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Properties

Label 405.3.h.d
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: \(\Q(\zeta_{12})\)
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: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{2} + 3 \beta_{2} q^{4} + ( - \beta_{3} + 4 \beta_{2} + \beta_1) q^{5} - 2 \beta_1 q^{7} - 7 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{2} + 3 \beta_{2} q^{4} + ( - \beta_{3} + 4 \beta_{2} + \beta_1) q^{5} - 2 \beta_1 q^{7} - 7 q^{8} + (\beta_{3} - 4) q^{10} - 7 \beta_1 q^{11} + (5 \beta_{3} - 5 \beta_1) q^{13} + ( - 2 \beta_{3} + 2 \beta_1) q^{14} + (5 \beta_{2} - 5) q^{16} - 23 q^{17} + 14 q^{19} + (12 \beta_{2} + 3 \beta_1 - 12) q^{20} + ( - 7 \beta_{3} + 7 \beta_1) q^{22} - 7 \beta_{2} q^{23} + (7 \beta_{2} + 8 \beta_1 - 7) q^{25} - 5 \beta_{3} q^{26} - 6 \beta_{3} q^{28} - \beta_1 q^{29} + 25 \beta_{2} q^{31} - 33 \beta_{2} q^{32} + ( - 23 \beta_{2} + 23) q^{34} + ( - 8 \beta_{3} - 18) q^{35} + 18 \beta_{3} q^{37} + (14 \beta_{2} - 14) q^{38} + (7 \beta_{3} - 28 \beta_{2} - 7 \beta_1) q^{40} + ( - 8 \beta_{3} + 8 \beta_1) q^{41} - 5 \beta_1 q^{43} - 21 \beta_{3} q^{44} + 7 q^{46} + (49 \beta_{2} - 49) q^{47} - 13 \beta_{2} q^{49} + (8 \beta_{3} - 7 \beta_{2} - 8 \beta_1) q^{50} - 15 \beta_1 q^{52} - 14 q^{53} + ( - 28 \beta_{3} - 63) q^{55} + 14 \beta_1 q^{56} + ( - \beta_{3} + \beta_1) q^{58} + (10 \beta_{3} - 10 \beta_1) q^{59} + (44 \beta_{2} - 44) q^{61} - 25 q^{62} + 13 q^{64} + (45 \beta_{2} - 20 \beta_1 - 45) q^{65} + ( - 22 \beta_{3} + 22 \beta_1) q^{67} - 69 \beta_{2} q^{68} + ( - 18 \beta_{2} + 8 \beta_1 + 18) q^{70} + 6 \beta_{3} q^{71} - 18 \beta_1 q^{74} + 42 \beta_{2} q^{76} + 126 \beta_{2} q^{77} + ( - 37 \beta_{2} + 37) q^{79} + (5 \beta_{3} - 20) q^{80} + 8 \beta_{3} q^{82} + ( - 116 \beta_{2} + 116) q^{83} + (23 \beta_{3} - 92 \beta_{2} - 23 \beta_1) q^{85} + ( - 5 \beta_{3} + 5 \beta_1) q^{86} + 49 \beta_1 q^{88} + 42 \beta_{3} q^{89} + 90 q^{91} + ( - 21 \beta_{2} + 21) q^{92} - 49 \beta_{2} q^{94} + ( - 14 \beta_{3} + 56 \beta_{2} + 14 \beta_1) q^{95} - 26 \beta_1 q^{97} + 13 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 6 q^{4} + 8 q^{5} - 28 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 6 q^{4} + 8 q^{5} - 28 q^{8} - 16 q^{10} - 10 q^{16} - 92 q^{17} + 56 q^{19} - 24 q^{20} - 14 q^{23} - 14 q^{25} + 50 q^{31} - 66 q^{32} + 46 q^{34} - 72 q^{35} - 28 q^{38} - 56 q^{40} + 28 q^{46} - 98 q^{47} - 26 q^{49} - 14 q^{50} - 56 q^{53} - 252 q^{55} - 88 q^{61} - 100 q^{62} + 52 q^{64} - 90 q^{65} - 138 q^{68} + 36 q^{70} + 84 q^{76} + 252 q^{77} + 74 q^{79} - 80 q^{80} + 232 q^{83} - 184 q^{85} + 360 q^{91} + 42 q^{92} - 98 q^{94} + 112 q^{95} + 52 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

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

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

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

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

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
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.500000 − 0.866025i 0 1.50000 − 2.59808i −0.598076 − 4.96410i 0 5.19615 − 3.00000i −7.00000 0 −4.00000 + 3.00000i
134.2 −0.500000 − 0.866025i 0 1.50000 − 2.59808i 4.59808 − 1.96410i 0 −5.19615 + 3.00000i −7.00000 0 −4.00000 − 3.00000i
269.1 −0.500000 + 0.866025i 0 1.50000 + 2.59808i −0.598076 + 4.96410i 0 5.19615 + 3.00000i −7.00000 0 −4.00000 − 3.00000i
269.2 −0.500000 + 0.866025i 0 1.50000 + 2.59808i 4.59808 + 1.96410i 0 −5.19615 − 3.00000i −7.00000 0 −4.00000 + 3.00000i
\(n\): 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.d 4
3.b odd 2 1 405.3.h.g 4
5.b even 2 1 405.3.h.g 4
9.c even 3 1 135.3.d.e yes 2
9.c even 3 1 inner 405.3.h.d 4
9.d odd 6 1 135.3.d.b ✓ 2
9.d odd 6 1 405.3.h.g 4
15.d odd 2 1 inner 405.3.h.d 4
36.f odd 6 1 2160.3.c.b 2
36.h even 6 1 2160.3.c.e 2
45.h odd 6 1 135.3.d.e yes 2
45.h odd 6 1 inner 405.3.h.d 4
45.j even 6 1 135.3.d.b ✓ 2
45.j even 6 1 405.3.h.g 4
45.k odd 12 1 675.3.c.l 2
45.k odd 12 1 675.3.c.m 2
45.l even 12 1 675.3.c.l 2
45.l even 12 1 675.3.c.m 2
180.n even 6 1 2160.3.c.b 2
180.p odd 6 1 2160.3.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.3.d.b ✓ 2 9.d odd 6 1
135.3.d.b ✓ 2 45.j even 6 1
135.3.d.e yes 2 9.c even 3 1
135.3.d.e yes 2 45.h odd 6 1
405.3.h.d 4 1.a even 1 1 trivial
405.3.h.d 4 9.c even 3 1 inner
405.3.h.d 4 15.d odd 2 1 inner
405.3.h.d 4 45.h odd 6 1 inner
405.3.h.g 4 3.b odd 2 1
405.3.h.g 4 5.b even 2 1
405.3.h.g 4 9.d odd 6 1
405.3.h.g 4 45.j even 6 1
675.3.c.l 2 45.k odd 12 1
675.3.c.l 2 45.l even 12 1
675.3.c.m 2 45.k odd 12 1
675.3.c.m 2 45.l even 12 1
2160.3.c.b 2 36.f odd 6 1
2160.3.c.b 2 180.n even 6 1
2160.3.c.e 2 36.h even 6 1
2160.3.c.e 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
\( T_{7}^{4} - 36T_{7}^{2} + 1296 \) 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
$5$ \( T^{4} - 8 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$11$ \( T^{4} - 441 T^{2} + 194481 \) Copy content Toggle raw display
$13$ \( T^{4} - 225 T^{2} + 50625 \) Copy content Toggle raw display
$17$ \( (T + 23)^{4} \) Copy content Toggle raw display
$19$ \( (T - 14)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$31$ \( (T^{2} - 25 T + 625)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2916)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 576 T^{2} + 331776 \) Copy content Toggle raw display
$43$ \( T^{4} - 225 T^{2} + 50625 \) Copy content Toggle raw display
$47$ \( (T^{2} + 49 T + 2401)^{2} \) Copy content Toggle raw display
$53$ \( (T + 14)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 900 T^{2} + 810000 \) Copy content Toggle raw display
$61$ \( (T^{2} + 44 T + 1936)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 4356 T^{2} + 18974736 \) Copy content Toggle raw display
$71$ \( (T^{2} + 324)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 37 T + 1369)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 116 T + 13456)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 15876)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 6084 T^{2} + 37015056 \) Copy content Toggle raw display
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