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Properties

Label 9680.2.a.bv.1.1
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 = [2,0,2,0,2,0,0,0,-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: \(1\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 2420)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} -1.73205 q^{7} -2.00000 q^{9} +1.00000 q^{15} +6.92820 q^{19} -1.73205 q^{21} +1.00000 q^{25} -5.00000 q^{27} +2.00000 q^{31} -1.73205 q^{35} -8.00000 q^{37} -5.19615 q^{41} -5.19615 q^{43} -2.00000 q^{45} +3.00000 q^{47} -4.00000 q^{49} -12.0000 q^{53} +6.92820 q^{57} +6.00000 q^{59} +12.1244 q^{61} +3.46410 q^{63} -7.00000 q^{67} -6.00000 q^{71} -6.92820 q^{73} +1.00000 q^{75} -3.46410 q^{79} +1.00000 q^{81} -10.3923 q^{83} -9.00000 q^{89} +2.00000 q^{93} +6.92820 q^{95} -4.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 4 q^{9} + 2 q^{15} + 2 q^{25} - 10 q^{27} + 4 q^{31} - 16 q^{37} - 4 q^{45} + 6 q^{47} - 8 q^{49} - 24 q^{53} + 12 q^{59} - 14 q^{67} - 12 q^{71} + 2 q^{75} + 2 q^{81} - 18 q^{89}+ \cdots - 8 q^{97}+O(q^{100}) \) 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 \(-0.606148\pi\)
−0.327327 + 0.944911i \(0.606148\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 6.92820 1.58944 0.794719 − 0.606977i \(-0.207618\pi\)
0.794719 + 0.606977i \(0.207618\pi\)
\(20\) 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
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 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
\(39\) 0 0
\(40\) 0 0
\(41\) −5.19615 −0.811503 −0.405751 − 0.913984i \(-0.632990\pi\)
−0.405751 + 0.913984i \(0.632990\pi\)
\(42\) 0 0
\(43\) −5.19615 −0.792406 −0.396203 − 0.918163i \(-0.629672\pi\)
−0.396203 + 0.918163i \(0.629672\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 3.00000 0.437595 0.218797 − 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) −4.00000 −0.571429
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 − 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.92820 0.917663
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 12.1244 1.55236 0.776182 − 0.630509i \(-0.217154\pi\)
0.776182 + 0.630509i \(0.217154\pi\)
\(62\) 0 0
\(63\) 3.46410 0.436436
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −7.00000 −0.855186 −0.427593 − 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −6.92820 −0.810885 −0.405442 − 0.914121i \(-0.632883\pi\)
−0.405442 + 0.914121i \(0.632883\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.46410 −0.389742 −0.194871 − 0.980829i \(-0.562429\pi\)
−0.194871 + 0.980829i \(0.562429\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.3923 −1.14070 −0.570352 − 0.821401i \(-0.693193\pi\)
−0.570352 + 0.821401i \(0.693193\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 − 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.00000 0.207390
\(94\) 0 0
\(95\) 6.92820 0.710819
\(96\) 0 0
\(97\) −4.00000 −0.406138 −0.203069 − 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 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.bv.1.1 2
4.3 odd 2 2420.2.a.h.1.2 yes 2
11.10 odd 2 inner 9680.2.a.bv.1.2 2
44.43 even 2 2420.2.a.h.1.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2420.2.a.h.1.1 ✓ 2 44.43 even 2
2420.2.a.h.1.2 yes 2 4.3 odd 2
9680.2.a.bv.1.1 2 1.1 even 1 trivial
9680.2.a.bv.1.2 2 11.10 odd 2 inner