Properties

Label 390.2.y.a
Level $390$
Weight $2$
Character orbit 390.y
Analytic conductor $3.114$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{12}^{2}\)

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
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i −1.00000 2.00000i −0.500000 0.866025i −3.46410 + 2.00000i 1.00000i 0.500000 + 0.866025i −0.133975 + 2.23205i
139.2 0.866025 + 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i −1.00000 + 2.00000i −0.500000 0.866025i 3.46410 2.00000i 1.00000i 0.500000 + 0.866025i −1.86603 + 1.23205i
289.1 −0.866025 + 0.500000i 0.866025 0.500000i 0.500000 0.866025i −1.00000 + 2.00000i −0.500000 + 0.866025i −3.46410 2.00000i 1.00000i 0.500000 0.866025i −0.133975 2.23205i
289.2 0.866025 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i −1.00000 2.00000i −0.500000 + 0.866025i 3.46410 + 2.00000i 1.00000i 0.500000 0.866025i −1.86603 1.23205i
\(n\): e.g. 2-40 or 990-1000
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.a 4
3.b odd 2 1 1170.2.bp.f 4
5.b even 2 1 inner 390.2.y.a 4
5.c odd 4 1 1950.2.i.g 2
5.c odd 4 1 1950.2.i.t 2
13.c even 3 1 inner 390.2.y.a 4
15.d odd 2 1 1170.2.bp.f 4
39.i odd 6 1 1170.2.bp.f 4
65.n even 6 1 inner 390.2.y.a 4
65.q odd 12 1 1950.2.i.g 2
65.q odd 12 1 1950.2.i.t 2
195.x odd 6 1 1170.2.bp.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.y.a 4 1.a even 1 1 trivial
390.2.y.a 4 5.b even 2 1 inner
390.2.y.a 4 13.c even 3 1 inner
390.2.y.a 4 65.n even 6 1 inner
1170.2.bp.f 4 3.b odd 2 1
1170.2.bp.f 4 15.d odd 2 1
1170.2.bp.f 4 39.i odd 6 1
1170.2.bp.f 4 195.x odd 6 1
1950.2.i.g 2 5.c odd 4 1
1950.2.i.g 2 65.q odd 12 1
1950.2.i.t 2 5.c odd 4 1
1950.2.i.t 2 65.q odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 16 T_{7}^{2} + 256 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( ( 5 + 2 T + T^{2} )^{2} \)
$7$ \( 256 - 16 T^{2} + T^{4} \)
$11$ \( ( 25 + 5 T + T^{2} )^{2} \)
$13$ \( 169 - 22 T^{2} + T^{4} \)
$17$ \( 16 - 4 T^{2} + T^{4} \)
$19$ \( ( 64 - 8 T + T^{2} )^{2} \)
$23$ \( 81 - 9 T^{2} + T^{4} \)
$29$ \( ( 49 - 7 T + T^{2} )^{2} \)
$31$ \( ( 7 + T )^{4} \)
$37$ \( 81 - 9 T^{2} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( 1 - T^{2} + T^{4} \)
$47$ \( ( 49 + T^{2} )^{2} \)
$53$ \( ( 100 + T^{2} )^{2} \)
$59$ \( ( 25 + 5 T + T^{2} )^{2} \)
$61$ \( ( 16 + 4 T + T^{2} )^{2} \)
$67$ \( 256 - 16 T^{2} + T^{4} \)
$71$ \( ( 64 + 8 T + T^{2} )^{2} \)
$73$ \( ( 16 + T^{2} )^{2} \)
$79$ \( ( 1 + T )^{4} \)
$83$ \( ( 144 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( 20736 - 144 T^{2} + T^{4} \)
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