Properties

Label 98.6.a.b
Level $98$
Weight $6$
Character orbit 98.a
Self dual yes
Analytic conductor $15.718$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [98,6,Mod(1,98)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(98, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(15.7176143417\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 4 q^{2} - 8 q^{3} + 16 q^{4} - 10 q^{5} - 32 q^{6} + 64 q^{8} - 179 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} - 8 q^{3} + 16 q^{4} - 10 q^{5} - 32 q^{6} + 64 q^{8} - 179 q^{9} - 40 q^{10} - 340 q^{11} - 128 q^{12} + 294 q^{13} + 80 q^{15} + 256 q^{16} - 1226 q^{17} - 716 q^{18} - 2432 q^{19} - 160 q^{20} - 1360 q^{22} + 2000 q^{23} - 512 q^{24} - 3025 q^{25} + 1176 q^{26} + 3376 q^{27} - 6746 q^{29} + 320 q^{30} - 8856 q^{31} + 1024 q^{32} + 2720 q^{33} - 4904 q^{34} - 2864 q^{36} + 9182 q^{37} - 9728 q^{38} - 2352 q^{39} - 640 q^{40} + 14574 q^{41} + 8108 q^{43} - 5440 q^{44} + 1790 q^{45} + 8000 q^{46} + 312 q^{47} - 2048 q^{48} - 12100 q^{50} + 9808 q^{51} + 4704 q^{52} - 14634 q^{53} + 13504 q^{54} + 3400 q^{55} + 19456 q^{57} - 26984 q^{58} + 27656 q^{59} + 1280 q^{60} - 34338 q^{61} - 35424 q^{62} + 4096 q^{64} - 2940 q^{65} + 10880 q^{66} + 12316 q^{67} - 19616 q^{68} - 16000 q^{69} + 36920 q^{71} - 11456 q^{72} + 61718 q^{73} + 36728 q^{74} + 24200 q^{75} - 38912 q^{76} - 9408 q^{78} - 64752 q^{79} - 2560 q^{80} + 16489 q^{81} + 58296 q^{82} + 77056 q^{83} + 12260 q^{85} + 32432 q^{86} + 53968 q^{87} - 21760 q^{88} + 8166 q^{89} + 7160 q^{90} + 32000 q^{92} + 70848 q^{93} + 1248 q^{94} + 24320 q^{95} - 8192 q^{96} - 20650 q^{97} + 60860 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
4.00000 −8.00000 16.0000 −10.0000 −32.0000 0 64.0000 −179.000 −40.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.6.a.b 1
3.b odd 2 1 882.6.a.g 1
4.b odd 2 1 784.6.a.h 1
7.b odd 2 1 14.6.a.b 1
7.c even 3 2 98.6.c.b 2
7.d odd 6 2 98.6.c.a 2
21.c even 2 1 126.6.a.c 1
28.d even 2 1 112.6.a.d 1
35.c odd 2 1 350.6.a.b 1
35.f even 4 2 350.6.c.f 2
56.e even 2 1 448.6.a.k 1
56.h odd 2 1 448.6.a.f 1
84.h odd 2 1 1008.6.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.b 1 7.b odd 2 1
98.6.a.b 1 1.a even 1 1 trivial
98.6.c.a 2 7.d odd 6 2
98.6.c.b 2 7.c even 3 2
112.6.a.d 1 28.d even 2 1
126.6.a.c 1 21.c even 2 1
350.6.a.b 1 35.c odd 2 1
350.6.c.f 2 35.f even 4 2
448.6.a.f 1 56.h odd 2 1
448.6.a.k 1 56.e even 2 1
784.6.a.h 1 4.b odd 2 1
882.6.a.g 1 3.b odd 2 1
1008.6.a.n 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 8 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(98))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T + 8 \) Copy content Toggle raw display
$5$ \( T + 10 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 340 \) Copy content Toggle raw display
$13$ \( T - 294 \) Copy content Toggle raw display
$17$ \( T + 1226 \) Copy content Toggle raw display
$19$ \( T + 2432 \) Copy content Toggle raw display
$23$ \( T - 2000 \) Copy content Toggle raw display
$29$ \( T + 6746 \) Copy content Toggle raw display
$31$ \( T + 8856 \) Copy content Toggle raw display
$37$ \( T - 9182 \) Copy content Toggle raw display
$41$ \( T - 14574 \) Copy content Toggle raw display
$43$ \( T - 8108 \) Copy content Toggle raw display
$47$ \( T - 312 \) Copy content Toggle raw display
$53$ \( T + 14634 \) Copy content Toggle raw display
$59$ \( T - 27656 \) Copy content Toggle raw display
$61$ \( T + 34338 \) Copy content Toggle raw display
$67$ \( T - 12316 \) Copy content Toggle raw display
$71$ \( T - 36920 \) Copy content Toggle raw display
$73$ \( T - 61718 \) Copy content Toggle raw display
$79$ \( T + 64752 \) Copy content Toggle raw display
$83$ \( T - 77056 \) Copy content Toggle raw display
$89$ \( T - 8166 \) Copy content Toggle raw display
$97$ \( T + 20650 \) Copy content Toggle raw display
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