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Properties

Label 6042.2.a.bb
Level $6042$
Weight $2$
Character orbit 6042.a
Self dual yes
Analytic conductor $48.246$
Analytic rank $1$
Dimension $9$
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] = [6042,2,Mod(1,6042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6042, 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("6042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.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: \(48.2456129013\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 16x^{7} + 76x^{6} + 30x^{5} - 366x^{4} + 300x^{3} + 101x^{2} - 106x + 16 \) 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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4x^{8} - 16x^{7} + 76x^{6} + 30x^{5} - 366x^{4} + 300x^{3} + 101x^{2} - 106x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 615 \nu^{8} + 2940 \nu^{7} + 11484 \nu^{6} - 57016 \nu^{5} - 46158 \nu^{4} + 285404 \nu^{3} + \cdots - 19336 ) / 26914 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 779 \nu^{8} + 3724 \nu^{7} + 11855 \nu^{6} - 70426 \nu^{5} - 12713 \nu^{4} + 336392 \nu^{3} + \cdots + 44587 ) / 13457 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1831 \nu^{8} - 4158 \nu^{7} - 33928 \nu^{6} + 78330 \nu^{5} + 148714 \nu^{4} - 371346 \nu^{3} + \cdots + 17350 ) / 26914 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2002 \nu^{8} + 7273 \nu^{7} + 34561 \nu^{6} - 138099 \nu^{5} - 110149 \nu^{4} + 678224 \nu^{3} + \cdots + 120443 ) / 13457 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4037 \nu^{8} - 12078 \nu^{7} - 74202 \nu^{6} + 227530 \nu^{5} + 299972 \nu^{4} - 1097896 \nu^{3} + \cdots - 166196 ) / 26914 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2164 \nu^{8} + 7391 \nu^{7} + 38702 \nu^{6} - 140186 \nu^{5} - 141607 \nu^{4} + 684937 \nu^{3} + \cdots + 63294 ) / 13457 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3474 \nu^{8} + 12997 \nu^{7} + 58897 \nu^{6} - 249600 \nu^{5} - 167719 \nu^{4} + 1236964 \nu^{3} + \cdots + 224837 ) / 13457 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{7} + \beta_{6} - \beta_{5} - \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - 2\beta_{2} + 9\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + 23\beta_{7} + 9\beta_{6} - 14\beta_{5} + 3\beta_{4} - 14\beta_{2} + \beta _1 + 44 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{8} + 28\beta_{7} + 11\beta_{6} + 14\beta_{4} - 8\beta_{3} - 29\beta_{2} + 91\beta _1 - 17 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 15\beta_{8} + 253\beta_{7} + 82\beta_{6} - 169\beta_{5} + 51\beta_{4} - \beta_{3} - 157\beta_{2} + 29\beta _1 + 436 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 63 \beta_{8} + 350 \beta_{7} + 124 \beta_{6} + 8 \beta_{5} + 170 \beta_{4} - 55 \beta_{3} - 344 \beta_{2} + \cdots - 74 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 182 \beta_{8} + 2770 \beta_{7} + 776 \beta_{6} - 1950 \beta_{5} + 706 \beta_{4} - 4 \beta_{3} + \cdots + 4518 \) 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
1.91020
1.49919
−3.25110
3.42636
−0.613026
0.217879
−2.88457
0.394407
3.30066
1.00000 −1.00000 1.00000 −4.16096 −1.00000 0.910204 1.00000 1.00000 −4.16096
1.2 1.00000 −1.00000 1.00000 −3.26675 −1.00000 0.499186 1.00000 1.00000 −3.26675
1.3 1.00000 −1.00000 1.00000 −2.51610 −1.00000 −4.25110 1.00000 1.00000 −2.51610
1.4 1.00000 −1.00000 1.00000 −2.03181 −1.00000 2.42636 1.00000 1.00000 −2.03181
1.5 1.00000 −1.00000 1.00000 −0.133530 −1.00000 −1.61303 1.00000 1.00000 −0.133530
1.6 1.00000 −1.00000 1.00000 0.926601 −1.00000 −0.782121 1.00000 1.00000 0.926601
1.7 1.00000 −1.00000 1.00000 1.64105 −1.00000 −3.88457 1.00000 1.00000 1.64105
1.8 1.00000 −1.00000 1.00000 1.69036 −1.00000 −0.605593 1.00000 1.00000 1.69036
1.9 1.00000 −1.00000 1.00000 2.85114 −1.00000 2.30066 1.00000 1.00000 2.85114
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(19\) \(1\)
\(53\) \(-1\)

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 6042.2.a.bb ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.bb ✓ 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{9} + 5T_{5}^{8} - 14T_{5}^{7} - 79T_{5}^{6} + 71T_{5}^{5} + 389T_{5}^{4} - 238T_{5}^{3} - 647T_{5}^{2} + 428T_{5} + 68 \) Copy content Toggle raw display
\( T_{7}^{9} + 5T_{7}^{8} - 12T_{7}^{7} - 64T_{7}^{6} + 52T_{7}^{5} + 210T_{7}^{4} - 44T_{7}^{3} - 167T_{7}^{2} + 3T_{7} + 32 \) Copy content Toggle raw display
\( T_{11}^{9} - 3 T_{11}^{8} - 38 T_{11}^{7} + 103 T_{11}^{6} + 424 T_{11}^{5} - 1052 T_{11}^{4} + \cdots - 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{9} \) Copy content Toggle raw display
$3$ \( (T + 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} + 5 T^{8} + \cdots + 68 \) Copy content Toggle raw display
$7$ \( T^{9} + 5 T^{8} + \cdots + 32 \) Copy content Toggle raw display
$11$ \( T^{9} - 3 T^{8} + \cdots - 1024 \) Copy content Toggle raw display
$13$ \( T^{9} + 4 T^{8} + \cdots + 50144 \) Copy content Toggle raw display
$17$ \( T^{9} + 16 T^{8} + \cdots + 8324 \) Copy content Toggle raw display
$19$ \( (T + 1)^{9} \) Copy content Toggle raw display
$23$ \( T^{9} - 161 T^{7} + \cdots + 72256 \) Copy content Toggle raw display
$29$ \( T^{9} + 2 T^{8} + \cdots + 32218 \) Copy content Toggle raw display
$31$ \( T^{9} + 10 T^{8} + \cdots - 608 \) Copy content Toggle raw display
$37$ \( T^{9} + 14 T^{8} + \cdots + 6052 \) Copy content Toggle raw display
$41$ \( T^{9} + 21 T^{8} + \cdots + 463648 \) Copy content Toggle raw display
$43$ \( T^{9} + 4 T^{8} + \cdots - 4167424 \) Copy content Toggle raw display
$47$ \( T^{9} + 8 T^{8} + \cdots - 4072 \) Copy content Toggle raw display
$53$ \( (T - 1)^{9} \) Copy content Toggle raw display
$59$ \( T^{9} + 12 T^{8} + \cdots - 186388 \) Copy content Toggle raw display
$61$ \( T^{9} + 13 T^{8} + \cdots - 2854856 \) Copy content Toggle raw display
$67$ \( T^{9} - 22 T^{8} + \cdots + 5381504 \) Copy content Toggle raw display
$71$ \( T^{9} + 13 T^{8} + \cdots - 5797376 \) Copy content Toggle raw display
$73$ \( T^{9} + 17 T^{8} + \cdots + 2548736 \) Copy content Toggle raw display
$79$ \( T^{9} - 506 T^{7} + \cdots + 62195968 \) Copy content Toggle raw display
$83$ \( T^{9} + 24 T^{8} + \cdots + 155888 \) Copy content Toggle raw display
$89$ \( T^{9} + 23 T^{8} + \cdots + 258898144 \) Copy content Toggle raw display
$97$ \( T^{9} + 29 T^{8} + \cdots - 50131456 \) Copy content Toggle raw display
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