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Properties

Label 105.2.g.b
Level $105$
Weight $2$
Character orbit 105.g
Analytic conductor $0.838$
Analytic rank $0$
Dimension $4$
CM discriminant -35
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [105,2,Mod(104,105)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(105, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("105.104");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 105.g (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(0.838429221223\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-5}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - 2 q^{4} + (\beta_{3} + \beta_1) q^{5} + (\beta_{3} - \beta_1) q^{7} + ( - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - 2 q^{4} + (\beta_{3} + \beta_1) q^{5} + (\beta_{3} - \beta_1) q^{7} + ( - \beta_{2} + 1) q^{9} + (2 \beta_{2} - 1) q^{11} - 2 \beta_{3} q^{12} + ( - \beta_{3} + \beta_1) q^{13} + ( - \beta_{2} - 2) q^{15} + 4 q^{16} + ( - \beta_{3} - \beta_1) q^{17} + ( - 2 \beta_{3} - 2 \beta_1) q^{20} + ( - \beta_{2} + 4) q^{21} - 5 q^{25} + (\beta_{3} + 3 \beta_1) q^{27} + ( - 2 \beta_{3} + 2 \beta_1) q^{28} + (2 \beta_{2} - 1) q^{29} + ( - \beta_{3} - 6 \beta_1) q^{33} + ( - 2 \beta_{2} + 1) q^{35} + (2 \beta_{2} - 2) q^{36} + (\beta_{2} - 4) q^{39} + ( - 4 \beta_{2} + 2) q^{44} + ( - 2 \beta_{3} + 3 \beta_1) q^{45} + (5 \beta_{3} + 5 \beta_1) q^{47} + 4 \beta_{3} q^{48} + 7 q^{49} + (\beta_{2} + 2) q^{51} + (2 \beta_{3} - 2 \beta_1) q^{52} + (5 \beta_{3} - 5 \beta_1) q^{55} + (2 \beta_{2} + 4) q^{60} + (4 \beta_{3} + 3 \beta_1) q^{63} - 8 q^{64} + (2 \beta_{2} - 1) q^{65} + (2 \beta_{3} + 2 \beta_1) q^{68} + ( - 4 \beta_{2} + 2) q^{71} + ( - 4 \beta_{3} + 4 \beta_1) q^{73} - 5 \beta_{3} q^{75} + ( - 7 \beta_{3} - 7 \beta_1) q^{77} - q^{79} + (4 \beta_{3} + 4 \beta_1) q^{80} + ( - \beta_{2} - 8) q^{81} + ( - 4 \beta_{3} - 4 \beta_1) q^{83} + (2 \beta_{2} - 8) q^{84} + 5 q^{85} + ( - \beta_{3} - 6 \beta_1) q^{87} - 7 q^{91} + ( - 7 \beta_{3} + 7 \beta_1) q^{97} + (\beta_{2} + 17) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 2 q^{9} - 10 q^{15} + 16 q^{16} + 14 q^{21} - 20 q^{25} - 4 q^{36} - 14 q^{39} + 28 q^{49} + 10 q^{51} + 20 q^{60} - 32 q^{64} - 4 q^{79} - 34 q^{81} - 28 q^{84} + 20 q^{85} - 28 q^{91} + 70 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - \nu ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} + \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
104.1
1.32288 − 1.11803i
1.32288 + 1.11803i
−1.32288 − 1.11803i
−1.32288 + 1.11803i
0 −1.32288 − 1.11803i −2.00000 − 2.23607i 0 −2.64575 0 0.500000 + 2.95804i 0
104.2 0 −1.32288 + 1.11803i −2.00000 2.23607i 0 −2.64575 0 0.500000 − 2.95804i 0
104.3 0 1.32288 − 1.11803i −2.00000 − 2.23607i 0 2.64575 0 0.500000 − 2.95804i 0
104.4 0 1.32288 + 1.11803i −2.00000 2.23607i 0 2.64575 0 0.500000 + 2.95804i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.2.g.b ✓ 4
3.b odd 2 1 inner 105.2.g.b ✓ 4
4.b odd 2 1 1680.2.k.b 4
5.b even 2 1 inner 105.2.g.b ✓ 4
5.c odd 4 2 525.2.b.f 4
7.b odd 2 1 inner 105.2.g.b ✓ 4
7.c even 3 2 735.2.p.b 8
7.d odd 6 2 735.2.p.b 8
12.b even 2 1 1680.2.k.b 4
15.d odd 2 1 inner 105.2.g.b ✓ 4
15.e even 4 2 525.2.b.f 4
20.d odd 2 1 1680.2.k.b 4
21.c even 2 1 inner 105.2.g.b ✓ 4
21.g even 6 2 735.2.p.b 8
21.h odd 6 2 735.2.p.b 8
28.d even 2 1 1680.2.k.b 4
35.c odd 2 1 CM 105.2.g.b ✓ 4
35.f even 4 2 525.2.b.f 4
35.i odd 6 2 735.2.p.b 8
35.j even 6 2 735.2.p.b 8
60.h even 2 1 1680.2.k.b 4
84.h odd 2 1 1680.2.k.b 4
105.g even 2 1 inner 105.2.g.b ✓ 4
105.k odd 4 2 525.2.b.f 4
105.o odd 6 2 735.2.p.b 8
105.p even 6 2 735.2.p.b 8
140.c even 2 1 1680.2.k.b 4
420.o odd 2 1 1680.2.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.g.b ✓ 4 1.a even 1 1 trivial
105.2.g.b ✓ 4 3.b odd 2 1 inner
105.2.g.b ✓ 4 5.b even 2 1 inner
105.2.g.b ✓ 4 7.b odd 2 1 inner
105.2.g.b ✓ 4 15.d odd 2 1 inner
105.2.g.b ✓ 4 21.c even 2 1 inner
105.2.g.b ✓ 4 35.c odd 2 1 CM
105.2.g.b ✓ 4 105.g even 2 1 inner
525.2.b.f 4 5.c odd 4 2
525.2.b.f 4 15.e even 4 2
525.2.b.f 4 35.f even 4 2
525.2.b.f 4 105.k odd 4 2
735.2.p.b 8 7.c even 3 2
735.2.p.b 8 7.d odd 6 2
735.2.p.b 8 21.g even 6 2
735.2.p.b 8 21.h odd 6 2
735.2.p.b 8 35.i odd 6 2
735.2.p.b 8 35.j even 6 2
735.2.p.b 8 105.o odd 6 2
735.2.p.b 8 105.p even 6 2
1680.2.k.b 4 4.b odd 2 1
1680.2.k.b 4 12.b even 2 1
1680.2.k.b 4 20.d odd 2 1
1680.2.k.b 4 28.d even 2 1
1680.2.k.b 4 60.h even 2 1
1680.2.k.b 4 84.h odd 2 1
1680.2.k.b 4 140.c even 2 1
1680.2.k.b 4 420.o odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(105, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{13}^{2} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 35)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 35)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 125)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 140)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 112)^{2} \) Copy content Toggle raw display
$79$ \( (T + 1)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 343)^{2} \) Copy content Toggle raw display
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