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Properties

Label 633.2.y.a
Level $633$
Weight $2$
Character orbit 633.y
Analytic conductor $5.055$
Analytic rank $0$
Dimension $12$
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] = [633,2,Mod(26,633)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(633, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 29]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("633.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 633 = 3 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 633.y (of order \(42\), degree \(12\), 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: \(5.05453044795\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{42}]$

$q$-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 \(\zeta_{21}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{21}^{8} + 2 \zeta_{21}) q^{3} + (2 \zeta_{21}^{9} + 2 \zeta_{21}^{2}) q^{4} + ( - 3 \zeta_{21}^{11} + 3 \zeta_{21}^{10} + \cdots + 3) q^{7}+ \cdots + (3 \zeta_{21}^{9} + 3 \zeta_{21}^{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{21}^{8} + 2 \zeta_{21}) q^{3} + (2 \zeta_{21}^{9} + 2 \zeta_{21}^{2}) q^{4} + ( - 3 \zeta_{21}^{11} + 3 \zeta_{21}^{10} + \cdots + 3) q^{7}+ \cdots + (8 \zeta_{21}^{10} + \cdots - 3 \zeta_{21}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} - 2 q^{4} - 9 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{3} - 2 q^{4} - 9 q^{7} - 3 q^{9} + 14 q^{13} + 4 q^{16} + q^{19} - 54 q^{21} - 10 q^{25} - 3 q^{31} + 6 q^{36} + 11 q^{37} + 63 q^{39} - 43 q^{43} - 12 q^{48} - 20 q^{49} + 14 q^{52} - 3 q^{57} - 27 q^{61} + 16 q^{64} + 77 q^{67} - 112 q^{73} + 15 q^{75} + 4 q^{76} + 8 q^{79} + 9 q^{81} - 18 q^{84} + 63 q^{91} + 3 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/633\mathbb{Z}\right)^\times\).

\(n\) \(212\) \(424\)
\(\chi(n)\) \(-1\) \(\zeta_{21}^{2} + \zeta_{21}^{9}\)

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
0.0747301 − 0.997204i
0.826239 + 0.563320i
0.955573 − 0.294755i
0.955573 + 0.294755i
−0.733052 − 0.680173i
0.365341 + 0.930874i
−0.733052 + 0.680173i
−0.988831 + 0.149042i
0.826239 − 0.563320i
0.0747301 + 0.997204i
−0.988831 − 0.149042i
0.365341 − 0.930874i
0 0.975699 − 1.43109i −0.730682 − 1.86175i 0 0 −5.18162 + 0.388309i 0 −1.09602 − 2.79262i 0
32.1 0 1.72721 + 0.129436i 1.97766 + 0.298085i 0 0 −2.92710 − 4.29326i 0 2.96649 + 0.447127i 0
38.1 0 1.17809 − 1.26968i −0.149460 − 1.99441i 0 0 1.53159 − 4.96530i 0 −0.224190 − 2.99161i 0
50.1 0 1.17809 + 1.26968i −0.149460 + 1.99441i 0 0 1.53159 + 4.96530i 0 −0.224190 + 2.99161i 0
110.1 0 −0.510531 − 1.65510i −1.65248 + 1.12664i 0 0 −3.53428 − 3.80905i 0 −2.47872 + 1.68996i 0
242.1 0 −0.258149 + 1.71271i −1.91115 − 0.589510i 0 0 4.83696 + 1.89837i 0 −2.86672 − 0.884266i 0
305.1 0 −0.510531 + 1.65510i −1.65248 − 1.12664i 0 0 −3.53428 + 3.80905i 0 −2.47872 − 1.68996i 0
368.1 0 −1.61232 − 0.632789i 1.46610 + 1.36035i 0 0 0.774446 − 5.13812i 0 2.19916 + 2.04052i 0
455.1 0 1.72721 − 0.129436i 1.97766 − 0.298085i 0 0 −2.92710 + 4.29326i 0 2.96649 − 0.447127i 0
560.1 0 0.975699 + 1.43109i −0.730682 + 1.86175i 0 0 −5.18162 − 0.388309i 0 −1.09602 + 2.79262i 0
590.1 0 −1.61232 + 0.632789i 1.46610 − 1.36035i 0 0 0.774446 + 5.13812i 0 2.19916 − 2.04052i 0
599.1 0 −0.258149 − 1.71271i −1.91115 + 0.589510i 0 0 4.83696 − 1.89837i 0 −2.86672 + 0.884266i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
211.m odd 42 1 inner
633.y even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 633.2.y.a ✓ 12
3.b odd 2 1 CM 633.2.y.a ✓ 12
211.m odd 42 1 inner 633.2.y.a ✓ 12
633.y even 42 1 inner 633.2.y.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.2.y.a ✓ 12 1.a even 1 1 trivial
633.2.y.a ✓ 12 3.b odd 2 1 CM
633.2.y.a ✓ 12 211.m odd 42 1 inner
633.2.y.a ✓ 12 633.y even 42 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(633, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 3 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 387420489 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} - 14 T^{11} + \cdots + 82369 \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 1856834281 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 28934350201 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 3129619249 \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 16745136409 \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 42008191681 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 30598055929 \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 31218295969 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 4799433759121 \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 902334707569 \) Copy content Toggle raw display
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