Properties

Label 6042.2.a.z
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: \( x^{9} - 4x^{8} - 11x^{7} + 51x^{6} + 25x^{5} - 180x^{4} + 29x^{3} + 119x^{2} - 8x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4x^{8} - 11x^{7} + 51x^{6} + 25x^{5} - 180x^{4} + 29x^{3} + 119x^{2} - 8x - 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{8} - 9\nu^{7} + 34\nu^{6} - 119\nu^{5} - 96\nu^{4} + 1732\nu^{3} - 755\nu^{2} - 3266\nu - 862 ) / 716 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{8} - 81\nu^{7} + 306\nu^{6} + 361\nu^{5} - 4444\nu^{4} + 1984\nu^{3} + 14685\nu^{2} - 6482\nu - 4894 ) / 1432 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 119 \nu^{8} - 355 \nu^{7} - 1682 \nu^{6} + 4455 \nu^{5} + 7192 \nu^{4} - 14420 \nu^{3} - 8937 \nu^{2} + 5146 \nu + 3390 ) / 1432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 119 \nu^{8} + 355 \nu^{7} + 1682 \nu^{6} - 4455 \nu^{5} - 7192 \nu^{4} + 14420 \nu^{3} + 7505 \nu^{2} - 3714 \nu + 906 ) / 1432 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 127 \nu^{8} - 427 \nu^{7} - 1410 \nu^{6} + 4935 \nu^{5} + 3560 \nu^{4} - 14884 \nu^{3} + 775 \nu^{2} + 4794 \nu + 3654 ) / 1432 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -42\nu^{8} + 199\nu^{7} + 362\nu^{6} - 2520\nu^{5} + 273\nu^{4} + 8343\nu^{3} - 5701\nu^{2} - 2806\nu + 1478 ) / 358 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 61\nu^{8} - 191\nu^{7} - 790\nu^{6} + 2407\nu^{5} + 2736\nu^{4} - 8192\nu^{3} - 1305\nu^{2} + 4476\nu + 402 ) / 358 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -36\nu^{8} + 145\nu^{7} + 387\nu^{6} - 1802\nu^{5} - 840\nu^{4} + 6026\nu^{3} - 1102\nu^{2} - 3070\nu + 244 ) / 179 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{5} + \beta_{4} + \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{5} - \beta_{4} - 2\beta_{3} + \beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{8} + 7\beta_{7} - 2\beta_{6} - 5\beta_{5} + 5\beta_{4} - 2\beta_{3} - 2\beta_{2} + 9\beta _1 + 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{8} + 11\beta_{7} - 2\beta_{6} - 5\beta_{5} - 7\beta_{4} - 22\beta_{3} - 6\beta_{2} + 15\beta _1 + 60 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 24\beta_{8} + 63\beta_{7} - 24\beta_{6} - 29\beta_{5} + 29\beta_{4} - 44\beta_{3} - 32\beta_{2} + 83\beta _1 + 146 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 42 \beta_{8} + 133 \beta_{7} - 40 \beta_{6} - 21 \beta_{5} - 45 \beta_{4} - 246 \beta_{3} - 96 \beta_{2} + 187 \beta _1 + 520 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 276 \beta_{8} + 633 \beta_{7} - 260 \beta_{6} - 161 \beta_{5} + 171 \beta_{4} - 638 \beta_{3} - 396 \beta_{2} + 819 \beta _1 + 1432 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 640 \beta_{8} + 1625 \beta_{7} - 564 \beta_{6} - 27 \beta_{5} - 301 \beta_{4} - 2772 \beta_{3} - 1220 \beta_{2} + 2195 \beta _1 + 4926 ) / 2 \) 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
−0.690250
−2.30368
2.71772
1.09225
0.407650
2.32668
3.37216
−2.56034
−0.362196
−1.00000 −1.00000 1.00000 −4.37472 1.00000 −0.773393 −1.00000 1.00000 4.37472
1.2 −1.00000 −1.00000 1.00000 −2.37696 1.00000 0.603136 −1.00000 1.00000 2.37696
1.3 −1.00000 −1.00000 1.00000 −2.02880 1.00000 −1.70084 −1.00000 1.00000 2.02880
1.4 −1.00000 −1.00000 1.00000 −0.410706 1.00000 4.57610 −1.00000 1.00000 0.410706
1.5 −1.00000 −1.00000 1.00000 −0.186875 1.00000 −2.28090 −1.00000 1.00000 0.186875
1.6 −1.00000 −1.00000 1.00000 0.442981 1.00000 −2.93910 −1.00000 1.00000 −0.442981
1.7 −1.00000 −1.00000 1.00000 0.912007 1.00000 3.48340 −1.00000 1.00000 −0.912007
1.8 −1.00000 −1.00000 1.00000 3.19328 1.00000 3.82171 −1.00000 1.00000 −3.19328
1.9 −1.00000 −1.00000 1.00000 3.82978 1.00000 −0.790105 −1.00000 1.00000 −3.82978
\(n\): 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.z 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.z 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{9} + T_{5}^{8} - 27T_{5}^{7} - 22T_{5}^{6} + 186T_{5}^{5} + 157T_{5}^{4} - 250T_{5}^{3} - 66T_{5}^{2} + 40T_{5} + 8 \) Copy content Toggle raw display
\( T_{7}^{9} - 4T_{7}^{8} - 25T_{7}^{7} + 71T_{7}^{6} + 265T_{7}^{5} - 274T_{7}^{4} - 1184T_{7}^{3} - 592T_{7}^{2} + 384T_{7} + 256 \) Copy content Toggle raw display
\( T_{11}^{9} - 12 T_{11}^{8} + 31 T_{11}^{7} + 107 T_{11}^{6} - 447 T_{11}^{5} - 232 T_{11}^{4} + 1402 T_{11}^{3} + 384 T_{11}^{2} - 272 T_{11} - 64 \) 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
$5$ \( T^{9} + T^{8} - 27 T^{7} - 22 T^{6} + \cdots + 8 \) Copy content Toggle raw display
$7$ \( T^{9} - 4 T^{8} - 25 T^{7} + 71 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{9} - 12 T^{8} + 31 T^{7} + 107 T^{6} + \cdots - 64 \) Copy content Toggle raw display
$13$ \( T^{9} + T^{8} - 62 T^{7} - 19 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{9} - 12 T^{8} + T^{7} + 387 T^{6} + \cdots - 1312 \) Copy content Toggle raw display
$19$ \( (T + 1)^{9} \) Copy content Toggle raw display
$23$ \( T^{9} - 11 T^{8} - 21 T^{7} + \cdots + 22912 \) Copy content Toggle raw display
$29$ \( T^{9} - 7 T^{8} - 109 T^{7} + \cdots - 41984 \) Copy content Toggle raw display
$31$ \( T^{9} + 12 T^{8} - 25 T^{7} + \cdots - 7168 \) Copy content Toggle raw display
$37$ \( T^{9} + 9 T^{8} - 92 T^{7} + \cdots - 145952 \) Copy content Toggle raw display
$41$ \( T^{9} + 4 T^{8} - 135 T^{7} + \cdots + 28672 \) Copy content Toggle raw display
$43$ \( T^{9} - 23 T^{8} + 101 T^{7} + \cdots - 204608 \) Copy content Toggle raw display
$47$ \( T^{9} - 35 T^{8} + 395 T^{7} + \cdots + 856288 \) Copy content Toggle raw display
$53$ \( (T - 1)^{9} \) Copy content Toggle raw display
$59$ \( T^{9} - 14 T^{8} - 176 T^{7} + \cdots + 345776 \) Copy content Toggle raw display
$61$ \( T^{9} - 14 T^{8} - 131 T^{7} + \cdots + 5183488 \) Copy content Toggle raw display
$67$ \( T^{9} + 10 T^{8} - 287 T^{7} + \cdots + 1878784 \) Copy content Toggle raw display
$71$ \( T^{9} - 4 T^{8} - 292 T^{7} + \cdots - 14360576 \) Copy content Toggle raw display
$73$ \( T^{9} + 5 T^{8} - 270 T^{7} + \cdots - 4339328 \) Copy content Toggle raw display
$79$ \( T^{9} + 14 T^{8} + \cdots - 117175808 \) Copy content Toggle raw display
$83$ \( T^{9} - 37 T^{8} + \cdots + 178515968 \) Copy content Toggle raw display
$89$ \( T^{9} + 20 T^{8} - 176 T^{7} + \cdots - 7273384 \) Copy content Toggle raw display
$97$ \( T^{9} + 2 T^{8} - 369 T^{7} + \cdots - 29410976 \) Copy content Toggle raw display
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