LMFDB - Embedded newform 2151.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$1$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.31397 q^{2} +20.2383 q^{4} -1.11401 q^{5} +6.72325 q^{7} -65.0342 q^{8} +O(q^{10})$	\(q-5.31397 q^{2} +20.2383 q^{4} -1.11401 q^{5} +6.72325 q^{7} -65.0342 q^{8} +5.91984 q^{10} +41.9133 q^{11} +35.6062 q^{13} -35.7272 q^{14} +183.683 q^{16} +52.6666 q^{17} +148.199 q^{19} -22.5458 q^{20} -222.726 q^{22} -139.223 q^{23} -123.759 q^{25} -189.211 q^{26} +136.067 q^{28} -77.5892 q^{29} -212.446 q^{31} -455.815 q^{32} -279.869 q^{34} -7.48979 q^{35} +216.186 q^{37} -787.528 q^{38} +72.4489 q^{40} -145.987 q^{41} -164.462 q^{43} +848.256 q^{44} +739.828 q^{46} +210.081 q^{47} -297.798 q^{49} +657.652 q^{50} +720.611 q^{52} -37.1674 q^{53} -46.6920 q^{55} -437.241 q^{56} +412.307 q^{58} +76.0266 q^{59} -666.103 q^{61} +1128.93 q^{62} +952.722 q^{64} -39.6658 q^{65} -538.598 q^{67} +1065.88 q^{68} +39.8005 q^{70} -241.054 q^{71} -774.882 q^{73} -1148.81 q^{74} +2999.31 q^{76} +281.794 q^{77} -481.251 q^{79} -204.626 q^{80} +775.770 q^{82} +254.277 q^{83} -58.6713 q^{85} +873.947 q^{86} -2725.80 q^{88} -793.585 q^{89} +239.389 q^{91} -2817.64 q^{92} -1116.37 q^{94} -165.096 q^{95} +1585.71 q^{97} +1582.49 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * q^98

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 $

$2$	−5.31397	−1.87877	−0.939387	−	0.342859i	$-0.888605\pi$
$2$	−5.31397	−1.87877	−0.939387	+	0.342859i	$0.888605\pi$
$3$	0	0
$3$	0	0
$4$	20.2383	2.52979
$5$	−1.11401	−0.0996404	−0.0498202	−	0.998758i	$-0.515865\pi$
$5$	−1.11401	−0.0996404	−0.0498202	+	0.998758i	$0.515865\pi$
$6$	0	0
$7$	6.72325	0.363021	0.181511	−	0.983389i	$-0.441901\pi$
$7$	6.72325	0.363021	0.181511	+	0.983389i	$0.441901\pi$
$8$	−65.0342	−2.87413
$9$	0	0
$10$	5.91984	0.187202
$11$	41.9133	1.14885	0.574425	−	0.818557i	$-0.305226\pi$
$11$	41.9133	1.14885	0.574425	+	0.818557i	$0.305226\pi$
$12$	0	0
$13$	35.6062	0.759646	0.379823	−	0.925059i	$-0.375985\pi$
$13$	35.6062	0.759646	0.379823	+	0.925059i	$0.375985\pi$
$14$	−35.7272	−0.682035
$15$	0	0
$16$	183.683	2.87005
$17$	52.6666	0.751384	0.375692	−	0.926745i	$-0.377405\pi$
$17$	52.6666	0.751384	0.375692	+	0.926745i	$0.377405\pi$
$18$	0	0
$19$	148.199	1.78944	0.894718	−	0.446632i	$-0.147377\pi$
$19$	148.199	1.78944	0.894718	+	0.446632i	$0.147377\pi$
$20$	−22.5458	−0.252069
$21$	0	0
$22$	−222.726	−2.15843
$23$	−139.223	−1.26217	−0.631087	−	0.775712i	$-0.717391\pi$
$23$	−139.223	−1.26217	−0.631087	+	0.775712i	$0.717391\pi$
$24$	0	0
$25$	−123.759	−0.990072
$26$	−189.211	−1.42720
$27$	0	0
$28$	136.067	0.918367
$29$	−77.5892	−0.496826	−0.248413	−	0.968654i	$-0.579909\pi$
$29$	−77.5892	−0.496826	−0.248413	+	0.968654i	$0.579909\pi$
$30$	0	0
$31$	−212.446	−1.23085	−0.615427	−	0.788194i	$-0.711017\pi$
$31$	−212.446	−1.23085	−0.615427	+	0.788194i	$0.711017\pi$
$32$	−455.815	−2.51804
$33$	0	0
$34$	−279.869	−1.41168
$35$	−7.48979	−0.0361716
$36$	0	0
$37$	216.186	0.960563	0.480281	−	0.877114i	$-0.340535\pi$
$37$	216.186	0.960563	0.480281	+	0.877114i	$0.340535\pi$
$38$	−787.528	−3.36194
$39$	0	0
$40$	72.4489	0.286379
$41$	−145.987	−0.556080	−0.278040	−	0.960569i	$-0.589685\pi$
$41$	−145.987	−0.556080	−0.278040	+	0.960569i	$0.589685\pi$
$42$	0	0
$43$	−164.462	−0.583261	−0.291630	−	0.956531i	$-0.594198\pi$
$43$	−164.462	−0.583261	−0.291630	+	0.956531i	$0.594198\pi$
$44$	848.256	2.90635
$45$	0	0
$46$	739.828	2.37134
$47$	210.081	0.651989	0.325995	−	0.945372i	$-0.394301\pi$
$47$	210.081	0.651989	0.325995	+	0.945372i	$0.394301\pi$
$48$	0	0
$49$	−297.798	−0.868216
$50$	657.652	1.86012
$51$	0	0
$52$	720.611	1.92174
$53$	−37.1674	−0.0963271	−0.0481636	−	0.998839i	$-0.515337\pi$
$53$	−37.1674	−0.0963271	−0.0481636	+	0.998839i	$0.515337\pi$
$54$	0	0
$55$	−46.6920	−0.114472
$56$	−437.241	−1.04337
$57$	0	0
$58$	412.307	0.933424
$59$	76.0266	0.167760	0.0838798	−	0.996476i	$-0.473269\pi$
$59$	76.0266	0.167760	0.0838798	+	0.996476i	$0.473269\pi$
$60$	0	0
$61$	−666.103	−1.39813	−0.699064	−	0.715060i	$-0.746400\pi$
$61$	−666.103	−1.39813	−0.699064	+	0.715060i	$0.746400\pi$
$62$	1128.93	2.31250
$63$	0	0
$64$	952.722	1.86079
$65$	−39.6658	−0.0756914
$66$	0	0
$67$	−538.598	−0.982093	−0.491046	−	0.871133i	$-0.663385\pi$
$67$	−538.598	−0.982093	−0.491046	+	0.871133i	$0.663385\pi$
$68$	1065.88	1.90084
$69$	0	0
$70$	39.8005	0.0679582
$71$	−241.054	−0.402927	−0.201464	−	0.979496i	$-0.564570\pi$
$71$	−241.054	−0.402927	−0.201464	+	0.979496i	$0.564570\pi$
$72$	0	0
$73$	−774.882	−1.24237	−0.621185	−	0.783664i	$-0.713349\pi$
$73$	−774.882	−1.24237	−0.621185	+	0.783664i	$0.713349\pi$
$74$	−1148.81	−1.80468
$75$	0	0
$76$	2999.31	4.52690
$77$	281.794	0.417057
$78$	0	0
$79$	−481.251	−0.685379	−0.342690	−	0.939449i	$-0.611338\pi$
$79$	−481.251	−0.685379	−0.342690	+	0.939449i	$0.611338\pi$
$80$	−204.626	−0.285973
$81$	0	0
$82$	775.770	1.04475
$83$	254.277	0.336271	0.168135	−	0.985764i	$-0.446225\pi$
$83$	254.277	0.336271	0.168135	+	0.985764i	$0.446225\pi$
$84$	0	0
$85$	−58.6713	−0.0748682
$86$	873.947	1.09581
$87$	0	0
$88$	−2725.80	−3.30194
$89$	−793.585	−0.945167	−0.472583	−	0.881286i	$-0.656678\pi$
$89$	−793.585	−0.945167	−0.472583	+	0.881286i	$0.656678\pi$
$90$	0	0
$91$	239.389	0.275767
$92$	−2817.64	−3.19304
$93$	0	0
$94$	−1116.37	−1.22494
$95$	−165.096	−0.178300
$96$	0	0
$97$	1585.71	1.65984	0.829919	−	0.557883i	$-0.188386\pi$
$97$	1585.71	1.65984	0.829919	+	0.557883i	$0.188386\pi$
$98$	1582.49	1.63118
$99$	0	0
$100$	−2504.67	−2.50467
$101$	−1102.76	−1.08642	−0.543211	−	0.839596i	$-0.682792\pi$
$101$	−1102.76	−1.08642	−0.543211	+	0.839596i	$0.682792\pi$
$102$	0	0
$103$	−1412.44	−1.35118	−0.675591	−	0.737276i	$-0.736112\pi$
$103$	−1412.44	−1.35118	−0.675591	+	0.737276i	$0.736112\pi$
$104$	−2315.62	−2.18332
$105$	0	0
$106$	197.507	0.180977
$107$	−1655.53	−1.49576	−0.747880	−	0.663834i	$-0.768928\pi$
$107$	−1655.53	−1.49576	−0.747880	+	0.663834i	$0.768928\pi$
$108$	0	0
$109$	−983.080	−0.863872	−0.431936	−	0.901904i	$-0.642169\pi$
$109$	−983.080	−0.863872	−0.431936	+	0.901904i	$0.642169\pi$
$110$	248.120	0.215067
$111$	0	0
$112$	1234.95	1.04189
$113$	286.160	0.238227	0.119114	−	0.992881i	$-0.461995\pi$
$113$	286.160	0.238227	0.119114	+	0.992881i	$0.461995\pi$
$114$	0	0
$115$	155.096	0.125764
$116$	−1570.28	−1.25687
$117$	0	0
$118$	−404.003	−0.315182
$119$	354.091	0.272768
$120$	0	0
$121$	425.727	0.319855
$122$	3539.65	2.62676
$123$	0	0
$124$	−4299.56	−3.11380
$125$	277.121	0.198292
$126$	0	0
$127$	−1024.11	−0.715554	−0.357777	−	0.933807i	$-0.616465\pi$
$127$	−1024.11	−0.715554	−0.357777	+	0.933807i	$0.616465\pi$
$128$	−1416.22	−0.977950
$129$	0	0
$130$	210.783	0.142207
$131$	56.9649	0.0379927	0.0189964	−	0.999820i	$-0.493953\pi$
$131$	56.9649	0.0379927	0.0189964	+	0.999820i	$0.493953\pi$
$132$	0	0
$133$	996.381	0.649603
$134$	2862.10	1.84513
$135$	0	0
$136$	−3425.13	−2.15958
$137$	1748.39	1.09033	0.545163	−	0.838330i	$-0.316468\pi$
$137$	1748.39	1.09033	0.545163	+	0.838330i	$0.316468\pi$
$138$	0	0
$139$	−2401.77	−1.46558	−0.732789	−	0.680456i	$-0.761782\pi$
$139$	−2401.77	−1.46558	−0.732789	+	0.680456i	$0.761782\pi$
$140$	−151.581	−0.0915065
$141$	0	0
$142$	1280.95	0.757009
$143$	1492.38	0.872718
$144$	0	0
$145$	86.4355	0.0495040
$146$	4117.70	2.33413
$147$	0	0
$148$	4375.25	2.43002
$149$	1504.90	0.827427	0.413713	−	0.910407i	$-0.364232\pi$
$149$	1504.90	0.827427	0.413713	+	0.910407i	$0.364232\pi$
$150$	0	0
$151$	3271.64	1.76320	0.881598	−	0.472002i	$-0.156468\pi$
$151$	3271.64	1.76320	0.881598	+	0.472002i	$0.156468\pi$
$152$	−9638.02	−5.14307
$153$	0	0
$154$	−1497.44	−0.783555
$155$	236.668	0.122643
$156$	0	0
$157$	1696.26	0.862272	0.431136	−	0.902287i	$-0.358113\pi$
$157$	1696.26	0.862272	0.431136	+	0.902287i	$0.358113\pi$
$158$	2557.35	1.28767
$159$	0	0
$160$	507.784	0.250899
$161$	−936.031	−0.458196
$162$	0	0
$163$	121.427	0.0583492	0.0291746	−	0.999574i	$-0.490712\pi$
$163$	121.427	0.0583492	0.0291746	+	0.999574i	$0.490712\pi$
$164$	−2954.53	−1.40677
$165$	0	0
$166$	−1351.22	−0.631777
$167$	−2294.07	−1.06300	−0.531498	−	0.847059i	$-0.678371\pi$
$167$	−2294.07	−1.06300	−0.531498	+	0.847059i	$0.678371\pi$
$168$	0	0
$169$	−929.196	−0.422939
$170$	311.778	0.140660
$171$	0	0
$172$	−3328.43	−1.47553
$173$	−1759.41	−0.773213	−0.386606	−	0.922245i	$-0.626353\pi$
$173$	−1759.41	−0.773213	−0.386606	+	0.922245i	$0.626353\pi$
$174$	0	0
$175$	−832.062	−0.359417
$176$	7698.77	3.29726
$177$	0	0
$178$	4217.09	1.77575
$179$	2743.05	1.14539	0.572697	−	0.819767i	$-0.305897\pi$
$179$	2743.05	1.14539	0.572697	+	0.819767i	$0.305897\pi$
$180$	0	0
$181$	−379.052	−0.155661	−0.0778306	−	0.996967i	$-0.524799\pi$
$181$	−379.052	−0.155661	−0.0778306	+	0.996967i	$0.524799\pi$
$182$	−1272.11	−0.518105
$183$	0	0
$184$	9054.25	3.62765
$185$	−240.835	−0.0957109
$186$	0	0
$187$	2207.43	0.863227
$188$	4251.69	1.64940
$189$	0	0
$190$	877.317	0.334985
$191$	−4015.18	−1.52109	−0.760544	−	0.649286i	$-0.775068\pi$
$191$	−4015.18	−1.52109	−0.760544	+	0.649286i	$0.775068\pi$
$192$	0	0
$193$	−2147.90	−0.801083	−0.400541	−	0.916279i	$-0.631178\pi$
$193$	−2147.90	−0.801083	−0.400541	+	0.916279i	$0.631178\pi$
$194$	−8426.42	−3.11846
$195$	0	0
$196$	−6026.93	−2.19640
$197$	1276.02	0.461485	0.230742	−	0.973015i	$-0.425885\pi$
$197$	1276.02	0.461485	0.230742	+	0.973015i	$0.425885\pi$
$198$	0	0
$199$	2972.19	1.05876	0.529379	−	0.848385i	$-0.322425\pi$
$199$	2972.19	1.05876	0.529379	+	0.848385i	$0.322425\pi$
$200$	8048.56	2.84560
$201$	0	0
$202$	5860.04	2.04114
$203$	−521.652	−0.180358
$204$	0	0
$205$	162.631	0.0554081
$206$	7505.67	2.53857
$207$	0	0
$208$	6540.27	2.18022
$209$	6211.53	2.05579
$210$	0	0
$211$	−5415.02	−1.76676	−0.883378	−	0.468661i	$-0.844737\pi$
$211$	−5415.02	−1.76676	−0.883378	+	0.468661i	$0.844737\pi$
$212$	−752.206	−0.243687
$213$	0	0
$214$	8797.45	2.81019
$215$	183.213	0.0581163
$216$	0	0
$217$	−1428.33	−0.446826
$218$	5224.06	1.62302
$219$	0	0
$220$	−944.968	−0.289590
$221$	1875.26	0.570786
$222$	0	0
$223$	4913.42	1.47546	0.737728	−	0.675098i	$-0.235899\pi$
$223$	4913.42	1.47546	0.737728	+	0.675098i	$0.235899\pi$
$224$	−3064.55	−0.914103
$225$	0	0
$226$	−1520.65	−0.447575
$227$	5911.74	1.72853	0.864264	−	0.503038i	$-0.167785\pi$
$227$	5911.74	1.72853	0.864264	+	0.503038i	$0.167785\pi$
$228$	0	0
$229$	−2524.06	−0.728361	−0.364181	−	0.931328i	$-0.618651\pi$
$229$	−2524.06	−0.728361	−0.364181	+	0.931328i	$0.618651\pi$
$230$	−824.178	−0.236281
$231$	0	0
$232$	5045.95	1.42794
$233$	−5280.44	−1.48469	−0.742346	−	0.670017i	$-0.766287\pi$
$233$	−5280.44	−1.48469	−0.742346	+	0.670017i	$0.766287\pi$
$234$	0	0
$235$	−234.033	−0.0649644
$236$	1538.65	0.424397
$237$	0	0
$238$	−1881.63	−0.512470
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	535.406	0.143106	0.0715530	−	0.997437i	$-0.477205\pi$
$241$	535.406	0.143106	0.0715530	+	0.997437i	$0.477205\pi$
$242$	−2262.30	−0.600935
$243$	0	0
$244$	−13480.8	−3.53697
$245$	331.751	0.0865093
$246$	0	0
$247$	5276.82	1.35934
$248$	13816.3	3.53764
$249$	0	0
$250$	−1472.61	−0.372545
$251$	1799.25	0.452462	0.226231	−	0.974074i	$-0.427360\pi$
$251$	1799.25	0.452462	0.226231	+	0.974074i	$0.427360\pi$
$252$	0	0
$253$	−5835.30	−1.45005
$254$	5442.11	1.34436
$255$	0	0
$256$	−96.0022	−0.0234380
$257$	5089.78	1.23538	0.617688	−	0.786423i	$-0.288070\pi$
$257$	5089.78	1.23538	0.617688	+	0.786423i	$0.288070\pi$
$258$	0	0
$259$	1453.47	0.348705
$260$	−802.770	−0.191483
$261$	0	0
$262$	−302.710	−0.0713798
$263$	−273.965	−0.0642335	−0.0321167	−	0.999484i	$-0.510225\pi$
$263$	−273.965	−0.0642335	−0.0321167	+	0.999484i	$0.510225\pi$
$264$	0	0
$265$	41.4050	0.00959807
$266$	−5294.74	−1.22046
$267$	0	0
$268$	−10900.3	−2.48449
$269$	−2859.66	−0.648166	−0.324083	−	0.946029i	$-0.605056\pi$
$269$	−2859.66	−0.648166	−0.324083	+	0.946029i	$0.605056\pi$
$270$	0	0
$271$	7800.41	1.74849	0.874246	−	0.485484i	$-0.161357\pi$
$271$	7800.41	1.74849	0.874246	+	0.485484i	$0.161357\pi$
$272$	9673.97	2.15651
$273$	0	0
$274$	−9290.88	−2.04848
$275$	−5187.15	−1.13744
$276$	0	0
$277$	6946.10	1.50668	0.753341	−	0.657630i	$-0.228441\pi$
$277$	6946.10	1.50668	0.753341	+	0.657630i	$0.228441\pi$
$278$	12762.9	2.75349
$279$	0	0
$280$	487.092	0.103962
$281$	−6446.54	−1.36857	−0.684285	−	0.729214i	$-0.739886\pi$
$281$	−6446.54	−1.36857	−0.684285	+	0.729214i	$0.739886\pi$
$282$	0	0
$283$	−2224.14	−0.467178	−0.233589	−	0.972335i	$-0.575047\pi$
$283$	−2224.14	−0.467178	−0.233589	+	0.972335i	$0.575047\pi$
$284$	−4878.53	−1.01932
$285$	0	0
$286$	−7930.45	−1.63964
$287$	−981.505	−0.201869
$288$	0	0
$289$	−2139.23	−0.435422
$290$	−459.316	−0.0930067
$291$	0	0
$292$	−15682.3	−3.14294
$293$	8483.02	1.69141	0.845706	−	0.533650i	$-0.179180\pi$
$293$	8483.02	1.69141	0.845706	+	0.533650i	$0.179180\pi$
$294$	0	0
$295$	−84.6946	−0.0167156
$296$	−14059.5	−2.76078
$297$	0	0
$298$	−7997.02	−1.55455
$299$	−4957.21	−0.958805
$300$	0	0
$301$	−1105.72	−0.211736
$302$	−17385.4	−3.31264
$303$	0	0
$304$	27221.7	5.13577
$305$	742.048	0.139310
$306$	0	0
$307$	−2920.34	−0.542908	−0.271454	−	0.962451i	$-0.587505\pi$
$307$	−2920.34	−0.542908	−0.271454	+	0.962451i	$0.587505\pi$
$308$	5703.03	1.05507
$309$	0	0
$310$	−1257.65	−0.230418
$311$	−3073.07	−0.560314	−0.280157	−	0.959954i	$-0.590386\pi$
$311$	−3073.07	−0.560314	−0.280157	+	0.959954i	$0.590386\pi$
$312$	0	0
$313$	5730.56	1.03486	0.517429	−	0.855726i	$-0.326889\pi$
$313$	5730.56	1.03486	0.517429	+	0.855726i	$0.326889\pi$
$314$	−9013.91	−1.62001
$315$	0	0
$316$	−9739.71	−1.73387
$317$	3606.48	0.638990	0.319495	−	0.947588i	$-0.396487\pi$
$317$	3606.48	0.638990	0.319495	+	0.947588i	$0.396487\pi$
$318$	0	0
$319$	−3252.02	−0.570779
$320$	−1061.35	−0.185409
$321$	0	0
$322$	4974.04	0.860847
$323$	7805.16	1.34455
$324$	0	0
$325$	−4406.59	−0.752104
$326$	−645.261	−0.109625
$327$	0	0
$328$	9494.12	1.59825
$329$	1412.43	0.236686
$330$	0	0
$331$	−973.514	−0.161659	−0.0808296	−	0.996728i	$-0.525757\pi$
$331$	−973.514	−0.161659	−0.0808296	+	0.996728i	$0.525757\pi$
$332$	5146.13	0.850695
$333$	0	0
$334$	12190.6	1.99713
$335$	600.005	0.0978561
$336$	0	0
$337$	5418.63	0.875880	0.437940	−	0.899004i	$-0.355708\pi$
$337$	5418.63	0.875880	0.437940	+	0.899004i	$0.355708\pi$
$338$	4937.73	0.794606
$339$	0	0
$340$	−1187.41	−0.189401
$341$	−8904.33	−1.41407
$342$	0	0
$343$	−4308.24	−0.678202
$344$	10695.6	1.67637
$345$	0	0
$346$	9349.48	1.45269
$347$	−1905.03	−0.294719	−0.147360	−	0.989083i	$-0.547077\pi$
$347$	−1905.03	−0.294719	−0.147360	+	0.989083i	$0.547077\pi$
$348$	0	0
$349$	−14.4977	−0.00222362	−0.00111181	−	0.999999i	$-0.500354\pi$
$349$	−14.4977	−0.00222362	−0.00111181	+	0.999999i	$0.500354\pi$
$350$	4421.56	0.675263
$351$	0	0
$352$	−19104.7	−2.89285
$353$	−2213.74	−0.333784	−0.166892	−	0.985975i	$-0.553373\pi$
$353$	−2213.74	−0.333784	−0.166892	+	0.985975i	$0.553373\pi$
$354$	0	0
$355$	268.537	0.0401478
$356$	−16060.8	−2.39107
$357$	0	0
$358$	−14576.5	−2.15194
$359$	−6943.88	−1.02085	−0.510423	−	0.859923i	$-0.670511\pi$
$359$	−6943.88	−1.02085	−0.510423	+	0.859923i	$0.670511\pi$
$360$	0	0
$361$	15104.1	2.20208
$362$	2014.27	0.292452
$363$	0	0
$364$	4844.84	0.697634
$365$	863.229	0.123790
$366$	0	0
$367$	6325.98	0.899765	0.449882	−	0.893088i	$-0.351466\pi$
$367$	6325.98	0.899765	0.449882	+	0.893088i	$0.351466\pi$
$368$	−25572.9	−3.62250
$369$	0	0
$370$	1279.79	0.179819
$371$	−249.886	−0.0349688
$372$	0	0
$373$	4984.06	0.691863	0.345932	−	0.938260i	$-0.387563\pi$
$373$	4984.06	0.691863	0.345932	+	0.938260i	$0.387563\pi$
$374$	−11730.2	−1.62181
$375$	0	0
$376$	−13662.4	−1.87390
$377$	−2762.66	−0.377412
$378$	0	0
$379$	−5710.84	−0.774000	−0.387000	−	0.922080i	$-0.626489\pi$
$379$	−5710.84	−0.774000	−0.387000	+	0.922080i	$0.626489\pi$
$380$	−3341.27	−0.451062
$381$	0	0
$382$	21336.5	2.85778
$383$	636.741	0.0849503	0.0424752	−	0.999098i	$-0.486476\pi$
$383$	636.741	0.0849503	0.0424752	+	0.999098i	$0.486476\pi$
$384$	0	0
$385$	−313.922	−0.0415557
$386$	11413.9	1.50505
$387$	0	0
$388$	32092.1	4.19904
$389$	−8941.36	−1.16541	−0.582706	−	0.812683i	$-0.698006\pi$
$389$	−8941.36	−1.16541	−0.582706	+	0.812683i	$0.698006\pi$
$390$	0	0
$391$	−7332.41	−0.948378
$392$	19367.0	2.49537
$393$	0	0
$394$	−6780.73	−0.867026
$395$	536.120	0.0682914
$396$	0	0
$397$	−7490.81	−0.946985	−0.473493	−	0.880798i	$-0.657007\pi$
$397$	−7490.81	−0.946985	−0.473493	+	0.880798i	$0.657007\pi$
$398$	−15794.1	−1.98917
$399$	0	0
$400$	−22732.4	−2.84156
$401$	−1596.09	−0.198765	−0.0993825	−	0.995049i	$-0.531687\pi$
$401$	−1596.09	−0.198765	−0.0993825	+	0.995049i	$0.531687\pi$
$402$	0	0
$403$	−7564.42	−0.935013
$404$	−22318.0	−2.74842
$405$	0	0
$406$	2772.04	0.338853
$407$	9061.09	1.10354
$408$	0	0
$409$	−6992.22	−0.845337	−0.422669	−	0.906284i	$-0.638907\pi$
$409$	−6992.22	−0.845337	−0.422669	+	0.906284i	$0.638907\pi$
$410$	−864.218	−0.104099
$411$	0	0
$412$	−28585.4	−3.41821
$413$	511.145	0.0609003
$414$	0	0
$415$	−283.268	−0.0335062
$416$	−16229.8	−1.91282
$417$	0	0
$418$	−33007.9	−3.86237
$419$	887.998	0.103536	0.0517680	−	0.998659i	$-0.483514\pi$
$419$	887.998	0.103536	0.0517680	+	0.998659i	$0.483514\pi$
$420$	0	0
$421$	6552.04	0.758497	0.379248	−	0.925295i	$-0.376183\pi$
$421$	6552.04	0.758497	0.379248	+	0.925295i	$0.376183\pi$
$422$	28775.3	3.31934
$423$	0	0
$424$	2417.15	0.276857
$425$	−6517.97	−0.743924
$426$	0	0
$427$	−4478.37	−0.507550
$428$	−33505.2	−3.78396
$429$	0	0
$430$	−973.588	−0.109187
$431$	−592.132	−0.0661763	−0.0330882	−	0.999452i	$-0.510534\pi$
$431$	−592.132	−0.0661763	−0.0330882	+	0.999452i	$0.510534\pi$
$432$	0	0
$433$	−16798.3	−1.86437	−0.932186	−	0.361980i	$-0.882101\pi$
$433$	−16798.3	−1.86437	−0.932186	+	0.361980i	$0.882101\pi$
$434$	7590.11	0.839485
$435$	0	0
$436$	−19895.9	−2.18541
$437$	−20632.8	−2.25858
$438$	0	0
$439$	−2655.38	−0.288688	−0.144344	−	0.989528i	$-0.546107\pi$
$439$	−2655.38	−0.288688	−0.144344	+	0.989528i	$0.546107\pi$
$440$	3036.58	0.329007
$441$	0	0
$442$	−9965.08	−1.07238
$443$	−14692.9	−1.57580	−0.787899	−	0.615805i	$-0.788831\pi$
$443$	−14692.9	−1.57580	−0.787899	+	0.615805i	$0.788831\pi$
$444$	0	0
$445$	884.064	0.0941768
$446$	−26109.8	−2.77205
$447$	0	0
$448$	6405.38	0.675504
$449$	−4379.64	−0.460329	−0.230165	−	0.973152i	$-0.573926\pi$
$449$	−4379.64	−0.460329	−0.230165	+	0.973152i	$0.573926\pi$
$450$	0	0
$451$	−6118.79	−0.638853
$452$	5791.40	0.602665
$453$	0	0
$454$	−31414.8	−3.24751
$455$	−266.683	−0.0274776
$456$	0	0
$457$	11367.4	1.16355	0.581777	−	0.813348i	$-0.302358\pi$
$457$	11367.4	1.16355	0.581777	+	0.813348i	$0.302358\pi$
$458$	13412.8	1.36843
$459$	0	0
$460$	3138.89	0.318155
$461$	14631.4	1.47820	0.739100	−	0.673596i	$-0.235251\pi$
$461$	14631.4	1.47820	0.739100	+	0.673596i	$0.235251\pi$
$462$	0	0
$463$	420.546	0.0422126	0.0211063	−	0.999777i	$-0.493281\pi$
$463$	420.546	0.0422126	0.0211063	+	0.999777i	$0.493281\pi$
$464$	−14251.8	−1.42592
$465$	0	0
$466$	28060.1	2.78940
$467$	−1196.27	−0.118537	−0.0592686	−	0.998242i	$-0.518877\pi$
$467$	−1196.27	−0.118537	−0.0592686	+	0.998242i	$0.518877\pi$
$468$	0	0
$469$	−3621.13	−0.356521
$470$	1243.65	0.122053
$471$	0	0
$472$	−4944.32	−0.482163
$473$	−6893.15	−0.670079
$474$	0	0
$475$	−18341.0	−1.77167
$476$	7166.20	0.690047
$477$	0	0
$478$	1270.04	0.121528
$479$	−1210.89	−0.115505	−0.0577525	−	0.998331i	$-0.518393\pi$
$479$	−1210.89	−0.115505	−0.0577525	+	0.998331i	$0.518393\pi$
$480$	0	0
$481$	7697.58	0.729687
$482$	−2845.13	−0.268864
$483$	0	0
$484$	8616.00	0.809166
$485$	−1766.50	−0.165387
$486$	0	0
$487$	−19207.2	−1.78719	−0.893596	−	0.448872i	$-0.851826\pi$
$487$	−19207.2	−1.78719	−0.893596	+	0.448872i	$0.851826\pi$
$488$	43319.4	4.01840
$489$	0	0
$490$	−1762.92	−0.162531
$491$	−10851.1	−0.997355	−0.498677	−	0.866788i	$-0.666181\pi$
$491$	−10851.1	−0.997355	−0.498677	+	0.866788i	$0.666181\pi$
$492$	0	0
$493$	−4086.36	−0.373307
$494$	−28040.9	−2.55389
$495$	0	0
$496$	−39022.8	−3.53262
$497$	−1620.66	−0.146271
$498$	0	0
$499$	5499.15	0.493338	0.246669	−	0.969100i	$-0.420664\pi$
$499$	5499.15	0.493338	0.246669	+	0.969100i	$0.420664\pi$
$500$	5608.46	0.501636
$501$	0	0
$502$	−9561.19	−0.850073
$503$	−3830.43	−0.339544	−0.169772	−	0.985483i	$-0.554303\pi$
$503$	−3830.43	−0.339544	−0.169772	+	0.985483i	$0.554303\pi$
$504$	0	0
$505$	1228.49	0.108252
$506$	31008.6	2.72431
$507$	0	0
$508$	−20726.3	−1.81020
$509$	−18785.7	−1.63588	−0.817939	−	0.575305i	$-0.804884\pi$
$509$	−18785.7	−1.63588	−0.817939	+	0.575305i	$0.804884\pi$
$510$	0	0
$511$	−5209.72	−0.451007
$512$	11839.9	1.02199
$513$	0	0
$514$	−27046.9	−2.32099
$515$	1573.48	0.134632
$516$	0	0
$517$	8805.20	0.749037
$518$	−7723.73	−0.655137
$519$	0	0
$520$	2579.63	0.217547
$521$	12134.3	1.02037	0.510184	−	0.860065i	$-0.329577\pi$
$521$	12134.3	1.02037	0.510184	+	0.860065i	$0.329577\pi$
$522$	0	0
$523$	−18010.4	−1.50581	−0.752905	−	0.658129i	$-0.771348\pi$
$523$	−18010.4	−1.50581	−0.752905	+	0.658129i	$0.771348\pi$
$524$	1152.87	0.0961137
$525$	0	0
$526$	1455.84	0.120680
$527$	−11188.8	−0.924845
$528$	0	0
$529$	7216.06	0.593084
$530$	−220.025	−0.0180326
$531$	0	0
$532$	20165.1	1.64336
$533$	−5198.04	−0.422424
$534$	0	0
$535$	1844.28	0.149038
$536$	35027.3	2.82266
$537$	0	0
$538$	15196.2	1.21776
$539$	−12481.7	−0.997449
$540$	0	0
$541$	19448.0	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$541$	19448.0	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$542$	−41451.2	−3.28502
$543$	0	0
$544$	−24006.2	−1.89202
$545$	1095.16	0.0860765
$546$	0	0
$547$	8265.78	0.646104	0.323052	−	0.946381i	$-0.395291\pi$
$547$	8265.78	0.646104	0.323052	+	0.946381i	$0.395291\pi$
$548$	35384.4	2.75830
$549$	0	0
$550$	27564.4	2.13700
$551$	−11498.7	−0.889039
$552$	0	0
$553$	−3235.57	−0.248807
$554$	−36911.4	−2.83072
$555$	0	0
$556$	−48607.7	−3.70760
$557$	−20442.6	−1.55508	−0.777540	−	0.628834i	$-0.783533\pi$
$557$	−20442.6	−1.55508	−0.777540	+	0.628834i	$0.783533\pi$
$558$	0	0
$559$	−5855.87	−0.443071
$560$	−1375.75	−0.103814
$561$	0	0
$562$	34256.8	2.57123
$563$	−16087.8	−1.20430	−0.602149	−	0.798384i	$-0.705689\pi$
$563$	−16087.8	−1.20430	−0.602149	+	0.798384i	$0.705689\pi$
$564$	0	0
$565$	−318.786	−0.0237371
$566$	11819.0	0.877721
$567$	0	0
$568$	15676.7	1.15806
$569$	−8849.31	−0.651990	−0.325995	−	0.945371i	$-0.605699\pi$
$569$	−8849.31	−0.651990	−0.325995	+	0.945371i	$0.605699\pi$
$570$	0	0
$571$	3179.82	0.233050	0.116525	−	0.993188i	$-0.462825\pi$
$571$	3179.82	0.233050	0.116525	+	0.993188i	$0.462825\pi$
$572$	30203.2	2.20779
$573$	0	0
$574$	5215.69	0.379266
$575$	17230.1	1.24964
$576$	0	0
$577$	−6653.23	−0.480030	−0.240015	−	0.970769i	$-0.577152\pi$
$577$	−6653.23	−0.480030	−0.240015	+	0.970769i	$0.577152\pi$
$578$	11367.8	0.818059
$579$	0	0
$580$	1749.31	0.125235
$581$	1709.56	0.122073
$582$	0	0
$583$	−1557.81	−0.110665
$584$	50393.8	3.57073
$585$	0	0
$586$	−45078.6	−3.17778
$587$	5889.25	0.414097	0.207049	−	0.978331i	$-0.433614\pi$
$587$	5889.25	0.414097	0.207049	+	0.978331i	$0.433614\pi$
$588$	0	0
$589$	−31484.4	−2.20254
$590$	450.065	0.0314049
$591$	0	0
$592$	39709.8	2.75686
$593$	16240.7	1.12466	0.562332	−	0.826912i	$-0.309904\pi$
$593$	16240.7	1.12466	0.562332	+	0.826912i	$0.309904\pi$
$594$	0	0
$595$	−394.462	−0.0271787
$596$	30456.7	2.09322
$597$	0	0
$598$	26342.5	1.80138
$599$	16782.0	1.14473	0.572365	−	0.819999i	$-0.306026\pi$
$599$	16782.0	1.14473	0.572365	+	0.819999i	$0.306026\pi$
$600$	0	0
$601$	2612.38	0.177306	0.0886531	−	0.996063i	$-0.471744\pi$
$601$	2612.38	0.177306	0.0886531	+	0.996063i	$0.471744\pi$
$602$	5875.76	0.397804
$603$	0	0
$604$	66212.6	4.46051
$605$	−474.266	−0.0318705
$606$	0	0
$607$	22628.5	1.51312	0.756559	−	0.653926i	$-0.226879\pi$
$607$	22628.5	1.51312	0.756559	+	0.653926i	$0.226879\pi$
$608$	−67551.5	−4.50588
$609$	0	0
$610$	−3943.22	−0.261732
$611$	7480.20	0.495281
$612$	0	0
$613$	4235.87	0.279095	0.139548	−	0.990215i	$-0.455435\pi$
$613$	4235.87	0.279095	0.139548	+	0.990215i	$0.455435\pi$
$614$	15518.6	1.02000
$615$	0	0
$616$	−18326.2	−1.19868
$617$	9762.25	0.636975	0.318487	−	0.947927i	$-0.396825\pi$
$617$	9762.25	0.636975	0.318487	+	0.947927i	$0.396825\pi$
$618$	0	0
$619$	8873.28	0.576167	0.288083	−	0.957605i	$-0.406982\pi$
$619$	8873.28	0.576167	0.288083	+	0.957605i	$0.406982\pi$
$620$	4789.77	0.310261
$621$	0	0
$622$	16330.2	1.05270
$623$	−5335.47	−0.343116
$624$	0	0
$625$	15161.2	0.970314
$626$	−30452.1	−1.94426
$627$	0	0
$628$	34329.6	2.18137
$629$	11385.8	0.721752
$630$	0	0
$631$	177.235	0.0111816	0.00559082	−	0.999984i	$-0.498220\pi$
$631$	177.235	0.0111816	0.00559082	+	0.999984i	$0.498220\pi$
$632$	31297.7	1.96987
$633$	0	0
$634$	−19164.7	−1.20052
$635$	1140.88	0.0712980
$636$	0	0
$637$	−10603.5	−0.659536
$638$	17281.2	1.07236
$639$	0	0
$640$	1577.69	0.0974434
$641$	−14284.5	−0.880196	−0.440098	−	0.897950i	$-0.645056\pi$
$641$	−14284.5	−0.880196	−0.440098	+	0.897950i	$0.645056\pi$
$642$	0	0
$643$	13860.2	0.850066	0.425033	−	0.905178i	$-0.360263\pi$
$643$	13860.2	0.850066	0.425033	+	0.905178i	$0.360263\pi$
$644$	−18943.7	−1.15914
$645$	0	0
$646$	−41476.4	−2.52611
$647$	29147.9	1.77113	0.885565	−	0.464516i	$-0.153772\pi$
$647$	29147.9	1.77113	0.885565	+	0.464516i	$0.153772\pi$
$648$	0	0
$649$	3186.53	0.192731
$650$	23416.5	1.41303
$651$	0	0
$652$	2457.48	0.147611
$653$	−5157.00	−0.309049	−0.154524	−	0.987989i	$-0.549385\pi$
$653$	−5157.00	−0.309049	−0.154524	+	0.987989i	$0.549385\pi$
$654$	0	0
$655$	−63.4597	−0.00378561
$656$	−26815.3	−1.59598
$657$	0	0
$658$	−7505.60	−0.444679
$659$	−22856.4	−1.35108	−0.675539	−	0.737324i	$-0.736089\pi$
$659$	−22856.4	−1.35108	−0.675539	+	0.737324i	$0.736089\pi$
$660$	0	0
$661$	−9522.37	−0.560329	−0.280164	−	0.959952i	$-0.590389\pi$
$661$	−9522.37	−0.560329	−0.280164	+	0.959952i	$0.590389\pi$
$662$	5173.23	0.303721
$663$	0	0
$664$	−16536.7	−0.966486
$665$	−1109.98	−0.0647267
$666$	0	0
$667$	10802.2	0.627081
$668$	−46428.1	−2.68916
$669$	0	0
$670$	−3188.41	−0.183850
$671$	−27918.6	−1.60624
$672$	0	0
$673$	21093.1	1.20814	0.604071	−	0.796931i	$-0.293544\pi$
$673$	21093.1	1.20814	0.604071	+	0.796931i	$0.293544\pi$
$674$	−28794.5	−1.64558
$675$	0	0
$676$	−18805.4	−1.06995
$677$	−21915.3	−1.24412	−0.622062	−	0.782968i	$-0.713705\pi$
$677$	−21915.3	−1.24412	−0.622062	+	0.782968i	$0.713705\pi$
$678$	0	0
$679$	10661.1	0.602557
$680$	3815.64	0.215181
$681$	0	0
$682$	47317.4	2.65671
$683$	−29193.9	−1.63554	−0.817770	−	0.575545i	$-0.804790\pi$
$683$	−29193.9	−1.63554	−0.817770	+	0.575545i	$0.804790\pi$
$684$	0	0
$685$	−1947.73	−0.108641
$686$	22893.9	1.27419
$687$	0	0
$688$	−30208.9	−1.67399
$689$	−1323.39	−0.0731745
$690$	0	0
$691$	−23231.2	−1.27895	−0.639475	−	0.768812i	$-0.720848\pi$
$691$	−23231.2	−1.27895	−0.639475	+	0.768812i	$0.720848\pi$
$692$	−35607.6	−1.95607
$693$	0	0
$694$	10123.3	0.553710
$695$	2675.60	0.146031
$696$	0	0
$697$	−7688.63	−0.417830
$698$	77.0403	0.00417768
$699$	0	0
$700$	−16839.5	−0.909250
$701$	−7456.24	−0.401738	−0.200869	−	0.979618i	$-0.564377\pi$
$701$	−7456.24	−0.401738	−0.200869	+	0.979618i	$0.564377\pi$
$702$	0	0
$703$	32038.7	1.71887
$704$	39931.8	2.13776
$705$	0	0
$706$	11763.8	0.627104
$707$	−7414.12	−0.394394
$708$	0	0
$709$	34605.3	1.83304	0.916521	−	0.399985i	$-0.130985\pi$
$709$	34605.3	1.83304	0.916521	+	0.399985i	$0.130985\pi$
$710$	−1427.00	−0.0754286
$711$	0	0
$712$	51610.1	2.71653
$713$	29577.4	1.55355
$714$	0	0
$715$	−1662.53	−0.0869580
$716$	55514.8	2.89761
$717$	0	0
$718$	36899.6	1.91794
$719$	−20072.9	−1.04116	−0.520578	−	0.853814i	$-0.674283\pi$
$719$	−20072.9	−1.04116	−0.520578	+	0.853814i	$0.674283\pi$
$720$	0	0
$721$	−9496.18	−0.490508
$722$	−80262.6	−4.13721
$723$	0	0
$724$	−7671.37	−0.393790
$725$	9602.37	0.491894
$726$	0	0
$727$	5173.76	0.263939	0.131970	−	0.991254i	$-0.457870\pi$
$727$	5173.76	0.263939	0.131970	+	0.991254i	$0.457870\pi$
$728$	−15568.5	−0.792591
$729$	0	0
$730$	−4587.17	−0.232574
$731$	−8661.66	−0.438253
$732$	0	0
$733$	−18481.4	−0.931279	−0.465640	−	0.884974i	$-0.654176\pi$
$733$	−18481.4	−0.931279	−0.465640	+	0.884974i	$0.654176\pi$
$734$	−33616.1	−1.69045
$735$	0	0
$736$	63459.9	3.17821
$737$	−22574.4	−1.12828
$738$	0	0
$739$	24072.1	1.19825	0.599125	−	0.800655i	$-0.295515\pi$
$739$	24072.1	1.19825	0.599125	+	0.800655i	$0.295515\pi$
$740$	−4874.09	−0.242128
$741$	0	0
$742$	1327.89	0.0656984
$743$	−26357.9	−1.30145	−0.650724	−	0.759314i	$-0.725535\pi$
$743$	−26357.9	−1.30145	−0.650724	+	0.759314i	$0.725535\pi$
$744$	0	0
$745$	−1676.48	−0.0824451
$746$	−26485.2	−1.29985
$747$	0	0
$748$	44674.8	2.18378
$749$	−11130.5	−0.542992
$750$	0	0
$751$	11031.1	0.535995	0.267997	−	0.963420i	$-0.413638\pi$
$751$	11031.1	0.535995	0.267997	+	0.963420i	$0.413638\pi$
$752$	38588.4	1.87124
$753$	0	0
$754$	14680.7	0.709071
$755$	−3644.65	−0.175685
$756$	0	0
$757$	−4197.65	−0.201541	−0.100770	−	0.994910i	$-0.532131\pi$
$757$	−4197.65	−0.201541	−0.100770	+	0.994910i	$0.532131\pi$
$758$	30347.3	1.45417
$759$	0	0
$760$	10736.9	0.512458
$761$	29622.2	1.41104	0.705521	−	0.708689i	$-0.250713\pi$
$761$	29622.2	1.41104	0.705521	+	0.708689i	$0.250713\pi$
$762$	0	0
$763$	−6609.49	−0.313604
$764$	−81260.5	−3.84804
$765$	0	0
$766$	−3383.63	−0.159602
$767$	2707.02	0.127438
$768$	0	0
$769$	−5771.20	−0.270631	−0.135315	−	0.990803i	$-0.543205\pi$
$769$	−5771.20	−0.270631	−0.135315	+	0.990803i	$0.543205\pi$
$770$	1668.17	0.0780737
$771$	0	0
$772$	−43469.8	−2.02657
$773$	−15686.2	−0.729873	−0.364936	−	0.931032i	$-0.618909\pi$
$773$	−15686.2	−0.729873	−0.364936	+	0.931032i	$0.618909\pi$
$774$	0	0
$775$	26292.1	1.21863
$776$	−103125.	−4.77059
$777$	0	0
$778$	47514.2	2.18954
$779$	−21635.1	−0.995070
$780$	0	0
$781$	−10103.4	−0.462903
$782$	38964.2	1.78179
$783$	0	0
$784$	−54700.5	−2.49182
$785$	−1889.66	−0.0859171
$786$	0	0
$787$	31204.6	1.41337	0.706686	−	0.707527i	$-0.250189\pi$
$787$	31204.6	1.41337	0.706686	+	0.707527i	$0.250189\pi$
$788$	25824.5	1.16746
$789$	0	0
$790$	−2848.93	−0.128304
$791$	1923.93	0.0864816
$792$	0	0
$793$	−23717.4	−1.06208
$794$	39806.0	1.77917
$795$	0	0
$796$	60152.2	2.67844
$797$	13565.4	0.602899	0.301450	−	0.953482i	$-0.402529\pi$
$797$	13565.4	0.602899	0.301450	+	0.953482i	$0.402529\pi$
$798$	0	0
$799$	11064.3	0.489894
$800$	56411.2	2.49304
$801$	0	0
$802$	8481.56	0.373434
$803$	−32477.9	−1.42730
$804$	0	0
$805$	1042.75	0.0456548
$806$	40197.1	1.75668
$807$	0	0
$808$	71717.0	3.12252
$809$	−4055.66	−0.176254	−0.0881269	−	0.996109i	$-0.528088\pi$
$809$	−4055.66	−0.176254	−0.0881269	+	0.996109i	$0.528088\pi$
$810$	0	0
$811$	1909.01	0.0826565	0.0413282	−	0.999146i	$-0.486841\pi$
$811$	1909.01	0.0826565	0.0413282	+	0.999146i	$0.486841\pi$
$812$	−10557.4	−0.456269
$813$	0	0
$814$	−48150.4	−2.07331
$815$	−135.272	−0.00581393
$816$	0	0
$817$	−24373.2	−1.04371
$818$	37156.5	1.58820
$819$	0	0
$820$	3291.38	0.140171
$821$	22512.2	0.956981	0.478490	−	0.878093i	$-0.341184\pi$
$821$	22512.2	0.956981	0.478490	+	0.878093i	$0.341184\pi$
$822$	0	0
$823$	−31791.4	−1.34651	−0.673256	−	0.739409i	$-0.735105\pi$
$823$	−31791.4	−1.34651	−0.673256	+	0.739409i	$0.735105\pi$
$824$	91856.8	3.88348
$825$	0	0
$826$	−2716.21	−0.114418
$827$	−46407.5	−1.95133	−0.975663	−	0.219275i	$-0.929631\pi$
$827$	−46407.5	−1.95133	−0.975663	+	0.219275i	$0.929631\pi$
$828$	0	0
$829$	7411.48	0.310508	0.155254	−	0.987875i	$-0.450380\pi$
$829$	7411.48	0.310508	0.155254	+	0.987875i	$0.450380\pi$
$830$	1505.28	0.0629505
$831$	0	0
$832$	33922.8	1.41354
$833$	−15684.0	−0.652364
$834$	0	0
$835$	2555.62	0.105917
$836$	125711.	5.20072
$837$	0	0
$838$	−4718.80	−0.194521
$839$	9032.53	0.371678	0.185839	−	0.982580i	$-0.440500\pi$
$839$	9032.53	0.371678	0.185839	+	0.982580i	$0.440500\pi$
$840$	0	0
$841$	−18368.9	−0.753164
$842$	−34817.4	−1.42504
$843$	0	0
$844$	−109591.	−4.46952
$845$	1035.14	0.0421418
$846$	0	0
$847$	2862.27	0.116114
$848$	−6827.03	−0.276464
$849$	0	0
$850$	34636.3	1.39767
$851$	−30098.1	−1.21240
$852$	0	0
$853$	−33178.4	−1.33178	−0.665890	−	0.746050i	$-0.731948\pi$
$853$	−33178.4	−1.33178	−0.665890	+	0.746050i	$0.731948\pi$
$854$	23798.0	0.953571
$855$	0	0
$856$	107666.	4.29901
$857$	−28183.5	−1.12337	−0.561686	−	0.827350i	$-0.689847\pi$
$857$	−28183.5	−1.12337	−0.561686	+	0.827350i	$0.689847\pi$
$858$	0	0
$859$	−26718.9	−1.06128	−0.530638	−	0.847599i	$-0.678048\pi$
$859$	−26718.9	−1.06128	−0.530638	+	0.847599i	$0.678048\pi$
$860$	3707.92	0.147022
$861$	0	0
$862$	3146.58	0.124330
$863$	−31780.1	−1.25354	−0.626772	−	0.779203i	$-0.715624\pi$
$863$	−31780.1	−1.25354	−0.626772	+	0.779203i	$0.715624\pi$
$864$	0	0
$865$	1960.01	0.0770432
$866$	89265.5	3.50273
$867$	0	0
$868$	−28907.0	−1.13038
$869$	−20170.8	−0.787397
$870$	0	0
$871$	−19177.4	−0.746043
$872$	63933.8	2.48288
$873$	0	0
$874$	109642.	4.24336
$875$	1863.15	0.0719840
$876$	0	0
$877$	−6407.36	−0.246706	−0.123353	−	0.992363i	$-0.539365\pi$
$877$	−6407.36	−0.246706	−0.123353	+	0.992363i	$0.539365\pi$
$878$	14110.6	0.542380
$879$	0	0
$880$	−8576.54	−0.328540
$881$	21945.3	0.839223	0.419612	−	0.907704i	$-0.362166\pi$
$881$	21945.3	0.839223	0.419612	+	0.907704i	$0.362166\pi$
$882$	0	0
$883$	19846.9	0.756401	0.378200	−	0.925724i	$-0.376543\pi$
$883$	19846.9	0.756401	0.378200	+	0.925724i	$0.376543\pi$
$884$	37952.1	1.44397
$885$	0	0
$886$	78077.4	2.96057
$887$	33667.1	1.27444	0.637222	−	0.770680i	$-0.280083\pi$
$887$	33667.1	1.27444	0.637222	+	0.770680i	$0.280083\pi$
$888$	0	0
$889$	−6885.36	−0.259761
$890$	−4697.90	−0.176937
$891$	0	0
$892$	99439.3	3.73259
$893$	31133.9	1.16669
$894$	0	0
$895$	−3055.80	−0.114127
$896$	−9521.62	−0.355017
$897$	0	0
$898$	23273.3	0.864854
$899$	16483.6	0.611521
$900$	0	0
$901$	−1957.48	−0.0723787
$902$	32515.1	1.20026
$903$	0	0
$904$	−18610.2	−0.684697
$905$	422.268	0.0155101
$906$	0	0
$907$	−16066.7	−0.588186	−0.294093	−	0.955777i	$-0.595018\pi$
$907$	−16066.7	−0.588186	−0.294093	+	0.955777i	$0.595018\pi$
$908$	119644.	4.37281
$909$	0	0
$910$	1417.15	0.0516241
$911$	−2596.44	−0.0944279	−0.0472139	−	0.998885i	$-0.515034\pi$
$911$	−2596.44	−0.0944279	−0.0472139	+	0.998885i	$0.515034\pi$
$912$	0	0
$913$	10657.6	0.386325
$914$	−60406.0	−2.18605
$915$	0	0
$916$	−51082.8	−1.84260
$917$	382.989	0.0137922
$918$	0	0
$919$	−12085.8	−0.433814	−0.216907	−	0.976192i	$-0.569597\pi$
$919$	−12085.8	−0.433814	−0.216907	+	0.976192i	$0.569597\pi$
$920$	−10086.6	−0.361461
$921$	0	0
$922$	−77750.6	−2.77720
$923$	−8583.02	−0.306082
$924$	0	0
$925$	−26755.0	−0.951026
$926$	−2234.77	−0.0793079
$927$	0	0
$928$	35366.3	1.25103
$929$	−33132.1	−1.17011	−0.585053	−	0.810995i	$-0.698926\pi$
$929$	−33132.1	−1.17011	−0.585053	+	0.810995i	$0.698926\pi$
$930$	0	0
$931$	−44133.5	−1.55362
$932$	−106867.	−3.75596
$933$	0	0
$934$	6356.96	0.222705
$935$	−2459.11	−0.0860123
$936$	0	0
$937$	−41224.0	−1.43728	−0.718640	−	0.695383i	$-0.755235\pi$
$937$	−41224.0	−1.43728	−0.718640	+	0.695383i	$0.755235\pi$
$938$	19242.6	0.669821
$939$	0	0
$940$	−4736.44	−0.164346
$941$	9212.73	0.319157	0.159578	−	0.987185i	$-0.448987\pi$
$941$	9212.73	0.319157	0.159578	+	0.987185i	$0.448987\pi$
$942$	0	0
$943$	20324.7	0.701871
$944$	13964.8	0.481479
$945$	0	0
$946$	36630.0	1.25893
$947$	27069.9	0.928885	0.464442	−	0.885603i	$-0.346255\pi$
$947$	27069.9	0.928885	0.464442	+	0.885603i	$0.346255\pi$
$948$	0	0
$949$	−27590.6	−0.943761
$950$	97463.6	3.32857
$951$	0	0
$952$	−23028.0	−0.783972
$953$	352.144	0.0119696	0.00598481	−	0.999982i	$-0.498095\pi$
$953$	352.144	0.0119696	0.00598481	+	0.999982i	$0.498095\pi$
$954$	0	0
$955$	4472.96	0.151562
$956$	−4836.96	−0.163639
$957$	0	0
$958$	6434.63	0.217008
$959$	11754.8	0.395811
$960$	0	0
$961$	15342.5	0.515003
$962$	−40904.8	−1.37092
$963$	0	0
$964$	10835.7	0.362028
$965$	2392.79	0.0798202
$966$	0	0
$967$	−1219.80	−0.0405649	−0.0202824	−	0.999794i	$-0.506457\pi$
$967$	−1219.80	−0.0405649	−0.0202824	+	0.999794i	$0.506457\pi$
$968$	−27686.8	−0.919305
$969$	0	0
$970$	9387.14	0.310725
$971$	−2421.28	−0.0800233	−0.0400116	−	0.999199i	$-0.512739\pi$
$971$	−2421.28	−0.0800233	−0.0400116	+	0.999199i	$0.512739\pi$
$972$	0	0
$973$	−16147.7	−0.532036
$974$	102067.	3.35773
$975$	0	0
$976$	−122352.	−4.01270
$977$	40023.5	1.31061	0.655304	−	0.755365i	$-0.272540\pi$
$977$	40023.5	1.31061	0.655304	+	0.755365i	$0.272540\pi$
$978$	0	0
$979$	−33261.8	−1.08585
$980$	6714.08	0.218851
$981$	0	0
$982$	57662.2	1.87380
$983$	33312.0	1.08086	0.540431	−	0.841388i	$-0.318261\pi$
$983$	33312.0	1.08086	0.540431	+	0.841388i	$0.318261\pi$
$984$	0	0
$985$	−1421.50	−0.0459825
$986$	21714.8	0.701360
$987$	0	0
$988$	106794.	3.43884
$989$	22896.9	0.736177
$990$	0	0
$991$	46018.8	1.47511	0.737556	−	0.675286i	$-0.235980\pi$
$991$	46018.8	1.47511	0.737556	+	0.675286i	$0.235980\pi$
$992$	96836.2	3.09935
$993$	0	0
$994$	8612.17	0.274810
$995$	−3311.06	−0.105495
$996$	0	0
$997$	−18810.8	−0.597535	−0.298768	−	0.954326i	$-0.596576\pi$
$997$	−18810.8	−0.597535	−0.298768	+	0.954326i	$0.596576\pi$
$998$	−29222.3	−0.926871
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.d.1.2		32
3.2	odd	2		717.4.a.d.1.31	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.d.1.31	✓	32	3.2	odd	2
2151.4.a.d.1.2		32	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-5.31397 q^{2} +20.2383 q^{4} -1.11401 q^{5} +6.72325 q^{7} -65.0342 q^{8} +O(q^{10})\)	\(q-5.31397 q^{2} +20.2383 q^{4} -1.11401 q^{5} +6.72325 q^{7} -65.0342 q^{8} +5.91984 q^{10} +41.9133 q^{11} +35.6062 q^{13} -35.7272 q^{14} +183.683 q^{16} +52.6666 q^{17} +148.199 q^{19} -22.5458 q^{20} -222.726 q^{22} -139.223 q^{23} -123.759 q^{25} -189.211 q^{26} +136.067 q^{28} -77.5892 q^{29} -212.446 q^{31} -455.815 q^{32} -279.869 q^{34} -7.48979 q^{35} +216.186 q^{37} -787.528 q^{38} +72.4489 q^{40} -145.987 q^{41} -164.462 q^{43} +848.256 q^{44} +739.828 q^{46} +210.081 q^{47} -297.798 q^{49} +657.652 q^{50} +720.611 q^{52} -37.1674 q^{53} -46.6920 q^{55} -437.241 q^{56} +412.307 q^{58} +76.0266 q^{59} -666.103 q^{61} +1128.93 q^{62} +952.722 q^{64} -39.6658 q^{65} -538.598 q^{67} +1065.88 q^{68} +39.8005 q^{70} -241.054 q^{71} -774.882 q^{73} -1148.81 q^{74} +2999.31 q^{76} +281.794 q^{77} -481.251 q^{79} -204.626 q^{80} +775.770 q^{82} +254.277 q^{83} -58.6713 q^{85} +873.947 q^{86} -2725.80 q^{88} -793.585 q^{89} +239.389 q^{91} -2817.64 q^{92} -1116.37 q^{94} -165.096 q^{95} +1585.71 q^{97} +1582.49 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * q^98

Introduction

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