↑

Properties

Label 3267.2.a.z.1.3
Level $3267$
Weight $2$
Character 3267.1
Self dual yes
Analytic conductor $26.087$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

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 \(-0.871816\pi\)
−0.920006 + 0.391905i \(0.871816\pi\)
\(6\) 0 0
\(7\) −4.46410 −1.68727 −0.843636 − 0.536916i \(-0.819589\pi\)
−0.843636 + 0.536916i \(0.819589\pi\)
\(8\) −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 \(-0.388994\pi\)
0.341709 + 0.939806i \(0.388994\pi\)
\(14\) −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 \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) −1.10245 −0.246515
\(21\) 0 0
\(22\) 0 0
\(23\) −1.10245 −0.229876 −0.114938 − 0.993373i \(-0.536667\pi\)
−0.114938 + 0.993373i \(0.536667\pi\)
\(24\) 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 \(-0.585087\pi\)
−0.264135 + 0.964486i \(0.585087\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.66025 −0.244791
\(47\) −10.1383 −1.47882 −0.739410 − 0.673256i \(-0.764895\pi\)
−0.739410 + 0.673256i \(0.764895\pi\)
\(48\) 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 \(-0.0655406\pi\)
0.978877 + 0.204450i \(0.0655406\pi\)
\(54\) 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 \(-0.729421\pi\)
−0.659945 + 0.751314i \(0.729421\pi\)
\(60\) 0 0
\(61\) 9.46410 1.21175 0.605877 − 0.795558i \(-0.292822\pi\)
0.605877 + 0.795558i \(0.292822\pi\)
\(62\) −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 \(-0.399817\pi\)
0.309562 + 0.950879i \(0.399817\pi\)
\(72\) 0 0
\(73\) 9.39230 1.09929 0.549643 − 0.835400i \(-0.314764\pi\)
0.549643 + 0.835400i \(0.314764\pi\)
\(74\) 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 \(-0.352849\pi\)
0.445996 + 0.895035i \(0.352849\pi\)
\(80\) 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 \(-0.589172\pi\)
−0.276492 + 0.961016i \(0.589172\pi\)
\(90\) 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.z.1.3 yes 4
3.2 odd 2 inner 3267.2.a.z.1.2 ✓ 4
11.10 odd 2 3267.2.a.ba.1.2 yes 4
33.32 even 2 3267.2.a.ba.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3267.2.a.z.1.2 ✓ 4 3.2 odd 2 inner
3267.2.a.z.1.3 yes 4 1.1 even 1 trivial
3267.2.a.ba.1.2 yes 4 11.10 odd 2
3267.2.a.ba.1.3 yes 4 33.32 even 2