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Properties

Label 3267.2.a.ba.1.3
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 = [4,0,0,8,0,0,4,0,0,4,0,0,4,0,0,-4,0,0,24,0,0,0,0,0,20,0,0,-16, 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: \(\Q(\sqrt{4 + \sqrt{3}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 8x^{2} + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.50597\) 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)
 
\(f(q)\) \(=\) \(q+1.50597 q^{2} +0.267949 q^{4} +4.11439 q^{5} +4.46410 q^{7} -2.60842 q^{8} +6.19615 q^{10} -2.46410 q^{13} +6.72281 q^{14} -4.46410 q^{16} -1.10245 q^{17} +6.00000 q^{19} +1.10245 q^{20} +1.10245 q^{23} +11.9282 q^{25} -3.71087 q^{26} +1.19615 q^{28} -5.21684 q^{29} -2.46410 q^{31} -1.50597 q^{32} -1.66025 q^{34} +18.3671 q^{35} +2.53590 q^{37} +9.03583 q^{38} -10.7321 q^{40} +4.11439 q^{41} +3.46410 q^{43} +1.66025 q^{46} +10.1383 q^{47} +12.9282 q^{49} +17.9635 q^{50} -0.660254 q^{52} -14.2527 q^{53} -11.6442 q^{56} -7.85641 q^{58} +10.1383 q^{59} -9.46410 q^{61} -3.71087 q^{62} +6.66025 q^{64} -10.1383 q^{65} +1.53590 q^{67} -0.295400 q^{68} +27.6603 q^{70} -5.21684 q^{71} -9.39230 q^{73} +3.81899 q^{74} +1.60770 q^{76} -7.92820 q^{79} -18.3671 q^{80} +6.19615 q^{82} -9.03583 q^{83} -4.53590 q^{85} +5.21684 q^{86} +5.21684 q^{89} -11.0000 q^{91} +0.295400 q^{92} +15.2679 q^{94} +24.6863 q^{95} -11.3923 q^{97} +19.4695 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} + 4 q^{7} + 4 q^{10} + 4 q^{13} - 4 q^{16} + 24 q^{19} + 20 q^{25} - 16 q^{28} + 4 q^{31} + 28 q^{34} + 24 q^{37} - 36 q^{40} - 28 q^{46} + 24 q^{49} + 32 q^{52} + 24 q^{58} - 24 q^{61} - 8 q^{64}+ \cdots - 4 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.50597 1.06488 0.532441 − 0.846467i \(-0.321275\pi\)
0.532441 + 0.846467i \(0.321275\pi\)
\(3\) 0 0
\(4\) 0.267949 0.133975
\(5\) 4.11439 1.84001 0.920006 − 0.391905i \(-0.128184\pi\)
0.920006 + 0.391905i \(0.128184\pi\)
\(6\) 0 0
\(7\) 4.46410 1.68727 0.843636 − 0.536916i \(-0.180411\pi\)
0.843636 + 0.536916i \(0.180411\pi\)
\(8\) −2.60842 −0.922215
\(9\) 0 0
\(10\) 6.19615 1.95940
\(11\) 0 0
\(12\) 0 0
\(13\) −2.46410 −0.683419 −0.341709 − 0.939806i \(-0.611006\pi\)
−0.341709 + 0.939806i \(0.611006\pi\)
\(14\) 6.72281 1.79675
\(15\) 0 0
\(16\) −4.46410 −1.11603
\(17\) −1.10245 −0.267383 −0.133691 − 0.991023i \(-0.542683\pi\)
−0.133691 + 0.991023i \(0.542683\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 1.10245 0.246515
\(21\) 0 0
\(22\) 0 0
\(23\) 1.10245 0.229876 0.114938 − 0.993373i \(-0.463333\pi\)
0.114938 + 0.993373i \(0.463333\pi\)
\(24\) 0 0
\(25\) 11.9282 2.38564
\(26\) −3.71087 −0.727761
\(27\) 0 0
\(28\) 1.19615 0.226052
\(29\) −5.21684 −0.968742 −0.484371 − 0.874863i \(-0.660952\pi\)
−0.484371 + 0.874863i \(0.660952\pi\)
\(30\) 0 0
\(31\) −2.46410 −0.442566 −0.221283 − 0.975210i \(-0.571024\pi\)
−0.221283 + 0.975210i \(0.571024\pi\)
\(32\) −1.50597 −0.266221
\(33\) 0 0
\(34\) −1.66025 −0.284731
\(35\) 18.3671 3.10460
\(36\) 0 0
\(37\) 2.53590 0.416899 0.208450 − 0.978033i \(-0.433158\pi\)
0.208450 + 0.978033i \(0.433158\pi\)
\(38\) 9.03583 1.46580
\(39\) 0 0
\(40\) −10.7321 −1.69689
\(41\) 4.11439 0.642560 0.321280 − 0.946984i \(-0.395887\pi\)
0.321280 + 0.946984i \(0.395887\pi\)
\(42\) 0 0
\(43\) 3.46410 0.528271 0.264135 − 0.964486i \(-0.414913\pi\)
0.264135 + 0.964486i \(0.414913\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.66025 0.244791
\(47\) 10.1383 1.47882 0.739410 − 0.673256i \(-0.235105\pi\)
0.739410 + 0.673256i \(0.235105\pi\)
\(48\) 0 0
\(49\) 12.9282 1.84689
\(50\) 17.9635 2.54043
\(51\) 0 0
\(52\) −0.660254 −0.0915608
\(53\) −14.2527 −1.95775 −0.978877 − 0.204450i \(-0.934459\pi\)
−0.978877 + 0.204450i \(0.934459\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −11.6442 −1.55603
\(57\) 0 0
\(58\) −7.85641 −1.03160
\(59\) 10.1383 1.31989 0.659945 − 0.751314i \(-0.270579\pi\)
0.659945 + 0.751314i \(0.270579\pi\)
\(60\) 0 0
\(61\) −9.46410 −1.21175 −0.605877 − 0.795558i \(-0.707178\pi\)
−0.605877 + 0.795558i \(0.707178\pi\)
\(62\) −3.71087 −0.471280
\(63\) 0 0
\(64\) 6.66025 0.832532
\(65\) −10.1383 −1.25750
\(66\) 0 0
\(67\) 1.53590 0.187640 0.0938199 − 0.995589i \(-0.470092\pi\)
0.0938199 + 0.995589i \(0.470092\pi\)
\(68\) −0.295400 −0.0358225
\(69\) 0 0
\(70\) 27.6603 3.30603
\(71\) −5.21684 −0.619125 −0.309562 − 0.950879i \(-0.600183\pi\)
−0.309562 + 0.950879i \(0.600183\pi\)
\(72\) 0 0
\(73\) −9.39230 −1.09929 −0.549643 − 0.835400i \(-0.685236\pi\)
−0.549643 + 0.835400i \(0.685236\pi\)
\(74\) 3.81899 0.443949
\(75\) 0 0
\(76\) 1.60770 0.184415
\(77\) 0 0
\(78\) 0 0
\(79\) −7.92820 −0.891993 −0.445996 − 0.895035i \(-0.647151\pi\)
−0.445996 + 0.895035i \(0.647151\pi\)
\(80\) −18.3671 −2.05350
\(81\) 0 0
\(82\) 6.19615 0.684251
\(83\) −9.03583 −0.991811 −0.495905 − 0.868377i \(-0.665164\pi\)
−0.495905 + 0.868377i \(0.665164\pi\)
\(84\) 0 0
\(85\) −4.53590 −0.491987
\(86\) 5.21684 0.562546
\(87\) 0 0
\(88\) 0 0
\(89\) 5.21684 0.552984 0.276492 − 0.961016i \(-0.410828\pi\)
0.276492 + 0.961016i \(0.410828\pi\)
\(90\) 0 0
\(91\) −11.0000 −1.15311
\(92\) 0.295400 0.0307976
\(93\) 0 0
\(94\) 15.2679 1.57477
\(95\) 24.6863 2.53276
\(96\) 0 0
\(97\) −11.3923 −1.15671 −0.578357 − 0.815784i \(-0.696306\pi\)
−0.578357 + 0.815784i \(0.696306\pi\)
\(98\) 19.4695 1.96672
\(99\) 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.ba.1.3 yes 4
3.2 odd 2 inner 3267.2.a.ba.1.2 yes 4
11.10 odd 2 3267.2.a.z.1.2 ✓ 4
33.32 even 2 3267.2.a.z.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3267.2.a.z.1.2 ✓ 4 11.10 odd 2
3267.2.a.z.1.3 yes 4 33.32 even 2
3267.2.a.ba.1.2 yes 4 3.2 odd 2 inner
3267.2.a.ba.1.3 yes 4 1.1 even 1 trivial