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Properties

Label 2400.3.j.a
Level $2400$
Weight $3$
Character orbit 2400.j
Analytic conductor $65.395$
Analytic rank $0$
Dimension $4$
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] = [2400,3,Mod(799,2400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2400.799");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2400.j (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(65.3952634465\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - 4 \beta_1 - 4) q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - 4 \beta_1 - 4) q^{7} + 3 q^{9} + ( - 4 \beta_{3} + 2 \beta_{2}) q^{11} + ( - 5 \beta_{3} + 4 \beta_{2}) q^{13} + ( - 3 \beta_{3} + 4 \beta_{2}) q^{17} + ( - 12 \beta_{3} + 2 \beta_{2}) q^{19} + ( - 4 \beta_1 - 12) q^{21} + 8 q^{23} + 3 \beta_1 q^{27} + ( - 4 \beta_1 + 10) q^{29} + (2 \beta_{3} - 10 \beta_{2}) q^{31} + (6 \beta_{3} - 4 \beta_{2}) q^{33} + (9 \beta_{3} + 8 \beta_{2}) q^{37} + (12 \beta_{3} - 5 \beta_{2}) q^{39} + ( - 8 \beta_1 - 22) q^{41} + (20 \beta_1 + 40) q^{43} + (8 \beta_1 - 56) q^{47} + (32 \beta_1 + 15) q^{49} + (12 \beta_{3} - 3 \beta_{2}) q^{51} + ( - 3 \beta_{3} - 2 \beta_{2}) q^{53} + (6 \beta_{3} - 12 \beta_{2}) q^{57} + (16 \beta_{3} - 22 \beta_{2}) q^{59} - 14 q^{61} + ( - 12 \beta_1 - 12) q^{63} + (28 \beta_1 + 32) q^{67} + 8 \beta_1 q^{69} + (20 \beta_{3} + 24 \beta_{2}) q^{71} + (15 \beta_{3} + 16 \beta_{2}) q^{73} + ( - 8 \beta_{3} + 8 \beta_{2}) q^{77} + (38 \beta_{3} - 6 \beta_{2}) q^{79} + 9 q^{81} + ( - 4 \beta_1 + 56) q^{83} + (10 \beta_1 - 12) q^{87} + (16 \beta_1 + 78) q^{89} + ( - 28 \beta_{3} + 4 \beta_{2}) q^{91} + ( - 30 \beta_{3} + 2 \beta_{2}) q^{93} + ( - 31 \beta_{3} + 24 \beta_{2}) q^{97} + ( - 12 \beta_{3} + 6 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{7} + 12 q^{9} - 48 q^{21} + 32 q^{23} + 40 q^{29} - 88 q^{41} + 160 q^{43} - 224 q^{47} + 60 q^{49} - 56 q^{61} - 48 q^{63} + 128 q^{67} + 36 q^{81} + 224 q^{83} - 48 q^{87} + 312 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\zeta_{12}^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(1951\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
−0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
0.866025 − 0.500000i
0 −1.73205 0 0 0 2.92820 0 3.00000 0
799.2 0 −1.73205 0 0 0 2.92820 0 3.00000 0
799.3 0 1.73205 0 0 0 −10.9282 0 3.00000 0
799.4 0 1.73205 0 0 0 −10.9282 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2400.3.j.a 4
4.b odd 2 1 2400.3.j.b 4
5.b even 2 1 2400.3.j.b 4
5.c odd 4 1 96.3.g.a ✓ 4
5.c odd 4 1 2400.3.e.a 4
15.e even 4 1 288.3.g.d 4
20.d odd 2 1 inner 2400.3.j.a 4
20.e even 4 1 96.3.g.a ✓ 4
20.e even 4 1 2400.3.e.a 4
40.i odd 4 1 192.3.g.c 4
40.k even 4 1 192.3.g.c 4
60.l odd 4 1 288.3.g.d 4
80.i odd 4 1 768.3.b.a 4
80.j even 4 1 768.3.b.a 4
80.s even 4 1 768.3.b.d 4
80.t odd 4 1 768.3.b.d 4
120.q odd 4 1 576.3.g.j 4
120.w even 4 1 576.3.g.j 4
240.z odd 4 1 2304.3.b.k 4
240.bb even 4 1 2304.3.b.o 4
240.bd odd 4 1 2304.3.b.o 4
240.bf even 4 1 2304.3.b.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.3.g.a ✓ 4 5.c odd 4 1
96.3.g.a ✓ 4 20.e even 4 1
192.3.g.c 4 40.i odd 4 1
192.3.g.c 4 40.k even 4 1
288.3.g.d 4 15.e even 4 1
288.3.g.d 4 60.l odd 4 1
576.3.g.j 4 120.q odd 4 1
576.3.g.j 4 120.w even 4 1
768.3.b.a 4 80.i odd 4 1
768.3.b.a 4 80.j even 4 1
768.3.b.d 4 80.s even 4 1
768.3.b.d 4 80.t odd 4 1
2304.3.b.k 4 240.z odd 4 1
2304.3.b.k 4 240.bf even 4 1
2304.3.b.o 4 240.bb even 4 1
2304.3.b.o 4 240.bd odd 4 1
2400.3.e.a 4 5.c odd 4 1
2400.3.e.a 4 20.e even 4 1
2400.3.j.a 4 1.a even 1 1 trivial
2400.3.j.a 4 20.d odd 2 1 inner
2400.3.j.b 4 4.b odd 2 1
2400.3.j.b 4 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 8T_{7} - 32 \) acting on \(S_{3}^{\mathrm{new}}(2400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 8 T - 32)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 224T^{2} + 256 \) Copy content Toggle raw display
$13$ \( T^{4} + 584T^{2} + 8464 \) Copy content Toggle raw display
$17$ \( T^{4} + 456 T^{2} + 24336 \) Copy content Toggle raw display
$19$ \( T^{4} + 1248 T^{2} + 278784 \) Copy content Toggle raw display
$23$ \( (T - 8)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 20 T + 52)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2432 T^{2} + 1401856 \) Copy content Toggle raw display
$37$ \( T^{4} + 2184 T^{2} + 197136 \) Copy content Toggle raw display
$41$ \( (T^{2} + 44 T + 292)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 80 T + 400)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 112 T + 2944)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$59$ \( T^{4} + 13664 T^{2} + 22886656 \) Copy content Toggle raw display
$61$ \( (T + 14)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 64 T - 1328)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 17024 T^{2} + 28217344 \) Copy content Toggle raw display
$73$ \( T^{4} + 7944 T^{2} + 4717584 \) Copy content Toggle raw display
$79$ \( T^{4} + 12416 T^{2} + 28558336 \) Copy content Toggle raw display
$83$ \( (T^{2} - 112 T + 3088)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 156 T + 5316)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 21512 T^{2} + 9412624 \) Copy content Toggle raw display
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