Properties

Label 105.4.g.a
Level $105$
Weight $4$
Character orbit 105.g
Analytic conductor $6.195$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.19520055060\)
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: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + ( 2 \beta_{1} - \beta_{3} ) q^{3} -8 q^{4} -5 \beta_{1} q^{5} + 7 \beta_{3} q^{7} + ( -13 - 4 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 2 \beta_{1} - \beta_{3} ) q^{3} -8 q^{4} -5 \beta_{1} q^{5} + 7 \beta_{3} q^{7} + ( -13 - 4 \beta_{2} ) q^{9} + 2 \beta_{2} q^{11} + ( -16 \beta_{1} + 8 \beta_{3} ) q^{12} + 32 \beta_{3} q^{13} + ( 50 + 5 \beta_{2} ) q^{15} + 64 q^{16} -46 \beta_{1} q^{17} + 40 \beta_{1} q^{20} + ( -49 + 14 \beta_{2} ) q^{21} -125 q^{25} + ( 2 \beta_{1} + 53 \beta_{3} ) q^{27} -56 \beta_{3} q^{28} -52 \beta_{2} q^{29} + ( -14 \beta_{1} - 20 \beta_{3} ) q^{33} -35 \beta_{2} q^{35} + ( 104 + 32 \beta_{2} ) q^{36} + ( -224 + 64 \beta_{2} ) q^{39} -16 \beta_{2} q^{44} + ( 65 \beta_{1} - 100 \beta_{3} ) q^{45} + 80 \beta_{1} q^{47} + ( 128 \beta_{1} - 64 \beta_{3} ) q^{48} + 343 q^{49} + ( 460 + 46 \beta_{2} ) q^{51} -256 \beta_{3} q^{52} + 50 \beta_{3} q^{55} + ( -400 - 40 \beta_{2} ) q^{60} + ( -196 \beta_{1} - 91 \beta_{3} ) q^{63} -512 q^{64} -160 \beta_{2} q^{65} + 368 \beta_{1} q^{68} + 146 \beta_{2} q^{71} + 428 \beta_{3} q^{73} + ( -250 \beta_{1} + 125 \beta_{3} ) q^{75} + 98 \beta_{1} q^{77} + 236 q^{79} -320 \beta_{1} q^{80} + ( -391 + 104 \beta_{2} ) q^{81} -676 \beta_{1} q^{83} + ( 392 - 112 \beta_{2} ) q^{84} -1150 q^{85} + ( 364 \beta_{1} + 520 \beta_{3} ) q^{87} + 1568 q^{91} -364 \beta_{3} q^{97} + ( 280 - 26 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 32q^{4} - 52q^{9} + O(q^{10}) \) \( 4q - 32q^{4} - 52q^{9} + 200q^{15} + 256q^{16} - 196q^{21} - 500q^{25} + 416q^{36} - 896q^{39} + 1372q^{49} + 1840q^{51} - 1600q^{60} - 2048q^{64} + 944q^{79} - 1564q^{81} + 1568q^{84} - 4600q^{85} + 6272q^{91} + 1120q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 1 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 1\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{3} + 2 \beta_{1}\)

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 −2.64575 4.47214i −8.00000 11.1803i 0 18.5203 0 −13.0000 + 23.6643i 0
104.2 0 −2.64575 + 4.47214i −8.00000 11.1803i 0 18.5203 0 −13.0000 23.6643i 0
104.3 0 2.64575 4.47214i −8.00000 11.1803i 0 −18.5203 0 −13.0000 23.6643i 0
104.4 0 2.64575 + 4.47214i −8.00000 11.1803i 0 −18.5203 0 −13.0000 + 23.6643i 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 105.4.g.a 4
3.b odd 2 1 inner 105.4.g.a 4
5.b even 2 1 inner 105.4.g.a 4
7.b odd 2 1 inner 105.4.g.a 4
15.d odd 2 1 inner 105.4.g.a 4
21.c even 2 1 inner 105.4.g.a 4
35.c odd 2 1 CM 105.4.g.a 4
105.g even 2 1 inner 105.4.g.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.g.a 4 1.a even 1 1 trivial
105.4.g.a 4 3.b odd 2 1 inner
105.4.g.a 4 5.b even 2 1 inner
105.4.g.a 4 7.b odd 2 1 inner
105.4.g.a 4 15.d odd 2 1 inner
105.4.g.a 4 21.c even 2 1 inner
105.4.g.a 4 35.c odd 2 1 CM
105.4.g.a 4 105.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{4}^{\mathrm{new}}(105, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 729 + 26 T^{2} + T^{4} \)
$5$ \( ( 125 + T^{2} )^{2} \)
$7$ \( ( -343 + T^{2} )^{2} \)
$11$ \( ( 140 + T^{2} )^{2} \)
$13$ \( ( -7168 + T^{2} )^{2} \)
$17$ \( ( 10580 + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( 94640 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( ( 32000 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( ( 746060 + T^{2} )^{2} \)
$73$ \( ( -1282288 + T^{2} )^{2} \)
$79$ \( ( -236 + T )^{4} \)
$83$ \( ( 2284880 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( ( -927472 + T^{2} )^{2} \)
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