Properties

Label 405.4.e.e
Level $405$
Weight $4$
Character orbit 405.e
Analytic conductor $23.896$
Analytic rank $0$
Dimension $2$
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] = [405,4,Mod(136,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.136");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.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: \(23.8957735523\)
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 135)
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.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} + 4 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} + 4 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} - 24 q^{8} - 10 q^{10} + ( - 10 \zeta_{6} + 10) q^{11} + 80 \zeta_{6} q^{13} + ( - 16 \zeta_{6} + 16) q^{16} - 7 q^{17} - 113 q^{19} + (20 \zeta_{6} - 20) q^{20} + 20 \zeta_{6} q^{22} - 81 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} - 160 q^{26} + (220 \zeta_{6} - 220) q^{29} + 189 \zeta_{6} q^{31} - 160 \zeta_{6} q^{32} + ( - 14 \zeta_{6} + 14) q^{34} + 170 q^{37} + ( - 226 \zeta_{6} + 226) q^{38} - 120 \zeta_{6} q^{40} - 130 \zeta_{6} q^{41} + (10 \zeta_{6} - 10) q^{43} + 40 q^{44} + 162 q^{46} + ( - 160 \zeta_{6} + 160) q^{47} + 343 \zeta_{6} q^{49} - 50 \zeta_{6} q^{50} + (320 \zeta_{6} - 320) q^{52} - 631 q^{53} + 50 q^{55} - 440 \zeta_{6} q^{58} - 560 \zeta_{6} q^{59} + (229 \zeta_{6} - 229) q^{61} - 378 q^{62} + 448 q^{64} + (400 \zeta_{6} - 400) q^{65} - 750 \zeta_{6} q^{67} - 28 \zeta_{6} q^{68} - 890 q^{71} - 890 q^{73} + (340 \zeta_{6} - 340) q^{74} - 452 \zeta_{6} q^{76} + ( - 27 \zeta_{6} + 27) q^{79} + 80 q^{80} + 260 q^{82} + ( - 429 \zeta_{6} + 429) q^{83} - 35 \zeta_{6} q^{85} - 20 \zeta_{6} q^{86} + (240 \zeta_{6} - 240) q^{88} + 750 q^{89} + ( - 324 \zeta_{6} + 324) q^{92} + 320 \zeta_{6} q^{94} - 565 \zeta_{6} q^{95} + ( - 1480 \zeta_{6} + 1480) q^{97} - 686 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 4 q^{4} + 5 q^{5} - 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 4 q^{4} + 5 q^{5} - 48 q^{8} - 20 q^{10} + 10 q^{11} + 80 q^{13} + 16 q^{16} - 14 q^{17} - 226 q^{19} - 20 q^{20} + 20 q^{22} - 81 q^{23} - 25 q^{25} - 320 q^{26} - 220 q^{29} + 189 q^{31} - 160 q^{32} + 14 q^{34} + 340 q^{37} + 226 q^{38} - 120 q^{40} - 130 q^{41} - 10 q^{43} + 80 q^{44} + 324 q^{46} + 160 q^{47} + 343 q^{49} - 50 q^{50} - 320 q^{52} - 1262 q^{53} + 100 q^{55} - 440 q^{58} - 560 q^{59} - 229 q^{61} - 756 q^{62} + 896 q^{64} - 400 q^{65} - 750 q^{67} - 28 q^{68} - 1780 q^{71} - 1780 q^{73} - 340 q^{74} - 452 q^{76} + 27 q^{79} + 160 q^{80} + 520 q^{82} + 429 q^{83} - 35 q^{85} - 20 q^{86} - 240 q^{88} + 1500 q^{89} + 324 q^{92} + 320 q^{94} - 565 q^{95} + 1480 q^{97} - 1372 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 + 1.73205i 0 2.00000 + 3.46410i 2.50000 + 4.33013i 0 0 −24.0000 0 −10.0000
271.1 −1.00000 1.73205i 0 2.00000 3.46410i 2.50000 4.33013i 0 0 −24.0000 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 405.4.e.e 2
3.b odd 2 1 405.4.e.j 2
9.c even 3 1 135.4.a.d yes 1
9.c even 3 1 inner 405.4.e.e 2
9.d odd 6 1 135.4.a.a 1
9.d odd 6 1 405.4.e.j 2
36.f odd 6 1 2160.4.a.d 1
36.h even 6 1 2160.4.a.n 1
45.h odd 6 1 675.4.a.i 1
45.j even 6 1 675.4.a.b 1
45.k odd 12 2 675.4.b.c 2
45.l even 12 2 675.4.b.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.a 1 9.d odd 6 1
135.4.a.d yes 1 9.c even 3 1
405.4.e.e 2 1.a even 1 1 trivial
405.4.e.e 2 9.c even 3 1 inner
405.4.e.j 2 3.b odd 2 1
405.4.e.j 2 9.d odd 6 1
675.4.a.b 1 45.j even 6 1
675.4.a.i 1 45.h odd 6 1
675.4.b.c 2 45.k odd 12 2
675.4.b.d 2 45.l even 12 2
2160.4.a.d 1 36.f odd 6 1
2160.4.a.n 1 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{2} + 2T_{2} + 4 \) Copy content Toggle raw display
\( T_{7} \) 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
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$13$ \( T^{2} - 80T + 6400 \) Copy content Toggle raw display
$17$ \( (T + 7)^{2} \) Copy content Toggle raw display
$19$ \( (T + 113)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 81T + 6561 \) Copy content Toggle raw display
$29$ \( T^{2} + 220T + 48400 \) Copy content Toggle raw display
$31$ \( T^{2} - 189T + 35721 \) Copy content Toggle raw display
$37$ \( (T - 170)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 130T + 16900 \) Copy content Toggle raw display
$43$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$47$ \( T^{2} - 160T + 25600 \) Copy content Toggle raw display
$53$ \( (T + 631)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 560T + 313600 \) Copy content Toggle raw display
$61$ \( T^{2} + 229T + 52441 \) Copy content Toggle raw display
$67$ \( T^{2} + 750T + 562500 \) Copy content Toggle raw display
$71$ \( (T + 890)^{2} \) Copy content Toggle raw display
$73$ \( (T + 890)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 27T + 729 \) Copy content Toggle raw display
$83$ \( T^{2} - 429T + 184041 \) Copy content Toggle raw display
$89$ \( (T - 750)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 1480 T + 2190400 \) Copy content Toggle raw display
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