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Properties

Label 9680.2.a.bg.1.2
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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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,-4,0,2,0,0,0,2,0,0,0,0,0,-4,0,0] 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: \(1\)
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: \( 2 \)
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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} +1.00000 q^{5} +3.46410 q^{7} +1.00000 q^{9} -2.00000 q^{15} +6.92820 q^{17} -6.92820 q^{19} -6.92820 q^{21} -6.00000 q^{23} +1.00000 q^{25} +4.00000 q^{27} -4.00000 q^{31} +3.46410 q^{35} +10.0000 q^{37} -6.92820 q^{41} -3.46410 q^{43} +1.00000 q^{45} +6.00000 q^{47} +5.00000 q^{49} -13.8564 q^{51} -6.00000 q^{53} +13.8564 q^{57} -6.92820 q^{61} +3.46410 q^{63} -10.0000 q^{67} +12.0000 q^{69} +6.92820 q^{73} -2.00000 q^{75} -6.92820 q^{79} -11.0000 q^{81} -17.3205 q^{83} +6.92820 q^{85} -6.00000 q^{89} +8.00000 q^{93} -6.92820 q^{95} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 2 q^{5} + 2 q^{9} - 4 q^{15} - 12 q^{23} + 2 q^{25} + 8 q^{27} - 8 q^{31} + 20 q^{37} + 2 q^{45} + 12 q^{47} + 10 q^{49} - 12 q^{53} - 20 q^{67} + 24 q^{69} - 4 q^{75} - 22 q^{81} - 12 q^{89}+ \cdots - 20 q^{97}+O(q^{100}) \) 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
\(3\) −2.00000 −1.15470 −0.577350 − 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.46410 1.30931 0.654654 − 0.755929i \(-0.272814\pi\)
0.654654 + 0.755929i \(0.272814\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) 6.92820 1.68034 0.840168 − 0.542326i \(-0.182456\pi\)
0.840168 + 0.542326i \(0.182456\pi\)
\(18\) 0 0
\(19\) −6.92820 −1.58944 −0.794719 − 0.606977i \(-0.792382\pi\)
−0.794719 + 0.606977i \(0.792382\pi\)
\(20\) 0 0
\(21\) −6.92820 −1.51186
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.46410 0.585540
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 − 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.92820 −1.08200 −0.541002 − 0.841021i \(-0.681955\pi\)
−0.541002 + 0.841021i \(0.681955\pi\)
\(42\) 0 0
\(43\) −3.46410 −0.528271 −0.264135 − 0.964486i \(-0.585087\pi\)
−0.264135 + 0.964486i \(0.585087\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) −13.8564 −1.94029
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 13.8564 1.83533
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −6.92820 −0.887066 −0.443533 − 0.896258i \(-0.646275\pi\)
−0.443533 + 0.896258i \(0.646275\pi\)
\(62\) 0 0
\(63\) 3.46410 0.436436
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 − 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) 0 0
\(69\) 12.0000 1.44463
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 6.92820 0.810885 0.405442 − 0.914121i \(-0.367117\pi\)
0.405442 + 0.914121i \(0.367117\pi\)
\(74\) 0 0
\(75\) −2.00000 −0.230940
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.92820 −0.779484 −0.389742 − 0.920924i \(-0.627436\pi\)
−0.389742 + 0.920924i \(0.627436\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −17.3205 −1.90117 −0.950586 − 0.310460i \(-0.899517\pi\)
−0.950586 + 0.310460i \(0.899517\pi\)
\(84\) 0 0
\(85\) 6.92820 0.751469
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) −6.92820 −0.710819
\(96\) 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.bg.1.2 2
4.3 odd 2 605.2.a.f.1.1 ✓ 2
11.10 odd 2 inner 9680.2.a.bg.1.1 2
12.11 even 2 5445.2.a.r.1.2 2
20.19 odd 2 3025.2.a.j.1.2 2
44.3 odd 10 605.2.g.h.251.1 8
44.7 even 10 605.2.g.h.511.2 8
44.15 odd 10 605.2.g.h.511.1 8
44.19 even 10 605.2.g.h.251.2 8
44.27 odd 10 605.2.g.h.366.2 8
44.31 odd 10 605.2.g.h.81.2 8
44.35 even 10 605.2.g.h.81.1 8
44.39 even 10 605.2.g.h.366.1 8
44.43 even 2 605.2.a.f.1.2 yes 2
132.131 odd 2 5445.2.a.r.1.1 2
220.219 even 2 3025.2.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
605.2.a.f.1.1 ✓ 2 4.3 odd 2
605.2.a.f.1.2 yes 2 44.43 even 2
605.2.g.h.81.1 8 44.35 even 10
605.2.g.h.81.2 8 44.31 odd 10
605.2.g.h.251.1 8 44.3 odd 10
605.2.g.h.251.2 8 44.19 even 10
605.2.g.h.366.1 8 44.39 even 10
605.2.g.h.366.2 8 44.27 odd 10
605.2.g.h.511.1 8 44.15 odd 10
605.2.g.h.511.2 8 44.7 even 10
3025.2.a.j.1.1 2 220.219 even 2
3025.2.a.j.1.2 2 20.19 odd 2
5445.2.a.r.1.1 2 132.131 odd 2
5445.2.a.r.1.2 2 12.11 even 2
9680.2.a.bg.1.1 2 11.10 odd 2 inner
9680.2.a.bg.1.2 2 1.1 even 1 trivial