Properties

Label 390.2.e.b
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.

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

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 4.00000i 1.00000i −1.00000 1.00000 2.00000i
79.2 1.00000i 1.00000i −1.00000 2.00000 1.00000i −1.00000 4.00000i 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.b 2
3.b odd 2 1 1170.2.e.b 2
4.b odd 2 1 3120.2.l.h 2
5.b even 2 1 inner 390.2.e.b 2
5.c odd 4 1 1950.2.a.g 1
5.c odd 4 1 1950.2.a.u 1
15.d odd 2 1 1170.2.e.b 2
15.e even 4 1 5850.2.a.x 1
15.e even 4 1 5850.2.a.bd 1
20.d odd 2 1 3120.2.l.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.e.b 2 1.a even 1 1 trivial
390.2.e.b 2 5.b even 2 1 inner
1170.2.e.b 2 3.b odd 2 1
1170.2.e.b 2 15.d odd 2 1
1950.2.a.g 1 5.c odd 4 1
1950.2.a.u 1 5.c odd 4 1
3120.2.l.h 2 4.b odd 2 1
3120.2.l.h 2 20.d odd 2 1
5850.2.a.x 1 15.e even 4 1
5850.2.a.bd 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 \(S_{2}^{\mathrm{new}}(390, [\chi])\):

\( T_{7}^{2} + 16 \)
\( T_{11} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 5 - 4 T + T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 16 + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 100 + T^{2} \)
$59$ \( ( -14 + T )^{2} \)
$61$ \( ( -10 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( -18 + T )^{2} \)
$97$ \( 36 + T^{2} \)
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