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Properties

Label 1380.4.a.i
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: \( x^{7} - 1584x^{5} - 4254x^{4} + 647195x^{3} + 3452070x^{2} - 15124380x - 76516056 \) 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.

\(f(q)\) \(=\) \( q + 3 q^{3} - 5 q^{5} + (\beta_1 - 1) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} - 5 q^{5} + (\beta_1 - 1) q^{7} + 9 q^{9} + ( - \beta_{5} - 3) q^{11} + (\beta_{3} + \beta_1 - 1) q^{13} - 15 q^{15} + ( - \beta_{6} - \beta_{3} + \beta_1) q^{17} + (\beta_{2} + \beta_1 + 6) q^{19} + (3 \beta_1 - 3) q^{21} - 23 q^{23} + 25 q^{25} + 27 q^{27} + (\beta_{6} - \beta_{5} + \beta_{4} + \cdots + 33) q^{29}+ \cdots + ( - 9 \beta_{5} - 27) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 21 q^{3} - 35 q^{5} - 7 q^{7} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 21 q^{3} - 35 q^{5} - 7 q^{7} + 63 q^{9} - 20 q^{11} - 6 q^{13} - 105 q^{15} - 3 q^{17} + 44 q^{19} - 21 q^{21} - 161 q^{23} + 175 q^{25} + 189 q^{27} + 229 q^{29} + 101 q^{31} - 60 q^{33} + 35 q^{35} + 467 q^{37} - 18 q^{39} + 657 q^{41} + 386 q^{43} - 315 q^{45} + 114 q^{47} + 774 q^{49} - 9 q^{51} - 141 q^{53} + 100 q^{55} + 132 q^{57} + 395 q^{59} + 808 q^{61} - 63 q^{63} + 30 q^{65} + 1449 q^{67} - 483 q^{69} + 1127 q^{71} + 1252 q^{73} + 525 q^{75} + 356 q^{77} + 886 q^{79} + 567 q^{81} + 1255 q^{83} + 15 q^{85} + 687 q^{87} + 1180 q^{89} + 3004 q^{91} + 303 q^{93} - 220 q^{95} + 2936 q^{97} - 180 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 1584x^{5} - 4254x^{4} + 647195x^{3} + 3452070x^{2} - 15124380x - 76516056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13019 \nu^{6} + 43617 \nu^{5} + 19949565 \nu^{4} - 139561469 \nu^{3} - 7709124338 \nu^{2} + \cdots + 331905933848 ) / 3724902400 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 24373 \nu^{6} - 33039 \nu^{5} - 32930355 \nu^{4} - 4921677 \nu^{3} + 10537696046 \nu^{2} + \cdots - 17836679016 ) / 5587353600 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 8591 \nu^{6} - 61187 \nu^{5} + 13017385 \nu^{4} + 74057559 \nu^{3} - 4995114282 \nu^{2} + \cdots + 135955756472 ) / 1862451200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17447 \nu^{6} - 148421 \nu^{5} - 26881745 \nu^{4} + 120374097 \nu^{3} + 10888747194 \nu^{2} + \cdots - 312277384824 ) / 1862451200 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6787 \nu^{6} - 30841 \nu^{5} - 11136245 \nu^{4} + 20118837 \nu^{3} + 4706788674 \nu^{2} + \cdots - 113173087704 ) / 465612800 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -3\beta_{6} + 5\beta_{5} - 5\beta_{4} - 6\beta_{3} + 3\beta _1 + 455 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -6\beta_{6} + 2\beta_{5} - 2\beta_{4} - 12\beta_{3} - 32\beta_{2} + 783\beta _1 + 1836 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2903\beta_{6} + 4513\beta_{5} - 4281\beta_{4} - 4518\beta_{3} + 4471\beta _1 + 349167 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9406\beta_{6} + 1434\beta_{5} - 22514\beta_{4} - 20916\beta_{3} - 31776\beta_{2} + 620847\beta _1 + 2362292 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2639151 \beta_{6} + 3938105 \beta_{5} - 3653225 \beta_{4} - 3311694 \beta_{3} - 49536 \beta_{2} + \cdots + 279344447 \) 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
−27.9985
−24.5037
−6.51528
−4.14015
4.87368
28.1942
30.0898
0 3.00000 0 −5.00000 0 −28.9985 0 9.00000 0
1.2 0 3.00000 0 −5.00000 0 −25.5037 0 9.00000 0
1.3 0 3.00000 0 −5.00000 0 −7.51528 0 9.00000 0
1.4 0 3.00000 0 −5.00000 0 −5.14015 0 9.00000 0
1.5 0 3.00000 0 −5.00000 0 3.87368 0 9.00000 0
1.6 0 3.00000 0 −5.00000 0 27.1942 0 9.00000 0
1.7 0 3.00000 0 −5.00000 0 29.0898 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.i ✓ 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.4.a.i ✓ 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{7} + 7T_{7}^{6} - 1563T_{7}^{5} - 12139T_{7}^{4} + 614374T_{7}^{3} + 5352312T_{7}^{2} - 6303584T_{7} - 87547008 \) 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
$7$ \( T^{7} + 7 T^{6} + \cdots - 87547008 \) Copy content Toggle raw display
$11$ \( T^{7} + \cdots + 120865167360 \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots + 195439101696 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots + 4624727555520 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots - 5660090597376 \) Copy content Toggle raw display
$23$ \( (T + 23)^{7} \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots - 27\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots + 870957915897856 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots - 10\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots + 23\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots + 37\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots - 167917598539776 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots + 12\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots + 510015282100224 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots + 71\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots + 38\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots + 18\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots - 32\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots + 38\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots + 87\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots - 14\!\cdots\!32 \) Copy content Toggle raw display
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