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Properties

Label 3096.2.a.u.1.4
Level $3096$
Weight $2$
Character 3096.1
Self dual yes
Analytic conductor $24.722$
Analytic rank $1$
Dimension $6$
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] = [3096,2,Mod(1,3096)] 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("3096.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3096, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3096 = 2^{3} \cdot 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3096.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-5,0,-5,0,0,0,1] 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: \(24.7216844658\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.824018032.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 21x^{4} - 3x^{3} + 76x^{2} + 16x - 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.310363\) of defining polynomial
Character \(\chi\) \(=\) 3096.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.689637 q^{5} -4.23877 q^{7} +4.98100 q^{11} -0.864210 q^{13} +5.09505 q^{17} +1.61805 q^{19} -5.47181 q^{23} -4.52440 q^{25} -5.59851 q^{29} -1.37456 q^{31} +2.92321 q^{35} +11.1424 q^{37} -8.29136 q^{41} -1.00000 q^{43} +0.114054 q^{47} +10.9672 q^{49} +0.291362 q^{53} -3.43508 q^{55} -2.81955 q^{59} +10.5669 q^{61} +0.595991 q^{65} -2.04986 q^{67} -13.3864 q^{71} +11.1424 q^{73} -21.1133 q^{77} -10.6605 q^{79} +5.88987 q^{83} -3.51374 q^{85} -3.80118 q^{89} +3.66319 q^{91} -1.11586 q^{95} -10.8103 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 5 q^{5} - 5 q^{7} + q^{11} + 5 q^{13} - 6 q^{17} - 9 q^{19} + 2 q^{23} + 17 q^{25} - 9 q^{29} + 2 q^{31} - 19 q^{35} + 4 q^{37} - 20 q^{41} - 6 q^{43} - 7 q^{47} + 25 q^{49} - 28 q^{53} - 7 q^{55}+ \cdots + 15 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) −0.689637 −0.308415 −0.154208 − 0.988038i \(-0.549282\pi\)
−0.154208 + 0.988038i \(0.549282\pi\)
\(6\) 0 0
\(7\) −4.23877 −1.60211 −0.801053 − 0.598594i \(-0.795726\pi\)
−0.801053 + 0.598594i \(0.795726\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.98100 1.50183 0.750914 − 0.660400i \(-0.229613\pi\)
0.750914 + 0.660400i \(0.229613\pi\)
\(12\) 0 0
\(13\) −0.864210 −0.239689 −0.119844 − 0.992793i \(-0.538240\pi\)
−0.119844 + 0.992793i \(0.538240\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.09505 1.23573 0.617866 − 0.786283i \(-0.287997\pi\)
0.617866 + 0.786283i \(0.287997\pi\)
\(18\) 0 0
\(19\) 1.61805 0.371205 0.185603 − 0.982625i \(-0.440576\pi\)
0.185603 + 0.982625i \(0.440576\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.47181 −1.14095 −0.570476 − 0.821314i \(-0.693241\pi\)
−0.570476 + 0.821314i \(0.693241\pi\)
\(24\) 0 0
\(25\) −4.52440 −0.904880
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.59851 −1.03962 −0.519808 − 0.854283i \(-0.673997\pi\)
−0.519808 + 0.854283i \(0.673997\pi\)
\(30\) 0 0
\(31\) −1.37456 −0.246879 −0.123439 − 0.992352i \(-0.539392\pi\)
−0.123439 + 0.992352i \(0.539392\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.92321 0.494113
\(36\) 0 0
\(37\) 11.1424 1.83181 0.915904 − 0.401398i \(-0.131476\pi\)
0.915904 + 0.401398i \(0.131476\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.29136 −1.29489 −0.647447 − 0.762111i \(-0.724163\pi\)
−0.647447 + 0.762111i \(0.724163\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.114054 0.0166366 0.00831828 − 0.999965i \(-0.497352\pi\)
0.00831828 + 0.999965i \(0.497352\pi\)
\(48\) 0 0
\(49\) 10.9672 1.56674
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.291362 0.0400216 0.0200108 − 0.999800i \(-0.493630\pi\)
0.0200108 + 0.999800i \(0.493630\pi\)
\(54\) 0 0
\(55\) −3.43508 −0.463186
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.81955 −0.367074 −0.183537 − 0.983013i \(-0.558755\pi\)
−0.183537 + 0.983013i \(0.558755\pi\)
\(60\) 0 0
\(61\) 10.5669 1.35295 0.676474 − 0.736466i \(-0.263507\pi\)
0.676474 + 0.736466i \(0.263507\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.595991 0.0739236
\(66\) 0 0
\(67\) −2.04986 −0.250430 −0.125215 − 0.992130i \(-0.539962\pi\)
−0.125215 + 0.992130i \(0.539962\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −13.3864 −1.58868 −0.794338 − 0.607477i \(-0.792182\pi\)
−0.794338 + 0.607477i \(0.792182\pi\)
\(72\) 0 0
\(73\) 11.1424 1.30412 0.652062 − 0.758165i \(-0.273904\pi\)
0.652062 + 0.758165i \(0.273904\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −21.1133 −2.40609
\(78\) 0 0
\(79\) −10.6605 −1.19940 −0.599701 − 0.800224i \(-0.704714\pi\)
−0.599701 + 0.800224i \(0.704714\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.88987 0.646497 0.323249 − 0.946314i \(-0.395225\pi\)
0.323249 + 0.946314i \(0.395225\pi\)
\(84\) 0 0
\(85\) −3.51374 −0.381118
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.80118 −0.402924 −0.201462 − 0.979496i \(-0.564569\pi\)
−0.201462 + 0.979496i \(0.564569\pi\)
\(90\) 0 0
\(91\) 3.66319 0.384007
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.11586 −0.114485
\(96\) 0 0
\(97\) −10.8103 −1.09762 −0.548812 − 0.835946i \(-0.684920\pi\)
−0.548812 + 0.835946i \(0.684920\pi\)
\(98\) 0 0
\(99\) 0 0
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 3096.2.a.u.1.4 ✓ 6
3.2 odd 2 3096.2.a.v.1.3 yes 6
4.3 odd 2 6192.2.a.cb.1.4 6
12.11 even 2 6192.2.a.cc.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3096.2.a.u.1.4 ✓ 6 1.1 even 1 trivial
3096.2.a.v.1.3 yes 6 3.2 odd 2
6192.2.a.cb.1.4 6 4.3 odd 2
6192.2.a.cc.1.3 6 12.11 even 2