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Properties

Label 405.3.i.a
Level $405$
Weight $3$
Character orbit 405.i
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(26,405)]
 
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([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.26");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.i (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: \(\Q(\sqrt{-3}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{3} + \beta_1) q^{5} + (12 \beta_{2} - 12) q^{7} - 3 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{3} + \beta_1) q^{5} + (12 \beta_{2} - 12) q^{7} - 3 \beta_{3} q^{8} + 5 q^{10} + 7 \beta_1 q^{11} + 11 \beta_{2} q^{13} + (12 \beta_{3} - 12 \beta_1) q^{14} + ( - 19 \beta_{2} + 19) q^{16} + 11 \beta_{3} q^{17} + 16 q^{19} + \beta_1 q^{20} + 35 \beta_{2} q^{22} + (3 \beta_{3} - 3 \beta_1) q^{23} + ( - 5 \beta_{2} + 5) q^{25} + 11 \beta_{3} q^{26} - 12 q^{28} + 5 \beta_1 q^{29} - 9 \beta_{2} q^{31} + ( - 7 \beta_{3} + 7 \beta_1) q^{32} + (55 \beta_{2} - 55) q^{34} + 12 \beta_{3} q^{35} + 2 q^{37} + 16 \beta_1 q^{38} - 15 \beta_{2} q^{40} + ( - 10 \beta_{3} + 10 \beta_1) q^{41} + ( - 47 \beta_{2} + 47) q^{43} + 7 \beta_{3} q^{44} - 15 q^{46} - 41 \beta_1 q^{47} - 95 \beta_{2} q^{49} + ( - 5 \beta_{3} + 5 \beta_1) q^{50} + (11 \beta_{2} - 11) q^{52} + 38 \beta_{3} q^{53} + 35 q^{55} + 36 \beta_1 q^{56} + 25 \beta_{2} q^{58} + ( - 2 \beta_{3} + 2 \beta_1) q^{59} + ( - 26 \beta_{2} + 26) q^{61} - 9 \beta_{3} q^{62} - 41 q^{64} + 11 \beta_1 q^{65} - 6 \beta_{2} q^{67} + (11 \beta_{3} - 11 \beta_1) q^{68} + (60 \beta_{2} - 60) q^{70} + 16 \beta_{3} q^{71} - 92 q^{73} + 2 \beta_1 q^{74} + 16 \beta_{2} q^{76} + (84 \beta_{3} - 84 \beta_1) q^{77} + ( - 69 \beta_{2} + 69) q^{79} - 19 \beta_{3} q^{80} + 50 q^{82} - 30 \beta_1 q^{83} + 55 \beta_{2} q^{85} + ( - 47 \beta_{3} + 47 \beta_1) q^{86} + ( - 105 \beta_{2} + 105) q^{88} - 24 \beta_{3} q^{89} - 132 q^{91} - 3 \beta_1 q^{92} - 205 \beta_{2} q^{94} + ( - 16 \beta_{3} + 16 \beta_1) q^{95} + ( - 58 \beta_{2} + 58) q^{97} - 95 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 24 q^{7} + 20 q^{10} + 22 q^{13} + 38 q^{16} + 64 q^{19} + 70 q^{22} + 10 q^{25} - 48 q^{28} - 18 q^{31} - 110 q^{34} + 8 q^{37} - 30 q^{40} + 94 q^{43} - 60 q^{46} - 190 q^{49} - 22 q^{52} + 140 q^{55} + 50 q^{58} + 52 q^{61} - 164 q^{64} - 12 q^{67} - 120 q^{70} - 368 q^{73} + 32 q^{76} + 138 q^{79} + 200 q^{82} + 110 q^{85} + 210 q^{88} - 528 q^{91} - 410 q^{94} + 116 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−1.93649 − 1.11803i
1.93649 + 1.11803i
−1.93649 + 1.11803i
1.93649 − 1.11803i
−1.93649 − 1.11803i 0 0.500000 + 0.866025i −1.93649 + 1.11803i 0 −6.00000 + 10.3923i 6.70820i 0 5.00000
26.2 1.93649 + 1.11803i 0 0.500000 + 0.866025i 1.93649 − 1.11803i 0 −6.00000 + 10.3923i − 6.70820i 0 5.00000
296.1 −1.93649 + 1.11803i 0 0.500000 − 0.866025i −1.93649 − 1.11803i 0 −6.00000 − 10.3923i − 6.70820i 0 5.00000
296.2 1.93649 − 1.11803i 0 0.500000 − 0.866025i 1.93649 + 1.11803i 0 −6.00000 − 10.3923i 6.70820i 0 5.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d 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.i.a 4
3.b odd 2 1 inner 405.3.i.a 4
9.c even 3 1 135.3.c.a ✓ 2
9.c even 3 1 inner 405.3.i.a 4
9.d odd 6 1 135.3.c.a ✓ 2
9.d odd 6 1 inner 405.3.i.a 4
36.f odd 6 1 2160.3.l.a 2
36.h even 6 1 2160.3.l.a 2
45.h odd 6 1 675.3.c.i 2
45.j even 6 1 675.3.c.i 2
45.k odd 12 2 675.3.d.f 4
45.l even 12 2 675.3.d.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.3.c.a ✓ 2 9.c even 3 1
135.3.c.a ✓ 2 9.d odd 6 1
405.3.i.a 4 1.a even 1 1 trivial
405.3.i.a 4 3.b odd 2 1 inner
405.3.i.a 4 9.c even 3 1 inner
405.3.i.a 4 9.d odd 6 1 inner
675.3.c.i 2 45.h odd 6 1
675.3.c.i 2 45.j even 6 1
675.3.d.f 4 45.k odd 12 2
675.3.d.f 4 45.l even 12 2
2160.3.l.a 2 36.f odd 6 1
2160.3.l.a 2 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 \(S_{3}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{4} - 5T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{7}^{2} + 12T_{7} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 5T^{2} + 25 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 5T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 12 T + 144)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 245 T^{2} + 60025 \) Copy content Toggle raw display
$13$ \( (T^{2} - 11 T + 121)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 605)^{2} \) Copy content Toggle raw display
$19$ \( (T - 16)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 45T^{2} + 2025 \) Copy content Toggle raw display
$29$ \( T^{4} - 125 T^{2} + 15625 \) Copy content Toggle raw display
$31$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 500 T^{2} + 250000 \) Copy content Toggle raw display
$43$ \( (T^{2} - 47 T + 2209)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 8405 T^{2} + 70644025 \) Copy content Toggle raw display
$53$ \( (T^{2} + 7220)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 20T^{2} + 400 \) Copy content Toggle raw display
$61$ \( (T^{2} - 26 T + 676)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 1280)^{2} \) Copy content Toggle raw display
$73$ \( (T + 92)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 69 T + 4761)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 4500 T^{2} + 20250000 \) Copy content Toggle raw display
$89$ \( (T^{2} + 2880)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 58 T + 3364)^{2} \) Copy content Toggle raw display
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