LMFDB - Embedded newform 2151.4.a.c.1.5

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:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	$q-3.82627 q^{2} +6.64037 q^{4} +6.06587 q^{5} +7.45896 q^{7} +5.20233 q^{8} +O(q^{10})$	\(q-3.82627 q^{2} +6.64037 q^{4} +6.06587 q^{5} +7.45896 q^{7} +5.20233 q^{8} -23.2097 q^{10} +54.8904 q^{11} -8.46396 q^{13} -28.5400 q^{14} -73.0285 q^{16} +75.7508 q^{17} +1.60821 q^{19} +40.2796 q^{20} -210.026 q^{22} +105.664 q^{23} -88.2052 q^{25} +32.3854 q^{26} +49.5302 q^{28} -94.0739 q^{29} +204.670 q^{31} +237.808 q^{32} -289.843 q^{34} +45.2451 q^{35} +10.6376 q^{37} -6.15345 q^{38} +31.5566 q^{40} +246.551 q^{41} +454.688 q^{43} +364.493 q^{44} -404.298 q^{46} +218.412 q^{47} -287.364 q^{49} +337.497 q^{50} -56.2038 q^{52} -26.8532 q^{53} +332.958 q^{55} +38.8040 q^{56} +359.952 q^{58} -110.168 q^{59} -218.689 q^{61} -783.123 q^{62} -325.692 q^{64} -51.3413 q^{65} +540.693 q^{67} +503.013 q^{68} -173.120 q^{70} +688.394 q^{71} +325.248 q^{73} -40.7022 q^{74} +10.6791 q^{76} +409.426 q^{77} -491.233 q^{79} -442.981 q^{80} -943.373 q^{82} +729.130 q^{83} +459.495 q^{85} -1739.76 q^{86} +285.558 q^{88} -676.629 q^{89} -63.1324 q^{91} +701.645 q^{92} -835.703 q^{94} +9.75519 q^{95} -759.513 q^{97} +1099.53 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8}+O(q^{10}) $ 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8	$ 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8} - 68 q^{10} + 258 q^{11} - 134 q^{13} + 292 q^{14} + 327 q^{16} + 364 q^{17} - 278 q^{19} + 986 q^{20} - 179 q^{22} + 668 q^{23} + 490 q^{25} + 760 q^{26} - 802 q^{28} + 714 q^{29} - 608 q^{31} + 918 q^{32} - 228 q^{34} + 934 q^{35} - 1080 q^{37} + 1395 q^{38} - 563 q^{40} + 1796 q^{41} - 1934 q^{43} + 3157 q^{44} - 940 q^{46} + 2032 q^{47} + 762 q^{49} + 1754 q^{50} - 2328 q^{52} + 1790 q^{53} - 478 q^{55} + 3557 q^{56} - 2626 q^{58} + 3622 q^{59} + 324 q^{61} + 796 q^{62} + 2023 q^{64} + 2200 q^{65} - 2444 q^{67} - 357 q^{68} + 4305 q^{70} + 1298 q^{71} - 1368 q^{73} - 813 q^{74} + 1390 q^{76} + 1408 q^{77} - 1378 q^{79} + 7684 q^{80} + 9001 q^{82} + 3524 q^{83} + 60 q^{85} + 2543 q^{86} + 1749 q^{88} + 7854 q^{89} + 850 q^{91} + 496 q^{92} + 6634 q^{94} + 3696 q^{95} - 1746 q^{97} + 4632 q^{98}+O(q^{100}) $ 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8 - 68 * q^10 + 258 * q^11 - 134 * q^13 + 292 * q^14 + 327 * q^16 + 364 * q^17 - 278 * q^19 + 986 * q^20 - 179 * q^22 + 668 * q^23 + 490 * q^25 + 760 * q^26 - 802 * q^28 + 714 * q^29 - 608 * q^31 + 918 * q^32 - 228 * q^34 + 934 * q^35 - 1080 * q^37 + 1395 * q^38 - 563 * q^40 + 1796 * q^41 - 1934 * q^43 + 3157 * q^44 - 940 * q^46 + 2032 * q^47 + 762 * q^49 + 1754 * q^50 - 2328 * q^52 + 1790 * q^53 - 478 * q^55 + 3557 * q^56 - 2626 * q^58 + 3622 * q^59 + 324 * q^61 + 796 * q^62 + 2023 * q^64 + 2200 * q^65 - 2444 * q^67 - 357 * q^68 + 4305 * q^70 + 1298 * q^71 - 1368 * q^73 - 813 * q^74 + 1390 * q^76 + 1408 * q^77 - 1378 * q^79 + 7684 * q^80 + 9001 * q^82 + 3524 * q^83 + 60 * q^85 + 2543 * q^86 + 1749 * q^88 + 7854 * q^89 + 850 * q^91 + 496 * q^92 + 6634 * q^94 + 3696 * q^95 - 1746 * q^97 + 4632 * 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 $

$2$	−3.82627	−1.35279	−0.676396	−	0.736538i	$-0.736459\pi$
$2$	−3.82627	−1.35279	−0.676396	+	0.736538i	$0.736459\pi$
$3$	0	0
$3$	0	0
$4$	6.64037	0.830046
$5$	6.06587	0.542548	0.271274	−	0.962502i	$-0.412555\pi$
$5$	6.06587	0.542548	0.271274	+	0.962502i	$0.412555\pi$
$6$	0	0
$7$	7.45896	0.402746	0.201373	−	0.979515i	$-0.435460\pi$
$7$	7.45896	0.402746	0.201373	+	0.979515i	$0.435460\pi$
$8$	5.20233	0.229913
$9$	0	0
$10$	−23.2097	−0.733954
$11$	54.8904	1.50455	0.752277	−	0.658847i	$-0.228956\pi$
$11$	54.8904	1.50455	0.752277	+	0.658847i	$0.228956\pi$
$12$	0	0
$13$	−8.46396	−0.180575	−0.0902877	−	0.995916i	$-0.528779\pi$
$13$	−8.46396	−0.180575	−0.0902877	+	0.995916i	$0.528779\pi$
$14$	−28.5400	−0.544832
$15$	0	0
$16$	−73.0285	−1.14107
$17$	75.7508	1.08072	0.540361	−	0.841433i	$-0.318288\pi$
$17$	75.7508	1.08072	0.540361	+	0.841433i	$0.318288\pi$
$18$	0	0
$19$	1.60821	0.0194183	0.00970917	−	0.999953i	$-0.496909\pi$
$19$	1.60821	0.0194183	0.00970917	+	0.999953i	$0.496909\pi$
$20$	40.2796	0.450339
$21$	0	0
$22$	−210.026	−2.03535
$23$	105.664	0.957930	0.478965	−	0.877834i	$-0.341012\pi$
$23$	105.664	0.957930	0.478965	+	0.877834i	$0.341012\pi$
$24$	0	0
$25$	−88.2052	−0.705642
$26$	32.3854	0.244281
$27$	0	0
$28$	49.5302	0.334298
$29$	−94.0739	−0.602382	−0.301191	−	0.953564i	$-0.597384\pi$
$29$	−94.0739	−0.602382	−0.301191	+	0.953564i	$0.597384\pi$
$30$	0	0
$31$	204.670	1.18580	0.592900	−	0.805276i	$-0.297983\pi$
$31$	204.670	1.18580	0.592900	+	0.805276i	$0.297983\pi$
$32$	237.808	1.31372
$33$	0	0
$34$	−289.843	−1.46199
$35$	45.2451	0.218509
$36$	0	0
$37$	10.6376	0.0472650	0.0236325	−	0.999721i	$-0.492477\pi$
$37$	10.6376	0.0472650	0.0236325	+	0.999721i	$0.492477\pi$
$38$	−6.15345	−0.0262690
$39$	0	0
$40$	31.5566	0.124739
$41$	246.551	0.939143	0.469571	−	0.882895i	$-0.344408\pi$
$41$	246.551	0.939143	0.469571	+	0.882895i	$0.344408\pi$
$42$	0	0
$43$	454.688	1.61254	0.806270	−	0.591548i	$-0.201483\pi$
$43$	454.688	1.61254	0.806270	+	0.591548i	$0.201483\pi$
$44$	364.493	1.24885
$45$	0	0
$46$	−404.298	−1.29588
$47$	218.412	0.677843	0.338921	−	0.940815i	$-0.389938\pi$
$47$	218.412	0.677843	0.338921	+	0.940815i	$0.389938\pi$
$48$	0	0
$49$	−287.364	−0.837796
$50$	337.497	0.954587
$51$	0	0
$52$	−56.2038	−0.149886
$53$	−26.8532	−0.0695958	−0.0347979	−	0.999394i	$-0.511079\pi$
$53$	−26.8532	−0.0695958	−0.0347979	+	0.999394i	$0.511079\pi$
$54$	0	0
$55$	332.958	0.816292
$56$	38.8040	0.0925964
$57$	0	0
$58$	359.952	0.814898
$59$	−110.168	−0.243095	−0.121548	−	0.992586i	$-0.538786\pi$
$59$	−110.168	−0.243095	−0.121548	+	0.992586i	$0.538786\pi$
$60$	0	0
$61$	−218.689	−0.459020	−0.229510	−	0.973306i	$-0.573712\pi$
$61$	−218.689	−0.459020	−0.229510	+	0.973306i	$0.573712\pi$
$62$	−783.123	−1.60414
$63$	0	0
$64$	−325.692	−0.636116
$65$	−51.3413	−0.0979708
$66$	0	0
$67$	540.693	0.985912	0.492956	−	0.870054i	$-0.335916\pi$
$67$	540.693	0.985912	0.492956	+	0.870054i	$0.335916\pi$
$68$	503.013	0.897049
$69$	0	0
$70$	−173.120	−0.295597
$71$	688.394	1.15067	0.575333	−	0.817919i	$-0.304872\pi$
$71$	688.394	1.15067	0.575333	+	0.817919i	$0.304872\pi$
$72$	0	0
$73$	325.248	0.521472	0.260736	−	0.965410i	$-0.416035\pi$
$73$	325.248	0.521472	0.260736	+	0.965410i	$0.416035\pi$
$74$	−40.7022	−0.0639397
$75$	0	0
$76$	10.6791	0.0161181
$77$	409.426	0.605953
$78$	0	0
$79$	−491.233	−0.699595	−0.349798	−	0.936825i	$-0.613750\pi$
$79$	−491.233	−0.699595	−0.349798	+	0.936825i	$0.613750\pi$
$80$	−442.981	−0.619085
$81$	0	0
$82$	−943.373	−1.27046
$83$	729.130	0.964246	0.482123	−	0.876103i	$-0.339866\pi$
$83$	729.130	0.964246	0.482123	+	0.876103i	$0.339866\pi$
$84$	0	0
$85$	459.495	0.586343
$86$	−1739.76	−2.18143
$87$	0	0
$88$	285.558	0.345916
$89$	−676.629	−0.805871	−0.402936	−	0.915228i	$-0.632010\pi$
$89$	−676.629	−0.805871	−0.402936	+	0.915228i	$0.632010\pi$
$90$	0	0
$91$	−63.1324	−0.0727260
$92$	701.645	0.795126
$93$	0	0
$94$	−835.703	−0.916980
$95$	9.75519	0.0105354
$96$	0	0
$97$	−759.513	−0.795019	−0.397509	−	0.917598i	$-0.630125\pi$
$97$	−759.513	−0.795019	−0.397509	+	0.917598i	$0.630125\pi$
$98$	1099.53	1.13336
$99$	0	0
$100$	−585.715	−0.585715
$101$	994.085	0.979358	0.489679	−	0.871903i	$-0.337114\pi$
$101$	994.085	0.979358	0.489679	+	0.871903i	$0.337114\pi$
$102$	0	0
$103$	−71.1650	−0.0680786	−0.0340393	−	0.999420i	$-0.510837\pi$
$103$	−71.1650	−0.0680786	−0.0340393	+	0.999420i	$0.510837\pi$
$104$	−44.0323	−0.0415166
$105$	0	0
$106$	102.748	0.0941486
$107$	813.520	0.735009	0.367504	−	0.930022i	$-0.380212\pi$
$107$	813.520	0.735009	0.367504	+	0.930022i	$0.380212\pi$
$108$	0	0
$109$	174.184	0.153062	0.0765312	−	0.997067i	$-0.475616\pi$
$109$	174.184	0.153062	0.0765312	+	0.997067i	$0.475616\pi$
$110$	−1273.99	−1.10427
$111$	0	0
$112$	−544.716	−0.459561
$113$	125.350	0.104353	0.0521766	−	0.998638i	$-0.483384\pi$
$113$	125.350	0.104353	0.0521766	+	0.998638i	$0.483384\pi$
$114$	0	0
$115$	640.941	0.519723
$116$	−624.685	−0.500005
$117$	0	0
$118$	421.532	0.328857
$119$	565.022	0.435256
$120$	0	0
$121$	1681.96	1.26368
$122$	836.762	0.620958
$123$	0	0
$124$	1359.08	0.984268
$125$	−1293.27	−0.925392
$126$	0	0
$127$	−1125.87	−0.786653	−0.393327	−	0.919399i	$-0.628676\pi$
$127$	−1125.87	−0.786653	−0.393327	+	0.919399i	$0.628676\pi$
$128$	−656.281	−0.453184
$129$	0	0
$130$	196.446	0.132534
$131$	−956.637	−0.638029	−0.319014	−	0.947750i	$-0.603352\pi$
$131$	−956.637	−0.638029	−0.319014	+	0.947750i	$0.603352\pi$
$132$	0	0
$133$	11.9956	0.00782066
$134$	−2068.84	−1.33373
$135$	0	0
$136$	394.081	0.248472
$137$	−1296.13	−0.808291	−0.404146	−	0.914695i	$-0.632431\pi$
$137$	−1296.13	−0.808291	−0.404146	+	0.914695i	$0.632431\pi$
$138$	0	0
$139$	1234.91	0.753553	0.376777	−	0.926304i	$-0.377032\pi$
$139$	1234.91	0.753553	0.376777	+	0.926304i	$0.377032\pi$
$140$	300.444	0.181372
$141$	0	0
$142$	−2633.98	−1.55661
$143$	−464.591	−0.271685
$144$	0	0
$145$	−570.640	−0.326821
$146$	−1244.49	−0.705443
$147$	0	0
$148$	70.6373	0.0392321
$149$	−2390.42	−1.31430	−0.657150	−	0.753760i	$-0.728238\pi$
$149$	−2390.42	−1.31430	−0.657150	+	0.753760i	$0.728238\pi$
$150$	0	0
$151$	1997.74	1.07665	0.538323	−	0.842739i	$-0.319058\pi$
$151$	1997.74	1.07665	0.538323	+	0.842739i	$0.319058\pi$
$152$	8.36643	0.00446452
$153$	0	0
$154$	−1566.57	−0.819728
$155$	1241.50	0.643353
$156$	0	0
$157$	−1647.80	−0.837633	−0.418817	−	0.908071i	$-0.637555\pi$
$157$	−1647.80	−0.837633	−0.418817	+	0.908071i	$0.637555\pi$
$158$	1879.59	0.946407
$159$	0	0
$160$	1442.51	0.712754
$161$	788.141	0.385802
$162$	0	0
$163$	2601.77	1.25022	0.625111	−	0.780536i	$-0.285054\pi$
$163$	2601.77	1.25022	0.625111	+	0.780536i	$0.285054\pi$
$164$	1637.19	0.779532
$165$	0	0
$166$	−2789.85	−1.30442
$167$	1064.39	0.493205	0.246603	−	0.969117i	$-0.420686\pi$
$167$	1064.39	0.493205	0.246603	+	0.969117i	$0.420686\pi$
$168$	0	0
$169$	−2125.36	−0.967393
$170$	−1758.15	−0.793200
$171$	0	0
$172$	3019.29	1.33848
$173$	−2001.82	−0.879743	−0.439872	−	0.898061i	$-0.644976\pi$
$173$	−2001.82	−0.879743	−0.439872	+	0.898061i	$0.644976\pi$
$174$	0	0
$175$	−657.919	−0.284194
$176$	−4008.56	−1.71680
$177$	0	0
$178$	2588.97	1.09018
$179$	−1919.02	−0.801311	−0.400655	−	0.916229i	$-0.631218\pi$
$179$	−1919.02	−0.801311	−0.400655	+	0.916229i	$0.631218\pi$
$180$	0	0
$181$	1017.57	0.417876	0.208938	−	0.977929i	$-0.432999\pi$
$181$	1017.57	0.417876	0.208938	+	0.977929i	$0.432999\pi$
$182$	241.562	0.0983832
$183$	0	0
$184$	549.697	0.220240
$185$	64.5261	0.0256435
$186$	0	0
$187$	4158.00	1.62600
$188$	1450.33	0.562641
$189$	0	0
$190$	−37.3260	−0.0142522
$191$	1030.19	0.390273	0.195137	−	0.980776i	$-0.437485\pi$
$191$	1030.19	0.390273	0.195137	+	0.980776i	$0.437485\pi$
$192$	0	0
$193$	−2868.67	−1.06991	−0.534953	−	0.844882i	$-0.679671\pi$
$193$	−2868.67	−1.06991	−0.534953	+	0.844882i	$0.679671\pi$
$194$	2906.10	1.07550
$195$	0	0
$196$	−1908.20	−0.695409
$197$	−665.517	−0.240691	−0.120346	−	0.992732i	$-0.538400\pi$
$197$	−665.517	−0.240691	−0.120346	+	0.992732i	$0.538400\pi$
$198$	0	0
$199$	−3650.05	−1.30023	−0.650114	−	0.759837i	$-0.725279\pi$
$199$	−3650.05	−1.30023	−0.650114	+	0.759837i	$0.725279\pi$
$200$	−458.873	−0.162236
$201$	0	0
$202$	−3803.64	−1.32487
$203$	−701.693	−0.242607
$204$	0	0
$205$	1495.55	0.509530
$206$	272.297	0.0920962
$207$	0	0
$208$	618.110	0.206049
$209$	88.2753	0.0292159
$210$	0	0
$211$	−220.967	−0.0720947	−0.0360474	−	0.999350i	$-0.511477\pi$
$211$	−220.967	−0.0720947	−0.0360474	+	0.999350i	$0.511477\pi$
$212$	−178.315	−0.0577677
$213$	0	0
$214$	−3112.75	−0.994314
$215$	2758.08	0.874880
$216$	0	0
$217$	1526.62	0.477576
$218$	−666.475	−0.207062
$219$	0	0
$220$	2210.96	0.677560
$221$	−641.152	−0.195152
$222$	0	0
$223$	−86.3501	−0.0259302	−0.0129651	−	0.999916i	$-0.504127\pi$
$223$	−86.3501	−0.0259302	−0.0129651	+	0.999916i	$0.504127\pi$
$224$	1773.80	0.529094
$225$	0	0
$226$	−479.622	−0.141168
$227$	626.043	0.183048	0.0915241	−	0.995803i	$-0.470826\pi$
$227$	626.043	0.183048	0.0915241	+	0.995803i	$0.470826\pi$
$228$	0	0
$229$	−5271.73	−1.52125	−0.760623	−	0.649193i	$-0.775107\pi$
$229$	−5271.73	−1.52125	−0.760623	+	0.649193i	$0.775107\pi$
$230$	−2452.42	−0.703076
$231$	0	0
$232$	−489.403	−0.138495
$233$	457.670	0.128682	0.0643411	−	0.997928i	$-0.479505\pi$
$233$	457.670	0.128682	0.0643411	+	0.997928i	$0.479505\pi$
$234$	0	0
$235$	1324.86	0.367762
$236$	−731.555	−0.201780
$237$	0	0
$238$	−2161.93	−0.588811
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	131.864	0.0352452	0.0176226	−	0.999845i	$-0.494390\pi$
$241$	131.864	0.0352452	0.0176226	+	0.999845i	$0.494390\pi$
$242$	−6435.64	−1.70950
$243$	0	0
$244$	−1452.17	−0.381007
$245$	−1743.11	−0.454544
$246$	0	0
$247$	−13.6118	−0.00350648
$248$	1064.76	0.272630
$249$	0	0
$250$	4948.42	1.25186
$251$	−3889.05	−0.977987	−0.488994	−	0.872287i	$-0.662636\pi$
$251$	−3889.05	−0.977987	−0.488994	+	0.872287i	$0.662636\pi$
$252$	0	0
$253$	5799.92	1.44126
$254$	4307.89	1.06418
$255$	0	0
$256$	5116.64	1.24918
$257$	2954.88	0.717199	0.358600	−	0.933491i	$-0.383254\pi$
$257$	2954.88	0.717199	0.358600	+	0.933491i	$0.383254\pi$
$258$	0	0
$259$	79.3452	0.0190358
$260$	−340.925	−0.0813203
$261$	0	0
$262$	3660.35	0.863120
$263$	−923.971	−0.216633	−0.108317	−	0.994116i	$-0.534546\pi$
$263$	−923.971	−0.216633	−0.108317	+	0.994116i	$0.534546\pi$
$264$	0	0
$265$	−162.888	−0.0377590
$266$	−45.8983	−0.0105797
$267$	0	0
$268$	3590.40	0.818352
$269$	6978.45	1.58172	0.790862	−	0.611995i	$-0.209633\pi$
$269$	6978.45	1.58172	0.790862	+	0.611995i	$0.209633\pi$
$270$	0	0
$271$	1687.79	0.378324	0.189162	−	0.981946i	$-0.439423\pi$
$271$	1687.79	0.378324	0.189162	+	0.981946i	$0.439423\pi$
$272$	−5531.97	−1.23318
$273$	0	0
$274$	4959.35	1.09345
$275$	−4841.62	−1.06168
$276$	0	0
$277$	−3993.95	−0.866329	−0.433165	−	0.901315i	$-0.642603\pi$
$277$	−3993.95	−0.866329	−0.433165	+	0.901315i	$0.642603\pi$
$278$	−4725.11	−1.01940
$279$	0	0
$280$	235.380	0.0502380
$281$	−4511.94	−0.957864	−0.478932	−	0.877852i	$-0.658976\pi$
$281$	−4511.94	−0.957864	−0.478932	+	0.877852i	$0.658976\pi$
$282$	0	0
$283$	−953.172	−0.200213	−0.100106	−	0.994977i	$-0.531918\pi$
$283$	−953.172	−0.200213	−0.100106	+	0.994977i	$0.531918\pi$
$284$	4571.19	0.955106
$285$	0	0
$286$	1777.65	0.367534
$287$	1839.02	0.378236
$288$	0	0
$289$	825.188	0.167960
$290$	2183.42	0.442121
$291$	0	0
$292$	2159.77	0.432846
$293$	5545.45	1.10569	0.552847	−	0.833282i	$-0.313541\pi$
$293$	5545.45	1.10569	0.552847	+	0.833282i	$0.313541\pi$
$294$	0	0
$295$	−668.263	−0.131891
$296$	55.3401	0.0108668
$297$	0	0
$298$	9146.39	1.77797
$299$	−894.333	−0.172979
$300$	0	0
$301$	3391.50	0.649444
$302$	−7643.88	−1.45648
$303$	0	0
$304$	−117.445	−0.0221577
$305$	−1326.54	−0.249040
$306$	0	0
$307$	−668.774	−0.124329	−0.0621644	−	0.998066i	$-0.519800\pi$
$307$	−668.774	−0.124329	−0.0621644	+	0.998066i	$0.519800\pi$
$308$	2718.74	0.502969
$309$	0	0
$310$	−4750.32	−0.870323
$311$	−1110.75	−0.202523	−0.101262	−	0.994860i	$-0.532288\pi$
$311$	−1110.75	−0.202523	−0.101262	+	0.994860i	$0.532288\pi$
$312$	0	0
$313$	−729.484	−0.131734	−0.0658672	−	0.997828i	$-0.520981\pi$
$313$	−729.484	−0.131734	−0.0658672	+	0.997828i	$0.520981\pi$
$314$	6304.92	1.13314
$315$	0	0
$316$	−3261.97	−0.580696
$317$	−5668.81	−1.00439	−0.502196	−	0.864754i	$-0.667474\pi$
$317$	−5668.81	−1.00439	−0.502196	+	0.864754i	$0.667474\pi$
$318$	0	0
$319$	−5163.76	−0.906316
$320$	−1975.60	−0.345123
$321$	0	0
$322$	−3015.64	−0.521910
$323$	121.823	0.0209858
$324$	0	0
$325$	746.566	0.127422
$326$	−9955.08	−1.69129
$327$	0	0
$328$	1282.64	0.215921
$329$	1629.12	0.272999
$330$	0	0
$331$	−3366.70	−0.559066	−0.279533	−	0.960136i	$-0.590180\pi$
$331$	−3366.70	−0.559066	−0.279533	+	0.960136i	$0.590180\pi$
$332$	4841.69	0.800369
$333$	0	0
$334$	−4072.66	−0.667204
$335$	3279.77	0.534904
$336$	0	0
$337$	7062.29	1.14157	0.570783	−	0.821101i	$-0.306640\pi$
$337$	7062.29	1.14157	0.570783	+	0.821101i	$0.306640\pi$
$338$	8132.21	1.30868
$339$	0	0
$340$	3051.21	0.486692
$341$	11234.4	1.78410
$342$	0	0
$343$	−4701.86	−0.740165
$344$	2365.43	0.370743
$345$	0	0
$346$	7659.51	1.19011
$347$	11754.1	1.81842	0.909212	−	0.416333i	$-0.136685\pi$
$347$	11754.1	1.81842	0.909212	+	0.416333i	$0.136685\pi$
$348$	0	0
$349$	2160.67	0.331398	0.165699	−	0.986176i	$-0.447012\pi$
$349$	2160.67	0.331398	0.165699	+	0.986176i	$0.447012\pi$
$350$	2517.38	0.384456
$351$	0	0
$352$	13053.4	1.97656
$353$	−4982.26	−0.751216	−0.375608	−	0.926779i	$-0.622566\pi$
$353$	−4982.26	−0.751216	−0.375608	+	0.926779i	$0.622566\pi$
$354$	0	0
$355$	4175.71	0.624291
$356$	−4493.06	−0.668910
$357$	0	0
$358$	7342.71	1.08401
$359$	3766.98	0.553798	0.276899	−	0.960899i	$-0.410693\pi$
$359$	3766.98	0.553798	0.276899	+	0.960899i	$0.410693\pi$
$360$	0	0
$361$	−6856.41	−0.999623
$362$	−3893.51	−0.565299
$363$	0	0
$364$	−419.222	−0.0603660
$365$	1972.91	0.282923
$366$	0	0
$367$	3984.48	0.566726	0.283363	−	0.959013i	$-0.408550\pi$
$367$	3984.48	0.566726	0.283363	+	0.959013i	$0.408550\pi$
$368$	−7716.45	−1.09306
$369$	0	0
$370$	−246.894	−0.0346903
$371$	−200.297	−0.0280294
$372$	0	0
$373$	−11613.4	−1.61212	−0.806060	−	0.591834i	$-0.798404\pi$
$373$	−11613.4	−1.61212	−0.806060	+	0.591834i	$0.798404\pi$
$374$	−15909.6	−2.19965
$375$	0	0
$376$	1136.25	0.155845
$377$	796.238	0.108775
$378$	0	0
$379$	4634.02	0.628057	0.314029	−	0.949413i	$-0.398321\pi$
$379$	4634.02	0.628057	0.314029	+	0.949413i	$0.398321\pi$
$380$	64.7780	0.00874485
$381$	0	0
$382$	−3941.80	−0.527958
$383$	−4688.57	−0.625522	−0.312761	−	0.949832i	$-0.601254\pi$
$383$	−4688.57	−0.625522	−0.312761	+	0.949832i	$0.601254\pi$
$384$	0	0
$385$	2483.52	0.328758
$386$	10976.3	1.44736
$387$	0	0
$388$	−5043.44	−0.659902
$389$	5152.83	0.671617	0.335809	−	0.941930i	$-0.390990\pi$
$389$	5152.83	0.671617	0.335809	+	0.941930i	$0.390990\pi$
$390$	0	0
$391$	8004.11	1.03526
$392$	−1494.96	−0.192620
$393$	0	0
$394$	2546.45	0.325605
$395$	−2979.75	−0.379564
$396$	0	0
$397$	−4644.58	−0.587165	−0.293583	−	0.955934i	$-0.594848\pi$
$397$	−4644.58	−0.587165	−0.293583	+	0.955934i	$0.594848\pi$
$398$	13966.1	1.75894
$399$	0	0
$400$	6441.49	0.805187
$401$	12972.0	1.61543	0.807717	−	0.589570i	$-0.200703\pi$
$401$	12972.0	1.61543	0.807717	+	0.589570i	$0.200703\pi$
$402$	0	0
$403$	−1732.32	−0.214126
$404$	6601.09	0.812912
$405$	0	0
$406$	2684.87	0.328197
$407$	583.901	0.0711127
$408$	0	0
$409$	9623.25	1.16342	0.581710	−	0.813396i	$-0.302384\pi$
$409$	9623.25	1.16342	0.581710	+	0.813396i	$0.302384\pi$
$410$	−5722.38	−0.689288
$411$	0	0
$412$	−472.562	−0.0565084
$413$	−821.737	−0.0979057
$414$	0	0
$415$	4422.81	0.523150
$416$	−2012.80	−0.237225
$417$	0	0
$418$	−337.765	−0.0395231
$419$	8532.97	0.994900	0.497450	−	0.867493i	$-0.334270\pi$
$419$	8532.97	0.994900	0.497450	+	0.867493i	$0.334270\pi$
$420$	0	0
$421$	16176.5	1.87267	0.936333	−	0.351114i	$-0.114197\pi$
$421$	16176.5	1.87267	0.936333	+	0.351114i	$0.114197\pi$
$422$	845.480	0.0975292
$423$	0	0
$424$	−139.699	−0.0160009
$425$	−6681.62	−0.762603
$426$	0	0
$427$	−1631.19	−0.184868
$428$	5402.07	0.610091
$429$	0	0
$430$	−10553.2	−1.18353
$431$	9701.05	1.08418	0.542092	−	0.840319i	$-0.317633\pi$
$431$	9701.05	1.08418	0.542092	+	0.840319i	$0.317633\pi$
$432$	0	0
$433$	10276.0	1.14049	0.570245	−	0.821475i	$-0.306848\pi$
$433$	10276.0	1.14049	0.570245	+	0.821475i	$0.306848\pi$
$434$	−5841.28	−0.646061
$435$	0	0
$436$	1156.65	0.127049
$437$	169.929	0.0186014
$438$	0	0
$439$	13363.7	1.45288	0.726442	−	0.687228i	$-0.241173\pi$
$439$	13363.7	1.45288	0.726442	+	0.687228i	$0.241173\pi$
$440$	1732.16	0.187676
$441$	0	0
$442$	2453.22	0.264000
$443$	4763.11	0.510840	0.255420	−	0.966830i	$-0.417786\pi$
$443$	4763.11	0.510840	0.255420	+	0.966830i	$0.417786\pi$
$444$	0	0
$445$	−4104.34	−0.437224
$446$	330.399	0.0350781
$447$	0	0
$448$	−2429.32	−0.256193
$449$	2222.58	0.233608	0.116804	−	0.993155i	$-0.462735\pi$
$449$	2222.58	0.233608	0.116804	+	0.993155i	$0.462735\pi$
$450$	0	0
$451$	13533.3	1.41299
$452$	832.368	0.0866179
$453$	0	0
$454$	−2395.41	−0.247626
$455$	−382.953	−0.0394573
$456$	0	0
$457$	−7222.03	−0.739239	−0.369620	−	0.929183i	$-0.620512\pi$
$457$	−7222.03	−0.739239	−0.369620	+	0.929183i	$0.620512\pi$
$458$	20171.1	2.05793
$459$	0	0
$460$	4256.09	0.431394
$461$	18699.3	1.88919	0.944593	−	0.328244i	$-0.106457\pi$
$461$	18699.3	1.88919	0.944593	+	0.328244i	$0.106457\pi$
$462$	0	0
$463$	15573.6	1.56321	0.781607	−	0.623771i	$-0.214400\pi$
$463$	15573.6	1.56321	0.781607	+	0.623771i	$0.214400\pi$
$464$	6870.07	0.687360
$465$	0	0
$466$	−1751.17	−0.174080
$467$	−369.281	−0.0365916	−0.0182958	−	0.999833i	$-0.505824\pi$
$467$	−369.281	−0.0365916	−0.0182958	+	0.999833i	$0.505824\pi$
$468$	0	0
$469$	4033.00	0.397072
$470$	−5069.26	−0.497506
$471$	0	0
$472$	−573.129	−0.0558907
$473$	24958.0	2.42615
$474$	0	0
$475$	−141.853	−0.0137024
$476$	3751.96	0.361283
$477$	0	0
$478$	−914.479	−0.0875048
$479$	−1971.88	−0.188095	−0.0940474	−	0.995568i	$-0.529981\pi$
$479$	−1971.88	−0.188095	−0.0940474	+	0.995568i	$0.529981\pi$
$480$	0	0
$481$	−90.0359	−0.00853490
$482$	−504.548	−0.0476795
$483$	0	0
$484$	11168.8	1.04891
$485$	−4607.10	−0.431336
$486$	0	0
$487$	−5885.38	−0.547623	−0.273811	−	0.961783i	$-0.588284\pi$
$487$	−5885.38	−0.547623	−0.273811	+	0.961783i	$0.588284\pi$
$488$	−1137.69	−0.105534
$489$	0	0
$490$	6669.62	0.614904
$491$	−831.565	−0.0764318	−0.0382159	−	0.999270i	$-0.512167\pi$
$491$	−831.565	−0.0764318	−0.0382159	+	0.999270i	$0.512167\pi$
$492$	0	0
$493$	−7126.17	−0.651008
$494$	52.0826	0.00474353
$495$	0	0
$496$	−14946.7	−1.35308
$497$	5134.70	0.463426
$498$	0	0
$499$	9225.09	0.827599	0.413799	−	0.910368i	$-0.364202\pi$
$499$	9225.09	0.827599	0.413799	+	0.910368i	$0.364202\pi$
$500$	−8587.82	−0.768118
$501$	0	0
$502$	14880.6	1.32301
$503$	8514.93	0.754796	0.377398	−	0.926051i	$-0.376819\pi$
$503$	8514.93	0.754796	0.377398	+	0.926051i	$0.376819\pi$
$504$	0	0
$505$	6029.99	0.531349
$506$	−22192.1	−1.94972
$507$	0	0
$508$	−7476.20	−0.652958
$509$	−3164.21	−0.275543	−0.137771	−	0.990464i	$-0.543994\pi$
$509$	−3164.21	−0.275543	−0.137771	+	0.990464i	$0.543994\pi$
$510$	0	0
$511$	2426.01	0.210021
$512$	−14327.4	−1.23670
$513$	0	0
$514$	−11306.2	−0.970222
$515$	−431.678	−0.0369359
$516$	0	0
$517$	11988.7	1.01985
$518$	−303.596	−0.0257515
$519$	0	0
$520$	−267.094	−0.0225247
$521$	−19979.4	−1.68007	−0.840033	−	0.542535i	$-0.817465\pi$
$521$	−19979.4	−1.68007	−0.840033	+	0.542535i	$0.817465\pi$
$522$	0	0
$523$	−16269.9	−1.36029	−0.680146	−	0.733077i	$-0.738084\pi$
$523$	−16269.9	−1.36029	−0.680146	+	0.733077i	$0.738084\pi$
$524$	−6352.42	−0.529593
$525$	0	0
$526$	3535.37	0.293060
$527$	15503.9	1.28152
$528$	0	0
$529$	−1002.20	−0.0823707
$530$	623.255	0.0510801
$531$	0	0
$532$	79.6550	0.00649151
$533$	−2086.80	−0.169586
$534$	0	0
$535$	4934.71	0.398777
$536$	2812.86	0.226674
$537$	0	0
$538$	−26701.5	−2.13974
$539$	−15773.5	−1.26051
$540$	0	0
$541$	−4641.92	−0.368894	−0.184447	−	0.982842i	$-0.559049\pi$
$541$	−4641.92	−0.368894	−0.184447	+	0.982842i	$0.559049\pi$
$542$	−6457.94	−0.511794
$543$	0	0
$544$	18014.2	1.41976
$545$	1056.58	0.0830436
$546$	0	0
$547$	−12241.6	−0.956879	−0.478440	−	0.878120i	$-0.658797\pi$
$547$	−12241.6	−0.956879	−0.478440	+	0.878120i	$0.658797\pi$
$548$	−8606.78	−0.670919
$549$	0	0
$550$	18525.4	1.43623
$551$	−151.291	−0.0116973
$552$	0	0
$553$	−3664.09	−0.281759
$554$	15281.9	1.17196
$555$	0	0
$556$	8200.27	0.625484
$557$	−4777.98	−0.363464	−0.181732	−	0.983348i	$-0.558170\pi$
$557$	−4777.98	−0.363464	−0.181732	+	0.983348i	$0.558170\pi$
$558$	0	0
$559$	−3848.46	−0.291185
$560$	−3304.18	−0.249334
$561$	0	0
$562$	17263.9	1.29579
$563$	6102.07	0.456788	0.228394	−	0.973569i	$-0.426653\pi$
$563$	6102.07	0.456788	0.228394	+	0.973569i	$0.426653\pi$
$564$	0	0
$565$	760.355	0.0566166
$566$	3647.10	0.270846
$567$	0	0
$568$	3581.25	0.264553
$569$	−13128.8	−0.967286	−0.483643	−	0.875265i	$-0.660687\pi$
$569$	−13128.8	−0.967286	−0.483643	+	0.875265i	$0.660687\pi$
$570$	0	0
$571$	1553.98	0.113891	0.0569456	−	0.998377i	$-0.481864\pi$
$571$	1553.98	0.113891	0.0569456	+	0.998377i	$0.481864\pi$
$572$	−3085.05	−0.225511
$573$	0	0
$574$	−7036.58	−0.511675
$575$	−9320.08	−0.675955
$576$	0	0
$577$	−16166.4	−1.16641	−0.583203	−	0.812326i	$-0.698201\pi$
$577$	−16166.4	−1.16641	−0.583203	+	0.812326i	$0.698201\pi$
$578$	−3157.40	−0.227215
$579$	0	0
$580$	−3789.26	−0.271276
$581$	5438.55	0.388346
$582$	0	0
$583$	−1473.99	−0.104711
$584$	1692.05	0.119893
$585$	0	0
$586$	−21218.4	−1.49578
$587$	13374.5	0.940420	0.470210	−	0.882555i	$-0.344178\pi$
$587$	13374.5	0.940420	0.470210	+	0.882555i	$0.344178\pi$
$588$	0	0
$589$	329.152	0.0230263
$590$	2556.96	0.178421
$591$	0	0
$592$	−776.845	−0.0539327
$593$	23813.0	1.64905	0.824523	−	0.565828i	$-0.191443\pi$
$593$	23813.0	1.64905	0.824523	+	0.565828i	$0.191443\pi$
$594$	0	0
$595$	3427.35	0.236147
$596$	−15873.2	−1.09093
$597$	0	0
$598$	3421.96	0.234004
$599$	11102.3	0.757307	0.378653	−	0.925539i	$-0.376387\pi$
$599$	11102.3	0.757307	0.378653	+	0.925539i	$0.376387\pi$
$600$	0	0
$601$	−1742.18	−0.118244	−0.0591222	−	0.998251i	$-0.518830\pi$
$601$	−1742.18	−0.118244	−0.0591222	+	0.998251i	$0.518830\pi$
$602$	−12976.8	−0.878563
$603$	0	0
$604$	13265.7	0.893665
$605$	10202.5	0.685607
$606$	0	0
$607$	27727.3	1.85406	0.927032	−	0.374982i	$-0.122351\pi$
$607$	27727.3	1.85406	0.927032	+	0.374982i	$0.122351\pi$
$608$	382.445	0.0255102
$609$	0	0
$610$	5075.69	0.336899
$611$	−1848.63	−0.122402
$612$	0	0
$613$	863.200	0.0568749	0.0284374	−	0.999596i	$-0.490947\pi$
$613$	863.200	0.0568749	0.0284374	+	0.999596i	$0.490947\pi$
$614$	2558.91	0.168191
$615$	0	0
$616$	2129.97	0.139316
$617$	4548.83	0.296806	0.148403	−	0.988927i	$-0.452587\pi$
$617$	4548.83	0.296806	0.148403	+	0.988927i	$0.452587\pi$
$618$	0	0
$619$	−18511.5	−1.20200	−0.601000	−	0.799249i	$-0.705231\pi$
$619$	−18511.5	−1.20200	−0.601000	+	0.799249i	$0.705231\pi$
$620$	8244.02	0.534013
$621$	0	0
$622$	4250.02	0.273972
$623$	−5046.95	−0.324561
$624$	0	0
$625$	3180.82	0.203573
$626$	2791.20	0.178209
$627$	0	0
$628$	−10942.0	−0.695274
$629$	805.804	0.0510803
$630$	0	0
$631$	−16755.2	−1.05707	−0.528537	−	0.848910i	$-0.677259\pi$
$631$	−16755.2	−1.05707	−0.528537	+	0.848910i	$0.677259\pi$
$632$	−2555.56	−0.160846
$633$	0	0
$634$	21690.4	1.35873
$635$	−6829.39	−0.426797
$636$	0	0
$637$	2432.24	0.151285
$638$	19757.9	1.22606
$639$	0	0
$640$	−3980.91	−0.245874
$641$	30888.3	1.90330	0.951650	−	0.307186i	$-0.0993873\pi$
$641$	30888.3	1.90330	0.951650	+	0.307186i	$0.0993873\pi$
$642$	0	0
$643$	27083.0	1.66104	0.830520	−	0.556989i	$-0.188043\pi$
$643$	27083.0	1.66104	0.830520	+	0.556989i	$0.188043\pi$
$644$	5233.54	0.320234
$645$	0	0
$646$	−466.129	−0.0283895
$647$	−2241.89	−0.136225	−0.0681125	−	0.997678i	$-0.521698\pi$
$647$	−2241.89	−0.136225	−0.0681125	+	0.997678i	$0.521698\pi$
$648$	0	0
$649$	−6047.16	−0.365750
$650$	−2856.57	−0.172375
$651$	0	0
$652$	17276.7	1.03774
$653$	−6333.09	−0.379530	−0.189765	−	0.981830i	$-0.560773\pi$
$653$	−6333.09	−0.379530	−0.189765	+	0.981830i	$0.560773\pi$
$654$	0	0
$655$	−5802.83	−0.346161
$656$	−18005.3	−1.07163
$657$	0	0
$658$	−6233.47	−0.369310
$659$	12904.2	0.762787	0.381394	−	0.924413i	$-0.375444\pi$
$659$	12904.2	0.762787	0.381394	+	0.924413i	$0.375444\pi$
$660$	0	0
$661$	−14993.5	−0.882268	−0.441134	−	0.897441i	$-0.645424\pi$
$661$	−14993.5	−0.882268	−0.441134	+	0.897441i	$0.645424\pi$
$662$	12881.9	0.756299
$663$	0	0
$664$	3793.17	0.221692
$665$	72.7635	0.00424308
$666$	0	0
$667$	−9940.19	−0.577040
$668$	7067.97	0.409383
$669$	0	0
$670$	−12549.3	−0.723614
$671$	−12003.9	−0.690620
$672$	0	0
$673$	20326.6	1.16424	0.582119	−	0.813104i	$-0.302224\pi$
$673$	20326.6	1.16424	0.582119	+	0.813104i	$0.302224\pi$
$674$	−27022.3	−1.54430
$675$	0	0
$676$	−14113.2	−0.802980
$677$	2353.44	0.133604	0.0668021	−	0.997766i	$-0.478720\pi$
$677$	2353.44	0.133604	0.0668021	+	0.997766i	$0.478720\pi$
$678$	0	0
$679$	−5665.18	−0.320191
$680$	2390.44	0.134808
$681$	0	0
$682$	−42986.0	−2.41352
$683$	10276.1	0.575700	0.287850	−	0.957676i	$-0.407060\pi$
$683$	10276.1	0.575700	0.287850	+	0.957676i	$0.407060\pi$
$684$	0	0
$685$	−7862.16	−0.438537
$686$	17990.6	1.00129
$687$	0	0
$688$	−33205.1	−1.84002
$689$	227.285	0.0125673
$690$	0	0
$691$	−19391.7	−1.06757	−0.533787	−	0.845619i	$-0.679232\pi$
$691$	−19391.7	−1.06757	−0.533787	+	0.845619i	$0.679232\pi$
$692$	−13292.8	−0.730227
$693$	0	0
$694$	−44974.4	−2.45995
$695$	7490.82	0.408839
$696$	0	0
$697$	18676.5	1.01495
$698$	−8267.30	−0.448312
$699$	0	0
$700$	−4368.83	−0.235894
$701$	7729.11	0.416440	0.208220	−	0.978082i	$-0.433233\pi$
$701$	7729.11	0.416440	0.208220	+	0.978082i	$0.433233\pi$
$702$	0	0
$703$	17.1074	0.000917808 0
$704$	−17877.4	−0.957071
$705$	0	0
$706$	19063.5	1.01624
$707$	7414.84	0.394433
$708$	0	0
$709$	−31851.4	−1.68717	−0.843585	−	0.536995i	$-0.819559\pi$
$709$	−31851.4	−1.68717	−0.843585	+	0.536995i	$0.819559\pi$
$710$	−15977.4	−0.844536
$711$	0	0
$712$	−3520.05	−0.185280
$713$	21626.2	1.13591
$714$	0	0
$715$	−2818.15	−0.147402
$716$	−12743.0	−0.665125
$717$	0	0
$718$	−14413.5	−0.749173
$719$	−14976.4	−0.776809	−0.388405	−	0.921489i	$-0.626974\pi$
$719$	−14976.4	−0.776809	−0.388405	+	0.921489i	$0.626974\pi$
$720$	0	0
$721$	−530.817	−0.0274184
$722$	26234.5	1.35228
$723$	0	0
$724$	6757.05	0.346856
$725$	8297.81	0.425066
$726$	0	0
$727$	15639.6	0.797855	0.398928	−	0.916982i	$-0.369382\pi$
$727$	15639.6	0.797855	0.398928	+	0.916982i	$0.369382\pi$
$728$	−328.435	−0.0167206
$729$	0	0
$730$	−7548.91	−0.382736
$731$	34443.0	1.74271
$732$	0	0
$733$	−33883.5	−1.70739	−0.853695	−	0.520774i	$-0.825644\pi$
$733$	−33883.5	−1.70739	−0.853695	+	0.520774i	$0.825644\pi$
$734$	−15245.7	−0.766662
$735$	0	0
$736$	25127.7	1.25845
$737$	29678.9	1.48336
$738$	0	0
$739$	−21564.3	−1.07342	−0.536708	−	0.843768i	$-0.680332\pi$
$739$	−21564.3	−1.07342	−0.536708	+	0.843768i	$0.680332\pi$
$740$	428.477	0.0212853
$741$	0	0
$742$	766.392	0.0379180
$743$	13389.0	0.661095	0.330548	−	0.943789i	$-0.392767\pi$
$743$	13389.0	0.661095	0.330548	+	0.943789i	$0.392767\pi$
$744$	0	0
$745$	−14500.0	−0.713070
$746$	44436.2	2.18086
$747$	0	0
$748$	27610.6	1.34966
$749$	6068.01	0.296022
$750$	0	0
$751$	18354.6	0.891836	0.445918	−	0.895074i	$-0.352877\pi$
$751$	18354.6	0.891836	0.445918	+	0.895074i	$0.352877\pi$
$752$	−15950.3	−0.773466
$753$	0	0
$754$	−3046.62	−0.147151
$755$	12118.0	0.584131
$756$	0	0
$757$	−27413.6	−1.31620	−0.658100	−	0.752930i	$-0.728640\pi$
$757$	−27413.6	−1.31620	−0.658100	+	0.752930i	$0.728640\pi$
$758$	−17731.0	−0.849631
$759$	0	0
$760$	50.7497	0.00242222
$761$	12956.8	0.617191	0.308595	−	0.951193i	$-0.400141\pi$
$761$	12956.8	0.617191	0.308595	+	0.951193i	$0.400141\pi$
$762$	0	0
$763$	1299.23	0.0616453
$764$	6840.86	0.323945
$765$	0	0
$766$	17939.8	0.846201
$767$	932.456	0.0438971
$768$	0	0
$769$	12896.9	0.604778	0.302389	−	0.953185i	$-0.402216\pi$
$769$	12896.9	0.604778	0.302389	+	0.953185i	$0.402216\pi$
$770$	−9502.63	−0.444742
$771$	0	0
$772$	−19049.1	−0.888070
$773$	−24265.7	−1.12908	−0.564538	−	0.825407i	$-0.690946\pi$
$773$	−24265.7	−1.12908	−0.564538	+	0.825407i	$0.690946\pi$
$774$	0	0
$775$	−18053.0	−0.836750
$776$	−3951.23	−0.182785
$777$	0	0
$778$	−19716.2	−0.908558
$779$	396.506	0.0182366
$780$	0	0
$781$	37786.2	1.73124
$782$	−30625.9	−1.40049
$783$	0	0
$784$	20985.7	0.955983
$785$	−9995.31	−0.454456
$786$	0	0
$787$	−27772.5	−1.25792	−0.628959	−	0.777438i	$-0.716519\pi$
$787$	−27772.5	−1.25792	−0.628959	+	0.777438i	$0.716519\pi$
$788$	−4419.28	−0.199785
$789$	0	0
$790$	11401.4	0.513471
$791$	934.978	0.0420278
$792$	0	0
$793$	1850.97	0.0828877
$794$	17771.4	0.794312
$795$	0	0
$796$	−24237.7	−1.07925
$797$	−7038.92	−0.312838	−0.156419	−	0.987691i	$-0.549995\pi$
$797$	−7038.92	−0.312838	−0.156419	+	0.987691i	$0.549995\pi$
$798$	0	0
$799$	16544.9	0.732560
$800$	−20975.9	−0.927014
$801$	0	0
$802$	−49634.3	−2.18535
$803$	17853.0	0.784582
$804$	0	0
$805$	4780.76	0.209316
$806$	6628.32	0.289668
$807$	0	0
$808$	5171.56	0.225167
$809$	12094.8	0.525625	0.262812	−	0.964847i	$-0.415350\pi$
$809$	12094.8	0.525625	0.262812	+	0.964847i	$0.415350\pi$
$810$	0	0
$811$	−41604.8	−1.80141	−0.900705	−	0.434431i	$-0.856949\pi$
$811$	−41604.8	−1.80141	−0.900705	+	0.434431i	$0.856949\pi$
$812$	−4659.50	−0.201375
$813$	0	0
$814$	−2234.16	−0.0962007
$815$	15782.0	0.678305
$816$	0	0
$817$	731.233	0.0313129
$818$	−36821.2	−1.57387
$819$	0	0
$820$	9930.99	0.422933
$821$	1556.82	0.0661796	0.0330898	−	0.999452i	$-0.489465\pi$
$821$	1556.82	0.0661796	0.0330898	+	0.999452i	$0.489465\pi$
$822$	0	0
$823$	−35472.9	−1.50244	−0.751221	−	0.660051i	$-0.770535\pi$
$823$	−35472.9	−1.50244	−0.751221	+	0.660051i	$0.770535\pi$
$824$	−370.224	−0.0156521
$825$	0	0
$826$	3144.19	0.132446
$827$	36998.7	1.55571	0.777854	−	0.628446i	$-0.216308\pi$
$827$	36998.7	1.55571	0.777854	+	0.628446i	$0.216308\pi$
$828$	0	0
$829$	19883.8	0.833045	0.416523	−	0.909125i	$-0.363249\pi$
$829$	19883.8	0.833045	0.416523	+	0.909125i	$0.363249\pi$
$830$	−16922.9	−0.707713
$831$	0	0
$832$	2756.64	0.114867
$833$	−21768.1	−0.905424
$834$	0	0
$835$	6456.47	0.267587
$836$	586.180	0.0242506
$837$	0	0
$838$	−32649.5	−1.34589
$839$	1979.00	0.0814334	0.0407167	−	0.999171i	$-0.487036\pi$
$839$	1979.00	0.0814334	0.0407167	+	0.999171i	$0.487036\pi$
$840$	0	0
$841$	−15539.1	−0.637136
$842$	−61895.5	−2.53333
$843$	0	0
$844$	−1467.30	−0.0598419
$845$	−12892.2	−0.524857
$846$	0	0
$847$	12545.7	0.508943
$848$	1961.05	0.0794136
$849$	0	0
$850$	25565.7	1.03164
$851$	1124.00	0.0452765
$852$	0	0
$853$	−15039.8	−0.603696	−0.301848	−	0.953356i	$-0.597603\pi$
$853$	−15039.8	−0.603696	−0.301848	+	0.953356i	$0.597603\pi$
$854$	6241.38	0.250088
$855$	0	0
$856$	4232.20	0.168988
$857$	−18738.8	−0.746915	−0.373457	−	0.927647i	$-0.621828\pi$
$857$	−18738.8	−0.746915	−0.373457	+	0.927647i	$0.621828\pi$
$858$	0	0
$859$	35255.5	1.40035	0.700175	−	0.713971i	$-0.253105\pi$
$859$	35255.5	1.40035	0.700175	+	0.713971i	$0.253105\pi$
$860$	18314.6	0.726190
$861$	0	0
$862$	−37118.9	−1.46667
$863$	−13110.9	−0.517151	−0.258575	−	0.965991i	$-0.583253\pi$
$863$	−13110.9	−0.517151	−0.258575	+	0.965991i	$0.583253\pi$
$864$	0	0
$865$	−12142.8	−0.477303
$866$	−39318.7	−1.54284
$867$	0	0
$868$	10137.3	0.396410
$869$	−26964.0	−1.05258
$870$	0	0
$871$	−4576.40	−0.178032
$872$	906.162	0.0351910
$873$	0	0
$874$	−650.195	−0.0251638
$875$	−9646.49	−0.372698
$876$	0	0
$877$	39974.1	1.53915	0.769573	−	0.638559i	$-0.220469\pi$
$877$	39974.1	1.53915	0.769573	+	0.638559i	$0.220469\pi$
$878$	−51133.3	−1.96545
$879$	0	0
$880$	−24315.4	−0.931446
$881$	−26773.1	−1.02385	−0.511924	−	0.859031i	$-0.671067\pi$
$881$	−26773.1	−1.02385	−0.511924	+	0.859031i	$0.671067\pi$
$882$	0	0
$883$	−23377.4	−0.890953	−0.445476	−	0.895294i	$-0.646966\pi$
$883$	−23377.4	−0.890953	−0.445476	+	0.895294i	$0.646966\pi$
$884$	−4257.49	−0.161985
$885$	0	0
$886$	−18224.9	−0.691060
$887$	−24665.0	−0.933676	−0.466838	−	0.884343i	$-0.654607\pi$
$887$	−24665.0	−0.933676	−0.466838	+	0.884343i	$0.654607\pi$
$888$	0	0
$889$	−8397.83	−0.316821
$890$	15704.3	0.591472
$891$	0	0
$892$	−573.396	−0.0215232
$893$	351.252	0.0131626
$894$	0	0
$895$	−11640.5	−0.434749
$896$	−4895.17	−0.182518
$897$	0	0
$898$	−8504.20	−0.316023
$899$	−19254.1	−0.714305
$900$	0	0
$901$	−2034.16	−0.0752137
$902$	−51782.1	−1.91148
$903$	0	0
$904$	652.110	0.0239921
$905$	6172.46	0.226718
$906$	0	0
$907$	50243.6	1.83937	0.919686	−	0.392655i	$-0.128443\pi$
$907$	50243.6	1.83937	0.919686	+	0.392655i	$0.128443\pi$
$908$	4157.15	0.151938
$909$	0	0
$910$	1465.28	0.0533776
$911$	13634.7	0.495870	0.247935	−	0.968777i	$-0.420248\pi$
$911$	13634.7	0.495870	0.247935	+	0.968777i	$0.420248\pi$
$912$	0	0
$913$	40022.3	1.45076
$914$	27633.5	1.00004
$915$	0	0
$916$	−35006.2	−1.26270
$917$	−7135.52	−0.256964
$918$	0	0
$919$	−42623.3	−1.52994	−0.764968	−	0.644068i	$-0.777245\pi$
$919$	−42623.3	−1.52994	−0.764968	+	0.644068i	$0.777245\pi$
$920$	3334.39	0.119491
$921$	0	0
$922$	−71548.8	−2.55568
$923$	−5826.54	−0.207782
$924$	0	0
$925$	−938.289	−0.0333522
$926$	−59589.0	−2.11470
$927$	0	0
$928$	−22371.5	−0.791360
$929$	−14828.0	−0.523670	−0.261835	−	0.965113i	$-0.584328\pi$
$929$	−14828.0	−0.523670	−0.261835	+	0.965113i	$0.584328\pi$
$930$	0	0
$931$	−462.141	−0.0162686
$932$	3039.10	0.106812
$933$	0	0
$934$	1412.97	0.0495008
$935$	25221.9	0.882185
$936$	0	0
$937$	50887.4	1.77419	0.887097	−	0.461583i	$-0.152718\pi$
$937$	50887.4	1.77419	0.887097	+	0.461583i	$0.152718\pi$
$938$	−15431.4	−0.537156
$939$	0	0
$940$	8797.53	0.305259
$941$	−21952.9	−0.760513	−0.380256	−	0.924881i	$-0.624164\pi$
$941$	−21952.9	−0.760513	−0.380256	+	0.924881i	$0.624164\pi$
$942$	0	0
$943$	26051.5	0.899633
$944$	8045.39	0.277389
$945$	0	0
$946$	−95496.1	−3.28208
$947$	−2982.20	−0.102332	−0.0511660	−	0.998690i	$-0.516294\pi$
$947$	−2982.20	−0.102332	−0.0511660	+	0.998690i	$0.516294\pi$
$948$	0	0
$949$	−2752.89	−0.0941650
$950$	542.766	0.0185365
$951$	0	0
$952$	2939.43	0.100071
$953$	−42394.2	−1.44101	−0.720505	−	0.693450i	$-0.756090\pi$
$953$	−42394.2	−1.44101	−0.720505	+	0.693450i	$0.756090\pi$
$954$	0	0
$955$	6249.02	0.211742
$956$	1587.05	0.0536912
$957$	0	0
$958$	7544.95	0.254453
$959$	−9667.79	−0.325536
$960$	0	0
$961$	12098.8	0.406121
$962$	344.502	0.0115459
$963$	0	0
$964$	875.625	0.0292552
$965$	−17401.0	−0.580475
$966$	0	0
$967$	11752.3	0.390827	0.195414	−	0.980721i	$-0.437395\pi$
$967$	11752.3	0.390827	0.195414	+	0.980721i	$0.437395\pi$
$968$	8750.11	0.290536
$969$	0	0
$970$	17628.0	0.583507
$971$	36922.8	1.22030	0.610149	−	0.792287i	$-0.291110\pi$
$971$	36922.8	1.22030	0.610149	+	0.792287i	$0.291110\pi$
$972$	0	0
$973$	9211.16	0.303491
$974$	22519.1	0.740820
$975$	0	0
$976$	15970.5	0.523773
$977$	−43565.9	−1.42661	−0.713304	−	0.700855i	$-0.752802\pi$
$977$	−43565.9	−1.42661	−0.713304	+	0.700855i	$0.752802\pi$
$978$	0	0
$979$	−37140.5	−1.21248
$980$	−11574.9	−0.377292
$981$	0	0
$982$	3181.79	0.103396
$983$	−25824.6	−0.837922	−0.418961	−	0.908004i	$-0.637606\pi$
$983$	−25824.6	−0.837922	−0.418961	+	0.908004i	$0.637606\pi$
$984$	0	0
$985$	−4036.94	−0.130586
$986$	27266.7	0.880678
$987$	0	0
$988$	−90.3875	−0.00291054
$989$	48043.9	1.54470
$990$	0	0
$991$	25511.5	0.817759	0.408880	−	0.912588i	$-0.365920\pi$
$991$	25511.5	0.817759	0.408880	+	0.912588i	$0.365920\pi$
$992$	48672.2	1.55781
$993$	0	0
$994$	−19646.8	−0.626919
$995$	−22140.7	−0.705435
$996$	0	0
$997$	28831.0	0.915834	0.457917	−	0.888995i	$-0.348596\pi$
$997$	28831.0	0.915834	0.457917	+	0.888995i	$0.348596\pi$
$998$	−35297.7	−1.11957
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.c.1.5		28
3.2	odd	2		717.4.a.a.1.24	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.a.1.24	✓	28	3.2	odd	2
2151.4.a.c.1.5		28	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)

\(f(q)\)	\(=\)	\(q-3.82627 q^{2} +6.64037 q^{4} +6.06587 q^{5} +7.45896 q^{7} +5.20233 q^{8} +O(q^{10})\)	\(q-3.82627 q^{2} +6.64037 q^{4} +6.06587 q^{5} +7.45896 q^{7} +5.20233 q^{8} -23.2097 q^{10} +54.8904 q^{11} -8.46396 q^{13} -28.5400 q^{14} -73.0285 q^{16} +75.7508 q^{17} +1.60821 q^{19} +40.2796 q^{20} -210.026 q^{22} +105.664 q^{23} -88.2052 q^{25} +32.3854 q^{26} +49.5302 q^{28} -94.0739 q^{29} +204.670 q^{31} +237.808 q^{32} -289.843 q^{34} +45.2451 q^{35} +10.6376 q^{37} -6.15345 q^{38} +31.5566 q^{40} +246.551 q^{41} +454.688 q^{43} +364.493 q^{44} -404.298 q^{46} +218.412 q^{47} -287.364 q^{49} +337.497 q^{50} -56.2038 q^{52} -26.8532 q^{53} +332.958 q^{55} +38.8040 q^{56} +359.952 q^{58} -110.168 q^{59} -218.689 q^{61} -783.123 q^{62} -325.692 q^{64} -51.3413 q^{65} +540.693 q^{67} +503.013 q^{68} -173.120 q^{70} +688.394 q^{71} +325.248 q^{73} -40.7022 q^{74} +10.6791 q^{76} +409.426 q^{77} -491.233 q^{79} -442.981 q^{80} -943.373 q^{82} +729.130 q^{83} +459.495 q^{85} -1739.76 q^{86} +285.558 q^{88} -676.629 q^{89} -63.1324 q^{91} +701.645 q^{92} -835.703 q^{94} +9.75519 q^{95} -759.513 q^{97} +1099.53 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8}+O(q^{10}) \) 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8	\( 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8} - 68 q^{10} + 258 q^{11} - 134 q^{13} + 292 q^{14} + 327 q^{16} + 364 q^{17} - 278 q^{19} + 986 q^{20} - 179 q^{22} + 668 q^{23} + 490 q^{25} + 760 q^{26} - 802 q^{28} + 714 q^{29} - 608 q^{31} + 918 q^{32} - 228 q^{34} + 934 q^{35} - 1080 q^{37} + 1395 q^{38} - 563 q^{40} + 1796 q^{41} - 1934 q^{43} + 3157 q^{44} - 940 q^{46} + 2032 q^{47} + 762 q^{49} + 1754 q^{50} - 2328 q^{52} + 1790 q^{53} - 478 q^{55} + 3557 q^{56} - 2626 q^{58} + 3622 q^{59} + 324 q^{61} + 796 q^{62} + 2023 q^{64} + 2200 q^{65} - 2444 q^{67} - 357 q^{68} + 4305 q^{70} + 1298 q^{71} - 1368 q^{73} - 813 q^{74} + 1390 q^{76} + 1408 q^{77} - 1378 q^{79} + 7684 q^{80} + 9001 q^{82} + 3524 q^{83} + 60 q^{85} + 2543 q^{86} + 1749 q^{88} + 7854 q^{89} + 850 q^{91} + 496 q^{92} + 6634 q^{94} + 3696 q^{95} - 1746 q^{97} + 4632 q^{98}+O(q^{100}) \) 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8 - 68 * q^10 + 258 * q^11 - 134 * q^13 + 292 * q^14 + 327 * q^16 + 364 * q^17 - 278 * q^19 + 986 * q^20 - 179 * q^22 + 668 * q^23 + 490 * q^25 + 760 * q^26 - 802 * q^28 + 714 * q^29 - 608 * q^31 + 918 * q^32 - 228 * q^34 + 934 * q^35 - 1080 * q^37 + 1395 * q^38 - 563 * q^40 + 1796 * q^41 - 1934 * q^43 + 3157 * q^44 - 940 * q^46 + 2032 * q^47 + 762 * q^49 + 1754 * q^50 - 2328 * q^52 + 1790 * q^53 - 478 * q^55 + 3557 * q^56 - 2626 * q^58 + 3622 * q^59 + 324 * q^61 + 796 * q^62 + 2023 * q^64 + 2200 * q^65 - 2444 * q^67 - 357 * q^68 + 4305 * q^70 + 1298 * q^71 - 1368 * q^73 - 813 * q^74 + 1390 * q^76 + 1408 * q^77 - 1378 * q^79 + 7684 * q^80 + 9001 * q^82 + 3524 * q^83 + 60 * q^85 + 2543 * q^86 + 1749 * q^88 + 7854 * q^89 + 850 * q^91 + 496 * q^92 + 6634 * q^94 + 3696 * q^95 - 1746 * q^97 + 4632 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more