Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1040,2,Mod(961,1040)] 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("1040.961"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,6] 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: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.9144576.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 12x^{4} + 36x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 961.3
Root \(0.339877i\) of defining polynomial
Character \(\chi\) \(=\) 1040.961
Dual form 1040.2.k.c.961.4

$q$-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)
 
\(f(q)\) \(=\) \(q+0.339877 q^{3} -1.00000i q^{5} +3.88448i q^{7} -2.88448 q^{9} +1.54461i q^{11} +(3.54461 - 0.660123i) q^{13} -0.339877i q^{15} +2.86485i q^{19} +1.32025i q^{21} -5.42909 q^{23} -1.00000 q^{25} -2.00000 q^{27} -5.20473 q^{29} +6.22436i q^{31} +0.524976i q^{33} +3.88448 q^{35} +8.56424i q^{37} +(1.20473 - 0.224361i) q^{39} +9.08921i q^{41} -0.980369 q^{43} +2.88448i q^{45} -6.52498i q^{47} -8.08921 q^{49} +6.44872 q^{53} +1.54461 q^{55} +0.973697i q^{57} +4.45539i q^{59} +9.65345 q^{61} -11.2047i q^{63} +(-0.660123 - 3.54461i) q^{65} +6.97370i q^{67} -1.84522 q^{69} -12.6731i q^{71} +3.43576i q^{73} -0.339877 q^{75} -6.00000 q^{77} +13.1285 q^{79} +7.97370 q^{81} +8.56424i q^{83} -1.76897 q^{87} +17.1285i q^{89} +(2.56424 + 13.7690i) q^{91} +2.11552i q^{93} +2.86485 q^{95} -13.7690i q^{97} -4.45539i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{9} + 12 q^{23} - 6 q^{25} - 12 q^{27} - 12 q^{29} - 12 q^{39} - 12 q^{43} - 6 q^{49} - 12 q^{53} - 12 q^{55} - 12 q^{61} - 6 q^{65} - 36 q^{77} + 24 q^{79} - 18 q^{81} + 36 q^{87} - 12 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 0.339877 0.196228 0.0981140 − 0.995175i \(-0.468719\pi\)
0.0981140 + 0.995175i \(0.468719\pi\)
\(4\) 0 0
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) 3.88448i 1.46820i 0.679043 + 0.734098i \(0.262395\pi\)
−0.679043 + 0.734098i \(0.737605\pi\)
\(8\) 0 0
\(9\) −2.88448 −0.961495
\(10\) 0 0
\(11\) 1.54461i 0.465716i 0.972511 + 0.232858i \(0.0748078\pi\)
−0.972511 + 0.232858i \(0.925192\pi\)
\(12\) 0 0
\(13\) 3.54461 − 0.660123i 0.983097 − 0.183085i
\(14\) 0 0
\(15\) − 0.339877i − 0.0877558i
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 2.86485i 0.657242i 0.944462 + 0.328621i \(0.106584\pi\)
−0.944462 + 0.328621i \(0.893416\pi\)
\(20\) 0 0
\(21\) 1.32025i 0.288101i
\(22\) 0 0
\(23\) −5.42909 −1.13204 −0.566022 − 0.824390i \(-0.691518\pi\)
−0.566022 + 0.824390i \(0.691518\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −2.00000 −0.384900
\(28\) 0 0
\(29\) −5.20473 −0.966494 −0.483247 − 0.875484i \(-0.660543\pi\)
−0.483247 + 0.875484i \(0.660543\pi\)
\(30\) 0 0
\(31\) 6.22436i 1.11793i 0.829192 + 0.558964i \(0.188801\pi\)
−0.829192 + 0.558964i \(0.811199\pi\)
\(32\) 0 0
\(33\) 0.524976i 0.0913866i
\(34\) 0 0
\(35\) 3.88448 0.656598
\(36\) 0 0
\(37\) 8.56424i 1.40795i 0.710224 + 0.703976i \(0.248594\pi\)
−0.710224 + 0.703976i \(0.751406\pi\)
\(38\) 0 0
\(39\) 1.20473 − 0.224361i 0.192911 − 0.0359264i
\(40\) 0 0
\(41\) 9.08921i 1.41950i 0.704455 + 0.709748i \(0.251191\pi\)
−0.704455 + 0.709748i \(0.748809\pi\)
\(42\) 0 0
\(43\) −0.980369 −0.149505 −0.0747525 − 0.997202i \(-0.523817\pi\)
−0.0747525 + 0.997202i \(0.523817\pi\)
\(44\) 0 0
\(45\) 2.88448i 0.429993i
\(46\) 0 0
\(47\) − 6.52498i − 0.951766i −0.879509 − 0.475883i \(-0.842129\pi\)
0.879509 − 0.475883i \(-0.157871\pi\)
\(48\) 0 0
\(49\) −8.08921 −1.15560
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.44872 0.885800 0.442900 − 0.896571i \(-0.353950\pi\)
0.442900 + 0.896571i \(0.353950\pi\)
\(54\) 0 0
\(55\) 1.54461 0.208275
\(56\) 0 0
\(57\) 0.973697i 0.128969i
\(58\) 0 0
\(59\) 4.45539i 0.580043i 0.957020 + 0.290021i \(0.0936624\pi\)
−0.957020 + 0.290021i \(0.906338\pi\)
\(60\) 0 0
\(61\) 9.65345 1.23600 0.617999 − 0.786179i \(-0.287944\pi\)
0.617999 + 0.786179i \(0.287944\pi\)
\(62\) 0 0
\(63\) − 11.2047i − 1.41166i
\(64\) 0 0
\(65\) −0.660123 − 3.54461i −0.0818782 − 0.439654i
\(66\) 0 0
\(67\) 6.97370i 0.851973i 0.904730 + 0.425986i \(0.140073\pi\)
−0.904730 + 0.425986i \(0.859927\pi\)
\(68\) 0 0
\(69\) −1.84522 −0.222139
\(70\) 0 0
\(71\) − 12.6731i − 1.50402i −0.659153 − 0.752009i \(-0.729085\pi\)
0.659153 − 0.752009i \(-0.270915\pi\)
\(72\) 0 0
\(73\) 3.43576i 0.402126i 0.979578 + 0.201063i \(0.0644395\pi\)
−0.979578 + 0.201063i \(0.935560\pi\)
\(74\) 0 0
\(75\) −0.339877 −0.0392456
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 13.1285 1.47707 0.738534 − 0.674216i \(-0.235518\pi\)
0.738534 + 0.674216i \(0.235518\pi\)
\(80\) 0 0
\(81\) 7.97370 0.885966
\(82\) 0 0
\(83\) 8.56424i 0.940047i 0.882654 + 0.470024i \(0.155755\pi\)
−0.882654 + 0.470024i \(0.844245\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.76897 −0.189653
\(88\) 0 0
\(89\) 17.1285i 1.81561i 0.419387 + 0.907807i \(0.362245\pi\)
−0.419387 + 0.907807i \(0.637755\pi\)
\(90\) 0 0
\(91\) 2.56424 + 13.7690i 0.268805 + 1.44338i
\(92\) 0 0
\(93\) 2.11552i 0.219369i
\(94\) 0 0
\(95\) 2.86485 0.293928
\(96\) 0 0
\(97\) − 13.7690i − 1.39803i −0.715109 − 0.699013i \(-0.753623\pi\)
0.715109 − 0.699013i \(-0.246377\pi\)
\(98\) 0 0
\(99\) − 4.45539i − 0.447784i
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 1040.2.k.c.961.3 6
4.3 odd 2 260.2.f.a.181.3 ✓ 6
12.11 even 2 2340.2.c.d.181.4 6
13.12 even 2 inner 1040.2.k.c.961.4 6
20.3 even 4 1300.2.d.d.649.3 6
20.7 even 4 1300.2.d.c.649.4 6
20.19 odd 2 1300.2.f.e.701.4 6
52.31 even 4 3380.2.a.m.1.2 3
52.47 even 4 3380.2.a.n.1.2 3
52.51 odd 2 260.2.f.a.181.4 yes 6
156.155 even 2 2340.2.c.d.181.3 6
260.103 even 4 1300.2.d.c.649.3 6
260.207 even 4 1300.2.d.d.649.4 6
260.259 odd 2 1300.2.f.e.701.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.f.a.181.3 ✓ 6 4.3 odd 2
260.2.f.a.181.4 yes 6 52.51 odd 2
1040.2.k.c.961.3 6 1.1 even 1 trivial
1040.2.k.c.961.4 6 13.12 even 2 inner
1300.2.d.c.649.3 6 260.103 even 4
1300.2.d.c.649.4 6 20.7 even 4
1300.2.d.d.649.3 6 20.3 even 4
1300.2.d.d.649.4 6 260.207 even 4
1300.2.f.e.701.3 6 260.259 odd 2
1300.2.f.e.701.4 6 20.19 odd 2
2340.2.c.d.181.3 6 156.155 even 2
2340.2.c.d.181.4 6 12.11 even 2
3380.2.a.m.1.2 3 52.31 even 4
3380.2.a.n.1.2 3 52.47 even 4