Properties

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

Newform invariants

Self dual: no
Analytic conductor: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 2.00000 1.00000i 1.00000 2.00000i 1.00000i −1.00000 −1.00000 2.00000i
79.2 1.00000i 1.00000i −1.00000 2.00000 + 1.00000i 1.00000 2.00000i 1.00000i −1.00000 −1.00000 + 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.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.e.c 2
3.b odd 2 1 1170.2.e.a 2
4.b odd 2 1 3120.2.l.g 2
5.b even 2 1 inner 390.2.e.c 2
5.c odd 4 1 1950.2.a.c 1
5.c odd 4 1 1950.2.a.z 1
15.d odd 2 1 1170.2.e.a 2
15.e even 4 1 5850.2.a.t 1
15.e even 4 1 5850.2.a.bj 1
20.d odd 2 1 3120.2.l.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.e.c 2 1.a even 1 1 trivial
390.2.e.c 2 5.b even 2 1 inner
1170.2.e.a 2 3.b odd 2 1
1170.2.e.a 2 15.d odd 2 1
1950.2.a.c 1 5.c odd 4 1
1950.2.a.z 1 5.c odd 4 1
3120.2.l.g 2 4.b odd 2 1
3120.2.l.g 2 20.d odd 2 1
5850.2.a.t 1 15.e even 4 1
5850.2.a.bj 1 15.e even 4 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 5 - 4 T + T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 4 + T )^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 4 + T^{2} \)
$59$ \( ( 10 + T )^{2} \)
$61$ \( ( 14 + T )^{2} \)
$67$ \( 256 + T^{2} \)
$71$ \( ( 4 + T )^{2} \)
$73$ \( 64 + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( 144 + T^{2} \)
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