Properties

Label 4030.2.a.j
Level $4030$
Weight $2$
Character orbit 4030.a
Self dual yes
Analytic conductor $32.180$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4030,2,Mod(1,4030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4030, 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("4030.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4030.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: \(32.1797120146\)
Analytic rank: \(1\)
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: \( x^{7} - 3x^{6} - 6x^{5} + 20x^{4} + 9x^{3} - 37x^{2} - 3x + 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + q^{2} - \beta_{3} q^{3} + q^{4} - q^{5} - \beta_{3} q^{6} + ( - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{7} + q^{8} + ( - \beta_{6} - \beta_{5} + \beta_{3} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_{3} q^{3} + q^{4} - q^{5} - \beta_{3} q^{6} + ( - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{7} + q^{8} + ( - \beta_{6} - \beta_{5} + \beta_{3} + \beta_1) q^{9} - q^{10} + (\beta_{5} + \beta_{4} - 2) q^{11} - \beta_{3} q^{12} - q^{13} + ( - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{14} + \beta_{3} q^{15} + q^{16} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2} - 2) q^{17} + ( - \beta_{6} - \beta_{5} + \beta_{3} + \beta_1) q^{18} + (\beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - 1) q^{19} - q^{20} + (\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{21} + (\beta_{5} + \beta_{4} - 2) q^{22} + (2 \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - 2) q^{23} - \beta_{3} q^{24} + q^{25} - q^{26} + (2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 - 3) q^{27} + ( - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{28} + ( - \beta_{6} - 3 \beta_{5} - 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{29} + \beta_{3} q^{30} + q^{31} + q^{32} + (\beta_{5} - \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{33} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2} - 2) q^{34} + (\beta_{4} - \beta_{3} + \beta_1 - 1) q^{35} + ( - \beta_{6} - \beta_{5} + \beta_{3} + \beta_1) q^{36} + (\beta_{6} - \beta_{3} + 2 \beta_{2}) q^{37} + (\beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - 1) q^{38} + \beta_{3} q^{39} - q^{40} + ( - \beta_{6} + \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + \beta_1 - 2) q^{41} + (\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{42} + ( - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{43} + (\beta_{5} + \beta_{4} - 2) q^{44} + (\beta_{6} + \beta_{5} - \beta_{3} - \beta_1) q^{45} + (2 \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - 2) q^{46} + ( - 2 \beta_{6} + 3 \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 2) q^{47} - \beta_{3} q^{48} + (\beta_{5} - 2 \beta_{4} + 4 \beta_{3}) q^{49} + q^{50} + (3 \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{2} - 2 \beta_1 - 4) q^{51} - q^{52} + ( - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + 3 \beta_{2} + 4 \beta_1 - 2) q^{53} + (2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 - 3) q^{54} + ( - \beta_{5} - \beta_{4} + 2) q^{55} + ( - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{56} + ( - 3 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{57} + ( - \beta_{6} - 3 \beta_{5} - 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{58} + (2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 2) q^{59} + \beta_{3} q^{60} + ( - 4 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 3) q^{61} + q^{62} + ( - 3 \beta_{6} - 2 \beta_{5} + \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 1) q^{63} + q^{64} + q^{65} + (\beta_{5} - \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{66} + (3 \beta_{5} - 2 \beta_{4} + \beta_{3} - 3 \beta_{2} - \beta_1 - 4) q^{67} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2} - 2) q^{68} + ( - 2 \beta_{6} - \beta_{5} + \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - 5 \beta_1 - 3) q^{69} + (\beta_{4} - \beta_{3} + \beta_1 - 1) q^{70} + (\beta_{6} + 4 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 - 3) q^{71} + ( - \beta_{6} - \beta_{5} + \beta_{3} + \beta_1) q^{72} + ( - \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} + 6 \beta_1) q^{73} + (\beta_{6} - \beta_{3} + 2 \beta_{2}) q^{74} - \beta_{3} q^{75} + (\beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - 1) q^{76} + ( - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 5 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{77} + \beta_{3} q^{78} + (\beta_{5} - 5 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{79} - q^{80} + ( - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 1) q^{81} + ( - \beta_{6} + \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + \beta_1 - 2) q^{82} + (\beta_{5} + 3 \beta_{3} + 5 \beta_{2} - \beta_1) q^{83} + (\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{84} + (\beta_{5} - 2 \beta_{4} + \beta_{2} + 2) q^{85} + ( - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{86} + (\beta_{6} - 2 \beta_{5} - \beta_{3} + 3 \beta_{2} + 7 \beta_1 + 3) q^{87} + (\beta_{5} + \beta_{4} - 2) q^{88} + (\beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + 6 \beta_1 - 2) q^{89} + (\beta_{6} + \beta_{5} - \beta_{3} - \beta_1) q^{90} + (\beta_{4} - \beta_{3} + \beta_1 - 1) q^{91} + (2 \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - 2) q^{92} - \beta_{3} q^{93} + ( - 2 \beta_{6} + 3 \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 2) q^{94} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + 1) q^{95} - \beta_{3} q^{96} + ( - 3 \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 2) q^{97} + (\beta_{5} - 2 \beta_{4} + 4 \beta_{3}) q^{98} + (4 \beta_{6} + 2 \beta_{5} - \beta_{3} + 3 \beta_{2} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{2} - 3 q^{3} + 7 q^{4} - 7 q^{5} - 3 q^{6} + 4 q^{7} + 7 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 7 q^{2} - 3 q^{3} + 7 q^{4} - 7 q^{5} - 3 q^{6} + 4 q^{7} + 7 q^{8} + 4 q^{9} - 7 q^{10} - 10 q^{11} - 3 q^{12} - 7 q^{13} + 4 q^{14} + 3 q^{15} + 7 q^{16} - 6 q^{17} + 4 q^{18} - 5 q^{19} - 7 q^{20} - 15 q^{21} - 10 q^{22} - 11 q^{23} - 3 q^{24} + 7 q^{25} - 7 q^{26} - 21 q^{27} + 4 q^{28} - 18 q^{29} + 3 q^{30} + 7 q^{31} + 7 q^{32} + 4 q^{33} - 6 q^{34} - 4 q^{35} + 4 q^{36} - 8 q^{37} - 5 q^{38} + 3 q^{39} - 7 q^{40} - 12 q^{41} - 15 q^{42} - 5 q^{43} - 10 q^{44} - 4 q^{45} - 11 q^{46} - 10 q^{47} - 3 q^{48} + 7 q^{49} + 7 q^{50} - 29 q^{51} - 7 q^{52} - 18 q^{53} - 21 q^{54} + 10 q^{55} + 4 q^{56} + 13 q^{57} - 18 q^{58} - 11 q^{59} + 3 q^{60} - 25 q^{61} + 7 q^{62} + 17 q^{63} + 7 q^{64} + 7 q^{65} + 4 q^{66} - 22 q^{67} - 6 q^{68} - 18 q^{69} - 4 q^{70} - 22 q^{71} + 4 q^{72} + 19 q^{73} - 8 q^{74} - 3 q^{75} - 5 q^{76} - 47 q^{77} + 3 q^{78} - 20 q^{79} - 7 q^{80} + 7 q^{81} - 12 q^{82} - 8 q^{83} - 15 q^{84} + 6 q^{85} - 5 q^{86} + 29 q^{87} - 10 q^{88} - 4 q^{89} - 4 q^{90} - 4 q^{91} - 11 q^{92} - 3 q^{93} - 10 q^{94} + 5 q^{95} - 3 q^{96} + 13 q^{97} + 7 q^{98} - 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 6x^{5} + 20x^{4} + 9x^{3} - 37x^{2} - 3x + 20 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{6} - 2\nu^{5} - 7\nu^{4} + 11\nu^{3} + 14\nu^{2} - 12\nu - 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{6} + 2\nu^{5} + 8\nu^{4} - 11\nu^{3} - 21\nu^{2} + 11\nu + 17 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{6} + 2\nu^{5} + 8\nu^{4} - 12\nu^{3} - 20\nu^{2} + 15\nu + 15 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - \nu^{5} - 10\nu^{4} + 6\nu^{3} + 30\nu^{2} - 7\nu - 24 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 7\beta_{2} + 8\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} - 5\beta_{5} + 8\beta_{4} + 2\beta_{3} + 10\beta_{2} + 28\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{6} + \beta_{5} + 12\beta_{4} + 12\beta_{3} + 44\beta_{2} + 55\beta _1 + 57 \) 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
−0.937425
2.64004
1.30511
−1.28870
1.13299
−1.98668
2.13467
1.00000 −3.21109 1.00000 −1.00000 −3.21109 3.80132 1.00000 7.31111 −1.00000
1.2 1.00000 −2.33934 1.00000 −1.00000 −2.33934 −3.11075 1.00000 2.47253 −1.00000
1.3 1.00000 −1.69801 1.00000 −1.00000 −1.69801 4.41789 1.00000 −0.116762 −1.00000
1.4 1.00000 0.444821 1.00000 −1.00000 0.444821 −0.0224858 1.00000 −2.80213 −1.00000
1.5 1.00000 0.779589 1.00000 −1.00000 0.779589 −2.22130 1.00000 −2.39224 −1.00000
1.6 1.00000 0.820963 1.00000 −1.00000 0.820963 2.40836 1.00000 −2.32602 −1.00000
1.7 1.00000 2.20307 1.00000 −1.00000 2.20307 −1.27303 1.00000 1.85352 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(13\) \(1\)
\(31\) \(-1\)

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 4030.2.a.j 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4030.2.a.j 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{7} + 3T_{3}^{6} - 8T_{3}^{5} - 20T_{3}^{4} + 22T_{3}^{3} + 24T_{3}^{2} - 31T_{3} + 8 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{7} + 3 T^{6} - 8 T^{5} - 20 T^{4} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( (T + 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{7} - 4 T^{6} - 20 T^{5} + 64 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$11$ \( T^{7} + 10 T^{6} + 9 T^{5} - 158 T^{4} + \cdots + 730 \) Copy content Toggle raw display
$13$ \( (T + 1)^{7} \) Copy content Toggle raw display
$17$ \( T^{7} + 6 T^{6} - 53 T^{5} + \cdots - 1780 \) Copy content Toggle raw display
$19$ \( T^{7} + 5 T^{6} - 59 T^{5} + \cdots + 1168 \) Copy content Toggle raw display
$23$ \( T^{7} + 11 T^{6} - 32 T^{5} + \cdots - 12898 \) Copy content Toggle raw display
$29$ \( T^{7} + 18 T^{6} + 15 T^{5} + \cdots + 17044 \) Copy content Toggle raw display
$31$ \( (T - 1)^{7} \) Copy content Toggle raw display
$37$ \( T^{7} + 8 T^{6} - 39 T^{5} - 280 T^{4} + \cdots - 718 \) Copy content Toggle raw display
$41$ \( T^{7} + 12 T^{6} - 155 T^{5} + \cdots - 727682 \) Copy content Toggle raw display
$43$ \( T^{7} + 5 T^{6} - 89 T^{5} - 116 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$47$ \( T^{7} + 10 T^{6} - 164 T^{5} + \cdots - 296060 \) Copy content Toggle raw display
$53$ \( T^{7} + 18 T^{6} - 79 T^{5} + \cdots - 1550 \) Copy content Toggle raw display
$59$ \( T^{7} + 11 T^{6} - 96 T^{5} + \cdots - 11432 \) Copy content Toggle raw display
$61$ \( T^{7} + 25 T^{6} - 9 T^{5} + \cdots + 169768 \) Copy content Toggle raw display
$67$ \( T^{7} + 22 T^{6} - 66 T^{5} + \cdots + 3085532 \) Copy content Toggle raw display
$71$ \( T^{7} + 22 T^{6} + 35 T^{5} + \cdots - 147320 \) Copy content Toggle raw display
$73$ \( T^{7} - 19 T^{6} - 219 T^{5} + \cdots - 1544548 \) Copy content Toggle raw display
$79$ \( T^{7} + 20 T^{6} - 169 T^{5} + \cdots + 150364 \) Copy content Toggle raw display
$83$ \( T^{7} + 8 T^{6} - 388 T^{5} + \cdots - 1088852 \) Copy content Toggle raw display
$89$ \( T^{7} + 4 T^{6} - 401 T^{5} + \cdots + 907502 \) Copy content Toggle raw display
$97$ \( T^{7} - 13 T^{6} - 232 T^{5} + \cdots + 967030 \) Copy content Toggle raw display
show more
show less