LMFDB - Embedded newform 2450.4.a.cq.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,4,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, 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("2450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2450.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:	$144.554679514$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 33x^{2} - 37x + 70 $ x^4 - 33x^2 - 37x + 70
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 350)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.01007$ of defining polynomial
Character	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +8.46241 q^{3} +4.00000 q^{4} +16.9248 q^{6} +8.00000 q^{8} +44.6125 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +8.46241 q^{3} +4.00000 q^{4} +16.9248 q^{6} +8.00000 q^{8} +44.6125 q^{9} +5.60896 q^{11} +33.8497 q^{12} +44.1803 q^{13} +16.0000 q^{16} -109.417 q^{17} +89.2249 q^{18} +136.728 q^{19} +11.2179 q^{22} -21.4055 q^{23} +67.6993 q^{24} +88.3605 q^{26} +149.044 q^{27} +99.6624 q^{29} +17.1338 q^{31} +32.0000 q^{32} +47.4653 q^{33} -218.834 q^{34} +178.450 q^{36} +3.20623 q^{37} +273.457 q^{38} +373.872 q^{39} +298.615 q^{41} +413.814 q^{43} +22.4358 q^{44} -42.8109 q^{46} -587.606 q^{47} +135.399 q^{48} -925.931 q^{51} +176.721 q^{52} -601.523 q^{53} +298.088 q^{54} +1157.05 q^{57} +199.325 q^{58} +611.971 q^{59} +696.931 q^{61} +34.2676 q^{62} +64.0000 q^{64} +94.9306 q^{66} +463.973 q^{67} -437.667 q^{68} -181.142 q^{69} +231.542 q^{71} +356.900 q^{72} +705.968 q^{73} +6.41246 q^{74} +546.914 q^{76} +747.743 q^{78} +1013.74 q^{79} +56.7347 q^{81} +597.230 q^{82} -476.913 q^{83} +827.628 q^{86} +843.385 q^{87} +44.8717 q^{88} -780.526 q^{89} -85.6219 q^{92} +144.993 q^{93} -1175.21 q^{94} +270.797 q^{96} -908.934 q^{97} +250.229 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} - q^{3} + 16 q^{4} - 2 q^{6} + 32 q^{8} + 73 q^{9}+O(q^{10}) $ 4 * q + 8 * q^2 - q^3 + 16 * q^4 - 2 * q^6 + 32 * q^8 + 73 * q^9	$ 4 q + 8 q^{2} - q^{3} + 16 q^{4} - 2 q^{6} + 32 q^{8} + 73 q^{9} - 10 q^{11} - 4 q^{12} + 94 q^{13} + 64 q^{16} + 99 q^{17} + 146 q^{18} + 246 q^{19} - 20 q^{22} - 2 q^{23} - 8 q^{24} + 188 q^{26} - 196 q^{27} - 98 q^{29} + 304 q^{31} + 128 q^{32} + 526 q^{33} + 198 q^{34} + 292 q^{36} - 82 q^{37} + 492 q^{38} - 214 q^{39} + 352 q^{41} + 131 q^{43} - 40 q^{44} - 4 q^{46} - 491 q^{47} - 16 q^{48} - 1437 q^{51} + 376 q^{52} - 140 q^{53} - 392 q^{54} + 219 q^{57} - 196 q^{58} + 673 q^{59} + 1425 q^{61} + 608 q^{62} + 256 q^{64} + 1052 q^{66} - 666 q^{67} + 396 q^{68} + 938 q^{69} - 6 q^{71} + 584 q^{72} - 78 q^{73} - 164 q^{74} + 984 q^{76} - 428 q^{78} + 1744 q^{79} + 1708 q^{81} + 704 q^{82} + 926 q^{83} + 262 q^{86} + 1931 q^{87} - 80 q^{88} - 871 q^{89} - 8 q^{92} - 3106 q^{93} - 982 q^{94} - 32 q^{96} - 1462 q^{97} - 5281 q^{99}+O(q^{100}) $ 4 * q + 8 * q^2 - q^3 + 16 * q^4 - 2 * q^6 + 32 * q^8 + 73 * q^9 - 10 * q^11 - 4 * q^12 + 94 * q^13 + 64 * q^16 + 99 * q^17 + 146 * q^18 + 246 * q^19 - 20 * q^22 - 2 * q^23 - 8 * q^24 + 188 * q^26 - 196 * q^27 - 98 * q^29 + 304 * q^31 + 128 * q^32 + 526 * q^33 + 198 * q^34 + 292 * q^36 - 82 * q^37 + 492 * q^38 - 214 * q^39 + 352 * q^41 + 131 * q^43 - 40 * q^44 - 4 * q^46 - 491 * q^47 - 16 * q^48 - 1437 * q^51 + 376 * q^52 - 140 * q^53 - 392 * q^54 + 219 * q^57 - 196 * q^58 + 673 * q^59 + 1425 * q^61 + 608 * q^62 + 256 * q^64 + 1052 * q^66 - 666 * q^67 + 396 * q^68 + 938 * q^69 - 6 * q^71 + 584 * q^72 - 78 * q^73 - 164 * q^74 + 984 * q^76 - 428 * q^78 + 1744 * q^79 + 1708 * q^81 + 704 * q^82 + 926 * q^83 + 262 * q^86 + 1931 * q^87 - 80 * q^88 - 871 * q^89 - 8 * q^92 - 3106 * q^93 - 982 * q^94 - 32 * q^96 - 1462 * q^97 - 5281 * q^99

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	8.46241	1.62859	0.814296	−	0.580450i	$-0.197123\pi$
$3$	8.46241	1.62859	0.814296	+	0.580450i	$0.197123\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	16.9248	1.15159
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	44.6125	1.65231
$10$	0	0
$11$	5.60896	0.153742	0.0768711	−	0.997041i	$-0.475507\pi$
$11$	5.60896	0.153742	0.0768711	+	0.997041i	$0.475507\pi$
$12$	33.8497	0.814296
$13$	44.1803	0.942569	0.471285	−	0.881981i	$-0.343790\pi$
$13$	44.1803	0.942569	0.471285	+	0.881981i	$0.343790\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−109.417	−1.56103	−0.780514	−	0.625138i	$-0.785043\pi$
$17$	−109.417	−1.56103	−0.780514	+	0.625138i	$0.785043\pi$
$18$	89.2249	1.16836
$19$	136.728	1.65093	0.825465	−	0.564454i	$-0.190913\pi$
$19$	136.728	1.65093	0.825465	+	0.564454i	$0.190913\pi$
$20$	0	0
$21$	0	0
$22$	11.2179	0.108712
$23$	−21.4055	−0.194059	−0.0970293	−	0.995282i	$-0.530934\pi$
$23$	−21.4055	−0.194059	−0.0970293	+	0.995282i	$0.530934\pi$
$24$	67.6993	0.575794
$25$	0	0
$26$	88.3605	0.666497
$27$	149.044	1.06235
$28$	0	0
$29$	99.6624	0.638167	0.319084	−	0.947727i	$-0.396625\pi$
$29$	99.6624	0.638167	0.319084	+	0.947727i	$0.396625\pi$
$30$	0	0
$31$	17.1338	0.0992684	0.0496342	−	0.998767i	$-0.484194\pi$
$31$	17.1338	0.0992684	0.0496342	+	0.998767i	$0.484194\pi$
$32$	32.0000	0.176777
$33$	47.4653	0.250383
$34$	−218.834	−1.10381
$35$	0	0
$36$	178.450	0.826157
$37$	3.20623	0.0142460	0.00712298	−	0.999975i	$-0.497733\pi$
$37$	3.20623	0.0142460	0.00712298	+	0.999975i	$0.497733\pi$
$38$	273.457	1.16738
$39$	373.872	1.53506
$40$	0	0
$41$	298.615	1.13746	0.568730	−	0.822524i	$-0.307435\pi$
$41$	298.615	1.13746	0.568730	+	0.822524i	$0.307435\pi$
$42$	0	0
$43$	413.814	1.46758	0.733791	−	0.679375i	$-0.237749\pi$
$43$	413.814	1.46758	0.733791	+	0.679375i	$0.237749\pi$
$44$	22.4358	0.0768711
$45$	0	0
$46$	−42.8109	−0.137220
$47$	−587.606	−1.82364	−0.911821	−	0.410589i	$-0.865323\pi$
$47$	−587.606	−1.82364	−0.911821	+	0.410589i	$0.865323\pi$
$48$	135.399	0.407148
$49$	0	0
$50$	0	0
$51$	−925.931	−2.54228
$52$	176.721	0.471285
$53$	−601.523	−1.55897	−0.779487	−	0.626419i	$-0.784520\pi$
$53$	−601.523	−1.55897	−0.779487	+	0.626419i	$0.784520\pi$
$54$	298.088	0.751196
$55$	0	0
$56$	0	0
$57$	1157.05	2.68869
$58$	199.325	0.451252
$59$	611.971	1.35037	0.675185	−	0.737648i	$-0.264064\pi$
$59$	611.971	1.35037	0.675185	+	0.737648i	$0.264064\pi$
$60$	0	0
$61$	696.931	1.46283	0.731417	−	0.681931i	$-0.238859\pi$
$61$	696.931	1.46283	0.731417	+	0.681931i	$0.238859\pi$
$62$	34.2676	0.0701934
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	94.9306	0.177048
$67$	463.973	0.846019	0.423010	−	0.906125i	$-0.360974\pi$
$67$	463.973	0.846019	0.423010	+	0.906125i	$0.360974\pi$
$68$	−437.667	−0.780514
$69$	−181.142	−0.316042
$70$	0	0
$71$	231.542	0.387028	0.193514	−	0.981098i	$-0.438011\pi$
$71$	231.542	0.387028	0.193514	+	0.981098i	$0.438011\pi$
$72$	356.900	0.584181
$73$	705.968	1.13188	0.565940	−	0.824446i	$-0.308513\pi$
$73$	705.968	1.13188	0.565940	+	0.824446i	$0.308513\pi$
$74$	6.41246	0.0100734
$75$	0	0
$76$	546.914	0.825465
$77$	0	0
$78$	747.743	1.08545
$79$	1013.74	1.44373	0.721867	−	0.692031i	$-0.243284\pi$
$79$	1013.74	1.44373	0.721867	+	0.692031i	$0.243284\pi$
$80$	0	0
$81$	56.7347	0.0778254
$82$	597.230	0.804306
$83$	−476.913	−0.630698	−0.315349	−	0.948976i	$-0.602122\pi$
$83$	−476.913	−0.630698	−0.315349	+	0.948976i	$0.602122\pi$
$84$	0	0
$85$	0	0
$86$	827.628	1.03774
$87$	843.385	1.03931
$88$	44.8717	0.0543561
$89$	−780.526	−0.929613	−0.464806	−	0.885412i	$-0.653876\pi$
$89$	−780.526	−0.929613	−0.464806	+	0.885412i	$0.653876\pi$
$90$	0	0
$91$	0	0
$92$	−85.6219	−0.0970293
$93$	144.993	0.161668
$94$	−1175.21	−1.28951
$95$	0	0
$96$	270.797	0.287897
$97$	−908.934	−0.951425	−0.475713	−	0.879601i	$-0.657810\pi$
$97$	−908.934	−0.951425	−0.475713	+	0.879601i	$0.657810\pi$
$98$	0	0
$99$	250.229	0.254030
$100$	0	0
$101$	−1512.69	−1.49028	−0.745142	−	0.666906i	$-0.767618\pi$
$101$	−1512.69	−1.49028	−0.745142	+	0.666906i	$0.767618\pi$
$102$	−1851.86	−1.79766
$103$	−1111.53	−1.06332	−0.531662	−	0.846957i	$-0.678432\pi$
$103$	−1111.53	−1.06332	−0.531662	+	0.846957i	$0.678432\pi$
$104$	353.442	0.333249
$105$	0	0
$106$	−1203.05	−1.10236
$107$	−534.959	−0.483331	−0.241666	−	0.970360i	$-0.577694\pi$
$107$	−534.959	−0.483331	−0.241666	+	0.970360i	$0.577694\pi$
$108$	596.175	0.531176
$109$	598.136	0.525606	0.262803	−	0.964850i	$-0.415353\pi$
$109$	598.136	0.525606	0.262803	+	0.964850i	$0.415353\pi$
$110$	0	0
$111$	27.1324	0.0232009
$112$	0	0
$113$	1216.27	1.01254	0.506271	−	0.862374i	$-0.331023\pi$
$113$	1216.27	1.01254	0.506271	+	0.862374i	$0.331023\pi$
$114$	2314.11	1.90119
$115$	0	0
$116$	398.650	0.319084
$117$	1970.99	1.55742
$118$	1223.94	0.954856
$119$	0	0
$120$	0	0
$121$	−1299.54	−0.976363
$122$	1393.86	1.03438
$123$	2527.01	1.85246
$124$	68.5352	0.0496342
$125$	0	0
$126$	0	0
$127$	855.997	0.598090	0.299045	−	0.954239i	$-0.403332\pi$
$127$	855.997	0.598090	0.299045	+	0.954239i	$0.403332\pi$
$128$	128.000	0.0883883
$129$	3501.86	2.39009
$130$	0	0
$131$	770.703	0.514020	0.257010	−	0.966409i	$-0.417263\pi$
$131$	770.703	0.514020	0.257010	+	0.966409i	$0.417263\pi$
$132$	189.861	0.125192
$133$	0	0
$134$	927.945	0.598226
$135$	0	0
$136$	−875.335	−0.551907
$137$	−794.894	−0.495711	−0.247855	−	0.968797i	$-0.579726\pi$
$137$	−794.894	−0.495711	−0.247855	+	0.968797i	$0.579726\pi$
$138$	−362.284	−0.223476
$139$	502.958	0.306909	0.153455	−	0.988156i	$-0.450960\pi$
$139$	502.958	0.306909	0.153455	+	0.988156i	$0.450960\pi$
$140$	0	0
$141$	−4972.56	−2.96997
$142$	463.084	0.273670
$143$	247.805	0.144913
$144$	713.799	0.413078
$145$	0	0
$146$	1411.94	0.800360
$147$	0	0
$148$	12.8249	0.00712298
$149$	−1869.86	−1.02809	−0.514044	−	0.857764i	$-0.671853\pi$
$149$	−1869.86	−1.02809	−0.514044	+	0.857764i	$0.671853\pi$
$150$	0	0
$151$	−1025.73	−0.552797	−0.276399	−	0.961043i	$-0.589141\pi$
$151$	−1025.73	−0.552797	−0.276399	+	0.961043i	$0.589141\pi$
$152$	1093.83	0.583692
$153$	−4881.35	−2.57931
$154$	0	0
$155$	0	0
$156$	1495.49	0.767530
$157$	2726.55	1.38600	0.693000	−	0.720937i	$-0.256288\pi$
$157$	2726.55	1.38600	0.693000	+	0.720937i	$0.256288\pi$
$158$	2027.49	1.02087
$159$	−5090.34	−2.53893
$160$	0	0
$161$	0	0
$162$	113.469	0.0550309
$163$	2167.85	1.04171	0.520857	−	0.853644i	$-0.325613\pi$
$163$	2167.85	1.04171	0.520857	+	0.853644i	$0.325613\pi$
$164$	1194.46	0.568730
$165$	0	0
$166$	−953.825	−0.445971
$167$	1584.77	0.734330	0.367165	−	0.930156i	$-0.380328\pi$
$167$	1584.77	0.734330	0.367165	+	0.930156i	$0.380328\pi$
$168$	0	0
$169$	−245.105	−0.111563
$170$	0	0
$171$	6099.79	2.72785
$172$	1655.26	0.733791
$173$	3362.63	1.47778	0.738890	−	0.673826i	$-0.235350\pi$
$173$	3362.63	1.47778	0.738890	+	0.673826i	$0.235350\pi$
$174$	1686.77	0.734906
$175$	0	0
$176$	89.7433	0.0384356
$177$	5178.75	2.19920
$178$	−1561.05	−0.657336
$179$	−3610.02	−1.50740	−0.753702	−	0.657217i	$-0.771734\pi$
$179$	−3610.02	−1.50740	−0.753702	+	0.657217i	$0.771734\pi$
$180$	0	0
$181$	−1653.09	−0.678856	−0.339428	−	0.940632i	$-0.610234\pi$
$181$	−1653.09	−0.678856	−0.339428	+	0.940632i	$0.610234\pi$
$182$	0	0
$183$	5897.72	2.38236
$184$	−171.244	−0.0686101
$185$	0	0
$186$	289.987	0.114316
$187$	−613.714	−0.239996
$188$	−2350.42	−0.911821
$189$	0	0
$190$	0	0
$191$	−2934.84	−1.11182	−0.555909	−	0.831243i	$-0.687630\pi$
$191$	−2934.84	−1.11182	−0.555909	+	0.831243i	$0.687630\pi$
$192$	541.595	0.203574
$193$	−3833.46	−1.42973	−0.714867	−	0.699261i	$-0.753513\pi$
$193$	−3833.46	−1.42973	−0.714867	+	0.699261i	$0.753513\pi$
$194$	−1817.87	−0.672759
$195$	0	0
$196$	0	0
$197$	−168.443	−0.0609190	−0.0304595	−	0.999536i	$-0.509697\pi$
$197$	−168.443	−0.0609190	−0.0304595	+	0.999536i	$0.509697\pi$
$198$	500.459	0.179627
$199$	578.476	0.206066	0.103033	−	0.994678i	$-0.467145\pi$
$199$	578.476	0.206066	0.103033	+	0.994678i	$0.467145\pi$
$200$	0	0
$201$	3926.33	1.37782
$202$	−3025.39	−1.05379
$203$	0	0
$204$	−3703.72	−1.27114
$205$	0	0
$206$	−2223.06	−0.751884
$207$	−954.951	−0.320646
$208$	706.884	0.235642
$209$	766.904	0.253818
$210$	0	0
$211$	4636.03	1.51260	0.756298	−	0.654228i	$-0.227006\pi$
$211$	4636.03	1.51260	0.756298	+	0.654228i	$0.227006\pi$
$212$	−2406.09	−0.779487
$213$	1959.41	0.630311
$214$	−1069.92	−0.341767
$215$	0	0
$216$	1192.35	0.375598
$217$	0	0
$218$	1196.27	0.371660
$219$	5974.19	1.84337
$220$	0	0
$221$	−4834.06	−1.47138
$222$	54.2649	0.0164055
$223$	−1171.13	−0.351679	−0.175840	−	0.984419i	$-0.556264\pi$
$223$	−1171.13	−0.351679	−0.175840	+	0.984419i	$0.556264\pi$
$224$	0	0
$225$	0	0
$226$	2432.55	0.715976
$227$	3940.96	1.15229	0.576147	−	0.817346i	$-0.304556\pi$
$227$	3940.96	1.15229	0.576147	+	0.817346i	$0.304556\pi$
$228$	4628.21	1.34435
$229$	6279.52	1.81206	0.906031	−	0.423212i	$-0.139097\pi$
$229$	6279.52	1.81206	0.906031	+	0.423212i	$0.139097\pi$
$230$	0	0
$231$	0	0
$232$	797.299	0.225626
$233$	−6952.55	−1.95484	−0.977418	−	0.211313i	$-0.932226\pi$
$233$	−6952.55	−1.95484	−0.977418	+	0.211313i	$0.932226\pi$
$234$	3941.98	1.10126
$235$	0	0
$236$	2447.88	0.675185
$237$	8578.72	2.35126
$238$	0	0
$239$	−2139.43	−0.579031	−0.289516	−	0.957173i	$-0.593494\pi$
$239$	−2139.43	−0.579031	−0.289516	+	0.957173i	$0.593494\pi$
$240$	0	0
$241$	3633.53	0.971188	0.485594	−	0.874184i	$-0.338603\pi$
$241$	3633.53	0.971188	0.485594	+	0.874184i	$0.338603\pi$
$242$	−2599.08	−0.690393
$243$	−3544.07	−0.935606
$244$	2787.72	0.731417
$245$	0	0
$246$	5054.01	1.30989
$247$	6040.70	1.55612
$248$	137.070	0.0350967
$249$	−4035.83	−1.02715
$250$	0	0
$251$	3155.90	0.793620	0.396810	−	0.917901i	$-0.370117\pi$
$251$	3155.90	0.793620	0.396810	+	0.917901i	$0.370117\pi$
$252$	0	0
$253$	−120.062	−0.0298350
$254$	1711.99	0.422914
$255$	0	0
$256$	256.000	0.0625000
$257$	266.895	0.0647800	0.0323900	−	0.999475i	$-0.489688\pi$
$257$	266.895	0.0647800	0.0323900	+	0.999475i	$0.489688\pi$
$258$	7003.73	1.69005
$259$	0	0
$260$	0	0
$261$	4446.18	1.05445
$262$	1541.41	0.363467
$263$	−4785.51	−1.12200	−0.561002	−	0.827814i	$-0.689584\pi$
$263$	−4785.51	−1.12200	−0.561002	+	0.827814i	$0.689584\pi$
$264$	379.723	0.0885239
$265$	0	0
$266$	0	0
$267$	−6605.13	−1.51396
$268$	1855.89	0.423010
$269$	−768.419	−0.174169	−0.0870843	−	0.996201i	$-0.527755\pi$
$269$	−768.419	−0.174169	−0.0870843	+	0.996201i	$0.527755\pi$
$270$	0	0
$271$	2199.28	0.492977	0.246489	−	0.969146i	$-0.420723\pi$
$271$	2199.28	0.492977	0.246489	+	0.969146i	$0.420723\pi$
$272$	−1750.67	−0.390257
$273$	0	0
$274$	−1589.79	−0.350520
$275$	0	0
$276$	−724.568	−0.158021
$277$	−4599.74	−0.997731	−0.498866	−	0.866679i	$-0.666250\pi$
$277$	−4599.74	−0.997731	−0.498866	+	0.866679i	$0.666250\pi$
$278$	1005.92	0.217018
$279$	764.381	0.164022
$280$	0	0
$281$	−32.6041	−0.00692170	−0.00346085	−	0.999994i	$-0.501102\pi$
$281$	−32.6041	−0.00692170	−0.00346085	+	0.999994i	$0.501102\pi$
$282$	−9945.13	−2.10008
$283$	−7967.95	−1.67366	−0.836830	−	0.547463i	$-0.815594\pi$
$283$	−7967.95	−1.67366	−0.836830	+	0.547463i	$0.815594\pi$
$284$	926.169	0.193514
$285$	0	0
$286$	495.610	0.102469
$287$	0	0
$288$	1427.60	0.292090
$289$	7059.05	1.43681
$290$	0	0
$291$	−7691.77	−1.54948
$292$	2823.87	0.565940
$293$	151.426	0.0301926	0.0150963	−	0.999886i	$-0.495195\pi$
$293$	151.426	0.0301926	0.0150963	+	0.999886i	$0.495195\pi$
$294$	0	0
$295$	0	0
$296$	25.6498	0.00503671
$297$	835.981	0.163328
$298$	−3739.72	−0.726968
$299$	−945.699	−0.182914
$300$	0	0
$301$	0	0
$302$	−2051.45	−0.390887
$303$	−12801.0	−2.42706
$304$	2187.66	0.412732
$305$	0	0
$306$	−9762.71	−1.82385
$307$	3177.58	0.590729	0.295365	−	0.955385i	$-0.404559\pi$
$307$	3177.58	0.590729	0.295365	+	0.955385i	$0.404559\pi$
$308$	0	0
$309$	−9406.23	−1.73172
$310$	0	0
$311$	918.536	0.167477	0.0837386	−	0.996488i	$-0.473314\pi$
$311$	918.536	0.167477	0.0837386	+	0.996488i	$0.473314\pi$
$312$	2990.97	0.542726
$313$	3317.72	0.599134	0.299567	−	0.954075i	$-0.403158\pi$
$313$	3317.72	0.599134	0.299567	+	0.954075i	$0.403158\pi$
$314$	5453.09	0.980051
$315$	0	0
$316$	4054.97	0.721867
$317$	−6414.18	−1.13645	−0.568227	−	0.822872i	$-0.692371\pi$
$317$	−6414.18	−1.13645	−0.568227	+	0.822872i	$0.692371\pi$
$318$	−10180.7	−1.79530
$319$	559.002	0.0981132
$320$	0	0
$321$	−4527.04	−0.787149
$322$	0	0
$323$	−14960.4	−2.57715
$324$	226.939	0.0389127
$325$	0	0
$326$	4335.70	0.736602
$327$	5061.68	0.855998
$328$	2388.92	0.402153
$329$	0	0
$330$	0	0
$331$	−6775.23	−1.12508	−0.562538	−	0.826771i	$-0.690175\pi$
$331$	−6775.23	−1.12508	−0.562538	+	0.826771i	$0.690175\pi$
$332$	−1907.65	−0.315349
$333$	143.038	0.0235388
$334$	3169.54	0.519250
$335$	0	0
$336$	0	0
$337$	9890.09	1.59866	0.799329	−	0.600894i	$-0.205188\pi$
$337$	9890.09	1.59866	0.799329	+	0.600894i	$0.205188\pi$
$338$	−490.209	−0.0788872
$339$	10292.6	1.64902
$340$	0	0
$341$	96.1027	0.0152617
$342$	12199.6	1.92888
$343$	0	0
$344$	3310.51	0.518868
$345$	0	0
$346$	6725.26	1.04495
$347$	3221.86	0.498440	0.249220	−	0.968447i	$-0.419826\pi$
$347$	3221.86	0.498440	0.249220	+	0.968447i	$0.419826\pi$
$348$	3373.54	0.519657
$349$	−4622.04	−0.708917	−0.354459	−	0.935072i	$-0.615335\pi$
$349$	−4622.04	−0.708917	−0.354459	+	0.935072i	$0.615335\pi$
$350$	0	0
$351$	6584.80	1.00134
$352$	179.487	0.0271780
$353$	−2168.33	−0.326936	−0.163468	−	0.986549i	$-0.552268\pi$
$353$	−2168.33	−0.326936	−0.163468	+	0.986549i	$0.552268\pi$
$354$	10357.5	1.55507
$355$	0	0
$356$	−3122.10	−0.464806
$357$	0	0
$358$	−7220.03	−1.06590
$359$	466.028	0.0685126	0.0342563	−	0.999413i	$-0.489094\pi$
$359$	466.028	0.0685126	0.0342563	+	0.999413i	$0.489094\pi$
$360$	0	0
$361$	11835.7	1.72557
$362$	−3306.17	−0.480024
$363$	−10997.2	−1.59010
$364$	0	0
$365$	0	0
$366$	11795.4	1.68458
$367$	−2892.81	−0.411454	−0.205727	−	0.978609i	$-0.565956\pi$
$367$	−2892.81	−0.411454	−0.205727	+	0.978609i	$0.565956\pi$
$368$	−342.488	−0.0485147
$369$	13322.0	1.87944
$370$	0	0
$371$	0	0
$372$	579.973	0.0808339
$373$	−1809.53	−0.251190	−0.125595	−	0.992082i	$-0.540084\pi$
$373$	−1809.53	−0.251190	−0.125595	+	0.992082i	$0.540084\pi$
$374$	−1227.43	−0.169703
$375$	0	0
$376$	−4700.85	−0.644754
$377$	4403.11	0.601517
$378$	0	0
$379$	−8994.04	−1.21898	−0.609489	−	0.792794i	$-0.708625\pi$
$379$	−8994.04	−1.21898	−0.609489	+	0.792794i	$0.708625\pi$
$380$	0	0
$381$	7243.80	0.974045
$382$	−5869.67	−0.786174
$383$	−12361.6	−1.64922	−0.824608	−	0.565704i	$-0.808605\pi$
$383$	−12361.6	−1.64922	−0.824608	+	0.565704i	$0.808605\pi$
$384$	1083.19	0.143949
$385$	0	0
$386$	−7666.92	−1.01097
$387$	18461.2	2.42490
$388$	−3635.73	−0.475713
$389$	−4785.58	−0.623749	−0.311875	−	0.950123i	$-0.600957\pi$
$389$	−4785.58	−0.623749	−0.311875	+	0.950123i	$0.600957\pi$
$390$	0	0
$391$	2342.12	0.302931
$392$	0	0
$393$	6522.01	0.837130
$394$	−336.885	−0.0430762
$395$	0	0
$396$	1000.92	0.127015
$397$	3999.06	0.505559	0.252780	−	0.967524i	$-0.418655\pi$
$397$	3999.06	0.505559	0.252780	+	0.967524i	$0.418655\pi$
$398$	1156.95	0.145711
$399$	0	0
$400$	0	0
$401$	5939.93	0.739715	0.369858	−	0.929088i	$-0.379406\pi$
$401$	5939.93	0.739715	0.369858	+	0.929088i	$0.379406\pi$
$402$	7852.66	0.974266
$403$	756.975	0.0935673
$404$	−6050.77	−0.745142
$405$	0	0
$406$	0	0
$407$	17.9836	0.00219021
$408$	−7407.45	−0.898831
$409$	−9268.66	−1.12055	−0.560276	−	0.828306i	$-0.689305\pi$
$409$	−9268.66	−1.12055	−0.560276	+	0.828306i	$0.689305\pi$
$410$	0	0
$411$	−6726.72	−0.807310
$412$	−4446.12	−0.531662
$413$	0	0
$414$	−1909.90	−0.226731
$415$	0	0
$416$	1413.77	0.166624
$417$	4256.24	0.499830
$418$	1533.81	0.179476
$419$	2058.22	0.239977	0.119989	−	0.992775i	$-0.461714\pi$
$419$	2058.22	0.239977	0.119989	+	0.992775i	$0.461714\pi$
$420$	0	0
$421$	−14211.3	−1.64517	−0.822583	−	0.568645i	$-0.807468\pi$
$421$	−14211.3	−1.64517	−0.822583	+	0.568645i	$0.807468\pi$
$422$	9272.07	1.06957
$423$	−26214.5	−3.01323
$424$	−4812.19	−0.551180
$425$	0	0
$426$	3918.81	0.445697
$427$	0	0
$428$	−2139.84	−0.241666
$429$	2097.03	0.236004
$430$	0	0
$431$	−87.6971	−0.00980098	−0.00490049	−	0.999988i	$-0.501560\pi$
$431$	−87.6971	−0.00980098	−0.00490049	+	0.999988i	$0.501560\pi$
$432$	2384.70	0.265588
$433$	−10482.6	−1.16342	−0.581709	−	0.813397i	$-0.697616\pi$
$433$	−10482.6	−1.16342	−0.581709	+	0.813397i	$0.697616\pi$
$434$	0	0
$435$	0	0
$436$	2392.54	0.262803
$437$	−2926.74	−0.320377
$438$	11948.4	1.30346
$439$	3849.34	0.418494	0.209247	−	0.977863i	$-0.432899\pi$
$439$	3849.34	0.418494	0.209247	+	0.977863i	$0.432899\pi$
$440$	0	0
$441$	0	0
$442$	−9668.13	−1.04042
$443$	−10511.8	−1.12738	−0.563690	−	0.825986i	$-0.690619\pi$
$443$	−10511.8	−1.12738	−0.563690	+	0.825986i	$0.690619\pi$
$444$	108.530	0.0116004
$445$	0	0
$446$	−2342.25	−0.248675
$447$	−15823.5	−1.67434
$448$	0	0
$449$	−6500.68	−0.683265	−0.341633	−	0.939834i	$-0.610980\pi$
$449$	−6500.68	−0.683265	−0.341633	+	0.939834i	$0.610980\pi$
$450$	0	0
$451$	1674.92	0.174876
$452$	4865.09	0.506271
$453$	−8680.12	−0.900282
$454$	7881.92	0.814795
$455$	0	0
$456$	9256.42	0.950596
$457$	−5003.21	−0.512123	−0.256061	−	0.966660i	$-0.582425\pi$
$457$	−5003.21	−0.512123	−0.256061	+	0.966660i	$0.582425\pi$
$458$	12559.0	1.28132
$459$	−16307.9	−1.65836
$460$	0	0
$461$	−6656.10	−0.672463	−0.336231	−	0.941779i	$-0.609152\pi$
$461$	−6656.10	−0.672463	−0.336231	+	0.941779i	$0.609152\pi$
$462$	0	0
$463$	−9725.18	−0.976172	−0.488086	−	0.872796i	$-0.662305\pi$
$463$	−9725.18	−0.976172	−0.488086	+	0.872796i	$0.662305\pi$
$464$	1594.60	0.159542
$465$	0	0
$466$	−13905.1	−1.38228
$467$	205.062	0.0203193	0.0101597	−	0.999948i	$-0.496766\pi$
$467$	205.062	0.0203193	0.0101597	+	0.999948i	$0.496766\pi$
$468$	7883.96	0.778710
$469$	0	0
$470$	0	0
$471$	23073.2	2.25723
$472$	4895.77	0.477428
$473$	2321.06	0.225629
$474$	17157.4	1.66259
$475$	0	0
$476$	0	0
$477$	−26835.4	−2.57591
$478$	−4278.87	−0.409437
$479$	10952.7	1.04476	0.522380	−	0.852713i	$-0.325044\pi$
$479$	10952.7	1.04476	0.522380	+	0.852713i	$0.325044\pi$
$480$	0	0
$481$	141.652	0.0134278
$482$	7267.07	0.686734
$483$	0	0
$484$	−5198.16	−0.488182
$485$	0	0
$486$	−7088.14	−0.661573
$487$	−12259.3	−1.14071	−0.570353	−	0.821400i	$-0.693194\pi$
$487$	−12259.3	−1.14071	−0.570353	+	0.821400i	$0.693194\pi$
$488$	5575.45	0.517190
$489$	18345.3	1.69653
$490$	0	0
$491$	1917.95	0.176285	0.0881425	−	0.996108i	$-0.471907\pi$
$491$	1917.95	0.176285	0.0881425	+	0.996108i	$0.471907\pi$
$492$	10108.0	0.926229
$493$	−10904.7	−0.996197
$494$	12081.4	1.10034
$495$	0	0
$496$	274.141	0.0248171
$497$	0	0
$498$	−8071.66	−0.726305
$499$	5128.16	0.460056	0.230028	−	0.973184i	$-0.426118\pi$
$499$	5128.16	0.460056	0.230028	+	0.973184i	$0.426118\pi$
$500$	0	0
$501$	13411.0	1.19592
$502$	6311.80	0.561174
$503$	16093.9	1.42662	0.713310	−	0.700849i	$-0.247195\pi$
$503$	16093.9	1.42662	0.713310	+	0.700849i	$0.247195\pi$
$504$	0	0
$505$	0	0
$506$	−240.125	−0.0210965
$507$	−2074.18	−0.181691
$508$	3423.99	0.299045
$509$	−4205.53	−0.366222	−0.183111	−	0.983092i	$-0.558617\pi$
$509$	−4205.53	−0.366222	−0.183111	+	0.983092i	$0.558617\pi$
$510$	0	0
$511$	0	0
$512$	512.000	0.0441942
$513$	20378.5	1.75387
$514$	533.791	0.0458064
$515$	0	0
$516$	14007.5	1.19505
$517$	−3295.86	−0.280371
$518$	0	0
$519$	28456.0	2.40670
$520$	0	0
$521$	1686.04	0.141779	0.0708896	−	0.997484i	$-0.477416\pi$
$521$	1686.04	0.141779	0.0708896	+	0.997484i	$0.477416\pi$
$522$	8892.37	0.745610
$523$	21840.6	1.82605	0.913023	−	0.407907i	$-0.133741\pi$
$523$	21840.6	1.82605	0.913023	+	0.407907i	$0.133741\pi$
$524$	3082.81	0.257010
$525$	0	0
$526$	−9571.02	−0.793377
$527$	−1874.73	−0.154961
$528$	759.445	0.0625958
$529$	−11708.8	−0.962341
$530$	0	0
$531$	27301.5	2.23123
$532$	0	0
$533$	13192.9	1.07213
$534$	−13210.3	−1.07053
$535$	0	0
$536$	3711.78	0.299113
$537$	−30549.4	−2.45495
$538$	−1536.84	−0.123156
$539$	0	0
$540$	0	0
$541$	−16573.7	−1.31712	−0.658558	−	0.752530i	$-0.728833\pi$
$541$	−16573.7	−1.31712	−0.658558	+	0.752530i	$0.728833\pi$
$542$	4398.56	0.348587
$543$	−13989.1	−1.10558
$544$	−3501.34	−0.275953
$545$	0	0
$546$	0	0
$547$	−11756.1	−0.918930	−0.459465	−	0.888196i	$-0.651959\pi$
$547$	−11756.1	−0.918930	−0.459465	+	0.888196i	$0.651959\pi$
$548$	−3179.57	−0.247855
$549$	31091.8	2.41706
$550$	0	0
$551$	13626.7	1.05357
$552$	−1449.14	−0.111738
$553$	0	0
$554$	−9199.48	−0.705502
$555$	0	0
$556$	2011.83	0.153455
$557$	−1851.29	−0.140829	−0.0704143	−	0.997518i	$-0.522432\pi$
$557$	−1851.29	−0.140829	−0.0704143	+	0.997518i	$0.522432\pi$
$558$	1528.76	0.115981
$559$	18282.4	1.38330
$560$	0	0
$561$	−5193.51	−0.390856
$562$	−65.2082	−0.00489438
$563$	−1673.85	−0.125301	−0.0626506	−	0.998036i	$-0.519955\pi$
$563$	−1673.85	−0.125301	−0.0626506	+	0.998036i	$0.519955\pi$
$564$	−19890.3	−1.48498
$565$	0	0
$566$	−15935.9	−1.18346
$567$	0	0
$568$	1852.34	0.136835
$569$	6225.01	0.458640	0.229320	−	0.973351i	$-0.426350\pi$
$569$	6225.01	0.458640	0.229320	+	0.973351i	$0.426350\pi$
$570$	0	0
$571$	11914.0	0.873177	0.436589	−	0.899661i	$-0.356187\pi$
$571$	11914.0	0.873177	0.436589	+	0.899661i	$0.356187\pi$
$572$	991.221	0.0724563
$573$	−24835.8	−1.81070
$574$	0	0
$575$	0	0
$576$	2855.20	0.206539
$577$	−628.857	−0.0453720	−0.0226860	−	0.999743i	$-0.507222\pi$
$577$	−628.857	−0.0453720	−0.0226860	+	0.999743i	$0.507222\pi$
$578$	14118.1	1.01598
$579$	−32440.3	−2.32845
$580$	0	0
$581$	0	0
$582$	−15383.5	−1.09565
$583$	−3373.92	−0.239680
$584$	5647.74	0.400180
$585$	0	0
$586$	302.853	0.0213494
$587$	−19553.2	−1.37487	−0.687434	−	0.726246i	$-0.741263\pi$
$587$	−19553.2	−1.37487	−0.687434	+	0.726246i	$0.741263\pi$
$588$	0	0
$589$	2342.68	0.163885
$590$	0	0
$591$	−1425.43	−0.0992122
$592$	51.2997	0.00356149
$593$	−3254.24	−0.225355	−0.112678	−	0.993632i	$-0.535943\pi$
$593$	−3254.24	−0.225355	−0.112678	+	0.993632i	$0.535943\pi$
$594$	1671.96	0.115491
$595$	0	0
$596$	−7479.45	−0.514044
$597$	4895.31	0.335597
$598$	−1891.40	−0.129340
$599$	19174.0	1.30790	0.653948	−	0.756540i	$-0.273112\pi$
$599$	19174.0	1.30790	0.653948	+	0.756540i	$0.273112\pi$
$600$	0	0
$601$	12137.0	0.823756	0.411878	−	0.911239i	$-0.364873\pi$
$601$	12137.0	0.823756	0.411878	+	0.911239i	$0.364873\pi$
$602$	0	0
$603$	20699.0	1.39789
$604$	−4102.90	−0.276399
$605$	0	0
$606$	−25602.1	−1.71619
$607$	−20458.8	−1.36803	−0.684017	−	0.729466i	$-0.739769\pi$
$607$	−20458.8	−1.36803	−0.684017	+	0.729466i	$0.739769\pi$
$608$	4375.31	0.291846
$609$	0	0
$610$	0	0
$611$	−25960.6	−1.71891
$612$	−19525.4	−1.28965
$613$	1234.16	0.0813169	0.0406585	−	0.999173i	$-0.487054\pi$
$613$	1234.16	0.0813169	0.0406585	+	0.999173i	$0.487054\pi$
$614$	6355.16	0.417709
$615$	0	0
$616$	0	0
$617$	−12020.4	−0.784314	−0.392157	−	0.919898i	$-0.628271\pi$
$617$	−12020.4	−0.784314	−0.392157	+	0.919898i	$0.628271\pi$
$618$	−18812.5	−1.22451
$619$	21110.1	1.37074	0.685368	−	0.728197i	$-0.259642\pi$
$619$	21110.1	1.37074	0.685368	+	0.728197i	$0.259642\pi$
$620$	0	0
$621$	−3190.35	−0.206159
$622$	1837.07	0.118424
$623$	0	0
$624$	5981.95	0.383765
$625$	0	0
$626$	6635.45	0.423651
$627$	6489.86	0.413365
$628$	10906.2	0.693000
$629$	−350.816	−0.0222384
$630$	0	0
$631$	−2345.47	−0.147974	−0.0739872	−	0.997259i	$-0.523572\pi$
$631$	−2345.47	−0.147974	−0.0739872	+	0.997259i	$0.523572\pi$
$632$	8109.95	0.510437
$633$	39232.0	2.46340
$634$	−12828.4	−0.803595
$635$	0	0
$636$	−20361.4	−1.26947
$637$	0	0
$638$	1118.00	0.0693765
$639$	10329.7	0.639492
$640$	0	0
$641$	−21519.6	−1.32601	−0.663005	−	0.748615i	$-0.730719\pi$
$641$	−21519.6	−1.32601	−0.663005	+	0.748615i	$0.730719\pi$
$642$	−9054.09	−0.556599
$643$	−29827.7	−1.82937	−0.914687	−	0.404162i	$-0.867563\pi$
$643$	−29827.7	−1.82937	−0.914687	+	0.404162i	$0.867563\pi$
$644$	0	0
$645$	0	0
$646$	−29920.8	−1.82232
$647$	27926.1	1.69689	0.848446	−	0.529283i	$-0.177539\pi$
$647$	27926.1	1.69689	0.848446	+	0.529283i	$0.177539\pi$
$648$	453.878	0.0275154
$649$	3432.52	0.207609
$650$	0	0
$651$	0	0
$652$	8671.41	0.520857
$653$	30177.9	1.80850	0.904252	−	0.427000i	$-0.140429\pi$
$653$	30177.9	1.80850	0.904252	+	0.427000i	$0.140429\pi$
$654$	10123.4	0.605282
$655$	0	0
$656$	4777.84	0.284365
$657$	31495.0	1.87022
$658$	0	0
$659$	11978.1	0.708044	0.354022	−	0.935237i	$-0.384814\pi$
$659$	11978.1	0.708044	0.354022	+	0.935237i	$0.384814\pi$
$660$	0	0
$661$	2018.93	0.118801	0.0594005	−	0.998234i	$-0.481081\pi$
$661$	2018.93	0.118801	0.0594005	+	0.998234i	$0.481081\pi$
$662$	−13550.5	−0.795549
$663$	−40907.9	−2.39627
$664$	−3815.30	−0.222985
$665$	0	0
$666$	286.075	0.0166444
$667$	−2133.32	−0.123842
$668$	6339.07	0.367165
$669$	−9910.56	−0.572742
$670$	0	0
$671$	3909.05	0.224899
$672$	0	0
$673$	11029.5	0.631733	0.315867	−	0.948804i	$-0.397705\pi$
$673$	11029.5	0.631733	0.315867	+	0.948804i	$0.397705\pi$
$674$	19780.2	1.13042
$675$	0	0
$676$	−980.419	−0.0557817
$677$	4517.02	0.256430	0.128215	−	0.991746i	$-0.459075\pi$
$677$	4517.02	0.256430	0.128215	+	0.991746i	$0.459075\pi$
$678$	20585.2	1.16603
$679$	0	0
$680$	0	0
$681$	33350.0	1.87662
$682$	192.205	0.0107917
$683$	7758.01	0.434630	0.217315	−	0.976102i	$-0.430270\pi$
$683$	7758.01	0.434630	0.217315	+	0.976102i	$0.430270\pi$
$684$	24399.2	1.36393
$685$	0	0
$686$	0	0
$687$	53139.9	2.95111
$688$	6621.02	0.366895
$689$	−26575.5	−1.46944
$690$	0	0
$691$	−13138.1	−0.723292	−0.361646	−	0.932315i	$-0.617785\pi$
$691$	−13138.1	−0.723292	−0.361646	+	0.932315i	$0.617785\pi$
$692$	13450.5	0.738890
$693$	0	0
$694$	6443.72	0.352450
$695$	0	0
$696$	6747.08	0.367453
$697$	−32673.5	−1.77561
$698$	−9244.08	−0.501280
$699$	−58835.4	−3.18363
$700$	0	0
$701$	−26118.1	−1.40723	−0.703614	−	0.710583i	$-0.748431\pi$
$701$	−26118.1	−1.40723	−0.703614	+	0.710583i	$0.748431\pi$
$702$	13169.6	0.708055
$703$	438.383	0.0235191
$704$	358.973	0.0192178
$705$	0	0
$706$	−4336.66	−0.231179
$707$	0	0
$708$	20715.0	1.09960
$709$	−7135.36	−0.377961	−0.188980	−	0.981981i	$-0.560518\pi$
$709$	−7135.36	−0.377961	−0.188980	+	0.981981i	$0.560518\pi$
$710$	0	0
$711$	45225.6	2.38550
$712$	−6244.20	−0.328668
$713$	−366.757	−0.0192639
$714$	0	0
$715$	0	0
$716$	−14440.1	−0.753702
$717$	−18104.8	−0.943006
$718$	932.056	0.0484457
$719$	13139.8	0.681548	0.340774	−	0.940145i	$-0.389311\pi$
$719$	13139.8	0.681548	0.340774	+	0.940145i	$0.389311\pi$
$720$	0	0
$721$	0	0
$722$	23671.3	1.22016
$723$	30748.5	1.58167
$724$	−6612.35	−0.339428
$725$	0	0
$726$	−21994.5	−1.12437
$727$	−29182.7	−1.48876	−0.744379	−	0.667757i	$-0.767254\pi$
$727$	−29182.7	−1.48876	−0.744379	+	0.667757i	$0.767254\pi$
$728$	0	0
$729$	−31523.2	−1.60155
$730$	0	0
$731$	−45278.2	−2.29094
$732$	23590.9	1.19118
$733$	−17530.6	−0.883366	−0.441683	−	0.897171i	$-0.645618\pi$
$733$	−17530.6	−0.883366	−0.441683	+	0.897171i	$0.645618\pi$
$734$	−5785.62	−0.290942
$735$	0	0
$736$	−684.975	−0.0343050
$737$	2602.40	0.130069
$738$	26643.9	1.32896
$739$	8788.44	0.437467	0.218733	−	0.975785i	$-0.429808\pi$
$739$	8788.44	0.437467	0.218733	+	0.975785i	$0.429808\pi$
$740$	0	0
$741$	51118.9	2.53428
$742$	0	0
$743$	23553.9	1.16300	0.581500	−	0.813546i	$-0.302466\pi$
$743$	23553.9	1.16300	0.581500	+	0.813546i	$0.302466\pi$
$744$	1159.95	0.0571582
$745$	0	0
$746$	−3619.05	−0.177618
$747$	−21276.2	−1.04211
$748$	−2454.86	−0.119998
$749$	0	0
$750$	0	0
$751$	2860.04	0.138967	0.0694836	−	0.997583i	$-0.477865\pi$
$751$	2860.04	0.138967	0.0694836	+	0.997583i	$0.477865\pi$
$752$	−9401.69	−0.455910
$753$	26706.5	1.29248
$754$	8806.22	0.425336
$755$	0	0
$756$	0	0
$757$	11181.8	0.536866	0.268433	−	0.963298i	$-0.413494\pi$
$757$	11181.8	0.536866	0.268433	+	0.963298i	$0.413494\pi$
$758$	−17988.1	−0.861948
$759$	−1016.02	−0.0485891
$760$	0	0
$761$	3241.06	0.154387	0.0771935	−	0.997016i	$-0.475404\pi$
$761$	3241.06	0.154387	0.0771935	+	0.997016i	$0.475404\pi$
$762$	14487.6	0.688754
$763$	0	0
$764$	−11739.3	−0.555909
$765$	0	0
$766$	−24723.3	−1.16617
$767$	27037.0	1.27282
$768$	2166.38	0.101787
$769$	−5261.79	−0.246743	−0.123371	−	0.992361i	$-0.539371\pi$
$769$	−5261.79	−0.246743	−0.123371	+	0.992361i	$0.539371\pi$
$770$	0	0
$771$	2258.58	0.105500
$772$	−15333.8	−0.714867
$773$	−4393.54	−0.204430	−0.102215	−	0.994762i	$-0.532593\pi$
$773$	−4393.54	−0.204430	−0.102215	+	0.994762i	$0.532593\pi$
$774$	36922.5	1.71467
$775$	0	0
$776$	−7271.47	−0.336380
$777$	0	0
$778$	−9571.15	−0.441057
$779$	40829.2	1.87787
$780$	0	0
$781$	1298.71	0.0595026
$782$	4684.24	0.214205
$783$	14854.1	0.677958
$784$	0	0
$785$	0	0
$786$	13044.0	0.591940
$787$	20812.5	0.942675	0.471338	−	0.881953i	$-0.343771\pi$
$787$	20812.5	0.942675	0.471338	+	0.881953i	$0.343771\pi$
$788$	−673.771	−0.0304595
$789$	−40497.0	−1.82729
$790$	0	0
$791$	0	0
$792$	2001.83	0.0898133
$793$	30790.6	1.37882
$794$	7998.12	0.357484
$795$	0	0
$796$	2313.91	0.103033
$797$	−27138.1	−1.20612	−0.603062	−	0.797694i	$-0.706053\pi$
$797$	−27138.1	−1.20612	−0.603062	+	0.797694i	$0.706053\pi$
$798$	0	0
$799$	64294.0	2.84676
$800$	0	0
$801$	−34821.2	−1.53601
$802$	11879.9	0.523058
$803$	3959.74	0.174018
$804$	15705.3	0.688910
$805$	0	0
$806$	1513.95	0.0661621
$807$	−6502.68	−0.283650
$808$	−12101.5	−0.526895
$809$	13393.0	0.582044	0.291022	−	0.956716i	$-0.406005\pi$
$809$	13393.0	0.582044	0.291022	+	0.956716i	$0.406005\pi$
$810$	0	0
$811$	7438.93	0.322092	0.161046	−	0.986947i	$-0.448513\pi$
$811$	7438.93	0.322092	0.161046	+	0.986947i	$0.448513\pi$
$812$	0	0
$813$	18611.2	0.802859
$814$	35.9672	0.00154871
$815$	0	0
$816$	−14814.9	−0.635570
$817$	56580.1	2.42287
$818$	−18537.3	−0.792350
$819$	0	0
$820$	0	0
$821$	1588.99	0.0675472	0.0337736	−	0.999430i	$-0.489247\pi$
$821$	1588.99	0.0675472	0.0337736	+	0.999430i	$0.489247\pi$
$822$	−13453.4	−0.570855
$823$	−21641.9	−0.916634	−0.458317	−	0.888789i	$-0.651548\pi$
$823$	−21641.9	−0.916634	−0.458317	+	0.888789i	$0.651548\pi$
$824$	−8892.25	−0.375942
$825$	0	0
$826$	0	0
$827$	16078.0	0.676041	0.338021	−	0.941139i	$-0.390243\pi$
$827$	16078.0	0.676041	0.338021	+	0.941139i	$0.390243\pi$
$828$	−3819.80	−0.160323
$829$	−28262.3	−1.18407	−0.592033	−	0.805914i	$-0.701674\pi$
$829$	−28262.3	−1.18407	−0.592033	+	0.805914i	$0.701674\pi$
$830$	0	0
$831$	−38924.9	−1.62490
$832$	2827.54	0.117821
$833$	0	0
$834$	8512.48	0.353433
$835$	0	0
$836$	3067.62	0.126909
$837$	2553.69	0.105458
$838$	4116.44	0.169690
$839$	1240.80	0.0510576	0.0255288	−	0.999674i	$-0.491873\pi$
$839$	1240.80	0.0510576	0.0255288	+	0.999674i	$0.491873\pi$
$840$	0	0
$841$	−14456.4	−0.592743
$842$	−28422.5	−1.16331
$843$	−275.910	−0.0112726
$844$	18544.1	0.756298
$845$	0	0
$846$	−52429.1	−2.13067
$847$	0	0
$848$	−9624.38	−0.389743
$849$	−67428.1	−2.72571
$850$	0	0
$851$	−68.6308	−0.00276455
$852$	7837.62	0.315156
$853$	23806.6	0.955595	0.477797	−	0.878470i	$-0.341435\pi$
$853$	23806.6	0.955595	0.477797	+	0.878470i	$0.341435\pi$
$854$	0	0
$855$	0	0
$856$	−4279.67	−0.170883
$857$	29907.5	1.19209	0.596045	−	0.802951i	$-0.296738\pi$
$857$	29907.5	1.19209	0.596045	+	0.802951i	$0.296738\pi$
$858$	4194.06	0.166880
$859$	12293.0	0.488277	0.244139	−	0.969740i	$-0.421495\pi$
$859$	12293.0	0.488277	0.244139	+	0.969740i	$0.421495\pi$
$860$	0	0
$861$	0	0
$862$	−175.394	−0.00693034
$863$	26135.3	1.03089	0.515444	−	0.856924i	$-0.327627\pi$
$863$	26135.3	1.03089	0.515444	+	0.856924i	$0.327627\pi$
$864$	4769.40	0.187799
$865$	0	0
$866$	−20965.1	−0.822660
$867$	59736.6	2.33998
$868$	0	0
$869$	5686.04	0.221963
$870$	0	0
$871$	20498.4	0.797432
$872$	4785.09	0.185830
$873$	−40549.8	−1.57205
$874$	−5853.47	−0.226541
$875$	0	0
$876$	23896.8	0.921686
$877$	−36508.9	−1.40572	−0.702861	−	0.711328i	$-0.748094\pi$
$877$	−36508.9	−1.40572	−0.702861	+	0.711328i	$0.748094\pi$
$878$	7698.68	0.295920
$879$	1281.43	0.0491714
$880$	0	0
$881$	−250.978	−0.00959780	−0.00479890	−	0.999988i	$-0.501528\pi$
$881$	−250.978	−0.00959780	−0.00479890	+	0.999988i	$0.501528\pi$
$882$	0	0
$883$	−35300.9	−1.34538	−0.672690	−	0.739924i	$-0.734861\pi$
$883$	−35300.9	−1.34538	−0.672690	+	0.739924i	$0.734861\pi$
$884$	−19336.3	−0.735689
$885$	0	0
$886$	−21023.6	−0.797178
$887$	−261.911	−0.00991444	−0.00495722	−	0.999988i	$-0.501578\pi$
$887$	−261.911	−0.00991444	−0.00495722	+	0.999988i	$0.501578\pi$
$888$	217.060	0.00820275
$889$	0	0
$890$	0	0
$891$	318.223	0.0119651
$892$	−4684.51	−0.175840
$893$	−80342.5	−3.01070
$894$	−31647.1	−1.18393
$895$	0	0
$896$	0	0
$897$	−8002.90	−0.297892
$898$	−13001.4	−0.483142
$899$	1707.60	0.0633498
$900$	0	0
$901$	65816.8	2.43360
$902$	3349.84	0.123656
$903$	0	0
$904$	9730.18	0.357988
$905$	0	0
$906$	−17360.2	−0.636595
$907$	4376.24	0.160210	0.0801052	−	0.996786i	$-0.474474\pi$
$907$	4376.24	0.160210	0.0801052	+	0.996786i	$0.474474\pi$
$908$	15763.8	0.576147
$909$	−67485.0	−2.46241
$910$	0	0
$911$	12958.0	0.471261	0.235630	−	0.971843i	$-0.424285\pi$
$911$	12958.0	0.471261	0.235630	+	0.971843i	$0.424285\pi$
$912$	18512.8	0.672173
$913$	−2674.98	−0.0969649
$914$	−10006.4	−0.362126
$915$	0	0
$916$	25118.1	0.906031
$917$	0	0
$918$	−32615.8	−1.17264
$919$	40791.9	1.46420	0.732101	−	0.681196i	$-0.238540\pi$
$919$	40791.9	1.46420	0.732101	+	0.681196i	$0.238540\pi$
$920$	0	0
$921$	26890.0	0.962057
$922$	−13312.2	−0.475503
$923$	10229.6	0.364801
$924$	0	0
$925$	0	0
$926$	−19450.4	−0.690258
$927$	−49588.1	−1.75694
$928$	3189.20	0.112813
$929$	−5214.24	−0.184148	−0.0920740	−	0.995752i	$-0.529350\pi$
$929$	−5214.24	−0.184148	−0.0920740	+	0.995752i	$0.529350\pi$
$930$	0	0
$931$	0	0
$932$	−27810.2	−0.977418
$933$	7773.03	0.272752
$934$	410.123	0.0143679
$935$	0	0
$936$	15767.9	0.550631
$937$	−52538.8	−1.83177	−0.915885	−	0.401441i	$-0.868509\pi$
$937$	−52538.8	−1.83177	−0.915885	+	0.401441i	$0.868509\pi$
$938$	0	0
$939$	28075.9	0.975744
$940$	0	0
$941$	−17883.3	−0.619533	−0.309766	−	0.950813i	$-0.600251\pi$
$941$	−17883.3	−0.619533	−0.309766	+	0.950813i	$0.600251\pi$
$942$	46146.3	1.59610
$943$	−6392.00	−0.220734
$944$	9791.54	0.337593
$945$	0	0
$946$	4642.13	0.159544
$947$	−20390.2	−0.699676	−0.349838	−	0.936810i	$-0.613763\pi$
$947$	−20390.2	−0.699676	−0.349838	+	0.936810i	$0.613763\pi$
$948$	34314.9	1.17563
$949$	31189.8	1.06688
$950$	0	0
$951$	−54279.4	−1.85082
$952$	0	0
$953$	−616.922	−0.0209696	−0.0104848	−	0.999945i	$-0.503337\pi$
$953$	−616.922	−0.0209696	−0.0104848	+	0.999945i	$0.503337\pi$
$954$	−53670.9	−1.82145
$955$	0	0
$956$	−8557.74	−0.289516
$957$	4730.51	0.159786
$958$	21905.3	0.738757
$959$	0	0
$960$	0	0
$961$	−29497.4	−0.990146
$962$	283.304	0.00949490
$963$	−23865.8	−0.798614
$964$	14534.1	0.485594
$965$	0	0
$966$	0	0
$967$	16415.7	0.545910	0.272955	−	0.962027i	$-0.411999\pi$
$967$	16415.7	0.545910	0.272955	+	0.962027i	$0.411999\pi$
$968$	−10396.3	−0.345197
$969$	−126601.	−4.19712
$970$	0	0
$971$	21173.5	0.699784	0.349892	−	0.936790i	$-0.386218\pi$
$971$	21173.5	0.699784	0.349892	+	0.936790i	$0.386218\pi$
$972$	−14176.3	−0.467803
$973$	0	0
$974$	−24518.7	−0.806601
$975$	0	0
$976$	11150.9	0.365708
$977$	−46189.6	−1.51252	−0.756261	−	0.654269i	$-0.772976\pi$
$977$	−46189.6	−1.51252	−0.756261	+	0.654269i	$0.772976\pi$
$978$	36690.5	1.19962
$979$	−4377.93	−0.142921
$980$	0	0
$981$	26684.3	0.868466
$982$	3835.90	0.124652
$983$	44166.1	1.43304	0.716520	−	0.697566i	$-0.245734\pi$
$983$	44166.1	1.43304	0.716520	+	0.697566i	$0.245734\pi$
$984$	20216.0	0.654943
$985$	0	0
$986$	−21809.5	−0.704418
$987$	0	0
$988$	24162.8	0.778058
$989$	−8857.88	−0.284797
$990$	0	0
$991$	−32606.3	−1.04518	−0.522590	−	0.852584i	$-0.675034\pi$
$991$	−32606.3	−1.04518	−0.522590	+	0.852584i	$0.675034\pi$
$992$	548.281	0.0175483
$993$	−57334.8	−1.83229
$994$	0	0
$995$	0	0
$996$	−16143.3	−0.513575
$997$	−28262.5	−0.897776	−0.448888	−	0.893588i	$-0.648180\pi$
$997$	−28262.5	−0.897776	−0.448888	+	0.893588i	$0.648180\pi$
$998$	10256.3	0.325309
$999$	477.869	0.0151342

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.cq.1.4		4
5.4	even	2		2450.4.a.co.1.1		4
7.2	even	3		350.4.e.l.151.1	yes	8
7.4	even	3		350.4.e.l.51.1	✓	8
7.6	odd	2		2450.4.a.cu.1.1		4
35.2	odd	12		350.4.j.j.249.8		16
35.4	even	6		350.4.e.m.51.4	yes	8
35.9	even	6		350.4.e.m.151.4	yes	8
35.18	odd	12		350.4.j.j.149.8		16
35.23	odd	12		350.4.j.j.249.1		16
35.32	odd	12		350.4.j.j.149.1		16
35.34	odd	2		2450.4.a.ck.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.4.e.l.51.1	✓	8	7.4	even	3
350.4.e.l.151.1	yes	8	7.2	even	3
350.4.e.m.51.4	yes	8	35.4	even	6
350.4.e.m.151.4	yes	8	35.9	even	6
350.4.j.j.149.1		16	35.32	odd	12
350.4.j.j.149.8		16	35.18	odd	12
350.4.j.j.249.1		16	35.23	odd	12
350.4.j.j.249.8		16	35.2	odd	12
2450.4.a.ck.1.4		4	35.34	odd	2
2450.4.a.co.1.1		4	5.4	even	2
2450.4.a.cq.1.4		4	1.1	even	1	trivial
2450.4.a.cu.1.1		4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +8.46241 q^{3} +4.00000 q^{4} +16.9248 q^{6} +8.00000 q^{8} +44.6125 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +8.46241 q^{3} +4.00000 q^{4} +16.9248 q^{6} +8.00000 q^{8} +44.6125 q^{9} +5.60896 q^{11} +33.8497 q^{12} +44.1803 q^{13} +16.0000 q^{16} -109.417 q^{17} +89.2249 q^{18} +136.728 q^{19} +11.2179 q^{22} -21.4055 q^{23} +67.6993 q^{24} +88.3605 q^{26} +149.044 q^{27} +99.6624 q^{29} +17.1338 q^{31} +32.0000 q^{32} +47.4653 q^{33} -218.834 q^{34} +178.450 q^{36} +3.20623 q^{37} +273.457 q^{38} +373.872 q^{39} +298.615 q^{41} +413.814 q^{43} +22.4358 q^{44} -42.8109 q^{46} -587.606 q^{47} +135.399 q^{48} -925.931 q^{51} +176.721 q^{52} -601.523 q^{53} +298.088 q^{54} +1157.05 q^{57} +199.325 q^{58} +611.971 q^{59} +696.931 q^{61} +34.2676 q^{62} +64.0000 q^{64} +94.9306 q^{66} +463.973 q^{67} -437.667 q^{68} -181.142 q^{69} +231.542 q^{71} +356.900 q^{72} +705.968 q^{73} +6.41246 q^{74} +546.914 q^{76} +747.743 q^{78} +1013.74 q^{79} +56.7347 q^{81} +597.230 q^{82} -476.913 q^{83} +827.628 q^{86} +843.385 q^{87} +44.8717 q^{88} -780.526 q^{89} -85.6219 q^{92} +144.993 q^{93} -1175.21 q^{94} +270.797 q^{96} -908.934 q^{97} +250.229 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} - q^{3} + 16 q^{4} - 2 q^{6} + 32 q^{8} + 73 q^{9}+O(q^{10}) \) 4 * q + 8 * q^2 - q^3 + 16 * q^4 - 2 * q^6 + 32 * q^8 + 73 * q^9	\( 4 q + 8 q^{2} - q^{3} + 16 q^{4} - 2 q^{6} + 32 q^{8} + 73 q^{9} - 10 q^{11} - 4 q^{12} + 94 q^{13} + 64 q^{16} + 99 q^{17} + 146 q^{18} + 246 q^{19} - 20 q^{22} - 2 q^{23} - 8 q^{24} + 188 q^{26} - 196 q^{27} - 98 q^{29} + 304 q^{31} + 128 q^{32} + 526 q^{33} + 198 q^{34} + 292 q^{36} - 82 q^{37} + 492 q^{38} - 214 q^{39} + 352 q^{41} + 131 q^{43} - 40 q^{44} - 4 q^{46} - 491 q^{47} - 16 q^{48} - 1437 q^{51} + 376 q^{52} - 140 q^{53} - 392 q^{54} + 219 q^{57} - 196 q^{58} + 673 q^{59} + 1425 q^{61} + 608 q^{62} + 256 q^{64} + 1052 q^{66} - 666 q^{67} + 396 q^{68} + 938 q^{69} - 6 q^{71} + 584 q^{72} - 78 q^{73} - 164 q^{74} + 984 q^{76} - 428 q^{78} + 1744 q^{79} + 1708 q^{81} + 704 q^{82} + 926 q^{83} + 262 q^{86} + 1931 q^{87} - 80 q^{88} - 871 q^{89} - 8 q^{92} - 3106 q^{93} - 982 q^{94} - 32 q^{96} - 1462 q^{97} - 5281 q^{99}+O(q^{100}) \) 4 * q + 8 * q^2 - q^3 + 16 * q^4 - 2 * q^6 + 32 * q^8 + 73 * q^9 - 10 * q^11 - 4 * q^12 + 94 * q^13 + 64 * q^16 + 99 * q^17 + 146 * q^18 + 246 * q^19 - 20 * q^22 - 2 * q^23 - 8 * q^24 + 188 * q^26 - 196 * q^27 - 98 * q^29 + 304 * q^31 + 128 * q^32 + 526 * q^33 + 198 * q^34 + 292 * q^36 - 82 * q^37 + 492 * q^38 - 214 * q^39 + 352 * q^41 + 131 * q^43 - 40 * q^44 - 4 * q^46 - 491 * q^47 - 16 * q^48 - 1437 * q^51 + 376 * q^52 - 140 * q^53 - 392 * q^54 + 219 * q^57 - 196 * q^58 + 673 * q^59 + 1425 * q^61 + 608 * q^62 + 256 * q^64 + 1052 * q^66 - 666 * q^67 + 396 * q^68 + 938 * q^69 - 6 * q^71 + 584 * q^72 - 78 * q^73 - 164 * q^74 + 984 * q^76 - 428 * q^78 + 1744 * q^79 + 1708 * q^81 + 704 * q^82 + 926 * q^83 + 262 * q^86 + 1931 * q^87 - 80 * q^88 - 871 * q^89 - 8 * q^92 - 3106 * q^93 - 982 * q^94 - 32 * q^96 - 1462 * q^97 - 5281 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more