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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Newspace parameters

Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 105.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.838429221223\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-5}, \sqrt{7})\)
Defining polynomial: \(x^{4} - x^{2} + 9\)
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

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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3} + \beta_{1}\)

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.

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 \(S_{2}^{\mathrm{new}}(105, [\chi])\):

\( T_{2} \)
\( T_{13}^{2} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} )^{4} \)
$3$ \( 1 - T^{2} + 9 T^{4} \)
$5$ \( ( 1 + 5 T^{2} )^{2} \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( ( 1 - 3 T + 11 T^{2} )^{2}( 1 + 3 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 + 19 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 29 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 19 T^{2} )^{4} \)
$23$ \( ( 1 + 23 T^{2} )^{4} \)
$29$ \( ( 1 - 9 T + 29 T^{2} )^{2}( 1 + 9 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 31 T^{2} )^{4} \)
$37$ \( ( 1 - 37 T^{2} )^{4} \)
$41$ \( ( 1 + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 43 T^{2} )^{4} \)
$47$ \( ( 1 + 31 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 53 T^{2} )^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{4} \)
$61$ \( ( 1 - 61 T^{2} )^{4} \)
$67$ \( ( 1 - 67 T^{2} )^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2}( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 34 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + T + 79 T^{2} )^{4} \)
$83$ \( ( 1 - 86 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 89 T^{2} )^{4} \)
$97$ \( ( 1 - 149 T^{2} + 9409 T^{4} )^{2} \)
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