Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2320,2,Mod(929,2320)] 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("2320.929"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2320.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,4,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.5252932689\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 145)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 929.1
Root \(0.517638i\) of defining polynomial
Character \(\chi\) \(=\) 2320.929
Dual form 2320.2.d.f.929.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421i q^{3} +(-1.73205 + 1.41421i) q^{5} -2.44949i q^{7} +1.00000 q^{9} +1.26795 q^{11} -1.79315i q^{13} +(2.00000 + 2.44949i) q^{15} -1.41421i q^{17} -3.26795 q^{19} -3.46410 q^{21} +6.31319i q^{23} +(1.00000 - 4.89898i) q^{25} -5.65685i q^{27} -1.00000 q^{29} +8.73205 q^{31} -1.79315i q^{33} +(3.46410 + 4.24264i) q^{35} -9.14162i q^{37} -2.53590 q^{39} -6.92820 q^{41} -9.14162i q^{43} +(-1.73205 + 1.41421i) q^{45} +1.41421i q^{47} +1.00000 q^{49} -2.00000 q^{51} +5.93426i q^{53} +(-2.19615 + 1.79315i) q^{55} +4.62158i q^{57} -10.3923 q^{59} +2.92820 q^{61} -2.44949i q^{63} +(2.53590 + 3.10583i) q^{65} +4.24264i q^{67} +8.92820 q^{69} -3.46410 q^{71} -7.34847i q^{73} +(-6.92820 - 1.41421i) q^{75} -3.10583i q^{77} -4.19615 q^{79} -5.00000 q^{81} -10.1769i q^{83} +(2.00000 + 2.44949i) q^{85} +1.41421i q^{87} -10.3923 q^{89} -4.39230 q^{91} -12.3490i q^{93} +(5.66025 - 4.62158i) q^{95} +10.9348i q^{97} +1.26795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9} + 12 q^{11} + 8 q^{15} - 20 q^{19} + 4 q^{25} - 4 q^{29} + 28 q^{31} - 24 q^{39} + 4 q^{49} - 8 q^{51} + 12 q^{55} - 16 q^{61} + 24 q^{65} + 8 q^{69} + 4 q^{79} - 20 q^{81} + 8 q^{85} + 24 q^{91}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(581\) \(1857\) \(2031\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) − 1.41421i − 0.816497i −0.912871 − 0.408248i \(-0.866140\pi\)
0.912871 − 0.408248i \(-0.133860\pi\)
\(4\) 0 0
\(5\) −1.73205 + 1.41421i −0.774597 + 0.632456i
\(6\) 0 0
\(7\) − 2.44949i − 0.925820i −0.886405 − 0.462910i \(-0.846805\pi\)
0.886405 − 0.462910i \(-0.153195\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.26795 0.382301 0.191151 − 0.981561i \(-0.438778\pi\)
0.191151 + 0.981561i \(0.438778\pi\)
\(12\) 0 0
\(13\) − 1.79315i − 0.497331i −0.968589 − 0.248665i \(-0.920008\pi\)
0.968589 − 0.248665i \(-0.0799919\pi\)
\(14\) 0 0
\(15\) 2.00000 + 2.44949i 0.516398 + 0.632456i
\(16\) 0 0
\(17\) − 1.41421i − 0.342997i −0.985184 − 0.171499i \(-0.945139\pi\)
0.985184 − 0.171499i \(-0.0548609\pi\)
\(18\) 0 0
\(19\) −3.26795 −0.749719 −0.374859 − 0.927082i \(-0.622309\pi\)
−0.374859 + 0.927082i \(0.622309\pi\)
\(20\) 0 0
\(21\) −3.46410 −0.755929
\(22\) 0 0
\(23\) 6.31319i 1.31639i 0.752847 + 0.658196i \(0.228680\pi\)
−0.752847 + 0.658196i \(0.771320\pi\)
\(24\) 0 0
\(25\) 1.00000 − 4.89898i 0.200000 − 0.979796i
\(26\) 0 0
\(27\) − 5.65685i − 1.08866i
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 8.73205 1.56832 0.784161 − 0.620557i \(-0.213093\pi\)
0.784161 + 0.620557i \(0.213093\pi\)
\(32\) 0 0
\(33\) − 1.79315i − 0.312148i
\(34\) 0 0
\(35\) 3.46410 + 4.24264i 0.585540 + 0.717137i
\(36\) 0 0
\(37\) − 9.14162i − 1.50287i −0.659805 − 0.751437i \(-0.729361\pi\)
0.659805 − 0.751437i \(-0.270639\pi\)
\(38\) 0 0
\(39\) −2.53590 −0.406069
\(40\) 0 0
\(41\) −6.92820 −1.08200 −0.541002 − 0.841021i \(-0.681955\pi\)
−0.541002 + 0.841021i \(0.681955\pi\)
\(42\) 0 0
\(43\) − 9.14162i − 1.39408i −0.717030 − 0.697042i \(-0.754499\pi\)
0.717030 − 0.697042i \(-0.245501\pi\)
\(44\) 0 0
\(45\) −1.73205 + 1.41421i −0.258199 + 0.210819i
\(46\) 0 0
\(47\) 1.41421i 0.206284i 0.994667 + 0.103142i \(0.0328896\pi\)
−0.994667 + 0.103142i \(0.967110\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 5.93426i 0.815133i 0.913176 + 0.407566i \(0.133622\pi\)
−0.913176 + 0.407566i \(0.866378\pi\)
\(54\) 0 0
\(55\) −2.19615 + 1.79315i −0.296129 + 0.241788i
\(56\) 0 0
\(57\) 4.62158i 0.612143i
\(58\) 0 0
\(59\) −10.3923 −1.35296 −0.676481 − 0.736460i \(-0.736496\pi\)
−0.676481 + 0.736460i \(0.736496\pi\)
\(60\) 0 0
\(61\) 2.92820 0.374918 0.187459 − 0.982272i \(-0.439975\pi\)
0.187459 + 0.982272i \(0.439975\pi\)
\(62\) 0 0
\(63\) − 2.44949i − 0.308607i
\(64\) 0 0
\(65\) 2.53590 + 3.10583i 0.314539 + 0.385231i
\(66\) 0 0
\(67\) 4.24264i 0.518321i 0.965834 + 0.259161i \(0.0834459\pi\)
−0.965834 + 0.259161i \(0.916554\pi\)
\(68\) 0 0
\(69\) 8.92820 1.07483
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 − 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 0 0
\(73\) − 7.34847i − 0.860073i −0.902811 − 0.430037i \(-0.858501\pi\)
0.902811 − 0.430037i \(-0.141499\pi\)
\(74\) 0 0
\(75\) −6.92820 − 1.41421i −0.800000 − 0.163299i
\(76\) 0 0
\(77\) − 3.10583i − 0.353942i
\(78\) 0 0
\(79\) −4.19615 −0.472104 −0.236052 − 0.971740i \(-0.575854\pi\)
−0.236052 + 0.971740i \(0.575854\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) − 10.1769i − 1.11706i −0.829484 − 0.558530i \(-0.811366\pi\)
0.829484 − 0.558530i \(-0.188634\pi\)
\(84\) 0 0
\(85\) 2.00000 + 2.44949i 0.216930 + 0.265684i
\(86\) 0 0
\(87\) 1.41421i 0.151620i
\(88\) 0 0
\(89\) −10.3923 −1.10158 −0.550791 − 0.834643i \(-0.685674\pi\)
−0.550791 + 0.834643i \(0.685674\pi\)
\(90\) 0 0
\(91\) −4.39230 −0.460439
\(92\) 0 0
\(93\) − 12.3490i − 1.28053i
\(94\) 0 0
\(95\) 5.66025 − 4.62158i 0.580730 − 0.474164i
\(96\) 0 0
\(97\) 10.9348i 1.11026i 0.831764 + 0.555129i \(0.187331\pi\)
−0.831764 + 0.555129i \(0.812669\pi\)
\(98\) 0 0
\(99\) 1.26795 0.127434
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 2320.2.d.f.929.1 4
4.3 odd 2 145.2.b.b.59.2 ✓ 4
5.4 even 2 inner 2320.2.d.f.929.3 4
12.11 even 2 1305.2.c.f.784.3 4
20.3 even 4 725.2.a.f.1.2 4
20.7 even 4 725.2.a.f.1.3 4
20.19 odd 2 145.2.b.b.59.3 yes 4
60.23 odd 4 6525.2.a.bj.1.3 4
60.47 odd 4 6525.2.a.bj.1.2 4
60.59 even 2 1305.2.c.f.784.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.b.b.59.2 ✓ 4 4.3 odd 2
145.2.b.b.59.3 yes 4 20.19 odd 2
725.2.a.f.1.2 4 20.3 even 4
725.2.a.f.1.3 4 20.7 even 4
1305.2.c.f.784.2 4 60.59 even 2
1305.2.c.f.784.3 4 12.11 even 2
2320.2.d.f.929.1 4 1.1 even 1 trivial
2320.2.d.f.929.3 4 5.4 even 2 inner
6525.2.a.bj.1.2 4 60.47 odd 4
6525.2.a.bj.1.3 4 60.23 odd 4