Properties

Label 405.4.e.p
Level $405$
Weight $4$
Character orbit 405.e
Analytic conductor $23.896$
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] = [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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
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 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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + (2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 4) q^{4} + ( - 5 \beta_1 + 5) q^{5} + 4 \beta_{2} q^{7} + ( - 10 \beta_{3} + 6) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + (2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 4) q^{4} + ( - 5 \beta_1 + 5) q^{5} + 4 \beta_{2} q^{7} + ( - 10 \beta_{3} + 6) q^{8} + ( - 5 \beta_{3} + 5) q^{10} + (28 \beta_{2} - 11 \beta_1) q^{11} + (4 \beta_{3} - 4 \beta_{2} - 32 \beta_1 + 32) q^{13} + ( - 4 \beta_{3} + 4 \beta_{2} - 12 \beta_1 + 12) q^{14} + ( - 32 \beta_{2} + 4 \beta_1) q^{16} + (56 \beta_{3} - 16) q^{17} + (68 \beta_{3} - 5) q^{19} + ( - 10 \beta_{2} - 20 \beta_1) q^{20} + ( - 39 \beta_{3} + 39 \beta_{2} - 95 \beta_1 + 95) q^{22} + (32 \beta_{3} - 32 \beta_{2} + 42 \beta_1 - 42) q^{23} - 25 \beta_1 q^{25} + ( - 28 \beta_{3} + 20) q^{26} + (16 \beta_{3} + 24) q^{28} + (24 \beta_{2} - 85 \beta_1) q^{29} + (12 \beta_{3} - 12 \beta_{2} - 129 \beta_1 + 129) q^{31} + ( - 44 \beta_{3} + 44 \beta_{2} + 52 \beta_1 - 52) q^{32} + (72 \beta_{2} - 184 \beta_1) q^{34} + 20 \beta_{3} q^{35} + ( - 148 \beta_{3} + 38) q^{37} + (73 \beta_{2} - 209 \beta_1) q^{38} + ( - 50 \beta_{3} + 50 \beta_{2} - 30 \beta_1 + 30) q^{40} + ( - 48 \beta_{3} + 48 \beta_{2} + 289 \beta_1 - 289) q^{41} + ( - 156 \beta_{2} - 190 \beta_1) q^{43} + (90 \beta_{3} + 124) q^{44} + (74 \beta_{3} - 138) q^{46} + (16 \beta_{2} - 242 \beta_1) q^{47} + ( - 295 \beta_1 + 295) q^{49} + ( - 25 \beta_{3} + 25 \beta_{2} - 25 \beta_1 + 25) q^{50} + ( - 80 \beta_{2} - 152 \beta_1) q^{52} + (100 \beta_{3} + 272) q^{53} + (140 \beta_{3} - 55) q^{55} + (24 \beta_{2} - 120 \beta_1) q^{56} + ( - 109 \beta_{3} + 109 \beta_{2} - 157 \beta_1 + 157) q^{58} + ( - 28 \beta_{3} + 28 \beta_{2} + 353 \beta_1 - 353) q^{59} + (192 \beta_{2} - 334 \beta_1) q^{61} + ( - 117 \beta_{3} + 93) q^{62} + ( - 248 \beta_{3} + 112) q^{64} + ( - 20 \beta_{2} - 160 \beta_1) q^{65} + (52 \beta_{3} - 52 \beta_{2} - 726 \beta_1 + 726) q^{67} + (192 \beta_{3} - 192 \beta_{2} - 272 \beta_1 + 272) q^{68} + (20 \beta_{2} - 60 \beta_1) q^{70} + ( - 196 \beta_{3} + 487) q^{71} + ( - 92 \beta_{3} + 592) q^{73} + ( - 186 \beta_{2} + 482 \beta_1) q^{74} + (262 \beta_{3} - 262 \beta_{2} - 388 \beta_1 + 388) q^{76} + (44 \beta_{3} - 44 \beta_{2} + 336 \beta_1 - 336) q^{77} + ( - 104 \beta_{2} - 204 \beta_1) q^{79} + ( - 160 \beta_{3} + 20) q^{80} + (241 \beta_{3} - 145) q^{82} + ( - 600 \beta_{2} - 222 \beta_1) q^{83} + (280 \beta_{3} - 280 \beta_{2} + 80 \beta_1 - 80) q^{85} + ( - 34 \beta_{3} + 34 \beta_{2} + 278 \beta_1 - 278) q^{86} + (278 \beta_{2} - 906 \beta_1) q^{88} + 513 q^{89} + (128 \beta_{3} + 48) q^{91} + ( - 44 \beta_{2} - 24 \beta_1) q^{92} + ( - 258 \beta_{3} + 258 \beta_{2} - 290 \beta_1 + 290) q^{94} + (340 \beta_{3} - 340 \beta_{2} + 25 \beta_1 - 25) q^{95} + (440 \beta_{2} + 334 \beta_1) q^{97} + ( - 295 \beta_{3} + 295) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 8 q^{4} + 10 q^{5} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 8 q^{4} + 10 q^{5} + 24 q^{8} + 20 q^{10} - 22 q^{11} + 64 q^{13} + 24 q^{14} + 8 q^{16} - 64 q^{17} - 20 q^{19} - 40 q^{20} + 190 q^{22} - 84 q^{23} - 50 q^{25} + 80 q^{26} + 96 q^{28} - 170 q^{29} + 258 q^{31} - 104 q^{32} - 368 q^{34} + 152 q^{37} - 418 q^{38} + 60 q^{40} - 578 q^{41} - 380 q^{43} + 496 q^{44} - 552 q^{46} - 484 q^{47} + 590 q^{49} + 50 q^{50} - 304 q^{52} + 1088 q^{53} - 220 q^{55} - 240 q^{56} + 314 q^{58} - 706 q^{59} - 668 q^{61} + 372 q^{62} + 448 q^{64} - 320 q^{65} + 1452 q^{67} + 544 q^{68} - 120 q^{70} + 1948 q^{71} + 2368 q^{73} + 964 q^{74} + 776 q^{76} - 672 q^{77} - 408 q^{79} + 80 q^{80} - 580 q^{82} - 444 q^{83} - 160 q^{85} - 556 q^{86} - 1812 q^{88} + 2052 q^{89} + 192 q^{91} - 48 q^{92} + 580 q^{94} - 50 q^{95} + 668 q^{97} + 1180 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(82\) \(326\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.366025 + 0.633975i 0 3.73205 + 6.46410i 2.50000 + 4.33013i 0 3.46410 6.00000i −11.3205 0 −3.66025
136.2 1.36603 2.36603i 0 0.267949 + 0.464102i 2.50000 + 4.33013i 0 −3.46410 + 6.00000i 23.3205 0 13.6603
271.1 −0.366025 0.633975i 0 3.73205 6.46410i 2.50000 4.33013i 0 3.46410 + 6.00000i −11.3205 0 −3.66025
271.2 1.36603 + 2.36603i 0 0.267949 0.464102i 2.50000 4.33013i 0 −3.46410 6.00000i 23.3205 0 13.6603
\(n\): 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.p 4
3.b odd 2 1 405.4.e.o 4
9.c even 3 1 405.4.a.c 2
9.c even 3 1 inner 405.4.e.p 4
9.d odd 6 1 405.4.a.f yes 2
9.d odd 6 1 405.4.e.o 4
45.h odd 6 1 2025.4.a.i 2
45.j even 6 1 2025.4.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.c 2 9.c even 3 1
405.4.a.f yes 2 9.d odd 6 1
405.4.e.o 4 3.b odd 2 1
405.4.e.o 4 9.d odd 6 1
405.4.e.p 4 1.a even 1 1 trivial
405.4.e.p 4 9.c even 3 1 inner
2025.4.a.i 2 45.h odd 6 1
2025.4.a.m 2 45.j even 6 1

Hecke kernels

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

\( T_{2}^{4} - 2T_{2}^{3} + 6T_{2}^{2} + 4T_{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} + 48T_{7}^{2} + 2304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$11$ \( T^{4} + 22 T^{3} + 2715 T^{2} + \cdots + 4977361 \) Copy content Toggle raw display
$13$ \( T^{4} - 64 T^{3} + 3120 T^{2} + \cdots + 952576 \) Copy content Toggle raw display
$17$ \( (T^{2} + 32 T - 9152)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 10 T - 13847)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 84 T^{3} + 8364 T^{2} + \cdots + 1710864 \) Copy content Toggle raw display
$29$ \( T^{4} + 170 T^{3} + \cdots + 30217009 \) Copy content Toggle raw display
$31$ \( T^{4} - 258 T^{3} + \cdots + 262731681 \) Copy content Toggle raw display
$37$ \( (T^{2} - 76 T - 64268)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 578 T^{3} + \cdots + 5868938881 \) Copy content Toggle raw display
$43$ \( T^{4} + 380 T^{3} + \cdots + 1362200464 \) Copy content Toggle raw display
$47$ \( T^{4} + 484 T^{3} + \cdots + 3340377616 \) Copy content Toggle raw display
$53$ \( (T^{2} - 544 T + 43984)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 706 T^{3} + \cdots + 14946774049 \) Copy content Toggle raw display
$61$ \( T^{4} + 668 T^{3} + 445260 T^{2} + \cdots + 929296 \) Copy content Toggle raw display
$67$ \( T^{4} - 1452 T^{3} + \cdots + 269323633296 \) Copy content Toggle raw display
$71$ \( (T^{2} - 974 T + 121921)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 1184 T + 325072)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 408 T^{3} + \cdots + 84052224 \) Copy content Toggle raw display
$83$ \( T^{4} + 444 T^{3} + \cdots + 1062375472656 \) Copy content Toggle raw display
$89$ \( (T - 513)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 668 T^{3} + \cdots + 220189931536 \) Copy content Toggle raw display
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