Properties

Label 810.4.e.i
Level $810$
Weight $4$
Character orbit 810.e
Analytic conductor $47.792$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [810,4,Mod(271,810)]
 
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([2, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.271");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 810.e (of order \(3\), 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: \(47.7915471046\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + 8 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + 8 q^{8} - 10 q^{10} + ( - 48 \zeta_{6} + 48) q^{11} - 2 \zeta_{6} q^{13} + 8 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} - 114 q^{17} + 140 q^{19} + ( - 20 \zeta_{6} + 20) q^{20} + 96 \zeta_{6} q^{22} - 72 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + 4 q^{26} - 16 q^{28} + (210 \zeta_{6} - 210) q^{29} - 272 \zeta_{6} q^{31} - 32 \zeta_{6} q^{32} + ( - 228 \zeta_{6} + 228) q^{34} + 20 q^{35} - 334 q^{37} + (280 \zeta_{6} - 280) q^{38} + 40 \zeta_{6} q^{40} + 198 \zeta_{6} q^{41} + ( - 268 \zeta_{6} + 268) q^{43} - 192 q^{44} + 144 q^{46} + (216 \zeta_{6} - 216) q^{47} + 327 \zeta_{6} q^{49} - 50 \zeta_{6} q^{50} + (8 \zeta_{6} - 8) q^{52} - 78 q^{53} + 240 q^{55} + ( - 32 \zeta_{6} + 32) q^{56} - 420 \zeta_{6} q^{58} - 240 \zeta_{6} q^{59} + (302 \zeta_{6} - 302) q^{61} + 544 q^{62} + 64 q^{64} + ( - 10 \zeta_{6} + 10) q^{65} - 596 \zeta_{6} q^{67} + 456 \zeta_{6} q^{68} + (40 \zeta_{6} - 40) q^{70} - 768 q^{71} - 478 q^{73} + ( - 668 \zeta_{6} + 668) q^{74} - 560 \zeta_{6} q^{76} - 192 \zeta_{6} q^{77} + ( - 640 \zeta_{6} + 640) q^{79} - 80 q^{80} - 396 q^{82} + ( - 348 \zeta_{6} + 348) q^{83} - 570 \zeta_{6} q^{85} + 536 \zeta_{6} q^{86} + ( - 384 \zeta_{6} + 384) q^{88} + 210 q^{89} - 8 q^{91} + (288 \zeta_{6} - 288) q^{92} - 432 \zeta_{6} q^{94} + 700 \zeta_{6} q^{95} + ( - 1534 \zeta_{6} + 1534) q^{97} - 654 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{4} + 5 q^{5} + 4 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{4} + 5 q^{5} + 4 q^{7} + 16 q^{8} - 20 q^{10} + 48 q^{11} - 2 q^{13} + 8 q^{14} - 16 q^{16} - 228 q^{17} + 280 q^{19} + 20 q^{20} + 96 q^{22} - 72 q^{23} - 25 q^{25} + 8 q^{26} - 32 q^{28} - 210 q^{29} - 272 q^{31} - 32 q^{32} + 228 q^{34} + 40 q^{35} - 668 q^{37} - 280 q^{38} + 40 q^{40} + 198 q^{41} + 268 q^{43} - 384 q^{44} + 288 q^{46} - 216 q^{47} + 327 q^{49} - 50 q^{50} - 8 q^{52} - 156 q^{53} + 480 q^{55} + 32 q^{56} - 420 q^{58} - 240 q^{59} - 302 q^{61} + 1088 q^{62} + 128 q^{64} + 10 q^{65} - 596 q^{67} + 456 q^{68} - 40 q^{70} - 1536 q^{71} - 956 q^{73} + 668 q^{74} - 560 q^{76} - 192 q^{77} + 640 q^{79} - 160 q^{80} - 792 q^{82} + 348 q^{83} - 570 q^{85} + 536 q^{86} + 384 q^{88} + 420 q^{89} - 16 q^{91} - 288 q^{92} - 432 q^{94} + 700 q^{95} + 1534 q^{97} - 1308 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(487\) \(731\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.00000 1.73205i 0 −2.00000 + 3.46410i 2.50000 4.33013i 0 2.00000 + 3.46410i 8.00000 0 −10.0000
541.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i 2.50000 + 4.33013i 0 2.00000 3.46410i 8.00000 0 −10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.4.e.i 2
3.b odd 2 1 810.4.e.p 2
9.c even 3 1 30.4.a.b 1
9.c even 3 1 inner 810.4.e.i 2
9.d odd 6 1 90.4.a.c 1
9.d odd 6 1 810.4.e.p 2
36.f odd 6 1 240.4.a.b 1
36.h even 6 1 720.4.a.y 1
45.h odd 6 1 450.4.a.r 1
45.j even 6 1 150.4.a.b 1
45.k odd 12 2 150.4.c.c 2
45.l even 12 2 450.4.c.j 2
63.l odd 6 1 1470.4.a.r 1
72.n even 6 1 960.4.a.n 1
72.p odd 6 1 960.4.a.bg 1
180.p odd 6 1 1200.4.a.ba 1
180.x even 12 2 1200.4.f.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.4.a.b 1 9.c even 3 1
90.4.a.c 1 9.d odd 6 1
150.4.a.b 1 45.j even 6 1
150.4.c.c 2 45.k odd 12 2
240.4.a.b 1 36.f odd 6 1
450.4.a.r 1 45.h odd 6 1
450.4.c.j 2 45.l even 12 2
720.4.a.y 1 36.h even 6 1
810.4.e.i 2 1.a even 1 1 trivial
810.4.e.i 2 9.c even 3 1 inner
810.4.e.p 2 3.b odd 2 1
810.4.e.p 2 9.d odd 6 1
960.4.a.n 1 72.n even 6 1
960.4.a.bg 1 72.p odd 6 1
1200.4.a.ba 1 180.p odd 6 1
1200.4.f.r 2 180.x even 12 2
1470.4.a.r 1 63.l odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{2} - 4T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} - 48T_{11} + 2304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$11$ \( T^{2} - 48T + 2304 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$17$ \( (T + 114)^{2} \) Copy content Toggle raw display
$19$ \( (T - 140)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 72T + 5184 \) Copy content Toggle raw display
$29$ \( T^{2} + 210T + 44100 \) Copy content Toggle raw display
$31$ \( T^{2} + 272T + 73984 \) Copy content Toggle raw display
$37$ \( (T + 334)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 198T + 39204 \) Copy content Toggle raw display
$43$ \( T^{2} - 268T + 71824 \) Copy content Toggle raw display
$47$ \( T^{2} + 216T + 46656 \) Copy content Toggle raw display
$53$ \( (T + 78)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 240T + 57600 \) Copy content Toggle raw display
$61$ \( T^{2} + 302T + 91204 \) Copy content Toggle raw display
$67$ \( T^{2} + 596T + 355216 \) Copy content Toggle raw display
$71$ \( (T + 768)^{2} \) Copy content Toggle raw display
$73$ \( (T + 478)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 640T + 409600 \) Copy content Toggle raw display
$83$ \( T^{2} - 348T + 121104 \) Copy content Toggle raw display
$89$ \( (T - 210)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 1534 T + 2353156 \) Copy content Toggle raw display
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