Properties

Label 810.3.h.a
Level $810$
Weight $3$
Character orbit 810.h
Analytic conductor $22.071$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 30)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{3}) q^{2} + ( - 2 \beta_{2} + 2) q^{4} + \beta_{5} q^{5} + ( - 3 \beta_{6} - 2 \beta_{2}) q^{7} + 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_{3}) q^{2} + ( - 2 \beta_{2} + 2) q^{4} + \beta_{5} q^{5} + ( - 3 \beta_{6} - 2 \beta_{2}) q^{7} + 2 \beta_{3} q^{8} + ( - \beta_{7} + \beta_{6}) q^{10} + (6 \beta_{4} - 6 \beta_{3}) q^{11} + ( - 10 \beta_{2} + 10) q^{13} + ( - 6 \beta_{5} + 2 \beta_{4}) q^{14} - 4 \beta_{2} q^{16} + ( - 6 \beta_{5} - 12 \beta_{3} - 6 \beta_1) q^{17} + (6 \beta_{7} - 6 \beta_{6} + 8) q^{19} - 2 \beta_1 q^{20} + (12 \beta_{2} - 12) q^{22} + ( - 6 \beta_{5} + 3 \beta_{4}) q^{23} + 5 \beta_{2} q^{25} + 10 \beta_{3} q^{26} + (6 \beta_{7} - 6 \beta_{6} - 4) q^{28} + 12 \beta_1 q^{29} + (8 \beta_{2} - 8) q^{31} + 4 \beta_{4} q^{32} + ( - 6 \beta_{6} + 24 \beta_{2}) q^{34} + ( - 2 \beta_{5} + 15 \beta_{3} - 2 \beta_1) q^{35} + (12 \beta_{7} - 12 \beta_{6} - 22) q^{37} + ( - 8 \beta_{4} + 8 \beta_{3} + 12 \beta_1) q^{38} - 2 \beta_{7} q^{40} + ( - 6 \beta_{5} - 24 \beta_{4}) q^{41} + ( - 9 \beta_{6} - 14 \beta_{2}) q^{43} - 12 \beta_{3} q^{44} + (6 \beta_{7} - 6 \beta_{6} - 6) q^{46} + ( - 15 \beta_{4} + 15 \beta_{3} - 30 \beta_1) q^{47} + (12 \beta_{7} + 45 \beta_{2} - 45) q^{49} - 5 \beta_{4} q^{50} - 20 \beta_{2} q^{52} + ( - 6 \beta_{5} - 12 \beta_{3} - 6 \beta_1) q^{53} + (6 \beta_{7} - 6 \beta_{6}) q^{55} + (4 \beta_{4} - 4 \beta_{3} + 12 \beta_1) q^{56} + 12 \beta_{7} q^{58} + ( - 12 \beta_{5} + 36 \beta_{4}) q^{59} + ( - 12 \beta_{6} + 16 \beta_{2}) q^{61} - 8 \beta_{3} q^{62} - 8 q^{64} - 10 \beta_1 q^{65} + (9 \beta_{7} - 82 \beta_{2} + 82) q^{67} + ( - 12 \beta_{5} - 24 \beta_{4}) q^{68} + ( - 2 \beta_{6} - 30 \beta_{2}) q^{70} + ( - 12 \beta_{5} + 30 \beta_{3} - 12 \beta_1) q^{71} + (12 \beta_{7} - 12 \beta_{6} + 50) q^{73} + (22 \beta_{4} - 22 \beta_{3} + 24 \beta_1) q^{74} + (12 \beta_{7} - 16 \beta_{2} + 16) q^{76} + (36 \beta_{5} - 12 \beta_{4}) q^{77} + ( - 6 \beta_{6} + 28 \beta_{2}) q^{79} + ( - 4 \beta_{5} - 4 \beta_1) q^{80} + (6 \beta_{7} - 6 \beta_{6} + 48) q^{82} + (9 \beta_{4} - 9 \beta_{3} - 6 \beta_1) q^{83} + (12 \beta_{7} - 30 \beta_{2} + 30) q^{85} + ( - 18 \beta_{5} + 14 \beta_{4}) q^{86} + 24 \beta_{2} q^{88} + (12 \beta_{5} + 24 \beta_{3} + 12 \beta_1) q^{89} + (30 \beta_{7} - 30 \beta_{6} - 20) q^{91} + (6 \beta_{4} - 6 \beta_{3} + 12 \beta_1) q^{92} + ( - 30 \beta_{7} - 30 \beta_{2} + 30) q^{94} + (8 \beta_{5} + 30 \beta_{4}) q^{95} + ( - 12 \beta_{6} - 74 \beta_{2}) q^{97} + (24 \beta_{5} - 45 \beta_{3} + 24 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} - 8 q^{7} + 40 q^{13} - 16 q^{16} + 64 q^{19} - 48 q^{22} + 20 q^{25} - 32 q^{28} - 32 q^{31} + 96 q^{34} - 176 q^{37} - 56 q^{43} - 48 q^{46} - 180 q^{49} - 80 q^{52} + 64 q^{61} - 64 q^{64} + 328 q^{67} - 120 q^{70} + 400 q^{73} + 64 q^{76} + 112 q^{79} + 384 q^{82} + 120 q^{85} + 96 q^{88} - 160 q^{91} + 120 q^{94} - 296 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} + 14\nu^{4} - 7\nu^{2} - 36 ) / 63 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{6} + 7\nu^{4} + 28\nu^{2} + 144 ) / 63 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{7} + 7\nu^{5} - 35\nu^{3} + 81\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} + 7\nu^{5} - 35\nu^{3} - 180\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -8\nu^{6} - 14\nu^{4} + 7\nu^{2} - 162 ) / 63 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 13\nu ) / 21 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\nu^{7} + 49\nu^{5} + 133\nu^{3} + 684\nu ) / 189 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} - \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - \beta_{6} - 7\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} + 19\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{5} - 7\beta _1 - 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -29\beta_{6} - 13\beta_{4} + 13\beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

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

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

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
0.178197 1.72286i
−1.40294 + 1.01575i
−0.178197 + 1.72286i
1.40294 1.01575i
0.178197 + 1.72286i
−1.40294 1.01575i
−0.178197 1.72286i
1.40294 + 1.01575i
−1.22474 0.707107i 0 1.00000 + 1.73205i −1.93649 + 1.11803i 0 −5.74342 + 9.94789i 2.82843i 0 3.16228
431.2 −1.22474 0.707107i 0 1.00000 + 1.73205i 1.93649 1.11803i 0 3.74342 6.48379i 2.82843i 0 −3.16228
431.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −1.93649 + 1.11803i 0 3.74342 6.48379i 2.82843i 0 −3.16228
431.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 1.93649 1.11803i 0 −5.74342 + 9.94789i 2.82843i 0 3.16228
701.1 −1.22474 + 0.707107i 0 1.00000 1.73205i −1.93649 1.11803i 0 −5.74342 9.94789i 2.82843i 0 3.16228
701.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 1.93649 + 1.11803i 0 3.74342 + 6.48379i 2.82843i 0 −3.16228
701.3 1.22474 0.707107i 0 1.00000 1.73205i −1.93649 1.11803i 0 3.74342 + 6.48379i 2.82843i 0 −3.16228
701.4 1.22474 0.707107i 0 1.00000 1.73205i 1.93649 + 1.11803i 0 −5.74342 9.94789i 2.82843i 0 3.16228
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.4
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.a 8
3.b odd 2 1 inner 810.3.h.a 8
9.c even 3 1 30.3.d.a 4
9.c even 3 1 inner 810.3.h.a 8
9.d odd 6 1 30.3.d.a 4
9.d odd 6 1 inner 810.3.h.a 8
36.f odd 6 1 240.3.l.c 4
36.h even 6 1 240.3.l.c 4
45.h odd 6 1 150.3.d.c 4
45.j even 6 1 150.3.d.c 4
45.k odd 12 2 150.3.b.b 8
45.l even 12 2 150.3.b.b 8
72.j odd 6 1 960.3.l.e 4
72.l even 6 1 960.3.l.f 4
72.n even 6 1 960.3.l.e 4
72.p odd 6 1 960.3.l.f 4
180.n even 6 1 1200.3.l.u 4
180.p odd 6 1 1200.3.l.u 4
180.v odd 12 2 1200.3.c.k 8
180.x even 12 2 1200.3.c.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.d.a 4 9.c even 3 1
30.3.d.a 4 9.d odd 6 1
150.3.b.b 8 45.k odd 12 2
150.3.b.b 8 45.l even 12 2
150.3.d.c 4 45.h odd 6 1
150.3.d.c 4 45.j even 6 1
240.3.l.c 4 36.f odd 6 1
240.3.l.c 4 36.h even 6 1
810.3.h.a 8 1.a even 1 1 trivial
810.3.h.a 8 3.b odd 2 1 inner
810.3.h.a 8 9.c even 3 1 inner
810.3.h.a 8 9.d odd 6 1 inner
960.3.l.e 4 72.j odd 6 1
960.3.l.e 4 72.n even 6 1
960.3.l.f 4 72.l even 6 1
960.3.l.f 4 72.p odd 6 1
1200.3.c.k 8 180.v odd 12 2
1200.3.c.k 8 180.x even 12 2
1200.3.l.u 4 180.n even 6 1
1200.3.l.u 4 180.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 4T_{7}^{3} + 102T_{7}^{2} - 344T_{7} + 7396 \) acting on \(S_{3}^{\mathrm{new}}(810, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 4 T^{3} + 102 T^{2} - 344 T + 7396)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 72 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 10 T + 100)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 936 T^{2} + 11664)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 16 T - 296)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 396 T^{6} + \cdots + 688747536 \) Copy content Toggle raw display
$29$ \( (T^{4} - 720 T^{2} + 518400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 64)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 44 T - 956)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} - 2664 T^{6} + \cdots + 892616806656 \) Copy content Toggle raw display
$43$ \( (T^{4} + 28 T^{3} + 1398 T^{2} + \cdots + 376996)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 269042006250000 \) Copy content Toggle raw display
$53$ \( (T^{4} + 936 T^{2} + 11664)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 6624 T^{6} + \cdots + 12280707219456 \) Copy content Toggle raw display
$61$ \( (T^{4} - 32 T^{3} + 2208 T^{2} + \cdots + 1401856)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 164 T^{3} + 20982 T^{2} + \cdots + 34975396)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 5040 T^{2} + 1166400)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 100 T + 1060)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 56 T^{3} + 2712 T^{2} + \cdots + 179776)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 684 T^{6} + 467532 T^{4} + \cdots + 104976 \) Copy content Toggle raw display
$89$ \( (T^{4} + 3744 T^{2} + 186624)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 148 T^{3} + 17868 T^{2} + \cdots + 16289296)^{2} \) Copy content Toggle raw display
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