LMFDB - Embedded newform 624.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(1,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, 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("624.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.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:	$36.8171918436$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} -16.0000 q^{5} +8.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} -16.0000 q^{5} +8.00000 q^{7} +9.00000 q^{9} +38.0000 q^{11} -13.0000 q^{13} +48.0000 q^{15} -78.0000 q^{17} +72.0000 q^{19} -24.0000 q^{21} +52.0000 q^{23} +131.000 q^{25} -27.0000 q^{27} +242.000 q^{29} -76.0000 q^{31} -114.000 q^{33} -128.000 q^{35} +342.000 q^{37} +39.0000 q^{39} -336.000 q^{41} -76.0000 q^{43} -144.000 q^{45} -94.0000 q^{47} -279.000 q^{49} +234.000 q^{51} -450.000 q^{53} -608.000 q^{55} -216.000 q^{57} -854.000 q^{59} -110.000 q^{61} +72.0000 q^{63} +208.000 q^{65} +908.000 q^{67} -156.000 q^{69} -838.000 q^{71} -970.000 q^{73} -393.000 q^{75} +304.000 q^{77} +352.000 q^{79} +81.0000 q^{81} -474.000 q^{83} +1248.00 q^{85} -726.000 q^{87} -1452.00 q^{89} -104.000 q^{91} +228.000 q^{93} -1152.00 q^{95} -562.000 q^{97} +342.000 q^{99} +O(q^{100})$

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
$5$	−16.0000	−1.43108	−0.715542	−	0.698570i	$-0.753820\pi$
$5$	−16.0000	−1.43108	−0.715542	+	0.698570i	$0.753820\pi$
$6$	0	0
$7$	8.00000	0.431959	0.215980	−	0.976398i	$-0.430705\pi$
$7$	8.00000	0.431959	0.215980	+	0.976398i	$0.430705\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	38.0000	1.04158	0.520792	−	0.853683i	$-0.325637\pi$
$11$	38.0000	1.04158	0.520792	+	0.853683i	$0.325637\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	48.0000	0.826236
$16$	0	0
$17$	−78.0000	−1.11281	−0.556405	−	0.830911i	$-0.687820\pi$
$17$	−78.0000	−1.11281	−0.556405	+	0.830911i	$0.687820\pi$
$18$	0	0
$19$	72.0000	0.869365	0.434682	−	0.900584i	$-0.356861\pi$
$19$	72.0000	0.869365	0.434682	+	0.900584i	$0.356861\pi$
$20$	0	0
$21$	−24.0000	−0.249392
$22$	0	0
$23$	52.0000	0.471424	0.235712	−	0.971823i	$-0.424258\pi$
$23$	52.0000	0.471424	0.235712	+	0.971823i	$0.424258\pi$
$24$	0	0
$25$	131.000	1.04800
$26$	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
$31$	−76.0000	−0.440323	−0.220161	−	0.975463i	$-0.570658\pi$
$31$	−76.0000	−0.440323	−0.220161	+	0.975463i	$0.570658\pi$
$32$	0	0
$33$	−114.000	−0.601359
$34$	0	0
$35$	−128.000	−0.618170
$36$	0	0
$37$	342.000	1.51958	0.759790	−	0.650169i	$-0.225302\pi$
$37$	342.000	1.51958	0.759790	+	0.650169i	$0.225302\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	−336.000	−1.27986	−0.639932	−	0.768432i	$-0.721037\pi$
$41$	−336.000	−1.27986	−0.639932	+	0.768432i	$0.721037\pi$
$42$	0	0
$43$	−76.0000	−0.269532	−0.134766	−	0.990877i	$-0.543028\pi$
$43$	−76.0000	−0.269532	−0.134766	+	0.990877i	$0.543028\pi$
$44$	0	0
$45$	−144.000	−0.477028
$46$	0	0
$47$	−94.0000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−94.0000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	0	0
$51$	234.000	0.642481
$52$	0	0
$53$	−450.000	−1.16627	−0.583134	−	0.812376i	$-0.698174\pi$
$53$	−450.000	−1.16627	−0.583134	+	0.812376i	$0.698174\pi$
$54$	0	0
$55$	−608.000	−1.49059
$56$	0	0
$57$	−216.000	−0.501928
$58$	0	0
$59$	−854.000	−1.88443	−0.942215	−	0.335010i	$-0.891260\pi$
$59$	−854.000	−1.88443	−0.942215	+	0.335010i	$0.891260\pi$
$60$	0	0
$61$	−110.000	−0.230886	−0.115443	−	0.993314i	$-0.536829\pi$
$61$	−110.000	−0.230886	−0.115443	+	0.993314i	$0.536829\pi$
$62$	0	0
$63$	72.0000	0.143986
$64$	0	0
$65$	208.000	0.396911
$66$	0	0
$67$	908.000	1.65567	0.827835	−	0.560972i	$-0.189572\pi$
$67$	908.000	1.65567	0.827835	+	0.560972i	$0.189572\pi$
$68$	0	0
$69$	−156.000	−0.272177
$70$	0	0
$71$	−838.000	−1.40074	−0.700368	−	0.713782i	$-0.746981\pi$
$71$	−838.000	−1.40074	−0.700368	+	0.713782i	$0.746981\pi$
$72$	0	0
$73$	−970.000	−1.55520	−0.777602	−	0.628757i	$-0.783564\pi$
$73$	−970.000	−1.55520	−0.777602	+	0.628757i	$0.783564\pi$
$74$	0	0
$75$	−393.000	−0.605063
$76$	0	0
$77$	304.000	0.449922
$78$	0	0
$79$	352.000	0.501305	0.250652	−	0.968077i	$-0.419355\pi$
$79$	352.000	0.501305	0.250652	+	0.968077i	$0.419355\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−474.000	−0.626846	−0.313423	−	0.949614i	$-0.601476\pi$
$83$	−474.000	−0.626846	−0.313423	+	0.949614i	$0.601476\pi$
$84$	0	0
$85$	1248.00	1.59252
$86$	0	0
$87$	−726.000	−0.894659
$88$	0	0
$89$	−1452.00	−1.72934	−0.864672	−	0.502336i	$-0.832474\pi$
$89$	−1452.00	−1.72934	−0.864672	+	0.502336i	$0.832474\pi$
$90$	0	0
$91$	−104.000	−0.119804
$92$	0	0
$93$	228.000	0.254220
$94$	0	0
$95$	−1152.00	−1.24413
$96$	0	0
$97$	−562.000	−0.588273	−0.294136	−	0.955763i	$-0.595032\pi$
$97$	−562.000	−0.588273	−0.294136	+	0.955763i	$0.595032\pi$
$98$	0	0
$99$	342.000	0.347195
$100$	0	0
$101$	466.000	0.459096	0.229548	−	0.973297i	$-0.426275\pi$
$101$	466.000	0.459096	0.229548	+	0.973297i	$0.426275\pi$
$102$	0	0
$103$	1448.00	1.38520	0.692600	−	0.721321i	$-0.256465\pi$
$103$	1448.00	1.38520	0.692600	+	0.721321i	$0.256465\pi$
$104$	0	0
$105$	384.000	0.356901
$106$	0	0
$107$	424.000	0.383081	0.191540	−	0.981485i	$-0.438652\pi$
$107$	424.000	0.383081	0.191540	+	0.981485i	$0.438652\pi$
$108$	0	0
$109$	−782.000	−0.687174	−0.343587	−	0.939121i	$-0.611642\pi$
$109$	−782.000	−0.687174	−0.343587	+	0.939121i	$0.611642\pi$
$110$	0	0
$111$	−1026.00	−0.877330
$112$	0	0
$113$	−634.000	−0.527803	−0.263901	−	0.964550i	$-0.585009\pi$
$113$	−634.000	−0.527803	−0.263901	+	0.964550i	$0.585009\pi$
$114$	0	0
$115$	−832.000	−0.674647
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	−624.000	−0.480689
$120$	0	0
$121$	113.000	0.0848986
$122$	0	0
$123$	1008.00	0.738929
$124$	0	0
$125$	−96.0000	−0.0686920
$126$	0	0
$127$	−256.000	−0.178869	−0.0894344	−	0.995993i	$-0.528506\pi$
$127$	−256.000	−0.178869	−0.0894344	+	0.995993i	$0.528506\pi$
$128$	0	0
$129$	228.000	0.155615
$130$	0	0
$131$	1360.00	0.907052	0.453526	−	0.891243i	$-0.350166\pi$
$131$	1360.00	0.907052	0.453526	+	0.891243i	$0.350166\pi$
$132$	0	0
$133$	576.000	0.375530
$134$	0	0
$135$	432.000	0.275412
$136$	0	0
$137$	2976.00	1.85589	0.927945	−	0.372718i	$-0.121574\pi$
$137$	2976.00	1.85589	0.927945	+	0.372718i	$0.121574\pi$
$138$	0	0
$139$	−2764.00	−1.68661	−0.843307	−	0.537432i	$-0.819395\pi$
$139$	−2764.00	−1.68661	−0.843307	+	0.537432i	$0.819395\pi$
$140$	0	0
$141$	282.000	0.168430
$142$	0	0
$143$	−494.000	−0.288884
$144$	0	0
$145$	−3872.00	−2.21760
$146$	0	0
$147$	837.000	0.469623
$148$	0	0
$149$	−2940.00	−1.61647	−0.808236	−	0.588859i	$-0.799577\pi$
$149$	−2940.00	−1.61647	−0.808236	+	0.588859i	$0.799577\pi$
$150$	0	0
$151$	−1188.00	−0.640252	−0.320126	−	0.947375i	$-0.603725\pi$
$151$	−1188.00	−0.640252	−0.320126	+	0.947375i	$0.603725\pi$
$152$	0	0
$153$	−702.000	−0.370937
$154$	0	0
$155$	1216.00	0.630139
$156$	0	0
$157$	−2410.00	−1.22509	−0.612544	−	0.790436i	$-0.709854\pi$
$157$	−2410.00	−1.22509	−0.612544	+	0.790436i	$0.709854\pi$
$158$	0	0
$159$	1350.00	0.673346
$160$	0	0
$161$	416.000	0.203636
$162$	0	0
$163$	2248.00	1.08023	0.540113	−	0.841592i	$-0.318381\pi$
$163$	2248.00	1.08023	0.540113	+	0.841592i	$0.318381\pi$
$164$	0	0
$165$	1824.00	0.860595
$166$	0	0
$167$	1530.00	0.708952	0.354476	−	0.935065i	$-0.384659\pi$
$167$	1530.00	0.708952	0.354476	+	0.935065i	$0.384659\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	648.000	0.289788
$172$	0	0
$173$	−1030.00	−0.452656	−0.226328	−	0.974051i	$-0.572672\pi$
$173$	−1030.00	−0.452656	−0.226328	+	0.974051i	$0.572672\pi$
$174$	0	0
$175$	1048.00	0.452693
$176$	0	0
$177$	2562.00	1.08798
$178$	0	0
$179$	−1380.00	−0.576235	−0.288117	−	0.957595i	$-0.593029\pi$
$179$	−1380.00	−0.576235	−0.288117	+	0.957595i	$0.593029\pi$
$180$	0	0
$181$	2286.00	0.938768	0.469384	−	0.882994i	$-0.344476\pi$
$181$	2286.00	0.938768	0.469384	+	0.882994i	$0.344476\pi$
$182$	0	0
$183$	330.000	0.133302
$184$	0	0
$185$	−5472.00	−2.17465
$186$	0	0
$187$	−2964.00	−1.15909
$188$	0	0
$189$	−216.000	−0.0831306
$190$	0	0
$191$	−4720.00	−1.78810	−0.894050	−	0.447967i	$-0.852148\pi$
$191$	−4720.00	−1.78810	−0.894050	+	0.447967i	$0.852148\pi$
$192$	0	0
$193$	2042.00	0.761587	0.380794	−	0.924660i	$-0.375651\pi$
$193$	2042.00	0.761587	0.380794	+	0.924660i	$0.375651\pi$
$194$	0	0
$195$	−624.000	−0.229157
$196$	0	0
$197$	−1512.00	−0.546830	−0.273415	−	0.961896i	$-0.588153\pi$
$197$	−1512.00	−0.546830	−0.273415	+	0.961896i	$0.588153\pi$
$198$	0	0
$199$	2224.00	0.792237	0.396119	−	0.918199i	$-0.370357\pi$
$199$	2224.00	0.792237	0.396119	+	0.918199i	$0.370357\pi$
$200$	0	0
$201$	−2724.00	−0.955901
$202$	0	0
$203$	1936.00	0.669362
$204$	0	0
$205$	5376.00	1.83159
$206$	0	0
$207$	468.000	0.157141
$208$	0	0
$209$	2736.00	0.905517
$210$	0	0
$211$	−4652.00	−1.51781	−0.758903	−	0.651204i	$-0.774264\pi$
$211$	−4652.00	−1.51781	−0.758903	+	0.651204i	$0.774264\pi$
$212$	0	0
$213$	2514.00	0.808716
$214$	0	0
$215$	1216.00	0.385723
$216$	0	0
$217$	−608.000	−0.190202
$218$	0	0
$219$	2910.00	0.897898
$220$	0	0
$221$	1014.00	0.308638
$222$	0	0
$223$	1812.00	0.544128	0.272064	−	0.962279i	$-0.412294\pi$
$223$	1812.00	0.544128	0.272064	+	0.962279i	$0.412294\pi$
$224$	0	0
$225$	1179.00	0.349333
$226$	0	0
$227$	−126.000	−0.0368410	−0.0184205	−	0.999830i	$-0.505864\pi$
$227$	−126.000	−0.0368410	−0.0184205	+	0.999830i	$0.505864\pi$
$228$	0	0
$229$	3186.00	0.919375	0.459687	−	0.888081i	$-0.347961\pi$
$229$	3186.00	0.919375	0.459687	+	0.888081i	$0.347961\pi$
$230$	0	0
$231$	−912.000	−0.259763
$232$	0	0
$233$	−2378.00	−0.668618	−0.334309	−	0.942464i	$-0.608503\pi$
$233$	−2378.00	−0.668618	−0.334309	+	0.942464i	$0.608503\pi$
$234$	0	0
$235$	1504.00	0.417490
$236$	0	0
$237$	−1056.00	−0.289429
$238$	0	0
$239$	1338.00	0.362126	0.181063	−	0.983472i	$-0.442046\pi$
$239$	1338.00	0.362126	0.181063	+	0.983472i	$0.442046\pi$
$240$	0	0
$241$	6870.00	1.83625	0.918124	−	0.396294i	$-0.129704\pi$
$241$	6870.00	1.83625	0.918124	+	0.396294i	$0.129704\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	4464.00	1.16406
$246$	0	0
$247$	−936.000	−0.241118
$248$	0	0
$249$	1422.00	0.361910
$250$	0	0
$251$	−6768.00	−1.70196	−0.850981	−	0.525197i	$-0.823992\pi$
$251$	−6768.00	−1.70196	−0.850981	+	0.525197i	$0.823992\pi$
$252$	0	0
$253$	1976.00	0.491028
$254$	0	0
$255$	−3744.00	−0.919445
$256$	0	0
$257$	−3546.00	−0.860675	−0.430337	−	0.902668i	$-0.641605\pi$
$257$	−3546.00	−0.860675	−0.430337	+	0.902668i	$0.641605\pi$
$258$	0	0
$259$	2736.00	0.656397
$260$	0	0
$261$	2178.00	0.516532
$262$	0	0
$263$	5340.00	1.25201	0.626005	−	0.779819i	$-0.284689\pi$
$263$	5340.00	1.25201	0.626005	+	0.779819i	$0.284689\pi$
$264$	0	0
$265$	7200.00	1.66903
$266$	0	0
$267$	4356.00	0.998438
$268$	0	0
$269$	−3486.00	−0.790131	−0.395065	−	0.918653i	$-0.629278\pi$
$269$	−3486.00	−0.790131	−0.395065	+	0.918653i	$0.629278\pi$
$270$	0	0
$271$	−256.000	−0.0573834	−0.0286917	−	0.999588i	$-0.509134\pi$
$271$	−256.000	−0.0573834	−0.0286917	+	0.999588i	$0.509134\pi$
$272$	0	0
$273$	312.000	0.0691689
$274$	0	0
$275$	4978.00	1.09158
$276$	0	0
$277$	−3354.00	−0.727517	−0.363759	−	0.931493i	$-0.618507\pi$
$277$	−3354.00	−0.727517	−0.363759	+	0.931493i	$0.618507\pi$
$278$	0	0
$279$	−684.000	−0.146774
$280$	0	0
$281$	−6608.00	−1.40285	−0.701424	−	0.712744i	$-0.747452\pi$
$281$	−6608.00	−1.40285	−0.701424	+	0.712744i	$0.747452\pi$
$282$	0	0
$283$	1148.00	0.241136	0.120568	−	0.992705i	$-0.461528\pi$
$283$	1148.00	0.241136	0.120568	+	0.992705i	$0.461528\pi$
$284$	0	0
$285$	3456.00	0.718301
$286$	0	0
$287$	−2688.00	−0.552849
$288$	0	0
$289$	1171.00	0.238347
$290$	0	0
$291$	1686.00	0.339639
$292$	0	0
$293$	1972.00	0.393193	0.196596	−	0.980485i	$-0.437011\pi$
$293$	1972.00	0.393193	0.196596	+	0.980485i	$0.437011\pi$
$294$	0	0
$295$	13664.0	2.69678
$296$	0	0
$297$	−1026.00	−0.200453
$298$	0	0
$299$	−676.000	−0.130749
$300$	0	0
$301$	−608.000	−0.116427
$302$	0	0
$303$	−1398.00	−0.265059
$304$	0	0
$305$	1760.00	0.330417
$306$	0	0
$307$	7876.00	1.46419	0.732096	−	0.681201i	$-0.238542\pi$
$307$	7876.00	1.46419	0.732096	+	0.681201i	$0.238542\pi$
$308$	0	0
$309$	−4344.00	−0.799746
$310$	0	0
$311$	6852.00	1.24933	0.624664	−	0.780893i	$-0.285236\pi$
$311$	6852.00	1.24933	0.624664	+	0.780893i	$0.285236\pi$
$312$	0	0
$313$	−4714.00	−0.851281	−0.425641	−	0.904892i	$-0.639951\pi$
$313$	−4714.00	−0.851281	−0.425641	+	0.904892i	$0.639951\pi$
$314$	0	0
$315$	−1152.00	−0.206057
$316$	0	0
$317$	−480.000	−0.0850457	−0.0425228	−	0.999095i	$-0.513540\pi$
$317$	−480.000	−0.0850457	−0.0425228	+	0.999095i	$0.513540\pi$
$318$	0	0
$319$	9196.00	1.61403
$320$	0	0
$321$	−1272.00	−0.221172
$322$	0	0
$323$	−5616.00	−0.967438
$324$	0	0
$325$	−1703.00	−0.290663
$326$	0	0
$327$	2346.00	0.396740
$328$	0	0
$329$	−752.000	−0.126016
$330$	0	0
$331$	−7628.00	−1.26669	−0.633343	−	0.773872i	$-0.718318\pi$
$331$	−7628.00	−1.26669	−0.633343	+	0.773872i	$0.718318\pi$
$332$	0	0
$333$	3078.00	0.506527
$334$	0	0
$335$	−14528.0	−2.36940
$336$	0	0
$337$	−9346.00	−1.51071	−0.755355	−	0.655316i	$-0.772535\pi$
$337$	−9346.00	−1.51071	−0.755355	+	0.655316i	$0.772535\pi$
$338$	0	0
$339$	1902.00	0.304727
$340$	0	0
$341$	−2888.00	−0.458633
$342$	0	0
$343$	−4976.00	−0.783320
$344$	0	0
$345$	2496.00	0.389508
$346$	0	0
$347$	492.000	0.0761151	0.0380576	−	0.999276i	$-0.487883\pi$
$347$	492.000	0.0761151	0.0380576	+	0.999276i	$0.487883\pi$
$348$	0	0
$349$	358.000	0.0549092	0.0274546	−	0.999623i	$-0.491260\pi$
$349$	358.000	0.0549092	0.0274546	+	0.999623i	$0.491260\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	1648.00	0.248482	0.124241	−	0.992252i	$-0.460350\pi$
$353$	1648.00	0.248482	0.124241	+	0.992252i	$0.460350\pi$
$354$	0	0
$355$	13408.0	2.00457
$356$	0	0
$357$	1872.00	0.277526
$358$	0	0
$359$	−9750.00	−1.43339	−0.716693	−	0.697389i	$-0.754345\pi$
$359$	−9750.00	−1.43339	−0.716693	+	0.697389i	$0.754345\pi$
$360$	0	0
$361$	−1675.00	−0.244205
$362$	0	0
$363$	−339.000	−0.0490162
$364$	0	0
$365$	15520.0	2.22563
$366$	0	0
$367$	−10856.0	−1.54408	−0.772042	−	0.635572i	$-0.780764\pi$
$367$	−10856.0	−1.54408	−0.772042	+	0.635572i	$0.780764\pi$
$368$	0	0
$369$	−3024.00	−0.426621
$370$	0	0
$371$	−3600.00	−0.503781
$372$	0	0
$373$	1826.00	0.253476	0.126738	−	0.991936i	$-0.459549\pi$
$373$	1826.00	0.253476	0.126738	+	0.991936i	$0.459549\pi$
$374$	0	0
$375$	288.000	0.0396593
$376$	0	0
$377$	−3146.00	−0.429780
$378$	0	0
$379$	896.000	0.121436	0.0607182	−	0.998155i	$-0.480661\pi$
$379$	896.000	0.121436	0.0607182	+	0.998155i	$0.480661\pi$
$380$	0	0
$381$	768.000	0.103270
$382$	0	0
$383$	−2826.00	−0.377028	−0.188514	−	0.982070i	$-0.560367\pi$
$383$	−2826.00	−0.377028	−0.188514	+	0.982070i	$0.560367\pi$
$384$	0	0
$385$	−4864.00	−0.643876
$386$	0	0
$387$	−684.000	−0.0898441
$388$	0	0
$389$	9846.00	1.28332	0.641661	−	0.766989i	$-0.278246\pi$
$389$	9846.00	1.28332	0.641661	+	0.766989i	$0.278246\pi$
$390$	0	0
$391$	−4056.00	−0.524605
$392$	0	0
$393$	−4080.00	−0.523686
$394$	0	0
$395$	−5632.00	−0.717409
$396$	0	0
$397$	8678.00	1.09707	0.548534	−	0.836128i	$-0.315186\pi$
$397$	8678.00	1.09707	0.548534	+	0.836128i	$0.315186\pi$
$398$	0	0
$399$	−1728.00	−0.216813
$400$	0	0
$401$	−9948.00	−1.23885	−0.619426	−	0.785055i	$-0.712634\pi$
$401$	−9948.00	−1.23885	−0.619426	+	0.785055i	$0.712634\pi$
$402$	0	0
$403$	988.000	0.122124
$404$	0	0
$405$	−1296.00	−0.159009
$406$	0	0
$407$	12996.0	1.58277
$408$	0	0
$409$	−98.0000	−0.0118479	−0.00592395	−	0.999982i	$-0.501886\pi$
$409$	−98.0000	−0.0118479	−0.00592395	+	0.999982i	$0.501886\pi$
$410$	0	0
$411$	−8928.00	−1.07150
$412$	0	0
$413$	−6832.00	−0.813997
$414$	0	0
$415$	7584.00	0.897070
$416$	0	0
$417$	8292.00	0.973767
$418$	0	0
$419$	3216.00	0.374969	0.187484	−	0.982268i	$-0.439967\pi$
$419$	3216.00	0.374969	0.187484	+	0.982268i	$0.439967\pi$
$420$	0	0
$421$	4738.00	0.548494	0.274247	−	0.961659i	$-0.411571\pi$
$421$	4738.00	0.548494	0.274247	+	0.961659i	$0.411571\pi$
$422$	0	0
$423$	−846.000	−0.0972433
$424$	0	0
$425$	−10218.0	−1.16623
$426$	0	0
$427$	−880.000	−0.0997335
$428$	0	0
$429$	1482.00	0.166787
$430$	0	0
$431$	2598.00	0.290351	0.145175	−	0.989406i	$-0.453625\pi$
$431$	2598.00	0.290351	0.145175	+	0.989406i	$0.453625\pi$
$432$	0	0
$433$	−7490.00	−0.831285	−0.415643	−	0.909528i	$-0.636443\pi$
$433$	−7490.00	−0.831285	−0.415643	+	0.909528i	$0.636443\pi$
$434$	0	0
$435$	11616.0	1.28033
$436$	0	0
$437$	3744.00	0.409839
$438$	0	0
$439$	17632.0	1.91692	0.958462	−	0.285221i	$-0.0920670\pi$
$439$	17632.0	1.91692	0.958462	+	0.285221i	$0.0920670\pi$
$440$	0	0
$441$	−2511.00	−0.271137
$442$	0	0
$443$	−9696.00	−1.03989	−0.519945	−	0.854200i	$-0.674047\pi$
$443$	−9696.00	−1.03989	−0.519945	+	0.854200i	$0.674047\pi$
$444$	0	0
$445$	23232.0	2.47484
$446$	0	0
$447$	8820.00	0.933270
$448$	0	0
$449$	−4436.00	−0.466253	−0.233127	−	0.972446i	$-0.574896\pi$
$449$	−4436.00	−0.466253	−0.233127	+	0.972446i	$0.574896\pi$
$450$	0	0
$451$	−12768.0	−1.33309
$452$	0	0
$453$	3564.00	0.369650
$454$	0	0
$455$	1664.00	0.171450
$456$	0	0
$457$	12862.0	1.31654	0.658270	−	0.752782i	$-0.271288\pi$
$457$	12862.0	1.31654	0.658270	+	0.752782i	$0.271288\pi$
$458$	0	0
$459$	2106.00	0.214160
$460$	0	0
$461$	9816.00	0.991707	0.495853	−	0.868406i	$-0.334855\pi$
$461$	9816.00	0.991707	0.495853	+	0.868406i	$0.334855\pi$
$462$	0	0
$463$	−10408.0	−1.04471	−0.522355	−	0.852728i	$-0.674946\pi$
$463$	−10408.0	−1.04471	−0.522355	+	0.852728i	$0.674946\pi$
$464$	0	0
$465$	−3648.00	−0.363811
$466$	0	0
$467$	−10472.0	−1.03766	−0.518829	−	0.854878i	$-0.673632\pi$
$467$	−10472.0	−1.03766	−0.518829	+	0.854878i	$0.673632\pi$
$468$	0	0
$469$	7264.00	0.715182
$470$	0	0
$471$	7230.00	0.707305
$472$	0	0
$473$	−2888.00	−0.280741
$474$	0	0
$475$	9432.00	0.911094
$476$	0	0
$477$	−4050.00	−0.388756
$478$	0	0
$479$	13398.0	1.27802	0.639009	−	0.769200i	$-0.279345\pi$
$479$	13398.0	1.27802	0.639009	+	0.769200i	$0.279345\pi$
$480$	0	0
$481$	−4446.00	−0.421456
$482$	0	0
$483$	−1248.00	−0.117569
$484$	0	0
$485$	8992.00	0.841867
$486$	0	0
$487$	−14780.0	−1.37525	−0.687624	−	0.726067i	$-0.741346\pi$
$487$	−14780.0	−1.37525	−0.687624	+	0.726067i	$0.741346\pi$
$488$	0	0
$489$	−6744.00	−0.623669
$490$	0	0
$491$	12632.0	1.16105	0.580524	−	0.814243i	$-0.302848\pi$
$491$	12632.0	1.16105	0.580524	+	0.814243i	$0.302848\pi$
$492$	0	0
$493$	−18876.0	−1.72441
$494$	0	0
$495$	−5472.00	−0.496865
$496$	0	0
$497$	−6704.00	−0.605061
$498$	0	0
$499$	−17260.0	−1.54842	−0.774212	−	0.632926i	$-0.781854\pi$
$499$	−17260.0	−1.54842	−0.774212	+	0.632926i	$0.781854\pi$
$500$	0	0
$501$	−4590.00	−0.409314
$502$	0	0
$503$	−76.0000	−0.00673692	−0.00336846	−	0.999994i	$-0.501072\pi$
$503$	−76.0000	−0.00673692	−0.00336846	+	0.999994i	$0.501072\pi$
$504$	0	0
$505$	−7456.00	−0.657005
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	−11144.0	−0.970430	−0.485215	−	0.874395i	$-0.661259\pi$
$509$	−11144.0	−0.970430	−0.485215	+	0.874395i	$0.661259\pi$
$510$	0	0
$511$	−7760.00	−0.671785
$512$	0	0
$513$	−1944.00	−0.167309
$514$	0	0
$515$	−23168.0	−1.98234
$516$	0	0
$517$	−3572.00	−0.303861
$518$	0	0
$519$	3090.00	0.261341
$520$	0	0
$521$	−4242.00	−0.356709	−0.178355	−	0.983966i	$-0.557077\pi$
$521$	−4242.00	−0.356709	−0.178355	+	0.983966i	$0.557077\pi$
$522$	0	0
$523$	9564.00	0.799626	0.399813	−	0.916597i	$-0.369075\pi$
$523$	9564.00	0.799626	0.399813	+	0.916597i	$0.369075\pi$
$524$	0	0
$525$	−3144.00	−0.261363
$526$	0	0
$527$	5928.00	0.489996
$528$	0	0
$529$	−9463.00	−0.777760
$530$	0	0
$531$	−7686.00	−0.628143
$532$	0	0
$533$	4368.00	0.354970
$534$	0	0
$535$	−6784.00	−0.548220
$536$	0	0
$537$	4140.00	0.332689
$538$	0	0
$539$	−10602.0	−0.847236
$540$	0	0
$541$	−16078.0	−1.27772	−0.638861	−	0.769322i	$-0.720594\pi$
$541$	−16078.0	−1.27772	−0.638861	+	0.769322i	$0.720594\pi$
$542$	0	0
$543$	−6858.00	−0.541998
$544$	0	0
$545$	12512.0	0.983404
$546$	0	0
$547$	−6292.00	−0.491822	−0.245911	−	0.969292i	$-0.579087\pi$
$547$	−6292.00	−0.491822	−0.245911	+	0.969292i	$0.579087\pi$
$548$	0	0
$549$	−990.000	−0.0769621
$550$	0	0
$551$	17424.0	1.34716
$552$	0	0
$553$	2816.00	0.216543
$554$	0	0
$555$	16416.0	1.25553
$556$	0	0
$557$	−3588.00	−0.272942	−0.136471	−	0.990644i	$-0.543576\pi$
$557$	−3588.00	−0.272942	−0.136471	+	0.990644i	$0.543576\pi$
$558$	0	0
$559$	988.000	0.0747548
$560$	0	0
$561$	8892.00	0.669199
$562$	0	0
$563$	5932.00	0.444057	0.222028	−	0.975040i	$-0.428732\pi$
$563$	5932.00	0.444057	0.222028	+	0.975040i	$0.428732\pi$
$564$	0	0
$565$	10144.0	0.755330
$566$	0	0
$567$	648.000	0.0479955
$568$	0	0
$569$	−1178.00	−0.0867914	−0.0433957	−	0.999058i	$-0.513818\pi$
$569$	−1178.00	−0.0867914	−0.0433957	+	0.999058i	$0.513818\pi$
$570$	0	0
$571$	−18444.0	−1.35176	−0.675882	−	0.737010i	$-0.736237\pi$
$571$	−18444.0	−1.35176	−0.675882	+	0.737010i	$0.736237\pi$
$572$	0	0
$573$	14160.0	1.03236
$574$	0	0
$575$	6812.00	0.494052
$576$	0	0
$577$	−2382.00	−0.171861	−0.0859306	−	0.996301i	$-0.527386\pi$
$577$	−2382.00	−0.171861	−0.0859306	+	0.996301i	$0.527386\pi$
$578$	0	0
$579$	−6126.00	−0.439703
$580$	0	0
$581$	−3792.00	−0.270772
$582$	0	0
$583$	−17100.0	−1.21477
$584$	0	0
$585$	1872.00	0.132304
$586$	0	0
$587$	−15698.0	−1.10379	−0.551896	−	0.833913i	$-0.686095\pi$
$587$	−15698.0	−1.10379	−0.551896	+	0.833913i	$0.686095\pi$
$588$	0	0
$589$	−5472.00	−0.382801
$590$	0	0
$591$	4536.00	0.315713
$592$	0	0
$593$	−8452.00	−0.585299	−0.292649	−	0.956220i	$-0.594537\pi$
$593$	−8452.00	−0.585299	−0.292649	+	0.956220i	$0.594537\pi$
$594$	0	0
$595$	9984.00	0.687906
$596$	0	0
$597$	−6672.00	−0.457398
$598$	0	0
$599$	−5836.00	−0.398084	−0.199042	−	0.979991i	$-0.563783\pi$
$599$	−5836.00	−0.398084	−0.199042	+	0.979991i	$0.563783\pi$
$600$	0	0
$601$	−25850.0	−1.75448	−0.877241	−	0.480051i	$-0.840618\pi$
$601$	−25850.0	−1.75448	−0.877241	+	0.480051i	$0.840618\pi$
$602$	0	0
$603$	8172.00	0.551890
$604$	0	0
$605$	−1808.00	−0.121497
$606$	0	0
$607$	−21624.0	−1.44595	−0.722975	−	0.690875i	$-0.757226\pi$
$607$	−21624.0	−1.44595	−0.722975	+	0.690875i	$0.757226\pi$
$608$	0	0
$609$	−5808.00	−0.386457
$610$	0	0
$611$	1222.00	0.0809113
$612$	0	0
$613$	3902.00	0.257097	0.128548	−	0.991703i	$-0.458968\pi$
$613$	3902.00	0.257097	0.128548	+	0.991703i	$0.458968\pi$
$614$	0	0
$615$	−16128.0	−1.05747
$616$	0	0
$617$	16888.0	1.10192	0.550961	−	0.834531i	$-0.314261\pi$
$617$	16888.0	1.10192	0.550961	+	0.834531i	$0.314261\pi$
$618$	0	0
$619$	27452.0	1.78253	0.891267	−	0.453478i	$-0.149817\pi$
$619$	27452.0	1.78253	0.891267	+	0.453478i	$0.149817\pi$
$620$	0	0
$621$	−1404.00	−0.0907256
$622$	0	0
$623$	−11616.0	−0.747007
$624$	0	0
$625$	−14839.0	−0.949696
$626$	0	0
$627$	−8208.00	−0.522801
$628$	0	0
$629$	−26676.0	−1.69100
$630$	0	0
$631$	−5548.00	−0.350020	−0.175010	−	0.984567i	$-0.555996\pi$
$631$	−5548.00	−0.350020	−0.175010	+	0.984567i	$0.555996\pi$
$632$	0	0
$633$	13956.0	0.876305
$634$	0	0
$635$	4096.00	0.255976
$636$	0	0
$637$	3627.00	0.225600
$638$	0	0
$639$	−7542.00	−0.466912
$640$	0	0
$641$	1618.00	0.0996992	0.0498496	−	0.998757i	$-0.484126\pi$
$641$	1618.00	0.0996992	0.0498496	+	0.998757i	$0.484126\pi$
$642$	0	0
$643$	19900.0	1.22050	0.610248	−	0.792210i	$-0.291070\pi$
$643$	19900.0	1.22050	0.610248	+	0.792210i	$0.291070\pi$
$644$	0	0
$645$	−3648.00	−0.222697
$646$	0	0
$647$	−18832.0	−1.14430	−0.572150	−	0.820149i	$-0.693891\pi$
$647$	−18832.0	−1.14430	−0.572150	+	0.820149i	$0.693891\pi$
$648$	0	0
$649$	−32452.0	−1.96279
$650$	0	0
$651$	1824.00	0.109813
$652$	0	0
$653$	−4542.00	−0.272193	−0.136097	−	0.990696i	$-0.543456\pi$
$653$	−4542.00	−0.272193	−0.136097	+	0.990696i	$0.543456\pi$
$654$	0	0
$655$	−21760.0	−1.29807
$656$	0	0
$657$	−8730.00	−0.518401
$658$	0	0
$659$	−8820.00	−0.521363	−0.260682	−	0.965425i	$-0.583947\pi$
$659$	−8820.00	−0.521363	−0.260682	+	0.965425i	$0.583947\pi$
$660$	0	0
$661$	21014.0	1.23654	0.618268	−	0.785968i	$-0.287835\pi$
$661$	21014.0	1.23654	0.618268	+	0.785968i	$0.287835\pi$
$662$	0	0
$663$	−3042.00	−0.178192
$664$	0	0
$665$	−9216.00	−0.537415
$666$	0	0
$667$	12584.0	0.730516
$668$	0	0
$669$	−5436.00	−0.314152
$670$	0	0
$671$	−4180.00	−0.240487
$672$	0	0
$673$	1714.00	0.0981721	0.0490861	−	0.998795i	$-0.484369\pi$
$673$	1714.00	0.0981721	0.0490861	+	0.998795i	$0.484369\pi$
$674$	0	0
$675$	−3537.00	−0.201688
$676$	0	0
$677$	−15114.0	−0.858018	−0.429009	−	0.903300i	$-0.641137\pi$
$677$	−15114.0	−0.858018	−0.429009	+	0.903300i	$0.641137\pi$
$678$	0	0
$679$	−4496.00	−0.254110
$680$	0	0
$681$	378.000	0.0212702
$682$	0	0
$683$	20486.0	1.14769	0.573847	−	0.818963i	$-0.305450\pi$
$683$	20486.0	1.14769	0.573847	+	0.818963i	$0.305450\pi$
$684$	0	0
$685$	−47616.0	−2.65593
$686$	0	0
$687$	−9558.00	−0.530801
$688$	0	0
$689$	5850.00	0.323465
$690$	0	0
$691$	8948.00	0.492616	0.246308	−	0.969192i	$-0.420782\pi$
$691$	8948.00	0.492616	0.246308	+	0.969192i	$0.420782\pi$
$692$	0	0
$693$	2736.00	0.149974
$694$	0	0
$695$	44224.0	2.41369
$696$	0	0
$697$	26208.0	1.42425
$698$	0	0
$699$	7134.00	0.386027
$700$	0	0
$701$	1350.00	0.0727372	0.0363686	−	0.999338i	$-0.488421\pi$
$701$	1350.00	0.0727372	0.0363686	+	0.999338i	$0.488421\pi$
$702$	0	0
$703$	24624.0	1.32107
$704$	0	0
$705$	−4512.00	−0.241038
$706$	0	0
$707$	3728.00	0.198311
$708$	0	0
$709$	19802.0	1.04891	0.524457	−	0.851437i	$-0.324268\pi$
$709$	19802.0	1.04891	0.524457	+	0.851437i	$0.324268\pi$
$710$	0	0
$711$	3168.00	0.167102
$712$	0	0
$713$	−3952.00	−0.207579
$714$	0	0
$715$	7904.00	0.413417
$716$	0	0
$717$	−4014.00	−0.209073
$718$	0	0
$719$	28204.0	1.46291	0.731455	−	0.681890i	$-0.238842\pi$
$719$	28204.0	1.46291	0.731455	+	0.681890i	$0.238842\pi$
$720$	0	0
$721$	11584.0	0.598350
$722$	0	0
$723$	−20610.0	−1.06016
$724$	0	0
$725$	31702.0	1.62398
$726$	0	0
$727$	−20992.0	−1.07091	−0.535454	−	0.844564i	$-0.679859\pi$
$727$	−20992.0	−1.07091	−0.535454	+	0.844564i	$0.679859\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	5928.00	0.299938
$732$	0	0
$733$	−19894.0	−1.00246	−0.501229	−	0.865315i	$-0.667119\pi$
$733$	−19894.0	−1.00246	−0.501229	+	0.865315i	$0.667119\pi$
$734$	0	0
$735$	−13392.0	−0.672070
$736$	0	0
$737$	34504.0	1.72452
$738$	0	0
$739$	−6252.00	−0.311209	−0.155605	−	0.987819i	$-0.549733\pi$
$739$	−6252.00	−0.311209	−0.155605	+	0.987819i	$0.549733\pi$
$740$	0	0
$741$	2808.00	0.139210
$742$	0	0
$743$	30938.0	1.52760	0.763799	−	0.645454i	$-0.223332\pi$
$743$	30938.0	1.52760	0.763799	+	0.645454i	$0.223332\pi$
$744$	0	0
$745$	47040.0	2.31331
$746$	0	0
$747$	−4266.00	−0.208949
$748$	0	0
$749$	3392.00	0.165475
$750$	0	0
$751$	−11328.0	−0.550419	−0.275209	−	0.961384i	$-0.588747\pi$
$751$	−11328.0	−0.550419	−0.275209	+	0.961384i	$0.588747\pi$
$752$	0	0
$753$	20304.0	0.982628
$754$	0	0
$755$	19008.0	0.916254
$756$	0	0
$757$	32754.0	1.57261	0.786304	−	0.617840i	$-0.211992\pi$
$757$	32754.0	1.57261	0.786304	+	0.617840i	$0.211992\pi$
$758$	0	0
$759$	−5928.00	−0.283495
$760$	0	0
$761$	−19776.0	−0.942023	−0.471011	−	0.882127i	$-0.656111\pi$
$761$	−19776.0	−0.942023	−0.471011	+	0.882127i	$0.656111\pi$
$762$	0	0
$763$	−6256.00	−0.296831
$764$	0	0
$765$	11232.0	0.530842
$766$	0	0
$767$	11102.0	0.522647
$768$	0	0
$769$	28362.0	1.32999	0.664993	−	0.746849i	$-0.268434\pi$
$769$	28362.0	1.32999	0.664993	+	0.746849i	$0.268434\pi$
$770$	0	0
$771$	10638.0	0.496911
$772$	0	0
$773$	17108.0	0.796031	0.398016	−	0.917379i	$-0.369699\pi$
$773$	17108.0	0.796031	0.398016	+	0.917379i	$0.369699\pi$
$774$	0	0
$775$	−9956.00	−0.461458
$776$	0	0
$777$	−8208.00	−0.378971
$778$	0	0
$779$	−24192.0	−1.11267
$780$	0	0
$781$	−31844.0	−1.45899
$782$	0	0
$783$	−6534.00	−0.298220
$784$	0	0
$785$	38560.0	1.75320
$786$	0	0
$787$	21364.0	0.967655	0.483827	−	0.875163i	$-0.339246\pi$
$787$	21364.0	0.967655	0.483827	+	0.875163i	$0.339246\pi$
$788$	0	0
$789$	−16020.0	−0.722848
$790$	0	0
$791$	−5072.00	−0.227989
$792$	0	0
$793$	1430.00	0.0640363
$794$	0	0
$795$	−21600.0	−0.963614
$796$	0	0
$797$	28542.0	1.26852	0.634259	−	0.773120i	$-0.281305\pi$
$797$	28542.0	1.26852	0.634259	+	0.773120i	$0.281305\pi$
$798$	0	0
$799$	7332.00	0.324640
$800$	0	0
$801$	−13068.0	−0.576448
$802$	0	0
$803$	−36860.0	−1.61988
$804$	0	0
$805$	−6656.00	−0.291420
$806$	0	0
$807$	10458.0	0.456182
$808$	0	0
$809$	26046.0	1.13193	0.565963	−	0.824430i	$-0.308504\pi$
$809$	26046.0	1.13193	0.565963	+	0.824430i	$0.308504\pi$
$810$	0	0
$811$	15352.0	0.664712	0.332356	−	0.943154i	$-0.392156\pi$
$811$	15352.0	0.664712	0.332356	+	0.943154i	$0.392156\pi$
$812$	0	0
$813$	768.000	0.0331303
$814$	0	0
$815$	−35968.0	−1.54589
$816$	0	0
$817$	−5472.00	−0.234322
$818$	0	0
$819$	−936.000	−0.0399347
$820$	0	0
$821$	−31972.0	−1.35911	−0.679556	−	0.733624i	$-0.737827\pi$
$821$	−31972.0	−1.35911	−0.679556	+	0.733624i	$0.737827\pi$
$822$	0	0
$823$	32208.0	1.36416	0.682078	−	0.731279i	$-0.261076\pi$
$823$	32208.0	1.36416	0.682078	+	0.731279i	$0.261076\pi$
$824$	0	0
$825$	−14934.0	−0.630224
$826$	0	0
$827$	27006.0	1.13554	0.567769	−	0.823188i	$-0.307807\pi$
$827$	27006.0	1.13554	0.567769	+	0.823188i	$0.307807\pi$
$828$	0	0
$829$	5818.00	0.243748	0.121874	−	0.992546i	$-0.461110\pi$
$829$	5818.00	0.243748	0.121874	+	0.992546i	$0.461110\pi$
$830$	0	0
$831$	10062.0	0.420032
$832$	0	0
$833$	21762.0	0.905172
$834$	0	0
$835$	−24480.0	−1.01457
$836$	0	0
$837$	2052.00	0.0847401
$838$	0	0
$839$	5926.00	0.243848	0.121924	−	0.992539i	$-0.461094\pi$
$839$	5926.00	0.243848	0.121924	+	0.992539i	$0.461094\pi$
$840$	0	0
$841$	34175.0	1.40125
$842$	0	0
$843$	19824.0	0.809935
$844$	0	0
$845$	−2704.00	−0.110083
$846$	0	0
$847$	904.000	0.0366727
$848$	0	0
$849$	−3444.00	−0.139220
$850$	0	0
$851$	17784.0	0.716366
$852$	0	0
$853$	−40874.0	−1.64068	−0.820339	−	0.571877i	$-0.806215\pi$
$853$	−40874.0	−1.64068	−0.820339	+	0.571877i	$0.806215\pi$
$854$	0	0
$855$	−10368.0	−0.414711
$856$	0	0
$857$	−3530.00	−0.140703	−0.0703515	−	0.997522i	$-0.522412\pi$
$857$	−3530.00	−0.140703	−0.0703515	+	0.997522i	$0.522412\pi$
$858$	0	0
$859$	34756.0	1.38051	0.690256	−	0.723565i	$-0.257498\pi$
$859$	34756.0	1.38051	0.690256	+	0.723565i	$0.257498\pi$
$860$	0	0
$861$	8064.00	0.319187
$862$	0	0
$863$	9878.00	0.389630	0.194815	−	0.980840i	$-0.437589\pi$
$863$	9878.00	0.389630	0.194815	+	0.980840i	$0.437589\pi$
$864$	0	0
$865$	16480.0	0.647788
$866$	0	0
$867$	−3513.00	−0.137610
$868$	0	0
$869$	13376.0	0.522152
$870$	0	0
$871$	−11804.0	−0.459200
$872$	0	0
$873$	−5058.00	−0.196091
$874$	0	0
$875$	−768.000	−0.0296722
$876$	0	0
$877$	150.000	0.00577553	0.00288777	−	0.999996i	$-0.499081\pi$
$877$	150.000	0.00577553	0.00288777	+	0.999996i	$0.499081\pi$
$878$	0	0
$879$	−5916.00	−0.227010
$880$	0	0
$881$	−3666.00	−0.140194	−0.0700969	−	0.997540i	$-0.522331\pi$
$881$	−3666.00	−0.140194	−0.0700969	+	0.997540i	$0.522331\pi$
$882$	0	0
$883$	−24316.0	−0.926725	−0.463363	−	0.886169i	$-0.653357\pi$
$883$	−24316.0	−0.926725	−0.463363	+	0.886169i	$0.653357\pi$
$884$	0	0
$885$	−40992.0	−1.55698
$886$	0	0
$887$	−2992.00	−0.113260	−0.0566299	−	0.998395i	$-0.518036\pi$
$887$	−2992.00	−0.113260	−0.0566299	+	0.998395i	$0.518036\pi$
$888$	0	0
$889$	−2048.00	−0.0772640
$890$	0	0
$891$	3078.00	0.115732
$892$	0	0
$893$	−6768.00	−0.253620
$894$	0	0
$895$	22080.0	0.824640
$896$	0	0
$897$	2028.00	0.0754882
$898$	0	0
$899$	−18392.0	−0.682322
$900$	0	0
$901$	35100.0	1.29784
$902$	0	0
$903$	1824.00	0.0672192
$904$	0	0
$905$	−36576.0	−1.34346
$906$	0	0
$907$	1956.00	0.0716074	0.0358037	−	0.999359i	$-0.488601\pi$
$907$	1956.00	0.0716074	0.0358037	+	0.999359i	$0.488601\pi$
$908$	0	0
$909$	4194.00	0.153032
$910$	0	0
$911$	38832.0	1.41225	0.706126	−	0.708086i	$-0.250441\pi$
$911$	38832.0	1.41225	0.706126	+	0.708086i	$0.250441\pi$
$912$	0	0
$913$	−18012.0	−0.652914
$914$	0	0
$915$	−5280.00	−0.190767
$916$	0	0
$917$	10880.0	0.391809
$918$	0	0
$919$	−504.000	−0.0180908	−0.00904539	−	0.999959i	$-0.502879\pi$
$919$	−504.000	−0.0180908	−0.00904539	+	0.999959i	$0.502879\pi$
$920$	0	0
$921$	−23628.0	−0.845352
$922$	0	0
$923$	10894.0	0.388494
$924$	0	0
$925$	44802.0	1.59252
$926$	0	0
$927$	13032.0	0.461734
$928$	0	0
$929$	−2976.00	−0.105102	−0.0525508	−	0.998618i	$-0.516735\pi$
$929$	−2976.00	−0.105102	−0.0525508	+	0.998618i	$0.516735\pi$
$930$	0	0
$931$	−20088.0	−0.707151
$932$	0	0
$933$	−20556.0	−0.721300
$934$	0	0
$935$	47424.0	1.65875
$936$	0	0
$937$	−14082.0	−0.490970	−0.245485	−	0.969400i	$-0.578947\pi$
$937$	−14082.0	−0.490970	−0.245485	+	0.969400i	$0.578947\pi$
$938$	0	0
$939$	14142.0	0.491487
$940$	0	0
$941$	−3260.00	−0.112936	−0.0564681	−	0.998404i	$-0.517984\pi$
$941$	−3260.00	−0.112936	−0.0564681	+	0.998404i	$0.517984\pi$
$942$	0	0
$943$	−17472.0	−0.603358
$944$	0	0
$945$	3456.00	0.118967
$946$	0	0
$947$	−5886.00	−0.201974	−0.100987	−	0.994888i	$-0.532200\pi$
$947$	−5886.00	−0.201974	−0.100987	+	0.994888i	$0.532200\pi$
$948$	0	0
$949$	12610.0	0.431336
$950$	0	0
$951$	1440.00	0.0491012
$952$	0	0
$953$	22574.0	0.767307	0.383654	−	0.923477i	$-0.374666\pi$
$953$	22574.0	0.767307	0.383654	+	0.923477i	$0.374666\pi$
$954$	0	0
$955$	75520.0	2.55892
$956$	0	0
$957$	−27588.0	−0.931864
$958$	0	0
$959$	23808.0	0.801669
$960$	0	0
$961$	−24015.0	−0.806116
$962$	0	0
$963$	3816.00	0.127694
$964$	0	0
$965$	−32672.0	−1.08990
$966$	0	0
$967$	−32996.0	−1.09729	−0.548645	−	0.836055i	$-0.684856\pi$
$967$	−32996.0	−1.09729	−0.548645	+	0.836055i	$0.684856\pi$
$968$	0	0
$969$	16848.0	0.558551
$970$	0	0
$971$	−14292.0	−0.472350	−0.236175	−	0.971711i	$-0.575894\pi$
$971$	−14292.0	−0.472350	−0.236175	+	0.971711i	$0.575894\pi$
$972$	0	0
$973$	−22112.0	−0.728549
$974$	0	0
$975$	5109.00	0.167814
$976$	0	0
$977$	48756.0	1.59656	0.798282	−	0.602284i	$-0.205743\pi$
$977$	48756.0	1.59656	0.798282	+	0.602284i	$0.205743\pi$
$978$	0	0
$979$	−55176.0	−1.80126
$980$	0	0
$981$	−7038.00	−0.229058
$982$	0	0
$983$	42022.0	1.36347	0.681736	−	0.731598i	$-0.261225\pi$
$983$	42022.0	1.36347	0.681736	+	0.731598i	$0.261225\pi$
$984$	0	0
$985$	24192.0	0.782560
$986$	0	0
$987$	2256.00	0.0727551
$988$	0	0
$989$	−3952.00	−0.127064
$990$	0	0
$991$	−46752.0	−1.49861	−0.749307	−	0.662223i	$-0.769613\pi$
$991$	−46752.0	−1.49861	−0.749307	+	0.662223i	$0.769613\pi$
$992$	0	0
$993$	22884.0	0.731321
$994$	0	0
$995$	−35584.0	−1.13376
$996$	0	0
$997$	−37666.0	−1.19648	−0.598242	−	0.801316i	$-0.704134\pi$
$997$	−37666.0	−1.19648	−0.598242	+	0.801316i	$0.704134\pi$
$998$	0	0
$999$	−9234.00	−0.292443

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.a.a.1.1		1
3.2	odd	2		1872.4.a.p.1.1		1
4.3	odd	2		78.4.a.b.1.1	✓	1
8.3	odd	2		2496.4.a.h.1.1		1
8.5	even	2		2496.4.a.p.1.1		1
12.11	even	2		234.4.a.j.1.1		1
20.19	odd	2		1950.4.a.k.1.1		1
52.31	even	4		1014.4.b.f.337.2		2
52.47	even	4		1014.4.b.f.337.1		2
52.51	odd	2		1014.4.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.4.a.b.1.1	✓	1	4.3	odd	2
234.4.a.j.1.1		1	12.11	even	2
624.4.a.a.1.1		1	1.1	even	1	trivial
1014.4.a.k.1.1		1	52.51	odd	2
1014.4.b.f.337.1		2	52.47	even	4
1014.4.b.f.337.2		2	52.31	even	4
1872.4.a.p.1.1		1	3.2	odd	2
1950.4.a.k.1.1		1	20.19	odd	2
2496.4.a.h.1.1		1	8.3	odd	2
2496.4.a.p.1.1		1	8.5	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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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