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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1520,1,Mod(189,1520)] 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("1520.189"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1520, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 2, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1520.bh (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0] 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: \(0.758578819202\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.972800.1

Embedding invariants

Embedding label 189.2
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1520.189
Dual form 1520.1.bh.a.949.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.707107 - 0.707107i) q^{2} +(-0.707107 - 0.707107i) q^{3} -1.00000i q^{4} -1.00000i q^{5} -1.00000 q^{6} -1.00000 q^{7} +(-0.707107 - 0.707107i) q^{8} +(-0.707107 - 0.707107i) q^{10} +(-0.707107 + 0.707107i) q^{12} +(0.707107 + 0.707107i) q^{13} +(-0.707107 + 0.707107i) q^{14} +(-0.707107 + 0.707107i) q^{15} -1.00000 q^{16} -1.00000i q^{17} +(-0.707107 + 0.707107i) q^{19} -1.00000 q^{20} +(0.707107 + 0.707107i) q^{21} +1.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{26} +(-0.707107 + 0.707107i) q^{27} +1.00000i q^{28} +(0.707107 - 0.707107i) q^{29} +1.00000i q^{30} +(-0.707107 + 0.707107i) q^{32} +(-0.707107 - 0.707107i) q^{34} +1.00000i q^{35} +1.00000i q^{38} -1.00000i q^{39} +(-0.707107 + 0.707107i) q^{40} +1.00000 q^{42} +(1.00000 + 1.00000i) q^{43} +(0.707107 - 0.707107i) q^{46} -2.00000i q^{47} +(0.707107 + 0.707107i) q^{48} +(-0.707107 + 0.707107i) q^{50} +(-0.707107 + 0.707107i) q^{51} +(0.707107 - 0.707107i) q^{52} +(-0.707107 + 0.707107i) q^{53} +1.00000i q^{54} +(0.707107 + 0.707107i) q^{56} +1.00000 q^{57} -1.00000i q^{58} +(0.707107 + 0.707107i) q^{59} +(0.707107 + 0.707107i) q^{60} +(-1.00000 - 1.00000i) q^{61} +1.00000i q^{64} +(0.707107 - 0.707107i) q^{65} +(-0.707107 - 0.707107i) q^{67} -1.00000 q^{68} +(-0.707107 - 0.707107i) q^{69} +(0.707107 + 0.707107i) q^{70} +1.00000 q^{73} +(0.707107 + 0.707107i) q^{75} +(0.707107 + 0.707107i) q^{76} +(-0.707107 - 0.707107i) q^{78} -1.41421i q^{79} +1.00000i q^{80} +1.00000 q^{81} +(0.707107 - 0.707107i) q^{84} -1.00000 q^{85} +1.41421 q^{86} -1.00000 q^{87} -1.41421 q^{89} +(-0.707107 - 0.707107i) q^{91} -1.00000i q^{92} +(-1.41421 - 1.41421i) q^{94} +(0.707107 + 0.707107i) q^{95} +1.00000 q^{96} +1.41421 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{6} - 4 q^{7} - 4 q^{16} - 4 q^{20} + 4 q^{23} - 4 q^{25} + 4 q^{26} + 4 q^{42} + 4 q^{43} + 4 q^{57} - 4 q^{61} - 4 q^{68} + 4 q^{73} + 4 q^{81} - 4 q^{85} - 4 q^{87} + 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(3\) −0.707107 − 0.707107i −0.707107 − 0.707107i 0.258819 − 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(4\) − 1.00000i − 1.00000i
\(5\) − 1.00000i − 1.00000i
\(6\) −1.00000 −1.00000
\(7\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(9\) 0 0
\(10\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(11\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(12\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(13\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 − 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(14\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(15\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(16\) −1.00000 −1.00000
\(17\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(18\) 0 0
\(19\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(20\) −1.00000 −1.00000
\(21\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(22\) 0 0
\(23\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 1.00000i 1.00000i
\(25\) −1.00000 −1.00000
\(26\) 1.00000 1.00000
\(27\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(28\) 1.00000i 1.00000i
\(29\) 0.707107 − 0.707107i 0.707107 − 0.707107i −0.258819 − 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(30\) 1.00000i 1.00000i
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(33\) 0 0
\(34\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(35\) 1.00000i 1.00000i
\(36\) 0 0
\(37\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 1.00000i 1.00000i
\(39\) − 1.00000i − 1.00000i
\(40\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 1.00000 1.00000
\(43\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(47\) − 2.00000i − 2.00000i − 1.00000i \(-0.5\pi\)
− 1.00000i \(-0.5\pi\)
\(48\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(49\) 0 0
\(50\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(51\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(52\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(53\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 − 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(54\) 1.00000i 1.00000i
\(55\) 0 0
\(56\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(57\) 1.00000 1.00000
\(58\) − 1.00000i − 1.00000i
\(59\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 − 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(60\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(61\) −1.00000 − 1.00000i −1.00000 − 1.00000i − 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000i 1.00000i
\(65\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(66\) 0 0
\(67\) −0.707107 − 0.707107i −0.707107 − 0.707107i 0.258819 − 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(68\) −1.00000 −1.00000
\(69\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(70\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(74\) 0 0
\(75\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(76\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(77\) 0 0
\(78\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(79\) − 1.41421i − 1.41421i −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 − 0.707107i \(-0.250000\pi\)
\(80\) 1.00000i 1.00000i
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(85\) −1.00000 −1.00000
\(86\) 1.41421 1.41421
\(87\) −1.00000 −1.00000
\(88\) 0 0
\(89\) −1.41421 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(90\) 0 0
\(91\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(92\) − 1.00000i − 1.00000i
\(93\) 0 0
\(94\) −1.41421 − 1.41421i −1.41421 − 1.41421i
\(95\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(96\) 1.00000 1.00000
\(97\) 1.41421 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(98\) 0 0
\(99\) 0 0
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 1520.1.bh.a.189.2 yes 4
5.4 even 2 1520.1.bh.b.189.1 yes 4
16.5 even 4 1520.1.bh.b.949.2 yes 4
19.18 odd 2 inner 1520.1.bh.a.189.1 ✓ 4
80.69 even 4 inner 1520.1.bh.a.949.1 yes 4
95.94 odd 2 1520.1.bh.b.189.2 yes 4
304.37 odd 4 1520.1.bh.b.949.1 yes 4
1520.949 odd 4 inner 1520.1.bh.a.949.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1520.1.bh.a.189.1 ✓ 4 19.18 odd 2 inner
1520.1.bh.a.189.2 yes 4 1.1 even 1 trivial
1520.1.bh.a.949.1 yes 4 80.69 even 4 inner
1520.1.bh.a.949.2 yes 4 1520.949 odd 4 inner
1520.1.bh.b.189.1 yes 4 5.4 even 2
1520.1.bh.b.189.2 yes 4 95.94 odd 2
1520.1.bh.b.949.1 yes 4 304.37 odd 4
1520.1.bh.b.949.2 yes 4 16.5 even 4