Properties

Label 29.8.a.b
Level $29$
Weight $8$
Character orbit 29.a
Self dual yes
Analytic conductor $9.059$
Analytic rank $0$
Dimension $10$
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] = [29,8,Mod(1,29)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(29, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("29.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 29.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: \(9.05916573904\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 1101 x^{8} - 1540 x^{7} + 405148 x^{6} + 870160 x^{5} - 54569376 x^{4} - 87078400 x^{3} + 2140673280 x^{2} - 1918315520 x - 9372051456 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 8) q^{3} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 92) q^{4} + ( - \beta_{7} + \beta_{2} - 4 \beta_1 + 18) q^{5} + (\beta_{8} + 2 \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - 26 \beta_1 + 36) q^{6} + (\beta_{9} - 2 \beta_{8} + 2 \beta_{5} - 3 \beta_{4} - \beta_{3} + 3 \beta_{2} - 22 \beta_1 + 104) q^{7} + ( - 5 \beta_{9} - 4 \beta_{8} + \beta_{7} + \beta_{6} + 3 \beta_{5} - 5 \beta_{3} + \cdots - 460) q^{8}+ \cdots + (7 \beta_{9} + 7 \beta_{8} - 6 \beta_{7} - 6 \beta_{6} - \beta_{5} + 5 \beta_{4} + \cdots + 1098) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 8) q^{3} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 92) q^{4} + ( - \beta_{7} + \beta_{2} - 4 \beta_1 + 18) q^{5} + (\beta_{8} + 2 \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - 26 \beta_1 + 36) q^{6} + (\beta_{9} - 2 \beta_{8} + 2 \beta_{5} - 3 \beta_{4} - \beta_{3} + 3 \beta_{2} - 22 \beta_1 + 104) q^{7} + ( - 5 \beta_{9} - 4 \beta_{8} + \beta_{7} + \beta_{6} + 3 \beta_{5} - 5 \beta_{3} + \cdots - 460) q^{8}+ \cdots + ( - 7974 \beta_{9} - 7837 \beta_{8} - 5399 \beta_{7} + \cdots - 1429911) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 80 q^{3} + 922 q^{4} + 180 q^{5} + 358 q^{6} + 1040 q^{7} - 4620 q^{8} + 10986 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 80 q^{3} + 922 q^{4} + 180 q^{5} + 358 q^{6} + 1040 q^{7} - 4620 q^{8} + 10986 q^{9} + 8496 q^{10} + 7384 q^{11} + 49720 q^{12} + 20820 q^{13} + 50976 q^{14} + 43516 q^{15} + 122082 q^{16} - 11620 q^{17} + 66060 q^{18} + 75068 q^{19} - 42914 q^{20} + 51480 q^{21} - 36950 q^{22} + 62040 q^{23} - 205942 q^{24} + 261022 q^{25} - 201528 q^{26} - 28060 q^{27} - 24980 q^{28} - 243890 q^{29} - 1284894 q^{30} + 200600 q^{31} - 1761460 q^{32} - 1068000 q^{33} - 503932 q^{34} + 107528 q^{35} - 26300 q^{36} - 367740 q^{37} + 766880 q^{38} + 392692 q^{39} - 865000 q^{40} + 932764 q^{41} - 2058060 q^{42} + 1443560 q^{43} - 1325912 q^{44} + 4245684 q^{45} + 1760460 q^{46} - 286960 q^{47} + 3187120 q^{48} + 4713194 q^{49} - 3682652 q^{50} + 1451016 q^{51} + 2560210 q^{52} + 3953220 q^{53} - 3147534 q^{54} + 3981316 q^{55} + 2082464 q^{56} + 2050640 q^{57} + 6712320 q^{59} + 7476756 q^{60} + 1905196 q^{61} - 8048490 q^{62} + 3643800 q^{63} + 8445458 q^{64} + 4667544 q^{65} - 12425580 q^{66} - 2718200 q^{67} - 17699740 q^{68} + 1109064 q^{69} - 30441624 q^{70} + 3447736 q^{71} - 22466840 q^{72} - 2554460 q^{73} - 4214584 q^{74} + 1088084 q^{75} - 8294848 q^{76} - 3967800 q^{77} - 24809970 q^{78} + 4187744 q^{79} - 17715290 q^{80} + 5161402 q^{81} + 7020500 q^{82} + 3498720 q^{83} + 22947224 q^{84} + 1817072 q^{85} - 361638 q^{86} - 1951120 q^{87} + 15118470 q^{88} - 303268 q^{89} - 28959160 q^{90} + 27215080 q^{91} - 10783380 q^{92} + 1097360 q^{93} + 55641726 q^{94} - 8810536 q^{95} - 53327238 q^{96} + 4908620 q^{97} + 40120080 q^{98} - 14408716 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 1101 x^{8} - 1540 x^{7} + 405148 x^{6} + 870160 x^{5} - 54569376 x^{4} - 87078400 x^{3} + 2140673280 x^{2} - 1918315520 x - 9372051456 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14005209807 \nu^{9} + 59170210660 \nu^{8} + 14265030073043 \nu^{7} - 19758503934520 \nu^{6} + \cdots - 10\!\cdots\!60 ) / 25\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14005209807 \nu^{9} - 59170210660 \nu^{8} - 14265030073043 \nu^{7} + 19758503934520 \nu^{6} + \cdots - 46\!\cdots\!40 ) / 25\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10384500596 \nu^{9} + 30939298131 \nu^{8} + 10038072254884 \nu^{7} - 2197921748399 \nu^{6} + \cdots - 53\!\cdots\!92 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 29512730997 \nu^{9} + 180010810744 \nu^{8} + 27002918567993 \nu^{7} - 47739245380996 \nu^{6} + \cdots - 60\!\cdots\!48 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11916632683 \nu^{9} - 174800952576 \nu^{8} + 14992620801047 \nu^{7} + 168334959164684 \nu^{6} + \cdots - 41\!\cdots\!88 ) / 43\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 51681392253 \nu^{9} + 157018466129 \nu^{8} + 49336581999097 \nu^{7} + 7370824996639 \nu^{6} + \cdots - 71\!\cdots\!48 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 151314526971 \nu^{9} + 647264453194 \nu^{8} - 164885198551999 \nu^{7} - 935728438657246 \nu^{6} + \cdots + 36\!\cdots\!92 ) / 25\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2454351473 \nu^{9} - 5753235209 \nu^{8} + 2590223575437 \nu^{7} + 11490332531601 \nu^{6} - 890885912097160 \nu^{5} + \cdots - 53\!\cdots\!12 ) / 33\!\cdots\!20 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 2\beta _1 + 220 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{9} + 4\beta_{8} - \beta_{7} - \beta_{6} - 3\beta_{5} + 5\beta_{3} + 5\beta_{2} + 366\beta _1 + 460 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{9} + 22 \beta_{8} + 14 \beta_{7} + 20 \beta_{6} + 14 \beta_{5} - 54 \beta_{4} + 437 \beta_{3} + 463 \beta_{2} + 1876 \beta _1 + 80282 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2889 \beta_{9} + 2588 \beta_{8} - 569 \beta_{7} - 289 \beta_{6} - 1419 \beta_{5} - 12 \beta_{4} + 3381 \beta_{3} + 3525 \beta_{2} + 149430 \beta _1 + 411260 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6974 \beta_{9} + 16526 \beta_{8} + 4002 \beta_{7} + 10272 \beta_{6} + 9226 \beta_{5} - 33054 \beta_{4} + 190377 \beta_{3} + 201755 \beta_{2} + 1189860 \beta _1 + 32692322 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1360817 \beta_{9} + 1284308 \beta_{8} - 308305 \beta_{7} - 60289 \beta_{6} - 559619 \beta_{5} - 63540 \beta_{4} + 1913513 \beta_{3} + 2224097 \beta_{2} + 64077830 \beta _1 + 259537956 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 6265666 \beta_{9} + 9944574 \beta_{8} - 491682 \beta_{7} + 4193788 \beta_{6} + 4829694 \beta_{5} - 16077006 \beta_{4} + 83942829 \beta_{3} + 91444191 \beta_{2} + \cdots + 13994765650 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 609645753 \beta_{9} + 591000076 \beta_{8} - 170764441 \beta_{7} - 5077777 \beta_{6} - 199397579 \beta_{5} - 64607340 \beta_{4} + 1013643517 \beta_{3} + \cdots + 145602030764 \) Copy content Toggle raw display

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
22.0686
21.4083
13.0110
4.34965
3.85204
−1.69492
−9.91427
−14.7228
−19.0438
−19.3138
−22.0686 73.8979 359.023 376.792 −1630.82 647.379 −5098.36 3273.90 −8315.26
1.2 −21.4083 −17.3673 330.317 −555.983 371.805 −1113.12 −4331.27 −1885.38 11902.7
1.3 −13.0110 25.6161 41.2859 124.561 −333.291 −962.222 1128.24 −1530.82 −1620.67
1.4 −4.34965 −40.5558 −109.081 −341.807 176.403 956.613 1031.22 −542.229 1486.74
1.5 −3.85204 −70.5970 −113.162 −69.8627 271.943 −1359.73 928.965 2796.94 269.114
1.6 1.69492 56.8671 −125.127 194.958 96.3850 1221.62 −429.029 1046.87 330.438
1.7 9.91427 −83.3983 −29.7073 326.241 −826.834 1247.39 −1563.55 4768.28 3234.44
1.8 14.7228 83.8824 88.7619 246.177 1234.99 −1219.72 −577.696 4849.26 3624.43
1.9 19.0438 0.836957 234.668 287.010 15.9389 −124.374 2031.37 −2186.30 5465.77
1.10 19.3138 50.8179 245.021 −408.086 981.485 1746.17 2260.12 395.462 −7881.68
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(29\) \(1\)

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 29.8.a.b 10
3.b odd 2 1 261.8.a.f 10
4.b odd 2 1 464.8.a.g 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.8.a.b 10 1.a even 1 1 trivial
261.8.a.f 10 3.b odd 2 1
464.8.a.g 10 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 1101 T_{2}^{8} + 1540 T_{2}^{7} + 405148 T_{2}^{6} - 870160 T_{2}^{5} - 54569376 T_{2}^{4} + 87078400 T_{2}^{3} + 2140673280 T_{2}^{2} + 1918315520 T_{2} - 9372051456 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(29))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 1101 T^{8} + \cdots - 9372051456 \) Copy content Toggle raw display
$3$ \( T^{10} - 80 T^{9} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$5$ \( T^{10} - 180 T^{9} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} - 1040 T^{9} + \cdots - 36\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{10} - 7384 T^{9} + \cdots - 55\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{10} - 20820 T^{9} + \cdots - 23\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{10} + 11620 T^{9} + \cdots - 19\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{10} - 75068 T^{9} + \cdots - 67\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{10} - 62040 T^{9} + \cdots - 91\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T + 24389)^{10} \) Copy content Toggle raw display
$31$ \( T^{10} - 200600 T^{9} + \cdots + 53\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{10} + 367740 T^{9} + \cdots + 44\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{10} - 932764 T^{9} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{10} - 1443560 T^{9} + \cdots - 15\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{10} + 286960 T^{9} + \cdots - 51\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{10} - 3953220 T^{9} + \cdots - 81\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{10} - 6712320 T^{9} + \cdots - 22\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} - 1905196 T^{9} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{10} + 2718200 T^{9} + \cdots - 70\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{10} - 3447736 T^{9} + \cdots - 20\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{10} + 2554460 T^{9} + \cdots - 10\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{10} - 4187744 T^{9} + \cdots - 93\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{10} - 3498720 T^{9} + \cdots - 42\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{10} + 303268 T^{9} + \cdots + 71\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{10} - 4908620 T^{9} + \cdots + 33\!\cdots\!76 \) Copy content Toggle raw display
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