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Properties

Label 141.8.a.c
Level $141$
Weight $8$
Character orbit 141.a
Self dual yes
Analytic conductor $44.046$
Analytic rank $0$
Dimension $14$
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 1413 x^{12} + 2251 x^{11} + 777636 x^{10} - 1689662 x^{9} - 211782922 x^{8} + \cdots - 510625049571000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{5} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} - 27 q^{3} + (\beta_{2} + \beta_1 + 75) q^{4} + ( - \beta_{4} - 2 \beta_1 - 10) q^{5} + (27 \beta_1 + 27) q^{6} + (\beta_{6} - 4 \beta_1 + 117) q^{7} + ( - \beta_{11} - \beta_{10} + \cdots - 149) q^{8}+ \cdots + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} - 27 q^{3} + (\beta_{2} + \beta_1 + 75) q^{4} + ( - \beta_{4} - 2 \beta_1 - 10) q^{5} + (27 \beta_1 + 27) q^{6} + (\beta_{6} - 4 \beta_1 + 117) q^{7} + ( - \beta_{11} - \beta_{10} + \cdots - 149) q^{8}+ \cdots + (729 \beta_{12} - 2187 \beta_{11} + \cdots + 386370) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 15 q^{2} - 378 q^{3} + 1051 q^{4} - 139 q^{5} + 405 q^{6} + 1633 q^{7} - 2145 q^{8} + 10206 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 15 q^{2} - 378 q^{3} + 1051 q^{4} - 139 q^{5} + 405 q^{6} + 1633 q^{7} - 2145 q^{8} + 10206 q^{9} + 5508 q^{10} + 7479 q^{11} - 28377 q^{12} + 17300 q^{13} + 8754 q^{14} + 3753 q^{15} + 23835 q^{16} - 46892 q^{17} - 10935 q^{18} + 70160 q^{19} - 73970 q^{20} - 44091 q^{21} - 123992 q^{22} + 38641 q^{23} + 57915 q^{24} + 141091 q^{25} - 142532 q^{26} - 275562 q^{27} + 56248 q^{28} + 21553 q^{29} - 148716 q^{30} + 419464 q^{31} - 707805 q^{32} - 201933 q^{33} + 325602 q^{34} + 365741 q^{35} + 766179 q^{36} - 90033 q^{37} - 1583280 q^{38} - 467100 q^{39} + 714690 q^{40} + 69686 q^{41} - 236358 q^{42} + 252156 q^{43} + 2033670 q^{44} - 101331 q^{45} + 5497298 q^{46} - 1453522 q^{47} - 643545 q^{48} + 9003275 q^{49} + 5182849 q^{50} + 1266084 q^{51} + 6026628 q^{52} - 596498 q^{53} + 295245 q^{54} + 5548431 q^{55} + 14323100 q^{56} - 1894320 q^{57} + 12221036 q^{58} + 5111116 q^{59} + 1997190 q^{60} + 8380664 q^{61} + 13345874 q^{62} + 1190457 q^{63} + 8635155 q^{64} + 2944970 q^{65} + 3347784 q^{66} + 3435336 q^{67} + 11173054 q^{68} - 1043307 q^{69} + 30560446 q^{70} + 132282 q^{71} - 1563705 q^{72} + 5232040 q^{73} + 7199232 q^{74} - 3809457 q^{75} + 26703280 q^{76} + 639899 q^{77} + 3848364 q^{78} - 5118575 q^{79} + 3228718 q^{80} + 7440174 q^{81} + 35279256 q^{82} + 19880604 q^{83} - 1518696 q^{84} - 4052778 q^{85} + 4713992 q^{86} - 581931 q^{87} + 16566114 q^{88} - 12109956 q^{89} + 4015332 q^{90} + 20023406 q^{91} - 5825168 q^{92} - 11325528 q^{93} + 1557345 q^{94} - 11526082 q^{95} + 19110735 q^{96} - 9018269 q^{97} - 9349693 q^{98} + 5452191 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 1413 x^{12} + 2251 x^{11} + 777636 x^{10} - 1689662 x^{9} - 211782922 x^{8} + \cdots - 510625049571000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 202 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 73\!\cdots\!69 \nu^{13} + \cdots - 41\!\cdots\!40 ) / 22\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 29\!\cdots\!03 \nu^{13} + \cdots - 12\!\cdots\!20 ) / 39\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 24\!\cdots\!49 \nu^{13} + \cdots - 11\!\cdots\!80 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 36\!\cdots\!81 \nu^{13} + \cdots - 15\!\cdots\!00 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 38\!\cdots\!37 \nu^{13} + \cdots + 15\!\cdots\!60 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10\!\cdots\!51 \nu^{13} + \cdots + 44\!\cdots\!40 ) / 39\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!23 \nu^{13} + \cdots + 44\!\cdots\!00 ) / 33\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 32\!\cdots\!73 \nu^{13} + \cdots - 14\!\cdots\!60 ) / 99\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 15\!\cdots\!47 \nu^{13} + \cdots - 63\!\cdots\!20 ) / 39\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 43\!\cdots\!00 \nu^{13} + \cdots + 18\!\cdots\!40 ) / 99\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 25\!\cdots\!07 \nu^{13} + \cdots - 10\!\cdots\!00 ) / 39\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 202 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{9} - \beta_{6} - \beta_{5} - \beta_{4} + 311\beta _1 - 202 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{13} - 7 \beta_{12} + 6 \beta_{11} + 5 \beta_{10} + 4 \beta_{9} + \beta_{8} + 5 \beta_{7} + \cdots + 62838 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 60 \beta_{13} - 20 \beta_{12} + 520 \beta_{11} + 500 \beta_{10} + 590 \beta_{9} - 40 \beta_{8} + \cdots - 105736 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4232 \beta_{13} - 4942 \beta_{12} + 3394 \beta_{11} + 2268 \beta_{10} + 1490 \beta_{9} + \cdots + 22920598 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 50464 \beta_{13} - 5334 \beta_{12} + 228943 \beta_{11} + 224201 \beta_{10} + 283383 \beta_{9} + \cdots - 47127834 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2398806 \beta_{13} - 2653209 \beta_{12} + 1518720 \beta_{11} + 691597 \beta_{10} + 187006 \beta_{9} + \cdots + 9022810374 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 31567676 \beta_{13} + 792174 \beta_{12} + 96298054 \beta_{11} + 99094356 \beta_{10} + 128786524 \beta_{9} + \cdots - 22058279048 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1258830136 \beta_{13} - 1303037304 \beta_{12} + 631323564 \beta_{11} + 127083196 \beta_{10} + \cdots + 3697076966886 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 17602583184 \beta_{13} + 1843854284 \beta_{12} + 39991830885 \beta_{11} + 43856422257 \beta_{10} + \cdots - 11027778381690 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 634229379014 \beta_{13} - 616402478587 \beta_{12} + 253492619706 \beta_{11} - 25757580891 \beta_{10} + \cdots + 15\!\cdots\!42 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 9250550273724 \beta_{13} + 1460026692816 \beta_{12} + 16591599460868 \beta_{11} + 19484495769044 \beta_{10} + \cdots - 57\!\cdots\!16 \) 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
20.4956
19.8036
12.9137
12.0506
11.9386
9.66549
4.37329
−2.77420
−5.66252
−10.7712
−14.0421
−16.6587
−18.7735
−21.5586
−21.4956 −27.0000 334.059 87.0950 580.380 −1686.68 −4429.36 729.000 −1872.16
1.2 −20.8036 −27.0000 304.790 −199.901 561.698 872.896 −3677.88 729.000 4158.66
1.3 −13.9137 −27.0000 65.5903 −464.494 375.669 1626.38 868.348 729.000 6462.81
1.4 −13.0506 −27.0000 42.3188 −230.213 352.367 −43.0201 1118.19 729.000 3004.42
1.5 −12.9386 −27.0000 39.4077 221.334 349.343 −1178.75 1146.26 729.000 −2863.76
1.6 −10.6655 −27.0000 −14.2472 286.798 287.968 1085.05 1517.14 729.000 −3058.84
1.7 −5.37329 −27.0000 −99.1278 67.2917 145.079 442.945 1220.42 729.000 −361.577
1.8 1.77420 −27.0000 −124.852 292.955 −47.9035 1758.37 −448.611 729.000 519.761
1.9 4.66252 −27.0000 −106.261 −65.0218 −125.888 −1486.71 −1092.25 729.000 −303.165
1.10 9.77121 −27.0000 −32.5235 −441.972 −263.823 −93.7419 −1568.51 729.000 −4318.60
1.11 13.0421 −27.0000 42.0974 331.229 −352.138 −675.493 −1120.35 729.000 4319.94
1.12 15.6587 −27.0000 117.195 −343.243 −422.785 −1561.03 −169.191 729.000 −5374.73
1.13 17.7735 −27.0000 187.899 490.425 −479.885 1251.24 1064.61 729.000 8716.59
1.14 20.5586 −27.0000 294.654 −171.284 −555.081 1321.54 3426.17 729.000 −3521.35
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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.c ✓ 14
3.b odd 2 1 423.8.a.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.8.a.c ✓ 14 1.a even 1 1 trivial
423.8.a.d 14 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 15 T_{2}^{13} - 1309 T_{2}^{12} - 18765 T_{2}^{11} + 660904 T_{2}^{10} + \cdots - 363172220125184 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(141))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots - 363172220125184 \) Copy content Toggle raw display
$3$ \( (T + 27)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots - 63\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots - 24\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 11\!\cdots\!60 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots - 21\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 42\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 96\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 29\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 38\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 18\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T + 103823)^{14} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 21\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 94\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 33\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 64\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 21\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 88\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 35\!\cdots\!36 \) Copy content Toggle raw display
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