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Properties

Label 4600.2.e.t.4049.8
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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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(4049,4600)] 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("4600.4049"); 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, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,-12,0,2,0,0,0,0,0,0,0,30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.61734359296.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 13x^{6} + 38x^{4} + 25x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.8
Root \(0.774457i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.t.4049.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.43375i q^{3} -2.58245i q^{7} -8.79066 q^{9} +1.03354 q^{11} +1.95669i q^{13} +1.54891i q^{17} +2.96646 q^{19} +8.86751 q^{21} +1.00000i q^{23} -19.8837i q^{27} -6.93936 q^{29} -8.05951 q^{31} +3.54891i q^{33} -5.70260i q^{37} -6.71881 q^{39} -2.85130 q^{41} -9.90105i q^{43} -0.625758i q^{47} +0.330937 q^{49} -5.31859 q^{51} -13.7350i q^{53} +10.1861i q^{57} -8.08549 q^{59} +8.80043 q^{61} +22.7015i q^{63} +10.8675i q^{67} -3.43375 q^{69} +2.69283 q^{71} +7.84153i q^{73} -2.66906i q^{77} -3.96812 q^{79} +41.9038 q^{81} -9.53657i q^{83} -23.8281i q^{87} +7.25152 q^{89} +5.05307 q^{91} -27.6744i q^{93} -13.6160i q^{97} -9.08549 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{9} + 2 q^{11} + 30 q^{19} + 16 q^{21} - 2 q^{29} - 24 q^{31} + 6 q^{39} - 18 q^{41} + 6 q^{49} - 4 q^{51} + 10 q^{59} + 28 q^{61} - 4 q^{71} + 42 q^{79} + 88 q^{81} + 32 q^{89} + 66 q^{91} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 3.43375i 1.98248i 0.132078 + 0.991239i \(0.457835\pi\)
−0.132078 + 0.991239i \(0.542165\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.58245i − 0.976075i −0.872823 − 0.488038i \(-0.837713\pi\)
0.872823 − 0.488038i \(-0.162287\pi\)
\(8\) 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
\(13\) 1.95669i 0.542689i 0.962482 + 0.271345i \(0.0874683\pi\)
−0.962482 + 0.271345i \(0.912532\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.54891i 0.375667i 0.982201 + 0.187834i \(0.0601465\pi\)
−0.982201 + 0.187834i \(0.939853\pi\)
\(18\) 0 0
\(19\) 2.96646 0.680553 0.340277 − 0.940325i \(-0.389479\pi\)
0.340277 + 0.940325i \(0.389479\pi\)
\(20\) 0 0
\(21\) 8.86751 1.93505
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 19.8837i − 3.82662i
\(28\) 0 0
\(29\) −6.93936 −1.28861 −0.644304 − 0.764770i \(-0.722853\pi\)
−0.644304 + 0.764770i \(0.722853\pi\)
\(30\) 0 0
\(31\) −8.05951 −1.44753 −0.723766 − 0.690046i \(-0.757590\pi\)
−0.723766 + 0.690046i \(0.757590\pi\)
\(32\) 0 0
\(33\) 3.54891i 0.617787i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 5.70260i − 0.937502i −0.883330 − 0.468751i \(-0.844704\pi\)
0.883330 − 0.468751i \(-0.155296\pi\)
\(38\) 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
\(43\) − 9.90105i − 1.50990i −0.655785 − 0.754948i \(-0.727662\pi\)
0.655785 − 0.754948i \(-0.272338\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 0.625758i − 0.0912762i −0.998958 − 0.0456381i \(-0.985468\pi\)
0.998958 − 0.0456381i \(-0.0145321\pi\)
\(48\) 0 0
\(49\) 0.330937 0.0472767
\(50\) 0 0
\(51\) −5.31859 −0.744752
\(52\) 0 0
\(53\) − 13.7350i − 1.88665i −0.331871 − 0.943325i \(-0.607680\pi\)
0.331871 − 0.943325i \(-0.392320\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 10.1861i 1.34918i
\(58\) 0 0
\(59\) −8.08549 −1.05264 −0.526320 − 0.850286i \(-0.676429\pi\)
−0.526320 + 0.850286i \(0.676429\pi\)
\(60\) 0 0
\(61\) 8.80043 1.12678 0.563390 − 0.826191i \(-0.309497\pi\)
0.563390 + 0.826191i \(0.309497\pi\)
\(62\) 0 0
\(63\) 22.7015i 2.86012i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 10.8675i 1.32768i 0.747876 + 0.663839i \(0.231074\pi\)
−0.747876 + 0.663839i \(0.768926\pi\)
\(68\) 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
\(73\) 7.84153i 0.917782i 0.888493 + 0.458891i \(0.151753\pi\)
−0.888493 + 0.458891i \(0.848247\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 2.66906i − 0.304168i
\(78\) 0 0
\(79\) −3.96812 −0.446449 −0.223224 − 0.974767i \(-0.571658\pi\)
−0.223224 + 0.974767i \(0.571658\pi\)
\(80\) 0 0
\(81\) 41.9038 4.65598
\(82\) 0 0
\(83\) − 9.53657i − 1.04677i −0.852095 − 0.523387i \(-0.824668\pi\)
0.852095 − 0.523387i \(-0.175332\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 23.8281i − 2.55464i
\(88\) 0 0
\(89\) 7.25152 0.768659 0.384330 − 0.923196i \(-0.374433\pi\)
0.384330 + 0.923196i \(0.374433\pi\)
\(90\) 0 0
\(91\) 5.05307 0.529706
\(92\) 0 0
\(93\) − 27.6744i − 2.86970i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 13.6160i − 1.38249i −0.722618 − 0.691247i \(-0.757062\pi\)
0.722618 − 0.691247i \(-0.242938\pi\)
\(98\) 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.e.t.4049.8 8
5.2 odd 4 4600.2.a.ba.1.4 ✓ 4
5.3 odd 4 4600.2.a.bb.1.1 yes 4
5.4 even 2 inner 4600.2.e.t.4049.1 8
20.3 even 4 9200.2.a.cn.1.4 4
20.7 even 4 9200.2.a.cp.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.ba.1.4 ✓ 4 5.2 odd 4
4600.2.a.bb.1.1 yes 4 5.3 odd 4
4600.2.e.t.4049.1 8 5.4 even 2 inner
4600.2.e.t.4049.8 8 1.1 even 1 trivial
9200.2.a.cn.1.4 4 20.3 even 4
9200.2.a.cp.1.1 4 20.7 even 4