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Properties

Label 704.2.e.c.703.4
Level $704$
Weight $2$
Character 704.703
Analytic conductor $5.621$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $4$

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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(703,704)] 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("704.703"); 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, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 704.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 176)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 703.4
Root \(-1.18614 - 1.26217i\) of defining polynomial
Character \(\chi\) \(=\) 704.703
Dual form 704.2.e.c.703.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.52434i q^{3} +4.37228 q^{5} -3.37228 q^{9} +3.31662i q^{11} +11.0371i q^{15} -9.45254i q^{23} +14.1168 q^{25} -0.939764i q^{27} +0.644810i q^{31} -8.37228 q^{33} -5.11684 q^{37} -14.7446 q^{45} +6.63325i q^{47} -7.00000 q^{49} -6.00000 q^{53} +14.5012i q^{55} -11.3321i q^{59} +6.28339i q^{67} +23.8614 q^{69} -5.69349i q^{71} +35.6357i q^{75} -7.74456 q^{81} -9.86141 q^{89} -1.62772 q^{93} +17.1168 q^{97} -11.1846i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{5} - 2 q^{9} + 22 q^{25} - 22 q^{33} + 14 q^{37} - 36 q^{45} - 28 q^{49} - 24 q^{53} + 38 q^{69} - 8 q^{81} + 18 q^{89} - 18 q^{93} + 34 q^{97}+O(q^{100}) \) 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
\(3\) 2.52434i 1.45743i 0.684819 + 0.728714i \(0.259881\pi\)
−0.684819 + 0.728714i \(0.740119\pi\)
\(4\) 0 0
\(5\) 4.37228 1.95534 0.977672 − 0.210138i \(-0.0673912\pi\)
0.977672 + 0.210138i \(0.0673912\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −3.37228 −1.12409
\(10\) 0 0
\(11\) 3.31662i 1.00000i
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 11.0371i 2.84977i
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 9.45254i − 1.97099i −0.169701 − 0.985496i \(-0.554280\pi\)
0.169701 − 0.985496i \(-0.445720\pi\)
\(24\) 0 0
\(25\) 14.1168 2.82337
\(26\) 0 0
\(27\) − 0.939764i − 0.180858i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0.644810i 0.115811i 0.998322 + 0.0579057i \(0.0184423\pi\)
−0.998322 + 0.0579057i \(0.981558\pi\)
\(32\) 0 0
\(33\) −8.37228 −1.45743
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.11684 −0.841204 −0.420602 − 0.907245i \(-0.638181\pi\)
−0.420602 + 0.907245i \(0.638181\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −14.7446 −2.19799
\(46\) 0 0
\(47\) 6.63325i 0.967559i 0.875190 + 0.483779i \(0.160736\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 14.5012i 1.95534i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 11.3321i − 1.47531i −0.675178 − 0.737655i \(-0.735933\pi\)
0.675178 − 0.737655i \(-0.264067\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.28339i 0.767639i 0.923408 + 0.383819i \(0.125391\pi\)
−0.923408 + 0.383819i \(0.874609\pi\)
\(68\) 0 0
\(69\) 23.8614 2.87258
\(70\) 0 0
\(71\) − 5.69349i − 0.675692i −0.941201 − 0.337846i \(-0.890302\pi\)
0.941201 − 0.337846i \(-0.109698\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 35.6357i 4.11485i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −7.74456 −0.860507
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.86141 −1.04531 −0.522654 − 0.852545i \(-0.675058\pi\)
−0.522654 + 0.852545i \(0.675058\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.62772 −0.168787
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 17.1168 1.73795 0.868976 − 0.494854i \(-0.164778\pi\)
0.868976 + 0.494854i \(0.164778\pi\)
\(98\) 0 0
\(99\) − 11.1846i − 1.12409i
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.e.c.703.4 4
4.3 odd 2 inner 704.2.e.c.703.1 4
8.3 odd 2 176.2.e.b.175.4 yes 4
8.5 even 2 176.2.e.b.175.1 ✓ 4
11.10 odd 2 CM 704.2.e.c.703.4 4
16.3 odd 4 2816.2.g.c.1407.7 8
16.5 even 4 2816.2.g.c.1407.8 8
16.11 odd 4 2816.2.g.c.1407.2 8
16.13 even 4 2816.2.g.c.1407.1 8
24.5 odd 2 1584.2.o.e.703.4 4
24.11 even 2 1584.2.o.e.703.3 4
44.43 even 2 inner 704.2.e.c.703.1 4
88.21 odd 2 176.2.e.b.175.1 ✓ 4
88.43 even 2 176.2.e.b.175.4 yes 4
176.21 odd 4 2816.2.g.c.1407.8 8
176.43 even 4 2816.2.g.c.1407.2 8
176.109 odd 4 2816.2.g.c.1407.1 8
176.131 even 4 2816.2.g.c.1407.7 8
264.131 odd 2 1584.2.o.e.703.3 4
264.197 even 2 1584.2.o.e.703.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
176.2.e.b.175.1 ✓ 4 8.5 even 2
176.2.e.b.175.1 ✓ 4 88.21 odd 2
176.2.e.b.175.4 yes 4 8.3 odd 2
176.2.e.b.175.4 yes 4 88.43 even 2
704.2.e.c.703.1 4 4.3 odd 2 inner
704.2.e.c.703.1 4 44.43 even 2 inner
704.2.e.c.703.4 4 1.1 even 1 trivial
704.2.e.c.703.4 4 11.10 odd 2 CM
1584.2.o.e.703.3 4 24.11 even 2
1584.2.o.e.703.3 4 264.131 odd 2
1584.2.o.e.703.4 4 24.5 odd 2
1584.2.o.e.703.4 4 264.197 even 2
2816.2.g.c.1407.1 8 16.13 even 4
2816.2.g.c.1407.1 8 176.109 odd 4
2816.2.g.c.1407.2 8 16.11 odd 4
2816.2.g.c.1407.2 8 176.43 even 4
2816.2.g.c.1407.7 8 16.3 odd 4
2816.2.g.c.1407.7 8 176.131 even 4
2816.2.g.c.1407.8 8 16.5 even 4
2816.2.g.c.1407.8 8 176.21 odd 4