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Properties

Label 6042.2.a.bg
Level $6042$
Weight $2$
Character orbit 6042.a
Self dual yes
Analytic conductor $48.246$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} - 34 x^{10} + 169 x^{9} + 453 x^{8} - 2095 x^{7} - 3056 x^{6} + 11545 x^{5} + \cdots + 7848 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} - 34 x^{10} + 169 x^{9} + 453 x^{8} - 2095 x^{7} - 3056 x^{6} + 11545 x^{5} + \cdots + 7848 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 204446 \nu^{11} + 2608739 \nu^{10} + 7340808 \nu^{9} - 137165906 \nu^{8} + 23174311 \nu^{7} + \cdots - 5802453752 ) / 669887500 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1345873 \nu^{11} + 12257282 \nu^{10} + 10822604 \nu^{9} - 340995353 \nu^{8} + \cdots - 13115755176 ) / 1339775000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4094329 \nu^{11} + 15259486 \nu^{10} + 135031392 \nu^{9} - 375232369 \nu^{8} + \cdots + 12411594952 ) / 1339775000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7020169 \nu^{11} - 62872796 \nu^{10} - 89204962 \nu^{9} + 1961538359 \nu^{8} + \cdots + 71971042128 ) / 1339775000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8194199 \nu^{11} + 58777566 \nu^{10} + 170893902 \nu^{9} - 1833840689 \nu^{8} + \cdots - 49602823488 ) / 1339775000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4234532 \nu^{11} - 27548113 \nu^{10} - 105119461 \nu^{9} + 886007227 \nu^{8} + \cdots + 20499466684 ) / 669887500 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5101356 \nu^{11} + 28749929 \nu^{10} + 131490613 \nu^{9} - 829014291 \nu^{8} + \cdots - 6270409172 ) / 669887500 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1085494 \nu^{11} + 5435946 \nu^{10} + 32741387 \nu^{9} - 164915409 \nu^{8} + \cdots - 2493542228 ) / 133977500 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1085494 \nu^{11} - 5435946 \nu^{10} - 32741387 \nu^{9} + 164915409 \nu^{8} + 359968771 \nu^{7} + \cdots + 1555699728 ) / 133977500 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11405033 \nu^{11} - 65451972 \nu^{10} - 329920034 \nu^{9} + 2197952363 \nu^{8} + \cdots + 61388109096 ) / 1339775000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} + \beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{10} + \beta_{9} - 2\beta_{7} + \beta_{5} + \beta_{3} - \beta_{2} + 12\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + 17 \beta_{10} + 15 \beta_{9} + \beta_{8} - 6 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} + \cdots + 76 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{11} + 56 \beta_{10} + 32 \beta_{9} + \beta_{8} - 43 \beta_{7} + 7 \beta_{6} + 23 \beta_{5} + \cdots + 100 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 23 \beta_{11} + 300 \beta_{10} + 237 \beta_{9} + 9 \beta_{8} - 162 \beta_{7} + 11 \beta_{6} + 52 \beta_{5} + \cdots + 997 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 66 \beta_{11} + 1223 \beta_{10} + 732 \beta_{9} + 8 \beta_{8} - 831 \beta_{7} + 190 \beta_{6} + \cdots + 2061 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 490 \beta_{11} + 5631 \beta_{10} + 4104 \beta_{9} - 37 \beta_{8} - 3455 \beta_{7} + 418 \beta_{6} + \cdots + 14695 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1785 \beta_{11} + 24783 \beta_{10} + 15275 \beta_{9} - 450 \beta_{8} - 15984 \beta_{7} + 4002 \beta_{6} + \cdots + 39988 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 10821 \beta_{11} + 108976 \beta_{10} + 75835 \beta_{9} - 4232 \beta_{8} - 69216 \beta_{7} + \cdots + 234569 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 44889 \beta_{11} + 489981 \beta_{10} + 309441 \beta_{9} - 22465 \beta_{8} - 309458 \beta_{7} + \cdots + 758488 \) 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
−3.46060
−2.85168
−2.80221
−1.58560
−1.52770
−0.365306
0.927745
2.31473
2.39115
3.49639
4.02498
4.43811
1.00000 1.00000 1.00000 −3.46060 1.00000 0.599762 1.00000 1.00000 −3.46060
1.2 1.00000 1.00000 1.00000 −2.85168 1.00000 3.66552 1.00000 1.00000 −2.85168
1.3 1.00000 1.00000 1.00000 −2.80221 1.00000 −4.03420 1.00000 1.00000 −2.80221
1.4 1.00000 1.00000 1.00000 −1.58560 1.00000 4.52427 1.00000 1.00000 −1.58560
1.5 1.00000 1.00000 1.00000 −1.52770 1.00000 −0.0681258 1.00000 1.00000 −1.52770
1.6 1.00000 1.00000 1.00000 −0.365306 1.00000 −3.59210 1.00000 1.00000 −0.365306
1.7 1.00000 1.00000 1.00000 0.927745 1.00000 1.16139 1.00000 1.00000 0.927745
1.8 1.00000 1.00000 1.00000 2.31473 1.00000 4.85590 1.00000 1.00000 2.31473
1.9 1.00000 1.00000 1.00000 2.39115 1.00000 −1.97837 1.00000 1.00000 2.39115
1.10 1.00000 1.00000 1.00000 3.49639 1.00000 1.05576 1.00000 1.00000 3.49639
1.11 1.00000 1.00000 1.00000 4.02498 1.00000 −3.74027 1.00000 1.00000 4.02498
1.12 1.00000 1.00000 1.00000 4.43811 1.00000 3.55047 1.00000 1.00000 4.43811
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.bg ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.bg ✓ 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{12} - 5 T_{5}^{11} - 34 T_{5}^{10} + 169 T_{5}^{9} + 453 T_{5}^{8} - 2095 T_{5}^{7} - 3056 T_{5}^{6} + \cdots + 7848 \) Copy content Toggle raw display
\( T_{7}^{12} - 6 T_{7}^{11} - 42 T_{7}^{10} + 280 T_{7}^{9} + 548 T_{7}^{8} - 4596 T_{7}^{7} - 1304 T_{7}^{6} + \cdots - 1536 \) Copy content Toggle raw display
\( T_{11}^{12} - 7 T_{11}^{11} - 84 T_{11}^{10} + 653 T_{11}^{9} + 2126 T_{11}^{8} - 21152 T_{11}^{7} + \cdots - 475136 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{12} \) Copy content Toggle raw display
$3$ \( (T - 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 5 T^{11} + \cdots + 7848 \) Copy content Toggle raw display
$7$ \( T^{12} - 6 T^{11} + \cdots - 1536 \) Copy content Toggle raw display
$11$ \( T^{12} - 7 T^{11} + \cdots - 475136 \) Copy content Toggle raw display
$13$ \( T^{12} - 9 T^{11} + \cdots - 256 \) Copy content Toggle raw display
$17$ \( T^{12} - 25 T^{11} + \cdots - 1613056 \) Copy content Toggle raw display
$19$ \( (T + 1)^{12} \) Copy content Toggle raw display
$23$ \( T^{12} - 22 T^{11} + \cdots + 768000 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots - 595056720 \) Copy content Toggle raw display
$31$ \( T^{12} - 23 T^{11} + \cdots - 5845248 \) Copy content Toggle raw display
$37$ \( T^{12} + T^{11} + \cdots + 41463808 \) Copy content Toggle raw display
$41$ \( T^{12} + 15 T^{11} + \cdots - 5838848 \) Copy content Toggle raw display
$43$ \( T^{12} - 2 T^{11} + \cdots + 1159168 \) Copy content Toggle raw display
$47$ \( T^{12} - 11 T^{11} + \cdots - 429568 \) Copy content Toggle raw display
$53$ \( (T - 1)^{12} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 45101343360 \) Copy content Toggle raw display
$61$ \( T^{12} - 16 T^{11} + \cdots - 36297088 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 2894979072 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 7738519552 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 805586944 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 28579207680 \) Copy content Toggle raw display
$83$ \( T^{12} - 39 T^{11} + \cdots - 4562944 \) Copy content Toggle raw display
$89$ \( T^{12} - 11 T^{11} + \cdots + 906240 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 127817142272 \) Copy content Toggle raw display
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