↑

Properties

Label 429.2.e.c
Level $429$
Weight $2$
Character orbit 429.e
Analytic conductor $3.426$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [429,2,Mod(428,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.428");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.e (of order \(2\), degree \(1\), 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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 20x^{14} + 164x^{12} - 666x^{10} + 1300x^{8} - 924x^{6} + 273x^{4} + 404x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 13^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{5} q^{3} + (\beta_{5} + \beta_{4} - 1) q^{4} + \beta_{10} q^{5} + (\beta_{8} - \beta_{6}) q^{6} - \beta_{8} q^{8} + (\beta_{9} + \beta_{5} + \beta_{4} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{5} q^{3} + (\beta_{5} + \beta_{4} - 1) q^{4} + \beta_{10} q^{5} + (\beta_{8} - \beta_{6}) q^{6} - \beta_{8} q^{8} + (\beta_{9} + \beta_{5} + \beta_{4} - 1) q^{9} + (\beta_{14} + \beta_{13} + \cdots - \beta_{3}) q^{10}+ \cdots + ( - \beta_{12} - \beta_{10} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} - 8 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{3} - 8 q^{4} - 12 q^{9} - 32 q^{12} - 8 q^{16} + 8 q^{22} + 40 q^{25} - 16 q^{27} + 40 q^{36} - 32 q^{48} - 112 q^{49} + 120 q^{55} + 88 q^{64} + 32 q^{66} + 60 q^{69} + 24 q^{75} - 92 q^{78} - 92 q^{81} - 112 q^{82} - 56 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 20x^{14} + 164x^{12} - 666x^{10} + 1300x^{8} - 924x^{6} + 273x^{4} + 404x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1017 \nu^{14} - 18889 \nu^{12} + 141809 \nu^{10} - 504639 \nu^{8} + 779299 \nu^{6} - 250803 \nu^{4} + \cdots + 1865592 ) / 695368 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 31233 \nu^{15} - 628770 \nu^{13} + 5186482 \nu^{11} - 21171500 \nu^{9} + 41344590 \nu^{7} + \cdots + 16866948 \nu ) / 695368 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5799 \nu^{14} - 117329 \nu^{12} + 976398 \nu^{10} - 4060171 \nu^{8} + 8332329 \nu^{6} + \cdots + 1437494 ) / 173842 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 36955 \nu^{14} + 751451 \nu^{12} - 6309143 \nu^{10} + 26657365 \nu^{8} - 56418201 \nu^{6} + \cdots - 9249288 ) / 695368 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 40155 \nu^{14} - 814827 \nu^{12} + 6814631 \nu^{10} - 28599597 \nu^{8} + 59716185 \nu^{6} + \cdots + 10837864 ) / 695368 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 53617 \nu^{14} + 1086197 \nu^{12} - 9069441 \nu^{10} + 37939335 \nu^{8} - 78591519 \nu^{6} + \cdots - 13405760 ) / 695368 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11595 \nu^{15} + 51170 \nu^{14} + 231812 \nu^{13} - 1036686 \nu^{12} - 1903314 \nu^{11} + \cdots + 13045736 ) / 695368 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 9893 \nu^{14} + 200507 \nu^{12} - 1675924 \nu^{10} + 7033280 \nu^{8} - 14711623 \nu^{6} + \cdots - 2542687 ) / 86921 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25297 \nu^{14} - 511003 \nu^{12} + 4256369 \nu^{10} - 17787087 \nu^{8} + 36994087 \nu^{6} + \cdots + 6255548 ) / 173842 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 155031 \nu^{15} - 3141792 \nu^{13} + 26258800 \nu^{11} - 110207086 \nu^{9} + 230644536 \nu^{7} + \cdots + 40253716 \nu ) / 695368 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 194169 \nu^{15} - 3937730 \nu^{13} + 32931622 \nu^{11} - 138302044 \nu^{9} + 289581422 \nu^{7} + \cdots + 46444516 \nu ) / 695368 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 213619 \nu^{15} + 4316546 \nu^{13} - 35936294 \nu^{11} + 149903292 \nu^{9} - 310245654 \nu^{7} + \cdots - 56266252 \nu ) / 695368 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 282598 \nu^{15} - 12255 \nu^{14} - 5719412 \nu^{13} + 249485 \nu^{12} + 47711512 \nu^{11} + \cdots - 2535072 ) / 695368 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 275554 \nu^{15} - 15719 \nu^{14} + 5581644 \nu^{13} + 317885 \nu^{12} - 46621406 \nu^{11} + \cdots - 4760688 ) / 695368 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 429155 \nu^{15} - 8692814 \nu^{13} + 72609066 \nu^{11} - 304468532 \nu^{9} + 636294706 \nu^{7} + \cdots + 110595980 \nu ) / 695368 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 7 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 3 \beta_{12} - 3 \beta_{11} + 26 \beta_{10} + \cdots + 2 \beta_1 ) / 52 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{8} - \beta_{5} + \beta_{4} - 2\beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 40 \beta_{15} - 35 \beta_{14} - 43 \beta_{13} - 37 \beta_{12} - 11 \beta_{11} + 91 \beta_{10} + \cdots + 4 \beta_1 ) / 52 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{9} - 11\beta_{8} - 7\beta_{5} + 9\beta_{4} - 2\beta_{3} - 8\beta _1 + 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 124 \beta_{15} - 297 \beta_{14} - 353 \beta_{13} - 223 \beta_{12} - 93 \beta_{11} + 273 \beta_{10} + \cdots + 28 \beta_1 ) / 52 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -20\beta_{9} - 81\beta_{8} + 4\beta_{6} - 53\beta_{5} + 51\beta_{4} - 6\beta_{3} - 20\beta _1 + 35 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 52 \beta_{15} - 1807 \beta_{14} - 2015 \beta_{13} - 1029 \beta_{12} - 743 \beta_{11} + \cdots + 104 \beta_1 ) / 52 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -142\beta_{9} - 467\beta_{8} + 16\beta_{6} - 349\beta_{5} + 219\beta_{4} + 14\beta_{3} + 48\beta _1 - 135 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2884 \beta_{15} - 9521 \beta_{14} - 8809 \beta_{13} - 3099 \beta_{12} - 5205 \beta_{11} + \cdots - 356 \beta_1 ) / 52 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -810\beta_{9} - 2169\beta_{8} - 96\beta_{6} - 2033\beta_{5} + 543\beta_{4} + 246\beta_{3} + 1128\beta _1 - 2305 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 29188 \beta_{15} - 44407 \beta_{14} - 25663 \beta_{13} + 2255 \beta_{12} - 33027 \beta_{11} + \cdots - 9372 \beta_1 ) / 52 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 3584 \beta_{9} - 7259 \beta_{8} - 2024 \beta_{6} - 10415 \beta_{5} - 1567 \beta_{4} + 1278 \beta_{3} + \cdots - 18635 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 202092 \beta_{15} - 176321 \beta_{14} + 11039 \beta_{13} + 123133 \beta_{12} - 190505 \beta_{11} + \cdots - 93680 \beta_1 ) / 52 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 9250 \beta_{9} - 5053 \beta_{8} - 19332 \beta_{6} - 44551 \beta_{5} - 32711 \beta_{4} + 486 \beta_{3} + \cdots - 116489 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1142500 \beta_{15} - 501039 \beta_{14} + 869017 \beta_{13} + 1171595 \beta_{12} - 976083 \beta_{11} + \cdots - 685028 \beta_1 ) / 52 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(67\) \(79\) \(287\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
428.1
−0.946412 + 0.500000i
0.946412 − 0.500000i
−2.14576 − 0.500000i
2.14576 + 0.500000i
−2.34517 + 0.500000i
2.34517 − 0.500000i
0.0131233 − 0.500000i
−0.0131233 + 0.500000i
0.0131233 + 0.500000i
−0.0131233 − 0.500000i
−2.34517 − 0.500000i
2.34517 + 0.500000i
−2.14576 + 0.500000i
2.14576 − 0.500000i
−0.946412 − 0.500000i
0.946412 + 0.500000i
− 2.13578i 0.780776 − 1.54609i −2.56155 −3.09218 −3.30210 − 1.66757i 0 1.19935i −1.78078 − 2.41430i 6.60421i
428.2 − 2.13578i 0.780776 − 1.54609i −2.56155 3.09218 −3.30210 − 1.66757i 0 1.19935i −1.78078 − 2.41430i − 6.60421i
428.3 − 2.13578i 0.780776 + 1.54609i −2.56155 −3.09218 3.30210 − 1.66757i 0 1.19935i −1.78078 + 2.41430i 6.60421i
428.4 − 2.13578i 0.780776 + 1.54609i −2.56155 3.09218 3.30210 − 1.66757i 0 1.19935i −1.78078 + 2.41430i − 6.60421i
428.5 − 0.662153i −1.28078 − 1.16602i 1.56155 −2.33205 −0.772087 + 0.848071i 0 − 2.35829i 0.280776 + 2.98683i 1.54417i
428.6 − 0.662153i −1.28078 − 1.16602i 1.56155 2.33205 −0.772087 + 0.848071i 0 − 2.35829i 0.280776 + 2.98683i − 1.54417i
428.7 − 0.662153i −1.28078 + 1.16602i 1.56155 −2.33205 0.772087 + 0.848071i 0 − 2.35829i 0.280776 − 2.98683i 1.54417i
428.8 − 0.662153i −1.28078 + 1.16602i 1.56155 2.33205 0.772087 + 0.848071i 0 − 2.35829i 0.280776 − 2.98683i − 1.54417i
428.9 0.662153i −1.28078 − 1.16602i 1.56155 −2.33205 0.772087 − 0.848071i 0 2.35829i 0.280776 + 2.98683i − 1.54417i
428.10 0.662153i −1.28078 − 1.16602i 1.56155 2.33205 0.772087 − 0.848071i 0 2.35829i 0.280776 + 2.98683i 1.54417i
428.11 0.662153i −1.28078 + 1.16602i 1.56155 −2.33205 −0.772087 − 0.848071i 0 2.35829i 0.280776 − 2.98683i − 1.54417i
428.12 0.662153i −1.28078 + 1.16602i 1.56155 2.33205 −0.772087 − 0.848071i 0 2.35829i 0.280776 − 2.98683i 1.54417i
428.13 2.13578i 0.780776 − 1.54609i −2.56155 −3.09218 3.30210 + 1.66757i 0 − 1.19935i −1.78078 − 2.41430i − 6.60421i
428.14 2.13578i 0.780776 − 1.54609i −2.56155 3.09218 3.30210 + 1.66757i 0 − 1.19935i −1.78078 − 2.41430i 6.60421i
428.15 2.13578i 0.780776 + 1.54609i −2.56155 −3.09218 −3.30210 + 1.66757i 0 − 1.19935i −1.78078 + 2.41430i − 6.60421i
428.16 2.13578i 0.780776 + 1.54609i −2.56155 3.09218 −3.30210 + 1.66757i 0 − 1.19935i −1.78078 + 2.41430i 6.60421i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 428.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
13.b even 2 1 inner
33.d even 2 1 inner
39.d odd 2 1 inner
143.d odd 2 1 inner
429.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.e.c ✓ 16
3.b odd 2 1 inner 429.2.e.c ✓ 16
11.b odd 2 1 inner 429.2.e.c ✓ 16
13.b even 2 1 inner 429.2.e.c ✓ 16
33.d even 2 1 inner 429.2.e.c ✓ 16
39.d odd 2 1 inner 429.2.e.c ✓ 16
143.d odd 2 1 inner 429.2.e.c ✓ 16
429.e even 2 1 inner 429.2.e.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.e.c ✓ 16 1.a even 1 1 trivial
429.2.e.c ✓ 16 3.b odd 2 1 inner
429.2.e.c ✓ 16 11.b odd 2 1 inner
429.2.e.c ✓ 16 13.b even 2 1 inner
429.2.e.c ✓ 16 33.d even 2 1 inner
429.2.e.c ✓ 16 39.d odd 2 1 inner
429.2.e.c ✓ 16 143.d odd 2 1 inner
429.2.e.c ✓ 16 429.e even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 5T_{2}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 5 T^{2} + 2)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + 2 T^{2} + \cdots + 9)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 15 T^{2} + 52)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 16 T^{6} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 6 T^{6} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 74 T^{2} + 1352)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 58 T^{2} + 416)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 15 T^{2} + 52)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 74 T^{2} + 1352)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 101 T^{2} + 676)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 121 T^{2} + 2704)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 46 T^{2} + 512)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 46 T^{2} + 104)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( (T^{4} + 60 T^{2} + 832)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 123 T^{2} + 208)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 58 T^{2} + 416)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 101 T^{2} + 676)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 59 T^{2} + 832)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 164 T^{2} + 6656)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 158 T^{2} + 104)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 250 T^{2} + 14792)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 247 T^{2} + 8788)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 417 T^{2} + 43264)^{4} \) Copy content Toggle raw display
show more
show less