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Properties

Label 29.5.c.a
Level $29$
Weight $5$
Character orbit 29.c
Analytic conductor $2.998$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [29,5,Mod(12,29)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(29, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("29.12");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 29.c (of order \(4\), degree \(2\), minimal)

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: no
Analytic conductor: \(2.99772892943\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 8 x^{17} + 32 x^{16} + 54 x^{15} + 2934 x^{14} - 22338 x^{13} + 86274 x^{12} + \cdots + 1032033312 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{13} + 6 \beta_{4} + \cdots + \beta_1) q^{4}+ \cdots + ( - \beta_{16} - \beta_{15} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{13} + 6 \beta_{4} + \cdots + \beta_1) q^{4}+ \cdots + (12 \beta_{17} + 12 \beta_{16} + \cdots - 378) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 8 q^{2} - 2 q^{3} - 4 q^{7} + 162 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 8 q^{2} - 2 q^{3} - 4 q^{7} + 162 q^{8} + 30 q^{10} - 62 q^{11} + 222 q^{12} - 712 q^{14} - 310 q^{15} - 152 q^{16} - 626 q^{17} + 1058 q^{18} + 84 q^{19} + 2172 q^{20} - 828 q^{21} + 248 q^{23} - 2760 q^{24} - 3414 q^{25} - 450 q^{26} + 2746 q^{27} - 1904 q^{29} + 10892 q^{30} - 2682 q^{31} + 1894 q^{32} - 5640 q^{36} + 3314 q^{37} - 1782 q^{39} - 4726 q^{40} - 1586 q^{41} - 2022 q^{43} + 4082 q^{44} + 8260 q^{45} + 676 q^{46} + 4594 q^{47} + 3018 q^{48} - 4002 q^{49} + 2482 q^{50} - 14184 q^{52} - 10012 q^{53} - 26696 q^{54} - 2374 q^{55} + 21120 q^{56} - 6168 q^{58} + 11576 q^{59} - 14226 q^{60} + 15846 q^{61} + 15228 q^{65} + 40794 q^{66} + 12020 q^{68} - 8424 q^{69} - 26204 q^{70} + 48528 q^{72} - 10190 q^{73} - 34348 q^{74} + 4440 q^{75} - 30072 q^{76} - 18724 q^{77} - 31632 q^{78} + 25770 q^{79} + 20378 q^{81} - 21296 q^{82} - 37480 q^{83} + 40148 q^{84} + 29056 q^{85} - 31766 q^{87} + 10516 q^{88} + 2698 q^{89} - 17248 q^{90} + 21812 q^{94} + 50820 q^{95} + 51946 q^{97} - 33672 q^{98} + 4284 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 8 x^{17} + 32 x^{16} + 54 x^{15} + 2934 x^{14} - 22338 x^{13} + 86274 x^{12} + \cdots + 1032033312 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 53\!\cdots\!51 \nu^{17} + \cdots + 92\!\cdots\!80 ) / 24\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 78\!\cdots\!11 \nu^{17} + \cdots - 54\!\cdots\!32 ) / 24\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 56\!\cdots\!15 \nu^{17} + \cdots - 72\!\cdots\!76 ) / 15\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 39\!\cdots\!63 \nu^{17} + \cdots - 56\!\cdots\!44 ) / 44\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 52\!\cdots\!56 \nu^{17} + \cdots + 73\!\cdots\!88 ) / 44\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11\!\cdots\!43 \nu^{17} + \cdots - 50\!\cdots\!92 ) / 44\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 89\!\cdots\!80 \nu^{17} + \cdots + 22\!\cdots\!44 ) / 21\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 79\!\cdots\!23 \nu^{17} + \cdots + 11\!\cdots\!20 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13\!\cdots\!53 \nu^{17} + \cdots + 11\!\cdots\!44 ) / 21\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 30\!\cdots\!44 \nu^{17} + \cdots + 43\!\cdots\!60 ) / 44\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!44 \nu^{17} + \cdots - 14\!\cdots\!96 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12\!\cdots\!11 \nu^{17} + \cdots + 16\!\cdots\!52 ) / 15\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 17\!\cdots\!10 \nu^{17} + \cdots + 37\!\cdots\!48 ) / 21\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 13\!\cdots\!35 \nu^{17} + \cdots - 20\!\cdots\!72 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 31\!\cdots\!27 \nu^{17} + \cdots - 42\!\cdots\!12 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 58\!\cdots\!27 \nu^{17} + \cdots - 11\!\cdots\!24 ) / 13\!\cdots\!08 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + 22\beta_{4} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{7} - \beta_{5} + 6\beta_{4} - \beta_{3} - 37\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{14} - \beta_{11} - 3\beta_{10} - 2\beta_{8} + \beta_{7} - 45\beta_{3} - 58\beta_{2} - 58\beta _1 - 800 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 5 \beta_{16} + 6 \beta_{15} + 6 \beta_{14} - 70 \beta_{13} + \beta_{12} - 53 \beta_{11} - 5 \beta_{10} + \cdots - 525 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{17} - 178 \beta_{16} + 148 \beta_{15} - 1971 \beta_{13} - \beta_{12} - 86 \beta_{11} + \cdots - 3120 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 46 \beta_{17} - 383 \beta_{16} + 570 \beta_{15} - 570 \beta_{14} - 4007 \beta_{13} + 383 \beta_{10} + \cdots + 37128 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 187 \beta_{17} - 8826 \beta_{14} + 187 \beta_{12} + 5343 \beta_{11} + 8887 \beta_{10} + 8090 \beta_{8} + \cdots + 1380266 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 22810 \beta_{16} - 38556 \beta_{15} - 38556 \beta_{14} + 213578 \beta_{13} - 797 \beta_{12} + \cdots + 2294733 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 18002 \beta_{17} + 427010 \beta_{16} - 493364 \beta_{15} + 3848135 \beta_{13} + 18002 \beta_{12} + \cdots + 8399939 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 58114 \beta_{17} + 1252372 \beta_{16} - 2308416 \beta_{15} + 2308416 \beta_{14} + 10980609 \beta_{13} + \cdots - 130671540 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1335836 \beta_{17} + 26824906 \beta_{14} - 1335836 \beta_{12} - 15793769 \beta_{11} + \cdots - 2684541860 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 66344769 \beta_{16} + 130448622 \beta_{15} + 130448622 \beta_{14} - 552504654 \beta_{13} + \cdots - 7073702985 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 86966637 \beta_{17} - 971602710 \beta_{16} + 1436009100 \beta_{15} - 7830110623 \beta_{13} + \cdots - 21167803048 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 637462182 \beta_{17} - 3447226875 \beta_{16} + 7135585794 \beta_{15} - 7135585794 \beta_{14} + \cdots + 370742497908 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 5252880903 \beta_{17} - 76048276490 \beta_{14} + 5252880903 \beta_{12} + 41447481691 \beta_{11} + \cdots + 5573937093074 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 176948994122 \beta_{16} - 382318829052 \beta_{15} - 382318829052 \beta_{14} + 1348156443730 \beta_{13} + \cdots + 19022467842237 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1
4.98352 + 4.98352i
4.72166 + 4.72166i
3.13199 + 3.13199i
2.30923 + 2.30923i
0.274526 + 0.274526i
−0.822421 − 0.822421i
−1.99862 − 1.99862i
−4.25890 − 4.25890i
−4.34098 − 4.34098i
4.98352 − 4.98352i
4.72166 − 4.72166i
3.13199 − 3.13199i
2.30923 − 2.30923i
0.274526 − 0.274526i
−0.822421 + 0.822421i
−1.99862 + 1.99862i
−4.25890 + 4.25890i
−4.34098 + 4.34098i
−4.98352 − 4.98352i −11.0004 − 11.0004i 33.6709i − 34.7532i 109.642i 45.3584 88.0634 − 88.0634i 161.019i −173.193 + 173.193i
12.2 −4.72166 − 4.72166i 6.00088 + 6.00088i 28.5881i 23.0126i − 56.6683i 55.1654 59.4369 − 59.4369i − 8.97877i 108.658 − 108.658i
12.3 −3.13199 − 3.13199i 2.96941 + 2.96941i 3.61872i − 27.2683i − 18.6003i −70.9422 −38.7780 + 38.7780i − 63.3653i −85.4042 + 85.4042i
12.4 −2.30923 − 2.30923i −6.71929 − 6.71929i − 5.33492i 39.5684i 31.0328i −37.5146 −49.2672 + 49.2672i 9.29780i 91.3726 − 91.3726i
12.5 −0.274526 − 0.274526i 0.717348 + 0.717348i − 15.8493i − 16.6184i − 0.393862i 59.7719 −8.74346 + 8.74346i − 79.9708i −4.56218 + 4.56218i
12.6 0.822421 + 0.822421i 9.75879 + 9.75879i − 14.6472i 22.1059i 16.0517i −5.60830 25.2049 − 25.2049i 109.468i −18.1804 + 18.1804i
12.7 1.99862 + 1.99862i −8.92850 − 8.92850i − 8.01101i − 13.1079i − 35.6894i −20.2472 47.9890 − 47.9890i 78.4361i 26.1978 − 26.1978i
12.8 4.25890 + 4.25890i 7.42039 + 7.42039i 20.2765i − 41.6756i 63.2054i −56.7798 −18.2133 + 18.2133i 29.1243i 177.493 − 177.493i
12.9 4.34098 + 4.34098i −1.21859 − 1.21859i 21.6881i 24.7365i − 10.5797i 28.7964 −24.6921 + 24.6921i − 78.0301i −107.380 + 107.380i
17.1 −4.98352 + 4.98352i −11.0004 + 11.0004i − 33.6709i 34.7532i − 109.642i 45.3584 88.0634 + 88.0634i − 161.019i −173.193 − 173.193i
17.2 −4.72166 + 4.72166i 6.00088 − 6.00088i − 28.5881i − 23.0126i 56.6683i 55.1654 59.4369 + 59.4369i 8.97877i 108.658 + 108.658i
17.3 −3.13199 + 3.13199i 2.96941 − 2.96941i − 3.61872i 27.2683i 18.6003i −70.9422 −38.7780 − 38.7780i 63.3653i −85.4042 − 85.4042i
17.4 −2.30923 + 2.30923i −6.71929 + 6.71929i 5.33492i − 39.5684i − 31.0328i −37.5146 −49.2672 − 49.2672i − 9.29780i 91.3726 + 91.3726i
17.5 −0.274526 + 0.274526i 0.717348 − 0.717348i 15.8493i 16.6184i 0.393862i 59.7719 −8.74346 − 8.74346i 79.9708i −4.56218 − 4.56218i
17.6 0.822421 − 0.822421i 9.75879 − 9.75879i 14.6472i − 22.1059i − 16.0517i −5.60830 25.2049 + 25.2049i − 109.468i −18.1804 − 18.1804i
17.7 1.99862 − 1.99862i −8.92850 + 8.92850i 8.01101i 13.1079i 35.6894i −20.2472 47.9890 + 47.9890i − 78.4361i 26.1978 + 26.1978i
17.8 4.25890 − 4.25890i 7.42039 − 7.42039i − 20.2765i 41.6756i − 63.2054i −56.7798 −18.2133 − 18.2133i − 29.1243i 177.493 + 177.493i
17.9 4.34098 − 4.34098i −1.21859 + 1.21859i − 21.6881i − 24.7365i 10.5797i 28.7964 −24.6921 − 24.6921i 78.0301i −107.380 − 107.380i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.5.c.a ✓ 18
3.b odd 2 1 261.5.f.a 18
29.c odd 4 1 inner 29.5.c.a ✓ 18
87.f even 4 1 261.5.f.a 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.5.c.a ✓ 18 1.a even 1 1 trivial
29.5.c.a ✓ 18 29.c odd 4 1 inner
261.5.f.a 18 3.b odd 2 1
261.5.f.a 18 87.f even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(29, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots + 1032033312 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 283715351603208 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$7$ \( (T^{9} + \cdots + 73901681366464)^{2} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 32\!\cdots\!92 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 12\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 10\!\cdots\!52 \) Copy content Toggle raw display
$23$ \( (T^{9} + \cdots - 37\!\cdots\!56)^{2} \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 44\!\cdots\!21 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 77\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 24\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 91\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 10\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 67\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( (T^{9} + \cdots - 79\!\cdots\!72)^{2} \) Copy content Toggle raw display
$59$ \( (T^{9} + \cdots + 12\!\cdots\!84)^{2} \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 67\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 11\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 51\!\cdots\!48 \) Copy content Toggle raw display
$83$ \( (T^{9} + \cdots - 29\!\cdots\!76)^{2} \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 13\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 48\!\cdots\!08 \) Copy content Toggle raw display
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