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Properties

Label 405.3.i.c
Level $405$
Weight $3$
Character orbit 405.i
Analytic conductor $11.035$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{6} - \beta_{3}) q^{4} + (\beta_{7} + \beta_{4} - \beta_{2}) q^{5} + ( - 2 \beta_{6} - 4 \beta_{5}) q^{7} + (\beta_{7} - \beta_{4} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{6} - \beta_{3}) q^{4} + (\beta_{7} + \beta_{4} - \beta_{2}) q^{5} + ( - 2 \beta_{6} - 4 \beta_{5}) q^{7} + (\beta_{7} - \beta_{4} + \beta_1) q^{8} + (\beta_{3} - 3) q^{10} + (4 \beta_{2} - 6 \beta_1) q^{11} + (2 \beta_{6} - 2 \beta_{3}) q^{13} + ( - 2 \beta_{7} - 10 \beta_{4} + 2 \beta_{2}) q^{14} + ( - 5 \beta_{6} - 5 \beta_{5}) q^{16} + (5 \beta_{7} + 3 \beta_{4} - 3 \beta_1) q^{17} + ( - 4 \beta_{3} + 21) q^{19} + (3 \beta_{2} + 2 \beta_1) q^{20} + (6 \beta_{6} + 20 \beta_{5} + \cdots + 20) q^{22}+ \cdots + ( - 12 \beta_{7} - 47 \beta_{4} + 47 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{4} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{4} + 12 q^{7} - 20 q^{10} - 4 q^{13} + 10 q^{16} + 152 q^{19} + 68 q^{22} + 20 q^{25} + 168 q^{28} - 36 q^{31} - 22 q^{34} - 320 q^{37} + 44 q^{43} - 60 q^{46} - 20 q^{49} - 92 q^{52} - 200 q^{55} - 32 q^{58} - 76 q^{61} + 296 q^{64} - 216 q^{67} - 120 q^{70} - 184 q^{73} + 142 q^{76} + 300 q^{79} - 776 q^{82} - 80 q^{85} + 36 q^{88} + 336 q^{91} + 332 q^{94} - 584 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 5\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 17\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{6} + 31 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} - 8\nu^{3} + 3\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{6} - 8\nu^{4} + 24\nu^{2} - 9 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{6} + 10\nu^{4} - 24\nu^{2} + 9 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15\nu^{7} - 40\nu^{5} + 104\nu^{3} - 5\nu ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 5\beta_{5} - \beta_{3} + 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 5\beta_{4} - 5\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{7} + 13\beta_{4} - 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8\beta_{3} - 31 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -5\beta_{2} + 34\beta_1 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(82\) \(326\)
\(\chi(n)\) \(1\) \(1 + \beta_{5}\)

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
−1.40126 − 0.809017i
0.535233 + 0.309017i
−0.535233 − 0.309017i
1.40126 + 0.809017i
−1.40126 + 0.809017i
0.535233 − 0.309017i
−0.535233 + 0.309017i
1.40126 − 0.809017i
−2.26728 − 1.30902i 0 1.42705 + 2.47172i 1.93649 − 1.11803i 0 4.85410 − 8.40755i 3.00000i 0 −5.85410
26.2 −0.330792 − 0.190983i 0 −1.92705 − 3.33775i −1.93649 + 1.11803i 0 −1.85410 + 3.21140i 3.00000i 0 0.854102
26.3 0.330792 + 0.190983i 0 −1.92705 − 3.33775i 1.93649 − 1.11803i 0 −1.85410 + 3.21140i − 3.00000i 0 0.854102
26.4 2.26728 + 1.30902i 0 1.42705 + 2.47172i −1.93649 + 1.11803i 0 4.85410 − 8.40755i − 3.00000i 0 −5.85410
296.1 −2.26728 + 1.30902i 0 1.42705 − 2.47172i 1.93649 + 1.11803i 0 4.85410 + 8.40755i − 3.00000i 0 −5.85410
296.2 −0.330792 + 0.190983i 0 −1.92705 + 3.33775i −1.93649 − 1.11803i 0 −1.85410 − 3.21140i − 3.00000i 0 0.854102
296.3 0.330792 − 0.190983i 0 −1.92705 + 3.33775i 1.93649 + 1.11803i 0 −1.85410 − 3.21140i 3.00000i 0 0.854102
296.4 2.26728 − 1.30902i 0 1.42705 − 2.47172i −1.93649 − 1.11803i 0 4.85410 + 8.40755i 3.00000i 0 −5.85410
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.4
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.c 8
3.b odd 2 1 inner 405.3.i.c 8
9.c even 3 1 135.3.c.c ✓ 4
9.c even 3 1 inner 405.3.i.c 8
9.d odd 6 1 135.3.c.c ✓ 4
9.d odd 6 1 inner 405.3.i.c 8
36.f odd 6 1 2160.3.l.g 4
36.h even 6 1 2160.3.l.g 4
45.h odd 6 1 675.3.c.p 4
45.j even 6 1 675.3.c.p 4
45.k odd 12 1 675.3.d.e 4
45.k odd 12 1 675.3.d.i 4
45.l even 12 1 675.3.d.e 4
45.l even 12 1 675.3.d.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.3.c.c ✓ 4 9.c even 3 1
135.3.c.c ✓ 4 9.d odd 6 1
405.3.i.c 8 1.a even 1 1 trivial
405.3.i.c 8 3.b odd 2 1 inner
405.3.i.c 8 9.c even 3 1 inner
405.3.i.c 8 9.d odd 6 1 inner
675.3.c.p 4 45.h odd 6 1
675.3.c.p 4 45.j even 6 1
675.3.d.e 4 45.k odd 12 1
675.3.d.e 4 45.l even 12 1
675.3.d.i 4 45.k odd 12 1
675.3.d.i 4 45.l even 12 1
2160.3.l.g 4 36.f odd 6 1
2160.3.l.g 4 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 \(S_{3}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{8} - 7T_{2}^{6} + 48T_{2}^{4} - 7T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} - 6T_{7}^{3} + 72T_{7}^{2} + 216T_{7} + 1296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 7 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 6 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 268 T^{6} + \cdots + 181063936 \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 178 T^{2} + 5041)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 38 T + 181)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 810 T^{6} + \cdots + 4100625 \) Copy content Toggle raw display
$29$ \( T^{8} - 3148 T^{6} + \cdots + 181063936 \) Copy content Toggle raw display
$31$ \( (T^{4} + 18 T^{3} + \cdots + 2368521)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 80 T + 1420)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 246907671183616 \) Copy content Toggle raw display
$43$ \( (T^{4} - 22 T^{3} + \cdots + 12418576)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 109656417116416 \) Copy content Toggle raw display
$53$ \( (T^{4} + 6202 T^{2} + 4414201)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 55\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{2} + 19 T + 361)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 108 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 12268 T^{2} + 37405456)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 46 T - 4916)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 150 T^{3} + \cdots + 24059025)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 61377839028801 \) Copy content Toggle raw display
$89$ \( (T^{4} + 8028 T^{2} + 15397776)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 292 T^{3} + \cdots + 446730496)^{2} \) Copy content Toggle raw display
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