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Properties

Label 704.2.g.a.351.1
Level $704$
Weight $2$
Character 704.351
Analytic conductor $5.621$
Analytic rank $0$
Dimension $4$
CM discriminant -88
Inner twists $8$

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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] = [704,2,Mod(351,704)] 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("704.351"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(704, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 704.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-12] 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: no
Analytic conductor: \(5.62146830230\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 351.1
Root \(1.65831 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 704.351
Dual form 704.2.g.a.351.3

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{9} -3.31662i q^{11} -6.63325 q^{13} -6.63325i q^{19} -2.00000i q^{23} +5.00000 q^{25} -6.63325 q^{29} +6.00000i q^{31} -6.63325i q^{43} -10.0000i q^{47} -7.00000 q^{49} -6.63325 q^{61} +14.0000i q^{71} +9.00000 q^{81} -6.63325i q^{83} +2.00000 q^{89} -6.00000 q^{97} +9.94987i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + 20 q^{25} - 28 q^{49} + 36 q^{81} + 8 q^{89} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/704\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(321\) \(639\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) − 3.31662i − 1.00000i
\(12\) 0 0
\(13\) −6.63325 −1.83973 −0.919866 − 0.392232i \(-0.871703\pi\)
−0.919866 + 0.392232i \(0.871703\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) − 6.63325i − 1.52177i −0.648886 − 0.760886i \(-0.724765\pi\)
0.648886 − 0.760886i \(-0.275235\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 2.00000i − 0.417029i −0.978019 − 0.208514i \(-0.933137\pi\)
0.978019 − 0.208514i \(-0.0668628\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.63325 −1.23176 −0.615882 − 0.787839i \(-0.711200\pi\)
−0.615882 + 0.787839i \(0.711200\pi\)
\(30\) 0 0
\(31\) 6.00000i 1.07763i 0.842424 + 0.538816i \(0.181128\pi\)
−0.842424 + 0.538816i \(0.818872\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) − 6.63325i − 1.01156i −0.862662 − 0.505781i \(-0.831205\pi\)
0.862662 − 0.505781i \(-0.168795\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 10.0000i − 1.45865i −0.684167 − 0.729325i \(-0.739834\pi\)
0.684167 − 0.729325i \(-0.260166\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −6.63325 −0.849301 −0.424650 − 0.905357i \(-0.639603\pi\)
−0.424650 + 0.905357i \(0.639603\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.0000i 1.66149i 0.556650 + 0.830747i \(0.312086\pi\)
−0.556650 + 0.830747i \(0.687914\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) − 6.63325i − 0.728094i −0.931381 − 0.364047i \(-0.881395\pi\)
0.931381 − 0.364047i \(-0.118605\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 9.94987i 1.00000i
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 704.2.g.a.351.1 ✓ 4
4.3 odd 2 inner 704.2.g.a.351.3 yes 4
8.3 odd 2 inner 704.2.g.a.351.2 yes 4
8.5 even 2 inner 704.2.g.a.351.4 yes 4
11.10 odd 2 inner 704.2.g.a.351.4 yes 4
16.3 odd 4 2816.2.e.e.2815.3 4
16.5 even 4 2816.2.e.e.2815.4 4
16.11 odd 4 2816.2.e.e.2815.2 4
16.13 even 4 2816.2.e.e.2815.1 4
44.43 even 2 inner 704.2.g.a.351.2 yes 4
88.21 odd 2 CM 704.2.g.a.351.1 ✓ 4
88.43 even 2 inner 704.2.g.a.351.3 yes 4
176.21 odd 4 2816.2.e.e.2815.1 4
176.43 even 4 2816.2.e.e.2815.3 4
176.109 odd 4 2816.2.e.e.2815.4 4
176.131 even 4 2816.2.e.e.2815.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
704.2.g.a.351.1 ✓ 4 1.1 even 1 trivial
704.2.g.a.351.1 ✓ 4 88.21 odd 2 CM
704.2.g.a.351.2 yes 4 8.3 odd 2 inner
704.2.g.a.351.2 yes 4 44.43 even 2 inner
704.2.g.a.351.3 yes 4 4.3 odd 2 inner
704.2.g.a.351.3 yes 4 88.43 even 2 inner
704.2.g.a.351.4 yes 4 8.5 even 2 inner
704.2.g.a.351.4 yes 4 11.10 odd 2 inner
2816.2.e.e.2815.1 4 16.13 even 4
2816.2.e.e.2815.1 4 176.21 odd 4
2816.2.e.e.2815.2 4 16.11 odd 4
2816.2.e.e.2815.2 4 176.131 even 4
2816.2.e.e.2815.3 4 16.3 odd 4
2816.2.e.e.2815.3 4 176.43 even 4
2816.2.e.e.2815.4 4 16.5 even 4
2816.2.e.e.2815.4 4 176.109 odd 4