Properties

Label 7098.2.a.bn
Level $7098$
Weight $2$
Character orbit 7098.a
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
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: \(0\)
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: \(\Z[a_1, \ldots, a_{23}]\)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + 2 \beta q^{5} - q^{6} + q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} + 2 \beta q^{5} - q^{6} + q^{7} - q^{8} + q^{9} - 2 \beta q^{10} + 4 q^{11} + q^{12} - q^{14} + 2 \beta q^{15} + q^{16} + ( - 2 \beta + 3) q^{17} - q^{18} + ( - 2 \beta + 4) q^{19} + 2 \beta q^{20} + q^{21} - 4 q^{22} + ( - \beta + 4) q^{23} - q^{24} + 7 q^{25} + q^{27} + q^{28} + 8 q^{29} - 2 \beta q^{30} + (2 \beta - 3) q^{31} - q^{32} + 4 q^{33} + (2 \beta - 3) q^{34} + 2 \beta q^{35} + q^{36} + (4 \beta + 2) q^{37} + (2 \beta - 4) q^{38} - 2 \beta q^{40} - 4 \beta q^{41} - q^{42} + ( - 2 \beta + 1) q^{43} + 4 q^{44} + 2 \beta q^{45} + (\beta - 4) q^{46} + ( - 2 \beta - 4) q^{47} + q^{48} + q^{49} - 7 q^{50} + ( - 2 \beta + 3) q^{51} + ( - \beta - 10) q^{53} - q^{54} + 8 \beta q^{55} - q^{56} + ( - 2 \beta + 4) q^{57} - 8 q^{58} + (\beta + 8) q^{59} + 2 \beta q^{60} + 3 \beta q^{61} + ( - 2 \beta + 3) q^{62} + q^{63} + q^{64} - 4 q^{66} + (3 \beta - 2) q^{67} + ( - 2 \beta + 3) q^{68} + ( - \beta + 4) q^{69} - 2 \beta q^{70} + ( - 4 \beta - 5) q^{71} - q^{72} + (4 \beta - 4) q^{73} + ( - 4 \beta - 2) q^{74} + 7 q^{75} + ( - 2 \beta + 4) q^{76} + 4 q^{77} + (2 \beta - 6) q^{79} + 2 \beta q^{80} + q^{81} + 4 \beta q^{82} - \beta q^{83} + q^{84} + (6 \beta - 12) q^{85} + (2 \beta - 1) q^{86} + 8 q^{87} - 4 q^{88} + (\beta - 10) q^{89} - 2 \beta q^{90} + ( - \beta + 4) q^{92} + (2 \beta - 3) q^{93} + (2 \beta + 4) q^{94} + (8 \beta - 12) q^{95} - q^{96} + ( - 6 \beta + 2) q^{97} - q^{98} + 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 8 q^{11} + 2 q^{12} - 2 q^{14} + 2 q^{16} + 6 q^{17} - 2 q^{18} + 8 q^{19} + 2 q^{21} - 8 q^{22} + 8 q^{23} - 2 q^{24} + 14 q^{25} + 2 q^{27} + 2 q^{28} + 16 q^{29} - 6 q^{31} - 2 q^{32} + 8 q^{33} - 6 q^{34} + 2 q^{36} + 4 q^{37} - 8 q^{38} - 2 q^{42} + 2 q^{43} + 8 q^{44} - 8 q^{46} - 8 q^{47} + 2 q^{48} + 2 q^{49} - 14 q^{50} + 6 q^{51} - 20 q^{53} - 2 q^{54} - 2 q^{56} + 8 q^{57} - 16 q^{58} + 16 q^{59} + 6 q^{62} + 2 q^{63} + 2 q^{64} - 8 q^{66} - 4 q^{67} + 6 q^{68} + 8 q^{69} - 10 q^{71} - 2 q^{72} - 8 q^{73} - 4 q^{74} + 14 q^{75} + 8 q^{76} + 8 q^{77} - 12 q^{79} + 2 q^{81} + 2 q^{84} - 24 q^{85} - 2 q^{86} + 16 q^{87} - 8 q^{88} - 20 q^{89} + 8 q^{92} - 6 q^{93} + 8 q^{94} - 24 q^{95} - 2 q^{96} + 4 q^{97} - 2 q^{98} + 8 q^{99}+O(q^{100}) \) 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 −3.46410 −1.00000 1.00000 −1.00000 1.00000 3.46410
1.2 −1.00000 1.00000 1.00000 3.46410 −1.00000 1.00000 −1.00000 1.00000 −3.46410
\(n\): 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.bn 2
13.b even 2 1 7098.2.a.bz 2
13.f odd 12 2 546.2.s.c 4
39.k even 12 2 1638.2.bj.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.c 4 13.f odd 12 2
1638.2.bj.e 4 39.k even 12 2
7098.2.a.bn 2 1.a even 1 1 trivial
7098.2.a.bz 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5}^{2} - 12 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} - 3 \) 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
$5$ \( T^{2} - 12 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T - 3 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} - 8T + 13 \) Copy content Toggle raw display
$29$ \( (T - 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 6T - 3 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
$41$ \( T^{2} - 48 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T - 11 \) Copy content Toggle raw display
$47$ \( T^{2} + 8T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 20T + 97 \) Copy content Toggle raw display
$59$ \( T^{2} - 16T + 61 \) Copy content Toggle raw display
$61$ \( T^{2} - 27 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 23 \) Copy content Toggle raw display
$71$ \( T^{2} + 10T - 23 \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 32 \) Copy content Toggle raw display
$79$ \( T^{2} + 12T + 24 \) Copy content Toggle raw display
$83$ \( T^{2} - 3 \) Copy content Toggle raw display
$89$ \( T^{2} + 20T + 97 \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 104 \) Copy content Toggle raw display
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