Properties

Label 33.4.d.a
Level $33$
Weight $4$
Character orbit 33.d
Analytic conductor $1.947$
Analytic rank $0$
Dimension $2$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 33.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.94706303019\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Defining polynomial: \(x^{2} - x + 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -4 - \beta ) q^{3} -8 q^{4} -4 \beta q^{5} + ( 5 + 8 \beta ) q^{9} +O(q^{10})\) \( q + ( -4 - \beta ) q^{3} -8 q^{4} -4 \beta q^{5} + ( 5 + 8 \beta ) q^{9} -11 \beta q^{11} + ( 32 + 8 \beta ) q^{12} + ( -44 + 16 \beta ) q^{15} + 64 q^{16} + 32 \beta q^{20} -58 \beta q^{23} -51 q^{25} + ( 68 - 37 \beta ) q^{27} -340 q^{31} + ( -121 + 44 \beta ) q^{33} + ( -40 - 64 \beta ) q^{36} + 434 q^{37} + 88 \beta q^{44} + ( 352 - 20 \beta ) q^{45} + 194 \beta q^{47} + ( -256 - 64 \beta ) q^{48} + 343 q^{49} + 68 \beta q^{53} -484 q^{55} -166 \beta q^{59} + ( 352 - 128 \beta ) q^{60} -512 q^{64} + 416 q^{67} + ( -638 + 232 \beta ) q^{69} -310 \beta q^{71} + ( 204 + 51 \beta ) q^{75} -256 \beta q^{80} + ( -679 + 80 \beta ) q^{81} -40 \beta q^{89} + 464 \beta q^{92} + ( 1360 + 340 \beta ) q^{93} -34 q^{97} + ( 968 - 55 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{3} - 16q^{4} + 10q^{9} + O(q^{10}) \) \( 2q - 8q^{3} - 16q^{4} + 10q^{9} + 64q^{12} - 88q^{15} + 128q^{16} - 102q^{25} + 136q^{27} - 680q^{31} - 242q^{33} - 80q^{36} + 868q^{37} + 704q^{45} - 512q^{48} + 686q^{49} - 968q^{55} + 704q^{60} - 1024q^{64} + 832q^{67} - 1276q^{69} + 408q^{75} - 1358q^{81} + 2720q^{93} - 68q^{97} + 1936q^{99} + O(q^{100}) \)

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.

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 −4.00000 3.31662i −8.00000 13.2665i 0 0 0 5.00000 + 26.5330i 0
32.2 0 −4.00000 + 3.31662i −8.00000 13.2665i 0 0 0 5.00000 26.5330i 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 33.4.d.a 2
3.b odd 2 1 inner 33.4.d.a 2
4.b odd 2 1 528.4.b.a 2
11.b odd 2 1 CM 33.4.d.a 2
12.b even 2 1 528.4.b.a 2
33.d even 2 1 inner 33.4.d.a 2
44.c even 2 1 528.4.b.a 2
132.d odd 2 1 528.4.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.d.a 2 1.a even 1 1 trivial
33.4.d.a 2 3.b odd 2 1 inner
33.4.d.a 2 11.b odd 2 1 CM
33.4.d.a 2 33.d even 2 1 inner
528.4.b.a 2 4.b odd 2 1
528.4.b.a 2 12.b even 2 1
528.4.b.a 2 44.c even 2 1
528.4.b.a 2 132.d odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 8 T^{2} )^{2} \)
$3$ \( 1 + 8 T + 27 T^{2} \)
$5$ \( ( 1 - 18 T + 125 T^{2} )( 1 + 18 T + 125 T^{2} ) \)
$7$ \( ( 1 - 343 T^{2} )^{2} \)
$11$ \( 1 + 1331 T^{2} \)
$13$ \( ( 1 - 2197 T^{2} )^{2} \)
$17$ \( ( 1 + 4913 T^{2} )^{2} \)
$19$ \( ( 1 - 6859 T^{2} )^{2} \)
$23$ \( ( 1 - 108 T + 12167 T^{2} )( 1 + 108 T + 12167 T^{2} ) \)
$29$ \( ( 1 + 24389 T^{2} )^{2} \)
$31$ \( ( 1 + 340 T + 29791 T^{2} )^{2} \)
$37$ \( ( 1 - 434 T + 50653 T^{2} )^{2} \)
$41$ \( ( 1 + 68921 T^{2} )^{2} \)
$43$ \( ( 1 - 79507 T^{2} )^{2} \)
$47$ \( ( 1 - 36 T + 103823 T^{2} )( 1 + 36 T + 103823 T^{2} ) \)
$53$ \( ( 1 - 738 T + 148877 T^{2} )( 1 + 738 T + 148877 T^{2} ) \)
$59$ \( ( 1 - 720 T + 205379 T^{2} )( 1 + 720 T + 205379 T^{2} ) \)
$61$ \( ( 1 - 226981 T^{2} )^{2} \)
$67$ \( ( 1 - 416 T + 300763 T^{2} )^{2} \)
$71$ \( ( 1 - 612 T + 357911 T^{2} )( 1 + 612 T + 357911 T^{2} ) \)
$73$ \( ( 1 - 389017 T^{2} )^{2} \)
$79$ \( ( 1 - 493039 T^{2} )^{2} \)
$83$ \( ( 1 + 571787 T^{2} )^{2} \)
$89$ \( ( 1 - 1674 T + 704969 T^{2} )( 1 + 1674 T + 704969 T^{2} ) \)
$97$ \( ( 1 + 34 T + 912673 T^{2} )^{2} \)
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