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Properties

Label 141.8.a.d
Level $141$
Weight $8$
Character orbit 141.a
Self dual yes
Analytic conductor $44.046$
Analytic rank $0$
Dimension $15$
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] = [141,8,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: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 5 x^{14} - 1549 x^{13} + 5499 x^{12} + 938696 x^{11} - 2239798 x^{10} - 282573164 x^{9} + \cdots + 103996455419904 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{2} + 27 q^{3} + (\beta_{2} - 2 \beta_1 + 84) q^{4} + (\beta_{5} + 33) q^{5} + ( - 27 \beta_1 + 54) q^{6} + ( - \beta_{7} + \beta_{2} - 5 \beta_1 + 107) q^{7} + ( - \beta_{6} - \beta_{5} + 2 \beta_{2} + \cdots + 401) q^{8}+ \cdots + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{2} + 27 q^{3} + (\beta_{2} - 2 \beta_1 + 84) q^{4} + (\beta_{5} + 33) q^{5} + ( - 27 \beta_1 + 54) q^{6} + ( - \beta_{7} + \beta_{2} - 5 \beta_1 + 107) q^{7} + ( - \beta_{6} - \beta_{5} + 2 \beta_{2} + \cdots + 401) q^{8}+ \cdots + ( - 729 \beta_{10} - 729 \beta_{7} + \cdots + 617463) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 25 q^{2} + 405 q^{3} + 1243 q^{4} + 501 q^{5} + 675 q^{6} + 1569 q^{7} + 5535 q^{8} + 10935 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 25 q^{2} + 405 q^{3} + 1243 q^{4} + 501 q^{5} + 675 q^{6} + 1569 q^{7} + 5535 q^{8} + 10935 q^{9} - 420 q^{10} + 12613 q^{11} + 33561 q^{12} + 1372 q^{13} + 17376 q^{14} + 13527 q^{15} + 140859 q^{16} + 61060 q^{17} + 18225 q^{18} + 109600 q^{19} + 177256 q^{20} + 42363 q^{21} + 57870 q^{22} + 130403 q^{23} + 149445 q^{24} + 412482 q^{25} + 343264 q^{26} + 295245 q^{27} + 452244 q^{28} + 144725 q^{29} - 11340 q^{30} + 53202 q^{31} + 578375 q^{32} + 340551 q^{33} + 466110 q^{34} + 604447 q^{35} + 906147 q^{36} - 526099 q^{37} - 1481454 q^{38} + 37044 q^{39} - 1903152 q^{40} + 1521072 q^{41} + 469152 q^{42} + 176914 q^{43} + 2191610 q^{44} + 365229 q^{45} - 2134612 q^{46} + 1557345 q^{47} + 3803193 q^{48} - 95212 q^{49} + 4023687 q^{50} + 1648620 q^{51} - 312420 q^{52} + 3097086 q^{53} + 492075 q^{54} + 450531 q^{55} + 3922840 q^{56} + 2959200 q^{57} - 482732 q^{58} + 7858234 q^{59} + 4785912 q^{60} + 4464770 q^{61} + 11609312 q^{62} + 1143801 q^{63} + 21509923 q^{64} + 8519960 q^{65} + 1562490 q^{66} + 3598330 q^{67} + 25451314 q^{68} + 3520881 q^{69} + 6727372 q^{70} + 6147430 q^{71} + 4035015 q^{72} + 7707828 q^{73} + 43246 q^{74} + 11137014 q^{75} + 17885454 q^{76} + 8004637 q^{77} + 9268128 q^{78} + 25507773 q^{79} + 19386676 q^{80} + 7971615 q^{81} - 7890462 q^{82} + 14084018 q^{83} + 12210588 q^{84} + 18360832 q^{85} + 22002738 q^{86} + 3907575 q^{87} - 28243134 q^{88} + 17235386 q^{89} - 306180 q^{90} + 10445016 q^{91} - 390544 q^{92} + 1436454 q^{93} + 2595575 q^{94} + 11582098 q^{95} + 15616125 q^{96} + 28803145 q^{97} + 18934353 q^{98} + 9194877 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 5 x^{14} - 1549 x^{13} + 5499 x^{12} + 938696 x^{11} - 2239798 x^{10} - 282573164 x^{9} + \cdots + 103996455419904 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 208 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23\!\cdots\!97 \nu^{14} + \cdots - 55\!\cdots\!48 ) / 68\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 70\!\cdots\!79 \nu^{14} + \cdots - 28\!\cdots\!16 ) / 64\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14\!\cdots\!47 \nu^{14} + \cdots - 52\!\cdots\!88 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14\!\cdots\!47 \nu^{14} + \cdots + 11\!\cdots\!48 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14\!\cdots\!03 \nu^{14} + \cdots + 85\!\cdots\!12 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 30\!\cdots\!91 \nu^{14} + \cdots - 31\!\cdots\!36 ) / 64\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 32\!\cdots\!01 \nu^{14} + \cdots + 42\!\cdots\!04 ) / 64\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 31\!\cdots\!57 \nu^{14} + \cdots + 45\!\cdots\!28 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!87 \nu^{14} + \cdots - 15\!\cdots\!48 ) / 80\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 50\!\cdots\!69 \nu^{14} + \cdots - 26\!\cdots\!84 ) / 25\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 61\!\cdots\!03 \nu^{14} + \cdots - 28\!\cdots\!12 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 43\!\cdots\!09 \nu^{14} + \cdots - 41\!\cdots\!36 ) / 64\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 208 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + 4\beta_{2} + 348\beta _1 + 343 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} + 3 \beta_{13} - 4 \beta_{12} - 7 \beta_{10} + 2 \beta_{9} + \beta_{8} + 4 \beta_{7} + \cdots + 72306 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 24 \beta_{14} + 18 \beta_{13} - 20 \beta_{12} - 57 \beta_{11} - 225 \beta_{10} + 31 \beta_{9} + \cdots + 314160 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1043 \beta_{14} + 2045 \beta_{13} - 3016 \beta_{12} - 656 \beta_{11} - 8889 \beta_{10} + \cdots + 29855514 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 25564 \beta_{14} + 20970 \beta_{13} - 23676 \beta_{12} - 59483 \beta_{11} - 238567 \beta_{10} + \cdots + 232851286 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 768019 \beta_{14} + 1180445 \beta_{13} - 1786288 \beta_{12} - 897356 \beta_{11} - 7166845 \beta_{10} + \cdots + 13696627858 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 19141776 \beta_{14} + 16710630 \beta_{13} - 19155852 \beta_{12} - 45007475 \beta_{11} + \cdots + 155895281386 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 502587863 \beta_{14} + 664645553 \beta_{13} - 991497312 \beta_{12} - 798386296 \beta_{11} + \cdots + 6769369941074 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 12627287828 \beta_{14} + 11521751210 \beta_{13} - 13361092588 \beta_{12} - 30224735135 \beta_{11} + \cdots + 99017666167458 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 312889454387 \beta_{14} + 375948130845 \beta_{13} - 543337924352 \beta_{12} - 594364957044 \beta_{11} + \cdots + 35\!\cdots\!14 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 7874941248184 \beta_{14} + 7412490633758 \beta_{13} - 8672054789356 \beta_{12} + \cdots + 61\!\cdots\!02 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 190277814751999 \beta_{14} + 214906122574393 \beta_{13} - 299643781939392 \beta_{12} + \cdots + 19\!\cdots\!54 \) 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
24.3008
19.6933
17.2341
13.3060
10.7660
10.1592
1.38516
0.236167
−3.88326
−7.16970
−9.52921
−15.7264
−16.3249
−19.3100
−20.1372
−22.3008 27.0000 369.324 283.064 −602.121 202.435 −5381.71 729.000 −6312.53
1.2 −17.6933 27.0000 185.054 314.670 −477.720 −467.359 −1009.47 729.000 −5567.56
1.3 −15.2341 27.0000 104.076 −252.869 −411.319 818.080 364.454 729.000 3852.22
1.4 −11.3060 27.0000 −0.173757 −340.464 −305.263 −1051.86 1449.14 729.000 3849.30
1.5 −8.76595 27.0000 −51.1581 −185.089 −236.681 1072.17 1570.49 729.000 1622.48
1.6 −8.15917 27.0000 −61.4279 357.157 −220.298 612.122 1545.58 729.000 −2914.11
1.7 0.614841 27.0000 −127.622 303.375 16.6007 −1499.42 −157.167 729.000 186.528
1.8 1.76383 27.0000 −124.889 −319.511 47.6235 −585.017 −446.054 729.000 −563.565
1.9 5.88326 27.0000 −93.3873 360.757 158.848 1533.90 −1302.48 729.000 2122.43
1.10 9.16970 27.0000 −43.9166 −485.319 247.582 −636.574 −1576.42 729.000 −4450.23
1.11 11.5292 27.0000 4.92270 −22.8692 311.289 817.529 −1418.98 729.000 −263.664
1.12 17.7264 27.0000 186.224 488.444 478.612 −973.812 1032.11 729.000 8658.34
1.13 18.3249 27.0000 207.802 100.017 494.772 536.140 1462.36 729.000 1832.79
1.14 21.3100 27.0000 326.117 303.084 575.370 1202.41 4221.87 729.000 6458.73
1.15 22.1372 27.0000 362.054 −403.447 597.704 −11.7433 5181.30 729.000 −8931.16
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.d ✓ 15
3.b odd 2 1 423.8.a.e 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.8.a.d ✓ 15 1.a even 1 1 trivial
423.8.a.e 15 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} - 25 T_{2}^{14} - 1269 T_{2}^{13} + 32955 T_{2}^{12} + 594664 T_{2}^{11} + \cdots - 502412602245120 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(141))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + \cdots - 502412602245120 \) Copy content Toggle raw display
$3$ \( (T - 27)^{15} \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots - 30\!\cdots\!80 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots - 35\!\cdots\!60 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 81\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 46\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 91\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 17\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 21\!\cdots\!40 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 12\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 42\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T - 103823)^{15} \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 19\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 51\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 28\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 92\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 57\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 15\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 17\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 32\!\cdots\!80 \) Copy content Toggle raw display
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