Properties

Label 7098.2.a.f
Level $7098$
Weight $2$
Character orbit 7098.a
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7098,2,Mod(1,7098)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7098, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7098.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.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: \(56.6778153547\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
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)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} + q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - 2 q^{10} + 4 q^{11} - q^{12} - q^{14} - 2 q^{15} + q^{16} + 2 q^{17} - q^{18} + 4 q^{19} + 2 q^{20} - q^{21} - 4 q^{22} + 8 q^{23} + q^{24} - q^{25} - q^{27} + q^{28} - 2 q^{29} + 2 q^{30} - q^{32} - 4 q^{33} - 2 q^{34} + 2 q^{35} + q^{36} + 10 q^{37} - 4 q^{38} - 2 q^{40} + 6 q^{41} + q^{42} - 4 q^{43} + 4 q^{44} + 2 q^{45} - 8 q^{46} - q^{48} + q^{49} + q^{50} - 2 q^{51} + 6 q^{53} + q^{54} + 8 q^{55} - q^{56} - 4 q^{57} + 2 q^{58} - 4 q^{59} - 2 q^{60} + 6 q^{61} + q^{63} + q^{64} + 4 q^{66} - 4 q^{67} + 2 q^{68} - 8 q^{69} - 2 q^{70} - 8 q^{71} - q^{72} - 10 q^{73} - 10 q^{74} + q^{75} + 4 q^{76} + 4 q^{77} + 2 q^{80} + q^{81} - 6 q^{82} + 4 q^{83} - q^{84} + 4 q^{85} + 4 q^{86} + 2 q^{87} - 4 q^{88} + 6 q^{89} - 2 q^{90} + 8 q^{92} + 8 q^{95} + q^{96} + 14 q^{97} - q^{98} + 4 q^{99}+O(q^{100}) \) 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
−1.00000 −1.00000 1.00000 2.00000 1.00000 1.00000 −1.00000 1.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(-1\)
\(13\) \(1\)

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 7098.2.a.f 1
13.b even 2 1 42.2.a.a 1
39.d odd 2 1 126.2.a.a 1
52.b odd 2 1 336.2.a.d 1
65.d even 2 1 1050.2.a.i 1
65.h odd 4 2 1050.2.g.a 2
91.b odd 2 1 294.2.a.g 1
91.r even 6 2 294.2.e.c 2
91.s odd 6 2 294.2.e.a 2
104.e even 2 1 1344.2.a.q 1
104.h odd 2 1 1344.2.a.i 1
117.n odd 6 2 1134.2.f.j 2
117.t even 6 2 1134.2.f.g 2
143.d odd 2 1 5082.2.a.d 1
156.h even 2 1 1008.2.a.j 1
195.e odd 2 1 3150.2.a.bo 1
195.s even 4 2 3150.2.g.r 2
208.o odd 4 2 5376.2.c.e 2
208.p even 4 2 5376.2.c.bc 2
260.g odd 2 1 8400.2.a.k 1
273.g even 2 1 882.2.a.b 1
273.w odd 6 2 882.2.g.h 2
273.ba even 6 2 882.2.g.j 2
312.b odd 2 1 4032.2.a.e 1
312.h even 2 1 4032.2.a.m 1
364.h even 2 1 2352.2.a.l 1
364.x even 6 2 2352.2.q.n 2
364.bl odd 6 2 2352.2.q.i 2
455.h odd 2 1 7350.2.a.f 1
728.b even 2 1 9408.2.a.bw 1
728.l odd 2 1 9408.2.a.n 1
1092.d odd 2 1 7056.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 13.b even 2 1
126.2.a.a 1 39.d odd 2 1
294.2.a.g 1 91.b odd 2 1
294.2.e.a 2 91.s odd 6 2
294.2.e.c 2 91.r even 6 2
336.2.a.d 1 52.b odd 2 1
882.2.a.b 1 273.g even 2 1
882.2.g.h 2 273.w odd 6 2
882.2.g.j 2 273.ba even 6 2
1008.2.a.j 1 156.h even 2 1
1050.2.a.i 1 65.d even 2 1
1050.2.g.a 2 65.h odd 4 2
1134.2.f.g 2 117.t even 6 2
1134.2.f.j 2 117.n odd 6 2
1344.2.a.i 1 104.h odd 2 1
1344.2.a.q 1 104.e even 2 1
2352.2.a.l 1 364.h even 2 1
2352.2.q.i 2 364.bl odd 6 2
2352.2.q.n 2 364.x even 6 2
3150.2.a.bo 1 195.e odd 2 1
3150.2.g.r 2 195.s even 4 2
4032.2.a.e 1 312.b odd 2 1
4032.2.a.m 1 312.h even 2 1
5082.2.a.d 1 143.d odd 2 1
5376.2.c.e 2 208.o odd 4 2
5376.2.c.bc 2 208.p even 4 2
7056.2.a.k 1 1092.d odd 2 1
7098.2.a.f 1 1.a even 1 1 trivial
7350.2.a.f 1 455.h odd 2 1
8400.2.a.k 1 260.g odd 2 1
9408.2.a.n 1 728.l odd 2 1
9408.2.a.bw 1 728.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{17} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T - 8 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 10 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 4 \) Copy content Toggle raw display
$61$ \( T - 6 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T + 8 \) Copy content Toggle raw display
$73$ \( T + 10 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 4 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
show more
show less