Properties

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

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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.

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

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 −2.23205 + 0.133975i −0.500000 0.866025i 2.36603 1.36603i 1.00000i 0.500000 + 0.866025i 2.00000 + 1.00000i
139.2 0.866025 + 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 1.23205 1.86603i −0.500000 0.866025i 0.633975 0.366025i 1.00000i 0.500000 + 0.866025i 2.00000 1.00000i
289.1 −0.866025 + 0.500000i 0.866025 0.500000i 0.500000 0.866025i −2.23205 0.133975i −0.500000 + 0.866025i 2.36603 + 1.36603i 1.00000i 0.500000 0.866025i 2.00000 1.00000i
289.2 0.866025 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 1.23205 + 1.86603i −0.500000 + 0.866025i 0.633975 + 0.366025i 1.00000i 0.500000 0.866025i 2.00000 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 6 T_{7}^{3} + 14 T_{7}^{2} - 12 T_{7} + 4 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( 25 + 10 T - T^{2} + 2 T^{3} + T^{4} \)
$7$ \( 4 - 12 T + 14 T^{2} - 6 T^{3} + T^{4} \)
$11$ \( 4 + 4 T + 6 T^{2} - 2 T^{3} + T^{4} \)
$13$ \( 169 - 52 T + 3 T^{2} - 4 T^{3} + T^{4} \)
$17$ \( 121 - 132 T + 59 T^{2} - 12 T^{3} + T^{4} \)
$19$ \( 36 - 36 T + 30 T^{2} - 6 T^{3} + T^{4} \)
$23$ \( 676 - 468 T + 134 T^{2} - 18 T^{3} + T^{4} \)
$29$ \( 9 - 18 T + 39 T^{2} + 6 T^{3} + T^{4} \)
$31$ \( ( -4 + T )^{4} \)
$37$ \( 2209 + 1128 T + 239 T^{2} + 24 T^{3} + T^{4} \)
$41$ \( 121 + 88 T + 75 T^{2} - 8 T^{3} + T^{4} \)
$43$ \( 484 - 132 T - 10 T^{2} + 6 T^{3} + T^{4} \)
$47$ \( 5476 + 152 T^{2} + T^{4} \)
$53$ \( 1089 + 78 T^{2} + T^{4} \)
$59$ \( 10816 - 416 T + 120 T^{2} + 4 T^{3} + T^{4} \)
$61$ \( 20449 - 572 T + 159 T^{2} + 4 T^{3} + T^{4} \)
$67$ \( 5476 + 2220 T + 374 T^{2} + 30 T^{3} + T^{4} \)
$71$ \( 36 + 36 T + 30 T^{2} + 6 T^{3} + T^{4} \)
$73$ \( 3481 + 182 T^{2} + T^{4} \)
$79$ \( ( 12 + T )^{4} \)
$83$ \( 2116 + 104 T^{2} + T^{4} \)
$89$ \( 1936 + 176 T + 60 T^{2} - 4 T^{3} + T^{4} \)
$97$ \( 10000 - 100 T^{2} + T^{4} \)
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