Properties

Label 405.4.e.a
Level $405$
Weight $4$
Character orbit 405.e
Analytic conductor $23.896$
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] = [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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (5 \zeta_{6} - 5) q^{2} - 17 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + (9 \zeta_{6} - 9) q^{7} + 45 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (5 \zeta_{6} - 5) q^{2} - 17 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + (9 \zeta_{6} - 9) q^{7} + 45 q^{8} - 25 q^{10} + (8 \zeta_{6} - 8) q^{11} - 43 \zeta_{6} q^{13} - 45 \zeta_{6} q^{14} + (89 \zeta_{6} - 89) q^{16} + 122 q^{17} - 59 q^{19} + ( - 85 \zeta_{6} + 85) q^{20} - 40 \zeta_{6} q^{22} - 213 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + 215 q^{26} + 153 q^{28} + ( - 224 \zeta_{6} + 224) q^{29} + 36 \zeta_{6} q^{31} - 85 \zeta_{6} q^{32} + (610 \zeta_{6} - 610) q^{34} - 45 q^{35} + 206 q^{37} + ( - 295 \zeta_{6} + 295) q^{38} + 225 \zeta_{6} q^{40} + 413 \zeta_{6} q^{41} + ( - 392 \zeta_{6} + 392) q^{43} + 136 q^{44} + 1065 q^{46} + (311 \zeta_{6} - 311) q^{47} + 262 \zeta_{6} q^{49} - 125 \zeta_{6} q^{50} + (731 \zeta_{6} - 731) q^{52} + 377 q^{53} - 40 q^{55} + (405 \zeta_{6} - 405) q^{56} + 1120 \zeta_{6} q^{58} + 337 \zeta_{6} q^{59} + (40 \zeta_{6} - 40) q^{61} - 180 q^{62} - 287 q^{64} + ( - 215 \zeta_{6} + 215) q^{65} - 348 \zeta_{6} q^{67} - 2074 \zeta_{6} q^{68} + ( - 225 \zeta_{6} + 225) q^{70} - 62 q^{71} - 1214 q^{73} + (1030 \zeta_{6} - 1030) q^{74} + 1003 \zeta_{6} q^{76} - 72 \zeta_{6} q^{77} + ( - 294 \zeta_{6} + 294) q^{79} - 445 q^{80} - 2065 q^{82} + ( - 534 \zeta_{6} + 534) q^{83} + 610 \zeta_{6} q^{85} + 1960 \zeta_{6} q^{86} + (360 \zeta_{6} - 360) q^{88} + 810 q^{89} + 387 q^{91} + (3621 \zeta_{6} - 3621) q^{92} - 1555 \zeta_{6} q^{94} - 295 \zeta_{6} q^{95} + ( - 928 \zeta_{6} + 928) q^{97} - 1310 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 5 q^{2} - 17 q^{4} + 5 q^{5} - 9 q^{7} + 90 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 5 q^{2} - 17 q^{4} + 5 q^{5} - 9 q^{7} + 90 q^{8} - 50 q^{10} - 8 q^{11} - 43 q^{13} - 45 q^{14} - 89 q^{16} + 244 q^{17} - 118 q^{19} + 85 q^{20} - 40 q^{22} - 213 q^{23} - 25 q^{25} + 430 q^{26} + 306 q^{28} + 224 q^{29} + 36 q^{31} - 85 q^{32} - 610 q^{34} - 90 q^{35} + 412 q^{37} + 295 q^{38} + 225 q^{40} + 413 q^{41} + 392 q^{43} + 272 q^{44} + 2130 q^{46} - 311 q^{47} + 262 q^{49} - 125 q^{50} - 731 q^{52} + 754 q^{53} - 80 q^{55} - 405 q^{56} + 1120 q^{58} + 337 q^{59} - 40 q^{61} - 360 q^{62} - 574 q^{64} + 215 q^{65} - 348 q^{67} - 2074 q^{68} + 225 q^{70} - 124 q^{71} - 2428 q^{73} - 1030 q^{74} + 1003 q^{76} - 72 q^{77} + 294 q^{79} - 890 q^{80} - 4130 q^{82} + 534 q^{83} + 610 q^{85} + 1960 q^{86} - 360 q^{88} + 1620 q^{89} + 774 q^{91} - 3621 q^{92} - 1555 q^{94} - 295 q^{95} + 928 q^{97} - 2620 q^{98}+O(q^{100}) \) 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
−2.50000 + 4.33013i 0 −8.50000 14.7224i 2.50000 + 4.33013i 0 −4.50000 + 7.79423i 45.0000 0 −25.0000
271.1 −2.50000 4.33013i 0 −8.50000 + 14.7224i 2.50000 4.33013i 0 −4.50000 7.79423i 45.0000 0 −25.0000
\(n\): 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.a 2
3.b odd 2 1 405.4.e.m 2
9.c even 3 1 405.4.a.b yes 1
9.c even 3 1 inner 405.4.e.a 2
9.d odd 6 1 405.4.a.a 1
9.d odd 6 1 405.4.e.m 2
45.h odd 6 1 2025.4.a.f 1
45.j even 6 1 2025.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.a 1 9.d odd 6 1
405.4.a.b yes 1 9.c even 3 1
405.4.e.a 2 1.a even 1 1 trivial
405.4.e.a 2 9.c even 3 1 inner
405.4.e.m 2 3.b odd 2 1
405.4.e.m 2 9.d odd 6 1
2025.4.a.a 1 45.j even 6 1
2025.4.a.f 1 45.h odd 6 1

Hecke kernels

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

\( T_{2}^{2} + 5T_{2} + 25 \) Copy content Toggle raw display
\( T_{7}^{2} + 9T_{7} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$11$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$13$ \( T^{2} + 43T + 1849 \) Copy content Toggle raw display
$17$ \( (T - 122)^{2} \) Copy content Toggle raw display
$19$ \( (T + 59)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 213T + 45369 \) Copy content Toggle raw display
$29$ \( T^{2} - 224T + 50176 \) Copy content Toggle raw display
$31$ \( T^{2} - 36T + 1296 \) Copy content Toggle raw display
$37$ \( (T - 206)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 413T + 170569 \) Copy content Toggle raw display
$43$ \( T^{2} - 392T + 153664 \) Copy content Toggle raw display
$47$ \( T^{2} + 311T + 96721 \) Copy content Toggle raw display
$53$ \( (T - 377)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 337T + 113569 \) Copy content Toggle raw display
$61$ \( T^{2} + 40T + 1600 \) Copy content Toggle raw display
$67$ \( T^{2} + 348T + 121104 \) Copy content Toggle raw display
$71$ \( (T + 62)^{2} \) Copy content Toggle raw display
$73$ \( (T + 1214)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 294T + 86436 \) Copy content Toggle raw display
$83$ \( T^{2} - 534T + 285156 \) Copy content Toggle raw display
$89$ \( (T - 810)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 928T + 861184 \) Copy content Toggle raw display
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