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Properties

Label 6030.2.a.bx
Level $6030$
Weight $2$
Character orbit 6030.a
Self dual yes
Analytic conductor $48.150$
Analytic rank $0$
Dimension $8$
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] = [6030,2,Mod(1,6030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6030, 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("6030.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6030.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: \(48.1497924188\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 22x^{6} + 8x^{5} + 149x^{4} + 118x^{3} - 152x^{2} - 180x - 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} + ( - \beta_{6} + 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - q^{5} + ( - \beta_{6} + 1) q^{7} - q^{8} + q^{10} + ( - \beta_{3} - 1) q^{11} + \beta_{2} q^{13} + (\beta_{6} - 1) q^{14} + q^{16} + \beta_{5} q^{17} + ( - \beta_1 + 2) q^{19} - q^{20} + (\beta_{3} + 1) q^{22} + (\beta_{7} - \beta_{6} - 1) q^{23} + q^{25} - \beta_{2} q^{26} + ( - \beta_{6} + 1) q^{28} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots - \beta_1) q^{29}+ \cdots + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{5} + 4 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{5} + 4 q^{7} - 8 q^{8} + 8 q^{10} - 6 q^{11} + 4 q^{13} - 4 q^{14} + 8 q^{16} - 4 q^{17} + 12 q^{19} - 8 q^{20} + 6 q^{22} - 12 q^{23} + 8 q^{25} - 4 q^{26} + 4 q^{28} - 4 q^{29} + 8 q^{31} - 8 q^{32} + 4 q^{34} - 4 q^{35} + 14 q^{37} - 12 q^{38} + 8 q^{40} - 8 q^{41} + 8 q^{43} - 6 q^{44} + 12 q^{46} - 12 q^{47} + 14 q^{49} - 8 q^{50} + 4 q^{52} - 8 q^{53} + 6 q^{55} - 4 q^{56} + 4 q^{58} + 8 q^{61} - 8 q^{62} + 8 q^{64} - 4 q^{65} + 8 q^{67} - 4 q^{68} + 4 q^{70} - 8 q^{71} + 16 q^{73} - 14 q^{74} + 12 q^{76} - 12 q^{77} + 20 q^{79} - 8 q^{80} + 8 q^{82} - 20 q^{83} + 4 q^{85} - 8 q^{86} + 6 q^{88} + 24 q^{89} + 32 q^{91} - 12 q^{92} + 12 q^{94} - 12 q^{95} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 22x^{6} + 8x^{5} + 149x^{4} + 118x^{3} - 152x^{2} - 180x - 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} + 25\nu^{5} - 26\nu^{4} - 158\nu^{3} + 20\nu^{2} + 152\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - 22\nu^{5} + 8\nu^{4} + 155\nu^{3} + 94\nu^{2} - 182\nu - 96 ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} - 7\nu^{6} - 35\nu^{5} + 73\nu^{4} + 217\nu^{3} - 124\nu^{2} - 298\nu + 12 ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 5\nu^{6} + 9\nu^{5} - 45\nu^{4} - 34\nu^{3} + 78\nu^{2} + 12\nu - 24 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} + 19\nu^{6} + 74\nu^{5} - 163\nu^{4} - 439\nu^{3} + 106\nu^{2} + 514\nu + 156 ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{7} + 26\nu^{6} + 106\nu^{5} - 224\nu^{4} - 617\nu^{3} + 122\nu^{2} + 662\nu + 228 ) / 12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} + 3\beta _1 + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{7} + 8\beta_{6} + 2\beta_{5} + 6\beta_{4} + 8\beta_{2} + 15\beta _1 + 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -42\beta_{7} + 54\beta_{6} + 13\beta_{5} + 33\beta_{4} - 5\beta_{3} + 46\beta_{2} + 63\beta _1 + 101 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -182\beta_{7} + 256\beta_{6} + 42\beta_{5} + 166\beta_{4} + 16\beta_{3} + 216\beta_{2} + 289\beta _1 + 344 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -950\beta_{7} + 1338\beta_{6} + 207\beta_{5} + 827\beta_{4} + 73\beta_{3} + 1066\beta_{2} + 1309\beta _1 + 1655 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4450 \beta_{7} + 6448 \beta_{6} + 830 \beta_{5} + 4018 \beta_{4} + 656 \beta_{3} + 5088 \beta_{2} + \cdots + 6936 ) / 2 \) Copy content Toggle raw display

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

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
−2.29420
1.20894
−0.270376
−1.60973
4.73096
3.28278
−2.15895
−0.889434
−1.00000 0 1.00000 −1.00000 0 −3.99708 −1.00000 0 1.00000
1.2 −1.00000 0 1.00000 −1.00000 0 −2.33384 −1.00000 0 1.00000
1.3 −1.00000 0 1.00000 −1.00000 0 −1.70690 −1.00000 0 1.00000
1.4 −1.00000 0 1.00000 −1.00000 0 0.104756 −1.00000 0 1.00000
1.5 −1.00000 0 1.00000 −1.00000 0 0.341555 −1.00000 0 1.00000
1.6 −1.00000 0 1.00000 −1.00000 0 3.17277 −1.00000 0 1.00000
1.7 −1.00000 0 1.00000 −1.00000 0 4.08421 −1.00000 0 1.00000
1.8 −1.00000 0 1.00000 −1.00000 0 4.33452 −1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(1\)
\(67\) \(-1\)

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 6030.2.a.bx ✓ 8
3.b odd 2 1 6030.2.a.by yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6030.2.a.bx ✓ 8 1.a even 1 1 trivial
6030.2.a.by yes 8 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6030))\):

\( T_{7}^{8} - 4T_{7}^{7} - 27T_{7}^{6} + 96T_{7}^{5} + 220T_{7}^{4} - 536T_{7}^{3} - 696T_{7}^{2} + 384T_{7} - 32 \) Copy content Toggle raw display
\( T_{11}^{8} + 6T_{11}^{7} - 45T_{11}^{6} - 244T_{11}^{5} + 842T_{11}^{4} + 3216T_{11}^{3} - 8096T_{11}^{2} - 13312T_{11} + 30336 \) Copy content Toggle raw display
\( T_{13}^{8} - 4T_{13}^{7} - 52T_{13}^{6} + 136T_{13}^{5} + 872T_{13}^{4} - 1472T_{13}^{3} - 5856T_{13}^{2} + 5120T_{13} + 13312 \) Copy content Toggle raw display
\( T_{17}^{8} + 4T_{17}^{7} - 66T_{17}^{6} - 232T_{17}^{5} + 1576T_{17}^{4} + 4320T_{17}^{3} - 16576T_{17}^{2} - 26112T_{17} + 66816 \) Copy content Toggle raw display
\( T_{23}^{8} + 12T_{23}^{7} - 32T_{23}^{6} - 744T_{23}^{5} - 728T_{23}^{4} + 10496T_{23}^{3} + 5312T_{23}^{2} - 59392T_{23} + 49152 \) Copy content Toggle raw display
\( T_{29}^{8} + 4 T_{29}^{7} - 128 T_{29}^{6} - 568 T_{29}^{5} + 4264 T_{29}^{4} + 21952 T_{29}^{3} + \cdots - 110592 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} + \cdots - 32 \) Copy content Toggle raw display
$11$ \( T^{8} + 6 T^{7} + \cdots + 30336 \) Copy content Toggle raw display
$13$ \( T^{8} - 4 T^{7} + \cdots + 13312 \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{7} + \cdots + 66816 \) Copy content Toggle raw display
$19$ \( T^{8} - 12 T^{7} + \cdots - 29696 \) Copy content Toggle raw display
$23$ \( T^{8} + 12 T^{7} + \cdots + 49152 \) Copy content Toggle raw display
$29$ \( T^{8} + 4 T^{7} + \cdots - 110592 \) Copy content Toggle raw display
$31$ \( T^{8} - 8 T^{7} + \cdots - 384 \) Copy content Toggle raw display
$37$ \( T^{8} - 14 T^{7} + \cdots + 7776 \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} + \cdots + 329088 \) Copy content Toggle raw display
$43$ \( T^{8} - 8 T^{7} + \cdots - 2224128 \) Copy content Toggle raw display
$47$ \( T^{8} + 12 T^{7} + \cdots - 76032 \) Copy content Toggle raw display
$53$ \( T^{8} + 8 T^{7} + \cdots - 884736 \) Copy content Toggle raw display
$59$ \( T^{8} - 176 T^{6} + \cdots - 63744 \) Copy content Toggle raw display
$61$ \( T^{8} - 8 T^{7} + \cdots + 82512 \) Copy content Toggle raw display
$67$ \( (T - 1)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} + 8 T^{7} + \cdots + 423456 \) Copy content Toggle raw display
$73$ \( T^{8} - 16 T^{7} + \cdots - 848384 \) Copy content Toggle raw display
$79$ \( T^{8} - 20 T^{7} + \cdots - 90432 \) Copy content Toggle raw display
$83$ \( T^{8} + 20 T^{7} + \cdots + 7776 \) Copy content Toggle raw display
$89$ \( T^{8} - 24 T^{7} + \cdots - 146976768 \) Copy content Toggle raw display
$97$ \( T^{8} - 591 T^{6} + \cdots + 7657792 \) Copy content Toggle raw display
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