Properties

Label 675.2.l.a
Level $675$
Weight $2$
Character orbit 675.l
Analytic conductor $5.390$
Analytic rank $1$
Dimension $6$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [675,2,Mod(76,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([14, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.76");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.l (of order \(9\), degree \(6\), 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: \(5.38990213644\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{18}^{5} + \zeta_{18}^{4} + \cdots - 1) q^{2}+ \cdots - 3 \zeta_{18}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{18}^{5} + \zeta_{18}^{4} + \cdots - 1) q^{2}+ \cdots + (9 \zeta_{18}^{4} + 9 \zeta_{18}^{3} - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} - 6 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} - 6 q^{7} + 6 q^{8} - 18 q^{11} - 9 q^{12} - 6 q^{14} - 18 q^{16} - 6 q^{17} + 18 q^{18} - 12 q^{19} + 36 q^{22} - 18 q^{23} + 18 q^{24} + 12 q^{26} - 27 q^{27} + 36 q^{28} - 21 q^{29} - 3 q^{31} + 9 q^{32} + 9 q^{34} - 18 q^{36} - 6 q^{37} - 12 q^{38} - 9 q^{39} - 18 q^{41} - 18 q^{42} + 6 q^{43} - 27 q^{44} - 9 q^{46} + 6 q^{47} - 9 q^{48} - 12 q^{49} + 9 q^{51} - 36 q^{52} - 48 q^{53} + 27 q^{54} - 48 q^{56} + 18 q^{57} + 27 q^{58} + 36 q^{61} + 15 q^{62} + 18 q^{63} - 12 q^{64} - 18 q^{67} + 12 q^{68} + 36 q^{69} + 3 q^{71} + 9 q^{72} + 6 q^{73} + 12 q^{74} - 6 q^{76} + 18 q^{77} + 27 q^{78} - 12 q^{79} + 36 q^{82} - 18 q^{83} + 36 q^{84} + 3 q^{86} + 18 q^{88} + 3 q^{89} + 63 q^{92} - 9 q^{93} + 27 q^{94} + 27 q^{96} - 3 q^{97} - 9 q^{98} - 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(-\zeta_{18}^{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
0.939693 + 0.342020i
0.939693 0.342020i
−0.173648 0.984808i
−0.766044 0.642788i
−0.766044 + 0.642788i
−0.173648 + 0.984808i
−1.93969 + 1.62760i 1.11334 1.32683i 0.766044 4.34445i 0 4.38571i 0.532089 + 3.01763i 3.05303 + 5.28801i −0.520945 2.95442i 0
151.1 −1.93969 1.62760i 1.11334 + 1.32683i 0.766044 + 4.34445i 0 4.38571i 0.532089 3.01763i 3.05303 5.28801i −0.520945 + 2.95442i 0
301.1 −0.826352 0.300767i 0.592396 + 1.62760i −0.939693 0.788496i 0 1.52314i −2.87939 + 2.41609i 1.41875 + 2.45734i −2.29813 + 1.92836i 0
376.1 −0.233956 + 1.32683i −1.70574 + 0.300767i 0.173648 + 0.0632028i 0 2.33359i −0.652704 + 0.237565i −1.47178 + 2.54920i 2.81908 1.02606i 0
526.1 −0.233956 1.32683i −1.70574 0.300767i 0.173648 0.0632028i 0 2.33359i −0.652704 0.237565i −1.47178 2.54920i 2.81908 + 1.02606i 0
601.1 −0.826352 + 0.300767i 0.592396 1.62760i −0.939693 + 0.788496i 0 1.52314i −2.87939 2.41609i 1.41875 2.45734i −2.29813 1.92836i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.l.a 6
5.b even 2 1 675.2.l.b yes 6
5.c odd 4 2 675.2.u.a 12
27.e even 9 1 inner 675.2.l.a 6
135.p even 18 1 675.2.l.b yes 6
135.r odd 36 2 675.2.u.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
675.2.l.a 6 1.a even 1 1 trivial
675.2.l.a 6 27.e even 9 1 inner
675.2.l.b yes 6 5.b even 2 1
675.2.l.b yes 6 135.p even 18 1
675.2.u.a 12 5.c odd 4 2
675.2.u.a 12 135.r odd 36 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 6T_{2}^{5} + 18T_{2}^{4} + 30T_{2}^{3} + 36T_{2}^{2} + 27T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 6 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} + 9T^{3} + 27 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 6 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{6} + 18 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$13$ \( T^{6} - 9 T^{4} + \cdots + 361 \) Copy content Toggle raw display
$17$ \( T^{6} + 6 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{6} + 12 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{6} + 18 T^{5} + \cdots + 23409 \) Copy content Toggle raw display
$29$ \( T^{6} + 21 T^{5} + \cdots + 47961 \) Copy content Toggle raw display
$31$ \( T^{6} + 3 T^{5} + \cdots + 289 \) Copy content Toggle raw display
$37$ \( T^{6} + 6 T^{5} + \cdots + 7921 \) Copy content Toggle raw display
$41$ \( T^{6} + 18 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{6} - 6 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$47$ \( T^{6} - 6 T^{5} + \cdots + 12321 \) Copy content Toggle raw display
$53$ \( (T^{3} + 24 T^{2} + \cdots + 327)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 144 T^{4} + \cdots + 331776 \) Copy content Toggle raw display
$61$ \( T^{6} - 36 T^{5} + \cdots + 1371241 \) Copy content Toggle raw display
$67$ \( T^{6} + 18 T^{5} + \cdots + 16129 \) Copy content Toggle raw display
$71$ \( T^{6} - 3 T^{5} + \cdots + 47961 \) Copy content Toggle raw display
$73$ \( T^{6} - 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{6} + 12 T^{5} + \cdots + 273529 \) Copy content Toggle raw display
$83$ \( T^{6} + 18 T^{5} + \cdots + 29241 \) Copy content Toggle raw display
$89$ \( T^{6} - 3 T^{5} + \cdots + 2601 \) Copy content Toggle raw display
$97$ \( T^{6} + 3 T^{5} + \cdots + 237169 \) Copy content Toggle raw display
show more
show less