Properties

Label 390.2.bb.a
Level $390$
Weight $2$
Character orbit 390.bb
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.bb (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.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 1.00000i −0.866025 + 0.500000i −2.59808 + 1.50000i 1.00000i −0.500000 0.866025i −0.500000 + 0.866025i
121.2 0.866025 + 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 1.00000i 0.866025 0.500000i 2.59808 1.50000i 1.00000i −0.500000 0.866025i −0.500000 + 0.866025i
361.1 −0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 1.00000i −0.866025 0.500000i −2.59808 1.50000i 1.00000i −0.500000 + 0.866025i −0.500000 0.866025i
361.2 0.866025 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 1.00000i 0.866025 + 0.500000i 2.59808 + 1.50000i 1.00000i −0.500000 + 0.866025i −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e 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.bb.a 4
3.b odd 2 1 1170.2.bs.d 4
5.b even 2 1 1950.2.bc.a 4
5.c odd 4 1 1950.2.y.d 4
5.c odd 4 1 1950.2.y.e 4
13.c even 3 1 5070.2.b.p 4
13.e even 6 1 inner 390.2.bb.a 4
13.e even 6 1 5070.2.b.p 4
13.f odd 12 1 5070.2.a.ba 2
13.f odd 12 1 5070.2.a.be 2
39.h odd 6 1 1170.2.bs.d 4
65.l even 6 1 1950.2.bc.a 4
65.r odd 12 1 1950.2.y.d 4
65.r odd 12 1 1950.2.y.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.a 4 1.a even 1 1 trivial
390.2.bb.a 4 13.e even 6 1 inner
1170.2.bs.d 4 3.b odd 2 1
1170.2.bs.d 4 39.h odd 6 1
1950.2.y.d 4 5.c odd 4 1
1950.2.y.d 4 65.r odd 12 1
1950.2.y.e 4 5.c odd 4 1
1950.2.y.e 4 65.r odd 12 1
1950.2.bc.a 4 5.b even 2 1
1950.2.bc.a 4 65.l even 6 1
5070.2.a.ba 2 13.f odd 12 1
5070.2.a.be 2 13.f odd 12 1
5070.2.b.p 4 13.c even 3 1
5070.2.b.p 4 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 9 T_{7}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( 81 - 9 T^{2} + T^{4} \)
$11$ \( 1 - 6 T + 11 T^{2} + 6 T^{3} + T^{4} \)
$13$ \( 169 + 23 T^{2} + T^{4} \)
$17$ \( ( 16 + 4 T + T^{2} )^{2} \)
$19$ \( 169 + 78 T - T^{2} - 6 T^{3} + T^{4} \)
$23$ \( 144 + 12 T^{2} + T^{4} \)
$29$ \( 64 + 32 T + 24 T^{2} - 4 T^{3} + T^{4} \)
$31$ \( 1936 + 104 T^{2} + T^{4} \)
$37$ \( 2209 + 1128 T + 239 T^{2} + 24 T^{3} + T^{4} \)
$41$ \( 256 - 16 T^{2} + T^{4} \)
$43$ \( ( 36 + 6 T + T^{2} )^{2} \)
$47$ \( 9 + 42 T^{2} + T^{4} \)
$53$ \( ( 1 + 4 T + T^{2} )^{2} \)
$59$ \( 2704 - 624 T - 4 T^{2} + 12 T^{3} + T^{4} \)
$61$ \( 16 - 32 T + 60 T^{2} - 8 T^{3} + T^{4} \)
$67$ \( 64 - 96 T + 56 T^{2} - 12 T^{3} + T^{4} \)
$71$ \( 144 + 288 T + 204 T^{2} + 24 T^{3} + T^{4} \)
$73$ \( ( 48 + T^{2} )^{2} \)
$79$ \( ( 52 - 20 T + T^{2} )^{2} \)
$83$ \( 576 + 96 T^{2} + T^{4} \)
$89$ \( 20449 + 6006 T + 731 T^{2} + 42 T^{3} + T^{4} \)
$97$ \( 10816 - 3744 T + 536 T^{2} - 36 T^{3} + T^{4} \)
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