Properties

Label 105.2.g.a
Level $105$
Weight $2$
Character orbit 105.g
Analytic conductor $0.838$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_1 - 1) q^{3} + q^{4} + ( - \beta_{2} + \beta_1) q^{5} + (\beta_{3} + \beta_{2}) q^{6} + (\beta_{3} + 1) q^{7} + \beta_{2} q^{8} + (2 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_1 - 1) q^{3} + q^{4} + ( - \beta_{2} + \beta_1) q^{5} + (\beta_{3} + \beta_{2}) q^{6} + (\beta_{3} + 1) q^{7} + \beta_{2} q^{8} + (2 \beta_1 - 1) q^{9} + ( - \beta_{3} + 3) q^{10} + 2 \beta_1 q^{11} + ( - \beta_1 - 1) q^{12} - 4 q^{13} + ( - \beta_{2} - 3 \beta_1) q^{14} + (\beta_{3} + \beta_{2} - \beta_1 + 2) q^{15} - 5 q^{16} + 2 \beta_1 q^{17} + ( - 2 \beta_{3} + \beta_{2}) q^{18} + ( - \beta_{2} + \beta_1) q^{20} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{21} - 2 \beta_{3} q^{22} - 2 \beta_{2} q^{23} + ( - \beta_{3} - \beta_{2}) q^{24} + ( - 2 \beta_{3} + 1) q^{25} + 4 \beta_{2} q^{26} + ( - \beta_1 + 5) q^{27} + (\beta_{3} + 1) q^{28} - 4 \beta_1 q^{29} + (\beta_{3} - 2 \beta_{2} - 3 \beta_1 - 3) q^{30} + 4 \beta_{3} q^{31} + 3 \beta_{2} q^{32} + ( - 2 \beta_1 + 4) q^{33} - 2 \beta_{3} q^{34} + ( - 3 \beta_{2} - 2 \beta_1) q^{35} + (2 \beta_1 - 1) q^{36} + (4 \beta_1 + 4) q^{39} + (\beta_{3} - 3) q^{40} - 2 \beta_{2} q^{41} + (\beta_{3} + \beta_{2} + 3 \beta_1 - 6) q^{42} + 2 \beta_{3} q^{43} + 2 \beta_1 q^{44} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 - 4) q^{45} + 6 q^{46} + 2 \beta_1 q^{47} + (5 \beta_1 + 5) q^{48} + (2 \beta_{3} - 5) q^{49} + ( - \beta_{2} + 6 \beta_1) q^{50} + ( - 2 \beta_1 + 4) q^{51} - 4 q^{52} + (\beta_{3} - 5 \beta_{2}) q^{54} + ( - 2 \beta_{3} - 4) q^{55} + (\beta_{2} + 3 \beta_1) q^{56} + 4 \beta_{3} q^{58} + 4 \beta_{2} q^{59} + (\beta_{3} + \beta_{2} - \beta_1 + 2) q^{60} - 4 \beta_{3} q^{61} - 12 \beta_1 q^{62} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 1) q^{63} + q^{64} + (4 \beta_{2} - 4 \beta_1) q^{65} + (2 \beta_{3} - 4 \beta_{2}) q^{66} - 2 \beta_{3} q^{67} + 2 \beta_1 q^{68} + (2 \beta_{3} + 2 \beta_{2}) q^{69} + (2 \beta_{3} + 9) q^{70} + 2 \beta_1 q^{71} + (2 \beta_{3} - \beta_{2}) q^{72} + 8 q^{73} + (2 \beta_{3} - 4 \beta_{2} - \beta_1 - 1) q^{75} + ( - 4 \beta_{2} + 2 \beta_1) q^{77} + ( - 4 \beta_{3} - 4 \beta_{2}) q^{78} + 8 q^{79} + (5 \beta_{2} - 5 \beta_1) q^{80} + ( - 4 \beta_1 - 7) q^{81} + 6 q^{82} + 2 \beta_1 q^{83} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{84} + ( - 2 \beta_{3} - 4) q^{85} - 6 \beta_1 q^{86} + (4 \beta_1 - 8) q^{87} + 2 \beta_{3} q^{88} + 6 \beta_{2} q^{89} + (\beta_{3} + 4 \beta_{2} + 6 \beta_1 - 3) q^{90} + ( - 4 \beta_{3} - 4) q^{91} - 2 \beta_{2} q^{92} + ( - 4 \beta_{3} + 8 \beta_{2}) q^{93} - 2 \beta_{3} q^{94} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{96} + 8 q^{97} + (5 \beta_{2} - 6 \beta_1) q^{98} + ( - 2 \beta_1 - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{4} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 4 q^{4} + 4 q^{7} - 4 q^{9} + 12 q^{10} - 4 q^{12} - 16 q^{13} + 8 q^{15} - 20 q^{16} - 4 q^{21} + 4 q^{25} + 20 q^{27} + 4 q^{28} - 12 q^{30} + 16 q^{33} - 4 q^{36} + 16 q^{39} - 12 q^{40} - 24 q^{42} - 16 q^{45} + 24 q^{46} + 20 q^{48} - 20 q^{49} + 16 q^{51} - 16 q^{52} - 16 q^{55} + 8 q^{60} - 4 q^{63} + 4 q^{64} + 36 q^{70} + 32 q^{73} - 4 q^{75} + 32 q^{79} - 28 q^{81} + 24 q^{82} - 4 q^{84} - 16 q^{85} - 32 q^{87} - 12 q^{90} - 16 q^{91} + 32 q^{97} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} + 5\beta_1 ) / 2 \) Copy content Toggle raw display

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
0.517638i
0.517638i
1.93185i
1.93185i
−1.73205 −1.00000 1.41421i 1.00000 −1.73205 + 1.41421i 1.73205 + 2.44949i 1.00000 + 2.44949i 1.73205 −1.00000 + 2.82843i 3.00000 2.44949i
104.2 −1.73205 −1.00000 + 1.41421i 1.00000 −1.73205 1.41421i 1.73205 2.44949i 1.00000 2.44949i 1.73205 −1.00000 2.82843i 3.00000 + 2.44949i
104.3 1.73205 −1.00000 1.41421i 1.00000 1.73205 + 1.41421i −1.73205 2.44949i 1.00000 2.44949i −1.73205 −1.00000 + 2.82843i 3.00000 + 2.44949i
104.4 1.73205 −1.00000 + 1.41421i 1.00000 1.73205 1.41421i −1.73205 + 2.44949i 1.00000 + 2.44949i −1.73205 −1.00000 2.82843i 3.00000 2.44949i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
35.c odd 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.a 4
3.b odd 2 1 inner 105.2.g.a 4
4.b odd 2 1 1680.2.k.c 4
5.b even 2 1 105.2.g.c yes 4
5.c odd 4 2 525.2.b.j 8
7.b odd 2 1 105.2.g.c yes 4
7.c even 3 2 735.2.p.c 8
7.d odd 6 2 735.2.p.a 8
12.b even 2 1 1680.2.k.c 4
15.d odd 2 1 105.2.g.c yes 4
15.e even 4 2 525.2.b.j 8
20.d odd 2 1 1680.2.k.a 4
21.c even 2 1 105.2.g.c yes 4
21.g even 6 2 735.2.p.a 8
21.h odd 6 2 735.2.p.c 8
28.d even 2 1 1680.2.k.a 4
35.c odd 2 1 inner 105.2.g.a 4
35.f even 4 2 525.2.b.j 8
35.i odd 6 2 735.2.p.c 8
35.j even 6 2 735.2.p.a 8
60.h even 2 1 1680.2.k.a 4
84.h odd 2 1 1680.2.k.a 4
105.g even 2 1 inner 105.2.g.a 4
105.k odd 4 2 525.2.b.j 8
105.o odd 6 2 735.2.p.a 8
105.p even 6 2 735.2.p.c 8
140.c even 2 1 1680.2.k.c 4
420.o odd 2 1 1680.2.k.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.g.a 4 1.a even 1 1 trivial
105.2.g.a 4 3.b odd 2 1 inner
105.2.g.a 4 35.c odd 2 1 inner
105.2.g.a 4 105.g even 2 1 inner
105.2.g.c yes 4 5.b even 2 1
105.2.g.c yes 4 7.b odd 2 1
105.2.g.c yes 4 15.d odd 2 1
105.2.g.c yes 4 21.c even 2 1
525.2.b.j 8 5.c odd 4 2
525.2.b.j 8 15.e even 4 2
525.2.b.j 8 35.f even 4 2
525.2.b.j 8 105.k odd 4 2
735.2.p.a 8 7.d odd 6 2
735.2.p.a 8 21.g even 6 2
735.2.p.a 8 35.j even 6 2
735.2.p.a 8 105.o odd 6 2
735.2.p.c 8 7.c even 3 2
735.2.p.c 8 21.h odd 6 2
735.2.p.c 8 35.i odd 6 2
735.2.p.c 8 105.p even 6 2
1680.2.k.a 4 20.d odd 2 1
1680.2.k.a 4 28.d even 2 1
1680.2.k.a 4 60.h even 2 1
1680.2.k.a 4 84.h odd 2 1
1680.2.k.c 4 4.b odd 2 1
1680.2.k.c 4 12.b even 2 1
1680.2.k.c 4 140.c even 2 1
1680.2.k.c 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 \(S_{2}^{\mathrm{new}}(105, [\chi])\):

\( T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 2T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$13$ \( (T + 4)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$73$ \( (T - 8)^{4} \) Copy content Toggle raw display
$79$ \( (T - 8)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$97$ \( (T - 8)^{4} \) Copy content Toggle raw display
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