Properties

Label 7098.2.a.bj
Level $7098$
Weight $2$
Character orbit 7098.a
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $2$
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] = [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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + (\beta + 1) q^{5} + q^{6} + q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + (\beta + 1) q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + ( - \beta - 1) q^{10} + ( - \beta + 2) q^{11} - q^{12} - q^{14} + ( - \beta - 1) q^{15} + q^{16} + (\beta + 2) q^{17} - q^{18} + (2 \beta - 1) q^{19} + (\beta + 1) q^{20} - q^{21} + (\beta - 2) q^{22} - 2 \beta q^{23} + q^{24} + (2 \beta - 1) q^{25} - q^{27} + q^{28} - 3 q^{29} + (\beta + 1) q^{30} + ( - 5 \beta - 1) q^{31} - q^{32} + (\beta - 2) q^{33} + ( - \beta - 2) q^{34} + (\beta + 1) q^{35} + q^{36} + ( - \beta - 3) q^{37} + ( - 2 \beta + 1) q^{38} + ( - \beta - 1) q^{40} - 7 q^{41} + q^{42} + ( - \beta - 1) q^{43} + ( - \beta + 2) q^{44} + (\beta + 1) q^{45} + 2 \beta q^{46} + ( - 2 \beta + 1) q^{47} - q^{48} + q^{49} + ( - 2 \beta + 1) q^{50} + ( - \beta - 2) q^{51} + (2 \beta - 7) q^{53} + q^{54} + (\beta - 1) q^{55} - q^{56} + ( - 2 \beta + 1) q^{57} + 3 q^{58} + ( - 4 \beta - 6) q^{59} + ( - \beta - 1) q^{60} + (\beta - 10) q^{61} + (5 \beta + 1) q^{62} + q^{63} + q^{64} + ( - \beta + 2) q^{66} + (4 \beta - 6) q^{67} + (\beta + 2) q^{68} + 2 \beta q^{69} + ( - \beta - 1) q^{70} + (3 \beta + 3) q^{71} - q^{72} + (2 \beta + 10) q^{73} + (\beta + 3) q^{74} + ( - 2 \beta + 1) q^{75} + (2 \beta - 1) q^{76} + ( - \beta + 2) q^{77} + ( - 8 \beta - 3) q^{79} + (\beta + 1) q^{80} + q^{81} + 7 q^{82} + ( - 5 \beta - 3) q^{83} - q^{84} + (3 \beta + 5) q^{85} + (\beta + 1) q^{86} + 3 q^{87} + (\beta - 2) q^{88} + (2 \beta - 3) q^{89} + ( - \beta - 1) q^{90} - 2 \beta q^{92} + (5 \beta + 1) q^{93} + (2 \beta - 1) q^{94} + (\beta + 5) q^{95} + q^{96} + (\beta + 1) q^{97} - q^{98} + ( - \beta + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 4 q^{11} - 2 q^{12} - 2 q^{14} - 2 q^{15} + 2 q^{16} + 4 q^{17} - 2 q^{18} - 2 q^{19} + 2 q^{20} - 2 q^{21} - 4 q^{22} + 2 q^{24} - 2 q^{25} - 2 q^{27} + 2 q^{28} - 6 q^{29} + 2 q^{30} - 2 q^{31} - 2 q^{32} - 4 q^{33} - 4 q^{34} + 2 q^{35} + 2 q^{36} - 6 q^{37} + 2 q^{38} - 2 q^{40} - 14 q^{41} + 2 q^{42} - 2 q^{43} + 4 q^{44} + 2 q^{45} + 2 q^{47} - 2 q^{48} + 2 q^{49} + 2 q^{50} - 4 q^{51} - 14 q^{53} + 2 q^{54} - 2 q^{55} - 2 q^{56} + 2 q^{57} + 6 q^{58} - 12 q^{59} - 2 q^{60} - 20 q^{61} + 2 q^{62} + 2 q^{63} + 2 q^{64} + 4 q^{66} - 12 q^{67} + 4 q^{68} - 2 q^{70} + 6 q^{71} - 2 q^{72} + 20 q^{73} + 6 q^{74} + 2 q^{75} - 2 q^{76} + 4 q^{77} - 6 q^{79} + 2 q^{80} + 2 q^{81} + 14 q^{82} - 6 q^{83} - 2 q^{84} + 10 q^{85} + 2 q^{86} + 6 q^{87} - 4 q^{88} - 6 q^{89} - 2 q^{90} + 2 q^{93} - 2 q^{94} + 10 q^{95} + 2 q^{96} + 2 q^{97} - 2 q^{98} + 4 q^{99}+O(q^{100}) \) 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
−1.73205
1.73205
−1.00000 −1.00000 1.00000 −0.732051 1.00000 1.00000 −1.00000 1.00000 0.732051
1.2 −1.00000 −1.00000 1.00000 2.73205 1.00000 1.00000 −1.00000 1.00000 −2.73205
\(n\): 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.bj 2
13.b even 2 1 7098.2.a.bs 2
13.f odd 12 2 546.2.s.d 4
39.k even 12 2 1638.2.bj.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.d 4 13.f odd 12 2
1638.2.bj.d 4 39.k even 12 2
7098.2.a.bj 2 1.a even 1 1 trivial
7098.2.a.bs 2 13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5}^{2} - 2T_{5} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} + 1 \) Copy content Toggle raw display
\( T_{17}^{2} - 4T_{17} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 4T + 1 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 11 \) Copy content Toggle raw display
$23$ \( T^{2} - 12 \) Copy content Toggle raw display
$29$ \( (T + 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 74 \) Copy content Toggle raw display
$37$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$41$ \( (T + 7)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 11 \) Copy content Toggle raw display
$53$ \( T^{2} + 14T + 37 \) Copy content Toggle raw display
$59$ \( T^{2} + 12T - 12 \) Copy content Toggle raw display
$61$ \( T^{2} + 20T + 97 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T - 12 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T - 18 \) Copy content Toggle raw display
$73$ \( T^{2} - 20T + 88 \) Copy content Toggle raw display
$79$ \( T^{2} + 6T - 183 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T - 66 \) Copy content Toggle raw display
$89$ \( T^{2} + 6T - 3 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
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