Properties

Label 3800.2.d.j
Level $3800$
Weight $2$
Character orbit 3800.d
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,2,Mod(3649,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3800.3649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.d (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: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.59105344.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 21x^{4} + 116x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 152)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - \beta_{4} + \beta_{2} + \beta_1) q^{7} + (\beta_{5} - 2 \beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - \beta_{4} + \beta_{2} + \beta_1) q^{7} + (\beta_{5} - 2 \beta_{3} - 3) q^{9} + (\beta_{5} - \beta_{3} - 1) q^{11} + (\beta_{4} - \beta_{2}) q^{13} + ( - \beta_{2} - \beta_1) q^{17} + q^{19} + ( - 3 \beta_{5} - 4) q^{21} + ( - 3 \beta_{2} - 2 \beta_1) q^{23} + ( - 3 \beta_{2} + 2 \beta_1) q^{27} + (\beta_{5} - 2 \beta_{3} + 4) q^{29} + ( - 2 \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{33} + \beta_{4} q^{37} + (\beta_{5} + 2 \beta_{3} + 6) q^{39} + ( - 2 \beta_{5} + 2) q^{41} + (3 \beta_{4} - 2 \beta_{2} + \beta_1) q^{43} + (\beta_{4} - 2 \beta_{2} - 3 \beta_1) q^{47} + ( - 2 \beta_{5} - \beta_{3} - 2) q^{49} + (\beta_{5} + 4) q^{51} + (\beta_{4} - \beta_{2} - 4 \beta_1) q^{53} + \beta_{2} q^{57} + (\beta_{5} + 8) q^{59} + ( - \beta_{5} - \beta_{3} + 1) q^{61} + (9 \beta_{4} - 4 \beta_{2} - 3 \beta_1) q^{63} + ( - 2 \beta_{4} - \beta_{2} - 2 \beta_1) q^{67} + (\beta_{5} + 2 \beta_{3} + 14) q^{69} + ( - 4 \beta_{5} + 2 \beta_{3} - 6) q^{71} + (3 \beta_{2} + \beta_1) q^{73} + (3 \beta_{4} - \beta_1) q^{77} + (2 \beta_{5} - 8) q^{79} + ( - 4 \beta_{5} + 4 \beta_{3} + 13) q^{81} + ( - 2 \beta_{4} + 2 \beta_{2} + 4 \beta_1) q^{83} + (\beta_{2} + 2 \beta_1) q^{87} + ( - 4 \beta_{5} + 2 \beta_{3} - 8) q^{89} + (3 \beta_{5} + 2 \beta_{3} + 6) q^{91} + (\beta_{4} + 2 \beta_1) q^{97} + ( - 7 \beta_{5} + 5 \beta_{3} + 13) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 24 q^{9} - 10 q^{11} + 6 q^{19} - 18 q^{21} + 18 q^{29} + 38 q^{39} + 16 q^{41} - 10 q^{49} + 22 q^{51} + 46 q^{59} + 6 q^{61} + 86 q^{69} - 24 q^{71} - 52 q^{79} + 94 q^{81} - 36 q^{89} + 34 q^{91} + 102 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 21x^{4} + 116x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 5\nu^{3} - 60\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} - 9\nu^{2} + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 13\nu^{3} + 28\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{4} + 13\nu^{2} + 20 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 2\beta_{2} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{5} - 13\beta_{3} + 71 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{4} + 26\beta_{2} + 115\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3649.1
3.08387i
0.786802i
3.29707i
3.29707i
0.786802i
3.08387i
0 3.29707i 0 0 0 1.78680i 0 −7.87067 0
3649.2 0 3.08387i 0 0 0 4.29707i 0 −6.51027 0
3649.3 0 0.786802i 0 0 0 2.08387i 0 2.38094 0
3649.4 0 0.786802i 0 0 0 2.08387i 0 2.38094 0
3649.5 0 3.08387i 0 0 0 4.29707i 0 −6.51027 0
3649.6 0 3.29707i 0 0 0 1.78680i 0 −7.87067 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3649.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.2.d.j 6
5.b even 2 1 inner 3800.2.d.j 6
5.c odd 4 1 152.2.a.c 3
5.c odd 4 1 3800.2.a.r 3
15.e even 4 1 1368.2.a.n 3
20.e even 4 1 304.2.a.g 3
20.e even 4 1 7600.2.a.bv 3
35.f even 4 1 7448.2.a.bf 3
40.i odd 4 1 1216.2.a.u 3
40.k even 4 1 1216.2.a.v 3
60.l odd 4 1 2736.2.a.bd 3
95.g even 4 1 2888.2.a.o 3
380.j odd 4 1 5776.2.a.bp 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.a.c 3 5.c odd 4 1
304.2.a.g 3 20.e even 4 1
1216.2.a.u 3 40.i odd 4 1
1216.2.a.v 3 40.k even 4 1
1368.2.a.n 3 15.e even 4 1
2736.2.a.bd 3 60.l odd 4 1
2888.2.a.o 3 95.g even 4 1
3800.2.a.r 3 5.c odd 4 1
3800.2.d.j 6 1.a even 1 1 trivial
3800.2.d.j 6 5.b even 2 1 inner
5776.2.a.bp 3 380.j odd 4 1
7448.2.a.bf 3 35.f even 4 1
7600.2.a.bv 3 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3800, [\chi])\):

\( T_{3}^{6} + 21T_{3}^{4} + 116T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{6} + 26T_{7}^{4} + 153T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{3} + 5T_{11}^{2} - 2T_{11} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 21 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 26 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{3} + 5 T^{2} - 2 T - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 29 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{6} + 22 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( (T - 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 153 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$29$ \( (T^{3} - 9 T^{2} - 4 T + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{3} \) Copy content Toggle raw display
$41$ \( (T^{3} - 8 T^{2} + \cdots + 128)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 241 T^{4} + \cdots + 135424 \) Copy content Toggle raw display
$47$ \( T^{6} + 145 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$53$ \( T^{6} + 269 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$59$ \( (T^{3} - 23 T^{2} + \cdots - 376)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 3 T^{2} - 28 T + 92)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 137 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( (T^{3} + 12 T^{2} + \cdots - 928)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 150 T^{4} + \cdots + 106276 \) Copy content Toggle raw display
$79$ \( (T^{3} + 26 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 260 T^{4} + \cdots + 541696 \) Copy content Toggle raw display
$89$ \( (T^{3} + 18 T^{2} + \cdots - 1024)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 104 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
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