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Properties

Label 3267.2.a.bb.1.4
Level $3267$
Weight $2$
Character 3267.1
Self dual yes
Analytic conductor $26.087$
Analytic rank $0$
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: \(0\)
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 1.4
Root \(2.57794\) 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)
 
\(f(q)\) \(=\) \(q+2.57794 q^{2} +4.64575 q^{4} +3.64575 q^{5} +2.57794 q^{7} +6.82058 q^{8} +9.39851 q^{10} -4.24264 q^{13} +6.64575 q^{14} +8.29150 q^{16} -6.82058 q^{17} -0.913230 q^{19} +16.9373 q^{20} +6.64575 q^{23} +8.29150 q^{25} -10.9373 q^{26} +11.9764 q^{28} -2.57794 q^{29} +1.64575 q^{31} +7.73381 q^{32} -17.5830 q^{34} +9.39851 q^{35} -8.64575 q^{37} -2.35425 q^{38} +24.8661 q^{40} -5.90735 q^{41} +8.64704 q^{43} +17.1323 q^{46} +1.70850 q^{47} -0.354249 q^{49} +21.3750 q^{50} -19.7103 q^{52} +6.00000 q^{53} +17.5830 q^{56} -6.64575 q^{58} -7.93725 q^{59} -9.39851 q^{61} +4.24264 q^{62} +3.35425 q^{64} -15.4676 q^{65} +1.64575 q^{67} -31.6867 q^{68} +24.2288 q^{70} +1.29150 q^{71} -3.32941 q^{73} -22.2882 q^{74} -4.24264 q^{76} +11.9764 q^{79} +30.2288 q^{80} -15.2288 q^{82} -6.06910 q^{83} -24.8661 q^{85} +22.2915 q^{86} -8.58301 q^{89} -10.9373 q^{91} +30.8745 q^{92} +4.40440 q^{94} -3.32941 q^{95} +9.93725 q^{97} -0.913230 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} + 4 q^{5} + 16 q^{14} + 12 q^{16} + 36 q^{20} + 16 q^{23} + 12 q^{25} - 12 q^{26} - 4 q^{31} - 28 q^{34} - 24 q^{37} - 20 q^{38} + 28 q^{47} - 12 q^{49} + 24 q^{53} + 28 q^{56} - 16 q^{58}+ \cdots + 8 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.57794 1.82288 0.911438 − 0.411438i \(-0.134973\pi\)
0.911438 + 0.411438i \(0.134973\pi\)
\(3\) 0 0
\(4\) 4.64575 2.32288
\(5\) 3.64575 1.63043 0.815215 − 0.579159i \(-0.196619\pi\)
0.815215 + 0.579159i \(0.196619\pi\)
\(6\) 0 0
\(7\) 2.57794 0.974368 0.487184 − 0.873299i \(-0.338024\pi\)
0.487184 + 0.873299i \(0.338024\pi\)
\(8\) 6.82058 2.41144
\(9\) 0 0
\(10\) 9.39851 2.97207
\(11\) 0 0
\(12\) 0 0
\(13\) −4.24264 −1.17670 −0.588348 − 0.808608i \(-0.700222\pi\)
−0.588348 + 0.808608i \(0.700222\pi\)
\(14\) 6.64575 1.77615
\(15\) 0 0
\(16\) 8.29150 2.07288
\(17\) −6.82058 −1.65423 −0.827116 − 0.562031i \(-0.810020\pi\)
−0.827116 + 0.562031i \(0.810020\pi\)
\(18\) 0 0
\(19\) −0.913230 −0.209509 −0.104755 − 0.994498i \(-0.533406\pi\)
−0.104755 + 0.994498i \(0.533406\pi\)
\(20\) 16.9373 3.78729
\(21\) 0 0
\(22\) 0 0
\(23\) 6.64575 1.38573 0.692867 − 0.721065i \(-0.256347\pi\)
0.692867 + 0.721065i \(0.256347\pi\)
\(24\) 0 0
\(25\) 8.29150 1.65830
\(26\) −10.9373 −2.14497
\(27\) 0 0
\(28\) 11.9764 2.26334
\(29\) −2.57794 −0.478711 −0.239355 − 0.970932i \(-0.576936\pi\)
−0.239355 + 0.970932i \(0.576936\pi\)
\(30\) 0 0
\(31\) 1.64575 0.295586 0.147793 − 0.989018i \(-0.452783\pi\)
0.147793 + 0.989018i \(0.452783\pi\)
\(32\) 7.73381 1.36716
\(33\) 0 0
\(34\) −17.5830 −3.01546
\(35\) 9.39851 1.58864
\(36\) 0 0
\(37\) −8.64575 −1.42135 −0.710676 − 0.703519i \(-0.751611\pi\)
−0.710676 + 0.703519i \(0.751611\pi\)
\(38\) −2.35425 −0.381910
\(39\) 0 0
\(40\) 24.8661 3.93168
\(41\) −5.90735 −0.922572 −0.461286 − 0.887251i \(-0.652612\pi\)
−0.461286 + 0.887251i \(0.652612\pi\)
\(42\) 0 0
\(43\) 8.64704 1.31866 0.659330 − 0.751853i \(-0.270840\pi\)
0.659330 + 0.751853i \(0.270840\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 17.1323 2.52602
\(47\) 1.70850 0.249210 0.124605 − 0.992206i \(-0.460234\pi\)
0.124605 + 0.992206i \(0.460234\pi\)
\(48\) 0 0
\(49\) −0.354249 −0.0506070
\(50\) 21.3750 3.02288
\(51\) 0 0
\(52\) −19.7103 −2.73332
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 17.5830 2.34963
\(57\) 0 0
\(58\) −6.64575 −0.872630
\(59\) −7.93725 −1.03334 −0.516671 − 0.856184i \(-0.672829\pi\)
−0.516671 + 0.856184i \(0.672829\pi\)
\(60\) 0 0
\(61\) −9.39851 −1.20336 −0.601678 − 0.798739i \(-0.705501\pi\)
−0.601678 + 0.798739i \(0.705501\pi\)
\(62\) 4.24264 0.538816
\(63\) 0 0
\(64\) 3.35425 0.419281
\(65\) −15.4676 −1.91852
\(66\) 0 0
\(67\) 1.64575 0.201061 0.100530 − 0.994934i \(-0.467946\pi\)
0.100530 + 0.994934i \(0.467946\pi\)
\(68\) −31.6867 −3.84258
\(69\) 0 0
\(70\) 24.2288 2.89589
\(71\) 1.29150 0.153273 0.0766366 − 0.997059i \(-0.475582\pi\)
0.0766366 + 0.997059i \(0.475582\pi\)
\(72\) 0 0
\(73\) −3.32941 −0.389678 −0.194839 − 0.980835i \(-0.562418\pi\)
−0.194839 + 0.980835i \(0.562418\pi\)
\(74\) −22.2882 −2.59095
\(75\) 0 0
\(76\) −4.24264 −0.486664
\(77\) 0 0
\(78\) 0 0
\(79\) 11.9764 1.34746 0.673728 − 0.738980i \(-0.264692\pi\)
0.673728 + 0.738980i \(0.264692\pi\)
\(80\) 30.2288 3.37968
\(81\) 0 0
\(82\) −15.2288 −1.68173
\(83\) −6.06910 −0.666170 −0.333085 − 0.942897i \(-0.608090\pi\)
−0.333085 + 0.942897i \(0.608090\pi\)
\(84\) 0 0
\(85\) −24.8661 −2.69711
\(86\) 22.2915 2.40375
\(87\) 0 0
\(88\) 0 0
\(89\) −8.58301 −0.909797 −0.454898 − 0.890543i \(-0.650324\pi\)
−0.454898 + 0.890543i \(0.650324\pi\)
\(90\) 0 0
\(91\) −10.9373 −1.14654
\(92\) 30.8745 3.21889
\(93\) 0 0
\(94\) 4.40440 0.454279
\(95\) −3.32941 −0.341590
\(96\) 0 0
\(97\) 9.93725 1.00898 0.504488 − 0.863419i \(-0.331681\pi\)
0.504488 + 0.863419i \(0.331681\pi\)
\(98\) −0.913230 −0.0922502
\(99\) 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.bb.1.4 yes 4
3.2 odd 2 3267.2.a.y.1.1 ✓ 4
11.10 odd 2 inner 3267.2.a.bb.1.1 yes 4
33.32 even 2 3267.2.a.y.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3267.2.a.y.1.1 ✓ 4 3.2 odd 2
3267.2.a.y.1.4 yes 4 33.32 even 2
3267.2.a.bb.1.1 yes 4 11.10 odd 2 inner
3267.2.a.bb.1.4 yes 4 1.1 even 1 trivial