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Properties

Label 3328.2.a.bp.1.6
Level $3328$
Weight $2$
Character 3328.1
Self dual yes
Analytic conductor $26.574$
Analytic rank $0$
Dimension $6$
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] = [3328,2,Mod(1,3328)] 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("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3328 = 2^{8} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3328.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,8,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.5742137927\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.10323968.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} + 25x^{2} - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1664)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(2.47817\) of defining polynomial
Character \(\chi\) \(=\) 3328.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.47817 q^{3} +4.14134 q^{5} -1.96435 q^{7} +3.14134 q^{9} +3.34225 q^{11} +1.00000 q^{13} +10.2629 q^{15} -4.86799 q^{17} -7.27095 q^{19} -4.86799 q^{21} +4.95634 q^{23} +12.1507 q^{25} +0.350255 q^{27} +2.00000 q^{29} +4.44252 q^{31} +8.28267 q^{33} -8.13503 q^{35} +2.58532 q^{37} +2.47817 q^{39} +11.7360 q^{41} +0.350255 q^{43} +13.0093 q^{45} -4.79278 q^{47} -3.14134 q^{49} -12.0637 q^{51} +13.7360 q^{53} +13.8414 q^{55} -18.0187 q^{57} -2.64174 q^{59} -1.45331 q^{61} -6.17068 q^{63} +4.14134 q^{65} +5.54279 q^{67} +12.2827 q^{69} +0.936701 q^{71} +0.829359 q^{73} +30.1114 q^{75} -6.56534 q^{77} +11.6408 q^{79} -8.55602 q^{81} -8.99911 q^{83} -20.1600 q^{85} +4.95634 q^{87} -6.28267 q^{89} -1.96435 q^{91} +11.0093 q^{93} -30.1114 q^{95} +14.5653 q^{97} +10.4991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{5} + 2 q^{9} + 6 q^{13} - 4 q^{17} - 4 q^{21} + 14 q^{25} + 12 q^{29} + 16 q^{33} + 24 q^{37} + 20 q^{41} + 36 q^{45} - 2 q^{49} + 32 q^{53} - 24 q^{57} + 8 q^{61} + 8 q^{65} + 40 q^{69} - 12 q^{73}+ \cdots + 20 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.47817 1.43077 0.715387 − 0.698729i \(-0.246251\pi\)
0.715387 + 0.698729i \(0.246251\pi\)
\(4\) 0 0
\(5\) 4.14134 1.85206 0.926031 − 0.377448i \(-0.123198\pi\)
0.926031 + 0.377448i \(0.123198\pi\)
\(6\) 0 0
\(7\) −1.96435 −0.742454 −0.371227 − 0.928542i \(-0.621063\pi\)
−0.371227 + 0.928542i \(0.621063\pi\)
\(8\) 0 0
\(9\) 3.14134 1.04711
\(10\) 0 0
\(11\) 3.34225 1.00773 0.503863 − 0.863783i \(-0.331912\pi\)
0.503863 + 0.863783i \(0.331912\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 10.2629 2.64988
\(16\) 0 0
\(17\) −4.86799 −1.18066 −0.590331 − 0.807161i \(-0.701003\pi\)
−0.590331 + 0.807161i \(0.701003\pi\)
\(18\) 0 0
\(19\) −7.27095 −1.66807 −0.834035 − 0.551712i \(-0.813975\pi\)
−0.834035 + 0.551712i \(0.813975\pi\)
\(20\) 0 0
\(21\) −4.86799 −1.06228
\(22\) 0 0
\(23\) 4.95634 1.03347 0.516735 − 0.856146i \(-0.327147\pi\)
0.516735 + 0.856146i \(0.327147\pi\)
\(24\) 0 0
\(25\) 12.1507 2.43013
\(26\) 0 0
\(27\) 0.350255 0.0674066
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 4.44252 0.797900 0.398950 − 0.916973i \(-0.369375\pi\)
0.398950 + 0.916973i \(0.369375\pi\)
\(32\) 0 0
\(33\) 8.28267 1.44183
\(34\) 0 0
\(35\) −8.13503 −1.37507
\(36\) 0 0
\(37\) 2.58532 0.425024 0.212512 − 0.977158i \(-0.431836\pi\)
0.212512 + 0.977158i \(0.431836\pi\)
\(38\) 0 0
\(39\) 2.47817 0.396825
\(40\) 0 0
\(41\) 11.7360 1.83285 0.916426 − 0.400203i \(-0.131060\pi\)
0.916426 + 0.400203i \(0.131060\pi\)
\(42\) 0 0
\(43\) 0.350255 0.0534134 0.0267067 − 0.999643i \(-0.491498\pi\)
0.0267067 + 0.999643i \(0.491498\pi\)
\(44\) 0 0
\(45\) 13.0093 1.93932
\(46\) 0 0
\(47\) −4.79278 −0.699098 −0.349549 − 0.936918i \(-0.613665\pi\)
−0.349549 + 0.936918i \(0.613665\pi\)
\(48\) 0 0
\(49\) −3.14134 −0.448762
\(50\) 0 0
\(51\) −12.0637 −1.68926
\(52\) 0 0
\(53\) 13.7360 1.88678 0.943391 − 0.331682i \(-0.107616\pi\)
0.943391 + 0.331682i \(0.107616\pi\)
\(54\) 0 0
\(55\) 13.8414 1.86637
\(56\) 0 0
\(57\) −18.0187 −2.38663
\(58\) 0 0
\(59\) −2.64174 −0.343925 −0.171963 − 0.985103i \(-0.555011\pi\)
−0.171963 + 0.985103i \(0.555011\pi\)
\(60\) 0 0
\(61\) −1.45331 −0.186078 −0.0930388 − 0.995662i \(-0.529658\pi\)
−0.0930388 + 0.995662i \(0.529658\pi\)
\(62\) 0 0
\(63\) −6.17068 −0.777432
\(64\) 0 0
\(65\) 4.14134 0.513670
\(66\) 0 0
\(67\) 5.54279 0.677160 0.338580 − 0.940938i \(-0.390054\pi\)
0.338580 + 0.940938i \(0.390054\pi\)
\(68\) 0 0
\(69\) 12.2827 1.47866
\(70\) 0 0
\(71\) 0.936701 0.111166 0.0555830 − 0.998454i \(-0.482298\pi\)
0.0555830 + 0.998454i \(0.482298\pi\)
\(72\) 0 0
\(73\) 0.829359 0.0970692 0.0485346 − 0.998822i \(-0.484545\pi\)
0.0485346 + 0.998822i \(0.484545\pi\)
\(74\) 0 0
\(75\) 30.1114 3.47697
\(76\) 0 0
\(77\) −6.56534 −0.748190
\(78\) 0 0
\(79\) 11.6408 1.30970 0.654849 − 0.755760i \(-0.272732\pi\)
0.654849 + 0.755760i \(0.272732\pi\)
\(80\) 0 0
\(81\) −8.55602 −0.950668
\(82\) 0 0
\(83\) −8.99911 −0.987780 −0.493890 − 0.869524i \(-0.664425\pi\)
−0.493890 + 0.869524i \(0.664425\pi\)
\(84\) 0 0
\(85\) −20.1600 −2.18666
\(86\) 0 0
\(87\) 4.95634 0.531376
\(88\) 0 0
\(89\) −6.28267 −0.665962 −0.332981 − 0.942934i \(-0.608054\pi\)
−0.332981 + 0.942934i \(0.608054\pi\)
\(90\) 0 0
\(91\) −1.96435 −0.205920
\(92\) 0 0
\(93\) 11.0093 1.14161
\(94\) 0 0
\(95\) −30.1114 −3.08937
\(96\) 0 0
\(97\) 14.5653 1.47889 0.739443 − 0.673219i \(-0.235089\pi\)
0.739443 + 0.673219i \(0.235089\pi\)
\(98\) 0 0
\(99\) 10.4991 1.05520
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 3328.2.a.bp.1.6 6
4.3 odd 2 inner 3328.2.a.bp.1.1 6
8.3 odd 2 3328.2.a.bo.1.6 6
8.5 even 2 3328.2.a.bo.1.1 6
16.3 odd 4 1664.2.b.k.833.1 ✓ 12
16.5 even 4 1664.2.b.k.833.2 yes 12
16.11 odd 4 1664.2.b.k.833.12 yes 12
16.13 even 4 1664.2.b.k.833.11 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1664.2.b.k.833.1 ✓ 12 16.3 odd 4
1664.2.b.k.833.2 yes 12 16.5 even 4
1664.2.b.k.833.11 yes 12 16.13 even 4
1664.2.b.k.833.12 yes 12 16.11 odd 4
3328.2.a.bo.1.1 6 8.5 even 2
3328.2.a.bo.1.6 6 8.3 odd 2
3328.2.a.bp.1.1 6 4.3 odd 2 inner
3328.2.a.bp.1.6 6 1.1 even 1 trivial