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Properties

Label 3240.2.a.q.1.1
Level $3240$
Weight $2$
Character 3240.1
Self dual yes
Analytic conductor $25.872$
Analytic rank $0$
Dimension $3$
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] = [3240,2,Mod(1,3240)] 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("3240.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3240.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,-3,0,-5,0,0,0,2] 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: \(25.8715302549\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 360)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.571993\) of defining polynomial
Character \(\chi\) \(=\) 3240.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} -5.24482 q^{7} -2.67282 q^{11} +3.81681 q^{13} -3.52884 q^{17} -4.67282 q^{19} +4.95684 q^{23} +1.00000 q^{25} +1.85601 q^{29} -8.67282 q^{31} +5.24482 q^{35} -2.67282 q^{37} +3.67282 q^{41} +3.52884 q^{43} -9.26329 q^{47} +20.5081 q^{49} +2.85601 q^{53} +2.67282 q^{55} -4.20166 q^{59} -7.96080 q^{61} -3.81681 q^{65} -0.859966 q^{67} +15.1625 q^{71} +6.28797 q^{73} +14.0185 q^{77} +5.63362 q^{79} +3.89917 q^{83} +3.52884 q^{85} +11.0000 q^{89} -20.0185 q^{91} +4.67282 q^{95} -3.83528 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} - 5 q^{7} + 2 q^{11} - 2 q^{17} - 4 q^{19} + 7 q^{23} + 3 q^{25} + 7 q^{29} - 16 q^{31} + 5 q^{35} + 2 q^{37} + q^{41} + 2 q^{43} + 13 q^{47} + 10 q^{49} + 10 q^{53} - 2 q^{55} + 6 q^{59}+ \cdots + 30 q^{97}+O(q^{100}) \) 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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −5.24482 −1.98235 −0.991177 − 0.132543i \(-0.957686\pi\)
−0.991177 + 0.132543i \(0.957686\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.67282 −0.805887 −0.402943 − 0.915225i \(-0.632013\pi\)
−0.402943 + 0.915225i \(0.632013\pi\)
\(12\) 0 0
\(13\) 3.81681 1.05859 0.529296 − 0.848437i \(-0.322456\pi\)
0.529296 + 0.848437i \(0.322456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.52884 −0.855869 −0.427934 − 0.903810i \(-0.640759\pi\)
−0.427934 + 0.903810i \(0.640759\pi\)
\(18\) 0 0
\(19\) −4.67282 −1.07202 −0.536010 − 0.844212i \(-0.680069\pi\)
−0.536010 + 0.844212i \(0.680069\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.95684 1.03357 0.516787 − 0.856114i \(-0.327128\pi\)
0.516787 + 0.856114i \(0.327128\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.85601 0.344653 0.172327 − 0.985040i \(-0.444872\pi\)
0.172327 + 0.985040i \(0.444872\pi\)
\(30\) 0 0
\(31\) −8.67282 −1.55769 −0.778843 − 0.627219i \(-0.784193\pi\)
−0.778843 + 0.627219i \(0.784193\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.24482 0.886536
\(36\) 0 0
\(37\) −2.67282 −0.439410 −0.219705 − 0.975566i \(-0.570509\pi\)
−0.219705 + 0.975566i \(0.570509\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.67282 0.573599 0.286799 − 0.957991i \(-0.407409\pi\)
0.286799 + 0.957991i \(0.407409\pi\)
\(42\) 0 0
\(43\) 3.52884 0.538143 0.269071 − 0.963120i \(-0.413283\pi\)
0.269071 + 0.963120i \(0.413283\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.26329 −1.35119 −0.675595 − 0.737273i \(-0.736113\pi\)
−0.675595 + 0.737273i \(0.736113\pi\)
\(48\) 0 0
\(49\) 20.5081 2.92973
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.85601 0.392304 0.196152 − 0.980574i \(-0.437155\pi\)
0.196152 + 0.980574i \(0.437155\pi\)
\(54\) 0 0
\(55\) 2.67282 0.360403
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.20166 −0.547010 −0.273505 − 0.961871i \(-0.588183\pi\)
−0.273505 + 0.961871i \(0.588183\pi\)
\(60\) 0 0
\(61\) −7.96080 −1.01928 −0.509638 − 0.860389i \(-0.670221\pi\)
−0.509638 + 0.860389i \(0.670221\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.81681 −0.473417
\(66\) 0 0
\(67\) −0.859966 −0.105062 −0.0525308 − 0.998619i \(-0.516729\pi\)
−0.0525308 + 0.998619i \(0.516729\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 15.1625 1.79945 0.899726 − 0.436454i \(-0.143766\pi\)
0.899726 + 0.436454i \(0.143766\pi\)
\(72\) 0 0
\(73\) 6.28797 0.735952 0.367976 − 0.929835i \(-0.380051\pi\)
0.367976 + 0.929835i \(0.380051\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 14.0185 1.59755
\(78\) 0 0
\(79\) 5.63362 0.633832 0.316916 − 0.948454i \(-0.397353\pi\)
0.316916 + 0.948454i \(0.397353\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.89917 0.427989 0.213995 − 0.976835i \(-0.431352\pi\)
0.213995 + 0.976835i \(0.431352\pi\)
\(84\) 0 0
\(85\) 3.52884 0.382756
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 11.0000 1.16600 0.582999 − 0.812473i \(-0.301879\pi\)
0.582999 + 0.812473i \(0.301879\pi\)
\(90\) 0 0
\(91\) −20.0185 −2.09851
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.67282 0.479422
\(96\) 0 0
\(97\) −3.83528 −0.389414 −0.194707 − 0.980861i \(-0.562376\pi\)
−0.194707 + 0.980861i \(0.562376\pi\)
\(98\) 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 3240.2.a.q.1.1 3
3.2 odd 2 3240.2.a.r.1.1 3
4.3 odd 2 6480.2.a.bu.1.3 3
9.2 odd 6 360.2.q.d.121.3 ✓ 6
9.4 even 3 1080.2.q.d.721.3 6
9.5 odd 6 360.2.q.d.241.3 yes 6
9.7 even 3 1080.2.q.d.361.3 6
12.11 even 2 6480.2.a.bx.1.3 3
36.7 odd 6 2160.2.q.j.1441.1 6
36.11 even 6 720.2.q.j.481.1 6
36.23 even 6 720.2.q.j.241.1 6
36.31 odd 6 2160.2.q.j.721.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.q.d.121.3 ✓ 6 9.2 odd 6
360.2.q.d.241.3 yes 6 9.5 odd 6
720.2.q.j.241.1 6 36.23 even 6
720.2.q.j.481.1 6 36.11 even 6
1080.2.q.d.361.3 6 9.7 even 3
1080.2.q.d.721.3 6 9.4 even 3
2160.2.q.j.721.1 6 36.31 odd 6
2160.2.q.j.1441.1 6 36.7 odd 6
3240.2.a.q.1.1 3 1.1 even 1 trivial
3240.2.a.r.1.1 3 3.2 odd 2
6480.2.a.bu.1.3 3 4.3 odd 2
6480.2.a.bx.1.3 3 12.11 even 2