Properties

Label 33.3.b.a
Level $33$
Weight $3$
Character orbit 33.b
Analytic conductor $0.899$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [33,3,Mod(23,33)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.23");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 33.b (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: \(0.899184872389\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

\(f(q)\) \(=\) \( q - \beta q^{2} + 3 q^{3} - 7 q^{4} + 2 \beta q^{5} - 3 \beta q^{6} - 8 q^{7} + 3 \beta q^{8} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + 3 q^{3} - 7 q^{4} + 2 \beta q^{5} - 3 \beta q^{6} - 8 q^{7} + 3 \beta q^{8} + 9 q^{9} + 22 q^{10} - \beta q^{11} - 21 q^{12} + 4 q^{13} + 8 \beta q^{14} + 6 \beta q^{15} + 5 q^{16} - 4 \beta q^{17} - 9 \beta q^{18} - 6 q^{19} - 14 \beta q^{20} - 24 q^{21} - 11 q^{22} - 2 \beta q^{23} + 9 \beta q^{24} - 19 q^{25} - 4 \beta q^{26} + 27 q^{27} + 56 q^{28} + 12 \beta q^{29} + 66 q^{30} - 26 q^{31} + 7 \beta q^{32} - 3 \beta q^{33} - 44 q^{34} - 16 \beta q^{35} - 63 q^{36} + 30 q^{37} + 6 \beta q^{38} + 12 q^{39} - 66 q^{40} - 4 \beta q^{41} + 24 \beta q^{42} + 42 q^{43} + 7 \beta q^{44} + 18 \beta q^{45} - 22 q^{46} - 26 \beta q^{47} + 15 q^{48} + 15 q^{49} + 19 \beta q^{50} - 12 \beta q^{51} - 28 q^{52} + 18 \beta q^{53} - 27 \beta q^{54} + 22 q^{55} - 24 \beta q^{56} - 18 q^{57} + 132 q^{58} - 20 \beta q^{59} - 42 \beta q^{60} + 12 q^{61} + 26 \beta q^{62} - 72 q^{63} + 97 q^{64} + 8 \beta q^{65} - 33 q^{66} + 2 q^{67} + 28 \beta q^{68} - 6 \beta q^{69} - 176 q^{70} + 18 \beta q^{71} + 27 \beta q^{72} - 74 q^{73} - 30 \beta q^{74} - 57 q^{75} + 42 q^{76} + 8 \beta q^{77} - 12 \beta q^{78} - 40 q^{79} + 10 \beta q^{80} + 81 q^{81} - 44 q^{82} + 12 \beta q^{83} + 168 q^{84} + 88 q^{85} - 42 \beta q^{86} + 36 \beta q^{87} + 33 q^{88} + 36 \beta q^{89} + 198 q^{90} - 32 q^{91} + 14 \beta q^{92} - 78 q^{93} - 286 q^{94} - 12 \beta q^{95} + 21 \beta q^{96} + 62 q^{97} - 15 \beta q^{98} - 9 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 14 q^{4} - 16 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 14 q^{4} - 16 q^{7} + 18 q^{9} + 44 q^{10} - 42 q^{12} + 8 q^{13} + 10 q^{16} - 12 q^{19} - 48 q^{21} - 22 q^{22} - 38 q^{25} + 54 q^{27} + 112 q^{28} + 132 q^{30} - 52 q^{31} - 88 q^{34} - 126 q^{36} + 60 q^{37} + 24 q^{39} - 132 q^{40} + 84 q^{43} - 44 q^{46} + 30 q^{48} + 30 q^{49} - 56 q^{52} + 44 q^{55} - 36 q^{57} + 264 q^{58} + 24 q^{61} - 144 q^{63} + 194 q^{64} - 66 q^{66} + 4 q^{67} - 352 q^{70} - 148 q^{73} - 114 q^{75} + 84 q^{76} - 80 q^{79} + 162 q^{81} - 88 q^{82} + 336 q^{84} + 176 q^{85} + 66 q^{88} + 396 q^{90} - 64 q^{91} - 156 q^{93} - 572 q^{94} + 124 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
0.500000 + 1.65831i
0.500000 1.65831i
3.31662i 3.00000 −7.00000 6.63325i 9.94987i −8.00000 9.94987i 9.00000 22.0000
23.2 3.31662i 3.00000 −7.00000 6.63325i 9.94987i −8.00000 9.94987i 9.00000 22.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.3.b.a 2
3.b odd 2 1 inner 33.3.b.a 2
4.b odd 2 1 528.3.i.a 2
11.b odd 2 1 363.3.b.d 2
11.c even 5 4 363.3.h.e 8
11.d odd 10 4 363.3.h.d 8
12.b even 2 1 528.3.i.a 2
33.d even 2 1 363.3.b.d 2
33.f even 10 4 363.3.h.d 8
33.h odd 10 4 363.3.h.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.b.a 2 1.a even 1 1 trivial
33.3.b.a 2 3.b odd 2 1 inner
363.3.b.d 2 11.b odd 2 1
363.3.b.d 2 33.d even 2 1
363.3.h.d 8 11.d odd 10 4
363.3.h.d 8 33.f even 10 4
363.3.h.e 8 11.c even 5 4
363.3.h.e 8 33.h odd 10 4
528.3.i.a 2 4.b odd 2 1
528.3.i.a 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 11 \) acting on \(S_{3}^{\mathrm{new}}(33, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 11 \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 44 \) Copy content Toggle raw display
$7$ \( (T + 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 11 \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 176 \) Copy content Toggle raw display
$19$ \( (T + 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 44 \) Copy content Toggle raw display
$29$ \( T^{2} + 1584 \) Copy content Toggle raw display
$31$ \( (T + 26)^{2} \) Copy content Toggle raw display
$37$ \( (T - 30)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 176 \) Copy content Toggle raw display
$43$ \( (T - 42)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 7436 \) Copy content Toggle raw display
$53$ \( T^{2} + 3564 \) Copy content Toggle raw display
$59$ \( T^{2} + 4400 \) Copy content Toggle raw display
$61$ \( (T - 12)^{2} \) Copy content Toggle raw display
$67$ \( (T - 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 3564 \) Copy content Toggle raw display
$73$ \( (T + 74)^{2} \) Copy content Toggle raw display
$79$ \( (T + 40)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1584 \) Copy content Toggle raw display
$89$ \( T^{2} + 14256 \) Copy content Toggle raw display
$97$ \( (T - 62)^{2} \) Copy content Toggle raw display
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