711.000	0.0384728
$700$	0	0
$701$	−16430.0	−0.885239	−0.442619	−	0.896710i	$-0.645951\pi$
$701$	−16430.0	−0.885239	−0.442619	+	0.896710i	$0.645951\pi$
$702$	0	0
$703$	4408.00	0.236488
$704$	0	0
$705$	12705.0	0.678721
$706$	0	0
$707$	−9570.00	−0.509076
$708$	0	0
$709$	−15882.0	−0.841271	−0.420635	−	0.907230i	$-0.638193\pi$
$709$	−15882.0	−0.841271	−0.420635	+	0.907230i	$0.638193\pi$
$710$	0	0
$711$	468.000	0.0246855
$712$	0	0
$713$	7000.00	0.367675
$714$	0	0
$715$	26158.0	1.36819
$716$	0	0
$717$	−4905.00	−0.255482
$718$	0	0
$719$	−19079.0	−0.989606	−0.494803	−	0.869005i	$-0.664760\pi$
$719$	−19079.0	−0.989606	−0.494803	+	0.869005i	$0.664760\pi$
$720$	0	0
$721$	−11790.0	−0.608991
$722$	0	0
$723$	−492.000	−0.0253080
$724$	0	0
$725$	472.000	0.0241788
$726$	0	0
$727$	−17275.0	−0.881285	−0.440643	−	0.897683i	$-0.645249\pi$
$727$	−17275.0	−0.881285	−0.440643	+	0.897683i	$0.645249\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	7749.00	0.392075
$732$	0	0
$733$	−26358.0	−1.32818	−0.664089	−	0.747653i	$-0.731181\pi$
$733$	−26358.0	−1.32818	−0.664089	+	0.747653i	$0.731181\pi$
$734$	0	0
$735$	3894.00	0.195418
$736$	0	0
$737$	−3828.00	−0.191325
$738$	0	0
$739$	1675.00	0.0833774	0.0416887	−	0.999131i	$-0.486726\pi$
$739$	1675.00	0.0833774	0.0416887	+	0.999131i	$0.486726\pi$
$740$	0	0
$741$	−4674.00	−0.231719
$742$	0	0
$743$	−26428.0	−1.30491	−0.652456	−	0.757827i	$-0.726261\pi$
$743$	−26428.0	−1.30491	−0.652456	+	0.757827i	$0.726261\pi$
$744$	0	0
$745$	−24585.0	−1.20903
$746$	0	0
$747$	−2484.00	−0.121666
$748$	0	0
$749$	19650.0	0.958605
$750$	0	0
$751$	21328.0	1.03631	0.518156	−	0.855286i	$-0.326619\pi$
$751$	21328.0	1.03631	0.518156	+	0.855286i	$0.326619\pi$
$752$	0	0
$753$	6297.00	0.304748
$754$	0	0
$755$	−30404.0	−1.46558
$756$	0	0
$757$	−16229.0	−0.779198	−0.389599	−	0.920985i	$-0.627386\pi$
$757$	−16229.0	−0.779198	−0.389599	+	0.920985i	$0.627386\pi$
$758$	0	0
$759$	−8700.00	−0.416061
$760$	0	0
$761$	−4965.00	−0.236506	−0.118253	−	0.992983i	$-0.537729\pi$
$761$	−4965.00	−0.236506	−0.118253	+	0.992983i	$0.537729\pi$
$762$	0	0
$763$	−19440.0	−0.922379
$764$	0	0
$765$	−2673.00	−0.126330
$766$	0	0
$767$	−24600.0	−1.15809
$768$	0	0
$769$	35275.0	1.65416	0.827080	−	0.562084i	$-0.190000\pi$
$769$	35275.0	1.65416	0.827080	+	0.562084i	$0.190000\pi$
$770$	0	0
$771$	16608.0	0.775775
$772$	0	0
$773$	4776.00	0.222226	0.111113	−	0.993808i	$-0.464558\pi$
$773$	4776.00	0.222226	0.111113	+	0.993808i	$0.464558\pi$
$774$	0	0
$775$	280.000	0.0129779
$776$	0	0
$777$	10440.0	0.482024
$778$	0	0
$779$	152.000	0.00699097
$780$	0	0
$781$	−13688.0	−0.627138
$782$	0	0
$783$	−3186.00	−0.145413
$784$	0	0
$785$	−18722.0	−0.851232
$786$	0	0
$787$	−21136.0	−0.957328	−0.478664	−	0.877998i	$-0.658879\pi$
$787$	−21136.0	−0.957328	−0.478664	+	0.877998i	$0.658879\pi$
$788$	0	0
$789$	−6219.00	−0.280611
$790$	0	0
$791$	−16950.0	−0.761912
$792$	0	0
$793$	73882.0	3.30848
$794$	0	0
$795$	−17754.0	−0.792037
$796$	0	0
$797$	38256.0	1.70025	0.850124	−	0.526583i	$-0.176527\pi$
$797$	38256.0	1.70025	0.850124	+	0.526583i	$0.176527\pi$
$798$	0	0
$799$	−10395.0	−0.460261
$800$	0	0
$801$	−11718.0	−0.516898
$802$	0	0
$803$	−32799.0	−1.44141
$804$	0	0
$805$	16500.0	0.722421
$806$	0	0
$807$	4446.00	0.193936
$808$	0	0
$809$	6189.00	0.268966	0.134483	−	0.990916i	$-0.457063\pi$
$809$	6189.00	0.268966	0.134483	+	0.990916i	$0.457063\pi$
$810$	0	0
$811$	3030.00	0.131193	0.0655966	−	0.997846i	$-0.479105\pi$
$811$	3030.00	0.131193	0.0655966	+	0.997846i	$0.479105\pi$
$812$	0	0
$813$	21804.0	0.940590
$814$	0	0
$815$	31240.0	1.34269
$816$	0	0
$817$	5453.00	0.233508
$818$	0	0
$819$	−11070.0	−0.472304
$820$	0	0
$821$	38571.0	1.63963	0.819816	−	0.572628i	$-0.194076\pi$
$821$	38571.0	1.63963	0.819816	+	0.572628i	$0.194076\pi$
$822$	0	0
$823$	14287.0	0.605120	0.302560	−	0.953130i	$-0.402159\pi$
$823$	14287.0	0.605120	0.302560	+	0.953130i	$0.402159\pi$
$824$	0	0
$825$	−348.000	−0.0146858
$826$	0	0
$827$	−23464.0	−0.986606	−0.493303	−	0.869858i	$-0.664211\pi$
$827$	−23464.0	−0.986606	−0.493303	+	0.869858i	$0.664211\pi$
$828$	0	0
$829$	−38700.0	−1.62136	−0.810679	−	0.585490i	$-0.800902\pi$
$829$	−38700.0	−1.62136	−0.810679	+	0.585490i	$0.800902\pi$
$830$	0	0
$831$	−13749.0	−0.573944
$832$	0	0
$833$	−3186.00	−0.132519
$834$	0	0
$835$	13134.0	0.544336
$836$	0	0
$837$	−1890.00	−0.0780501
$838$	0	0
$839$	21758.0	0.895315	0.447658	−	0.894205i	$-0.352258\pi$
$839$	21758.0	0.895315	0.447658	+	0.894205i	$0.352258\pi$
$840$	0	0
$841$	−10465.0	−0.429087
$842$	0	0
$843$	17286.0	0.706241
$844$	0	0
$845$	−49797.0	−2.02730
$846$	0	0
$847$	−7350.00	−0.298169
$848$	0	0
$849$	−11727.0	−0.474051
$850$	0	0
$851$	−23200.0	−0.934531
$852$	0	0
$853$	22770.0	0.913986	0.456993	−	0.889470i	$-0.348926\pi$
$853$	22770.0	0.913986	0.456993	+	0.889470i	$0.348926\pi$
$854$	0	0
$855$	−1881.00	−0.0752384
$856$	0	0
$857$	−38484.0	−1.53394	−0.766971	−	0.641682i	$-0.778237\pi$
$857$	−38484.0	−1.53394	−0.766971	+	0.641682i	$0.778237\pi$
$858$	0	0
$859$	−23653.0	−0.939499	−0.469750	−	0.882800i	$-0.655656\pi$
$859$	−23653.0	−0.939499	−0.469750	+	0.882800i	$0.655656\pi$
$860$	0	0
$861$	360.000	0.0142494
$862$	0	0
$863$	−29988.0	−1.18285	−0.591427	−	0.806358i	$-0.701435\pi$
$863$	−29988.0	−1.18285	−0.591427	+	0.806358i	$0.701435\pi$
$864$	0	0
$865$	41646.0	1.63700
$866$	0	0
$867$	−12552.0	−0.491682
$868$	0	0
$869$	1508.00	0.0588670
$870$	0	0
$871$	10824.0	0.421076
$872$	0	0
$873$	−11790.0	−0.457080
$874$	0	0
$875$	21285.0	0.822359
$876$	0	0
$877$	−29726.0	−1.14456	−0.572278	−	0.820060i	$-0.693940\pi$
$877$	−29726.0	−1.14456	−0.572278	+	0.820060i	$0.693940\pi$
$878$	0	0
$879$	−24444.0	−0.937970
$880$	0	0
$881$	−28713.0	−1.09803	−0.549016	−	0.835812i	$-0.684997\pi$
$881$	−28713.0	−1.09803	−0.549016	+	0.835812i	$0.684997\pi$
$882$	0	0
$883$	2771.00	0.105608	0.0528038	−	0.998605i	$-0.483184\pi$
$883$	2771.00	0.105608	0.0528038	+	0.998605i	$0.483184\pi$
$884$	0	0
$885$	−9900.00	−0.376028
$886$	0	0
$887$	−23000.0	−0.870648	−0.435324	−	0.900274i	$-0.643366\pi$
$887$	−23000.0	−0.870648	−0.435324	+	0.900274i	$0.643366\pi$
$888$	0	0
$889$	−11250.0	−0.424424
$890$	0	0
$891$	2349.00	0.0883215
$892$	0	0
$893$	−7315.00	−0.274118
$894$	0	0
$895$	23034.0	0.860270
$896$	0	0
$897$	24600.0	0.915686
$898$	0	0
$899$	8260.00	0.306437
$900$	0	0
$901$	14526.0	0.537105
$902$	0	0
$903$	12915.0	0.475952
$904$	0	0
$905$	3850.00	0.141413
$906$	0	0
$907$	−42074.0	−1.54029	−0.770146	−	0.637868i	$-0.779817\pi$
$907$	−42074.0	−1.54029	−0.770146	+	0.637868i	$0.779817\pi$
$908$	0	0
$909$	−5742.00	−0.209516
$910$	0	0
$911$	−6462.00	−0.235012	−0.117506	−	0.993072i	$-0.537490\pi$
$911$	−6462.00	−0.235012	−0.117506	+	0.993072i	$0.537490\pi$
$912$	0	0
$913$	−8004.00	−0.290136
$914$	0	0
$915$	29733.0	1.07425
$916$	0	0
$917$	19125.0	0.688728
$918$	0	0
$919$	−24776.0	−0.889320	−0.444660	−	0.895700i	$-0.646675\pi$
$919$	−24776.0	−0.889320	−0.444660	+	0.895700i	$0.646675\pi$
$920$	0	0
$921$	−1800.00	−0.0643996
$922$	0	0
$923$	38704.0	1.38024
$924$	0	0
$925$	−928.000	−0.0329864
$926$	0	0
$927$	−7074.00	−0.250637
$928$	0	0
$929$	15022.0	0.530523	0.265261	−	0.964177i	$-0.414542\pi$
$929$	15022.0	0.530523	0.265261	+	0.964177i	$0.414542\pi$
$930$	0	0
$931$	−2242.00	−0.0789244
$932$	0	0
$933$	14889.0	0.522448
$934$	0	0
$935$	−8613.00	−0.301257
$936$	0	0
$937$	26285.0	0.916429	0.458214	−	0.888842i	$-0.348489\pi$
$937$	26285.0	0.916429	0.458214	+	0.888842i	$0.348489\pi$
$938$	0	0
$939$	16386.0	0.569475
$940$	0	0
$941$	−18622.0	−0.645122	−0.322561	−	0.946549i	$-0.604544\pi$
$941$	−18622.0	−0.645122	−0.322561	+	0.946549i	$0.604544\pi$
$942$	0	0
$943$	−800.000	−0.0276263
$944$	0	0
$945$	−4455.00	−0.153356
$946$	0	0
$947$	3660.00	0.125590	0.0627952	−	0.998026i	$-0.479999\pi$
$947$	3660.00	0.125590	0.0627952	+	0.998026i	$0.479999\pi$
$948$	0	0
$949$	92742.0	3.17232
$950$	0	0
$951$	−2952.00	−0.100657
$952$	0	0
$953$	−30044.0	−1.02122	−0.510609	−	0.859813i	$-0.670580\pi$
$953$	−30044.0	−1.02122	−0.510609	+	0.859813i	$0.670580\pi$
$954$	0	0
$955$	−16379.0	−0.554986
$956$	0	0
$957$	−10266.0	−0.346763
$958$	0	0
$959$	31845.0	1.07229
$960$	0	0
$961$	−24891.0	−0.835521
$962$	0	0
$963$	11790.0	0.394525
$964$	0	0
$965$	−6600.00	−0.220167
$966$	0	0
$967$	6288.00	0.209109	0.104555	−	0.994519i	$-0.466658\pi$
$967$	6288.00	0.209109	0.104555	+	0.994519i	$0.466658\pi$
$968$	0	0
$969$	1539.00	0.0510215
$970$	0	0
$971$	−39028.0	−1.28987	−0.644937	−	0.764236i	$-0.723117\pi$
$971$	−39028.0	−1.28987	−0.644937	+	0.764236i	$0.723117\pi$
$972$	0	0
$973$	−34155.0	−1.12534
$974$	0	0
$975$	984.000	0.0323213
$976$	0	0
$977$	1678.00	0.0549478	0.0274739	−	0.999623i	$-0.491254\pi$
$977$	1678.00	0.0549478	0.0274739	+	0.999623i	$0.491254\pi$
$978$	0	0
$979$	−37758.0	−1.23264
$980$	0	0
$981$	−11664.0	−0.379616
$982$	0	0
$983$	39972.0	1.29696	0.648479	−	0.761233i	$-0.275406\pi$
$983$	39972.0	1.29696	0.648479	+	0.761233i	$0.275406\pi$
$984$	0	0
$985$	3190.00	0.103190
$986$	0	0
$987$	−17325.0	−0.558724
$988$	0	0
$989$	−28700.0	−0.922757
$990$	0	0
$991$	23672.0	0.758795	0.379398	−	0.925234i	$-0.376131\pi$
$991$	23672.0	0.758795	0.379398	+	0.925234i	$0.376131\pi$
$992$	0	0
$993$	7896.00	0.252338
$994$	0	0
$995$	45859.0	1.46113
$996$	0	0
$997$	−36899.0	−1.17212	−0.586060	−	0.810268i	$-0.699322\pi$
$997$	−36899.0	−1.17212	−0.586060	+	0.810268i	$0.699322\pi$
$998$	0	0
$999$	6264.00	0.198383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.4.a.f.1.1		1
4.3	odd	2		114.4.a.d.1.1	✓	1
12.11	even	2		342.4.a.a.1.1		1
76.75	even	2		2166.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.a.d.1.1	✓	1	4.3	odd	2
342.4.a.a.1.1		1	12.11	even	2
912.4.a.f.1.1		1	1.1	even	1	trivial
2166.4.a.a.1.1		1	76.75	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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	912.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more