Properties

Label 33.6.a.e
Level $33$
Weight $6$
Character orbit 33.a
Self dual yes
Analytic conductor $5.293$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 33.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.29266605383\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 7) q^{2} + 9 q^{3} + ( - 13 \beta + 25) q^{4} + (10 \beta + 24) q^{5} + ( - 9 \beta + 63) q^{6} + (62 \beta + 42) q^{7} + ( - 71 \beta + 55) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 7) q^{2} + 9 q^{3} + ( - 13 \beta + 25) q^{4} + (10 \beta + 24) q^{5} + ( - 9 \beta + 63) q^{6} + (62 \beta + 42) q^{7} + ( - 71 \beta + 55) q^{8} + 81 q^{9} + (36 \beta + 88) q^{10} - 121 q^{11} + ( - 117 \beta + 225) q^{12} + ( - 74 \beta - 28) q^{13} + (330 \beta - 202) q^{14} + (90 \beta + 216) q^{15} + ( - 65 \beta + 153) q^{16} + (372 \beta - 550) q^{17} + ( - 81 \beta + 567) q^{18} + ( - 852 \beta + 12) q^{19} + ( - 192 \beta - 440) q^{20} + (558 \beta + 378) q^{21} + (121 \beta - 847) q^{22} + ( - 330 \beta + 46) q^{23} + ( - 639 \beta + 495) q^{24} + (580 \beta - 1749) q^{25} + ( - 416 \beta + 396) q^{26} + 729 q^{27} + (198 \beta - 5398) q^{28} + ( - 1492 \beta + 1094) q^{29} + (324 \beta + 792) q^{30} + (1600 \beta - 6040) q^{31} + (1729 \beta - 169) q^{32} - 1089 q^{33} + (2782 \beta - 6826) q^{34} + (2528 \beta + 5968) q^{35} + ( - 1053 \beta + 2025) q^{36} + ( - 2816 \beta + 454) q^{37} + ( - 5124 \beta + 6900) q^{38} + ( - 666 \beta - 252) q^{39} + ( - 1864 \beta - 4360) q^{40} + ( - 8 \beta + 18246) q^{41} + (2970 \beta - 1818) q^{42} + ( - 3112 \beta + 6440) q^{43} + (1573 \beta - 3025) q^{44} + (810 \beta + 1944) q^{45} + ( - 2026 \beta + 2962) q^{46} + ( - 390 \beta + 22066) q^{47} + ( - 585 \beta + 1377) q^{48} + (9052 \beta + 15709) q^{49} + (5229 \beta - 16883) q^{50} + (3348 \beta - 4950) q^{51} + ( - 524 \beta + 6996) q^{52} + ( - 7102 \beta - 2536) q^{53} + ( - 729 \beta + 5103) q^{54} + ( - 1210 \beta - 2904) q^{55} + ( - 3974 \beta - 32906) q^{56} + ( - 7668 \beta + 108) q^{57} + ( - 10046 \beta + 19594) q^{58} + (1980 \beta - 2384) q^{59} + ( - 1728 \beta - 3960) q^{60} + (2026 \beta - 13664) q^{61} + (15640 \beta - 55080) q^{62} + (5022 \beta + 3402) q^{63} + (12623 \beta - 19911) q^{64} + ( - 2796 \beta - 6592) q^{65} + (1089 \beta - 7623) q^{66} + ( - 12704 \beta - 13908) q^{67} + (11614 \beta - 52438) q^{68} + ( - 2970 \beta + 414) q^{69} + (9200 \beta + 21552) q^{70} + ( - 4354 \beta + 17870) q^{71} + ( - 5751 \beta + 4455) q^{72} + (5568 \beta - 26174) q^{73} + ( - 17350 \beta + 25706) q^{74} + (5220 \beta - 15741) q^{75} + ( - 10380 \beta + 88908) q^{76} + ( - 7502 \beta - 5082) q^{77} + ( - 3744 \beta + 3564) q^{78} + (11426 \beta - 14138) q^{79} + ( - 680 \beta - 1528) q^{80} + 6561 q^{81} + ( - 18294 \beta + 127786) q^{82} + (21960 \beta + 28740) q^{83} + (1782 \beta - 48582) q^{84} + (7148 \beta + 16560) q^{85} + ( - 25112 \beta + 69976) q^{86} + ( - 13428 \beta + 9846) q^{87} + (8591 \beta - 6655) q^{88} + (26704 \beta - 40454) q^{89} + (2916 \beta + 7128) q^{90} + ( - 9432 \beta - 37880) q^{91} + ( - 4558 \beta + 35470) q^{92} + (14400 \beta - 54360) q^{93} + ( - 24406 \beta + 157582) q^{94} + ( - 28848 \beta - 67872) q^{95} + (15561 \beta - 1521) q^{96} + (9924 \beta - 125746) q^{97} + (38603 \beta + 37547) q^{98} - 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 13 q^{2} + 18 q^{3} + 37 q^{4} + 58 q^{5} + 117 q^{6} + 146 q^{7} + 39 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 13 q^{2} + 18 q^{3} + 37 q^{4} + 58 q^{5} + 117 q^{6} + 146 q^{7} + 39 q^{8} + 162 q^{9} + 212 q^{10} - 242 q^{11} + 333 q^{12} - 130 q^{13} - 74 q^{14} + 522 q^{15} + 241 q^{16} - 728 q^{17} + 1053 q^{18} - 828 q^{19} - 1072 q^{20} + 1314 q^{21} - 1573 q^{22} - 238 q^{23} + 351 q^{24} - 2918 q^{25} + 376 q^{26} + 1458 q^{27} - 10598 q^{28} + 696 q^{29} + 1908 q^{30} - 10480 q^{31} + 1391 q^{32} - 2178 q^{33} - 10870 q^{34} + 14464 q^{35} + 2997 q^{36} - 1908 q^{37} + 8676 q^{38} - 1170 q^{39} - 10584 q^{40} + 36484 q^{41} - 666 q^{42} + 9768 q^{43} - 4477 q^{44} + 4698 q^{45} + 3898 q^{46} + 43742 q^{47} + 2169 q^{48} + 40470 q^{49} - 28537 q^{50} - 6552 q^{51} + 13468 q^{52} - 12174 q^{53} + 9477 q^{54} - 7018 q^{55} - 69786 q^{56} - 7452 q^{57} + 29142 q^{58} - 2788 q^{59} - 9648 q^{60} - 25302 q^{61} - 94520 q^{62} + 11826 q^{63} - 27199 q^{64} - 15980 q^{65} - 14157 q^{66} - 40520 q^{67} - 93262 q^{68} - 2142 q^{69} + 52304 q^{70} + 31386 q^{71} + 3159 q^{72} - 46780 q^{73} + 34062 q^{74} - 26262 q^{75} + 167436 q^{76} - 17666 q^{77} + 3384 q^{78} - 16850 q^{79} - 3736 q^{80} + 13122 q^{81} + 237278 q^{82} + 79440 q^{83} - 95382 q^{84} + 40268 q^{85} + 114840 q^{86} + 6264 q^{87} - 4719 q^{88} - 54204 q^{89} + 17172 q^{90} - 85192 q^{91} + 66382 q^{92} - 94320 q^{93} + 290758 q^{94} - 164592 q^{95} + 12519 q^{96} - 241568 q^{97} + 113697 q^{98} - 19602 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.37228
−2.37228
3.62772 9.00000 −18.8397 57.7228 32.6495 251.081 −184.432 81.0000 209.402
1.2 9.37228 9.00000 55.8397 0.277187 84.3505 −105.081 223.432 81.0000 2.59787
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.6.a.e 2
3.b odd 2 1 99.6.a.d 2
4.b odd 2 1 528.6.a.o 2
5.b even 2 1 825.6.a.c 2
11.b odd 2 1 363.6.a.f 2
33.d even 2 1 1089.6.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.e 2 1.a even 1 1 trivial
99.6.a.d 2 3.b odd 2 1
363.6.a.f 2 11.b odd 2 1
528.6.a.o 2 4.b odd 2 1
825.6.a.c 2 5.b even 2 1
1089.6.a.p 2 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 13T_{2} + 34 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(33))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 13T + 34 \) Copy content Toggle raw display
$3$ \( (T - 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 58T + 16 \) Copy content Toggle raw display
$7$ \( T^{2} - 146T - 26384 \) Copy content Toggle raw display
$11$ \( (T + 121)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 130T - 40952 \) Copy content Toggle raw display
$17$ \( T^{2} + 728 T - 1009172 \) Copy content Toggle raw display
$19$ \( T^{2} + 828 T - 5817312 \) Copy content Toggle raw display
$23$ \( T^{2} + 238T - 884264 \) Copy content Toggle raw display
$29$ \( T^{2} - 696 T - 18243924 \) Copy content Toggle raw display
$31$ \( T^{2} + 10480 T + 6337600 \) Copy content Toggle raw display
$37$ \( T^{2} + 1908 T - 64511196 \) Copy content Toggle raw display
$41$ \( T^{2} - 36484 T + 332770036 \) Copy content Toggle raw display
$43$ \( T^{2} - 9768 T - 56044032 \) Copy content Toggle raw display
$47$ \( T^{2} - 43742 T + 477085816 \) Copy content Toggle raw display
$53$ \( T^{2} + 12174 T - 379065264 \) Copy content Toggle raw display
$59$ \( T^{2} + 2788 T - 30400064 \) Copy content Toggle raw display
$61$ \( T^{2} + 25302 T + 126184224 \) Copy content Toggle raw display
$67$ \( T^{2} + 40520 T - 921013232 \) Copy content Toggle raw display
$71$ \( T^{2} - 31386 T + 89872392 \) Copy content Toggle raw display
$73$ \( T^{2} + 46780 T + 291320452 \) Copy content Toggle raw display
$79$ \( T^{2} + 16850 T - 1006085552 \) Copy content Toggle raw display
$83$ \( T^{2} - 79440 T - 2400814800 \) Copy content Toggle raw display
$89$ \( T^{2} + 54204 T - 5148586428 \) Copy content Toggle raw display
$97$ \( T^{2} + 241568 T + 13776267004 \) Copy content Toggle raw display
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