Properties

Label 3800.2.d.k
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: 6.0.5161984.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + ( - \beta_{5} + \beta_{3}) q^{7} + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + ( - \beta_{5} + \beta_{3}) q^{7} + ( - \beta_{2} - 1) q^{9} + (\beta_{2} - \beta_1 - 2) q^{11} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3}) q^{13} + ( - \beta_{5} + 2 \beta_{3}) q^{17} + q^{19} + (\beta_{2} - 2 \beta_1 - 2) q^{21} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{23} + ( - \beta_{5} - 2 \beta_{4} + 3 \beta_{3}) q^{27} + 3 \beta_1 q^{29} + (2 \beta_1 - 4) q^{31} + (2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3}) q^{33} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3}) q^{37} + (2 \beta_{2} - 3 \beta_1) q^{39} + ( - \beta_{2} + 3 \beta_1 - 2) q^{41} + (4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{43} + (2 \beta_{5} + 2 \beta_{4}) q^{47} + (\beta_{2} - 4 \beta_1 - 3) q^{49} + (2 \beta_{2} - 3 \beta_1 - 2) q^{51} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3}) q^{53} + \beta_{4} q^{57} + ( - \beta_1 + 2) q^{59} + ( - 2 \beta_{2} + 2 \beta_1 + 8) q^{61} + (2 \beta_{4} - 2 \beta_{3}) q^{63} + (4 \beta_{5} + 3 \beta_{4}) q^{67} + ( - 3 \beta_{2} - 10) q^{69} + (2 \beta_{2} - 2 \beta_1 - 4) q^{71} + (3 \beta_{5} + 4 \beta_{4} - 2 \beta_{3}) q^{73} + (4 \beta_{5} - 4 \beta_{4} - 4 \beta_{3}) q^{77} + (\beta_{2} + 3 \beta_1 - 4) q^{79} + (2 \beta_{2} - 4 \beta_1 + 3) q^{81} + ( - 2 \beta_{5} - 4 \beta_{3}) q^{83} + ( - 3 \beta_{5} + 3 \beta_{3}) q^{87} + (\beta_{2} + \beta_1 - 6) q^{89} + (\beta_{2} - 4 \beta_1 - 14) q^{91} + ( - 2 \beta_{5} - 4 \beta_{4} + 2 \beta_{3}) q^{93} + (\beta_{5} + 7 \beta_{4} + 2 \beta_{3}) q^{97} + ( - 3 \beta_{2} + 3 \beta_1 - 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 8 q^{9} - 8 q^{11} + 6 q^{19} - 6 q^{21} - 6 q^{29} - 28 q^{31} + 10 q^{39} - 20 q^{41} - 8 q^{49} - 2 q^{51} + 14 q^{59} + 40 q^{61} - 66 q^{69} - 16 q^{71} - 28 q^{79} + 30 q^{81} - 36 q^{89} - 74 q^{91} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{5} + 25\nu^{4} + 10\nu^{3} - 4\nu^{2} + 323 ) / 121 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -12\nu^{5} - 29\nu^{4} - 60\nu^{3} + 24\nu^{2} - 123 ) / 121 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25\nu^{5} + 10\nu^{4} + 4\nu^{3} - 50\nu^{2} + 605\nu - 258 ) / 121 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 65\nu^{5} + 26\nu^{4} - 38\nu^{3} - 372\nu^{2} + 1331\nu - 574 ) / 242 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -75\nu^{5} - 30\nu^{4} - 12\nu^{3} + 392\nu^{2} - 1815\nu + 774 ) / 242 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{5} - 2\beta_{4} + \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 3\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -10\beta_{5} - 10\beta_{4} - 4\beta_{3} - 5\beta_{2} - 5\beta _1 + 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{2} + 6\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 58\beta_{5} + 50\beta_{4} + 32\beta_{3} - 25\beta_{2} - 33\beta _1 + 64 ) / 4 \) 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
1.32001 + 1.32001i
0.432320 0.432320i
−1.75233 + 1.75233i
−1.75233 1.75233i
0.432320 + 0.432320i
1.32001 1.32001i
0 3.12489i 0 0 0 1.51514i 0 −6.76491 0
3649.2 0 1.76156i 0 0 0 4.62620i 0 −0.103084 0
3649.3 0 0.363328i 0 0 0 1.14134i 0 2.86799 0
3649.4 0 0.363328i 0 0 0 1.14134i 0 2.86799 0
3649.5 0 1.76156i 0 0 0 4.62620i 0 −0.103084 0
3649.6 0 3.12489i 0 0 0 1.51514i 0 −6.76491 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.k 6
5.b even 2 1 inner 3800.2.d.k 6
5.c odd 4 1 760.2.a.h 3
5.c odd 4 1 3800.2.a.y 3
15.e even 4 1 6840.2.a.bj 3
20.e even 4 1 1520.2.a.r 3
20.e even 4 1 7600.2.a.bo 3
40.i odd 4 1 6080.2.a.bw 3
40.k even 4 1 6080.2.a.bs 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.h 3 5.c odd 4 1
1520.2.a.r 3 20.e even 4 1
3800.2.a.y 3 5.c odd 4 1
3800.2.d.k 6 1.a even 1 1 trivial
3800.2.d.k 6 5.b even 2 1 inner
6080.2.a.bs 3 40.k even 4 1
6080.2.a.bw 3 40.i odd 4 1
6840.2.a.bj 3 15.e even 4 1
7600.2.a.bo 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 \(S_{2}^{\mathrm{new}}(3800, [\chi])\):

\( T_{3}^{6} + 13T_{3}^{4} + 32T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{6} + 25T_{7}^{4} + 80T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{3} + 4T_{11}^{2} - 20T_{11} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 13 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 25 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T^{3} + 4 T^{2} - 20 T - 64)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 69 T^{4} + \cdots + 11236 \) Copy content Toggle raw display
$17$ \( T^{6} + 57 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$19$ \( (T - 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 113 T^{4} + \cdots + 25600 \) Copy content Toggle raw display
$29$ \( (T^{3} + 3 T^{2} + \cdots - 108)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 14 T^{2} + \cdots - 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 104 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$41$ \( (T^{3} + 10 T^{2} + \cdots - 472)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 276 T^{4} + \cdots + 719104 \) Copy content Toggle raw display
$47$ \( T^{6} + 80 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{6} + 77 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$59$ \( (T^{3} - 7 T^{2} + 8 T + 8)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 20 T^{2} + \cdots + 640)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 269 T^{4} + \cdots + 280900 \) Copy content Toggle raw display
$71$ \( (T^{3} + 8 T^{2} + \cdots - 512)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 297 T^{4} + \cdots + 250000 \) Copy content Toggle raw display
$79$ \( (T^{3} + 14 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 292 T^{4} + \cdots + 123904 \) Copy content Toggle raw display
$89$ \( (T^{3} + 18 T^{2} + \cdots + 40)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 584 T^{4} + \cdots + 5326864 \) Copy content Toggle raw display
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