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Properties

Label 3312.2.a.x.1.2
Level $3312$
Weight $2$
Character 3312.1
Self dual yes
Analytic conductor $26.446$
Analytic rank $1$
Dimension $2$
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] = [3312,2,Mod(1,3312)] 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("3312.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3312.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-2,0,2,0,0,0,-4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.4464531494\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1656)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 3312.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.732051 q^{5} -0.732051 q^{7} +1.46410 q^{11} -5.46410 q^{13} -0.732051 q^{17} +4.73205 q^{19} -1.00000 q^{23} -4.46410 q^{25} -5.46410 q^{29} +2.53590 q^{31} -0.535898 q^{35} -0.535898 q^{37} +6.92820 q^{41} -0.732051 q^{43} -7.46410 q^{47} -6.46410 q^{49} -14.1962 q^{53} +1.07180 q^{55} +0.535898 q^{59} +6.39230 q^{61} -4.00000 q^{65} +6.19615 q^{67} -10.9282 q^{71} -6.00000 q^{73} -1.07180 q^{77} +9.12436 q^{79} -16.3923 q^{83} -0.535898 q^{85} +6.19615 q^{89} +4.00000 q^{91} +3.46410 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 2 q^{7} - 4 q^{11} - 4 q^{13} + 2 q^{17} + 6 q^{19} - 2 q^{23} - 2 q^{25} - 4 q^{29} + 12 q^{31} - 8 q^{35} - 8 q^{37} + 2 q^{43} - 8 q^{47} - 6 q^{49} - 18 q^{53} + 16 q^{55} + 8 q^{59}+ \cdots - 4 q^{97}+O(q^{100}) \) 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
\(5\) 0.732051 0.327383 0.163692 − 0.986512i \(-0.447660\pi\)
0.163692 + 0.986512i \(0.447660\pi\)
\(6\) 0 0
\(7\) −0.732051 −0.276689 −0.138345 − 0.990384i \(-0.544178\pi\)
−0.138345 + 0.990384i \(0.544178\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.46410 0.441443 0.220722 − 0.975337i \(-0.429159\pi\)
0.220722 + 0.975337i \(0.429159\pi\)
\(12\) 0 0
\(13\) −5.46410 −1.51547 −0.757735 − 0.652563i \(-0.773694\pi\)
−0.757735 + 0.652563i \(0.773694\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.732051 −0.177548 −0.0887742 − 0.996052i \(-0.528295\pi\)
−0.0887742 + 0.996052i \(0.528295\pi\)
\(18\) 0 0
\(19\) 4.73205 1.08561 0.542803 − 0.839860i \(-0.317363\pi\)
0.542803 + 0.839860i \(0.317363\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −4.46410 −0.892820
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.46410 −1.01466 −0.507329 − 0.861752i \(-0.669367\pi\)
−0.507329 + 0.861752i \(0.669367\pi\)
\(30\) 0 0
\(31\) 2.53590 0.455461 0.227730 − 0.973724i \(-0.426870\pi\)
0.227730 + 0.973724i \(0.426870\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.535898 −0.0905834
\(36\) 0 0
\(37\) −0.535898 −0.0881012 −0.0440506 − 0.999029i \(-0.514026\pi\)
−0.0440506 + 0.999029i \(0.514026\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820 1.08200 0.541002 − 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) 0 0
\(43\) −0.732051 −0.111637 −0.0558184 − 0.998441i \(-0.517777\pi\)
−0.0558184 + 0.998441i \(0.517777\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.46410 −1.08875 −0.544376 − 0.838842i \(-0.683233\pi\)
−0.544376 + 0.838842i \(0.683233\pi\)
\(48\) 0 0
\(49\) −6.46410 −0.923443
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −14.1962 −1.94999 −0.974996 − 0.222224i \(-0.928669\pi\)
−0.974996 + 0.222224i \(0.928669\pi\)
\(54\) 0 0
\(55\) 1.07180 0.144521
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.535898 0.0697680 0.0348840 − 0.999391i \(-0.488894\pi\)
0.0348840 + 0.999391i \(0.488894\pi\)
\(60\) 0 0
\(61\) 6.39230 0.818451 0.409225 − 0.912433i \(-0.365799\pi\)
0.409225 + 0.912433i \(0.365799\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 6.19615 0.756980 0.378490 − 0.925605i \(-0.376443\pi\)
0.378490 + 0.925605i \(0.376443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.9282 −1.29694 −0.648470 − 0.761241i \(-0.724591\pi\)
−0.648470 + 0.761241i \(0.724591\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.07180 −0.122143
\(78\) 0 0
\(79\) 9.12436 1.02657 0.513285 − 0.858218i \(-0.328428\pi\)
0.513285 + 0.858218i \(0.328428\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −16.3923 −1.79929 −0.899645 − 0.436623i \(-0.856174\pi\)
−0.899645 + 0.436623i \(0.856174\pi\)
\(84\) 0 0
\(85\) −0.535898 −0.0581263
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.19615 0.656791 0.328395 − 0.944540i \(-0.393492\pi\)
0.328395 + 0.944540i \(0.393492\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.46410 0.355409
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 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 3312.2.a.x.1.2 2
3.2 odd 2 3312.2.a.bd.1.1 2
4.3 odd 2 1656.2.a.k.1.2 ✓ 2
12.11 even 2 1656.2.a.m.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1656.2.a.k.1.2 ✓ 2 4.3 odd 2
1656.2.a.m.1.1 yes 2 12.11 even 2
3312.2.a.x.1.2 2 1.1 even 1 trivial
3312.2.a.bd.1.1 2 3.2 odd 2