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Properties

Label 6042.2.a.ba
Level $6042$
Weight $2$
Character orbit 6042.a
Self dual yes
Analytic conductor $48.246$
Analytic rank $0$
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: \(0\)
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: \( x^{9} - x^{8} - 25x^{7} + 62x^{6} + 76x^{5} - 360x^{4} + 182x^{3} + 459x^{2} - 595x + 199 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 25x^{7} + 62x^{6} + 76x^{5} - 360x^{4} + 182x^{3} + 459x^{2} - 595x + 199 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -10\nu^{8} - 7\nu^{7} + 239\nu^{6} - 212\nu^{5} - 1140\nu^{4} + 1658\nu^{3} + 1099\nu^{2} - 2760\nu + 1112 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -21\nu^{8} - 13\nu^{7} + 504\nu^{6} - 486\nu^{5} - 2384\nu^{4} + 3702\nu^{3} + 2176\nu^{2} - 6126\nu + 2578 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -30\nu^{8} - 17\nu^{7} + 723\nu^{6} - 728\nu^{5} - 3413\nu^{4} + 5454\nu^{3} + 3052\nu^{2} - 8976\nu + 3824 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -34\nu^{8} - 21\nu^{7} + 816\nu^{6} - 788\nu^{5} - 3858\nu^{4} + 5998\nu^{3} + 3512\nu^{2} - 9919\nu + 4186 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 58\nu^{8} + 34\nu^{7} - 1396\nu^{6} + 1382\nu^{5} + 6599\nu^{4} - 10417\nu^{3} - 5959\nu^{2} + 17178\nu - 7279 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -66\nu^{8} - 37\nu^{7} + 1592\nu^{6} - 1608\nu^{5} - 7520\nu^{4} + 12025\nu^{3} + 6728\nu^{2} - 19783\nu + 8433 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 68\nu^{8} + 36\nu^{7} - 1645\nu^{6} + 1700\nu^{5} + 7769\nu^{4} - 12598\nu^{3} - 6893\nu^{2} + 20670\nu - 8850 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -98\nu^{8} - 57\nu^{7} + 2360\nu^{6} - 2343\nu^{5} - 11157\nu^{4} + 17632\nu^{3} + 10069\nu^{2} - 29060\nu + 12319 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{8} + 2\beta_{7} + \beta_{6} - \beta_{5} - 3\beta_{4} - \beta_{3} - 2\beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{8} - 6\beta_{7} - 9\beta_{6} + \beta_{5} + 7\beta_{4} + 5\beta_{3} + 2\beta_{2} + \beta _1 + 24 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 14\beta_{8} + 18\beta_{7} + 17\beta_{6} - 3\beta_{5} - 27\beta_{4} - 11\beta_{3} - 6\beta_{2} - 7\beta _1 - 26 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 86 \beta_{8} - 170 \beta_{7} - 219 \beta_{6} + 19 \beta_{5} + 229 \beta_{4} + 115 \beta_{3} + \cdots + 376 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 590 \beta_{8} + 898 \beta_{7} + 1005 \beta_{6} - 85 \beta_{5} - 1291 \beta_{4} - 569 \beta_{3} + \cdots - 1580 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 648 \beta_{8} - 1133 \beta_{7} - 1378 \beta_{6} + 104 \beta_{5} + 1576 \beta_{4} + 731 \beta_{3} + \cdots + 2200 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 14474 \beta_{8} + 23410 \beta_{7} + 27309 \beta_{6} - 2005 \beta_{5} - 33163 \beta_{4} - 14901 \beta_{3} + \cdots - 42976 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 70914 \beta_{8} - 119602 \beta_{7} - 142821 \beta_{6} + 10373 \beta_{5} + 167787 \beta_{4} + \cdots + 225496 ) / 4 \) 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
2.76936
−5.12868
1.36790
1.58250
−1.61632
−2.00915
1.74814
0.734954
1.55130
−1.00000 1.00000 1.00000 −4.25796 −1.00000 −1.07291 −1.00000 1.00000 4.25796
1.2 −1.00000 1.00000 1.00000 −2.21991 −1.00000 3.78890 −1.00000 1.00000 2.21991
1.3 −1.00000 1.00000 1.00000 −1.75621 −1.00000 2.24076 −1.00000 1.00000 1.75621
1.4 −1.00000 1.00000 1.00000 −1.57851 −1.00000 4.87562 −1.00000 1.00000 1.57851
1.5 −1.00000 1.00000 1.00000 0.436408 −1.00000 −4.48433 −1.00000 1.00000 −0.436408
1.6 −1.00000 1.00000 1.00000 1.50179 −1.00000 −2.36178 −1.00000 1.00000 −1.50179
1.7 −1.00000 1.00000 1.00000 2.77115 −1.00000 −0.160214 −1.00000 1.00000 −2.77115
1.8 −1.00000 1.00000 1.00000 3.47154 −1.00000 3.94390 −1.00000 1.00000 −3.47154
1.9 −1.00000 1.00000 1.00000 3.63169 −1.00000 3.23005 −1.00000 1.00000 −3.63169
\(n\): 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.ba ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.ba ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{9} - 2T_{5}^{8} - 30T_{5}^{7} + 57T_{5}^{6} + 272T_{5}^{5} - 401T_{5}^{4} - 999T_{5}^{3} + 898T_{5}^{2} + 1196T_{5} - 600 \) Copy content Toggle raw display
\( T_{7}^{9} - 10T_{7}^{8} + 2T_{7}^{7} + 256T_{7}^{6} - 654T_{7}^{5} - 1110T_{7}^{4} + 4510T_{7}^{3} + 149T_{7}^{2} - 6088T_{7} - 960 \) Copy content Toggle raw display
\( T_{11}^{9} - 7 T_{11}^{8} - 36 T_{11}^{7} + 245 T_{11}^{6} + 620 T_{11}^{5} - 2772 T_{11}^{4} + \cdots + 8192 \) 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
$5$ \( T^{9} - 2 T^{8} + \cdots - 600 \) Copy content Toggle raw display
$7$ \( T^{9} - 10 T^{8} + \cdots - 960 \) Copy content Toggle raw display
$11$ \( T^{9} - 7 T^{8} + \cdots + 8192 \) Copy content Toggle raw display
$13$ \( T^{9} - 2 T^{8} + \cdots - 36 \) Copy content Toggle raw display
$17$ \( T^{9} - 4 T^{8} + \cdots + 3252 \) Copy content Toggle raw display
$19$ \( (T + 1)^{9} \) Copy content Toggle raw display
$23$ \( T^{9} - 15 T^{8} + \cdots - 357376 \) Copy content Toggle raw display
$29$ \( T^{9} - 5 T^{8} + \cdots + 26588 \) Copy content Toggle raw display
$31$ \( T^{9} - T^{8} + \cdots + 7808 \) Copy content Toggle raw display
$37$ \( T^{9} + 4 T^{8} + \cdots + 1718148 \) Copy content Toggle raw display
$41$ \( T^{9} - 5 T^{8} + \cdots - 232 \) Copy content Toggle raw display
$43$ \( T^{9} - 27 T^{8} + \cdots - 6097408 \) Copy content Toggle raw display
$47$ \( T^{9} - 171 T^{7} + \cdots + 90048 \) Copy content Toggle raw display
$53$ \( (T + 1)^{9} \) Copy content Toggle raw display
$59$ \( T^{9} - 7 T^{8} + \cdots + 13191536 \) Copy content Toggle raw display
$61$ \( T^{9} - 5 T^{8} + \cdots - 5851948 \) Copy content Toggle raw display
$67$ \( T^{9} - 19 T^{8} + \cdots - 1147392 \) Copy content Toggle raw display
$71$ \( T^{9} + 11 T^{8} + \cdots + 38653440 \) Copy content Toggle raw display
$73$ \( T^{9} - 25 T^{8} + \cdots + 9370624 \) Copy content Toggle raw display
$79$ \( T^{9} - 28 T^{8} + \cdots + 3744512 \) Copy content Toggle raw display
$83$ \( T^{9} + 8 T^{8} + \cdots + 825370800 \) Copy content Toggle raw display
$89$ \( T^{9} - 22 T^{8} + \cdots - 22852800 \) Copy content Toggle raw display
$97$ \( T^{9} - 13 T^{8} + \cdots - 120320 \) Copy content Toggle raw display
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