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Properties

Label 810.3.h.b
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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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: \(8\)
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 270)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{3}) q^{2} + ( - 2 \beta_{2} + 2) q^{4} + \beta_{5} q^{5} + \beta_{2} q^{7} + 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_{3}) q^{2} + ( - 2 \beta_{2} + 2) q^{4} + \beta_{5} q^{5} + \beta_{2} q^{7} + 2 \beta_{3} q^{8} + ( - \beta_{7} + \beta_{6}) q^{10} + 6 \beta_1 q^{11} + (6 \beta_{7} + 5 \beta_{2} - 5) q^{13} - \beta_{4} q^{14} - 4 \beta_{2} q^{16} - 6 \beta_{3} q^{17} + ( - 6 \beta_{7} + 6 \beta_{6} - 7) q^{19} - 2 \beta_1 q^{20} + 6 \beta_{7} q^{22} + (6 \beta_{5} - 12 \beta_{4}) q^{23} + 5 \beta_{2} q^{25} + (12 \beta_{5} - 5 \beta_{3} + 12 \beta_1) q^{26} + 2 q^{28} + (6 \beta_{4} - 6 \beta_{3} + 6 \beta_1) q^{29} + (12 \beta_{7} + 2 \beta_{2} - 2) q^{31} + 4 \beta_{4} q^{32} + 12 \beta_{2} q^{34} + (\beta_{5} + \beta_1) q^{35} + ( - 6 \beta_{7} + 6 \beta_{6} + 5) q^{37} + (7 \beta_{4} - 7 \beta_{3} - 12 \beta_1) q^{38} - 2 \beta_{7} q^{40} + ( - 6 \beta_{5} + 30 \beta_{4}) q^{41} + (12 \beta_{6} - 38 \beta_{2}) q^{43} + (12 \beta_{5} + 12 \beta_1) q^{44} + ( - 6 \beta_{7} + 6 \beta_{6} + 24) q^{46} + ( - 30 \beta_{4} + 30 \beta_{3} + 12 \beta_1) q^{47} + ( - 48 \beta_{2} + 48) q^{49} - 5 \beta_{4} q^{50} + (12 \beta_{6} + 10 \beta_{2}) q^{52} + (6 \beta_{5} + 54 \beta_{3} + 6 \beta_1) q^{53} - 30 q^{55} + ( - 2 \beta_{4} + 2 \beta_{3}) q^{56} + (6 \beta_{7} + 12 \beta_{2} - 12) q^{58} - 6 \beta_{4} q^{59} + (24 \beta_{6} + 25 \beta_{2}) q^{61} + (24 \beta_{5} - 2 \beta_{3} + 24 \beta_1) q^{62} - 8 q^{64} + (30 \beta_{4} - 30 \beta_{3} + 5 \beta_1) q^{65} + (71 \beta_{2} - 71) q^{67} - 12 \beta_{4} q^{68} + \beta_{6} q^{70} + ( - 30 \beta_{5} - 42 \beta_{3} - 30 \beta_1) q^{71} + ( - 6 \beta_{7} + 6 \beta_{6} + 41) q^{73} + ( - 5 \beta_{4} + 5 \beta_{3} - 12 \beta_1) q^{74} + ( - 12 \beta_{7} + 14 \beta_{2} - 14) q^{76} - 6 \beta_{5} q^{77} + (30 \beta_{6} + 19 \beta_{2}) q^{79} + ( - 4 \beta_{5} - 4 \beta_1) q^{80} + (6 \beta_{7} - 6 \beta_{6} - 60) q^{82} + (42 \beta_{4} - 42 \beta_{3} + 42 \beta_1) q^{83} + 6 \beta_{7} q^{85} + (24 \beta_{5} + 38 \beta_{4}) q^{86} + 12 \beta_{6} q^{88} + ( - 24 \beta_{5} - 66 \beta_{3} - 24 \beta_1) q^{89} + (6 \beta_{7} - 6 \beta_{6} - 5) q^{91} + ( - 24 \beta_{4} + 24 \beta_{3} - 12 \beta_1) q^{92} + (12 \beta_{7} - 60 \beta_{2} + 60) q^{94} + ( - 7 \beta_{5} - 30 \beta_{4}) q^{95} + (30 \beta_{6} - 5 \beta_{2}) q^{97} + 48 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} + 4 q^{7} - 20 q^{13} - 16 q^{16} - 56 q^{19} + 20 q^{25} + 16 q^{28} - 8 q^{31} + 48 q^{34} + 40 q^{37} - 152 q^{43} + 192 q^{46} + 192 q^{49} + 40 q^{52} - 240 q^{55} - 48 q^{58} + 100 q^{61} - 64 q^{64} - 284 q^{67} + 328 q^{73} - 56 q^{76} + 76 q^{79} - 480 q^{82} - 40 q^{91} + 240 q^{94} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{4} + \beta_{3} ) / 2 \) 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

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
0.178197 − 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 0.500000 − 0.866025i − 2.82843i 0 3.16228
431.2 −1.22474 − 0.707107i 0 1.00000 + 1.73205i 1.93649 − 1.11803i 0 0.500000 − 0.866025i − 2.82843i 0 −3.16228
431.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −1.93649 + 1.11803i 0 0.500000 − 0.866025i 2.82843i 0 −3.16228
431.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 1.93649 − 1.11803i 0 0.500000 − 0.866025i 2.82843i 0 3.16228
701.1 −1.22474 + 0.707107i 0 1.00000 − 1.73205i −1.93649 − 1.11803i 0 0.500000 + 0.866025i 2.82843i 0 3.16228
701.2 −1.22474 + 0.707107i 0 1.00000 − 1.73205i 1.93649 + 1.11803i 0 0.500000 + 0.866025i 2.82843i 0 −3.16228
701.3 1.22474 − 0.707107i 0 1.00000 − 1.73205i −1.93649 − 1.11803i 0 0.500000 + 0.866025i − 2.82843i 0 −3.16228
701.4 1.22474 − 0.707107i 0 1.00000 − 1.73205i 1.93649 + 1.11803i 0 0.500000 + 0.866025i − 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.b 8
3.b odd 2 1 inner 810.3.h.b 8
9.c even 3 1 270.3.d.a ✓ 4
9.c even 3 1 inner 810.3.h.b 8
9.d odd 6 1 270.3.d.a ✓ 4
9.d odd 6 1 inner 810.3.h.b 8
36.f odd 6 1 2160.3.l.f 4
36.h even 6 1 2160.3.l.f 4
45.h odd 6 1 1350.3.d.m 4
45.j even 6 1 1350.3.d.m 4
45.k odd 12 2 1350.3.b.h 8
45.l even 12 2 1350.3.b.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.d.a ✓ 4 9.c even 3 1
270.3.d.a ✓ 4 9.d odd 6 1
810.3.h.b 8 1.a even 1 1 trivial
810.3.h.b 8 3.b odd 2 1 inner
810.3.h.b 8 9.c even 3 1 inner
810.3.h.b 8 9.d odd 6 1 inner
1350.3.b.h 8 45.k odd 12 2
1350.3.b.h 8 45.l even 12 2
1350.3.d.m 4 45.h odd 6 1
1350.3.d.m 4 45.j even 6 1
2160.3.l.f 4 36.f odd 6 1
2160.3.l.f 4 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - T_{7} + 1 \) 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
$7$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 180 T^{2} + 32400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 10 T^{3} + \cdots + 112225)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 14 T - 311)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 936 T^{6} + \cdots + 136048896 \) Copy content Toggle raw display
$29$ \( T^{8} - 504 T^{6} + \cdots + 136048896 \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{3} + \cdots + 2062096)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 10 T - 335)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 6887475360000 \) Copy content Toggle raw display
$43$ \( (T^{4} + 76 T^{3} + \cdots + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 1360488960000 \) Copy content Toggle raw display
$53$ \( (T^{4} + 12024 T^{2} + 31945104)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 72 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 50 T^{3} + \cdots + 26368225)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 71 T + 5041)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 16056 T^{2} + 944784)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 82 T + 1321)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 38 T^{3} + \cdots + 74632321)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 784294811709696 \) Copy content Toggle raw display
$89$ \( (T^{4} + 23184 T^{2} + 34012224)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 10 T^{3} + \cdots + 80550625)^{2} \) Copy content Toggle raw display
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