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Properties

Label 3267.2.a.j.1.1
Level $3267$
Weight $2$
Character 3267.1
Self dual yes
Analytic conductor $26.087$
Analytic rank $0$
Dimension $1$
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] = [3267,2,Mod(1,3267)] 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("3267.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3267, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3267 = 3^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3267.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,0,-1,2,0,5,-3,0,2,0,0,2,5,0,-1,7,0,0,-2,0,0,1,0,-1,2,0,-5, 3,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.0871263404\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 297)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3267.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} +5.00000 q^{7} -3.00000 q^{8} +2.00000 q^{10} +2.00000 q^{13} +5.00000 q^{14} -1.00000 q^{16} +7.00000 q^{17} -2.00000 q^{20} +1.00000 q^{23} -1.00000 q^{25} +2.00000 q^{26} -5.00000 q^{28} +3.00000 q^{29} -8.00000 q^{31} +5.00000 q^{32} +7.00000 q^{34} +10.0000 q^{35} -3.00000 q^{37} -6.00000 q^{40} -11.0000 q^{41} +9.00000 q^{43} +1.00000 q^{46} +1.00000 q^{47} +18.0000 q^{49} -1.00000 q^{50} -2.00000 q^{52} +12.0000 q^{53} -15.0000 q^{56} +3.00000 q^{58} -5.00000 q^{59} -6.00000 q^{61} -8.00000 q^{62} +7.00000 q^{64} +4.00000 q^{65} -4.00000 q^{67} -7.00000 q^{68} +10.0000 q^{70} -4.00000 q^{73} -3.00000 q^{74} -5.00000 q^{79} -2.00000 q^{80} -11.0000 q^{82} -6.00000 q^{83} +14.0000 q^{85} +9.00000 q^{86} +6.00000 q^{89} +10.0000 q^{91} -1.00000 q^{92} +1.00000 q^{94} +11.0000 q^{97} +18.0000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 − 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 5.00000 1.88982 0.944911 − 0.327327i \(-0.106148\pi\)
0.944911 + 0.327327i \(0.106148\pi\)
\(8\) −3.00000 −1.06066
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 0 0
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 5.00000 1.33631
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 7.00000 1.69775 0.848875 − 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514 0.104257 − 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) −5.00000 −0.944911
\(29\) 3.00000 0.557086 0.278543 − 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 7.00000 1.20049
\(35\) 10.0000 1.69031
\(36\) 0 0
\(37\) −3.00000 −0.493197 −0.246598 − 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −6.00000 −0.948683
\(41\) −11.0000 −1.71791 −0.858956 − 0.512050i \(-0.828886\pi\)
−0.858956 + 0.512050i \(0.828886\pi\)
\(42\) 0 0
\(43\) 9.00000 1.37249 0.686244 − 0.727372i \(-0.259258\pi\)
0.686244 + 0.727372i \(0.259258\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 1.00000 0.145865 0.0729325 − 0.997337i \(-0.476764\pi\)
0.0729325 + 0.997337i \(0.476764\pi\)
\(48\) 0 0
\(49\) 18.0000 2.57143
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 12.0000 1.64833 0.824163 − 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −15.0000 −2.00446
\(57\) 0 0
\(58\) 3.00000 0.393919
\(59\) −5.00000 −0.650945 −0.325472 − 0.945552i \(-0.605523\pi\)
−0.325472 + 0.945552i \(0.605523\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −7.00000 −0.848875
\(69\) 0 0
\(70\) 10.0000 1.19523
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −3.00000 −0.348743
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −5.00000 −0.562544 −0.281272 − 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) −11.0000 −1.21475
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 14.0000 1.51851
\(86\) 9.00000 0.970495
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 10.0000 1.04828
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 1.00000 0.103142
\(95\) 0 0
\(96\) 0 0
\(97\) 11.0000 1.11688 0.558440 − 0.829545i \(-0.311400\pi\)
0.558440 + 0.829545i \(0.311400\pi\)
\(98\) 18.0000 1.81827
\(99\) 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 3267.2.a.j.1.1 1
3.2 odd 2 3267.2.a.c.1.1 1
11.10 odd 2 297.2.a.b.1.1 ✓ 1
33.32 even 2 297.2.a.c.1.1 yes 1
44.43 even 2 4752.2.a.r.1.1 1
55.54 odd 2 7425.2.a.s.1.1 1
99.32 even 6 891.2.e.f.298.1 2
99.43 odd 6 891.2.e.h.595.1 2
99.65 even 6 891.2.e.f.595.1 2
99.76 odd 6 891.2.e.h.298.1 2
132.131 odd 2 4752.2.a.g.1.1 1
165.164 even 2 7425.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
297.2.a.b.1.1 ✓ 1 11.10 odd 2
297.2.a.c.1.1 yes 1 33.32 even 2
891.2.e.f.298.1 2 99.32 even 6
891.2.e.f.595.1 2 99.65 even 6
891.2.e.h.298.1 2 99.76 odd 6
891.2.e.h.595.1 2 99.43 odd 6
3267.2.a.c.1.1 1 3.2 odd 2
3267.2.a.j.1.1 1 1.1 even 1 trivial
4752.2.a.g.1.1 1 132.131 odd 2
4752.2.a.r.1.1 1 44.43 even 2
7425.2.a.k.1.1 1 165.164 even 2
7425.2.a.s.1.1 1 55.54 odd 2