↑

Properties

Label 810.3.h.e
Level $810$
Weight $3$
Character orbit 810.h
Analytic conductor $22.071$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [810,3,Mod(431,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.431");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 810.h (of order \(6\), degree \(2\), not 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: \(22.0709014132\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.11007531417600000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{13} q^{2} + ( - 2 \beta_{2} + 2) q^{4} + ( - \beta_{9} - \beta_1) q^{5} + ( - 2 \beta_{15} + \beta_{12} + \beta_{2}) q^{7} + 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{13} q^{2} + ( - 2 \beta_{2} + 2) q^{4} + ( - \beta_{9} - \beta_1) q^{5} + ( - 2 \beta_{15} + \beta_{12} + \beta_{2}) q^{7} + 2 \beta_{3} q^{8} + \beta_{6} q^{10} + ( - 2 \beta_{14} - \beta_{13} + \cdots - 3 \beta_1) q^{11}+ \cdots + (8 \beta_{14} - 22 \beta_{13} + \cdots - 5 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{4} + 8 q^{7} + 8 q^{13} - 32 q^{16} - 64 q^{19} + 24 q^{22} + 40 q^{25} + 32 q^{28} - 88 q^{31} + 24 q^{34} + 80 q^{37} - 160 q^{43} + 48 q^{49} - 16 q^{52} + 240 q^{55} - 168 q^{58} + 248 q^{61} - 128 q^{64} + 104 q^{67} + 120 q^{70} - 784 q^{73} - 64 q^{76} - 88 q^{79} + 48 q^{82} - 48 q^{88} - 320 q^{91} + 216 q^{94} - 256 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{14} + 281\nu^{2} ) / 48 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{12} + 48\nu^{8} - 336\nu^{4} + 49 ) / 48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{15} - 17\nu^{13} + 120\nu^{9} - 816\nu^{5} - 305\nu^{3} + 119\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{15} - \nu^{13} - 51\nu^{12} + 336\nu^{8} - 2256\nu^{4} - 610\nu^{3} - 233\nu - 315 ) / 144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2 \nu^{15} + 35 \nu^{13} + 45 \nu^{12} - 240 \nu^{9} - 336 \nu^{8} + 1632 \nu^{5} + 2256 \nu^{4} + \cdots - 651 ) / 144 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{15} - 19\nu^{13} + 132\nu^{9} - 912\nu^{5} + 341\nu^{3} + 133\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4\nu^{15} + 111\nu^{14} + \nu^{13} - 768\nu^{10} + 5280\nu^{6} - 1364\nu^{3} - 393\nu^{2} + 521\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4 \nu^{15} + 57 \nu^{14} - 77 \nu^{13} - 384 \nu^{10} + 528 \nu^{9} + 2640 \nu^{6} - 3648 \nu^{5} + \cdots + 11 \nu ) / 144 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -7\nu^{14} + 48\nu^{10} - 328\nu^{6} + \nu^{2} ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4 \nu^{15} + 114 \nu^{14} + 77 \nu^{13} - 768 \nu^{10} - 528 \nu^{9} + 5280 \nu^{6} + 3648 \nu^{5} + \cdots - 11 \nu ) / 144 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 85\nu^{15} + 104\nu^{13} - 624\nu^{11} - 720\nu^{9} + 4272\nu^{7} + 4896\nu^{5} - 1843\nu^{3} - 248\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -4\nu^{15} - 222\nu^{14} + \nu^{13} + 1536\nu^{10} - 10560\nu^{6} - 1364\nu^{3} + 786\nu^{2} + 521\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -89\nu^{15} + 35\nu^{13} + 624\nu^{11} - 240\nu^{9} - 4272\nu^{7} + 1632\nu^{5} + 623\nu^{3} - 5\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 95\nu^{15} - 32\nu^{13} - 624\nu^{11} + 240\nu^{9} + 4272\nu^{7} - 1632\nu^{5} + 1207\nu^{3} + 704\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 199\nu^{15} + 77\nu^{13} - 1392\nu^{11} - 528\nu^{9} + 9552\nu^{7} + 3648\nu^{5} - 1393\nu^{3} - 11\nu ) / 144 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 2 \beta_{14} - 3 \beta_{13} + 2 \beta_{12} - \beta_{11} + \beta_{10} - 2 \beta_{8} + \cdots - 4 \beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{12} + 2\beta_{10} + 2\beta_{8} - \beta_{7} - 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{14} - \beta_{12} + 2\beta_{11} + \beta_{10} - 2\beta_{8} - 2\beta_{7} + 3\beta_{6} + 8\beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{14} + 3\beta_{13} + \beta_{11} - 3\beta_{5} + 6\beta_{4} + \beta_{3} - 21\beta_{2} + 21 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 11 \beta_{14} - 33 \beta_{13} + 5 \beta_{12} - 22 \beta_{11} + 10 \beta_{10} - 20 \beta_{8} + \cdots + 11 \beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{12} + \beta_{10} + 4\beta_{9} + \beta_{8} + \beta_{7} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 117 \beta_{15} - 58 \beta_{14} + 174 \beta_{13} - 26 \beta_{12} - 29 \beta_{11} - 13 \beta_{10} + \cdots + 145 \beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 7\beta_{14} + 21\beta_{13} + 14\beta_{11} - 42\beta_{5} + 21\beta_{4} + 14\beta_{3} - 141\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 38 \beta_{14} - 17 \beta_{12} - 38 \beta_{11} + 17 \beta_{10} - 34 \beta_{8} - 34 \beta_{7} + \cdots + 190 \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -82\beta_{12} - 41\beta_{10} + 165\beta_{9} - 41\beta_{8} + 82\beta_{7} + 165\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 801 \beta_{15} - 199 \beta_{14} + 1194 \beta_{13} - 89 \beta_{12} - 398 \beta_{11} + \cdots + 199 \beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -8\beta_{14} + 8\beta_{11} - 24\beta_{5} - 24\beta_{4} + 8\beta_{3} - 161 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1042 \beta_{14} + 1563 \beta_{13} - 466 \beta_{12} + 521 \beta_{11} - 233 \beta_{10} + \cdots + 2084 \beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -281\beta_{12} - 562\beta_{10} - 562\beta_{8} + 281\beta_{7} + 1131\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 682 \beta_{14} + 305 \beta_{12} - 682 \beta_{11} - 305 \beta_{10} + 610 \beta_{8} + \cdots - 2728 \beta_{3} ) / 9 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(731\)
\(\chi(n)\) \(1\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1
−0.418778 + 1.56290i
0.596975 + 0.159959i
0.159959 − 0.596975i
−1.56290 − 0.418778i
0.418778 − 1.56290i
−0.596975 − 0.159959i
−0.159959 + 0.596975i
1.56290 + 0.418778i
−0.418778 − 1.56290i
0.596975 − 0.159959i
0.159959 + 0.596975i
−1.56290 + 0.418778i
0.418778 + 1.56290i
−0.596975 + 0.159959i
−0.159959 − 0.596975i
1.56290 − 0.418778i
−1.22474 − 0.707107i 0 1.00000 + 1.73205i −1.93649 + 1.11803i 0 −0.229649 + 0.397764i − 2.82843i 0 3.16228
431.2 −1.22474 − 0.707107i 0 1.00000 + 1.73205i −1.93649 + 1.11803i 0 5.97307 − 10.3457i − 2.82843i 0 3.16228
431.3 −1.22474 − 0.707107i 0 1.00000 + 1.73205i 1.93649 − 1.11803i 0 −2.23445 + 3.87019i − 2.82843i 0 −3.16228
431.4 −1.22474 − 0.707107i 0 1.00000 + 1.73205i 1.93649 − 1.11803i 0 −1.50896 + 2.61360i − 2.82843i 0 −3.16228
431.5 1.22474 + 0.707107i 0 1.00000 + 1.73205i −1.93649 + 1.11803i 0 −2.23445 + 3.87019i 2.82843i 0 −3.16228
431.6 1.22474 + 0.707107i 0 1.00000 + 1.73205i −1.93649 + 1.11803i 0 −1.50896 + 2.61360i 2.82843i 0 −3.16228
431.7 1.22474 + 0.707107i 0 1.00000 + 1.73205i 1.93649 − 1.11803i 0 −0.229649 + 0.397764i 2.82843i 0 3.16228
431.8 1.22474 + 0.707107i 0 1.00000 + 1.73205i 1.93649 − 1.11803i 0 5.97307 − 10.3457i 2.82843i 0 3.16228
701.1 −1.22474 + 0.707107i 0 1.00000 − 1.73205i −1.93649 − 1.11803i 0 −0.229649 − 0.397764i 2.82843i 0 3.16228
701.2 −1.22474 + 0.707107i 0 1.00000 − 1.73205i −1.93649 − 1.11803i 0 5.97307 + 10.3457i 2.82843i 0 3.16228
701.3 −1.22474 + 0.707107i 0 1.00000 − 1.73205i 1.93649 + 1.11803i 0 −2.23445 − 3.87019i 2.82843i 0 −3.16228
701.4 −1.22474 + 0.707107i 0 1.00000 − 1.73205i 1.93649 + 1.11803i 0 −1.50896 − 2.61360i 2.82843i 0 −3.16228
701.5 1.22474 − 0.707107i 0 1.00000 − 1.73205i −1.93649 − 1.11803i 0 −2.23445 − 3.87019i − 2.82843i 0 −3.16228
701.6 1.22474 − 0.707107i 0 1.00000 − 1.73205i −1.93649 − 1.11803i 0 −1.50896 − 2.61360i − 2.82843i 0 −3.16228
701.7 1.22474 − 0.707107i 0 1.00000 − 1.73205i 1.93649 + 1.11803i 0 −0.229649 − 0.397764i − 2.82843i 0 3.16228
701.8 1.22474 − 0.707107i 0 1.00000 − 1.73205i 1.93649 + 1.11803i 0 5.97307 + 10.3457i − 2.82843i 0 3.16228
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.3.h.e 16
3.b odd 2 1 inner 810.3.h.e 16
9.c even 3 1 810.3.d.a ✓ 8
9.c even 3 1 inner 810.3.h.e 16
9.d odd 6 1 810.3.d.a ✓ 8
9.d odd 6 1 inner 810.3.h.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
810.3.d.a ✓ 8 9.c even 3 1
810.3.d.a ✓ 8 9.d odd 6 1
810.3.h.e 16 1.a even 1 1 trivial
810.3.h.e 16 3.b odd 2 1 inner
810.3.h.e 16 9.c even 3 1 inner
810.3.h.e 16 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 4T_{7}^{7} + 94T_{7}^{6} + 704T_{7}^{5} + 5374T_{7}^{4} + 14696T_{7}^{3} + 32644T_{7}^{2} + 14504T_{7} + 5476 \) acting on \(S_{3}^{\mathrm{new}}(810, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{4} \) Copy content Toggle raw display
$7$ \( (T^{8} - 4 T^{7} + \cdots + 5476)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 10\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( (T^{8} - 4 T^{7} + \cdots + 440896)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 84 T^{6} + \cdots + 2916)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 16 T^{3} + \cdots - 71)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 850305600000000 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 38\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( (T^{8} + 44 T^{7} + \cdots + 140535764161)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 20 T^{3} + \cdots - 59306)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 20\!\cdots\!41 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 9299242885156)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{8} + 5772 T^{6} + \cdots + 428980461156)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 66\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( (T^{8} - 124 T^{7} + \cdots + 424014554896)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 49339610414656)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 7848 T^{6} + \cdots + 15946385841)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 196 T^{3} + \cdots - 230954)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 76\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 72157396691601)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 35579864992996)^{2} \) Copy content Toggle raw display
show more
show less