Properties

Label 29.3.c.a
Level $29$
Weight $3$
Character orbit 29.c
Analytic conductor $0.790$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [29,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("29.12");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 3 \)
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: \(0.790192766645\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 18x^{6} + 91x^{4} + 126x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + (\beta_{6} + \beta_{2}) q^{3} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3}) q^{4} + ( - \beta_{4} - \beta_{3}) q^{5} + (\beta_{6} + \beta_{5} - 5 \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1) q^{6} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{2} + \beta_1) q^{7} + ( - 2 \beta_{7} - \beta_{5} + 5 \beta_{4} + 2 \beta_{3} - \beta_1 - 5) q^{8} + (\beta_{6} + \beta_{5} + 6 \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + (\beta_{6} + \beta_{2}) q^{3} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3}) q^{4} + ( - \beta_{4} - \beta_{3}) q^{5} + (\beta_{6} + \beta_{5} - 5 \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1) q^{6} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{2} + \beta_1) q^{7} + ( - 2 \beta_{7} - \beta_{5} + 5 \beta_{4} + 2 \beta_{3} - \beta_1 - 5) q^{8} + (\beta_{6} + \beta_{5} + 6 \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1) q^{9} + (\beta_{7} + 3 \beta_{5} - \beta_{3} + \beta_1) q^{10} + ( - \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} - 4 \beta_{2} - 1) q^{11} + (2 \beta_{7} + 7 \beta_{5} - 5 \beta_{4} - 2 \beta_{3} + 5) q^{12} + ( - \beta_{6} - \beta_{5} + 5 \beta_{4} + 2 \beta_{2} + 2 \beta_1) q^{13} + (\beta_{7} - 5 \beta_{4} + \beta_{3} + \beta_{2} - 5) q^{14} + ( - 5 \beta_{5} - 2 \beta_1) q^{15} + ( - 2 \beta_{7} + 6 \beta_{6} - 6 \beta_{5} + \beta_{2} - \beta_1 - 1) q^{16} + ( - 6 \beta_{6} + \beta_{2}) q^{17} + ( - \beta_{7} - 10 \beta_{5} - 5 \beta_{4} + \beta_{3} - 5 \beta_1 + 5) q^{18} + ( - 4 \beta_{7} + 4 \beta_{6} - \beta_{4} - 4 \beta_{3} - \beta_{2} - 1) q^{19} + (2 \beta_{7} - 5 \beta_{6} + 5 \beta_{5} + 11) q^{20} + (\beta_{7} - 2 \beta_{6} - 5 \beta_{4} + \beta_{3} + 3 \beta_{2} - 5) q^{21} + (2 \beta_{6} + 2 \beta_{5} + 5 \beta_{4} + 3 \beta_{3} + 5 \beta_{2} + 5 \beta_1) q^{22} + (3 \beta_{7} + 2 \beta_{2} - 2 \beta_1) q^{23} + (3 \beta_{7} - 12 \beta_{6} + 12 \beta_{5} - 6 \beta_{2} + 6 \beta_1 + 15) q^{24} + (2 \beta_{6} - 2 \beta_{5} - \beta_{2} + \beta_1 + 14) q^{25} + (\beta_{7} - 7 \beta_{5} + 5 \beta_{4} - \beta_{3} - 4 \beta_1 - 5) q^{26} + ( - 3 \beta_{7} + 11 \beta_{5} + 15 \beta_{4} + 3 \beta_{3} + 2 \beta_1 - 15) q^{27} + ( - \beta_{6} - \beta_{5} - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{28} + (3 \beta_{6} + 7 \beta_{5} + 6 \beta_{4} + 2 \beta_{2} - 5 \beta_1 + 15) q^{29} + ( - 7 \beta_{7} + 5 \beta_{6} - 5 \beta_{5} - 2 \beta_{2} + 2 \beta_1 - 25) q^{30} + (4 \beta_{7} + 9 \beta_{6} + \beta_{4} + 4 \beta_{3} - 7 \beta_{2} + 1) q^{31} + ( - \beta_{7} + 13 \beta_{6} - 10 \beta_{4} - \beta_{3} - 4 \beta_{2} - 10) q^{32} + ( - 12 \beta_{6} - 12 \beta_{5} - 45 \beta_{4} - 6 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{33} + ( - 6 \beta_{6} - 6 \beta_{5} + 30 \beta_{4} + 5 \beta_{3} - \beta_{2} - \beta_1) q^{34} + (\beta_{6} + \beta_{5} - 10 \beta_{4} + 5 \beta_{3} - \beta_{2} - \beta_1) q^{35} + ( - 5 \beta_{7} + 11 \beta_{6} - 11 \beta_{5} - 26) q^{36} + (8 \beta_{7} - 2 \beta_{5} + 10 \beta_{4} - 8 \beta_{3} - 6 \beta_1 - 10) q^{37} + (13 \beta_{6} + 13 \beta_{5} - 20 \beta_{4} - 11 \beta_{3} + 5 \beta_{2} + 5 \beta_1) q^{38} + (15 \beta_{5} + 15 \beta_{4} + 3 \beta_1 - 15) q^{39} + (3 \beta_{7} - 13 \beta_{6} + 25 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 25) q^{40} + ( - 5 \beta_{7} + 6 \beta_{5} - \beta_{4} + 5 \beta_{3} + 7 \beta_1 + 1) q^{41} + (\beta_{6} + \beta_{5} + 10 \beta_{4} + \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{42} + (\beta_{7} - 5 \beta_{6} - 25 \beta_{4} + \beta_{3} + 6 \beta_{2} - 25) q^{43} + ( - 3 \beta_{5} - 6 \beta_{4} + 3 \beta_1 + 6) q^{44} + (3 \beta_{7} - \beta_{6} + \beta_{5} + 5 \beta_{2} - 5 \beta_1 + 36) q^{45} + (\beta_{7} - 6 \beta_{6} + \beta_{3} - 7 \beta_{2}) q^{46} + ( - 2 \beta_{7} - 11 \beta_{5} - 10 \beta_{4} + 2 \beta_{3} + 6 \beta_1 + 10) q^{47} + (13 \beta_{7} - 17 \beta_{6} + 40 \beta_{4} + 13 \beta_{3} + 9 \beta_{2} + \cdots + 40) q^{48}+ \cdots + (18 \beta_{7} - 48 \beta_{5} - 81 \beta_{4} - 18 \beta_{3} - 39 \beta_1 + 81) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 2 q^{3} - 4 q^{7} - 42 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 2 q^{3} - 4 q^{7} - 42 q^{8} + 6 q^{10} - 6 q^{11} + 54 q^{12} - 40 q^{14} - 10 q^{15} - 32 q^{16} + 12 q^{17} + 20 q^{18} - 16 q^{19} + 108 q^{20} - 36 q^{21} + 168 q^{24} + 104 q^{25} - 54 q^{26} - 98 q^{27} + 128 q^{29} - 220 q^{30} - 10 q^{31} - 106 q^{32} - 252 q^{36} - 84 q^{37} - 90 q^{39} + 226 q^{40} + 20 q^{41} - 190 q^{43} + 42 q^{44} + 292 q^{45} + 12 q^{46} + 58 q^{47} + 354 q^{48} - 72 q^{49} - 60 q^{50} - 144 q^{52} + 252 q^{53} + 400 q^{54} - 74 q^{55} - 192 q^{56} + 326 q^{58} - 40 q^{59} - 258 q^{60} - 208 q^{61} + 36 q^{65} - 414 q^{66} - 296 q^{68} + 120 q^{69} + 44 q^{70} - 636 q^{72} - 188 q^{73} - 64 q^{74} - 12 q^{75} + 592 q^{76} + 180 q^{77} + 600 q^{78} - 382 q^{79} - 124 q^{81} + 228 q^{82} + 280 q^{83} - 124 q^{84} + 32 q^{85} + 34 q^{87} + 20 q^{88} - 64 q^{89} + 128 q^{90} - 460 q^{94} - 380 q^{95} - 44 q^{97} - 66 q^{98} + 552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 18x^{6} + 91x^{4} + 126x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} + 9\nu^{2} + 6\nu + 5 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{4} - 9\nu^{2} + 6\nu - 5 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} - 15\nu^{3} - 47\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 18\nu^{5} - 86\nu^{3} - 81\nu ) / 30 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 17\nu^{5} + \nu^{4} + 77\nu^{3} + 15\nu^{2} + 82\nu + 35 ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 17\nu^{5} - \nu^{4} + 77\nu^{3} - 15\nu^{2} + 82\nu - 35 ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 16\nu^{4} + 62\nu^{2} + 35 ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{6} + 2\beta_{5} + \beta_{2} - \beta _1 - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + 5\beta_{4} - \beta_{3} - 4\beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 18\beta_{6} - 18\beta_{5} - 15\beta_{2} + 15\beta _1 + 80 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -30\beta_{6} - 30\beta_{5} - 150\beta_{4} + 18\beta_{3} + 73\beta_{2} + 73\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6\beta_{7} - 82\beta_{6} + 82\beta_{5} + 89\beta_{2} - 89\beta _1 - 365 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 368\beta_{6} + 368\beta_{5} + 1780\beta_{4} - 152\beta_{3} - 707\beta_{2} - 707\beta_1 ) / 2 \) Copy content Toggle raw display

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
2.35663i
3.22189i
1.35225i
0.486981i
2.35663i
3.22189i
1.35225i
0.486981i
−1.45515 1.45515i 3.81178 + 3.81178i 0.234947i 3.14526i 11.0935i 0.342313 −5.47873 + 5.47873i 20.0593i −4.57683 + 4.57683i
12.2 −1.07935 1.07935i −2.14254 2.14254i 1.67001i 0.488689i 4.62511i 8.09117 −6.11992 + 6.11992i 0.180982i −0.527467 + 0.527467i
12.3 0.909588 + 0.909588i 0.442660 + 0.442660i 2.34530i 4.16447i 0.805276i −9.68815 5.77161 5.77161i 8.60810i −3.78796 + 3.78796i
12.4 2.62492 + 2.62492i −3.11190 3.11190i 9.78036i 4.53053i 16.3369i −0.745339 −15.1730 + 15.1730i 10.3678i 11.8923 11.8923i
17.1 −1.45515 + 1.45515i 3.81178 3.81178i 0.234947i 3.14526i 11.0935i 0.342313 −5.47873 5.47873i 20.0593i −4.57683 4.57683i
17.2 −1.07935 + 1.07935i −2.14254 + 2.14254i 1.67001i 0.488689i 4.62511i 8.09117 −6.11992 6.11992i 0.180982i −0.527467 0.527467i
17.3 0.909588 0.909588i 0.442660 0.442660i 2.34530i 4.16447i 0.805276i −9.68815 5.77161 + 5.77161i 8.60810i −3.78796 3.78796i
17.4 2.62492 2.62492i −3.11190 + 3.11190i 9.78036i 4.53053i 16.3369i −0.745339 −15.1730 15.1730i 10.3678i 11.8923 + 11.8923i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.4
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 29.3.c.a 8
3.b odd 2 1 261.3.f.a 8
4.b odd 2 1 464.3.l.c 8
29.c odd 4 1 inner 29.3.c.a 8
87.f even 4 1 261.3.f.a 8
116.e even 4 1 464.3.l.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.3.c.a 8 1.a even 1 1 trivial
29.3.c.a 8 29.c odd 4 1 inner
261.3.f.a 8 3.b odd 2 1
261.3.f.a 8 87.f even 4 1
464.3.l.c 8 4.b odd 2 1
464.3.l.c 8 116.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(29, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + 2 T^{6} + 18 T^{5} + \cdots + 225 \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + 2 T^{6} + 22 T^{5} + \cdots + 2025 \) Copy content Toggle raw display
$5$ \( T^{8} + 48 T^{6} + 742 T^{4} + \cdots + 841 \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} - 78 T^{2} - 32 T + 20)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 6 T^{7} + 18 T^{6} - 1710 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{8} + 452 T^{6} + \cdots + 137945025 \) Copy content Toggle raw display
$17$ \( T^{8} - 12 T^{7} + \cdots + 660490000 \) Copy content Toggle raw display
$19$ \( T^{8} + 16 T^{7} + \cdots + 5996334096 \) Copy content Toggle raw display
$23$ \( (T^{4} - 318 T^{2} - 204 T + 19940)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 128 T^{7} + \cdots + 500246412961 \) Copy content Toggle raw display
$31$ \( T^{8} + 10 T^{7} + \cdots + 21518249481 \) Copy content Toggle raw display
$37$ \( T^{8} + 84 T^{7} + \cdots + 1885238841600 \) Copy content Toggle raw display
$41$ \( T^{8} - 20 T^{7} + \cdots + 24625141776 \) Copy content Toggle raw display
$43$ \( T^{8} + 190 T^{7} + \cdots + 401582027025 \) Copy content Toggle raw display
$47$ \( T^{8} - 58 T^{7} + \cdots + 4236057225 \) Copy content Toggle raw display
$53$ \( (T^{4} - 126 T^{3} + 884 T^{2} + \cdots - 3983085)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 20 T^{3} - 9886 T^{2} + \cdots - 2998700)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 208 T^{7} + \cdots + 8832379812624 \) Copy content Toggle raw display
$67$ \( T^{8} + 24904 T^{6} + \cdots + 32439859360000 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 163478420513424 \) Copy content Toggle raw display
$73$ \( T^{8} + 188 T^{7} + \cdots + 23666745600 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 101810622515625 \) Copy content Toggle raw display
$83$ \( (T^{4} - 140 T^{3} - 4838 T^{2} + \cdots - 10359540)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 622709901672336 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 502668058467600 \) Copy content Toggle raw display
show more
show less