Properties

Label 1323.4.a.s
Level $1323$
Weight $4$
Character orbit 1323.a
Self dual yes
Analytic conductor $78.060$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1323,4,Mod(1,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1323.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1323.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: \(78.0595269376\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 189)
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 \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{4} + \beta q^{5} - 9 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - q^{4} + \beta q^{5} - 9 \beta q^{8} + 7 q^{10} - 7 \beta q^{11} + 26 q^{13} - 55 q^{16} + 24 \beta q^{17} + 35 q^{19} - \beta q^{20} - 49 q^{22} + 39 \beta q^{23} - 118 q^{25} + 26 \beta q^{26} + 2 \beta q^{29} + 75 q^{31} + 17 \beta q^{32} + 168 q^{34} - 111 q^{37} + 35 \beta q^{38} - 63 q^{40} - 181 \beta q^{41} - 328 q^{43} + 7 \beta q^{44} + 273 q^{46} - 144 \beta q^{47} - 118 \beta q^{50} - 26 q^{52} - 48 \beta q^{53} - 49 q^{55} + 14 q^{58} - 48 \beta q^{59} + 152 q^{61} + 75 \beta q^{62} + 559 q^{64} + 26 \beta q^{65} + 202 q^{67} - 24 \beta q^{68} - 399 \beta q^{71} - 672 q^{73} - 111 \beta q^{74} - 35 q^{76} - 988 q^{79} - 55 \beta q^{80} - 1267 q^{82} + 54 \beta q^{83} + 168 q^{85} - 328 \beta q^{86} + 441 q^{88} - 365 \beta q^{89} - 39 \beta q^{92} - 1008 q^{94} + 35 \beta q^{95} + 492 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 14 q^{10} + 52 q^{13} - 110 q^{16} + 70 q^{19} - 98 q^{22} - 236 q^{25} + 150 q^{31} + 336 q^{34} - 222 q^{37} - 126 q^{40} - 656 q^{43} + 546 q^{46} - 52 q^{52} - 98 q^{55} + 28 q^{58} + 304 q^{61} + 1118 q^{64} + 404 q^{67} - 1344 q^{73} - 70 q^{76} - 1976 q^{79} - 2534 q^{82} + 336 q^{85} + 882 q^{88} - 2016 q^{94} + 984 q^{97}+O(q^{100}) \) 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
−2.64575
2.64575
−2.64575 0 −1.00000 −2.64575 0 0 23.8118 0 7.00000
1.2 2.64575 0 −1.00000 2.64575 0 0 −23.8118 0 7.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.4.a.s 2
3.b odd 2 1 inner 1323.4.a.s 2
7.b odd 2 1 189.4.a.g 2
21.c even 2 1 189.4.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.a.g 2 7.b odd 2 1
189.4.a.g 2 21.c even 2 1
1323.4.a.s 2 1.a even 1 1 trivial
1323.4.a.s 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1323))\):

\( T_{2}^{2} - 7 \) Copy content Toggle raw display
\( T_{5}^{2} - 7 \) Copy content Toggle raw display
\( T_{13} - 26 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 7 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 7 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 343 \) Copy content Toggle raw display
$13$ \( (T - 26)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 4032 \) Copy content Toggle raw display
$19$ \( (T - 35)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 10647 \) Copy content Toggle raw display
$29$ \( T^{2} - 28 \) Copy content Toggle raw display
$31$ \( (T - 75)^{2} \) Copy content Toggle raw display
$37$ \( (T + 111)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 229327 \) Copy content Toggle raw display
$43$ \( (T + 328)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 145152 \) Copy content Toggle raw display
$53$ \( T^{2} - 16128 \) Copy content Toggle raw display
$59$ \( T^{2} - 16128 \) Copy content Toggle raw display
$61$ \( (T - 152)^{2} \) Copy content Toggle raw display
$67$ \( (T - 202)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 1114407 \) Copy content Toggle raw display
$73$ \( (T + 672)^{2} \) Copy content Toggle raw display
$79$ \( (T + 988)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 20412 \) Copy content Toggle raw display
$89$ \( T^{2} - 932575 \) Copy content Toggle raw display
$97$ \( (T - 492)^{2} \) Copy content Toggle raw display
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