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Properties

Label 7360.2.a.bw.1.1
Level $7360$
Weight $2$
Character 7360.1
Self dual yes
Analytic conductor $58.770$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7360,2,Mod(1,7360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7360.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7360.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-2,0,-3,0,7,0,3,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(58.7698958877\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.11491\) of defining polynomial
Character \(\chi\) \(=\) 7360.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.47283 q^{3} -1.00000 q^{5} +0.527166 q^{7} +3.11491 q^{9} -3.11491 q^{11} -4.11491 q^{13} +2.47283 q^{15} -4.39905 q^{17} +3.70265 q^{19} -1.30359 q^{21} -1.00000 q^{23} +1.00000 q^{25} -0.284147 q^{27} +9.10170 q^{29} +4.83076 q^{31} +7.70265 q^{33} -0.527166 q^{35} -9.74378 q^{37} +10.1755 q^{39} +6.93246 q^{41} +4.45963 q^{43} -3.11491 q^{45} +0.642074 q^{47} -6.72210 q^{49} +10.8781 q^{51} +3.89134 q^{53} +3.11491 q^{55} -9.15604 q^{57} -8.79811 q^{59} +3.45339 q^{61} +1.64207 q^{63} +4.11491 q^{65} +8.60719 q^{67} +2.47283 q^{69} +12.3642 q^{71} -5.81756 q^{73} -2.47283 q^{75} -1.64207 q^{77} -3.17548 q^{79} -8.64207 q^{81} -4.71585 q^{83} +4.39905 q^{85} -22.5070 q^{87} -5.43171 q^{89} -2.16924 q^{91} -11.9457 q^{93} -3.70265 q^{95} -4.06058 q^{97} -9.70265 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 3 q^{5} + 7 q^{7} + 3 q^{9} - 3 q^{11} - 6 q^{13} + 2 q^{15} - 5 q^{17} - 7 q^{19} + 6 q^{21} - 3 q^{23} + 3 q^{25} + q^{27} + q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} - 2 q^{37} + 7 q^{39}+ \cdots - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.47283 −1.42769 −0.713846 − 0.700303i \(-0.753048\pi\)
−0.713846 + 0.700303i \(0.753048\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.527166 0.199250 0.0996250 − 0.995025i \(-0.468236\pi\)
0.0996250 + 0.995025i \(0.468236\pi\)
\(8\) 0 0
\(9\) 3.11491 1.03830
\(10\) 0 0
\(11\) −3.11491 −0.939180 −0.469590 − 0.882885i \(-0.655598\pi\)
−0.469590 + 0.882885i \(0.655598\pi\)
\(12\) 0 0
\(13\) −4.11491 −1.14127 −0.570635 − 0.821204i \(-0.693303\pi\)
−0.570635 + 0.821204i \(0.693303\pi\)
\(14\) 0 0
\(15\) 2.47283 0.638483
\(16\) 0 0
\(17\) −4.39905 −1.06693 −0.533464 − 0.845823i \(-0.679110\pi\)
−0.533464 + 0.845823i \(0.679110\pi\)
\(18\) 0 0
\(19\) 3.70265 0.849446 0.424723 − 0.905323i \(-0.360372\pi\)
0.424723 + 0.905323i \(0.360372\pi\)
\(20\) 0 0
\(21\) −1.30359 −0.284468
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −0.284147 −0.0546842
\(28\) 0 0
\(29\) 9.10170 1.69014 0.845072 − 0.534653i \(-0.179558\pi\)
0.845072 + 0.534653i \(0.179558\pi\)
\(30\) 0 0
\(31\) 4.83076 0.867630 0.433815 − 0.901002i \(-0.357167\pi\)
0.433815 + 0.901002i \(0.357167\pi\)
\(32\) 0 0
\(33\) 7.70265 1.34086
\(34\) 0 0
\(35\) −0.527166 −0.0891073
\(36\) 0 0
\(37\) −9.74378 −1.60187 −0.800934 − 0.598753i \(-0.795663\pi\)
−0.800934 + 0.598753i \(0.795663\pi\)
\(38\) 0 0
\(39\) 10.1755 1.62938
\(40\) 0 0
\(41\) 6.93246 1.08267 0.541334 − 0.840807i \(-0.317919\pi\)
0.541334 + 0.840807i \(0.317919\pi\)
\(42\) 0 0
\(43\) 4.45963 0.680087 0.340044 − 0.940410i \(-0.389558\pi\)
0.340044 + 0.940410i \(0.389558\pi\)
\(44\) 0 0
\(45\) −3.11491 −0.464343
\(46\) 0 0
\(47\) 0.642074 0.0936561 0.0468280 − 0.998903i \(-0.485089\pi\)
0.0468280 + 0.998903i \(0.485089\pi\)
\(48\) 0 0
\(49\) −6.72210 −0.960299
\(50\) 0 0
\(51\) 10.8781 1.52324
\(52\) 0 0
\(53\) 3.89134 0.534516 0.267258 − 0.963625i \(-0.413882\pi\)
0.267258 + 0.963625i \(0.413882\pi\)
\(54\) 0 0
\(55\) 3.11491 0.420014
\(56\) 0 0
\(57\) −9.15604 −1.21275
\(58\) 0 0
\(59\) −8.79811 −1.14542 −0.572708 − 0.819759i \(-0.694107\pi\)
−0.572708 + 0.819759i \(0.694107\pi\)
\(60\) 0 0
\(61\) 3.45339 0.442161 0.221080 − 0.975256i \(-0.429042\pi\)
0.221080 + 0.975256i \(0.429042\pi\)
\(62\) 0 0
\(63\) 1.64207 0.206882
\(64\) 0 0
\(65\) 4.11491 0.510391
\(66\) 0 0
\(67\) 8.60719 1.05154 0.525768 − 0.850628i \(-0.323778\pi\)
0.525768 + 0.850628i \(0.323778\pi\)
\(68\) 0 0
\(69\) 2.47283 0.297694
\(70\) 0 0
\(71\) 12.3642 1.46736 0.733678 − 0.679497i \(-0.237802\pi\)
0.733678 + 0.679497i \(0.237802\pi\)
\(72\) 0 0
\(73\) −5.81756 −0.680893 −0.340447 − 0.940264i \(-0.610578\pi\)
−0.340447 + 0.940264i \(0.610578\pi\)
\(74\) 0 0
\(75\) −2.47283 −0.285538
\(76\) 0 0
\(77\) −1.64207 −0.187132
\(78\) 0 0
\(79\) −3.17548 −0.357270 −0.178635 − 0.983915i \(-0.557168\pi\)
−0.178635 + 0.983915i \(0.557168\pi\)
\(80\) 0 0
\(81\) −8.64207 −0.960230
\(82\) 0 0
\(83\) −4.71585 −0.517632 −0.258816 − 0.965927i \(-0.583332\pi\)
−0.258816 + 0.965927i \(0.583332\pi\)
\(84\) 0 0
\(85\) 4.39905 0.477144
\(86\) 0 0
\(87\) −22.5070 −2.41300
\(88\) 0 0
\(89\) −5.43171 −0.575760 −0.287880 − 0.957667i \(-0.592950\pi\)
−0.287880 + 0.957667i \(0.592950\pi\)
\(90\) 0 0
\(91\) −2.16924 −0.227398
\(92\) 0 0
\(93\) −11.9457 −1.23871
\(94\) 0 0
\(95\) −3.70265 −0.379884
\(96\) 0 0
\(97\) −4.06058 −0.412289 −0.206144 − 0.978522i \(-0.566092\pi\)
−0.206144 + 0.978522i \(0.566092\pi\)
\(98\) 0 0
\(99\) −9.70265 −0.975153
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7360.2.a.bw.1.1 3
4.3 odd 2 7360.2.a.cf.1.3 3
8.3 odd 2 1840.2.a.q.1.1 3
8.5 even 2 920.2.a.i.1.3 ✓ 3
24.5 odd 2 8280.2.a.bl.1.1 3
40.13 odd 4 4600.2.e.q.4049.6 6
40.19 odd 2 9200.2.a.ci.1.3 3
40.29 even 2 4600.2.a.v.1.1 3
40.37 odd 4 4600.2.e.q.4049.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.i.1.3 ✓ 3 8.5 even 2
1840.2.a.q.1.1 3 8.3 odd 2
4600.2.a.v.1.1 3 40.29 even 2
4600.2.e.q.4049.1 6 40.37 odd 4
4600.2.e.q.4049.6 6 40.13 odd 4
7360.2.a.bw.1.1 3 1.1 even 1 trivial
7360.2.a.cf.1.3 3 4.3 odd 2
8280.2.a.bl.1.1 3 24.5 odd 2
9200.2.a.ci.1.3 3 40.19 odd 2