Properties

Label 1380.4.a.g
Level $1380$
Weight $4$
Character orbit 1380.a
Self dual yes
Analytic conductor $81.423$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1380,4,Mod(1,1380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1380.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1380.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(81.4226358079\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 345x^{3} - 1858x^{2} + 6144x + 4608 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} - 5 q^{5} + ( - \beta_1 - 1) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} - 5 q^{5} + ( - \beta_1 - 1) q^{7} + 9 q^{9} + (\beta_{4} + \beta_{2} - 4) q^{11} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 + 2) q^{13} - 15 q^{15} + ( - 4 \beta_{4} + \beta_{3} + \beta_1 - 5) q^{17} + (2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 18) q^{19} + ( - 3 \beta_1 - 3) q^{21} + 23 q^{23} + 25 q^{25} + 27 q^{27} + (3 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 15) q^{29} + ( - 9 \beta_{4} + \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 13) q^{31} + (3 \beta_{4} + 3 \beta_{2} - 12) q^{33} + (5 \beta_1 + 5) q^{35} + (4 \beta_{4} + 2 \beta_{3} + 8 \beta_{2} - 3 \beta_1 - 77) q^{37} + ( - 3 \beta_{3} - 3 \beta_{2} + 9 \beta_1 + 6) q^{39} + ( - \beta_{4} - 6 \beta_{3} - 7 \beta_{2} - \beta_1 - 65) q^{41} + ( - 6 \beta_{4} + 8 \beta_{3} - 9 \beta_{2} + \beta_1 - 160) q^{43} - 45 q^{45} + (6 \beta_{4} - 17 \beta_{3} - 10 \beta_{2} + 12 \beta_1 - 72) q^{47} + ( - 7 \beta_{4} - 3 \beta_{3} + 9 \beta_{2} + 19 \beta_1 - 32) q^{49} + ( - 12 \beta_{4} + 3 \beta_{3} + 3 \beta_1 - 15) q^{51} + (19 \beta_{4} + 8 \beta_{3} + 4 \beta_{2} - 16 \beta_1 - 79) q^{53} + ( - 5 \beta_{4} - 5 \beta_{2} + 20) q^{55} + (6 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 54) q^{57} + (\beta_{4} - 10 \beta_{3} + 3 \beta_{2} + 21 \beta_1 + 95) q^{59} + (11 \beta_{4} + 4 \beta_{3} + 24 \beta_{2} - 7 \beta_1 - 162) q^{61} + ( - 9 \beta_1 - 9) q^{63} + (5 \beta_{3} + 5 \beta_{2} - 15 \beta_1 - 10) q^{65} + (15 \beta_{4} + 3 \beta_{3} + 34 \beta_{2} + 8 \beta_1 - 67) q^{67} + 69 q^{69} + ( - 27 \beta_{4} + 10 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 9) q^{71} + (16 \beta_{4} - 17 \beta_{3} - 19 \beta_{2} - 27 \beta_1 - 218) q^{73} + 75 q^{75} + (2 \beta_{4} + 9 \beta_{3} - 2 \beta_{2} - 10 \beta_1 - 98) q^{77} + ( - 3 \beta_{4} - 17 \beta_{3} - 35 \beta_{2} + 10 \beta_1 - 250) q^{79} + 81 q^{81} + ( - 4 \beta_{4} + 17 \beta_{3} + 5 \beta_{2} - 20 \beta_1 - 137) q^{83} + (20 \beta_{4} - 5 \beta_{3} - 5 \beta_1 + 25) q^{85} + (9 \beta_{4} + 6 \beta_{3} - 12 \beta_{2} - 45) q^{87} + ( - 9 \beta_{4} - 13 \beta_{3} - 13 \beta_{2} - 32 \beta_1 - 306) q^{89} + (22 \beta_{4} - \beta_{3} - 43 \beta_{2} - 51 \beta_1 - 822) q^{91} + ( - 27 \beta_{4} + 3 \beta_{3} + 9 \beta_{2} + 9 \beta_1 - 39) q^{93} + ( - 10 \beta_{4} - 5 \beta_{3} + 5 \beta_{2} + 5 \beta_1 + 90) q^{95} + (14 \beta_{4} - 10 \beta_{3} + 27 \beta_{2} - 15 \beta_1 - 294) q^{97} + (9 \beta_{4} + 9 \beta_{2} - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 15 q^{3} - 25 q^{5} - 5 q^{7} + 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 15 q^{3} - 25 q^{5} - 5 q^{7} + 45 q^{9} - 18 q^{11} + 8 q^{13} - 75 q^{15} - 31 q^{17} - 84 q^{19} - 15 q^{21} + 115 q^{23} + 125 q^{25} + 135 q^{27} - 65 q^{29} - 81 q^{31} - 54 q^{33} + 25 q^{35} - 373 q^{37} + 24 q^{39} - 339 q^{41} - 796 q^{43} - 225 q^{45} - 382 q^{47} - 180 q^{49} - 93 q^{51} - 341 q^{53} + 90 q^{55} - 252 q^{57} + 457 q^{59} - 780 q^{61} - 45 q^{63} - 40 q^{65} - 299 q^{67} + 345 q^{69} + 11 q^{71} - 1092 q^{73} + 375 q^{75} - 468 q^{77} - 1290 q^{79} + 405 q^{81} - 659 q^{83} + 155 q^{85} - 195 q^{87} - 1574 q^{89} - 4068 q^{91} - 243 q^{93} + 420 q^{95} - 1462 q^{97} - 162 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 345x^{3} - 1858x^{2} + 6144x + 4608 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{4} + 96\nu^{3} - 231\nu^{2} - 19166\nu - 32448 ) / 4320 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 96\nu^{3} + 231\nu^{2} + 27806\nu + 32448 ) / 4320 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} + 16\nu^{3} + 169\nu^{2} - 1726\nu + 1632 ) / 240 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{4} + 24\nu^{3} - 3939\nu^{2} - 25694\nu + 48048 ) / 2160 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{4} - 6\beta_{3} - \beta_{2} + 19\beta _1 + 280 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -20\beta_{4} - 36\beta_{3} + 213\beta_{2} + 421\beta _1 + 2252 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -996\beta_{4} - 2070\beta_{3} + 1513\beta_{2} + 8221\beta _1 + 86616 ) / 2 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
20.5052
−9.77455
−0.640634
−12.8770
2.78694
0 3.00000 0 −5.00000 0 −30.7027 0 9.00000 0
1.2 0 3.00000 0 −5.00000 0 −8.87971 0 9.00000 0
1.3 0 3.00000 0 −5.00000 0 3.69672 0 9.00000 0
1.4 0 3.00000 0 −5.00000 0 12.0619 0 9.00000 0
1.5 0 3.00000 0 −5.00000 0 18.8238 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1380.4.a.g 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.4.a.g 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{5} + 5T_{7}^{4} - 755T_{7}^{3} + 3239T_{7}^{2} + 59806T_{7} - 228832 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1380))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( (T - 3)^{5} \) Copy content Toggle raw display
$5$ \( (T + 5)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + 5 T^{4} - 755 T^{3} + \cdots - 228832 \) Copy content Toggle raw display
$11$ \( T^{5} + 18 T^{4} - 1218 T^{3} + \cdots + 1134912 \) Copy content Toggle raw display
$13$ \( T^{5} - 8 T^{4} - 7138 T^{3} + \cdots - 26116032 \) Copy content Toggle raw display
$17$ \( T^{5} + 31 T^{4} - 9533 T^{3} + \cdots + 69622200 \) Copy content Toggle raw display
$19$ \( T^{5} + 84 T^{4} + \cdots + 532012032 \) Copy content Toggle raw display
$23$ \( (T - 23)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} + 65 T^{4} + \cdots + 11943825132 \) Copy content Toggle raw display
$31$ \( T^{5} + 81 T^{4} + \cdots + 2185574592 \) Copy content Toggle raw display
$37$ \( T^{5} + 373 T^{4} + \cdots + 27197717200 \) Copy content Toggle raw display
$41$ \( T^{5} + 339 T^{4} + \cdots - 177333137460 \) Copy content Toggle raw display
$43$ \( T^{5} + 796 T^{4} + \cdots - 4086918988800 \) Copy content Toggle raw display
$47$ \( T^{5} + 382 T^{4} + \cdots + 8652627173376 \) Copy content Toggle raw display
$53$ \( T^{5} + 341 T^{4} + \cdots + 991870895052 \) Copy content Toggle raw display
$59$ \( T^{5} - 457 T^{4} + \cdots + 640764426672 \) Copy content Toggle raw display
$61$ \( T^{5} + 780 T^{4} + \cdots + 1681676978400 \) Copy content Toggle raw display
$67$ \( T^{5} + 299 T^{4} + \cdots - 55666166592992 \) Copy content Toggle raw display
$71$ \( T^{5} - 11 T^{4} + \cdots + 4782857500224 \) Copy content Toggle raw display
$73$ \( T^{5} + 1092 T^{4} + \cdots + 29125225312416 \) Copy content Toggle raw display
$79$ \( T^{5} + 1290 T^{4} + \cdots + 6401513062400 \) Copy content Toggle raw display
$83$ \( T^{5} + 659 T^{4} + \cdots - 3169995431040 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots + 182366718082560 \) Copy content Toggle raw display
$97$ \( T^{5} + 1462 T^{4} + \cdots + 4996197291840 \) Copy content Toggle raw display
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