Properties

Label 380.2.i.b
Level $380$
Weight $2$
Character orbit 380.i
Analytic conductor $3.034$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1783323.2
Defining polynomial: \(x^{6} - x^{5} + 5 x^{4} - 2 x^{3} + 19 x^{2} - 12 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{3} - \beta_{5} ) q^{3} + ( 1 + \beta_{4} ) q^{5} + ( 1 - \beta_{3} ) q^{7} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( \beta_{3} - \beta_{5} ) q^{3} + ( 1 + \beta_{4} ) q^{5} + ( 1 - \beta_{3} ) q^{7} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{9} + ( 2 - \beta_{3} ) q^{11} -\beta_{4} q^{13} -\beta_{5} q^{15} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{19} + ( -6 - 2 \beta_{1} - \beta_{2} - 6 \beta_{4} + \beta_{5} ) q^{21} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - 5 \beta_{4} + 2 \beta_{5} ) q^{23} + \beta_{4} q^{25} + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{27} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{29} + ( 4 + \beta_{1} - \beta_{2} ) q^{31} + ( -6 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 6 \beta_{4} ) q^{33} + ( 1 - \beta_{3} + \beta_{4} + \beta_{5} ) q^{35} + ( 1 - \beta_{3} ) q^{37} + \beta_{3} q^{39} + ( -3 + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{41} + ( 3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{43} + ( -3 - \beta_{1} + \beta_{2} ) q^{45} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{47} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{49} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 9 \beta_{4} ) q^{51} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{53} + ( 2 - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{55} + ( 9 + 3 \beta_{1} + \beta_{2} + \beta_{3} + 9 \beta_{4} - 2 \beta_{5} ) q^{57} + ( 3 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{59} + ( 3 \beta_{4} + 2 \beta_{5} ) q^{61} + 5 \beta_{5} q^{63} + q^{65} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{67} + ( 3 + \beta_{1} - \beta_{2} + 7 \beta_{3} ) q^{69} + ( 3 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{71} + ( 7 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 7 \beta_{4} ) q^{73} -\beta_{3} q^{75} + ( 8 + \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{77} + ( -10 - 2 \beta_{3} - 10 \beta_{4} + 2 \beta_{5} ) q^{79} + ( -6 - 5 \beta_{3} - 6 \beta_{4} + 5 \beta_{5} ) q^{81} + ( -10 + \beta_{1} - \beta_{2} ) q^{83} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{85} + ( -\beta_{1} + \beta_{2} - 7 \beta_{3} ) q^{87} + ( 2 \beta_{4} + 4 \beta_{5} ) q^{89} + ( -\beta_{4} - \beta_{5} ) q^{91} + ( 3 + 2 \beta_{1} + \beta_{2} + 7 \beta_{3} + 3 \beta_{4} - 8 \beta_{5} ) q^{93} + ( -1 - \beta_{2} + \beta_{3} - \beta_{5} ) q^{95} + ( -4 - 4 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} + 5 \beta_{5} ) q^{97} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + q^{3} + 3q^{5} + 4q^{7} - 8q^{9} + O(q^{10}) \) \( 6q + q^{3} + 3q^{5} + 4q^{7} - 8q^{9} + 10q^{11} + 3q^{13} - q^{15} + 3q^{17} - 16q^{21} + 14q^{23} - 3q^{25} - 20q^{27} - 6q^{29} + 22q^{31} - 15q^{33} + 2q^{35} + 4q^{37} + 2q^{39} - 6q^{41} + 5q^{43} - 16q^{45} + 6q^{47} - 6q^{49} - 24q^{51} + 13q^{53} + 5q^{55} + 25q^{57} + 6q^{59} - 7q^{61} + 5q^{63} + 6q^{65} - 4q^{67} + 30q^{69} + 9q^{71} + 18q^{73} - 2q^{75} + 40q^{77} - 32q^{79} - 23q^{81} - 62q^{83} - 3q^{85} - 12q^{87} - 2q^{89} + 2q^{91} + 14q^{93} - 6q^{95} - 11q^{97} - 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 5 x^{4} - 2 x^{3} + 19 x^{2} - 12 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -3 \nu^{5} + 15 \nu^{4} + 8 \nu^{3} + 57 \nu^{2} + 47 \nu + 180 \)\()/83\)
\(\beta_{2}\)\(=\)\((\)\( 4 \nu^{5} - 20 \nu^{4} + 17 \nu^{3} - 76 \nu^{2} + 131 \nu - 240 \)\()/83\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{5} + 25 \nu^{4} - 42 \nu^{3} + 95 \nu^{2} - 60 \nu + 300 \)\()/83\)
\(\beta_{4}\)\(=\)\((\)\( -20 \nu^{5} + 17 \nu^{4} - 85 \nu^{3} - 35 \nu^{2} - 323 \nu - 45 \)\()/249\)
\(\beta_{5}\)\(=\)\((\)\( 16 \nu^{5} + 3 \nu^{4} + 68 \nu^{3} + 28 \nu^{2} + 358 \nu + 36 \)\()/83\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 2 \beta_{2} + \beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{5} - 9 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_{1} - 9\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-7 \beta_{3} - 5 \beta_{2} + 5 \beta_{1}\)\()/3\)
\(\nu^{4}\)\(=\)\(5 \beta_{5} + 12 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_{1}\)
\(\nu^{5}\)\(=\)\((\)\(18 \beta_{5} + 9 \beta_{4} + 5 \beta_{3} - 23 \beta_{2} - 46 \beta_{1} + 9\)\()/3\)

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(1\)

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
−0.956115 + 1.65604i
1.09935 1.90412i
0.356769 0.617942i
−0.956115 1.65604i
1.09935 + 1.90412i
0.356769 + 0.617942i
0 −1.28442 2.22469i 0 0.500000 + 0.866025i 0 3.56885 0 −1.79949 + 3.11682i 0
121.2 0 0.182224 + 0.315621i 0 0.500000 + 0.866025i 0 0.635552 0 1.43359 2.48305i 0
121.3 0 1.60220 + 2.77509i 0 0.500000 + 0.866025i 0 −2.20440 0 −3.63409 + 6.29444i 0
201.1 0 −1.28442 + 2.22469i 0 0.500000 0.866025i 0 3.56885 0 −1.79949 3.11682i 0
201.2 0 0.182224 0.315621i 0 0.500000 0.866025i 0 0.635552 0 1.43359 + 2.48305i 0
201.3 0 1.60220 2.77509i 0 0.500000 0.866025i 0 −2.20440 0 −3.63409 6.29444i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 201.3
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.b 6
3.b odd 2 1 3420.2.t.v 6
4.b odd 2 1 1520.2.q.i 6
5.b even 2 1 1900.2.i.c 6
5.c odd 4 2 1900.2.s.c 12
19.c even 3 1 inner 380.2.i.b 6
19.c even 3 1 7220.2.a.n 3
19.d odd 6 1 7220.2.a.o 3
57.h odd 6 1 3420.2.t.v 6
76.g odd 6 1 1520.2.q.i 6
95.i even 6 1 1900.2.i.c 6
95.m odd 12 2 1900.2.s.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.i.b 6 1.a even 1 1 trivial
380.2.i.b 6 19.c even 3 1 inner
1520.2.q.i 6 4.b odd 2 1
1520.2.q.i 6 76.g odd 6 1
1900.2.i.c 6 5.b even 2 1
1900.2.i.c 6 95.i even 6 1
1900.2.s.c 12 5.c odd 4 2
1900.2.s.c 12 95.m odd 12 2
3420.2.t.v 6 3.b odd 2 1
3420.2.t.v 6 57.h odd 6 1
7220.2.a.n 3 19.c even 3 1
7220.2.a.o 3 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - T_{3}^{5} + 9 T_{3}^{4} + 2 T_{3}^{3} + 67 T_{3}^{2} - 24 T_{3} + 9 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 9 - 24 T + 67 T^{2} + 2 T^{3} + 9 T^{4} - T^{5} + T^{6} \)
$5$ \( ( 1 - T + T^{2} )^{3} \)
$7$ \( ( 5 - 7 T - 2 T^{2} + T^{3} )^{2} \)
$11$ \( ( 9 - 5 T^{2} + T^{3} )^{2} \)
$13$ \( ( 1 - T + T^{2} )^{3} \)
$17$ \( 6561 - 2916 T + 1539 T^{2} - 54 T^{3} + 45 T^{4} - 3 T^{5} + T^{6} \)
$19$ \( 6859 - 342 T^{2} + 7 T^{3} - 18 T^{4} + T^{6} \)
$23$ \( 2025 + 1755 T + 2151 T^{2} - 636 T^{3} + 157 T^{4} - 14 T^{5} + T^{6} \)
$29$ \( 263169 + 41553 T + 9639 T^{2} + 540 T^{3} + 117 T^{4} + 6 T^{5} + T^{6} \)
$31$ \( ( 71 + 14 T - 11 T^{2} + T^{3} )^{2} \)
$37$ \( ( 5 - 7 T - 2 T^{2} + T^{3} )^{2} \)
$41$ \( 18225 + 8505 T + 4779 T^{2} - 108 T^{3} + 99 T^{4} + 6 T^{5} + T^{6} \)
$43$ \( 11025 + 6510 T + 3319 T^{2} + 520 T^{3} + 87 T^{4} - 5 T^{5} + T^{6} \)
$47$ \( 263169 - 41553 T + 9639 T^{2} - 540 T^{3} + 117 T^{4} - 6 T^{5} + T^{6} \)
$53$ \( 81 + 270 T + 1017 T^{2} - 408 T^{3} + 139 T^{4} - 13 T^{5} + T^{6} \)
$59$ \( 263169 - 41553 T + 9639 T^{2} - 540 T^{3} + 117 T^{4} - 6 T^{5} + T^{6} \)
$61$ \( 3969 + 1071 T + 730 T^{2} + 7 T^{3} + 66 T^{4} + 7 T^{5} + T^{6} \)
$67$ \( 28224 - 15456 T + 7792 T^{2} - 704 T^{3} + 108 T^{4} + 4 T^{5} + T^{6} \)
$71$ \( 729 - 486 T + 567 T^{2} + 108 T^{3} + 99 T^{4} - 9 T^{5} + T^{6} \)
$73$ \( 625 - 1725 T + 4311 T^{2} - 1192 T^{3} + 255 T^{4} - 18 T^{5} + T^{6} \)
$79$ \( 732736 + 263648 T + 67472 T^{2} + 8144 T^{3} + 716 T^{4} + 32 T^{5} + T^{6} \)
$83$ \( ( 855 + 294 T + 31 T^{2} + T^{3} )^{2} \)
$89$ \( 5184 + 9504 T + 17568 T^{2} - 120 T^{3} + 136 T^{4} + 2 T^{5} + T^{6} \)
$97$ \( 93025 + 28670 T + 12191 T^{2} - 424 T^{3} + 215 T^{4} + 11 T^{5} + T^{6} \)
show more
show less