Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5376,2,Mod(2689,5376)] 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("5376.2689"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5376, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5376.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-2,0,-2,0,0,0,0,0,-4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(42.9275761266\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2689.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5376.2689
Dual form 5376.2.c.e.2689.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -2.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} +4.00000i q^{11} -6.00000i q^{13} -2.00000 q^{15} +2.00000 q^{17} -4.00000i q^{19} +1.00000i q^{21} +8.00000 q^{23} +1.00000 q^{25} +1.00000i q^{27} +2.00000i q^{29} +4.00000 q^{33} +2.00000i q^{35} -10.0000i q^{37} -6.00000 q^{39} +6.00000 q^{41} +4.00000i q^{43} +2.00000i q^{45} +1.00000 q^{49} -2.00000i q^{51} +6.00000i q^{53} +8.00000 q^{55} -4.00000 q^{57} -4.00000i q^{59} -6.00000i q^{61} +1.00000 q^{63} -12.0000 q^{65} +4.00000i q^{67} -8.00000i q^{69} +8.00000 q^{71} -10.0000 q^{73} -1.00000i q^{75} -4.00000i q^{77} +1.00000 q^{81} -4.00000i q^{83} -4.00000i q^{85} +2.00000 q^{87} +6.00000 q^{89} +6.00000i q^{91} -8.00000 q^{95} -14.0000 q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7} - 2 q^{9} - 4 q^{15} + 4 q^{17} + 16 q^{23} + 2 q^{25} + 8 q^{33} - 12 q^{39} + 12 q^{41} + 2 q^{49} + 16 q^{55} - 8 q^{57} + 2 q^{63} - 24 q^{65} + 16 q^{71} - 20 q^{73} + 2 q^{81} + 4 q^{87}+ \cdots - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times\).

\(n\) \(1793\) \(2815\) \(4609\) \(5125\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) − 2.00000i − 0.894427i −0.894427 − 0.447214i \(-0.852416\pi\)
0.894427 − 0.447214i \(-0.147584\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.00000i 1.20605i 0.797724 + 0.603023i \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 0 0
\(13\) − 6.00000i − 1.66410i −0.554700 − 0.832050i \(-0.687167\pi\)
0.554700 − 0.832050i \(-0.312833\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) − 4.00000i − 0.917663i −0.888523 − 0.458831i \(-0.848268\pi\)
0.888523 − 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 1.00000i 0.218218i
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 2.00000i 0.371391i 0.982607 + 0.185695i \(0.0594537\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) 2.00000i 0.338062i
\(36\) 0 0
\(37\) − 10.0000i − 1.64399i −0.569495 − 0.821995i \(-0.692861\pi\)
0.569495 − 0.821995i \(-0.307139\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 2.00000i 0.298142i
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) − 2.00000i − 0.280056i
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) − 4.00000i − 0.520756i −0.965507 − 0.260378i \(-0.916153\pi\)
0.965507 − 0.260378i \(-0.0838471\pi\)
\(60\) 0 0
\(61\) − 6.00000i − 0.768221i −0.923287 − 0.384111i \(-0.874508\pi\)
0.923287 − 0.384111i \(-0.125492\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) − 8.00000i − 0.963087i
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) − 1.00000i − 0.115470i
\(76\) 0 0
\(77\) − 4.00000i − 0.455842i
\(78\) 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
\(83\) − 4.00000i − 0.439057i −0.975606 − 0.219529i \(-0.929548\pi\)
0.975606 − 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) − 4.00000i − 0.433861i
\(86\) 0 0
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 6.00000i 0.628971i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) − 4.00000i − 0.402015i
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 5376.2.c.e.2689.1 2
4.3 odd 2 5376.2.c.bc.2689.2 2
8.3 odd 2 5376.2.c.bc.2689.1 2
8.5 even 2 inner 5376.2.c.e.2689.2 2
16.3 odd 4 42.2.a.a.1.1 ✓ 1
16.5 even 4 1344.2.a.i.1.1 1
16.11 odd 4 1344.2.a.q.1.1 1
16.13 even 4 336.2.a.d.1.1 1
48.5 odd 4 4032.2.a.m.1.1 1
48.11 even 4 4032.2.a.e.1.1 1
48.29 odd 4 1008.2.a.j.1.1 1
48.35 even 4 126.2.a.a.1.1 1
80.3 even 4 1050.2.g.a.799.1 2
80.19 odd 4 1050.2.a.i.1.1 1
80.29 even 4 8400.2.a.k.1.1 1
80.67 even 4 1050.2.g.a.799.2 2
112.3 even 12 294.2.e.a.79.1 2
112.13 odd 4 2352.2.a.l.1.1 1
112.19 even 12 294.2.e.a.67.1 2
112.27 even 4 9408.2.a.n.1.1 1
112.45 odd 12 2352.2.q.n.961.1 2
112.51 odd 12 294.2.e.c.67.1 2
112.61 odd 12 2352.2.q.n.1537.1 2
112.67 odd 12 294.2.e.c.79.1 2
112.69 odd 4 9408.2.a.bw.1.1 1
112.83 even 4 294.2.a.g.1.1 1
112.93 even 12 2352.2.q.i.1537.1 2
112.109 even 12 2352.2.q.i.961.1 2
144.67 odd 12 1134.2.f.g.379.1 2
144.83 even 12 1134.2.f.j.757.1 2
144.115 odd 12 1134.2.f.g.757.1 2
144.131 even 12 1134.2.f.j.379.1 2
176.131 even 4 5082.2.a.d.1.1 1
208.51 odd 4 7098.2.a.f.1.1 1
240.83 odd 4 3150.2.g.r.2899.2 2
240.179 even 4 3150.2.a.bo.1.1 1
240.227 odd 4 3150.2.g.r.2899.1 2
336.83 odd 4 882.2.a.b.1.1 1
336.125 even 4 7056.2.a.k.1.1 1
336.131 odd 12 882.2.g.j.361.1 2
336.179 even 12 882.2.g.h.667.1 2
336.227 odd 12 882.2.g.j.667.1 2
336.275 even 12 882.2.g.h.361.1 2
560.419 even 4 7350.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.a.a.1.1 ✓ 1 16.3 odd 4
126.2.a.a.1.1 1 48.35 even 4
294.2.a.g.1.1 1 112.83 even 4
294.2.e.a.67.1 2 112.19 even 12
294.2.e.a.79.1 2 112.3 even 12
294.2.e.c.67.1 2 112.51 odd 12
294.2.e.c.79.1 2 112.67 odd 12
336.2.a.d.1.1 1 16.13 even 4
882.2.a.b.1.1 1 336.83 odd 4
882.2.g.h.361.1 2 336.275 even 12
882.2.g.h.667.1 2 336.179 even 12
882.2.g.j.361.1 2 336.131 odd 12
882.2.g.j.667.1 2 336.227 odd 12
1008.2.a.j.1.1 1 48.29 odd 4
1050.2.a.i.1.1 1 80.19 odd 4
1050.2.g.a.799.1 2 80.3 even 4
1050.2.g.a.799.2 2 80.67 even 4
1134.2.f.g.379.1 2 144.67 odd 12
1134.2.f.g.757.1 2 144.115 odd 12
1134.2.f.j.379.1 2 144.131 even 12
1134.2.f.j.757.1 2 144.83 even 12
1344.2.a.i.1.1 1 16.5 even 4
1344.2.a.q.1.1 1 16.11 odd 4
2352.2.a.l.1.1 1 112.13 odd 4
2352.2.q.i.961.1 2 112.109 even 12
2352.2.q.i.1537.1 2 112.93 even 12
2352.2.q.n.961.1 2 112.45 odd 12
2352.2.q.n.1537.1 2 112.61 odd 12
3150.2.a.bo.1.1 1 240.179 even 4
3150.2.g.r.2899.1 2 240.227 odd 4
3150.2.g.r.2899.2 2 240.83 odd 4
4032.2.a.e.1.1 1 48.11 even 4
4032.2.a.m.1.1 1 48.5 odd 4
5082.2.a.d.1.1 1 176.131 even 4
5376.2.c.e.2689.1 2 1.1 even 1 trivial
5376.2.c.e.2689.2 2 8.5 even 2 inner
5376.2.c.bc.2689.1 2 8.3 odd 2
5376.2.c.bc.2689.2 2 4.3 odd 2
7056.2.a.k.1.1 1 336.125 even 4
7098.2.a.f.1.1 1 208.51 odd 4
7350.2.a.f.1.1 1 560.419 even 4
8400.2.a.k.1.1 1 80.29 even 4
9408.2.a.n.1.1 1 112.27 even 4
9408.2.a.bw.1.1 1 112.69 odd 4