Properties

Label 2352.4.a.bn
Level $2352$
Weight $4$
Character orbit 2352.a
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 294)
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 \(\beta = 7\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + (\beta - 6) q^{5} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + (\beta - 6) q^{5} + 9 q^{9} + (6 \beta + 2) q^{11} + ( - 3 \beta - 24) q^{13} + ( - 3 \beta + 18) q^{15} + (\beta - 42) q^{17} + ( - 2 \beta - 36) q^{19} + ( - 6 \beta + 154) q^{23} + ( - 12 \beta + 9) q^{25} - 27 q^{27} + (18 \beta - 40) q^{29} + ( - 6 \beta + 192) q^{31} + ( - 18 \beta - 6) q^{33} + ( - 12 \beta + 268) q^{37} + (9 \beta + 72) q^{39} + ( - 5 \beta - 378) q^{41} + ( - 24 \beta - 200) q^{43} + (9 \beta - 54) q^{45} + ( - 10 \beta + 156) q^{47} + ( - 3 \beta + 126) q^{51} + ( - 24 \beta - 26) q^{53} + ( - 34 \beta + 576) q^{55} + (6 \beta + 108) q^{57} + (2 \beta + 432) q^{59} + (13 \beta - 708) q^{61} + ( - 6 \beta - 150) q^{65} + ( - 24 \beta - 72) q^{67} + (18 \beta - 462) q^{69} + (30 \beta + 762) q^{71} + ( - 83 \beta - 372) q^{73} + (36 \beta - 27) q^{75} + ( - 84 \beta - 488) q^{79} + 81 q^{81} + (136 \beta - 156) q^{83} + ( - 48 \beta + 350) q^{85} + ( - 54 \beta + 120) q^{87} + (29 \beta - 54) q^{89} + (18 \beta - 576) q^{93} + ( - 24 \beta + 20) q^{95} + (125 \beta + 372) q^{97} + (54 \beta + 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 12 q^{5} + 18 q^{9} + 4 q^{11} - 48 q^{13} + 36 q^{15} - 84 q^{17} - 72 q^{19} + 308 q^{23} + 18 q^{25} - 54 q^{27} - 80 q^{29} + 384 q^{31} - 12 q^{33} + 536 q^{37} + 144 q^{39} - 756 q^{41} - 400 q^{43} - 108 q^{45} + 312 q^{47} + 252 q^{51} - 52 q^{53} + 1152 q^{55} + 216 q^{57} + 864 q^{59} - 1416 q^{61} - 300 q^{65} - 144 q^{67} - 924 q^{69} + 1524 q^{71} - 744 q^{73} - 54 q^{75} - 976 q^{79} + 162 q^{81} - 312 q^{83} + 700 q^{85} + 240 q^{87} - 108 q^{89} - 1152 q^{93} + 40 q^{95} + 744 q^{97} + 36 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
0 −3.00000 0 −15.8995 0 0 0 9.00000 0
1.2 0 −3.00000 0 3.89949 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.bn 2
4.b odd 2 1 294.4.a.k yes 2
7.b odd 2 1 2352.4.a.cd 2
12.b even 2 1 882.4.a.bi 2
28.d even 2 1 294.4.a.j 2
28.f even 6 2 294.4.e.o 4
28.g odd 6 2 294.4.e.n 4
84.h odd 2 1 882.4.a.bc 2
84.j odd 6 2 882.4.g.bd 4
84.n even 6 2 882.4.g.y 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.j 2 28.d even 2 1
294.4.a.k yes 2 4.b odd 2 1
294.4.e.n 4 28.g odd 6 2
294.4.e.o 4 28.f even 6 2
882.4.a.bc 2 84.h odd 2 1
882.4.a.bi 2 12.b even 2 1
882.4.g.y 4 84.n even 6 2
882.4.g.bd 4 84.j odd 6 2
2352.4.a.bn 2 1.a even 1 1 trivial
2352.4.a.cd 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5}^{2} + 12T_{5} - 62 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} - 3524 \) 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
$5$ \( T^{2} + 12T - 62 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T - 3524 \) Copy content Toggle raw display
$13$ \( T^{2} + 48T - 306 \) Copy content Toggle raw display
$17$ \( T^{2} + 84T + 1666 \) Copy content Toggle raw display
$19$ \( T^{2} + 72T + 904 \) Copy content Toggle raw display
$23$ \( T^{2} - 308T + 20188 \) Copy content Toggle raw display
$29$ \( T^{2} + 80T - 30152 \) Copy content Toggle raw display
$31$ \( T^{2} - 384T + 33336 \) Copy content Toggle raw display
$37$ \( T^{2} - 536T + 57712 \) Copy content Toggle raw display
$41$ \( T^{2} + 756T + 140434 \) Copy content Toggle raw display
$43$ \( T^{2} + 400T - 16448 \) Copy content Toggle raw display
$47$ \( T^{2} - 312T + 14536 \) Copy content Toggle raw display
$53$ \( T^{2} + 52T - 55772 \) Copy content Toggle raw display
$59$ \( T^{2} - 864T + 186232 \) Copy content Toggle raw display
$61$ \( T^{2} + 1416 T + 484702 \) Copy content Toggle raw display
$67$ \( T^{2} + 144T - 51264 \) Copy content Toggle raw display
$71$ \( T^{2} - 1524 T + 492444 \) Copy content Toggle raw display
$73$ \( T^{2} + 744T - 536738 \) Copy content Toggle raw display
$79$ \( T^{2} + 976T - 453344 \) Copy content Toggle raw display
$83$ \( T^{2} + 312 T - 1788272 \) Copy content Toggle raw display
$89$ \( T^{2} + 108T - 79502 \) Copy content Toggle raw display
$97$ \( T^{2} - 744 T - 1392866 \) Copy content Toggle raw display
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