Properties

Label 390.2.i.g
Level $390$
Weight $2$
Character orbit 390.i
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial: \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 5 \nu^{2} - 5 \nu + 16 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 4 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 4 \beta_{2} + \beta_{1} - 4\)
\(\nu^{3}\)\(=\)\(5 \beta_{3} - 4\)

Character values

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

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
1.28078 + 2.21837i
−0.780776 1.35234i
1.28078 2.21837i
−0.780776 + 1.35234i
−0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 1.00000 −0.500000 0.866025i −1.78078 3.08440i 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
61.2 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 1.00000 −0.500000 0.866025i 0.280776 + 0.486319i 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
211.1 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −1.78078 + 3.08440i 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
211.2 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 0.280776 0.486319i 1.00000 −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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.i.g 4
3.b odd 2 1 1170.2.i.o 4
5.b even 2 1 1950.2.i.bi 4
5.c odd 4 2 1950.2.z.n 8
13.c even 3 1 inner 390.2.i.g 4
13.c even 3 1 5070.2.a.bi 2
13.e even 6 1 5070.2.a.bb 2
13.f odd 12 2 5070.2.b.r 4
39.i odd 6 1 1170.2.i.o 4
65.n even 6 1 1950.2.i.bi 4
65.q odd 12 2 1950.2.z.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.g 4 1.a even 1 1 trivial
390.2.i.g 4 13.c even 3 1 inner
1170.2.i.o 4 3.b odd 2 1
1170.2.i.o 4 39.i odd 6 1
1950.2.i.bi 4 5.b even 2 1
1950.2.i.bi 4 65.n even 6 1
1950.2.z.n 8 5.c odd 4 2
1950.2.z.n 8 65.q odd 12 2
5070.2.a.bb 2 13.e even 6 1
5070.2.a.bi 2 13.c even 3 1
5070.2.b.r 4 13.f odd 12 2

Hecke kernels

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

\( T_{7}^{4} + 3 T_{7}^{3} + 11 T_{7}^{2} - 6 T_{7} + 4 \)
\( T_{11}^{4} + 17 T_{11}^{2} + 289 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( ( -1 + T )^{4} \)
$7$ \( 4 - 6 T + 11 T^{2} + 3 T^{3} + T^{4} \)
$11$ \( 289 + 17 T^{2} + T^{4} \)
$13$ \( 169 + 13 T - 12 T^{2} + T^{3} + T^{4} \)
$17$ \( 256 - 32 T + 20 T^{2} + 2 T^{3} + T^{4} \)
$19$ \( 4 + 6 T + 11 T^{2} - 3 T^{3} + T^{4} \)
$23$ \( 1296 + 108 T + 45 T^{2} - 3 T^{3} + T^{4} \)
$29$ \( 256 + 144 T + 65 T^{2} + 9 T^{3} + T^{4} \)
$31$ \( ( -38 + T + T^{2} )^{2} \)
$37$ \( 289 + 17 T^{2} + T^{4} \)
$41$ \( 2704 - 416 T + 116 T^{2} + 8 T^{3} + T^{4} \)
$43$ \( 4 + 10 T + 23 T^{2} + 5 T^{3} + T^{4} \)
$47$ \( ( -7 + T )^{4} \)
$53$ \( ( 38 - 13 T + T^{2} )^{2} \)
$59$ \( 4624 + 1156 T + 221 T^{2} + 17 T^{3} + T^{4} \)
$61$ \( ( 36 + 6 T + T^{2} )^{2} \)
$67$ \( 1024 - 384 T + 176 T^{2} + 12 T^{3} + T^{4} \)
$71$ \( 4096 - 1152 T + 260 T^{2} - 18 T^{3} + T^{4} \)
$73$ \( ( -144 + 6 T + T^{2} )^{2} \)
$79$ \( ( 86 - 19 T + T^{2} )^{2} \)
$83$ \( ( -8 + 6 T + T^{2} )^{2} \)
$89$ \( 1156 + 578 T + 323 T^{2} - 17 T^{3} + T^{4} \)
$97$ \( 64 + 48 T + 44 T^{2} - 6 T^{3} + T^{4} \)
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