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Properties

Label 3267.2.a.y.1.3
Level $3267$
Weight $2$
Character 3267.1
Self dual yes
Analytic conductor $26.087$
Analytic rank $1$
Dimension $4$
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] = [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 = [4,0,0,8,-4,0,0,0,0,0,0,0,0,-16,0,12,0,0,0,-36,0,0,-16,0,12,12, 0,0,0,0,-4] 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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 8x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.16372\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+1.16372 q^{2} -0.645751 q^{4} +1.64575 q^{5} -1.16372 q^{7} -3.07892 q^{8} +1.91520 q^{10} -4.24264 q^{13} -1.35425 q^{14} -2.29150 q^{16} +3.07892 q^{17} +6.57008 q^{19} -1.06275 q^{20} -1.35425 q^{23} -2.29150 q^{25} -4.93725 q^{26} +0.751475 q^{28} -1.16372 q^{29} -3.64575 q^{31} +3.49117 q^{32} +3.58301 q^{34} -1.91520 q^{35} -3.35425 q^{37} +7.64575 q^{38} -5.06713 q^{40} +9.64900 q^{41} -10.0613 q^{43} -1.57597 q^{46} -12.2915 q^{47} -5.64575 q^{49} -2.66667 q^{50} +2.73969 q^{52} -6.00000 q^{53} +3.58301 q^{56} -1.35425 q^{58} -7.93725 q^{59} -1.91520 q^{61} -4.24264 q^{62} +8.64575 q^{64} -6.98233 q^{65} -3.64575 q^{67} -1.98822 q^{68} -2.22876 q^{70} +9.29150 q^{71} -10.8127 q^{73} -3.90341 q^{74} -4.24264 q^{76} +0.751475 q^{79} -3.77124 q^{80} +11.2288 q^{82} -8.89753 q^{83} +5.06713 q^{85} -11.7085 q^{86} -12.5830 q^{89} +4.93725 q^{91} +0.874508 q^{92} -14.3039 q^{94} +10.8127 q^{95} -5.93725 q^{97} -6.57008 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} - 4 q^{5} - 16 q^{14} + 12 q^{16} - 36 q^{20} - 16 q^{23} + 12 q^{25} + 12 q^{26} - 4 q^{31} - 28 q^{34} - 24 q^{37} + 20 q^{38} - 28 q^{47} - 12 q^{49} - 24 q^{53} - 28 q^{56} - 16 q^{58}+ \cdots + 8 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.16372 0.822876 0.411438 − 0.911438i \(-0.365027\pi\)
0.411438 + 0.911438i \(0.365027\pi\)
\(3\) 0 0
\(4\) −0.645751 −0.322876
\(5\) 1.64575 0.736002 0.368001 − 0.929825i \(-0.380042\pi\)
0.368001 + 0.929825i \(0.380042\pi\)
\(6\) 0 0
\(7\) −1.16372 −0.439846 −0.219923 − 0.975517i \(-0.570581\pi\)
−0.219923 + 0.975517i \(0.570581\pi\)
\(8\) −3.07892 −1.08856
\(9\) 0 0
\(10\) 1.91520 0.605638
\(11\) 0 0
\(12\) 0 0
\(13\) −4.24264 −1.17670 −0.588348 − 0.808608i \(-0.700222\pi\)
−0.588348 + 0.808608i \(0.700222\pi\)
\(14\) −1.35425 −0.361938
\(15\) 0 0
\(16\) −2.29150 −0.572876
\(17\) 3.07892 0.746747 0.373374 − 0.927681i \(-0.378201\pi\)
0.373374 + 0.927681i \(0.378201\pi\)
\(18\) 0 0
\(19\) 6.57008 1.50728 0.753640 − 0.657287i \(-0.228296\pi\)
0.753640 + 0.657287i \(0.228296\pi\)
\(20\) −1.06275 −0.237637
\(21\) 0 0
\(22\) 0 0
\(23\) −1.35425 −0.282380 −0.141190 − 0.989982i \(-0.545093\pi\)
−0.141190 + 0.989982i \(0.545093\pi\)
\(24\) 0 0
\(25\) −2.29150 −0.458301
\(26\) −4.93725 −0.968275
\(27\) 0 0
\(28\) 0.751475 0.142015
\(29\) −1.16372 −0.216098 −0.108049 − 0.994146i \(-0.534460\pi\)
−0.108049 + 0.994146i \(0.534460\pi\)
\(30\) 0 0
\(31\) −3.64575 −0.654796 −0.327398 − 0.944886i \(-0.606172\pi\)
−0.327398 + 0.944886i \(0.606172\pi\)
\(32\) 3.49117 0.617157
\(33\) 0 0
\(34\) 3.58301 0.614480
\(35\) −1.91520 −0.323727
\(36\) 0 0
\(37\) −3.35425 −0.551435 −0.275718 − 0.961239i \(-0.588915\pi\)
−0.275718 + 0.961239i \(0.588915\pi\)
\(38\) 7.64575 1.24030
\(39\) 0 0
\(40\) −5.06713 −0.801184
\(41\) 9.64900 1.50692 0.753461 − 0.657493i \(-0.228383\pi\)
0.753461 + 0.657493i \(0.228383\pi\)
\(42\) 0 0
\(43\) −10.0613 −1.53433 −0.767163 − 0.641452i \(-0.778332\pi\)
−0.767163 + 0.641452i \(0.778332\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.57597 −0.232364
\(47\) −12.2915 −1.79290 −0.896450 − 0.443145i \(-0.853863\pi\)
−0.896450 + 0.443145i \(0.853863\pi\)
\(48\) 0 0
\(49\) −5.64575 −0.806536
\(50\) −2.66667 −0.377124
\(51\) 0 0
\(52\) 2.73969 0.379927
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.58301 0.478799
\(57\) 0 0
\(58\) −1.35425 −0.177822
\(59\) −7.93725 −1.03334 −0.516671 − 0.856184i \(-0.672829\pi\)
−0.516671 + 0.856184i \(0.672829\pi\)
\(60\) 0 0
\(61\) −1.91520 −0.245216 −0.122608 − 0.992455i \(-0.539126\pi\)
−0.122608 + 0.992455i \(0.539126\pi\)
\(62\) −4.24264 −0.538816
\(63\) 0 0
\(64\) 8.64575 1.08072
\(65\) −6.98233 −0.866052
\(66\) 0 0
\(67\) −3.64575 −0.445399 −0.222700 − 0.974887i \(-0.571487\pi\)
−0.222700 + 0.974887i \(0.571487\pi\)
\(68\) −1.98822 −0.241107
\(69\) 0 0
\(70\) −2.22876 −0.266387
\(71\) 9.29150 1.10270 0.551349 − 0.834275i \(-0.314113\pi\)
0.551349 + 0.834275i \(0.314113\pi\)
\(72\) 0 0
\(73\) −10.8127 −1.26553 −0.632767 − 0.774342i \(-0.718081\pi\)
−0.632767 + 0.774342i \(0.718081\pi\)
\(74\) −3.90341 −0.453763
\(75\) 0 0
\(76\) −4.24264 −0.486664
\(77\) 0 0
\(78\) 0 0
\(79\) 0.751475 0.0845475 0.0422738 − 0.999106i \(-0.486540\pi\)
0.0422738 + 0.999106i \(0.486540\pi\)
\(80\) −3.77124 −0.421638
\(81\) 0 0
\(82\) 11.2288 1.24001
\(83\) −8.89753 −0.976631 −0.488315 − 0.872667i \(-0.662388\pi\)
−0.488315 + 0.872667i \(0.662388\pi\)
\(84\) 0 0
\(85\) 5.06713 0.549608
\(86\) −11.7085 −1.26256
\(87\) 0 0
\(88\) 0 0
\(89\) −12.5830 −1.33380 −0.666898 − 0.745149i \(-0.732378\pi\)
−0.666898 + 0.745149i \(0.732378\pi\)
\(90\) 0 0
\(91\) 4.93725 0.517565
\(92\) 0.874508 0.0911737
\(93\) 0 0
\(94\) −14.3039 −1.47533
\(95\) 10.8127 1.10936
\(96\) 0 0
\(97\) −5.93725 −0.602837 −0.301418 − 0.953492i \(-0.597460\pi\)
−0.301418 + 0.953492i \(0.597460\pi\)
\(98\) −6.57008 −0.663679
\(99\) 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.y.1.3 yes 4
3.2 odd 2 3267.2.a.bb.1.2 yes 4
11.10 odd 2 inner 3267.2.a.y.1.2 ✓ 4
33.32 even 2 3267.2.a.bb.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3267.2.a.y.1.2 ✓ 4 11.10 odd 2 inner
3267.2.a.y.1.3 yes 4 1.1 even 1 trivial
3267.2.a.bb.1.2 yes 4 3.2 odd 2
3267.2.a.bb.1.3 yes 4 33.32 even 2