Properties

Label 33.3.b.b
Level $33$
Weight $3$
Character orbit 33.b
Analytic conductor $0.899$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_{2} - \beta_1) q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{3} + \beta_1 + 1) q^{4} + (\beta_{3} - \beta_1) q^{5} + ( - \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{6} + ( - 2 \beta_{3} - 2 \beta_1) q^{7} + (3 \beta_{3} - \beta_{2} - 3 \beta_1) q^{8} + ( - 5 \beta_{3} + 3 \beta_{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_{2} - \beta_1) q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{3} + \beta_1 + 1) q^{4} + (\beta_{3} - \beta_1) q^{5} + ( - \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{6} + ( - 2 \beta_{3} - 2 \beta_1) q^{7} + (3 \beta_{3} - \beta_{2} - 3 \beta_1) q^{8} + ( - 5 \beta_{3} + 3 \beta_{2} - 3) q^{9} - 2 q^{10} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{11} + ( - 5 \beta_{3} + 3 \beta_1 + 3) q^{12} + ( - 4 \beta_{3} - 4 \beta_1 - 4) q^{13} + (4 \beta_{3} - 8 \beta_{2} - 4 \beta_1) q^{14} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 4) q^{15} + (5 \beta_{3} + 5 \beta_1 - 3) q^{16} + ( - 2 \beta_{3} + 16 \beta_{2} + 2 \beta_1) q^{17} + ( - \beta_{3} - 7 \beta_{2} - 5 \beta_1 + 8) q^{18} + (6 \beta_{3} + 6 \beta_1 - 6) q^{19} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{20} + (10 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 8) q^{21} + ( - \beta_{3} - \beta_1 + 5) q^{22} + ( - \beta_{3} - 16 \beta_{2} + \beta_1) q^{23} + (\beta_{3} + 5 \beta_{2} + 7 \beta_1 + 8) q^{24} + ( - \beta_{3} - \beta_1 + 21) q^{25} + (4 \beta_{3} - 12 \beta_{2} - 4 \beta_1) q^{26} + (16 \beta_{3} - 8 \beta_{2} - 16 \beta_1 - 5) q^{27} - 16 q^{28} + ( - 24 \beta_{3} + 4 \beta_{2} + 24 \beta_1) q^{29} + (2 \beta_{2} - 2 \beta_1 + 2) q^{30} + (5 \beta_{3} + 5 \beta_1 + 14) q^{31} + ( - \beta_{3} + 19 \beta_{2} + \beta_1) q^{32} + ( - 4 \beta_{2} - 5 \beta_1 - 4) q^{33} + ( - 16 \beta_{3} - 16 \beta_1 + 20) q^{34} + ( - 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{35} + (\beta_{3} - 8 \beta_{2} - 7 \beta_1 - 23) q^{36} + ( - 7 \beta_{3} - 7 \beta_1 - 26) q^{37} + ( - 18 \beta_{3} + 30 \beta_{2} + 18 \beta_1) q^{38} + (20 \beta_{3} - 12 \beta_1 - 12) q^{39} + ( - 2 \beta_{3} - 2 \beta_1 - 10) q^{40} + (4 \beta_{3} + 12 \beta_{2} - 4 \beta_1) q^{41} + ( - 12 \beta_{3} + 20 \beta_{2} + 16 \beta_1 - 16) q^{42} + (4 \beta_{3} + 4 \beta_1 + 26) q^{43} + ( - \beta_{3} - 5 \beta_{2} + \beta_1) q^{44} + ( - 6 \beta_{3} - 5 \beta_{2} + 5 \beta_1 + 4) q^{45} + (16 \beta_{3} + 16 \beta_1 - 14) q^{46} + ( - 22 \beta_{3} + 14 \beta_{2} + 22 \beta_1) q^{47} + ( - 25 \beta_{3} + 8 \beta_{2} + 7 \beta_1 + 23) q^{48} + ( - 4 \beta_{3} - 4 \beta_1 - 17) q^{49} + (23 \beta_{3} - 25 \beta_{2} - 23 \beta_1) q^{50} + (30 \beta_{3} - 34 \beta_{2} - 20 \beta_1 + 56) q^{51} + ( - 4 \beta_{3} - 4 \beta_1 - 36) q^{52} + (30 \beta_{3} - 38 \beta_{2} - 30 \beta_1) q^{53} + (3 \beta_{3} + 5 \beta_{2} + 13 \beta_1 - 40) q^{54} + (\beta_{3} + \beta_1 + 6) q^{55} - 16 \beta_{2} q^{56} + ( - 30 \beta_{3} + 12 \beta_{2} + 6 \beta_1 + 30) q^{57} + ( - 4 \beta_{3} - 4 \beta_1 + 52) q^{58} + (23 \beta_{3} + 2 \beta_{2} - 23 \beta_1) q^{59} + (6 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 16) q^{60} + (20 \beta_{3} + 20 \beta_1 + 4) q^{61} + (4 \beta_{3} + 6 \beta_{2} - 4 \beta_1) q^{62} + ( - 12 \beta_{3} + 22 \beta_{2} + 14 \beta_1 + 40) q^{63} + (\beta_{3} + \beta_1 + 9) q^{64} + ( - 8 \beta_{3} - 8 \beta_{2} + 8 \beta_1) q^{65} + (5 \beta_{3} - 6 \beta_{2} + 3 \beta_1 - 9) q^{66} + (17 \beta_{3} + 17 \beta_1 - 6) q^{67} + (44 \beta_{3} - 20 \beta_{2} - 44 \beta_1) q^{68} + ( - 33 \beta_{3} + 31 \beta_{2} + 14 \beta_1 - 68) q^{69} + (4 \beta_{3} + 4 \beta_1) q^{70} + (3 \beta_{3} + 4 \beta_{2} - 3 \beta_1) q^{71} + ( - 13 \beta_{3} - 17 \beta_{2} + 5 \beta_1 + 16) q^{72} + ( - 6 \beta_{3} - 6 \beta_1 - 74) q^{73} + ( - 12 \beta_{3} - 2 \beta_{2} + 12 \beta_1) q^{74} + (5 \beta_{3} - 22 \beta_{2} + 19 \beta_1 - 25) q^{75} + ( - 6 \beta_{3} - 6 \beta_1 + 42) q^{76} + ( - 2 \beta_{3} + 12 \beta_{2} + 2 \beta_1) q^{77} + ( - 20 \beta_{3} + 28 \beta_{2} + 20 \beta_1 - 32) q^{78} + ( - 26 \beta_{3} - 26 \beta_1 - 32) q^{79} + (2 \beta_{3} + 10 \beta_{2} - 2 \beta_1) q^{80} + (37 \beta_{2} + 35 \beta_1 + 37) q^{81} + ( - 12 \beta_{3} - 12 \beta_1 + 4) q^{82} + ( - 30 \beta_{3} + 24 \beta_{2} + 30 \beta_1) q^{83} + (16 \beta_{2} - 16 \beta_1 + 16) q^{84} + ( - 14 \beta_{3} - 14 \beta_1 - 24) q^{85} + (18 \beta_{3} - 10 \beta_{2} - 18 \beta_1) q^{86} + ( - 16 \beta_{3} - 32 \beta_{2} - 52 \beta_1 - 80) q^{87} + (\beta_{3} + \beta_1 + 17) q^{88} + (39 \beta_{3} + 26 \beta_{2} - 39 \beta_1) q^{89} + (10 \beta_{3} - 6 \beta_{2} + 6) q^{90} + 64 q^{91} + ( - 50 \beta_{3} + 14 \beta_{2} + 50 \beta_1) q^{92} + ( - 25 \beta_{3} - 9 \beta_{2} + 24 \beta_1 + 6) q^{93} + ( - 14 \beta_{3} - 14 \beta_1 + 58) q^{94} + 12 \beta_{2} q^{95} + (37 \beta_{3} - 39 \beta_{2} - 21 \beta_1 + 72) q^{96} + ( - 33 \beta_{3} - 33 \beta_1 + 38) q^{97} + ( - 9 \beta_{3} + \beta_{2} + 9 \beta_1) q^{98} + (7 \beta_{3} + 12 \beta_{2} - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 5 q^{3} + 2 q^{4} - 2 q^{6} + 4 q^{7} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 5 q^{3} + 2 q^{4} - 2 q^{6} + 4 q^{7} - 7 q^{9} - 8 q^{10} + 14 q^{12} - 8 q^{13} + 13 q^{15} - 22 q^{16} + 38 q^{18} - 36 q^{19} - 38 q^{21} + 22 q^{22} + 24 q^{24} + 86 q^{25} - 20 q^{27} - 64 q^{28} + 10 q^{30} + 46 q^{31} - 11 q^{33} + 112 q^{34} - 86 q^{36} - 90 q^{37} - 56 q^{39} - 36 q^{40} - 68 q^{42} + 96 q^{43} + 17 q^{45} - 88 q^{46} + 110 q^{48} - 60 q^{49} + 214 q^{51} - 136 q^{52} - 176 q^{54} + 22 q^{55} + 144 q^{57} + 216 q^{58} + 56 q^{60} - 24 q^{61} + 158 q^{63} + 34 q^{64} - 44 q^{66} - 58 q^{67} - 253 q^{69} - 8 q^{70} + 72 q^{72} - 284 q^{73} - 124 q^{75} + 180 q^{76} - 128 q^{78} - 76 q^{79} + 113 q^{81} + 40 q^{82} + 80 q^{84} - 68 q^{85} - 252 q^{87} + 66 q^{88} + 14 q^{90} + 256 q^{91} + 25 q^{93} + 260 q^{94} + 272 q^{96} + 218 q^{97} - 55 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} + 4\nu - 9 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} - 2\nu^{2} + 2\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} - 3\beta_{2} + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + \beta_{2} + 2\beta _1 + 4 \) 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
−1.18614 + 1.26217i
1.68614 + 0.396143i
1.68614 0.396143i
−1.18614 1.26217i
2.52434i −2.68614 1.33591i −2.37228 0.792287i −3.37228 + 6.78073i 6.74456 4.10891i 5.43070 + 7.17687i −2.00000
23.2 0.792287i 0.186141 + 2.99422i 3.37228 2.52434i 2.37228 0.147477i −4.74456 5.84096i −8.93070 + 1.11469i −2.00000
23.3 0.792287i 0.186141 2.99422i 3.37228 2.52434i 2.37228 + 0.147477i −4.74456 5.84096i −8.93070 1.11469i −2.00000
23.4 2.52434i −2.68614 + 1.33591i −2.37228 0.792287i −3.37228 6.78073i 6.74456 4.10891i 5.43070 7.17687i −2.00000
\(n\): 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.b 4
3.b odd 2 1 inner 33.3.b.b 4
4.b odd 2 1 528.3.i.d 4
11.b odd 2 1 363.3.b.h 4
11.c even 5 4 363.3.h.m 16
11.d odd 10 4 363.3.h.l 16
12.b even 2 1 528.3.i.d 4
33.d even 2 1 363.3.b.h 4
33.f even 10 4 363.3.h.l 16
33.h odd 10 4 363.3.h.m 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.b.b 4 1.a even 1 1 trivial
33.3.b.b 4 3.b odd 2 1 inner
363.3.b.h 4 11.b odd 2 1
363.3.b.h 4 33.d even 2 1
363.3.h.l 16 11.d odd 10 4
363.3.h.l 16 33.f even 10 4
363.3.h.m 16 11.c even 5 4
363.3.h.m 16 33.h odd 10 4
528.3.i.d 4 4.b odd 2 1
528.3.i.d 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 7T_{2}^{2} + 4 \) acting on \(S_{3}^{\mathrm{new}}(33, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 7T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} + 5 T^{3} + 16 T^{2} + 45 T + 81 \) Copy content Toggle raw display
$5$ \( T^{4} + 7T^{2} + 4 \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T - 32)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 11)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T - 128)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 1372 T^{2} + 440896 \) Copy content Toggle raw display
$19$ \( (T^{2} + 18 T - 216)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1639 T^{2} + 662596 \) Copy content Toggle raw display
$29$ \( T^{4} + 3552 T^{2} + \cdots + 1937664 \) Copy content Toggle raw display
$31$ \( (T^{2} - 23 T - 74)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 45 T + 102)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 1264 T^{2} + 295936 \) Copy content Toggle raw display
$43$ \( (T^{2} - 48 T + 444)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 2716 T^{2} + \cdots + 1700416 \) Copy content Toggle raw display
$53$ \( T^{4} + 8124 T^{2} + 788544 \) Copy content Toggle raw display
$59$ \( T^{4} + 4003 T^{2} + 824464 \) Copy content Toggle raw display
$61$ \( (T^{2} + 12 T - 3264)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 29 T - 2174)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 231T^{2} + 4356 \) Copy content Toggle raw display
$73$ \( (T^{2} + 142 T + 4744)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 38 T - 5216)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 5436 T^{2} + \cdots + 4981824 \) Copy content Toggle raw display
$89$ \( T^{4} + 20787 T^{2} + \cdots + 4112784 \) Copy content Toggle raw display
$97$ \( (T^{2} - 109 T - 6014)^{2} \) Copy content Toggle raw display
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