Properties

Label 33.3.c.a
Level $33$
Weight $3$
Character orbit 33.c
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(10,33)]
 
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([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.10");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 33.c (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: 4.0.39744.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 12x^{2} + 4x + 22 \) 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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_1 q^{3} + ( - 2 \beta_1 - 5) q^{4} + (3 \beta_1 + 1) q^{5} + (\beta_{3} - \beta_{2}) q^{6} + \beta_{2} q^{7} + (3 \beta_{3} - 2 \beta_{2}) q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_1 q^{3} + ( - 2 \beta_1 - 5) q^{4} + (3 \beta_1 + 1) q^{5} + (\beta_{3} - \beta_{2}) q^{6} + \beta_{2} q^{7} + (3 \beta_{3} - 2 \beta_{2}) q^{8} + 3 q^{9} + ( - 4 \beta_{3} + 3 \beta_{2}) q^{10} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 5) q^{11} + (5 \beta_1 + 6) q^{12} + (6 \beta_{3} - \beta_{2}) q^{13} + ( - 7 \beta_1 + 3) q^{14} + ( - \beta_1 - 9) q^{15} + (12 \beta_1 + 1) q^{16} + (2 \beta_{3} - 2 \beta_{2}) q^{17} - 3 \beta_{3} q^{18} + ( - 2 \beta_{3} - 4 \beta_{2}) q^{19} + ( - 17 \beta_1 - 23) q^{20} + (2 \beta_{3} + \beta_{2}) q^{21} + ( - 6 \beta_{3} + \beta_{2} - 16 \beta_1 - 3) q^{22} + (5 \beta_1 - 23) q^{23} + ( - 7 \beta_{3} + \beta_{2}) q^{24} + (6 \beta_1 + 3) q^{25} + (19 \beta_1 + 51) q^{26} - 3 \beta_1 q^{27} + (4 \beta_{3} - 3 \beta_{2}) q^{28} + ( - 8 \beta_{3} + 4 \beta_{2}) q^{29} + (10 \beta_{3} - \beta_{2}) q^{30} + ( - 18 \beta_1 + 20) q^{31} + ( - \beta_{3} + 4 \beta_{2}) q^{32} + (5 \beta_{3} + \beta_{2} - 5 \beta_1 - 3) q^{33} + (18 \beta_1 + 12) q^{34} + ( - 6 \beta_{3} - 2 \beta_{2}) q^{35} + ( - 6 \beta_1 - 15) q^{36} + (2 \beta_1 - 16) q^{37} + (24 \beta_1 - 30) q^{38} + ( - 8 \beta_{3} + 5 \beta_{2}) q^{39} + (24 \beta_{3} - 5 \beta_{2}) q^{40} + (4 \beta_{3} + 6 \beta_{2}) q^{41} + ( - 3 \beta_1 + 21) q^{42} + ( - 2 \beta_{3} - 12 \beta_{2}) q^{43} + (15 \beta_{3} - 8 \beta_{2} - 15 \beta_1 - 31) q^{44} + (9 \beta_1 + 3) q^{45} + (18 \beta_{3} + 5 \beta_{2}) q^{46} + ( - 3 \beta_1 + 25) q^{47} + ( - \beta_1 - 36) q^{48} + (10 \beta_1 + 25) q^{49} + ( - 9 \beta_{3} + 6 \beta_{2}) q^{50} - 6 \beta_{3} q^{51} + ( - 46 \beta_{3} + 15 \beta_{2}) q^{52} + ( - 11 \beta_1 + 7) q^{53} + (3 \beta_{3} - 3 \beta_{2}) q^{54} + ( - 16 \beta_{3} - \beta_{2} + 16 \beta_1 + 14) q^{55} + (\beta_1 + 39) q^{56} + ( - 6 \beta_{3} - 6 \beta_{2}) q^{57} + ( - 44 \beta_1 - 60) q^{58} + ( - 42 \beta_1 + 10) q^{59} + (23 \beta_1 + 51) q^{60} + (2 \beta_{3} + 11 \beta_{2}) q^{61} + ( - 2 \beta_{3} - 18 \beta_{2}) q^{62} + 3 \beta_{2} q^{63} + (18 \beta_1 + 7) q^{64} + (30 \beta_{3} - 16 \beta_{2}) q^{65} + (8 \beta_{3} - 5 \beta_{2} + 3 \beta_1 + 48) q^{66} - 34 q^{67} + ( - 22 \beta_{3} + 10 \beta_{2}) q^{68} + (23 \beta_1 - 15) q^{69} + (2 \beta_1 - 60) q^{70} + (\beta_1 - 71) q^{71} + (9 \beta_{3} - 6 \beta_{2}) q^{72} + ( - 4 \beta_{3} + 10 \beta_{2}) q^{73} + (14 \beta_{3} + 2 \beta_{2}) q^{74} + ( - 3 \beta_1 - 18) q^{75} + ( - 2 \beta_{3} + 8 \beta_{2}) q^{76} + ( - 2 \beta_{3} + 4 \beta_{2} + 13 \beta_1 - 45) q^{77} + ( - 51 \beta_1 - 57) q^{78} + (24 \beta_{3} - 5 \beta_{2}) q^{79} + (15 \beta_1 + 109) q^{80} + 9 q^{81} + ( - 34 \beta_1 + 54) q^{82} + ( - 16 \beta_{3} + 6 \beta_{2}) q^{83} + ( - 10 \beta_{3} + \beta_{2}) q^{84} + (20 \beta_{3} - 2 \beta_{2}) q^{85} + (80 \beta_1 - 54) q^{86} + (16 \beta_{3} - 4 \beta_{2}) q^{87} + (22 \beta_{3} - 11 \beta_{2} + 22 \beta_1 + 99) q^{88} + (18 \beta_1 + 76) q^{89} + ( - 12 \beta_{3} + 9 \beta_{2}) q^{90} + (32 \beta_1 + 6) q^{91} + (21 \beta_1 + 85) q^{92} + ( - 20 \beta_1 + 54) q^{93} + ( - 22 \beta_{3} - 3 \beta_{2}) q^{94} + (16 \beta_{3} + 14 \beta_{2}) q^{95} + (9 \beta_{3} + 3 \beta_{2}) q^{96} + ( - 42 \beta_1 - 94) q^{97} + ( - 35 \beta_{3} + 10 \beta_{2}) q^{98} + ( - 3 \beta_{3} + 6 \beta_{2} + 3 \beta_1 + 15) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{4} + 4 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{4} + 4 q^{5} + 12 q^{9} + 20 q^{11} + 24 q^{12} + 12 q^{14} - 36 q^{15} + 4 q^{16} - 92 q^{20} - 12 q^{22} - 92 q^{23} + 12 q^{25} + 204 q^{26} + 80 q^{31} - 12 q^{33} + 48 q^{34} - 60 q^{36} - 64 q^{37} - 120 q^{38} + 84 q^{42} - 124 q^{44} + 12 q^{45} + 100 q^{47} - 144 q^{48} + 100 q^{49} + 28 q^{53} + 56 q^{55} + 156 q^{56} - 240 q^{58} + 40 q^{59} + 204 q^{60} + 28 q^{64} + 192 q^{66} - 136 q^{67} - 60 q^{69} - 240 q^{70} - 284 q^{71} - 72 q^{75} - 180 q^{77} - 228 q^{78} + 436 q^{80} + 36 q^{81} + 216 q^{82} - 216 q^{86} + 396 q^{88} + 304 q^{89} + 24 q^{91} + 340 q^{92} + 216 q^{93} - 376 q^{97} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} + 12x^{2} + 4x + 22 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu^{2} + 2\nu + 23 ) / 13 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + \nu^{2} + 28\nu + 10 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{3} + 10\nu^{2} - 32\nu + 9 ) / 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 2\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} - 2\beta_{2} + 12\beta _1 - 19 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
−0.366025 1.29224i
1.36603 + 3.21405i
1.36603 3.21405i
−0.366025 + 1.29224i
3.53045i −1.73205 −8.46410 6.19615 6.11492i 2.58447i 15.7603i 3.00000 21.8752i
10.2 2.35285i 1.73205 −1.53590 −4.19615 4.07525i 6.42810i 5.79766i 3.00000 9.87291i
10.3 2.35285i 1.73205 −1.53590 −4.19615 4.07525i 6.42810i 5.79766i 3.00000 9.87291i
10.4 3.53045i −1.73205 −8.46410 6.19615 6.11492i 2.58447i 15.7603i 3.00000 21.8752i
\(n\): 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.3.c.a 4
3.b odd 2 1 99.3.c.b 4
4.b odd 2 1 528.3.j.c 4
5.b even 2 1 825.3.b.a 4
5.c odd 4 2 825.3.h.a 8
8.b even 2 1 2112.3.j.a 4
8.d odd 2 1 2112.3.j.d 4
11.b odd 2 1 inner 33.3.c.a 4
11.c even 5 4 363.3.g.e 16
11.d odd 10 4 363.3.g.e 16
12.b even 2 1 1584.3.j.f 4
33.d even 2 1 99.3.c.b 4
44.c even 2 1 528.3.j.c 4
55.d odd 2 1 825.3.b.a 4
55.e even 4 2 825.3.h.a 8
88.b odd 2 1 2112.3.j.a 4
88.g even 2 1 2112.3.j.d 4
132.d odd 2 1 1584.3.j.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.c.a 4 1.a even 1 1 trivial
33.3.c.a 4 11.b odd 2 1 inner
99.3.c.b 4 3.b odd 2 1
99.3.c.b 4 33.d even 2 1
363.3.g.e 16 11.c even 5 4
363.3.g.e 16 11.d odd 10 4
528.3.j.c 4 4.b odd 2 1
528.3.j.c 4 44.c even 2 1
825.3.b.a 4 5.b even 2 1
825.3.b.a 4 55.d odd 2 1
825.3.h.a 8 5.c odd 4 2
825.3.h.a 8 55.e even 4 2
1584.3.j.f 4 12.b even 2 1
1584.3.j.f 4 132.d odd 2 1
2112.3.j.a 4 8.b even 2 1
2112.3.j.a 4 88.b odd 2 1
2112.3.j.d 4 8.d odd 2 1
2112.3.j.d 4 88.g even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(33, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 18T^{2} + 69 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 2 T - 26)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 48T^{2} + 276 \) Copy content Toggle raw display
$11$ \( T^{4} - 20 T^{3} + 330 T^{2} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{4} + 624 T^{2} + 33396 \) Copy content Toggle raw display
$17$ \( T^{4} + 216T^{2} + 9936 \) Copy content Toggle raw display
$19$ \( T^{4} + 936T^{2} + 9936 \) Copy content Toggle raw display
$23$ \( (T^{2} + 46 T + 454)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 1536 T^{2} + 70656 \) Copy content Toggle raw display
$31$ \( (T^{2} - 40 T - 572)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 32 T + 244)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 2304 T^{2} + 4416 \) Copy content Toggle raw display
$43$ \( T^{4} + 7272 T^{2} + \cdots + 3843024 \) Copy content Toggle raw display
$47$ \( (T^{2} - 50 T + 598)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 14 T - 314)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 20 T - 5192)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 6144 T^{2} + \cdots + 2596884 \) Copy content Toggle raw display
$67$ \( (T + 34)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 142 T + 5038)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 4608 T^{2} + \cdots + 4809024 \) Copy content Toggle raw display
$79$ \( T^{4} + 10128 T^{2} + \cdots + 5643924 \) Copy content Toggle raw display
$83$ \( T^{4} + 5184 T^{2} + 4416 \) Copy content Toggle raw display
$89$ \( (T^{2} - 152 T + 4804)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 188 T + 3544)^{2} \) Copy content Toggle raw display
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