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Properties

Label 3800.2.d.n
Level $3800$
Weight $2$
Character orbit 3800.d
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
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] = [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: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) 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 760)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + ( - \beta_{5} + \beta_{4}) q^{7} - \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + ( - \beta_{5} + \beta_{4}) q^{7} - \beta_1 q^{9} + 2 \beta_1 q^{11} + (2 \beta_{5} + \beta_{4} + \beta_{2}) q^{13} + (\beta_{5} + \beta_{4} - \beta_{2}) q^{17} - q^{19} + ( - \beta_{3} - 4) q^{21} + ( - \beta_{5} - \beta_{4} + 2 \beta_{2}) q^{23} + ( - \beta_{5} + \beta_{4}) q^{27} + ( - 2 \beta_{3} + \beta_1 + 3) q^{29} + ( - 2 \beta_1 + 2) q^{31} + (2 \beta_{5} + 4 \beta_{4}) q^{33} + ( - \beta_{5} - 3 \beta_{2}) q^{37} + (3 \beta_{3} - 4 \beta_1 - 2) q^{39} + ( - \beta_{3} + 3 \beta_1 - 7) q^{41} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2}) q^{43} + ( - 4 \beta_{5} - 4 \beta_{4} + 2 \beta_{2}) q^{47} + ( - 3 \beta_{3} + 2 \beta_1 - 3) q^{49} + ( - \beta_1 - 1) q^{51} + ( - 2 \beta_{5} + 5 \beta_{4} + \beta_{2}) q^{53} - \beta_{4} q^{57} + ( - 2 \beta_{3} + 5 \beta_1 - 3) q^{59} + ( - \beta_{3} + 3 \beta_1 - 5) q^{61} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2}) q^{63} + (4 \beta_{5} + 3 \beta_{4} - 2 \beta_{2}) q^{67} + \beta_{3} q^{69} + ( - 3 \beta_{5} - 7 \beta_{4} + 3 \beta_{2}) q^{73} + (4 \beta_{5} + 4 \beta_{4} - 4 \beta_{2}) q^{77} + (\beta_{3} - \beta_1 - 9) q^{79} + ( - \beta_{3} - 3 \beta_1 - 4) q^{81} + (2 \beta_{4} + 2 \beta_{2}) q^{83} + (3 \beta_{5} + 3 \beta_{4} + 4 \beta_{2}) q^{87} + (3 \beta_{3} + 3 \beta_1 + 1) q^{89} + (5 \beta_{3} - 4 \beta_1 + 8) q^{91} + ( - 2 \beta_{5} - 2 \beta_{4}) q^{93} + ( - 3 \beta_{5} - 4 \beta_{4} + 3 \beta_{2}) q^{97} + (2 \beta_{3} - 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{19} - 26 q^{21} + 14 q^{29} + 12 q^{31} - 6 q^{39} - 44 q^{41} - 24 q^{49} - 6 q^{51} - 22 q^{59} - 32 q^{61} + 2 q^{69} - 52 q^{79} - 26 q^{81} + 12 q^{89} + 58 q^{91} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 3\nu^{3} - 4\nu^{2} + 2\nu - 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + \nu^{4} - \nu^{3} + 5\nu^{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{5} + 2\nu^{4} - 5\nu^{3} + 6\nu^{2} - 2\nu + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{5} - 4\nu^{4} + 11\nu^{3} - 16\nu^{2} + 14\nu - 28 ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + \beta_{3} - \beta_{2} - \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{4} + 2\beta_{3} - \beta_{2} - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{5} - 2\beta_{4} + \beta_{3} + 3\beta_{2} + 3\beta _1 + 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{4} + 2\beta_{3} + 5\beta_{2} - 4\beta _1 + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{5} - 6\beta_{4} + 3\beta_{3} - 3\beta_{2} - 7\beta _1 + 7 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−0.671462 − 1.24464i
1.40680 + 0.144584i
0.264658 + 1.38923i
0.264658 − 1.38923i
1.40680 − 0.144584i
−0.671462 + 1.24464i
0 − 2.34292i 0 0 0 − 1.19656i 0 −2.48929 0
3649.2 0 − 1.81361i 0 0 0 − 4.91638i 0 −0.289169 0
3649.3 0 − 0.470683i 0 0 0 − 2.71982i 0 2.77846 0
3649.4 0 0.470683i 0 0 0 2.71982i 0 2.77846 0
3649.5 0 1.81361i 0 0 0 4.91638i 0 −0.289169 0
3649.6 0 2.34292i 0 0 0 1.19656i 0 −2.48929 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.n 6
5.b even 2 1 inner 3800.2.d.n 6
5.c odd 4 1 760.2.a.i ✓ 3
5.c odd 4 1 3800.2.a.w 3
15.e even 4 1 6840.2.a.bm 3
20.e even 4 1 1520.2.a.q 3
20.e even 4 1 7600.2.a.bp 3
40.i odd 4 1 6080.2.a.bx 3
40.k even 4 1 6080.2.a.br 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.i ✓ 3 5.c odd 4 1
1520.2.a.q 3 20.e even 4 1
3800.2.a.w 3 5.c odd 4 1
3800.2.d.n 6 1.a even 1 1 trivial
3800.2.d.n 6 5.b even 2 1 inner
6080.2.a.br 3 40.k even 4 1
6080.2.a.bx 3 40.i odd 4 1
6840.2.a.bm 3 15.e even 4 1
7600.2.a.bp 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 \(S_{2}^{\mathrm{new}}(3800, [\chi])\):

\( T_{3}^{6} + 9T_{3}^{4} + 20T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{6} + 33T_{7}^{4} + 224T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{3} - 28T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 9 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 33 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{3} - 28 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 89 T^{4} + \cdots + 7396 \) Copy content Toggle raw display
$17$ \( T^{6} + 17 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 41 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( (T^{3} - 7 T^{2} + \cdots + 292)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 6 T^{2} - 16 T + 32)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 148 T^{4} + \cdots + 59536 \) Copy content Toggle raw display
$41$ \( (T^{3} + 22 T^{2} + \cdots - 232)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 164 T^{4} + \cdots + 123904 \) Copy content Toggle raw display
$47$ \( T^{6} + 224 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$53$ \( T^{6} + 385 T^{4} + \cdots + 1800964 \) Copy content Toggle raw display
$59$ \( (T^{3} + 11 T^{2} + \cdots - 1544)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 16 T^{2} + \cdots - 352)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 201 T^{4} + \cdots + 68644 \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} + 369 T^{4} + \cdots + 1507984 \) Copy content Toggle raw display
$79$ \( (T^{3} + 26 T^{2} + \cdots + 496)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 100 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$89$ \( (T^{3} - 6 T^{2} + \cdots + 1256)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 180 T^{4} + \cdots + 85264 \) Copy content Toggle raw display
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