Properties

Label 33.6.d.a
Level 33
Weight 6
Character orbit 33.d
Analytic conductor 5.293
Analytic rank 0
Dimension 2
CM discriminant -11
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 33.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.29266605383\)
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: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( 16 - \beta ) q^{3} -32 q^{4} + ( 29 - 58 \beta ) q^{5} + ( 253 - 31 \beta ) q^{9} +O(q^{10})\) \( q + ( 16 - \beta ) q^{3} -32 q^{4} + ( 29 - 58 \beta ) q^{5} + ( 253 - 31 \beta ) q^{9} + ( 121 - 242 \beta ) q^{11} + ( -512 + 32 \beta ) q^{12} + ( 290 - 899 \beta ) q^{15} + 1024 q^{16} + ( -928 + 1856 \beta ) q^{20} + ( -1501 + 3002 \beta ) q^{23} -6126 q^{25} + ( 3955 - 718 \beta ) q^{27} + 7775 q^{31} + ( 1210 - 3751 \beta ) q^{33} + ( -8096 + 992 \beta ) q^{36} -1267 q^{37} + ( -3872 + 7744 \beta ) q^{44} + ( 1943 - 13775 \beta ) q^{45} + ( 5282 - 10564 \beta ) q^{47} + ( 16384 - 1024 \beta ) q^{48} + 16807 q^{49} + ( 6476 - 12952 \beta ) q^{53} -38599 q^{55} + ( -14281 + 28562 \beta ) q^{59} + ( -9280 + 28768 \beta ) q^{60} -32768 q^{64} + 72917 q^{67} + ( -15010 + 46531 \beta ) q^{69} + ( 16025 - 32050 \beta ) q^{71} + ( -98016 + 6126 \beta ) q^{75} + ( 29696 - 59392 \beta ) q^{80} + ( 61126 - 14725 \beta ) q^{81} + ( -35725 + 71450 \beta ) q^{89} + ( 48032 - 96064 \beta ) q^{92} + ( 124400 - 7775 \beta ) q^{93} -163183 q^{97} + ( 8107 - 57475 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 31q^{3} - 64q^{4} + 475q^{9} + O(q^{10}) \) \( 2q + 31q^{3} - 64q^{4} + 475q^{9} - 992q^{12} - 319q^{15} + 2048q^{16} - 12252q^{25} + 7192q^{27} + 15550q^{31} - 1331q^{33} - 15200q^{36} - 2534q^{37} - 9889q^{45} + 31744q^{48} + 33614q^{49} - 77198q^{55} + 10208q^{60} - 65536q^{64} + 145834q^{67} + 16511q^{69} - 189906q^{75} + 107527q^{81} + 241025q^{93} - 326366q^{97} - 41261q^{99} + O(q^{100}) \)

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 15.5000 1.65831i −32.0000 96.1821i 0 0 0 237.500 51.4077i 0
32.2 0 15.5000 + 1.65831i −32.0000 96.1821i 0 0 0 237.500 + 51.4077i 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 33.6.d.a 2
3.b odd 2 1 inner 33.6.d.a 2
11.b odd 2 1 CM 33.6.d.a 2
33.d even 2 1 inner 33.6.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.d.a 2 1.a even 1 1 trivial
33.6.d.a 2 3.b odd 2 1 inner
33.6.d.a 2 11.b odd 2 1 CM
33.6.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 \(S_{6}^{\mathrm{new}}(33, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 32 T^{2} )^{2} \)
$3$ \( 1 - 31 T + 243 T^{2} \)
$5$ \( ( 1 - 57 T + 3125 T^{2} )( 1 + 57 T + 3125 T^{2} ) \)
$7$ \( ( 1 - 16807 T^{2} )^{2} \)
$11$ \( 1 + 161051 T^{2} \)
$13$ \( ( 1 - 371293 T^{2} )^{2} \)
$17$ \( ( 1 + 1419857 T^{2} )^{2} \)
$19$ \( ( 1 - 2476099 T^{2} )^{2} \)
$23$ \( ( 1 - 981 T + 6436343 T^{2} )( 1 + 981 T + 6436343 T^{2} ) \)
$29$ \( ( 1 + 20511149 T^{2} )^{2} \)
$31$ \( ( 1 - 7775 T + 28629151 T^{2} )^{2} \)
$37$ \( ( 1 + 1267 T + 69343957 T^{2} )^{2} \)
$41$ \( ( 1 + 115856201 T^{2} )^{2} \)
$43$ \( ( 1 - 147008443 T^{2} )^{2} \)
$47$ \( ( 1 - 24708 T + 229345007 T^{2} )( 1 + 24708 T + 229345007 T^{2} ) \)
$53$ \( ( 1 - 34806 T + 418195493 T^{2} )( 1 + 34806 T + 418195493 T^{2} ) \)
$59$ \( ( 1 - 24825 T + 714924299 T^{2} )( 1 + 24825 T + 714924299 T^{2} ) \)
$61$ \( ( 1 - 844596301 T^{2} )^{2} \)
$67$ \( ( 1 - 72917 T + 1350125107 T^{2} )^{2} \)
$71$ \( ( 1 - 66273 T + 1804229351 T^{2} )( 1 + 66273 T + 1804229351 T^{2} ) \)
$73$ \( ( 1 - 2073071593 T^{2} )^{2} \)
$79$ \( ( 1 - 3077056399 T^{2} )^{2} \)
$83$ \( ( 1 + 3939040643 T^{2} )^{2} \)
$89$ \( ( 1 - 91089 T + 5584059449 T^{2} )( 1 + 91089 T + 5584059449 T^{2} ) \)
$97$ \( ( 1 + 163183 T + 8587340257 T^{2} )^{2} \)
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