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Properties

Label 141.10.a.a
Level $141$
Weight $10$
Character orbit 141.a
Self dual yes
Analytic conductor $72.620$
Analytic rank $1$
Dimension $16$
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: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} - 5776 x^{14} - 2252 x^{13} + 12928477 x^{12} + 18688139 x^{11} - 14180744788 x^{10} + \cdots + 12\!\cdots\!08 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{5} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 3) q^{2} + 81 q^{3} + (\beta_{2} - 4 \beta_1 + 219) q^{4} + ( - \beta_{4} - 3 \beta_1 - 78) q^{5} + (81 \beta_1 - 243) q^{6} + (\beta_{11} + \beta_{4} - 3 \beta_{2} + \cdots - 1051) q^{7}+ \cdots + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 3) q^{2} + 81 q^{3} + (\beta_{2} - 4 \beta_1 + 219) q^{4} + ( - \beta_{4} - 3 \beta_1 - 78) q^{5} + (81 \beta_1 - 243) q^{6} + (\beta_{11} + \beta_{4} - 3 \beta_{2} + \cdots - 1051) q^{7}+ \cdots + ( - 13122 \beta_{14} + 13122 \beta_{12} + \cdots - 81986256) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 47 q^{2} + 1296 q^{3} + 3499 q^{4} - 1254 q^{5} - 3807 q^{6} - 16826 q^{7} - 32169 q^{8} + 104976 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 47 q^{2} + 1296 q^{3} + 3499 q^{4} - 1254 q^{5} - 3807 q^{6} - 16826 q^{7} - 32169 q^{8} + 104976 q^{9} - 29868 q^{10} - 200120 q^{11} + 283419 q^{12} - 95502 q^{13} - 177594 q^{14} - 101574 q^{15} + 1615515 q^{16} - 690496 q^{17} - 308367 q^{18} - 1278736 q^{19} - 2356390 q^{20} - 1362906 q^{21} - 1458880 q^{22} - 1399562 q^{23} - 2605689 q^{24} + 726622 q^{25} - 17122512 q^{26} + 8503056 q^{27} - 20990864 q^{28} - 13003242 q^{29} - 2419308 q^{30} - 7478442 q^{31} - 25301845 q^{32} - 16209720 q^{33} - 25410462 q^{34} - 11824534 q^{35} + 22956939 q^{36} - 50074630 q^{37} - 81697476 q^{38} - 7735662 q^{39} - 94310538 q^{40} - 54106182 q^{41} - 14385114 q^{42} + 16409280 q^{43} - 62098030 q^{44} - 8227494 q^{45} + 113690318 q^{46} + 78074896 q^{47} + 130856715 q^{48} - 96307122 q^{49} + 108684653 q^{50} - 55930176 q^{51} + 99873960 q^{52} - 106888332 q^{53} - 24977727 q^{54} + 38221686 q^{55} + 204712532 q^{56} - 103577616 q^{57} + 309096804 q^{58} - 186026440 q^{59} - 190867590 q^{60} - 38459056 q^{61} - 889878 q^{62} - 110395386 q^{63} + 291910595 q^{64} - 155786144 q^{65} - 118169280 q^{66} + 159171012 q^{67} - 1088766466 q^{68} - 113364522 q^{69} + 185962258 q^{70} - 228485180 q^{71} - 211060809 q^{72} + 207306648 q^{73} - 300916052 q^{74} + 58856382 q^{75} - 310314044 q^{76} - 478954510 q^{77} - 1386923472 q^{78} - 1294130282 q^{79} - 1669184374 q^{80} + 688747536 q^{81} - 562966728 q^{82} - 1677738604 q^{83} - 1700259984 q^{84} - 1200731300 q^{85} - 2666467868 q^{86} - 1053262602 q^{87} - 680788314 q^{88} - 2359609872 q^{89} - 195963948 q^{90} - 1767901468 q^{91} - 1970519048 q^{92} - 605753802 q^{93} - 229345007 q^{94} - 4512883876 q^{95} - 2049449445 q^{96} - 4639693838 q^{97} - 3279528953 q^{98} - 1312987320 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} - 5776 x^{14} - 2252 x^{13} + 12928477 x^{12} + 18688139 x^{11} - 14180744788 x^{10} + \cdots + 12\!\cdots\!08 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 722 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 77\!\cdots\!91 \nu^{15} + \cdots + 51\!\cdots\!72 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 38\!\cdots\!67 \nu^{15} + \cdots + 23\!\cdots\!64 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 66\!\cdots\!77 \nu^{15} + \cdots - 29\!\cdots\!96 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16\!\cdots\!13 \nu^{15} + \cdots + 11\!\cdots\!44 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 81\!\cdots\!53 \nu^{15} + \cdots - 11\!\cdots\!56 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 26\!\cdots\!89 \nu^{15} + \cdots + 33\!\cdots\!08 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 23\!\cdots\!13 \nu^{15} + \cdots + 88\!\cdots\!04 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 83\!\cdots\!81 \nu^{15} + \cdots + 31\!\cdots\!92 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11\!\cdots\!43 \nu^{15} + \cdots - 25\!\cdots\!16 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!23 \nu^{15} + \cdots + 20\!\cdots\!84 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 22\!\cdots\!03 \nu^{15} + \cdots + 10\!\cdots\!76 ) / 69\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 25\!\cdots\!91 \nu^{15} + \cdots + 64\!\cdots\!72 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 68\!\cdots\!19 \nu^{15} + \cdots + 28\!\cdots\!72 ) / 13\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 722 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{9} - \beta_{8} + 3\beta_{4} + 4\beta_{2} + 1302\beta _1 + 1424 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{15} + 6 \beta_{13} + 26 \beta_{12} - 35 \beta_{11} - 3 \beta_{10} + \beta_{9} - 16 \beta_{8} + \cdots + 939884 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 100 \beta_{15} - 484 \beta_{14} - 56 \beta_{13} + 344 \beta_{12} - 782 \beta_{11} + 2588 \beta_{10} + \cdots + 4490884 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3788 \beta_{15} - 7764 \beta_{14} + 24784 \beta_{13} + 77224 \beta_{12} - 99406 \beta_{11} + \cdots + 1419017842 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 155788 \beta_{15} - 1562124 \beta_{14} + 6000 \beta_{13} + 1403448 \beta_{12} - 3039786 \beta_{11} + \cdots + 11449839312 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 8603102 \beta_{15} - 28083124 \beta_{14} + 64075366 \beta_{13} + 179595314 \beta_{12} + \cdots + 2320530524556 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 58323808 \beta_{15} - 3701155696 \beta_{14} + 399641448 \beta_{13} + 4009048416 \beta_{12} + \cdots + 27524298547124 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 22068920568 \beta_{15} - 74882181864 \beta_{14} + 142878402368 \beta_{13} + 385502931056 \beta_{12} + \cdots + 40\!\cdots\!46 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 410650136232 \beta_{15} - 7889308132952 \beta_{14} + 1630276022752 \beta_{13} + 9969351784432 \beta_{12} + \cdots + 63\!\cdots\!48 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 55445817152634 \beta_{15} - 179212382379688 \beta_{14} + 299769863324998 \beta_{13} + \cdots + 71\!\cdots\!96 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 17\!\cdots\!08 \beta_{15} + \cdots + 14\!\cdots\!88 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 13\!\cdots\!40 \beta_{15} + \cdots + 13\!\cdots\!70 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 48\!\cdots\!48 \beta_{15} + \cdots + 32\!\cdots\!52 \) 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
−40.5327
−38.8451
−32.6937
−24.5037
−23.5979
−12.8121
−8.93848
−0.0848664
2.64646
4.07067
13.2009
17.3734
31.1019
32.3646
37.0090
45.2416
−43.5327 81.0000 1383.09 702.320 −3526.15 −698.132 −37921.1 6561.00 −30573.9
1.2 −41.8451 81.0000 1239.02 −1515.58 −3389.46 −6028.49 −30422.0 6561.00 63419.5
1.3 −35.6937 81.0000 762.038 1080.13 −2891.19 −9795.47 −8924.79 6561.00 −38554.0
1.4 −27.5037 81.0000 244.453 −742.902 −2227.80 8446.28 7358.54 6561.00 20432.6
1.5 −26.5979 81.0000 195.449 1057.16 −2154.43 7092.42 8419.59 6561.00 −28118.3
1.6 −15.8121 81.0000 −261.978 −43.2767 −1280.78 −6351.87 12238.2 6561.00 684.294
1.7 −11.9385 81.0000 −369.473 −2551.28 −967.017 5690.41 10523.4 6561.00 30458.4
1.8 −3.08487 81.0000 −502.484 −1913.20 −249.874 −3551.44 3129.55 6561.00 5901.96
1.9 −0.353538 81.0000 −511.875 1702.06 −28.6366 7267.55 361.979 6561.00 −601.741
1.10 1.07067 81.0000 −510.854 1130.88 86.7243 −5595.70 −1095.14 6561.00 1210.80
1.11 10.2009 81.0000 −407.941 1936.66 826.275 −4264.75 −9384.25 6561.00 19755.7
1.12 14.3734 81.0000 −305.407 −939.597 1164.24 1791.32 −11748.9 6561.00 −13505.2
1.13 28.1019 81.0000 277.717 −265.647 2276.25 −195.085 −6583.80 6561.00 −7465.19
1.14 29.3646 81.0000 350.281 1830.18 2378.53 −5674.11 −4748.82 6561.00 53742.5
1.15 34.0090 81.0000 644.609 −1010.91 2754.73 2976.37 4509.90 6561.00 −34380.1
1.16 42.2416 81.0000 1272.35 −1711.00 3421.57 −7935.29 32118.6 6561.00 −72275.5
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
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.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.10.a.a ✓ 16 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 47 T_{2}^{15} - 4741 T_{2}^{14} - 230669 T_{2}^{13} + 8245240 T_{2}^{12} + \cdots - 18\!\cdots\!84 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(141))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots - 18\!\cdots\!84 \) Copy content Toggle raw display
$3$ \( (T - 81)^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots - 56\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 25\!\cdots\!20 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots - 34\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 76\!\cdots\!48 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots - 27\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots - 55\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 15\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 84\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 18\!\cdots\!60 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots - 13\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T - 4879681)^{16} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots - 69\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots - 51\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 29\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 72\!\cdots\!20 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots - 16\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
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