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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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.899184872389\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Defining polynomial: \(x^{2} - x + 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 \(S_{3}^{\mathrm{new}}(33, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T^{2} + 16 T^{4} \)
$3$ \( ( 1 - 3 T )^{2} \)
$5$ \( 1 - 6 T^{2} + 625 T^{4} \)
$7$ \( ( 1 + 8 T + 49 T^{2} )^{2} \)
$11$ \( 1 + 11 T^{2} \)
$13$ \( ( 1 - 4 T + 169 T^{2} )^{2} \)
$17$ \( 1 - 402 T^{2} + 83521 T^{4} \)
$19$ \( ( 1 + 6 T + 361 T^{2} )^{2} \)
$23$ \( 1 - 1014 T^{2} + 279841 T^{4} \)
$29$ \( 1 - 98 T^{2} + 707281 T^{4} \)
$31$ \( ( 1 + 26 T + 961 T^{2} )^{2} \)
$37$ \( ( 1 - 30 T + 1369 T^{2} )^{2} \)
$41$ \( 1 - 3186 T^{2} + 2825761 T^{4} \)
$43$ \( ( 1 - 42 T + 1849 T^{2} )^{2} \)
$47$ \( 1 + 3018 T^{2} + 4879681 T^{4} \)
$53$ \( 1 - 2054 T^{2} + 7890481 T^{4} \)
$59$ \( 1 - 2562 T^{2} + 12117361 T^{4} \)
$61$ \( ( 1 - 12 T + 3721 T^{2} )^{2} \)
$67$ \( ( 1 - 2 T + 4489 T^{2} )^{2} \)
$71$ \( 1 - 6518 T^{2} + 25411681 T^{4} \)
$73$ \( ( 1 + 74 T + 5329 T^{2} )^{2} \)
$79$ \( ( 1 + 40 T + 6241 T^{2} )^{2} \)
$83$ \( 1 - 12194 T^{2} + 47458321 T^{4} \)
$89$ \( 1 - 1586 T^{2} + 62742241 T^{4} \)
$97$ \( ( 1 - 62 T + 9409 T^{2} )^{2} \)
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