LMFDB - Embedded newform 207.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,4,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, 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("207.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	207.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:	$12.2133953712$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 26x^{3} + 10x^{2} + 144x + 56 $ x^5 - x^4 - 26x^3 + 10x^2 + 144*x + 56
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.06335$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.06335 q^{2} +17.6375 q^{4} -2.15617 q^{5} +2.11437 q^{7} -48.7982 q^{8} +O(q^{10})$	\(q-5.06335 q^{2} +17.6375 q^{4} -2.15617 q^{5} +2.11437 q^{7} -48.7982 q^{8} +10.9174 q^{10} -33.1380 q^{11} +6.68057 q^{13} -10.7058 q^{14} +105.982 q^{16} +55.3297 q^{17} +77.1072 q^{19} -38.0295 q^{20} +167.789 q^{22} +23.0000 q^{23} -120.351 q^{25} -33.8261 q^{26} +37.2923 q^{28} +14.9375 q^{29} -40.4477 q^{31} -146.240 q^{32} -280.154 q^{34} -4.55894 q^{35} -223.461 q^{37} -390.421 q^{38} +105.217 q^{40} -29.5816 q^{41} -430.366 q^{43} -584.473 q^{44} -116.457 q^{46} -498.856 q^{47} -338.529 q^{49} +609.379 q^{50} +117.829 q^{52} -485.561 q^{53} +71.4511 q^{55} -103.178 q^{56} -75.6336 q^{58} +425.760 q^{59} +185.041 q^{61} +204.801 q^{62} -107.394 q^{64} -14.4044 q^{65} +20.9209 q^{67} +975.879 q^{68} +23.0835 q^{70} +839.195 q^{71} -830.042 q^{73} +1131.46 q^{74} +1359.98 q^{76} -70.0661 q^{77} -392.651 q^{79} -228.516 q^{80} +149.782 q^{82} -932.749 q^{83} -119.300 q^{85} +2179.09 q^{86} +1617.08 q^{88} -411.955 q^{89} +14.1252 q^{91} +405.663 q^{92} +2525.88 q^{94} -166.256 q^{95} +1608.89 q^{97} +1714.09 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{2} + 16 q^{4} - 20 q^{5} - 10 q^{7} - 48 q^{8}+O(q^{10}) $ 5 * q - 4 * q^2 + 16 * q^4 - 20 * q^5 - 10 * q^7 - 48 * q^8	$ 5 q - 4 q^{2} + 16 q^{4} - 20 q^{5} - 10 q^{7} - 48 q^{8} + 50 q^{10} - 46 q^{11} + 54 q^{13} - 164 q^{14} - 60 q^{16} - 250 q^{17} - 28 q^{19} - 242 q^{20} - 10 q^{22} + 115 q^{23} + 239 q^{25} - 368 q^{26} + 224 q^{28} - 460 q^{29} - 360 q^{31} - 268 q^{32} - 20 q^{34} - 964 q^{35} - 92 q^{37} - 266 q^{38} - 278 q^{40} - 788 q^{41} + 108 q^{43} - 354 q^{44} - 92 q^{46} - 724 q^{47} + 149 q^{49} - 128 q^{50} - 288 q^{52} - 580 q^{53} - 672 q^{55} + 1172 q^{56} - 48 q^{58} + 408 q^{59} - 620 q^{61} + 1088 q^{62} + 696 q^{64} - 184 q^{65} + 100 q^{67} + 1584 q^{68} + 728 q^{70} + 124 q^{71} - 1946 q^{73} + 2866 q^{74} + 1362 q^{76} + 396 q^{77} - 198 q^{79} + 2398 q^{80} + 76 q^{82} + 578 q^{83} + 376 q^{85} + 3910 q^{86} + 1366 q^{88} - 930 q^{89} + 1972 q^{91} + 368 q^{92} + 976 q^{94} + 908 q^{95} + 2618 q^{97} + 2400 q^{98}+O(q^{100}) $ 5 * q - 4 * q^2 + 16 * q^4 - 20 * q^5 - 10 * q^7 - 48 * q^8 + 50 * q^10 - 46 * q^11 + 54 * q^13 - 164 * q^14 - 60 * q^16 - 250 * q^17 - 28 * q^19 - 242 * q^20 - 10 * q^22 + 115 * q^23 + 239 * q^25 - 368 * q^26 + 224 * q^28 - 460 * q^29 - 360 * q^31 - 268 * q^32 - 20 * q^34 - 964 * q^35 - 92 * q^37 - 266 * q^38 - 278 * q^40 - 788 * q^41 + 108 * q^43 - 354 * q^44 - 92 * q^46 - 724 * q^47 + 149 * q^49 - 128 * q^50 - 288 * q^52 - 580 * q^53 - 672 * q^55 + 1172 * q^56 - 48 * q^58 + 408 * q^59 - 620 * q^61 + 1088 * q^62 + 696 * q^64 - 184 * q^65 + 100 * q^67 + 1584 * q^68 + 728 * q^70 + 124 * q^71 - 1946 * q^73 + 2866 * q^74 + 1362 * q^76 + 396 * q^77 - 198 * q^79 + 2398 * q^80 + 76 * q^82 + 578 * q^83 + 376 * q^85 + 3910 * q^86 + 1366 * q^88 - 930 * q^89 + 1972 * q^91 + 368 * q^92 + 976 * q^94 + 908 * q^95 + 2618 * q^97 + 2400 * 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 $

$2$	−5.06335	−1.79017	−0.895083	−	0.445900i	$-0.852884\pi$
$2$	−5.06335	−1.79017	−0.895083	+	0.445900i	$0.852884\pi$
$3$	0	0
$3$	0	0
$4$	17.6375	2.20469
$5$	−2.15617	−0.192853	−0.0964267	−	0.995340i	$-0.530741\pi$
$5$	−2.15617	−0.192853	−0.0964267	+	0.995340i	$0.530741\pi$
$6$	0	0
$7$	2.11437	0.114165	0.0570827	−	0.998369i	$-0.481820\pi$
$7$	2.11437	0.114165	0.0570827	+	0.998369i	$0.481820\pi$
$8$	−48.7982	−2.15660
$9$	0	0
$10$	10.9174	0.345239
$11$	−33.1380	−0.908317	−0.454159	−	0.890921i	$-0.650060\pi$
$11$	−33.1380	−0.908317	−0.454159	+	0.890921i	$0.650060\pi$
$12$	0	0
$13$	6.68057	0.142527	0.0712637	−	0.997458i	$-0.477297\pi$
$13$	6.68057	0.142527	0.0712637	+	0.997458i	$0.477297\pi$
$14$	−10.7058	−0.204375
$15$	0	0
$16$	105.982	1.65597
$17$	55.3297	0.789378	0.394689	−	0.918815i	$-0.370852\pi$
$17$	55.3297	0.789378	0.394689	+	0.918815i	$0.370852\pi$
$18$	0	0
$19$	77.1072	0.931032	0.465516	−	0.885040i	$-0.345869\pi$
$19$	77.1072	0.931032	0.465516	+	0.885040i	$0.345869\pi$
$20$	−38.0295	−0.425182
$21$	0	0
$22$	167.789	1.62604
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−120.351	−0.962808
$26$	−33.8261	−0.255148
$27$	0	0
$28$	37.2923	0.251700
$29$	14.9375	0.0956488	0.0478244	−	0.998856i	$-0.484771\pi$
$29$	14.9375	0.0956488	0.0478244	+	0.998856i	$0.484771\pi$
$30$	0	0
$31$	−40.4477	−0.234342	−0.117171	−	0.993112i	$-0.537383\pi$
$31$	−40.4477	−0.234342	−0.117171	+	0.993112i	$0.537383\pi$
$32$	−146.240	−0.807870
$33$	0	0
$34$	−280.154	−1.41312
$35$	−4.55894	−0.0220172
$36$	0	0
$37$	−223.461	−0.992884	−0.496442	−	0.868070i	$-0.665360\pi$
$37$	−223.461	−0.992884	−0.496442	+	0.868070i	$0.665360\pi$
$38$	−390.421	−1.66670
$39$	0	0
$40$	105.217	0.415907
$41$	−29.5816	−0.112680	−0.0563399	−	0.998412i	$-0.517943\pi$
$41$	−29.5816	−0.112680	−0.0563399	+	0.998412i	$0.517943\pi$
$42$	0	0
$43$	−430.366	−1.52628	−0.763141	−	0.646232i	$-0.776344\pi$
$43$	−430.366	−1.52628	−0.763141	+	0.646232i	$0.776344\pi$
$44$	−584.473	−2.00256
$45$	0	0
$46$	−116.457	−0.373275
$47$	−498.856	−1.54820	−0.774102	−	0.633061i	$-0.781798\pi$
$47$	−498.856	−1.54820	−0.774102	+	0.633061i	$0.781798\pi$
$48$	0	0
$49$	−338.529	−0.986966
$50$	609.379	1.72358
$51$	0	0
$52$	117.829	0.314229
$53$	−485.561	−1.25843	−0.629216	−	0.777231i	$-0.716624\pi$
$53$	−485.561	−1.25843	−0.629216	+	0.777231i	$0.716624\pi$
$54$	0	0
$55$	71.4511	0.175172
$56$	−103.178	−0.246209
$57$	0	0
$58$	−75.6336	−0.171227
$59$	425.760	0.939478	0.469739	−	0.882805i	$-0.344348\pi$
$59$	425.760	0.939478	0.469739	+	0.882805i	$0.344348\pi$
$60$	0	0
$61$	185.041	0.388394	0.194197	−	0.980963i	$-0.437790\pi$
$61$	185.041	0.388394	0.194197	+	0.980963i	$0.437790\pi$
$62$	204.801	0.419512
$63$	0	0
$64$	−107.394	−0.209753
$65$	−14.4044	−0.0274869
$66$	0	0
$67$	20.9209	0.0381477	0.0190738	−	0.999818i	$-0.493928\pi$
$67$	20.9209	0.0381477	0.0190738	+	0.999818i	$0.493928\pi$
$68$	975.879	1.74033
$69$	0	0
$70$	23.0835	0.0394144
$71$	839.195	1.40273	0.701367	−	0.712800i	$-0.252573\pi$
$71$	839.195	1.40273	0.701367	+	0.712800i	$0.252573\pi$
$72$	0	0
$73$	−830.042	−1.33081	−0.665404	−	0.746483i	$-0.731741\pi$
$73$	−830.042	−1.33081	−0.665404	+	0.746483i	$0.731741\pi$
$74$	1131.46	1.77743
$75$	0	0
$76$	1359.98	2.05264
$77$	−70.0661	−0.103698
$78$	0	0
$79$	−392.651	−0.559198	−0.279599	−	0.960117i	$-0.590202\pi$
$79$	−392.651	−0.559198	−0.279599	+	0.960117i	$0.590202\pi$
$80$	−228.516	−0.319360
$81$	0	0
$82$	149.782	0.201716
$83$	−932.749	−1.23352	−0.616762	−	0.787150i	$-0.711556\pi$
$83$	−932.749	−1.23352	−0.616762	+	0.787150i	$0.711556\pi$
$84$	0	0
$85$	−119.300	−0.152234
$86$	2179.09	2.73230
$87$	0	0
$88$	1617.08	1.95887
$89$	−411.955	−0.490642	−0.245321	−	0.969442i	$-0.578893\pi$
$89$	−411.955	−0.490642	−0.245321	+	0.969442i	$0.578893\pi$
$90$	0	0
$91$	14.1252	0.0162717
$92$	405.663	0.459710
$93$	0	0
$94$	2525.88	2.77154
$95$	−166.256	−0.179553
$96$	0	0
$97$	1608.89	1.68411	0.842054	−	0.539393i	$-0.181346\pi$
$97$	1608.89	1.68411	0.842054	+	0.539393i	$0.181346\pi$
$98$	1714.09	1.76683
$99$	0	0
$100$	−2122.69	−2.12269
$101$	−487.980	−0.480751	−0.240376	−	0.970680i	$-0.577271\pi$
$101$	−487.980	−0.480751	−0.240376	+	0.970680i	$0.577271\pi$
$102$	0	0
$103$	−865.083	−0.827565	−0.413782	−	0.910376i	$-0.635793\pi$
$103$	−865.083	−0.827565	−0.413782	+	0.910376i	$0.635793\pi$
$104$	−326.000	−0.307374
$105$	0	0
$106$	2458.56	2.25280
$107$	−1776.61	−1.60516	−0.802578	−	0.596547i	$-0.796539\pi$
$107$	−1776.61	−1.60516	−0.802578	+	0.596547i	$0.796539\pi$
$108$	0	0
$109$	127.044	0.111639	0.0558193	−	0.998441i	$-0.482223\pi$
$109$	127.044	0.111639	0.0558193	+	0.998441i	$0.482223\pi$
$110$	−361.782	−0.313587
$111$	0	0
$112$	224.086	0.189055
$113$	−1384.34	−1.15246	−0.576229	−	0.817288i	$-0.695477\pi$
$113$	−1384.34	−1.15246	−0.576229	+	0.817288i	$0.695477\pi$
$114$	0	0
$115$	−49.5918	−0.0402127
$116$	263.460	0.210876
$117$	0	0
$118$	−2155.77	−1.68182
$119$	116.988	0.0901196
$120$	0	0
$121$	−232.872	−0.174960
$122$	−936.925	−0.695289
$123$	0	0
$124$	−713.397	−0.516653
$125$	529.018	0.378534
$126$	0	0
$127$	−2433.91	−1.70058	−0.850292	−	0.526311i	$-0.823575\pi$
$127$	−2433.91	−1.70058	−0.850292	+	0.526311i	$0.823575\pi$
$128$	1713.69	1.18336
$129$	0	0
$130$	72.9347	0.0492061
$131$	739.866	0.493453	0.246727	−	0.969085i	$-0.420645\pi$
$131$	739.866	0.493453	0.246727	+	0.969085i	$0.420645\pi$
$132$	0	0
$133$	163.033	0.106292
$134$	−105.930	−0.0682907
$135$	0	0
$136$	−2699.99	−1.70237
$137$	749.250	0.467246	0.233623	−	0.972327i	$-0.424942\pi$
$137$	749.250	0.467246	0.233623	+	0.972327i	$0.424942\pi$
$138$	0	0
$139$	2693.60	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$139$	2693.60	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$140$	−80.4085	−0.0485411
$141$	0	0
$142$	−4249.14	−2.51113
$143$	−221.381	−0.129460
$144$	0	0
$145$	−32.2076	−0.0184462
$146$	4202.79	2.38237
$147$	0	0
$148$	−3941.30	−2.18900
$149$	−1253.10	−0.688978	−0.344489	−	0.938790i	$-0.611948\pi$
$149$	−1253.10	−0.688978	−0.344489	+	0.938790i	$0.611948\pi$
$150$	0	0
$151$	696.459	0.375344	0.187672	−	0.982232i	$-0.439906\pi$
$151$	696.459	0.375344	0.187672	+	0.982232i	$0.439906\pi$
$152$	−3762.69	−2.00786
$153$	0	0
$154$	354.770	0.185637
$155$	87.2119	0.0451937
$156$	0	0
$157$	3144.36	1.59839	0.799195	−	0.601071i	$-0.205259\pi$
$157$	3144.36	1.59839	0.799195	+	0.601071i	$0.205259\pi$
$158$	1988.13	1.00106
$159$	0	0
$160$	315.318	0.155800
$161$	48.6306	0.0238051
$162$	0	0
$163$	2733.73	1.31363	0.656816	−	0.754051i	$-0.271903\pi$
$163$	2733.73	1.31363	0.656816	+	0.754051i	$0.271903\pi$
$164$	−521.747	−0.248424
$165$	0	0
$166$	4722.83	2.20821
$167$	341.032	0.158023	0.0790115	−	0.996874i	$-0.474824\pi$
$167$	341.032	0.158023	0.0790115	+	0.996874i	$0.474824\pi$
$168$	0	0
$169$	−2152.37	−0.979686
$170$	604.058	0.272524
$171$	0	0
$172$	−7590.59	−3.36498
$173$	−1277.25	−0.561314	−0.280657	−	0.959808i	$-0.590552\pi$
$173$	−1277.25	−0.561314	−0.280657	+	0.959808i	$0.590552\pi$
$174$	0	0
$175$	−254.467	−0.109919
$176$	−3512.04	−1.50415
$177$	0	0
$178$	2085.87	0.878331
$179$	−1177.28	−0.491588	−0.245794	−	0.969322i	$-0.579049\pi$
$179$	−1177.28	−0.491588	−0.245794	+	0.969322i	$0.579049\pi$
$180$	0	0
$181$	−367.909	−0.151085	−0.0755427	−	0.997143i	$-0.524069\pi$
$181$	−367.909	−0.151085	−0.0755427	+	0.997143i	$0.524069\pi$
$182$	−71.5210	−0.0291290
$183$	0	0
$184$	−1122.36	−0.449682
$185$	481.818	0.191481
$186$	0	0
$187$	−1833.52	−0.717005
$188$	−8798.58	−3.41331
$189$	0	0
$190$	841.812	0.321429
$191$	1892.89	0.717091	0.358546	−	0.933512i	$-0.383273\pi$
$191$	1892.89	0.717091	0.358546	+	0.933512i	$0.383273\pi$
$192$	0	0
$193$	3305.10	1.23268	0.616338	−	0.787481i	$-0.288615\pi$
$193$	3305.10	1.23268	0.616338	+	0.787481i	$0.288615\pi$
$194$	−8146.40	−3.01483
$195$	0	0
$196$	−5970.82	−2.17596
$197$	−4083.31	−1.47677	−0.738385	−	0.674380i	$-0.764411\pi$
$197$	−4083.31	−1.47677	−0.738385	+	0.674380i	$0.764411\pi$
$198$	0	0
$199$	2885.38	1.02783	0.513917	−	0.857840i	$-0.328194\pi$
$199$	2885.38	1.02783	0.513917	+	0.857840i	$0.328194\pi$
$200$	5872.91	2.07639
$201$	0	0
$202$	2470.82	0.860624
$203$	31.5834	0.0109198
$204$	0	0
$205$	63.7829	0.0217307
$206$	4380.22	1.48148
$207$	0	0
$208$	708.023	0.236022
$209$	−2555.18	−0.845672
$210$	0	0
$211$	−898.025	−0.292998	−0.146499	−	0.989211i	$-0.546801\pi$
$211$	−898.025	−0.292998	−0.146499	+	0.989211i	$0.546801\pi$
$212$	−8564.09	−2.77445
$213$	0	0
$214$	8995.62	2.87349
$215$	927.940	0.294349
$216$	0	0
$217$	−85.5214	−0.0267538
$218$	−643.268	−0.199851
$219$	0	0
$220$	1260.22	0.386200
$221$	369.634	0.112508
$222$	0	0
$223$	3934.09	1.18137	0.590686	−	0.806901i	$-0.298857\pi$
$223$	3934.09	1.18137	0.590686	+	0.806901i	$0.298857\pi$
$224$	−309.206	−0.0922308
$225$	0	0
$226$	7009.41	2.06309
$227$	5362.67	1.56799	0.783993	−	0.620770i	$-0.213180\pi$
$227$	5362.67	1.56799	0.783993	+	0.620770i	$0.213180\pi$
$228$	0	0
$229$	3830.28	1.10529	0.552646	−	0.833416i	$-0.313618\pi$
$229$	3830.28	1.10529	0.552646	+	0.833416i	$0.313618\pi$
$230$	251.101	0.0719874
$231$	0	0
$232$	−728.921	−0.206276
$233$	1997.73	0.561697	0.280848	−	0.959752i	$-0.409384\pi$
$233$	1997.73	0.561697	0.280848	+	0.959752i	$0.409384\pi$
$234$	0	0
$235$	1075.62	0.298576
$236$	7509.36	2.07126
$237$	0	0
$238$	−592.349	−0.161329
$239$	−3145.12	−0.851217	−0.425609	−	0.904907i	$-0.639940\pi$
$239$	−3145.12	−0.851217	−0.425609	+	0.904907i	$0.639940\pi$
$240$	0	0
$241$	−3189.67	−0.852552	−0.426276	−	0.904593i	$-0.640175\pi$
$241$	−3189.67	−0.852552	−0.426276	+	0.904593i	$0.640175\pi$
$242$	1179.11	0.313207
$243$	0	0
$244$	3263.66	0.856288
$245$	729.926	0.190340
$246$	0	0
$247$	515.120	0.132698
$248$	1973.77	0.505382
$249$	0	0
$250$	−2678.60	−0.677639
$251$	4270.59	1.07393	0.536967	−	0.843603i	$-0.319570\pi$
$251$	4270.59	1.07393	0.536967	+	0.843603i	$0.319570\pi$
$252$	0	0
$253$	−762.174	−0.189397
$254$	12323.7	3.04433
$255$	0	0
$256$	−7817.88	−1.90866
$257$	−7585.83	−1.84121	−0.920606	−	0.390493i	$-0.872304\pi$
$257$	−7585.83	−1.84121	−0.920606	+	0.390493i	$0.872304\pi$
$258$	0	0
$259$	−472.479	−0.113353
$260$	−254.059	−0.0606002
$261$	0	0
$262$	−3746.20	−0.883363
$263$	−961.653	−0.225468	−0.112734	−	0.993625i	$-0.535961\pi$
$263$	−961.653	−0.225468	−0.112734	+	0.993625i	$0.535961\pi$
$264$	0	0
$265$	1046.95	0.242693
$266$	−825.495	−0.190280
$267$	0	0
$268$	368.993	0.0841039
$269$	7770.51	1.76125	0.880625	−	0.473814i	$-0.157123\pi$
$269$	7770.51	1.76125	0.880625	+	0.473814i	$0.157123\pi$
$270$	0	0
$271$	−3812.06	−0.854487	−0.427243	−	0.904137i	$-0.640515\pi$
$271$	−3812.06	−0.854487	−0.427243	+	0.904137i	$0.640515\pi$
$272$	5863.97	1.30719
$273$	0	0
$274$	−3793.72	−0.836448
$275$	3988.19	0.874535
$276$	0	0
$277$	−6781.71	−1.47102	−0.735512	−	0.677512i	$-0.763058\pi$
$277$	−6781.71	−1.47102	−0.735512	+	0.677512i	$0.763058\pi$
$278$	−13638.6	−2.94241
$279$	0	0
$280$	222.468	0.0474822
$281$	−5389.07	−1.14408	−0.572038	−	0.820227i	$-0.693847\pi$
$281$	−5389.07	−1.14408	−0.572038	+	0.820227i	$0.693847\pi$
$282$	0	0
$283$	3747.11	0.787076	0.393538	−	0.919308i	$-0.371251\pi$
$283$	3747.11	0.787076	0.393538	+	0.919308i	$0.371251\pi$
$284$	14801.3	3.09260
$285$	0	0
$286$	1120.93	0.231755
$287$	−62.5466	−0.0128641
$288$	0	0
$289$	−1851.63	−0.376883
$290$	163.079	0.0330217
$291$	0	0
$292$	−14639.9	−2.93402
$293$	−2580.83	−0.514585	−0.257293	−	0.966334i	$-0.582830\pi$
$293$	−2580.83	−0.514585	−0.257293	+	0.966334i	$0.582830\pi$
$294$	0	0
$295$	−918.009	−0.181182
$296$	10904.5	2.14125
$297$	0	0
$298$	6344.87	1.23338
$299$	153.653	0.0297190
$300$	0	0
$301$	−909.954	−0.174249
$302$	−3526.42	−0.671929
$303$	0	0
$304$	8172.00	1.54176
$305$	−398.978	−0.0749030
$306$	0	0
$307$	−2665.09	−0.495455	−0.247727	−	0.968830i	$-0.579684\pi$
$307$	−2665.09	−0.495455	−0.247727	+	0.968830i	$0.579684\pi$
$308$	−1235.79	−0.228623
$309$	0	0
$310$	−441.584	−0.0809042
$311$	5643.13	1.02892	0.514458	−	0.857516i	$-0.327993\pi$
$311$	5643.13	1.02892	0.514458	+	0.857516i	$0.327993\pi$
$312$	0	0
$313$	4589.28	0.828759	0.414380	−	0.910104i	$-0.363999\pi$
$313$	4589.28	0.828759	0.414380	+	0.910104i	$0.363999\pi$
$314$	−15921.0	−2.86138
$315$	0	0
$316$	−6925.39	−1.23286
$317$	−1268.43	−0.224738	−0.112369	−	0.993667i	$-0.535844\pi$
$317$	−1268.43	−0.224738	−0.112369	+	0.993667i	$0.535844\pi$
$318$	0	0
$319$	−494.998	−0.0868794
$320$	231.559	0.0404516
$321$	0	0
$322$	−246.234	−0.0426151
$323$	4266.32	0.734936
$324$	0	0
$325$	−804.013	−0.137227
$326$	−13841.8	−2.35162
$327$	0	0
$328$	1443.53	0.243005
$329$	−1054.77	−0.176751
$330$	0	0
$331$	−5744.29	−0.953881	−0.476940	−	0.878936i	$-0.658254\pi$
$331$	−5744.29	−0.953881	−0.476940	+	0.878936i	$0.658254\pi$
$332$	−16451.4	−2.71954
$333$	0	0
$334$	−1726.76	−0.282887
$335$	−45.1090	−0.00735691
$336$	0	0
$337$	−5584.45	−0.902684	−0.451342	−	0.892351i	$-0.649055\pi$
$337$	−5584.45	−0.902684	−0.451342	+	0.892351i	$0.649055\pi$
$338$	10898.2	1.75380
$339$	0	0
$340$	−2104.16	−0.335629
$341$	1340.36	0.212857
$342$	0	0
$343$	−1441.01	−0.226843
$344$	21001.1	3.29158
$345$	0	0
$346$	6467.15	1.00484
$347$	4329.48	0.669795	0.334897	−	0.942255i	$-0.391298\pi$
$347$	4329.48	0.669795	0.334897	+	0.942255i	$0.391298\pi$
$348$	0	0
$349$	−1605.98	−0.246322	−0.123161	−	0.992387i	$-0.539303\pi$
$349$	−1605.98	−0.246322	−0.123161	+	0.992387i	$0.539303\pi$
$350$	1288.46	0.196774
$351$	0	0
$352$	4846.11	0.733802
$353$	5975.48	0.900970	0.450485	−	0.892784i	$-0.351251\pi$
$353$	5975.48	0.900970	0.450485	+	0.892784i	$0.351251\pi$
$354$	0	0
$355$	−1809.44	−0.270522
$356$	−7265.88	−1.08172
$357$	0	0
$358$	5961.00	0.880024
$359$	1513.76	0.222544	0.111272	−	0.993790i	$-0.464508\pi$
$359$	1513.76	0.222544	0.111272	+	0.993790i	$0.464508\pi$
$360$	0	0
$361$	−913.481	−0.133180
$362$	1862.85	0.270468
$363$	0	0
$364$	249.134	0.0358741
$365$	1789.71	0.256651
$366$	0	0
$367$	−9072.68	−1.29044	−0.645218	−	0.763999i	$-0.723233\pi$
$367$	−9072.68	−1.29044	−0.645218	+	0.763999i	$0.723233\pi$
$368$	2437.59	0.345294
$369$	0	0
$370$	−2439.62	−0.342783
$371$	−1026.66	−0.143669
$372$	0	0
$373$	−5292.60	−0.734693	−0.367347	−	0.930084i	$-0.619734\pi$
$373$	−5292.60	−0.734693	−0.367347	+	0.930084i	$0.619734\pi$
$374$	9283.74	1.28356
$375$	0	0
$376$	24343.3	3.33885
$377$	99.7907	0.0136326
$378$	0	0
$379$	7061.29	0.957029	0.478515	−	0.878080i	$-0.341175\pi$
$379$	7061.29	0.957029	0.478515	+	0.878080i	$0.341175\pi$
$380$	−2932.34	−0.395858
$381$	0	0
$382$	−9584.35	−1.28371
$383$	1213.73	0.161929	0.0809644	−	0.996717i	$-0.474200\pi$
$383$	1213.73	0.161929	0.0809644	+	0.996717i	$0.474200\pi$
$384$	0	0
$385$	151.074	0.0199986
$386$	−16734.9	−2.20670
$387$	0	0
$388$	28376.9	3.71294
$389$	−1193.72	−0.155588	−0.0777942	−	0.996969i	$-0.524788\pi$
$389$	−1193.72	−0.155588	−0.0777942	+	0.996969i	$0.524788\pi$
$390$	0	0
$391$	1272.58	0.164597
$392$	16519.6	2.12849
$393$	0	0
$394$	20675.2	2.64366
$395$	846.620	0.107843
$396$	0	0
$397$	3390.50	0.428625	0.214312	−	0.976765i	$-0.431249\pi$
$397$	3390.50	0.428625	0.214312	+	0.976765i	$0.431249\pi$
$398$	−14609.7	−1.83999
$399$	0	0
$400$	−12755.1	−1.59438
$401$	−2342.79	−0.291754	−0.145877	−	0.989303i	$-0.546600\pi$
$401$	−2342.79	−0.291754	−0.145877	+	0.989303i	$0.546600\pi$
$402$	0	0
$403$	−270.213	−0.0334002
$404$	−8606.77	−1.05991
$405$	0	0
$406$	−159.918	−0.0195482
$407$	7405.04	0.901854
$408$	0	0
$409$	12070.1	1.45923	0.729617	−	0.683856i	$-0.239698\pi$
$409$	12070.1	1.45923	0.729617	+	0.683856i	$0.239698\pi$
$410$	−322.955	−0.0389015
$411$	0	0
$412$	−15257.9	−1.82453
$413$	900.216	0.107256
$414$	0	0
$415$	2011.16	0.237889
$416$	−976.967	−0.115144
$417$	0	0
$418$	12937.8	1.51389
$419$	1561.32	0.182042	0.0910211	−	0.995849i	$-0.470987\pi$
$419$	1561.32	0.182042	0.0910211	+	0.995849i	$0.470987\pi$
$420$	0	0
$421$	−7096.40	−0.821514	−0.410757	−	0.911745i	$-0.634736\pi$
$421$	−7096.40	−0.821514	−0.410757	+	0.911745i	$0.634736\pi$
$422$	4547.02	0.524515
$423$	0	0
$424$	23694.5	2.71393
$425$	−6658.98	−0.760019
$426$	0	0
$427$	391.245	0.0443411
$428$	−31335.1	−3.53887
$429$	0	0
$430$	−4698.49	−0.526933
$431$	14072.8	1.57277	0.786386	−	0.617736i	$-0.211950\pi$
$431$	14072.8	1.57277	0.786386	+	0.617736i	$0.211950\pi$
$432$	0	0
$433$	−12446.4	−1.38138	−0.690690	−	0.723151i	$-0.742693\pi$
$433$	−12446.4	−1.38138	−0.690690	+	0.723151i	$0.742693\pi$
$434$	433.025	0.0478937
$435$	0	0
$436$	2240.74	0.246129
$437$	1773.47	0.194134
$438$	0	0
$439$	−9441.14	−1.02643	−0.513213	−	0.858261i	$-0.671545\pi$
$439$	−9441.14	−1.02643	−0.513213	+	0.858261i	$0.671545\pi$
$440$	−3486.69	−0.377776
$441$	0	0
$442$	−1871.59	−0.201408
$443$	8932.41	0.957994	0.478997	−	0.877817i	$-0.341000\pi$
$443$	8932.41	0.957994	0.478997	+	0.877817i	$0.341000\pi$
$444$	0	0
$445$	888.244	0.0946221
$446$	−19919.7	−2.11485
$447$	0	0
$448$	−227.070	−0.0239466
$449$	−8402.96	−0.883208	−0.441604	−	0.897210i	$-0.645590\pi$
$449$	−8402.96	−0.883208	−0.441604	+	0.897210i	$0.645590\pi$
$450$	0	0
$451$	980.276	0.102349
$452$	−24416.4	−2.54082
$453$	0	0
$454$	−27153.1	−2.80695
$455$	−30.4563	−0.00313805
$456$	0	0
$457$	3059.29	0.313146	0.156573	−	0.987666i	$-0.449955\pi$
$457$	3059.29	0.313146	0.156573	+	0.987666i	$0.449955\pi$
$458$	−19394.0	−1.97866
$459$	0	0
$460$	−874.678	−0.0886566
$461$	−18346.1	−1.85350	−0.926751	−	0.375677i	$-0.877410\pi$
$461$	−18346.1	−1.85350	−0.926751	+	0.375677i	$0.877410\pi$
$462$	0	0
$463$	11616.6	1.16602	0.583010	−	0.812465i	$-0.301875\pi$
$463$	11616.6	1.16602	0.583010	+	0.812465i	$0.301875\pi$
$464$	1583.11	0.158392
$465$	0	0
$466$	−10115.2	−1.00553
$467$	−5595.70	−0.554471	−0.277236	−	0.960802i	$-0.589418\pi$
$467$	−5595.70	−0.554471	−0.277236	+	0.960802i	$0.589418\pi$
$468$	0	0
$469$	44.2346	0.00435515
$470$	−5446.22	−0.534501
$471$	0	0
$472$	−20776.3	−2.02608
$473$	14261.5	1.38635
$474$	0	0
$475$	−9279.92	−0.896404
$476$	2063.37	0.198686
$477$	0	0
$478$	15924.9	1.52382
$479$	−2323.55	−0.221640	−0.110820	−	0.993840i	$-0.535348\pi$
$479$	−2323.55	−0.221640	−0.110820	+	0.993840i	$0.535348\pi$
$480$	0	0
$481$	−1492.85	−0.141513
$482$	16150.4	1.52621
$483$	0	0
$484$	−4107.28	−0.385733
$485$	−3469.04	−0.324786
$486$	0	0
$487$	8265.12	0.769052	0.384526	−	0.923114i	$-0.374365\pi$
$487$	8265.12	0.769052	0.384526	+	0.923114i	$0.374365\pi$
$488$	−9029.65	−0.837609
$489$	0	0
$490$	−3695.87	−0.340740
$491$	10618.1	0.975947	0.487973	−	0.872859i	$-0.337736\pi$
$491$	10618.1	0.975947	0.487973	+	0.872859i	$0.337736\pi$
$492$	0	0
$493$	826.485	0.0755030
$494$	−2608.23	−0.237551
$495$	0	0
$496$	−4286.74	−0.388065
$497$	1774.37	0.160144
$498$	0	0
$499$	−5503.12	−0.493695	−0.246847	−	0.969054i	$-0.579395\pi$
$499$	−5503.12	−0.493695	−0.246847	+	0.969054i	$0.579395\pi$
$500$	9330.56	0.834551
$501$	0	0
$502$	−21623.5	−1.92252
$503$	15861.1	1.40599	0.702993	−	0.711197i	$-0.251846\pi$
$503$	15861.1	1.40599	0.702993	+	0.711197i	$0.251846\pi$
$504$	0	0
$505$	1052.17	0.0927145
$506$	3859.16	0.339052
$507$	0	0
$508$	−42928.1	−3.74926
$509$	−7085.11	−0.616979	−0.308489	−	0.951228i	$-0.599823\pi$
$509$	−7085.11	−0.616979	−0.308489	+	0.951228i	$0.599823\pi$
$510$	0	0
$511$	−1755.02	−0.151932
$512$	25875.1	2.23346
$513$	0	0
$514$	38409.8	3.29607
$515$	1865.26	0.159599
$516$	0	0
$517$	16531.1	1.40626
$518$	2392.33	0.202921
$519$	0	0
$520$	702.910	0.0592782
$521$	−22508.5	−1.89273	−0.946367	−	0.323093i	$-0.895277\pi$
$521$	−22508.5	−1.89273	−0.946367	+	0.323093i	$0.895277\pi$
$522$	0	0
$523$	722.225	0.0603838	0.0301919	−	0.999544i	$-0.490388\pi$
$523$	722.225	0.0603838	0.0301919	+	0.999544i	$0.490388\pi$
$524$	13049.4	1.08791
$525$	0	0
$526$	4869.19	0.403625
$527$	−2237.96	−0.184985
$528$	0	0
$529$	529.000	0.0434783
$530$	−5301.08	−0.434460
$531$	0	0
$532$	2875.51	0.234340
$533$	−197.622	−0.0160600
$534$	0	0
$535$	3830.67	0.309560
$536$	−1020.90	−0.0822692
$537$	0	0
$538$	−39344.8	−3.15293
$539$	11218.2	0.896478
$540$	0	0
$541$	16933.0	1.34566	0.672832	−	0.739795i	$-0.265077\pi$
$541$	16933.0	1.34566	0.672832	+	0.739795i	$0.265077\pi$
$542$	19301.8	1.52967
$543$	0	0
$544$	−8091.42	−0.637714
$545$	−273.928	−0.0215299
$546$	0	0
$547$	−19402.6	−1.51663	−0.758316	−	0.651888i	$-0.773977\pi$
$547$	−19402.6	−1.51663	−0.758316	+	0.651888i	$0.773977\pi$
$548$	13214.9	1.03013
$549$	0	0
$550$	−20193.6	−1.56556
$551$	1151.79	0.0890521
$552$	0	0
$553$	−830.210	−0.0638411
$554$	34338.2	2.63337
$555$	0	0
$556$	47508.4	3.62375
$557$	19903.5	1.51408	0.757038	−	0.653371i	$-0.226646\pi$
$557$	19903.5	1.51408	0.757038	+	0.653371i	$0.226646\pi$
$558$	0	0
$559$	−2875.09	−0.217537
$560$	−483.167	−0.0364599
$561$	0	0
$562$	27286.8	2.04808
$563$	−20255.6	−1.51629	−0.758147	−	0.652084i	$-0.773895\pi$
$563$	−20255.6	−1.51629	−0.758147	+	0.652084i	$0.773895\pi$
$564$	0	0
$565$	2984.87	0.222256
$566$	−18972.9	−1.40900
$567$	0	0
$568$	−40951.2	−3.02513
$569$	17124.4	1.26167	0.630835	−	0.775917i	$-0.282712\pi$
$569$	17124.4	1.26167	0.630835	+	0.775917i	$0.282712\pi$
$570$	0	0
$571$	−2476.48	−0.181502	−0.0907508	−	0.995874i	$-0.528927\pi$
$571$	−2476.48	−0.181502	−0.0907508	+	0.995874i	$0.528927\pi$
$572$	−3904.61	−0.285420
$573$	0	0
$574$	316.695	0.0230289
$575$	−2768.07	−0.200759
$576$	0	0
$577$	−5158.81	−0.372208	−0.186104	−	0.982530i	$-0.559586\pi$
$577$	−5158.81	−0.372208	−0.186104	+	0.982530i	$0.559586\pi$
$578$	9375.44	0.674683
$579$	0	0
$580$	−568.063	−0.0406682
$581$	−1972.18	−0.140826
$582$	0	0
$583$	16090.5	1.14306
$584$	40504.6	2.87002
$585$	0	0
$586$	13067.6	0.921192
$587$	1429.09	0.100485	0.0502425	−	0.998737i	$-0.484001\pi$
$587$	1429.09	0.100485	0.0502425	+	0.998737i	$0.484001\pi$
$588$	0	0
$589$	−3118.80	−0.218180
$590$	4648.21	0.324345
$591$	0	0
$592$	−23682.9	−1.64419
$593$	−19207.1	−1.33008	−0.665041	−	0.746807i	$-0.731586\pi$
$593$	−19207.1	−1.33008	−0.665041	+	0.746807i	$0.731586\pi$
$594$	0	0
$595$	−252.245	−0.0173799
$596$	−22101.5	−1.51898
$597$	0	0
$598$	−778.000	−0.0532020
$599$	−4323.19	−0.294893	−0.147447	−	0.989070i	$-0.547105\pi$
$599$	−4323.19	−0.294893	−0.147447	+	0.989070i	$0.547105\pi$
$600$	0	0
$601$	5603.82	0.380340	0.190170	−	0.981751i	$-0.439096\pi$
$601$	5603.82	0.380340	0.190170	+	0.981751i	$0.439096\pi$
$602$	4607.42	0.311934
$603$	0	0
$604$	12283.8	0.827519
$605$	502.110	0.0337416
$606$	0	0
$607$	15551.9	1.03992	0.519961	−	0.854190i	$-0.325947\pi$
$607$	15551.9	1.03992	0.519961	+	0.854190i	$0.325947\pi$
$608$	−11276.2	−0.752153
$609$	0	0
$610$	2020.17	0.134089
$611$	−3332.64	−0.220662
$612$	0	0
$613$	1211.55	0.0798275	0.0399137	−	0.999203i	$-0.487292\pi$
$613$	1211.55	0.0798275	0.0399137	+	0.999203i	$0.487292\pi$
$614$	13494.3	0.886946
$615$	0	0
$616$	3419.10	0.223636
$617$	−29762.6	−1.94197	−0.970987	−	0.239131i	$-0.923138\pi$
$617$	−29762.6	−1.94197	−0.970987	+	0.239131i	$0.923138\pi$
$618$	0	0
$619$	28628.5	1.85893	0.929464	−	0.368912i	$-0.120270\pi$
$619$	28628.5	1.85893	0.929464	+	0.368912i	$0.120270\pi$
$620$	1538.20	0.0996382
$621$	0	0
$622$	−28573.2	−1.84193
$623$	−871.027	−0.0560144
$624$	0	0
$625$	13903.2	0.889806
$626$	−23237.2	−1.48362
$627$	0	0
$628$	55458.8	3.52396
$629$	−12364.0	−0.783760
$630$	0	0
$631$	18457.5	1.16447	0.582234	−	0.813021i	$-0.302179\pi$
$631$	18457.5	1.16447	0.582234	+	0.813021i	$0.302179\pi$
$632$	19160.7	1.20596
$633$	0	0
$634$	6422.50	0.402319
$635$	5247.90	0.327963
$636$	0	0
$637$	−2261.57	−0.140670
$638$	2506.35	0.155529
$639$	0	0
$640$	−3695.01	−0.228216
$641$	−12136.0	−0.747808	−0.373904	−	0.927467i	$-0.621981\pi$
$641$	−12136.0	−0.747808	−0.373904	+	0.927467i	$0.621981\pi$
$642$	0	0
$643$	10954.8	0.671877	0.335939	−	0.941884i	$-0.390947\pi$
$643$	10954.8	0.671877	0.335939	+	0.941884i	$0.390947\pi$
$644$	857.724	0.0524830
$645$	0	0
$646$	−21601.9	−1.31566
$647$	22621.7	1.37457	0.687287	−	0.726386i	$-0.258801\pi$
$647$	22621.7	1.37457	0.687287	+	0.726386i	$0.258801\pi$
$648$	0	0
$649$	−14108.8	−0.853344
$650$	4071.00	0.245658
$651$	0	0
$652$	48216.2	2.89615
$653$	−29747.8	−1.78273	−0.891363	−	0.453291i	$-0.850250\pi$
$653$	−29747.8	−1.78273	−0.891363	+	0.453291i	$0.850250\pi$
$654$	0	0
$655$	−1595.27	−0.0951642
$656$	−3135.13	−0.186595
$657$	0	0
$658$	5340.66	0.316414
$659$	19334.4	1.14289	0.571443	−	0.820642i	$-0.306384\pi$
$659$	19334.4	1.14289	0.571443	+	0.820642i	$0.306384\pi$
$660$	0	0
$661$	2973.50	0.174971	0.0874854	−	0.996166i	$-0.472117\pi$
$661$	2973.50	0.174971	0.0874854	+	0.996166i	$0.472117\pi$
$662$	29085.3	1.70760
$663$	0	0
$664$	45516.5	2.66021
$665$	−351.527	−0.0204987
$666$	0	0
$667$	343.561	0.0199442
$668$	6014.96	0.348392
$669$	0	0
$670$	228.403	0.0131701
$671$	−6131.88	−0.352785
$672$	0	0
$673$	5190.74	0.297308	0.148654	−	0.988889i	$-0.452506\pi$
$673$	5190.74	0.297308	0.148654	+	0.988889i	$0.452506\pi$
$674$	28276.1	1.61595
$675$	0	0
$676$	−37962.5	−2.15991
$677$	−4033.92	−0.229005	−0.114502	−	0.993423i	$-0.536527\pi$
$677$	−4033.92	−0.229005	−0.114502	+	0.993423i	$0.536527\pi$
$678$	0	0
$679$	3401.80	0.192267
$680$	5821.63	0.328308
$681$	0	0
$682$	−6786.69	−0.381050
$683$	443.990	0.0248738	0.0124369	−	0.999923i	$-0.496041\pi$
$683$	443.990	0.0248738	0.0124369	+	0.999923i	$0.496041\pi$
$684$	0	0
$685$	−1615.51	−0.0901101
$686$	7296.33	0.406086
$687$	0	0
$688$	−45611.2	−2.52748
$689$	−3243.82	−0.179361
$690$	0	0
$691$	1305.21	0.0718562	0.0359281	−	0.999354i	$-0.488561\pi$
$691$	1305.21	0.0718562	0.0359281	+	0.999354i	$0.488561\pi$
$692$	−22527.5	−1.23752
$693$	0	0
$694$	−21921.7	−1.19904
$695$	−5807.84	−0.316984
$696$	0	0
$697$	−1636.74	−0.0889470
$698$	8131.66	0.440957
$699$	0	0
$700$	−4488.17	−0.242338
$701$	21638.2	1.16585	0.582926	−	0.812525i	$-0.301908\pi$
$701$	21638.2	1.16585	0.582926	+	0.812525i	$0.301908\pi$
$702$	0	0
$703$	−17230.4	−0.924407
$704$	3558.81	0.190523
$705$	0	0
$706$	−30255.9	−1.61289
$707$	−1031.77	−0.0548852
$708$	0	0
$709$	11914.6	0.631116	0.315558	−	0.948906i	$-0.397808\pi$
$709$	11914.6	0.631116	0.315558	+	0.948906i	$0.397808\pi$
$710$	9161.86	0.484279
$711$	0	0
$712$	20102.7	1.05812
$713$	−930.296	−0.0488638
$714$	0	0
$715$	477.334	0.0249668
$716$	−20764.4	−1.08380
$717$	0	0
$718$	−7664.70	−0.398390
$719$	10462.6	0.542683	0.271341	−	0.962483i	$-0.412533\pi$
$719$	10462.6	0.542683	0.271341	+	0.962483i	$0.412533\pi$
$720$	0	0
$721$	−1829.11	−0.0944793
$722$	4625.28	0.238414
$723$	0	0
$724$	−6489.01	−0.333097
$725$	−1797.74	−0.0920914
$726$	0	0
$727$	−31769.4	−1.62072	−0.810358	−	0.585935i	$-0.800727\pi$
$727$	−31769.4	−1.62072	−0.810358	+	0.585935i	$0.800727\pi$
$728$	−689.286	−0.0350915
$729$	0	0
$730$	−9061.92	−0.459448
$731$	−23812.0	−1.20481
$732$	0	0
$733$	9298.69	0.468561	0.234280	−	0.972169i	$-0.424727\pi$
$733$	9298.69	0.468561	0.234280	+	0.972169i	$0.424727\pi$
$734$	45938.2	2.31009
$735$	0	0
$736$	−3363.52	−0.168453
$737$	−693.277	−0.0346502
$738$	0	0
$739$	−19146.9	−0.953084	−0.476542	−	0.879152i	$-0.658110\pi$
$739$	−19146.9	−0.953084	−0.476542	+	0.879152i	$0.658110\pi$
$740$	8498.09	0.422157
$741$	0	0
$742$	5198.32	0.257192
$743$	−14937.9	−0.737574	−0.368787	−	0.929514i	$-0.620227\pi$
$743$	−14937.9	−0.737574	−0.368787	+	0.929514i	$0.620227\pi$
$744$	0	0
$745$	2701.89	0.132872
$746$	26798.3	1.31522
$747$	0	0
$748$	−32338.7	−1.58078
$749$	−3756.42	−0.183253
$750$	0	0
$751$	6447.80	0.313294	0.156647	−	0.987655i	$-0.449932\pi$
$751$	6447.80	0.313294	0.156647	+	0.987655i	$0.449932\pi$
$752$	−52869.9	−2.56378
$753$	0	0
$754$	−505.276	−0.0244046
$755$	−1501.68	−0.0723865
$756$	0	0
$757$	−39280.8	−1.88598	−0.942989	−	0.332824i	$-0.891998\pi$
$757$	−39280.8	−1.88598	−0.942989	+	0.332824i	$0.891998\pi$
$758$	−35753.8	−1.71324
$759$	0	0
$760$	8113.00	0.387223
$761$	−16581.0	−0.789830	−0.394915	−	0.918718i	$-0.629226\pi$
$761$	−16581.0	−0.789830	−0.394915	+	0.918718i	$0.629226\pi$
$762$	0	0
$763$	268.618	0.0127453
$764$	33385.8	1.58097
$765$	0	0
$766$	−6145.55	−0.289879
$767$	2844.32	0.133901
$768$	0	0
$769$	−2007.30	−0.0941289	−0.0470645	−	0.998892i	$-0.514987\pi$
$769$	−2007.30	−0.0941289	−0.0470645	+	0.998892i	$0.514987\pi$
$770$	−764.942	−0.0358008
$771$	0	0
$772$	58293.9	2.71767
$773$	27280.6	1.26936	0.634679	−	0.772776i	$-0.281132\pi$
$773$	27280.6	1.26936	0.634679	+	0.772776i	$0.281132\pi$
$774$	0	0
$775$	4867.91	0.225627
$776$	−78511.2	−3.63194
$777$	0	0
$778$	6044.21	0.278529
$779$	−2280.96	−0.104909
$780$	0	0
$781$	−27809.3	−1.27413
$782$	−6443.53	−0.294655
$783$	0	0
$784$	−35878.1	−1.63439
$785$	−6779.77	−0.308255
$786$	0	0
$787$	31184.4	1.41246	0.706228	−	0.707984i	$-0.250395\pi$
$787$	31184.4	1.41246	0.706228	+	0.707984i	$0.250395\pi$
$788$	−72019.5	−3.25582
$789$	0	0
$790$	−4286.73	−0.193057
$791$	−2927.01	−0.131571
$792$	0	0
$793$	1236.18	0.0553568
$794$	−17167.3	−0.767309
$795$	0	0
$796$	50891.0	2.26606
$797$	32389.4	1.43951	0.719756	−	0.694228i	$-0.244254\pi$
$797$	32389.4	1.43951	0.719756	+	0.694228i	$0.244254\pi$
$798$	0	0
$799$	−27601.5	−1.22212
$800$	17600.1	0.777823
$801$	0	0
$802$	11862.4	0.522288
$803$	27505.9	1.20880
$804$	0	0
$805$	−104.856	−0.00459090
$806$	1368.19	0.0597919
$807$	0	0
$808$	23812.6	1.03679
$809$	−43432.4	−1.88752	−0.943759	−	0.330634i	$-0.892738\pi$
$809$	−43432.4	−1.88752	−0.943759	+	0.330634i	$0.892738\pi$
$810$	0	0
$811$	−34577.5	−1.49714	−0.748570	−	0.663055i	$-0.769259\pi$
$811$	−34577.5	−1.49714	−0.748570	+	0.663055i	$0.769259\pi$
$812$	557.052	0.0240748
$813$	0	0
$814$	−37494.3	−1.61447
$815$	−5894.37	−0.253338
$816$	0	0
$817$	−33184.3	−1.42102
$818$	−61115.1	−2.61227
$819$	0	0
$820$	1124.97	0.0479095
$821$	11403.8	0.484769	0.242384	−	0.970180i	$-0.422071\pi$
$821$	11403.8	0.484769	0.242384	+	0.970180i	$0.422071\pi$
$822$	0	0
$823$	9128.97	0.386654	0.193327	−	0.981134i	$-0.438072\pi$
$823$	9128.97	0.386654	0.193327	+	0.981134i	$0.438072\pi$
$824$	42214.5	1.78472
$825$	0	0
$826$	−4558.11	−0.192006
$827$	−35646.9	−1.49887	−0.749434	−	0.662079i	$-0.769674\pi$
$827$	−35646.9	−1.49887	−0.749434	+	0.662079i	$0.769674\pi$
$828$	0	0
$829$	27340.1	1.14543	0.572714	−	0.819755i	$-0.305891\pi$
$829$	27340.1	1.14543	0.572714	+	0.819755i	$0.305891\pi$
$830$	−10183.2	−0.425861
$831$	0	0
$832$	−717.451	−0.0298956
$833$	−18730.7	−0.779089
$834$	0	0
$835$	−735.321	−0.0304753
$836$	−45067.1	−1.86445
$837$	0	0
$838$	−7905.53	−0.325886
$839$	−36805.4	−1.51450	−0.757248	−	0.653127i	$-0.773457\pi$
$839$	−36805.4	−1.51450	−0.757248	+	0.653127i	$0.773457\pi$
$840$	0	0
$841$	−24165.9	−0.990851
$842$	35931.6	1.47065
$843$	0	0
$844$	−15839.0	−0.645971
$845$	4640.87	0.188936
$846$	0	0
$847$	−492.378	−0.0199744
$848$	−51460.9	−2.08393
$849$	0	0
$850$	33716.8	1.36056
$851$	−5139.60	−0.207031
$852$	0	0
$853$	9553.72	0.383485	0.191743	−	0.981445i	$-0.438586\pi$
$853$	9553.72	0.383485	0.191743	+	0.981445i	$0.438586\pi$
$854$	−1981.01	−0.0793779
$855$	0	0
$856$	86695.6	3.46167
$857$	−37407.4	−1.49103	−0.745515	−	0.666489i	$-0.767796\pi$
$857$	−37407.4	−1.49103	−0.745515	+	0.666489i	$0.767796\pi$
$858$	0	0
$859$	−10460.8	−0.415506	−0.207753	−	0.978181i	$-0.566615\pi$
$859$	−10460.8	−0.415506	−0.207753	+	0.978181i	$0.566615\pi$
$860$	16366.6	0.648948
$861$	0	0
$862$	−71255.7	−2.81552
$863$	28475.5	1.12319	0.561597	−	0.827411i	$-0.310187\pi$
$863$	28475.5	1.12319	0.561597	+	0.827411i	$0.310187\pi$
$864$	0	0
$865$	2753.96	0.108251
$866$	63020.7	2.47290
$867$	0	0
$868$	−1508.39	−0.0589839
$869$	13011.7	0.507929
$870$	0	0
$871$	139.764	0.00543709
$872$	−6199.52	−0.240759
$873$	0	0
$874$	−8979.68	−0.347531
$875$	1118.54	0.0432155
$876$	0	0
$877$	9710.29	0.373880	0.186940	−	0.982371i	$-0.440143\pi$
$877$	9710.29	0.373880	0.186940	+	0.982371i	$0.440143\pi$
$878$	47803.8	1.83747
$879$	0	0
$880$	7572.55	0.290080
$881$	14335.7	0.548220	0.274110	−	0.961698i	$-0.411617\pi$
$881$	14335.7	0.548220	0.274110	+	0.961698i	$0.411617\pi$
$882$	0	0
$883$	12712.0	0.484475	0.242238	−	0.970217i	$-0.422119\pi$
$883$	12712.0	0.484475	0.242238	+	0.970217i	$0.422119\pi$
$884$	6519.43	0.248045
$885$	0	0
$886$	−45227.9	−1.71497
$887$	14044.4	0.531640	0.265820	−	0.964023i	$-0.414357\pi$
$887$	14044.4	0.531640	0.265820	+	0.964023i	$0.414357\pi$
$888$	0	0
$889$	−5146.18	−0.194148
$890$	−4497.49	−0.169389
$891$	0	0
$892$	69387.6	2.60456
$893$	−38465.3	−1.44143
$894$	0	0
$895$	2538.42	0.0948044
$896$	3623.39	0.135099
$897$	0	0
$898$	42547.2	1.58109
$899$	−604.185	−0.0224146
$900$	0	0
$901$	−26865.9	−0.993378
$902$	−4963.49	−0.183222
$903$	0	0
$904$	67553.4	2.48539
$905$	793.273	0.0291373
$906$	0	0
$907$	−21352.9	−0.781710	−0.390855	−	0.920452i	$-0.627821\pi$
$907$	−21352.9	−0.781710	−0.390855	+	0.920452i	$0.627821\pi$
$908$	94584.2	3.45692
$909$	0	0
$910$	154.211	0.00561764
$911$	51282.0	1.86504	0.932519	−	0.361121i	$-0.117606\pi$
$911$	51282.0	1.86504	0.932519	+	0.361121i	$0.117606\pi$
$912$	0	0
$913$	30909.4	1.12043
$914$	−15490.3	−0.560583
$915$	0	0
$916$	67556.6	2.43683
$917$	1564.35	0.0563353
$918$	0	0
$919$	−10026.8	−0.359905	−0.179953	−	0.983675i	$-0.557594\pi$
$919$	−10026.8	−0.359905	−0.179953	+	0.983675i	$0.557594\pi$
$920$	2419.99	0.0867226
$921$	0	0
$922$	92892.9	3.31807
$923$	5606.30	0.199928
$924$	0	0
$925$	26893.7	0.955956
$926$	−58818.7	−2.08737
$927$	0	0
$928$	−2184.45	−0.0772718
$929$	−12614.6	−0.445501	−0.222750	−	0.974875i	$-0.571504\pi$
$929$	−12614.6	−0.445501	−0.222750	+	0.974875i	$0.571504\pi$
$930$	0	0
$931$	−26103.1	−0.918897
$932$	35235.0	1.23837
$933$	0	0
$934$	28333.0	0.992595
$935$	3953.37	0.138277
$936$	0	0
$937$	47032.5	1.63979	0.819896	−	0.572513i	$-0.194031\pi$
$937$	47032.5	1.63979	0.819896	+	0.572513i	$0.194031\pi$
$938$	−223.975	−0.00779643
$939$	0	0
$940$	18971.2	0.658269
$941$	40313.9	1.39660	0.698298	−	0.715807i	$-0.253941\pi$
$941$	40313.9	1.39660	0.698298	+	0.715807i	$0.253941\pi$
$942$	0	0
$943$	−680.377	−0.0234954
$944$	45123.0	1.55575
$945$	0	0
$946$	−72210.8	−2.48179
$947$	2843.88	0.0975857	0.0487928	−	0.998809i	$-0.484463\pi$
$947$	2843.88	0.0975857	0.0487928	+	0.998809i	$0.484463\pi$
$948$	0	0
$949$	−5545.15	−0.189677
$950$	46987.5	1.60471
$951$	0	0
$952$	−5708.79	−0.194352
$953$	47683.6	1.62080	0.810401	−	0.585876i	$-0.199249\pi$
$953$	47683.6	1.62080	0.810401	+	0.585876i	$0.199249\pi$
$954$	0	0
$955$	−4081.38	−0.138294
$956$	−55472.2	−1.87667
$957$	0	0
$958$	11765.0	0.396773
$959$	1584.19	0.0533434
$960$	0	0
$961$	−28155.0	−0.945084
$962$	7558.80	0.253332
$963$	0	0
$964$	−56258.0	−1.87961
$965$	−7126.36	−0.237726
$966$	0	0
$967$	−29531.0	−0.982061	−0.491031	−	0.871142i	$-0.663380\pi$
$967$	−29531.0	−0.982061	−0.491031	+	0.871142i	$0.663380\pi$
$968$	11363.7	0.377318
$969$	0	0
$970$	17565.0	0.581420
$971$	36965.3	1.22170	0.610851	−	0.791746i	$-0.290828\pi$
$971$	36965.3	1.22170	0.610851	+	0.791746i	$0.290828\pi$
$972$	0	0
$973$	5695.27	0.187648
$974$	−41849.2	−1.37673
$975$	0	0
$976$	19611.0	0.643170
$977$	−7366.55	−0.241225	−0.120612	−	0.992700i	$-0.538486\pi$
$977$	−7366.55	−0.241225	−0.120612	+	0.992700i	$0.538486\pi$
$978$	0	0
$979$	13651.4	0.445659
$980$	12874.1	0.419641
$981$	0	0
$982$	−53763.4	−1.74711
$983$	−16255.6	−0.527440	−0.263720	−	0.964599i	$-0.584949\pi$
$983$	−16255.6	−0.527440	−0.263720	+	0.964599i	$0.584949\pi$
$984$	0	0
$985$	8804.29	0.284800
$986$	−4184.78	−0.135163
$987$	0	0
$988$	9085.45	0.292557
$989$	−9898.41	−0.318252
$990$	0	0
$991$	13730.4	0.440121	0.220061	−	0.975486i	$-0.429374\pi$
$991$	13730.4	0.440121	0.220061	+	0.975486i	$0.429374\pi$
$992$	5915.07	0.189318
$993$	0	0
$994$	−8984.27	−0.286684
$995$	−6221.36	−0.198221
$996$	0	0
$997$	−49655.8	−1.57735	−0.788673	−	0.614813i	$-0.789232\pi$
$997$	−49655.8	−1.57735	−0.788673	+	0.614813i	$0.789232\pi$
$998$	27864.3	0.883795
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.a.g.1.1	✓	5
3.2	odd	2		207.4.a.h.1.5	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.4.a.g.1.1	✓	5	1.1	even	1	trivial
207.4.a.h.1.5	yes	5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-5.06335 q^{2} +17.6375 q^{4} -2.15617 q^{5} +2.11437 q^{7} -48.7982 q^{8} +O(q^{10})\)	\(q-5.06335 q^{2} +17.6375 q^{4} -2.15617 q^{5} +2.11437 q^{7} -48.7982 q^{8} +10.9174 q^{10} -33.1380 q^{11} +6.68057 q^{13} -10.7058 q^{14} +105.982 q^{16} +55.3297 q^{17} +77.1072 q^{19} -38.0295 q^{20} +167.789 q^{22} +23.0000 q^{23} -120.351 q^{25} -33.8261 q^{26} +37.2923 q^{28} +14.9375 q^{29} -40.4477 q^{31} -146.240 q^{32} -280.154 q^{34} -4.55894 q^{35} -223.461 q^{37} -390.421 q^{38} +105.217 q^{40} -29.5816 q^{41} -430.366 q^{43} -584.473 q^{44} -116.457 q^{46} -498.856 q^{47} -338.529 q^{49} +609.379 q^{50} +117.829 q^{52} -485.561 q^{53} +71.4511 q^{55} -103.178 q^{56} -75.6336 q^{58} +425.760 q^{59} +185.041 q^{61} +204.801 q^{62} -107.394 q^{64} -14.4044 q^{65} +20.9209 q^{67} +975.879 q^{68} +23.0835 q^{70} +839.195 q^{71} -830.042 q^{73} +1131.46 q^{74} +1359.98 q^{76} -70.0661 q^{77} -392.651 q^{79} -228.516 q^{80} +149.782 q^{82} -932.749 q^{83} -119.300 q^{85} +2179.09 q^{86} +1617.08 q^{88} -411.955 q^{89} +14.1252 q^{91} +405.663 q^{92} +2525.88 q^{94} -166.256 q^{95} +1608.89 q^{97} +1714.09 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{2} + 16 q^{4} - 20 q^{5} - 10 q^{7} - 48 q^{8}+O(q^{10}) \) 5 * q - 4 * q^2 + 16 * q^4 - 20 * q^5 - 10 * q^7 - 48 * q^8	\( 5 q - 4 q^{2} + 16 q^{4} - 20 q^{5} - 10 q^{7} - 48 q^{8} + 50 q^{10} - 46 q^{11} + 54 q^{13} - 164 q^{14} - 60 q^{16} - 250 q^{17} - 28 q^{19} - 242 q^{20} - 10 q^{22} + 115 q^{23} + 239 q^{25} - 368 q^{26} + 224 q^{28} - 460 q^{29} - 360 q^{31} - 268 q^{32} - 20 q^{34} - 964 q^{35} - 92 q^{37} - 266 q^{38} - 278 q^{40} - 788 q^{41} + 108 q^{43} - 354 q^{44} - 92 q^{46} - 724 q^{47} + 149 q^{49} - 128 q^{50} - 288 q^{52} - 580 q^{53} - 672 q^{55} + 1172 q^{56} - 48 q^{58} + 408 q^{59} - 620 q^{61} + 1088 q^{62} + 696 q^{64} - 184 q^{65} + 100 q^{67} + 1584 q^{68} + 728 q^{70} + 124 q^{71} - 1946 q^{73} + 2866 q^{74} + 1362 q^{76} + 396 q^{77} - 198 q^{79} + 2398 q^{80} + 76 q^{82} + 578 q^{83} + 376 q^{85} + 3910 q^{86} + 1366 q^{88} - 930 q^{89} + 1972 q^{91} + 368 q^{92} + 976 q^{94} + 908 q^{95} + 2618 q^{97} + 2400 q^{98}+O(q^{100}) \) 5 * q - 4 * q^2 + 16 * q^4 - 20 * q^5 - 10 * q^7 - 48 * q^8 + 50 * q^10 - 46 * q^11 + 54 * q^13 - 164 * q^14 - 60 * q^16 - 250 * q^17 - 28 * q^19 - 242 * q^20 - 10 * q^22 + 115 * q^23 + 239 * q^25 - 368 * q^26 + 224 * q^28 - 460 * q^29 - 360 * q^31 - 268 * q^32 - 20 * q^34 - 964 * q^35 - 92 * q^37 - 266 * q^38 - 278 * q^40 - 788 * q^41 + 108 * q^43 - 354 * q^44 - 92 * q^46 - 724 * q^47 + 149 * q^49 - 128 * q^50 - 288 * q^52 - 580 * q^53 - 672 * q^55 + 1172 * q^56 - 48 * q^58 + 408 * q^59 - 620 * q^61 + 1088 * q^62 + 696 * q^64 - 184 * q^65 + 100 * q^67 + 1584 * q^68 + 728 * q^70 + 124 * q^71 - 1946 * q^73 + 2866 * q^74 + 1362 * q^76 + 396 * q^77 - 198 * q^79 + 2398 * q^80 + 76 * q^82 + 578 * q^83 + 376 * q^85 + 3910 * q^86 + 1366 * q^88 - 930 * q^89 + 1972 * q^91 + 368 * q^92 + 976 * q^94 + 908 * q^95 + 2618 * q^97 + 2400 * q^98

Introduction

L-functions

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