Properties

Label 2352.4.a.bp
Level $2352$
Weight $4$
Character orbit 2352.a
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
Dimension $2$
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] = [2352,4,Mod(1,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, 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("2352.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.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: \(138.772492334\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 84)
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 \(\beta = \frac{1}{2}(-1 + 3\sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + (\beta + 2) q^{5} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + (\beta + 2) q^{5} + 9 q^{9} + (\beta + 26) q^{11} + ( - 5 \beta - 33) q^{13} + ( - 3 \beta - 6) q^{15} + ( - 8 \beta - 16) q^{17} + (\beta + 85) q^{19} + 96 q^{23} + (3 \beta + 7) q^{25} - 27 q^{27} + ( - 17 \beta - 28) q^{29} + ( - 10 \beta - 51) q^{31} + ( - 3 \beta - 78) q^{33} + (19 \beta - 77) q^{37} + (15 \beta + 99) q^{39} + (34 \beta - 70) q^{41} + ( - 19 \beta + 239) q^{43} + (9 \beta + 18) q^{45} + (16 \beta + 98) q^{47} + (24 \beta + 48) q^{51} + (27 \beta + 156) q^{53} + (27 \beta + 180) q^{55} + ( - 3 \beta - 255) q^{57} + ( - 3 \beta - 636) q^{59} + (36 \beta - 146) q^{61} + ( - 38 \beta - 706) q^{65} + ( - 9 \beta + 433) q^{67} - 288 q^{69} + ( - 32 \beta + 686) q^{71} + (21 \beta + 691) q^{73} + ( - 9 \beta - 21) q^{75} + (4 \beta + 93) q^{79} + 81 q^{81} + (43 \beta - 178) q^{83} + ( - 24 \beta - 1056) q^{85} + (51 \beta + 84) q^{87} + (22 \beta - 400) q^{89} + (30 \beta + 153) q^{93} + (86 \beta + 298) q^{95} + (21 \beta - 410) q^{97} + (9 \beta + 234) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 3 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 3 q^{5} + 18 q^{9} + 51 q^{11} - 61 q^{13} - 9 q^{15} - 24 q^{17} + 169 q^{19} + 192 q^{23} + 11 q^{25} - 54 q^{27} - 39 q^{29} - 92 q^{31} - 153 q^{33} - 173 q^{37} + 183 q^{39} - 174 q^{41} + 497 q^{43} + 27 q^{45} + 180 q^{47} + 72 q^{51} + 285 q^{53} + 333 q^{55} - 507 q^{57} - 1269 q^{59} - 328 q^{61} - 1374 q^{65} + 875 q^{67} - 576 q^{69} + 1404 q^{71} + 1361 q^{73} - 33 q^{75} + 182 q^{79} + 162 q^{81} - 399 q^{83} - 2088 q^{85} + 117 q^{87} - 822 q^{89} + 276 q^{93} + 510 q^{95} - 841 q^{97} + 459 q^{99}+O(q^{100}) \) 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
−3.27492
4.27492
0 −3.00000 0 −9.82475 0 0 0 9.00000 0
1.2 0 −3.00000 0 12.8248 0 0 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)

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 2352.4.a.bp 2
4.b odd 2 1 588.4.a.h 2
7.b odd 2 1 2352.4.a.cb 2
7.d odd 6 2 336.4.q.h 4
12.b even 2 1 1764.4.a.p 2
28.d even 2 1 588.4.a.g 2
28.f even 6 2 84.4.i.b 4
28.g odd 6 2 588.4.i.i 4
84.h odd 2 1 1764.4.a.x 2
84.j odd 6 2 252.4.k.d 4
84.n even 6 2 1764.4.k.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.b 4 28.f even 6 2
252.4.k.d 4 84.j odd 6 2
336.4.q.h 4 7.d odd 6 2
588.4.a.g 2 28.d even 2 1
588.4.a.h 2 4.b odd 2 1
588.4.i.i 4 28.g odd 6 2
1764.4.a.p 2 12.b even 2 1
1764.4.a.x 2 84.h odd 2 1
1764.4.k.z 4 84.n even 6 2
2352.4.a.bp 2 1.a even 1 1 trivial
2352.4.a.cb 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5}^{2} - 3T_{5} - 126 \) Copy content Toggle raw display
\( T_{11}^{2} - 51T_{11} + 522 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3T - 126 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 51T + 522 \) Copy content Toggle raw display
$13$ \( T^{2} + 61T - 2276 \) Copy content Toggle raw display
$17$ \( T^{2} + 24T - 8064 \) Copy content Toggle raw display
$19$ \( T^{2} - 169T + 7012 \) Copy content Toggle raw display
$23$ \( (T - 96)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 39T - 36684 \) Copy content Toggle raw display
$31$ \( T^{2} + 92T - 10709 \) Copy content Toggle raw display
$37$ \( T^{2} + 173T - 38816 \) Copy content Toggle raw display
$41$ \( T^{2} + 174T - 140688 \) Copy content Toggle raw display
$43$ \( T^{2} - 497T + 15454 \) Copy content Toggle raw display
$47$ \( T^{2} - 180T - 24732 \) Copy content Toggle raw display
$53$ \( T^{2} - 285T - 73188 \) Copy content Toggle raw display
$59$ \( T^{2} + 1269 T + 401436 \) Copy content Toggle raw display
$61$ \( T^{2} + 328T - 139316 \) Copy content Toggle raw display
$67$ \( T^{2} - 875T + 181018 \) Copy content Toggle raw display
$71$ \( T^{2} - 1404 T + 361476 \) Copy content Toggle raw display
$73$ \( T^{2} - 1361 T + 406522 \) Copy content Toggle raw display
$79$ \( T^{2} - 182T + 6229 \) Copy content Toggle raw display
$83$ \( T^{2} + 399T - 197334 \) Copy content Toggle raw display
$89$ \( T^{2} + 822T + 106848 \) Copy content Toggle raw display
$97$ \( T^{2} + 841T + 120262 \) Copy content Toggle raw display
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