Properties

Label 33.8.d.a
Level $33$
Weight $8$
Character orbit 33.d
Analytic conductor $10.309$
Analytic rank $0$
Dimension $2$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [33,8,Mod(32,33)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(33, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.32");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 33.d (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: \(10.3087058410\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 13 \beta - 35) q^{3} - 128 q^{4} + ( - 142 \beta + 71) q^{5} + (1079 \beta + 718) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 13 \beta - 35) q^{3} - 128 q^{4} + ( - 142 \beta + 71) q^{5} + (1079 \beta + 718) q^{9} + (2662 \beta - 1331) q^{11} + (1664 \beta + 4480) q^{12} + (5893 \beta - 8023) q^{15} + 16384 q^{16} + (18176 \beta - 9088) q^{20} + (43706 \beta - 21853) q^{23} + 22674 q^{25} + ( - 61126 \beta + 16951) q^{27} + 39065 q^{31} + ( - 110473 \beta + 150403) q^{33} + ( - 138112 \beta - 91904) q^{36} - 562471 q^{37} + ( - 340736 \beta + 170368) q^{44} + ( - 178565 \beta + 510632) q^{45} + (328892 \beta - 164446) q^{47} + ( - 212992 \beta - 573440) q^{48} + 823543 q^{49} + (1288664 \beta - 644332) q^{53} + 1039511 q^{55} + (1900442 \beta - 950221) q^{59} + ( - 754304 \beta + 1026944) q^{60} - 2097152 q^{64} + 684671 q^{67} + ( - 1813799 \beta + 2469389) q^{69} + (1137230 \beta - 568615) q^{71} + ( - 294762 \beta - 793590) q^{75} + ( - 2326528 \beta + 1163264) q^{80} + (2713685 \beta - 2977199) q^{81} + ( - 7564330 \beta + 3782165) q^{89} + ( - 5594368 \beta + 2797184) q^{92} + ( - 507845 \beta - 1367275) q^{93} - 15182479 q^{97} + (3347465 \beta - 9572552) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 83 q^{3} - 256 q^{4} + 2515 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 83 q^{3} - 256 q^{4} + 2515 q^{9} + 10624 q^{12} - 10153 q^{15} + 32768 q^{16} + 45348 q^{25} - 27224 q^{27} + 78130 q^{31} + 190333 q^{33} - 321920 q^{36} - 1124942 q^{37} + 842699 q^{45} - 1359872 q^{48} + 1647086 q^{49} + 2079022 q^{55} + 1299584 q^{60} - 4194304 q^{64} + 1369342 q^{67} + 3124979 q^{69} - 1881942 q^{75} - 3240713 q^{81} - 3242395 q^{93} - 30364958 q^{97} - 15797639 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
0.500000 + 1.65831i
0.500000 1.65831i
0 −41.5000 21.5581i −128.000 235.480i 0 0 0 1257.50 + 1789.32i 0
32.2 0 −41.5000 + 21.5581i −128.000 235.480i 0 0 0 1257.50 1789.32i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
3.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.8.d.a 2
3.b odd 2 1 inner 33.8.d.a 2
11.b odd 2 1 CM 33.8.d.a 2
33.d even 2 1 inner 33.8.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.d.a 2 1.a even 1 1 trivial
33.8.d.a 2 3.b odd 2 1 inner
33.8.d.a 2 11.b odd 2 1 CM
33.8.d.a 2 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{8}^{\mathrm{new}}(33, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 83T + 2187 \) Copy content Toggle raw display
$5$ \( T^{2} + 55451 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 19487171 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 5253089699 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 39065)^{2} \) Copy content Toggle raw display
$37$ \( (T + 562471)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 297467356076 \) Copy content Toggle raw display
$53$ \( T^{2} + 4566800988464 \) Copy content Toggle raw display
$59$ \( T^{2} + 9932119437251 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( (T - 684671)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 3556553200475 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 157352492959475 \) Copy content Toggle raw display
$97$ \( (T + 15182479)^{2} \) Copy content Toggle raw display
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