Properties

Label 4100.2.a.j
Level $4100$
Weight $2$
Character orbit 4100.a
Self dual yes
Analytic conductor $32.739$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4100,2,Mod(1,4100)]
 
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, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4100.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4100.a (trivial)

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: yes
Analytic conductor: \(32.7386648287\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 9x^{5} + 23x^{4} + 21x^{3} - 30x^{2} - 27x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{4} q^{7} + (\beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{4} q^{7} + (\beta_{2} + \beta_1) q^{9} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + \beta_1) q^{11} + ( - \beta_{5} + \beta_{2} - \beta_1 + 1) q^{13} + ( - \beta_{6} - \beta_{3} + \beta_1 + 2) q^{17} + ( - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{19} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{21} + ( - \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - \beta_{2} + \beta_1 - 1) q^{23} + (\beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 1) q^{27} + ( - \beta_{5} - \beta_{3}) q^{29} + (\beta_{6} - \beta_{5} + \beta_{3}) q^{31} + ( - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{33} + (\beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_1 + 3) q^{37} + (\beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{39} + q^{41} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{3}) q^{43} + ( - \beta_{6} - 2 \beta_{5} - \beta_{3} - \beta_1 + 3) q^{47} + (2 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} - \beta_1) q^{49} + (2 \beta_{5} - \beta_{4} + 3 \beta_1 + 2) q^{51} + (2 \beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{53} + (\beta_{4} + \beta_{3} + 3 \beta_1 + 2) q^{57} + ( - 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - \beta_{2} + 5 \beta_1 - 4) q^{59} + ( - \beta_{6} - \beta_{4} + 2 \beta_{2} + 1) q^{61} + ( - \beta_{5} - 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{63} + (2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_1 + 3) q^{67} + ( - 2 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - \beta_{2} - 1) q^{69} + (3 \beta_{6} + \beta_{4} - \beta_1 + 3) q^{71} + (\beta_{6} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{73} + (\beta_{6} - \beta_{5} + 2 \beta_{4} + 4) q^{77} + (4 \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + 4 \beta_1 - 3) q^{79} + (\beta_{6} + 6 \beta_{4} + 4 \beta_{3} + 4 \beta_1) q^{81} + ( - \beta_{6} + \beta_{5} - \beta_{2}) q^{83} + ( - 2 \beta_{4} - \beta_{3} - 2 \beta_1 + 3) q^{87} + ( - 2 \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{89} + (\beta_{6} - 3 \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_1 + 3) q^{91} + (\beta_{6} - 4 \beta_{5} + \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 3) q^{93} + ( - 2 \beta_{6} + \beta_{5} - 5 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 4) q^{97} + (\beta_{6} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 3 q^{3} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 3 q^{3} + 6 q^{9} - q^{11} + 5 q^{13} + 11 q^{17} - q^{19} - q^{21} - 4 q^{23} + 21 q^{27} - 5 q^{29} + 4 q^{31} + 5 q^{33} + 32 q^{37} - 11 q^{39} + 7 q^{41} + 4 q^{43} + 8 q^{47} - 3 q^{49} + 27 q^{51} + 8 q^{53} + 26 q^{57} - 16 q^{59} + 10 q^{61} - 13 q^{63} + 16 q^{67} - 19 q^{69} + 27 q^{71} + 27 q^{73} + 29 q^{77} - q^{79} + 27 q^{81} - 4 q^{83} + 12 q^{87} - 16 q^{89} + 9 q^{91} + 16 q^{93} + 27 q^{97} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 9x^{5} + 23x^{4} + 21x^{3} - 30x^{2} - 27x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 2\nu^{5} - 11\nu^{4} + 15\nu^{3} + 30\nu^{2} - 24\nu - 18 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 5\nu^{5} - 2\nu^{4} + 36\nu^{3} - 27\nu^{2} - 33\nu + 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{6} + 14\nu^{5} + 26\nu^{4} - 99\nu^{3} - 6\nu^{2} + 90\nu + 15 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -10\nu^{6} + 38\nu^{5} + 59\nu^{4} - 276\nu^{3} + 15\nu^{2} + 282\nu + 45 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 2\beta_{4} + 2\beta_{3} + 8\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 6\beta_{4} + 4\beta_{3} + 9\beta_{2} + 13\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{6} + 7\beta_{5} + 31\beta_{4} + 27\beta_{3} + 8\beta_{2} + 73\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17\beta_{6} - \beta_{5} + 98\beta_{4} + 71\beta_{3} + 85\beta_{2} + 163\beta _1 + 137 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.47795
−0.894124
−0.806545
−0.131983
1.68828
2.20421
3.41812
0 −2.47795 0 0 0 −0.154217 0 3.14025 0
1.2 0 −0.894124 0 0 0 2.24085 0 −2.20054 0
1.3 0 −0.806545 0 0 0 −0.0998195 0 −2.34949 0
1.4 0 −0.131983 0 0 0 −4.26732 0 −2.98258 0
1.5 0 1.68828 0 0 0 4.03576 0 −0.149714 0
1.6 0 2.20421 0 0 0 0.688745 0 1.85852 0
1.7 0 3.41812 0 0 0 −2.44400 0 8.68355 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(41\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4100.2.a.j yes 7
5.b even 2 1 4100.2.a.g 7
5.c odd 4 2 4100.2.d.g 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4100.2.a.g 7 5.b even 2 1
4100.2.a.j yes 7 1.a even 1 1 trivial
4100.2.d.g 14 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{7} - 3T_{3}^{6} - 9T_{3}^{5} + 23T_{3}^{4} + 21T_{3}^{3} - 30T_{3}^{2} - 27T_{3} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4100))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} - 3 T^{6} - 9 T^{5} + 23 T^{4} + \cdots - 3 \) Copy content Toggle raw display
$5$ \( T^{7} \) Copy content Toggle raw display
$7$ \( T^{7} - 23 T^{5} + 5 T^{4} + 100 T^{3} + \cdots - 1 \) Copy content Toggle raw display
$11$ \( T^{7} + T^{6} - 42 T^{5} - 51 T^{4} + \cdots + 432 \) Copy content Toggle raw display
$13$ \( T^{7} - 5 T^{6} - 31 T^{5} + \cdots + 1617 \) Copy content Toggle raw display
$17$ \( T^{7} - 11 T^{6} - 6 T^{5} + 481 T^{4} + \cdots + 108 \) Copy content Toggle raw display
$19$ \( T^{7} + T^{6} - 61 T^{5} + 80 T^{4} + \cdots + 1269 \) Copy content Toggle raw display
$23$ \( T^{7} + 4 T^{6} - 111 T^{5} + \cdots - 12447 \) Copy content Toggle raw display
$29$ \( T^{7} + 5 T^{6} - 33 T^{5} + \cdots - 2673 \) Copy content Toggle raw display
$31$ \( T^{7} - 4 T^{6} - 71 T^{5} + \cdots - 7857 \) Copy content Toggle raw display
$37$ \( T^{7} - 32 T^{6} + 281 T^{5} + \cdots + 16431 \) Copy content Toggle raw display
$41$ \( (T - 1)^{7} \) Copy content Toggle raw display
$43$ \( T^{7} - 4 T^{6} - 171 T^{5} + \cdots + 2468 \) Copy content Toggle raw display
$47$ \( T^{7} - 8 T^{6} - 105 T^{5} + \cdots + 13131 \) Copy content Toggle raw display
$53$ \( T^{7} - 8 T^{6} - 168 T^{5} + \cdots - 9153 \) Copy content Toggle raw display
$59$ \( T^{7} + 16 T^{6} - 138 T^{5} + \cdots - 270621 \) Copy content Toggle raw display
$61$ \( T^{7} - 10 T^{6} - 171 T^{5} + \cdots - 5747 \) Copy content Toggle raw display
$67$ \( T^{7} - 16 T^{6} - 138 T^{5} + \cdots - 1500443 \) Copy content Toggle raw display
$71$ \( T^{7} - 27 T^{6} - 10 T^{5} + \cdots - 55479 \) Copy content Toggle raw display
$73$ \( T^{7} - 27 T^{6} + 72 T^{5} + \cdots - 655971 \) Copy content Toggle raw display
$79$ \( T^{7} + T^{6} - 298 T^{5} + \cdots - 2704419 \) Copy content Toggle raw display
$83$ \( T^{7} + 4 T^{6} - 36 T^{5} - 179 T^{4} + \cdots - 108 \) Copy content Toggle raw display
$89$ \( T^{7} + 16 T^{6} - 188 T^{5} + \cdots + 979827 \) Copy content Toggle raw display
$97$ \( T^{7} - 27 T^{6} - 66 T^{5} + \cdots + 167007 \) Copy content Toggle raw display
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