Properties

Label 6042.2.a.y
Level $6042$
Weight $2$
Character orbit 6042.a
Self dual yes
Analytic conductor $48.246$
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] = [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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 14x^{5} + 29x^{4} + 48x^{3} - 14x^{2} - 35x - 10 \) 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 14x^{5} + 29x^{4} + 48x^{3} - 14x^{2} - 35x - 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\nu^{6} - 51\nu^{5} - 216\nu^{4} + 517\nu^{3} + 682\nu^{2} - 480\nu - 445 ) / 25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 24\nu^{6} - 89\nu^{5} - 274\nu^{4} + 888\nu^{3} + 548\nu^{2} - 695\nu - 405 ) / 25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -28\nu^{6} + 108\nu^{5} + 303\nu^{4} - 1086\nu^{3} - 456\nu^{2} + 915\nu + 335 ) / 25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{6} + 34\nu^{5} + 99\nu^{4} - 338\nu^{3} - 163\nu^{2} + 260\nu + 115 ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -93\nu^{6} + 348\nu^{5} + 1043\nu^{4} - 3466\nu^{3} - 1886\nu^{2} + 2665\nu + 1285 ) / 25 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} + 2\beta_{3} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{6} - 2\beta_{5} - 2\beta_{4} + \beta_{3} + \beta_{2} + 11\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 16\beta_{6} - 20\beta_{5} - \beta_{4} + 22\beta_{3} + 2\beta_{2} + 21\beta _1 + 43 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 44\beta_{6} - 60\beta_{5} - 22\beta_{4} + 19\beta_{3} + 20\beta_{2} + 147\beta _1 + 77 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 249\beta_{6} - 354\beta_{5} - 19\beta_{4} + 240\beta_{3} + 60\beta_{2} + 384\beta _1 + 554 \) 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
4.05485
3.00316
0.982384
−0.456607
−0.724990
−0.835135
−3.02367
−1.00000 1.00000 1.00000 −3.05485 −1.00000 −2.36098 −1.00000 1.00000 3.05485
1.2 −1.00000 1.00000 1.00000 −2.00316 −1.00000 −4.20447 −1.00000 1.00000 2.00316
1.3 −1.00000 1.00000 1.00000 0.0176159 −1.00000 0.282315 −1.00000 1.00000 −0.0176159
1.4 −1.00000 1.00000 1.00000 1.45661 −1.00000 1.71471 −1.00000 1.00000 −1.45661
1.5 −1.00000 1.00000 1.00000 1.72499 −1.00000 −0.765322 −1.00000 1.00000 −1.72499
1.6 −1.00000 1.00000 1.00000 1.83513 −1.00000 0.943730 −1.00000 1.00000 −1.83513
1.7 −1.00000 1.00000 1.00000 4.02367 −1.00000 −4.60998 −1.00000 1.00000 −4.02367
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

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

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.y 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.y 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{7} - 4T_{5}^{6} - 11T_{5}^{5} + 51T_{5}^{4} - T_{5}^{3} - 140T_{5}^{2} + 116T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{7} + 9T_{7}^{6} + 16T_{7}^{5} - 39T_{7}^{4} - 80T_{7}^{3} + 56T_{7}^{2} + 48T_{7} - 16 \) Copy content Toggle raw display
\( T_{11}^{7} + 2T_{11}^{6} - 29T_{11}^{5} - 11T_{11}^{4} + 193T_{11}^{3} - 14T_{11}^{2} - 330T_{11} + 190 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{7} \) Copy content Toggle raw display
$3$ \( (T - 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} - 4 T^{6} - 11 T^{5} + 51 T^{4} + \cdots - 2 \) Copy content Toggle raw display
$7$ \( T^{7} + 9 T^{6} + 16 T^{5} - 39 T^{4} + \cdots - 16 \) Copy content Toggle raw display
$11$ \( T^{7} + 2 T^{6} - 29 T^{5} - 11 T^{4} + \cdots + 190 \) Copy content Toggle raw display
$13$ \( T^{7} + 7 T^{6} - 4 T^{5} - 73 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$17$ \( T^{7} + 10 T^{6} - 7 T^{5} - 231 T^{4} + \cdots + 40 \) Copy content Toggle raw display
$19$ \( (T + 1)^{7} \) Copy content Toggle raw display
$23$ \( T^{7} - 2 T^{6} - 83 T^{5} + \cdots + 1220 \) Copy content Toggle raw display
$29$ \( T^{7} + 6 T^{6} - 93 T^{5} + \cdots - 91232 \) Copy content Toggle raw display
$31$ \( T^{7} - 3 T^{6} - 58 T^{5} + 132 T^{4} + \cdots - 800 \) Copy content Toggle raw display
$37$ \( T^{7} + 3 T^{6} - 78 T^{5} + \cdots + 6536 \) Copy content Toggle raw display
$41$ \( T^{7} + 12 T^{6} - 57 T^{5} + \cdots - 133120 \) Copy content Toggle raw display
$43$ \( T^{7} + 26 T^{6} + 167 T^{5} + \cdots - 108904 \) Copy content Toggle raw display
$47$ \( T^{7} - 17 T^{6} - 73 T^{5} + \cdots - 49660 \) Copy content Toggle raw display
$53$ \( (T - 1)^{7} \) Copy content Toggle raw display
$59$ \( T^{7} - 7 T^{6} - 195 T^{5} + \cdots - 384464 \) Copy content Toggle raw display
$61$ \( T^{7} + 18 T^{6} - 195 T^{5} + \cdots - 457360 \) Copy content Toggle raw display
$67$ \( T^{7} + 3 T^{6} - 350 T^{5} + \cdots + 1247872 \) Copy content Toggle raw display
$71$ \( T^{7} - 2 T^{6} - 216 T^{5} + \cdots + 5120 \) Copy content Toggle raw display
$73$ \( T^{7} - T^{6} - 130 T^{5} + 487 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$79$ \( T^{7} + 18 T^{6} - 257 T^{5} + \cdots - 4444100 \) Copy content Toggle raw display
$83$ \( T^{7} + 17 T^{6} - 98 T^{5} + \cdots - 466432 \) Copy content Toggle raw display
$89$ \( T^{7} + 13 T^{6} - 245 T^{5} + \cdots - 44776 \) Copy content Toggle raw display
$97$ \( T^{7} + 20 T^{6} - 157 T^{5} + \cdots + 1473896 \) Copy content Toggle raw display
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