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Properties

Label 4100.2.g.c
Level $4100$
Weight $2$
Character orbit 4100.g
Analytic conductor $32.739$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4100,2,Mod(2049,4100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4100.2049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4100.g (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: \(32.7386648287\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.629407744.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 164)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{5} q^{7} - \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{5} q^{7} - \beta_{7} q^{9} + ( - 2 \beta_{3} - \beta_{2}) q^{11} - 2 \beta_{5} q^{13} + ( - 2 \beta_{5} + 2 \beta_1) q^{17} + ( - 3 \beta_{3} - 2 \beta_{2}) q^{19} + (\beta_{7} - 1) q^{21} + 2 \beta_{6} q^{23} + (\beta_{5} + \beta_1) q^{27} + 2 \beta_{3} q^{29} + ( - 2 \beta_{7} - 2) q^{31} + ( - \beta_{6} - 2 \beta_{4}) q^{33} + ( - \beta_{6} + 2 \beta_{4}) q^{37} + (2 \beta_{7} - 2) q^{39} + (2 \beta_{7} + 2 \beta_{3} + 1) q^{41} + (2 \beta_{6} - 2 \beta_{4}) q^{43} + (2 \beta_{5} - \beta_1) q^{47} + (\beta_{7} - 2) q^{49} + (4 \beta_{7} - 8) q^{51} + (2 \beta_{5} + 4 \beta_1) q^{53} + ( - \beta_{6} - 3 \beta_{4}) q^{57} + ( - 2 \beta_{7} + 2) q^{59} + (2 \beta_{7} - 6) q^{61} + (2 \beta_{5} + 3 \beta_1) q^{63} + (\beta_{5} + 6 \beta_1) q^{67} + (6 \beta_{3} - 2 \beta_{2}) q^{69} + \beta_{3} q^{71} + (3 \beta_{6} - 2 \beta_{4}) q^{73} + (3 \beta_{6} - 3 \beta_{4}) q^{77} + (3 \beta_{3} - 2 \beta_{2}) q^{79} + (3 \beta_{7} - 2) q^{81} + 6 \beta_{4} q^{83} + (2 \beta_{6} + 2 \beta_{4}) q^{87} + (2 \beta_{3} + 6 \beta_{2}) q^{89} + (2 \beta_{7} + 10) q^{91} + (2 \beta_{5} - 2 \beta_1) q^{93} - 4 \beta_{5} q^{97} + ( - \beta_{3} + 4 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{21} - 16 q^{31} - 16 q^{39} + 8 q^{41} - 16 q^{49} - 64 q^{51} + 16 q^{59} - 48 q^{61} - 16 q^{81} + 80 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - 2\nu^{5} + 14\nu^{3} - 8\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 2\nu^{5} + 2\nu^{3} - 8\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 2\nu^{5} + 2\nu^{3} ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 2\nu^{4} + 2\nu^{2} + 4 ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 2\nu^{5} - 2\nu^{3} + 16\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 2\nu^{4} + 6\nu^{2} - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} + 2\nu^{4} + 2\nu^{2} + 4 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + \beta_{3} + \beta_{2} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{7} - 3\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - \beta_{3} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} + \beta_{6} - 3\beta_{4} + 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{5} + 5\beta_{3} + 3\beta_{2} - 3\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4100\mathbb{Z}\right)^\times\).

\(n\) \(1477\) \(2051\) \(3901\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2049.1
−0.767178 + 1.18804i
−0.767178 − 1.18804i
1.38255 + 0.297594i
1.38255 − 0.297594i
−1.38255 + 0.297594i
−1.38255 − 0.297594i
0.767178 + 1.18804i
0.767178 − 1.18804i
0 −2.37608 0 0 0 1.53436 0 2.64575 0
2049.2 0 −2.37608 0 0 0 1.53436 0 2.64575 0
2049.3 0 −0.595188 0 0 0 −2.76510 0 −2.64575 0
2049.4 0 −0.595188 0 0 0 −2.76510 0 −2.64575 0
2049.5 0 0.595188 0 0 0 2.76510 0 −2.64575 0
2049.6 0 0.595188 0 0 0 2.76510 0 −2.64575 0
2049.7 0 2.37608 0 0 0 −1.53436 0 2.64575 0
2049.8 0 2.37608 0 0 0 −1.53436 0 2.64575 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2049.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4100.2.g.c 8
5.b even 2 1 inner 4100.2.g.c 8
5.c odd 4 1 164.2.b.a ✓ 4
5.c odd 4 1 4100.2.b.e 4
15.e even 4 1 1476.2.f.d 4
20.e even 4 1 656.2.d.d 4
40.i odd 4 1 2624.2.d.m 4
40.k even 4 1 2624.2.d.l 4
41.b even 2 1 inner 4100.2.g.c 8
205.c even 2 1 inner 4100.2.g.c 8
205.f odd 4 1 6724.2.a.d 4
205.g odd 4 1 164.2.b.a ✓ 4
205.g odd 4 1 4100.2.b.e 4
205.i odd 4 1 6724.2.a.d 4
615.p even 4 1 1476.2.f.d 4
820.r even 4 1 656.2.d.d 4
1640.u even 4 1 2624.2.d.l 4
1640.bg odd 4 1 2624.2.d.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.2.b.a ✓ 4 5.c odd 4 1
164.2.b.a ✓ 4 205.g odd 4 1
656.2.d.d 4 20.e even 4 1
656.2.d.d 4 820.r even 4 1
1476.2.f.d 4 15.e even 4 1
1476.2.f.d 4 615.p even 4 1
2624.2.d.l 4 40.k even 4 1
2624.2.d.l 4 1640.u even 4 1
2624.2.d.m 4 40.i odd 4 1
2624.2.d.m 4 1640.bg odd 4 1
4100.2.b.e 4 5.c odd 4 1
4100.2.b.e 4 205.g odd 4 1
4100.2.g.c 8 1.a even 1 1 trivial
4100.2.g.c 8 5.b even 2 1 inner
4100.2.g.c 8 41.b even 2 1 inner
4100.2.g.c 8 205.c even 2 1 inner
6724.2.a.d 4 205.f odd 4 1
6724.2.a.d 4 205.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 6T_{3}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(4100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 6 T^{2} + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 10 T^{2} + 18)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 26 T^{2} + 162)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 40 T^{2} + 288)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 80 T^{2} + 1152)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 70 T^{2} + 882)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 64 T^{2} + 576)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 24 T^{2} + 32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T - 24)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 32 T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} + \cdots + 1681)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 64 T^{2} + 576)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 54 T^{2} + 722)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 104 T^{2} + 2592)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 4 T - 24)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 12 T + 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 202 T^{2} + 6498)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 6 T^{2} + 2)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 128 T^{2} + 3844)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 118 T^{2} + 1458)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 336 T^{2} + 6272)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 160 T^{2} + 4608)^{2} \) Copy content Toggle raw display
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