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Properties

Label 98.4.a.b
Level $98$
Weight $4$
Character orbit 98.a
Self dual yes
Analytic conductor $5.782$
Analytic rank $0$
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] = [98,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(5.78218718056\)
Analytic rank: \(0\)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} - q^{3} + 4 q^{4} + 7 q^{5} + 2 q^{6} - 8 q^{8} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - q^{3} + 4 q^{4} + 7 q^{5} + 2 q^{6} - 8 q^{8} - 26 q^{9} - 14 q^{10} + 35 q^{11} - 4 q^{12} + 66 q^{13} - 7 q^{15} + 16 q^{16} + 59 q^{17} + 52 q^{18} + 137 q^{19} + 28 q^{20} - 70 q^{22} - 7 q^{23} + 8 q^{24} - 76 q^{25} - 132 q^{26} + 53 q^{27} + 106 q^{29} + 14 q^{30} + 75 q^{31} - 32 q^{32} - 35 q^{33} - 118 q^{34} - 104 q^{36} + 11 q^{37} - 274 q^{38} - 66 q^{39} - 56 q^{40} - 498 q^{41} + 260 q^{43} + 140 q^{44} - 182 q^{45} + 14 q^{46} - 171 q^{47} - 16 q^{48} + 152 q^{50} - 59 q^{51} + 264 q^{52} - 417 q^{53} - 106 q^{54} + 245 q^{55} - 137 q^{57} - 212 q^{58} - 17 q^{59} - 28 q^{60} + 51 q^{61} - 150 q^{62} + 64 q^{64} + 462 q^{65} + 70 q^{66} + 439 q^{67} + 236 q^{68} + 7 q^{69} - 784 q^{71} + 208 q^{72} + 295 q^{73} - 22 q^{74} + 76 q^{75} + 548 q^{76} + 132 q^{78} - 495 q^{79} + 112 q^{80} + 649 q^{81} + 996 q^{82} + 932 q^{83} + 413 q^{85} - 520 q^{86} - 106 q^{87} - 280 q^{88} - 873 q^{89} + 364 q^{90} - 28 q^{92} - 75 q^{93} + 342 q^{94} + 959 q^{95} + 32 q^{96} - 290 q^{97} - 910 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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
−2.00000 −1.00000 4.00000 7.00000 2.00000 0 −8.00000 −26.0000 −14.0000
\(n\): 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 98.4.a.b 1
3.b odd 2 1 882.4.a.k 1
4.b odd 2 1 784.4.a.l 1
5.b even 2 1 2450.4.a.bh 1
7.b odd 2 1 98.4.a.c 1
7.c even 3 2 14.4.c.b ✓ 2
7.d odd 6 2 98.4.c.e 2
21.c even 2 1 882.4.a.p 1
21.g even 6 2 882.4.g.d 2
21.h odd 6 2 126.4.g.c 2
28.d even 2 1 784.4.a.j 1
28.g odd 6 2 112.4.i.b 2
35.c odd 2 1 2450.4.a.bf 1
35.j even 6 2 350.4.e.b 2
35.l odd 12 4 350.4.j.d 4
56.k odd 6 2 448.4.i.d 2
56.p even 6 2 448.4.i.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b ✓ 2 7.c even 3 2
98.4.a.b 1 1.a even 1 1 trivial
98.4.a.c 1 7.b odd 2 1
98.4.c.e 2 7.d odd 6 2
112.4.i.b 2 28.g odd 6 2
126.4.g.c 2 21.h odd 6 2
350.4.e.b 2 35.j even 6 2
350.4.j.d 4 35.l odd 12 4
448.4.i.c 2 56.p even 6 2
448.4.i.d 2 56.k odd 6 2
784.4.a.j 1 28.d even 2 1
784.4.a.l 1 4.b odd 2 1
882.4.a.k 1 3.b odd 2 1
882.4.a.p 1 21.c even 2 1
882.4.g.d 2 21.g even 6 2
2450.4.a.bf 1 35.c odd 2 1
2450.4.a.bh 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(98))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T - 7 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 35 \) Copy content Toggle raw display
$13$ \( T - 66 \) Copy content Toggle raw display
$17$ \( T - 59 \) Copy content Toggle raw display
$19$ \( T - 137 \) Copy content Toggle raw display
$23$ \( T + 7 \) Copy content Toggle raw display
$29$ \( T - 106 \) Copy content Toggle raw display
$31$ \( T - 75 \) Copy content Toggle raw display
$37$ \( T - 11 \) Copy content Toggle raw display
$41$ \( T + 498 \) Copy content Toggle raw display
$43$ \( T - 260 \) Copy content Toggle raw display
$47$ \( T + 171 \) Copy content Toggle raw display
$53$ \( T + 417 \) Copy content Toggle raw display
$59$ \( T + 17 \) Copy content Toggle raw display
$61$ \( T - 51 \) Copy content Toggle raw display
$67$ \( T - 439 \) Copy content Toggle raw display
$71$ \( T + 784 \) Copy content Toggle raw display
$73$ \( T - 295 \) Copy content Toggle raw display
$79$ \( T + 495 \) Copy content Toggle raw display
$83$ \( T - 932 \) Copy content Toggle raw display
$89$ \( T + 873 \) Copy content Toggle raw display
$97$ \( T + 290 \) Copy content Toggle raw display
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