Properties

Label 675.2.u.a
Level $675$
Weight $2$
Character orbit 675.u
Analytic conductor $5.390$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [675,2,Mod(49,675)]
 
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, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.u (of order \(18\), degree \(6\), 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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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 \(\zeta_{36}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{36}^{9} + \cdots - \zeta_{36}^{5}) q^{2}+ \cdots + 3 \zeta_{36}^{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{36}^{9} + \cdots - \zeta_{36}^{5}) q^{2}+ \cdots + (9 \zeta_{36}^{6} - 9 \zeta_{36}^{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 36 q^{11} + 12 q^{14} - 36 q^{16} + 24 q^{19} - 36 q^{24} + 24 q^{26} + 42 q^{29} - 6 q^{31} - 18 q^{34} - 36 q^{36} + 18 q^{39} - 36 q^{41} + 54 q^{44} - 18 q^{46} + 24 q^{49} + 18 q^{51} - 54 q^{54} - 96 q^{56} + 72 q^{61} + 24 q^{64} - 72 q^{69} + 6 q^{71} - 24 q^{74} - 12 q^{76} + 24 q^{79} - 72 q^{84} + 6 q^{86} - 6 q^{89} - 54 q^{94} + 54 q^{96} + 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(326\) \(352\)
\(\chi(n)\) \(\zeta_{36}^{2} - \zeta_{36}^{8}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.342020 0.939693i
−0.342020 + 0.939693i
0.342020 + 0.939693i
−0.342020 0.939693i
0.984808 0.173648i
−0.984808 + 0.173648i
−0.642788 + 0.766044i
0.642788 0.766044i
−0.642788 0.766044i
0.642788 + 0.766044i
0.984808 + 0.173648i
−0.984808 0.173648i
−1.62760 1.93969i −1.32683 1.11334i −0.766044 + 4.34445i 0 4.38571i −3.01763 + 0.532089i 5.28801 3.05303i 0.520945 + 2.95442i 0
49.2 1.62760 + 1.93969i 1.32683 + 1.11334i −0.766044 + 4.34445i 0 4.38571i 3.01763 0.532089i −5.28801 + 3.05303i 0.520945 + 2.95442i 0
124.1 −1.62760 + 1.93969i −1.32683 + 1.11334i −0.766044 4.34445i 0 4.38571i −3.01763 0.532089i 5.28801 + 3.05303i 0.520945 2.95442i 0
124.2 1.62760 1.93969i 1.32683 1.11334i −0.766044 4.34445i 0 4.38571i 3.01763 + 0.532089i −5.28801 3.05303i 0.520945 2.95442i 0
274.1 −0.300767 + 0.826352i −1.62760 + 0.592396i 0.939693 + 0.788496i 0 1.52314i 2.41609 + 2.87939i −2.45734 + 1.41875i 2.29813 1.92836i 0
274.2 0.300767 0.826352i 1.62760 0.592396i 0.939693 + 0.788496i 0 1.52314i −2.41609 2.87939i 2.45734 1.41875i 2.29813 1.92836i 0
349.1 −1.32683 0.233956i 0.300767 + 1.70574i −0.173648 0.0632028i 0 2.33359i −0.237565 0.652704i 2.54920 + 1.47178i −2.81908 + 1.02606i 0
349.2 1.32683 + 0.233956i −0.300767 1.70574i −0.173648 0.0632028i 0 2.33359i 0.237565 + 0.652704i −2.54920 1.47178i −2.81908 + 1.02606i 0
499.1 −1.32683 + 0.233956i 0.300767 1.70574i −0.173648 + 0.0632028i 0 2.33359i −0.237565 + 0.652704i 2.54920 1.47178i −2.81908 1.02606i 0
499.2 1.32683 0.233956i −0.300767 + 1.70574i −0.173648 + 0.0632028i 0 2.33359i 0.237565 0.652704i −2.54920 + 1.47178i −2.81908 1.02606i 0
574.1 −0.300767 0.826352i −1.62760 0.592396i 0.939693 0.788496i 0 1.52314i 2.41609 2.87939i −2.45734 1.41875i 2.29813 + 1.92836i 0
574.2 0.300767 + 0.826352i 1.62760 + 0.592396i 0.939693 0.788496i 0 1.52314i −2.41609 + 2.87939i 2.45734 + 1.41875i 2.29813 + 1.92836i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
27.e even 9 1 inner
135.p even 18 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 36T_{2}^{8} - 90T_{2}^{6} + 81T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 36 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{12} + 27T^{6} + 729 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 12 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$11$ \( (T^{6} + 18 T^{5} + \cdots + 729)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 18 T^{10} + \cdots + 130321 \) Copy content Toggle raw display
$17$ \( T^{12} - 18 T^{10} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( (T^{6} - 12 T^{5} + \cdots + 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 547981281 \) Copy content Toggle raw display
$29$ \( (T^{6} - 21 T^{5} + \cdots + 47961)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 3 T^{5} + \cdots + 289)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 90 T^{10} + \cdots + 62742241 \) Copy content Toggle raw display
$41$ \( (T^{6} + 18 T^{5} + \cdots + 81)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - 276 T^{10} + \cdots + 130321 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 151807041 \) Copy content Toggle raw display
$53$ \( (T^{6} + 234 T^{4} + \cdots + 106929)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 144 T^{4} + \cdots + 331776)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 36 T^{5} + \cdots + 1371241)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 260144641 \) Copy content Toggle raw display
$71$ \( (T^{6} - 3 T^{5} + \cdots + 47961)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 30 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( (T^{6} - 12 T^{5} + \cdots + 273529)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 855036081 \) Copy content Toggle raw display
$89$ \( (T^{6} + 3 T^{5} + \cdots + 2601)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 56249134561 \) Copy content Toggle raw display
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