LMFDB - Embedded newform 6025.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.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:	$48.1098672178$
Analytic rank:	$0$
Dimension:	$66$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.81437 q^{2} +0.650611 q^{3} +5.92068 q^{4} -1.83106 q^{6} -2.61686 q^{7} -11.0342 q^{8} -2.57671 q^{9} +O(q^{10})$	\(q-2.81437 q^{2} +0.650611 q^{3} +5.92068 q^{4} -1.83106 q^{6} -2.61686 q^{7} -11.0342 q^{8} -2.57671 q^{9} +5.41331 q^{11} +3.85206 q^{12} +1.23017 q^{13} +7.36481 q^{14} +19.2131 q^{16} -3.00034 q^{17} +7.25180 q^{18} -6.30332 q^{19} -1.70256 q^{21} -15.2351 q^{22} -4.83187 q^{23} -7.17899 q^{24} -3.46214 q^{26} -3.62827 q^{27} -15.4936 q^{28} -0.808277 q^{29} -4.46822 q^{31} -32.0042 q^{32} +3.52196 q^{33} +8.44408 q^{34} -15.2558 q^{36} +3.55882 q^{37} +17.7399 q^{38} +0.800360 q^{39} -1.83696 q^{41} +4.79162 q^{42} -3.25301 q^{43} +32.0504 q^{44} +13.5987 q^{46} +9.92958 q^{47} +12.5002 q^{48} -0.152052 q^{49} -1.95206 q^{51} +7.28342 q^{52} +4.28280 q^{53} +10.2113 q^{54} +28.8750 q^{56} -4.10101 q^{57} +2.27479 q^{58} +8.63214 q^{59} -3.96432 q^{61} +12.5752 q^{62} +6.74287 q^{63} +51.6455 q^{64} -9.91209 q^{66} -1.74963 q^{67} -17.7641 q^{68} -3.14367 q^{69} -6.96965 q^{71} +28.4320 q^{72} -14.4128 q^{73} -10.0158 q^{74} -37.3199 q^{76} -14.1659 q^{77} -2.25251 q^{78} +3.91235 q^{79} +5.36953 q^{81} +5.16987 q^{82} -2.24169 q^{83} -10.0803 q^{84} +9.15517 q^{86} -0.525874 q^{87} -59.7317 q^{88} -0.00669658 q^{89} -3.21917 q^{91} -28.6080 q^{92} -2.90707 q^{93} -27.9455 q^{94} -20.8223 q^{96} +11.4628 q^{97} +0.427931 q^{98} -13.9485 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9}+O(q^{10}) $ 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9	$ 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9} + 48 q^{11} + 30 q^{14} + 98 q^{16} + 12 q^{19} + 18 q^{21} + 42 q^{24} + 48 q^{26} + 56 q^{29} + 48 q^{31} + 8 q^{34} + 158 q^{36} + 84 q^{39} + 56 q^{41} + 144 q^{44} + 36 q^{46} + 98 q^{49} + 44 q^{51} + 86 q^{54} + 104 q^{56} + 108 q^{59} + 22 q^{61} + 136 q^{64} + 74 q^{66} + 20 q^{69} + 212 q^{71} + 84 q^{74} + 6 q^{76} + 66 q^{79} + 162 q^{81} - 52 q^{84} + 100 q^{86} + 54 q^{89} + 72 q^{91} - 96 q^{94} + 122 q^{96} + 112 q^{99}+O(q^{100}) $ 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9 + 48 * q^11 + 30 * q^14 + 98 * q^16 + 12 * q^19 + 18 * q^21 + 42 * q^24 + 48 * q^26 + 56 * q^29 + 48 * q^31 + 8 * q^34 + 158 * q^36 + 84 * q^39 + 56 * q^41 + 144 * q^44 + 36 * q^46 + 98 * q^49 + 44 * q^51 + 86 * q^54 + 104 * q^56 + 108 * q^59 + 22 * q^61 + 136 * q^64 + 74 * q^66 + 20 * q^69 + 212 * q^71 + 84 * q^74 + 6 * q^76 + 66 * q^79 + 162 * q^81 - 52 * q^84 + 100 * q^86 + 54 * q^89 + 72 * q^91 - 96 * q^94 + 122 * q^96 + 112 * 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 $

$2$	−2.81437	−1.99006	−0.995030	−	0.0995765i	$-0.968251\pi$
$2$	−2.81437	−1.99006	−0.995030	+	0.0995765i	$0.968251\pi$
$3$	0.650611	0.375630	0.187815	−	0.982204i	$-0.439859\pi$
$3$	0.650611	0.375630	0.187815	+	0.982204i	$0.439859\pi$
$4$	5.92068	2.96034
$5$	0	0
$5$	0	0
$6$	−1.83106	−0.747527
$7$	−2.61686	−0.989079	−0.494540	−	0.869155i	$-0.664663\pi$
$7$	−2.61686	−0.989079	−0.494540	+	0.869155i	$0.664663\pi$
$8$	−11.0342	−3.90119
$9$	−2.57671	−0.858902
$10$	0	0
$11$	5.41331	1.63217	0.816087	−	0.577929i	$-0.196139\pi$
$11$	5.41331	1.63217	0.816087	+	0.577929i	$0.196139\pi$
$12$	3.85206	1.11199
$13$	1.23017	0.341187	0.170593	−	0.985342i	$-0.445432\pi$
$13$	1.23017	0.341187	0.170593	+	0.985342i	$0.445432\pi$
$14$	7.36481	1.96833
$15$	0	0
$16$	19.2131	4.80326
$17$	−3.00034	−0.727691	−0.363845	−	0.931459i	$-0.618536\pi$
$17$	−3.00034	−0.727691	−0.363845	+	0.931459i	$0.618536\pi$
$18$	7.25180	1.70927
$19$	−6.30332	−1.44608	−0.723041	−	0.690805i	$-0.757256\pi$
$19$	−6.30332	−1.44608	−0.723041	+	0.690805i	$0.757256\pi$
$20$	0	0
$21$	−1.70256	−0.371528
$22$	−15.2351	−3.24812
$23$	−4.83187	−1.00752	−0.503758	−	0.863845i	$-0.668050\pi$
$23$	−4.83187	−1.00752	−0.503758	+	0.863845i	$0.668050\pi$
$24$	−7.17899	−1.46541
$25$	0	0
$26$	−3.46214	−0.678982
$27$	−3.62827	−0.698260
$28$	−15.4936	−2.92801
$29$	−0.808277	−0.150093	−0.0750466	−	0.997180i	$-0.523911\pi$
$29$	−0.808277	−0.150093	−0.0750466	+	0.997180i	$0.523911\pi$
$30$	0	0
$31$	−4.46822	−0.802516	−0.401258	−	0.915965i	$-0.631427\pi$
$31$	−4.46822	−0.802516	−0.401258	+	0.915965i	$0.631427\pi$
$32$	−32.0042	−5.65759
$33$	3.52196	0.613094
$34$	8.44408	1.44815
$35$	0	0
$36$	−15.2558	−2.54264
$37$	3.55882	0.585066	0.292533	−	0.956255i	$-0.405502\pi$
$37$	3.55882	0.585066	0.292533	+	0.956255i	$0.405502\pi$
$38$	17.7399	2.87779
$39$	0.800360	0.128160
$40$	0	0
$41$	−1.83696	−0.286884	−0.143442	−	0.989659i	$-0.545817\pi$
$41$	−1.83696	−0.286884	−0.143442	+	0.989659i	$0.545817\pi$
$42$	4.79162	0.739364
$43$	−3.25301	−0.496079	−0.248039	−	0.968750i	$-0.579786\pi$
$43$	−3.25301	−0.496079	−0.248039	+	0.968750i	$0.579786\pi$
$44$	32.0504	4.83179
$45$	0	0
$46$	13.5987	2.00502
$47$	9.92958	1.44838	0.724189	−	0.689602i	$-0.242214\pi$
$47$	9.92958	1.44838	0.724189	+	0.689602i	$0.242214\pi$
$48$	12.5002	1.80425
$49$	−0.152052	−0.0217218
$50$	0	0
$51$	−1.95206	−0.273343
$52$	7.28342	1.01003
$53$	4.28280	0.588288	0.294144	−	0.955761i	$-0.404966\pi$
$53$	4.28280	0.588288	0.294144	+	0.955761i	$0.404966\pi$
$54$	10.2113	1.38958
$55$	0	0
$56$	28.8750	3.85859
$57$	−4.10101	−0.543192
$58$	2.27479	0.298695
$59$	8.63214	1.12381	0.561904	−	0.827202i	$-0.310069\pi$
$59$	8.63214	1.12381	0.561904	+	0.827202i	$0.310069\pi$
$60$	0	0
$61$	−3.96432	−0.507579	−0.253790	−	0.967259i	$-0.581677\pi$
$61$	−3.96432	−0.507579	−0.253790	+	0.967259i	$0.581677\pi$
$62$	12.5752	1.59706
$63$	6.74287	0.849522
$64$	51.6455	6.45568
$65$	0	0
$66$	−9.91209	−1.22009
$67$	−1.74963	−0.213751	−0.106876	−	0.994272i	$-0.534085\pi$
$67$	−1.74963	−0.213751	−0.106876	+	0.994272i	$0.534085\pi$
$68$	−17.7641	−2.15421
$69$	−3.14367	−0.378453
$70$	0	0
$71$	−6.96965	−0.827145	−0.413573	−	0.910471i	$-0.635719\pi$
$71$	−6.96965	−0.827145	−0.413573	+	0.910471i	$0.635719\pi$
$72$	28.4320	3.35074
$73$	−14.4128	−1.68689	−0.843444	−	0.537216i	$-0.819476\pi$
$73$	−14.4128	−1.68689	−0.843444	+	0.537216i	$0.819476\pi$
$74$	−10.0158	−1.16432
$75$	0	0
$76$	−37.3199	−4.28089
$77$	−14.1659	−1.61435
$78$	−2.25251	−0.255046
$79$	3.91235	0.440174	0.220087	−	0.975480i	$-0.429366\pi$
$79$	3.91235	0.440174	0.220087	+	0.975480i	$0.429366\pi$
$80$	0	0
$81$	5.36953	0.596614
$82$	5.16987	0.570917
$83$	−2.24169	−0.246058	−0.123029	−	0.992403i	$-0.539261\pi$
$83$	−2.24169	−0.246058	−0.123029	+	0.992403i	$0.539261\pi$
$84$	−10.0803	−1.09985
$85$	0	0
$86$	9.15517	0.987227
$87$	−0.525874	−0.0563796
$88$	−59.7317	−6.36742
$89$	−0.00669658	−0.000709836 0	−0.000354918	−	1.00000i	$-0.500113\pi$
$89$	−0.00669658	−0.000709836 0	−0.000354918		1.00000i	$0.500113\pi$
$90$	0	0
$91$	−3.21917	−0.337461
$92$	−28.6080	−2.98259
$93$	−2.90707	−0.301449
$94$	−27.9455	−2.88236
$95$	0	0
$96$	−20.8223	−2.12516
$97$	11.4628	1.16387	0.581934	−	0.813236i	$-0.302296\pi$
$97$	11.4628	1.16387	0.581934	+	0.813236i	$0.302296\pi$
$98$	0.427931	0.0432276
$99$	−13.9485	−1.40188
$100$	0	0
$101$	−15.2533	−1.51776	−0.758879	−	0.651231i	$-0.774253\pi$
$101$	−15.2533	−1.51776	−0.758879	+	0.651231i	$0.774253\pi$
$102$	5.49381	0.543968
$103$	10.6257	1.04698	0.523490	−	0.852032i	$-0.324630\pi$
$103$	10.6257	1.04698	0.523490	+	0.852032i	$0.324630\pi$
$104$	−13.5739	−1.33103
$105$	0	0
$106$	−12.0534	−1.17073
$107$	12.1123	1.17094	0.585472	−	0.810693i	$-0.300909\pi$
$107$	12.1123	1.17094	0.585472	+	0.810693i	$0.300909\pi$
$108$	−21.4818	−2.06709
$109$	15.2837	1.46392	0.731959	−	0.681349i	$-0.238606\pi$
$109$	15.2837	1.46392	0.731959	+	0.681349i	$0.238606\pi$
$110$	0	0
$111$	2.31541	0.219769
$112$	−50.2778	−4.75081
$113$	20.2175	1.90190	0.950951	−	0.309341i	$-0.100109\pi$
$113$	20.2175	1.90190	0.950951	+	0.309341i	$0.100109\pi$
$114$	11.5418	1.08099
$115$	0	0
$116$	−4.78555	−0.444327
$117$	−3.16978	−0.293046
$118$	−24.2940	−2.23645
$119$	7.85148	0.719744
$120$	0	0
$121$	18.3039	1.66399
$122$	11.1571	1.01011
$123$	−1.19514	−0.107762
$124$	−26.4549	−2.37572
$125$	0	0
$126$	−18.9769	−1.69060
$127$	−10.7914	−0.957583	−0.478791	−	0.877929i	$-0.658925\pi$
$127$	−10.7914	−0.957583	−0.478791	+	0.877929i	$0.658925\pi$
$128$	−81.3411	−7.18960
$129$	−2.11644	−0.186342
$130$	0	0
$131$	4.42934	0.386993	0.193497	−	0.981101i	$-0.438017\pi$
$131$	4.42934	0.386993	0.193497	+	0.981101i	$0.438017\pi$
$132$	20.8524	1.81497
$133$	16.4949	1.43029
$134$	4.92410	0.425378
$135$	0	0
$136$	33.1065	2.83886
$137$	−5.24958	−0.448502	−0.224251	−	0.974531i	$-0.571994\pi$
$137$	−5.24958	−0.448502	−0.224251	+	0.974531i	$0.571994\pi$
$138$	8.84745	0.753145
$139$	6.16732	0.523105	0.261553	−	0.965189i	$-0.415765\pi$
$139$	6.16732	0.523105	0.261553	+	0.965189i	$0.415765\pi$
$140$	0	0
$141$	6.46029	0.544055
$142$	19.6152	1.64607
$143$	6.65927	0.556876
$144$	−49.5064	−4.12553
$145$	0	0
$146$	40.5629	3.35701
$147$	−0.0989269	−0.00815935
$148$	21.0706	1.73199
$149$	10.5225	0.862034	0.431017	−	0.902344i	$-0.358155\pi$
$149$	10.5225	0.862034	0.431017	+	0.902344i	$0.358155\pi$
$150$	0	0
$151$	−20.7257	−1.68663	−0.843315	−	0.537420i	$-0.819399\pi$
$151$	−20.7257	−1.68663	−0.843315	+	0.537420i	$0.819399\pi$
$152$	69.5523	5.64144
$153$	7.73100	0.625015
$154$	39.8680	3.21265
$155$	0	0
$156$	4.73867	0.379397
$157$	−17.8432	−1.42404	−0.712019	−	0.702160i	$-0.752219\pi$
$157$	−17.8432	−1.42404	−0.712019	+	0.702160i	$0.752219\pi$
$158$	−11.0108	−0.875972
$159$	2.78644	0.220979
$160$	0	0
$161$	12.6443	0.996513
$162$	−15.1118	−1.18730
$163$	−4.59778	−0.360126	−0.180063	−	0.983655i	$-0.557630\pi$
$163$	−4.59778	−0.360126	−0.180063	+	0.983655i	$0.557630\pi$
$164$	−10.8760	−0.849275
$165$	0	0
$166$	6.30895	0.489670
$167$	17.5052	1.35459	0.677297	−	0.735710i	$-0.263151\pi$
$167$	17.5052	1.35459	0.677297	+	0.735710i	$0.263151\pi$
$168$	18.7864	1.44940
$169$	−11.4867	−0.883592
$170$	0	0
$171$	16.2418	1.24204
$172$	−19.2600	−1.46856
$173$	−7.56238	−0.574957	−0.287478	−	0.957787i	$-0.592817\pi$
$173$	−7.56238	−0.574957	−0.287478	+	0.957787i	$0.592817\pi$
$174$	1.48000	0.112199
$175$	0	0
$176$	104.006	7.83976
$177$	5.61616	0.422137
$178$	0.0188467	0.00141262
$179$	18.9722	1.41805	0.709024	−	0.705184i	$-0.249136\pi$
$179$	18.9722	1.41805	0.709024	+	0.705184i	$0.249136\pi$
$180$	0	0
$181$	23.4516	1.74314	0.871572	−	0.490268i	$-0.163101\pi$
$181$	23.4516	1.74314	0.871572	+	0.490268i	$0.163101\pi$
$182$	9.05994	0.671567
$183$	−2.57923	−0.190662
$184$	53.3160	3.93051
$185$	0	0
$186$	8.18158	0.599902
$187$	−16.2418	−1.18772
$188$	58.7898	4.28769
$189$	9.49466	0.690635
$190$	0	0
$191$	1.45986	0.105632	0.0528159	−	0.998604i	$-0.483180\pi$
$191$	1.45986	0.105632	0.0528159	+	0.998604i	$0.483180\pi$
$192$	33.6011	2.42495
$193$	0.501079	0.0360684	0.0180342	−	0.999837i	$-0.494259\pi$
$193$	0.501079	0.0360684	0.0180342	+	0.999837i	$0.494259\pi$
$194$	−32.2605	−2.31617
$195$	0	0
$196$	−0.900253	−0.0643038
$197$	−3.38682	−0.241301	−0.120650	−	0.992695i	$-0.538498\pi$
$197$	−3.38682	−0.241301	−0.120650	+	0.992695i	$0.538498\pi$
$198$	39.2562	2.78982
$199$	−8.11615	−0.575339	−0.287669	−	0.957730i	$-0.592880\pi$
$199$	−8.11615	−0.575339	−0.287669	+	0.957730i	$0.592880\pi$
$200$	0	0
$201$	−1.13833	−0.0802914
$202$	42.9284	3.02043
$203$	2.11515	0.148454
$204$	−11.5575	−0.809187
$205$	0	0
$206$	−29.9046	−2.08355
$207$	12.4503	0.865357
$208$	23.6353	1.63881
$209$	−34.1218	−2.36026
$210$	0	0
$211$	−19.3941	−1.33514	−0.667571	−	0.744546i	$-0.732666\pi$
$211$	−19.3941	−1.33514	−0.667571	+	0.744546i	$0.732666\pi$
$212$	25.3571	1.74153
$213$	−4.53453	−0.310701
$214$	−34.0886	−2.33025
$215$	0	0
$216$	40.0351	2.72405
$217$	11.6927	0.793752
$218$	−43.0141	−2.91328
$219$	−9.37712	−0.633647
$220$	0	0
$221$	−3.69092	−0.248278
$222$	−6.51641	−0.437353
$223$	−10.0804	−0.675036	−0.337518	−	0.941319i	$-0.609587\pi$
$223$	−10.0804	−0.675036	−0.337518	+	0.941319i	$0.609587\pi$
$224$	83.7504	5.59581
$225$	0	0
$226$	−56.8995	−3.78490
$227$	18.2708	1.21267	0.606337	−	0.795208i	$-0.292638\pi$
$227$	18.2708	1.21267	0.606337	+	0.795208i	$0.292638\pi$
$228$	−24.2808	−1.60803
$229$	8.47594	0.560106	0.280053	−	0.959985i	$-0.409648\pi$
$229$	8.47594	0.560106	0.280053	+	0.959985i	$0.409648\pi$
$230$	0	0
$231$	−9.21646	−0.606399
$232$	8.91872	0.585542
$233$	6.84820	0.448640	0.224320	−	0.974516i	$-0.427984\pi$
$233$	6.84820	0.448640	0.224320	+	0.974516i	$0.427984\pi$
$234$	8.92092	0.583179
$235$	0	0
$236$	51.1081	3.32685
$237$	2.54542	0.165343
$238$	−22.0970	−1.43233
$239$	22.5107	1.45609	0.728047	−	0.685527i	$-0.240428\pi$
$239$	22.5107	1.45609	0.728047	+	0.685527i	$0.240428\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	−51.5140	−3.31144
$243$	14.3783	0.922366
$244$	−23.4715	−1.50261
$245$	0	0
$246$	3.36358	0.214454
$247$	−7.75414	−0.493384
$248$	49.3034	3.13077
$249$	−1.45847	−0.0924268
$250$	0	0
$251$	−16.1052	−1.01655	−0.508276	−	0.861194i	$-0.669717\pi$
$251$	−16.1052	−1.01655	−0.508276	+	0.861194i	$0.669717\pi$
$252$	39.9224	2.51487
$253$	−26.1564	−1.64444
$254$	30.3710	1.90565
$255$	0	0
$256$	125.633	7.85206
$257$	−10.7289	−0.669251	−0.334626	−	0.942351i	$-0.608610\pi$
$257$	−10.7289	−0.669251	−0.334626	+	0.942351i	$0.608610\pi$
$258$	5.95645	0.370832
$259$	−9.31293	−0.578677
$260$	0	0
$261$	2.08269	0.128915
$262$	−12.4658	−0.770140
$263$	21.5665	1.32985	0.664924	−	0.746911i	$-0.268464\pi$
$263$	21.5665	1.32985	0.664924	+	0.746911i	$0.268464\pi$
$264$	−38.8621	−2.39180
$265$	0	0
$266$	−46.4228	−2.84636
$267$	−0.00435687	−0.000266636 0
$268$	−10.3590	−0.632776
$269$	2.92655	0.178435	0.0892174	−	0.996012i	$-0.471563\pi$
$269$	2.92655	0.178435	0.0892174	+	0.996012i	$0.471563\pi$
$270$	0	0
$271$	−1.75309	−0.106493	−0.0532463	−	0.998581i	$-0.516957\pi$
$271$	−1.75309	−0.106493	−0.0532463	+	0.998581i	$0.516957\pi$
$272$	−57.6458	−3.49529
$273$	−2.09443	−0.126761
$274$	14.7743	0.892546
$275$	0	0
$276$	−18.6127	−1.12035
$277$	−9.59302	−0.576389	−0.288194	−	0.957572i	$-0.593055\pi$
$277$	−9.59302	−0.576389	−0.288194	+	0.957572i	$0.593055\pi$
$278$	−17.3571	−1.04101
$279$	11.5133	0.689283
$280$	0	0
$281$	−2.01836	−0.120406	−0.0602028	−	0.998186i	$-0.519175\pi$
$281$	−2.01836	−0.120406	−0.0602028	+	0.998186i	$0.519175\pi$
$282$	−18.1817	−1.08270
$283$	1.20732	0.0717678	0.0358839	−	0.999356i	$-0.488575\pi$
$283$	1.20732	0.0717678	0.0358839	+	0.999356i	$0.488575\pi$
$284$	−41.2650	−2.44863
$285$	0	0
$286$	−18.7416	−1.10822
$287$	4.80705	0.283751
$288$	82.4653	4.85932
$289$	−7.99793	−0.470467
$290$	0	0
$291$	7.45780	0.437184
$292$	−85.3334	−4.99376
$293$	11.5977	0.677547	0.338774	−	0.940868i	$-0.389988\pi$
$293$	11.5977	0.677547	0.338774	+	0.940868i	$0.389988\pi$
$294$	0.278417	0.0162376
$295$	0	0
$296$	−39.2688	−2.28246
$297$	−19.6409	−1.13968
$298$	−29.6141	−1.71550
$299$	−5.94401	−0.343751
$300$	0	0
$301$	8.51266	0.490662
$302$	58.3297	3.35649
$303$	−9.92395	−0.570116
$304$	−121.106	−6.94591
$305$	0	0
$306$	−21.7579	−1.24382
$307$	29.1382	1.66301	0.831503	−	0.555520i	$-0.187481\pi$
$307$	29.1382	1.66301	0.831503	+	0.555520i	$0.187481\pi$
$308$	−83.8715	−4.77902
$309$	6.91319	0.393278
$310$	0	0
$311$	6.98179	0.395901	0.197950	−	0.980212i	$-0.436571\pi$
$311$	6.98179	0.395901	0.197950	+	0.980212i	$0.436571\pi$
$312$	−8.83136	−0.499977
$313$	16.8525	0.952560	0.476280	−	0.879294i	$-0.341985\pi$
$313$	16.8525	0.952560	0.476280	+	0.879294i	$0.341985\pi$
$314$	50.2172	2.83392
$315$	0	0
$316$	23.1637	1.30306
$317$	10.8529	0.609562	0.304781	−	0.952423i	$-0.401417\pi$
$317$	10.8529	0.609562	0.304781	+	0.952423i	$0.401417\pi$
$318$	−7.84206	−0.439761
$319$	−4.37545	−0.244978
$320$	0	0
$321$	7.88042	0.439842
$322$	−35.5858	−1.98312
$323$	18.9121	1.05230
$324$	31.7912	1.76618
$325$	0	0
$326$	12.9399	0.716672
$327$	9.94377	0.549892
$328$	20.2694	1.11919
$329$	−25.9843	−1.43256
$330$	0	0
$331$	−13.0540	−0.717512	−0.358756	−	0.933431i	$-0.616799\pi$
$331$	−13.0540	−0.717512	−0.358756	+	0.933431i	$0.616799\pi$
$332$	−13.2723	−0.728414
$333$	−9.17003	−0.502515
$334$	−49.2661	−2.69572
$335$	0	0
$336$	−32.7113	−1.78455
$337$	−21.4023	−1.16586	−0.582929	−	0.812523i	$-0.698093\pi$
$337$	−21.4023	−1.16586	−0.582929	+	0.812523i	$0.698093\pi$
$338$	32.3278	1.75840
$339$	13.1537	0.714412
$340$	0	0
$341$	−24.1879	−1.30985
$342$	−45.7105	−2.47174
$343$	18.7159	1.01056
$344$	35.8944	1.93530
$345$	0	0
$346$	21.2833	1.14420
$347$	16.0213	0.860066	0.430033	−	0.902813i	$-0.358502\pi$
$347$	16.0213	0.860066	0.430033	+	0.902813i	$0.358502\pi$
$348$	−3.11353	−0.166903
$349$	−3.02544	−0.161948	−0.0809740	−	0.996716i	$-0.525803\pi$
$349$	−3.02544	−0.161948	−0.0809740	+	0.996716i	$0.525803\pi$
$350$	0	0
$351$	−4.46337	−0.238237
$352$	−173.248	−9.23417
$353$	17.6346	0.938593	0.469297	−	0.883041i	$-0.344508\pi$
$353$	17.6346	0.938593	0.469297	+	0.883041i	$0.344508\pi$
$354$	−15.8060	−0.840077
$355$	0	0
$356$	−0.0396483	−0.00210135
$357$	5.10826	0.270358
$358$	−53.3948	−2.82200
$359$	28.6242	1.51073	0.755363	−	0.655307i	$-0.227461\pi$
$359$	28.6242	1.51073	0.755363	+	0.655307i	$0.227461\pi$
$360$	0	0
$361$	20.7319	1.09115
$362$	−66.0015	−3.46896
$363$	11.9087	0.625046
$364$	−19.0597	−0.998998
$365$	0	0
$366$	7.25891	0.379429
$367$	−3.90612	−0.203898	−0.101949	−	0.994790i	$-0.532508\pi$
$367$	−3.90612	−0.203898	−0.101949	+	0.994790i	$0.532508\pi$
$368$	−92.8350	−4.83936
$369$	4.73329	0.246405
$370$	0	0
$371$	−11.2075	−0.581863
$372$	−17.2118	−0.892392
$373$	7.22326	0.374006	0.187003	−	0.982359i	$-0.440123\pi$
$373$	7.22326	0.374006	0.187003	+	0.982359i	$0.440123\pi$
$374$	45.7104	2.36363
$375$	0	0
$376$	−109.565	−5.65040
$377$	−0.994315	−0.0512098
$378$	−26.7215	−1.37440
$379$	−9.64189	−0.495271	−0.247635	−	0.968853i	$-0.579654\pi$
$379$	−9.64189	−0.495271	−0.247635	+	0.968853i	$0.579654\pi$
$380$	0	0
$381$	−7.02101	−0.359697
$382$	−4.10859	−0.210214
$383$	−32.2148	−1.64610	−0.823050	−	0.567969i	$-0.807729\pi$
$383$	−32.2148	−1.64610	−0.823050	+	0.567969i	$0.807729\pi$
$384$	−52.9214	−2.70063
$385$	0	0
$386$	−1.41022	−0.0717783
$387$	8.38204	0.426083
$388$	67.8673	3.44544
$389$	−28.7781	−1.45911	−0.729554	−	0.683923i	$-0.760272\pi$
$389$	−28.7781	−1.45911	−0.729554	+	0.683923i	$0.760272\pi$
$390$	0	0
$391$	14.4973	0.733159
$392$	1.67778	0.0847407
$393$	2.88178	0.145366
$394$	9.53176	0.480203
$395$	0	0
$396$	−82.5846	−4.15003
$397$	16.4418	0.825188	0.412594	−	0.910915i	$-0.364623\pi$
$397$	16.4418	0.825188	0.412594	+	0.910915i	$0.364623\pi$
$398$	22.8418	1.14496
$399$	10.7318	0.537260
$400$	0	0
$401$	37.9686	1.89606	0.948030	−	0.318180i	$-0.103072\pi$
$401$	37.9686	1.89606	0.948030	+	0.318180i	$0.103072\pi$
$402$	3.20367	0.159785
$403$	−5.49666	−0.273808
$404$	−90.3097	−4.49308
$405$	0	0
$406$	−5.95280	−0.295433
$407$	19.2650	0.954930
$408$	21.5395	1.06636
$409$	5.16290	0.255289	0.127644	−	0.991820i	$-0.459258\pi$
$409$	5.16290	0.255289	0.127644	+	0.991820i	$0.459258\pi$
$410$	0	0
$411$	−3.41543	−0.168471
$412$	62.9113	3.09942
$413$	−22.5891	−1.11154
$414$	−35.0398	−1.72211
$415$	0	0
$416$	−39.3705	−1.93030
$417$	4.01253	0.196494
$418$	96.0315	4.69705
$419$	−28.7636	−1.40519	−0.702597	−	0.711588i	$-0.747976\pi$
$419$	−28.7636	−1.40519	−0.702597	+	0.711588i	$0.747976\pi$
$420$	0	0
$421$	20.3201	0.990343	0.495171	−	0.868795i	$-0.335105\pi$
$421$	20.3201	0.990343	0.495171	+	0.868795i	$0.335105\pi$
$422$	54.5820	2.65701
$423$	−25.5856	−1.24401
$424$	−47.2574	−2.29502
$425$	0	0
$426$	12.7618	0.618313
$427$	10.3741	0.502036
$428$	71.7132	3.46639
$429$	4.33259	0.209180
$430$	0	0
$431$	−11.8735	−0.571924	−0.285962	−	0.958241i	$-0.592313\pi$
$431$	−11.8735	−0.571924	−0.285962	+	0.958241i	$0.592313\pi$
$432$	−69.7101	−3.35393
$433$	−27.7828	−1.33515	−0.667577	−	0.744540i	$-0.732669\pi$
$433$	−27.7828	−1.33515	−0.667577	+	0.744540i	$0.732669\pi$
$434$	−32.9076	−1.57961
$435$	0	0
$436$	90.4901	4.33369
$437$	30.4569	1.45695
$438$	26.3907	1.26100
$439$	19.8875	0.949177	0.474589	−	0.880208i	$-0.342597\pi$
$439$	19.8875	0.949177	0.474589	+	0.880208i	$0.342597\pi$
$440$	0	0
$441$	0.391794	0.0186569
$442$	10.3876	0.494089
$443$	12.6572	0.601362	0.300681	−	0.953725i	$-0.402786\pi$
$443$	12.6572	0.601362	0.300681	+	0.953725i	$0.402786\pi$
$444$	13.7088	0.650590
$445$	0	0
$446$	28.3701	1.34336
$447$	6.84603	0.323806
$448$	−135.149	−6.38518
$449$	−16.8009	−0.792882	−0.396441	−	0.918060i	$-0.629755\pi$
$449$	−16.8009	−0.792882	−0.396441	+	0.918060i	$0.629755\pi$
$450$	0	0
$451$	−9.94401	−0.468245
$452$	119.701	5.63027
$453$	−13.4843	−0.633550
$454$	−51.4207	−2.41329
$455$	0	0
$456$	45.2515	2.11910
$457$	25.2275	1.18009	0.590047	−	0.807369i	$-0.299109\pi$
$457$	25.2275	1.18009	0.590047	+	0.807369i	$0.299109\pi$
$458$	−23.8544	−1.11464
$459$	10.8860	0.508117
$460$	0	0
$461$	5.43924	0.253331	0.126665	−	0.991946i	$-0.459573\pi$
$461$	5.43924	0.253331	0.126665	+	0.991946i	$0.459573\pi$
$462$	25.9385	1.20677
$463$	7.84739	0.364699	0.182350	−	0.983234i	$-0.441630\pi$
$463$	7.84739	0.364699	0.182350	+	0.983234i	$0.441630\pi$
$464$	−15.5295	−0.720938
$465$	0	0
$466$	−19.2734	−0.892821
$467$	21.2589	0.983744	0.491872	−	0.870668i	$-0.336313\pi$
$467$	21.2589	0.983744	0.491872	+	0.870668i	$0.336313\pi$
$468$	−18.7672	−0.867515
$469$	4.57853	0.211417
$470$	0	0
$471$	−11.6090	−0.534912
$472$	−95.2490	−4.38419
$473$	−17.6095	−0.809687
$474$	−7.16374	−0.329042
$475$	0	0
$476$	46.4861	2.13068
$477$	−11.0355	−0.505281
$478$	−63.3533	−2.89771
$479$	−18.2829	−0.835367	−0.417684	−	0.908593i	$-0.637158\pi$
$479$	−18.2829	−0.835367	−0.417684	+	0.908593i	$0.637158\pi$
$480$	0	0
$481$	4.37794	0.199617
$482$	2.81437	0.128191
$483$	8.22654	0.374320
$484$	108.372	4.92598
$485$	0	0
$486$	−40.4658	−1.83556
$487$	−12.8478	−0.582191	−0.291095	−	0.956694i	$-0.594020\pi$
$487$	−12.8478	−0.582191	−0.291095	+	0.956694i	$0.594020\pi$
$488$	43.7432	1.98016
$489$	−2.99137	−0.135274
$490$	0	0
$491$	30.3931	1.37162	0.685812	−	0.727779i	$-0.259447\pi$
$491$	30.3931	1.37162	0.685812	+	0.727779i	$0.259447\pi$
$492$	−7.07606	−0.319013
$493$	2.42511	0.109221
$494$	21.8230	0.981864
$495$	0	0
$496$	−85.8482	−3.85470
$497$	18.2386	0.818112
$498$	4.10467	0.183935
$499$	3.88505	0.173919	0.0869593	−	0.996212i	$-0.472285\pi$
$499$	3.88505	0.173919	0.0869593	+	0.996212i	$0.472285\pi$
$500$	0	0
$501$	11.3891	0.508826
$502$	45.3260	2.02300
$503$	−28.7969	−1.28399	−0.641994	−	0.766709i	$-0.721893\pi$
$503$	−28.7969	−1.28399	−0.641994	+	0.766709i	$0.721893\pi$
$504$	−74.4024	−3.31415
$505$	0	0
$506$	73.6138	3.27253
$507$	−7.47337	−0.331904
$508$	−63.8924	−2.83477
$509$	−10.6962	−0.474099	−0.237050	−	0.971498i	$-0.576180\pi$
$509$	−10.6962	−0.474099	−0.237050	+	0.971498i	$0.576180\pi$
$510$	0	0
$511$	37.7162	1.66847
$512$	−190.895	−8.43646
$513$	22.8701	1.00974
$514$	30.1951	1.33185
$515$	0	0
$516$	−12.5308	−0.551636
$517$	53.7519	2.36400
$518$	26.2100	1.15160
$519$	−4.92017	−0.215971
$520$	0	0
$521$	11.8613	0.519652	0.259826	−	0.965656i	$-0.416335\pi$
$521$	11.8613	0.519652	0.259826	+	0.965656i	$0.416335\pi$
$522$	−5.86146	−0.256549
$523$	5.93707	0.259610	0.129805	−	0.991540i	$-0.458565\pi$
$523$	5.93707	0.259610	0.129805	+	0.991540i	$0.458565\pi$
$524$	26.2247	1.14563
$525$	0	0
$526$	−60.6961	−2.64648
$527$	13.4062	0.583983
$528$	67.6676	2.94485
$529$	0.347002	0.0150871
$530$	0	0
$531$	−22.2425	−0.965241
$532$	97.6610	4.23414
$533$	−2.25976	−0.0978812
$534$	0.0122618	0.000530622 0
$535$	0	0
$536$	19.3058	0.833884
$537$	12.3435	0.532662
$538$	−8.23639	−0.355096
$539$	−0.823106	−0.0354537
$540$	0	0
$541$	−20.2259	−0.869580	−0.434790	−	0.900532i	$-0.643177\pi$
$541$	−20.2259	−0.869580	−0.434790	+	0.900532i	$0.643177\pi$
$542$	4.93385	0.211927
$543$	15.2579	0.654778
$544$	96.0235	4.11698
$545$	0	0
$546$	5.89450	0.252261
$547$	−43.0374	−1.84014	−0.920072	−	0.391749i	$-0.871870\pi$
$547$	−43.0374	−1.84014	−0.920072	+	0.391749i	$0.871870\pi$
$548$	−31.0811	−1.32772
$549$	10.2149	0.435961
$550$	0	0
$551$	5.09483	0.217047
$552$	34.6880	1.47642
$553$	−10.2381	−0.435367
$554$	26.9983	1.14705
$555$	0	0
$556$	36.5147	1.54857
$557$	27.1432	1.15009	0.575047	−	0.818121i	$-0.304984\pi$
$557$	27.1432	1.15009	0.575047	+	0.818121i	$0.304984\pi$
$558$	−32.4026	−1.37171
$559$	−4.00174	−0.169256
$560$	0	0
$561$	−10.5671	−0.446143
$562$	5.68042	0.239614
$563$	−6.85771	−0.289018	−0.144509	−	0.989503i	$-0.546160\pi$
$563$	−6.85771	−0.289018	−0.144509	+	0.989503i	$0.546160\pi$
$564$	38.2493	1.61059
$565$	0	0
$566$	−3.39785	−0.142822
$567$	−14.0513	−0.590099
$568$	76.9047	3.22685
$569$	34.1608	1.43210	0.716048	−	0.698051i	$-0.245949\pi$
$569$	34.1608	1.43210	0.716048	+	0.698051i	$0.245949\pi$
$570$	0	0
$571$	13.6990	0.573284	0.286642	−	0.958038i	$-0.407461\pi$
$571$	13.6990	0.573284	0.286642	+	0.958038i	$0.407461\pi$
$572$	39.4274	1.64854
$573$	0.949801	0.0396785
$574$	−13.5288	−0.564682
$575$	0	0
$576$	−133.075	−5.54480
$577$	−10.4072	−0.433259	−0.216629	−	0.976254i	$-0.569506\pi$
$577$	−10.4072	−0.433259	−0.216629	+	0.976254i	$0.569506\pi$
$578$	22.5091	0.936257
$579$	0.326007	0.0135484
$580$	0	0
$581$	5.86619	0.243371
$582$	−20.9890	−0.870022
$583$	23.1841	0.960188
$584$	159.034	6.58087
$585$	0	0
$586$	−32.6403	−1.34836
$587$	38.7926	1.60114	0.800572	−	0.599237i	$-0.204529\pi$
$587$	38.7926	1.60114	0.800572	+	0.599237i	$0.204529\pi$
$588$	−0.585714	−0.0241544
$589$	28.1646	1.16050
$590$	0	0
$591$	−2.20350	−0.0906400
$592$	68.3758	2.81023
$593$	−40.6345	−1.66866	−0.834328	−	0.551268i	$-0.814144\pi$
$593$	−40.6345	−1.66866	−0.834328	+	0.551268i	$0.814144\pi$
$594$	55.2768	2.26804
$595$	0	0
$596$	62.3001	2.55191
$597$	−5.28046	−0.216115
$598$	16.7286	0.684085
$599$	−10.5304	−0.430261	−0.215131	−	0.976585i	$-0.569018\pi$
$599$	−10.5304	−0.430261	−0.215131	+	0.976585i	$0.569018\pi$
$600$	0	0
$601$	14.2969	0.583184	0.291592	−	0.956543i	$-0.405815\pi$
$601$	14.2969	0.583184	0.291592	+	0.956543i	$0.405815\pi$
$602$	−23.9578	−0.976446
$603$	4.50828	0.183591
$604$	−122.710	−4.99299
$605$	0	0
$606$	27.9297	1.13457
$607$	−23.0505	−0.935591	−0.467795	−	0.883837i	$-0.654952\pi$
$607$	−23.0505	−0.935591	−0.467795	+	0.883837i	$0.654952\pi$
$608$	201.733	8.18134
$609$	1.37614	0.0557639
$610$	0	0
$611$	12.2150	0.494167
$612$	45.7728	1.85025
$613$	20.9443	0.845932	0.422966	−	0.906146i	$-0.360989\pi$
$613$	20.9443	0.845932	0.422966	+	0.906146i	$0.360989\pi$
$614$	−82.0058	−3.30948
$615$	0	0
$616$	156.309	6.29788
$617$	34.0970	1.37269	0.686346	−	0.727275i	$-0.259214\pi$
$617$	34.0970	1.37269	0.686346	+	0.727275i	$0.259214\pi$
$618$	−19.4563	−0.782646
$619$	5.94768	0.239058	0.119529	−	0.992831i	$-0.461862\pi$
$619$	5.94768	0.239058	0.119529	+	0.992831i	$0.461862\pi$
$620$	0	0
$621$	17.5313	0.703508
$622$	−19.6493	−0.787867
$623$	0.0175240	0.000702084 0
$624$	15.3774	0.615587
$625$	0	0
$626$	−47.4292	−1.89565
$627$	−22.2000	−0.886584
$628$	−105.644	−4.21564
$629$	−10.6777	−0.425747
$630$	0	0
$631$	31.1383	1.23960	0.619798	−	0.784761i	$-0.287215\pi$
$631$	31.1383	1.23960	0.619798	+	0.784761i	$0.287215\pi$
$632$	−43.1698	−1.71720
$633$	−12.6180	−0.501520
$634$	−30.5442	−1.21306
$635$	0	0
$636$	16.4976	0.654172
$637$	−0.187050	−0.00741118
$638$	12.3141	0.487521
$639$	17.9587	0.710436
$640$	0	0
$641$	−41.4089	−1.63555	−0.817776	−	0.575536i	$-0.804793\pi$
$641$	−41.4089	−1.63555	−0.817776	+	0.575536i	$0.804793\pi$
$642$	−22.1784	−0.875312
$643$	38.2661	1.50907	0.754534	−	0.656261i	$-0.227863\pi$
$643$	38.2661	1.50907	0.754534	+	0.656261i	$0.227863\pi$
$644$	74.8630	2.95001
$645$	0	0
$646$	−53.2258	−2.09414
$647$	−38.1859	−1.50124	−0.750622	−	0.660732i	$-0.770246\pi$
$647$	−38.1859	−1.50124	−0.750622	+	0.660732i	$0.770246\pi$
$648$	−59.2486	−2.32750
$649$	46.7284	1.83425
$650$	0	0
$651$	7.60740	0.298157
$652$	−27.2220	−1.06609
$653$	0.323660	0.0126658	0.00633290	−	0.999980i	$-0.497984\pi$
$653$	0.323660	0.0126658	0.00633290	+	0.999980i	$0.497984\pi$
$654$	−27.9854	−1.09432
$655$	0	0
$656$	−35.2935	−1.37798
$657$	37.1375	1.44887
$658$	73.1294	2.85088
$659$	−23.9845	−0.934303	−0.467151	−	0.884177i	$-0.654720\pi$
$659$	−23.9845	−0.934303	−0.467151	+	0.884177i	$0.654720\pi$
$660$	0	0
$661$	12.3973	0.482198	0.241099	−	0.970501i	$-0.422492\pi$
$661$	12.3973	0.482198	0.241099	+	0.970501i	$0.422492\pi$
$662$	36.7388	1.42789
$663$	−2.40136	−0.0932609
$664$	24.7354	0.959918
$665$	0	0
$666$	25.8079	1.00003
$667$	3.90549	0.151221
$668$	103.643	4.01005
$669$	−6.55844	−0.253564
$670$	0	0
$671$	−21.4601	−0.828458
$672$	54.4889	2.10196
$673$	30.2704	1.16684	0.583418	−	0.812172i	$-0.301715\pi$
$673$	30.2704	1.16684	0.583418	+	0.812172i	$0.301715\pi$
$674$	60.2340	2.32013
$675$	0	0
$676$	−68.0090	−2.61573
$677$	10.3794	0.398912	0.199456	−	0.979907i	$-0.436083\pi$
$677$	10.3794	0.398912	0.199456	+	0.979907i	$0.436083\pi$
$678$	−37.0194	−1.42172
$679$	−29.9964	−1.15116
$680$	0	0
$681$	11.8872	0.455517
$682$	68.0736	2.60667
$683$	42.6642	1.63250	0.816251	−	0.577697i	$-0.196049\pi$
$683$	42.6642	1.63250	0.816251	+	0.577697i	$0.196049\pi$
$684$	96.1625	3.67687
$685$	0	0
$686$	−52.6735	−2.01108
$687$	5.51454	0.210393
$688$	−62.5002	−2.38280
$689$	5.26856	0.200716
$690$	0	0
$691$	−44.7595	−1.70273	−0.851366	−	0.524573i	$-0.824225\pi$
$691$	−44.7595	−1.70273	−0.851366	+	0.524573i	$0.824225\pi$
$692$	−44.7744	−1.70207
$693$	36.5012	1.38657
$694$	−45.0897	−1.71158
$695$	0	0
$696$	5.80261	0.219948
$697$	5.51150	0.208763
$698$	8.51471	0.322286
$699$	4.45551	0.168523
$700$	0	0
$701$	14.4117	0.544323	0.272162	−	0.962252i	$-0.412261\pi$
$701$	14.4117	0.544323	0.272162	+	0.962252i	$0.412261\pi$
$702$	12.5616	0.474106
$703$	−22.4324	−0.846054
$704$	279.573	10.5368
$705$	0	0
$706$	−49.6302	−1.86786
$707$	39.9157	1.50118
$708$	33.2515	1.24967
$709$	−14.0009	−0.525817	−0.262908	−	0.964821i	$-0.584682\pi$
$709$	−14.0009	−0.525817	−0.262908	+	0.964821i	$0.584682\pi$
$710$	0	0
$711$	−10.0810	−0.378066
$712$	0.0738916	0.00276921
$713$	21.5899	0.808547
$714$	−14.3765	−0.538028
$715$	0	0
$716$	112.328	4.19790
$717$	14.6457	0.546953
$718$	−80.5590	−3.00643
$719$	2.38506	0.0889476	0.0444738	−	0.999011i	$-0.485839\pi$
$719$	2.38506	0.0889476	0.0444738	+	0.999011i	$0.485839\pi$
$720$	0	0
$721$	−27.8059	−1.03555
$722$	−58.3472	−2.17146
$723$	−0.650611	−0.0241965
$724$	138.849	5.16029
$725$	0	0
$726$	−33.5155	−1.24388
$727$	18.9362	0.702303	0.351151	−	0.936319i	$-0.385790\pi$
$727$	18.9362	0.702303	0.351151	+	0.936319i	$0.385790\pi$
$728$	35.5211	1.31650
$729$	−6.75392	−0.250145
$730$	0	0
$731$	9.76014	0.360992
$732$	−15.2708	−0.564425
$733$	9.15528	0.338158	0.169079	−	0.985602i	$-0.445921\pi$
$733$	9.15528	0.338158	0.169079	+	0.985602i	$0.445921\pi$
$734$	10.9933	0.405768
$735$	0	0
$736$	154.640	5.70011
$737$	−9.47128	−0.348879
$738$	−13.3212	−0.490362
$739$	27.9642	1.02868	0.514341	−	0.857586i	$-0.328037\pi$
$739$	27.9642	1.02868	0.514341	+	0.857586i	$0.328037\pi$
$740$	0	0
$741$	−5.04493	−0.185330
$742$	31.5420	1.15794
$743$	26.4697	0.971081	0.485540	−	0.874214i	$-0.338623\pi$
$743$	26.4697	0.971081	0.485540	+	0.874214i	$0.338623\pi$
$744$	32.0773	1.17601
$745$	0	0
$746$	−20.3289	−0.744295
$747$	5.77618	0.211339
$748$	−96.1624	−3.51605
$749$	−31.6963	−1.15816
$750$	0	0
$751$	3.79196	0.138370	0.0691852	−	0.997604i	$-0.477960\pi$
$751$	3.79196	0.138370	0.0691852	+	0.997604i	$0.477960\pi$
$752$	190.778	6.95694
$753$	−10.4782	−0.381848
$754$	2.79837	0.101911
$755$	0	0
$756$	56.2148	2.04451
$757$	45.5341	1.65496	0.827482	−	0.561492i	$-0.189772\pi$
$757$	45.5341	1.65496	0.827482	+	0.561492i	$0.189772\pi$
$758$	27.1359	0.985619
$759$	−17.0177	−0.617702
$760$	0	0
$761$	−6.64478	−0.240873	−0.120436	−	0.992721i	$-0.538429\pi$
$761$	−6.64478	−0.240873	−0.120436	+	0.992721i	$0.538429\pi$
$762$	19.7597	0.715819
$763$	−39.9954	−1.44793
$764$	8.64336	0.312706
$765$	0	0
$766$	90.6644	3.27584
$767$	10.6190	0.383429
$768$	81.7381	2.94947
$769$	15.0592	0.543049	0.271525	−	0.962431i	$-0.412472\pi$
$769$	15.0592	0.543049	0.271525	+	0.962431i	$0.412472\pi$
$770$	0	0
$771$	−6.98035	−0.251391
$772$	2.96672	0.106775
$773$	7.96647	0.286534	0.143267	−	0.989684i	$-0.454239\pi$
$773$	7.96647	0.286534	0.143267	+	0.989684i	$0.454239\pi$
$774$	−23.5902	−0.847931
$775$	0	0
$776$	−126.483	−4.54047
$777$	−6.05909	−0.217369
$778$	80.9923	2.90371
$779$	11.5789	0.414858
$780$	0	0
$781$	−37.7289	−1.35004
$782$	−40.8007	−1.45903
$783$	2.93264	0.104804
$784$	−2.92139	−0.104335
$785$	0	0
$786$	−8.11039	−0.289288
$787$	20.7723	0.740451	0.370226	−	0.928942i	$-0.379280\pi$
$787$	20.7723	0.740451	0.370226	+	0.928942i	$0.379280\pi$
$788$	−20.0523	−0.714332
$789$	14.0314	0.499531
$790$	0	0
$791$	−52.9063	−1.88113
$792$	153.911	5.46899
$793$	−4.87678	−0.173179
$794$	−46.2732	−1.64217
$795$	0	0
$796$	−48.0531	−1.70320
$797$	−2.33880	−0.0828444	−0.0414222	−	0.999142i	$-0.513189\pi$
$797$	−2.33880	−0.0828444	−0.0414222	+	0.999142i	$0.513189\pi$
$798$	−30.2032	−1.06918
$799$	−29.7922	−1.05397
$800$	0	0
$801$	0.0172551	0.000609679 0
$802$	−106.858	−3.77327
$803$	−78.0209	−2.75330
$804$	−6.73967	−0.237690
$805$	0	0
$806$	15.4696	0.544894
$807$	1.90404	0.0670256
$808$	168.308	5.92106
$809$	−50.5106	−1.77586	−0.887928	−	0.459982i	$-0.847856\pi$
$809$	−50.5106	−1.77586	−0.887928	+	0.459982i	$0.847856\pi$
$810$	0	0
$811$	−20.6544	−0.725274	−0.362637	−	0.931930i	$-0.618124\pi$
$811$	−20.6544	−0.725274	−0.362637	+	0.931930i	$0.618124\pi$
$812$	12.5231	0.439475
$813$	−1.14058	−0.0400019
$814$	−54.2188	−1.90037
$815$	0	0
$816$	−37.5050	−1.31294
$817$	20.5048	0.717371
$818$	−14.5303	−0.508040
$819$	8.29486	0.289846
$820$	0	0
$821$	9.30646	0.324798	0.162399	−	0.986725i	$-0.448077\pi$
$821$	9.30646	0.324798	0.162399	+	0.986725i	$0.448077\pi$
$822$	9.61230	0.335267
$823$	−19.6925	−0.686436	−0.343218	−	0.939256i	$-0.611517\pi$
$823$	−19.6925	−0.686436	−0.343218	+	0.939256i	$0.611517\pi$
$824$	−117.246	−4.08447
$825$	0	0
$826$	63.5740	2.21202
$827$	−13.4233	−0.466774	−0.233387	−	0.972384i	$-0.574981\pi$
$827$	−13.4233	−0.466774	−0.233387	+	0.972384i	$0.574981\pi$
$828$	73.7143	2.56175
$829$	40.6611	1.41222	0.706109	−	0.708103i	$-0.250449\pi$
$829$	40.6611	1.41222	0.706109	+	0.708103i	$0.250449\pi$
$830$	0	0
$831$	−6.24132	−0.216509
$832$	63.5325	2.20259
$833$	0.456209	0.0158067
$834$	−11.2927	−0.391035
$835$	0	0
$836$	−202.024	−6.98716
$837$	16.2119	0.560365
$838$	80.9515	2.79642
$839$	−31.2319	−1.07824	−0.539122	−	0.842228i	$-0.681244\pi$
$839$	−31.2319	−1.07824	−0.539122	+	0.842228i	$0.681244\pi$
$840$	0	0
$841$	−28.3467	−0.977472
$842$	−57.1884	−1.97084
$843$	−1.31317	−0.0452280
$844$	−114.826	−3.95247
$845$	0	0
$846$	72.0073	2.47566
$847$	−47.8987	−1.64582
$848$	82.2856	2.82570
$849$	0.785497	0.0269582
$850$	0	0
$851$	−17.1958	−0.589463
$852$	−26.8475	−0.919780
$853$	15.6453	0.535684	0.267842	−	0.963463i	$-0.413690\pi$
$853$	15.6453	0.535684	0.267842	+	0.963463i	$0.413690\pi$
$854$	−29.1965	−0.999082
$855$	0	0
$856$	−133.650	−4.56808
$857$	11.5099	0.393172	0.196586	−	0.980487i	$-0.437014\pi$
$857$	11.5099	0.393172	0.196586	+	0.980487i	$0.437014\pi$
$858$	−12.1935	−0.416280
$859$	18.9398	0.646218	0.323109	−	0.946362i	$-0.395272\pi$
$859$	18.9398	0.646218	0.323109	+	0.946362i	$0.395272\pi$
$860$	0	0
$861$	3.12752	0.106586
$862$	33.4163	1.13816
$863$	−5.57862	−0.189899	−0.0949493	−	0.995482i	$-0.530269\pi$
$863$	−5.57862	−0.189899	−0.0949493	+	0.995482i	$0.530269\pi$
$864$	116.120	3.95047
$865$	0	0
$866$	78.1910	2.65704
$867$	−5.20354	−0.176722
$868$	69.2287	2.34978
$869$	21.1787	0.718440
$870$	0	0
$871$	−2.15233	−0.0729291
$872$	−168.644	−5.71102
$873$	−29.5362	−0.999648
$874$	−85.7169	−2.89942
$875$	0	0
$876$	−55.5189	−1.87581
$877$	−38.6574	−1.30537	−0.652684	−	0.757631i	$-0.726357\pi$
$877$	−38.6574	−1.30537	−0.652684	+	0.757631i	$0.726357\pi$
$878$	−55.9707	−1.88892
$879$	7.54562	0.254507
$880$	0	0
$881$	16.3265	0.550055	0.275028	−	0.961436i	$-0.411313\pi$
$881$	16.3265	0.550055	0.275028	+	0.961436i	$0.411313\pi$
$882$	−1.10265	−0.0371283
$883$	31.6712	1.06582	0.532911	−	0.846171i	$-0.321098\pi$
$883$	31.6712	1.06582	0.532911	+	0.846171i	$0.321098\pi$
$884$	−21.8528	−0.734988
$885$	0	0
$886$	−35.6221	−1.19675
$887$	19.9371	0.669423	0.334711	−	0.942321i	$-0.391361\pi$
$887$	19.9371	0.669423	0.334711	+	0.942321i	$0.391361\pi$
$888$	−25.5487	−0.857360
$889$	28.2396	0.947125
$890$	0	0
$891$	29.0669	0.973778
$892$	−59.6830	−1.99833
$893$	−62.5894	−2.09447
$894$	−19.2673	−0.644394
$895$	0	0
$896$	212.858	7.11109
$897$	−3.86724	−0.129123
$898$	47.2838	1.57788
$899$	3.61156	0.120452
$900$	0	0
$901$	−12.8499	−0.428091
$902$	27.9861	0.931836
$903$	5.53843	0.184307
$904$	−223.085	−7.41968
$905$	0	0
$906$	37.9499	1.26080
$907$	−4.26367	−0.141573	−0.0707865	−	0.997491i	$-0.522551\pi$
$907$	−4.26367	−0.141573	−0.0707865	+	0.997491i	$0.522551\pi$
$908$	108.175	3.58992
$909$	39.3032	1.30361
$910$	0	0
$911$	3.46818	0.114906	0.0574530	−	0.998348i	$-0.481702\pi$
$911$	3.46818	0.114906	0.0574530	+	0.998348i	$0.481702\pi$
$912$	−78.7930	−2.60910
$913$	−12.1350	−0.401609
$914$	−70.9996	−2.34846
$915$	0	0
$916$	50.1833	1.65810
$917$	−11.5910	−0.382767
$918$	−30.6374	−1.01118
$919$	−2.81869	−0.0929800	−0.0464900	−	0.998919i	$-0.514804\pi$
$919$	−2.81869	−0.0929800	−0.0464900	+	0.998919i	$0.514804\pi$
$920$	0	0
$921$	18.9577	0.624676
$922$	−15.3080	−0.504143
$923$	−8.57383	−0.282211
$924$	−54.5677	−1.79515
$925$	0	0
$926$	−22.0855	−0.725773
$927$	−27.3793	−0.899254
$928$	25.8682	0.849166
$929$	43.0563	1.41263	0.706316	−	0.707897i	$-0.250356\pi$
$929$	43.0563	1.41263	0.706316	+	0.707897i	$0.250356\pi$
$930$	0	0
$931$	0.958435	0.0314114
$932$	40.5459	1.32813
$933$	4.54243	0.148712
$934$	−59.8304	−1.95771
$935$	0	0
$936$	34.9761	1.14323
$937$	−19.1635	−0.626044	−0.313022	−	0.949746i	$-0.601341\pi$
$937$	−19.1635	−0.626044	−0.313022	+	0.949746i	$0.601341\pi$
$938$	−12.8857	−0.420732
$939$	10.9644	0.357810
$940$	0	0
$941$	36.7480	1.19795	0.598975	−	0.800767i	$-0.295575\pi$
$941$	36.7480	1.19795	0.598975	+	0.800767i	$0.295575\pi$
$942$	32.6719	1.06451
$943$	8.87594	0.289040
$944$	165.850	5.39795
$945$	0	0
$946$	49.5597	1.61133
$947$	6.22386	0.202248	0.101124	−	0.994874i	$-0.467756\pi$
$947$	6.22386	0.202248	0.101124	+	0.994874i	$0.467756\pi$
$948$	15.0706	0.489470
$949$	−17.7301	−0.575544
$950$	0	0
$951$	7.06104	0.228970
$952$	−86.6350	−2.80786
$953$	52.3602	1.69611	0.848056	−	0.529906i	$-0.177773\pi$
$953$	52.3602	1.69611	0.848056	+	0.529906i	$0.177773\pi$
$954$	31.0580	1.00554
$955$	0	0
$956$	133.278	4.31053
$957$	−2.84672	−0.0920213
$958$	51.4548	1.66243
$959$	13.7374	0.443604
$960$	0	0
$961$	−11.0350	−0.355968
$962$	−12.3211	−0.397250
$963$	−31.2099	−1.00573
$964$	−5.92068	−0.190692
$965$	0	0
$966$	−23.1525	−0.744920
$967$	23.5339	0.756800	0.378400	−	0.925642i	$-0.376474\pi$
$967$	23.5339	0.756800	0.378400	+	0.925642i	$0.376474\pi$
$968$	−201.970	−6.49155
$969$	12.3045	0.395276
$970$	0	0
$971$	44.8945	1.44073	0.720366	−	0.693594i	$-0.243974\pi$
$971$	44.8945	1.44073	0.720366	+	0.693594i	$0.243974\pi$
$972$	85.1291	2.73052
$973$	−16.1390	−0.517393
$974$	36.1585	1.15859
$975$	0	0
$976$	−76.1667	−2.43804
$977$	−54.7799	−1.75257	−0.876283	−	0.481798i	$-0.839984\pi$
$977$	−54.7799	−1.75257	−0.876283	+	0.481798i	$0.839984\pi$
$978$	8.41881	0.269204
$979$	−0.0362507	−0.00115858
$980$	0	0
$981$	−39.3817	−1.25736
$982$	−85.5375	−2.72961
$983$	−34.4464	−1.09867	−0.549335	−	0.835602i	$-0.685119\pi$
$983$	−34.4464	−1.09867	−0.549335	+	0.835602i	$0.685119\pi$
$984$	13.1875	0.420402
$985$	0	0
$986$	−6.82516	−0.217357
$987$	−16.9057	−0.538113
$988$	−45.9097	−1.46058
$989$	15.7181	0.499807
$990$	0	0
$991$	−38.9650	−1.23776	−0.618882	−	0.785484i	$-0.712414\pi$
$991$	−38.9650	−1.23776	−0.618882	+	0.785484i	$0.712414\pi$
$992$	143.002	4.54031
$993$	−8.49307	−0.269519
$994$	−51.3301	−1.62809
$995$	0	0
$996$	−8.63513	−0.273614
$997$	−13.8322	−0.438070	−0.219035	−	0.975717i	$-0.570291\pi$
$997$	−13.8322	−0.438070	−0.219035	+	0.975717i	$0.570291\pi$
$998$	−10.9340	−0.346108
$999$	−12.9123	−0.408528

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.q.1.1		66
5.2	odd	4		1205.2.b.d.724.1	✓	66
5.3	odd	4		1205.2.b.d.724.66	yes	66
5.4	even	2	inner	6025.2.a.q.1.66		66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.d.724.1	✓	66	5.2	odd	4
1205.2.b.d.724.66	yes	66	5.3	odd	4
6025.2.a.q.1.1		66	1.1	even	1	trivial
6025.2.a.q.1.66		66	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q-2.81437 q^{2} +0.650611 q^{3} +5.92068 q^{4} -1.83106 q^{6} -2.61686 q^{7} -11.0342 q^{8} -2.57671 q^{9} +O(q^{10})\)	\(q-2.81437 q^{2} +0.650611 q^{3} +5.92068 q^{4} -1.83106 q^{6} -2.61686 q^{7} -11.0342 q^{8} -2.57671 q^{9} +5.41331 q^{11} +3.85206 q^{12} +1.23017 q^{13} +7.36481 q^{14} +19.2131 q^{16} -3.00034 q^{17} +7.25180 q^{18} -6.30332 q^{19} -1.70256 q^{21} -15.2351 q^{22} -4.83187 q^{23} -7.17899 q^{24} -3.46214 q^{26} -3.62827 q^{27} -15.4936 q^{28} -0.808277 q^{29} -4.46822 q^{31} -32.0042 q^{32} +3.52196 q^{33} +8.44408 q^{34} -15.2558 q^{36} +3.55882 q^{37} +17.7399 q^{38} +0.800360 q^{39} -1.83696 q^{41} +4.79162 q^{42} -3.25301 q^{43} +32.0504 q^{44} +13.5987 q^{46} +9.92958 q^{47} +12.5002 q^{48} -0.152052 q^{49} -1.95206 q^{51} +7.28342 q^{52} +4.28280 q^{53} +10.2113 q^{54} +28.8750 q^{56} -4.10101 q^{57} +2.27479 q^{58} +8.63214 q^{59} -3.96432 q^{61} +12.5752 q^{62} +6.74287 q^{63} +51.6455 q^{64} -9.91209 q^{66} -1.74963 q^{67} -17.7641 q^{68} -3.14367 q^{69} -6.96965 q^{71} +28.4320 q^{72} -14.4128 q^{73} -10.0158 q^{74} -37.3199 q^{76} -14.1659 q^{77} -2.25251 q^{78} +3.91235 q^{79} +5.36953 q^{81} +5.16987 q^{82} -2.24169 q^{83} -10.0803 q^{84} +9.15517 q^{86} -0.525874 q^{87} -59.7317 q^{88} -0.00669658 q^{89} -3.21917 q^{91} -28.6080 q^{92} -2.90707 q^{93} -27.9455 q^{94} -20.8223 q^{96} +11.4628 q^{97} +0.427931 q^{98} -13.9485 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9}+O(q^{10}) \) 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9	\( 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9} + 48 q^{11} + 30 q^{14} + 98 q^{16} + 12 q^{19} + 18 q^{21} + 42 q^{24} + 48 q^{26} + 56 q^{29} + 48 q^{31} + 8 q^{34} + 158 q^{36} + 84 q^{39} + 56 q^{41} + 144 q^{44} + 36 q^{46} + 98 q^{49} + 44 q^{51} + 86 q^{54} + 104 q^{56} + 108 q^{59} + 22 q^{61} + 136 q^{64} + 74 q^{66} + 20 q^{69} + 212 q^{71} + 84 q^{74} + 6 q^{76} + 66 q^{79} + 162 q^{81} - 52 q^{84} + 100 q^{86} + 54 q^{89} + 72 q^{91} - 96 q^{94} + 122 q^{96} + 112 q^{99}+O(q^{100}) \) 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9 + 48 * q^11 + 30 * q^14 + 98 * q^16 + 12 * q^19 + 18 * q^21 + 42 * q^24 + 48 * q^26 + 56 * q^29 + 48 * q^31 + 8 * q^34 + 158 * q^36 + 84 * q^39 + 56 * q^41 + 144 * q^44 + 36 * q^46 + 98 * q^49 + 44 * q^51 + 86 * q^54 + 104 * q^56 + 108 * q^59 + 22 * q^61 + 136 * q^64 + 74 * q^66 + 20 * q^69 + 212 * q^71 + 84 * q^74 + 6 * q^76 + 66 * q^79 + 162 * q^81 - 52 * q^84 + 100 * q^86 + 54 * q^89 + 72 * q^91 - 96 * q^94 + 122 * q^96 + 112 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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