Properties

Label 390.2.i.f
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.

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

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 1.00000 + 1.73205i −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 1.00000 1.73205i −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.f 2
3.b odd 2 1 1170.2.i.a 2
5.b even 2 1 1950.2.i.d 2
5.c odd 4 2 1950.2.z.e 4
13.c even 3 1 inner 390.2.i.f 2
13.c even 3 1 5070.2.a.d 1
13.e even 6 1 5070.2.a.p 1
13.f odd 12 2 5070.2.b.h 2
39.i odd 6 1 1170.2.i.a 2
65.n even 6 1 1950.2.i.d 2
65.q odd 12 2 1950.2.z.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.f 2 1.a even 1 1 trivial
390.2.i.f 2 13.c even 3 1 inner
1170.2.i.a 2 3.b odd 2 1
1170.2.i.a 2 39.i odd 6 1
1950.2.i.d 2 5.b even 2 1
1950.2.i.d 2 65.n even 6 1
1950.2.z.e 4 5.c odd 4 2
1950.2.z.e 4 65.q odd 12 2
5070.2.a.d 1 13.c even 3 1
5070.2.a.p 1 13.e even 6 1
5070.2.b.h 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 \(S_{2}^{\mathrm{new}}(390, [\chi])\):

\( T_{7}^{2} - 2 T_{7} + 4 \)
\( T_{11}^{2} - 5 T_{11} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( 4 - 2 T + T^{2} \)
$11$ \( 25 - 5 T + T^{2} \)
$13$ \( 13 + 2 T + T^{2} \)
$17$ \( 4 - 2 T + T^{2} \)
$19$ \( 4 - 2 T + T^{2} \)
$23$ \( 1 - T + T^{2} \)
$29$ \( 25 + 5 T + T^{2} \)
$31$ \( ( 11 + T )^{2} \)
$37$ \( 9 + 3 T + T^{2} \)
$41$ \( 4 - 2 T + T^{2} \)
$43$ \( 121 - 11 T + T^{2} \)
$47$ \( ( -9 + T )^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( 225 - 15 T + T^{2} \)
$61$ \( 100 + 10 T + T^{2} \)
$67$ \( 256 + 16 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 6 + T )^{2} \)
$79$ \( ( 11 + T )^{2} \)
$83$ \( ( -6 + T )^{2} \)
$89$ \( 4 + 2 T + T^{2} \)
$97$ \( 4 - 2 T + T^{2} \)
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