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 = [2,0,2,0,-2,0,0,0,-4,0,0,0,0,0,-2,0,0] 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: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 605)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.73205\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +1.73205 q^{7} -2.00000 q^{9} +3.46410 q^{13} -1.00000 q^{15} -6.92820 q^{17} +3.46410 q^{19} +1.73205 q^{21} +1.00000 q^{25} -5.00000 q^{27} +8.00000 q^{31} -1.73205 q^{35} -8.00000 q^{37} +3.46410 q^{39} +12.1244 q^{41} -8.66025 q^{43} +2.00000 q^{45} -9.00000 q^{47} -4.00000 q^{49} -6.92820 q^{51} +6.00000 q^{53} +3.46410 q^{57} +12.0000 q^{59} +8.66025 q^{61} -3.46410 q^{63} -3.46410 q^{65} +5.00000 q^{67} +12.0000 q^{71} +1.00000 q^{75} +10.3923 q^{79} +1.00000 q^{81} -3.46410 q^{83} +6.92820 q^{85} +3.00000 q^{89} +6.00000 q^{91} +8.00000 q^{93} -3.46410 q^{95} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{9} - 2 q^{15} + 2 q^{25} - 10 q^{27} + 16 q^{31} - 16 q^{37} + 4 q^{45} - 18 q^{47} - 8 q^{49} + 12 q^{53} + 24 q^{59} + 10 q^{67} + 24 q^{71} + 2 q^{75} + 2 q^{81} + 6 q^{89}+ \cdots - 20 q^{97}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350 0.288675 − 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.73205 0.654654 0.327327 − 0.944911i \(-0.393852\pi\)
0.327327 + 0.944911i \(0.393852\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 3.46410 0.960769 0.480384 − 0.877058i \(-0.340497\pi\)
0.480384 + 0.877058i \(0.340497\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −6.92820 −1.68034 −0.840168 − 0.542326i \(-0.817544\pi\)
−0.840168 + 0.542326i \(0.817544\pi\)
\(18\) 0 0
\(19\) 3.46410 0.794719 0.397360 − 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) 0 0
\(21\) 1.73205 0.377964
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.73205 −0.292770
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 3.46410 0.554700
\(40\) 0 0
\(41\) 12.1244 1.89351 0.946753 − 0.321960i \(-0.104342\pi\)
0.946753 + 0.321960i \(0.104342\pi\)
\(42\) 0 0
\(43\) −8.66025 −1.32068 −0.660338 − 0.750968i \(-0.729587\pi\)
−0.660338 + 0.750968i \(0.729587\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) −9.00000 −1.31278 −0.656392 − 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 0 0
\(49\) −4.00000 −0.571429
\(50\) 0 0
\(51\) −6.92820 −0.970143
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.46410 0.458831
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 8.66025 1.10883 0.554416 − 0.832240i \(-0.312942\pi\)
0.554416 + 0.832240i \(0.312942\pi\)
\(62\) 0 0
\(63\) −3.46410 −0.436436
\(64\) 0 0
\(65\) −3.46410 −0.429669
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 − 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.3923 1.16923 0.584613 − 0.811312i \(-0.301246\pi\)
0.584613 + 0.811312i \(0.301246\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −3.46410 −0.380235 −0.190117 − 0.981761i \(-0.560887\pi\)
−0.190117 + 0.981761i \(0.560887\pi\)
\(84\) 0 0
\(85\) 6.92820 0.751469
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) −3.46410 −0.355409
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 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.bu.1.2 2
4.3 odd 2 605.2.a.e.1.2 yes 2
11.10 odd 2 inner 9680.2.a.bu.1.1 2
12.11 even 2 5445.2.a.u.1.1 2
20.19 odd 2 3025.2.a.l.1.1 2
44.3 odd 10 605.2.g.i.251.2 8
44.7 even 10 605.2.g.i.511.1 8
44.15 odd 10 605.2.g.i.511.2 8
44.19 even 10 605.2.g.i.251.1 8
44.27 odd 10 605.2.g.i.366.1 8
44.31 odd 10 605.2.g.i.81.1 8
44.35 even 10 605.2.g.i.81.2 8
44.39 even 10 605.2.g.i.366.2 8
44.43 even 2 605.2.a.e.1.1 ✓ 2
132.131 odd 2 5445.2.a.u.1.2 2
220.219 even 2 3025.2.a.l.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
605.2.a.e.1.1 ✓ 2 44.43 even 2
605.2.a.e.1.2 yes 2 4.3 odd 2
605.2.g.i.81.1 8 44.31 odd 10
605.2.g.i.81.2 8 44.35 even 10
605.2.g.i.251.1 8 44.19 even 10
605.2.g.i.251.2 8 44.3 odd 10
605.2.g.i.366.1 8 44.27 odd 10
605.2.g.i.366.2 8 44.39 even 10
605.2.g.i.511.1 8 44.7 even 10
605.2.g.i.511.2 8 44.15 odd 10
3025.2.a.l.1.1 2 20.19 odd 2
3025.2.a.l.1.2 2 220.219 even 2
5445.2.a.u.1.1 2 12.11 even 2
5445.2.a.u.1.2 2 132.131 odd 2
9680.2.a.bu.1.1 2 11.10 odd 2 inner
9680.2.a.bu.1.2 2 1.1 even 1 trivial