Properties

Label 390.2.y.g
Level $390$
Weight $2$
Character orbit 390.y
Analytic conductor $3.114$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.y (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.303595776.1
Defining polynomial: \(x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 32 \nu^{4} + 16 \nu^{2} + 45 \)\()/144\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 32 \nu^{5} + 16 \nu^{3} + 45 \nu \)\()/432\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{6} - 16 \nu^{4} - 80 \nu^{2} - 225 \)\()/144\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 13 \nu \)\()/48\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} - 13 \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( 5 \nu^{7} + 16 \nu^{5} + 80 \nu^{3} + 225 \nu \)\()/144\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - 2 \beta_{4} - \beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(2 \beta_{7} - 3 \beta_{5} - 3 \beta_{3} - 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 5 \beta_{2}\)
\(\nu^{5}\)\(=\)\(-\beta_{7} + 15 \beta_{3}\)
\(\nu^{6}\)\(=\)\(-16 \beta_{6} - 13\)
\(\nu^{7}\)\(=\)\(48 \beta_{5} - 13 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(-1\) \(-1 - \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1
−0.396143 + 1.68614i
1.26217 1.18614i
0.396143 1.68614i
−1.26217 + 1.18614i
1.26217 + 1.18614i
−0.396143 1.68614i
−1.26217 1.18614i
0.396143 + 1.68614i
−0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i −1.50000 1.65831i 0.500000 + 0.866025i 2.00626 1.15831i 1.00000i 0.500000 + 0.866025i 0.469882 + 2.18614i
139.2 −0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i −1.50000 + 1.65831i 0.500000 + 0.866025i −3.73831 + 2.15831i 1.00000i 0.500000 + 0.866025i 2.12819 0.686141i
139.3 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i −1.50000 1.65831i 0.500000 + 0.866025i 3.73831 2.15831i 1.00000i 0.500000 + 0.866025i −0.469882 2.18614i
139.4 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i −1.50000 + 1.65831i 0.500000 + 0.866025i −2.00626 + 1.15831i 1.00000i 0.500000 + 0.866025i −2.12819 + 0.686141i
289.1 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i −1.50000 1.65831i 0.500000 0.866025i −3.73831 2.15831i 1.00000i 0.500000 0.866025i 2.12819 + 0.686141i
289.2 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i −1.50000 + 1.65831i 0.500000 0.866025i 2.00626 + 1.15831i 1.00000i 0.500000 0.866025i 0.469882 2.18614i
289.3 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i −1.50000 1.65831i 0.500000 0.866025i −2.00626 1.15831i 1.00000i 0.500000 0.866025i −2.12819 0.686141i
289.4 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i −1.50000 + 1.65831i 0.500000 0.866025i 3.73831 + 2.15831i 1.00000i 0.500000 0.866025i −0.469882 + 2.18614i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.y.g 8
3.b odd 2 1 1170.2.bp.g 8
5.b even 2 1 inner 390.2.y.g 8
5.c odd 4 1 1950.2.i.bb 4
5.c odd 4 1 1950.2.i.be 4
13.c even 3 1 inner 390.2.y.g 8
15.d odd 2 1 1170.2.bp.g 8
39.i odd 6 1 1170.2.bp.g 8
65.n even 6 1 inner 390.2.y.g 8
65.q odd 12 1 1950.2.i.bb 4
65.q odd 12 1 1950.2.i.be 4
195.x odd 6 1 1170.2.bp.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.y.g 8 1.a even 1 1 trivial
390.2.y.g 8 5.b even 2 1 inner
390.2.y.g 8 13.c even 3 1 inner
390.2.y.g 8 65.n even 6 1 inner
1170.2.bp.g 8 3.b odd 2 1
1170.2.bp.g 8 15.d odd 2 1
1170.2.bp.g 8 39.i odd 6 1
1170.2.bp.g 8 195.x odd 6 1
1950.2.i.bb 4 5.c odd 4 1
1950.2.i.bb 4 65.q odd 12 1
1950.2.i.be 4 5.c odd 4 1
1950.2.i.be 4 65.q odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 24 T_{7}^{6} + 476 T_{7}^{4} - 2400 T_{7}^{2} + 10000 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$3$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$5$ \( ( 5 + 3 T + T^{2} )^{4} \)
$7$ \( 10000 - 2400 T^{2} + 476 T^{4} - 24 T^{6} + T^{8} \)
$11$ \( ( 4 + 12 T + 38 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$13$ \( ( 169 + 23 T^{2} + T^{4} )^{2} \)
$17$ \( 3418801 - 166410 T^{2} + 6251 T^{4} - 90 T^{6} + T^{8} \)
$19$ \( ( 100 + 20 T + 14 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$23$ \( 38416 - 14112 T^{2} + 4988 T^{4} - 72 T^{6} + T^{8} \)
$29$ \( ( 9 - 3 T + T^{2} )^{4} \)
$31$ \( ( 4 + T )^{8} \)
$37$ \( 3418801 - 166410 T^{2} + 6251 T^{4} - 90 T^{6} + T^{8} \)
$41$ \( ( 25 - 40 T + 59 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$43$ \( 10000 - 2400 T^{2} + 476 T^{4} - 24 T^{6} + T^{8} \)
$47$ \( ( 196 + 72 T^{2} + T^{4} )^{2} \)
$53$ \( ( 2809 + 150 T^{2} + T^{4} )^{2} \)
$59$ \( ( 1600 + 160 T + 56 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$61$ \( ( 25 + 40 T + 59 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$67$ \( 65610000 - 1749600 T^{2} + 38556 T^{4} - 216 T^{6} + T^{8} \)
$71$ \( ( 196 + 140 T + 86 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$73$ \( ( 25 + 54 T^{2} + T^{4} )^{2} \)
$79$ \( ( -4 + T )^{8} \)
$83$ \( ( 196 + 72 T^{2} + T^{4} )^{2} \)
$89$ \( ( 29584 + 688 T + 188 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$97$ \( ( 1296 - 36 T^{2} + T^{4} )^{2} \)
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