Properties

Label 27.7.b.b
Level $27$
Weight $7$
Character orbit 27.b
Analytic conductor $6.211$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [27,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.26");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 7 \)
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: \(6.21146025774\)
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: \( 2\cdot 3 \)
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 \(\beta = 6i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 28 q^{4} - 40 \beta q^{5} + 299 q^{7} + 92 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 28 q^{4} - 40 \beta q^{5} + 299 q^{7} + 92 \beta q^{8} + 1440 q^{10} - 104 \beta q^{11} + 2495 q^{13} + 299 \beta q^{14} - 1520 q^{16} + 312 \beta q^{17} - 2509 q^{19} - 1120 \beta q^{20} + 3744 q^{22} - 2392 \beta q^{23} - 41975 q^{25} + 2495 \beta q^{26} + 8372 q^{28} + 3952 \beta q^{29} + 5330 q^{31} + 4368 \beta q^{32} - 11232 q^{34} - 11960 \beta q^{35} + 32591 q^{37} - 2509 \beta q^{38} + 132480 q^{40} + 11024 \beta q^{41} - 70630 q^{43} - 2912 \beta q^{44} + 86112 q^{46} + 664 \beta q^{47} - 28248 q^{49} - 41975 \beta q^{50} + 69860 q^{52} + 31824 \beta q^{53} - 149760 q^{55} + 27508 \beta q^{56} - 142272 q^{58} + 39560 \beta q^{59} - 61801 q^{61} + 5330 \beta q^{62} - 254528 q^{64} - 99800 \beta q^{65} - 430261 q^{67} + 8736 \beta q^{68} + 430560 q^{70} - 41952 \beta q^{71} + 251615 q^{73} + 32591 \beta q^{74} - 70252 q^{76} - 31096 \beta q^{77} + 660827 q^{79} + 60800 \beta q^{80} - 396864 q^{82} + 132976 \beta q^{83} + 449280 q^{85} - 70630 \beta q^{86} + 344448 q^{88} - 45096 \beta q^{89} + 746005 q^{91} - 66976 \beta q^{92} - 23904 q^{94} + 100360 \beta q^{95} + 220727 q^{97} - 28248 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 56 q^{4} + 598 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 56 q^{4} + 598 q^{7} + 2880 q^{10} + 4990 q^{13} - 3040 q^{16} - 5018 q^{19} + 7488 q^{22} - 83950 q^{25} + 16744 q^{28} + 10660 q^{31} - 22464 q^{34} + 65182 q^{37} + 264960 q^{40} - 141260 q^{43} + 172224 q^{46} - 56496 q^{49} + 139720 q^{52} - 299520 q^{55} - 284544 q^{58} - 123602 q^{61} - 509056 q^{64} - 860522 q^{67} + 861120 q^{70} + 503230 q^{73} - 140504 q^{76} + 1321654 q^{79} - 793728 q^{82} + 898560 q^{85} + 688896 q^{88} + 1492010 q^{91} - 47808 q^{94} + 441454 q^{97}+O(q^{100}) \) 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
6.00000i 0 28.0000 240.000i 0 299.000 552.000i 0 1440.00
26.2 6.00000i 0 28.0000 240.000i 0 299.000 552.000i 0 1440.00
\(n\): 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 27.7.b.b 2
3.b odd 2 1 inner 27.7.b.b 2
4.b odd 2 1 432.7.e.d 2
9.c even 3 2 81.7.d.b 4
9.d odd 6 2 81.7.d.b 4
12.b even 2 1 432.7.e.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.7.b.b 2 1.a even 1 1 trivial
27.7.b.b 2 3.b odd 2 1 inner
81.7.d.b 4 9.c even 3 2
81.7.d.b 4 9.d odd 6 2
432.7.e.d 2 4.b odd 2 1
432.7.e.d 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 36 \) acting on \(S_{7}^{\mathrm{new}}(27, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 36 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 57600 \) Copy content Toggle raw display
$7$ \( (T - 299)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 389376 \) Copy content Toggle raw display
$13$ \( (T - 2495)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3504384 \) Copy content Toggle raw display
$19$ \( (T + 2509)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 205979904 \) Copy content Toggle raw display
$29$ \( T^{2} + 562258944 \) Copy content Toggle raw display
$31$ \( (T - 5330)^{2} \) Copy content Toggle raw display
$37$ \( (T - 32591)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 4375028736 \) Copy content Toggle raw display
$43$ \( (T + 70630)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 15872256 \) Copy content Toggle raw display
$53$ \( T^{2} + 36459611136 \) Copy content Toggle raw display
$59$ \( T^{2} + 56339769600 \) Copy content Toggle raw display
$61$ \( (T + 61801)^{2} \) Copy content Toggle raw display
$67$ \( (T + 430261)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 63358930944 \) Copy content Toggle raw display
$73$ \( (T - 251615)^{2} \) Copy content Toggle raw display
$79$ \( (T - 660827)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 636574196736 \) Copy content Toggle raw display
$89$ \( T^{2} + 73211371776 \) Copy content Toggle raw display
$97$ \( (T - 220727)^{2} \) Copy content Toggle raw display
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