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Properties

Label 4600.2.a.be.1.2
Level $4600$
Weight $2$
Character 4600.1
Self dual yes
Analytic conductor $36.731$
Analytic rank $0$
Dimension $5$
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] = [4600,2,Mod(1,4600)] 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("4600.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,0,0,0,0,2,0,13,0,-1] 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: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.13955077.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 14x^{3} - x^{2} + 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 1.2
Root \(1.93283\) of defining polynomial
Character \(\chi\) \(=\) 4600.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.93283 q^{3} -2.38236 q^{7} +0.735829 q^{9} +5.33368 q^{11} +4.53752 q^{13} -1.81464 q^{17} +7.00233 q^{19} +4.60469 q^{21} +1.00000 q^{23} +4.37626 q^{27} -0.118188 q^{29} -0.884147 q^{31} -10.3091 q^{33} -7.51903 q^{37} -8.77026 q^{39} -1.45186 q^{41} -10.4389 q^{47} -1.32437 q^{49} +3.50739 q^{51} +9.42167 q^{53} -13.5343 q^{57} +7.79239 q^{59} -2.80533 q^{61} -1.75301 q^{63} +3.11134 q^{67} -1.93283 q^{69} -13.5909 q^{71} -12.4389 q^{73} -12.7067 q^{77} +6.80169 q^{79} -10.6660 q^{81} +13.5190 q^{83} +0.228437 q^{87} +2.89906 q^{89} -10.8100 q^{91} +1.70890 q^{93} +1.97774 q^{97} +3.92468 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{7} + 13 q^{9} - q^{11} - 4 q^{13} - 4 q^{17} + 7 q^{19} + 6 q^{21} + 5 q^{23} - 3 q^{27} + 4 q^{29} + 19 q^{31} - 17 q^{33} - 15 q^{37} + 19 q^{39} + 25 q^{41} + 11 q^{47} + 25 q^{49} + 19 q^{51}+ \cdots - 65 q^{99}+O(q^{100}) \) 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
\(3\) −1.93283 −1.11592 −0.557960 − 0.829868i \(-0.688416\pi\)
−0.557960 + 0.829868i \(0.688416\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.38236 −0.900447 −0.450223 − 0.892916i \(-0.648656\pi\)
−0.450223 + 0.892916i \(0.648656\pi\)
\(8\) 0 0
\(9\) 0.735829 0.245276
\(10\) 0 0
\(11\) 5.33368 1.60816 0.804082 − 0.594519i \(-0.202657\pi\)
0.804082 + 0.594519i \(0.202657\pi\)
\(12\) 0 0
\(13\) 4.53752 1.25848 0.629241 − 0.777210i \(-0.283366\pi\)
0.629241 + 0.777210i \(0.283366\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.81464 −0.440115 −0.220058 − 0.975487i \(-0.570625\pi\)
−0.220058 + 0.975487i \(0.570625\pi\)
\(18\) 0 0
\(19\) 7.00233 1.60645 0.803223 − 0.595679i \(-0.203117\pi\)
0.803223 + 0.595679i \(0.203117\pi\)
\(20\) 0 0
\(21\) 4.60469 1.00483
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.37626 0.842211
\(28\) 0 0
\(29\) −0.118188 −0.0219469 −0.0109735 − 0.999940i \(-0.503493\pi\)
−0.0109735 + 0.999940i \(0.503493\pi\)
\(30\) 0 0
\(31\) −0.884147 −0.158797 −0.0793987 − 0.996843i \(-0.525300\pi\)
−0.0793987 + 0.996843i \(0.525300\pi\)
\(32\) 0 0
\(33\) −10.3091 −1.79458
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.51903 −1.23612 −0.618061 − 0.786130i \(-0.712081\pi\)
−0.618061 + 0.786130i \(0.712081\pi\)
\(38\) 0 0
\(39\) −8.77026 −1.40436
\(40\) 0 0
\(41\) −1.45186 −0.226743 −0.113371 − 0.993553i \(-0.536165\pi\)
−0.113371 + 0.993553i \(0.536165\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.4389 −1.52267 −0.761336 − 0.648357i \(-0.775456\pi\)
−0.761336 + 0.648357i \(0.775456\pi\)
\(48\) 0 0
\(49\) −1.32437 −0.189195
\(50\) 0 0
\(51\) 3.50739 0.491133
\(52\) 0 0
\(53\) 9.42167 1.29417 0.647083 − 0.762420i \(-0.275989\pi\)
0.647083 + 0.762420i \(0.275989\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −13.5343 −1.79266
\(58\) 0 0
\(59\) 7.79239 1.01448 0.507241 − 0.861804i \(-0.330665\pi\)
0.507241 + 0.861804i \(0.330665\pi\)
\(60\) 0 0
\(61\) −2.80533 −0.359186 −0.179593 − 0.983741i \(-0.557478\pi\)
−0.179593 + 0.983741i \(0.557478\pi\)
\(62\) 0 0
\(63\) −1.75301 −0.220858
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.11134 0.380111 0.190055 − 0.981773i \(-0.439133\pi\)
0.190055 + 0.981773i \(0.439133\pi\)
\(68\) 0 0
\(69\) −1.93283 −0.232685
\(70\) 0 0
\(71\) −13.5909 −1.61294 −0.806470 − 0.591275i \(-0.798625\pi\)
−0.806470 + 0.591275i \(0.798625\pi\)
\(72\) 0 0
\(73\) −12.4389 −1.45586 −0.727932 − 0.685649i \(-0.759519\pi\)
−0.727932 + 0.685649i \(0.759519\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.7067 −1.44807
\(78\) 0 0
\(79\) 6.80169 0.765250 0.382625 − 0.923904i \(-0.375020\pi\)
0.382625 + 0.923904i \(0.375020\pi\)
\(80\) 0 0
\(81\) −10.6660 −1.18512
\(82\) 0 0
\(83\) 13.5190 1.48391 0.741953 − 0.670451i \(-0.233900\pi\)
0.741953 + 0.670451i \(0.233900\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.228437 0.0244910
\(88\) 0 0
\(89\) 2.89906 0.307300 0.153650 − 0.988125i \(-0.450897\pi\)
0.153650 + 0.988125i \(0.450897\pi\)
\(90\) 0 0
\(91\) −10.8100 −1.13320
\(92\) 0 0
\(93\) 1.70890 0.177205
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.97774 0.200809 0.100405 − 0.994947i \(-0.467986\pi\)
0.100405 + 0.994947i \(0.467986\pi\)
\(98\) 0 0
\(99\) 3.92468 0.394445
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 4600.2.a.be.1.2 5
4.3 odd 2 9200.2.a.cu.1.4 5
5.2 odd 4 4600.2.e.u.4049.8 10
5.3 odd 4 4600.2.e.u.4049.3 10
5.4 even 2 920.2.a.j.1.4 ✓ 5
15.14 odd 2 8280.2.a.bs.1.4 5
20.19 odd 2 1840.2.a.v.1.2 5
40.19 odd 2 7360.2.a.cp.1.4 5
40.29 even 2 7360.2.a.co.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.4 ✓ 5 5.4 even 2
1840.2.a.v.1.2 5 20.19 odd 2
4600.2.a.be.1.2 5 1.1 even 1 trivial
4600.2.e.u.4049.3 10 5.3 odd 4
4600.2.e.u.4049.8 10 5.2 odd 4
7360.2.a.co.1.2 5 40.29 even 2
7360.2.a.cp.1.4 5 40.19 odd 2
8280.2.a.bs.1.4 5 15.14 odd 2
9200.2.a.cu.1.4 5 4.3 odd 2