Properties

Label 390.2.e.d
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: \( x^{2} + 1 \) Copy content Toggle raw display
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.

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

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

\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T - 10)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( (T + 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T + 14)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
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