LMFDB - Embedded newform 2401.4.a.c.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2401,4,Mod(1,2401)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2401.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2401, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2401 = 7^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2401.a (trivial)

Newform invariants

comment:select newform

sage:traces = [39,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$141.663585924$
Analytic rank:	$1$
Dimension:	$39$
Twist minimal:	no (minimal twist has level 49)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	2401.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.57033 q^{2} +9.05210 q^{3} +4.74728 q^{4} -15.7342 q^{5} -32.3190 q^{6} +11.6133 q^{8} +54.9406 q^{9} +56.1764 q^{10} +33.4349 q^{11} +42.9729 q^{12} +58.2042 q^{13} -142.428 q^{15} -79.4416 q^{16} +5.45990 q^{17} -196.156 q^{18} -30.1574 q^{19} -74.6947 q^{20} -119.374 q^{22} -121.066 q^{23} +105.125 q^{24} +122.565 q^{25} -207.809 q^{26} +252.921 q^{27} -169.051 q^{29} +508.514 q^{30} +98.7312 q^{31} +190.727 q^{32} +302.656 q^{33} -19.4937 q^{34} +260.819 q^{36} -374.320 q^{37} +107.672 q^{38} +526.871 q^{39} -182.726 q^{40} +8.20760 q^{41} -415.395 q^{43} +158.725 q^{44} -864.446 q^{45} +432.247 q^{46} +89.0617 q^{47} -719.113 q^{48} -437.598 q^{50} +49.4236 q^{51} +276.312 q^{52} +12.4428 q^{53} -903.013 q^{54} -526.071 q^{55} -272.988 q^{57} +603.568 q^{58} -427.574 q^{59} -676.145 q^{60} -342.910 q^{61} -352.503 q^{62} -45.4254 q^{64} -915.797 q^{65} -1080.58 q^{66} +286.141 q^{67} +25.9197 q^{68} -1095.90 q^{69} -57.1851 q^{71} +638.040 q^{72} -397.212 q^{73} +1336.45 q^{74} +1109.47 q^{75} -143.166 q^{76} -1881.10 q^{78} +1157.33 q^{79} +1249.95 q^{80} +806.072 q^{81} -29.3039 q^{82} +539.941 q^{83} -85.9072 q^{85} +1483.10 q^{86} -1530.26 q^{87} +388.288 q^{88} -624.657 q^{89} +3086.36 q^{90} -574.736 q^{92} +893.725 q^{93} -317.980 q^{94} +474.502 q^{95} +1726.48 q^{96} -147.308 q^{97} +1836.93 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 39 q + q^{2} - q^{3} + 145 q^{4} - 27 q^{5} - 41 q^{6} - 12 q^{8} + 312 q^{9} - 78 q^{10} + q^{11} + 91 q^{12} - 77 q^{13} - 161 q^{15} + 461 q^{16} - 211 q^{17} + 8 q^{18} - 314 q^{19} - 476 q^{20}+ \cdots + 5384 q^{99}+O(q^{100}) $ 39 * q + q^2 - q^3 + 145 * q^4 - 27 * q^5 - 41 * q^6 - 12 * q^8 + 312 * q^9 - 78 * q^10 + q^11 + 91 * q^12 - 77 * q^13 - 161 * q^15 + 461 * q^16 - 211 * q^17 + 8 * q^18 - 314 * q^19 - 476 * q^20 - 61 * q^22 - 69 * q^23 - 330 * q^24 + 606 * q^25 - 504 * q^26 + 50 * q^27 + 57 * q^29 + 42 * q^30 - 638 * q^31 - 1600 * q^32 - 1574 * q^33 - 1343 * q^34 + 782 * q^36 + 71 * q^37 - 1359 * q^38 - 84 * q^39 + 155 * q^40 - 1393 * q^41 - 125 * q^43 + 52 * q^44 - 1129 * q^45 - 1454 * q^46 - 1483 * q^47 + 974 * q^48 + 3074 * q^50 - 2044 * q^51 - 3899 * q^52 + 2213 * q^53 - 1142 * q^54 - 1604 * q^55 + 98 * q^57 + 2403 * q^58 - 2073 * q^59 - 1519 * q^60 - 2575 * q^61 - 1742 * q^62 + 1358 * q^64 + 1876 * q^65 + 48 * q^66 + 176 * q^67 - 3038 * q^68 + 638 * q^69 - 1259 * q^71 - 3799 * q^72 + 307 * q^73 + 3845 * q^74 - 131 * q^75 - 1974 * q^76 - 6041 * q^78 + 22 * q^79 + 804 * q^80 + 795 * q^81 - 8043 * q^82 - 6349 * q^83 + 3094 * q^85 + 1745 * q^86 - 9508 * q^87 + 1299 * q^88 - 2253 * q^89 - 11156 * q^90 + 1284 * q^92 - 3430 * q^93 - 2738 * q^94 - 3290 * q^95 - 3031 * q^96 + 770 * q^97 + 5384 * q^99

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 $

$2$	−3.57033	−1.26230	−0.631152	−	0.775659i	$-0.717418\pi$
$2$	−3.57033	−1.26230	−0.631152	+	0.775659i	$0.717418\pi$
$3$	9.05210	1.74208	0.871039	−	0.491214i	$-0.163447\pi$
$3$	9.05210	1.74208	0.871039	+	0.491214i	$0.163447\pi$
$4$	4.74728	0.593411
$5$	−15.7342	−1.40731	−0.703655	−	0.710542i	$-0.748450\pi$
$5$	−15.7342	−1.40731	−0.703655	+	0.710542i	$0.748450\pi$
$6$	−32.3190	−2.19903
$7$	0	0
$7$	0	0
$8$	11.6133	0.513239
$9$	54.9406	2.03484
$10$	56.1764	1.77645
$11$	33.4349	0.916454	0.458227	−	0.888835i	$-0.348485\pi$
$11$	33.4349	0.916454	0.458227	+	0.888835i	$0.348485\pi$
$12$	42.9729	1.03377
$13$	58.2042	1.24177	0.620883	−	0.783904i	$-0.286774\pi$
$13$	58.2042	1.24177	0.620883	+	0.783904i	$0.286774\pi$
$14$	0	0
$15$	−142.428	−2.45164
$16$	−79.4416	−1.24127
$17$	5.45990	0.0778953	0.0389477	−	0.999241i	$-0.487599\pi$
$17$	5.45990	0.0778953	0.0389477	+	0.999241i	$0.487599\pi$
$18$	−196.156	−2.56858
$19$	−30.1574	−0.364136	−0.182068	−	0.983286i	$-0.558279\pi$
$19$	−30.1574	−0.364136	−0.182068	+	0.983286i	$0.558279\pi$
$20$	−74.6947	−0.835113
$21$	0	0
$22$	−119.374	−1.15684
$23$	−121.066	−1.09757	−0.548784	−	0.835964i	$-0.684909\pi$
$23$	−121.066	−1.09757	−0.548784	+	0.835964i	$0.684909\pi$
$24$	105.125	0.894103
$25$	122.565	0.980521
$26$	−207.809	−1.56748
$27$	252.921	1.80277
$28$	0	0
$29$	−169.051	−1.08248	−0.541240	−	0.840868i	$-0.682045\pi$
$29$	−169.051	−1.08248	−0.541240	+	0.840868i	$0.682045\pi$
$30$	508.514	3.09472
$31$	98.7312	0.572021	0.286010	−	0.958227i	$-0.407671\pi$
$31$	98.7312	0.572021	0.286010	+	0.958227i	$0.407671\pi$
$32$	190.727	1.05363
$33$	302.656	1.59653
$34$	−19.4937	−0.0983276
$35$	0	0
$36$	260.819	1.20749
$37$	−374.320	−1.66318	−0.831592	−	0.555388i	$-0.812570\pi$
$37$	−374.320	−1.66318	−0.831592	+	0.555388i	$0.812570\pi$
$38$	107.672	0.459650
$39$	526.871	2.16325
$40$	−182.726	−0.722287
$41$	8.20760	0.0312637	0.0156318	−	0.999878i	$-0.495024\pi$
$41$	8.20760	0.0312637	0.0156318	+	0.999878i	$0.495024\pi$
$42$	0	0
$43$	−415.395	−1.47319	−0.736594	−	0.676335i	$-0.763567\pi$
$43$	−415.395	−1.47319	−0.736594	+	0.676335i	$0.763567\pi$
$44$	158.725	0.543833
$45$	−864.446	−2.86365
$46$	432.247	1.38546
$47$	89.0617	0.276404	0.138202	−	0.990404i	$-0.455868\pi$
$47$	89.0617	0.276404	0.138202	+	0.990404i	$0.455868\pi$
$48$	−719.113	−2.16240
$49$	0	0
$50$	−437.598	−1.23772
$51$	49.4236	0.135700
$52$	276.312	0.736877
$53$	12.4428	0.0322480	0.0161240	−	0.999870i	$-0.494867\pi$
$53$	12.4428	0.0322480	0.0161240	+	0.999870i	$0.494867\pi$
$54$	−903.013	−2.27564
$55$	−526.071	−1.28973
$56$	0	0
$57$	−272.988	−0.634353
$58$	603.568	1.36642
$59$	−427.574	−0.943482	−0.471741	−	0.881737i	$-0.656374\pi$
$59$	−427.574	−0.943482	−0.471741	+	0.881737i	$0.656374\pi$
$60$	−676.145	−1.45483
$61$	−342.910	−0.719757	−0.359879	−	0.932999i	$-0.617182\pi$
$61$	−342.910	−0.719757	−0.359879	+	0.932999i	$0.617182\pi$
$62$	−352.503	−0.722064
$63$	0	0
$64$	−45.4254	−0.0887216
$65$	−915.797	−1.74755
$66$	−1080.58	−2.01531
$67$	286.141	0.521757	0.260878	−	0.965372i	$-0.415988\pi$
$67$	286.141	0.521757	0.260878	+	0.965372i	$0.415988\pi$
$68$	25.9197	0.0462239
$69$	−1095.90	−1.91205
$70$	0	0
$71$	−57.1851	−0.0955863	−0.0477931	−	0.998857i	$-0.515219\pi$
$71$	−57.1851	−0.0955863	−0.0477931	+	0.998857i	$0.515219\pi$
$72$	638.040	1.04436
$73$	−397.212	−0.636851	−0.318425	−	0.947948i	$-0.603154\pi$
$73$	−397.212	−0.636851	−0.318425	+	0.947948i	$0.603154\pi$
$74$	1336.45	2.09944
$75$	1109.47	1.70814
$76$	−143.166	−0.216082
$77$	0	0
$78$	−1881.10	−2.73068
$79$	1157.33	1.64823	0.824116	−	0.566421i	$-0.191672\pi$
$79$	1157.33	1.64823	0.824116	+	0.566421i	$0.191672\pi$
$80$	1249.95	1.74686
$81$	806.072	1.10572
$82$	−29.3039	−0.0394643
$83$	539.941	0.714050	0.357025	−	0.934095i	$-0.383791\pi$
$83$	539.941	0.714050	0.357025	+	0.934095i	$0.383791\pi$
$84$	0	0
$85$	−85.9072	−0.109623
$86$	1483.10	1.85961
$87$	−1530.26	−1.88577
$88$	388.288	0.470360
$89$	−624.657	−0.743972	−0.371986	−	0.928238i	$-0.621323\pi$
$89$	−624.657	−0.743972	−0.371986	+	0.928238i	$0.621323\pi$
$90$	3086.36	3.61479
$91$	0	0
$92$	−574.736	−0.651308
$93$	893.725	0.996505
$94$	−317.980	−0.348906
$95$	474.502	0.512452
$96$	1726.48	1.83550
$97$	−147.308	−0.154194	−0.0770970	−	0.997024i	$-0.524565\pi$
$97$	−147.308	−0.154194	−0.0770970	+	0.997024i	$0.524565\pi$
$98$	0	0
$99$	1836.93	1.86483
$100$	581.852	0.581852
$101$	593.756	0.584960	0.292480	−	0.956272i	$-0.405520\pi$
$101$	593.756	0.584960	0.292480	+	0.956272i	$0.405520\pi$
$102$	−176.459	−0.171294
$103$	1863.22	1.78241	0.891206	−	0.453600i	$-0.149860\pi$
$103$	1863.22	1.78241	0.891206	+	0.453600i	$0.149860\pi$
$104$	675.942	0.637323
$105$	0	0
$106$	−44.4248	−0.0407068
$107$	230.126	0.207917	0.103959	−	0.994582i	$-0.466849\pi$
$107$	230.126	0.207917	0.103959	+	0.994582i	$0.466849\pi$
$108$	1200.69	1.06978
$109$	−0.700065	−0.000615175 0	−0.000307587	−	1.00000i	$-0.500098\pi$
$109$	−0.700065	−0.000615175 0	−0.000307587		1.00000i	$0.500098\pi$
$110$	1878.25	1.62804
$111$	−3388.38	−2.89739
$112$	0	0
$113$	−934.188	−0.777708	−0.388854	−	0.921299i	$-0.627129\pi$
$113$	−934.188	−0.777708	−0.388854	+	0.921299i	$0.627129\pi$
$114$	974.657	0.800746
$115$	1904.88	1.54462
$116$	−802.532	−0.642355
$117$	3197.77	2.52679
$118$	1526.58	1.19096
$119$	0	0
$120$	−1654.05	−1.25828
$121$	−213.110	−0.160113
$122$	1224.30	0.908552
$123$	74.2960	0.0544638
$124$	468.705	0.339443
$125$	38.3108	0.0274130
$126$	0	0
$127$	1447.87	1.01163	0.505816	−	0.862641i	$-0.331191\pi$
$127$	1447.87	1.01163	0.505816	+	0.862641i	$0.331191\pi$
$128$	−1363.63	−0.941632
$129$	−3760.20	−2.56641
$130$	3269.70	2.20594
$131$	1260.85	0.840925	0.420463	−	0.907310i	$-0.361868\pi$
$131$	1260.85	0.840925	0.420463	+	0.907310i	$0.361868\pi$
$132$	1436.79	0.947400
$133$	0	0
$134$	−1021.62	−0.658615
$135$	−3979.51	−2.53705
$136$	63.4074	0.0399789
$137$	−192.057	−0.119771	−0.0598853	−	0.998205i	$-0.519073\pi$
$137$	−192.057	−0.119771	−0.0598853	+	0.998205i	$0.519073\pi$
$138$	3912.74	2.41359
$139$	188.714	0.115155	0.0575774	−	0.998341i	$-0.481662\pi$
$139$	188.714	0.115155	0.0575774	+	0.998341i	$0.481662\pi$
$140$	0	0
$141$	806.196	0.481517
$142$	204.170	0.120659
$143$	1946.05	1.13802
$144$	−4364.57	−2.52579
$145$	2659.88	1.52339
$146$	1418.18	0.803899
$147$	0	0
$148$	−1777.00	−0.986950
$149$	−3277.65	−1.80212	−0.901058	−	0.433699i	$-0.857208\pi$
$149$	−3277.65	−1.80212	−0.901058	+	0.433699i	$0.857208\pi$
$150$	−3961.19	−2.15620
$151$	−1952.70	−1.05237	−0.526187	−	0.850369i	$-0.676379\pi$
$151$	−1952.70	−1.05237	−0.526187	+	0.850369i	$0.676379\pi$
$152$	−350.226	−0.186889
$153$	299.970	0.158504
$154$	0	0
$155$	−1553.46	−0.805010
$156$	2501.21	1.28370
$157$	−354.446	−0.180178	−0.0900888	−	0.995934i	$-0.528715\pi$
$157$	−354.446	−0.180178	−0.0900888	+	0.995934i	$0.528715\pi$
$158$	−4132.07	−2.08057
$159$	112.633	0.0561786
$160$	−3000.93	−1.48278
$161$	0	0
$162$	−2877.95	−1.39576
$163$	1179.57	0.566817	0.283408	−	0.958999i	$-0.408535\pi$
$163$	1179.57	0.566817	0.283408	+	0.958999i	$0.408535\pi$
$164$	38.9638	0.0185522
$165$	−4762.05	−2.24682
$166$	−1927.77	−0.901348
$167$	−3764.36	−1.74428	−0.872140	−	0.489256i	$-0.837268\pi$
$167$	−3764.36	−1.74428	−0.872140	+	0.489256i	$0.837268\pi$
$168$	0	0
$169$	1190.73	0.541981
$170$	306.717	0.138377
$171$	−1656.86	−0.740956
$172$	−1972.00	−0.874206
$173$	−2932.66	−1.28882	−0.644411	−	0.764679i	$-0.722897\pi$
$173$	−2932.66	−1.28882	−0.644411	+	0.764679i	$0.722897\pi$
$174$	5463.56	2.38041
$175$	0	0
$176$	−2656.12	−1.13757
$177$	−3870.45	−1.64362
$178$	2230.23	0.939118
$179$	−617.174	−0.257708	−0.128854	−	0.991664i	$-0.541130\pi$
$179$	−617.174	−0.257708	−0.128854	+	0.991664i	$0.541130\pi$
$180$	−4103.77	−1.69932
$181$	−3489.72	−1.43309	−0.716544	−	0.697542i	$-0.754277\pi$
$181$	−3489.72	−1.43309	−0.716544	+	0.697542i	$0.754277\pi$
$182$	0	0
$183$	−3104.06	−1.25387
$184$	−1405.98	−0.563315
$185$	5889.62	2.34061
$186$	−3190.90	−1.25789
$187$	182.551	0.0713875
$188$	422.801	0.164021
$189$	0	0
$190$	−1694.13	−0.646870
$191$	2043.44	0.774128	0.387064	−	0.922053i	$-0.373489\pi$
$191$	2043.44	0.774128	0.387064	+	0.922053i	$0.373489\pi$
$192$	−411.196	−0.154560
$193$	5150.60	1.92098	0.960489	−	0.278319i	$-0.0897773\pi$
$193$	5150.60	1.92098	0.960489	+	0.278319i	$0.0897773\pi$
$194$	525.937	0.194640
$195$	−8289.89	−3.04437
$196$	0	0
$197$	−4426.06	−1.60073	−0.800365	−	0.599513i	$-0.795361\pi$
$197$	−4426.06	−1.60073	−0.800365	+	0.599513i	$0.795361\pi$
$198$	−6558.46	−2.35399
$199$	−4352.49	−1.55045	−0.775226	−	0.631684i	$-0.782364\pi$
$199$	−4352.49	−1.55045	−0.775226	+	0.631684i	$0.782364\pi$
$200$	1423.38	0.503242
$201$	2590.18	0.908941
$202$	−2119.91	−0.738397
$203$	0	0
$204$	234.628	0.0805257
$205$	−129.140	−0.0439977
$206$	−6652.31	−2.24994
$207$	−6651.45	−2.23337
$208$	−4623.84	−1.54137
$209$	−1008.31	−0.333713
$210$	0	0
$211$	4201.16	1.37071	0.685354	−	0.728210i	$-0.259647\pi$
$211$	4201.16	1.37071	0.685354	+	0.728210i	$0.259647\pi$
$212$	59.0693	0.0191363
$213$	−517.646	−0.166519
$214$	−821.627	−0.262455
$215$	6535.90	2.07323
$216$	2937.24	0.925250
$217$	0	0
$218$	2.49946	0.000776537 0
$219$	−3595.60	−1.10944
$220$	−2497.41	−0.765342
$221$	317.789	0.0967277
$222$	12097.7	3.65739
$223$	524.091	0.157380	0.0786900	−	0.996899i	$-0.474926\pi$
$223$	524.091	0.157380	0.0786900	+	0.996899i	$0.474926\pi$
$224$	0	0
$225$	6733.80	1.99520
$226$	3335.36	0.981704
$227$	−5501.25	−1.60851	−0.804253	−	0.594287i	$-0.797434\pi$
$227$	−5501.25	−1.60851	−0.804253	+	0.594287i	$0.797434\pi$
$228$	−1295.95	−0.376432
$229$	−4777.44	−1.37861	−0.689306	−	0.724470i	$-0.742084\pi$
$229$	−4777.44	−1.37861	−0.689306	+	0.724470i	$0.742084\pi$
$230$	−6801.06	−1.94978
$231$	0	0
$232$	−1963.23	−0.555571
$233$	−1483.72	−0.417175	−0.208587	−	0.978004i	$-0.566887\pi$
$233$	−1483.72	−0.417175	−0.208587	+	0.978004i	$0.566887\pi$
$234$	−11417.1	−3.18958
$235$	−1401.31	−0.388986
$236$	−2029.82	−0.559872
$237$	10476.3	2.87135
$238$	0	0
$239$	296.412	0.0802231	0.0401115	−	0.999195i	$-0.487229\pi$
$239$	296.412	0.0802231	0.0401115	+	0.999195i	$0.487229\pi$
$240$	11314.7	3.04316
$241$	2661.86	0.711476	0.355738	−	0.934586i	$-0.384230\pi$
$241$	2661.86	0.711476	0.355738	+	0.934586i	$0.384230\pi$
$242$	760.874	0.202111
$243$	467.780	0.123490
$244$	−1627.89	−0.427111
$245$	0	0
$246$	−265.262	−0.0687499
$247$	−1755.29	−0.452171
$248$	1146.59	0.293584
$249$	4887.60	1.24393
$250$	−136.782	−0.0346035
$251$	4220.16	1.06125	0.530625	−	0.847607i	$-0.321957\pi$
$251$	4220.16	1.06125	0.530625	+	0.847607i	$0.321957\pi$
$252$	0	0
$253$	−4047.83	−1.00587
$254$	−5169.37	−1.27699
$255$	−777.641	−0.190972
$256$	5232.02	1.27735
$257$	−3005.41	−0.729464	−0.364732	−	0.931112i	$-0.618839\pi$
$257$	−3005.41	−0.729464	−0.364732	+	0.931112i	$0.618839\pi$
$258$	13425.2	3.23959
$259$	0	0
$260$	−4347.55	−1.03701
$261$	−9287.75	−2.20267
$262$	−4501.67	−1.06150
$263$	−2239.53	−0.525078	−0.262539	−	0.964921i	$-0.584560\pi$
$263$	−2239.53	−0.525078	−0.262539	+	0.964921i	$0.584560\pi$
$264$	3514.83	0.819404
$265$	−195.777	−0.0453830
$266$	0	0
$267$	−5654.46	−1.29606
$268$	1358.39	0.309616
$269$	−3679.12	−0.833904	−0.416952	−	0.908928i	$-0.636902\pi$
$269$	−3679.12	−0.833904	−0.416952	+	0.908928i	$0.636902\pi$
$270$	14208.2	3.20253
$271$	−3652.63	−0.818751	−0.409376	−	0.912366i	$-0.634253\pi$
$271$	−3652.63	−0.818751	−0.409376	+	0.912366i	$0.634253\pi$
$272$	−433.743	−0.0966895
$273$	0	0
$274$	685.709	0.151187
$275$	4097.95	0.898602
$276$	−5202.57	−1.13463
$277$	551.737	0.119678	0.0598388	−	0.998208i	$-0.480941\pi$
$277$	551.737	0.119678	0.0598388	+	0.998208i	$0.480941\pi$
$278$	−673.773	−0.145360
$279$	5424.35	1.16397
$280$	0	0
$281$	597.604	0.126869	0.0634343	−	0.997986i	$-0.479795\pi$
$281$	597.604	0.126869	0.0634343	+	0.997986i	$0.479795\pi$
$282$	−2878.39	−0.607821
$283$	−1120.91	−0.235445	−0.117723	−	0.993047i	$-0.537559\pi$
$283$	−1120.91	−0.235445	−0.117723	+	0.993047i	$0.537559\pi$
$284$	−271.474	−0.0567219
$285$	4295.24	0.892731
$286$	−6948.05	−1.43653
$287$	0	0
$288$	10478.6	2.14396
$289$	−4883.19	−0.993932
$290$	−9496.65	−1.92297
$291$	−1333.44	−0.268618
$292$	−1885.68	−0.377914
$293$	4458.67	0.889005	0.444502	−	0.895778i	$-0.353381\pi$
$293$	4458.67	0.889005	0.444502	+	0.895778i	$0.353381\pi$
$294$	0	0
$295$	6727.54	1.32777
$296$	−4347.08	−0.853611
$297$	8456.38	1.65215
$298$	11702.3	2.27482
$299$	−7046.57	−1.36292
$300$	5266.98	1.01363
$301$	0	0
$302$	6971.79	1.32841
$303$	5374.74	1.01905
$304$	2395.75	0.451992
$305$	5395.42	1.01292
$306$	−1070.99	−0.200081
$307$	4282.26	0.796097	0.398048	−	0.917364i	$-0.369688\pi$
$307$	4282.26	0.796097	0.398048	+	0.917364i	$0.369688\pi$
$308$	0	0
$309$	16866.0	3.10510
$310$	5546.36	1.01617
$311$	−2991.35	−0.545414	−0.272707	−	0.962097i	$-0.587919\pi$
$311$	−2991.35	−0.545414	−0.272707	+	0.962097i	$0.587919\pi$
$312$	6118.70	1.11027
$313$	−3715.35	−0.670940	−0.335470	−	0.942051i	$-0.608895\pi$
$313$	−3715.35	−0.670940	−0.335470	+	0.942051i	$0.608895\pi$
$314$	1265.49	0.227439
$315$	0	0
$316$	5494.20	0.978078
$317$	−3217.96	−0.570154	−0.285077	−	0.958505i	$-0.592019\pi$
$317$	−3217.96	−0.570154	−0.285077	+	0.958505i	$0.592019\pi$
$318$	−402.138	−0.0709144
$319$	−5652.19	−0.992043
$320$	714.733	0.124859
$321$	2083.13	0.362208
$322$	0	0
$323$	−164.656	−0.0283645
$324$	3826.65	0.656148
$325$	7133.81	1.21758
$326$	−4211.46	−0.715495
$327$	−6.33706	−0.00107168
$328$	95.3171	0.0160458
$329$	0	0
$330$	17002.1	2.83617
$331$	2969.05	0.493032	0.246516	−	0.969139i	$-0.420714\pi$
$331$	2969.05	0.493032	0.246516	+	0.969139i	$0.420714\pi$
$332$	2563.25	0.423725
$333$	−20565.3	−3.38431
$334$	13440.0	2.20181
$335$	−4502.20	−0.734273
$336$	0	0
$337$	−5535.92	−0.894838	−0.447419	−	0.894324i	$-0.647657\pi$
$337$	−5535.92	−0.894838	−0.447419	+	0.894324i	$0.647657\pi$
$338$	−4251.31	−0.684145
$339$	−8456.37	−1.35483
$340$	−407.826	−0.0650514
$341$	3301.06	0.524231
$342$	5915.56	0.935312
$343$	0	0
$344$	−4824.09	−0.756098
$345$	17243.2	2.69085
$346$	10470.6	1.62688
$347$	5908.94	0.914146	0.457073	−	0.889429i	$-0.348898\pi$
$347$	5908.94	0.914146	0.457073	+	0.889429i	$0.348898\pi$
$348$	−7264.60	−1.11903
$349$	7089.91	1.08743	0.543717	−	0.839269i	$-0.317017\pi$
$349$	7089.91	1.08743	0.543717	+	0.839269i	$0.317017\pi$
$350$	0	0
$351$	14721.1	2.23861
$352$	6376.92	0.965599
$353$	3735.91	0.563294	0.281647	−	0.959518i	$-0.409119\pi$
$353$	3735.91	0.563294	0.281647	+	0.959518i	$0.409119\pi$
$354$	13818.8	2.07475
$355$	899.762	0.134519
$356$	−2965.42	−0.441481
$357$	0	0
$358$	2203.52	0.325306
$359$	−6143.56	−0.903189	−0.451594	−	0.892223i	$-0.649145\pi$
$359$	−6143.56	−0.903189	−0.451594	+	0.892223i	$0.649145\pi$
$360$	−10039.1	−1.46974
$361$	−5949.53	−0.867405
$362$	12459.5	1.80899
$363$	−1929.09	−0.278929
$364$	0	0
$365$	6249.81	0.896246
$366$	11082.5	1.58277
$367$	−7746.50	−1.10181	−0.550904	−	0.834568i	$-0.685717\pi$
$367$	−7746.50	−1.10181	−0.550904	+	0.834568i	$0.685717\pi$
$368$	9617.69	1.36238
$369$	450.930	0.0636165
$370$	−21027.9	−2.95457
$371$	0	0
$372$	4242.77	0.591337
$373$	8451.96	1.17326	0.586630	−	0.809855i	$-0.300454\pi$
$373$	8451.96	1.17326	0.586630	+	0.809855i	$0.300454\pi$
$374$	−651.768	−0.0901127
$375$	346.793	0.0477555
$376$	1034.30	0.141861
$377$	−9839.47	−1.34419
$378$	0	0
$379$	−884.442	−0.119870	−0.0599350	−	0.998202i	$-0.519089\pi$
$379$	−884.442	−0.119870	−0.0599350	+	0.998202i	$0.519089\pi$
$380$	2252.60	0.304094
$381$	13106.2	1.76234
$382$	−7295.78	−0.977184
$383$	−10090.8	−1.34626	−0.673130	−	0.739524i	$-0.735050\pi$
$383$	−10090.8	−1.34626	−0.673130	+	0.739524i	$0.735050\pi$
$384$	−12343.7	−1.64040
$385$	0	0
$386$	−18389.4	−2.42486
$387$	−22822.0	−2.99770
$388$	−699.311	−0.0915003
$389$	−12217.3	−1.59239	−0.796197	−	0.605037i	$-0.793158\pi$
$389$	−12217.3	−1.59239	−0.796197	+	0.605037i	$0.793158\pi$
$390$	29597.7	3.84291
$391$	−661.010	−0.0854954
$392$	0	0
$393$	11413.4	1.46496
$394$	15802.5	2.02061
$395$	−18209.7	−2.31957
$396$	8720.43	1.10661
$397$	4402.84	0.556605	0.278303	−	0.960493i	$-0.410228\pi$
$397$	4402.84	0.556605	0.278303	+	0.960493i	$0.410228\pi$
$398$	15539.8	1.95714
$399$	0	0
$400$	−9736.77	−1.21710
$401$	2205.42	0.274647	0.137324	−	0.990526i	$-0.456150\pi$
$401$	2205.42	0.274647	0.137324	+	0.990526i	$0.456150\pi$
$402$	−9247.80	−1.14736
$403$	5746.57	0.710316
$404$	2818.73	0.347121
$405$	−12682.9	−1.55610
$406$	0	0
$407$	−12515.3	−1.52423
$408$	573.970	0.0696464
$409$	−13478.3	−1.62948	−0.814741	−	0.579826i	$-0.803121\pi$
$409$	−13478.3	−1.62948	−0.814741	+	0.579826i	$0.803121\pi$
$410$	461.073	0.0555385
$411$	−1738.52	−0.208650
$412$	8845.23	1.05770
$413$	0	0
$414$	23747.9	2.81919
$415$	−8495.53	−1.00489
$416$	11101.1	1.30836
$417$	1708.26	0.200609
$418$	3600.00	0.421248
$419$	−9113.61	−1.06260	−0.531300	−	0.847184i	$-0.678296\pi$
$419$	−9113.61	−1.06260	−0.531300	+	0.847184i	$0.678296\pi$
$420$	0	0
$421$	−4084.96	−0.472895	−0.236448	−	0.971644i	$-0.575983\pi$
$421$	−4084.96	−0.472895	−0.236448	+	0.971644i	$0.575983\pi$
$422$	−14999.5	−1.73025
$423$	4893.10	0.562437
$424$	144.501	0.0165510
$425$	669.193	0.0763780
$426$	1848.17	0.210197
$427$	0	0
$428$	1092.47	0.123380
$429$	17615.9	1.98252
$430$	−23335.4	−2.61705
$431$	13099.8	1.46402	0.732011	−	0.681293i	$-0.238582\pi$
$431$	13099.8	1.46402	0.732011	+	0.681293i	$0.238582\pi$
$432$	−20092.4	−2.23773
$433$	−1469.20	−0.163061	−0.0815303	−	0.996671i	$-0.525981\pi$
$433$	−1469.20	−0.163061	−0.0815303	+	0.996671i	$0.525981\pi$
$434$	0	0
$435$	24077.5	2.65386
$436$	−3.32341	−0.000365051 0
$437$	3651.04	0.399663
$438$	12837.5	1.40045
$439$	3633.33	0.395010	0.197505	−	0.980302i	$-0.436716\pi$
$439$	3633.33	0.395010	0.197505	+	0.980302i	$0.436716\pi$
$440$	−6109.41	−0.661942
$441$	0	0
$442$	−1134.61	−0.122100
$443$	−2425.80	−0.260165	−0.130083	−	0.991503i	$-0.541524\pi$
$443$	−2425.80	−0.260165	−0.130083	+	0.991503i	$0.541524\pi$
$444$	−16085.6	−1.71934
$445$	9828.48	1.04700
$446$	−1871.18	−0.198661
$447$	−29669.6	−3.13943
$448$	0	0
$449$	6523.46	0.685659	0.342830	−	0.939398i	$-0.388615\pi$
$449$	6523.46	0.685659	0.342830	+	0.939398i	$0.388615\pi$
$450$	−24041.9	−2.51855
$451$	274.420	0.0286517
$452$	−4434.86	−0.461500
$453$	−17676.0	−1.83332
$454$	19641.3	2.03042
$455$	0	0
$456$	−3170.28	−0.325575
$457$	−3857.27	−0.394826	−0.197413	−	0.980320i	$-0.563254\pi$
$457$	−3857.27	−0.394826	−0.197413	+	0.980320i	$0.563254\pi$
$458$	17057.1	1.74023
$459$	1380.92	0.140427
$460$	9043.01	0.916593
$461$	12782.5	1.29141	0.645704	−	0.763588i	$-0.276564\pi$
$461$	12782.5	1.29141	0.645704	+	0.763588i	$0.276564\pi$
$462$	0	0
$463$	−16260.6	−1.63217	−0.816084	−	0.577933i	$-0.803860\pi$
$463$	−16260.6	−1.63217	−0.816084	+	0.577933i	$0.803860\pi$
$464$	13429.7	1.34366
$465$	−14062.0	−1.40239
$466$	5297.38	0.526601
$467$	−6051.48	−0.599634	−0.299817	−	0.953997i	$-0.596926\pi$
$467$	−6051.48	−0.599634	−0.299817	+	0.953997i	$0.596926\pi$
$468$	15180.7	1.49942
$469$	0	0
$470$	5003.16	0.491018
$471$	−3208.48	−0.313883
$472$	−4965.54	−0.484232
$473$	−13888.7	−1.35011
$474$	−37403.9	−3.62451
$475$	−3696.24	−0.357043
$476$	0	0
$477$	683.613	0.0656195
$478$	−1058.29	−0.101266
$479$	−6373.71	−0.607980	−0.303990	−	0.952675i	$-0.598319\pi$
$479$	−6373.71	−0.607980	−0.303990	+	0.952675i	$0.598319\pi$
$480$	−27164.8	−2.58312
$481$	−21787.0	−2.06528
$482$	−9503.75	−0.898099
$483$	0	0
$484$	−1011.69	−0.0950126
$485$	2317.77	0.216999
$486$	−1670.13	−0.155882
$487$	16775.9	1.56096	0.780480	−	0.625180i	$-0.214975\pi$
$487$	16775.9	1.56096	0.780480	+	0.625180i	$0.214975\pi$
$488$	−3982.31	−0.369408
$489$	10677.6	0.987440
$490$	0	0
$491$	20005.9	1.83881	0.919404	−	0.393314i	$-0.128672\pi$
$491$	20005.9	1.83881	0.919404	+	0.393314i	$0.128672\pi$
$492$	352.704	0.0323194
$493$	−923.000	−0.0843202
$494$	6266.96	0.570777
$495$	−28902.6	−2.62440
$496$	−7843.36	−0.710035
$497$	0	0
$498$	−17450.4	−1.57022
$499$	16434.6	1.47438	0.737189	−	0.675687i	$-0.236153\pi$
$499$	16434.6	1.47438	0.737189	+	0.675687i	$0.236153\pi$
$500$	181.872	0.0162671
$501$	−34075.4	−3.03867
$502$	−15067.4	−1.33962
$503$	13761.0	1.21982	0.609912	−	0.792469i	$-0.291205\pi$
$503$	13761.0	1.21982	0.609912	+	0.792469i	$0.291205\pi$
$504$	0	0
$505$	−9342.28	−0.823220
$506$	14452.1	1.26971
$507$	10778.6	0.944173
$508$	6873.43	0.600314
$509$	−1983.85	−0.172756	−0.0863778	−	0.996262i	$-0.527529\pi$
$509$	−1983.85	−0.172756	−0.0863778	+	0.996262i	$0.527529\pi$
$510$	2776.44	0.241064
$511$	0	0
$512$	−7771.01	−0.670768
$513$	−7627.44	−0.656451
$514$	10730.3	0.920805
$515$	−29316.3	−2.50840
$516$	−17850.7	−1.52293
$517$	2977.76	0.253311
$518$	0	0
$519$	−26546.8	−2.24523
$520$	−10635.4	−0.896911
$521$	−6683.75	−0.562035	−0.281018	−	0.959703i	$-0.590672\pi$
$521$	−6683.75	−0.562035	−0.281018	+	0.959703i	$0.590672\pi$
$522$	33160.4	2.78044
$523$	1301.86	0.108845	0.0544227	−	0.998518i	$-0.482668\pi$
$523$	1301.86	0.108845	0.0544227	+	0.998518i	$0.482668\pi$
$524$	5985.63	0.499014
$525$	0	0
$526$	7995.88	0.662808
$527$	539.062	0.0445577
$528$	−24043.5	−1.98174
$529$	2490.04	0.204655
$530$	698.989	0.0572871
$531$	−23491.2	−1.91983
$532$	0	0
$533$	477.717	0.0388222
$534$	20188.3	1.63602
$535$	−3620.85	−0.292604
$536$	3323.04	0.267786
$537$	−5586.72	−0.448947
$538$	13135.7	1.05264
$539$	0	0
$540$	−18891.9	−1.50551
$541$	4070.73	0.323501	0.161751	−	0.986832i	$-0.448286\pi$
$541$	4070.73	0.323501	0.161751	+	0.986832i	$0.448286\pi$
$542$	13041.1	1.03351
$543$	−31589.3	−2.49655
$544$	1041.35	0.0820725
$545$	11.0150	0.000865741 0
$546$	0	0
$547$	−5842.19	−0.456661	−0.228331	−	0.973584i	$-0.573327\pi$
$547$	−5842.19	−0.456661	−0.228331	+	0.973584i	$0.573327\pi$
$548$	−911.751	−0.0710732
$549$	−18839.7	−1.46459
$550$	−14631.0	−1.13431
$551$	5098.13	0.394170
$552$	−12727.0	−0.981339
$553$	0	0
$554$	−1969.89	−0.151069
$555$	53313.5	4.07753
$556$	895.880	0.0683341
$557$	−20828.1	−1.58440	−0.792202	−	0.610258i	$-0.791066\pi$
$557$	−20828.1	−1.58440	−0.792202	+	0.610258i	$0.791066\pi$
$558$	−19366.7	−1.46928
$559$	−24177.7	−1.82935
$560$	0	0
$561$	1652.47	0.124363
$562$	−2133.65	−0.160147
$563$	−9297.26	−0.695973	−0.347987	−	0.937500i	$-0.613135\pi$
$563$	−9297.26	−0.695973	−0.347987	+	0.937500i	$0.613135\pi$
$564$	3827.24	0.285737
$565$	14698.7	1.09448
$566$	4002.02	0.297204
$567$	0	0
$568$	−664.107	−0.0490586
$569$	22042.2	1.62400	0.812002	−	0.583654i	$-0.198378\pi$
$569$	22042.2	1.62400	0.812002	+	0.583654i	$0.198378\pi$
$570$	−15335.5	−1.12690
$571$	−1800.08	−0.131928	−0.0659641	−	0.997822i	$-0.521012\pi$
$571$	−1800.08	−0.131928	−0.0659641	+	0.997822i	$0.521012\pi$
$572$	9238.46	0.675313
$573$	18497.5	1.34859
$574$	0	0
$575$	−14838.5	−1.07619
$576$	−2495.70	−0.180534
$577$	26384.5	1.90364	0.951821	−	0.306655i	$-0.0992097\pi$
$577$	26384.5	1.90364	0.951821	+	0.306655i	$0.0992097\pi$
$578$	17434.6	1.25464
$579$	46623.8	3.34649
$580$	12627.2	0.903993
$581$	0	0
$582$	4760.84	0.339077
$583$	416.022	0.0295538
$584$	−4612.93	−0.326857
$585$	−50314.4	−3.55598
$586$	−15918.9	−1.12219
$587$	7714.83	0.542462	0.271231	−	0.962514i	$-0.412569\pi$
$587$	7714.83	0.542462	0.271231	+	0.962514i	$0.412569\pi$
$588$	0	0
$589$	−2977.47	−0.208293
$590$	−24019.6	−1.67605
$591$	−40065.2	−2.78860
$592$	29736.5	2.06447
$593$	16086.7	1.11400	0.556998	−	0.830514i	$-0.311953\pi$
$593$	16086.7	1.11400	0.556998	+	0.830514i	$0.311953\pi$
$594$	−30192.1	−2.08552
$595$	0	0
$596$	−15559.9	−1.06939
$597$	−39399.2	−2.70101
$598$	25158.6	1.72042
$599$	6392.53	0.436046	0.218023	−	0.975944i	$-0.430039\pi$
$599$	6392.53	0.436046	0.218023	+	0.975944i	$0.430039\pi$
$600$	12884.6	0.876687
$601$	2990.17	0.202948	0.101474	−	0.994838i	$-0.467644\pi$
$601$	2990.17	0.202948	0.101474	+	0.994838i	$0.467644\pi$
$602$	0	0
$603$	15720.8	1.06169
$604$	−9270.02	−0.624490
$605$	3353.12	0.225328
$606$	−19189.6	−1.28635
$607$	−4184.97	−0.279840	−0.139920	−	0.990163i	$-0.544684\pi$
$607$	−4184.97	−0.279840	−0.139920	+	0.990163i	$0.544684\pi$
$608$	−5751.82	−0.383663
$609$	0	0
$610$	−19263.5	−1.27861
$611$	5183.77	0.343229
$612$	1424.04	0.0940581
$613$	17371.7	1.14460	0.572299	−	0.820045i	$-0.306052\pi$
$613$	17371.7	1.14460	0.572299	+	0.820045i	$0.306052\pi$
$614$	−15289.1	−1.00492
$615$	−1168.99	−0.0766474
$616$	0	0
$617$	−14811.1	−0.966405	−0.483202	−	0.875509i	$-0.660526\pi$
$617$	−14811.1	−0.966405	−0.483202	+	0.875509i	$0.660526\pi$
$618$	−60217.4	−3.91958
$619$	994.347	0.0645657	0.0322829	−	0.999479i	$-0.489722\pi$
$619$	994.347	0.0645657	0.0322829	+	0.999479i	$0.489722\pi$
$620$	−7374.70	−0.477702
$621$	−30620.2	−1.97866
$622$	10680.1	0.688478
$623$	0	0
$624$	−41855.4	−2.68519
$625$	−15923.4	−1.01910
$626$	13265.1	0.846930
$627$	−9127.31	−0.581355
$628$	−1682.66	−0.106919
$629$	−2043.75	−0.129554
$630$	0	0
$631$	−22236.5	−1.40289	−0.701443	−	0.712725i	$-0.747461\pi$
$631$	−22236.5	−1.40289	−0.701443	+	0.712725i	$0.747461\pi$
$632$	13440.5	0.845937
$633$	38029.3	2.38788
$634$	11489.2	0.719707
$635$	−22781.0	−1.42368
$636$	534.702	0.0333370
$637$	0	0
$638$	20180.2	1.25226
$639$	−3141.78	−0.194502
$640$	21455.6	1.32517
$641$	−5824.34	−0.358889	−0.179444	−	0.983768i	$-0.557430\pi$
$641$	−5824.34	−0.358889	−0.179444	+	0.983768i	$0.557430\pi$
$642$	−7437.46	−0.457217
$643$	3182.80	0.195206	0.0976029	−	0.995225i	$-0.468882\pi$
$643$	3182.80	0.195206	0.0976029	+	0.995225i	$0.468882\pi$
$644$	0	0
$645$	59163.7	3.61173
$646$	587.878	0.0358046
$647$	−23020.5	−1.39881	−0.699405	−	0.714726i	$-0.746551\pi$
$647$	−23020.5	−1.39881	−0.699405	+	0.714726i	$0.746551\pi$
$648$	9361.14	0.567501
$649$	−14295.9	−0.864658
$650$	−25470.1	−1.53695
$651$	0	0
$652$	5599.76	0.336355
$653$	−4902.87	−0.293819	−0.146910	−	0.989150i	$-0.546933\pi$
$653$	−4902.87	−0.293819	−0.146910	+	0.989150i	$0.546933\pi$
$654$	22.6254	0.00135279
$655$	−19838.5	−1.18344
$656$	−652.024	−0.0388068
$657$	−21823.0	−1.29589
$658$	0	0
$659$	−9050.77	−0.535004	−0.267502	−	0.963557i	$-0.586198\pi$
$659$	−9050.77	−0.535004	−0.267502	+	0.963557i	$0.586198\pi$
$660$	−22606.8	−1.33329
$661$	23731.2	1.39642	0.698211	−	0.715892i	$-0.253980\pi$
$661$	23731.2	1.39642	0.698211	+	0.715892i	$0.253980\pi$
$662$	−10600.5	−0.622357
$663$	2876.66	0.168507
$664$	6270.48	0.366479
$665$	0	0
$666$	73425.1	4.27202
$667$	20466.3	1.18810
$668$	−17870.5	−1.03507
$669$	4744.13	0.274168
$670$	16074.4	0.926876
$671$	−11465.2	−0.659624
$672$	0	0
$673$	9470.05	0.542413	0.271206	−	0.962521i	$-0.412577\pi$
$673$	9470.05	0.542413	0.271206	+	0.962521i	$0.412577\pi$
$674$	19765.1	1.12956
$675$	30999.3	1.76765
$676$	5652.75	0.321617
$677$	−26650.1	−1.51292	−0.756459	−	0.654041i	$-0.773072\pi$
$677$	−26650.1	−1.51292	−0.756459	+	0.654041i	$0.773072\pi$
$678$	30192.1	1.71020
$679$	0	0
$680$	−997.664	−0.0562628
$681$	−49797.9	−2.80214
$682$	−11785.9	−0.661738
$683$	20278.9	1.13609	0.568045	−	0.822997i	$-0.307700\pi$
$683$	20278.9	1.13609	0.568045	+	0.822997i	$0.307700\pi$
$684$	−7865.61	−0.439691
$685$	3021.87	0.168554
$686$	0	0
$687$	−43245.9	−2.40165
$688$	32999.6	1.82863
$689$	724.222	0.0400445
$690$	−61563.9	−3.39666
$691$	−1153.80	−0.0635206	−0.0317603	−	0.999496i	$-0.510111\pi$
$691$	−1153.80	−0.0635206	−0.0317603	+	0.999496i	$0.510111\pi$
$692$	−13922.2	−0.764800
$693$	0	0
$694$	−21096.9	−1.15393
$695$	−2969.27	−0.162059
$696$	−17771.4	−0.967849
$697$	44.8127	0.00243530
$698$	−25313.4	−1.37267
$699$	−13430.8	−0.726751
$700$	0	0
$701$	−13298.1	−0.716493	−0.358247	−	0.933627i	$-0.616625\pi$
$701$	−13298.1	−0.716493	−0.358247	+	0.933627i	$0.616625\pi$
$702$	−52559.2	−2.82581
$703$	11288.5	0.605624
$704$	−1518.79	−0.0813092
$705$	−12684.8	−0.677644
$706$	−13338.5	−0.711048
$707$	0	0
$708$	−18374.1	−0.975341
$709$	16084.4	0.851993	0.425996	−	0.904725i	$-0.359924\pi$
$709$	16084.4	0.851993	0.425996	+	0.904725i	$0.359924\pi$
$710$	−3212.45	−0.169804
$711$	63584.7	3.35388
$712$	−7254.31	−0.381835
$713$	−11953.0	−0.627832
$714$	0	0
$715$	−30619.6	−1.60155
$716$	−2929.90	−0.152927
$717$	2683.16	0.139755
$718$	21934.6	1.14010
$719$	−20047.4	−1.03983	−0.519917	−	0.854217i	$-0.674037\pi$
$719$	−20047.4	−1.03983	−0.519917	+	0.854217i	$0.674037\pi$
$720$	68673.0	3.55457
$721$	0	0
$722$	21241.8	1.09493
$723$	24095.5	1.23945
$724$	−16566.7	−0.850409
$725$	−20719.7	−1.06139
$726$	6887.51	0.352093
$727$	8839.43	0.450944	0.225472	−	0.974250i	$-0.427608\pi$
$727$	8839.43	0.450944	0.225472	+	0.974250i	$0.427608\pi$
$728$	0	0
$729$	−17529.6	−0.890594
$730$	−22313.9	−1.13133
$731$	−2268.01	−0.114754
$732$	−14735.9	−0.744062
$733$	−652.185	−0.0328636	−0.0164318	−	0.999865i	$-0.505231\pi$
$733$	−652.185	−0.0328636	−0.0164318	+	0.999865i	$0.505231\pi$
$734$	27657.6	1.39082
$735$	0	0
$736$	−23090.6	−1.15643
$737$	9567.09	0.478166
$738$	−1609.97	−0.0803034
$739$	2514.37	0.125159	0.0625796	−	0.998040i	$-0.480067\pi$
$739$	2514.37	0.125159	0.0625796	+	0.998040i	$0.480067\pi$
$740$	27959.7	1.38895
$741$	−15889.0	−0.787717
$742$	0	0
$743$	−33049.2	−1.63184	−0.815919	−	0.578166i	$-0.803769\pi$
$743$	−33049.2	−1.63184	−0.815919	+	0.578166i	$0.803769\pi$
$744$	10379.1	0.511445
$745$	51571.1	2.53613
$746$	−30176.3	−1.48101
$747$	29664.6	1.45298
$748$	866.622	0.0423621
$749$	0	0
$750$	−1238.17	−0.0602820
$751$	−23623.7	−1.14786	−0.573930	−	0.818904i	$-0.694582\pi$
$751$	−23623.7	−1.14786	−0.573930	+	0.818904i	$0.694582\pi$
$752$	−7075.20	−0.343093
$753$	38201.3	1.84878
$754$	35130.2	1.69677
$755$	30724.2	1.48102
$756$	0	0
$757$	18319.8	0.879584	0.439792	−	0.898100i	$-0.355052\pi$
$757$	18319.8	0.879584	0.439792	+	0.898100i	$0.355052\pi$
$758$	3157.75	0.151312
$759$	−36641.4	−1.75230
$760$	5510.53	0.263010
$761$	6334.09	0.301722	0.150861	−	0.988555i	$-0.451795\pi$
$761$	6334.09	0.301722	0.150861	+	0.988555i	$0.451795\pi$
$762$	−46793.6	−2.22461
$763$	0	0
$764$	9700.81	0.459376
$765$	−4719.79	−0.223065
$766$	36027.6	1.69939
$767$	−24886.6	−1.17158
$768$	47360.8	2.22524
$769$	−3468.39	−0.162644	−0.0813221	−	0.996688i	$-0.525914\pi$
$769$	−3468.39	−0.162644	−0.0813221	+	0.996688i	$0.525914\pi$
$770$	0	0
$771$	−27205.3	−1.27078
$772$	24451.4	1.13993
$773$	−35906.2	−1.67071	−0.835353	−	0.549714i	$-0.814737\pi$
$773$	−35906.2	−1.67071	−0.835353	+	0.549714i	$0.814737\pi$
$774$	81482.3	3.78400
$775$	12101.0	0.560878
$776$	−1710.72	−0.0791384
$777$	0	0
$778$	43619.8	2.01009
$779$	−247.520	−0.0113842
$780$	−39354.5	−1.80656
$781$	−1911.98	−0.0876004
$782$	2360.03	0.107921
$783$	−42756.5	−1.95146
$784$	0	0
$785$	5576.93	0.253566
$786$	−40749.6	−1.84922
$787$	39103.9	1.77116	0.885580	−	0.464487i	$-0.153761\pi$
$787$	39103.9	1.77116	0.885580	+	0.464487i	$0.153761\pi$
$788$	−21011.8	−0.949890
$789$	−20272.5	−0.914726
$790$	65014.9	2.92801
$791$	0	0
$792$	21332.8	0.957106
$793$	−19958.8	−0.893769
$794$	−15719.6	−0.702605
$795$	−1772.19	−0.0790607
$796$	−20662.5	−0.920054
$797$	−19145.5	−0.850903	−0.425452	−	0.904981i	$-0.639885\pi$
$797$	−19145.5	−0.850903	−0.425452	+	0.904981i	$0.639885\pi$
$798$	0	0
$799$	486.268	0.0215306
$800$	23376.4	1.03310
$801$	−34319.0	−1.51386
$802$	−7874.10	−0.346688
$803$	−13280.7	−0.583644
$804$	12296.3	0.539375
$805$	0	0
$806$	−20517.2	−0.896634
$807$	−33303.8	−1.45273
$808$	6895.46	0.300224
$809$	−27091.2	−1.17735	−0.588675	−	0.808370i	$-0.700350\pi$
$809$	−27091.2	−1.17735	−0.588675	+	0.808370i	$0.700350\pi$
$810$	45282.2	1.96426
$811$	16236.9	0.703025	0.351513	−	0.936183i	$-0.385667\pi$
$811$	16236.9	0.703025	0.351513	+	0.936183i	$0.385667\pi$
$812$	0	0
$813$	−33064.0	−1.42633
$814$	44683.9	1.92404
$815$	−18559.6	−0.797687
$816$	−3926.29	−0.168441
$817$	12527.2	0.536440
$818$	48121.9	2.05690
$819$	0	0
$820$	−613.064	−0.0261087
$821$	−8059.42	−0.342601	−0.171301	−	0.985219i	$-0.554797\pi$
$821$	−8059.42	−0.342601	−0.171301	+	0.985219i	$0.554797\pi$
$822$	6207.11	0.263379
$823$	29296.3	1.24083	0.620417	−	0.784272i	$-0.286963\pi$
$823$	29296.3	1.24083	0.620417	+	0.784272i	$0.286963\pi$
$824$	21638.1	0.914803
$825$	37095.1	1.56544
$826$	0	0
$827$	15296.2	0.643171	0.321585	−	0.946881i	$-0.395784\pi$
$827$	15296.2	0.643171	0.321585	+	0.946881i	$0.395784\pi$
$828$	−31576.3	−1.32531
$829$	−20509.1	−0.859241	−0.429621	−	0.903010i	$-0.641353\pi$
$829$	−20509.1	−0.859241	−0.429621	+	0.903010i	$0.641353\pi$
$830$	30331.9	1.26848
$831$	4994.38	0.208488
$832$	−2643.95	−0.110171
$833$	0	0
$834$	−6099.06	−0.253229
$835$	59229.2	2.45474
$836$	−4786.72	−0.198029
$837$	24971.2	1.03122
$838$	32538.6	1.34132
$839$	3957.65	0.162853	0.0814263	−	0.996679i	$-0.474052\pi$
$839$	3957.65	0.162853	0.0814263	+	0.996679i	$0.474052\pi$
$840$	0	0
$841$	4189.15	0.171764
$842$	14584.7	0.596938
$843$	5409.57	0.221015
$844$	19944.1	0.813393
$845$	−18735.2	−0.762735
$846$	−17470.0	−0.709966
$847$	0	0
$848$	−988.473	−0.0400286
$849$	−10146.6	−0.410164
$850$	−2389.24	−0.0964122
$851$	45317.5	1.82546
$852$	−2457.41	−0.0988140
$853$	12448.9	0.499697	0.249849	−	0.968285i	$-0.419619\pi$
$853$	12448.9	0.499697	0.249849	+	0.968285i	$0.419619\pi$
$854$	0	0
$855$	26069.4	1.04276
$856$	2672.52	0.106711
$857$	−7452.00	−0.297031	−0.148516	−	0.988910i	$-0.547449\pi$
$857$	−7452.00	−0.297031	−0.148516	+	0.988910i	$0.547449\pi$
$858$	−62894.5	−2.50254
$859$	31526.7	1.25224	0.626122	−	0.779725i	$-0.284641\pi$
$859$	31526.7	1.25224	0.626122	+	0.779725i	$0.284641\pi$
$860$	31027.8	1.23028
$861$	0	0
$862$	−46770.6	−1.84804
$863$	7037.95	0.277607	0.138803	−	0.990320i	$-0.455674\pi$
$863$	7037.95	0.277607	0.138803	+	0.990320i	$0.455674\pi$
$864$	48238.8	1.89944
$865$	46143.1	1.81377
$866$	5245.54	0.205832
$867$	−44203.1	−1.73151
$868$	0	0
$869$	38695.3	1.51053
$870$	−85964.7	−3.34997
$871$	16654.6	0.647899
$872$	−8.13005	−0.000315732 0
$873$	−8093.16	−0.313760
$874$	−13035.4	−0.504497
$875$	0	0
$876$	−17069.3	−0.658356
$877$	19331.9	0.744346	0.372173	−	0.928163i	$-0.378613\pi$
$877$	19331.9	0.744346	0.372173	+	0.928163i	$0.378613\pi$
$878$	−12972.2	−0.498623
$879$	40360.4	1.54872
$880$	41791.9	1.60091
$881$	−47384.8	−1.81207	−0.906036	−	0.423201i	$-0.860906\pi$
$881$	−47384.8	−1.81207	−0.906036	+	0.423201i	$0.860906\pi$
$882$	0	0
$883$	−8680.62	−0.330834	−0.165417	−	0.986224i	$-0.552897\pi$
$883$	−8680.62	−0.330834	−0.165417	+	0.986224i	$0.552897\pi$
$884$	1508.64	0.0573993
$885$	60898.4	2.31308
$886$	8660.92	0.328408
$887$	−39192.5	−1.48360	−0.741801	−	0.670620i	$-0.766028\pi$
$887$	−39192.5	−1.48360	−0.741801	+	0.670620i	$0.766028\pi$
$888$	−39350.2	−1.48706
$889$	0	0
$890$	−35090.9	−1.32163
$891$	26950.9	1.01334
$892$	2488.01	0.0933909
$893$	−2685.87	−0.100648
$894$	105930.	3.96291
$895$	9710.74	0.362675
$896$	0	0
$897$	−63786.3	−2.37432
$898$	−23290.9	−0.865510
$899$	−16690.6	−0.619201
$900$	31967.3	1.18397
$901$	67.9363	0.00251197
$902$	−979.771	−0.0361672
$903$	0	0
$904$	−10849.0	−0.399150
$905$	54908.0	2.01680
$906$	63109.3	2.31420
$907$	9670.59	0.354032	0.177016	−	0.984208i	$-0.443356\pi$
$907$	9670.59	0.354032	0.177016	+	0.984208i	$0.443356\pi$
$908$	−26116.0	−0.954504
$909$	32621.3	1.19030
$910$	0	0
$911$	51397.9	1.86925	0.934627	−	0.355630i	$-0.115734\pi$
$911$	51397.9	1.86925	0.934627	+	0.355630i	$0.115734\pi$
$912$	21686.6	0.787406
$913$	18052.8	0.654394
$914$	13771.7	0.498390
$915$	48839.9	1.76459
$916$	−22679.9	−0.818083
$917$	0	0
$918$	−4930.36	−0.177262
$919$	16443.5	0.590228	0.295114	−	0.955462i	$-0.404642\pi$
$919$	16443.5	0.590228	0.295114	+	0.955462i	$0.404642\pi$
$920$	22121.9	0.792759
$921$	38763.5	1.38686
$922$	−45637.7	−1.63015
$923$	−3328.42	−0.118696
$924$	0	0
$925$	−45878.5	−1.63079
$926$	58055.8	2.06029
$927$	102366.	3.62692
$928$	−32242.5	−1.14053
$929$	−55223.0	−1.95028	−0.975139	−	0.221593i	$-0.928874\pi$
$929$	−55223.0	−1.95028	−0.975139	+	0.221593i	$0.928874\pi$
$930$	50206.2	1.77024
$931$	0	0
$932$	−7043.64	−0.247556
$933$	−27078.0	−0.950154
$934$	21605.8	0.756920
$935$	−2872.30	−0.100464
$936$	37136.6	1.29685
$937$	−24930.8	−0.869216	−0.434608	−	0.900620i	$-0.643113\pi$
$937$	−24930.8	−0.869216	−0.434608	+	0.900620i	$0.643113\pi$
$938$	0	0
$939$	−33631.8	−1.16883
$940$	−6652.44	−0.230828
$941$	−13779.5	−0.477364	−0.238682	−	0.971098i	$-0.576715\pi$
$941$	−13779.5	−0.477364	−0.238682	+	0.971098i	$0.576715\pi$
$942$	11455.4	0.396216
$943$	−993.663	−0.0343140
$944$	33967.2	1.17112
$945$	0	0
$946$	49587.2	1.70425
$947$	3174.72	0.108938	0.0544691	−	0.998515i	$-0.482653\pi$
$947$	3174.72	0.108938	0.0544691	+	0.998515i	$0.482653\pi$
$948$	49734.0	1.70389
$949$	−23119.4	−0.790819
$950$	13196.8	0.450696
$951$	−29129.3	−0.993252
$952$	0	0
$953$	−29455.4	−1.00121	−0.500606	−	0.865675i	$-0.666889\pi$
$953$	−29455.4	−1.00121	−0.500606	+	0.865675i	$0.666889\pi$
$954$	−2440.73	−0.0828317
$955$	−32152.0	−1.08944
$956$	1407.15	0.0476052
$957$	−51164.2	−1.72822
$958$	22756.3	0.767455
$959$	0	0
$960$	6469.84	0.217514
$961$	−20043.2	−0.672792
$962$	77786.8	2.60701
$963$	12643.3	0.423078
$964$	12636.6	0.422198
$965$	−81040.7	−2.70341
$966$	0	0
$967$	−41062.1	−1.36553	−0.682766	−	0.730637i	$-0.739223\pi$
$967$	−41062.1	−1.36553	−0.682766	+	0.730637i	$0.739223\pi$
$968$	−2474.91	−0.0821761
$969$	−1490.49	−0.0494131
$970$	−8275.20	−0.273918
$971$	33782.3	1.11651	0.558253	−	0.829671i	$-0.311472\pi$
$971$	33782.3	1.11651	0.558253	+	0.829671i	$0.311472\pi$
$972$	2220.68	0.0732803
$973$	0	0
$974$	−59895.5	−1.97041
$975$	64576.0	2.12111
$976$	27241.3	0.893416
$977$	−21387.9	−0.700367	−0.350183	−	0.936681i	$-0.613881\pi$
$977$	−21387.9	−0.700367	−0.350183	+	0.936681i	$0.613881\pi$
$978$	−38122.6	−1.24645
$979$	−20885.3	−0.681816
$980$	0	0
$981$	−38.4620	−0.00125178
$982$	−71427.8	−2.32113
$983$	−20322.8	−0.659405	−0.329703	−	0.944085i	$-0.606949\pi$
$983$	−20322.8	−0.659405	−0.329703	+	0.944085i	$0.606949\pi$
$984$	862.821	0.0279530
$985$	69640.5	2.25272
$986$	3295.42	0.106438
$987$	0	0
$988$	−8332.85	−0.268323
$989$	50290.3	1.61692
$990$	103192.	3.31279
$991$	50990.0	1.63446	0.817230	−	0.576311i	$-0.195508\pi$
$991$	50990.0	1.63446	0.817230	+	0.576311i	$0.195508\pi$
$992$	18830.7	0.602696
$993$	26876.1	0.858901
$994$	0	0
$995$	68483.0	2.18197
$996$	23202.8	0.738162
$997$	−30493.6	−0.968648	−0.484324	−	0.874889i	$-0.660934\pi$
$997$	−30493.6	−0.968648	−0.484324	+	0.874889i	$0.660934\pi$
$998$	−58677.1	−1.86111
$999$	−94673.3	−2.99833

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2401.4.a.c.1.9		39
7.6	odd	2		2401.4.a.d.1.9		39
49.13	odd	14		49.4.e.a.22.11	✓	78
49.34	odd	14		49.4.e.a.29.11	yes	78

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.22.11	✓	78	49.13	odd	14
49.4.e.a.29.11	yes	78	49.34	odd	14
2401.4.a.c.1.9		39	1.1	even	1	trivial
2401.4.a.d.1.9		39	7.6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2401 = 7^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2401.a (trivial)

\(f(q)\)	\(=\)	\(q-3.57033 q^{2} +9.05210 q^{3} +4.74728 q^{4} -15.7342 q^{5} -32.3190 q^{6} +11.6133 q^{8} +54.9406 q^{9} +56.1764 q^{10} +33.4349 q^{11} +42.9729 q^{12} +58.2042 q^{13} -142.428 q^{15} -79.4416 q^{16} +5.45990 q^{17} -196.156 q^{18} -30.1574 q^{19} -74.6947 q^{20} -119.374 q^{22} -121.066 q^{23} +105.125 q^{24} +122.565 q^{25} -207.809 q^{26} +252.921 q^{27} -169.051 q^{29} +508.514 q^{30} +98.7312 q^{31} +190.727 q^{32} +302.656 q^{33} -19.4937 q^{34} +260.819 q^{36} -374.320 q^{37} +107.672 q^{38} +526.871 q^{39} -182.726 q^{40} +8.20760 q^{41} -415.395 q^{43} +158.725 q^{44} -864.446 q^{45} +432.247 q^{46} +89.0617 q^{47} -719.113 q^{48} -437.598 q^{50} +49.4236 q^{51} +276.312 q^{52} +12.4428 q^{53} -903.013 q^{54} -526.071 q^{55} -272.988 q^{57} +603.568 q^{58} -427.574 q^{59} -676.145 q^{60} -342.910 q^{61} -352.503 q^{62} -45.4254 q^{64} -915.797 q^{65} -1080.58 q^{66} +286.141 q^{67} +25.9197 q^{68} -1095.90 q^{69} -57.1851 q^{71} +638.040 q^{72} -397.212 q^{73} +1336.45 q^{74} +1109.47 q^{75} -143.166 q^{76} -1881.10 q^{78} +1157.33 q^{79} +1249.95 q^{80} +806.072 q^{81} -29.3039 q^{82} +539.941 q^{83} -85.9072 q^{85} +1483.10 q^{86} -1530.26 q^{87} +388.288 q^{88} -624.657 q^{89} +3086.36 q^{90} -574.736 q^{92} +893.725 q^{93} -317.980 q^{94} +474.502 q^{95} +1726.48 q^{96} -147.308 q^{97} +1836.93 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 39 q + q^{2} - q^{3} + 145 q^{4} - 27 q^{5} - 41 q^{6} - 12 q^{8} + 312 q^{9} - 78 q^{10} + q^{11} + 91 q^{12} - 77 q^{13} - 161 q^{15} + 461 q^{16} - 211 q^{17} + 8 q^{18} - 314 q^{19} - 476 q^{20}+ \cdots + 5384 q^{99}+O(q^{100}) \) 39 * q + q^2 - q^3 + 145 * q^4 - 27 * q^5 - 41 * q^6 - 12 * q^8 + 312 * q^9 - 78 * q^10 + q^11 + 91 * q^12 - 77 * q^13 - 161 * q^15 + 461 * q^16 - 211 * q^17 + 8 * q^18 - 314 * q^19 - 476 * q^20 - 61 * q^22 - 69 * q^23 - 330 * q^24 + 606 * q^25 - 504 * q^26 + 50 * q^27 + 57 * q^29 + 42 * q^30 - 638 * q^31 - 1600 * q^32 - 1574 * q^33 - 1343 * q^34 + 782 * q^36 + 71 * q^37 - 1359 * q^38 - 84 * q^39 + 155 * q^40 - 1393 * q^41 - 125 * q^43 + 52 * q^44 - 1129 * q^45 - 1454 * q^46 - 1483 * q^47 + 974 * q^48 + 3074 * q^50 - 2044 * q^51 - 3899 * q^52 + 2213 * q^53 - 1142 * q^54 - 1604 * q^55 + 98 * q^57 + 2403 * q^58 - 2073 * q^59 - 1519 * q^60 - 2575 * q^61 - 1742 * q^62 + 1358 * q^64 + 1876 * q^65 + 48 * q^66 + 176 * q^67 - 3038 * q^68 + 638 * q^69 - 1259 * q^71 - 3799 * q^72 + 307 * q^73 + 3845 * q^74 - 131 * q^75 - 1974 * q^76 - 6041 * q^78 + 22 * q^79 + 804 * q^80 + 795 * q^81 - 8043 * q^82 - 6349 * q^83 + 3094 * q^85 + 1745 * q^86 - 9508 * q^87 + 1299 * q^88 - 2253 * q^89 - 11156 * q^90 + 1284 * q^92 - 3430 * q^93 - 2738 * q^94 - 3290 * q^95 - 3031 * q^96 + 770 * q^97 + 5384 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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