Properties

Label 380.3.g.a
Level $380$
Weight $3$
Character orbit 380.g
Analytic conductor $10.354$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-6}, \sqrt{14})\)
Defining polynomial: \( x^{4} - 4x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - 5 q^{5} + \beta_{2} q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - 5 q^{5} + \beta_{2} q^{7} + 5 q^{9} + 4 q^{11} - 3 \beta_1 q^{13} + 5 \beta_1 q^{15} + 2 \beta_{2} q^{17} + ( - \beta_{3} - 5) q^{19} - 2 \beta_{3} q^{21} - \beta_{2} q^{23} + 25 q^{25} + 4 \beta_1 q^{27} - 2 \beta_{3} q^{29} - 2 \beta_{3} q^{31} - 4 \beta_1 q^{33} - 5 \beta_{2} q^{35} - 9 \beta_1 q^{37} + 42 q^{39} + 2 \beta_{3} q^{41} - 7 \beta_{2} q^{43} - 25 q^{45} - \beta_{2} q^{47} - 47 q^{49} - 4 \beta_{3} q^{51} + 15 \beta_1 q^{53} - 20 q^{55} + (7 \beta_{2} + 5 \beta_1) q^{57} + 4 \beta_{3} q^{59} + 100 q^{61} + 5 \beta_{2} q^{63} + 15 \beta_1 q^{65} - 3 \beta_1 q^{67} + 2 \beta_{3} q^{69} - 2 \beta_{3} q^{71} - 2 \beta_{2} q^{73} - 25 \beta_1 q^{75} + 4 \beta_{2} q^{77} - 6 \beta_{3} q^{79} - 101 q^{81} - 3 \beta_{2} q^{83} - 10 \beta_{2} q^{85} + 14 \beta_{2} q^{87} + 8 \beta_{3} q^{89} - 6 \beta_{3} q^{91} + 14 \beta_{2} q^{93} + (5 \beta_{3} + 25) q^{95} - 33 \beta_1 q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{5} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{5} + 20 q^{9} + 16 q^{11} - 20 q^{19} + 100 q^{25} + 168 q^{39} - 100 q^{45} - 188 q^{49} - 80 q^{55} + 400 q^{61} - 404 q^{81} + 100 q^{95} + 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 9\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{3} + 4\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} - 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 4\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{2} - 4\beta_1 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
189.1
1.87083 1.22474i
1.87083 + 1.22474i
−1.87083 1.22474i
−1.87083 + 1.22474i
0 −3.74166 0 −5.00000 0 9.79796i 0 5.00000 0
189.2 0 −3.74166 0 −5.00000 0 9.79796i 0 5.00000 0
189.3 0 3.74166 0 −5.00000 0 9.79796i 0 5.00000 0
189.4 0 3.74166 0 −5.00000 0 9.79796i 0 5.00000 0
\(n\): 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
19.b odd 2 1 inner
95.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.3.g.a 4
3.b odd 2 1 3420.3.h.c 4
5.b even 2 1 inner 380.3.g.a 4
5.c odd 4 2 1900.3.e.c 4
15.d odd 2 1 3420.3.h.c 4
19.b odd 2 1 inner 380.3.g.a 4
57.d even 2 1 3420.3.h.c 4
95.d odd 2 1 inner 380.3.g.a 4
95.g even 4 2 1900.3.e.c 4
285.b even 2 1 3420.3.h.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.3.g.a 4 1.a even 1 1 trivial
380.3.g.a 4 5.b even 2 1 inner
380.3.g.a 4 19.b odd 2 1 inner
380.3.g.a 4 95.d odd 2 1 inner
1900.3.e.c 4 5.c odd 4 2
1900.3.e.c 4 95.g even 4 2
3420.3.h.c 4 3.b odd 2 1
3420.3.h.c 4 15.d odd 2 1
3420.3.h.c 4 57.d even 2 1
3420.3.h.c 4 285.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 14 \) acting on \(S_{3}^{\mathrm{new}}(380, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 14)^{2} \) Copy content Toggle raw display
$5$ \( (T + 5)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$11$ \( (T - 4)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 126)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 384)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 10 T + 361)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1344)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 1344)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 1134)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 1344)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4704)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 3150)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 5376)^{2} \) Copy content Toggle raw display
$61$ \( (T - 100)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 126)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 1344)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 384)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 12096)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 864)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 21504)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 15246)^{2} \) Copy content Toggle raw display
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