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Properties

Label 141.8.a.b
Level $141$
Weight $8$
Character orbit 141.a
Self dual yes
Analytic conductor $44.046$
Analytic rank $1$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [141,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("141.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 141 = 3 \cdot 47 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(44.0462885933\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 3 x^{12} - 1237 x^{11} + 1921 x^{10} + 583972 x^{9} - 96794 x^{8} - 129937452 x^{7} + \cdots + 169129082880 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 2) q^{2} + 27 q^{3} + (\beta_{2} - 2 \beta_1 + 67) q^{4} + ( - \beta_{7} - \beta_{2} - 4 \beta_1 - 57) q^{5} + (27 \beta_1 - 54) q^{6} + (\beta_{12} + \beta_{8} + 2 \beta_{7} + \cdots - 90) q^{7}+ \cdots + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 2) q^{2} + 27 q^{3} + (\beta_{2} - 2 \beta_1 + 67) q^{4} + ( - \beta_{7} - \beta_{2} - 4 \beta_1 - 57) q^{5} + (27 \beta_1 - 54) q^{6} + (\beta_{12} + \beta_{8} + 2 \beta_{7} + \cdots - 90) q^{7}+ \cdots + ( - 1458 \beta_{12} + 729 \beta_{11} + \cdots - 917082) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 23 q^{2} + 351 q^{3} + 859 q^{4} - 749 q^{5} - 621 q^{6} - 1175 q^{7} - 3681 q^{8} + 9477 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 23 q^{2} + 351 q^{3} + 859 q^{4} - 749 q^{5} - 621 q^{6} - 1175 q^{7} - 3681 q^{8} + 9477 q^{9} - 8420 q^{10} - 16669 q^{11} + 23193 q^{12} - 412 q^{13} - 15552 q^{14} - 20223 q^{15} + 6699 q^{16} - 86280 q^{17} - 16767 q^{18} - 135268 q^{19} - 243176 q^{20} - 31725 q^{21} - 245904 q^{22} - 90671 q^{23} - 99387 q^{24} + 116108 q^{25} - 240670 q^{26} + 255879 q^{27} - 213632 q^{28} - 406669 q^{29} - 227340 q^{30} - 510950 q^{31} - 797881 q^{32} - 450063 q^{33} + 256714 q^{34} - 569323 q^{35} + 626211 q^{36} + 244209 q^{37} + 1377210 q^{38} - 11124 q^{39} + 1857252 q^{40} - 912484 q^{41} - 419904 q^{42} - 269094 q^{43} - 1082840 q^{44} - 546021 q^{45} + 827048 q^{46} - 1349699 q^{47} + 180873 q^{48} + 3116274 q^{49} - 3187617 q^{50} - 2329560 q^{51} + 1921270 q^{52} - 2033122 q^{53} - 452709 q^{54} + 830987 q^{55} - 7222612 q^{56} - 3652236 q^{57} - 8791636 q^{58} - 5679186 q^{59} - 6565752 q^{60} - 318234 q^{61} - 4750958 q^{62} - 856575 q^{63} - 6300037 q^{64} - 10669552 q^{65} - 6639408 q^{66} - 10009978 q^{67} - 24092042 q^{68} - 2448117 q^{69} - 7769804 q^{70} - 14591494 q^{71} - 2683449 q^{72} - 6287132 q^{73} - 7574146 q^{74} + 3134916 q^{75} - 10814534 q^{76} - 23151217 q^{77} - 6498090 q^{78} - 30795171 q^{79} - 24835652 q^{80} + 6908733 q^{81} - 3264508 q^{82} - 18871754 q^{83} - 5768064 q^{84} + 5520008 q^{85} - 8927750 q^{86} - 10980063 q^{87} - 24798204 q^{88} - 23797654 q^{89} - 6138180 q^{90} - 32747080 q^{91} - 13653824 q^{92} - 13795650 q^{93} + 2387929 q^{94} - 45827182 q^{95} - 21542787 q^{96} - 2789891 q^{97} + 19626133 q^{98} - 12151701 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 3 x^{12} - 1237 x^{11} + 1921 x^{10} + 583972 x^{9} - 96794 x^{8} - 129937452 x^{7} + \cdots + 169129082880 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 191 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11\!\cdots\!37 \nu^{12} + \cdots - 11\!\cdots\!72 ) / 21\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14\!\cdots\!17 \nu^{12} + \cdots + 21\!\cdots\!64 ) / 21\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 54\!\cdots\!45 \nu^{12} + \cdots - 66\!\cdots\!72 ) / 35\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 33\!\cdots\!31 \nu^{12} + \cdots - 40\!\cdots\!44 ) / 21\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 34\!\cdots\!23 \nu^{12} + \cdots + 16\!\cdots\!28 ) / 21\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 62\!\cdots\!71 \nu^{12} + \cdots + 29\!\cdots\!84 ) / 21\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 75\!\cdots\!37 \nu^{12} + \cdots + 21\!\cdots\!68 ) / 21\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 99\!\cdots\!65 \nu^{12} + \cdots + 19\!\cdots\!52 ) / 21\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 26\!\cdots\!47 \nu^{12} + \cdots - 14\!\cdots\!80 ) / 35\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 26\!\cdots\!59 \nu^{12} + \cdots - 75\!\cdots\!80 ) / 23\!\cdots\!48 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 191 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - 2\beta_{8} - 3\beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + 300\beta _1 + 347 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{12} - 6 \beta_{10} - 2 \beta_{9} - \beta_{8} + 19 \beta_{7} + 10 \beta_{6} - 12 \beta_{5} + \cdots + 57318 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 22 \beta_{12} - 41 \beta_{11} - 138 \beta_{10} - 335 \beta_{9} - 1050 \beta_{8} - 1438 \beta_{7} + \cdots + 198782 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 230 \beta_{12} + 126 \beta_{11} - 5302 \beta_{10} - 936 \beta_{9} - 1953 \beta_{8} + 6279 \beta_{7} + \cdots + 19259472 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7014 \beta_{12} - 35969 \beta_{11} - 124722 \beta_{10} - 99317 \beta_{9} - 450912 \beta_{8} + \cdots + 95338064 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 221166 \beta_{12} - 66564 \beta_{11} - 3368946 \beta_{10} - 302202 \beta_{9} - 1568091 \beta_{8} + \cdots + 6832281196 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 35018 \beta_{12} - 22239291 \beta_{11} - 77767886 \beta_{10} - 29149315 \beta_{9} - 183325242 \beta_{8} + \cdots + 43495824652 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 209505490 \beta_{12} - 117370470 \beta_{11} - 1853233030 \beta_{10} - 77496404 \beta_{9} + \cdots + 2507739490852 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1440729002 \beta_{12} - 11807424865 \beta_{11} - 41694587698 \beta_{10} - 8519337937 \beta_{9} + \cdots + 19460059169252 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 127665488614 \beta_{12} - 92514496680 \beta_{11} - 940150762074 \beta_{10} - 12835393710 \beta_{9} + \cdots + 943841880004612 \) 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
−18.6527
−18.4008
−14.4664
−12.5330
−7.55617
−6.02722
−0.0372409
2.14705
8.24664
13.2752
18.0358
18.2023
20.7665
−20.6527 27.0000 298.534 −514.818 −557.623 −596.926 −3521.98 729.000 10632.4
1.2 −20.4008 27.0000 288.193 −39.5404 −550.822 1565.73 −3268.06 729.000 806.656
1.3 −16.4664 27.0000 143.142 20.9664 −444.593 −51.7194 −249.341 729.000 −345.241
1.4 −14.5330 27.0000 83.2078 444.446 −392.391 −1064.81 650.965 729.000 −6459.12
1.5 −9.55617 27.0000 −36.6796 −204.236 −258.017 −1518.40 1573.71 729.000 1951.71
1.6 −8.02722 27.0000 −63.5637 404.472 −216.735 347.573 1537.72 729.000 −3246.79
1.7 −2.03724 27.0000 −123.850 67.7306 −55.0055 597.334 513.078 729.000 −137.984
1.8 0.147054 27.0000 −127.978 −361.707 3.97045 1100.20 −37.6425 729.000 −53.1904
1.9 6.24664 27.0000 −88.9795 149.982 168.659 −629.131 −1355.39 729.000 936.883
1.10 11.2752 27.0000 −0.870911 93.8239 304.429 −8.24518 −1453.04 729.000 1057.88
1.11 16.0358 27.0000 129.147 −136.392 432.967 −775.792 18.3915 729.000 −2187.16
1.12 16.2023 27.0000 134.516 −494.306 437.463 1491.13 105.567 729.000 −8008.91
1.13 18.7665 27.0000 224.183 −179.421 506.697 −1631.94 1805.02 729.000 −3367.11
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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 141.8.a.b ✓ 13
3.b odd 2 1 423.8.a.c 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.8.a.b ✓ 13 1.a even 1 1 trivial
423.8.a.c 13 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} + 23 T_{2}^{12} - 997 T_{2}^{11} - 23797 T_{2}^{10} + 356412 T_{2}^{9} + 9145066 T_{2}^{8} + \cdots + 795729825792 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(141))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} + \cdots + 795729825792 \) Copy content Toggle raw display
$3$ \( (T - 27)^{13} \) Copy content Toggle raw display
$5$ \( T^{13} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{13} + \cdots - 17\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{13} + \cdots + 16\!\cdots\!80 \) Copy content Toggle raw display
$13$ \( T^{13} + \cdots - 53\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{13} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{13} + \cdots + 41\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{13} + \cdots + 27\!\cdots\!48 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots - 11\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots - 16\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots - 13\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( (T + 103823)^{13} \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots + 39\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots + 67\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots - 60\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots - 49\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots + 31\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots - 96\!\cdots\!80 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots + 15\!\cdots\!28 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 23\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 28\!\cdots\!28 \) Copy content Toggle raw display
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