Properties

Label 8030.2.a.z
Level $8030$
Weight $2$
Character orbit 8030.a
Self dual yes
Analytic conductor $64.120$
Analytic rank $1$
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] = [8030,2,Mod(1,8030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8030, 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("8030.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8030.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: \(64.1198728231\)
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: \( x^{7} - 2x^{6} - 11x^{5} + 17x^{4} + 25x^{3} - 9x^{2} - 14x - 3 \) 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 2x^{6} - 11x^{5} + 17x^{4} + 25x^{3} - 9x^{2} - 14x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{6} + 5\nu^{5} + 19\nu^{4} - 43\nu^{3} - 23\nu^{2} + 25\nu + 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{6} - 5\nu^{5} - 19\nu^{4} + 43\nu^{3} + 25\nu^{2} - 27\nu - 15 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{6} - 8\nu^{5} - 28\nu^{4} + 70\nu^{3} + 32\nu^{2} - 50\nu - 15 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{6} - 7\nu^{5} - 31\nu^{4} + 61\nu^{3} + 57\nu^{2} - 43\nu - 26 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{6} - 9\nu^{5} - 41\nu^{4} + 79\nu^{3} + 73\nu^{2} - 61\nu - 33 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - 2\beta_{5} + 9\beta_{3} + 8\beta_{2} + 9\beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{6} - 13\beta_{5} + 7\beta_{4} - 7\beta_{3} + 8\beta_{2} + 67\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 18\beta_{6} - 30\beta_{5} - 4\beta_{4} + 78\beta_{3} + 62\beta_{2} + 82\beta _1 + 245 \) 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
3.07415
2.55939
0.914270
−0.335989
−0.496547
−0.882415
−2.83286
−1.00000 −3.07415 1.00000 1.00000 3.07415 −2.88946 −1.00000 6.45040 −1.00000
1.2 −1.00000 −2.55939 1.00000 1.00000 2.55939 2.15359 −1.00000 3.55049 −1.00000
1.3 −1.00000 −0.914270 1.00000 1.00000 0.914270 2.28524 −1.00000 −2.16411 −1.00000
1.4 −1.00000 0.335989 1.00000 1.00000 −0.335989 −3.89614 −1.00000 −2.88711 −1.00000
1.5 −1.00000 0.496547 1.00000 1.00000 −0.496547 0.154508 −1.00000 −2.75344 −1.00000
1.6 −1.00000 0.882415 1.00000 1.00000 −0.882415 0.390090 −1.00000 −2.22134 −1.00000
1.7 −1.00000 2.83286 1.00000 1.00000 −2.83286 −1.19783 −1.00000 5.02511 −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
\(2\) \(1\)
\(5\) \(-1\)
\(11\) \(1\)
\(73\) \(-1\)

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 8030.2.a.z 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8030.2.a.z 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

\( T_{3}^{7} + 2T_{3}^{6} - 11T_{3}^{5} - 17T_{3}^{4} + 25T_{3}^{3} + 9T_{3}^{2} - 14T_{3} + 3 \) Copy content Toggle raw display
\( T_{7}^{7} + 3T_{7}^{6} - 13T_{7}^{5} - 27T_{7}^{4} + 53T_{7}^{3} + 45T_{7}^{2} - 34T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{7} + 2 T^{6} - 11 T^{5} - 17 T^{4} + \cdots + 3 \) Copy content Toggle raw display
$5$ \( (T - 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{7} + 3 T^{6} - 13 T^{5} - 27 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( (T + 1)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} + 3 T^{6} - 40 T^{5} - 21 T^{4} + \cdots + 338 \) Copy content Toggle raw display
$17$ \( T^{7} + 11 T^{6} - T^{5} - 189 T^{4} + \cdots + 48 \) Copy content Toggle raw display
$19$ \( T^{7} + 8 T^{6} - 52 T^{5} + \cdots + 7356 \) Copy content Toggle raw display
$23$ \( T^{7} - 8 T^{6} - 129 T^{5} + \cdots + 321624 \) Copy content Toggle raw display
$29$ \( T^{7} + 9 T^{6} - 47 T^{5} - 619 T^{4} + \cdots + 156 \) Copy content Toggle raw display
$31$ \( T^{7} + 9 T^{6} - 64 T^{5} + \cdots - 6448 \) Copy content Toggle raw display
$37$ \( T^{7} - 21 T^{6} + 27 T^{5} + \cdots - 46500 \) Copy content Toggle raw display
$41$ \( T^{7} + 10 T^{6} - 162 T^{5} + \cdots - 2976 \) Copy content Toggle raw display
$43$ \( T^{7} + 5 T^{6} - 161 T^{5} + \cdots - 39232 \) Copy content Toggle raw display
$47$ \( T^{7} - 19 T^{6} + 15 T^{5} + \cdots + 35104 \) Copy content Toggle raw display
$53$ \( T^{7} + 13 T^{6} - 80 T^{5} + \cdots - 102656 \) Copy content Toggle raw display
$59$ \( T^{7} - 5 T^{6} - 197 T^{5} + \cdots + 111224 \) Copy content Toggle raw display
$61$ \( T^{7} + 12 T^{6} + 12 T^{5} + \cdots + 236 \) Copy content Toggle raw display
$67$ \( T^{7} + 23 T^{6} + 172 T^{5} + \cdots - 208 \) Copy content Toggle raw display
$71$ \( T^{7} - 22 T^{6} - 113 T^{5} + \cdots + 457131 \) Copy content Toggle raw display
$73$ \( (T - 1)^{7} \) Copy content Toggle raw display
$79$ \( T^{7} + 32 T^{6} + 336 T^{5} + \cdots + 174208 \) Copy content Toggle raw display
$83$ \( T^{7} - T^{6} - 254 T^{5} + \cdots - 127234 \) Copy content Toggle raw display
$89$ \( T^{7} - 20 T^{6} - 205 T^{5} + \cdots + 62981 \) Copy content Toggle raw display
$97$ \( T^{7} + 2 T^{6} - 391 T^{5} + \cdots - 1443584 \) Copy content Toggle raw display
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