Properties

Label 4030.2.a.e
Level $4030$
Weight $2$
Character orbit 4030.a
Self dual yes
Analytic conductor $32.180$
Analytic rank $1$
Dimension $6$
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] = [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: \(6\)
Coefficient field: 6.6.6550837.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 8x^{4} + 6x^{3} + 17x^{2} - 9x - 5 \) 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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 8x^{4} + 6x^{3} + 17x^{2} - 9x - 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - \nu^{3} - 5\nu^{2} + 3\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{4} + 2\nu^{3} + 4\nu^{2} - 7\nu - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 2\nu^{4} - 4\nu^{3} + 7\nu^{2} + \nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 2\beta_{3} + 6\beta_{2} + \beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 6\beta_{4} + 8\beta_{3} + 9\beta_{2} + 17\beta _1 + 10 \) 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
2.39285
1.80051
0.920227
−0.357882
−1.82857
−1.92714
1.00000 −3.39285 1.00000 1.00000 −3.39285 −3.19544 1.00000 8.51145 1.00000
1.2 1.00000 −2.80051 1.00000 1.00000 −2.80051 3.22197 1.00000 4.84285 1.00000
1.3 1.00000 −1.92023 1.00000 1.00000 −1.92023 −3.60391 1.00000 0.687271 1.00000
1.4 1.00000 −0.642118 1.00000 1.00000 −0.642118 1.07525 1.00000 −2.58768 1.00000
1.5 1.00000 0.828569 1.00000 1.00000 0.828569 −0.496108 1.00000 −2.31347 1.00000
1.6 1.00000 0.927138 1.00000 1.00000 0.927138 −5.00176 1.00000 −2.14042 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4030.2.a.e 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 7T_{3}^{5} + 12T_{3}^{4} - 8T_{3}^{3} - 24T_{3}^{2} + 4T_{3} + 9 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 7 T^{5} + 12 T^{4} - 8 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 8 T^{5} + 2 T^{4} - 98 T^{3} + \cdots + 99 \) Copy content Toggle raw display
$11$ \( T^{6} + 10 T^{5} + 3 T^{4} - 194 T^{3} + \cdots - 9 \) Copy content Toggle raw display
$13$ \( (T - 1)^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 14 T^{5} + 15 T^{4} + \cdots + 4001 \) Copy content Toggle raw display
$19$ \( T^{6} + 3 T^{5} - 19 T^{4} - 30 T^{3} + \cdots - 1 \) Copy content Toggle raw display
$23$ \( T^{6} + 13 T^{5} - 14 T^{4} + \cdots + 2231 \) Copy content Toggle raw display
$29$ \( T^{6} + 8 T^{5} - 25 T^{4} - 228 T^{3} + \cdots + 659 \) Copy content Toggle raw display
$31$ \( (T - 1)^{6} \) Copy content Toggle raw display
$37$ \( T^{6} + 16 T^{5} - 63 T^{4} + \cdots - 5667 \) Copy content Toggle raw display
$41$ \( T^{6} + 10 T^{5} - 153 T^{4} + \cdots + 4637 \) Copy content Toggle raw display
$43$ \( T^{6} + 17 T^{5} - 9 T^{4} + \cdots + 19575 \) Copy content Toggle raw display
$47$ \( T^{6} + 14 T^{5} - 42 T^{4} + \cdots + 10211 \) Copy content Toggle raw display
$53$ \( T^{6} + 22 T^{5} + 39 T^{4} + \cdots + 49611 \) Copy content Toggle raw display
$59$ \( T^{6} + 23 T^{5} + 156 T^{4} + \cdots - 545 \) Copy content Toggle raw display
$61$ \( T^{6} - 17 T^{5} + 13 T^{4} + \cdots - 19811 \) Copy content Toggle raw display
$67$ \( T^{6} + 16 T^{5} - 8 T^{4} + \cdots + 4219 \) Copy content Toggle raw display
$71$ \( T^{6} - 2 T^{5} - 193 T^{4} + \cdots + 25113 \) Copy content Toggle raw display
$73$ \( T^{6} + 7 T^{5} - 53 T^{4} + \cdots - 6875 \) Copy content Toggle raw display
$79$ \( T^{6} + 8 T^{5} - 137 T^{4} + \cdots - 8693 \) Copy content Toggle raw display
$83$ \( T^{6} + 16 T^{5} - 370 T^{4} + \cdots + 1013007 \) Copy content Toggle raw display
$89$ \( T^{6} + 32 T^{5} + 173 T^{4} + \cdots - 12799 \) Copy content Toggle raw display
$97$ \( T^{6} - 31 T^{5} + 180 T^{4} + \cdots + 13849 \) Copy content Toggle raw display
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