↑

Properties

Label 429.2.i.f
Level $429$
Weight $2$
Character orbit 429.i
Analytic conductor $3.426$
Analytic rank $0$
Dimension $10$
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] = [429,2,Mod(100,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.100");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.i (of order \(3\), 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: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 7x^{8} + 4x^{7} + 32x^{6} + 3x^{5} + 30x^{4} - 7x^{3} + 26x^{2} - 5x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_1 + 1) q^{2} + (\beta_{6} - 1) q^{3} + (\beta_{9} - \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{4}+ \cdots - \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_1 + 1) q^{2} + (\beta_{6} - 1) q^{3} + (\beta_{9} - \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{4}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{2} - 5 q^{3} - 6 q^{4} - 8 q^{5} + 4 q^{6} + 3 q^{7} - 18 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 4 q^{2} - 5 q^{3} - 6 q^{4} - 8 q^{5} + 4 q^{6} + 3 q^{7} - 18 q^{8} - 5 q^{9} + 5 q^{10} - 5 q^{11} + 12 q^{12} + 5 q^{13} + 14 q^{14} + 4 q^{15} - 12 q^{16} - 9 q^{17} - 8 q^{18} + 9 q^{19} + 6 q^{20} - 6 q^{21} + 4 q^{22} + 9 q^{23} + 9 q^{24} + 2 q^{25} - 12 q^{26} + 10 q^{27} - q^{28} - 8 q^{29} + 5 q^{30} - 12 q^{31} + 27 q^{32} - 5 q^{33} - 14 q^{34} + 2 q^{35} - 6 q^{36} + 17 q^{37} - 2 q^{38} + 2 q^{39} + 46 q^{40} + 6 q^{41} - 7 q^{42} - 13 q^{43} + 12 q^{44} + 4 q^{45} - 6 q^{46} - 32 q^{47} - 12 q^{48} + 18 q^{49} - 8 q^{50} + 18 q^{51} + 7 q^{52} - 26 q^{53} + 4 q^{54} + 4 q^{55} - q^{56} - 18 q^{57} + 11 q^{58} + 8 q^{59} - 12 q^{60} - 4 q^{61} + 22 q^{62} + 3 q^{63} + 22 q^{64} + 26 q^{65} - 8 q^{66} + 5 q^{67} - 34 q^{68} + 9 q^{69} + 8 q^{70} + 19 q^{71} + 9 q^{72} - 4 q^{73} - 27 q^{74} - q^{75} - 6 q^{76} - 6 q^{77} - 3 q^{78} - 16 q^{79} + 32 q^{80} - 5 q^{81} - 16 q^{82} - 20 q^{83} - q^{84} + 21 q^{85} + 8 q^{86} - 8 q^{87} + 9 q^{88} + 32 q^{89} - 10 q^{90} - 17 q^{91} - 84 q^{92} + 6 q^{93} - 66 q^{94} - 11 q^{95} - 54 q^{96} - 5 q^{97} - 18 q^{98} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 7x^{8} + 4x^{7} + 32x^{6} + 3x^{5} + 30x^{4} - 7x^{3} + 26x^{2} - 5x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 911 \nu^{9} + 318 \nu^{8} - 583 \nu^{7} + 16404 \nu^{6} - 265 \nu^{5} + 8533 \nu^{4} - 159430 \nu^{3} + \cdots + 31405 ) / 160271 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4571 \nu^{9} + 29016 \nu^{8} - 53196 \nu^{7} + 154140 \nu^{6} - 24180 \nu^{5} + 778596 \nu^{4} + \cdots + 319366 ) / 160271 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4889 \nu^{9} - 36294 \nu^{8} + 66539 \nu^{7} - 199961 \nu^{6} + 30245 \nu^{5} - 973889 \nu^{4} + \cdots - 672224 ) / 160271 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5191 \nu^{9} + 12966 \nu^{8} - 23771 \nu^{7} + 9622 \nu^{6} - 10805 \nu^{5} + 347921 \nu^{4} + \cdots + 388420 ) / 160271 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 31405 \nu^{9} + 32316 \nu^{8} - 219517 \nu^{7} - 126203 \nu^{6} - 988556 \nu^{5} - 94480 \nu^{4} + \cdots + 155382 ) / 160271 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 62396 \nu^{9} + 33989 \nu^{8} - 409567 \nu^{7} - 433019 \nu^{6} - 2138559 \nu^{5} - 984502 \nu^{4} + \cdots - 15562 ) / 160271 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 91217 \nu^{9} + 91837 \nu^{8} - 622469 \nu^{7} - 394293 \nu^{6} - 2774426 \nu^{5} + \cdots + 373160 ) / 160271 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 128866 \nu^{9} + 127955 \nu^{8} - 902380 \nu^{7} - 514881 \nu^{6} - 4140116 \nu^{5} + \cdots + 645973 ) / 160271 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{7} - 2\beta_{6} - \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} - 2\beta_{4} - 2\beta_{3} - 5\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{9} + \beta_{8} + 8\beta_{7} + 9\beta_{6} - \beta_{5} - 7\beta_{4} - 10\beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -18\beta_{9} + 7\beta_{8} + 19\beta_{7} + 14\beta_{6} + 19\beta_{3} + 34\beta_{2} - 34\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{5} + 53\beta_{4} + 60\beta_{3} + 85\beta_{2} + 57 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 145\beta_{9} - 48\beta_{8} - 157\beta_{7} - 129\beta_{6} + 48\beta_{5} + 145\beta_{4} + 255\beta _1 + 129 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 412\beta_{9} - 109\beta_{8} - 460\beta_{7} - 413\beta_{6} - 460\beta_{3} - 686\beta_{2} + 686\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -351\beta_{5} - 1146\beta_{4} - 1255\beta_{3} - 1971\beta_{2} - 1069 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(79\) \(287\)
\(\chi(n)\) \(-\beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
100.1
1.40131 − 2.42714i
0.440981 − 0.763802i
0.100998 − 0.174933i
−0.580000 + 1.00459i
−0.863288 + 1.49526i
1.40131 + 2.42714i
0.440981 + 0.763802i
0.100998 + 0.174933i
−0.580000 − 1.00459i
−0.863288 − 1.49526i
−0.901309 + 1.56111i −0.500000 + 0.866025i −0.624717 − 1.08204i −2.90617 −0.901309 − 1.56111i −0.704431 − 1.22011i −1.35298 −0.500000 − 0.866025i 2.61936 − 4.53686i
100.2 0.0590186 − 0.102223i −0.500000 + 0.866025i 0.993034 + 1.71998i −2.39753 0.0590186 + 0.102223i 1.48513 + 2.57233i 0.470505 −0.500000 − 0.866025i −0.141499 + 0.245083i
100.3 0.399002 − 0.691092i −0.500000 + 0.866025i 0.681594 + 1.18056i 1.11866 0.399002 + 0.691092i −0.394706 − 0.683651i 2.68384 −0.500000 − 0.866025i 0.446346 − 0.773094i
100.4 1.08000 − 1.87062i −0.500000 + 0.866025i −1.33280 − 2.30848i 2.38790 1.08000 + 1.87062i 1.17823 + 2.04076i −1.43770 −0.500000 − 0.866025i 2.57893 − 4.46684i
100.5 1.36329 − 2.36128i −0.500000 + 0.866025i −2.71711 − 4.70617i −2.20286 1.36329 + 2.36128i −0.0642304 − 0.111250i −9.36366 −0.500000 − 0.866025i −3.00314 + 5.20159i
133.1 −0.901309 − 1.56111i −0.500000 − 0.866025i −0.624717 + 1.08204i −2.90617 −0.901309 + 1.56111i −0.704431 + 1.22011i −1.35298 −0.500000 + 0.866025i 2.61936 + 4.53686i
133.2 0.0590186 + 0.102223i −0.500000 − 0.866025i 0.993034 − 1.71998i −2.39753 0.0590186 − 0.102223i 1.48513 − 2.57233i 0.470505 −0.500000 + 0.866025i −0.141499 − 0.245083i
133.3 0.399002 + 0.691092i −0.500000 − 0.866025i 0.681594 − 1.18056i 1.11866 0.399002 − 0.691092i −0.394706 + 0.683651i 2.68384 −0.500000 + 0.866025i 0.446346 + 0.773094i
133.4 1.08000 + 1.87062i −0.500000 − 0.866025i −1.33280 + 2.30848i 2.38790 1.08000 − 1.87062i 1.17823 − 2.04076i −1.43770 −0.500000 + 0.866025i 2.57893 + 4.46684i
133.5 1.36329 + 2.36128i −0.500000 − 0.866025i −2.71711 + 4.70617i −2.20286 1.36329 − 2.36128i −0.0642304 + 0.111250i −9.36366 −0.500000 + 0.866025i −3.00314 − 5.20159i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 100.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.i.f ✓ 10
13.c even 3 1 inner 429.2.i.f ✓ 10
13.c even 3 1 5577.2.a.n 5
13.e even 6 1 5577.2.a.w 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.i.f ✓ 10 1.a even 1 1 trivial
429.2.i.f ✓ 10 13.c even 3 1 inner
5577.2.a.n 5 13.c even 3 1
5577.2.a.w 5 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\):

\( T_{2}^{10} - 4T_{2}^{9} + 16T_{2}^{8} - 26T_{2}^{7} + 62T_{2}^{6} - 79T_{2}^{5} + 173T_{2}^{4} - 130T_{2}^{3} + 87T_{2}^{2} - 10T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{10} - 3T_{7}^{9} + 13T_{7}^{8} - 6T_{7}^{7} + 34T_{7}^{6} + 19T_{7}^{5} + 120T_{7}^{4} + 89T_{7}^{3} + 72T_{7}^{2} + 9T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 4 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$5$ \( (T^{5} + 4 T^{4} - 5 T^{3} + \cdots + 41)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} - 3 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{10} - 5 T^{9} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( T^{10} + 9 T^{9} + \cdots + 2968729 \) Copy content Toggle raw display
$19$ \( T^{10} - 9 T^{9} + \cdots + 81 \) Copy content Toggle raw display
$23$ \( T^{10} - 9 T^{9} + \cdots + 130321 \) Copy content Toggle raw display
$29$ \( T^{10} + 8 T^{9} + \cdots + 29241 \) Copy content Toggle raw display
$31$ \( (T^{5} + 6 T^{4} + \cdots + 3087)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} - 17 T^{9} + \cdots + 2809 \) Copy content Toggle raw display
$41$ \( T^{10} - 6 T^{9} + \cdots + 6561 \) Copy content Toggle raw display
$43$ \( T^{10} + 13 T^{9} + \cdots + 36638809 \) Copy content Toggle raw display
$47$ \( (T^{5} + 16 T^{4} + \cdots + 36279)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + 13 T^{4} + \cdots + 361)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - 8 T^{9} + \cdots + 10374841 \) Copy content Toggle raw display
$61$ \( T^{10} + 4 T^{9} + \cdots + 124609 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 2107452649 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 682097689 \) Copy content Toggle raw display
$73$ \( (T^{5} + 2 T^{4} + \cdots + 14879)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + 8 T^{4} + \cdots - 7457)^{2} \) Copy content Toggle raw display
$83$ \( (T^{5} + 10 T^{4} + \cdots + 198059)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} - 32 T^{9} + \cdots + 49 \) Copy content Toggle raw display
$97$ \( T^{10} + 5 T^{9} + \cdots + 32001649 \) Copy content Toggle raw display
show more
show less