LMFDB - Embedded newform 1950.4.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,4,Mod(1,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1950.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1950.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:	$115.053724511$
Analytic rank:	$0$
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	$\chi$	$=$	1950.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{6} +8.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{6} +8.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -38.0000 q^{11} -12.0000 q^{12} +13.0000 q^{13} +16.0000 q^{14} +16.0000 q^{16} +78.0000 q^{17} +18.0000 q^{18} -72.0000 q^{19} -24.0000 q^{21} -76.0000 q^{22} +52.0000 q^{23} -24.0000 q^{24} +26.0000 q^{26} -27.0000 q^{27} +32.0000 q^{28} +242.000 q^{29} +76.0000 q^{31} +32.0000 q^{32} +114.000 q^{33} +156.000 q^{34} +36.0000 q^{36} -342.000 q^{37} -144.000 q^{38} -39.0000 q^{39} -336.000 q^{41} -48.0000 q^{42} -76.0000 q^{43} -152.000 q^{44} +104.000 q^{46} -94.0000 q^{47} -48.0000 q^{48} -279.000 q^{49} -234.000 q^{51} +52.0000 q^{52} +450.000 q^{53} -54.0000 q^{54} +64.0000 q^{56} +216.000 q^{57} +484.000 q^{58} +854.000 q^{59} -110.000 q^{61} +152.000 q^{62} +72.0000 q^{63} +64.0000 q^{64} +228.000 q^{66} +908.000 q^{67} +312.000 q^{68} -156.000 q^{69} +838.000 q^{71} +72.0000 q^{72} +970.000 q^{73} -684.000 q^{74} -288.000 q^{76} -304.000 q^{77} -78.0000 q^{78} -352.000 q^{79} +81.0000 q^{81} -672.000 q^{82} -474.000 q^{83} -96.0000 q^{84} -152.000 q^{86} -726.000 q^{87} -304.000 q^{88} -1452.00 q^{89} +104.000 q^{91} +208.000 q^{92} -228.000 q^{93} -188.000 q^{94} -96.0000 q^{96} +562.000 q^{97} -558.000 q^{98} -342.000 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−6.00000	−0.408248
$7$	8.00000	0.431959	0.215980	−	0.976398i	$-0.430705\pi$
$7$	8.00000	0.431959	0.215980	+	0.976398i	$0.430705\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	0	0
$11$	−38.0000	−1.04158	−0.520792	−	0.853683i	$-0.674363\pi$
$11$	−38.0000	−1.04158	−0.520792	+	0.853683i	$0.674363\pi$
$12$	−12.0000	−0.288675
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	16.0000	0.305441
$15$	0	0
$16$	16.0000	0.250000
$17$	78.0000	1.11281	0.556405	−	0.830911i	$-0.312180\pi$
$17$	78.0000	1.11281	0.556405	+	0.830911i	$0.312180\pi$
$18$	18.0000	0.235702
$19$	−72.0000	−0.869365	−0.434682	−	0.900584i	$-0.643139\pi$
$19$	−72.0000	−0.869365	−0.434682	+	0.900584i	$0.643139\pi$
$20$	0	0
$21$	−24.0000	−0.249392
$22$	−76.0000	−0.736512
$23$	52.0000	0.471424	0.235712	−	0.971823i	$-0.424258\pi$
$23$	52.0000	0.471424	0.235712	+	0.971823i	$0.424258\pi$
$24$	−24.0000	−0.204124
$25$	0	0
$26$	26.0000	0.196116
$27$	−27.0000	−0.192450
$28$	32.0000	0.215980
$29$	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	$-0.429342\pi$
$31$	76.0000	0.440323	0.220161	+	0.975463i	$0.429342\pi$
$32$	32.0000	0.176777
$33$	114.000	0.601359
$34$	156.000	0.786876
$35$	0	0
$36$	36.0000	0.166667
$37$	−342.000	−1.51958	−0.759790	−	0.650169i	$-0.774698\pi$
$37$	−342.000	−1.51958	−0.759790	+	0.650169i	$0.774698\pi$
$38$	−144.000	−0.614734
$39$	−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	$0.721037\pi$
$42$	−48.0000	−0.176347
$43$	−76.0000	−0.269532	−0.134766	−	0.990877i	$-0.543028\pi$
$43$	−76.0000	−0.269532	−0.134766	+	0.990877i	$0.543028\pi$
$44$	−152.000	−0.520792
$45$	0	0
$46$	104.000	0.333347
$47$	−94.0000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−94.0000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	−48.0000	−0.144338
$49$	−279.000	−0.813411
$50$	0	0
$51$	−234.000	−0.642481
$52$	52.0000	0.138675
$53$	450.000	1.16627	0.583134	−	0.812376i	$-0.301826\pi$
$53$	450.000	1.16627	0.583134	+	0.812376i	$0.301826\pi$
$54$	−54.0000	−0.136083
$55$	0	0
$56$	64.0000	0.152721
$57$	216.000	0.501928
$58$	484.000	1.09573
$59$	854.000	1.88443	0.942215	−	0.335010i	$-0.108740\pi$
$59$	854.000	1.88443	0.942215	+	0.335010i	$0.108740\pi$
$60$	0	0
$61$	−110.000	−0.230886	−0.115443	−	0.993314i	$-0.536829\pi$
$61$	−110.000	−0.230886	−0.115443	+	0.993314i	$0.536829\pi$
$62$	152.000	0.311355
$63$	72.0000	0.143986
$64$	64.0000	0.125000
$65$	0	0
$66$	228.000	0.425225
$67$	908.000	1.65567	0.827835	−	0.560972i	$-0.189572\pi$
$67$	908.000	1.65567	0.827835	+	0.560972i	$0.189572\pi$
$68$	312.000	0.556405
$69$	−156.000	−0.272177
$70$	0	0
$71$	838.000	1.40074	0.700368	−	0.713782i	$-0.253019\pi$
$71$	838.000	1.40074	0.700368	+	0.713782i	$0.253019\pi$
$72$	72.0000	0.117851
$73$	970.000	1.55520	0.777602	−	0.628757i	$-0.216436\pi$
$73$	970.000	1.55520	0.777602	+	0.628757i	$0.216436\pi$
$74$	−684.000	−1.07451
$75$	0	0
$76$	−288.000	−0.434682
$77$	−304.000	−0.449922
$78$	−78.0000	−0.113228
$79$	−352.000	−0.501305	−0.250652	−	0.968077i	$-0.580645\pi$
$79$	−352.000	−0.501305	−0.250652	+	0.968077i	$0.580645\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−672.000	−0.905000
$83$	−474.000	−0.626846	−0.313423	−	0.949614i	$-0.601476\pi$
$83$	−474.000	−0.626846	−0.313423	+	0.949614i	$0.601476\pi$
$84$	−96.0000	−0.124696
$85$	0	0
$86$	−152.000	−0.190588
$87$	−726.000	−0.894659
$88$	−304.000	−0.368256
$89$	−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
$92$	208.000	0.235712
$93$	−228.000	−0.254220
$94$	−188.000	−0.206284
$95$	0	0
$96$	−96.0000	−0.102062
$97$	562.000	0.588273	0.294136	−	0.955763i	$-0.404968\pi$
$97$	562.000	0.588273	0.294136	+	0.955763i	$0.404968\pi$
$98$	−558.000	−0.575168
$99$	−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	$0.426275\pi$
$102$	−468.000	−0.454303
$103$	1448.00	1.38520	0.692600	−	0.721321i	$-0.256465\pi$
$103$	1448.00	1.38520	0.692600	+	0.721321i	$0.256465\pi$
$104$	104.000	0.0980581
$105$	0	0
$106$	900.000	0.824677
$107$	424.000	0.383081	0.191540	−	0.981485i	$-0.438652\pi$
$107$	424.000	0.383081	0.191540	+	0.981485i	$0.438652\pi$
$108$	−108.000	−0.0962250
$109$	−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
$112$	128.000	0.107990
$113$	634.000	0.527803	0.263901	−	0.964550i	$-0.414991\pi$
$113$	634.000	0.527803	0.263901	+	0.964550i	$0.414991\pi$
$114$	432.000	0.354917
$115$	0	0
$116$	968.000	0.774798
$117$	117.000	0.0924500
$118$	1708.00	1.33249
$119$	624.000	0.480689
$120$	0	0
$121$	113.000	0.0848986
$122$	−220.000	−0.163261
$123$	1008.00	0.738929
$124$	304.000	0.220161
$125$	0	0
$126$	144.000	0.101814
$127$	−256.000	−0.178869	−0.0894344	−	0.995993i	$-0.528506\pi$
$127$	−256.000	−0.178869	−0.0894344	+	0.995993i	$0.528506\pi$
$128$	128.000	0.0883883
$129$	228.000	0.155615
$130$	0	0
$131$	−1360.00	−0.907052	−0.453526	−	0.891243i	$-0.649834\pi$
$131$	−1360.00	−0.907052	−0.453526	+	0.891243i	$0.649834\pi$
$132$	456.000	0.300680
$133$	−576.000	−0.375530
$134$	1816.00	1.17074
$135$	0	0
$136$	624.000	0.393438
$137$	−2976.00	−1.85589	−0.927945	−	0.372718i	$-0.878426\pi$
$137$	−2976.00	−1.85589	−0.927945	+	0.372718i	$0.878426\pi$
$138$	−312.000	−0.192458
$139$	2764.00	1.68661	0.843307	−	0.537432i	$-0.180605\pi$
$139$	2764.00	1.68661	0.843307	+	0.537432i	$0.180605\pi$
$140$	0	0
$141$	282.000	0.168430
$142$	1676.00	0.990470
$143$	−494.000	−0.288884
$144$	144.000	0.0833333
$145$	0	0
$146$	1940.00	1.09970
$147$	837.000	0.469623
$148$	−1368.00	−0.759790
$149$	−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	$-0.396275\pi$
$151$	1188.00	0.640252	0.320126	+	0.947375i	$0.396275\pi$
$152$	−576.000	−0.307367
$153$	702.000	0.370937
$154$	−608.000	−0.318143
$155$	0	0
$156$	−156.000	−0.0800641
$157$	2410.00	1.22509	0.612544	−	0.790436i	$-0.290146\pi$
$157$	2410.00	1.22509	0.612544	+	0.790436i	$0.290146\pi$
$158$	−704.000	−0.354476
$159$	−1350.00	−0.673346
$160$	0	0
$161$	416.000	0.203636
$162$	162.000	0.0785674
$163$	2248.00	1.08023	0.540113	−	0.841592i	$-0.318381\pi$
$163$	2248.00	1.08023	0.540113	+	0.841592i	$0.318381\pi$
$164$	−1344.00	−0.639932
$165$	0	0
$166$	−948.000	−0.443247
$167$	1530.00	0.708952	0.354476	−	0.935065i	$-0.384659\pi$
$167$	1530.00	0.708952	0.354476	+	0.935065i	$0.384659\pi$
$168$	−192.000	−0.0881733
$169$	169.000	0.0769231
$170$	0	0
$171$	−648.000	−0.289788
$172$	−304.000	−0.134766
$173$	1030.00	0.452656	0.226328	−	0.974051i	$-0.427328\pi$
$173$	1030.00	0.452656	0.226328	+	0.974051i	$0.427328\pi$
$174$	−1452.00	−0.632620
$175$	0	0
$176$	−608.000	−0.260396
$177$	−2562.00	−1.08798
$178$	−2904.00	−1.22283
$179$	1380.00	0.576235	0.288117	−	0.957595i	$-0.406971\pi$
$179$	1380.00	0.576235	0.288117	+	0.957595i	$0.406971\pi$
$180$	0	0
$181$	2286.00	0.938768	0.469384	−	0.882994i	$-0.344476\pi$
$181$	2286.00	0.938768	0.469384	+	0.882994i	$0.344476\pi$
$182$	208.000	0.0847142
$183$	330.000	0.133302
$184$	416.000	0.166674
$185$	0	0
$186$	−456.000	−0.179761
$187$	−2964.00	−1.15909
$188$	−376.000	−0.145865
$189$	−216.000	−0.0831306
$190$	0	0
$191$	4720.00	1.78810	0.894050	−	0.447967i	$-0.147852\pi$
$191$	4720.00	1.78810	0.894050	+	0.447967i	$0.147852\pi$
$192$	−192.000	−0.0721688
$193$	−2042.00	−0.761587	−0.380794	−	0.924660i	$-0.624349\pi$
$193$	−2042.00	−0.761587	−0.380794	+	0.924660i	$0.624349\pi$
$194$	1124.00	0.415972
$195$	0	0
$196$	−1116.00	−0.406706
$197$	1512.00	0.546830	0.273415	−	0.961896i	$-0.411847\pi$
$197$	1512.00	0.546830	0.273415	+	0.961896i	$0.411847\pi$
$198$	−684.000	−0.245504
$199$	−2224.00	−0.792237	−0.396119	−	0.918199i	$-0.629643\pi$
$199$	−2224.00	−0.792237	−0.396119	+	0.918199i	$0.629643\pi$
$200$	0	0
$201$	−2724.00	−0.955901
$202$	932.000	0.324630
$203$	1936.00	0.669362
$204$	−936.000	−0.321241
$205$	0	0
$206$	2896.00	0.979485
$207$	468.000	0.157141
$208$	208.000	0.0693375
$209$	2736.00	0.905517
$210$	0	0
$211$	4652.00	1.51781	0.758903	−	0.651204i	$-0.225736\pi$
$211$	4652.00	1.51781	0.758903	+	0.651204i	$0.225736\pi$
$212$	1800.00	0.583134
$213$	−2514.00	−0.808716
$214$	848.000	0.270879
$215$	0	0
$216$	−216.000	−0.0680414
$217$	608.000	0.190202
$218$	−1564.00	−0.485906
$219$	−2910.00	−0.897898
$220$	0	0
$221$	1014.00	0.308638
$222$	2052.00	0.620366
$223$	1812.00	0.544128	0.272064	−	0.962279i	$-0.412294\pi$
$223$	1812.00	0.544128	0.272064	+	0.962279i	$0.412294\pi$
$224$	256.000	0.0763604
$225$	0	0
$226$	1268.00	0.373213
$227$	−126.000	−0.0368410	−0.0184205	−	0.999830i	$-0.505864\pi$
$227$	−126.000	−0.0368410	−0.0184205	+	0.999830i	$0.505864\pi$
$228$	864.000	0.250964
$229$	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
$232$	1936.00	0.547865
$233$	2378.00	0.668618	0.334309	−	0.942464i	$-0.391497\pi$
$233$	2378.00	0.668618	0.334309	+	0.942464i	$0.391497\pi$
$234$	234.000	0.0653720
$235$	0	0
$236$	3416.00	0.942215
$237$	1056.00	0.289429
$238$	1248.00	0.339898
$239$	−1338.00	−0.362126	−0.181063	−	0.983472i	$-0.557954\pi$
$239$	−1338.00	−0.362126	−0.181063	+	0.983472i	$0.557954\pi$
$240$	0	0
$241$	6870.00	1.83625	0.918124	−	0.396294i	$-0.129704\pi$
$241$	6870.00	1.83625	0.918124	+	0.396294i	$0.129704\pi$
$242$	226.000	0.0600324
$243$	−243.000	−0.0641500
$244$	−440.000	−0.115443
$245$	0	0
$246$	2016.00	0.522502
$247$	−936.000	−0.241118
$248$	608.000	0.155678
$249$	1422.00	0.361910
$250$	0	0
$251$	6768.00	1.70196	0.850981	−	0.525197i	$-0.176008\pi$
$251$	6768.00	1.70196	0.850981	+	0.525197i	$0.176008\pi$
$252$	288.000	0.0719932
$253$	−1976.00	−0.491028
$254$	−512.000	−0.126479
$255$	0	0
$256$	256.000	0.0625000
$257$	3546.00	0.860675	0.430337	−	0.902668i	$-0.358395\pi$
$257$	3546.00	0.860675	0.430337	+	0.902668i	$0.358395\pi$
$258$	456.000	0.110036
$259$	−2736.00	−0.656397
$260$	0	0
$261$	2178.00	0.516532
$262$	−2720.00	−0.641382
$263$	5340.00	1.25201	0.626005	−	0.779819i	$-0.284689\pi$
$263$	5340.00	1.25201	0.626005	+	0.779819i	$0.284689\pi$
$264$	912.000	0.212613
$265$	0	0
$266$	−1152.00	−0.265540
$267$	4356.00	0.998438
$268$	3632.00	0.827835
$269$	−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	$-0.490866\pi$
$271$	256.000	0.0573834	0.0286917	+	0.999588i	$0.490866\pi$
$272$	1248.00	0.278203
$273$	−312.000	−0.0691689
$274$	−5952.00	−1.31231
$275$	0	0
$276$	−624.000	−0.136088
$277$	3354.00	0.727517	0.363759	−	0.931493i	$-0.381493\pi$
$277$	3354.00	0.727517	0.363759	+	0.931493i	$0.381493\pi$
$278$	5528.00	1.19262
$279$	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	$0.747452\pi$
$282$	564.000	0.119098
$283$	1148.00	0.241136	0.120568	−	0.992705i	$-0.461528\pi$
$283$	1148.00	0.241136	0.120568	+	0.992705i	$0.461528\pi$
$284$	3352.00	0.700368
$285$	0	0
$286$	−988.000	−0.204272
$287$	−2688.00	−0.552849
$288$	288.000	0.0589256
$289$	1171.00	0.238347
$290$	0	0
$291$	−1686.00	−0.339639
$292$	3880.00	0.777602
$293$	−1972.00	−0.393193	−0.196596	−	0.980485i	$-0.562989\pi$
$293$	−1972.00	−0.393193	−0.196596	+	0.980485i	$0.562989\pi$
$294$	1674.00	0.332074
$295$	0	0
$296$	−2736.00	−0.537253
$297$	1026.00	0.200453
$298$	−5880.00	−1.14302
$299$	676.000	0.130749
$300$	0	0
$301$	−608.000	−0.116427
$302$	2376.00	0.452727
$303$	−1398.00	−0.265059
$304$	−1152.00	−0.217341
$305$	0	0
$306$	1404.00	0.262292
$307$	7876.00	1.46419	0.732096	−	0.681201i	$-0.238542\pi$
$307$	7876.00	1.46419	0.732096	+	0.681201i	$0.238542\pi$
$308$	−1216.00	−0.224961
$309$	−4344.00	−0.799746
$310$	0	0
$311$	−6852.00	−1.24933	−0.624664	−	0.780893i	$-0.714764\pi$
$311$	−6852.00	−1.24933	−0.624664	+	0.780893i	$0.714764\pi$
$312$	−312.000	−0.0566139
$313$	4714.00	0.851281	0.425641	−	0.904892i	$-0.360049\pi$
$313$	4714.00	0.851281	0.425641	+	0.904892i	$0.360049\pi$
$314$	4820.00	0.866269
$315$	0	0
$316$	−1408.00	−0.250652
$317$	480.000	0.0850457	0.0425228	−	0.999095i	$-0.486460\pi$
$317$	480.000	0.0850457	0.0425228	+	0.999095i	$0.486460\pi$
$318$	−2700.00	−0.476127
$319$	−9196.00	−1.61403
$320$	0	0
$321$	−1272.00	−0.221172
$322$	832.000	0.143992
$323$	−5616.00	−0.967438
$324$	324.000	0.0555556
$325$	0	0
$326$	4496.00	0.763836
$327$	2346.00	0.396740
$328$	−2688.00	−0.452500
$329$	−752.000	−0.126016
$330$	0	0
$331$	7628.00	1.26669	0.633343	−	0.773872i	$-0.281682\pi$
$331$	7628.00	1.26669	0.633343	+	0.773872i	$0.281682\pi$
$332$	−1896.00	−0.313423
$333$	−3078.00	−0.506527
$334$	3060.00	0.501305
$335$	0	0
$336$	−384.000	−0.0623480
$337$	9346.00	1.51071	0.755355	−	0.655316i	$-0.227465\pi$
$337$	9346.00	1.51071	0.755355	+	0.655316i	$0.227465\pi$
$338$	338.000	0.0543928
$339$	−1902.00	−0.304727
$340$	0	0
$341$	−2888.00	−0.458633
$342$	−1296.00	−0.204911
$343$	−4976.00	−0.783320
$344$	−608.000	−0.0952941
$345$	0	0
$346$	2060.00	0.320076
$347$	492.000	0.0761151	0.0380576	−	0.999276i	$-0.487883\pi$
$347$	492.000	0.0761151	0.0380576	+	0.999276i	$0.487883\pi$
$348$	−2904.00	−0.447330
$349$	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
$352$	−1216.00	−0.184128
$353$	−1648.00	−0.248482	−0.124241	−	0.992252i	$-0.539650\pi$
$353$	−1648.00	−0.248482	−0.124241	+	0.992252i	$0.539650\pi$
$354$	−5124.00	−0.769315
$355$	0	0
$356$	−5808.00	−0.864672
$357$	−1872.00	−0.277526
$358$	2760.00	0.407460
$359$	9750.00	1.43339	0.716693	−	0.697389i	$-0.245655\pi$
$359$	9750.00	1.43339	0.716693	+	0.697389i	$0.245655\pi$
$360$	0	0
$361$	−1675.00	−0.244205
$362$	4572.00	0.663809
$363$	−339.000	−0.0490162
$364$	416.000	0.0599020
$365$	0	0
$366$	660.000	0.0942589
$367$	−10856.0	−1.54408	−0.772042	−	0.635572i	$-0.780764\pi$
$367$	−10856.0	−1.54408	−0.772042	+	0.635572i	$0.780764\pi$
$368$	832.000	0.117856
$369$	−3024.00	−0.426621
$370$	0	0
$371$	3600.00	0.503781
$372$	−912.000	−0.127110
$373$	−1826.00	−0.253476	−0.126738	−	0.991936i	$-0.540451\pi$
$373$	−1826.00	−0.253476	−0.126738	+	0.991936i	$0.540451\pi$
$374$	−5928.00	−0.819598
$375$	0	0
$376$	−752.000	−0.103142
$377$	3146.00	0.429780
$378$	−432.000	−0.0587822
$379$	−896.000	−0.121436	−0.0607182	−	0.998155i	$-0.519339\pi$
$379$	−896.000	−0.121436	−0.0607182	+	0.998155i	$0.519339\pi$
$380$	0	0
$381$	768.000	0.103270
$382$	9440.00	1.26438
$383$	−2826.00	−0.377028	−0.188514	−	0.982070i	$-0.560367\pi$
$383$	−2826.00	−0.377028	−0.188514	+	0.982070i	$0.560367\pi$
$384$	−384.000	−0.0510310
$385$	0	0
$386$	−4084.00	−0.538524
$387$	−684.000	−0.0898441
$388$	2248.00	0.294136
$389$	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
$392$	−2232.00	−0.287584
$393$	4080.00	0.523686
$394$	3024.00	0.386667
$395$	0	0
$396$	−1368.00	−0.173597
$397$	−8678.00	−1.09707	−0.548534	−	0.836128i	$-0.684814\pi$
$397$	−8678.00	−1.09707	−0.548534	+	0.836128i	$0.684814\pi$
$398$	−4448.00	−0.560196
$399$	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	$0.712634\pi$
$402$	−5448.00	−0.675924
$403$	988.000	0.122124
$404$	1864.00	0.229548
$405$	0	0
$406$	3872.00	0.473311
$407$	12996.0	1.58277
$408$	−1872.00	−0.227151
$409$	−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
$412$	5792.00	0.692600
$413$	6832.00	0.813997
$414$	936.000	0.111116
$415$	0	0
$416$	416.000	0.0490290
$417$	−8292.00	−0.973767
$418$	5472.00	0.640297
$419$	−3216.00	−0.374969	−0.187484	−	0.982268i	$-0.560033\pi$
$419$	−3216.00	−0.374969	−0.187484	+	0.982268i	$0.560033\pi$
$420$	0	0
$421$	4738.00	0.548494	0.274247	−	0.961659i	$-0.411571\pi$
$421$	4738.00	0.548494	0.274247	+	0.961659i	$0.411571\pi$
$422$	9304.00	1.07325
$423$	−846.000	−0.0972433
$424$	3600.00	0.412338
$425$	0	0
$426$	−5028.00	−0.571848
$427$	−880.000	−0.0997335
$428$	1696.00	0.191540
$429$	1482.00	0.166787
$430$	0	0
$431$	−2598.00	−0.290351	−0.145175	−	0.989406i	$-0.546375\pi$
$431$	−2598.00	−0.290351	−0.145175	+	0.989406i	$0.546375\pi$
$432$	−432.000	−0.0481125
$433$	7490.00	0.831285	0.415643	−	0.909528i	$-0.363557\pi$
$433$	7490.00	0.831285	0.415643	+	0.909528i	$0.363557\pi$
$434$	1216.00	0.134493
$435$	0	0
$436$	−3128.00	−0.343587
$437$	−3744.00	−0.409839
$438$	−5820.00	−0.634909
$439$	−17632.0	−1.91692	−0.958462	−	0.285221i	$-0.907933\pi$
$439$	−17632.0	−1.91692	−0.958462	+	0.285221i	$0.907933\pi$
$440$	0	0
$441$	−2511.00	−0.271137
$442$	2028.00	0.218240
$443$	−9696.00	−1.03989	−0.519945	−	0.854200i	$-0.674047\pi$
$443$	−9696.00	−1.03989	−0.519945	+	0.854200i	$0.674047\pi$
$444$	4104.00	0.438665
$445$	0	0
$446$	3624.00	0.384756
$447$	8820.00	0.933270
$448$	512.000	0.0539949
$449$	−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
$452$	2536.00	0.263901
$453$	−3564.00	−0.369650
$454$	−252.000	−0.0260505
$455$	0	0
$456$	1728.00	0.177458
$457$	−12862.0	−1.31654	−0.658270	−	0.752782i	$-0.728712\pi$
$457$	−12862.0	−1.31654	−0.658270	+	0.752782i	$0.728712\pi$
$458$	6372.00	0.650096
$459$	−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	$0.334855\pi$
$462$	1824.00	0.183680
$463$	−10408.0	−1.04471	−0.522355	−	0.852728i	$-0.674946\pi$
$463$	−10408.0	−1.04471	−0.522355	+	0.852728i	$0.674946\pi$
$464$	3872.00	0.387399
$465$	0	0
$466$	4756.00	0.472784
$467$	−10472.0	−1.03766	−0.518829	−	0.854878i	$-0.673632\pi$
$467$	−10472.0	−1.03766	−0.518829	+	0.854878i	$0.673632\pi$
$468$	468.000	0.0462250
$469$	7264.00	0.715182
$470$	0	0
$471$	−7230.00	−0.707305
$472$	6832.00	0.666246
$473$	2888.00	0.280741
$474$	2112.00	0.204657
$475$	0	0
$476$	2496.00	0.240344
$477$	4050.00	0.388756
$478$	−2676.00	−0.256061
$479$	−13398.0	−1.27802	−0.639009	−	0.769200i	$-0.720655\pi$
$479$	−13398.0	−1.27802	−0.639009	+	0.769200i	$0.720655\pi$
$480$	0	0
$481$	−4446.00	−0.421456
$482$	13740.0	1.29842
$483$	−1248.00	−0.117569
$484$	452.000	0.0424493
$485$	0	0
$486$	−486.000	−0.0453609
$487$	−14780.0	−1.37525	−0.687624	−	0.726067i	$-0.741346\pi$
$487$	−14780.0	−1.37525	−0.687624	+	0.726067i	$0.741346\pi$
$488$	−880.000	−0.0816306
$489$	−6744.00	−0.623669
$490$	0	0
$491$	−12632.0	−1.16105	−0.580524	−	0.814243i	$-0.697152\pi$
$491$	−12632.0	−1.16105	−0.580524	+	0.814243i	$0.697152\pi$
$492$	4032.00	0.369465
$493$	18876.0	1.72441
$494$	−1872.00	−0.170496
$495$	0	0
$496$	1216.00	0.110081
$497$	6704.00	0.605061
$498$	2844.00	0.255909
$499$	17260.0	1.54842	0.774212	−	0.632926i	$-0.218146\pi$
$499$	17260.0	1.54842	0.774212	+	0.632926i	$0.218146\pi$
$500$	0	0
$501$	−4590.00	−0.409314
$502$	13536.0	1.20347
$503$	−76.0000	−0.00673692	−0.00336846	−	0.999994i	$-0.501072\pi$
$503$	−76.0000	−0.00673692	−0.00336846	+	0.999994i	$0.501072\pi$
$504$	576.000	0.0509069
$505$	0	0
$506$	−3952.00	−0.347209
$507$	−507.000	−0.0444116
$508$	−1024.00	−0.0894344
$509$	−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
$512$	512.000	0.0441942
$513$	1944.00	0.167309
$514$	7092.00	0.608589
$515$	0	0
$516$	912.000	0.0778073
$517$	3572.00	0.303861
$518$	−5472.00	−0.464143
$519$	−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	$0.557077\pi$
$522$	4356.00	0.365243
$523$	9564.00	0.799626	0.399813	−	0.916597i	$-0.369075\pi$
$523$	9564.00	0.799626	0.399813	+	0.916597i	$0.369075\pi$
$524$	−5440.00	−0.453526
$525$	0	0
$526$	10680.0	0.885304
$527$	5928.00	0.489996
$528$	1824.00	0.150340
$529$	−9463.00	−0.777760
$530$	0	0
$531$	7686.00	0.628143
$532$	−2304.00	−0.187765
$533$	−4368.00	−0.354970
$534$	8712.00	0.706002
$535$	0	0
$536$	7264.00	0.585368
$537$	−4140.00	−0.332689
$538$	−6972.00	−0.558707
$539$	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	$0.720594\pi$
$542$	512.000	0.0405762
$543$	−6858.00	−0.541998
$544$	2496.00	0.196719
$545$	0	0
$546$	−624.000	−0.0489098
$547$	−6292.00	−0.491822	−0.245911	−	0.969292i	$-0.579087\pi$
$547$	−6292.00	−0.491822	−0.245911	+	0.969292i	$0.579087\pi$
$548$	−11904.0	−0.927945
$549$	−990.000	−0.0769621
$550$	0	0
$551$	−17424.0	−1.34716
$552$	−1248.00	−0.0962290
$553$	−2816.00	−0.216543
$554$	6708.00	0.514433
$555$	0	0
$556$	11056.0	0.843307
$557$	3588.00	0.272942	0.136471	−	0.990644i	$-0.456424\pi$
$557$	3588.00	0.272942	0.136471	+	0.990644i	$0.456424\pi$
$558$	1368.00	0.103785
$559$	−988.000	−0.0747548
$560$	0	0
$561$	8892.00	0.669199
$562$	−13216.0	−0.991963
$563$	5932.00	0.444057	0.222028	−	0.975040i	$-0.428732\pi$
$563$	5932.00	0.444057	0.222028	+	0.975040i	$0.428732\pi$
$564$	1128.00	0.0842152
$565$	0	0
$566$	2296.00	0.170509
$567$	648.000	0.0479955
$568$	6704.00	0.495235
$569$	−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	$-0.263763\pi$
$571$	18444.0	1.35176	0.675882	+	0.737010i	$0.263763\pi$
$572$	−1976.00	−0.144442
$573$	−14160.0	−1.03236
$574$	−5376.00	−0.390923
$575$	0	0
$576$	576.000	0.0416667
$577$	2382.00	0.171861	0.0859306	−	0.996301i	$-0.472614\pi$
$577$	2382.00	0.171861	0.0859306	+	0.996301i	$0.472614\pi$
$578$	2342.00	0.168537
$579$	6126.00	0.439703
$580$	0	0
$581$	−3792.00	−0.270772
$582$	−3372.00	−0.240161
$583$	−17100.0	−1.21477
$584$	7760.00	0.549848
$585$	0	0
$586$	−3944.00	−0.278029
$587$	−15698.0	−1.10379	−0.551896	−	0.833913i	$-0.686095\pi$
$587$	−15698.0	−1.10379	−0.551896	+	0.833913i	$0.686095\pi$
$588$	3348.00	0.234812
$589$	−5472.00	−0.382801
$590$	0	0
$591$	−4536.00	−0.315713
$592$	−5472.00	−0.379895
$593$	8452.00	0.585299	0.292649	−	0.956220i	$-0.405463\pi$
$593$	8452.00	0.585299	0.292649	+	0.956220i	$0.405463\pi$
$594$	2052.00	0.141742
$595$	0	0
$596$	−11760.0	−0.808236
$597$	6672.00	0.457398
$598$	1352.00	0.0924538
$599$	5836.00	0.398084	0.199042	−	0.979991i	$-0.436217\pi$
$599$	5836.00	0.398084	0.199042	+	0.979991i	$0.436217\pi$
$600$	0	0
$601$	−25850.0	−1.75448	−0.877241	−	0.480051i	$-0.840618\pi$
$601$	−25850.0	−1.75448	−0.877241	+	0.480051i	$0.840618\pi$
$602$	−1216.00	−0.0823263
$603$	8172.00	0.551890
$604$	4752.00	0.320126
$605$	0	0
$606$	−2796.00	−0.187425
$607$	−21624.0	−1.44595	−0.722975	−	0.690875i	$-0.757226\pi$
$607$	−21624.0	−1.44595	−0.722975	+	0.690875i	$0.757226\pi$
$608$	−2304.00	−0.153683
$609$	−5808.00	−0.386457
$610$	0	0
$611$	−1222.00	−0.0809113
$612$	2808.00	0.185468
$613$	−3902.00	−0.257097	−0.128548	−	0.991703i	$-0.541032\pi$
$613$	−3902.00	−0.257097	−0.128548	+	0.991703i	$0.541032\pi$
$614$	15752.0	1.03534
$615$	0	0
$616$	−2432.00	−0.159072
$617$	−16888.0	−1.10192	−0.550961	−	0.834531i	$-0.685739\pi$
$617$	−16888.0	−1.10192	−0.550961	+	0.834531i	$0.685739\pi$
$618$	−8688.00	−0.565506
$619$	−27452.0	−1.78253	−0.891267	−	0.453478i	$-0.850183\pi$
$619$	−27452.0	−1.78253	−0.891267	+	0.453478i	$0.850183\pi$
$620$	0	0
$621$	−1404.00	−0.0907256
$622$	−13704.0	−0.883409
$623$	−11616.0	−0.747007
$624$	−624.000	−0.0400320
$625$	0	0
$626$	9428.00	0.601947
$627$	−8208.00	−0.522801
$628$	9640.00	0.612544
$629$	−26676.0	−1.69100
$630$	0	0
$631$	5548.00	0.350020	0.175010	−	0.984567i	$-0.444004\pi$
$631$	5548.00	0.350020	0.175010	+	0.984567i	$0.444004\pi$
$632$	−2816.00	−0.177238
$633$	−13956.0	−0.876305
$634$	960.000	0.0601364
$635$	0	0
$636$	−5400.00	−0.336673
$637$	−3627.00	−0.225600
$638$	−18392.0	−1.14130
$639$	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	$0.484126\pi$
$642$	−2544.00	−0.156392
$643$	19900.0	1.22050	0.610248	−	0.792210i	$-0.291070\pi$
$643$	19900.0	1.22050	0.610248	+	0.792210i	$0.291070\pi$
$644$	1664.00	0.101818
$645$	0	0
$646$	−11232.0	−0.684082
$647$	−18832.0	−1.14430	−0.572150	−	0.820149i	$-0.693891\pi$
$647$	−18832.0	−1.14430	−0.572150	+	0.820149i	$0.693891\pi$
$648$	648.000	0.0392837
$649$	−32452.0	−1.96279
$650$	0	0
$651$	−1824.00	−0.109813
$652$	8992.00	0.540113
$653$	4542.00	0.272193	0.136097	−	0.990696i	$-0.456544\pi$
$653$	4542.00	0.272193	0.136097	+	0.990696i	$0.456544\pi$
$654$	4692.00	0.280538
$655$	0	0
$656$	−5376.00	−0.319966
$657$	8730.00	0.518401
$658$	−1504.00	−0.0891064
$659$	8820.00	0.521363	0.260682	−	0.965425i	$-0.416053\pi$
$659$	8820.00	0.521363	0.260682	+	0.965425i	$0.416053\pi$
$660$	0	0
$661$	21014.0	1.23654	0.618268	−	0.785968i	$-0.287835\pi$
$661$	21014.0	1.23654	0.618268	+	0.785968i	$0.287835\pi$
$662$	15256.0	0.895682
$663$	−3042.00	−0.178192
$664$	−3792.00	−0.221624
$665$	0	0
$666$	−6156.00	−0.358168
$667$	12584.0	0.730516
$668$	6120.00	0.354476
$669$	−5436.00	−0.314152
$670$	0	0
$671$	4180.00	0.240487
$672$	−768.000	−0.0440867
$673$	−1714.00	−0.0981721	−0.0490861	−	0.998795i	$-0.515631\pi$
$673$	−1714.00	−0.0981721	−0.0490861	+	0.998795i	$0.515631\pi$
$674$	18692.0	1.06823
$675$	0	0
$676$	676.000	0.0384615
$677$	15114.0	0.858018	0.429009	−	0.903300i	$-0.358863\pi$
$677$	15114.0	0.858018	0.429009	+	0.903300i	$0.358863\pi$
$678$	−3804.00	−0.215475
$679$	4496.00	0.254110
$680$	0	0
$681$	378.000	0.0212702
$682$	−5776.00	−0.324303
$683$	20486.0	1.14769	0.573847	−	0.818963i	$-0.305450\pi$
$683$	20486.0	1.14769	0.573847	+	0.818963i	$0.305450\pi$
$684$	−2592.00	−0.144894
$685$	0	0
$686$	−9952.00	−0.553891
$687$	−9558.00	−0.530801
$688$	−1216.00	−0.0673831
$689$	5850.00	0.323465
$690$	0	0
$691$	−8948.00	−0.492616	−0.246308	−	0.969192i	$-0.579218\pi$
$691$	−8948.00	−0.492616	−0.246308	+	0.969192i	$0.579218\pi$
$692$	4120.00	0.226328
$693$	−2736.00	−0.149974
$694$	984.000	0.0538215
$695$	0	0
$696$	−5808.00	−0.316310
$697$	−26208.0	−1.42425
$698$	716.000	0.0388266
$699$	−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	$0.488421\pi$
$702$	−702.000	−0.0377426
$703$	24624.0	1.32107
$704$	−2432.00	−0.130198
$705$	0	0
$706$	−3296.00	−0.175703
$707$	3728.00	0.198311
$708$	−10248.0	−0.543988
$709$	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
$712$	−11616.0	−0.611416
$713$	3952.00	0.207579
$714$	−3744.00	−0.196240
$715$	0	0
$716$	5520.00	0.288117
$717$	4014.00	0.209073
$718$	19500.0	1.01356
$719$	−28204.0	−1.46291	−0.731455	−	0.681890i	$-0.761158\pi$
$719$	−28204.0	−1.46291	−0.731455	+	0.681890i	$0.761158\pi$
$720$	0	0
$721$	11584.0	0.598350
$722$	−3350.00	−0.172679
$723$	−20610.0	−1.06016
$724$	9144.00	0.469384
$725$	0	0
$726$	−678.000	−0.0346597
$727$	−20992.0	−1.07091	−0.535454	−	0.844564i	$-0.679859\pi$
$727$	−20992.0	−1.07091	−0.535454	+	0.844564i	$0.679859\pi$
$728$	832.000	0.0423571
$729$	729.000	0.0370370
$730$	0	0
$731$	−5928.00	−0.299938
$732$	1320.00	0.0666511
$733$	19894.0	1.00246	0.501229	−	0.865315i	$-0.332881\pi$
$733$	19894.0	1.00246	0.501229	+	0.865315i	$0.332881\pi$
$734$	−21712.0	−1.09183
$735$	0	0
$736$	1664.00	0.0833368
$737$	−34504.0	−1.72452
$738$	−6048.00	−0.301667
$739$	6252.00	0.311209	0.155605	−	0.987819i	$-0.450267\pi$
$739$	6252.00	0.311209	0.155605	+	0.987819i	$0.450267\pi$
$740$	0	0
$741$	2808.00	0.139210
$742$	7200.00	0.356227
$743$	30938.0	1.52760	0.763799	−	0.645454i	$-0.223332\pi$
$743$	30938.0	1.52760	0.763799	+	0.645454i	$0.223332\pi$
$744$	−1824.00	−0.0898805
$745$	0	0
$746$	−3652.00	−0.179235
$747$	−4266.00	−0.208949
$748$	−11856.0	−0.579543
$749$	3392.00	0.165475
$750$	0	0
$751$	11328.0	0.550419	0.275209	−	0.961384i	$-0.411253\pi$
$751$	11328.0	0.550419	0.275209	+	0.961384i	$0.411253\pi$
$752$	−1504.00	−0.0729325
$753$	−20304.0	−0.982628
$754$	6292.00	0.303901
$755$	0	0
$756$	−864.000	−0.0415653
$757$	−32754.0	−1.57261	−0.786304	−	0.617840i	$-0.788008\pi$
$757$	−32754.0	−1.57261	−0.786304	+	0.617840i	$0.788008\pi$
$758$	−1792.00	−0.0858686
$759$	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	$0.656111\pi$
$762$	1536.00	0.0730228
$763$	−6256.00	−0.296831
$764$	18880.0	0.894050
$765$	0	0
$766$	−5652.00	−0.266599
$767$	11102.0	0.522647
$768$	−768.000	−0.0360844
$769$	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
$772$	−8168.00	−0.380794
$773$	−17108.0	−0.796031	−0.398016	−	0.917379i	$-0.630301\pi$
$773$	−17108.0	−0.796031	−0.398016	+	0.917379i	$0.630301\pi$
$774$	−1368.00	−0.0635294
$775$	0	0
$776$	4496.00	0.207986
$777$	8208.00	0.378971
$778$	19692.0	0.907445
$779$	24192.0	1.11267
$780$	0	0
$781$	−31844.0	−1.45899
$782$	8112.00	0.370952
$783$	−6534.00	−0.298220
$784$	−4464.00	−0.203353
$785$	0	0
$786$	8160.00	0.370302
$787$	21364.0	0.967655	0.483827	−	0.875163i	$-0.339246\pi$
$787$	21364.0	0.967655	0.483827	+	0.875163i	$0.339246\pi$
$788$	6048.00	0.273415
$789$	−16020.0	−0.722848
$790$	0	0
$791$	5072.00	0.227989
$792$	−2736.00	−0.122752
$793$	−1430.00	−0.0640363
$794$	−17356.0	−0.775745
$795$	0	0
$796$	−8896.00	−0.396119
$797$	−28542.0	−1.26852	−0.634259	−	0.773120i	$-0.718695\pi$
$797$	−28542.0	−1.26852	−0.634259	+	0.773120i	$0.718695\pi$
$798$	3456.00	0.153310
$799$	−7332.00	−0.324640
$800$	0	0
$801$	−13068.0	−0.576448
$802$	−19896.0	−0.876000
$803$	−36860.0	−1.61988
$804$	−10896.0	−0.477951
$805$	0	0
$806$	1976.00	0.0863544
$807$	10458.0	0.456182
$808$	3728.00	0.162315
$809$	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	$-0.607844\pi$
$811$	−15352.0	−0.664712	−0.332356	+	0.943154i	$0.607844\pi$
$812$	7744.00	0.334681
$813$	−768.000	−0.0331303
$814$	25992.0	1.11919
$815$	0	0
$816$	−3744.00	−0.160620
$817$	5472.00	0.234322
$818$	−196.000	−0.00837773
$819$	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	$0.737827\pi$
$822$	17856.0	0.757664
$823$	32208.0	1.36416	0.682078	−	0.731279i	$-0.261076\pi$
$823$	32208.0	1.36416	0.682078	+	0.731279i	$0.261076\pi$
$824$	11584.0	0.489742
$825$	0	0
$826$	13664.0	0.575583
$827$	27006.0	1.13554	0.567769	−	0.823188i	$-0.307807\pi$
$827$	27006.0	1.13554	0.567769	+	0.823188i	$0.307807\pi$
$828$	1872.00	0.0785706
$829$	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
$832$	832.000	0.0346688
$833$	−21762.0	−0.905172
$834$	−16584.0	−0.688557
$835$	0	0
$836$	10944.0	0.452759
$837$	−2052.00	−0.0847401
$838$	−6432.00	−0.265143
$839$	−5926.00	−0.243848	−0.121924	−	0.992539i	$-0.538906\pi$
$839$	−5926.00	−0.243848	−0.121924	+	0.992539i	$0.538906\pi$
$840$	0	0
$841$	34175.0	1.40125
$842$	9476.00	0.387844
$843$	19824.0	0.809935
$844$	18608.0	0.758903
$845$	0	0
$846$	−1692.00	−0.0687614
$847$	904.000	0.0366727
$848$	7200.00	0.291567
$849$	−3444.00	−0.139220
$850$	0	0
$851$	−17784.0	−0.716366
$852$	−10056.0	−0.404358
$853$	40874.0	1.64068	0.820339	−	0.571877i	$-0.193785\pi$
$853$	40874.0	1.64068	0.820339	+	0.571877i	$0.193785\pi$
$854$	−1760.00	−0.0705222
$855$	0	0
$856$	3392.00	0.135439
$857$	3530.00	0.140703	0.0703515	−	0.997522i	$-0.477588\pi$
$857$	3530.00	0.140703	0.0703515	+	0.997522i	$0.477588\pi$
$858$	2964.00	0.117936
$859$	−34756.0	−1.38051	−0.690256	−	0.723565i	$-0.742502\pi$
$859$	−34756.0	−1.38051	−0.690256	+	0.723565i	$0.742502\pi$
$860$	0	0
$861$	8064.00	0.319187
$862$	−5196.00	−0.205309
$863$	9878.00	0.389630	0.194815	−	0.980840i	$-0.437589\pi$
$863$	9878.00	0.389630	0.194815	+	0.980840i	$0.437589\pi$
$864$	−864.000	−0.0340207
$865$	0	0
$866$	14980.0	0.587807
$867$	−3513.00	−0.137610
$868$	2432.00	0.0951008
$869$	13376.0	0.522152
$870$	0	0
$871$	11804.0	0.459200
$872$	−6256.00	−0.242953
$873$	5058.00	0.196091
$874$	−7488.00	−0.289800
$875$	0	0
$876$	−11640.0	−0.448949
$877$	−150.000	−0.00577553	−0.00288777	−	0.999996i	$-0.500919\pi$
$877$	−150.000	−0.00577553	−0.00288777	+	0.999996i	$0.500919\pi$
$878$	−35264.0	−1.35547
$879$	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	$0.522331\pi$
$882$	−5022.00	−0.191723
$883$	−24316.0	−0.926725	−0.463363	−	0.886169i	$-0.653357\pi$
$883$	−24316.0	−0.926725	−0.463363	+	0.886169i	$0.653357\pi$
$884$	4056.00	0.154319
$885$	0	0
$886$	−19392.0	−0.735313
$887$	−2992.00	−0.113260	−0.0566299	−	0.998395i	$-0.518036\pi$
$887$	−2992.00	−0.113260	−0.0566299	+	0.998395i	$0.518036\pi$
$888$	8208.00	0.310183
$889$	−2048.00	−0.0772640
$890$	0	0
$891$	−3078.00	−0.115732
$892$	7248.00	0.272064
$893$	6768.00	0.253620
$894$	17640.0	0.659922
$895$	0	0
$896$	1024.00	0.0381802
$897$	−2028.00	−0.0754882
$898$	−8872.00	−0.329691
$899$	18392.0	0.682322
$900$	0	0
$901$	35100.0	1.29784
$902$	25536.0	0.942634
$903$	1824.00	0.0672192
$904$	5072.00	0.186606
$905$	0	0
$906$	−7128.00	−0.261382
$907$	1956.00	0.0716074	0.0358037	−	0.999359i	$-0.488601\pi$
$907$	1956.00	0.0716074	0.0358037	+	0.999359i	$0.488601\pi$
$908$	−504.000	−0.0184205
$909$	4194.00	0.153032
$910$	0	0
$911$	−38832.0	−1.41225	−0.706126	−	0.708086i	$-0.749559\pi$
$911$	−38832.0	−1.41225	−0.706126	+	0.708086i	$0.749559\pi$
$912$	3456.00	0.125482
$913$	18012.0	0.652914
$914$	−25724.0	−0.930935
$915$	0	0
$916$	12744.0	0.459687
$917$	−10880.0	−0.391809
$918$	−4212.00	−0.151434
$919$	504.000	0.0180908	0.00904539	−	0.999959i	$-0.497121\pi$
$919$	504.000	0.0180908	0.00904539	+	0.999959i	$0.497121\pi$
$920$	0	0
$921$	−23628.0	−0.845352
$922$	19632.0	0.701242
$923$	10894.0	0.388494
$924$	3648.00	0.129881
$925$	0	0
$926$	−20816.0	−0.738722
$927$	13032.0	0.461734
$928$	7744.00	0.273932
$929$	−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
$932$	9512.00	0.334309
$933$	20556.0	0.721300
$934$	−20944.0	−0.733735
$935$	0	0
$936$	936.000	0.0326860
$937$	14082.0	0.490970	0.245485	−	0.969400i	$-0.421053\pi$
$937$	14082.0	0.490970	0.245485	+	0.969400i	$0.421053\pi$
$938$	14528.0	0.505710
$939$	−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	$0.517984\pi$
$942$	−14460.0	−0.500140
$943$	−17472.0	−0.603358
$944$	13664.0	0.471107
$945$	0	0
$946$	5776.00	0.198514
$947$	−5886.00	−0.201974	−0.100987	−	0.994888i	$-0.532200\pi$
$947$	−5886.00	−0.201974	−0.100987	+	0.994888i	$0.532200\pi$
$948$	4224.00	0.144714
$949$	12610.0	0.431336
$950$	0	0
$951$	−1440.00	−0.0491012
$952$	4992.00	0.169949
$953$	−22574.0	−0.767307	−0.383654	−	0.923477i	$-0.625334\pi$
$953$	−22574.0	−0.767307	−0.383654	+	0.923477i	$0.625334\pi$
$954$	8100.00	0.274892
$955$	0	0
$956$	−5352.00	−0.181063
$957$	27588.0	0.931864
$958$	−26796.0	−0.903695
$959$	−23808.0	−0.801669
$960$	0	0
$961$	−24015.0	−0.806116
$962$	−8892.00	−0.298014
$963$	3816.00	0.127694
$964$	27480.0	0.918124
$965$	0	0
$966$	−2496.00	−0.0831340
$967$	−32996.0	−1.09729	−0.548645	−	0.836055i	$-0.684856\pi$
$967$	−32996.0	−1.09729	−0.548645	+	0.836055i	$0.684856\pi$
$968$	904.000	0.0300162
$969$	16848.0	0.558551
$970$	0	0
$971$	14292.0	0.472350	0.236175	−	0.971711i	$-0.424106\pi$
$971$	14292.0	0.472350	0.236175	+	0.971711i	$0.424106\pi$
$972$	−972.000	−0.0320750
$973$	22112.0	0.728549
$974$	−29560.0	−0.972447
$975$	0	0
$976$	−1760.00	−0.0577215
$977$	−48756.0	−1.59656	−0.798282	−	0.602284i	$-0.794257\pi$
$977$	−48756.0	−1.59656	−0.798282	+	0.602284i	$0.794257\pi$
$978$	−13488.0	−0.441001
$979$	55176.0	1.80126
$980$	0	0
$981$	−7038.00	−0.229058
$982$	−25264.0	−0.820984
$983$	42022.0	1.36347	0.681736	−	0.731598i	$-0.261225\pi$
$983$	42022.0	1.36347	0.681736	+	0.731598i	$0.261225\pi$
$984$	8064.00	0.261251
$985$	0	0
$986$	37752.0	1.21934
$987$	2256.00	0.0727551
$988$	−3744.00	−0.120559
$989$	−3952.00	−0.127064
$990$	0	0
$991$	46752.0	1.49861	0.749307	−	0.662223i	$-0.230387\pi$
$991$	46752.0	1.49861	0.749307	+	0.662223i	$0.230387\pi$
$992$	2432.00	0.0778388
$993$	−22884.0	−0.731321
$994$	13408.0	0.427843
$995$	0	0
$996$	5688.00	0.180955
$997$	37666.0	1.19648	0.598242	−	0.801316i	$-0.295866\pi$
$997$	37666.0	1.19648	0.598242	+	0.801316i	$0.295866\pi$
$998$	34520.0	1.09490
$999$	9234.00	0.292443

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.4.a.k.1.1		1
5.4	even	2		78.4.a.b.1.1	✓	1
15.14	odd	2		234.4.a.j.1.1		1
20.19	odd	2		624.4.a.a.1.1		1
40.19	odd	2		2496.4.a.p.1.1		1
40.29	even	2		2496.4.a.h.1.1		1
60.59	even	2		1872.4.a.p.1.1		1
65.34	odd	4		1014.4.b.f.337.1		2
65.44	odd	4		1014.4.b.f.337.2		2
65.64	even	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	5.4	even	2
234.4.a.j.1.1		1	15.14	odd	2
624.4.a.a.1.1		1	20.19	odd	2
1014.4.a.k.1.1		1	65.64	even	2
1014.4.b.f.337.1		2	65.34	odd	4
1014.4.b.f.337.2		2	65.44	odd	4
1872.4.a.p.1.1		1	60.59	even	2
1950.4.a.k.1.1		1	1.1	even	1	trivial
2496.4.a.h.1.1		1	40.29	even	2
2496.4.a.p.1.1		1	40.19	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:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1950.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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