LMFDB - Embedded newform 912.4.a.a.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
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)

$f(q)$

$=$

$q-3.00000 q^{3} -12.0000 q^{5} +20.0000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} -12.0000 q^{5} +20.0000 q^{7} +9.00000 q^{9} +4.00000 q^{11} -76.0000 q^{13} +36.0000 q^{15} +22.0000 q^{17} +19.0000 q^{19} -60.0000 q^{21} -82.0000 q^{23} +19.0000 q^{25} -27.0000 q^{27} +242.000 q^{29} +126.000 q^{31} -12.0000 q^{33} -240.000 q^{35} -180.000 q^{37} +228.000 q^{39} -390.000 q^{41} -308.000 q^{43} -108.000 q^{45} +522.000 q^{47} +57.0000 q^{49} -66.0000 q^{51} -70.0000 q^{53} -48.0000 q^{55} -57.0000 q^{57} -188.000 q^{59} -706.000 q^{61} +180.000 q^{63} +912.000 q^{65} -104.000 q^{67} +246.000 q^{69} +432.000 q^{71} +718.000 q^{73} -57.0000 q^{75} +80.0000 q^{77} -94.0000 q^{79} +81.0000 q^{81} +1296.00 q^{83} -264.000 q^{85} -726.000 q^{87} +846.000 q^{89} -1520.00 q^{91} -378.000 q^{93} -228.000 q^{95} +830.000 q^{97} +36.0000 q^{99} +O(q^{100})$

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
$5$	−12.0000	−1.07331	−0.536656	−	0.843801i	$-0.680313\pi$
$5$	−12.0000	−1.07331	−0.536656	+	0.843801i	$0.680313\pi$
$6$	0	0
$7$	20.0000	1.07990	0.539949	−	0.841698i	$-0.318443\pi$
$7$	20.0000	1.07990	0.539949	+	0.841698i	$0.318443\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	4.00000	0.109640	0.0548202	−	0.998496i	$-0.482541\pi$
$11$	4.00000	0.109640	0.0548202	+	0.998496i	$0.482541\pi$
$12$	0	0
$13$	−76.0000	−1.62143	−0.810716	−	0.585440i	$-0.800922\pi$
$13$	−76.0000	−1.62143	−0.810716	+	0.585440i	$0.800922\pi$
$14$	0	0
$15$	36.0000	0.619677
$16$	0	0
$17$	22.0000	0.313870	0.156935	−	0.987609i	$-0.449839\pi$
$17$	22.0000	0.313870	0.156935	+	0.987609i	$0.449839\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	−60.0000	−0.623480
$22$	0	0
$23$	−82.0000	−0.743399	−0.371700	−	0.928353i	$-0.621225\pi$
$23$	−82.0000	−0.743399	−0.371700	+	0.928353i	$0.621225\pi$
$24$	0	0
$25$	19.0000	0.152000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	242.000	1.54960	0.774798	−	0.632209i	$-0.217852\pi$
$29$	242.000	1.54960	0.774798	+	0.632209i	$0.217852\pi$
$30$	0	0
$31$	126.000	0.730009	0.365004	−	0.931006i	$-0.381068\pi$
$31$	126.000	0.730009	0.365004	+	0.931006i	$0.381068\pi$
$32$	0	0
$33$	−12.0000	−0.0633010
$34$	0	0
$35$	−240.000	−1.15907
$36$	0	0
$37$	−180.000	−0.799779	−0.399889	−	0.916563i	$-0.630951\pi$
$37$	−180.000	−0.799779	−0.399889	+	0.916563i	$0.630951\pi$
$38$	0	0
$39$	228.000	0.936134
$40$	0	0
$41$	−390.000	−1.48556	−0.742778	−	0.669538i	$-0.766492\pi$
$41$	−390.000	−1.48556	−0.742778	+	0.669538i	$0.766492\pi$
$42$	0	0
$43$	−308.000	−1.09232	−0.546158	−	0.837682i	$-0.683910\pi$
$43$	−308.000	−1.09232	−0.546158	+	0.837682i	$0.683910\pi$
$44$	0	0
$45$	−108.000	−0.357771
$46$	0	0
$47$	522.000	1.62003	0.810016	−	0.586407i	$-0.199458\pi$
$47$	522.000	1.62003	0.810016	+	0.586407i	$0.199458\pi$
$48$	0	0
$49$	57.0000	0.166181
$50$	0	0
$51$	−66.0000	−0.181213
$52$	0	0
$53$	−70.0000	−0.181420	−0.0907098	−	0.995877i	$-0.528914\pi$
$53$	−70.0000	−0.181420	−0.0907098	+	0.995877i	$0.528914\pi$
$54$	0	0
$55$	−48.0000	−0.117679
$56$	0	0
$57$	−57.0000	−0.132453
$58$	0	0
$59$	−188.000	−0.414839	−0.207420	−	0.978252i	$-0.566507\pi$
$59$	−188.000	−0.414839	−0.207420	+	0.978252i	$0.566507\pi$
$60$	0	0
$61$	−706.000	−1.48187	−0.740935	−	0.671577i	$-0.765617\pi$
$61$	−706.000	−1.48187	−0.740935	+	0.671577i	$0.765617\pi$
$62$	0	0
$63$	180.000	0.359966
$64$	0	0
$65$	912.000	1.74030
$66$	0	0
$67$	−104.000	−0.189636	−0.0948181	−	0.995495i	$-0.530227\pi$
$67$	−104.000	−0.189636	−0.0948181	+	0.995495i	$0.530227\pi$
$68$	0	0
$69$	246.000	0.429202
$70$	0	0
$71$	432.000	0.722098	0.361049	−	0.932547i	$-0.382419\pi$
$71$	432.000	0.722098	0.361049	+	0.932547i	$0.382419\pi$
$72$	0	0
$73$	718.000	1.15117	0.575586	−	0.817741i	$-0.304774\pi$
$73$	718.000	1.15117	0.575586	+	0.817741i	$0.304774\pi$
$74$	0	0
$75$	−57.0000	−0.0877572
$76$	0	0
$77$	80.0000	0.118401
$78$	0	0
$79$	−94.0000	−0.133871	−0.0669356	−	0.997757i	$-0.521322\pi$
$79$	−94.0000	−0.133871	−0.0669356	+	0.997757i	$0.521322\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1296.00	1.71391	0.856955	−	0.515392i	$-0.172354\pi$
$83$	1296.00	1.71391	0.856955	+	0.515392i	$0.172354\pi$
$84$	0	0
$85$	−264.000	−0.336880
$86$	0	0
$87$	−726.000	−0.894659
$88$	0	0
$89$	846.000	1.00759	0.503797	−	0.863822i	$-0.331936\pi$
$89$	846.000	1.00759	0.503797	+	0.863822i	$0.331936\pi$
$90$	0	0
$91$	−1520.00	−1.75098
$92$	0	0
$93$	−378.000	−0.421471
$94$	0	0
$95$	−228.000	−0.246235
$96$	0	0
$97$	830.000	0.868801	0.434401	−	0.900720i	$-0.356960\pi$
$97$	830.000	0.868801	0.434401	+	0.900720i	$0.356960\pi$
$98$	0	0
$99$	36.0000	0.0365468
$100$	0	0
$101$	1612.00	1.58812	0.794059	−	0.607840i	$-0.207964\pi$
$101$	1612.00	1.58812	0.794059	+	0.607840i	$0.207964\pi$
$102$	0	0
$103$	1874.00	1.79273	0.896363	−	0.443322i	$-0.146200\pi$
$103$	1874.00	1.79273	0.896363	+	0.443322i	$0.146200\pi$
$104$	0	0
$105$	720.000	0.669189
$106$	0	0
$107$	1932.00	1.74555	0.872773	−	0.488126i	$-0.162319\pi$
$107$	1932.00	1.74555	0.872773	+	0.488126i	$0.162319\pi$
$108$	0	0
$109$	1096.00	0.963099	0.481549	−	0.876419i	$-0.340074\pi$
$109$	1096.00	0.963099	0.481549	+	0.876419i	$0.340074\pi$
$110$	0	0
$111$	540.000	0.461753
$112$	0	0
$113$	1474.00	1.22710	0.613550	−	0.789656i	$-0.289741\pi$
$113$	1474.00	1.22710	0.613550	+	0.789656i	$0.289741\pi$
$114$	0	0
$115$	984.000	0.797900
$116$	0	0
$117$	−684.000	−0.540477
$118$	0	0
$119$	440.000	0.338947
$120$	0	0
$121$	−1315.00	−0.987979
$122$	0	0
$123$	1170.00	0.857686
$124$	0	0
$125$	1272.00	0.910169
$126$	0	0
$127$	1166.00	0.814691	0.407346	−	0.913274i	$-0.366454\pi$
$127$	1166.00	0.814691	0.407346	+	0.913274i	$0.366454\pi$
$128$	0	0
$129$	924.000	0.630649
$130$	0	0
$131$	−2192.00	−1.46195	−0.730977	−	0.682402i	$-0.760935\pi$
$131$	−2192.00	−1.46195	−0.730977	+	0.682402i	$0.760935\pi$
$132$	0	0
$133$	380.000	0.247746
$134$	0	0
$135$	324.000	0.206559
$136$	0	0
$137$	558.000	0.347979	0.173990	−	0.984747i	$-0.444334\pi$
$137$	558.000	0.347979	0.173990	+	0.984747i	$0.444334\pi$
$138$	0	0
$139$	−68.0000	−0.0414941	−0.0207471	−	0.999785i	$-0.506604\pi$
$139$	−68.0000	−0.0414941	−0.0207471	+	0.999785i	$0.506604\pi$
$140$	0	0
$141$	−1566.00	−0.935326
$142$	0	0
$143$	−304.000	−0.177775
$144$	0	0
$145$	−2904.00	−1.66320
$146$	0	0
$147$	−171.000	−0.0959445
$148$	0	0
$149$	576.000	0.316696	0.158348	−	0.987383i	$-0.449383\pi$
$149$	576.000	0.316696	0.158348	+	0.987383i	$0.449383\pi$
$150$	0	0
$151$	−990.000	−0.533543	−0.266772	−	0.963760i	$-0.585957\pi$
$151$	−990.000	−0.533543	−0.266772	+	0.963760i	$0.585957\pi$
$152$	0	0
$153$	198.000	0.104623
$154$	0	0
$155$	−1512.00	−0.783528
$156$	0	0
$157$	−654.000	−0.332451	−0.166226	−	0.986088i	$-0.553158\pi$
$157$	−654.000	−0.332451	−0.166226	+	0.986088i	$0.553158\pi$
$158$	0	0
$159$	210.000	0.104743
$160$	0	0
$161$	−1640.00	−0.802796
$162$	0	0
$163$	900.000	0.432475	0.216238	−	0.976341i	$-0.430621\pi$
$163$	900.000	0.432475	0.216238	+	0.976341i	$0.430621\pi$
$164$	0	0
$165$	144.000	0.0679417
$166$	0	0
$167$	−740.000	−0.342892	−0.171446	−	0.985194i	$-0.554844\pi$
$167$	−740.000	−0.342892	−0.171446	+	0.985194i	$0.554844\pi$
$168$	0	0
$169$	3579.00	1.62904
$170$	0	0
$171$	171.000	0.0764719
$172$	0	0
$173$	582.000	0.255772	0.127886	−	0.991789i	$-0.459181\pi$
$173$	582.000	0.255772	0.127886	+	0.991789i	$0.459181\pi$
$174$	0	0
$175$	380.000	0.164145
$176$	0	0
$177$	564.000	0.239508
$178$	0	0
$179$	−2748.00	−1.14746	−0.573730	−	0.819045i	$-0.694504\pi$
$179$	−2748.00	−1.14746	−0.573730	+	0.819045i	$0.694504\pi$
$180$	0	0
$181$	1336.00	0.548641	0.274321	−	0.961638i	$-0.411547\pi$
$181$	1336.00	0.548641	0.274321	+	0.961638i	$0.411547\pi$
$182$	0	0
$183$	2118.00	0.855558
$184$	0	0
$185$	2160.00	0.858413
$186$	0	0
$187$	88.0000	0.0344128
$188$	0	0
$189$	−540.000	−0.207827
$190$	0	0
$191$	606.000	0.229574	0.114787	−	0.993390i	$-0.463381\pi$
$191$	606.000	0.229574	0.114787	+	0.993390i	$0.463381\pi$
$192$	0	0
$193$	−3002.00	−1.11963	−0.559815	−	0.828617i	$-0.689128\pi$
$193$	−3002.00	−1.11963	−0.559815	+	0.828617i	$0.689128\pi$
$194$	0	0
$195$	−2736.00	−1.00476
$196$	0	0
$197$	−4456.00	−1.61156	−0.805779	−	0.592217i	$-0.798253\pi$
$197$	−4456.00	−1.61156	−0.805779	+	0.592217i	$0.798253\pi$
$198$	0	0
$199$	2844.00	1.01309	0.506547	−	0.862212i	$-0.330922\pi$
$199$	2844.00	1.01309	0.506547	+	0.862212i	$0.330922\pi$
$200$	0	0
$201$	312.000	0.109486
$202$	0	0
$203$	4840.00	1.67341
$204$	0	0
$205$	4680.00	1.59447
$206$	0	0
$207$	−738.000	−0.247800
$208$	0	0
$209$	76.0000	0.0251533
$210$	0	0
$211$	3108.00	1.01405	0.507023	−	0.861933i	$-0.330746\pi$
$211$	3108.00	1.01405	0.507023	+	0.861933i	$0.330746\pi$
$212$	0	0
$213$	−1296.00	−0.416904
$214$	0	0
$215$	3696.00	1.17240
$216$	0	0
$217$	2520.00	0.788335
$218$	0	0
$219$	−2154.00	−0.664629
$220$	0	0
$221$	−1672.00	−0.508918
$222$	0	0
$223$	4686.00	1.40716	0.703582	−	0.710614i	$-0.251583\pi$
$223$	4686.00	1.40716	0.703582	+	0.710614i	$0.251583\pi$
$224$	0	0
$225$	171.000	0.0506667
$226$	0	0
$227$	3036.00	0.887693	0.443847	−	0.896103i	$-0.353613\pi$
$227$	3036.00	0.887693	0.443847	+	0.896103i	$0.353613\pi$
$228$	0	0
$229$	−4970.00	−1.43418	−0.717089	−	0.696981i	$-0.754526\pi$
$229$	−4970.00	−1.43418	−0.717089	+	0.696981i	$0.754526\pi$
$230$	0	0
$231$	−240.000	−0.0683586
$232$	0	0
$233$	−2982.00	−0.838443	−0.419222	−	0.907884i	$-0.637697\pi$
$233$	−2982.00	−0.838443	−0.419222	+	0.907884i	$0.637697\pi$
$234$	0	0
$235$	−6264.00	−1.73880
$236$	0	0
$237$	282.000	0.0772906
$238$	0	0
$239$	522.000	0.141278	0.0706389	−	0.997502i	$-0.477496\pi$
$239$	522.000	0.141278	0.0706389	+	0.997502i	$0.477496\pi$
$240$	0	0
$241$	3350.00	0.895404	0.447702	−	0.894183i	$-0.352242\pi$
$241$	3350.00	0.895404	0.447702	+	0.894183i	$0.352242\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−684.000	−0.178364
$246$	0	0
$247$	−1444.00	−0.371982
$248$	0	0
$249$	−3888.00	−0.989526
$250$	0	0
$251$	2968.00	0.746369	0.373184	−	0.927757i	$-0.378266\pi$
$251$	2968.00	0.746369	0.373184	+	0.927757i	$0.378266\pi$
$252$	0	0
$253$	−328.000	−0.0815067
$254$	0	0
$255$	792.000	0.194498
$256$	0	0
$257$	−1234.00	−0.299513	−0.149756	−	0.988723i	$-0.547849\pi$
$257$	−1234.00	−0.299513	−0.149756	+	0.988723i	$0.547849\pi$
$258$	0	0
$259$	−3600.00	−0.863680
$260$	0	0
$261$	2178.00	0.516532
$262$	0	0
$263$	1994.00	0.467511	0.233755	−	0.972295i	$-0.424899\pi$
$263$	1994.00	0.467511	0.233755	+	0.972295i	$0.424899\pi$
$264$	0	0
$265$	840.000	0.194720
$266$	0	0
$267$	−2538.00	−0.581734
$268$	0	0
$269$	7214.00	1.63511	0.817556	−	0.575849i	$-0.195328\pi$
$269$	7214.00	1.63511	0.817556	+	0.575849i	$0.195328\pi$
$270$	0	0
$271$	−7572.00	−1.69729	−0.848646	−	0.528961i	$-0.822582\pi$
$271$	−7572.00	−1.69729	−0.848646	+	0.528961i	$0.822582\pi$
$272$	0	0
$273$	4560.00	1.01093
$274$	0	0
$275$	76.0000	0.0166654
$276$	0	0
$277$	6262.00	1.35829	0.679146	−	0.734003i	$-0.262350\pi$
$277$	6262.00	1.35829	0.679146	+	0.734003i	$0.262350\pi$
$278$	0	0
$279$	1134.00	0.243336
$280$	0	0
$281$	−2710.00	−0.575320	−0.287660	−	0.957733i	$-0.592877\pi$
$281$	−2710.00	−0.575320	−0.287660	+	0.957733i	$0.592877\pi$
$282$	0	0
$283$	556.000	0.116787	0.0583936	−	0.998294i	$-0.481402\pi$
$283$	556.000	0.116787	0.0583936	+	0.998294i	$0.481402\pi$
$284$	0	0
$285$	684.000	0.142164
$286$	0	0
$287$	−7800.00	−1.60425
$288$	0	0
$289$	−4429.00	−0.901486
$290$	0	0
$291$	−2490.00	−0.501603
$292$	0	0
$293$	3694.00	0.736539	0.368269	−	0.929719i	$-0.379950\pi$
$293$	3694.00	0.736539	0.368269	+	0.929719i	$0.379950\pi$
$294$	0	0
$295$	2256.00	0.445252
$296$	0	0
$297$	−108.000	−0.0211003
$298$	0	0
$299$	6232.00	1.20537
$300$	0	0
$301$	−6160.00	−1.17959
$302$	0	0
$303$	−4836.00	−0.916901
$304$	0	0
$305$	8472.00	1.59051
$306$	0	0
$307$	−3384.00	−0.629104	−0.314552	−	0.949240i	$-0.601854\pi$
$307$	−3384.00	−0.629104	−0.314552	+	0.949240i	$0.601854\pi$
$308$	0	0
$309$	−5622.00	−1.03503
$310$	0	0
$311$	9666.00	1.76241	0.881203	−	0.472737i	$-0.156734\pi$
$311$	9666.00	1.76241	0.881203	+	0.472737i	$0.156734\pi$
$312$	0	0
$313$	−6794.00	−1.22690	−0.613450	−	0.789734i	$-0.710219\pi$
$313$	−6794.00	−1.22690	−0.613450	+	0.789734i	$0.710219\pi$
$314$	0	0
$315$	−2160.00	−0.386356
$316$	0	0
$317$	−3242.00	−0.574413	−0.287206	−	0.957869i	$-0.592727\pi$
$317$	−3242.00	−0.574413	−0.287206	+	0.957869i	$0.592727\pi$
$318$	0	0
$319$	968.000	0.169898
$320$	0	0
$321$	−5796.00	−1.00779
$322$	0	0
$323$	418.000	0.0720066
$324$	0	0
$325$	−1444.00	−0.246458
$326$	0	0
$327$	−3288.00	−0.556045
$328$	0	0
$329$	10440.0	1.74947
$330$	0	0
$331$	−176.000	−0.0292261	−0.0146130	−	0.999893i	$-0.504652\pi$
$331$	−176.000	−0.0292261	−0.0146130	+	0.999893i	$0.504652\pi$
$332$	0	0
$333$	−1620.00	−0.266593
$334$	0	0
$335$	1248.00	0.203539
$336$	0	0
$337$	−4262.00	−0.688920	−0.344460	−	0.938801i	$-0.611938\pi$
$337$	−4262.00	−0.688920	−0.344460	+	0.938801i	$0.611938\pi$
$338$	0	0
$339$	−4422.00	−0.708466
$340$	0	0
$341$	504.000	0.0800385
$342$	0	0
$343$	−5720.00	−0.900440
$344$	0	0
$345$	−2952.00	−0.460668
$346$	0	0
$347$	−7060.00	−1.09222	−0.546110	−	0.837713i	$-0.683892\pi$
$347$	−7060.00	−1.09222	−0.546110	+	0.837713i	$0.683892\pi$
$348$	0	0
$349$	4746.00	0.727930	0.363965	−	0.931413i	$-0.381423\pi$
$349$	4746.00	0.727930	0.363965	+	0.931413i	$0.381423\pi$
$350$	0	0
$351$	2052.00	0.312045
$352$	0	0
$353$	2546.00	0.383881	0.191940	−	0.981407i	$-0.438522\pi$
$353$	2546.00	0.383881	0.191940	+	0.981407i	$0.438522\pi$
$354$	0	0
$355$	−5184.00	−0.775037
$356$	0	0
$357$	−1320.00	−0.195691
$358$	0	0
$359$	1702.00	0.250218	0.125109	−	0.992143i	$-0.460072\pi$
$359$	1702.00	0.250218	0.125109	+	0.992143i	$0.460072\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	3945.00	0.570410
$364$	0	0
$365$	−8616.00	−1.23557
$366$	0	0
$367$	7844.00	1.11568	0.557839	−	0.829950i	$-0.311631\pi$
$367$	7844.00	1.11568	0.557839	+	0.829950i	$0.311631\pi$
$368$	0	0
$369$	−3510.00	−0.495185
$370$	0	0
$371$	−1400.00	−0.195915
$372$	0	0
$373$	13612.0	1.88955	0.944776	−	0.327718i	$-0.106280\pi$
$373$	13612.0	1.88955	0.944776	+	0.327718i	$0.106280\pi$
$374$	0	0
$375$	−3816.00	−0.525486
$376$	0	0
$377$	−18392.0	−2.51256
$378$	0	0
$379$	−976.000	−0.132279	−0.0661395	−	0.997810i	$-0.521068\pi$
$379$	−976.000	−0.132279	−0.0661395	+	0.997810i	$0.521068\pi$
$380$	0	0
$381$	−3498.00	−0.470362
$382$	0	0
$383$	−2152.00	−0.287107	−0.143554	−	0.989643i	$-0.545853\pi$
$383$	−2152.00	−0.287107	−0.143554	+	0.989643i	$0.545853\pi$
$384$	0	0
$385$	−960.000	−0.127081
$386$	0	0
$387$	−2772.00	−0.364105
$388$	0	0
$389$	10572.0	1.37795	0.688974	−	0.724786i	$-0.258061\pi$
$389$	10572.0	1.37795	0.688974	+	0.724786i	$0.258061\pi$
$390$	0	0
$391$	−1804.00	−0.233330
$392$	0	0
$393$	6576.00	0.844059
$394$	0	0
$395$	1128.00	0.143686
$396$	0	0
$397$	−10910.0	−1.37924	−0.689619	−	0.724173i	$-0.742222\pi$
$397$	−10910.0	−1.37924	−0.689619	+	0.724173i	$0.742222\pi$
$398$	0	0
$399$	−1140.00	−0.143036
$400$	0	0
$401$	10146.0	1.26351	0.631754	−	0.775169i	$-0.282335\pi$
$401$	10146.0	1.26351	0.631754	+	0.775169i	$0.282335\pi$
$402$	0	0
$403$	−9576.00	−1.18366
$404$	0	0
$405$	−972.000	−0.119257
$406$	0	0
$407$	−720.000	−0.0876881
$408$	0	0
$409$	−13706.0	−1.65701	−0.828506	−	0.559980i	$-0.810809\pi$
$409$	−13706.0	−1.65701	−0.828506	+	0.559980i	$0.810809\pi$
$410$	0	0
$411$	−1674.00	−0.200906
$412$	0	0
$413$	−3760.00	−0.447984
$414$	0	0
$415$	−15552.0	−1.83956
$416$	0	0
$417$	204.000	0.0239566
$418$	0	0
$419$	−6812.00	−0.794243	−0.397122	−	0.917766i	$-0.629991\pi$
$419$	−6812.00	−0.794243	−0.397122	+	0.917766i	$0.629991\pi$
$420$	0	0
$421$	−6724.00	−0.778403	−0.389202	−	0.921153i	$-0.627249\pi$
$421$	−6724.00	−0.778403	−0.389202	+	0.921153i	$0.627249\pi$
$422$	0	0
$423$	4698.00	0.540011
$424$	0	0
$425$	418.000	0.0477082
$426$	0	0
$427$	−14120.0	−1.60027
$428$	0	0
$429$	912.000	0.102638
$430$	0	0
$431$	−13876.0	−1.55077	−0.775387	−	0.631487i	$-0.782445\pi$
$431$	−13876.0	−1.55077	−0.775387	+	0.631487i	$0.782445\pi$
$432$	0	0
$433$	342.000	0.0379572	0.0189786	−	0.999820i	$-0.493959\pi$
$433$	342.000	0.0379572	0.0189786	+	0.999820i	$0.493959\pi$
$434$	0	0
$435$	8712.00	0.960249
$436$	0	0
$437$	−1558.00	−0.170547
$438$	0	0
$439$	6526.00	0.709497	0.354748	−	0.934962i	$-0.384567\pi$
$439$	6526.00	0.709497	0.354748	+	0.934962i	$0.384567\pi$
$440$	0	0
$441$	513.000	0.0553936
$442$	0	0
$443$	5020.00	0.538391	0.269196	−	0.963085i	$-0.413242\pi$
$443$	5020.00	0.538391	0.269196	+	0.963085i	$0.413242\pi$
$444$	0	0
$445$	−10152.0	−1.08146
$446$	0	0
$447$	−1728.00	−0.182845
$448$	0	0
$449$	9486.00	0.997042	0.498521	−	0.866878i	$-0.333877\pi$
$449$	9486.00	0.997042	0.498521	+	0.866878i	$0.333877\pi$
$450$	0	0
$451$	−1560.00	−0.162877
$452$	0	0
$453$	2970.00	0.308041
$454$	0	0
$455$	18240.0	1.87935
$456$	0	0
$457$	−7262.00	−0.743330	−0.371665	−	0.928367i	$-0.621213\pi$
$457$	−7262.00	−0.743330	−0.371665	+	0.928367i	$0.621213\pi$
$458$	0	0
$459$	−594.000	−0.0604042
$460$	0	0
$461$	−13968.0	−1.41118	−0.705591	−	0.708620i	$-0.749318\pi$
$461$	−13968.0	−1.41118	−0.705591	+	0.708620i	$0.749318\pi$
$462$	0	0
$463$	−4604.00	−0.462130	−0.231065	−	0.972938i	$-0.574221\pi$
$463$	−4604.00	−0.462130	−0.231065	+	0.972938i	$0.574221\pi$
$464$	0	0
$465$	4536.00	0.452370
$466$	0	0
$467$	19480.0	1.93025	0.965125	−	0.261789i	$-0.0843124\pi$
$467$	19480.0	1.93025	0.965125	+	0.261789i	$0.0843124\pi$
$468$	0	0
$469$	−2080.00	−0.204788
$470$	0	0
$471$	1962.00	0.191941
$472$	0	0
$473$	−1232.00	−0.119762
$474$	0	0
$475$	361.000	0.0348712
$476$	0	0
$477$	−630.000	−0.0604732
$478$	0	0
$479$	12134.0	1.15745	0.578723	−	0.815524i	$-0.303551\pi$
$479$	12134.0	1.15745	0.578723	+	0.815524i	$0.303551\pi$
$480$	0	0
$481$	13680.0	1.29679
$482$	0	0
$483$	4920.00	0.463494
$484$	0	0
$485$	−9960.00	−0.932495
$486$	0	0
$487$	15658.0	1.45694	0.728472	−	0.685076i	$-0.240231\pi$
$487$	15658.0	1.45694	0.728472	+	0.685076i	$0.240231\pi$
$488$	0	0
$489$	−2700.00	−0.249690
$490$	0	0
$491$	−2520.00	−0.231621	−0.115811	−	0.993271i	$-0.536947\pi$
$491$	−2520.00	−0.231621	−0.115811	+	0.993271i	$0.536947\pi$
$492$	0	0
$493$	5324.00	0.486371
$494$	0	0
$495$	−432.000	−0.0392262
$496$	0	0
$497$	8640.00	0.779793
$498$	0	0
$499$	9460.00	0.848673	0.424336	−	0.905505i	$-0.360507\pi$
$499$	9460.00	0.848673	0.424336	+	0.905505i	$0.360507\pi$
$500$	0	0
$501$	2220.00	0.197969
$502$	0	0
$503$	12178.0	1.07950	0.539752	−	0.841824i	$-0.318518\pi$
$503$	12178.0	1.07950	0.539752	+	0.841824i	$0.318518\pi$
$504$	0	0
$505$	−19344.0	−1.70455
$506$	0	0
$507$	−10737.0	−0.940526
$508$	0	0
$509$	−4746.00	−0.413286	−0.206643	−	0.978416i	$-0.566254\pi$
$509$	−4746.00	−0.413286	−0.206643	+	0.978416i	$0.566254\pi$
$510$	0	0
$511$	14360.0	1.24315
$512$	0	0
$513$	−513.000	−0.0441511
$514$	0	0
$515$	−22488.0	−1.92415
$516$	0	0
$517$	2088.00	0.177621
$518$	0	0
$519$	−1746.00	−0.147670
$520$	0	0
$521$	4326.00	0.363773	0.181886	−	0.983320i	$-0.441780\pi$
$521$	4326.00	0.363773	0.181886	+	0.983320i	$0.441780\pi$
$522$	0	0
$523$	6328.00	0.529071	0.264535	−	0.964376i	$-0.414781\pi$
$523$	6328.00	0.529071	0.264535	+	0.964376i	$0.414781\pi$
$524$	0	0
$525$	−1140.00	−0.0947689
$526$	0	0
$527$	2772.00	0.229128
$528$	0	0
$529$	−5443.00	−0.447358
$530$	0	0
$531$	−1692.00	−0.138280
$532$	0	0
$533$	29640.0	2.40873
$534$	0	0
$535$	−23184.0	−1.87352
$536$	0	0
$537$	8244.00	0.662486
$538$	0	0
$539$	228.000	0.0182201
$540$	0	0
$541$	9378.00	0.745271	0.372636	−	0.927978i	$-0.378454\pi$
$541$	9378.00	0.745271	0.372636	+	0.927978i	$0.378454\pi$
$542$	0	0
$543$	−4008.00	−0.316758
$544$	0	0
$545$	−13152.0	−1.03371
$546$	0	0
$547$	5048.00	0.394583	0.197291	−	0.980345i	$-0.436785\pi$
$547$	5048.00	0.394583	0.197291	+	0.980345i	$0.436785\pi$
$548$	0	0
$549$	−6354.00	−0.493956
$550$	0	0
$551$	4598.00	0.355502
$552$	0	0
$553$	−1880.00	−0.144567
$554$	0	0
$555$	−6480.00	−0.495605
$556$	0	0
$557$	752.000	0.0572051	0.0286026	−	0.999591i	$-0.490894\pi$
$557$	752.000	0.0572051	0.0286026	+	0.999591i	$0.490894\pi$
$558$	0	0
$559$	23408.0	1.77111
$560$	0	0
$561$	−264.000	−0.0198683
$562$	0	0
$563$	−18156.0	−1.35912	−0.679560	−	0.733620i	$-0.737829\pi$
$563$	−18156.0	−1.35912	−0.679560	+	0.733620i	$0.737829\pi$
$564$	0	0
$565$	−17688.0	−1.31706
$566$	0	0
$567$	1620.00	0.119989
$568$	0	0
$569$	1398.00	0.103000	0.0515002	−	0.998673i	$-0.483600\pi$
$569$	1398.00	0.103000	0.0515002	+	0.998673i	$0.483600\pi$
$570$	0	0
$571$	−21180.0	−1.55229	−0.776143	−	0.630557i	$-0.782827\pi$
$571$	−21180.0	−1.55229	−0.776143	+	0.630557i	$0.782827\pi$
$572$	0	0
$573$	−1818.00	−0.132545
$574$	0	0
$575$	−1558.00	−0.112997
$576$	0	0
$577$	27186.0	1.96147	0.980735	−	0.195344i	$-0.0625823\pi$
$577$	27186.0	1.96147	0.980735	+	0.195344i	$0.0625823\pi$
$578$	0	0
$579$	9006.00	0.646419
$580$	0	0
$581$	25920.0	1.85085
$582$	0	0
$583$	−280.000	−0.0198909
$584$	0	0
$585$	8208.00	0.580101
$586$	0	0
$587$	10204.0	0.717486	0.358743	−	0.933436i	$-0.383205\pi$
$587$	10204.0	0.717486	0.358743	+	0.933436i	$0.383205\pi$
$588$	0	0
$589$	2394.00	0.167475
$590$	0	0
$591$	13368.0	0.930433
$592$	0	0
$593$	9978.00	0.690974	0.345487	−	0.938424i	$-0.387714\pi$
$593$	9978.00	0.690974	0.345487	+	0.938424i	$0.387714\pi$
$594$	0	0
$595$	−5280.00	−0.363796
$596$	0	0
$597$	−8532.00	−0.584910
$598$	0	0
$599$	11100.0	0.757151	0.378576	−	0.925570i	$-0.376414\pi$
$599$	11100.0	0.757151	0.378576	+	0.925570i	$0.376414\pi$
$600$	0	0
$601$	−3030.00	−0.205651	−0.102826	−	0.994699i	$-0.532788\pi$
$601$	−3030.00	−0.205651	−0.102826	+	0.994699i	$0.532788\pi$
$602$	0	0
$603$	−936.000	−0.0632121
$604$	0	0
$605$	15780.0	1.06041
$606$	0	0
$607$	−10478.0	−0.700641	−0.350320	−	0.936630i	$-0.613927\pi$
$607$	−10478.0	−0.700641	−0.350320	+	0.936630i	$0.613927\pi$
$608$	0	0
$609$	−14520.0	−0.966141
$610$	0	0
$611$	−39672.0	−2.62677
$612$	0	0
$613$	2706.00	0.178294	0.0891471	−	0.996018i	$-0.471586\pi$
$613$	2706.00	0.178294	0.0891471	+	0.996018i	$0.471586\pi$
$614$	0	0
$615$	−14040.0	−0.920565
$616$	0	0
$617$	−19734.0	−1.28762	−0.643810	−	0.765186i	$-0.722647\pi$
$617$	−19734.0	−1.28762	−0.643810	+	0.765186i	$0.722647\pi$
$618$	0	0
$619$	−21196.0	−1.37632	−0.688158	−	0.725561i	$-0.741580\pi$
$619$	−21196.0	−1.37632	−0.688158	+	0.725561i	$0.741580\pi$
$620$	0	0
$621$	2214.00	0.143067
$622$	0	0
$623$	16920.0	1.08810
$624$	0	0
$625$	−17639.0	−1.12890
$626$	0	0
$627$	−228.000	−0.0145222
$628$	0	0
$629$	−3960.00	−0.251026
$630$	0	0
$631$	−5040.00	−0.317970	−0.158985	−	0.987281i	$-0.550822\pi$
$631$	−5040.00	−0.317970	−0.158985	+	0.987281i	$0.550822\pi$
$632$	0	0
$633$	−9324.00	−0.585459
$634$	0	0
$635$	−13992.0	−0.874418
$636$	0	0
$637$	−4332.00	−0.269451
$638$	0	0
$639$	3888.00	0.240699
$640$	0	0
$641$	−13602.0	−0.838138	−0.419069	−	0.907954i	$-0.637644\pi$
$641$	−13602.0	−0.838138	−0.419069	+	0.907954i	$0.637644\pi$
$642$	0	0
$643$	−4628.00	−0.283842	−0.141921	−	0.989878i	$-0.545328\pi$
$643$	−4628.00	−0.283842	−0.141921	+	0.989878i	$0.545328\pi$
$644$	0	0
$645$	−11088.0	−0.676883
$646$	0	0
$647$	14142.0	0.859319	0.429659	−	0.902991i	$-0.358634\pi$
$647$	14142.0	0.859319	0.429659	+	0.902991i	$0.358634\pi$
$648$	0	0
$649$	−752.000	−0.0454832
$650$	0	0
$651$	−7560.00	−0.455146
$652$	0	0
$653$	−14424.0	−0.864402	−0.432201	−	0.901777i	$-0.642263\pi$
$653$	−14424.0	−0.864402	−0.432201	+	0.901777i	$0.642263\pi$
$654$	0	0
$655$	26304.0	1.56913
$656$	0	0
$657$	6462.00	0.383724
$658$	0	0
$659$	−24044.0	−1.42128	−0.710638	−	0.703558i	$-0.751594\pi$
$659$	−24044.0	−1.42128	−0.710638	+	0.703558i	$0.751594\pi$
$660$	0	0
$661$	10092.0	0.593848	0.296924	−	0.954901i	$-0.404039\pi$
$661$	10092.0	0.593848	0.296924	+	0.954901i	$0.404039\pi$
$662$	0	0
$663$	5016.00	0.293824
$664$	0	0
$665$	−4560.00	−0.265909
$666$	0	0
$667$	−19844.0	−1.15197
$668$	0	0
$669$	−14058.0	−0.812427
$670$	0	0
$671$	−2824.00	−0.162473
$672$	0	0
$673$	−7098.00	−0.406549	−0.203275	−	0.979122i	$-0.565158\pi$
$673$	−7098.00	−0.406549	−0.203275	+	0.979122i	$0.565158\pi$
$674$	0	0
$675$	−513.000	−0.0292524
$676$	0	0
$677$	29762.0	1.68958	0.844791	−	0.535097i	$-0.179725\pi$
$677$	29762.0	1.68958	0.844791	+	0.535097i	$0.179725\pi$
$678$	0	0
$679$	16600.0	0.938217
$680$	0	0
$681$	−9108.00	−0.512510
$682$	0	0
$683$	−11748.0	−0.658162	−0.329081	−	0.944302i	$-0.606739\pi$
$683$	−11748.0	−0.658162	−0.329081	+	0.944302i	$0.606739\pi$
$684$	0	0
$685$	−6696.00	−0.373491
$686$	0	0
$687$	14910.0	0.828023
$688$	0	0
$689$	5320.00	0.294159
$690$	0	0
$691$	30676.0	1.68881	0.844407	−	0.535703i	$-0.179953\pi$
$691$	30676.0	1.68881	0.844407	+	0.535703i	$0.179953\pi$
$692$	0	0
$693$	720.000	0.0394669
$694$	0	0
$695$	816.000	0.0445362
$696$	0	0
$697$	−8580.00	−0.466271
$698$	0	0
$699$	8946.00	0.484076
$700$	0	0
$701$	31228.0	1.68255	0.841273	−	0.540610i	$-0.181806\pi$
$701$	31228.0	1.68255	0.841273	+	0.540610i	$0.181806\pi$
$702$	0	0
$703$	−3420.00	−0.183482
$704$	0	0
$705$	18792.0	1.00390
$706$	0	0
$707$	32240.0	1.71501
$708$	0	0
$709$	−14658.0	−0.776435	−0.388218	−	0.921568i	$-0.626909\pi$
$709$	−14658.0	−0.776435	−0.388218	+	0.921568i	$0.626909\pi$
$710$	0	0
$711$	−846.000	−0.0446237
$712$	0	0
$713$	−10332.0	−0.542688
$714$	0	0
$715$	3648.00	0.190808
$716$	0	0
$717$	−1566.00	−0.0815667
$718$	0	0
$719$	−5502.00	−0.285382	−0.142691	−	0.989767i	$-0.545576\pi$
$719$	−5502.00	−0.285382	−0.142691	+	0.989767i	$0.545576\pi$
$720$	0	0
$721$	37480.0	1.93596
$722$	0	0
$723$	−10050.0	−0.516962
$724$	0	0
$725$	4598.00	0.235539
$726$	0	0
$727$	−6136.00	−0.313028	−0.156514	−	0.987676i	$-0.550026\pi$
$727$	−6136.00	−0.313028	−0.156514	+	0.987676i	$0.550026\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−6776.00	−0.342845
$732$	0	0
$733$	24442.0	1.23163	0.615816	−	0.787890i	$-0.288827\pi$
$733$	24442.0	1.23163	0.615816	+	0.787890i	$0.288827\pi$
$734$	0	0
$735$	2052.00	0.102978
$736$	0	0
$737$	−416.000	−0.0207918
$738$	0	0
$739$	−11980.0	−0.596335	−0.298167	−	0.954514i	$-0.596375\pi$
$739$	−11980.0	−0.596335	−0.298167	+	0.954514i	$0.596375\pi$
$740$	0	0
$741$	4332.00	0.214764
$742$	0	0
$743$	−15524.0	−0.766515	−0.383257	−	0.923642i	$-0.625198\pi$
$743$	−15524.0	−0.766515	−0.383257	+	0.923642i	$0.625198\pi$
$744$	0	0
$745$	−6912.00	−0.339914
$746$	0	0
$747$	11664.0	0.571303
$748$	0	0
$749$	38640.0	1.88501
$750$	0	0
$751$	−10494.0	−0.509895	−0.254948	−	0.966955i	$-0.582058\pi$
$751$	−10494.0	−0.509895	−0.254948	+	0.966955i	$0.582058\pi$
$752$	0	0
$753$	−8904.00	−0.430916
$754$	0	0
$755$	11880.0	0.572659
$756$	0	0
$757$	−24446.0	−1.17372	−0.586859	−	0.809689i	$-0.699636\pi$
$757$	−24446.0	−1.17372	−0.586859	+	0.809689i	$0.699636\pi$
$758$	0	0
$759$	984.000	0.0470579
$760$	0	0
$761$	−28650.0	−1.36473	−0.682366	−	0.731010i	$-0.739049\pi$
$761$	−28650.0	−1.36473	−0.682366	+	0.731010i	$0.739049\pi$
$762$	0	0
$763$	21920.0	1.04005
$764$	0	0
$765$	−2376.00	−0.112293
$766$	0	0
$767$	14288.0	0.672633
$768$	0	0
$769$	6974.00	0.327034	0.163517	−	0.986541i	$-0.447716\pi$
$769$	6974.00	0.327034	0.163517	+	0.986541i	$0.447716\pi$
$770$	0	0
$771$	3702.00	0.172924
$772$	0	0
$773$	−6170.00	−0.287089	−0.143544	−	0.989644i	$-0.545850\pi$
$773$	−6170.00	−0.287089	−0.143544	+	0.989644i	$0.545850\pi$
$774$	0	0
$775$	2394.00	0.110961
$776$	0	0
$777$	10800.0	0.498646
$778$	0	0
$779$	−7410.00	−0.340810
$780$	0	0
$781$	1728.00	0.0791712
$782$	0	0
$783$	−6534.00	−0.298220
$784$	0	0
$785$	7848.00	0.356824
$786$	0	0
$787$	17900.0	0.810757	0.405379	−	0.914149i	$-0.367140\pi$
$787$	17900.0	0.810757	0.405379	+	0.914149i	$0.367140\pi$
$788$	0	0
$789$	−5982.00	−0.269917
$790$	0	0
$791$	29480.0	1.32514
$792$	0	0
$793$	53656.0	2.40275
$794$	0	0
$795$	−2520.00	−0.112422
$796$	0	0
$797$	31358.0	1.39367	0.696836	−	0.717230i	$-0.254590\pi$
$797$	31358.0	1.39367	0.696836	+	0.717230i	$0.254590\pi$
$798$	0	0
$799$	11484.0	0.508479
$800$	0	0
$801$	7614.00	0.335864
$802$	0	0
$803$	2872.00	0.126215
$804$	0	0
$805$	19680.0	0.861651
$806$	0	0
$807$	−21642.0	−0.944033
$808$	0	0
$809$	−20210.0	−0.878301	−0.439151	−	0.898413i	$-0.644721\pi$
$809$	−20210.0	−0.878301	−0.439151	+	0.898413i	$0.644721\pi$
$810$	0	0
$811$	8648.00	0.374442	0.187221	−	0.982318i	$-0.440052\pi$
$811$	8648.00	0.374442	0.187221	+	0.982318i	$0.440052\pi$
$812$	0	0
$813$	22716.0	0.979932
$814$	0	0
$815$	−10800.0	−0.464181
$816$	0	0
$817$	−5852.00	−0.250594
$818$	0	0
$819$	−13680.0	−0.583660
$820$	0	0
$821$	−8940.00	−0.380034	−0.190017	−	0.981781i	$-0.560854\pi$
$821$	−8940.00	−0.380034	−0.190017	+	0.981781i	$0.560854\pi$
$822$	0	0
$823$	17504.0	0.741374	0.370687	−	0.928758i	$-0.379122\pi$
$823$	17504.0	0.741374	0.370687	+	0.928758i	$0.379122\pi$
$824$	0	0
$825$	−228.000	−0.00962175
$826$	0	0
$827$	−4356.00	−0.183160	−0.0915798	−	0.995798i	$-0.529192\pi$
$827$	−4356.00	−0.183160	−0.0915798	+	0.995798i	$0.529192\pi$
$828$	0	0
$829$	1528.00	0.0640164	0.0320082	−	0.999488i	$-0.489810\pi$
$829$	1528.00	0.0640164	0.0320082	+	0.999488i	$0.489810\pi$
$830$	0	0
$831$	−18786.0	−0.784211
$832$	0	0
$833$	1254.00	0.0521591
$834$	0	0
$835$	8880.00	0.368030
$836$	0	0
$837$	−3402.00	−0.140490
$838$	0	0
$839$	30204.0	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	30204.0	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	34175.0	1.40125
$842$	0	0
$843$	8130.00	0.332161
$844$	0	0
$845$	−42948.0	−1.74847
$846$	0	0
$847$	−26300.0	−1.06692
$848$	0	0
$849$	−1668.00	−0.0674271
$850$	0	0
$851$	14760.0	0.594555
$852$	0	0
$853$	−9218.00	−0.370010	−0.185005	−	0.982738i	$-0.559230\pi$
$853$	−9218.00	−0.370010	−0.185005	+	0.982738i	$0.559230\pi$
$854$	0	0
$855$	−2052.00	−0.0820783
$856$	0	0
$857$	−44554.0	−1.77589	−0.887944	−	0.459952i	$-0.847867\pi$
$857$	−44554.0	−1.77589	−0.887944	+	0.459952i	$0.847867\pi$
$858$	0	0
$859$	−9828.00	−0.390369	−0.195185	−	0.980767i	$-0.562531\pi$
$859$	−9828.00	−0.390369	−0.195185	+	0.980767i	$0.562531\pi$
$860$	0	0
$861$	23400.0	0.926214
$862$	0	0
$863$	11668.0	0.460236	0.230118	−	0.973163i	$-0.426089\pi$
$863$	11668.0	0.460236	0.230118	+	0.973163i	$0.426089\pi$
$864$	0	0
$865$	−6984.00	−0.274524
$866$	0	0
$867$	13287.0	0.520473
$868$	0	0
$869$	−376.000	−0.0146777
$870$	0	0
$871$	7904.00	0.307482
$872$	0	0
$873$	7470.00	0.289600
$874$	0	0
$875$	25440.0	0.982890
$876$	0	0
$877$	17292.0	0.665803	0.332902	−	0.942962i	$-0.391972\pi$
$877$	17292.0	0.665803	0.332902	+	0.942962i	$0.391972\pi$
$878$	0	0
$879$	−11082.0	−0.425241
$880$	0	0
$881$	4618.00	0.176600	0.0882999	−	0.996094i	$-0.471857\pi$
$881$	4618.00	0.176600	0.0882999	+	0.996094i	$0.471857\pi$
$882$	0	0
$883$	17740.0	0.676103	0.338051	−	0.941128i	$-0.390232\pi$
$883$	17740.0	0.676103	0.338051	+	0.941128i	$0.390232\pi$
$884$	0	0
$885$	−6768.00	−0.257066
$886$	0	0
$887$	24516.0	0.928035	0.464017	−	0.885826i	$-0.346408\pi$
$887$	24516.0	0.928035	0.464017	+	0.885826i	$0.346408\pi$
$888$	0	0
$889$	23320.0	0.879784
$890$	0	0
$891$	324.000	0.0121823
$892$	0	0
$893$	9918.00	0.371661
$894$	0	0
$895$	32976.0	1.23158
$896$	0	0
$897$	−18696.0	−0.695921
$898$	0	0
$899$	30492.0	1.13122
$900$	0	0
$901$	−1540.00	−0.0569421
$902$	0	0
$903$	18480.0	0.681036
$904$	0	0
$905$	−16032.0	−0.588864
$906$	0	0
$907$	−8392.00	−0.307224	−0.153612	−	0.988131i	$-0.549091\pi$
$907$	−8392.00	−0.307224	−0.153612	+	0.988131i	$0.549091\pi$
$908$	0	0
$909$	14508.0	0.529373
$910$	0	0
$911$	−17004.0	−0.618406	−0.309203	−	0.950996i	$-0.600062\pi$
$911$	−17004.0	−0.618406	−0.309203	+	0.950996i	$0.600062\pi$
$912$	0	0
$913$	5184.00	0.187914
$914$	0	0
$915$	−25416.0	−0.918281
$916$	0	0
$917$	−43840.0	−1.57876
$918$	0	0
$919$	46288.0	1.66148	0.830740	−	0.556661i	$-0.187918\pi$
$919$	46288.0	1.66148	0.830740	+	0.556661i	$0.187918\pi$
$920$	0	0
$921$	10152.0	0.363214
$922$	0	0
$923$	−32832.0	−1.17083
$924$	0	0
$925$	−3420.00	−0.121566
$926$	0	0
$927$	16866.0	0.597575
$928$	0	0
$929$	−4978.00	−0.175805	−0.0879025	−	0.996129i	$-0.528016\pi$
$929$	−4978.00	−0.175805	−0.0879025	+	0.996129i	$0.528016\pi$
$930$	0	0
$931$	1083.00	0.0381245
$932$	0	0
$933$	−28998.0	−1.01753
$934$	0	0
$935$	−1056.00	−0.0369357
$936$	0	0
$937$	39798.0	1.38756	0.693780	−	0.720187i	$-0.255944\pi$
$937$	39798.0	1.38756	0.693780	+	0.720187i	$0.255944\pi$
$938$	0	0
$939$	20382.0	0.708351
$940$	0	0
$941$	31662.0	1.09687	0.548433	−	0.836194i	$-0.315224\pi$
$941$	31662.0	1.09687	0.548433	+	0.836194i	$0.315224\pi$
$942$	0	0
$943$	31980.0	1.10436
$944$	0	0
$945$	6480.00	0.223063
$946$	0	0
$947$	44744.0	1.53536	0.767679	−	0.640834i	$-0.221411\pi$
$947$	44744.0	1.53536	0.767679	+	0.640834i	$0.221411\pi$
$948$	0	0
$949$	−54568.0	−1.86655
$950$	0	0
$951$	9726.00	0.331637
$952$	0	0
$953$	18626.0	0.633112	0.316556	−	0.948574i	$-0.397474\pi$
$953$	18626.0	0.633112	0.316556	+	0.948574i	$0.397474\pi$
$954$	0	0
$955$	−7272.00	−0.246405
$956$	0	0
$957$	−2904.00	−0.0980909
$958$	0	0
$959$	11160.0	0.375782
$960$	0	0
$961$	−13915.0	−0.467087
$962$	0	0
$963$	17388.0	0.581849
$964$	0	0
$965$	36024.0	1.20171
$966$	0	0
$967$	−30244.0	−1.00577	−0.502886	−	0.864353i	$-0.667728\pi$
$967$	−30244.0	−1.00577	−0.502886	+	0.864353i	$0.667728\pi$
$968$	0	0
$969$	−1254.00	−0.0415730
$970$	0	0
$971$	−46572.0	−1.53920	−0.769602	−	0.638524i	$-0.779545\pi$
$971$	−46572.0	−1.53920	−0.769602	+	0.638524i	$0.779545\pi$
$972$	0	0
$973$	−1360.00	−0.0448095
$974$	0	0
$975$	4332.00	0.142292
$976$	0	0
$977$	30162.0	0.987685	0.493842	−	0.869551i	$-0.335592\pi$
$977$	30162.0	0.987685	0.493842	+	0.869551i	$0.335592\pi$
$978$	0	0
$979$	3384.00	0.110473
$980$	0	0
$981$	9864.00	0.321033
$982$	0	0
$983$	−17792.0	−0.577291	−0.288645	−	0.957436i	$-0.593205\pi$
$983$	−17792.0	−0.577291	−0.288645	+	0.957436i	$0.593205\pi$
$984$	0	0
$985$	53472.0	1.72971
$986$	0	0
$987$	−31320.0	−1.01006
$988$	0	0
$989$	25256.0	0.812026
$990$	0	0
$991$	−9434.00	−0.302403	−0.151201	−	0.988503i	$-0.548314\pi$
$991$	−9434.00	−0.302403	−0.151201	+	0.988503i	$0.548314\pi$
$992$	0	0
$993$	528.000	0.0168737
$994$	0	0
$995$	−34128.0	−1.08737
$996$	0	0
$997$	−61286.0	−1.94679	−0.973394	−	0.229139i	$-0.926409\pi$
$997$	−61286.0	−1.94679	−0.973394	+	0.229139i	$0.926409\pi$
$998$	0	0
$999$	4860.00	0.153918

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.4.a.a.1.1		1
4.3	odd	2		57.4.a.a.1.1	✓	1
12.11	even	2		171.4.a.b.1.1		1
20.19	odd	2		1425.4.a.c.1.1		1
76.75	even	2		1083.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.4.a.a.1.1	✓	1	4.3	odd	2
171.4.a.b.1.1		1	12.11	even	2
912.4.a.a.1.1		1	1.1	even	1	trivial
1083.4.a.a.1.1		1	76.75	even	2
1425.4.a.c.1.1		1	20.19	odd	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

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