Properties

Label 810.3.d.b
Level $810$
Weight $3$
Character orbit 810.d
Analytic conductor $22.071$
Analytic rank $0$
Dimension $8$
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] = [810,3,Mod(161,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(2))
 
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("810.161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 810.d (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: \(22.0709014132\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 13x^{4} + 12x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 2 q^{4} - \beta_{2} q^{5} + ( - \beta_{6} - 2 \beta_{3} - 1) q^{7} + 2 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 2 q^{4} - \beta_{2} q^{5} + ( - \beta_{6} - 2 \beta_{3} - 1) q^{7} + 2 \beta_1 q^{8} - \beta_{3} q^{10} + (2 \beta_{7} + \beta_{4} - 3 \beta_{2}) q^{11} + ( - \beta_{6} + 3 \beta_{5} + \beta_{3} + 11) q^{13} + (\beta_{7} + \beta_{4} + 3 \beta_{2}) q^{14} + 4 q^{16} + (2 \beta_{7} - 7 \beta_{4} + 4 \beta_1) q^{17} + ( - \beta_{6} - \beta_{5} + 5 \beta_{3} - 10) q^{19} + 2 \beta_{2} q^{20} + (3 \beta_{6} - \beta_{5} - 2 \beta_{3} + 3) q^{22} + ( - 3 \beta_{7} - \beta_{4} + 9 \beta_{2} + 2 \beta_1) q^{23} - 5 q^{25} + (4 \beta_{7} - 2 \beta_{4} - 12 \beta_1) q^{26} + (2 \beta_{6} + 4 \beta_{3} + 2) q^{28} + ( - 5 \beta_{7} - 4 \beta_{4} - 6 \beta_{2} - 6 \beta_1) q^{29} + (2 \beta_{6} - 4 \beta_{5} - \beta_{3} - 19) q^{31} - 4 \beta_1 q^{32} + ( - 5 \beta_{6} - 9 \beta_{5} - 7 \beta_{3} + 3) q^{34} + ( - \beta_{7} - 2 \beta_{4} + \beta_{2} + 9 \beta_1) q^{35} + ( - 3 \beta_{6} - 12 \beta_{5} - 12 \beta_{3} + 17) q^{37} + (2 \beta_{4} - 12 \beta_{2} + 9 \beta_1) q^{38} + 2 \beta_{3} q^{40} + (7 \beta_{7} + 6 \beta_{2} - 5 \beta_1) q^{41} + ( - 2 \beta_{6} - 11 \beta_{5} - 11 \beta_{3} - 4) q^{43} + ( - 4 \beta_{7} - 2 \beta_{4} + 6 \beta_{2}) q^{44} + ( - 4 \beta_{6} + 2 \beta_{5} + 8 \beta_{3}) q^{46} + (4 \beta_{7} - 5 \beta_{4} - 18 \beta_{2} + 20 \beta_1) q^{47} + ( - 7 \beta_{6} - 3 \beta_{5} - 8 \beta_{3} - 6) q^{49} + 5 \beta_1 q^{50} + (2 \beta_{6} - 6 \beta_{5} - 2 \beta_{3} - 22) q^{52} + (9 \beta_{7} - 14 \beta_{4} + 9 \beta_{2} - 8 \beta_1) q^{53} + ( - 4 \beta_{6} - 3 \beta_{5} - 2 \beta_{3} - 15) q^{55} + ( - 2 \beta_{7} - 2 \beta_{4} - 6 \beta_{2}) q^{56} + ( - 9 \beta_{6} + \beta_{5} - 10 \beta_{3} - 21) q^{58} + ( - 10 \beta_{7} + 11 \beta_{4} + 3 \beta_{2} + 7 \beta_1) q^{59} + ( - \beta_{6} - 11 \beta_{5} - 12 \beta_{3} + 29) q^{61} + ( - 6 \beta_{7} + 2 \beta_{4} + 21 \beta_1) q^{62} - 8 q^{64} + (5 \beta_{7} - 5 \beta_{4} - 11 \beta_{2}) q^{65} + ( - 3 \beta_{6} - 9 \beta_{5} + 15 \beta_{3} + 23) q^{67} + ( - 4 \beta_{7} + 14 \beta_{4} - 8 \beta_1) q^{68} + ( - 3 \beta_{6} - \beta_{5} - \beta_{3} + 15) q^{70} + (6 \beta_{7} - 15 \beta_{4} + 9 \beta_{2} + 15 \beta_1) q^{71} + ( - 11 \beta_{6} - 10 \beta_{5} - 12 \beta_{3} + 35) q^{73} + ( - 9 \beta_{7} + 15 \beta_{4} + 9 \beta_{2} - 20 \beta_1) q^{74} + (2 \beta_{6} + 2 \beta_{5} - 10 \beta_{3} + 20) q^{76} + ( - 5 \beta_{7} + 2 \beta_{4} + 15 \beta_{2} - 6 \beta_1) q^{77} + ( - 13 \beta_{6} + 13 \beta_{5} - 6 \beta_{3} + 23) q^{79} - 4 \beta_{2} q^{80} + (7 \beta_{6} - 7 \beta_{5} + 6 \beta_{3} - 3) q^{82} + (5 \beta_{7} + 2 \beta_{4} - 3 \beta_{2} - 50 \beta_1) q^{83} + (12 \beta_{6} - 11 \beta_{5} + 2 \beta_{3}) q^{85} + ( - 9 \beta_{7} + 13 \beta_{4} + 9 \beta_{2} + 2 \beta_1) q^{86} + ( - 6 \beta_{6} + 2 \beta_{5} + 4 \beta_{3} - 6) q^{88} + (11 \beta_{7} - 18 \beta_{4} + 18 \beta_{2} - 31 \beta_1) q^{89} + ( - 10 \beta_{6} - 18 \beta_{5} - 32 \beta_{3} + 4) q^{91} + (6 \beta_{7} + 2 \beta_{4} - 18 \beta_{2} - 4 \beta_1) q^{92} + ( - \beta_{6} - 9 \beta_{5} - 23 \beta_{3} + 39) q^{94} + ( - 3 \beta_{7} - \beta_{4} + 10 \beta_{2} - 28 \beta_1) q^{95} + (8 \beta_{6} - \beta_{5} + 17 \beta_{3} - 64) q^{97} + (4 \beta_{7} + 10 \beta_{4} + 6 \beta_{2} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} - 8 q^{7} + 88 q^{13} + 32 q^{16} - 80 q^{19} + 24 q^{22} - 40 q^{25} + 16 q^{28} - 152 q^{31} + 24 q^{34} + 136 q^{37} - 32 q^{43} - 48 q^{49} - 176 q^{52} - 120 q^{55} - 168 q^{58} + 232 q^{61} - 64 q^{64} + 184 q^{67} + 120 q^{70} + 280 q^{73} + 160 q^{76} + 184 q^{79} - 24 q^{82} - 48 q^{88} + 32 q^{91} + 312 q^{94} - 512 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 13x^{4} + 12x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 14\nu^{5} - 25\nu^{3} - 78\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 6\nu^{4} + 9\nu^{2} - 2 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 14\nu^{5} + 61\nu^{3} - 30\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{7} - 22\nu^{5} - 96\nu^{4} + 5\nu^{3} + 384\nu^{2} + 30\nu + 144 ) / 144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{7} + 6\nu^{6} - 50\nu^{5} - 36\nu^{4} + 55\nu^{3} - 42\nu^{2} - 102\nu + 180 ) / 144 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{7} - 12\nu^{6} - 50\nu^{5} + 72\nu^{4} + 55\nu^{3} + 84\nu^{2} - 102\nu - 360 ) / 144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\nu^{7} - 94\nu^{5} + 65\nu^{3} + 246\nu ) / 144 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - 2\beta_{5} - 2\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - \beta_{5} + \beta_{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - \beta_{6} - 2\beta_{5} + 2\beta_{3} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} + 4\beta_{6} - 4\beta_{5} - 3\beta_{4} + 4\beta_{2} + \beta _1 + 19 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{7} - 5\beta_{6} - 10\beta_{5} + 5\beta_{3} + 17\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 6\beta_{7} + 15\beta_{6} - 15\beta_{5} - 18\beta_{4} + 31\beta_{2} + 6\beta _1 + 82 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 47\beta_{7} - 19\beta_{6} - 38\beta_{5} + 20\beta_{3} + 96\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(487\) \(731\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
2.15988 + 0.258819i
−0.578737 0.965926i
−2.15988 + 0.258819i
0.578737 0.965926i
−2.15988 0.258819i
0.578737 + 0.965926i
2.15988 0.258819i
−0.578737 + 0.965926i
1.41421i 0 −2.00000 2.23607i 0 −6.46891 2.82843i 0 −3.16228
161.2 1.41421i 0 −2.00000 2.23607i 0 −5.01793 2.82843i 0 −3.16228
161.3 1.41421i 0 −2.00000 2.23607i 0 −2.45930 2.82843i 0 3.16228
161.4 1.41421i 0 −2.00000 2.23607i 0 9.94613 2.82843i 0 3.16228
161.5 1.41421i 0 −2.00000 2.23607i 0 −2.45930 2.82843i 0 3.16228
161.6 1.41421i 0 −2.00000 2.23607i 0 9.94613 2.82843i 0 3.16228
161.7 1.41421i 0 −2.00000 2.23607i 0 −6.46891 2.82843i 0 −3.16228
161.8 1.41421i 0 −2.00000 2.23607i 0 −5.01793 2.82843i 0 −3.16228
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 4T_{7}^{3} - 78T_{7}^{2} - 524T_{7} - 794 \) acting on \(S_{3}^{\mathrm{new}}(810, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 4 T^{3} - 78 T^{2} - 524 T - 794)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 552 T^{6} + 46098 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$13$ \( (T^{4} - 44 T^{3} + 516 T^{2} - 704 T - 8744)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 2964 T^{6} + \cdots + 282400839396 \) Copy content Toggle raw display
$19$ \( (T^{4} + 40 T^{3} - 186 T^{2} + \cdots + 52369)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 2280 T^{6} + \cdots + 2361960000 \) Copy content Toggle raw display
$29$ \( T^{8} + 4728 T^{6} + \cdots + 1227661784001 \) Copy content Toggle raw display
$31$ \( (T^{4} + 76 T^{3} + 1722 T^{2} + \cdots - 3719)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 68 T^{3} - 2262 T^{2} + \cdots - 88874)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 3384 T^{6} + \cdots + 50305555521 \) Copy content Toggle raw display
$43$ \( (T^{4} + 16 T^{3} - 3138 T^{2} + \cdots + 868726)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 11316 T^{6} + \cdots + 7727654977956 \) Copy content Toggle raw display
$53$ \( T^{8} + 15276 T^{6} + \cdots + 1421755755876 \) Copy content Toggle raw display
$59$ \( T^{8} + 10056 T^{6} + \cdots + 2097749586321 \) Copy content Toggle raw display
$61$ \( (T^{4} - 116 T^{3} + 1680 T^{2} + \cdots - 4500956)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 92 T^{3} - 7860 T^{2} + \cdots + 7322584)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 16056 T^{6} + \cdots + 57836161890081 \) Copy content Toggle raw display
$73$ \( (T^{4} - 140 T^{3} - 174 T^{2} + \cdots + 1972054)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 92 T^{3} - 6672 T^{2} + \cdots + 12736516)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 184192832358756 \) Copy content Toggle raw display
$89$ \( T^{8} + 36792 T^{6} + \cdots + 16\!\cdots\!61 \) Copy content Toggle raw display
$97$ \( (T^{4} + 256 T^{3} + 18462 T^{2} + \cdots + 3143446)^{2} \) Copy content Toggle raw display
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