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Properties

Label 1380.4.a.j
Level $1380$
Weight $4$
Character orbit 1380.a
Self dual yes
Analytic conductor $81.423$
Analytic rank $0$
Dimension $7$
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: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 1420x^{5} - 7866x^{4} + 519199x^{3} + 5329890x^{2} - 8528484x - 84125016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + 5 q^{5} + ( - \beta_1 + 5) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + 5 q^{5} + ( - \beta_1 + 5) q^{7} + 9 q^{9} + (\beta_{3} + 7) q^{11} + (\beta_{2} + 13) q^{13} + 15 q^{15} + ( - \beta_{4} + 23) q^{17} + ( - \beta_{5} - \beta_1 + 29) q^{19} + ( - 3 \beta_1 + 15) q^{21} + 23 q^{23} + 25 q^{25} + 27 q^{27} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 12) q^{29}+ \cdots + (9 \beta_{3} + 63) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 21 q^{3} + 35 q^{5} + 35 q^{7} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 21 q^{3} + 35 q^{5} + 35 q^{7} + 63 q^{9} + 48 q^{11} + 90 q^{13} + 105 q^{15} + 163 q^{17} + 200 q^{19} + 105 q^{21} + 161 q^{23} + 175 q^{25} + 189 q^{27} + 81 q^{29} + 125 q^{31} + 144 q^{33} + 175 q^{35} + 5 q^{37} + 270 q^{39} + 369 q^{41} + 462 q^{43} + 315 q^{45} + 134 q^{47} + 614 q^{49} + 489 q^{51} + 561 q^{53} + 240 q^{55} + 600 q^{57} + 951 q^{59} + 860 q^{61} + 315 q^{63} + 450 q^{65} + 447 q^{67} + 483 q^{69} + 735 q^{71} + 1460 q^{73} + 525 q^{75} + 496 q^{77} + 18 q^{79} + 567 q^{81} + 261 q^{83} + 815 q^{85} + 243 q^{87} + 2024 q^{89} + 692 q^{91} + 375 q^{93} + 1000 q^{95} + 668 q^{97} + 432 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 1420x^{5} - 7866x^{4} + 519199x^{3} + 5329890x^{2} - 8528484x - 84125016 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6651 \nu^{6} - 334402 \nu^{5} + 16340291 \nu^{4} + 383441948 \nu^{3} - 7104688228 \nu^{2} + \cdots - 119303897808 ) / 7723549500 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 241709 \nu^{6} - 4782307 \nu^{5} - 243520969 \nu^{4} + 2995561043 \nu^{3} + \cdots + 3312910929072 ) / 77235495000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 324903 \nu^{6} + 4276769 \nu^{5} + 408624073 \nu^{4} - 2008023181 \nu^{3} + \cdots + 2905006697976 ) / 30894198000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 326149 \nu^{6} - 5568927 \nu^{5} - 383660159 \nu^{4} + 3908514123 \nu^{3} + \cdots - 2038890253608 ) / 12357679200 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9836199 \nu^{6} + 119468077 \nu^{5} + 12016045309 \nu^{4} - 64210886273 \nu^{3} + \cdots + 75257434726008 ) / 308941980000 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} - \beta_{5} - \beta_{4} - 4\beta_{3} + 4\beta_{2} + 8\beta _1 + 406 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -12\beta_{6} + 8\beta_{5} + 38\beta_{4} - 58\beta_{3} + 14\beta_{2} + 673\beta _1 + 3374 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -1205\beta_{6} - 1133\beta_{5} - 185\beta_{4} - 2300\beta_{3} + 3728\beta_{2} + 9772\beta _1 + 280238 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -24792\beta_{6} - 3308\beta_{5} + 46294\beta_{4} - 61598\beta_{3} + 26014\beta_{2} + 517841\beta _1 + 4184410 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1337569 \beta_{6} - 1087825 \beta_{5} + 476879 \beta_{4} - 1624576 \beta_{3} + 3224080 \beta_{2} + \cdots + 220992514 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
31.2782
25.9626
4.05705
−3.94389
−14.9962
−17.0788
−25.2790
0 3.00000 0 5.00000 0 −26.2782 0 9.00000 0
1.2 0 3.00000 0 5.00000 0 −20.9626 0 9.00000 0
1.3 0 3.00000 0 5.00000 0 0.942946 0 9.00000 0
1.4 0 3.00000 0 5.00000 0 8.94389 0 9.00000 0
1.5 0 3.00000 0 5.00000 0 19.9962 0 9.00000 0
1.6 0 3.00000 0 5.00000 0 22.0788 0 9.00000 0
1.7 0 3.00000 0 5.00000 0 30.2790 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.j ✓ 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.4.a.j ✓ 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{7} - 35T_{7}^{6} - 895T_{7}^{5} + 38991T_{7}^{4} + 28754T_{7}^{3} - 10228600T_{7}^{2} + 75449216T_{7} - 62104064 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1380))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( (T - 3)^{7} \) Copy content Toggle raw display
$5$ \( (T - 5)^{7} \) Copy content Toggle raw display
$7$ \( T^{7} - 35 T^{6} + \cdots - 62104064 \) Copy content Toggle raw display
$11$ \( T^{7} + \cdots - 4129719552 \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots + 83838958848 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots + 5166038757600 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots - 23206752976896 \) Copy content Toggle raw display
$23$ \( (T - 23)^{7} \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots + 1460109360096 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots - 64423357857792 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots - 53\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots - 71\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots - 65\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots - 22\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots - 14\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots - 11\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots - 52\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots - 25\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots - 56\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots + 74\!\cdots\!04 \) Copy content Toggle raw display
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