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Properties

Label 4600.2.a.ba.1.4
Level $4600$
Weight $2$
Character 4600.1
Self dual yes
Analytic conductor $36.731$
Analytic rank $0$
Dimension $4$
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 = [4,0,0,0,0,0,-1,0,6,0,1] 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: \(4\)
Coefficient field: 4.4.15529.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.774457\) 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)
 
\(f(q)\) \(=\) \(q+3.43375 q^{3} +2.58245 q^{7} +8.79066 q^{9} +1.03354 q^{11} +1.95669 q^{13} -1.54891 q^{17} -2.96646 q^{19} +8.86751 q^{21} +1.00000 q^{23} +19.8837 q^{27} +6.93936 q^{29} -8.05951 q^{31} +3.54891 q^{33} +5.70260 q^{37} +6.71881 q^{39} -2.85130 q^{41} -9.90105 q^{43} +0.625758 q^{47} -0.330937 q^{49} -5.31859 q^{51} -13.7350 q^{53} -10.1861 q^{57} +8.08549 q^{59} +8.80043 q^{61} +22.7015 q^{63} -10.8675 q^{67} +3.43375 q^{69} +2.69283 q^{71} +7.84153 q^{73} +2.66906 q^{77} +3.96812 q^{79} +41.9038 q^{81} -9.53657 q^{83} +23.8281 q^{87} -7.25152 q^{89} +5.05307 q^{91} -27.6744 q^{93} +13.6160 q^{97} +9.08549 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{7} + 6 q^{9} + q^{11} + 3 q^{13} + 2 q^{17} - 15 q^{19} + 8 q^{21} + 4 q^{23} + 27 q^{27} + q^{29} - 12 q^{31} + 6 q^{33} + 18 q^{37} - 3 q^{39} - 9 q^{41} - 9 q^{43} - 4 q^{47} - 3 q^{49} - 2 q^{51}+ \cdots - q^{99}+O(q^{100}) \) 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
\(3\) 3.43375 1.98248 0.991239 − 0.132078i \(-0.0421650\pi\)
0.991239 + 0.132078i \(0.0421650\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.58245 0.976075 0.488038 − 0.872823i \(-0.337713\pi\)
0.488038 + 0.872823i \(0.337713\pi\)
\(8\) 0 0
\(9\) 8.79066 2.93022
\(10\) 0 0
\(11\) 1.03354 0.311623 0.155812 − 0.987787i \(-0.450201\pi\)
0.155812 + 0.987787i \(0.450201\pi\)
\(12\) 0 0
\(13\) 1.95669 0.542689 0.271345 − 0.962482i \(-0.412532\pi\)
0.271345 + 0.962482i \(0.412532\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.54891 −0.375667 −0.187834 − 0.982201i \(-0.560147\pi\)
−0.187834 + 0.982201i \(0.560147\pi\)
\(18\) 0 0
\(19\) −2.96646 −0.680553 −0.340277 − 0.940325i \(-0.610521\pi\)
−0.340277 + 0.940325i \(0.610521\pi\)
\(20\) 0 0
\(21\) 8.86751 1.93505
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 19.8837 3.82662
\(28\) 0 0
\(29\) 6.93936 1.28861 0.644304 − 0.764770i \(-0.277147\pi\)
0.644304 + 0.764770i \(0.277147\pi\)
\(30\) 0 0
\(31\) −8.05951 −1.44753 −0.723766 − 0.690046i \(-0.757590\pi\)
−0.723766 + 0.690046i \(0.757590\pi\)
\(32\) 0 0
\(33\) 3.54891 0.617787
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.70260 0.937502 0.468751 − 0.883330i \(-0.344704\pi\)
0.468751 + 0.883330i \(0.344704\pi\)
\(38\) 0 0
\(39\) 6.71881 1.07587
\(40\) 0 0
\(41\) −2.85130 −0.445298 −0.222649 − 0.974899i \(-0.571470\pi\)
−0.222649 + 0.974899i \(0.571470\pi\)
\(42\) 0 0
\(43\) −9.90105 −1.50990 −0.754948 − 0.655785i \(-0.772338\pi\)
−0.754948 + 0.655785i \(0.772338\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.625758 0.0912762 0.0456381 − 0.998958i \(-0.485468\pi\)
0.0456381 + 0.998958i \(0.485468\pi\)
\(48\) 0 0
\(49\) −0.330937 −0.0472767
\(50\) 0 0
\(51\) −5.31859 −0.744752
\(52\) 0 0
\(53\) −13.7350 −1.88665 −0.943325 − 0.331871i \(-0.892320\pi\)
−0.943325 + 0.331871i \(0.892320\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −10.1861 −1.34918
\(58\) 0 0
\(59\) 8.08549 1.05264 0.526320 − 0.850286i \(-0.323571\pi\)
0.526320 + 0.850286i \(0.323571\pi\)
\(60\) 0 0
\(61\) 8.80043 1.12678 0.563390 − 0.826191i \(-0.309497\pi\)
0.563390 + 0.826191i \(0.309497\pi\)
\(62\) 0 0
\(63\) 22.7015 2.86012
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −10.8675 −1.32768 −0.663839 − 0.747876i \(-0.731074\pi\)
−0.663839 + 0.747876i \(0.731074\pi\)
\(68\) 0 0
\(69\) 3.43375 0.413375
\(70\) 0 0
\(71\) 2.69283 0.319581 0.159790 − 0.987151i \(-0.448918\pi\)
0.159790 + 0.987151i \(0.448918\pi\)
\(72\) 0 0
\(73\) 7.84153 0.917782 0.458891 − 0.888493i \(-0.348247\pi\)
0.458891 + 0.888493i \(0.348247\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.66906 0.304168
\(78\) 0 0
\(79\) 3.96812 0.446449 0.223224 − 0.974767i \(-0.428342\pi\)
0.223224 + 0.974767i \(0.428342\pi\)
\(80\) 0 0
\(81\) 41.9038 4.65598
\(82\) 0 0
\(83\) −9.53657 −1.04677 −0.523387 − 0.852095i \(-0.675332\pi\)
−0.523387 + 0.852095i \(0.675332\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 23.8281 2.55464
\(88\) 0 0
\(89\) −7.25152 −0.768659 −0.384330 − 0.923196i \(-0.625567\pi\)
−0.384330 + 0.923196i \(0.625567\pi\)
\(90\) 0 0
\(91\) 5.05307 0.529706
\(92\) 0 0
\(93\) −27.6744 −2.86970
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.6160 1.38249 0.691247 − 0.722618i \(-0.257062\pi\)
0.691247 + 0.722618i \(0.257062\pi\)
\(98\) 0 0
\(99\) 9.08549 0.913126
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.ba.1.4 ✓ 4
4.3 odd 2 9200.2.a.cp.1.1 4
5.2 odd 4 4600.2.e.t.4049.1 8
5.3 odd 4 4600.2.e.t.4049.8 8
5.4 even 2 4600.2.a.bb.1.1 yes 4
20.19 odd 2 9200.2.a.cn.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.ba.1.4 ✓ 4 1.1 even 1 trivial
4600.2.a.bb.1.1 yes 4 5.4 even 2
4600.2.e.t.4049.1 8 5.2 odd 4
4600.2.e.t.4049.8 8 5.3 odd 4
9200.2.a.cn.1.4 4 20.19 odd 2
9200.2.a.cp.1.1 4 4.3 odd 2