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Properties

Label 141.10.a.d
Level $141$
Weight $10$
Character orbit 141.a
Self dual yes
Analytic conductor $72.620$
Analytic rank $0$
Dimension $18$
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] = [141,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("141.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 141 = 3 \cdot 47 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(72.6200528991\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 5 x^{17} - 7047 x^{16} + 23381 x^{15} + 20549391 x^{14} - 37501757 x^{13} + \cdots - 13\!\cdots\!88 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 3) q^{2} + 81 q^{3} + (\beta_{2} - 4 \beta_1 + 281) q^{4} + ( - \beta_{4} - 3 \beta_1 + 278) q^{5} + ( - 81 \beta_1 + 243) q^{6} + (\beta_{6} - \beta_{4} + 3 \beta_{2} + \cdots + 138) q^{7}+ \cdots + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 3) q^{2} + 81 q^{3} + (\beta_{2} - 4 \beta_1 + 281) q^{4} + ( - \beta_{4} - 3 \beta_1 + 278) q^{5} + ( - 81 \beta_1 + 243) q^{6} + (\beta_{6} - \beta_{4} + 3 \beta_{2} + \cdots + 138) q^{7}+ \cdots + (6561 \beta_{14} + 13122 \beta_{6} + \cdots + 44765703) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 49 q^{2} + 1458 q^{3} + 5035 q^{4} + 4996 q^{5} + 3969 q^{6} + 2382 q^{7} + 41559 q^{8} + 118098 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 49 q^{2} + 1458 q^{3} + 5035 q^{4} + 4996 q^{5} + 3969 q^{6} + 2382 q^{7} + 41559 q^{8} + 118098 q^{9} + 50132 q^{10} + 121982 q^{11} + 407835 q^{12} + 131012 q^{13} + 283398 q^{14} + 404676 q^{15} + 524715 q^{16} + 814688 q^{17} + 321489 q^{18} + 1746898 q^{19} + 2098986 q^{20} + 192942 q^{21} + 2790206 q^{22} + 3585782 q^{23} + 3366279 q^{24} + 10235744 q^{25} + 6167742 q^{26} + 9565938 q^{27} + 20297700 q^{28} + 6215332 q^{29} + 4060692 q^{30} + 24366890 q^{31} + 18738347 q^{32} + 9880542 q^{33} + 33415958 q^{34} + 50023426 q^{35} + 33034635 q^{36} + 7986130 q^{37} + 72692436 q^{38} + 10611972 q^{39} + 22393854 q^{40} + 49629458 q^{41} + 22955238 q^{42} + 31281766 q^{43} - 32676380 q^{44} + 32778756 q^{45} + 39151046 q^{46} - 87834258 q^{47} + 42501915 q^{48} + 133927784 q^{49} - 29324267 q^{50} + 65989728 q^{51} - 17154094 q^{52} + 20466824 q^{53} + 26040609 q^{54} - 163723706 q^{55} + 101877448 q^{56} + 141498738 q^{57} - 176713756 q^{58} + 62345180 q^{59} + 170017866 q^{60} - 126946304 q^{61} - 3606912 q^{62} + 15628302 q^{63} - 142654709 q^{64} + 514251252 q^{65} + 226006686 q^{66} - 155847646 q^{67} + 814444818 q^{68} + 290448342 q^{69} + 348266734 q^{70} + 496675604 q^{71} + 272668599 q^{72} + 592924388 q^{73} + 439637972 q^{74} + 829095264 q^{75} + 1008739980 q^{76} + 1266391210 q^{77} + 499587102 q^{78} + 324168046 q^{79} + 1051441818 q^{80} + 774840978 q^{81} + 549460510 q^{82} + 1773714244 q^{83} + 1644113700 q^{84} + 263759316 q^{85} + 1808772684 q^{86} + 503441892 q^{87} + 1293126912 q^{88} + 2835516280 q^{89} + 328916052 q^{90} + 2379318264 q^{91} + 4938039392 q^{92} + 1973718090 q^{93} - 239104369 q^{94} + 3725840596 q^{95} + 1517806107 q^{96} + 399369274 q^{97} + 5645215371 q^{98} + 800323902 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 5 x^{17} - 7047 x^{16} + 23381 x^{15} + 20549391 x^{14} - 37501757 x^{13} + \cdots - 13\!\cdots\!88 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 784 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu^{2} - 1220\nu + 1494 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22\!\cdots\!97 \nu^{17} + \cdots - 22\!\cdots\!08 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\!\cdots\!79 \nu^{17} + \cdots + 13\!\cdots\!20 ) / 25\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19\!\cdots\!27 \nu^{17} + \cdots - 24\!\cdots\!48 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 46\!\cdots\!97 \nu^{17} + \cdots + 90\!\cdots\!12 ) / 16\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 65\!\cdots\!97 \nu^{17} + \cdots + 24\!\cdots\!40 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 87\!\cdots\!46 \nu^{17} + \cdots - 34\!\cdots\!24 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17\!\cdots\!59 \nu^{17} + \cdots - 12\!\cdots\!88 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 20\!\cdots\!29 \nu^{17} + \cdots - 35\!\cdots\!24 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 48\!\cdots\!71 \nu^{17} + \cdots + 49\!\cdots\!76 ) / 20\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 27\!\cdots\!31 \nu^{17} + \cdots - 39\!\cdots\!28 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 29\!\cdots\!49 \nu^{17} + \cdots + 21\!\cdots\!56 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 42\!\cdots\!87 \nu^{17} + \cdots + 41\!\cdots\!68 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 11\!\cdots\!95 \nu^{17} + \cdots + 90\!\cdots\!04 ) / 25\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 35\!\cdots\!99 \nu^{17} + \cdots + 20\!\cdots\!56 ) / 69\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 784 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_{2} + 1228\beta _1 + 1642 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} - 2 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 3 \beta_{11} + \beta_{9} + 3 \beta_{8} + \cdots + 963140 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 28 \beta_{17} - 27 \beta_{16} - 31 \beta_{15} + 67 \beta_{14} - 43 \beta_{13} + 31 \beta_{12} + \cdots + 4683915 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2700 \beta_{17} - 84 \beta_{16} - 5166 \beta_{15} + 5112 \beta_{14} - 5182 \beta_{13} + \cdots + 1357329310 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 62256 \beta_{17} - 58906 \beta_{16} - 91480 \beta_{15} + 177310 \beta_{14} - 132856 \beta_{13} + \cdots + 10611361970 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5484277 \beta_{17} - 468458 \beta_{16} - 10435178 \beta_{15} + 10239560 \beta_{14} + \cdots + 2038248947404 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 100560484 \beta_{17} - 92933393 \beta_{16} - 199367819 \beta_{15} + 357848829 \beta_{14} + \cdots + 22370861294215 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 10158426248 \beta_{17} - 1511500270 \beta_{16} - 19418404582 \beta_{15} + 18988690398 \beta_{14} + \cdots + 31\!\cdots\!38 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 144429107552 \beta_{17} - 128732993176 \beta_{16} - 392423562668 \beta_{15} + 662553918160 \beta_{14} + \cdots + 45\!\cdots\!90 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 18056536922665 \beta_{17} - 3785791672908 \beta_{16} - 34846071200282 \beta_{15} + 34143805638182 \beta_{14} + \cdots + 51\!\cdots\!80 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 198641764420588 \beta_{17} - 166444573753367 \beta_{16} - 741590055810407 \beta_{15} + \cdots + 89\!\cdots\!27 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 31\!\cdots\!92 \beta_{17} + \cdots + 83\!\cdots\!02 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 27\!\cdots\!56 \beta_{17} + \cdots + 17\!\cdots\!18 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 53\!\cdots\!77 \beta_{17} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 39\!\cdots\!68 \beta_{17} + \cdots + 32\!\cdots\!83 \) 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
42.1475
41.9260
33.6498
32.8056
29.4459
18.8257
15.0433
9.91427
7.09264
−6.14715
−13.0032
−17.2646
−19.9491
−25.3973
−30.5923
−36.6333
−38.2375
−38.6263
−39.1475 81.0000 1020.53 −631.309 −3170.95 −549.791 −19907.5 6561.00 24714.2
1.2 −38.9260 81.0000 1003.24 2170.75 −3153.01 7701.27 −19121.9 6561.00 −84498.7
1.3 −30.6498 81.0000 427.412 −845.954 −2482.64 3640.11 2592.59 6561.00 25928.3
1.4 −29.8056 81.0000 376.374 −2286.68 −2414.25 −2711.93 4042.42 6561.00 68155.9
1.5 −26.4459 81.0000 187.386 2218.24 −2142.12 −8114.99 8584.72 6561.00 −58663.4
1.6 −15.8257 81.0000 −261.548 −301.207 −1281.88 −6359.73 12241.9 6561.00 4766.80
1.7 −12.0433 81.0000 −366.958 2456.82 −975.510 934.928 10585.6 6561.00 −29588.3
1.8 −6.91427 81.0000 −464.193 56.6234 −560.056 10254.8 6749.66 6561.00 −391.509
1.9 −4.09264 81.0000 −495.250 −767.584 −331.504 −5819.18 4122.31 6561.00 3141.44
1.10 9.14715 81.0000 −428.330 269.245 740.919 4176.07 −8601.33 6561.00 2462.83
1.11 16.0032 81.0000 −255.897 −1780.98 1296.26 −12661.0 −12288.8 6561.00 −28501.4
1.12 20.2646 81.0000 −101.347 516.034 1641.43 −8920.93 −12429.2 6561.00 10457.2
1.13 22.9491 81.0000 14.6608 2311.68 1858.88 1160.60 −11413.5 6561.00 53051.0
1.14 28.3973 81.0000 294.406 2112.48 2300.18 11467.3 −6179.07 6561.00 59988.7
1.15 33.5923 81.0000 616.441 −2380.56 2720.97 −1320.20 3508.41 6561.00 −79968.4
1.16 39.6333 81.0000 1058.80 −642.263 3210.30 11027.9 21671.4 6561.00 −25455.0
1.17 41.2375 81.0000 1188.53 1011.83 3340.24 2336.01 27898.6 6561.00 41725.4
1.18 41.6263 81.0000 1220.75 1508.83 3371.73 −3859.27 29502.7 6561.00 62806.9
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
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 141.10.a.d ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.10.a.d ✓ 18 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 49 T_{2}^{17} - 5925 T_{2}^{16} + 298963 T_{2}^{15} + 14146836 T_{2}^{14} + \cdots - 87\!\cdots\!40 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(141))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots - 87\!\cdots\!40 \) Copy content Toggle raw display
$3$ \( (T - 81)^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots - 99\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 43\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 94\!\cdots\!40 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 64\!\cdots\!28 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 79\!\cdots\!88 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T + 4879681)^{18} \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 84\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 50\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 30\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 39\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 68\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 48\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 41\!\cdots\!16 \) Copy content Toggle raw display
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