Properties

Label 29.10.a.a
Level $29$
Weight $10$
Character orbit 29.a
Self dual yes
Analytic conductor $14.936$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [29,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("29.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(14.9360392488\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2901 x^{7} - 4592 x^{6} + 2830996 x^{5} + 7409504 x^{4} - 1038861888 x^{3} - 2974719488 x^{2} + 115773751296 x + 456378417152 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 7 \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{4} - 27) q^{3} + (\beta_{2} + 2 \beta_1 + 133) q^{4} + (\beta_{6} + 4 \beta_{4} - \beta_{2} - 5 \beta_1 - 83) q^{5} + ( - 3 \beta_{6} - \beta_{5} + 11 \beta_{4} - 5 \beta_{2} + 51 \beta_1 - 261) q^{6} + (\beta_{7} - 2 \beta_{6} + 4 \beta_{5} + 16 \beta_{4} - 8 \beta_{2} + 81 \beta_1 - 796) q^{7} + (\beta_{8} - 5 \beta_{7} + \beta_{6} + \beta_{5} + 27 \beta_{4} - \beta_{3} - 5 \beta_{2} + \cdots - 1538) q^{8}+ \cdots + ( - 7 \beta_{8} + \beta_{7} + 4 \beta_{6} - 10 \beta_{5} + 44 \beta_{4} + \cdots + 3670) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{4} - 27) q^{3} + (\beta_{2} + 2 \beta_1 + 133) q^{4} + (\beta_{6} + 4 \beta_{4} - \beta_{2} - 5 \beta_1 - 83) q^{5} + ( - 3 \beta_{6} - \beta_{5} + 11 \beta_{4} - 5 \beta_{2} + 51 \beta_1 - 261) q^{6} + (\beta_{7} - 2 \beta_{6} + 4 \beta_{5} + 16 \beta_{4} - 8 \beta_{2} + 81 \beta_1 - 796) q^{7} + (\beta_{8} - 5 \beta_{7} + \beta_{6} + \beta_{5} + 27 \beta_{4} - \beta_{3} - 5 \beta_{2} + \cdots - 1538) q^{8}+ \cdots + ( - 27859 \beta_{8} + 388654 \beta_{7} - 456137 \beta_{6} + \cdots + 34295796) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 244 q^{3} + 1194 q^{4} - 738 q^{5} - 2330 q^{6} - 7128 q^{7} - 13776 q^{8} + 33017 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 244 q^{3} + 1194 q^{4} - 738 q^{5} - 2330 q^{6} - 7128 q^{7} - 13776 q^{8} + 33017 q^{9} + 37812 q^{10} - 59512 q^{11} - 127348 q^{12} - 165758 q^{13} - 406080 q^{14} - 693178 q^{15} - 1044958 q^{16} - 394814 q^{17} - 1676576 q^{18} - 2256606 q^{19} - 2237578 q^{20} - 1750168 q^{21} - 5311718 q^{22} - 1699500 q^{23} - 4446318 q^{24} - 983481 q^{25} - 4264740 q^{26} - 6987958 q^{27} - 8491636 q^{28} - 6365529 q^{29} - 16907854 q^{30} - 11929632 q^{31} - 1346192 q^{32} + 1750252 q^{33} + 8655764 q^{34} - 3275324 q^{35} + 29848532 q^{36} + 14454898 q^{37} + 14709736 q^{38} + 41155042 q^{39} + 45167060 q^{40} + 52495202 q^{41} + 103102340 q^{42} + 21819888 q^{43} + 70837004 q^{44} + 61248326 q^{45} + 20628012 q^{46} + 44968948 q^{47} + 122982540 q^{48} - 26826775 q^{49} + 155997680 q^{50} - 28882428 q^{51} + 29562122 q^{52} - 111394302 q^{53} + 70575802 q^{54} - 173560742 q^{55} + 67419136 q^{56} + 85769252 q^{57} - 236142720 q^{59} - 47991000 q^{60} - 241129054 q^{61} + 261343278 q^{62} - 328513060 q^{63} - 333112958 q^{64} - 625660884 q^{65} + 223958776 q^{66} - 672046492 q^{67} - 63179948 q^{68} - 705827600 q^{69} - 366389016 q^{70} - 475841956 q^{71} - 18937608 q^{72} - 424813822 q^{73} - 532689728 q^{74} - 913708498 q^{75} - 552478056 q^{76} - 182224776 q^{77} + 928127886 q^{78} - 170801148 q^{79} + 562655678 q^{80} - 914585851 q^{81} + 1468192652 q^{82} - 468898296 q^{83} + 952386216 q^{84} - 271552972 q^{85} + 1462277802 q^{86} + 172576564 q^{87} + 1176890862 q^{88} - 676036598 q^{89} + 4017858752 q^{90} + 9763884 q^{91} + 2724990708 q^{92} - 858755220 q^{93} + 2429128614 q^{94} + 69331732 q^{95} + 3111862050 q^{96} + 170708754 q^{97} + 3278517600 q^{98} + 305494078 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2901 x^{7} - 4592 x^{6} + 2830996 x^{5} + 7409504 x^{4} - 1038861888 x^{3} - 2974719488 x^{2} + 115773751296 x + 456378417152 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 645 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 24206118505 \nu^{8} + 3628782688552 \nu^{7} - 51380960778211 \nu^{6} + \cdots - 31\!\cdots\!12 ) / 62\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 30645811893 \nu^{8} + 46114646956 \nu^{7} + 72238283274393 \nu^{6} + 39514496384020 \nu^{5} + \cdots - 56\!\cdots\!84 ) / 31\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 145943679107 \nu^{8} + 3948353615584 \nu^{7} + 511126104077119 \nu^{6} + \cdots - 23\!\cdots\!72 ) / 62\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 206831158817 \nu^{8} + 10132200890616 \nu^{7} + 426846090538933 \nu^{6} + \cdots + 59\!\cdots\!96 ) / 62\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 40920390157 \nu^{8} + 422294149298 \nu^{7} + 97500534227889 \nu^{6} - 819303572581354 \nu^{5} + \cdots - 10\!\cdots\!40 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 346626955361 \nu^{8} + 3949803218648 \nu^{7} - 990199083095925 \nu^{6} + \cdots + 15\!\cdots\!12 ) / 62\!\cdots\!24 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 645 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + 5\beta_{7} - \beta_{6} - \beta_{5} - 27\beta_{4} + \beta_{3} + 5\beta_{2} + 937\beta _1 + 1538 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 20 \beta_{8} + 52 \beta_{7} + 54 \beta_{6} - 66 \beta_{5} - 378 \beta_{4} - 96 \beta_{3} + 1353 \beta_{2} + 4364 \beta _1 + 612497 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2589 \beta_{8} + 6537 \beta_{7} - 1437 \beta_{6} - 1373 \beta_{5} - 42751 \beta_{4} + 3325 \beta_{3} + 10801 \beta_{2} + 987777 \beta _1 + 3296234 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 52276 \beta_{8} + 84564 \beta_{7} + 95262 \beta_{6} - 94106 \beta_{5} - 779282 \beta_{4} - 165616 \beta_{3} + 1633593 \beta_{2} + 7544132 \beta _1 + 650555745 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 4158157 \beta_{8} + 7523417 \beta_{7} - 1124589 \beta_{6} - 1517101 \beta_{5} - 58467439 \beta_{4} + 5509229 \beta_{3} + 17313905 \beta_{2} + 1093110241 \beta _1 + 5557992858 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 93877892 \beta_{8} + 106963044 \beta_{7} + 132621294 \beta_{6} - 107787978 \beta_{5} - 1308606562 \beta_{4} - 217160544 \beta_{3} + 1939154697 \beta_{2} + \cdots + 724198092865 \) 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
35.6552
33.5115
19.7716
15.4003
−4.12402
−12.4485
−24.3559
−30.2923
−33.1179
−35.6552 190.523 759.292 −886.258 −6793.13 −3878.27 −8817.24 16616.0 31599.7
1.2 −33.5115 −215.845 611.021 −1510.04 7233.29 2235.57 −3318.35 26906.1 50603.9
1.3 −19.7716 15.1371 −121.083 339.780 −299.284 3392.81 12517.1 −19453.9 −6718.00
1.4 −15.4003 −207.293 −274.831 1546.00 3192.38 3958.22 12117.4 23287.4 −23808.9
1.5 4.12402 158.305 −494.992 522.723 652.853 −9369.41 −4152.85 5377.54 2155.72
1.6 12.4485 −4.54310 −357.034 208.913 −56.5549 10778.0 −10818.2 −19662.4 2600.65
1.7 24.3559 100.461 81.2084 −2213.18 2446.80 −1575.25 −10492.3 −9590.68 −53903.9
1.8 30.2923 −209.271 405.624 2211.65 −6339.31 −7440.75 −3222.36 24111.4 66995.9
1.9 33.1179 −71.4733 584.794 −957.580 −2367.05 −5228.90 2410.80 −14574.6 −31713.0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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 29.10.a.a 9
3.b odd 2 1 261.10.a.b 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.10.a.a 9 1.a even 1 1 trivial
261.10.a.b 9 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} - 2901 T_{2}^{7} + 4592 T_{2}^{6} + 2830996 T_{2}^{5} - 7409504 T_{2}^{4} - 1038861888 T_{2}^{3} + 2974719488 T_{2}^{2} + 115773751296 T_{2} - 456378417152 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(29))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 2901 T^{7} + \cdots - 456378417152 \) Copy content Toggle raw display
$3$ \( T^{9} + 244 T^{8} + \cdots + 13\!\cdots\!70 \) Copy content Toggle raw display
$5$ \( T^{9} + 738 T^{8} + \cdots - 35\!\cdots\!50 \) Copy content Toggle raw display
$7$ \( T^{9} + 7128 T^{8} + \cdots + 72\!\cdots\!48 \) Copy content Toggle raw display
$11$ \( T^{9} + 59512 T^{8} + \cdots + 40\!\cdots\!78 \) Copy content Toggle raw display
$13$ \( T^{9} + 165758 T^{8} + \cdots - 14\!\cdots\!94 \) Copy content Toggle raw display
$17$ \( T^{9} + 394814 T^{8} + \cdots - 19\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{9} + 2256606 T^{8} + \cdots + 11\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{9} + 1699500 T^{8} + \cdots + 27\!\cdots\!32 \) Copy content Toggle raw display
$29$ \( (T + 707281)^{9} \) Copy content Toggle raw display
$31$ \( T^{9} + 11929632 T^{8} + \cdots - 54\!\cdots\!58 \) Copy content Toggle raw display
$37$ \( T^{9} - 14454898 T^{8} + \cdots + 15\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{9} - 52495202 T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{9} - 21819888 T^{8} + \cdots + 15\!\cdots\!26 \) Copy content Toggle raw display
$47$ \( T^{9} - 44968948 T^{8} + \cdots - 26\!\cdots\!66 \) Copy content Toggle raw display
$53$ \( T^{9} + 111394302 T^{8} + \cdots - 22\!\cdots\!22 \) Copy content Toggle raw display
$59$ \( T^{9} + 236142720 T^{8} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{9} + 241129054 T^{8} + \cdots - 35\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{9} + 672046492 T^{8} + \cdots - 13\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{9} + 475841956 T^{8} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{9} + 424813822 T^{8} + \cdots - 71\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{9} + 170801148 T^{8} + \cdots - 17\!\cdots\!50 \) Copy content Toggle raw display
$83$ \( T^{9} + 468898296 T^{8} + \cdots - 96\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{9} + 676036598 T^{8} + \cdots - 53\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{9} - 170708754 T^{8} + \cdots - 49\!\cdots\!32 \) Copy content Toggle raw display
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