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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1408,2,Mod(703,1408)] 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("1408.703"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1408, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1408 = 2^{7} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1408.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,20] 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: \(11.2429366046\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 703.4
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1408.703
Dual form 1408.2.g.c.703.3

$q$-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)
 
\(f(q)\) \(=\) \(q+2.82843 q^{3} +5.00000 q^{9} +(1.41421 + 3.00000i) q^{11} +5.65685i q^{17} +2.00000i q^{19} +5.00000 q^{25} +5.65685 q^{27} +(4.00000 + 8.48528i) q^{33} -11.3137i q^{41} -10.0000i q^{43} -7.00000 q^{49} +16.0000i q^{51} +5.65685i q^{57} +14.1421 q^{59} +8.48528 q^{67} -16.9706i q^{73} +14.1421 q^{75} +1.00000 q^{81} +18.0000i q^{83} -18.0000 q^{89} -10.0000 q^{97} +(7.07107 + 15.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 20 q^{9} + 20 q^{25} + 16 q^{33} - 28 q^{49} + 4 q^{81} - 72 q^{89} - 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1408\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(639\) \(1025\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) 2.82843 1.63299 0.816497 − 0.577350i \(-0.195913\pi\)
0.816497 + 0.577350i \(0.195913\pi\)
\(4\) 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
\(9\) 5.00000 1.66667
\(10\) 0 0
\(11\) 1.41421 + 3.00000i 0.426401 + 0.904534i
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.65685i 1.37199i 0.727607 + 0.685994i \(0.240633\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 2.00000i 0.458831i 0.973329 + 0.229416i \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 4.00000 + 8.48528i 0.696311 + 1.47710i
\(34\) 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
\(41\) − 11.3137i − 1.76690i −0.468521 − 0.883452i \(-0.655213\pi\)
0.468521 − 0.883452i \(-0.344787\pi\)
\(42\) 0 0
\(43\) − 10.0000i − 1.52499i −0.646997 − 0.762493i \(-0.723975\pi\)
0.646997 − 0.762493i \(-0.276025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 16.0000i 2.24045i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.65685i 0.749269i
\(58\) 0 0
\(59\) 14.1421 1.84115 0.920575 − 0.390567i \(-0.127721\pi\)
0.920575 + 0.390567i \(0.127721\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.48528 1.03664 0.518321 − 0.855186i \(-0.326557\pi\)
0.518321 + 0.855186i \(0.326557\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) − 16.9706i − 1.98625i −0.117041 − 0.993127i \(-0.537341\pi\)
0.117041 − 0.993127i \(-0.462659\pi\)
\(74\) 0 0
\(75\) 14.1421 1.63299
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 18.0000i 1.97576i 0.155230 + 0.987878i \(0.450388\pi\)
−0.155230 + 0.987878i \(0.549612\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −18.0000 −1.90800 −0.953998 − 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 7.07107 + 15.0000i 0.710669 + 1.50756i
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 1408.2.g.c.703.4 yes 4
4.3 odd 2 inner 1408.2.g.c.703.1 ✓ 4
8.3 odd 2 CM 1408.2.g.c.703.4 yes 4
8.5 even 2 inner 1408.2.g.c.703.1 ✓ 4
11.10 odd 2 inner 1408.2.g.c.703.3 yes 4
16.3 odd 4 2816.2.e.a.2815.1 2
16.5 even 4 2816.2.e.a.2815.1 2
16.11 odd 4 2816.2.e.b.2815.2 2
16.13 even 4 2816.2.e.b.2815.2 2
44.43 even 2 inner 1408.2.g.c.703.2 yes 4
88.21 odd 2 inner 1408.2.g.c.703.2 yes 4
88.43 even 2 inner 1408.2.g.c.703.3 yes 4
176.21 odd 4 2816.2.e.b.2815.1 2
176.43 even 4 2816.2.e.a.2815.2 2
176.109 odd 4 2816.2.e.a.2815.2 2
176.131 even 4 2816.2.e.b.2815.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1408.2.g.c.703.1 ✓ 4 4.3 odd 2 inner
1408.2.g.c.703.1 ✓ 4 8.5 even 2 inner
1408.2.g.c.703.2 yes 4 44.43 even 2 inner
1408.2.g.c.703.2 yes 4 88.21 odd 2 inner
1408.2.g.c.703.3 yes 4 11.10 odd 2 inner
1408.2.g.c.703.3 yes 4 88.43 even 2 inner
1408.2.g.c.703.4 yes 4 1.1 even 1 trivial
1408.2.g.c.703.4 yes 4 8.3 odd 2 CM
2816.2.e.a.2815.1 2 16.3 odd 4
2816.2.e.a.2815.1 2 16.5 even 4
2816.2.e.a.2815.2 2 176.43 even 4
2816.2.e.a.2815.2 2 176.109 odd 4
2816.2.e.b.2815.1 2 176.21 odd 4
2816.2.e.b.2815.1 2 176.131 even 4
2816.2.e.b.2815.2 2 16.11 odd 4
2816.2.e.b.2815.2 2 16.13 even 4