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Properties

Label 2793.1.er.a
Level $2793$
Weight $1$
Character orbit 2793.er
Analytic conductor $1.394$
Analytic rank $0$
Dimension $36$
Projective image $D_{126}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2793,1,Mod(59,2793)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2793, base_ring=CyclotomicField(126))
 
chi = DirichletCharacter(H, H._module([63, 39, 7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2793.59");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2793 = 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2793.er (of order \(126\), degree \(36\), 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: \(1.39388858028\)
Analytic rank: \(0\)
Dimension: \(36\)
Coefficient field: \(\Q(\zeta_{63})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{36} - x^{33} + x^{27} - x^{24} + x^{18} - x^{12} + x^{9} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{126}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{126} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{126}^{17} q^{3} - \zeta_{126}^{16} q^{4} + \zeta_{126}^{22} q^{7} + \zeta_{126}^{34} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{126}^{17} q^{3} - \zeta_{126}^{16} q^{4} + \zeta_{126}^{22} q^{7} + \zeta_{126}^{34} q^{9} - \zeta_{126}^{33} q^{12} + ( - \zeta_{126}^{47} - \zeta_{126}^{39}) q^{13} + \zeta_{126}^{32} q^{16} - \zeta_{126}^{36} q^{19} + \zeta_{126}^{39} q^{21} + \zeta_{126}^{13} q^{25} + \zeta_{126}^{51} q^{27} - \zeta_{126}^{38} q^{28} + ( - \zeta_{126}^{37} + \zeta_{126}^{26}) q^{31} - \zeta_{126}^{50} q^{36} + (\zeta_{126}^{59} + \zeta_{126}^{28}) q^{37} + ( - \zeta_{126}^{56} + \zeta_{126}) q^{39} + ( - \zeta_{126}^{24} + \zeta_{126}^{19}) q^{43} + \zeta_{126}^{49} q^{48} + \zeta_{126}^{44} q^{49} + (\zeta_{126}^{55} - 1) q^{52} - \zeta_{126}^{53} q^{57} + (\zeta_{126}^{14} + \zeta_{126}^{3}) q^{61} + \zeta_{126}^{56} q^{63} - \zeta_{126}^{48} q^{64} + ( - \zeta_{126}^{32} + \zeta_{126}^{24}) q^{67} + ( - \zeta_{126}^{57} - \zeta_{126}^{4}) q^{73} + \zeta_{126}^{30} q^{75} + \zeta_{126}^{52} q^{76} + (\zeta_{126}^{58} + \zeta_{126}^{33}) q^{79} - \zeta_{126}^{5} q^{81} - \zeta_{126}^{55} q^{84} + ( - \zeta_{126}^{61} + \zeta_{126}^{6}) q^{91} + ( - \zeta_{126}^{54} + \zeta_{126}^{43}) q^{93} + ( - \zeta_{126}^{62} + \zeta_{126}^{29}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q+O(q^{10}) \) Copy content Toggle raw display \( 36 q + 3 q^{12} + 3 q^{13} + 6 q^{19} - 3 q^{21} - 3 q^{27} - 3 q^{43} - 36 q^{52} - 3 q^{61} - 3 q^{64} + 3 q^{67} + 3 q^{73} + 3 q^{75} - 3 q^{79} + 3 q^{91} + 6 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2793\mathbb{Z}\right)^\times\).

\(n\) \(932\) \(2110\) \(2206\)
\(\chi(n)\) \(-1\) \(-\zeta_{126}^{48}\) \(-\zeta_{126}^{28}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
−0.853291 + 0.521435i
0.921476 + 0.388435i
−0.411287 − 0.911506i
−0.797133 − 0.603804i
0.995031 + 0.0995678i
−0.998757 − 0.0498459i
0.698237 + 0.715867i
−0.998757 + 0.0498459i
−0.411287 + 0.911506i
0.542546 − 0.840026i
0.878222 − 0.478254i
−0.969077 − 0.246757i
0.980172 + 0.198146i
0.698237 − 0.715867i
0.921476 − 0.388435i
−0.583744 + 0.811938i
0.980172 − 0.198146i
−0.0249307 − 0.999689i
−0.318487 + 0.947927i
−0.583744 − 0.811938i
0 −0.995031 − 0.0995678i 0.797133 + 0.603804i 0 0 0.878222 + 0.478254i 0 0.980172 + 0.198146i 0
89.1 0 −0.878222 − 0.478254i −0.995031 − 0.0995678i 0 0 −0.797133 + 0.603804i 0 0.542546 + 0.840026i 0
110.1 0 0.797133 + 0.603804i −0.878222 + 0.478254i 0 0 0.995031 + 0.0995678i 0 0.270840 + 0.962624i 0
185.1 0 0.0249307 − 0.999689i 0.583744 + 0.811938i 0 0 −0.124344 + 0.992239i 0 −0.998757 − 0.0498459i 0
257.1 0 0.124344 − 0.992239i 0.0249307 − 0.999689i 0 0 −0.583744 + 0.811938i 0 −0.969077 − 0.246757i 0
269.1 0 0.661686 + 0.749781i −0.698237 − 0.715867i 0 0 0.456211 + 0.889872i 0 −0.124344 + 0.992239i 0
458.1 0 −0.542546 − 0.840026i −0.980172 − 0.198146i 0 0 0.270840 − 0.962624i 0 −0.411287 + 0.911506i 0
488.1 0 0.661686 − 0.749781i −0.698237 + 0.715867i 0 0 0.456211 − 0.889872i 0 −0.124344 − 0.992239i 0
584.1 0 0.797133 − 0.603804i −0.878222 − 0.478254i 0 0 0.995031 − 0.0995678i 0 0.270840 − 0.962624i 0
857.1 0 0.318487 − 0.947927i 0.969077 − 0.246757i 0 0 −0.998757 − 0.0498459i 0 −0.797133 − 0.603804i 0
887.1 0 0.583744 + 0.811938i 0.124344 + 0.992239i 0 0 −0.0249307 + 0.999689i 0 −0.318487 + 0.947927i 0
908.1 0 −0.456211 − 0.889872i 0.661686 + 0.749781i 0 0 0.698237 − 0.715867i 0 −0.583744 + 0.811938i 0
983.1 0 0.969077 + 0.246757i 0.998757 + 0.0498459i 0 0 −0.318487 − 0.947927i 0 0.878222 + 0.478254i 0
1055.1 0 −0.542546 + 0.840026i −0.980172 + 0.198146i 0 0 0.270840 + 0.962624i 0 −0.411287 − 0.911506i 0
1067.1 0 −0.878222 + 0.478254i −0.995031 + 0.0995678i 0 0 −0.797133 − 0.603804i 0 0.542546 − 0.840026i 0
1286.1 0 −0.921476 + 0.388435i 0.853291 + 0.521435i 0 0 −0.411287 − 0.911506i 0 0.698237 − 0.715867i 0
1307.1 0 0.969077 − 0.246757i 0.998757 − 0.0498459i 0 0 −0.318487 + 0.947927i 0 0.878222 − 0.478254i 0
1382.1 0 0.411287 + 0.911506i −0.921476 + 0.388435i 0 0 −0.853291 + 0.521435i 0 −0.661686 + 0.749781i 0
1454.1 0 −0.698237 − 0.715867i −0.456211 + 0.889872i 0 0 −0.661686 − 0.749781i 0 −0.0249307 + 0.999689i 0
1466.1 0 −0.921476 − 0.388435i 0.853291 − 0.521435i 0 0 −0.411287 + 0.911506i 0 0.698237 + 0.715867i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
931.ci even 126 1 inner
2793.er odd 126 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2793.1.er.a ✓ 36
3.b odd 2 1 CM 2793.1.er.a ✓ 36
19.f odd 18 1 2793.1.ew.a yes 36
49.h odd 42 1 2793.1.ew.a yes 36
57.j even 18 1 2793.1.ew.a yes 36
147.o even 42 1 2793.1.ew.a yes 36
931.ci even 126 1 inner 2793.1.er.a ✓ 36
2793.er odd 126 1 inner 2793.1.er.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2793.1.er.a ✓ 36 1.a even 1 1 trivial
2793.1.er.a ✓ 36 3.b odd 2 1 CM
2793.1.er.a ✓ 36 931.ci even 126 1 inner
2793.1.er.a ✓ 36 2793.er odd 126 1 inner
2793.1.ew.a yes 36 19.f odd 18 1
2793.1.ew.a yes 36 49.h odd 42 1
2793.1.ew.a yes 36 57.j even 18 1
2793.1.ew.a yes 36 147.o even 42 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2793, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{36} \) Copy content Toggle raw display
$3$ \( T^{36} + T^{33} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{36} \) Copy content Toggle raw display
$7$ \( T^{36} - T^{33} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{36} \) Copy content Toggle raw display
$13$ \( T^{36} - 3 T^{35} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{36} \) Copy content Toggle raw display
$19$ \( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{36} \) Copy content Toggle raw display
$29$ \( T^{36} \) Copy content Toggle raw display
$31$ \( (T^{18} - 18 T^{16} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{36} + 3 T^{34} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{36} \) Copy content Toggle raw display
$43$ \( T^{36} + 3 T^{35} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{36} \) Copy content Toggle raw display
$53$ \( T^{36} \) Copy content Toggle raw display
$59$ \( T^{36} \) Copy content Toggle raw display
$61$ \( T^{36} + 3 T^{35} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{36} - 3 T^{35} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{36} \) Copy content Toggle raw display
$73$ \( T^{36} - 3 T^{35} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{36} + 3 T^{35} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{36} \) Copy content Toggle raw display
$89$ \( T^{36} \) Copy content Toggle raw display
$97$ \( T^{36} - T^{33} + \cdots + 1 \) Copy content Toggle raw display
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