LMFDB - Embedded newform 2166.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2166,4,Mod(1,2166)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2166, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2166.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2166 = 2 \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2166.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:	$127.798137072$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2166.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +2.00000 q^{5} -6.00000 q^{6} -21.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +2.00000 q^{5} -6.00000 q^{6} -21.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +4.00000 q^{10} -40.0000 q^{11} -12.0000 q^{12} -17.0000 q^{13} -42.0000 q^{14} -6.00000 q^{15} +16.0000 q^{16} +36.0000 q^{17} +18.0000 q^{18} +8.00000 q^{20} +63.0000 q^{21} -80.0000 q^{22} +74.0000 q^{23} -24.0000 q^{24} -121.000 q^{25} -34.0000 q^{26} -27.0000 q^{27} -84.0000 q^{28} -100.000 q^{29} -12.0000 q^{30} -103.000 q^{31} +32.0000 q^{32} +120.000 q^{33} +72.0000 q^{34} -42.0000 q^{35} +36.0000 q^{36} -187.000 q^{37} +51.0000 q^{39} +16.0000 q^{40} +128.000 q^{41} +126.000 q^{42} +121.000 q^{43} -160.000 q^{44} +18.0000 q^{45} +148.000 q^{46} +410.000 q^{47} -48.0000 q^{48} +98.0000 q^{49} -242.000 q^{50} -108.000 q^{51} -68.0000 q^{52} -230.000 q^{53} -54.0000 q^{54} -80.0000 q^{55} -168.000 q^{56} -200.000 q^{58} +744.000 q^{59} -24.0000 q^{60} -277.000 q^{61} -206.000 q^{62} -189.000 q^{63} +64.0000 q^{64} -34.0000 q^{65} +240.000 q^{66} +231.000 q^{67} +144.000 q^{68} -222.000 q^{69} -84.0000 q^{70} -578.000 q^{71} +72.0000 q^{72} +609.000 q^{73} -374.000 q^{74} +363.000 q^{75} +840.000 q^{77} +102.000 q^{78} -1259.00 q^{79} +32.0000 q^{80} +81.0000 q^{81} +256.000 q^{82} -696.000 q^{83} +252.000 q^{84} +72.0000 q^{85} +242.000 q^{86} +300.000 q^{87} -320.000 q^{88} +612.000 q^{89} +36.0000 q^{90} +357.000 q^{91} +296.000 q^{92} +309.000 q^{93} +820.000 q^{94} -96.0000 q^{96} +1550.00 q^{97} +196.000 q^{98} -360.000 q^{99} +O(q^{100})\)

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
$5$	2.00000	0.178885	0.0894427	−	0.995992i	$-0.471491\pi$
$5$	2.00000	0.178885	0.0894427	+	0.995992i	$0.471491\pi$
$6$	−6.00000	−0.408248
$7$	−21.0000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−21.0000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	4.00000	0.126491
$11$	−40.0000	−1.09640	−0.548202	−	0.836346i	$-0.684688\pi$
$11$	−40.0000	−1.09640	−0.548202	+	0.836346i	$0.684688\pi$
$12$	−12.0000	−0.288675
$13$	−17.0000	−0.362689	−0.181344	−	0.983420i	$-0.558045\pi$
$13$	−17.0000	−0.362689	−0.181344	+	0.983420i	$0.558045\pi$
$14$	−42.0000	−0.801784
$15$	−6.00000	−0.103280
$16$	16.0000	0.250000
$17$	36.0000	0.513605	0.256802	−	0.966464i	$-0.417331\pi$
$17$	36.0000	0.513605	0.256802	+	0.966464i	$0.417331\pi$
$18$	18.0000	0.235702
$19$	0	0
$19$	0	0
$20$	8.00000	0.0894427
$21$	63.0000	0.654654
$22$	−80.0000	−0.775275
$23$	74.0000	0.670872	0.335436	−	0.942063i	$-0.391116\pi$
$23$	74.0000	0.670872	0.335436	+	0.942063i	$0.391116\pi$
$24$	−24.0000	−0.204124
$25$	−121.000	−0.968000
$26$	−34.0000	−0.256460
$27$	−27.0000	−0.192450
$28$	−84.0000	−0.566947
$29$	−100.000	−0.640329	−0.320164	−	0.947362i	$-0.603738\pi$
$29$	−100.000	−0.640329	−0.320164	+	0.947362i	$0.603738\pi$
$30$	−12.0000	−0.0730297
$31$	−103.000	−0.596753	−0.298377	−	0.954448i	$-0.596445\pi$
$31$	−103.000	−0.596753	−0.298377	+	0.954448i	$0.596445\pi$
$32$	32.0000	0.176777
$33$	120.000	0.633010
$34$	72.0000	0.363173
$35$	−42.0000	−0.202837
$36$	36.0000	0.166667
$37$	−187.000	−0.830881	−0.415441	−	0.909620i	$-0.636373\pi$
$37$	−187.000	−0.830881	−0.415441	+	0.909620i	$0.636373\pi$
$38$	0	0
$39$	51.0000	0.209398
$40$	16.0000	0.0632456
$41$	128.000	0.487567	0.243783	−	0.969830i	$-0.421611\pi$
$41$	128.000	0.487567	0.243783	+	0.969830i	$0.421611\pi$
$42$	126.000	0.462910
$43$	121.000	0.429124	0.214562	−	0.976710i	$-0.431168\pi$
$43$	121.000	0.429124	0.214562	+	0.976710i	$0.431168\pi$
$44$	−160.000	−0.548202
$45$	18.0000	0.0596285
$46$	148.000	0.474378
$47$	410.000	1.27244	0.636220	−	0.771508i	$-0.280497\pi$
$47$	410.000	1.27244	0.636220	+	0.771508i	$0.280497\pi$
$48$	−48.0000	−0.144338
$49$	98.0000	0.285714
$50$	−242.000	−0.684479
$51$	−108.000	−0.296530
$52$	−68.0000	−0.181344
$53$	−230.000	−0.596093	−0.298047	−	0.954551i	$-0.596335\pi$
$53$	−230.000	−0.596093	−0.298047	+	0.954551i	$0.596335\pi$
$54$	−54.0000	−0.136083
$55$	−80.0000	−0.196131
$56$	−168.000	−0.400892
$57$	0	0
$58$	−200.000	−0.452781
$59$	744.000	1.64170	0.820852	−	0.571141i	$-0.193499\pi$
$59$	744.000	1.64170	0.820852	+	0.571141i	$0.193499\pi$
$60$	−24.0000	−0.0516398
$61$	−277.000	−0.581413	−0.290707	−	0.956812i	$-0.593890\pi$
$61$	−277.000	−0.581413	−0.290707	+	0.956812i	$0.593890\pi$
$62$	−206.000	−0.421968
$63$	−189.000	−0.377964
$64$	64.0000	0.125000
$65$	−34.0000	−0.0648797
$66$	240.000	0.447605
$67$	231.000	0.421211	0.210606	−	0.977571i	$-0.432456\pi$
$67$	231.000	0.421211	0.210606	+	0.977571i	$0.432456\pi$
$68$	144.000	0.256802
$69$	−222.000	−0.387328
$70$	−84.0000	−0.143427
$71$	−578.000	−0.966141	−0.483070	−	0.875582i	$-0.660478\pi$
$71$	−578.000	−0.966141	−0.483070	+	0.875582i	$0.660478\pi$
$72$	72.0000	0.117851
$73$	609.000	0.976412	0.488206	−	0.872728i	$-0.337652\pi$
$73$	609.000	0.976412	0.488206	+	0.872728i	$0.337652\pi$
$74$	−374.000	−0.587522
$75$	363.000	0.558875
$76$	0	0
$77$	840.000	1.24321
$78$	102.000	0.148067
$79$	−1259.00	−1.79302	−0.896510	−	0.443024i	$-0.853906\pi$
$79$	−1259.00	−1.79302	−0.896510	+	0.443024i	$0.853906\pi$
$80$	32.0000	0.0447214
$81$	81.0000	0.111111
$82$	256.000	0.344762
$83$	−696.000	−0.920433	−0.460216	−	0.887807i	$-0.652228\pi$
$83$	−696.000	−0.920433	−0.460216	+	0.887807i	$0.652228\pi$
$84$	252.000	0.327327
$85$	72.0000	0.0918764
$86$	242.000	0.303436
$87$	300.000	0.369694
$88$	−320.000	−0.387638
$89$	612.000	0.728897	0.364449	−	0.931223i	$-0.381257\pi$
$89$	612.000	0.728897	0.364449	+	0.931223i	$0.381257\pi$
$90$	36.0000	0.0421637
$91$	357.000	0.411250
$92$	296.000	0.335436
$93$	309.000	0.344536
$94$	820.000	0.899750
$95$	0	0
$96$	−96.0000	−0.102062
$97$	1550.00	1.62246	0.811230	−	0.584727i	$-0.198798\pi$
$97$	1550.00	1.62246	0.811230	+	0.584727i	$0.198798\pi$
$98$	196.000	0.202031
$99$	−360.000	−0.365468
$100$	−484.000	−0.484000
$101$	−418.000	−0.411807	−0.205904	−	0.978572i	$-0.566013\pi$
$101$	−418.000	−0.411807	−0.205904	+	0.978572i	$0.566013\pi$
$102$	−216.000	−0.209678
$103$	1701.00	1.62723	0.813614	−	0.581405i	$-0.197497\pi$
$103$	1701.00	1.62723	0.813614	+	0.581405i	$0.197497\pi$
$104$	−136.000	−0.128230
$105$	126.000	0.117108
$106$	−460.000	−0.421501
$107$	1690.00	1.52690	0.763451	−	0.645866i	$-0.223504\pi$
$107$	1690.00	1.52690	0.763451	+	0.645866i	$0.223504\pi$
$108$	−108.000	−0.0962250
$109$	−42.0000	−0.0369071	−0.0184535	−	0.999830i	$-0.505874\pi$
$109$	−42.0000	−0.0369071	−0.0184535	+	0.999830i	$0.505874\pi$
$110$	−160.000	−0.138685
$111$	561.000	0.479710
$112$	−336.000	−0.283473
$113$	1828.00	1.52180	0.760902	−	0.648867i	$-0.224757\pi$
$113$	1828.00	1.52180	0.760902	+	0.648867i	$0.224757\pi$
$114$	0	0
$115$	148.000	0.120009
$116$	−400.000	−0.320164
$117$	−153.000	−0.120896
$118$	1488.00	1.16086
$119$	−756.000	−0.582373
$120$	−48.0000	−0.0365148
$121$	269.000	0.202104
$122$	−554.000	−0.411121
$123$	−384.000	−0.281497
$124$	−412.000	−0.298377
$125$	−492.000	−0.352047
$126$	−378.000	−0.267261
$127$	2124.00	1.48405	0.742026	−	0.670371i	$-0.233865\pi$
$127$	2124.00	1.48405	0.742026	+	0.670371i	$0.233865\pi$
$128$	128.000	0.0883883
$129$	−363.000	−0.247755
$130$	−68.0000	−0.0458769
$131$	1206.00	0.804341	0.402171	−	0.915565i	$-0.368256\pi$
$131$	1206.00	0.804341	0.402171	+	0.915565i	$0.368256\pi$
$132$	480.000	0.316505
$133$	0	0
$134$	462.000	0.297841
$135$	−54.0000	−0.0344265
$136$	288.000	0.181587
$137$	310.000	0.193322	0.0966609	−	0.995317i	$-0.469184\pi$
$137$	310.000	0.193322	0.0966609	+	0.995317i	$0.469184\pi$
$138$	−444.000	−0.273883
$139$	1335.00	0.814627	0.407314	−	0.913288i	$-0.366466\pi$
$139$	1335.00	0.814627	0.407314	+	0.913288i	$0.366466\pi$
$140$	−168.000	−0.101419
$141$	−1230.00	−0.734643
$142$	−1156.00	−0.683165
$143$	680.000	0.397654
$144$	144.000	0.0833333
$145$	−200.000	−0.114545
$146$	1218.00	0.690427
$147$	−294.000	−0.164957
$148$	−748.000	−0.415441
$149$	−1116.00	−0.613599	−0.306800	−	0.951774i	$-0.599258\pi$
$149$	−1116.00	−0.613599	−0.306800	+	0.951774i	$0.599258\pi$
$150$	726.000	0.395184
$151$	−1328.00	−0.715703	−0.357851	−	0.933779i	$-0.616491\pi$
$151$	−1328.00	−0.715703	−0.357851	+	0.933779i	$0.616491\pi$
$152$	0	0
$153$	324.000	0.171202
$154$	1680.00	0.879080
$155$	−206.000	−0.106750
$156$	204.000	0.104699
$157$	−2537.00	−1.28965	−0.644824	−	0.764331i	$-0.723069\pi$
$157$	−2537.00	−1.28965	−0.644824	+	0.764331i	$0.723069\pi$
$158$	−2518.00	−1.26786
$159$	690.000	0.344154
$160$	64.0000	0.0316228
$161$	−1554.00	−0.760698
$162$	162.000	0.0785674
$163$	101.000	0.0485333	0.0242667	−	0.999706i	$-0.492275\pi$
$163$	101.000	0.0485333	0.0242667	+	0.999706i	$0.492275\pi$
$164$	512.000	0.243783
$165$	240.000	0.113236
$166$	−1392.00	−0.650844
$167$	−240.000	−0.111208	−0.0556041	−	0.998453i	$-0.517708\pi$
$167$	−240.000	−0.111208	−0.0556041	+	0.998453i	$0.517708\pi$
$168$	504.000	0.231455
$169$	−1908.00	−0.868457
$170$	144.000	0.0649664
$171$	0	0
$172$	484.000	0.214562
$173$	1428.00	0.627565	0.313783	−	0.949495i	$-0.398404\pi$
$173$	1428.00	0.627565	0.313783	+	0.949495i	$0.398404\pi$
$174$	600.000	0.261413
$175$	2541.00	1.09761
$176$	−640.000	−0.274101
$177$	−2232.00	−0.947838
$178$	1224.00	0.515408
$179$	2046.00	0.854331	0.427165	−	0.904173i	$-0.359512\pi$
$179$	2046.00	0.854331	0.427165	+	0.904173i	$0.359512\pi$
$180$	72.0000	0.0298142
$181$	−1510.00	−0.620096	−0.310048	−	0.950721i	$-0.600345\pi$
$181$	−1510.00	−0.620096	−0.310048	+	0.950721i	$0.600345\pi$
$182$	714.000	0.290798
$183$	831.000	0.335679
$184$	592.000	0.237189
$185$	−374.000	−0.148633
$186$	618.000	0.243623
$187$	−1440.00	−0.563119
$188$	1640.00	0.636220
$189$	567.000	0.218218
$190$	0	0
$191$	2290.00	0.867532	0.433766	−	0.901026i	$-0.357184\pi$
$191$	2290.00	0.867532	0.433766	+	0.901026i	$0.357184\pi$
$192$	−192.000	−0.0721688
$193$	−627.000	−0.233847	−0.116923	−	0.993141i	$-0.537303\pi$
$193$	−627.000	−0.233847	−0.116923	+	0.993141i	$0.537303\pi$
$194$	3100.00	1.14725
$195$	102.000	0.0374583
$196$	392.000	0.142857
$197$	−748.000	−0.270522	−0.135261	−	0.990810i	$-0.543187\pi$
$197$	−748.000	−0.270522	−0.135261	+	0.990810i	$0.543187\pi$
$198$	−720.000	−0.258425
$199$	2171.00	0.773357	0.386679	−	0.922215i	$-0.373622\pi$
$199$	2171.00	0.773357	0.386679	+	0.922215i	$0.373622\pi$
$200$	−968.000	−0.342240
$201$	−693.000	−0.243186
$202$	−836.000	−0.291192
$203$	2100.00	0.726065
$204$	−432.000	−0.148265
$205$	256.000	0.0872186
$206$	3402.00	1.15062
$207$	666.000	0.223624
$208$	−272.000	−0.0906721
$209$	0	0
$210$	252.000	0.0828079
$211$	3177.00	1.03656	0.518279	−	0.855212i	$-0.326573\pi$
$211$	3177.00	1.03656	0.518279	+	0.855212i	$0.326573\pi$
$212$	−920.000	−0.298047
$213$	1734.00	0.557802
$214$	3380.00	1.07968
$215$	242.000	0.0767640
$216$	−216.000	−0.0680414
$217$	2163.00	0.676654
$218$	−84.0000	−0.0260972
$219$	−1827.00	−0.563732
$220$	−320.000	−0.0980654
$221$	−612.000	−0.186279
$222$	1122.00	0.339206
$223$	−4751.00	−1.42668	−0.713342	−	0.700816i	$-0.752819\pi$
$223$	−4751.00	−1.42668	−0.713342	+	0.700816i	$0.752819\pi$
$224$	−672.000	−0.200446
$225$	−1089.00	−0.322667
$226$	3656.00	1.07608
$227$	4706.00	1.37598	0.687992	−	0.725719i	$-0.258493\pi$
$227$	4706.00	1.37598	0.687992	+	0.725719i	$0.258493\pi$
$228$	0	0
$229$	−4807.00	−1.38714	−0.693571	−	0.720388i	$-0.743964\pi$
$229$	−4807.00	−1.38714	−0.693571	+	0.720388i	$0.743964\pi$
$230$	296.000	0.0848594
$231$	−2520.00	−0.717765
$232$	−800.000	−0.226390
$233$	546.000	0.153518	0.0767589	−	0.997050i	$-0.475543\pi$
$233$	546.000	0.153518	0.0767589	+	0.997050i	$0.475543\pi$
$234$	−306.000	−0.0854865
$235$	820.000	0.227621
$236$	2976.00	0.820852
$237$	3777.00	1.03520
$238$	−1512.00	−0.411800
$239$	3966.00	1.07339	0.536693	−	0.843778i	$-0.319673\pi$
$239$	3966.00	1.07339	0.536693	+	0.843778i	$0.319673\pi$
$240$	−96.0000	−0.0258199
$241$	5051.00	1.35006	0.675028	−	0.737792i	$-0.264132\pi$
$241$	5051.00	1.35006	0.675028	+	0.737792i	$0.264132\pi$
$242$	538.000	0.142909
$243$	−243.000	−0.0641500
$244$	−1108.00	−0.290707
$245$	196.000	0.0511101
$246$	−768.000	−0.199048
$247$	0	0
$248$	−824.000	−0.210984
$249$	2088.00	0.531412
$250$	−984.000	−0.248934
$251$	4370.00	1.09893	0.549466	−	0.835516i	$-0.314831\pi$
$251$	4370.00	1.09893	0.549466	+	0.835516i	$0.314831\pi$
$252$	−756.000	−0.188982
$253$	−2960.00	−0.735548
$254$	4248.00	1.04938
$255$	−216.000	−0.0530449
$256$	256.000	0.0625000
$257$	670.000	0.162620	0.0813102	−	0.996689i	$-0.474090\pi$
$257$	670.000	0.162620	0.0813102	+	0.996689i	$0.474090\pi$
$258$	−726.000	−0.175189
$259$	3927.00	0.942131
$260$	−136.000	−0.0324399
$261$	−900.000	−0.213443
$262$	2412.00	0.568755
$263$	−7002.00	−1.64168	−0.820840	−	0.571158i	$-0.806494\pi$
$263$	−7002.00	−1.64168	−0.820840	+	0.571158i	$0.806494\pi$
$264$	960.000	0.223803
$265$	−460.000	−0.106632
$266$	0	0
$267$	−1836.00	−0.420829
$268$	924.000	0.210606
$269$	−2484.00	−0.563019	−0.281510	−	0.959558i	$-0.590835\pi$
$269$	−2484.00	−0.563019	−0.281510	+	0.959558i	$0.590835\pi$
$270$	−108.000	−0.0243432
$271$	3736.00	0.837439	0.418719	−	0.908116i	$-0.362479\pi$
$271$	3736.00	0.837439	0.418719	+	0.908116i	$0.362479\pi$
$272$	576.000	0.128401
$273$	−1071.00	−0.237435
$274$	620.000	0.136699
$275$	4840.00	1.06132
$276$	−888.000	−0.193664
$277$	574.000	0.124507	0.0622533	−	0.998060i	$-0.480171\pi$
$277$	574.000	0.124507	0.0622533	+	0.998060i	$0.480171\pi$
$278$	2670.00	0.576029
$279$	−927.000	−0.198918
$280$	−336.000	−0.0717137
$281$	9278.00	1.96968	0.984838	−	0.173475i	$-0.0554997\pi$
$281$	9278.00	1.96968	0.984838	+	0.173475i	$0.0554997\pi$
$282$	−2460.00	−0.519471
$283$	5244.00	1.10150	0.550748	−	0.834671i	$-0.314342\pi$
$283$	5244.00	1.10150	0.550748	+	0.834671i	$0.314342\pi$
$284$	−2312.00	−0.483070
$285$	0	0
$286$	1360.00	0.281184
$287$	−2688.00	−0.552849
$288$	288.000	0.0589256
$289$	−3617.00	−0.736210
$290$	−400.000	−0.0809959
$291$	−4650.00	−0.936728
$292$	2436.00	0.488206
$293$	666.000	0.132792	0.0663961	−	0.997793i	$-0.478850\pi$
$293$	666.000	0.132792	0.0663961	+	0.997793i	$0.478850\pi$
$294$	−588.000	−0.116642
$295$	1488.00	0.293677
$296$	−1496.00	−0.293761
$297$	1080.00	0.211003
$298$	−2232.00	−0.433880
$299$	−1258.00	−0.243318
$300$	1452.00	0.279438
$301$	−2541.00	−0.486581
$302$	−2656.00	−0.506078
$303$	1254.00	0.237757
$304$	0	0
$305$	−554.000	−0.104006
$306$	648.000	0.121058
$307$	−2388.00	−0.443943	−0.221971	−	0.975053i	$-0.571249\pi$
$307$	−2388.00	−0.443943	−0.221971	+	0.975053i	$0.571249\pi$
$308$	3360.00	0.621603
$309$	−5103.00	−0.939481
$310$	−412.000	−0.0754840
$311$	9460.00	1.72485	0.862423	−	0.506188i	$-0.168946\pi$
$311$	9460.00	1.72485	0.862423	+	0.506188i	$0.168946\pi$
$312$	408.000	0.0740335
$313$	10682.0	1.92902	0.964509	−	0.264052i	$-0.0850590\pi$
$313$	10682.0	1.92902	0.964509	+	0.264052i	$0.0850590\pi$
$314$	−5074.00	−0.911918
$315$	−378.000	−0.0676123
$316$	−5036.00	−0.896510
$317$	4542.00	0.804745	0.402372	−	0.915476i	$-0.368186\pi$
$317$	4542.00	0.804745	0.402372	+	0.915476i	$0.368186\pi$
$318$	1380.00	0.243354
$319$	4000.00	0.702060
$320$	128.000	0.0223607
$321$	−5070.00	−0.881557
$322$	−3108.00	−0.537895
$323$	0	0
$324$	324.000	0.0555556
$325$	2057.00	0.351083
$326$	202.000	0.0343182
$327$	126.000	0.0213083
$328$	1024.00	0.172381
$329$	−8610.00	−1.44281
$330$	480.000	0.0800701
$331$	−1313.00	−0.218033	−0.109017	−	0.994040i	$-0.534770\pi$
$331$	−1313.00	−0.218033	−0.109017	+	0.994040i	$0.534770\pi$
$332$	−2784.00	−0.460216
$333$	−1683.00	−0.276960
$334$	−480.000	−0.0786360
$335$	462.000	0.0753485
$336$	1008.00	0.163663
$337$	−6649.00	−1.07476	−0.537380	−	0.843340i	$-0.680586\pi$
$337$	−6649.00	−1.07476	−0.537380	+	0.843340i	$0.680586\pi$
$338$	−3816.00	−0.614092
$339$	−5484.00	−0.878614
$340$	288.000	0.0459382
$341$	4120.00	0.654283
$342$	0	0
$343$	5145.00	0.809924
$344$	968.000	0.151718
$345$	−444.000	−0.0692874
$346$	2856.00	0.443756
$347$	−10706.0	−1.65628	−0.828139	−	0.560523i	$-0.810600\pi$
$347$	−10706.0	−1.65628	−0.828139	+	0.560523i	$0.810600\pi$
$348$	1200.00	0.184847
$349$	3799.00	0.582681	0.291341	−	0.956619i	$-0.405899\pi$
$349$	3799.00	0.582681	0.291341	+	0.956619i	$0.405899\pi$
$350$	5082.00	0.776127
$351$	459.000	0.0697995
$352$	−1280.00	−0.193819
$353$	−10926.0	−1.64740	−0.823700	−	0.567026i	$-0.808094\pi$
$353$	−10926.0	−1.64740	−0.823700	+	0.567026i	$0.808094\pi$
$354$	−4464.00	−0.670223
$355$	−1156.00	−0.172828
$356$	2448.00	0.364449
$357$	2268.00	0.336233
$358$	4092.00	0.604103
$359$	204.000	0.0299908	0.0149954	−	0.999888i	$-0.495227\pi$
$359$	204.000	0.0299908	0.0149954	+	0.999888i	$0.495227\pi$
$360$	144.000	0.0210819
$361$	0	0
$362$	−3020.00	−0.438474
$363$	−807.000	−0.116685
$364$	1428.00	0.205625
$365$	1218.00	0.174666
$366$	1662.00	0.237361
$367$	−9119.00	−1.29702	−0.648512	−	0.761204i	$-0.724608\pi$
$367$	−9119.00	−1.29702	−0.648512	+	0.761204i	$0.724608\pi$
$368$	1184.00	0.167718
$369$	1152.00	0.162522
$370$	−748.000	−0.105099
$371$	4830.00	0.675906
$372$	1236.00	0.172268
$373$	13102.0	1.81876	0.909378	−	0.415971i	$-0.136558\pi$
$373$	13102.0	1.81876	0.909378	+	0.415971i	$0.136558\pi$
$374$	−2880.00	−0.398185
$375$	1476.00	0.203254
$376$	3280.00	0.449875
$377$	1700.00	0.232240
$378$	1134.00	0.154303
$379$	−2841.00	−0.385046	−0.192523	−	0.981292i	$-0.561667\pi$
$379$	−2841.00	−0.385046	−0.192523	+	0.981292i	$0.561667\pi$
$380$	0	0
$381$	−6372.00	−0.856817
$382$	4580.00	0.613438
$383$	14024.0	1.87100	0.935500	−	0.353327i	$-0.114950\pi$
$383$	14024.0	1.87100	0.935500	+	0.353327i	$0.114950\pi$
$384$	−384.000	−0.0510310
$385$	1680.00	0.222392
$386$	−1254.00	−0.165355
$387$	1089.00	0.143041
$388$	6200.00	0.811230
$389$	−9438.00	−1.23014	−0.615071	−	0.788471i	$-0.710873\pi$
$389$	−9438.00	−1.23014	−0.615071	+	0.788471i	$0.710873\pi$
$390$	204.000	0.0264870
$391$	2664.00	0.344563
$392$	784.000	0.101015
$393$	−3618.00	−0.464387
$394$	−1496.00	−0.191288
$395$	−2518.00	−0.320745
$396$	−1440.00	−0.182734
$397$	8267.00	1.04511	0.522555	−	0.852605i	$-0.324979\pi$
$397$	8267.00	1.04511	0.522555	+	0.852605i	$0.324979\pi$
$398$	4342.00	0.546846
$399$	0	0
$400$	−1936.00	−0.242000
$401$	−6664.00	−0.829886	−0.414943	−	0.909847i	$-0.636198\pi$
$401$	−6664.00	−0.829886	−0.414943	+	0.909847i	$0.636198\pi$
$402$	−1386.00	−0.171959
$403$	1751.00	0.216436
$404$	−1672.00	−0.205904
$405$	162.000	0.0198762
$406$	4200.00	0.513405
$407$	7480.00	0.910982
$408$	−864.000	−0.104839
$409$	−2038.00	−0.246388	−0.123194	−	0.992383i	$-0.539314\pi$
$409$	−2038.00	−0.246388	−0.123194	+	0.992383i	$0.539314\pi$
$410$	512.000	0.0616729
$411$	−930.000	−0.111614
$412$	6804.00	0.813614
$413$	−15624.0	−1.86152
$414$	1332.00	0.158126
$415$	−1392.00	−0.164652
$416$	−544.000	−0.0641149
$417$	−4005.00	−0.470325
$418$	0	0
$419$	8832.00	1.02976	0.514882	−	0.857261i	$-0.327836\pi$
$419$	8832.00	1.02976	0.514882	+	0.857261i	$0.327836\pi$
$420$	504.000	0.0585540
$421$	9826.00	1.13751	0.568753	−	0.822508i	$-0.307426\pi$
$421$	9826.00	1.13751	0.568753	+	0.822508i	$0.307426\pi$
$422$	6354.00	0.732957
$423$	3690.00	0.424146
$424$	−1840.00	−0.210751
$425$	−4356.00	−0.497169
$426$	3468.00	0.394425
$427$	5817.00	0.659261
$428$	6760.00	0.763451
$429$	−2040.00	−0.229585
$430$	484.000	0.0542804
$431$	8850.00	0.989071	0.494535	−	0.869157i	$-0.335338\pi$
$431$	8850.00	0.989071	0.494535	+	0.869157i	$0.335338\pi$
$432$	−432.000	−0.0481125
$433$	−10643.0	−1.18122	−0.590612	−	0.806956i	$-0.701114\pi$
$433$	−10643.0	−1.18122	−0.590612	+	0.806956i	$0.701114\pi$
$434$	4326.00	0.478467
$435$	600.000	0.0661329
$436$	−168.000	−0.0184535
$437$	0	0
$438$	−3654.00	−0.398618
$439$	10387.0	1.12926	0.564629	−	0.825345i	$-0.309019\pi$
$439$	10387.0	1.12926	0.564629	+	0.825345i	$0.309019\pi$
$440$	−640.000	−0.0693427
$441$	882.000	0.0952381
$442$	−1224.00	−0.131719
$443$	6652.00	0.713422	0.356711	−	0.934215i	$-0.383898\pi$
$443$	6652.00	0.713422	0.356711	+	0.934215i	$0.383898\pi$
$444$	2244.00	0.239855
$445$	1224.00	0.130389
$446$	−9502.00	−1.00882
$447$	3348.00	0.354262
$448$	−1344.00	−0.141737
$449$	16976.0	1.78429	0.892146	−	0.451747i	$-0.149199\pi$
$449$	16976.0	1.78429	0.892146	+	0.451747i	$0.149199\pi$
$450$	−2178.00	−0.228160
$451$	−5120.00	−0.534571
$452$	7312.00	0.760902
$453$	3984.00	0.413211
$454$	9412.00	0.972967
$455$	714.000	0.0735667
$456$	0	0
$457$	−6367.00	−0.651719	−0.325860	−	0.945418i	$-0.605654\pi$
$457$	−6367.00	−0.651719	−0.325860	+	0.945418i	$0.605654\pi$
$458$	−9614.00	−0.980857
$459$	−972.000	−0.0988433
$460$	592.000	0.0600047
$461$	−14008.0	−1.41522	−0.707611	−	0.706602i	$-0.750227\pi$
$461$	−14008.0	−1.41522	−0.707611	+	0.706602i	$0.750227\pi$
$462$	−5040.00	−0.507537
$463$	14161.0	1.42142	0.710710	−	0.703485i	$-0.248374\pi$
$463$	14161.0	1.42142	0.710710	+	0.703485i	$0.248374\pi$
$464$	−1600.00	−0.160082
$465$	618.000	0.0616324
$466$	1092.00	0.108553
$467$	−12356.0	−1.22434	−0.612171	−	0.790726i	$-0.709703\pi$
$467$	−12356.0	−1.22434	−0.612171	+	0.790726i	$0.709703\pi$
$468$	−612.000	−0.0604481
$469$	−4851.00	−0.477608
$470$	1640.00	0.160952
$471$	7611.00	0.744578
$472$	5952.00	0.580430
$473$	−4840.00	−0.470494
$474$	7554.00	0.731997
$475$	0	0
$476$	−3024.00	−0.291187
$477$	−2070.00	−0.198698
$478$	7932.00	0.758998
$479$	−10094.0	−0.962853	−0.481427	−	0.876486i	$-0.659881\pi$
$479$	−10094.0	−0.962853	−0.481427	+	0.876486i	$0.659881\pi$
$480$	−192.000	−0.0182574
$481$	3179.00	0.301351
$482$	10102.0	0.954634
$483$	4662.00	0.439189
$484$	1076.00	0.101052
$485$	3100.00	0.290235
$486$	−486.000	−0.0453609
$487$	7580.00	0.705303	0.352652	−	0.935755i	$-0.385280\pi$
$487$	7580.00	0.705303	0.352652	+	0.935755i	$0.385280\pi$
$488$	−2216.00	−0.205561
$489$	−303.000	−0.0280207
$490$	392.000	0.0361403
$491$	−19462.0	−1.78881	−0.894407	−	0.447254i	$-0.852402\pi$
$491$	−19462.0	−1.78881	−0.894407	+	0.447254i	$0.852402\pi$
$492$	−1536.00	−0.140748
$493$	−3600.00	−0.328876
$494$	0	0
$495$	−720.000	−0.0653770
$496$	−1648.00	−0.149188
$497$	12138.0	1.09550
$498$	4176.00	0.375765
$499$	14383.0	1.29032	0.645162	−	0.764046i	$-0.276790\pi$
$499$	14383.0	1.29032	0.645162	+	0.764046i	$0.276790\pi$
$500$	−1968.00	−0.176023
$501$	720.000	0.0642060
$502$	8740.00	0.777062
$503$	1716.00	0.152113	0.0760563	−	0.997104i	$-0.475767\pi$
$503$	1716.00	0.152113	0.0760563	+	0.997104i	$0.475767\pi$
$504$	−1512.00	−0.133631
$505$	−836.000	−0.0736664
$506$	−5920.00	−0.520111
$507$	5724.00	0.501404
$508$	8496.00	0.742026
$509$	−3954.00	−0.344318	−0.172159	−	0.985069i	$-0.555074\pi$
$509$	−3954.00	−0.344318	−0.172159	+	0.985069i	$0.555074\pi$
$510$	−432.000	−0.0375084
$511$	−12789.0	−1.10715
$512$	512.000	0.0441942
$513$	0	0
$514$	1340.00	0.114990
$515$	3402.00	0.291087
$516$	−1452.00	−0.123877
$517$	−16400.0	−1.39511
$518$	7854.00	0.666187
$519$	−4284.00	−0.362325
$520$	−272.000	−0.0229384
$521$	11356.0	0.954924	0.477462	−	0.878652i	$-0.341557\pi$
$521$	11356.0	0.954924	0.477462	+	0.878652i	$0.341557\pi$
$522$	−1800.00	−0.150927
$523$	−14627.0	−1.22293	−0.611467	−	0.791270i	$-0.709420\pi$
$523$	−14627.0	−1.22293	−0.611467	+	0.791270i	$0.709420\pi$
$524$	4824.00	0.402171
$525$	−7623.00	−0.633705
$526$	−14004.0	−1.16084
$527$	−3708.00	−0.306495
$528$	1920.00	0.158252
$529$	−6691.00	−0.549930
$530$	−920.000	−0.0754005
$531$	6696.00	0.547235
$532$	0	0
$533$	−2176.00	−0.176835
$534$	−3672.00	−0.297571
$535$	3380.00	0.273140
$536$	1848.00	0.148921
$537$	−6138.00	−0.493248
$538$	−4968.00	−0.398115
$539$	−3920.00	−0.313259
$540$	−216.000	−0.0172133
$541$	−6355.00	−0.505033	−0.252516	−	0.967593i	$-0.581258\pi$
$541$	−6355.00	−0.505033	−0.252516	+	0.967593i	$0.581258\pi$
$542$	7472.00	0.592158
$543$	4530.00	0.358013
$544$	1152.00	0.0907934
$545$	−84.0000	−0.00660214
$546$	−2142.00	−0.167892
$547$	−8077.00	−0.631348	−0.315674	−	0.948868i	$-0.602231\pi$
$547$	−8077.00	−0.631348	−0.315674	+	0.948868i	$0.602231\pi$
$548$	1240.00	0.0966609
$549$	−2493.00	−0.193804
$550$	9680.00	0.750467
$551$	0	0
$552$	−1776.00	−0.136941
$553$	26439.0	2.03309
$554$	1148.00	0.0880394
$555$	1122.00	0.0858131
$556$	5340.00	0.407314
$557$	−10346.0	−0.787027	−0.393514	−	0.919319i	$-0.628741\pi$
$557$	−10346.0	−0.787027	−0.393514	+	0.919319i	$0.628741\pi$
$558$	−1854.00	−0.140656
$559$	−2057.00	−0.155638
$560$	−672.000	−0.0507093
$561$	4320.00	0.325117
$562$	18556.0	1.39277
$563$	6126.00	0.458579	0.229290	−	0.973358i	$-0.426360\pi$
$563$	6126.00	0.458579	0.229290	+	0.973358i	$0.426360\pi$
$564$	−4920.00	−0.367322
$565$	3656.00	0.272228
$566$	10488.0	0.778875
$567$	−1701.00	−0.125988
$568$	−4624.00	−0.341582
$569$	−1444.00	−0.106390	−0.0531948	−	0.998584i	$-0.516940\pi$
$569$	−1444.00	−0.106390	−0.0531948	+	0.998584i	$0.516940\pi$
$570$	0	0
$571$	20951.0	1.53550	0.767751	−	0.640748i	$-0.221376\pi$
$571$	20951.0	1.53550	0.767751	+	0.640748i	$0.221376\pi$
$572$	2720.00	0.198827
$573$	−6870.00	−0.500870
$574$	−5376.00	−0.390923
$575$	−8954.00	−0.649405
$576$	576.000	0.0416667
$577$	8678.00	0.626118	0.313059	−	0.949734i	$-0.398646\pi$
$577$	8678.00	0.626118	0.313059	+	0.949734i	$0.398646\pi$
$578$	−7234.00	−0.520579
$579$	1881.00	0.135012
$580$	−800.000	−0.0572727
$581$	14616.0	1.04367
$582$	−9300.00	−0.662367
$583$	9200.00	0.653559
$584$	4872.00	0.345214
$585$	−306.000	−0.0216266
$586$	1332.00	0.0938983
$587$	−11700.0	−0.822676	−0.411338	−	0.911483i	$-0.634938\pi$
$587$	−11700.0	−0.822676	−0.411338	+	0.911483i	$0.634938\pi$
$588$	−1176.00	−0.0824786
$589$	0	0
$590$	2976.00	0.207661
$591$	2244.00	0.156186
$592$	−2992.00	−0.207720
$593$	−6968.00	−0.482532	−0.241266	−	0.970459i	$-0.577563\pi$
$593$	−6968.00	−0.482532	−0.241266	+	0.970459i	$0.577563\pi$
$594$	2160.00	0.149202
$595$	−1512.00	−0.104178
$596$	−4464.00	−0.306800
$597$	−6513.00	−0.446498
$598$	−2516.00	−0.172052
$599$	10248.0	0.699035	0.349517	−	0.936930i	$-0.386346\pi$
$599$	10248.0	0.699035	0.349517	+	0.936930i	$0.386346\pi$
$600$	2904.00	0.197592
$601$	1505.00	0.102147	0.0510734	−	0.998695i	$-0.483736\pi$
$601$	1505.00	0.102147	0.0510734	+	0.998695i	$0.483736\pi$
$602$	−5082.00	−0.344065
$603$	2079.00	0.140404
$604$	−5312.00	−0.357851
$605$	538.000	0.0361534
$606$	2508.00	0.168120
$607$	−22187.0	−1.48360	−0.741798	−	0.670624i	$-0.766027\pi$
$607$	−22187.0	−1.48360	−0.741798	+	0.670624i	$0.766027\pi$
$608$	0	0
$609$	−6300.00	−0.419194
$610$	−1108.00	−0.0735436
$611$	−6970.00	−0.461499
$612$	1296.00	0.0856008
$613$	9682.00	0.637932	0.318966	−	0.947766i	$-0.396664\pi$
$613$	9682.00	0.637932	0.318966	+	0.947766i	$0.396664\pi$
$614$	−4776.00	−0.313915
$615$	−768.000	−0.0503557
$616$	6720.00	0.439540
$617$	2844.00	0.185567	0.0927837	−	0.995686i	$-0.470423\pi$
$617$	2844.00	0.185567	0.0927837	+	0.995686i	$0.470423\pi$
$618$	−10206.0	−0.664313
$619$	−27809.0	−1.80572	−0.902858	−	0.429939i	$-0.858535\pi$
$619$	−27809.0	−1.80572	−0.902858	+	0.429939i	$0.858535\pi$
$620$	−824.000	−0.0533752
$621$	−1998.00	−0.129109
$622$	18920.0	1.21965
$623$	−12852.0	−0.826492
$624$	816.000	0.0523496
$625$	14141.0	0.905024
$626$	21364.0	1.36402
$627$	0	0
$628$	−10148.0	−0.644824
$629$	−6732.00	−0.426745
$630$	−756.000	−0.0478091
$631$	−28477.0	−1.79660	−0.898298	−	0.439388i	$-0.855195\pi$
$631$	−28477.0	−1.79660	−0.898298	+	0.439388i	$0.855195\pi$
$632$	−10072.0	−0.633928
$633$	−9531.00	−0.598457
$634$	9084.00	0.569041
$635$	4248.00	0.265475
$636$	2760.00	0.172077
$637$	−1666.00	−0.103625
$638$	8000.00	0.496431
$639$	−5202.00	−0.322047
$640$	256.000	0.0158114
$641$	−10926.0	−0.673247	−0.336623	−	0.941639i	$-0.609285\pi$
$641$	−10926.0	−0.673247	−0.336623	+	0.941639i	$0.609285\pi$
$642$	−10140.0	−0.623355
$643$	−5465.00	−0.335177	−0.167588	−	0.985857i	$-0.553598\pi$
$643$	−5465.00	−0.335177	−0.167588	+	0.985857i	$0.553598\pi$
$644$	−6216.00	−0.380349
$645$	−726.000	−0.0443197
$646$	0	0
$647$	−25876.0	−1.57232	−0.786160	−	0.618024i	$-0.787934\pi$
$647$	−25876.0	−1.57232	−0.786160	+	0.618024i	$0.787934\pi$
$648$	648.000	0.0392837
$649$	−29760.0	−1.79997
$650$	4114.00	0.248253
$651$	−6489.00	−0.390667
$652$	404.000	0.0242667
$653$	−21594.0	−1.29409	−0.647043	−	0.762453i	$-0.723995\pi$
$653$	−21594.0	−1.29409	−0.647043	+	0.762453i	$0.723995\pi$
$654$	252.000	0.0150672
$655$	2412.00	0.143885
$656$	2048.00	0.121892
$657$	5481.00	0.325471
$658$	−17220.0	−1.02022
$659$	−6392.00	−0.377841	−0.188920	−	0.981992i	$-0.560499\pi$
$659$	−6392.00	−0.377841	−0.188920	+	0.981992i	$0.560499\pi$
$660$	960.000	0.0566181
$661$	−346.000	−0.0203598	−0.0101799	−	0.999948i	$-0.503240\pi$
$661$	−346.000	−0.0203598	−0.0101799	+	0.999948i	$0.503240\pi$
$662$	−2626.00	−0.154173
$663$	1836.00	0.107548
$664$	−5568.00	−0.325422
$665$	0	0
$666$	−3366.00	−0.195841
$667$	−7400.00	−0.429579
$668$	−960.000	−0.0556041
$669$	14253.0	0.823696
$670$	924.000	0.0532795
$671$	11080.0	0.637464
$672$	2016.00	0.115728
$673$	−20603.0	−1.18007	−0.590035	−	0.807378i	$-0.700886\pi$
$673$	−20603.0	−1.18007	−0.590035	+	0.807378i	$0.700886\pi$
$674$	−13298.0	−0.759970
$675$	3267.00	0.186292
$676$	−7632.00	−0.434228
$677$	14342.0	0.814192	0.407096	−	0.913385i	$-0.366541\pi$
$677$	14342.0	0.814192	0.407096	+	0.913385i	$0.366541\pi$
$678$	−10968.0	−0.621274
$679$	−32550.0	−1.83970
$680$	576.000	0.0324832
$681$	−14118.0	−0.794424
$682$	8240.00	0.462648
$683$	15530.0	0.870042	0.435021	−	0.900420i	$-0.356741\pi$
$683$	15530.0	0.870042	0.435021	+	0.900420i	$0.356741\pi$
$684$	0	0
$685$	620.000	0.0345825
$686$	10290.0	0.572703
$687$	14421.0	0.800867
$688$	1936.00	0.107281
$689$	3910.00	0.216196
$690$	−888.000	−0.0489936
$691$	−24108.0	−1.32722	−0.663612	−	0.748077i	$-0.730977\pi$
$691$	−24108.0	−1.32722	−0.663612	+	0.748077i	$0.730977\pi$
$692$	5712.00	0.313783
$693$	7560.00	0.414402
$694$	−21412.0	−1.17116
$695$	2670.00	0.145725
$696$	2400.00	0.130707
$697$	4608.00	0.250417
$698$	7598.00	0.412018
$699$	−1638.00	−0.0886335
$700$	10164.0	0.548804
$701$	−2854.00	−0.153772	−0.0768859	−	0.997040i	$-0.524498\pi$
$701$	−2854.00	−0.153772	−0.0768859	+	0.997040i	$0.524498\pi$
$702$	918.000	0.0493557
$703$	0	0
$704$	−2560.00	−0.137051
$705$	−2460.00	−0.131417
$706$	−21852.0	−1.16489
$707$	8778.00	0.466946
$708$	−8928.00	−0.473919
$709$	18033.0	0.955209	0.477605	−	0.878575i	$-0.341505\pi$
$709$	18033.0	0.955209	0.477605	+	0.878575i	$0.341505\pi$
$710$	−2312.00	−0.122208
$711$	−11331.0	−0.597673
$712$	4896.00	0.257704
$713$	−7622.00	−0.400345
$714$	4536.00	0.237753
$715$	1360.00	0.0711344
$716$	8184.00	0.427165
$717$	−11898.0	−0.619720
$718$	408.000	0.0212067
$719$	11878.0	0.616098	0.308049	−	0.951370i	$-0.400324\pi$
$719$	11878.0	0.616098	0.308049	+	0.951370i	$0.400324\pi$
$720$	288.000	0.0149071
$721$	−35721.0	−1.84510
$722$	0	0
$723$	−15153.0	−0.779455
$724$	−6040.00	−0.310048
$725$	12100.0	0.619838
$726$	−1614.00	−0.0825085
$727$	17323.0	0.883734	0.441867	−	0.897081i	$-0.354316\pi$
$727$	17323.0	0.883734	0.441867	+	0.897081i	$0.354316\pi$
$728$	2856.00	0.145399
$729$	729.000	0.0370370
$730$	2436.00	0.123507
$731$	4356.00	0.220400
$732$	3324.00	0.167840
$733$	−5082.00	−0.256082	−0.128041	−	0.991769i	$-0.540869\pi$
$733$	−5082.00	−0.256082	−0.128041	+	0.991769i	$0.540869\pi$
$734$	−18238.0	−0.917135
$735$	−588.000	−0.0295084
$736$	2368.00	0.118595
$737$	−9240.00	−0.461818
$738$	2304.00	0.114921
$739$	1673.00	0.0832778	0.0416389	−	0.999133i	$-0.486742\pi$
$739$	1673.00	0.0832778	0.0416389	+	0.999133i	$0.486742\pi$
$740$	−1496.00	−0.0743163
$741$	0	0
$742$	9660.00	0.477938
$743$	−31172.0	−1.53915	−0.769576	−	0.638555i	$-0.779532\pi$
$743$	−31172.0	−1.53915	−0.769576	+	0.638555i	$0.779532\pi$
$744$	2472.00	0.121812
$745$	−2232.00	−0.109764
$746$	26204.0	1.28605
$747$	−6264.00	−0.306811
$748$	−5760.00	−0.281559
$749$	−35490.0	−1.73134
$750$	2952.00	0.143722
$751$	2941.00	0.142901	0.0714505	−	0.997444i	$-0.477237\pi$
$751$	2941.00	0.142901	0.0714505	+	0.997444i	$0.477237\pi$
$752$	6560.00	0.318110
$753$	−13110.0	−0.634469
$754$	3400.00	0.164218
$755$	−2656.00	−0.128029
$756$	2268.00	0.109109
$757$	6061.00	0.291005	0.145503	−	0.989358i	$-0.453520\pi$
$757$	6061.00	0.291005	0.145503	+	0.989358i	$0.453520\pi$
$758$	−5682.00	−0.272269
$759$	8880.00	0.424669
$760$	0	0
$761$	26004.0	1.23869	0.619346	−	0.785118i	$-0.287398\pi$
$761$	26004.0	1.23869	0.619346	+	0.785118i	$0.287398\pi$
$762$	−12744.0	−0.605861
$763$	882.000	0.0418487
$764$	9160.00	0.433766
$765$	648.000	0.0306255
$766$	28048.0	1.32300
$767$	−12648.0	−0.595427
$768$	−768.000	−0.0360844
$769$	−20837.0	−0.977115	−0.488558	−	0.872532i	$-0.662477\pi$
$769$	−20837.0	−0.977115	−0.488558	+	0.872532i	$0.662477\pi$
$770$	3360.00	0.157255
$771$	−2010.00	−0.0938890
$772$	−2508.00	−0.116923
$773$	−33162.0	−1.54302	−0.771510	−	0.636217i	$-0.780498\pi$
$773$	−33162.0	−1.54302	−0.771510	+	0.636217i	$0.780498\pi$
$774$	2178.00	0.101145
$775$	12463.0	0.577657
$776$	12400.0	0.573626
$777$	−11781.0	−0.543940
$778$	−18876.0	−0.869842
$779$	0	0
$780$	408.000	0.0187292
$781$	23120.0	1.05928
$782$	5328.00	0.243643
$783$	2700.00	0.123231
$784$	1568.00	0.0714286
$785$	−5074.00	−0.230699
$786$	−7236.00	−0.328371
$787$	497.000	0.0225110	0.0112555	−	0.999937i	$-0.496417\pi$
$787$	497.000	0.0225110	0.0112555	+	0.999937i	$0.496417\pi$
$788$	−2992.00	−0.135261
$789$	21006.0	0.947824
$790$	−5036.00	−0.226801
$791$	−38388.0	−1.72556
$792$	−2880.00	−0.129213
$793$	4709.00	0.210872
$794$	16534.0	0.739005
$795$	1380.00	0.0615642
$796$	8684.00	0.386679
$797$	−26466.0	−1.17625	−0.588127	−	0.808769i	$-0.700134\pi$
$797$	−26466.0	−1.17625	−0.588127	+	0.808769i	$0.700134\pi$
$798$	0	0
$799$	14760.0	0.653531
$800$	−3872.00	−0.171120
$801$	5508.00	0.242966
$802$	−13328.0	−0.586818
$803$	−24360.0	−1.07054
$804$	−2772.00	−0.121593
$805$	−3108.00	−0.136078
$806$	3502.00	0.153043
$807$	7452.00	0.325059
$808$	−3344.00	−0.145596
$809$	27090.0	1.17730	0.588649	−	0.808389i	$-0.299660\pi$
$809$	27090.0	1.17730	0.588649	+	0.808389i	$0.299660\pi$
$810$	324.000	0.0140546
$811$	29376.0	1.27192	0.635962	−	0.771720i	$-0.280603\pi$
$811$	29376.0	1.27192	0.635962	+	0.771720i	$0.280603\pi$
$812$	8400.00	0.363032
$813$	−11208.0	−0.483495
$814$	14960.0	0.644162
$815$	202.000	0.00868190
$816$	−1728.00	−0.0741325
$817$	0	0
$818$	−4076.00	−0.174222
$819$	3213.00	0.137083
$820$	1024.00	0.0436093
$821$	5916.00	0.251486	0.125743	−	0.992063i	$-0.459869\pi$
$821$	5916.00	0.251486	0.125743	+	0.992063i	$0.459869\pi$
$822$	−1860.00	−0.0789233
$823$	40784.0	1.72739	0.863694	−	0.504016i	$-0.168145\pi$
$823$	40784.0	1.72739	0.863694	+	0.504016i	$0.168145\pi$
$824$	13608.0	0.575312
$825$	−14520.0	−0.612753
$826$	−31248.0	−1.31629
$827$	−5768.00	−0.242531	−0.121265	−	0.992620i	$-0.538695\pi$
$827$	−5768.00	−0.242531	−0.121265	+	0.992620i	$0.538695\pi$
$828$	2664.00	0.111812
$829$	30735.0	1.28766	0.643830	−	0.765168i	$-0.277344\pi$
$829$	30735.0	1.28766	0.643830	+	0.765168i	$0.277344\pi$
$830$	−2784.00	−0.116427
$831$	−1722.00	−0.0718839
$832$	−1088.00	−0.0453361
$833$	3528.00	0.146744
$834$	−8010.00	−0.332570
$835$	−480.000	−0.0198935
$836$	0	0
$837$	2781.00	0.114845
$838$	17664.0	0.728154
$839$	19234.0	0.791456	0.395728	−	0.918368i	$-0.370492\pi$
$839$	19234.0	0.791456	0.395728	+	0.918368i	$0.370492\pi$
$840$	1008.00	0.0414039
$841$	−14389.0	−0.589979
$842$	19652.0	0.804338
$843$	−27834.0	−1.13719
$844$	12708.0	0.518279
$845$	−3816.00	−0.155354
$846$	7380.00	0.299917
$847$	−5649.00	−0.229164
$848$	−3680.00	−0.149023
$849$	−15732.0	−0.635949
$850$	−8712.00	−0.351552
$851$	−13838.0	−0.557415
$852$	6936.00	0.278901
$853$	−22293.0	−0.894839	−0.447420	−	0.894324i	$-0.647657\pi$
$853$	−22293.0	−0.894839	−0.447420	+	0.894324i	$0.647657\pi$
$854$	11634.0	0.466168
$855$	0	0
$856$	13520.0	0.539841
$857$	−29226.0	−1.16493	−0.582463	−	0.812857i	$-0.697911\pi$
$857$	−29226.0	−1.16493	−0.582463	+	0.812857i	$0.697911\pi$
$858$	−4080.00	−0.162341
$859$	−8315.00	−0.330273	−0.165136	−	0.986271i	$-0.552806\pi$
$859$	−8315.00	−0.330273	−0.165136	+	0.986271i	$0.552806\pi$
$860$	968.000	0.0383820
$861$	8064.00	0.319187
$862$	17700.0	0.699379
$863$	−45930.0	−1.81167	−0.905837	−	0.423626i	$-0.860757\pi$
$863$	−45930.0	−1.81167	−0.905837	+	0.423626i	$0.860757\pi$
$864$	−864.000	−0.0340207
$865$	2856.00	0.112262
$866$	−21286.0	−0.835251
$867$	10851.0	0.425051
$868$	8652.00	0.338327
$869$	50360.0	1.96588
$870$	1200.00	0.0467630
$871$	−3927.00	−0.152768
$872$	−336.000	−0.0130486
$873$	13950.0	0.540820
$874$	0	0
$875$	10332.0	0.399183
$876$	−7308.00	−0.281866
$877$	−26815.0	−1.03247	−0.516236	−	0.856446i	$-0.672667\pi$
$877$	−26815.0	−1.03247	−0.516236	+	0.856446i	$0.672667\pi$
$878$	20774.0	0.798506
$879$	−1998.00	−0.0766677
$880$	−1280.00	−0.0490327
$881$	−37662.0	−1.44026	−0.720128	−	0.693842i	$-0.755917\pi$
$881$	−37662.0	−1.44026	−0.720128	+	0.693842i	$0.755917\pi$
$882$	1764.00	0.0673435
$883$	14881.0	0.567141	0.283571	−	0.958951i	$-0.408481\pi$
$883$	14881.0	0.567141	0.283571	+	0.958951i	$0.408481\pi$
$884$	−2448.00	−0.0931393
$885$	−4464.00	−0.169554
$886$	13304.0	0.504466
$887$	14684.0	0.555852	0.277926	−	0.960603i	$-0.410353\pi$
$887$	14684.0	0.555852	0.277926	+	0.960603i	$0.410353\pi$
$888$	4488.00	0.169603
$889$	−44604.0	−1.68276
$890$	2448.00	0.0921990
$891$	−3240.00	−0.121823
$892$	−19004.0	−0.713342
$893$	0	0
$894$	6696.00	0.250501
$895$	4092.00	0.152827
$896$	−2688.00	−0.100223
$897$	3774.00	0.140480
$898$	33952.0	1.26168
$899$	10300.0	0.382118
$900$	−4356.00	−0.161333
$901$	−8280.00	−0.306156
$902$	−10240.0	−0.377999
$903$	7623.00	0.280928
$904$	14624.0	0.538039
$905$	−3020.00	−0.110926
$906$	7968.00	0.292184
$907$	51484.0	1.88478	0.942391	−	0.334512i	$-0.108571\pi$
$907$	51484.0	1.88478	0.942391	+	0.334512i	$0.108571\pi$
$908$	18824.0	0.687992
$909$	−3762.00	−0.137269
$910$	1428.00	0.0520195
$911$	38568.0	1.40265	0.701325	−	0.712841i	$-0.252592\pi$
$911$	38568.0	1.40265	0.701325	+	0.712841i	$0.252592\pi$
$912$	0	0
$913$	27840.0	1.00917
$914$	−12734.0	−0.460835
$915$	1662.00	0.0600481
$916$	−19228.0	−0.693571
$917$	−25326.0	−0.912037
$918$	−1944.00	−0.0698928
$919$	−37591.0	−1.34931	−0.674653	−	0.738135i	$-0.735707\pi$
$919$	−37591.0	−1.34931	−0.674653	+	0.738135i	$0.735707\pi$
$920$	1184.00	0.0424297
$921$	7164.00	0.256310
$922$	−28016.0	−1.00071
$923$	9826.00	0.350408
$924$	−10080.0	−0.358883
$925$	22627.0	0.804293
$926$	28322.0	1.00510
$927$	15309.0	0.542409
$928$	−3200.00	−0.113195
$929$	8138.00	0.287405	0.143702	−	0.989621i	$-0.454099\pi$
$929$	8138.00	0.287405	0.143702	+	0.989621i	$0.454099\pi$
$930$	1236.00	0.0435807
$931$	0	0
$932$	2184.00	0.0767589
$933$	−28380.0	−0.995841
$934$	−24712.0	−0.865740
$935$	−2880.00	−0.100734
$936$	−1224.00	−0.0427433
$937$	35543.0	1.23921	0.619605	−	0.784914i	$-0.287293\pi$
$937$	35543.0	1.23921	0.619605	+	0.784914i	$0.287293\pi$
$938$	−9702.00	−0.337720
$939$	−32046.0	−1.11372
$940$	3280.00	0.113810
$941$	14696.0	0.509114	0.254557	−	0.967058i	$-0.418070\pi$
$941$	14696.0	0.509114	0.254557	+	0.967058i	$0.418070\pi$
$942$	15222.0	0.526496
$943$	9472.00	0.327095
$944$	11904.0	0.410426
$945$	1134.00	0.0390360
$946$	−9680.00	−0.332689
$947$	47286.0	1.62259	0.811293	−	0.584640i	$-0.198764\pi$
$947$	47286.0	1.62259	0.811293	+	0.584640i	$0.198764\pi$
$948$	15108.0	0.517600
$949$	−10353.0	−0.354133
$950$	0	0
$951$	−13626.0	−0.464620
$952$	−6048.00	−0.205900
$953$	−20816.0	−0.707551	−0.353776	−	0.935330i	$-0.615102\pi$
$953$	−20816.0	−0.707551	−0.353776	+	0.935330i	$0.615102\pi$
$954$	−4140.00	−0.140500
$955$	4580.00	0.155189
$956$	15864.0	0.536693
$957$	−12000.0	−0.405334
$958$	−20188.0	−0.680840
$959$	−6510.00	−0.219206
$960$	−384.000	−0.0129099
$961$	−19182.0	−0.643886
$962$	6358.00	0.213087
$963$	15210.0	0.508967
$964$	20204.0	0.675028
$965$	−1254.00	−0.0418318
$966$	9324.00	0.310554
$967$	−7839.00	−0.260688	−0.130344	−	0.991469i	$-0.541608\pi$
$967$	−7839.00	−0.260688	−0.130344	+	0.991469i	$0.541608\pi$
$968$	2152.00	0.0714544
$969$	0	0
$970$	6200.00	0.205227
$971$	−38390.0	−1.26879	−0.634394	−	0.773010i	$-0.718750\pi$
$971$	−38390.0	−1.26879	−0.634394	+	0.773010i	$0.718750\pi$
$972$	−972.000	−0.0320750
$973$	−28035.0	−0.923701
$974$	15160.0	0.498725
$975$	−6171.00	−0.202698
$976$	−4432.00	−0.145353
$977$	41686.0	1.36505	0.682525	−	0.730863i	$-0.260882\pi$
$977$	41686.0	1.36505	0.682525	+	0.730863i	$0.260882\pi$
$978$	−606.000	−0.0198136
$979$	−24480.0	−0.799167
$980$	784.000	0.0255551
$981$	−378.000	−0.0123024
$982$	−38924.0	−1.26488
$983$	−34692.0	−1.12564	−0.562819	−	0.826580i	$-0.690283\pi$
$983$	−34692.0	−1.12564	−0.562819	+	0.826580i	$0.690283\pi$
$984$	−3072.00	−0.0995242
$985$	−1496.00	−0.0483924
$986$	−7200.00	−0.232550
$987$	25830.0	0.833007
$988$	0	0
$989$	8954.00	0.287887
$990$	−1440.00	−0.0462285
$991$	32375.0	1.03777	0.518883	−	0.854845i	$-0.326348\pi$
$991$	32375.0	1.03777	0.518883	+	0.854845i	$0.326348\pi$
$992$	−3296.00	−0.105492
$993$	3939.00	0.125882
$994$	24276.0	0.774636
$995$	4342.00	0.138342
$996$	8352.00	0.265706
$997$	29131.0	0.925364	0.462682	−	0.886524i	$-0.346887\pi$
$997$	29131.0	0.925364	0.462682	+	0.886524i	$0.346887\pi$
$998$	28766.0	0.912397
$999$	5049.00	0.159903

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.f.1.1		1
19.8	odd	6		114.4.e.a.7.1	✓	2
19.12	odd	6		114.4.e.a.49.1	yes	2
19.18	odd	2		2166.4.a.c.1.1		1
57.8	even	6		342.4.g.b.235.1		2
57.50	even	6		342.4.g.b.163.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.e.a.7.1	✓	2	19.8	odd	6
114.4.e.a.49.1	yes	2	19.12	odd	6
342.4.g.b.163.1		2	57.50	even	6
342.4.g.b.235.1		2	57.8	even	6
2166.4.a.c.1.1		1	19.18	odd	2
2166.4.a.f.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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