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Properties

Label 810.2.c.e
Level $810$
Weight $2$
Character orbit 810.c
Analytic conductor $6.468$
Analytic rank $0$
Dimension $4$
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] = [810,2,Mod(649,810)]
 
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([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 810.c (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: \(6.46788256372\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.22474 − 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
−1.22474 − 1.22474i
− 1.00000i 0 −1.00000 −2.22474 + 0.224745i 0 0.449490i 1.00000i 0 0.224745 + 2.22474i
649.2 − 1.00000i 0 −1.00000 0.224745 − 2.22474i 0 − 4.44949i 1.00000i 0 −2.22474 − 0.224745i
649.3 1.00000i 0 −1.00000 −2.22474 − 0.224745i 0 − 0.449490i − 1.00000i 0 0.224745 − 2.22474i
649.4 1.00000i 0 −1.00000 0.224745 + 2.22474i 0 4.44949i − 1.00000i 0 −2.22474 + 0.224745i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.2.c.e 4
3.b odd 2 1 810.2.c.f 4
5.b even 2 1 inner 810.2.c.e 4
5.c odd 4 1 4050.2.a.bm 2
5.c odd 4 1 4050.2.a.bz 2
9.c even 3 2 270.2.i.b 8
9.d odd 6 2 90.2.i.b ✓ 8
15.d odd 2 1 810.2.c.f 4
15.e even 4 1 4050.2.a.bq 2
15.e even 4 1 4050.2.a.bs 2
36.f odd 6 2 2160.2.by.d 8
36.h even 6 2 720.2.by.c 8
45.h odd 6 2 90.2.i.b ✓ 8
45.j even 6 2 270.2.i.b 8
45.k odd 12 2 1350.2.e.j 4
45.k odd 12 2 1350.2.e.m 4
45.l even 12 2 450.2.e.k 4
45.l even 12 2 450.2.e.n 4
180.n even 6 2 720.2.by.c 8
180.p odd 6 2 2160.2.by.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.i.b ✓ 8 9.d odd 6 2
90.2.i.b ✓ 8 45.h odd 6 2
270.2.i.b 8 9.c even 3 2
270.2.i.b 8 45.j even 6 2
450.2.e.k 4 45.l even 12 2
450.2.e.n 4 45.l even 12 2
720.2.by.c 8 36.h even 6 2
720.2.by.c 8 180.n even 6 2
810.2.c.e 4 1.a even 1 1 trivial
810.2.c.e 4 5.b even 2 1 inner
810.2.c.f 4 3.b odd 2 1
810.2.c.f 4 15.d odd 2 1
1350.2.e.j 4 45.k odd 12 2
1350.2.e.m 4 45.k odd 12 2
2160.2.by.d 8 36.f odd 6 2
2160.2.by.d 8 180.p odd 6 2
4050.2.a.bm 2 5.c odd 4 1
4050.2.a.bq 2 15.e even 4 1
4050.2.a.bs 2 15.e even 4 1
4050.2.a.bz 2 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{4} + 20T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} + 20T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T - 5)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 50T^{2} + 529 \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T + 3)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 56T^{2} + 400 \) Copy content Toggle raw display
$29$ \( (T + 6)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T + 10)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T - 1)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 62T^{2} + 361 \) Copy content Toggle raw display
$47$ \( T^{4} + 20T^{2} + 4 \) Copy content Toggle raw display
$53$ \( T^{4} + 84T^{2} + 900 \) Copy content Toggle raw display
$59$ \( (T^{2} - 2 T - 149)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 110T^{2} + 1849 \) Copy content Toggle raw display
$71$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 242T^{2} + 5041 \) Copy content Toggle raw display
$79$ \( (T^{2} - 54)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 16 T + 40)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 169)^{2} \) Copy content Toggle raw display
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