LMFDB - Embedded newform 2151.4.a.e.1.4

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:	$32$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-4.79937 q^{2} +15.0339 q^{4} -7.96880 q^{5} -5.93046 q^{7} -33.7584 q^{8} +O(q^{10})$	\(q-4.79937 q^{2} +15.0339 q^{4} -7.96880 q^{5} -5.93046 q^{7} -33.7584 q^{8} +38.2452 q^{10} -14.7981 q^{11} +19.5964 q^{13} +28.4624 q^{14} +41.7474 q^{16} -50.1371 q^{17} +139.829 q^{19} -119.802 q^{20} +71.0214 q^{22} +130.735 q^{23} -61.4982 q^{25} -94.0505 q^{26} -89.1580 q^{28} -204.489 q^{29} +141.379 q^{31} +69.7060 q^{32} +240.626 q^{34} +47.2586 q^{35} +300.801 q^{37} -671.091 q^{38} +269.014 q^{40} +7.52727 q^{41} +26.6198 q^{43} -222.473 q^{44} -627.447 q^{46} -495.608 q^{47} -307.830 q^{49} +295.152 q^{50} +294.611 q^{52} +430.509 q^{53} +117.923 q^{55} +200.202 q^{56} +981.418 q^{58} -200.816 q^{59} +269.379 q^{61} -678.531 q^{62} -668.523 q^{64} -156.160 q^{65} +162.214 q^{67} -753.757 q^{68} -226.812 q^{70} -639.615 q^{71} -312.142 q^{73} -1443.65 q^{74} +2102.18 q^{76} +87.7593 q^{77} -396.840 q^{79} -332.676 q^{80} -36.1261 q^{82} +451.541 q^{83} +399.532 q^{85} -127.758 q^{86} +499.558 q^{88} +561.912 q^{89} -116.216 q^{91} +1965.47 q^{92} +2378.61 q^{94} -1114.27 q^{95} -1458.44 q^{97} +1477.39 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8}+O(q^{10}) $ 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8	$ 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8} + 32 q^{10} - 154 q^{11} + 100 q^{13} - 42 q^{14} + 719 q^{16} - 32 q^{17} + 202 q^{19} + 132 q^{20} + 265 q^{22} - 552 q^{23} + 1086 q^{25} + 280 q^{26} + 390 q^{28} + 154 q^{29} + 560 q^{31} - 444 q^{32} + 156 q^{34} - 394 q^{35} + 914 q^{37} - 111 q^{38} + 257 q^{40} + 914 q^{41} + 1722 q^{43} - 1243 q^{44} + 584 q^{46} - 380 q^{47} + 2446 q^{49} + 454 q^{50} + 1552 q^{52} - 370 q^{53} + 918 q^{55} + 499 q^{56} + 2446 q^{58} - 492 q^{59} + 668 q^{61} - 578 q^{62} + 6475 q^{64} - 736 q^{65} + 4548 q^{67} - 5253 q^{68} + 7793 q^{70} - 258 q^{71} + 3096 q^{73} - 449 q^{74} + 6814 q^{76} - 3804 q^{77} + 2864 q^{79} + 1052 q^{80} + 14145 q^{82} - 2364 q^{83} + 3088 q^{85} - 2811 q^{86} + 8329 q^{88} + 4172 q^{89} + 7350 q^{91} - 13644 q^{92} + 6122 q^{94} - 3336 q^{95} + 6370 q^{97} - 1572 q^{98}+O(q^{100}) $ 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8 + 32 * q^10 - 154 * q^11 + 100 * q^13 - 42 * q^14 + 719 * q^16 - 32 * q^17 + 202 * q^19 + 132 * q^20 + 265 * q^22 - 552 * q^23 + 1086 * q^25 + 280 * q^26 + 390 * q^28 + 154 * q^29 + 560 * q^31 - 444 * q^32 + 156 * q^34 - 394 * q^35 + 914 * q^37 - 111 * q^38 + 257 * q^40 + 914 * q^41 + 1722 * q^43 - 1243 * q^44 + 584 * q^46 - 380 * q^47 + 2446 * q^49 + 454 * q^50 + 1552 * q^52 - 370 * q^53 + 918 * q^55 + 499 * q^56 + 2446 * q^58 - 492 * q^59 + 668 * q^61 - 578 * q^62 + 6475 * q^64 - 736 * q^65 + 4548 * q^67 - 5253 * q^68 + 7793 * q^70 - 258 * q^71 + 3096 * q^73 - 449 * q^74 + 6814 * q^76 - 3804 * q^77 + 2864 * q^79 + 1052 * q^80 + 14145 * q^82 - 2364 * q^83 + 3088 * q^85 - 2811 * q^86 + 8329 * q^88 + 4172 * q^89 + 7350 * q^91 - 13644 * q^92 + 6122 * q^94 - 3336 * q^95 + 6370 * q^97 - 1572 * 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 $

$2$	−4.79937	−1.69683	−0.848416	−	0.529330i	$-0.822443\pi$
$2$	−4.79937	−1.69683	−0.848416	+	0.529330i	$0.822443\pi$
$3$	0	0
$3$	0	0
$4$	15.0339	1.87924
$5$	−7.96880	−0.712751	−0.356376	−	0.934343i	$-0.615988\pi$
$5$	−7.96880	−0.712751	−0.356376	+	0.934343i	$0.615988\pi$
$6$	0	0
$7$	−5.93046	−0.320215	−0.160107	−	0.987100i	$-0.551184\pi$
$7$	−5.93046	−0.320215	−0.160107	+	0.987100i	$0.551184\pi$
$8$	−33.7584	−1.49192
$9$	0	0
$10$	38.2452	1.20942
$11$	−14.7981	−0.405617	−0.202808	−	0.979218i	$-0.565007\pi$
$11$	−14.7981	−0.405617	−0.202808	+	0.979218i	$0.565007\pi$
$12$	0	0
$13$	19.5964	0.418083	0.209041	−	0.977907i	$-0.432966\pi$
$13$	19.5964	0.418083	0.209041	+	0.977907i	$0.432966\pi$
$14$	28.4624	0.543351
$15$	0	0
$16$	41.7474	0.652302
$17$	−50.1371	−0.715296	−0.357648	−	0.933856i	$-0.616421\pi$
$17$	−50.1371	−0.715296	−0.357648	+	0.933856i	$0.616421\pi$
$18$	0	0
$19$	139.829	1.68837	0.844183	−	0.536054i	$-0.180086\pi$
$19$	139.829	1.68837	0.844183	+	0.536054i	$0.180086\pi$
$20$	−119.802	−1.33943
$21$	0	0
$22$	71.0214	0.688264
$23$	130.735	1.18523	0.592614	−	0.805487i	$-0.298096\pi$
$23$	130.735	1.18523	0.592614	+	0.805487i	$0.298096\pi$
$24$	0	0
$25$	−61.4982	−0.491986
$26$	−94.0505	−0.709416
$27$	0	0
$28$	−89.1580	−0.601760
$29$	−204.489	−1.30940	−0.654701	−	0.755888i	$-0.727205\pi$
$29$	−204.489	−1.30940	−0.654701	+	0.755888i	$0.727205\pi$
$30$	0	0
$31$	141.379	0.819112	0.409556	−	0.912285i	$-0.365684\pi$
$31$	141.379	0.819112	0.409556	+	0.912285i	$0.365684\pi$
$32$	69.7060	0.385075
$33$	0	0
$34$	240.626	1.21374
$35$	47.2586	0.228233
$36$	0	0
$37$	300.801	1.33652	0.668262	−	0.743926i	$-0.267039\pi$
$37$	300.801	1.33652	0.668262	+	0.743926i	$0.267039\pi$
$38$	−671.091	−2.86488
$39$	0	0
$40$	269.014	1.06337
$41$	7.52727	0.0286723	0.0143361	−	0.999897i	$-0.495437\pi$
$41$	7.52727	0.0286723	0.0143361	+	0.999897i	$0.495437\pi$
$42$	0	0
$43$	26.6198	0.0944065	0.0472032	−	0.998885i	$-0.484969\pi$
$43$	26.6198	0.0944065	0.0472032	+	0.998885i	$0.484969\pi$
$44$	−222.473	−0.762251
$45$	0	0
$46$	−627.447	−2.01113
$47$	−495.608	−1.53813	−0.769063	−	0.639173i	$-0.779277\pi$
$47$	−495.608	−1.53813	−0.769063	+	0.639173i	$0.779277\pi$
$48$	0	0
$49$	−307.830	−0.897463
$50$	295.152	0.834817
$51$	0	0
$52$	294.611	0.785678
$53$	430.509	1.11575	0.557877	−	0.829924i	$-0.311616\pi$
$53$	430.509	1.11575	0.557877	+	0.829924i	$0.311616\pi$
$54$	0	0
$55$	117.923	0.289104
$56$	200.202	0.477735
$57$	0	0
$58$	981.418	2.22184
$59$	−200.816	−0.443118	−0.221559	−	0.975147i	$-0.571115\pi$
$59$	−200.816	−0.443118	−0.221559	+	0.975147i	$0.571115\pi$
$60$	0	0
$61$	269.379	0.565418	0.282709	−	0.959206i	$-0.408767\pi$
$61$	269.379	0.565418	0.282709	+	0.959206i	$0.408767\pi$
$62$	−678.531	−1.38990
$63$	0	0
$64$	−668.523	−1.30571
$65$	−156.160	−0.297989
$66$	0	0
$67$	162.214	0.295784	0.147892	−	0.989003i	$-0.452751\pi$
$67$	162.214	0.295784	0.147892	+	0.989003i	$0.452751\pi$
$68$	−753.757	−1.34421
$69$	0	0
$70$	−226.812	−0.387274
$71$	−639.615	−1.06913	−0.534566	−	0.845127i	$-0.679525\pi$
$71$	−639.615	−1.06913	−0.534566	+	0.845127i	$0.679525\pi$
$72$	0	0
$73$	−312.142	−0.500458	−0.250229	−	0.968187i	$-0.580506\pi$
$73$	−312.142	−0.500458	−0.250229	+	0.968187i	$0.580506\pi$
$74$	−1443.65	−2.26786
$75$	0	0
$76$	2102.18	3.17285
$77$	87.7593	0.129884
$78$	0	0
$79$	−396.840	−0.565164	−0.282582	−	0.959243i	$-0.591191\pi$
$79$	−396.840	−0.565164	−0.282582	+	0.959243i	$0.591191\pi$
$80$	−332.676	−0.464929
$81$	0	0
$82$	−36.1261	−0.0486520
$83$	451.541	0.597145	0.298573	−	0.954387i	$-0.403489\pi$
$83$	451.541	0.597145	0.298573	+	0.954387i	$0.403489\pi$
$84$	0	0
$85$	399.532	0.509828
$86$	−127.758	−0.160192
$87$	0	0
$88$	499.558	0.605149
$89$	561.912	0.669242	0.334621	−	0.942353i	$-0.391392\pi$
$89$	561.912	0.669242	0.334621	+	0.942353i	$0.391392\pi$
$90$	0	0
$91$	−116.216	−0.133876
$92$	1965.47	2.22733
$93$	0	0
$94$	2378.61	2.60994
$95$	−1114.27	−1.20339
$96$	0	0
$97$	−1458.44	−1.52662	−0.763310	−	0.646033i	$-0.776427\pi$
$97$	−1458.44	−1.52662	−0.763310	+	0.646033i	$0.776427\pi$
$98$	1477.39	1.52284
$99$	0	0
$100$	−924.559	−0.924559
$101$	683.890	0.673759	0.336879	−	0.941548i	$-0.390629\pi$
$101$	683.890	0.673759	0.336879	+	0.941548i	$0.390629\pi$
$102$	0	0
$103$	1439.94	1.37749	0.688743	−	0.725005i	$-0.258163\pi$
$103$	1439.94	1.37749	0.688743	+	0.725005i	$0.258163\pi$
$104$	−661.544	−0.623747
$105$	0	0
$106$	−2066.17	−1.89325
$107$	92.7831	0.0838288	0.0419144	−	0.999121i	$-0.486654\pi$
$107$	92.7831	0.0838288	0.0419144	+	0.999121i	$0.486654\pi$
$108$	0	0
$109$	−472.221	−0.414959	−0.207480	−	0.978239i	$-0.566526\pi$
$109$	−472.221	−0.414959	−0.207480	+	0.978239i	$0.566526\pi$
$110$	−565.955	−0.490561
$111$	0	0
$112$	−247.581	−0.208877
$113$	−1209.10	−1.00657	−0.503287	−	0.864119i	$-0.667876\pi$
$113$	−1209.10	−1.00657	−0.503287	+	0.864119i	$0.667876\pi$
$114$	0	0
$115$	−1041.80	−0.844772
$116$	−3074.27	−2.46068
$117$	0	0
$118$	963.789	0.751898
$119$	297.336	0.229048
$120$	0	0
$121$	−1112.02	−0.835475
$122$	−1292.85	−0.959419
$123$	0	0
$124$	2125.48	1.53931
$125$	1486.17	1.06341
$126$	0	0
$127$	1307.27	0.913397	0.456698	−	0.889622i	$-0.349032\pi$
$127$	1307.27	0.913397	0.456698	+	0.889622i	$0.349032\pi$
$128$	2650.84	1.83050
$129$	0	0
$130$	749.470	0.505637
$131$	2635.48	1.75773	0.878865	−	0.477070i	$-0.158301\pi$
$131$	2635.48	1.75773	0.878865	+	0.477070i	$0.158301\pi$
$132$	0	0
$133$	−829.250	−0.540640
$134$	−778.523	−0.501896
$135$	0	0
$136$	1692.55	1.06717
$137$	−1954.58	−1.21891	−0.609457	−	0.792819i	$-0.708612\pi$
$137$	−1954.58	−1.21891	−0.609457	+	0.792819i	$0.708612\pi$
$138$	0	0
$139$	−1133.73	−0.691810	−0.345905	−	0.938270i	$-0.612428\pi$
$139$	−1133.73	−0.691810	−0.345905	+	0.938270i	$0.612428\pi$
$140$	710.483	0.428905
$141$	0	0
$142$	3069.75	1.81414
$143$	−289.990	−0.169581
$144$	0	0
$145$	1629.53	0.933278
$146$	1498.08	0.849193
$147$	0	0
$148$	4522.22	2.51165
$149$	700.779	0.385303	0.192651	−	0.981267i	$-0.438291\pi$
$149$	700.779	0.385303	0.192651	+	0.981267i	$0.438291\pi$
$150$	0	0
$151$	2528.69	1.36280	0.681398	−	0.731913i	$-0.261372\pi$
$151$	2528.69	1.36280	0.681398	+	0.731913i	$0.261372\pi$
$152$	−4720.40	−2.51891
$153$	0	0
$154$	−421.189	−0.220392
$155$	−1126.62	−0.583823
$156$	0	0
$157$	1686.19	0.857150	0.428575	−	0.903506i	$-0.359016\pi$
$157$	1686.19	0.857150	0.428575	+	0.903506i	$0.359016\pi$
$158$	1904.58	0.958988
$159$	0	0
$160$	−555.473	−0.274462
$161$	−775.321	−0.379527
$162$	0	0
$163$	−3591.23	−1.72569	−0.862844	−	0.505470i	$-0.831319\pi$
$163$	−3591.23	−1.72569	−0.862844	+	0.505470i	$0.831319\pi$
$164$	113.164	0.0538820
$165$	0	0
$166$	−2167.11	−1.01326
$167$	−3756.35	−1.74057	−0.870284	−	0.492550i	$-0.836065\pi$
$167$	−3756.35	−1.74057	−0.870284	+	0.492550i	$0.836065\pi$
$168$	0	0
$169$	−1812.98	−0.825207
$170$	−1917.50	−0.865093
$171$	0	0
$172$	400.200	0.177412
$173$	−1560.40	−0.685752	−0.342876	−	0.939381i	$-0.611401\pi$
$173$	−1560.40	−0.685752	−0.342876	+	0.939381i	$0.611401\pi$
$174$	0	0
$175$	364.713	0.157541
$176$	−617.780	−0.264585
$177$	0	0
$178$	−2696.82	−1.13559
$179$	4071.41	1.70006	0.850032	−	0.526731i	$-0.176583\pi$
$179$	4071.41	1.70006	0.850032	+	0.526731i	$0.176583\pi$
$180$	0	0
$181$	−3572.08	−1.46691	−0.733455	−	0.679738i	$-0.762093\pi$
$181$	−3572.08	−1.46691	−0.733455	+	0.679738i	$0.762093\pi$
$182$	557.763	0.227165
$183$	0	0
$184$	−4413.41	−1.76827
$185$	−2397.02	−0.952609
$186$	0	0
$187$	741.932	0.290136
$188$	−7450.94	−2.89051
$189$	0	0
$190$	5347.79	2.04194
$191$	1628.01	0.616745	0.308373	−	0.951266i	$-0.400216\pi$
$191$	1628.01	0.616745	0.308373	+	0.951266i	$0.400216\pi$
$192$	0	0
$193$	−2503.14	−0.933576	−0.466788	−	0.884369i	$-0.654589\pi$
$193$	−2503.14	−0.933576	−0.466788	+	0.884369i	$0.654589\pi$
$194$	6999.59	2.59042
$195$	0	0
$196$	−4627.89	−1.68655
$197$	1253.10	0.453197	0.226598	−	0.973988i	$-0.427240\pi$
$197$	1253.10	0.453197	0.226598	+	0.973988i	$0.427240\pi$
$198$	0	0
$199$	−753.695	−0.268482	−0.134241	−	0.990949i	$-0.542860\pi$
$199$	−753.695	−0.268482	−0.134241	+	0.990949i	$0.542860\pi$
$200$	2076.08	0.734005
$201$	0	0
$202$	−3282.24	−1.14326
$203$	1212.71	0.419290
$204$	0	0
$205$	−59.9833	−0.0204362
$206$	−6910.78	−2.33736
$207$	0	0
$208$	818.100	0.272716
$209$	−2069.20	−0.684830
$210$	0	0
$211$	1238.11	0.403958	0.201979	−	0.979390i	$-0.435263\pi$
$211$	1238.11	0.403958	0.201979	+	0.979390i	$0.435263\pi$
$212$	6472.24	2.09677
$213$	0	0
$214$	−445.300	−0.142243
$215$	−212.128	−0.0672883
$216$	0	0
$217$	−838.444	−0.262292
$218$	2266.36	0.704116
$219$	0	0
$220$	1772.84	0.543296
$221$	−982.509	−0.299053
$222$	0	0
$223$	−4996.16	−1.50030	−0.750152	−	0.661266i	$-0.770020\pi$
$223$	−4996.16	−1.50030	−0.750152	+	0.661266i	$0.770020\pi$
$224$	−413.388	−0.123307
$225$	0	0
$226$	5802.93	1.70799
$227$	−532.618	−0.155732	−0.0778659	−	0.996964i	$-0.524811\pi$
$227$	−532.618	−0.155732	−0.0778659	+	0.996964i	$0.524811\pi$
$228$	0	0
$229$	−678.270	−0.195726	−0.0978632	−	0.995200i	$-0.531201\pi$
$229$	−678.270	−0.195726	−0.0978632	+	0.995200i	$0.531201\pi$
$230$	5000.00	1.43344
$231$	0	0
$232$	6903.21	1.95353
$233$	−1088.23	−0.305975	−0.152988	−	0.988228i	$-0.548889\pi$
$233$	−1088.23	−0.305975	−0.152988	+	0.988228i	$0.548889\pi$
$234$	0	0
$235$	3949.40	1.09630
$236$	−3019.05	−0.832726
$237$	0	0
$238$	−1427.02	−0.388656
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−6692.78	−1.78888	−0.894439	−	0.447190i	$-0.852425\pi$
$241$	−6692.78	−1.78888	−0.894439	+	0.447190i	$0.852425\pi$
$242$	5336.98	1.41766
$243$	0	0
$244$	4049.83	1.06256
$245$	2453.03	0.639668
$246$	0	0
$247$	2740.15	0.705877
$248$	−4772.73	−1.22205
$249$	0	0
$250$	−7132.66	−1.80444
$251$	6105.48	1.53536	0.767679	−	0.640835i	$-0.221412\pi$
$251$	6105.48	1.53536	0.767679	+	0.640835i	$0.221412\pi$
$252$	0	0
$253$	−1934.63	−0.480748
$254$	−6274.06	−1.54988
$255$	0	0
$256$	−7374.17	−1.80033
$257$	−946.716	−0.229784	−0.114892	−	0.993378i	$-0.536652\pi$
$257$	−946.716	−0.229784	−0.114892	+	0.993378i	$0.536652\pi$
$258$	0	0
$259$	−1783.89	−0.427974
$260$	−2347.70	−0.559993
$261$	0	0
$262$	−12648.6	−2.98257
$263$	2466.18	0.578218	0.289109	−	0.957296i	$-0.406641\pi$
$263$	2466.18	0.578218	0.289109	+	0.957296i	$0.406641\pi$
$264$	0	0
$265$	−3430.64	−0.795255
$266$	3979.87	0.917375
$267$	0	0
$268$	2438.71	0.555850
$269$	7026.71	1.59266	0.796331	−	0.604861i	$-0.206771\pi$
$269$	7026.71	1.59266	0.796331	+	0.604861i	$0.206771\pi$
$270$	0	0
$271$	2359.35	0.528858	0.264429	−	0.964405i	$-0.414817\pi$
$271$	2359.35	0.528858	0.264429	+	0.964405i	$0.414817\pi$
$272$	−2093.09	−0.466589
$273$	0	0
$274$	9380.75	2.06829
$275$	910.055	0.199558
$276$	0	0
$277$	−2191.79	−0.475422	−0.237711	−	0.971336i	$-0.576397\pi$
$277$	−2191.79	−0.475422	−0.237711	+	0.971336i	$0.576397\pi$
$278$	5441.18	1.17389
$279$	0	0
$280$	−1595.37	−0.340506
$281$	−1652.78	−0.350878	−0.175439	−	0.984490i	$-0.556135\pi$
$281$	−1652.78	−0.350878	−0.175439	+	0.984490i	$0.556135\pi$
$282$	0	0
$283$	−6248.26	−1.31244	−0.656220	−	0.754570i	$-0.727846\pi$
$283$	−6248.26	−1.31244	−0.656220	+	0.754570i	$0.727846\pi$
$284$	−9615.92	−2.00915
$285$	0	0
$286$	1391.77	0.287751
$287$	−44.6402	−0.00918128
$288$	0	0
$289$	−2399.27	−0.488352
$290$	−7820.72	−1.58362
$291$	0	0
$292$	−4692.71	−0.940480
$293$	−996.397	−0.198669	−0.0993347	−	0.995054i	$-0.531671\pi$
$293$	−996.397	−0.198669	−0.0993347	+	0.995054i	$0.531671\pi$
$294$	0	0
$295$	1600.26	0.315833
$296$	−10154.5	−1.99399
$297$	0	0
$298$	−3363.30	−0.653794
$299$	2561.95	0.495523
$300$	0	0
$301$	−157.868	−0.0302303
$302$	−12136.1	−2.31244
$303$	0	0
$304$	5837.49	1.10133
$305$	−2146.63	−0.403002
$306$	0	0
$307$	8312.28	1.54530	0.772649	−	0.634833i	$-0.218931\pi$
$307$	8312.28	1.54530	0.772649	+	0.634833i	$0.218931\pi$
$308$	1319.37	0.244084
$309$	0	0
$310$	5407.08	0.990650
$311$	506.441	0.0923397	0.0461699	−	0.998934i	$-0.485298\pi$
$311$	506.441	0.0923397	0.0461699	+	0.998934i	$0.485298\pi$
$312$	0	0
$313$	−8812.47	−1.59141	−0.795703	−	0.605687i	$-0.792898\pi$
$313$	−8812.47	−1.59141	−0.795703	+	0.605687i	$0.792898\pi$
$314$	−8092.64	−1.45444
$315$	0	0
$316$	−5966.05	−1.06208
$317$	−2368.23	−0.419600	−0.209800	−	0.977744i	$-0.567281\pi$
$317$	−2368.23	−0.419600	−0.209800	+	0.977744i	$0.567281\pi$
$318$	0	0
$319$	3026.04	0.531116
$320$	5327.33	0.930646
$321$	0	0
$322$	3721.05	0.643994
$323$	−7010.62	−1.20768
$324$	0	0
$325$	−1205.15	−0.205691
$326$	17235.6	2.92820
$327$	0	0
$328$	−254.108	−0.0427768
$329$	2939.18	0.492530
$330$	0	0
$331$	8783.98	1.45864	0.729322	−	0.684170i	$-0.239835\pi$
$331$	8783.98	1.45864	0.729322	+	0.684170i	$0.239835\pi$
$332$	6788.43	1.12218
$333$	0	0
$334$	18028.1	2.95345
$335$	−1292.65	−0.210821
$336$	0	0
$337$	9461.91	1.52944	0.764722	−	0.644360i	$-0.222876\pi$
$337$	9461.91	1.52944	0.764722	+	0.644360i	$0.222876\pi$
$338$	8701.15	1.40024
$339$	0	0
$340$	6006.54	0.958089
$341$	−2092.14	−0.332246
$342$	0	0
$343$	3859.72	0.607595
$344$	−898.640	−0.140847
$345$	0	0
$346$	7488.94	1.16361
$347$	−10085.6	−1.56029	−0.780146	−	0.625597i	$-0.784855\pi$
$347$	−10085.6	−1.56029	−0.780146	+	0.625597i	$0.784855\pi$
$348$	0	0
$349$	−5563.96	−0.853387	−0.426693	−	0.904396i	$-0.640322\pi$
$349$	−5563.96	−0.853387	−0.426693	+	0.904396i	$0.640322\pi$
$350$	−1750.39	−0.267321
$351$	0	0
$352$	−1031.51	−0.156193
$353$	6239.98	0.940851	0.470426	−	0.882440i	$-0.344100\pi$
$353$	6239.98	0.940851	0.470426	+	0.882440i	$0.344100\pi$
$354$	0	0
$355$	5096.97	0.762025
$356$	8447.74	1.25767
$357$	0	0
$358$	−19540.2	−2.88472
$359$	−1598.16	−0.234952	−0.117476	−	0.993076i	$-0.537480\pi$
$359$	−1598.16	−0.234952	−0.117476	+	0.993076i	$0.537480\pi$
$360$	0	0
$361$	12693.1	1.85058
$362$	17143.7	2.48910
$363$	0	0
$364$	−1747.18	−0.251586
$365$	2487.39	0.356702
$366$	0	0
$367$	9915.59	1.41033	0.705163	−	0.709045i	$-0.250874\pi$
$367$	9915.59	1.41033	0.705163	+	0.709045i	$0.250874\pi$
$368$	5457.86	0.773127
$369$	0	0
$370$	11504.2	1.61642
$371$	−2553.12	−0.357281
$372$	0	0
$373$	1496.68	0.207762	0.103881	−	0.994590i	$-0.466874\pi$
$373$	1496.68	0.207762	0.103881	+	0.994590i	$0.466874\pi$
$374$	−3560.80	−0.492312
$375$	0	0
$376$	16730.9	2.29476
$377$	−4007.26	−0.547438
$378$	0	0
$379$	13121.3	1.77836	0.889178	−	0.457562i	$-0.151277\pi$
$379$	13121.3	1.77836	0.889178	+	0.457562i	$0.151277\pi$
$380$	−16751.8	−2.26145
$381$	0	0
$382$	−7813.40	−1.04651
$383$	1943.56	0.259299	0.129649	−	0.991560i	$-0.458615\pi$
$383$	1943.56	0.259299	0.129649	+	0.991560i	$0.458615\pi$
$384$	0	0
$385$	−699.337	−0.0925753
$386$	12013.5	1.58412
$387$	0	0
$388$	−21926.1	−2.86888
$389$	−643.786	−0.0839107	−0.0419554	−	0.999119i	$-0.513359\pi$
$389$	−643.786	−0.0839107	−0.0419554	+	0.999119i	$0.513359\pi$
$390$	0	0
$391$	−6554.70	−0.847788
$392$	10391.8	1.33894
$393$	0	0
$394$	−6014.09	−0.768999
$395$	3162.34	0.402821
$396$	0	0
$397$	7166.39	0.905972	0.452986	−	0.891518i	$-0.350359\pi$
$397$	7166.39	0.905972	0.452986	+	0.891518i	$0.350359\pi$
$398$	3617.26	0.455570
$399$	0	0
$400$	−2567.39	−0.320923
$401$	−178.047	−0.0221727	−0.0110863	−	0.999939i	$-0.503529\pi$
$401$	−178.047	−0.0221727	−0.0110863	+	0.999939i	$0.503529\pi$
$402$	0	0
$403$	2770.53	0.342457
$404$	10281.5	1.26615
$405$	0	0
$406$	−5820.26	−0.711464
$407$	−4451.27	−0.542117
$408$	0	0
$409$	5254.79	0.635288	0.317644	−	0.948210i	$-0.397108\pi$
$409$	5254.79	0.635288	0.317644	+	0.948210i	$0.397108\pi$
$410$	287.882	0.0346768
$411$	0	0
$412$	21647.9	2.58863
$413$	1190.93	0.141893
$414$	0	0
$415$	−3598.24	−0.425616
$416$	1365.99	0.160993
$417$	0	0
$418$	9930.84	1.16204
$419$	2968.16	0.346072	0.173036	−	0.984915i	$-0.444642\pi$
$419$	2968.16	0.346072	0.173036	+	0.984915i	$0.444642\pi$
$420$	0	0
$421$	11547.2	1.33676	0.668382	−	0.743818i	$-0.266987\pi$
$421$	11547.2	1.33676	0.668382	+	0.743818i	$0.266987\pi$
$422$	−5942.16	−0.685449
$423$	0	0
$424$	−14533.3	−1.66462
$425$	3083.34	0.351915
$426$	0	0
$427$	−1597.54	−0.181055
$428$	1394.89	0.157534
$429$	0	0
$430$	1018.08	0.114177
$431$	10508.9	1.17447	0.587233	−	0.809418i	$-0.300217\pi$
$431$	10508.9	1.17447	0.587233	+	0.809418i	$0.300217\pi$
$432$	0	0
$433$	9318.75	1.03425	0.517125	−	0.855910i	$-0.327002\pi$
$433$	9318.75	1.03425	0.517125	+	0.855910i	$0.327002\pi$
$434$	4024.00	0.445065
$435$	0	0
$436$	−7099.33	−0.779808
$437$	18280.6	2.00110
$438$	0	0
$439$	12269.5	1.33392	0.666961	−	0.745093i	$-0.267595\pi$
$439$	12269.5	1.33392	0.666961	+	0.745093i	$0.267595\pi$
$440$	−3980.88	−0.431321
$441$	0	0
$442$	4715.42	0.507443
$443$	8637.15	0.926328	0.463164	−	0.886273i	$-0.346714\pi$
$443$	8637.15	0.926328	0.463164	+	0.886273i	$0.346714\pi$
$444$	0	0
$445$	−4477.76	−0.477003
$446$	23978.4	2.54576
$447$	0	0
$448$	3964.65	0.418107
$449$	−4980.91	−0.523527	−0.261764	−	0.965132i	$-0.584304\pi$
$449$	−4980.91	−0.523527	−0.261764	+	0.965132i	$0.584304\pi$
$450$	0	0
$451$	−111.389	−0.0116300
$452$	−18177.6	−1.89159
$453$	0	0
$454$	2556.23	0.264251
$455$	926.101	0.0954204
$456$	0	0
$457$	15624.8	1.59934	0.799669	−	0.600441i	$-0.205008\pi$
$457$	15624.8	1.59934	0.799669	+	0.600441i	$0.205008\pi$
$458$	3255.27	0.332115
$459$	0	0
$460$	−15662.4	−1.58753
$461$	10846.0	1.09577	0.547886	−	0.836553i	$-0.315433\pi$
$461$	10846.0	1.09577	0.547886	+	0.836553i	$0.315433\pi$
$462$	0	0
$463$	−3083.37	−0.309496	−0.154748	−	0.987954i	$-0.549457\pi$
$463$	−3083.37	−0.309496	−0.154748	+	0.987954i	$0.549457\pi$
$464$	−8536.88	−0.854126
$465$	0	0
$466$	5222.81	0.519189
$467$	6154.69	0.609861	0.304931	−	0.952375i	$-0.401367\pi$
$467$	6154.69	0.609861	0.304931	+	0.952375i	$0.401367\pi$
$468$	0	0
$469$	−962.001	−0.0947145
$470$	−18954.6	−1.86024
$471$	0	0
$472$	6779.21	0.661098
$473$	−393.921	−0.0382929
$474$	0	0
$475$	−8599.23	−0.830652
$476$	4470.12	0.430437
$477$	0	0
$478$	−1147.05	−0.109759
$479$	7600.68	0.725018	0.362509	−	0.931980i	$-0.381920\pi$
$479$	7600.68	0.725018	0.362509	+	0.931980i	$0.381920\pi$
$480$	0	0
$481$	5894.63	0.558778
$482$	32121.1	3.03543
$483$	0	0
$484$	−16718.0	−1.57006
$485$	11622.0	1.08810
$486$	0	0
$487$	−3333.20	−0.310147	−0.155073	−	0.987903i	$-0.549561\pi$
$487$	−3333.20	−0.310147	−0.155073	+	0.987903i	$0.549561\pi$
$488$	−9093.80	−0.843560
$489$	0	0
$490$	−11773.0	−1.08541
$491$	7014.91	0.644762	0.322381	−	0.946610i	$-0.395517\pi$
$491$	7014.91	0.644762	0.322381	+	0.946610i	$0.395517\pi$
$492$	0	0
$493$	10252.5	0.936610
$494$	−13151.0	−1.19775
$495$	0	0
$496$	5902.21	0.534309
$497$	3793.21	0.342352
$498$	0	0
$499$	14793.4	1.32714	0.663570	−	0.748114i	$-0.269040\pi$
$499$	14793.4	1.32714	0.663570	+	0.748114i	$0.269040\pi$
$500$	22342.9	1.99841
$501$	0	0
$502$	−29302.4	−2.60524
$503$	−5736.95	−0.508544	−0.254272	−	0.967133i	$-0.581836\pi$
$503$	−5736.95	−0.508544	−0.254272	+	0.967133i	$0.581836\pi$
$504$	0	0
$505$	−5449.78	−0.480222
$506$	9285.01	0.815749
$507$	0	0
$508$	19653.4	1.71649
$509$	5145.84	0.448105	0.224052	−	0.974577i	$-0.428071\pi$
$509$	5145.84	0.448105	0.224052	+	0.974577i	$0.428071\pi$
$510$	0	0
$511$	1851.14	0.160254
$512$	14184.6	1.22437
$513$	0	0
$514$	4543.63	0.389905
$515$	−11474.6	−0.981805
$516$	0	0
$517$	7334.05	0.623890
$518$	8561.53	0.726201
$519$	0	0
$520$	5271.71	0.444576
$521$	−14008.0	−1.17793	−0.588964	−	0.808159i	$-0.700464\pi$
$521$	−14008.0	−1.17793	−0.588964	+	0.808159i	$0.700464\pi$
$522$	0	0
$523$	1981.54	0.165672	0.0828361	−	0.996563i	$-0.473602\pi$
$523$	1981.54	0.165672	0.0828361	+	0.996563i	$0.473602\pi$
$524$	39621.5	3.30320
$525$	0	0
$526$	−11836.1	−0.981138
$527$	−7088.35	−0.585907
$528$	0	0
$529$	4924.76	0.404764
$530$	16464.9	1.34941
$531$	0	0
$532$	−12466.9	−1.01599
$533$	147.508	0.0119874
$534$	0	0
$535$	−739.370	−0.0597491
$536$	−5476.07	−0.441287
$537$	0	0
$538$	−33723.8	−2.70248
$539$	4555.28	0.364026
$540$	0	0
$541$	8645.44	0.687054	0.343527	−	0.939143i	$-0.388378\pi$
$541$	8645.44	0.687054	0.343527	+	0.939143i	$0.388378\pi$
$542$	−11323.4	−0.897383
$543$	0	0
$544$	−3494.85	−0.275442
$545$	3763.03	0.295763
$546$	0	0
$547$	21697.6	1.69602	0.848010	−	0.529980i	$-0.177801\pi$
$547$	21697.6	1.69602	0.848010	+	0.529980i	$0.177801\pi$
$548$	−29385.0	−2.29063
$549$	0	0
$550$	−4367.69	−0.338616
$551$	−28593.5	−2.21075
$552$	0	0
$553$	2353.44	0.180974
$554$	10519.2	0.806712
$555$	0	0
$556$	−17044.4	−1.30008
$557$	20419.4	1.55332	0.776659	−	0.629922i	$-0.216913\pi$
$557$	20419.4	1.55332	0.776659	+	0.629922i	$0.216913\pi$
$558$	0	0
$559$	521.653	0.0394697
$560$	1972.92	0.148877
$561$	0	0
$562$	7932.32	0.595382
$563$	−16375.9	−1.22587	−0.612933	−	0.790135i	$-0.710011\pi$
$563$	−16375.9	−1.22587	−0.612933	+	0.790135i	$0.710011\pi$
$564$	0	0
$565$	9635.11	0.717437
$566$	29987.7	2.22699
$567$	0	0
$568$	21592.4	1.59506
$569$	−630.713	−0.0464690	−0.0232345	−	0.999730i	$-0.507396\pi$
$569$	−630.713	−0.0464690	−0.0232345	+	0.999730i	$0.507396\pi$
$570$	0	0
$571$	4887.47	0.358204	0.179102	−	0.983831i	$-0.442681\pi$
$571$	4887.47	0.358204	0.179102	+	0.983831i	$0.442681\pi$
$572$	−4359.68	−0.318684
$573$	0	0
$574$	214.245	0.0155791
$575$	−8040.00	−0.583115
$576$	0	0
$577$	−4680.58	−0.337704	−0.168852	−	0.985641i	$-0.554006\pi$
$577$	−4680.58	−0.337704	−0.168852	+	0.985641i	$0.554006\pi$
$578$	11515.0	0.828651
$579$	0	0
$580$	24498.3	1.75385
$581$	−2677.84	−0.191215
$582$	0	0
$583$	−6370.70	−0.452569
$584$	10537.4	0.746644
$585$	0	0
$586$	4782.08	0.337109
$587$	8756.13	0.615680	0.307840	−	0.951438i	$-0.400394\pi$
$587$	8756.13	0.615680	0.307840	+	0.951438i	$0.400394\pi$
$588$	0	0
$589$	19768.9	1.38296
$590$	−7680.24	−0.535916
$591$	0	0
$592$	12557.6	0.871818
$593$	−9965.71	−0.690123	−0.345061	−	0.938580i	$-0.612142\pi$
$593$	−9965.71	−0.690123	−0.345061	+	0.938580i	$0.612142\pi$
$594$	0	0
$595$	−2369.41	−0.163254
$596$	10535.5	0.724076
$597$	0	0
$598$	−12295.7	−0.840820
$599$	5647.62	0.385235	0.192617	−	0.981274i	$-0.438302\pi$
$599$	5647.62	0.385235	0.192617	+	0.981274i	$0.438302\pi$
$600$	0	0
$601$	−13770.1	−0.934598	−0.467299	−	0.884099i	$-0.654773\pi$
$601$	−13770.1	−0.934598	−0.467299	+	0.884099i	$0.654773\pi$
$602$	757.664	0.0512958
$603$	0	0
$604$	38016.2	2.56102
$605$	8861.44	0.595486
$606$	0	0
$607$	−10769.4	−0.720125	−0.360062	−	0.932928i	$-0.617245\pi$
$607$	−10769.4	−0.720125	−0.360062	+	0.932928i	$0.617245\pi$
$608$	9746.91	0.650147
$609$	0	0
$610$	10302.5	0.683827
$611$	−9712.16	−0.643064
$612$	0	0
$613$	−29506.6	−1.94414	−0.972071	−	0.234686i	$-0.924594\pi$
$613$	−29506.6	−1.94414	−0.972071	+	0.234686i	$0.924594\pi$
$614$	−39893.7	−2.62211
$615$	0	0
$616$	−2962.61	−0.193778
$617$	16927.5	1.10450	0.552248	−	0.833680i	$-0.313770\pi$
$617$	16927.5	1.10450	0.552248	+	0.833680i	$0.313770\pi$
$618$	0	0
$619$	3434.78	0.223030	0.111515	−	0.993763i	$-0.464430\pi$
$619$	3434.78	0.223030	0.111515	+	0.993763i	$0.464430\pi$
$620$	−16937.6	−1.09714
$621$	0	0
$622$	−2430.60	−0.156685
$623$	−3332.39	−0.214301
$624$	0	0
$625$	−4155.69	−0.265964
$626$	42294.3	2.70035
$627$	0	0
$628$	25350.0	1.61079
$629$	−15081.3	−0.956010
$630$	0	0
$631$	10560.6	0.666262	0.333131	−	0.942881i	$-0.391895\pi$
$631$	10560.6	0.666262	0.333131	+	0.942881i	$0.391895\pi$
$632$	13396.7	0.843181
$633$	0	0
$634$	11366.0	0.711990
$635$	−10417.4	−0.651025
$636$	0	0
$637$	−6032.37	−0.375214
$638$	−14523.1	−0.901214
$639$	0	0
$640$	−21124.0	−1.30469
$641$	8592.73	0.529473	0.264737	−	0.964321i	$-0.414715\pi$
$641$	8592.73	0.529473	0.264737	+	0.964321i	$0.414715\pi$
$642$	0	0
$643$	24046.9	1.47483	0.737416	−	0.675439i	$-0.236046\pi$
$643$	24046.9	1.47483	0.737416	+	0.675439i	$0.236046\pi$
$644$	−11656.1	−0.713223
$645$	0	0
$646$	33646.5	2.04923
$647$	10379.2	0.630677	0.315338	−	0.948979i	$-0.397882\pi$
$647$	10379.2	0.630677	0.315338	+	0.948979i	$0.397882\pi$
$648$	0	0
$649$	2971.69	0.179736
$650$	5783.94	0.349023
$651$	0	0
$652$	−53990.3	−3.24298
$653$	−4946.74	−0.296448	−0.148224	−	0.988954i	$-0.547356\pi$
$653$	−4946.74	−0.296448	−0.148224	+	0.988954i	$0.547356\pi$
$654$	0	0
$655$	−21001.6	−1.25282
$656$	314.244	0.0187030
$657$	0	0
$658$	−14106.2	−0.835742
$659$	−4569.28	−0.270097	−0.135049	−	0.990839i	$-0.543119\pi$
$659$	−4569.28	−0.270097	−0.135049	+	0.990839i	$0.543119\pi$
$660$	0	0
$661$	−4510.45	−0.265410	−0.132705	−	0.991156i	$-0.542366\pi$
$661$	−4510.45	−0.265410	−0.132705	+	0.991156i	$0.542366\pi$
$662$	−42157.6	−2.47508
$663$	0	0
$664$	−15243.3	−0.890894
$665$	6608.13	0.385342
$666$	0	0
$667$	−26734.0	−1.55194
$668$	−56472.6	−3.27095
$669$	0	0
$670$	6203.89	0.357727
$671$	−3986.29	−0.229343
$672$	0	0
$673$	22212.1	1.27223	0.636116	−	0.771593i	$-0.280540\pi$
$673$	22212.1	1.27223	0.636116	+	0.771593i	$0.280540\pi$
$674$	−45411.2	−2.59521
$675$	0	0
$676$	−27256.2	−1.55076
$677$	−27258.3	−1.54745	−0.773723	−	0.633524i	$-0.781608\pi$
$677$	−27258.3	−1.54745	−0.773723	+	0.633524i	$0.781608\pi$
$678$	0	0
$679$	8649.22	0.488846
$680$	−13487.6	−0.760624
$681$	0	0
$682$	10040.9	0.563765
$683$	−18692.4	−1.04721	−0.523606	−	0.851960i	$-0.675414\pi$
$683$	−18692.4	−1.04721	−0.523606	+	0.851960i	$0.675414\pi$
$684$	0	0
$685$	15575.7	0.868782
$686$	−18524.2	−1.03099
$687$	0	0
$688$	1111.31	0.0615816
$689$	8436.45	0.466477
$690$	0	0
$691$	15574.9	0.857447	0.428724	−	0.903436i	$-0.358963\pi$
$691$	15574.9	0.857447	0.428724	+	0.903436i	$0.358963\pi$
$692$	−23458.9	−1.28869
$693$	0	0
$694$	48404.3	2.64755
$695$	9034.45	0.493088
$696$	0	0
$697$	−377.396	−0.0205091
$698$	26703.5	1.44805
$699$	0	0
$700$	5483.06	0.296057
$701$	17574.8	0.946920	0.473460	−	0.880815i	$-0.343005\pi$
$701$	17574.8	0.946920	0.473460	+	0.880815i	$0.343005\pi$
$702$	0	0
$703$	42060.7	2.25654
$704$	9892.85	0.529618
$705$	0	0
$706$	−29947.9	−1.59647
$707$	−4055.78	−0.215747
$708$	0	0
$709$	−13293.2	−0.704143	−0.352072	−	0.935973i	$-0.614523\pi$
$709$	−13293.2	−0.704143	−0.352072	+	0.935973i	$0.614523\pi$
$710$	−24462.2	−1.29303
$711$	0	0
$712$	−18969.2	−0.998457
$713$	18483.3	0.970834
$714$	0	0
$715$	2310.87	0.120869
$716$	61209.2	3.19483
$717$	0	0
$718$	7670.17	0.398674
$719$	7311.94	0.379262	0.189631	−	0.981855i	$-0.439271\pi$
$719$	7311.94	0.379262	0.189631	+	0.981855i	$0.439271\pi$
$720$	0	0
$721$	−8539.48	−0.441091
$722$	−60919.1	−3.14013
$723$	0	0
$724$	−53702.3	−2.75667
$725$	12575.7	0.644207
$726$	0	0
$727$	−21400.7	−1.09176	−0.545880	−	0.837863i	$-0.683804\pi$
$727$	−21400.7	−1.09176	−0.545880	+	0.837863i	$0.683804\pi$
$728$	3923.26	0.199733
$729$	0	0
$730$	−11937.9	−0.605263
$731$	−1334.64	−0.0675286
$732$	0	0
$733$	6703.65	0.337796	0.168898	−	0.985633i	$-0.445979\pi$
$733$	6703.65	0.337796	0.168898	+	0.985633i	$0.445979\pi$
$734$	−47588.6	−2.39309
$735$	0	0
$736$	9113.04	0.456401
$737$	−2400.45	−0.119975
$738$	0	0
$739$	−32207.0	−1.60318	−0.801592	−	0.597872i	$-0.796013\pi$
$739$	−32207.0	−1.60318	−0.801592	+	0.597872i	$0.796013\pi$
$740$	−36036.7	−1.79018
$741$	0	0
$742$	12253.3	0.606245
$743$	8238.09	0.406765	0.203382	−	0.979099i	$-0.434807\pi$
$743$	8238.09	0.406765	0.203382	+	0.979099i	$0.434807\pi$
$744$	0	0
$745$	−5584.37	−0.274625
$746$	−7183.13	−0.352538
$747$	0	0
$748$	11154.1	0.545235
$749$	−550.246	−0.0268432
$750$	0	0
$751$	−33744.1	−1.63960	−0.819799	−	0.572651i	$-0.805915\pi$
$751$	−33744.1	−1.63960	−0.819799	+	0.572651i	$0.805915\pi$
$752$	−20690.3	−1.00332
$753$	0	0
$754$	19232.3	0.928911
$755$	−20150.7	−0.971335
$756$	0	0
$757$	−3150.48	−0.151263	−0.0756314	−	0.997136i	$-0.524097\pi$
$757$	−3150.48	−0.151263	−0.0756314	+	0.997136i	$0.524097\pi$
$758$	−62974.0	−3.01757
$759$	0	0
$760$	37615.9	1.79536
$761$	4779.28	0.227660	0.113830	−	0.993500i	$-0.463688\pi$
$761$	4779.28	0.227660	0.113830	+	0.993500i	$0.463688\pi$
$762$	0	0
$763$	2800.48	0.132876
$764$	24475.3	1.15901
$765$	0	0
$766$	−9327.86	−0.439986
$767$	−3935.28	−0.185260
$768$	0	0
$769$	28562.8	1.33940	0.669702	−	0.742630i	$-0.266422\pi$
$769$	28562.8	1.33940	0.669702	+	0.742630i	$0.266422\pi$
$770$	3356.37	0.157085
$771$	0	0
$772$	−37632.1	−1.75441
$773$	−6341.09	−0.295049	−0.147525	−	0.989058i	$-0.547131\pi$
$773$	−6341.09	−0.295049	−0.147525	+	0.989058i	$0.547131\pi$
$774$	0	0
$775$	−8694.57	−0.402991
$776$	49234.5	2.27760
$777$	0	0
$778$	3089.77	0.142382
$779$	1052.53	0.0484093
$780$	0	0
$781$	9465.07	0.433658
$782$	31458.4	1.43855
$783$	0	0
$784$	−12851.1	−0.585417
$785$	−13436.9	−0.610935
$786$	0	0
$787$	6673.65	0.302275	0.151137	−	0.988513i	$-0.451706\pi$
$787$	6673.65	0.302275	0.151137	+	0.988513i	$0.451706\pi$
$788$	18839.0	0.851665
$789$	0	0
$790$	−15177.2	−0.683520
$791$	7170.54	0.322320
$792$	0	0
$793$	5278.88	0.236392
$794$	−34394.1	−1.53728
$795$	0	0
$796$	−11331.0	−0.504543
$797$	29450.2	1.30888	0.654441	−	0.756113i	$-0.272904\pi$
$797$	29450.2	1.30888	0.654441	+	0.756113i	$0.272904\pi$
$798$	0	0
$799$	24848.4	1.10022
$800$	−4286.79	−0.189451
$801$	0	0
$802$	854.513	0.0376233
$803$	4619.09	0.202994
$804$	0	0
$805$	6178.38	0.270508
$806$	−13296.8	−0.581091
$807$	0	0
$808$	−23087.0	−1.00520
$809$	11052.9	0.480346	0.240173	−	0.970730i	$-0.422796\pi$
$809$	11052.9	0.480346	0.240173	+	0.970730i	$0.422796\pi$
$810$	0	0
$811$	−3548.93	−0.153662	−0.0768310	−	0.997044i	$-0.524480\pi$
$811$	−3548.93	−0.153662	−0.0768310	+	0.997044i	$0.524480\pi$
$812$	18231.8	0.787946
$813$	0	0
$814$	21363.3	0.919881
$815$	28617.8	1.22999
$816$	0	0
$817$	3722.22	0.159393
$818$	−25219.7	−1.07798
$819$	0	0
$820$	−901.785	−0.0384045
$821$	16741.7	0.711681	0.355841	−	0.934547i	$-0.384195\pi$
$821$	16741.7	0.711681	0.355841	+	0.934547i	$0.384195\pi$
$822$	0	0
$823$	29065.1	1.23104	0.615519	−	0.788122i	$-0.288946\pi$
$823$	29065.1	1.23104	0.615519	+	0.788122i	$0.288946\pi$
$824$	−48609.9	−2.05510
$825$	0	0
$826$	−5715.71	−0.240769
$827$	−30609.4	−1.28705	−0.643527	−	0.765423i	$-0.722530\pi$
$827$	−30609.4	−1.28705	−0.643527	+	0.765423i	$0.722530\pi$
$828$	0	0
$829$	6713.68	0.281273	0.140637	−	0.990061i	$-0.455085\pi$
$829$	6713.68	0.281273	0.140637	+	0.990061i	$0.455085\pi$
$830$	17269.3	0.722199
$831$	0	0
$832$	−13100.7	−0.545895
$833$	15433.7	0.641951
$834$	0	0
$835$	29933.6	1.24059
$836$	−31108.2	−1.28696
$837$	0	0
$838$	−14245.3	−0.587227
$839$	−17776.1	−0.731466	−0.365733	−	0.930720i	$-0.619182\pi$
$839$	−17776.1	−0.731466	−0.365733	+	0.930720i	$0.619182\pi$
$840$	0	0
$841$	17426.8	0.714533
$842$	−55419.4	−2.26827
$843$	0	0
$844$	18613.7	0.759134
$845$	14447.3	0.588167
$846$	0	0
$847$	6594.77	0.267531
$848$	17972.6	0.727809
$849$	0	0
$850$	−14798.1	−0.597141
$851$	39325.4	1.58408
$852$	0	0
$853$	−8596.72	−0.345072	−0.172536	−	0.985003i	$-0.555196\pi$
$853$	−8596.72	−0.345072	−0.172536	+	0.985003i	$0.555196\pi$
$854$	7667.19	0.307220
$855$	0	0
$856$	−3132.20	−0.125066
$857$	46319.8	1.84627	0.923136	−	0.384473i	$-0.125617\pi$
$857$	46319.8	1.84627	0.923136	+	0.384473i	$0.125617\pi$
$858$	0	0
$859$	4410.44	0.175183	0.0875915	−	0.996156i	$-0.472083\pi$
$859$	4410.44	0.175183	0.0875915	+	0.996156i	$0.472083\pi$
$860$	−3189.11	−0.126451
$861$	0	0
$862$	−50435.9	−1.99287
$863$	−10317.3	−0.406958	−0.203479	−	0.979079i	$-0.565225\pi$
$863$	−10317.3	−0.406958	−0.203479	+	0.979079i	$0.565225\pi$
$864$	0	0
$865$	12434.5	0.488771
$866$	−44724.1	−1.75495
$867$	0	0
$868$	−12605.1	−0.492909
$869$	5872.46	0.229240
$870$	0	0
$871$	3178.81	0.123662
$872$	15941.4	0.619087
$873$	0	0
$874$	−87735.3	−3.39553
$875$	−8813.65	−0.340521
$876$	0	0
$877$	37559.1	1.44616	0.723079	−	0.690765i	$-0.242726\pi$
$877$	37559.1	1.44616	0.723079	+	0.690765i	$0.242726\pi$
$878$	−58885.9	−2.26344
$879$	0	0
$880$	4922.97	0.188583
$881$	−4692.16	−0.179436	−0.0897178	−	0.995967i	$-0.528597\pi$
$881$	−4692.16	−0.179436	−0.0897178	+	0.995967i	$0.528597\pi$
$882$	0	0
$883$	−6238.34	−0.237754	−0.118877	−	0.992909i	$-0.537929\pi$
$883$	−6238.34	−0.237754	−0.118877	+	0.992909i	$0.537929\pi$
$884$	−14771.0	−0.561992
$885$	0	0
$886$	−41452.9	−1.57182
$887$	33127.8	1.25403	0.627014	−	0.779008i	$-0.284277\pi$
$887$	33127.8	1.25403	0.627014	+	0.779008i	$0.284277\pi$
$888$	0	0
$889$	−7752.71	−0.292483
$890$	21490.4	0.809394
$891$	0	0
$892$	−75111.9	−2.81943
$893$	−69300.4	−2.59692
$894$	0	0
$895$	−32444.2	−1.21172
$896$	−15720.7	−0.586151
$897$	0	0
$898$	23905.2	0.888338
$899$	−28910.5	−1.07255
$900$	0	0
$901$	−21584.5	−0.798094
$902$	534.597	0.0197341
$903$	0	0
$904$	40817.3	1.50173
$905$	28465.2	1.04554
$906$	0	0
$907$	−32941.7	−1.20597	−0.602983	−	0.797754i	$-0.706021\pi$
$907$	−32941.7	−1.20597	−0.602983	+	0.797754i	$0.706021\pi$
$908$	−8007.34	−0.292657
$909$	0	0
$910$	−4444.70	−0.161912
$911$	15568.5	0.566199	0.283099	−	0.959091i	$-0.408637\pi$
$911$	15568.5	0.566199	0.283099	+	0.959091i	$0.408637\pi$
$912$	0	0
$913$	−6681.93	−0.242212
$914$	−74989.2	−2.71381
$915$	0	0
$916$	−10197.1	−0.367817
$917$	−15629.6	−0.562851
$918$	0	0
$919$	−9900.07	−0.355357	−0.177679	−	0.984089i	$-0.556859\pi$
$919$	−9900.07	−0.355357	−0.177679	+	0.984089i	$0.556859\pi$
$920$	35169.6	1.26033
$921$	0	0
$922$	−52054.1	−1.85934
$923$	−12534.2	−0.446985
$924$	0	0
$925$	−18498.7	−0.657551
$926$	14798.2	0.525162
$927$	0	0
$928$	−14254.1	−0.504218
$929$	27348.8	0.965862	0.482931	−	0.875658i	$-0.339572\pi$
$929$	27348.8	0.965862	0.482931	+	0.875658i	$0.339572\pi$
$930$	0	0
$931$	−43043.5	−1.51525
$932$	−16360.4	−0.575001
$933$	0	0
$934$	−29538.6	−1.03483
$935$	−5912.31	−0.206795
$936$	0	0
$937$	1347.64	0.0469855	0.0234928	−	0.999724i	$-0.492521\pi$
$937$	1347.64	0.0469855	0.0234928	+	0.999724i	$0.492521\pi$
$938$	4617.00	0.160715
$939$	0	0
$940$	59375.0	2.06021
$941$	−6042.24	−0.209321	−0.104661	−	0.994508i	$-0.533376\pi$
$941$	−6042.24	−0.209321	−0.104661	+	0.994508i	$0.533376\pi$
$942$	0	0
$943$	984.082	0.0339831
$944$	−8383.53	−0.289047
$945$	0	0
$946$	1890.57	0.0649766
$947$	8434.37	0.289419	0.144710	−	0.989474i	$-0.453775\pi$
$947$	8434.37	0.289419	0.144710	+	0.989474i	$0.453775\pi$
$948$	0	0
$949$	−6116.87	−0.209233
$950$	41270.9	1.40948
$951$	0	0
$952$	−10037.6	−0.341722
$953$	−22019.9	−0.748474	−0.374237	−	0.927333i	$-0.622095\pi$
$953$	−22019.9	−0.748474	−0.374237	+	0.927333i	$0.622095\pi$
$954$	0	0
$955$	−12973.3	−0.439586
$956$	3593.11	0.121558
$957$	0	0
$958$	−36478.4	−1.23023
$959$	11591.6	0.390314
$960$	0	0
$961$	−9802.90	−0.329056
$962$	−28290.5	−0.948152
$963$	0	0
$964$	−100619.	−3.36173
$965$	19947.1	0.665408
$966$	0	0
$967$	37835.9	1.25824	0.629121	−	0.777307i	$-0.283415\pi$
$967$	37835.9	1.25824	0.629121	+	0.777307i	$0.283415\pi$
$968$	37539.9	1.24646
$969$	0	0
$970$	−55778.3	−1.84632
$971$	52181.1	1.72458	0.862292	−	0.506411i	$-0.169028\pi$
$971$	52181.1	1.72458	0.862292	+	0.506411i	$0.169028\pi$
$972$	0	0
$973$	6723.53	0.221528
$974$	15997.2	0.526267
$975$	0	0
$976$	11245.9	0.368823
$977$	20379.7	0.667353	0.333676	−	0.942688i	$-0.391711\pi$
$977$	20379.7	0.667353	0.333676	+	0.942688i	$0.391711\pi$
$978$	0	0
$979$	−8315.21	−0.271456
$980$	36878.7	1.20209
$981$	0	0
$982$	−33667.1	−1.09405
$983$	−26580.1	−0.862436	−0.431218	−	0.902248i	$-0.641916\pi$
$983$	−26580.1	−0.862436	−0.431218	+	0.902248i	$0.641916\pi$
$984$	0	0
$985$	−9985.71	−0.323016
$986$	−49205.4	−1.58927
$987$	0	0
$988$	41195.2	1.32651
$989$	3480.15	0.111893
$990$	0	0
$991$	3798.20	0.121749	0.0608747	−	0.998145i	$-0.480611\pi$
$991$	3798.20	0.121749	0.0608747	+	0.998145i	$0.480611\pi$
$992$	9854.98	0.315419
$993$	0	0
$994$	−18205.0	−0.580913
$995$	6006.04	0.191361
$996$	0	0
$997$	44535.8	1.41471	0.707353	−	0.706860i	$-0.249889\pi$
$997$	44535.8	1.41471	0.707353	+	0.706860i	$0.249889\pi$
$998$	−70998.9	−2.25193
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.e.1.4		32
3.2	odd	2		717.4.a.c.1.29	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.c.1.29	✓	32	3.2	odd	2
2151.4.a.e.1.4		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)

\(f(q)\)	\(=\)	\(q-4.79937 q^{2} +15.0339 q^{4} -7.96880 q^{5} -5.93046 q^{7} -33.7584 q^{8} +O(q^{10})\)	\(q-4.79937 q^{2} +15.0339 q^{4} -7.96880 q^{5} -5.93046 q^{7} -33.7584 q^{8} +38.2452 q^{10} -14.7981 q^{11} +19.5964 q^{13} +28.4624 q^{14} +41.7474 q^{16} -50.1371 q^{17} +139.829 q^{19} -119.802 q^{20} +71.0214 q^{22} +130.735 q^{23} -61.4982 q^{25} -94.0505 q^{26} -89.1580 q^{28} -204.489 q^{29} +141.379 q^{31} +69.7060 q^{32} +240.626 q^{34} +47.2586 q^{35} +300.801 q^{37} -671.091 q^{38} +269.014 q^{40} +7.52727 q^{41} +26.6198 q^{43} -222.473 q^{44} -627.447 q^{46} -495.608 q^{47} -307.830 q^{49} +295.152 q^{50} +294.611 q^{52} +430.509 q^{53} +117.923 q^{55} +200.202 q^{56} +981.418 q^{58} -200.816 q^{59} +269.379 q^{61} -678.531 q^{62} -668.523 q^{64} -156.160 q^{65} +162.214 q^{67} -753.757 q^{68} -226.812 q^{70} -639.615 q^{71} -312.142 q^{73} -1443.65 q^{74} +2102.18 q^{76} +87.7593 q^{77} -396.840 q^{79} -332.676 q^{80} -36.1261 q^{82} +451.541 q^{83} +399.532 q^{85} -127.758 q^{86} +499.558 q^{88} +561.912 q^{89} -116.216 q^{91} +1965.47 q^{92} +2378.61 q^{94} -1114.27 q^{95} -1458.44 q^{97} +1477.39 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8}+O(q^{10}) \) 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8	\( 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8} + 32 q^{10} - 154 q^{11} + 100 q^{13} - 42 q^{14} + 719 q^{16} - 32 q^{17} + 202 q^{19} + 132 q^{20} + 265 q^{22} - 552 q^{23} + 1086 q^{25} + 280 q^{26} + 390 q^{28} + 154 q^{29} + 560 q^{31} - 444 q^{32} + 156 q^{34} - 394 q^{35} + 914 q^{37} - 111 q^{38} + 257 q^{40} + 914 q^{41} + 1722 q^{43} - 1243 q^{44} + 584 q^{46} - 380 q^{47} + 2446 q^{49} + 454 q^{50} + 1552 q^{52} - 370 q^{53} + 918 q^{55} + 499 q^{56} + 2446 q^{58} - 492 q^{59} + 668 q^{61} - 578 q^{62} + 6475 q^{64} - 736 q^{65} + 4548 q^{67} - 5253 q^{68} + 7793 q^{70} - 258 q^{71} + 3096 q^{73} - 449 q^{74} + 6814 q^{76} - 3804 q^{77} + 2864 q^{79} + 1052 q^{80} + 14145 q^{82} - 2364 q^{83} + 3088 q^{85} - 2811 q^{86} + 8329 q^{88} + 4172 q^{89} + 7350 q^{91} - 13644 q^{92} + 6122 q^{94} - 3336 q^{95} + 6370 q^{97} - 1572 q^{98}+O(q^{100}) \) 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8 + 32 * q^10 - 154 * q^11 + 100 * q^13 - 42 * q^14 + 719 * q^16 - 32 * q^17 + 202 * q^19 + 132 * q^20 + 265 * q^22 - 552 * q^23 + 1086 * q^25 + 280 * q^26 + 390 * q^28 + 154 * q^29 + 560 * q^31 - 444 * q^32 + 156 * q^34 - 394 * q^35 + 914 * q^37 - 111 * q^38 + 257 * q^40 + 914 * q^41 + 1722 * q^43 - 1243 * q^44 + 584 * q^46 - 380 * q^47 + 2446 * q^49 + 454 * q^50 + 1552 * q^52 - 370 * q^53 + 918 * q^55 + 499 * q^56 + 2446 * q^58 - 492 * q^59 + 668 * q^61 - 578 * q^62 + 6475 * q^64 - 736 * q^65 + 4548 * q^67 - 5253 * q^68 + 7793 * q^70 - 258 * q^71 + 3096 * q^73 - 449 * q^74 + 6814 * q^76 - 3804 * q^77 + 2864 * q^79 + 1052 * q^80 + 14145 * q^82 - 2364 * q^83 + 3088 * q^85 - 2811 * q^86 + 8329 * q^88 + 4172 * q^89 + 7350 * q^91 - 13644 * q^92 + 6122 * q^94 - 3336 * q^95 + 6370 * q^97 - 1572 * q^98

Introduction

L-functions

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