Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2352,2,Mod(961,2352)] 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("2352.961"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2352, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-1,0,2,0,0,0,-1,0,-4,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1537.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2352.1537
Dual form 2352.2.q.i.961.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{3} +(1.00000 + 1.73205i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(-2.00000 + 3.46410i) q^{11} +6.00000 q^{13} -2.00000 q^{15} +(-1.00000 + 1.73205i) q^{17} +(-2.00000 - 3.46410i) q^{19} +(4.00000 + 6.92820i) q^{23} +(0.500000 - 0.866025i) q^{25} +1.00000 q^{27} -2.00000 q^{29} +(-2.00000 - 3.46410i) q^{33} +(5.00000 + 8.66025i) q^{37} +(-3.00000 + 5.19615i) q^{39} -6.00000 q^{41} +4.00000 q^{43} +(1.00000 - 1.73205i) q^{45} +(-1.00000 - 1.73205i) q^{51} +(-3.00000 + 5.19615i) q^{53} -8.00000 q^{55} +4.00000 q^{57} +(2.00000 - 3.46410i) q^{59} +(-3.00000 - 5.19615i) q^{61} +(6.00000 + 10.3923i) q^{65} +(2.00000 - 3.46410i) q^{67} -8.00000 q^{69} -8.00000 q^{71} +(-5.00000 + 8.66025i) q^{73} +(0.500000 + 0.866025i) q^{75} +(-0.500000 + 0.866025i) q^{81} +4.00000 q^{83} -4.00000 q^{85} +(1.00000 - 1.73205i) q^{87} +(3.00000 + 5.19615i) q^{89} +(4.00000 - 6.92820i) q^{95} -14.0000 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{5} - q^{9} - 4 q^{11} + 12 q^{13} - 4 q^{15} - 2 q^{17} - 4 q^{19} + 8 q^{23} + q^{25} + 2 q^{27} - 4 q^{29} - 4 q^{33} + 10 q^{37} - 6 q^{39} - 12 q^{41} + 8 q^{43} + 2 q^{45} - 2 q^{51}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) 0 0
\(5\) 1.00000 + 1.73205i 0.447214 + 0.774597i 0.998203 − 0.0599153i \(-0.0190830\pi\)
−0.550990 + 0.834512i \(0.685750\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) −2.00000 + 3.46410i −0.603023 + 1.04447i 0.389338 + 0.921095i \(0.372704\pi\)
−0.992361 + 0.123371i \(0.960630\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) −1.00000 + 1.73205i −0.242536 + 0.420084i −0.961436 − 0.275029i \(-0.911312\pi\)
0.718900 + 0.695113i \(0.244646\pi\)
\(18\) 0 0
\(19\) −2.00000 − 3.46410i −0.458831 − 0.794719i 0.540068 − 0.841621i \(-0.318398\pi\)
−0.998899 + 0.0469020i \(0.985065\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 + 6.92820i 0.834058 + 1.44463i 0.894795 + 0.446476i \(0.147321\pi\)
−0.0607377 + 0.998154i \(0.519345\pi\)
\(24\) 0 0
\(25\) 0.500000 − 0.866025i 0.100000 − 0.173205i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) 0 0
\(33\) −2.00000 − 3.46410i −0.348155 − 0.603023i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.00000 + 8.66025i 0.821995 + 1.42374i 0.904194 + 0.427121i \(0.140472\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 0 0
\(39\) −3.00000 + 5.19615i −0.480384 + 0.832050i
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 1.00000 − 1.73205i 0.149071 − 0.258199i
\(46\) 0 0
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −1.00000 − 1.73205i −0.140028 − 0.242536i
\(52\) 0 0
\(53\) −3.00000 + 5.19615i −0.412082 + 0.713746i −0.995117 − 0.0987002i \(-0.968532\pi\)
0.583036 + 0.812447i \(0.301865\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 2.00000 − 3.46410i 0.260378 − 0.450988i −0.705965 − 0.708247i \(-0.749486\pi\)
0.966342 + 0.257260i \(0.0828195\pi\)
\(60\) 0 0
\(61\) −3.00000 − 5.19615i −0.384111 − 0.665299i 0.607535 − 0.794293i \(-0.292159\pi\)
−0.991645 + 0.128994i \(0.958825\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000 + 10.3923i 0.744208 + 1.28901i
\(66\) 0 0
\(67\) 2.00000 − 3.46410i 0.244339 − 0.423207i −0.717607 − 0.696449i \(-0.754762\pi\)
0.961946 + 0.273241i \(0.0880957\pi\)
\(68\) 0 0
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −5.00000 + 8.66025i −0.585206 + 1.01361i 0.409644 + 0.912245i \(0.365653\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) 0.500000 + 0.866025i 0.0577350 + 0.100000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) 1.00000 − 1.73205i 0.107211 − 0.185695i
\(88\) 0 0
\(89\) 3.00000 + 5.19615i 0.317999 + 0.550791i 0.980071 − 0.198650i \(-0.0636557\pi\)
−0.662071 + 0.749441i \(0.730322\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 − 6.92820i 0.410391 − 0.710819i
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
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 2352.2.q.i.1537.1 2
4.3 odd 2 294.2.e.c.67.1 2
7.2 even 3 inner 2352.2.q.i.961.1 2
7.3 odd 6 2352.2.a.l.1.1 1
7.4 even 3 336.2.a.d.1.1 1
7.5 odd 6 2352.2.q.n.961.1 2
7.6 odd 2 2352.2.q.n.1537.1 2
12.11 even 2 882.2.g.h.361.1 2
21.11 odd 6 1008.2.a.j.1.1 1
21.17 even 6 7056.2.a.k.1.1 1
28.3 even 6 294.2.a.g.1.1 1
28.11 odd 6 42.2.a.a.1.1 ✓ 1
28.19 even 6 294.2.e.a.79.1 2
28.23 odd 6 294.2.e.c.79.1 2
28.27 even 2 294.2.e.a.67.1 2
35.4 even 6 8400.2.a.k.1.1 1
56.3 even 6 9408.2.a.n.1.1 1
56.11 odd 6 1344.2.a.q.1.1 1
56.45 odd 6 9408.2.a.bw.1.1 1
56.53 even 6 1344.2.a.i.1.1 1
84.11 even 6 126.2.a.a.1.1 1
84.23 even 6 882.2.g.h.667.1 2
84.47 odd 6 882.2.g.j.667.1 2
84.59 odd 6 882.2.a.b.1.1 1
84.83 odd 2 882.2.g.j.361.1 2
112.11 odd 12 5376.2.c.bc.2689.2 2
112.53 even 12 5376.2.c.e.2689.1 2
112.67 odd 12 5376.2.c.bc.2689.1 2
112.109 even 12 5376.2.c.e.2689.2 2
140.39 odd 6 1050.2.a.i.1.1 1
140.59 even 6 7350.2.a.f.1.1 1
140.67 even 12 1050.2.g.a.799.2 2
140.123 even 12 1050.2.g.a.799.1 2
168.11 even 6 4032.2.a.e.1.1 1
168.53 odd 6 4032.2.a.m.1.1 1
252.11 even 6 1134.2.f.j.757.1 2
252.67 odd 6 1134.2.f.g.379.1 2
252.95 even 6 1134.2.f.j.379.1 2
252.151 odd 6 1134.2.f.g.757.1 2
308.263 even 6 5082.2.a.d.1.1 1
364.207 odd 6 7098.2.a.f.1.1 1
420.179 even 6 3150.2.a.bo.1.1 1
420.263 odd 12 3150.2.g.r.2899.2 2
420.347 odd 12 3150.2.g.r.2899.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.a.a.1.1 ✓ 1 28.11 odd 6
126.2.a.a.1.1 1 84.11 even 6
294.2.a.g.1.1 1 28.3 even 6
294.2.e.a.67.1 2 28.27 even 2
294.2.e.a.79.1 2 28.19 even 6
294.2.e.c.67.1 2 4.3 odd 2
294.2.e.c.79.1 2 28.23 odd 6
336.2.a.d.1.1 1 7.4 even 3
882.2.a.b.1.1 1 84.59 odd 6
882.2.g.h.361.1 2 12.11 even 2
882.2.g.h.667.1 2 84.23 even 6
882.2.g.j.361.1 2 84.83 odd 2
882.2.g.j.667.1 2 84.47 odd 6
1008.2.a.j.1.1 1 21.11 odd 6
1050.2.a.i.1.1 1 140.39 odd 6
1050.2.g.a.799.1 2 140.123 even 12
1050.2.g.a.799.2 2 140.67 even 12
1134.2.f.g.379.1 2 252.67 odd 6
1134.2.f.g.757.1 2 252.151 odd 6
1134.2.f.j.379.1 2 252.95 even 6
1134.2.f.j.757.1 2 252.11 even 6
1344.2.a.i.1.1 1 56.53 even 6
1344.2.a.q.1.1 1 56.11 odd 6
2352.2.a.l.1.1 1 7.3 odd 6
2352.2.q.i.961.1 2 7.2 even 3 inner
2352.2.q.i.1537.1 2 1.1 even 1 trivial
2352.2.q.n.961.1 2 7.5 odd 6
2352.2.q.n.1537.1 2 7.6 odd 2
3150.2.a.bo.1.1 1 420.179 even 6
3150.2.g.r.2899.1 2 420.347 odd 12
3150.2.g.r.2899.2 2 420.263 odd 12
4032.2.a.e.1.1 1 168.11 even 6
4032.2.a.m.1.1 1 168.53 odd 6
5082.2.a.d.1.1 1 308.263 even 6
5376.2.c.e.2689.1 2 112.53 even 12
5376.2.c.e.2689.2 2 112.109 even 12
5376.2.c.bc.2689.1 2 112.67 odd 12
5376.2.c.bc.2689.2 2 112.11 odd 12
7056.2.a.k.1.1 1 21.17 even 6
7098.2.a.f.1.1 1 364.207 odd 6
7350.2.a.f.1.1 1 140.59 even 6
8400.2.a.k.1.1 1 35.4 even 6
9408.2.a.n.1.1 1 56.3 even 6
9408.2.a.bw.1.1 1 56.45 odd 6