Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [105,2,Mod(41,105)]
 
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, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("105.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 105.b (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: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) 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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_{2} - \beta_1) q^{2} + \beta_1 q^{3} + (\beta_{3} + \beta_1 - 1) q^{4} - q^{5} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{6} + ( - \beta_{2} - 2) q^{7} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{8} + ( - \beta_{3} - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_{2} - \beta_1) q^{2} + \beta_1 q^{3} + (\beta_{3} + \beta_1 - 1) q^{4} - q^{5} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{6} + ( - \beta_{2} - 2) q^{7} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{8} + ( - \beta_{3} - \beta_{2} + 1) q^{9} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{10} + (\beta_{3} - \beta_1) q^{11} + ( - \beta_{3} - \beta_{2} - \beta_1 + 4) q^{12} + (3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{13} + ( - \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 1) q^{14} - \beta_1 q^{15} + ( - \beta_{3} - \beta_1 + 3) q^{16} + ( - \beta_{3} - \beta_1 - 2) q^{17} + (3 \beta_{3} - 3 \beta_{2} - \beta_1) q^{18} + 2 \beta_{2} q^{19} + ( - \beta_{3} - \beta_1 + 1) q^{20} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{21} - 2 q^{22} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{23} + (5 \beta_{3} - 4 \beta_{2} - 3 \beta_1 + 1) q^{24} + q^{25} + (2 \beta_{3} + 2 \beta_1 - 8) q^{26} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{27} + ( - 5 \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{28} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{29} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{30} - 2 \beta_{2} q^{31} + (3 \beta_{3} - \beta_{2} - 3 \beta_1) q^{32} + (\beta_{3} + \beta_{2} + 2) q^{33} - 2 \beta_{2} q^{34} + (\beta_{2} + 2) q^{35} + ( - \beta_{3} + 2 \beta_{2} + 5 \beta_1 - 5) q^{36} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{37} + ( - 2 \beta_{3} - 2 \beta_1 + 2) q^{38} + ( - \beta_{3} + 5 \beta_{2} + 2 \beta_1 + 4) q^{39} + (\beta_{3} - 3 \beta_{2} - \beta_1) q^{40} + 6 q^{41} + (\beta_{3} - 5 \beta_{2} - 3 \beta_1 + 2) q^{42} + (2 \beta_{3} + 2 \beta_1) q^{43} + 2 \beta_{2} q^{44} + (\beta_{3} + \beta_{2} - 1) q^{45} + (2 \beta_{3} + 2 \beta_1 + 2) q^{46} + ( - \beta_{3} - \beta_1 + 4) q^{47} + (\beta_{3} + \beta_{2} + 3 \beta_1 - 4) q^{48} + (4 \beta_{2} + 1) q^{49} + (\beta_{3} - \beta_{2} - \beta_1) q^{50} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 4) q^{51} + ( - 6 \beta_{3} + 12 \beta_{2} + 6 \beta_1) q^{52} + ( - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{53} + ( - 5 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 5) q^{54} + ( - \beta_{3} + \beta_1) q^{55} + (\beta_{3} - 6 \beta_{2} - 3 \beta_1 + 7) q^{56} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{57} + (2 \beta_{3} + 2 \beta_1 - 4) q^{58} + (2 \beta_{3} + 2 \beta_1 + 4) q^{59} + (\beta_{3} + \beta_{2} + \beta_1 - 4) q^{60} - 4 \beta_{2} q^{61} + (2 \beta_{3} + 2 \beta_1 - 2) q^{62} + (3 \beta_{3} - 2 \beta_1 - 6) q^{63} + ( - \beta_{3} - \beta_1 - 1) q^{64} + ( - 3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{65} - 2 \beta_1 q^{66} + (2 \beta_{3} + 2 \beta_1) q^{67} - 6 q^{68} + ( - 6 \beta_{3} + 2 \beta_1 - 6) q^{69} + (\beta_{3} - 2 \beta_{2} - 3 \beta_1 + 1) q^{70} + (4 \beta_{3} - 6 \beta_{2} - 4 \beta_1) q^{71} + ( - 5 \beta_{3} + 7 \beta_{2} + 5 \beta_1 + 8) q^{72} - 4 \beta_{2} q^{73} + (2 \beta_{3} - 6 \beta_{2} - 2 \beta_1) q^{74} + \beta_1 q^{75} + (6 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{76} + ( - \beta_{3} + 3 \beta_1 + 2) q^{77} + ( - 2 \beta_{3} - 2 \beta_{2} - 8 \beta_1 + 8) q^{78} + ( - \beta_{3} - \beta_1) q^{79} + (\beta_{3} + \beta_1 - 3) q^{80} + ( - 3 \beta_{2} - 5 \beta_1 - 3) q^{81} + (6 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{82} + (4 \beta_{3} + 4 \beta_1 + 8) q^{83} + (5 \beta_{3} - 4 \beta_{2} - \beta_1 - 11) q^{84} + (\beta_{3} + \beta_1 + 2) q^{85} + ( - 4 \beta_{3} + 8 \beta_{2} + 4 \beta_1) q^{86} + ( - 3 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{87} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{88} + ( - 2 \beta_{3} - 2 \beta_1 - 10) q^{89} + ( - 3 \beta_{3} + 3 \beta_{2} + \beta_1) q^{90} + ( - 3 \beta_{3} + 4 \beta_{2} + 9 \beta_1) q^{91} + ( - 6 \beta_{3} + 2 \beta_{2} + 6 \beta_1) q^{92} + ( - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{93} + (6 \beta_{3} - 8 \beta_{2} - 6 \beta_1) q^{94} - 2 \beta_{2} q^{95} + (\beta_{3} + 4 \beta_{2} + \beta_1 + 5) q^{96} + (3 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{97} + ( - 3 \beta_{3} - \beta_{2} - 5 \beta_1 + 4) q^{98} + (2 \beta_{3} - \beta_{2} + \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} - 6 q^{4} - 4 q^{5} + 4 q^{6} - 8 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} - 6 q^{4} - 4 q^{5} + 4 q^{6} - 8 q^{7} + 5 q^{9} + 18 q^{12} - 6 q^{14} + q^{15} + 14 q^{16} - 6 q^{17} - 2 q^{18} + 6 q^{20} - q^{21} - 8 q^{22} + 2 q^{24} + 4 q^{25} - 36 q^{26} - 16 q^{27} + 12 q^{28} - 4 q^{30} + 7 q^{33} + 8 q^{35} - 24 q^{36} - 4 q^{37} + 12 q^{38} + 15 q^{39} + 24 q^{41} + 10 q^{42} - 4 q^{43} - 5 q^{45} + 4 q^{46} + 18 q^{47} - 20 q^{48} + 4 q^{49} - 15 q^{51} + 22 q^{54} + 30 q^{56} + 6 q^{57} - 20 q^{58} + 12 q^{59} - 18 q^{60} - 12 q^{62} - 25 q^{63} - 2 q^{64} + 2 q^{66} - 4 q^{67} - 24 q^{68} - 20 q^{69} + 6 q^{70} + 32 q^{72} - q^{75} + 6 q^{77} + 42 q^{78} + 2 q^{79} - 14 q^{80} - 7 q^{81} + 24 q^{83} - 48 q^{84} + 6 q^{85} + q^{87} - 4 q^{88} - 36 q^{89} + 2 q^{90} - 6 q^{91} - 6 q^{93} + 18 q^{96} + 24 q^{98} + 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} + 4\nu - 9 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} - 2\nu^{2} + 2\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} - 3\beta_{2} + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + \beta_{2} + 2\beta _1 + 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
−1.18614 + 1.26217i
1.68614 + 0.396143i
1.68614 0.396143i
−1.18614 1.26217i
2.52434i −1.68614 + 0.396143i −4.37228 −1.00000 1.00000 + 4.25639i −2.00000 1.73205i 5.98844i 2.68614 1.33591i 2.52434i
41.2 0.792287i 1.18614 + 1.26217i 1.37228 −1.00000 1.00000 0.939764i −2.00000 + 1.73205i 2.67181i −0.186141 + 2.99422i 0.792287i
41.3 0.792287i 1.18614 1.26217i 1.37228 −1.00000 1.00000 + 0.939764i −2.00000 1.73205i 2.67181i −0.186141 2.99422i 0.792287i
41.4 2.52434i −1.68614 0.396143i −4.37228 −1.00000 1.00000 4.25639i −2.00000 + 1.73205i 5.98844i 2.68614 + 1.33591i 2.52434i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c 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.b.c 4
3.b odd 2 1 105.2.b.d yes 4
4.b odd 2 1 1680.2.f.h 4
5.b even 2 1 525.2.b.g 4
5.c odd 4 2 525.2.g.e 8
7.b odd 2 1 105.2.b.d yes 4
7.c even 3 1 735.2.s.h 4
7.c even 3 1 735.2.s.i 4
7.d odd 6 1 735.2.s.g 4
7.d odd 6 1 735.2.s.j 4
12.b even 2 1 1680.2.f.g 4
15.d odd 2 1 525.2.b.e 4
15.e even 4 2 525.2.g.d 8
21.c even 2 1 inner 105.2.b.c 4
21.g even 6 1 735.2.s.h 4
21.g even 6 1 735.2.s.i 4
21.h odd 6 1 735.2.s.g 4
21.h odd 6 1 735.2.s.j 4
28.d even 2 1 1680.2.f.g 4
35.c odd 2 1 525.2.b.e 4
35.f even 4 2 525.2.g.d 8
84.h odd 2 1 1680.2.f.h 4
105.g even 2 1 525.2.b.g 4
105.k odd 4 2 525.2.g.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.b.c 4 1.a even 1 1 trivial
105.2.b.c 4 21.c even 2 1 inner
105.2.b.d yes 4 3.b odd 2 1
105.2.b.d yes 4 7.b odd 2 1
525.2.b.e 4 15.d odd 2 1
525.2.b.e 4 35.c odd 2 1
525.2.b.g 4 5.b even 2 1
525.2.b.g 4 105.g even 2 1
525.2.g.d 8 15.e even 4 2
525.2.g.d 8 35.f even 4 2
525.2.g.e 8 5.c odd 4 2
525.2.g.e 8 105.k odd 4 2
735.2.s.g 4 7.d odd 6 1
735.2.s.g 4 21.h odd 6 1
735.2.s.h 4 7.c even 3 1
735.2.s.h 4 21.g even 6 1
735.2.s.i 4 7.c even 3 1
735.2.s.i 4 21.g even 6 1
735.2.s.j 4 7.d odd 6 1
735.2.s.j 4 21.h odd 6 1
1680.2.f.g 4 12.b even 2 1
1680.2.f.g 4 28.d even 2 1
1680.2.f.h 4 4.b odd 2 1
1680.2.f.h 4 84.h odd 2 1

Hecke kernels

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

\( T_{2}^{4} + 7T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 3T_{17} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 7T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} - 2 T^{2} + 3 T + 9 \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 7T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{4} + 51T^{2} + 576 \) Copy content Toggle raw display
$17$ \( (T^{2} + 3 T - 6)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 76T^{2} + 256 \) Copy content Toggle raw display
$29$ \( T^{4} + 19T^{2} + 16 \) Copy content Toggle raw display
$31$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T - 32)^{2} \) Copy content Toggle raw display
$41$ \( (T - 6)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T - 32)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 9 T + 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 76T^{2} + 256 \) Copy content Toggle raw display
$59$ \( (T^{2} - 6 T - 24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T - 32)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 184T^{2} + 16 \) Copy content Toggle raw display
$73$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - T - 8)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 12 T - 96)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 18 T + 48)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 123T^{2} + 144 \) Copy content Toggle raw display
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