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Properties

Label 141.6.a.e
Level $141$
Weight $6$
Character orbit 141.a
Self dual yes
Analytic conductor $22.614$
Analytic rank $0$
Dimension $11$
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] = [141,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("141.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 141 = 3 \cdot 47 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(22.6141185936\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - x^{10} - 293 x^{9} + 291 x^{8} + 30852 x^{7} - 29294 x^{6} - 1418020 x^{5} + 1461392 x^{4} + \cdots + 359412480 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{11}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 9 q^{3} + (\beta_{2} + 21) q^{4} + (\beta_{3} + 1) q^{5} - 9 \beta_1 q^{6} + ( - \beta_{8} - \beta_{7} + \beta_{2} + 12) q^{7} + (\beta_{9} + \beta_{8} - \beta_{7} + \cdots - 6) q^{8}+ \cdots + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 9 q^{3} + (\beta_{2} + 21) q^{4} + (\beta_{3} + 1) q^{5} - 9 \beta_1 q^{6} + ( - \beta_{8} - \beta_{7} + \beta_{2} + 12) q^{7} + (\beta_{9} + \beta_{8} - \beta_{7} + \cdots - 6) q^{8}+ \cdots + ( - 162 \beta_{10} - 162 \beta_{9} + \cdots + 3402) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + q^{2} - 99 q^{3} + 235 q^{4} + 11 q^{5} - 9 q^{6} + 137 q^{7} - 57 q^{8} + 891 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + q^{2} - 99 q^{3} + 235 q^{4} + 11 q^{5} - 9 q^{6} + 137 q^{7} - 57 q^{8} + 891 q^{9} + 212 q^{10} + 473 q^{11} - 2115 q^{12} + 2 q^{13} + 308 q^{14} - 99 q^{15} + 3211 q^{16} - 1762 q^{17} + 81 q^{18} + 3532 q^{19} - 124 q^{20} - 1233 q^{21} + 1016 q^{22} + 187 q^{23} + 513 q^{24} + 19244 q^{25} + 13750 q^{26} - 8019 q^{27} + 22088 q^{28} - 6577 q^{29} - 1908 q^{30} + 19232 q^{31} - 2673 q^{32} - 4257 q^{33} + 36058 q^{34} + 18931 q^{35} + 19035 q^{36} + 23011 q^{37} + 77438 q^{38} - 18 q^{39} + 128264 q^{40} + 26568 q^{41} - 2772 q^{42} + 42440 q^{43} + 35620 q^{44} + 891 q^{45} + 116612 q^{46} + 24299 q^{47} - 28899 q^{48} + 16254 q^{49} + 73923 q^{50} + 15858 q^{51} + 87290 q^{52} - 36 q^{53} - 729 q^{54} - 83681 q^{55} + 47412 q^{56} - 31788 q^{57} - 14340 q^{58} + 8212 q^{59} + 1116 q^{60} + 26246 q^{61} - 115390 q^{62} + 11097 q^{63} + 90251 q^{64} - 102438 q^{65} - 9144 q^{66} + 49536 q^{67} - 261242 q^{68} - 1683 q^{69} - 25016 q^{70} + 19670 q^{71} - 4617 q^{72} + 34874 q^{73} + 105674 q^{74} - 173196 q^{75} + 25038 q^{76} - 48477 q^{77} - 123750 q^{78} + 93937 q^{79} - 174232 q^{80} + 72171 q^{81} - 123268 q^{82} + 42812 q^{83} - 198792 q^{84} - 9054 q^{85} - 214946 q^{86} + 59193 q^{87} - 255752 q^{88} + 22258 q^{89} + 17172 q^{90} + 397186 q^{91} - 218088 q^{92} - 173088 q^{93} + 2209 q^{94} + 35566 q^{95} + 24057 q^{96} + 241687 q^{97} - 274239 q^{98} + 38313 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - x^{10} - 293 x^{9} + 291 x^{8} + 30852 x^{7} - 29294 x^{6} - 1418020 x^{5} + 1461392 x^{4} + \cdots + 359412480 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 53 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 83343967 \nu^{10} - 390369431 \nu^{9} + 23976224789 \nu^{8} + 107256927765 \nu^{7} + \cdots + 96\!\cdots\!12 ) / 7739876818944 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 93187625 \nu^{10} - 96747697 \nu^{9} + 26576732227 \nu^{8} + 32216900163 \nu^{7} + \cdots + 17\!\cdots\!84 ) / 7739876818944 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 131313085 \nu^{10} + 315385733 \nu^{9} - 37195523519 \nu^{8} - 87684882879 \nu^{7} + \cdots - 14\!\cdots\!68 ) / 1934969204736 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24593773 \nu^{10} - 69339109 \nu^{9} + 7056136191 \nu^{8} + 19151971071 \nu^{7} + \cdots + 30\!\cdots\!04 ) / 322494867456 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 38849209 \nu^{10} - 80230193 \nu^{9} + 11079092963 \nu^{8} + 22534354147 \nu^{7} + \cdots + 41\!\cdots\!44 ) / 429993156608 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 207275365 \nu^{10} + 490804565 \nu^{9} - 59125402463 \nu^{8} - 140207370591 \nu^{7} + \cdots - 23\!\cdots\!36 ) / 1934969204736 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 919790915 \nu^{10} - 1755449371 \nu^{9} + 263166964321 \nu^{8} + 508451397729 \nu^{7} + \cdots + 99\!\cdots\!16 ) / 7739876818944 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1469465857 \nu^{10} + 2893996649 \nu^{9} - 421497477035 \nu^{8} - 830047349163 \nu^{7} + \cdots - 17\!\cdots\!04 ) / 7739876818944 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 53 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{3} + 82\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{10} - 9 \beta_{9} - 4 \beta_{8} + 10 \beta_{7} - 5 \beta_{6} + 6 \beta_{5} + 5 \beta_{4} + \cdots + 4342 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 40 \beta_{10} + 203 \beta_{9} + 159 \beta_{8} - 143 \beta_{7} + 40 \beta_{6} - 131 \beta_{5} + \cdots - 926 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 470 \beta_{10} - 1831 \beta_{9} - 798 \beta_{8} + 1912 \beta_{7} - 1065 \beta_{6} + 1120 \beta_{5} + \cdots + 409944 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8056 \beta_{10} + 30135 \beta_{9} + 20465 \beta_{8} - 18873 \beta_{7} + 8226 \beta_{6} - 15561 \beta_{5} + \cdots - 283706 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 77382 \beta_{10} - 275451 \beta_{9} - 119820 \beta_{8} + 272350 \beta_{7} - 159207 \beta_{6} + \cdots + 41273376 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1186616 \beta_{10} + 4004087 \beta_{9} + 2475651 \beta_{8} - 2444467 \beta_{7} + 1238072 \beta_{6} + \cdots - 59474934 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 11100622 \beta_{10} - 37250483 \beta_{9} - 16368450 \beta_{8} + 35056212 \beta_{7} + \cdots + 4331234828 \) 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
−11.0014
−8.34006
−7.81243
−4.95783
−4.22854
2.82034
3.12202
3.37256
8.19175
9.75832
10.0753
−11.0014 −9.00000 89.0319 −78.1888 99.0130 1.71764 −627.433 81.0000 860.191
1.2 −8.34006 −9.00000 37.5566 −17.3747 75.0606 214.237 −46.3426 81.0000 144.906
1.3 −7.81243 −9.00000 29.0340 65.4396 70.3118 −49.6251 23.1717 81.0000 −511.242
1.4 −4.95783 −9.00000 −7.41995 3.64574 44.6204 −207.591 195.437 81.0000 −18.0749
1.5 −4.22854 −9.00000 −14.1195 102.816 38.0568 170.907 195.018 81.0000 −434.762
1.6 2.82034 −9.00000 −24.0457 −88.3738 −25.3831 −162.838 −158.068 81.0000 −249.244
1.7 3.12202 −9.00000 −22.2530 −41.0567 −28.0982 −99.1368 −169.379 81.0000 −128.180
1.8 3.37256 −9.00000 −20.6259 42.3034 −30.3530 29.0501 −177.484 81.0000 142.670
1.9 8.19175 −9.00000 35.1048 −103.426 −73.7258 181.625 25.4341 81.0000 −847.239
1.10 9.75832 −9.00000 63.2247 27.1436 −87.8248 −36.1283 304.701 81.0000 264.875
1.11 10.0753 −9.00000 69.5119 98.0713 −90.6778 94.7823 377.945 81.0000 988.099
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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 141.6.a.e ✓ 11
3.b odd 2 1 423.6.a.f 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.6.a.e ✓ 11 1.a even 1 1 trivial
423.6.a.f 11 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{11} - T_{2}^{10} - 293 T_{2}^{9} + 291 T_{2}^{8} + 30852 T_{2}^{7} - 29294 T_{2}^{6} + \cdots + 359412480 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(141))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} + \cdots + 359412480 \) Copy content Toggle raw display
$3$ \( (T + 9)^{11} \) Copy content Toggle raw display
$5$ \( T^{11} + \cdots + 14\!\cdots\!68 \) Copy content Toggle raw display
$7$ \( T^{11} + \cdots + 18\!\cdots\!40 \) Copy content Toggle raw display
$11$ \( T^{11} + \cdots + 43\!\cdots\!88 \) Copy content Toggle raw display
$13$ \( T^{11} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{11} + \cdots - 39\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{11} + \cdots + 47\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( T^{11} + \cdots + 38\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{11} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{11} + \cdots + 26\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{11} + \cdots - 98\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{11} + \cdots - 40\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{11} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T - 2209)^{11} \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots + 43\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{11} + \cdots - 48\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots + 47\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{11} + \cdots + 61\!\cdots\!40 \) Copy content Toggle raw display
$71$ \( T^{11} + \cdots - 36\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots + 30\!\cdots\!20 \) Copy content Toggle raw display
$79$ \( T^{11} + \cdots + 35\!\cdots\!32 \) Copy content Toggle raw display
$83$ \( T^{11} + \cdots + 33\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots + 64\!\cdots\!28 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots + 79\!\cdots\!12 \) Copy content Toggle raw display
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