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Properties

Label 3267.2.a.y.1.4
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 1.4
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 \(-0.803381\pi\)
−0.815215 + 0.579159i \(0.803381\pi\)
\(6\) 0 0
\(7\) −2.57794 −0.974368 −0.487184 − 0.873299i \(-0.661976\pi\)
−0.487184 + 0.873299i \(0.661976\pi\)
\(8\) 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 \(-0.299778\pi\)
0.588348 + 0.808608i \(0.299778\pi\)
\(14\) −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 \(-0.466594\pi\)
0.104755 + 0.994498i \(0.466594\pi\)
\(20\) −16.9373 −3.78729
\(21\) 0 0
\(22\) 0 0
\(23\) −6.64575 −1.38573 −0.692867 − 0.721065i \(-0.743653\pi\)
−0.692867 + 0.721065i \(0.743653\pi\)
\(24\) 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 \(-0.729160\pi\)
−0.659330 + 0.751853i \(0.729160\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −17.1323 −2.52602
\(47\) −1.70850 −0.249210 −0.124605 − 0.992206i \(-0.539766\pi\)
−0.124605 + 0.992206i \(0.539766\pi\)
\(48\) 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 \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 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 \(-0.327171\pi\)
0.516671 + 0.856184i \(0.327171\pi\)
\(60\) 0 0
\(61\) 9.39851 1.20336 0.601678 − 0.798739i \(-0.294499\pi\)
0.601678 + 0.798739i \(0.294499\pi\)
\(62\) 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 \(-0.524418\pi\)
−0.0766366 + 0.997059i \(0.524418\pi\)
\(72\) 0 0
\(73\) 3.32941 0.389678 0.194839 − 0.980835i \(-0.437582\pi\)
0.194839 + 0.980835i \(0.437582\pi\)
\(74\) −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 \(-0.735308\pi\)
−0.673728 + 0.738980i \(0.735308\pi\)
\(80\) −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 \(-0.349676\pi\)
0.454898 + 0.890543i \(0.349676\pi\)
\(90\) 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.y.1.4 yes 4
3.2 odd 2 3267.2.a.bb.1.1 yes 4
11.10 odd 2 inner 3267.2.a.y.1.1 ✓ 4
33.32 even 2 3267.2.a.bb.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3267.2.a.y.1.1 ✓ 4 11.10 odd 2 inner
3267.2.a.y.1.4 yes 4 1.1 even 1 trivial
3267.2.a.bb.1.1 yes 4 3.2 odd 2
3267.2.a.bb.1.4 yes 4 33.32 even 2