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Properties

Label 4600.2.a.be.1.1
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 1.1
Root \(3.30649\) 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)
 
\(f(q)\) \(=\) \(q-3.30649 q^{3} +2.55040 q^{7} +7.93288 q^{9} -2.72314 q^{11} -7.12637 q^{13} -0.924010 q^{17} +7.51623 q^{19} -8.43286 q^{21} +1.00000 q^{23} -16.3105 q^{27} -2.38248 q^{29} +0.866248 q^{31} +9.00402 q^{33} -0.352855 q^{37} +23.5633 q^{39} +4.34066 q^{41} +13.3239 q^{47} -0.495474 q^{49} +3.05523 q^{51} -3.99262 q^{53} -24.8523 q^{57} -3.84064 q^{59} -9.14262 q^{61} +20.2320 q^{63} +3.15933 q^{67} -3.30649 q^{69} -6.07883 q^{71} +11.3239 q^{73} -6.94508 q^{77} -12.0593 q^{79} +30.1319 q^{81} +6.35285 q^{83} +7.87765 q^{87} -9.71377 q^{89} -18.1751 q^{91} -2.86424 q^{93} -8.76465 q^{97} -21.6023 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{7} + 13 q^{9} - q^{11} - 4 q^{13} - 4 q^{17} + 7 q^{19} + 6 q^{21} + 5 q^{23} - 3 q^{27} + 4 q^{29} + 19 q^{31} - 17 q^{33} - 15 q^{37} + 19 q^{39} + 25 q^{41} + 11 q^{47} + 25 q^{49} + 19 q^{51}+ \cdots - 65 q^{99}+O(q^{100}) \) 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
\(3\) −3.30649 −1.90900 −0.954501 − 0.298206i \(-0.903612\pi\)
−0.954501 + 0.298206i \(0.903612\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.55040 0.963960 0.481980 − 0.876182i \(-0.339918\pi\)
0.481980 + 0.876182i \(0.339918\pi\)
\(8\) 0 0
\(9\) 7.93288 2.64429
\(10\) 0 0
\(11\) −2.72314 −0.821056 −0.410528 − 0.911848i \(-0.634656\pi\)
−0.410528 + 0.911848i \(0.634656\pi\)
\(12\) 0 0
\(13\) −7.12637 −1.97650 −0.988250 − 0.152845i \(-0.951156\pi\)
−0.988250 + 0.152845i \(0.951156\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.924010 −0.224105 −0.112053 − 0.993702i \(-0.535743\pi\)
−0.112053 + 0.993702i \(0.535743\pi\)
\(18\) 0 0
\(19\) 7.51623 1.72434 0.862171 − 0.506617i \(-0.169104\pi\)
0.862171 + 0.506617i \(0.169104\pi\)
\(20\) 0 0
\(21\) −8.43286 −1.84020
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −16.3105 −3.13896
\(28\) 0 0
\(29\) −2.38248 −0.442415 −0.221208 − 0.975227i \(-0.571000\pi\)
−0.221208 + 0.975227i \(0.571000\pi\)
\(30\) 0 0
\(31\) 0.866248 0.155583 0.0777913 − 0.996970i \(-0.475213\pi\)
0.0777913 + 0.996970i \(0.475213\pi\)
\(32\) 0 0
\(33\) 9.00402 1.56740
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.352855 −0.0580089 −0.0290045 − 0.999579i \(-0.509234\pi\)
−0.0290045 + 0.999579i \(0.509234\pi\)
\(38\) 0 0
\(39\) 23.5633 3.77315
\(40\) 0 0
\(41\) 4.34066 0.677896 0.338948 − 0.940805i \(-0.389929\pi\)
0.338948 + 0.940805i \(0.389929\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 13.3239 1.94349 0.971746 − 0.236027i \(-0.0758454\pi\)
0.971746 + 0.236027i \(0.0758454\pi\)
\(48\) 0 0
\(49\) −0.495474 −0.0707819
\(50\) 0 0
\(51\) 3.05523 0.427818
\(52\) 0 0
\(53\) −3.99262 −0.548429 −0.274214 − 0.961669i \(-0.588418\pi\)
−0.274214 + 0.961669i \(0.588418\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −24.8523 −3.29177
\(58\) 0 0
\(59\) −3.84064 −0.500009 −0.250004 − 0.968245i \(-0.580432\pi\)
−0.250004 + 0.968245i \(0.580432\pi\)
\(60\) 0 0
\(61\) −9.14262 −1.17059 −0.585296 − 0.810820i \(-0.699022\pi\)
−0.585296 + 0.810820i \(0.699022\pi\)
\(62\) 0 0
\(63\) 20.2320 2.54899
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.15933 0.385974 0.192987 − 0.981201i \(-0.438183\pi\)
0.192987 + 0.981201i \(0.438183\pi\)
\(68\) 0 0
\(69\) −3.30649 −0.398055
\(70\) 0 0
\(71\) −6.07883 −0.721424 −0.360712 − 0.932677i \(-0.617466\pi\)
−0.360712 + 0.932677i \(0.617466\pi\)
\(72\) 0 0
\(73\) 11.3239 1.32536 0.662682 − 0.748901i \(-0.269418\pi\)
0.662682 + 0.748901i \(0.269418\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.94508 −0.791465
\(78\) 0 0
\(79\) −12.0593 −1.35677 −0.678386 − 0.734706i \(-0.737320\pi\)
−0.678386 + 0.734706i \(0.737320\pi\)
\(80\) 0 0
\(81\) 30.1319 3.34799
\(82\) 0 0
\(83\) 6.35285 0.697316 0.348658 − 0.937250i \(-0.386637\pi\)
0.348658 + 0.937250i \(0.386637\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 7.87765 0.844572
\(88\) 0 0
\(89\) −9.71377 −1.02966 −0.514829 − 0.857293i \(-0.672145\pi\)
−0.514829 + 0.857293i \(0.672145\pi\)
\(90\) 0 0
\(91\) −18.1751 −1.90527
\(92\) 0 0
\(93\) −2.86424 −0.297008
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.76465 −0.889916 −0.444958 − 0.895552i \(-0.646781\pi\)
−0.444958 + 0.895552i \(0.646781\pi\)
\(98\) 0 0
\(99\) −21.6023 −2.17111
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.1.1 5
4.3 odd 2 9200.2.a.cu.1.5 5
5.2 odd 4 4600.2.e.u.4049.9 10
5.3 odd 4 4600.2.e.u.4049.2 10
5.4 even 2 920.2.a.j.1.5 ✓ 5
15.14 odd 2 8280.2.a.bs.1.2 5
20.19 odd 2 1840.2.a.v.1.1 5
40.19 odd 2 7360.2.a.cp.1.5 5
40.29 even 2 7360.2.a.co.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.5 ✓ 5 5.4 even 2
1840.2.a.v.1.1 5 20.19 odd 2
4600.2.a.be.1.1 5 1.1 even 1 trivial
4600.2.e.u.4049.2 10 5.3 odd 4
4600.2.e.u.4049.9 10 5.2 odd 4
7360.2.a.co.1.1 5 40.29 even 2
7360.2.a.cp.1.5 5 40.19 odd 2
8280.2.a.bs.1.2 5 15.14 odd 2
9200.2.a.cu.1.5 5 4.3 odd 2