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Properties

Label 98.2.g.a
Level $98$
Weight $2$
Character orbit 98.g
Analytic conductor $0.783$
Analytic rank $0$
Dimension $24$
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] = [98,2,Mod(9,98)] 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("98.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 98.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.782533939809\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} + 7 q^{3} + 2 q^{4} - 7 q^{6} + 4 q^{8} - 33 q^{9} - 7 q^{11} + 14 q^{13} - 7 q^{15} + 2 q^{16} - 7 q^{17} + 5 q^{18} - 7 q^{20} + 7 q^{21} - 7 q^{22} - 21 q^{23} + 4 q^{25} - 7 q^{26} - 35 q^{27}+ \cdots - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0.733052 + 0.680173i 0.532240 + 0.0802223i 0.0747301 + 0.997204i 0.459870 + 1.17173i 0.335594 + 0.420822i −1.68832 − 2.03705i −0.623490 + 0.781831i −2.58987 − 0.798871i −0.459870 + 1.17173i
9.2 0.733052 + 0.680173i 1.17911 + 0.177723i 0.0747301 + 0.997204i −1.10349 − 2.81164i 0.743468 + 0.932280i 1.42353 + 2.23015i −0.623490 + 0.781831i −1.50800 − 0.465156i 1.10349 − 2.81164i
11.1 0.733052 − 0.680173i 0.532240 − 0.0802223i 0.0747301 − 0.997204i 0.459870 − 1.17173i 0.335594 − 0.420822i −1.68832 + 2.03705i −0.623490 − 0.781831i −2.58987 + 0.798871i −0.459870 − 1.17173i
11.2 0.733052 − 0.680173i 1.17911 − 0.177723i 0.0747301 − 0.997204i −1.10349 + 2.81164i 0.743468 − 0.932280i 1.42353 − 2.23015i −0.623490 − 0.781831i −1.50800 + 0.465156i 1.10349 + 2.81164i
23.1 −0.0747301 + 0.997204i −1.08331 + 0.334156i −0.988831 − 0.149042i 2.80532 + 2.60296i −0.252266 − 1.10525i −2.61464 + 0.404557i 0.222521 − 0.974928i −1.41682 + 0.965972i −2.80532 + 2.60296i
23.2 −0.0747301 + 0.997204i 1.52870 − 0.471543i −0.988831 − 0.149042i −0.144243 − 0.133838i 0.355984 + 1.55967i 2.44775 + 1.00425i 0.222521 − 0.974928i −0.364133 + 0.248262i 0.144243 − 0.133838i
25.1 −0.955573 + 0.294755i −0.692327 − 1.76402i 0.826239 − 0.563320i −4.01065 − 0.604508i 1.18152 + 1.48158i −2.17740 + 1.50297i −0.623490 + 0.781831i −0.433294 + 0.402038i 4.01065 − 0.604508i
25.2 −0.955573 + 0.294755i 1.04951 + 2.67410i 0.826239 − 0.563320i 0.926902 + 0.139708i −1.79108 − 2.24595i −2.29875 − 1.30987i −0.623490 + 0.781831i −3.85019 + 3.57245i −0.926902 + 0.139708i
37.1 −0.826239 + 0.563320i 0.0153718 − 0.0142629i 0.365341 − 0.930874i 1.15218 + 0.355400i −0.00466617 + 0.0204438i 2.64164 + 0.147471i 0.222521 + 0.974928i −0.224157 + 2.99117i −1.15218 + 0.355400i
37.2 −0.826239 + 0.563320i 2.11865 − 1.96582i 0.365341 − 0.930874i −0.596780 − 0.184082i −0.643125 + 2.81771i −2.13962 − 1.55628i 0.222521 + 0.974928i 0.400039 − 5.33815i 0.596780 − 0.184082i
39.1 0.988831 + 0.149042i −1.46985 + 1.00212i 0.955573 + 0.294755i 0.169008 + 2.25526i −1.60279 + 0.771862i 2.40601 − 1.10050i 0.900969 + 0.433884i 0.0601727 − 0.153317i −0.169008 + 2.25526i
39.2 0.988831 + 0.149042i 0.520118 − 0.354611i 0.955573 + 0.294755i −0.0807922 − 1.07810i 0.567161 − 0.273130i −2.64324 − 0.115218i 0.900969 + 0.433884i −0.951249 + 2.42374i 0.0807922 − 1.07810i
51.1 −0.955573 − 0.294755i −0.692327 + 1.76402i 0.826239 + 0.563320i −4.01065 + 0.604508i 1.18152 − 1.48158i −2.17740 − 1.50297i −0.623490 − 0.781831i −0.433294 − 0.402038i 4.01065 + 0.604508i
51.2 −0.955573 − 0.294755i 1.04951 − 2.67410i 0.826239 + 0.563320i 0.926902 − 0.139708i −1.79108 + 2.24595i −2.29875 + 1.30987i −0.623490 − 0.781831i −3.85019 − 3.57245i −0.926902 − 0.139708i
53.1 −0.826239 − 0.563320i 0.0153718 + 0.0142629i 0.365341 + 0.930874i 1.15218 − 0.355400i −0.00466617 − 0.0204438i 2.64164 − 0.147471i 0.222521 − 0.974928i −0.224157 − 2.99117i −1.15218 − 0.355400i
53.2 −0.826239 − 0.563320i 2.11865 + 1.96582i 0.365341 + 0.930874i −0.596780 + 0.184082i −0.643125 − 2.81771i −2.13962 + 1.55628i 0.222521 − 0.974928i 0.400039 + 5.33815i 0.596780 + 0.184082i
65.1 −0.365341 − 0.930874i −0.250133 − 3.33779i −0.733052 + 0.680173i −1.09780 + 0.748464i −3.01568 + 1.45228i 2.60370 + 0.469838i 0.900969 + 0.433884i −8.11181 + 1.22266i 1.09780 + 0.748464i
65.2 −0.365341 − 0.930874i 0.0519131 + 0.692733i −0.733052 + 0.680173i 1.52046 − 1.03663i 0.625881 − 0.301408i 2.03934 − 1.68555i 0.900969 + 0.433884i 2.48931 − 0.375203i −1.52046 − 1.03663i
81.1 −0.0747301 − 0.997204i −1.08331 − 0.334156i −0.988831 + 0.149042i 2.80532 − 2.60296i −0.252266 + 1.10525i −2.61464 − 0.404557i 0.222521 + 0.974928i −1.41682 − 0.965972i −2.80532 − 2.60296i
81.2 −0.0747301 − 0.997204i 1.52870 + 0.471543i −0.988831 + 0.149042i −0.144243 + 0.133838i 0.355984 − 1.55967i 2.44775 − 1.00425i 0.222521 + 0.974928i −0.364133 − 0.248262i 0.144243 + 0.133838i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.2
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.2.g.a ✓ 24
3.b odd 2 1 882.2.z.d 24
4.b odd 2 1 784.2.bg.a 24
7.b odd 2 1 686.2.g.b 24
7.c even 3 1 686.2.e.f 24
7.c even 3 1 686.2.g.a 24
7.d odd 6 1 686.2.e.e 24
7.d odd 6 1 686.2.g.c 24
49.e even 7 1 686.2.g.a 24
49.f odd 14 1 686.2.g.c 24
49.g even 21 1 inner 98.2.g.a ✓ 24
49.g even 21 1 686.2.e.f 24
49.g even 21 1 4802.2.a.i 12
49.h odd 42 1 686.2.e.e 24
49.h odd 42 1 686.2.g.b 24
49.h odd 42 1 4802.2.a.k 12
147.n odd 42 1 882.2.z.d 24
196.o odd 42 1 784.2.bg.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.g.a ✓ 24 1.a even 1 1 trivial
98.2.g.a ✓ 24 49.g even 21 1 inner
686.2.e.e 24 7.d odd 6 1
686.2.e.e 24 49.h odd 42 1
686.2.e.f 24 7.c even 3 1
686.2.e.f 24 49.g even 21 1
686.2.g.a 24 7.c even 3 1
686.2.g.a 24 49.e even 7 1
686.2.g.b 24 7.b odd 2 1
686.2.g.b 24 49.h odd 42 1
686.2.g.c 24 7.d odd 6 1
686.2.g.c 24 49.f odd 14 1
784.2.bg.a 24 4.b odd 2 1
784.2.bg.a 24 196.o odd 42 1
882.2.z.d 24 3.b odd 2 1
882.2.z.d 24 147.n odd 42 1
4802.2.a.i 12 49.g even 21 1
4802.2.a.k 12 49.h odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 7 T_{3}^{23} + 38 T_{3}^{22} - 140 T_{3}^{21} + 413 T_{3}^{20} - 861 T_{3}^{19} + 1387 T_{3}^{18} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display