Properties

Label 380.2.i.a
Level $380$
Weight $2$
Character orbit 380.i
Analytic conductor $3.034$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{3} + ( - \zeta_{6} + 1) q^{5} - 4 q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{3} + ( - \zeta_{6} + 1) q^{5} - 4 q^{7} - \zeta_{6} q^{9} - 3 q^{11} - 6 \zeta_{6} q^{13} + 2 \zeta_{6} q^{15} + (2 \zeta_{6} - 2) q^{17} + (3 \zeta_{6} + 2) q^{19} + ( - 8 \zeta_{6} + 8) q^{21} - 4 \zeta_{6} q^{23} - \zeta_{6} q^{25} - 4 q^{27} - \zeta_{6} q^{29} - 5 q^{31} + ( - 6 \zeta_{6} + 6) q^{33} + (4 \zeta_{6} - 4) q^{35} - 4 q^{37} + 12 q^{39} + (2 \zeta_{6} - 2) q^{41} - q^{45} - 6 \zeta_{6} q^{47} + 9 q^{49} - 4 \zeta_{6} q^{51} + 6 \zeta_{6} q^{53} + (3 \zeta_{6} - 3) q^{55} + (4 \zeta_{6} - 10) q^{57} + ( - \zeta_{6} + 1) q^{59} + 7 \zeta_{6} q^{61} + 4 \zeta_{6} q^{63} - 6 q^{65} + 14 \zeta_{6} q^{67} + 8 q^{69} + (15 \zeta_{6} - 15) q^{71} + (12 \zeta_{6} - 12) q^{73} + 2 q^{75} + 12 q^{77} + ( - \zeta_{6} + 1) q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + 16 q^{83} + 2 \zeta_{6} q^{85} + 2 q^{87} - 17 \zeta_{6} q^{89} + 24 \zeta_{6} q^{91} + ( - 10 \zeta_{6} + 10) q^{93} + ( - 2 \zeta_{6} + 5) q^{95} + (12 \zeta_{6} - 12) q^{97} + 3 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + q^{5} - 8 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + q^{5} - 8 q^{7} - q^{9} - 6 q^{11} - 6 q^{13} + 2 q^{15} - 2 q^{17} + 7 q^{19} + 8 q^{21} - 4 q^{23} - q^{25} - 8 q^{27} - q^{29} - 10 q^{31} + 6 q^{33} - 4 q^{35} - 8 q^{37} + 24 q^{39} - 2 q^{41} - 2 q^{45} - 6 q^{47} + 18 q^{49} - 4 q^{51} + 6 q^{53} - 3 q^{55} - 16 q^{57} + q^{59} + 7 q^{61} + 4 q^{63} - 12 q^{65} + 14 q^{67} + 16 q^{69} - 15 q^{71} - 12 q^{73} + 4 q^{75} + 24 q^{77} + q^{79} + 11 q^{81} + 32 q^{83} + 2 q^{85} + 4 q^{87} - 17 q^{89} + 24 q^{91} + 10 q^{93} + 8 q^{95} - 12 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.00000 1.73205i 0 0.500000 + 0.866025i 0 −4.00000 0 −0.500000 + 0.866025i 0
201.1 0 −1.00000 + 1.73205i 0 0.500000 0.866025i 0 −4.00000 0 −0.500000 0.866025i 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
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.i.a 2
3.b odd 2 1 3420.2.t.b 2
4.b odd 2 1 1520.2.q.g 2
5.b even 2 1 1900.2.i.b 2
5.c odd 4 2 1900.2.s.b 4
19.c even 3 1 inner 380.2.i.a 2
19.c even 3 1 7220.2.a.e 1
19.d odd 6 1 7220.2.a.a 1
57.h odd 6 1 3420.2.t.b 2
76.g odd 6 1 1520.2.q.g 2
95.i even 6 1 1900.2.i.b 2
95.m odd 12 2 1900.2.s.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.i.a 2 1.a even 1 1 trivial
380.2.i.a 2 19.c even 3 1 inner
1520.2.q.g 2 4.b odd 2 1
1520.2.q.g 2 76.g odd 6 1
1900.2.i.b 2 5.b even 2 1
1900.2.i.b 2 95.i even 6 1
1900.2.s.b 4 5.c odd 4 2
1900.2.s.b 4 95.m odd 12 2
3420.2.t.b 2 3.b odd 2 1
3420.2.t.b 2 57.h odd 6 1
7220.2.a.a 1 19.d odd 6 1
7220.2.a.e 1 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 2T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( (T + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 7T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$31$ \( (T + 5)^{2} \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$61$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$71$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$73$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$79$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$83$ \( (T - 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
$97$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
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