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Properties

Label 141.6.a.c
Level $141$
Weight $6$
Character orbit 141.a
Self dual yes
Analytic conductor $22.614$
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] = [141,6,Mod(1,141)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(141, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("141.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 141 = 3 \cdot 47 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 141.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: \(22.6141185936\)
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: \( x^{9} - 4 x^{8} - 213 x^{7} + 712 x^{6} + 13331 x^{5} - 34292 x^{4} - 212515 x^{3} + 298480 x^{2} + \cdots - 750384 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + 9 q^{3} + (\beta_{2} - \beta_1 + 18) q^{4} + (\beta_{7} + 10) q^{5} + ( - 9 \beta_1 + 9) q^{6} + (\beta_{8} - \beta_{7} - \beta_{4} + \cdots + 1) q^{7}+ \cdots + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + 9 q^{3} + (\beta_{2} - \beta_1 + 18) q^{4} + (\beta_{7} + 10) q^{5} + ( - 9 \beta_1 + 9) q^{6} + (\beta_{8} - \beta_{7} - \beta_{4} + \cdots + 1) q^{7}+ \cdots + (81 \beta_{8} + 162 \beta_{6} + \cdots + 19035) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 5 q^{2} + 81 q^{3} + 155 q^{4} + 86 q^{5} + 45 q^{6} + 42 q^{7} + 519 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 5 q^{2} + 81 q^{3} + 155 q^{4} + 86 q^{5} + 45 q^{6} + 42 q^{7} + 519 q^{8} + 729 q^{9} - 28 q^{10} + 2134 q^{11} + 1395 q^{12} + 754 q^{13} - 2138 q^{14} + 774 q^{15} + 7307 q^{16} + 1574 q^{17} + 405 q^{18} + 5606 q^{19} + 5666 q^{20} + 378 q^{21} - 1258 q^{22} + 3314 q^{23} + 4671 q^{24} + 3305 q^{25} - 2002 q^{26} + 6561 q^{27} - 19020 q^{28} + 3902 q^{29} - 252 q^{30} - 7666 q^{31} + 40923 q^{32} + 19206 q^{33} - 11906 q^{34} - 2314 q^{35} + 12555 q^{36} + 5896 q^{37} + 40636 q^{38} + 6786 q^{39} + 41462 q^{40} + 50488 q^{41} - 19242 q^{42} + 20558 q^{43} + 78236 q^{44} + 6966 q^{45} - 21130 q^{46} - 19881 q^{47} + 65763 q^{48} + 81643 q^{49} + 23841 q^{50} + 14166 q^{51} - 15542 q^{52} + 35606 q^{53} + 3645 q^{54} + 74602 q^{55} - 39048 q^{56} + 50454 q^{57} + 37188 q^{58} + 181980 q^{59} + 50994 q^{60} + 36082 q^{61} - 19136 q^{62} + 3402 q^{63} + 67275 q^{64} + 18848 q^{65} - 11322 q^{66} - 14718 q^{67} - 103086 q^{68} + 29826 q^{69} - 387218 q^{70} + 22672 q^{71} + 42039 q^{72} + 19598 q^{73} - 55604 q^{74} + 29745 q^{75} - 31388 q^{76} + 43050 q^{77} - 18018 q^{78} - 181450 q^{79} + 195410 q^{80} + 59049 q^{81} - 282538 q^{82} + 151468 q^{83} - 171180 q^{84} - 256300 q^{85} - 7612 q^{86} + 35118 q^{87} - 172904 q^{88} - 55094 q^{89} - 2268 q^{90} + 95772 q^{91} - 413712 q^{92} - 68994 q^{93} - 11045 q^{94} + 43976 q^{95} + 368307 q^{96} - 385596 q^{97} - 289185 q^{98} + 172854 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4 x^{8} - 213 x^{7} + 712 x^{6} + 13331 x^{5} - 34292 x^{4} - 212515 x^{3} + 298480 x^{2} + \cdots - 750384 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 49 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 100 \nu^{8} - 2623 \nu^{7} - 3840 \nu^{6} + 526294 \nu^{5} - 1544758 \nu^{4} - 28753643 \nu^{3} + \cdots - 477391968 ) / 4770720 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2725 \nu^{8} - 19297 \nu^{7} - 581712 \nu^{6} + 3980104 \nu^{5} + 35504087 \nu^{4} + \cdots + 1406514960 ) / 38165760 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1079 \nu^{8} + 1813 \nu^{7} - 256116 \nu^{6} - 516664 \nu^{5} + 19245445 \nu^{4} + \cdots + 1637322192 ) / 9541440 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5765 \nu^{8} - 51329 \nu^{7} - 1175520 \nu^{6} + 9579272 \nu^{5} + 66115351 \nu^{4} + \cdots + 92282256 ) / 38165760 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13211 \nu^{8} - 45671 \nu^{7} - 2797248 \nu^{6} + 7508744 \nu^{5} + 172514905 \nu^{4} + \cdots + 4967624304 ) / 38165760 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 155 \nu^{8} - 587 \nu^{7} - 32456 \nu^{6} + 103792 \nu^{5} + 1920145 \nu^{4} - 4745721 \nu^{3} + \cdots + 15952432 ) / 424064 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 49 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - 2\beta_{7} + \beta_{6} + 2\beta_{5} - \beta_{4} + \beta_{3} + 89\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{8} + 2\beta_{7} + 8\beta_{5} + 6\beta_{4} + 8\beta_{3} + 107\beta_{2} + 143\beta _1 + 4415 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 137 \beta_{8} - 280 \beta_{7} + 105 \beta_{6} + 280 \beta_{5} - 55 \beta_{4} + 155 \beta_{3} + \cdots + 3466 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 918 \beta_{8} + 352 \beta_{7} - 154 \beta_{6} + 1268 \beta_{5} + 846 \beta_{4} + 1582 \beta_{3} + \cdots + 434149 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 17161 \beta_{8} - 34210 \beta_{7} + 8869 \beta_{6} + 34150 \beta_{5} - 805 \beta_{4} + 20281 \beta_{3} + \cdots + 598558 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 111290 \beta_{8} + 45634 \beta_{7} - 38352 \beta_{6} + 169476 \beta_{5} + 105982 \beta_{4} + \cdots + 43803175 \) 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
10.6909
9.71526
5.29688
2.08268
0.960505
−2.16493
−3.44918
−9.11870
−10.0134
−9.69086 9.00000 61.9127 −14.4251 −87.2177 −191.960 −289.879 81.0000 139.791
1.2 −8.71526 9.00000 43.9558 51.5411 −78.4373 191.351 −104.198 81.0000 −449.194
1.3 −4.29688 9.00000 −13.5368 37.4997 −38.6720 223.966 195.666 81.0000 −161.132
1.4 −1.08268 9.00000 −30.8278 −81.5768 −9.74416 −106.605 68.0227 81.0000 88.3220
1.5 0.0394955 9.00000 −31.9984 100.490 0.355459 −9.42123 −2.52765 81.0000 3.96890
1.6 3.16493 9.00000 −21.9832 −66.2746 28.4844 188.997 −170.853 81.0000 −209.754
1.7 4.44918 9.00000 −12.2048 17.1311 40.0426 −119.397 −196.675 81.0000 76.2192
1.8 10.1187 9.00000 70.3880 −28.4598 91.0683 71.7275 388.436 81.0000 −287.976
1.9 11.0134 9.00000 89.2946 70.0743 99.1204 −206.658 631.007 81.0000 771.755
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(47\) \(1\)

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 141.6.a.c ✓ 9
3.b odd 2 1 423.6.a.e 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.6.a.c ✓ 9 1.a even 1 1 trivial
423.6.a.e 9 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} - 5 T_{2}^{8} - 209 T_{2}^{7} + 807 T_{2}^{6} + 13032 T_{2}^{5} - 35434 T_{2}^{4} + \cdots - 24352 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(141))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 5 T^{8} + \cdots - 24352 \) Copy content Toggle raw display
$3$ \( (T - 9)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} + \cdots - 517501402558080 \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots + 27\!\cdots\!40 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots - 23\!\cdots\!60 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots + 54\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots - 13\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots - 73\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots - 28\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots - 16\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots - 13\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( (T + 2209)^{9} \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots - 62\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots - 29\!\cdots\!08 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots - 39\!\cdots\!88 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots + 57\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots - 46\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots - 24\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots + 33\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots - 11\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots + 53\!\cdots\!44 \) Copy content Toggle raw display
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