Properties

Label 27.5.b.c
Level $27$
Weight $5$
Character orbit 27.b
Analytic conductor $2.791$
Analytic rank $0$
Dimension $2$
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] = [27,5,Mod(26,27)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(27, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.26");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 27.b (of order \(2\), degree \(1\), 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: \(2.79098900326\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 3i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 7 q^{4} + 11 \beta q^{5} - 19 q^{7} + 23 \beta q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 7 q^{4} + 11 \beta q^{5} - 19 q^{7} + 23 \beta q^{8} - 99 q^{10} - 41 \beta q^{11} + 302 q^{13} - 19 \beta q^{14} - 95 q^{16} - 138 \beta q^{17} - 304 q^{19} + 77 \beta q^{20} + 369 q^{22} - 100 \beta q^{23} - 464 q^{25} + 302 \beta q^{26} - 133 q^{28} + 226 \beta q^{29} + 239 q^{31} + 273 \beta q^{32} + 1242 q^{34} - 209 \beta q^{35} + 740 q^{37} - 304 \beta q^{38} - 2277 q^{40} - 76 \beta q^{41} - 982 q^{43} - 287 \beta q^{44} + 900 q^{46} - 722 \beta q^{47} - 2040 q^{49} - 464 \beta q^{50} + 2114 q^{52} + 531 \beta q^{53} + 4059 q^{55} - 437 \beta q^{56} - 2034 q^{58} + 974 \beta q^{59} - 316 q^{61} + 239 \beta q^{62} - 3977 q^{64} + 3322 \beta q^{65} + 4622 q^{67} - 966 \beta q^{68} + 1881 q^{70} - 606 \beta q^{71} - 3031 q^{73} + 740 \beta q^{74} - 2128 q^{76} + 779 \beta q^{77} - 10450 q^{79} - 1045 \beta q^{80} + 684 q^{82} - 4211 \beta q^{83} + 13662 q^{85} - 982 \beta q^{86} + 8487 q^{88} + 2334 \beta q^{89} - 5738 q^{91} - 700 \beta q^{92} + 6498 q^{94} - 3344 \beta q^{95} - 6517 q^{97} - 2040 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{4} - 38 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 14 q^{4} - 38 q^{7} - 198 q^{10} + 604 q^{13} - 190 q^{16} - 608 q^{19} + 738 q^{22} - 928 q^{25} - 266 q^{28} + 478 q^{31} + 2484 q^{34} + 1480 q^{37} - 4554 q^{40} - 1964 q^{43} + 1800 q^{46} - 4080 q^{49} + 4228 q^{52} + 8118 q^{55} - 4068 q^{58} - 632 q^{61} - 7954 q^{64} + 9244 q^{67} + 3762 q^{70} - 6062 q^{73} - 4256 q^{76} - 20900 q^{79} + 1368 q^{82} + 27324 q^{85} + 16974 q^{88} - 11476 q^{91} + 12996 q^{94} - 13034 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
1.00000i
1.00000i
3.00000i 0 7.00000 33.0000i 0 −19.0000 69.0000i 0 −99.0000
26.2 3.00000i 0 7.00000 33.0000i 0 −19.0000 69.0000i 0 −99.0000
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.5.b.c 2
3.b odd 2 1 inner 27.5.b.c 2
4.b odd 2 1 432.5.e.e 2
5.b even 2 1 675.5.c.h 2
5.c odd 4 1 675.5.d.a 2
5.c odd 4 1 675.5.d.d 2
9.c even 3 2 81.5.d.b 4
9.d odd 6 2 81.5.d.b 4
12.b even 2 1 432.5.e.e 2
15.d odd 2 1 675.5.c.h 2
15.e even 4 1 675.5.d.a 2
15.e even 4 1 675.5.d.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.5.b.c 2 1.a even 1 1 trivial
27.5.b.c 2 3.b odd 2 1 inner
81.5.d.b 4 9.c even 3 2
81.5.d.b 4 9.d odd 6 2
432.5.e.e 2 4.b odd 2 1
432.5.e.e 2 12.b even 2 1
675.5.c.h 2 5.b even 2 1
675.5.c.h 2 15.d odd 2 1
675.5.d.a 2 5.c odd 4 1
675.5.d.a 2 15.e even 4 1
675.5.d.d 2 5.c odd 4 1
675.5.d.d 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 9 \) acting on \(S_{5}^{\mathrm{new}}(27, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1089 \) Copy content Toggle raw display
$7$ \( (T + 19)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 15129 \) Copy content Toggle raw display
$13$ \( (T - 302)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 171396 \) Copy content Toggle raw display
$19$ \( (T + 304)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 90000 \) Copy content Toggle raw display
$29$ \( T^{2} + 459684 \) Copy content Toggle raw display
$31$ \( (T - 239)^{2} \) Copy content Toggle raw display
$37$ \( (T - 740)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 51984 \) Copy content Toggle raw display
$43$ \( (T + 982)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 4691556 \) Copy content Toggle raw display
$53$ \( T^{2} + 2537649 \) Copy content Toggle raw display
$59$ \( T^{2} + 8538084 \) Copy content Toggle raw display
$61$ \( (T + 316)^{2} \) Copy content Toggle raw display
$67$ \( (T - 4622)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 3305124 \) Copy content Toggle raw display
$73$ \( (T + 3031)^{2} \) Copy content Toggle raw display
$79$ \( (T + 10450)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 159592689 \) Copy content Toggle raw display
$89$ \( T^{2} + 49028004 \) Copy content Toggle raw display
$97$ \( (T + 6517)^{2} \) Copy content Toggle raw display
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