Properties

Label 390.2.b.c
Level $390$
Weight $2$
Character orbit 390.b
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} - q^{4} + \beta_1 q^{5} - \beta_1 q^{6} + (\beta_{2} - \beta_1) q^{7} - \beta_1 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} - q^{4} + \beta_1 q^{5} - \beta_1 q^{6} + (\beta_{2} - \beta_1) q^{7} - \beta_1 q^{8} + q^{9} - q^{10} + q^{12} - \beta_{3} q^{13} + ( - \beta_{3} + 1) q^{14} - \beta_1 q^{15} + q^{16} + ( - \beta_{3} + 1) q^{17} + \beta_1 q^{18} + (\beta_{2} - \beta_1) q^{19} - \beta_1 q^{20} + ( - \beta_{2} + \beta_1) q^{21} + ( - \beta_{3} - 5) q^{23} + \beta_1 q^{24} - q^{25} - \beta_{2} q^{26} - q^{27} + ( - \beta_{2} + \beta_1) q^{28} + ( - \beta_{3} + 1) q^{29} + q^{30} + 6 \beta_1 q^{31} + \beta_1 q^{32} + ( - \beta_{2} + \beta_1) q^{34} + ( - \beta_{3} + 1) q^{35} - q^{36} + (2 \beta_{2} - 2 \beta_1) q^{37} + ( - \beta_{3} + 1) q^{38} + \beta_{3} q^{39} + q^{40} + ( - 2 \beta_{2} - 4 \beta_1) q^{41} + (\beta_{3} - 1) q^{42} + 8 q^{43} + \beta_1 q^{45} + ( - \beta_{2} - 5 \beta_1) q^{46} + (2 \beta_{2} - 2 \beta_1) q^{47} - q^{48} + (2 \beta_{3} - 7) q^{49} - \beta_1 q^{50} + (\beta_{3} - 1) q^{51} + \beta_{3} q^{52} + 6 q^{53} - \beta_1 q^{54} + (\beta_{3} - 1) q^{56} + ( - \beta_{2} + \beta_1) q^{57} + ( - \beta_{2} + \beta_1) q^{58} + (2 \beta_{2} - 2 \beta_1) q^{59} + \beta_1 q^{60} + (2 \beta_{3} - 4) q^{61} - 6 q^{62} + (\beta_{2} - \beta_1) q^{63} - q^{64} - \beta_{2} q^{65} + (2 \beta_{2} + 4 \beta_1) q^{67} + (\beta_{3} - 1) q^{68} + (\beta_{3} + 5) q^{69} + ( - \beta_{2} + \beta_1) q^{70} + ( - 2 \beta_{2} + 2 \beta_1) q^{71} - \beta_1 q^{72} + ( - \beta_{2} - 5 \beta_1) q^{73} + ( - 2 \beta_{3} + 2) q^{74} + q^{75} + ( - \beta_{2} + \beta_1) q^{76} + \beta_{2} q^{78} + 4 \beta_{3} q^{79} + \beta_1 q^{80} + q^{81} + (2 \beta_{3} + 4) q^{82} + (2 \beta_{2} + 10 \beta_1) q^{83} + (\beta_{2} - \beta_1) q^{84} + ( - \beta_{2} + \beta_1) q^{85} + 8 \beta_1 q^{86} + (\beta_{3} - 1) q^{87} + ( - 2 \beta_{2} + 8 \beta_1) q^{89} - q^{90} + (\beta_{2} - 13 \beta_1) q^{91} + (\beta_{3} + 5) q^{92} - 6 \beta_1 q^{93} + ( - 2 \beta_{3} + 2) q^{94} + ( - \beta_{3} + 1) q^{95} - \beta_1 q^{96} + (\beta_{2} + 5 \beta_1) q^{97} + (2 \beta_{2} - 7 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{10} + 4 q^{12} + 4 q^{14} + 4 q^{16} + 4 q^{17} - 20 q^{23} - 4 q^{25} - 4 q^{27} + 4 q^{29} + 4 q^{30} + 4 q^{35} - 4 q^{36} + 4 q^{38} + 4 q^{40} - 4 q^{42} + 32 q^{43} - 4 q^{48} - 28 q^{49} - 4 q^{51} + 24 q^{53} - 4 q^{56} - 16 q^{61} - 24 q^{62} - 4 q^{64} - 4 q^{68} + 20 q^{69} + 8 q^{74} + 4 q^{75} + 4 q^{81} + 16 q^{82} - 4 q^{87} - 4 q^{90} + 20 q^{92} + 8 q^{94} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 10\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{2} + 5\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
1.30278i
2.30278i
2.30278i
1.30278i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 2.60555i 1.00000i 1.00000 −1.00000
181.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 4.60555i 1.00000i 1.00000 −1.00000
181.3 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 4.60555i 1.00000i 1.00000 −1.00000
181.4 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 2.60555i 1.00000i 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.b.c 4
3.b odd 2 1 1170.2.b.d 4
4.b odd 2 1 3120.2.g.q 4
5.b even 2 1 1950.2.b.k 4
5.c odd 4 1 1950.2.f.m 4
5.c odd 4 1 1950.2.f.n 4
13.b even 2 1 inner 390.2.b.c 4
13.d odd 4 1 5070.2.a.z 2
13.d odd 4 1 5070.2.a.bf 2
39.d odd 2 1 1170.2.b.d 4
52.b odd 2 1 3120.2.g.q 4
65.d even 2 1 1950.2.b.k 4
65.h odd 4 1 1950.2.f.m 4
65.h odd 4 1 1950.2.f.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.b.c 4 1.a even 1 1 trivial
390.2.b.c 4 13.b even 2 1 inner
1170.2.b.d 4 3.b odd 2 1
1170.2.b.d 4 39.d odd 2 1
1950.2.b.k 4 5.b even 2 1
1950.2.b.k 4 65.d even 2 1
1950.2.f.m 4 5.c odd 4 1
1950.2.f.m 4 65.h odd 4 1
1950.2.f.n 4 5.c odd 4 1
1950.2.f.n 4 65.h odd 4 1
3120.2.g.q 4 4.b odd 2 1
3120.2.g.q 4 52.b odd 2 1
5070.2.a.z 2 13.d odd 4 1
5070.2.a.bf 2 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\):

\( T_{7}^{4} + 28T_{7}^{2} + 144 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 28T^{2} + 144 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 13)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T - 12)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 28T^{2} + 144 \) Copy content Toggle raw display
$23$ \( (T^{2} + 10 T + 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 2 T - 12)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 112T^{2} + 2304 \) Copy content Toggle raw display
$41$ \( T^{4} + 136T^{2} + 1296 \) Copy content Toggle raw display
$43$ \( (T - 8)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 112T^{2} + 2304 \) Copy content Toggle raw display
$53$ \( (T - 6)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 112T^{2} + 2304 \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T - 36)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 136T^{2} + 1296 \) Copy content Toggle raw display
$71$ \( T^{4} + 112T^{2} + 2304 \) Copy content Toggle raw display
$73$ \( T^{4} + 76T^{2} + 144 \) Copy content Toggle raw display
$79$ \( (T^{2} - 208)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 304T^{2} + 2304 \) Copy content Toggle raw display
$89$ \( T^{4} + 232T^{2} + 144 \) Copy content Toggle raw display
$97$ \( T^{4} + 76T^{2} + 144 \) Copy content Toggle raw display
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