Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9680,2,Mod(1,9680)] 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("9680.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,2,0,1,0,11,0,0,0,-4,0,1,0,3] 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: yes
Analytic conductor: \(77.2951891566\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) 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 110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.37228 q^{3} +1.00000 q^{5} -2.37228 q^{7} +2.62772 q^{9} -2.00000 q^{13} -2.37228 q^{15} +4.37228 q^{17} +6.37228 q^{19} +5.62772 q^{21} +8.74456 q^{23} +1.00000 q^{25} +0.883156 q^{27} +4.37228 q^{29} +2.37228 q^{31} -2.37228 q^{35} +3.62772 q^{37} +4.74456 q^{39} -11.4891 q^{41} -4.00000 q^{43} +2.62772 q^{45} +8.74456 q^{47} -1.37228 q^{49} -10.3723 q^{51} +13.1168 q^{53} -15.1168 q^{57} -8.74456 q^{59} -0.372281 q^{61} -6.23369 q^{63} -2.00000 q^{65} -8.00000 q^{67} -20.7446 q^{69} +7.11684 q^{71} -7.48913 q^{73} -2.37228 q^{75} -12.7446 q^{79} -9.97825 q^{81} +8.74456 q^{83} +4.37228 q^{85} -10.3723 q^{87} +4.37228 q^{89} +4.74456 q^{91} -5.62772 q^{93} +6.37228 q^{95} -1.25544 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 2 q^{5} + q^{7} + 11 q^{9} - 4 q^{13} + q^{15} + 3 q^{17} + 7 q^{19} + 17 q^{21} + 6 q^{23} + 2 q^{25} + 19 q^{27} + 3 q^{29} - q^{31} + q^{35} + 13 q^{37} - 2 q^{39} - 8 q^{43} + 11 q^{45}+ \cdots - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −2.37228 −1.36964 −0.684819 − 0.728714i \(-0.740119\pi\)
−0.684819 + 0.728714i \(0.740119\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.37228 −0.896638 −0.448319 − 0.893874i \(-0.647977\pi\)
−0.448319 + 0.893874i \(0.647977\pi\)
\(8\) 0 0
\(9\) 2.62772 0.875906
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −2.37228 −0.612520
\(16\) 0 0
\(17\) 4.37228 1.06043 0.530217 − 0.847862i \(-0.322110\pi\)
0.530217 + 0.847862i \(0.322110\pi\)
\(18\) 0 0
\(19\) 6.37228 1.46190 0.730951 − 0.682430i \(-0.239077\pi\)
0.730951 + 0.682430i \(0.239077\pi\)
\(20\) 0 0
\(21\) 5.62772 1.22807
\(22\) 0 0
\(23\) 8.74456 1.82337 0.911684 − 0.410893i \(-0.134783\pi\)
0.911684 + 0.410893i \(0.134783\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0.883156 0.169963
\(28\) 0 0
\(29\) 4.37228 0.811912 0.405956 − 0.913893i \(-0.366939\pi\)
0.405956 + 0.913893i \(0.366939\pi\)
\(30\) 0 0
\(31\) 2.37228 0.426074 0.213037 − 0.977044i \(-0.431664\pi\)
0.213037 + 0.977044i \(0.431664\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.37228 −0.400989
\(36\) 0 0
\(37\) 3.62772 0.596393 0.298197 − 0.954504i \(-0.403615\pi\)
0.298197 + 0.954504i \(0.403615\pi\)
\(38\) 0 0
\(39\) 4.74456 0.759738
\(40\) 0 0
\(41\) −11.4891 −1.79430 −0.897150 − 0.441726i \(-0.854366\pi\)
−0.897150 + 0.441726i \(0.854366\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 2.62772 0.391717
\(46\) 0 0
\(47\) 8.74456 1.27553 0.637763 − 0.770233i \(-0.279860\pi\)
0.637763 + 0.770233i \(0.279860\pi\)
\(48\) 0 0
\(49\) −1.37228 −0.196040
\(50\) 0 0
\(51\) −10.3723 −1.45241
\(52\) 0 0
\(53\) 13.1168 1.80174 0.900869 − 0.434092i \(-0.142931\pi\)
0.900869 + 0.434092i \(0.142931\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −15.1168 −2.00227
\(58\) 0 0
\(59\) −8.74456 −1.13845 −0.569223 − 0.822183i \(-0.692756\pi\)
−0.569223 + 0.822183i \(0.692756\pi\)
\(60\) 0 0
\(61\) −0.372281 −0.0476657 −0.0238329 − 0.999716i \(-0.507587\pi\)
−0.0238329 + 0.999716i \(0.507587\pi\)
\(62\) 0 0
\(63\) −6.23369 −0.785371
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −20.7446 −2.49735
\(70\) 0 0
\(71\) 7.11684 0.844614 0.422307 − 0.906453i \(-0.361220\pi\)
0.422307 + 0.906453i \(0.361220\pi\)
\(72\) 0 0
\(73\) −7.48913 −0.876536 −0.438268 − 0.898844i \(-0.644408\pi\)
−0.438268 + 0.898844i \(0.644408\pi\)
\(74\) 0 0
\(75\) −2.37228 −0.273927
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −12.7446 −1.43388 −0.716938 − 0.697137i \(-0.754457\pi\)
−0.716938 + 0.697137i \(0.754457\pi\)
\(80\) 0 0
\(81\) −9.97825 −1.10869
\(82\) 0 0
\(83\) 8.74456 0.959840 0.479920 − 0.877312i \(-0.340666\pi\)
0.479920 + 0.877312i \(0.340666\pi\)
\(84\) 0 0
\(85\) 4.37228 0.474240
\(86\) 0 0
\(87\) −10.3723 −1.11203
\(88\) 0 0
\(89\) 4.37228 0.463461 0.231730 − 0.972780i \(-0.425561\pi\)
0.231730 + 0.972780i \(0.425561\pi\)
\(90\) 0 0
\(91\) 4.74456 0.497365
\(92\) 0 0
\(93\) −5.62772 −0.583567
\(94\) 0 0
\(95\) 6.37228 0.653782
\(96\) 0 0
\(97\) −1.25544 −0.127470 −0.0637352 − 0.997967i \(-0.520301\pi\)
−0.0637352 + 0.997967i \(0.520301\pi\)
\(98\) 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 9680.2.a.bt.1.1 2
4.3 odd 2 1210.2.a.r.1.2 2
11.10 odd 2 880.2.a.n.1.1 2
20.19 odd 2 6050.2.a.cb.1.1 2
33.32 even 2 7920.2.a.bq.1.2 2
44.43 even 2 110.2.a.d.1.2 ✓ 2
55.32 even 4 4400.2.b.p.4049.3 4
55.43 even 4 4400.2.b.p.4049.2 4
55.54 odd 2 4400.2.a.bl.1.2 2
88.21 odd 2 3520.2.a.bj.1.2 2
88.43 even 2 3520.2.a.bq.1.1 2
132.131 odd 2 990.2.a.m.1.1 2
220.43 odd 4 550.2.b.f.199.4 4
220.87 odd 4 550.2.b.f.199.1 4
220.219 even 2 550.2.a.n.1.1 2
308.307 odd 2 5390.2.a.bp.1.1 2
660.263 even 4 4950.2.c.bc.199.2 4
660.527 even 4 4950.2.c.bc.199.3 4
660.659 odd 2 4950.2.a.bw.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.a.d.1.2 ✓ 2 44.43 even 2
550.2.a.n.1.1 2 220.219 even 2
550.2.b.f.199.1 4 220.87 odd 4
550.2.b.f.199.4 4 220.43 odd 4
880.2.a.n.1.1 2 11.10 odd 2
990.2.a.m.1.1 2 132.131 odd 2
1210.2.a.r.1.2 2 4.3 odd 2
3520.2.a.bj.1.2 2 88.21 odd 2
3520.2.a.bq.1.1 2 88.43 even 2
4400.2.a.bl.1.2 2 55.54 odd 2
4400.2.b.p.4049.2 4 55.43 even 4
4400.2.b.p.4049.3 4 55.32 even 4
4950.2.a.bw.1.2 2 660.659 odd 2
4950.2.c.bc.199.2 4 660.263 even 4
4950.2.c.bc.199.3 4 660.527 even 4
5390.2.a.bp.1.1 2 308.307 odd 2
6050.2.a.cb.1.1 2 20.19 odd 2
7920.2.a.bq.1.2 2 33.32 even 2
9680.2.a.bt.1.1 2 1.1 even 1 trivial