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Properties

Label 9800.2.a.dc.1.3
Level $9800$
Weight $2$
Character 9800.1
Self dual yes
Analytic conductor $78.253$
Analytic rank $0$
Dimension $10$
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] = [9800,2,Mod(1,9800)] 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("9800.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9800.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,0,0,0,0,14,0,-12,0,0,0,0,0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(78.2533939809\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 22x^{8} - 16x^{7} + 146x^{6} + 200x^{5} - 206x^{4} - 440x^{3} - 124x^{2} + 72x + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 1960)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.609577\) of defining polynomial
Character \(\chi\) \(=\) 9800.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.35056 q^{3} -1.17599 q^{9} +3.15296 q^{11} +1.17296 q^{13} +2.07395 q^{17} +1.92745 q^{19} -4.51311 q^{23} +5.63992 q^{27} -6.97293 q^{29} +3.07731 q^{31} -4.25827 q^{33} +11.4901 q^{37} -1.58416 q^{39} -2.79587 q^{41} +3.22814 q^{43} -1.89635 q^{47} -2.80099 q^{51} +6.64710 q^{53} -2.60313 q^{57} -7.93376 q^{59} -8.15581 q^{61} +8.63580 q^{67} +6.09523 q^{69} -2.84298 q^{71} +5.35195 q^{73} +15.5300 q^{79} -4.08910 q^{81} -18.0496 q^{83} +9.41736 q^{87} +2.01829 q^{89} -4.15609 q^{93} +7.00968 q^{97} -3.70784 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 14 q^{9} - 12 q^{11} + 16 q^{23} + 12 q^{29} + 36 q^{37} - 20 q^{39} + 24 q^{43} - 36 q^{51} + 8 q^{53} + 16 q^{57} + 40 q^{67} - 8 q^{71} - 4 q^{79} + 50 q^{81} + 48 q^{93} - 72 q^{99}+O(q^{100}) \) 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
\(3\) −1.35056 −0.779746 −0.389873 − 0.920869i \(-0.627481\pi\)
−0.389873 + 0.920869i \(0.627481\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.17599 −0.391995
\(10\) 0 0
\(11\) 3.15296 0.950654 0.475327 − 0.879809i \(-0.342330\pi\)
0.475327 + 0.879809i \(0.342330\pi\)
\(12\) 0 0
\(13\) 1.17296 0.325322 0.162661 − 0.986682i \(-0.447992\pi\)
0.162661 + 0.986682i \(0.447992\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.07395 0.503006 0.251503 − 0.967857i \(-0.419075\pi\)
0.251503 + 0.967857i \(0.419075\pi\)
\(18\) 0 0
\(19\) 1.92745 0.442186 0.221093 − 0.975253i \(-0.429038\pi\)
0.221093 + 0.975253i \(0.429038\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.51311 −0.941049 −0.470524 − 0.882387i \(-0.655935\pi\)
−0.470524 + 0.882387i \(0.655935\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.63992 1.08540
\(28\) 0 0
\(29\) −6.97293 −1.29484 −0.647420 − 0.762134i \(-0.724152\pi\)
−0.647420 + 0.762134i \(0.724152\pi\)
\(30\) 0 0
\(31\) 3.07731 0.552701 0.276351 − 0.961057i \(-0.410875\pi\)
0.276351 + 0.961057i \(0.410875\pi\)
\(32\) 0 0
\(33\) −4.25827 −0.741269
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 11.4901 1.88896 0.944480 − 0.328570i \(-0.106567\pi\)
0.944480 + 0.328570i \(0.106567\pi\)
\(38\) 0 0
\(39\) −1.58416 −0.253669
\(40\) 0 0
\(41\) −2.79587 −0.436641 −0.218321 − 0.975877i \(-0.570058\pi\)
−0.218321 + 0.975877i \(0.570058\pi\)
\(42\) 0 0
\(43\) 3.22814 0.492286 0.246143 − 0.969234i \(-0.420837\pi\)
0.246143 + 0.969234i \(0.420837\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.89635 −0.276611 −0.138306 − 0.990390i \(-0.544166\pi\)
−0.138306 + 0.990390i \(0.544166\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −2.80099 −0.392217
\(52\) 0 0
\(53\) 6.64710 0.913050 0.456525 − 0.889711i \(-0.349094\pi\)
0.456525 + 0.889711i \(0.349094\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.60313 −0.344793
\(58\) 0 0
\(59\) −7.93376 −1.03289 −0.516444 − 0.856321i \(-0.672745\pi\)
−0.516444 + 0.856321i \(0.672745\pi\)
\(60\) 0 0
\(61\) −8.15581 −1.04424 −0.522122 − 0.852871i \(-0.674860\pi\)
−0.522122 + 0.852871i \(0.674860\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.63580 1.05503 0.527515 − 0.849545i \(-0.323124\pi\)
0.527515 + 0.849545i \(0.323124\pi\)
\(68\) 0 0
\(69\) 6.09523 0.733780
\(70\) 0 0
\(71\) −2.84298 −0.337400 −0.168700 − 0.985667i \(-0.553957\pi\)
−0.168700 + 0.985667i \(0.553957\pi\)
\(72\) 0 0
\(73\) 5.35195 0.626399 0.313199 − 0.949687i \(-0.398599\pi\)
0.313199 + 0.949687i \(0.398599\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 15.5300 1.74726 0.873631 − 0.486588i \(-0.161759\pi\)
0.873631 + 0.486588i \(0.161759\pi\)
\(80\) 0 0
\(81\) −4.08910 −0.454344
\(82\) 0 0
\(83\) −18.0496 −1.98120 −0.990599 − 0.136795i \(-0.956320\pi\)
−0.990599 + 0.136795i \(0.956320\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.41736 1.00965
\(88\) 0 0
\(89\) 2.01829 0.213938 0.106969 − 0.994262i \(-0.465885\pi\)
0.106969 + 0.994262i \(0.465885\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.15609 −0.430967
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.00968 0.711725 0.355862 − 0.934538i \(-0.384187\pi\)
0.355862 + 0.934538i \(0.384187\pi\)
\(98\) 0 0
\(99\) −3.70784 −0.372652
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 9800.2.a.dc.1.3 10
5.2 odd 4 1960.2.g.g.1569.15 yes 20
5.3 odd 4 1960.2.g.g.1569.5 ✓ 20
5.4 even 2 9800.2.a.db.1.8 10
7.6 odd 2 inner 9800.2.a.dc.1.8 10
35.13 even 4 1960.2.g.g.1569.16 yes 20
35.27 even 4 1960.2.g.g.1569.6 yes 20
35.34 odd 2 9800.2.a.db.1.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1960.2.g.g.1569.5 ✓ 20 5.3 odd 4
1960.2.g.g.1569.6 yes 20 35.27 even 4
1960.2.g.g.1569.15 yes 20 5.2 odd 4
1960.2.g.g.1569.16 yes 20 35.13 even 4
9800.2.a.db.1.3 10 35.34 odd 2
9800.2.a.db.1.8 10 5.4 even 2
9800.2.a.dc.1.3 10 1.1 even 1 trivial
9800.2.a.dc.1.8 10 7.6 odd 2 inner