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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 = [2,0,0,0,-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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 176)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 703.2
Root \(0.500000 - 1.65831i\) of defining polynomial
Character \(\chi\) \(=\) 704.703
Dual form 704.2.e.a.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)
 
\(f(q)\) \(=\) \(q+3.31662i q^{3} -3.00000 q^{5} -8.00000 q^{9} -3.31662i q^{11} -9.94987i q^{15} -3.31662i q^{23} +4.00000 q^{25} -16.5831i q^{27} +9.94987i q^{31} +11.0000 q^{33} -7.00000 q^{37} +24.0000 q^{45} -6.63325i q^{47} -7.00000 q^{49} -6.00000 q^{53} +9.94987i q^{55} +3.31662i q^{59} -9.94987i q^{67} +11.0000 q^{69} -16.5831i q^{71} +13.2665i q^{75} +31.0000 q^{81} -9.00000 q^{89} -33.0000 q^{93} -17.0000 q^{97} +26.5330i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{5} - 16 q^{9} + 8 q^{25} + 22 q^{33} - 14 q^{37} + 48 q^{45} - 14 q^{49} - 12 q^{53} + 22 q^{69} + 62 q^{81} - 18 q^{89} - 66 q^{93} - 34 q^{97}+O(q^{100}) \) 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
\(3\) 3.31662i 1.91485i 0.288675 + 0.957427i \(0.406785\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) −3.00000 −1.34164 −0.670820 − 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −8.00000 −2.66667
\(10\) 0 0
\(11\) − 3.31662i − 1.00000i
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) − 9.94987i − 2.56905i
\(16\) 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
\(23\) − 3.31662i − 0.691564i −0.938315 − 0.345782i \(-0.887614\pi\)
0.938315 − 0.345782i \(-0.112386\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) − 16.5831i − 3.19142i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 9.94987i 1.78705i 0.449013 + 0.893525i \(0.351776\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) 11.0000 1.91485
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 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
\(45\) 24.0000 3.57771
\(46\) 0 0
\(47\) − 6.63325i − 0.967559i −0.875190 − 0.483779i \(-0.839264\pi\)
0.875190 − 0.483779i \(-0.160736\pi\)
\(48\) 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
\(55\) 9.94987i 1.34164i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.31662i 0.431788i 0.976417 + 0.215894i \(0.0692665\pi\)
−0.976417 + 0.215894i \(0.930733\pi\)
\(60\) 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
\(67\) − 9.94987i − 1.21557i −0.794101 − 0.607785i \(-0.792058\pi\)
0.794101 − 0.607785i \(-0.207942\pi\)
\(68\) 0 0
\(69\) 11.0000 1.32424
\(70\) 0 0
\(71\) − 16.5831i − 1.96805i −0.178017 − 0.984027i \(-0.556968\pi\)
0.178017 − 0.984027i \(-0.443032\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 13.2665i 1.53188i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 31.0000 3.44444
\(82\) 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
\(89\) −9.00000 −0.953998 −0.476999 − 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −33.0000 −3.42194
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −17.0000 −1.72609 −0.863044 − 0.505128i \(-0.831445\pi\)
−0.863044 + 0.505128i \(0.831445\pi\)
\(98\) 0 0
\(99\) 26.5330i 2.66667i
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.a.703.2 2
4.3 odd 2 inner 704.2.e.a.703.1 2
8.3 odd 2 176.2.e.a.175.2 yes 2
8.5 even 2 176.2.e.a.175.1 ✓ 2
11.10 odd 2 CM 704.2.e.a.703.2 2
16.3 odd 4 2816.2.g.a.1407.4 4
16.5 even 4 2816.2.g.a.1407.3 4
16.11 odd 4 2816.2.g.a.1407.1 4
16.13 even 4 2816.2.g.a.1407.2 4
24.5 odd 2 1584.2.o.a.703.1 2
24.11 even 2 1584.2.o.a.703.2 2
44.43 even 2 inner 704.2.e.a.703.1 2
88.21 odd 2 176.2.e.a.175.1 ✓ 2
88.43 even 2 176.2.e.a.175.2 yes 2
176.21 odd 4 2816.2.g.a.1407.3 4
176.43 even 4 2816.2.g.a.1407.1 4
176.109 odd 4 2816.2.g.a.1407.2 4
176.131 even 4 2816.2.g.a.1407.4 4
264.131 odd 2 1584.2.o.a.703.2 2
264.197 even 2 1584.2.o.a.703.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
176.2.e.a.175.1 ✓ 2 8.5 even 2
176.2.e.a.175.1 ✓ 2 88.21 odd 2
176.2.e.a.175.2 yes 2 8.3 odd 2
176.2.e.a.175.2 yes 2 88.43 even 2
704.2.e.a.703.1 2 4.3 odd 2 inner
704.2.e.a.703.1 2 44.43 even 2 inner
704.2.e.a.703.2 2 1.1 even 1 trivial
704.2.e.a.703.2 2 11.10 odd 2 CM
1584.2.o.a.703.1 2 24.5 odd 2
1584.2.o.a.703.1 2 264.197 even 2
1584.2.o.a.703.2 2 24.11 even 2
1584.2.o.a.703.2 2 264.131 odd 2
2816.2.g.a.1407.1 4 16.11 odd 4
2816.2.g.a.1407.1 4 176.43 even 4
2816.2.g.a.1407.2 4 16.13 even 4
2816.2.g.a.1407.2 4 176.109 odd 4
2816.2.g.a.1407.3 4 16.5 even 4
2816.2.g.a.1407.3 4 176.21 odd 4
2816.2.g.a.1407.4 4 16.3 odd 4
2816.2.g.a.1407.4 4 176.131 even 4