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Properties

Label 405.3.h.h
Level $405$
Weight $3$
Character orbit 405.h
Analytic conductor $11.035$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [405,3,Mod(134,405)]
 
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([5, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.134");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.h (of order \(6\), 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: \(11.0354507066\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} - 3 \beta_{2} q^{4} + ( - \beta_{2} - \beta_1) q^{5} + ( - 2 \beta_{3} - \beta_{2} - 1) q^{7} + 7 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} - 3 \beta_{2} q^{4} + ( - \beta_{2} - \beta_1) q^{5} + ( - 2 \beta_{3} - \beta_{2} - 1) q^{7} + 7 q^{8} + (\beta_{3} - \beta_1 + 1) q^{10} + ( - 2 \beta_{3} - \beta_{2} - 1) q^{11} + (2 \beta_{2} + 4 \beta_1) q^{13} + ( - \beta_{2} - 2 \beta_1) q^{14} + ( - 5 \beta_{2} - 5) q^{16} - 22 q^{17} - 4 q^{19} + ( - 3 \beta_{3} - 3 \beta_{2} - 3) q^{20} + ( - \beta_{2} - 2 \beta_1) q^{22} + 20 \beta_{2} q^{23} + ( - \beta_{3} + 24 \beta_{2} + 24) q^{25} + ( - 4 \beta_{3} + 4 \beta_1 - 2) q^{26} + ( - 6 \beta_{3} + 6 \beta_1 - 3) q^{28} + ( - 8 \beta_{3} - 4 \beta_{2} - 4) q^{29} + 29 \beta_{2} q^{31} - 33 \beta_{2} q^{32} + ( - 22 \beta_{2} - 22) q^{34} + ( - \beta_{3} + \beta_1 + 49) q^{35} + ( - 4 \beta_{2} - 4) q^{38} + ( - 7 \beta_{2} - 7 \beta_1) q^{40} + ( - 4 \beta_{2} - 8 \beta_1) q^{41} + (4 \beta_{3} + 2 \beta_{2} + 2) q^{43} + ( - 6 \beta_{3} + 6 \beta_1 - 3) q^{44} - 20 q^{46} + (58 \beta_{2} + 58) q^{47} - 50 \beta_{2} q^{49} + (24 \beta_{2} - \beta_1) q^{50} + (12 \beta_{3} + 6 \beta_{2} + 6) q^{52} - 31 q^{53} + ( - \beta_{3} + \beta_1 + 49) q^{55} + ( - 14 \beta_{3} - 7 \beta_{2} - 7) q^{56} + ( - 4 \beta_{2} - 8 \beta_1) q^{58} + ( - 4 \beta_{2} - 8 \beta_1) q^{59} + ( - 44 \beta_{2} - 44) q^{61} - 29 q^{62} + 13 q^{64} + (2 \beta_{3} - 98 \beta_{2} - 98) q^{65} + (2 \beta_{2} + 4 \beta_1) q^{67} + 66 \beta_{2} q^{68} + ( - \beta_{3} + 49 \beta_{2} + 49) q^{70} + (12 \beta_{3} - 12 \beta_1 + 6) q^{71} + (18 \beta_{3} - 18 \beta_1 + 9) q^{73} + 12 \beta_{2} q^{76} - 99 \beta_{2} q^{77} + (10 \beta_{2} + 10) q^{79} + ( - 5 \beta_{3} + 5 \beta_1 - 5) q^{80} + (8 \beta_{3} - 8 \beta_1 + 4) q^{82} + (19 \beta_{2} + 19) q^{83} + (22 \beta_{2} + 22 \beta_1) q^{85} + (2 \beta_{2} + 4 \beta_1) q^{86} + ( - 14 \beta_{3} - 7 \beta_{2} - 7) q^{88} + (12 \beta_{3} - 12 \beta_1 + 6) q^{89} - 198 q^{91} + (60 \beta_{2} + 60) q^{92} + 58 \beta_{2} q^{94} + (4 \beta_{2} + 4 \beta_1) q^{95} + ( - 26 \beta_{3} - 13 \beta_{2} - 13) q^{97} + 50 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 6 q^{4} + q^{5} + 28 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 6 q^{4} + q^{5} + 28 q^{8} + 2 q^{10} - 10 q^{16} - 88 q^{17} - 16 q^{19} - 3 q^{20} - 40 q^{23} + 49 q^{25} - 58 q^{31} + 66 q^{32} - 44 q^{34} + 198 q^{35} - 8 q^{38} + 7 q^{40} - 80 q^{46} + 116 q^{47} + 100 q^{49} - 49 q^{50} - 124 q^{53} + 198 q^{55} - 88 q^{61} - 116 q^{62} + 52 q^{64} - 198 q^{65} - 132 q^{68} + 99 q^{70} - 24 q^{76} + 198 q^{77} + 20 q^{79} - 10 q^{80} + 38 q^{83} - 22 q^{85} - 792 q^{91} + 120 q^{92} - 116 q^{94} - 4 q^{95} + 200 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} + 16\nu - 9 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 9 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{3} + \nu^{2} + 8\nu + 12 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 8\beta_{2} + 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + 2\beta _1 + 11 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(82\) \(326\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
134.1
1.68614 + 0.396143i
−1.18614 − 1.26217i
1.68614 − 0.396143i
−1.18614 + 1.26217i
0.500000 + 0.866025i 0 1.50000 − 2.59808i −4.05842 − 2.92048i 0 −8.61684 + 4.97494i 7.00000 0 0.500000 − 4.97494i
134.2 0.500000 + 0.866025i 0 1.50000 − 2.59808i 4.55842 + 2.05446i 0 8.61684 − 4.97494i 7.00000 0 0.500000 + 4.97494i
269.1 0.500000 − 0.866025i 0 1.50000 + 2.59808i −4.05842 + 2.92048i 0 −8.61684 − 4.97494i 7.00000 0 0.500000 + 4.97494i
269.2 0.500000 − 0.866025i 0 1.50000 + 2.59808i 4.55842 − 2.05446i 0 8.61684 + 4.97494i 7.00000 0 0.500000 − 4.97494i
\(n\): 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
15.d odd 2 1 inner
45.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.3.h.h 4
3.b odd 2 1 405.3.h.c 4
5.b even 2 1 405.3.h.c 4
9.c even 3 1 135.3.d.a ✓ 2
9.c even 3 1 inner 405.3.h.h 4
9.d odd 6 1 135.3.d.f yes 2
9.d odd 6 1 405.3.h.c 4
15.d odd 2 1 inner 405.3.h.h 4
36.f odd 6 1 2160.3.c.c 2
36.h even 6 1 2160.3.c.d 2
45.h odd 6 1 135.3.d.a ✓ 2
45.h odd 6 1 inner 405.3.h.h 4
45.j even 6 1 135.3.d.f yes 2
45.j even 6 1 405.3.h.c 4
45.k odd 12 2 675.3.c.q 4
45.l even 12 2 675.3.c.q 4
180.n even 6 1 2160.3.c.c 2
180.p odd 6 1 2160.3.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.3.d.a ✓ 2 9.c even 3 1
135.3.d.a ✓ 2 45.h odd 6 1
135.3.d.f yes 2 9.d odd 6 1
135.3.d.f yes 2 45.j even 6 1
405.3.h.c 4 3.b odd 2 1
405.3.h.c 4 5.b even 2 1
405.3.h.c 4 9.d odd 6 1
405.3.h.c 4 45.j even 6 1
405.3.h.h 4 1.a even 1 1 trivial
405.3.h.h 4 9.c even 3 1 inner
405.3.h.h 4 15.d odd 2 1 inner
405.3.h.h 4 45.h odd 6 1 inner
675.3.c.q 4 45.k odd 12 2
675.3.c.q 4 45.l even 12 2
2160.3.c.c 2 36.f odd 6 1
2160.3.c.c 2 180.n even 6 1
2160.3.c.d 2 36.h even 6 1
2160.3.c.d 2 180.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} - 99T_{7}^{2} + 9801 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - T^{3} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{4} - 99T^{2} + 9801 \) Copy content Toggle raw display
$11$ \( T^{4} - 99T^{2} + 9801 \) Copy content Toggle raw display
$13$ \( T^{4} - 396 T^{2} + 156816 \) Copy content Toggle raw display
$17$ \( (T + 22)^{4} \) Copy content Toggle raw display
$19$ \( (T + 4)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 20 T + 400)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 1584 T^{2} + 2509056 \) Copy content Toggle raw display
$31$ \( (T^{2} + 29 T + 841)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 1584 T^{2} + 2509056 \) Copy content Toggle raw display
$43$ \( T^{4} - 396 T^{2} + 156816 \) Copy content Toggle raw display
$47$ \( (T^{2} - 58 T + 3364)^{2} \) Copy content Toggle raw display
$53$ \( (T + 31)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 1584 T^{2} + 2509056 \) Copy content Toggle raw display
$61$ \( (T^{2} + 44 T + 1936)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 396 T^{2} + 156816 \) Copy content Toggle raw display
$71$ \( (T^{2} + 3564)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 8019)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 19 T + 361)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 3564)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 16731 T^{2} + 279926361 \) Copy content Toggle raw display
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