Properties

Label 390.2.i.d
Level $390$
Weight $2$
Character orbit 390.i
Analytic conductor $3.114$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −1.00000 0.500000 + 0.866025i 2.50000 + 4.33013i −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 2.50000 4.33013i −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.d 2
3.b odd 2 1 1170.2.i.g 2
5.b even 2 1 1950.2.i.i 2
5.c odd 4 2 1950.2.z.j 4
13.c even 3 1 inner 390.2.i.d 2
13.c even 3 1 5070.2.a.i 1
13.e even 6 1 5070.2.a.x 1
13.f odd 12 2 5070.2.b.l 2
39.i odd 6 1 1170.2.i.g 2
65.n even 6 1 1950.2.i.i 2
65.q odd 12 2 1950.2.z.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.d 2 1.a even 1 1 trivial
390.2.i.d 2 13.c even 3 1 inner
1170.2.i.g 2 3.b odd 2 1
1170.2.i.g 2 39.i odd 6 1
1950.2.i.i 2 5.b even 2 1
1950.2.i.i 2 65.n even 6 1
1950.2.z.j 4 5.c odd 4 2
1950.2.z.j 4 65.q odd 12 2
5070.2.a.i 1 13.c even 3 1
5070.2.a.x 1 13.e even 6 1
5070.2.b.l 2 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 \(S_{2}^{\mathrm{new}}(390, [\chi])\):

\( T_{7}^{2} - 5 T_{7} + 25 \)
\( T_{11}^{2} - 3 T_{11} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( 25 - 5 T + T^{2} \)
$11$ \( 9 - 3 T + T^{2} \)
$13$ \( 13 + 5 T + T^{2} \)
$17$ \( 64 - 8 T + T^{2} \)
$19$ \( 25 - 5 T + T^{2} \)
$23$ \( 16 - 4 T + T^{2} \)
$29$ \( 16 - 4 T + T^{2} \)
$31$ \( ( 2 + T )^{2} \)
$37$ \( 49 - 7 T + T^{2} \)
$41$ \( 36 + 6 T + T^{2} \)
$43$ \( 36 + 6 T + T^{2} \)
$47$ \( ( 3 + T )^{2} \)
$53$ \( ( -1 + T )^{2} \)
$59$ \( 144 + 12 T + T^{2} \)
$61$ \( 4 + 2 T + T^{2} \)
$67$ \( 64 + 8 T + T^{2} \)
$71$ \( 4 + 2 T + T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( 2 + T )^{2} \)
$83$ \( ( -8 + T )^{2} \)
$89$ \( 121 - 11 T + T^{2} \)
$97$ \( T^{2} \)
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