LMFDB - Embedded newform 2842.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2842,2,Mod(1,2842)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2842, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2842 = 2 \cdot 7^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2842.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:	$22.6934842544$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.9248.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 2 $ x^4 - 5*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.662153$ of defining polynomial
Character	$\chi$	$=$	2842.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.35829 q^{3} +1.00000 q^{4} +0.662153 q^{5} +2.35829 q^{6} -1.00000 q^{8} +2.56155 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.35829 q^{3} +1.00000 q^{4} +0.662153 q^{5} +2.35829 q^{6} -1.00000 q^{8} +2.56155 q^{9} -0.662153 q^{10} -0.438447 q^{11} -2.35829 q^{12} +4.05444 q^{13} -1.56155 q^{15} +1.00000 q^{16} +4.34475 q^{17} -2.56155 q^{18} -7.36520 q^{19} +0.662153 q^{20} +0.438447 q^{22} -8.24621 q^{23} +2.35829 q^{24} -4.56155 q^{25} -4.05444 q^{26} +1.03399 q^{27} +1.00000 q^{29} +1.56155 q^{30} +4.05444 q^{31} -1.00000 q^{32} +1.03399 q^{33} -4.34475 q^{34} +2.56155 q^{36} +7.12311 q^{37} +7.36520 q^{38} -9.56155 q^{39} -0.662153 q^{40} +4.34475 q^{41} -4.68466 q^{43} -0.438447 q^{44} +1.69614 q^{45} +8.24621 q^{46} +0.662153 q^{47} -2.35829 q^{48} +4.56155 q^{50} -10.2462 q^{51} +4.05444 q^{52} -11.5616 q^{53} -1.03399 q^{54} -0.290319 q^{55} +17.3693 q^{57} -1.00000 q^{58} -4.34475 q^{59} -1.56155 q^{60} -1.32431 q^{61} -4.05444 q^{62} +1.00000 q^{64} +2.68466 q^{65} -1.03399 q^{66} -4.87689 q^{67} +4.34475 q^{68} +19.4470 q^{69} +11.3693 q^{71} -2.56155 q^{72} +8.48071 q^{73} -7.12311 q^{74} +10.7575 q^{75} -7.36520 q^{76} +9.56155 q^{78} +7.80776 q^{79} +0.662153 q^{80} -10.1231 q^{81} -4.34475 q^{82} +5.08842 q^{83} +2.87689 q^{85} +4.68466 q^{86} -2.35829 q^{87} +0.438447 q^{88} +9.06134 q^{89} -1.69614 q^{90} -8.24621 q^{92} -9.56155 q^{93} -0.662153 q^{94} -4.87689 q^{95} +2.35829 q^{96} +6.99337 q^{97} -1.12311 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 2 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 2 * q^9	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 2 q^{9} - 10 q^{11} + 2 q^{15} + 4 q^{16} - 2 q^{18} + 10 q^{22} - 10 q^{25} + 4 q^{29} - 2 q^{30} - 4 q^{32} + 2 q^{36} + 12 q^{37} - 30 q^{39} + 6 q^{43} - 10 q^{44} + 10 q^{50} - 8 q^{51} - 38 q^{53} + 20 q^{57} - 4 q^{58} + 2 q^{60} + 4 q^{64} - 14 q^{65} - 36 q^{67} - 4 q^{71} - 2 q^{72} - 12 q^{74} + 30 q^{78} - 10 q^{79} - 24 q^{81} + 28 q^{85} - 6 q^{86} + 10 q^{88} - 30 q^{93} - 36 q^{95} + 12 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 2 * q^9 - 10 * q^11 + 2 * q^15 + 4 * q^16 - 2 * q^18 + 10 * q^22 - 10 * q^25 + 4 * q^29 - 2 * q^30 - 4 * q^32 + 2 * q^36 + 12 * q^37 - 30 * q^39 + 6 * q^43 - 10 * q^44 + 10 * q^50 - 8 * q^51 - 38 * q^53 + 20 * q^57 - 4 * q^58 + 2 * q^60 + 4 * q^64 - 14 * q^65 - 36 * q^67 - 4 * q^71 - 2 * q^72 - 12 * q^74 + 30 * q^78 - 10 * q^79 - 24 * q^81 + 28 * q^85 - 6 * q^86 + 10 * q^88 - 30 * q^93 - 36 * q^95 + 12 * q^99

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.35829	−1.36156	−0.680781	−	0.732487i	$-0.738359\pi$
$3$	−2.35829	−1.36156	−0.680781	+	0.732487i	$0.738359\pi$
$4$	1.00000	0.500000
$5$	0.662153	0.296124	0.148062	−	0.988978i	$-0.452696\pi$
$5$	0.662153	0.296124	0.148062	+	0.988978i	$0.452696\pi$
$6$	2.35829	0.962770
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	2.56155	0.853851
$10$	−0.662153	−0.209391
$11$	−0.438447	−0.132197	−0.0660984	−	0.997813i	$-0.521055\pi$
$11$	−0.438447	−0.132197	−0.0660984	+	0.997813i	$0.521055\pi$
$12$	−2.35829	−0.680781
$13$	4.05444	1.12450	0.562249	−	0.826968i	$-0.309936\pi$
$13$	4.05444	1.12450	0.562249	+	0.826968i	$0.309936\pi$
$14$	0	0
$15$	−1.56155	−0.403191
$16$	1.00000	0.250000
$17$	4.34475	1.05376	0.526879	−	0.849940i	$-0.323362\pi$
$17$	4.34475	1.05376	0.526879	+	0.849940i	$0.323362\pi$
$18$	−2.56155	−0.603764
$19$	−7.36520	−1.68969	−0.844847	−	0.535008i	$-0.820308\pi$
$19$	−7.36520	−1.68969	−0.844847	+	0.535008i	$0.820308\pi$
$20$	0.662153	0.148062
$21$	0	0
$22$	0.438447	0.0934773
$23$	−8.24621	−1.71945	−0.859727	−	0.510754i	$-0.829366\pi$
$23$	−8.24621	−1.71945	−0.859727	+	0.510754i	$0.829366\pi$
$24$	2.35829	0.481385
$25$	−4.56155	−0.912311
$26$	−4.05444	−0.795140
$27$	1.03399	0.198991
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	1.56155	0.285099
$31$	4.05444	0.728198	0.364099	−	0.931360i	$-0.381377\pi$
$31$	4.05444	0.728198	0.364099	+	0.931360i	$0.381377\pi$
$32$	−1.00000	−0.176777
$33$	1.03399	0.179994
$34$	−4.34475	−0.745119
$35$	0	0
$36$	2.56155	0.426925
$37$	7.12311	1.17103	0.585516	−	0.810661i	$-0.300892\pi$
$37$	7.12311	1.17103	0.585516	+	0.810661i	$0.300892\pi$
$38$	7.36520	1.19479
$39$	−9.56155	−1.53107
$40$	−0.662153	−0.104696
$41$	4.34475	0.678537	0.339268	−	0.940690i	$-0.389821\pi$
$41$	4.34475	0.678537	0.339268	+	0.940690i	$0.389821\pi$
$42$	0	0
$43$	−4.68466	−0.714404	−0.357202	−	0.934027i	$-0.616269\pi$
$43$	−4.68466	−0.714404	−0.357202	+	0.934027i	$0.616269\pi$
$44$	−0.438447	−0.0660984
$45$	1.69614	0.252846
$46$	8.24621	1.21584
$47$	0.662153	0.0965850	0.0482925	−	0.998833i	$-0.484622\pi$
$47$	0.662153	0.0965850	0.0482925	+	0.998833i	$0.484622\pi$
$48$	−2.35829	−0.340390
$49$	0	0
$50$	4.56155	0.645101
$51$	−10.2462	−1.43476
$52$	4.05444	0.562249
$53$	−11.5616	−1.58810	−0.794051	−	0.607852i	$-0.792032\pi$
$53$	−11.5616	−1.58810	−0.794051	+	0.607852i	$0.792032\pi$
$54$	−1.03399	−0.140708
$55$	−0.290319	−0.0391466
$56$	0	0
$57$	17.3693	2.30062
$58$	−1.00000	−0.131306
$59$	−4.34475	−0.565639	−0.282819	−	0.959173i	$-0.591270\pi$
$59$	−4.34475	−0.565639	−0.282819	+	0.959173i	$0.591270\pi$
$60$	−1.56155	−0.201596
$61$	−1.32431	−0.169560	−0.0847801	−	0.996400i	$-0.527019\pi$
$61$	−1.32431	−0.169560	−0.0847801	+	0.996400i	$0.527019\pi$
$62$	−4.05444	−0.514914
$63$	0	0
$64$	1.00000	0.125000
$65$	2.68466	0.332991
$66$	−1.03399	−0.127275
$67$	−4.87689	−0.595807	−0.297904	−	0.954596i	$-0.596287\pi$
$67$	−4.87689	−0.595807	−0.297904	+	0.954596i	$0.596287\pi$
$68$	4.34475	0.526879
$69$	19.4470	2.34114
$70$	0	0
$71$	11.3693	1.34929	0.674645	−	0.738142i	$-0.264297\pi$
$71$	11.3693	1.34929	0.674645	+	0.738142i	$0.264297\pi$
$72$	−2.56155	−0.301882
$73$	8.48071	0.992591	0.496296	−	0.868154i	$-0.334693\pi$
$73$	8.48071	0.992591	0.496296	+	0.868154i	$0.334693\pi$
$74$	−7.12311	−0.828044
$75$	10.7575	1.24217
$76$	−7.36520	−0.844847
$77$	0	0
$78$	9.56155	1.08263
$79$	7.80776	0.878442	0.439221	−	0.898379i	$-0.355254\pi$
$79$	7.80776	0.878442	0.439221	+	0.898379i	$0.355254\pi$
$80$	0.662153	0.0740310
$81$	−10.1231	−1.12479
$82$	−4.34475	−0.479798
$83$	5.08842	0.558527	0.279263	−	0.960215i	$-0.409910\pi$
$83$	5.08842	0.558527	0.279263	+	0.960215i	$0.409910\pi$
$84$	0	0
$85$	2.87689	0.312043
$86$	4.68466	0.505160
$87$	−2.35829	−0.252836
$88$	0.438447	0.0467386
$89$	9.06134	0.960501	0.480250	−	0.877132i	$-0.340546\pi$
$89$	9.06134	0.960501	0.480250	+	0.877132i	$0.340546\pi$
$90$	−1.69614	−0.178789
$91$	0	0
$92$	−8.24621	−0.859727
$93$	−9.56155	−0.991487
$94$	−0.662153	−0.0682959
$95$	−4.87689	−0.500359
$96$	2.35829	0.240692
$97$	6.99337	0.710069	0.355034	−	0.934853i	$-0.384469\pi$
$97$	6.99337	0.710069	0.355034	+	0.934853i	$0.384469\pi$
$98$	0	0
$99$	−1.12311	−0.112876
$100$	−4.56155	−0.456155
$101$	−11.3381	−1.12819	−0.564093	−	0.825711i	$-0.690774\pi$
$101$	−11.3381	−1.12819	−0.564093	+	0.825711i	$0.690774\pi$
$102$	10.2462	1.01453
$103$	−12.8255	−1.26373	−0.631865	−	0.775078i	$-0.717710\pi$
$103$	−12.8255	−1.26373	−0.631865	+	0.775078i	$0.717710\pi$
$104$	−4.05444	−0.397570
$105$	0	0
$106$	11.5616	1.12296
$107$	−0.876894	−0.0847726	−0.0423863	−	0.999101i	$-0.513496\pi$
$107$	−0.876894	−0.0847726	−0.0423863	+	0.999101i	$0.513496\pi$
$108$	1.03399	0.0994955
$109$	10.6847	1.02340	0.511702	−	0.859163i	$-0.329015\pi$
$109$	10.6847	1.02340	0.511702	+	0.859163i	$0.329015\pi$
$110$	0.290319	0.0276809
$111$	−16.7984	−1.59443
$112$	0	0
$113$	−19.3693	−1.82211	−0.911056	−	0.412283i	$-0.864732\pi$
$113$	−19.3693	−1.82211	−0.911056	+	0.412283i	$0.864732\pi$
$114$	−17.3693	−1.62679
$115$	−5.46026	−0.509172
$116$	1.00000	0.0928477
$117$	10.3857	0.960154
$118$	4.34475	0.399967
$119$	0	0
$120$	1.56155	0.142550
$121$	−10.8078	−0.982524
$122$	1.32431	0.119897
$123$	−10.2462	−0.923870
$124$	4.05444	0.364099
$125$	−6.33122	−0.566281
$126$	0	0
$127$	−2.24621	−0.199319	−0.0996595	−	0.995022i	$-0.531775\pi$
$127$	−2.24621	−0.199319	−0.0996595	+	0.995022i	$0.531775\pi$
$128$	−1.00000	−0.0883883
$129$	11.0478	0.972705
$130$	−2.68466	−0.235460
$131$	2.06798	0.180680	0.0903399	−	0.995911i	$-0.471205\pi$
$131$	2.06798	0.180680	0.0903399	+	0.995911i	$0.471205\pi$
$132$	1.03399	0.0899971
$133$	0	0
$134$	4.87689	0.421300
$135$	0.684658	0.0589260
$136$	−4.34475	−0.372560
$137$	−8.24621	−0.704521	−0.352261	−	0.935902i	$-0.614587\pi$
$137$	−8.24621	−0.704521	−0.352261	+	0.935902i	$0.614587\pi$
$138$	−19.4470	−1.65544
$139$	8.31768	0.705496	0.352748	−	0.935718i	$-0.385247\pi$
$139$	8.31768	0.705496	0.352748	+	0.935718i	$0.385247\pi$
$140$	0	0
$141$	−1.56155	−0.131506
$142$	−11.3693	−0.954092
$143$	−1.77766	−0.148655
$144$	2.56155	0.213463
$145$	0.662153	0.0549889
$146$	−8.48071	−0.701868
$147$	0	0
$148$	7.12311	0.585516
$149$	2.68466	0.219936	0.109968	−	0.993935i	$-0.464925\pi$
$149$	2.68466	0.219936	0.109968	+	0.993935i	$0.464925\pi$
$150$	−10.7575	−0.878345
$151$	4.24621	0.345552	0.172776	−	0.984961i	$-0.444726\pi$
$151$	4.24621	0.345552	0.172776	+	0.984961i	$0.444726\pi$
$152$	7.36520	0.597397
$153$	11.1293	0.899752
$154$	0	0
$155$	2.68466	0.215637
$156$	−9.56155	−0.765537
$157$	17.3790	1.38700	0.693498	−	0.720458i	$-0.256068\pi$
$157$	17.3790	1.38700	0.693498	+	0.720458i	$0.256068\pi$
$158$	−7.80776	−0.621152
$159$	27.2655	2.16230
$160$	−0.662153	−0.0523478
$161$	0	0
$162$	10.1231	0.795346
$163$	−4.43845	−0.347646	−0.173823	−	0.984777i	$-0.555612\pi$
$163$	−4.43845	−0.347646	−0.173823	+	0.984777i	$0.555612\pi$
$164$	4.34475	0.339268
$165$	0.684658	0.0533006
$166$	−5.08842	−0.394938
$167$	−9.43318	−0.729961	−0.364981	−	0.931015i	$-0.618924\pi$
$167$	−9.43318	−0.729961	−0.364981	+	0.931015i	$0.618924\pi$
$168$	0	0
$169$	3.43845	0.264496
$170$	−2.87689	−0.220648
$171$	−18.8664	−1.44275
$172$	−4.68466	−0.357202
$173$	−11.1293	−0.846146	−0.423073	−	0.906095i	$-0.639049\pi$
$173$	−11.1293	−0.846146	−0.423073	+	0.906095i	$0.639049\pi$
$174$	2.35829	0.178782
$175$	0	0
$176$	−0.438447	−0.0330492
$177$	10.2462	0.770152
$178$	−9.06134	−0.679176
$179$	−1.75379	−0.131084	−0.0655422	−	0.997850i	$-0.520878\pi$
$179$	−1.75379	−0.131084	−0.0655422	+	0.997850i	$0.520878\pi$
$180$	1.69614	0.126423
$181$	−22.1771	−1.64841	−0.824206	−	0.566290i	$-0.808378\pi$
$181$	−22.1771	−1.64841	−0.824206	+	0.566290i	$0.808378\pi$
$182$	0	0
$183$	3.12311	0.230867
$184$	8.24621	0.607919
$185$	4.71659	0.346771
$186$	9.56155	0.701087
$187$	−1.90495	−0.139303
$188$	0.662153	0.0482925
$189$	0	0
$190$	4.87689	0.353807
$191$	−10.2462	−0.741390	−0.370695	−	0.928755i	$-0.620880\pi$
$191$	−10.2462	−0.741390	−0.370695	+	0.928755i	$0.620880\pi$
$192$	−2.35829	−0.170195
$193$	−3.36932	−0.242529	−0.121264	−	0.992620i	$-0.538695\pi$
$193$	−3.36932	−0.242529	−0.121264	+	0.992620i	$0.538695\pi$
$194$	−6.99337	−0.502095
$195$	−6.33122	−0.453388
$196$	0	0
$197$	−3.75379	−0.267446	−0.133723	−	0.991019i	$-0.542693\pi$
$197$	−3.75379	−0.267446	−0.133723	+	0.991019i	$0.542693\pi$
$198$	1.12311	0.0798156
$199$	−20.7713	−1.47244	−0.736219	−	0.676743i	$-0.763391\pi$
$199$	−20.7713	−1.47244	−0.736219	+	0.676743i	$0.763391\pi$
$200$	4.56155	0.322550
$201$	11.5012	0.811229
$202$	11.3381	0.797748
$203$	0	0
$204$	−10.2462	−0.717378
$205$	2.87689	0.200931
$206$	12.8255	0.893592
$207$	−21.1231	−1.46816
$208$	4.05444	0.281125
$209$	3.22925	0.223372
$210$	0	0
$211$	4.68466	0.322505	0.161253	−	0.986913i	$-0.448447\pi$
$211$	4.68466	0.322505	0.161253	+	0.986913i	$0.448447\pi$
$212$	−11.5616	−0.794051
$213$	−26.8122	−1.83714
$214$	0.876894	0.0599433
$215$	−3.10196	−0.211552
$216$	−1.03399	−0.0703539
$217$	0	0
$218$	−10.6847	−0.723656
$219$	−20.0000	−1.35147
$220$	−0.290319	−0.0195733
$221$	17.6155	1.18495
$222$	16.7984	1.12743
$223$	−3.39228	−0.227164	−0.113582	−	0.993529i	$-0.536232\pi$
$223$	−3.39228	−0.227164	−0.113582	+	0.993529i	$0.536232\pi$
$224$	0	0
$225$	−11.6847	−0.778977
$226$	19.3693	1.28843
$227$	−12.4536	−0.826576	−0.413288	−	0.910600i	$-0.635620\pi$
$227$	−12.4536	−0.826576	−0.413288	+	0.910600i	$0.635620\pi$
$228$	17.3693	1.15031
$229$	−23.5829	−1.55840	−0.779202	−	0.626772i	$-0.784376\pi$
$229$	−23.5829	−1.55840	−0.779202	+	0.626772i	$0.784376\pi$
$230$	5.46026	0.360039
$231$	0	0
$232$	−1.00000	−0.0656532
$233$	−12.9309	−0.847129	−0.423565	−	0.905866i	$-0.639221\pi$
$233$	−12.9309	−0.847129	−0.423565	+	0.905866i	$0.639221\pi$
$234$	−10.3857	−0.678931
$235$	0.438447	0.0286011
$236$	−4.34475	−0.282819
$237$	−18.4130	−1.19605
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	−1.56155	−0.100798
$241$	24.6169	1.58572	0.792858	−	0.609406i	$-0.208592\pi$
$241$	24.6169	1.58572	0.792858	+	0.609406i	$0.208592\pi$
$242$	10.8078	0.694749
$243$	20.7713	1.33248
$244$	−1.32431	−0.0847801
$245$	0	0
$246$	10.2462	0.653275
$247$	−29.8617	−1.90006
$248$	−4.05444	−0.257457
$249$	−12.0000	−0.760469
$250$	6.33122	0.400421
$251$	−22.5490	−1.42328	−0.711639	−	0.702546i	$-0.752047\pi$
$251$	−22.5490	−1.42328	−0.711639	+	0.702546i	$0.752047\pi$
$252$	0	0
$253$	3.61553	0.227306
$254$	2.24621	0.140940
$255$	−6.78456	−0.424866
$256$	1.00000	0.0625000
$257$	−13.8594	−0.864529	−0.432264	−	0.901747i	$-0.642285\pi$
$257$	−13.8594	−0.864529	−0.432264	+	0.901747i	$0.642285\pi$
$258$	−11.0478	−0.687806
$259$	0	0
$260$	2.68466	0.166495
$261$	2.56155	0.158556
$262$	−2.06798	−0.127760
$263$	−15.8078	−0.974748	−0.487374	−	0.873193i	$-0.662045\pi$
$263$	−15.8078	−0.974748	−0.487374	+	0.873193i	$0.662045\pi$
$264$	−1.03399	−0.0636375
$265$	−7.65552	−0.470275
$266$	0	0
$267$	−21.3693	−1.30778
$268$	−4.87689	−0.297904
$269$	−24.9073	−1.51862	−0.759311	−	0.650728i	$-0.774464\pi$
$269$	−24.9073	−1.51862	−0.759311	+	0.650728i	$0.774464\pi$
$270$	−0.684658	−0.0416670
$271$	12.0003	0.728965	0.364482	−	0.931210i	$-0.381246\pi$
$271$	12.0003	0.728965	0.364482	+	0.931210i	$0.381246\pi$
$272$	4.34475	0.263439
$273$	0	0
$274$	8.24621	0.498172
$275$	2.00000	0.120605
$276$	19.4470	1.17057
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	−8.31768	−0.498861
$279$	10.3857	0.621773
$280$	0	0
$281$	−29.1771	−1.74056	−0.870279	−	0.492558i	$-0.836062\pi$
$281$	−29.1771	−1.74056	−0.870279	+	0.492558i	$0.836062\pi$
$282$	1.56155	0.0929891
$283$	−10.5487	−0.627054	−0.313527	−	0.949579i	$-0.601511\pi$
$283$	−10.5487	−0.627054	−0.313527	+	0.949579i	$0.601511\pi$
$284$	11.3693	0.674645
$285$	11.5012	0.681270
$286$	1.77766	0.105115
$287$	0	0
$288$	−2.56155	−0.150941
$289$	1.87689	0.110406
$290$	−0.662153	−0.0388830
$291$	−16.4924	−0.966803
$292$	8.48071	0.496296
$293$	11.5012	0.671905	0.335952	−	0.941879i	$-0.390942\pi$
$293$	11.5012	0.671905	0.335952	+	0.941879i	$0.390942\pi$
$294$	0	0
$295$	−2.87689	−0.167499
$296$	−7.12311	−0.414022
$297$	−0.453349	−0.0263060
$298$	−2.68466	−0.155518
$299$	−33.4337	−1.93352
$300$	10.7575	0.621084
$301$	0	0
$302$	−4.24621	−0.244342
$303$	26.7386	1.53609
$304$	−7.36520	−0.422423
$305$	−0.876894	−0.0502108
$306$	−11.1293	−0.636221
$307$	−11.2108	−0.639836	−0.319918	−	0.947445i	$-0.603655\pi$
$307$	−11.2108	−0.639836	−0.319918	+	0.947445i	$0.603655\pi$
$308$	0	0
$309$	30.2462	1.72065
$310$	−2.68466	−0.152478
$311$	9.80501	0.555991	0.277996	−	0.960582i	$-0.410330\pi$
$311$	9.80501	0.555991	0.277996	+	0.960582i	$0.410330\pi$
$312$	9.56155	0.541316
$313$	−27.2655	−1.54114	−0.770570	−	0.637356i	$-0.780028\pi$
$313$	−27.2655	−1.54114	−0.770570	+	0.637356i	$0.780028\pi$
$314$	−17.3790	−0.980755
$315$	0	0
$316$	7.80776	0.439221
$317$	14.4924	0.813976	0.406988	−	0.913434i	$-0.366579\pi$
$317$	14.4924	0.813976	0.406988	+	0.913434i	$0.366579\pi$
$318$	−27.2655	−1.52898
$319$	−0.438447	−0.0245483
$320$	0.662153	0.0370155
$321$	2.06798	0.115423
$322$	0	0
$323$	−32.0000	−1.78053
$324$	−10.1231	−0.562395
$325$	−18.4945	−1.02589
$326$	4.43845	0.245823
$327$	−25.1976	−1.39343
$328$	−4.34475	−0.239899
$329$	0	0
$330$	−0.684658	−0.0376892
$331$	36.3002	1.99524	0.997619	−	0.0689611i	$-0.0219684\pi$
$331$	36.3002	1.99524	0.997619	+	0.0689611i	$0.0219684\pi$
$332$	5.08842	0.279263
$333$	18.2462	0.999886
$334$	9.43318	0.516161
$335$	−3.22925	−0.176433
$336$	0	0
$337$	−24.2462	−1.32078	−0.660388	−	0.750925i	$-0.729608\pi$
$337$	−24.2462	−1.32078	−0.660388	+	0.750925i	$0.729608\pi$
$338$	−3.43845	−0.187027
$339$	45.6786	2.48092
$340$	2.87689	0.156022
$341$	−1.77766	−0.0962655
$342$	18.8664	1.02018
$343$	0	0
$344$	4.68466	0.252580
$345$	12.8769	0.693269
$346$	11.1293	0.598316
$347$	−31.1231	−1.67078	−0.835388	−	0.549661i	$-0.814757\pi$
$347$	−31.1231	−1.67078	−0.835388	+	0.549661i	$0.814757\pi$
$348$	−2.35829	−0.126418
$349$	22.9208	1.22692	0.613461	−	0.789725i	$-0.289777\pi$
$349$	22.9208	1.22692	0.613461	+	0.789725i	$0.289777\pi$
$350$	0	0
$351$	4.19224	0.223765
$352$	0.438447	0.0233693
$353$	9.43318	0.502077	0.251039	−	0.967977i	$-0.419228\pi$
$353$	9.43318	0.502077	0.251039	+	0.967977i	$0.419228\pi$
$354$	−10.2462	−0.544580
$355$	7.52823	0.399557
$356$	9.06134	0.480250
$357$	0	0
$358$	1.75379	0.0926906
$359$	−28.6847	−1.51392	−0.756959	−	0.653462i	$-0.773316\pi$
$359$	−28.6847	−1.51392	−0.756959	+	0.653462i	$0.773316\pi$
$360$	−1.69614	−0.0893945
$361$	35.2462	1.85506
$362$	22.1771	1.16560
$363$	25.4879	1.33777
$364$	0	0
$365$	5.61553	0.293930
$366$	−3.12311	−0.163247
$367$	0.371834	0.0194096	0.00970479	−	0.999953i	$-0.496911\pi$
$367$	0.371834	0.0194096	0.00970479	+	0.999953i	$0.496911\pi$
$368$	−8.24621	−0.429863
$369$	11.1293	0.579369
$370$	−4.71659	−0.245204
$371$	0	0
$372$	−9.56155	−0.495743
$373$	−21.8078	−1.12916	−0.564582	−	0.825377i	$-0.690962\pi$
$373$	−21.8078	−1.12916	−0.564582	+	0.825377i	$0.690962\pi$
$374$	1.90495	0.0985024
$375$	14.9309	0.771027
$376$	−0.662153	−0.0341480
$377$	4.05444	0.208814
$378$	0	0
$379$	32.4924	1.66902	0.834512	−	0.550990i	$-0.185750\pi$
$379$	32.4924	1.66902	0.834512	+	0.550990i	$0.185750\pi$
$380$	−4.87689	−0.250179
$381$	5.29723	0.271385
$382$	10.2462	0.524242
$383$	7.36520	0.376344	0.188172	−	0.982136i	$-0.439744\pi$
$383$	7.36520	0.376344	0.188172	+	0.982136i	$0.439744\pi$
$384$	2.35829	0.120346
$385$	0	0
$386$	3.36932	0.171494
$387$	−12.0000	−0.609994
$388$	6.99337	0.355034
$389$	−32.9848	−1.67240	−0.836199	−	0.548426i	$-0.815227\pi$
$389$	−32.9848	−1.67240	−0.836199	+	0.548426i	$0.815227\pi$
$390$	6.33122	0.320594
$391$	−35.8278	−1.81189
$392$	0	0
$393$	−4.87689	−0.246007
$394$	3.75379	0.189113
$395$	5.16994	0.260128
$396$	−1.12311	−0.0564382
$397$	−10.2584	−0.514852	−0.257426	−	0.966298i	$-0.582874\pi$
$397$	−10.2584	−0.514852	−0.257426	+	0.966298i	$0.582874\pi$
$398$	20.7713	1.04117
$399$	0	0
$400$	−4.56155	−0.228078
$401$	14.1922	0.708726	0.354363	−	0.935108i	$-0.384698\pi$
$401$	14.1922	0.708726	0.354363	+	0.935108i	$0.384698\pi$
$402$	−11.5012	−0.573625
$403$	16.4384	0.818857
$404$	−11.3381	−0.564093
$405$	−6.70305	−0.333077
$406$	0	0
$407$	−3.12311	−0.154807
$408$	10.2462	0.507263
$409$	29.9957	1.48319	0.741595	−	0.670848i	$-0.234069\pi$
$409$	29.9957	1.48319	0.741595	+	0.670848i	$0.234069\pi$
$410$	−2.87689	−0.142080
$411$	19.4470	0.959249
$412$	−12.8255	−0.631865
$413$	0	0
$414$	21.1231	1.03814
$415$	3.36932	0.165393
$416$	−4.05444	−0.198785
$417$	−19.6155	−0.960577
$418$	−3.22925	−0.157948
$419$	5.83209	0.284916	0.142458	−	0.989801i	$-0.454499\pi$
$419$	5.83209	0.284916	0.142458	+	0.989801i	$0.454499\pi$
$420$	0	0
$421$	−31.1231	−1.51685	−0.758424	−	0.651762i	$-0.774030\pi$
$421$	−31.1231	−1.51685	−0.758424	+	0.651762i	$0.774030\pi$
$422$	−4.68466	−0.228046
$423$	1.69614	0.0824692
$424$	11.5616	0.561479
$425$	−19.8188	−0.961354
$426$	26.8122	1.29906
$427$	0	0
$428$	−0.876894	−0.0423863
$429$	4.19224	0.202403
$430$	3.10196	0.149590
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	1.03399	0.0497478
$433$	−16.5896	−0.797244	−0.398622	−	0.917115i	$-0.630511\pi$
$433$	−16.5896	−0.797244	−0.398622	+	0.917115i	$0.630511\pi$
$434$	0	0
$435$	−1.56155	−0.0748707
$436$	10.6847	0.511702
$437$	60.7350	2.90535
$438$	20.0000	0.955637
$439$	35.6647	1.70218	0.851092	−	0.525016i	$-0.175941\pi$
$439$	35.6647	1.70218	0.851092	+	0.525016i	$0.175941\pi$
$440$	0.290319	0.0138404
$441$	0	0
$442$	−17.6155	−0.837885
$443$	33.1231	1.57373	0.786863	−	0.617128i	$-0.211704\pi$
$443$	33.1231	1.57373	0.786863	+	0.617128i	$0.211704\pi$
$444$	−16.7984	−0.797216
$445$	6.00000	0.284427
$446$	3.39228	0.160629
$447$	−6.33122	−0.299456
$448$	0	0
$449$	−19.3693	−0.914095	−0.457047	−	0.889442i	$-0.651093\pi$
$449$	−19.3693	−0.914095	−0.457047	+	0.889442i	$0.651093\pi$
$450$	11.6847	0.550820
$451$	−1.90495	−0.0897004
$452$	−19.3693	−0.911056
$453$	−10.0138	−0.470490
$454$	12.4536	0.584478
$455$	0	0
$456$	−17.3693	−0.813393
$457$	29.3693	1.37384	0.686919	−	0.726734i	$-0.258963\pi$
$457$	29.3693	1.37384	0.686919	+	0.726734i	$0.258963\pi$
$458$	23.5829	1.10196
$459$	4.49242	0.209688
$460$	−5.46026	−0.254586
$461$	23.4199	1.09077	0.545387	−	0.838184i	$-0.316383\pi$
$461$	23.4199	1.09077	0.545387	+	0.838184i	$0.316383\pi$
$462$	0	0
$463$	−23.3693	−1.08606	−0.543032	−	0.839712i	$-0.682724\pi$
$463$	−23.3693	−1.08606	−0.543032	+	0.839712i	$0.682724\pi$
$464$	1.00000	0.0464238
$465$	−6.33122	−0.293603
$466$	12.9309	0.599011
$467$	6.33122	0.292974	0.146487	−	0.989213i	$-0.453203\pi$
$467$	6.33122	0.292974	0.146487	+	0.989213i	$0.453203\pi$
$468$	10.3857	0.480077
$469$	0	0
$470$	−0.438447	−0.0202241
$471$	−40.9848	−1.88848
$472$	4.34475	0.199984
$473$	2.05398	0.0944419
$474$	18.4130	0.845737
$475$	33.5968	1.54153
$476$	0	0
$477$	−29.6155	−1.35600
$478$	0	0
$479$	40.2998	1.84135	0.920673	−	0.390336i	$-0.127641\pi$
$479$	40.2998	1.84135	0.920673	+	0.390336i	$0.127641\pi$
$480$	1.56155	0.0712748
$481$	28.8802	1.31682
$482$	−24.6169	−1.12127
$483$	0	0
$484$	−10.8078	−0.491262
$485$	4.63068	0.210268
$486$	−20.7713	−0.942205
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	1.32431	0.0599486
$489$	10.4672	0.473342
$490$	0	0
$491$	−25.1771	−1.13623	−0.568113	−	0.822951i	$-0.692326\pi$
$491$	−25.1771	−1.13623	−0.568113	+	0.822951i	$0.692326\pi$
$492$	−10.2462	−0.461935
$493$	4.34475	0.195678
$494$	29.8617	1.34354
$495$	−0.743668	−0.0334254
$496$	4.05444	0.182050
$497$	0	0
$498$	12.0000	0.537733
$499$	3.12311	0.139809	0.0699047	−	0.997554i	$-0.477730\pi$
$499$	3.12311	0.139809	0.0699047	+	0.997554i	$0.477730\pi$
$500$	−6.33122	−0.283141
$501$	22.2462	0.993887
$502$	22.5490	1.00641
$503$	−33.6783	−1.50164	−0.750820	−	0.660507i	$-0.770341\pi$
$503$	−33.6783	−1.50164	−0.750820	+	0.660507i	$0.770341\pi$
$504$	0	0
$505$	−7.50758	−0.334083
$506$	−3.61553	−0.160730
$507$	−8.10887	−0.360128
$508$	−2.24621	−0.0996595
$509$	−34.4219	−1.52573	−0.762863	−	0.646560i	$-0.776207\pi$
$509$	−34.4219	−1.52573	−0.762863	+	0.646560i	$0.776207\pi$
$510$	6.78456	0.300426
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−7.61553	−0.336234
$514$	13.8594	0.611314
$515$	−8.49242	−0.374221
$516$	11.0478	0.486352
$517$	−0.290319	−0.0127682
$518$	0	0
$519$	26.2462	1.15208
$520$	−2.68466	−0.117730
$521$	24.4539	1.07134	0.535672	−	0.844426i	$-0.320058\pi$
$521$	24.4539	1.07134	0.535672	+	0.844426i	$0.320058\pi$
$522$	−2.56155	−0.112116
$523$	−43.8194	−1.91609	−0.958044	−	0.286621i	$-0.907468\pi$
$523$	−43.8194	−1.91609	−0.958044	+	0.286621i	$0.907468\pi$
$524$	2.06798	0.0903399
$525$	0	0
$526$	15.8078	0.689251
$527$	17.6155	0.767344
$528$	1.03399	0.0449985
$529$	45.0000	1.95652
$530$	7.65552	0.332535
$531$	−11.1293	−0.482971
$532$	0	0
$533$	17.6155	0.763013
$534$	21.3693	0.924741
$535$	−0.580639	−0.0251032
$536$	4.87689	0.210650
$537$	4.13595	0.178479
$538$	24.9073	1.07383
$539$	0	0
$540$	0.684658	0.0294630
$541$	9.12311	0.392233	0.196116	−	0.980581i	$-0.437167\pi$
$541$	9.12311	0.392233	0.196116	+	0.980581i	$0.437167\pi$
$542$	−12.0003	−0.515456
$543$	52.3002	2.24442
$544$	−4.34475	−0.186280
$545$	7.07488	0.303055
$546$	0	0
$547$	7.61553	0.325616	0.162808	−	0.986658i	$-0.447945\pi$
$547$	7.61553	0.325616	0.162808	+	0.986658i	$0.447945\pi$
$548$	−8.24621	−0.352261
$549$	−3.39228	−0.144779
$550$	−2.00000	−0.0852803
$551$	−7.36520	−0.313768
$552$	−19.4470	−0.827719
$553$	0	0
$554$	10.0000	0.424859
$555$	−11.1231	−0.472150
$556$	8.31768	0.352748
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	−10.3857	−0.439660
$559$	−18.9936	−0.803346
$560$	0	0
$561$	4.49242	0.189670
$562$	29.1771	1.23076
$563$	−18.4130	−0.776016	−0.388008	−	0.921656i	$-0.626837\pi$
$563$	−18.4130	−0.776016	−0.388008	+	0.921656i	$0.626837\pi$
$564$	−1.56155	−0.0657532
$565$	−12.8255	−0.539571
$566$	10.5487	0.443394
$567$	0	0
$568$	−11.3693	−0.477046
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	−11.5012	−0.481730
$571$	16.8769	0.706276	0.353138	−	0.935571i	$-0.385115\pi$
$571$	16.8769	0.706276	0.353138	+	0.935571i	$0.385115\pi$
$572$	−1.77766	−0.0743275
$573$	24.1636	1.00945
$574$	0	0
$575$	37.6155	1.56868
$576$	2.56155	0.106731
$577$	−4.18173	−0.174087	−0.0870437	−	0.996204i	$-0.527742\pi$
$577$	−4.18173	−0.174087	−0.0870437	+	0.996204i	$0.527742\pi$
$578$	−1.87689	−0.0780685
$579$	7.94584	0.330218
$580$	0.662153	0.0274944
$581$	0	0
$582$	16.4924	0.683633
$583$	5.06913	0.209942
$584$	−8.48071	−0.350934
$585$	6.87689	0.284325
$586$	−11.5012	−0.475108
$587$	−35.8735	−1.48066	−0.740330	−	0.672244i	$-0.765331\pi$
$587$	−35.8735	−1.48066	−0.740330	+	0.672244i	$0.765331\pi$
$588$	0	0
$589$	−29.8617	−1.23043
$590$	2.87689	0.118440
$591$	8.85254	0.364145
$592$	7.12311	0.292758
$593$	11.6284	0.477523	0.238761	−	0.971078i	$-0.423259\pi$
$593$	11.6284	0.477523	0.238761	+	0.971078i	$0.423259\pi$
$594$	0.453349	0.0186011
$595$	0	0
$596$	2.68466	0.109968
$597$	48.9848	2.00482
$598$	33.4337	1.36721
$599$	−24.6847	−1.00859	−0.504294	−	0.863532i	$-0.668247\pi$
$599$	−24.6847	−1.00859	−0.504294	+	0.863532i	$0.668247\pi$
$600$	−10.7575	−0.439172
$601$	−5.83209	−0.237896	−0.118948	−	0.992900i	$-0.537952\pi$
$601$	−5.83209	−0.237896	−0.118948	+	0.992900i	$0.537952\pi$
$602$	0	0
$603$	−12.4924	−0.508731
$604$	4.24621	0.172776
$605$	−7.15640	−0.290949
$606$	−26.7386	−1.08618
$607$	−32.9346	−1.33677	−0.668387	−	0.743813i	$-0.733015\pi$
$607$	−32.9346	−1.33677	−0.668387	+	0.743813i	$0.733015\pi$
$608$	7.36520	0.298698
$609$	0	0
$610$	0.876894	0.0355044
$611$	2.68466	0.108610
$612$	11.1293	0.449876
$613$	20.0540	0.809972	0.404986	−	0.914323i	$-0.367276\pi$
$613$	20.0540	0.809972	0.404986	+	0.914323i	$0.367276\pi$
$614$	11.2108	0.452432
$615$	−6.78456	−0.273580
$616$	0	0
$617$	7.36932	0.296678	0.148339	−	0.988937i	$-0.452607\pi$
$617$	7.36932	0.296678	0.148339	+	0.988937i	$0.452607\pi$
$618$	−30.2462	−1.21668
$619$	−31.4015	−1.26213	−0.631066	−	0.775729i	$-0.717382\pi$
$619$	−31.4015	−1.26213	−0.631066	+	0.775729i	$0.717382\pi$
$620$	2.68466	0.107818
$621$	−8.52648	−0.342156
$622$	−9.80501	−0.393145
$623$	0	0
$624$	−9.56155	−0.382768
$625$	18.6155	0.744621
$626$	27.2655	1.08975
$627$	−7.61553	−0.304135
$628$	17.3790	0.693498
$629$	30.9481	1.23398
$630$	0	0
$631$	22.2462	0.885608	0.442804	−	0.896619i	$-0.353984\pi$
$631$	22.2462	0.885608	0.442804	+	0.896619i	$0.353984\pi$
$632$	−7.80776	−0.310576
$633$	−11.0478	−0.439111
$634$	−14.4924	−0.575568
$635$	−1.48734	−0.0590231
$636$	27.2655	1.08115
$637$	0	0
$638$	0.438447	0.0173583
$639$	29.1231	1.15209
$640$	−0.662153	−0.0261739
$641$	7.36932	0.291071	0.145535	−	0.989353i	$-0.453510\pi$
$641$	7.36932	0.291071	0.145535	+	0.989353i	$0.453510\pi$
$642$	−2.06798	−0.0816165
$643$	13.0343	0.514021	0.257011	−	0.966409i	$-0.417262\pi$
$643$	13.0343	0.514021	0.257011	+	0.966409i	$0.417262\pi$
$644$	0	0
$645$	7.31534	0.288041
$646$	32.0000	1.25902
$647$	−0.743668	−0.0292366	−0.0146183	−	0.999893i	$-0.504653\pi$
$647$	−0.743668	−0.0292366	−0.0146183	+	0.999893i	$0.504653\pi$
$648$	10.1231	0.397673
$649$	1.90495	0.0747757
$650$	18.4945	0.725415
$651$	0	0
$652$	−4.43845	−0.173823
$653$	47.8617	1.87297	0.936487	−	0.350701i	$-0.114057\pi$
$653$	47.8617	1.87297	0.936487	+	0.350701i	$0.114057\pi$
$654$	25.1976	0.985303
$655$	1.36932	0.0535036
$656$	4.34475	0.169634
$657$	21.7238	0.847525
$658$	0	0
$659$	−34.3002	−1.33615	−0.668073	−	0.744096i	$-0.732881\pi$
$659$	−34.3002	−1.33615	−0.668073	+	0.744096i	$0.732881\pi$
$660$	0.684658	0.0266503
$661$	25.8597	1.00583	0.502913	−	0.864337i	$-0.332261\pi$
$661$	25.8597	1.00583	0.502913	+	0.864337i	$0.332261\pi$
$662$	−36.3002	−1.41085
$663$	−41.5426	−1.61338
$664$	−5.08842	−0.197469
$665$	0	0
$666$	−18.2462	−0.707026
$667$	−8.24621	−0.319295
$668$	−9.43318	−0.364981
$669$	8.00000	0.309298
$670$	3.22925	0.124757
$671$	0.580639	0.0224153
$672$	0	0
$673$	41.6695	1.60624	0.803121	−	0.595816i	$-0.203171\pi$
$673$	41.6695	1.60624	0.803121	+	0.595816i	$0.203171\pi$
$674$	24.2462	0.933929
$675$	−4.71659	−0.181542
$676$	3.43845	0.132248
$677$	−1.16128	−0.0446315	−0.0223158	−	0.999751i	$-0.507104\pi$
$677$	−1.16128	−0.0446315	−0.0223158	+	0.999751i	$0.507104\pi$
$678$	−45.6786	−1.75427
$679$	0	0
$680$	−2.87689	−0.110324
$681$	29.3693	1.12543
$682$	1.77766	0.0680700
$683$	−22.2462	−0.851228	−0.425614	−	0.904905i	$-0.639942\pi$
$683$	−22.2462	−0.851228	−0.425614	+	0.904905i	$0.639942\pi$
$684$	−18.8664	−0.721373
$685$	−5.46026	−0.208626
$686$	0	0
$687$	55.6155	2.12186
$688$	−4.68466	−0.178601
$689$	−46.8756	−1.78582
$690$	−12.8769	−0.490215
$691$	29.9957	1.14109	0.570545	−	0.821267i	$-0.306732\pi$
$691$	29.9957	1.14109	0.570545	+	0.821267i	$0.306732\pi$
$692$	−11.1293	−0.423073
$693$	0	0
$694$	31.1231	1.18142
$695$	5.50758	0.208914
$696$	2.35829	0.0893909
$697$	18.8769	0.715013
$698$	−22.9208	−0.867565
$699$	30.4948	1.15342
$700$	0	0
$701$	−12.0540	−0.455272	−0.227636	−	0.973746i	$-0.573100\pi$
$701$	−12.0540	−0.455272	−0.227636	+	0.973746i	$0.573100\pi$
$702$	−4.19224	−0.158226
$703$	−52.4631	−1.97868
$704$	−0.438447	−0.0165246
$705$	−1.03399	−0.0389422
$706$	−9.43318	−0.355022
$707$	0	0
$708$	10.2462	0.385076
$709$	3.06913	0.115264	0.0576318	−	0.998338i	$-0.481645\pi$
$709$	3.06913	0.115264	0.0576318	+	0.998338i	$0.481645\pi$
$710$	−7.52823	−0.282530
$711$	20.0000	0.750059
$712$	−9.06134	−0.339588
$713$	−33.4337	−1.25210
$714$	0	0
$715$	−1.17708	−0.0440203
$716$	−1.75379	−0.0655422
$717$	0	0
$718$	28.6847	1.07050
$719$	29.6238	1.10478	0.552391	−	0.833585i	$-0.313715\pi$
$719$	29.6238	1.10478	0.552391	+	0.833585i	$0.313715\pi$
$720$	1.69614	0.0632114
$721$	0	0
$722$	−35.2462	−1.31173
$723$	−58.0540	−2.15905
$724$	−22.1771	−0.824206
$725$	−4.56155	−0.169412
$726$	−25.4879	−0.945944
$727$	34.7123	1.28741	0.643703	−	0.765275i	$-0.277397\pi$
$727$	34.7123	1.28741	0.643703	+	0.765275i	$0.277397\pi$
$728$	0	0
$729$	−18.6155	−0.689464
$730$	−5.61553	−0.207840
$731$	−20.3537	−0.752809
$732$	3.12311	0.115433
$733$	−24.3266	−0.898524	−0.449262	−	0.893400i	$-0.648313\pi$
$733$	−24.3266	−0.898524	−0.449262	+	0.893400i	$0.648313\pi$
$734$	−0.371834	−0.0137246
$735$	0	0
$736$	8.24621	0.303959
$737$	2.13826	0.0787638
$738$	−11.1293	−0.409676
$739$	−22.0540	−0.811269	−0.405634	−	0.914035i	$-0.632949\pi$
$739$	−22.0540	−0.811269	−0.405634	+	0.914035i	$0.632949\pi$
$740$	4.71659	0.173385
$741$	70.4228	2.58705
$742$	0	0
$743$	44.9848	1.65033	0.825167	−	0.564889i	$-0.191081\pi$
$743$	44.9848	1.65033	0.825167	+	0.564889i	$0.191081\pi$
$744$	9.56155	0.350544
$745$	1.77766	0.0651283
$746$	21.8078	0.798439
$747$	13.0343	0.476899
$748$	−1.90495	−0.0696517
$749$	0	0
$750$	−14.9309	−0.545198
$751$	30.7386	1.12167	0.560834	−	0.827928i	$-0.310480\pi$
$751$	30.7386	1.12167	0.560834	+	0.827928i	$0.310480\pi$
$752$	0.662153	0.0241463
$753$	53.1771	1.93788
$754$	−4.05444	−0.147654
$755$	2.81164	0.102326
$756$	0	0
$757$	−0.492423	−0.0178974	−0.00894870	−	0.999960i	$-0.502848\pi$
$757$	−0.492423	−0.0178974	−0.00894870	+	0.999960i	$0.502848\pi$
$758$	−32.4924	−1.18018
$759$	−8.52648	−0.309492
$760$	4.87689	0.176904
$761$	27.1383	0.983761	0.491881	−	0.870663i	$-0.336310\pi$
$761$	27.1383	0.983761	0.491881	+	0.870663i	$0.336310\pi$
$762$	−5.29723	−0.191898
$763$	0	0
$764$	−10.2462	−0.370695
$765$	7.36932	0.266438
$766$	−7.36520	−0.266116
$767$	−17.6155	−0.636060
$768$	−2.35829	−0.0850976
$769$	24.6984	0.890649	0.445324	−	0.895369i	$-0.353088\pi$
$769$	24.6984	0.890649	0.445324	+	0.895369i	$0.353088\pi$
$770$	0	0
$771$	32.6847	1.17711
$772$	−3.36932	−0.121264
$773$	−43.0299	−1.54768	−0.773840	−	0.633382i	$-0.781666\pi$
$773$	−43.0299	−1.54768	−0.773840	+	0.633382i	$0.781666\pi$
$774$	12.0000	0.431331
$775$	−18.4945	−0.664343
$776$	−6.99337	−0.251047
$777$	0	0
$778$	32.9848	1.18256
$779$	−32.0000	−1.14652
$780$	−6.33122	−0.226694
$781$	−4.98485	−0.178372
$782$	35.8278	1.28120
$783$	1.03399	0.0369517
$784$	0	0
$785$	11.5076	0.410723
$786$	4.87689	0.173953
$787$	−29.9957	−1.06923	−0.534615	−	0.845096i	$-0.679543\pi$
$787$	−29.9957	−1.06923	−0.534615	+	0.845096i	$0.679543\pi$
$788$	−3.75379	−0.133723
$789$	37.2794	1.32718
$790$	−5.16994	−0.183938
$791$	0	0
$792$	1.12311	0.0399078
$793$	−5.36932	−0.190670
$794$	10.2584	0.364056
$795$	18.0540	0.640309
$796$	−20.7713	−0.736219
$797$	−47.1659	−1.67070	−0.835351	−	0.549717i	$-0.814735\pi$
$797$	−47.1659	−1.67070	−0.835351	+	0.549717i	$0.814735\pi$
$798$	0	0
$799$	2.87689	0.101777
$800$	4.56155	0.161275
$801$	23.2111	0.820124
$802$	−14.1922	−0.501145
$803$	−3.71834	−0.131217
$804$	11.5012	0.405614
$805$	0	0
$806$	−16.4384	−0.579020
$807$	58.7386	2.06770
$808$	11.3381	0.398874
$809$	42.9848	1.51127	0.755633	−	0.654995i	$-0.227329\pi$
$809$	42.9848	1.51127	0.755633	+	0.654995i	$0.227329\pi$
$810$	6.70305	0.235521
$811$	−54.7399	−1.92218	−0.961089	−	0.276239i	$-0.910912\pi$
$811$	−54.7399	−1.92218	−0.961089	+	0.276239i	$0.910912\pi$
$812$	0	0
$813$	−28.3002	−0.992531
$814$	3.12311	0.109465
$815$	−2.93893	−0.102946
$816$	−10.2462	−0.358689
$817$	34.5035	1.20712
$818$	−29.9957	−1.04877
$819$	0	0
$820$	2.87689	0.100466
$821$	−16.9309	−0.590891	−0.295446	−	0.955360i	$-0.595468\pi$
$821$	−16.9309	−0.590891	−0.295446	+	0.955360i	$0.595468\pi$
$822$	−19.4470	−0.678292
$823$	26.2462	0.914885	0.457443	−	0.889239i	$-0.348765\pi$
$823$	26.2462	0.914885	0.457443	+	0.889239i	$0.348765\pi$
$824$	12.8255	0.446796
$825$	−4.71659	−0.164211
$826$	0	0
$827$	−25.3153	−0.880301	−0.440150	−	0.897924i	$-0.645075\pi$
$827$	−25.3153	−0.880301	−0.440150	+	0.897924i	$0.645075\pi$
$828$	−21.1231	−0.734079
$829$	−22.2586	−0.773074	−0.386537	−	0.922274i	$-0.626329\pi$
$829$	−22.2586	−0.773074	−0.386537	+	0.922274i	$0.626329\pi$
$830$	−3.36932	−0.116951
$831$	23.5829	0.818083
$832$	4.05444	0.140562
$833$	0	0
$834$	19.6155	0.679230
$835$	−6.24621	−0.216159
$836$	3.22925	0.111686
$837$	4.19224	0.144905
$838$	−5.83209	−0.201466
$839$	−21.4335	−0.739965	−0.369983	−	0.929039i	$-0.620636\pi$
$839$	−21.4335	−0.739965	−0.369983	+	0.929039i	$0.620636\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	31.1231	1.07257
$843$	68.8081	2.36988
$844$	4.68466	0.161253
$845$	2.27678	0.0783236
$846$	−1.69614	−0.0583145
$847$	0	0
$848$	−11.5616	−0.397025
$849$	24.8769	0.853773
$850$	19.8188	0.679780
$851$	−58.7386	−2.01353
$852$	−26.8122	−0.918571
$853$	−36.0823	−1.23544	−0.617718	−	0.786400i	$-0.711943\pi$
$853$	−36.0823	−1.23544	−0.617718	+	0.786400i	$0.711943\pi$
$854$	0	0
$855$	−12.4924	−0.427232
$856$	0.876894	0.0299716
$857$	32.5628	1.11232	0.556162	−	0.831074i	$-0.312274\pi$
$857$	32.5628	1.11232	0.556162	+	0.831074i	$0.312274\pi$
$858$	−4.19224	−0.143121
$859$	33.8871	1.15621	0.578106	−	0.815962i	$-0.303792\pi$
$859$	33.8871	1.15621	0.578106	+	0.815962i	$0.303792\pi$
$860$	−3.10196	−0.105776
$861$	0	0
$862$	−8.00000	−0.272481
$863$	−14.0000	−0.476566	−0.238283	−	0.971196i	$-0.576585\pi$
$863$	−14.0000	−0.476566	−0.238283	+	0.971196i	$0.576585\pi$
$864$	−1.03399	−0.0351770
$865$	−7.36932	−0.250564
$866$	16.5896	0.563737
$867$	−4.42627	−0.150324
$868$	0	0
$869$	−3.42329	−0.116127
$870$	1.56155	0.0529416
$871$	−19.7731	−0.669984
$872$	−10.6847	−0.361828
$873$	17.9139	0.606293
$874$	−60.7350	−2.05439
$875$	0	0
$876$	−20.0000	−0.675737
$877$	−30.6847	−1.03615	−0.518074	−	0.855336i	$-0.673351\pi$
$877$	−30.6847	−1.03615	−0.518074	+	0.855336i	$0.673351\pi$
$878$	−35.6647	−1.20363
$879$	−27.1231	−0.914840
$880$	−0.290319	−0.00978666
$881$	−32.4813	−1.09432	−0.547161	−	0.837028i	$-0.684291\pi$
$881$	−32.4813	−1.09432	−0.547161	+	0.837028i	$0.684291\pi$
$882$	0	0
$883$	−35.1231	−1.18199	−0.590993	−	0.806676i	$-0.701264\pi$
$883$	−35.1231	−1.18199	−0.590993	+	0.806676i	$0.701264\pi$
$884$	17.6155	0.592474
$885$	6.78456	0.228061
$886$	−33.1231	−1.11279
$887$	6.28544	0.211044	0.105522	−	0.994417i	$-0.466349\pi$
$887$	6.28544	0.211044	0.105522	+	0.994417i	$0.466349\pi$
$888$	16.7984	0.563717
$889$	0	0
$890$	−6.00000	−0.201120
$891$	4.43845	0.148694
$892$	−3.39228	−0.113582
$893$	−4.87689	−0.163199
$894$	6.33122	0.211748
$895$	−1.16128	−0.0388172
$896$	0	0
$897$	78.8466	2.63261
$898$	19.3693	0.646362
$899$	4.05444	0.135223
$900$	−11.6847	−0.389489
$901$	−50.2321	−1.67347
$902$	1.90495	0.0634277
$903$	0	0
$904$	19.3693	0.644214
$905$	−14.6847	−0.488135
$906$	10.0138	0.332687
$907$	19.3693	0.643148	0.321574	−	0.946885i	$-0.395788\pi$
$907$	19.3693	0.643148	0.321574	+	0.946885i	$0.395788\pi$
$908$	−12.4536	−0.413288
$909$	−29.0432	−0.963302
$910$	0	0
$911$	0.300187	0.00994562	0.00497281	−	0.999988i	$-0.498417\pi$
$911$	0.300187	0.00994562	0.00497281	+	0.999988i	$0.498417\pi$
$912$	17.3693	0.575156
$913$	−2.23100	−0.0738355
$914$	−29.3693	−0.971451
$915$	2.06798	0.0683651
$916$	−23.5829	−0.779202
$917$	0	0
$918$	−4.49242	−0.148272
$919$	18.0000	0.593765	0.296883	−	0.954914i	$-0.404053\pi$
$919$	18.0000	0.593765	0.296883	+	0.954914i	$0.404053\pi$
$920$	5.46026	0.180019
$921$	26.4384	0.871176
$922$	−23.4199	−0.771294
$923$	46.0962	1.51727
$924$	0	0
$925$	−32.4924	−1.06834
$926$	23.3693	0.767963
$927$	−32.8531	−1.07904
$928$	−1.00000	−0.0328266
$929$	24.1636	0.792781	0.396391	−	0.918082i	$-0.370263\pi$
$929$	24.1636	0.792781	0.396391	+	0.918082i	$0.370263\pi$
$930$	6.33122	0.207609
$931$	0	0
$932$	−12.9309	−0.423565
$933$	−23.1231	−0.757016
$934$	−6.33122	−0.207164
$935$	−1.26137	−0.0412511
$936$	−10.3857	−0.339466
$937$	15.0565	0.491873	0.245937	−	0.969286i	$-0.420904\pi$
$937$	15.0565	0.491873	0.245937	+	0.969286i	$0.420904\pi$
$938$	0	0
$939$	64.3002	2.09836
$940$	0.438447	0.0143006
$941$	−10.0953	−0.329098	−0.164549	−	0.986369i	$-0.552617\pi$
$941$	−10.0953	−0.329098	−0.164549	+	0.986369i	$0.552617\pi$
$942$	40.9848	1.33536
$943$	−35.8278	−1.16671
$944$	−4.34475	−0.141410
$945$	0	0
$946$	−2.05398	−0.0667805
$947$	−52.6847	−1.71202	−0.856011	−	0.516958i	$-0.827064\pi$
$947$	−52.6847	−1.71202	−0.856011	+	0.516958i	$0.827064\pi$
$948$	−18.4130	−0.598027
$949$	34.3845	1.11617
$950$	−33.5968	−1.09002
$951$	−34.1774	−1.10828
$952$	0	0
$953$	−38.0540	−1.23269	−0.616345	−	0.787477i	$-0.711387\pi$
$953$	−38.0540	−1.23269	−0.616345	+	0.787477i	$0.711387\pi$
$954$	29.6155	0.958838
$955$	−6.78456	−0.219543
$956$	0	0
$957$	1.03399	0.0334241
$958$	−40.2998	−1.30203
$959$	0	0
$960$	−1.56155	−0.0503989
$961$	−14.5616	−0.469728
$962$	−28.8802	−0.931134
$963$	−2.24621	−0.0723831
$964$	24.6169	0.792858
$965$	−2.23100	−0.0718186
$966$	0	0
$967$	−22.9309	−0.737407	−0.368704	−	0.929547i	$-0.620198\pi$
$967$	−22.9309	−0.737407	−0.368704	+	0.929547i	$0.620198\pi$
$968$	10.8078	0.347375
$969$	75.4654	2.42430
$970$	−4.63068	−0.148682
$971$	11.5012	0.369090	0.184545	−	0.982824i	$-0.440919\pi$
$971$	11.5012	0.369090	0.184545	+	0.982824i	$0.440919\pi$
$972$	20.7713	0.666240
$973$	0	0
$974$	8.00000	0.256337
$975$	43.6155	1.39681
$976$	−1.32431	−0.0423900
$977$	17.6695	0.565297	0.282649	−	0.959223i	$-0.408787\pi$
$977$	17.6695	0.565297	0.282649	+	0.959223i	$0.408787\pi$
$978$	−10.4672	−0.334703
$979$	−3.97292	−0.126975
$980$	0	0
$981$	27.3693	0.873835
$982$	25.1771	0.803433
$983$	−39.1385	−1.24833	−0.624163	−	0.781294i	$-0.714560\pi$
$983$	−39.1385	−1.24833	−0.624163	+	0.781294i	$0.714560\pi$
$984$	10.2462	0.326637
$985$	−2.48558	−0.0791973
$986$	−4.34475	−0.138365
$987$	0	0
$988$	−29.8617	−0.950028
$989$	38.6307	1.22838
$990$	0.743668	0.0236353
$991$	−39.6155	−1.25843	−0.629214	−	0.777232i	$-0.716623\pi$
$991$	−39.6155	−1.25843	−0.629214	+	0.777232i	$0.716623\pi$
$992$	−4.05444	−0.128728
$993$	−85.6065	−2.71664
$994$	0	0
$995$	−13.7538	−0.436024
$996$	−12.0000	−0.380235
$997$	25.4879	0.807210	0.403605	−	0.914933i	$-0.367757\pi$
$997$	25.4879	0.807210	0.403605	+	0.914933i	$0.367757\pi$
$998$	−3.12311	−0.0988602
$999$	7.36520	0.233025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2842.2.a.p.1.1	✓	4
7.6	odd	2	inner	2842.2.a.p.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2842.2.a.p.1.1	✓	4	1.1	even	1	trivial
2842.2.a.p.1.4	yes	4	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2842.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.35829 q^{3} +1.00000 q^{4} +0.662153 q^{5} +2.35829 q^{6} -1.00000 q^{8} +2.56155 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.35829 q^{3} +1.00000 q^{4} +0.662153 q^{5} +2.35829 q^{6} -1.00000 q^{8} +2.56155 q^{9} -0.662153 q^{10} -0.438447 q^{11} -2.35829 q^{12} +4.05444 q^{13} -1.56155 q^{15} +1.00000 q^{16} +4.34475 q^{17} -2.56155 q^{18} -7.36520 q^{19} +0.662153 q^{20} +0.438447 q^{22} -8.24621 q^{23} +2.35829 q^{24} -4.56155 q^{25} -4.05444 q^{26} +1.03399 q^{27} +1.00000 q^{29} +1.56155 q^{30} +4.05444 q^{31} -1.00000 q^{32} +1.03399 q^{33} -4.34475 q^{34} +2.56155 q^{36} +7.12311 q^{37} +7.36520 q^{38} -9.56155 q^{39} -0.662153 q^{40} +4.34475 q^{41} -4.68466 q^{43} -0.438447 q^{44} +1.69614 q^{45} +8.24621 q^{46} +0.662153 q^{47} -2.35829 q^{48} +4.56155 q^{50} -10.2462 q^{51} +4.05444 q^{52} -11.5616 q^{53} -1.03399 q^{54} -0.290319 q^{55} +17.3693 q^{57} -1.00000 q^{58} -4.34475 q^{59} -1.56155 q^{60} -1.32431 q^{61} -4.05444 q^{62} +1.00000 q^{64} +2.68466 q^{65} -1.03399 q^{66} -4.87689 q^{67} +4.34475 q^{68} +19.4470 q^{69} +11.3693 q^{71} -2.56155 q^{72} +8.48071 q^{73} -7.12311 q^{74} +10.7575 q^{75} -7.36520 q^{76} +9.56155 q^{78} +7.80776 q^{79} +0.662153 q^{80} -10.1231 q^{81} -4.34475 q^{82} +5.08842 q^{83} +2.87689 q^{85} +4.68466 q^{86} -2.35829 q^{87} +0.438447 q^{88} +9.06134 q^{89} -1.69614 q^{90} -8.24621 q^{92} -9.56155 q^{93} -0.662153 q^{94} -4.87689 q^{95} +2.35829 q^{96} +6.99337 q^{97} -1.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 2 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 2 * q^9	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 2 q^{9} - 10 q^{11} + 2 q^{15} + 4 q^{16} - 2 q^{18} + 10 q^{22} - 10 q^{25} + 4 q^{29} - 2 q^{30} - 4 q^{32} + 2 q^{36} + 12 q^{37} - 30 q^{39} + 6 q^{43} - 10 q^{44} + 10 q^{50} - 8 q^{51} - 38 q^{53} + 20 q^{57} - 4 q^{58} + 2 q^{60} + 4 q^{64} - 14 q^{65} - 36 q^{67} - 4 q^{71} - 2 q^{72} - 12 q^{74} + 30 q^{78} - 10 q^{79} - 24 q^{81} + 28 q^{85} - 6 q^{86} + 10 q^{88} - 30 q^{93} - 36 q^{95} + 12 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 2 * q^9 - 10 * q^11 + 2 * q^15 + 4 * q^16 - 2 * q^18 + 10 * q^22 - 10 * q^25 + 4 * q^29 - 2 * q^30 - 4 * q^32 + 2 * q^36 + 12 * q^37 - 30 * q^39 + 6 * q^43 - 10 * q^44 + 10 * q^50 - 8 * q^51 - 38 * q^53 + 20 * q^57 - 4 * q^58 + 2 * q^60 + 4 * q^64 - 14 * q^65 - 36 * q^67 - 4 * q^71 - 2 * q^72 - 12 * q^74 + 30 * q^78 - 10 * q^79 - 24 * q^81 + 28 * q^85 - 6 * q^86 + 10 * q^88 - 30 * q^93 - 36 * q^95 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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