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: \(x^{2} - x - 8\)
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.

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

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 \( T_{2}^{2} - 13 T_{2} + 34 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(33))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 13 T + 98 T^{2} - 416 T^{3} + 1024 T^{4} \)
$3$ \( ( 1 - 9 T )^{2} \)
$5$ \( 1 - 58 T + 6266 T^{2} - 181250 T^{3} + 9765625 T^{4} \)
$7$ \( 1 - 146 T + 7230 T^{2} - 2453822 T^{3} + 282475249 T^{4} \)
$11$ \( ( 1 + 121 T )^{2} \)
$13$ \( 1 + 130 T + 701634 T^{2} + 48268090 T^{3} + 137858491849 T^{4} \)
$17$ \( 1 + 728 T + 1830542 T^{2} + 1033655896 T^{3} + 2015993900449 T^{4} \)
$19$ \( 1 + 828 T - 865114 T^{2} + 2050209972 T^{3} + 6131066257801 T^{4} \)
$23$ \( 1 + 238 T + 11988422 T^{2} + 1531849634 T^{3} + 41426511213649 T^{4} \)
$29$ \( 1 - 696 T + 22778374 T^{2} - 14275759704 T^{3} + 420707233300201 T^{4} \)
$31$ \( 1 + 10480 T + 63595902 T^{2} + 300033502480 T^{3} + 819628286980801 T^{4} \)
$37$ \( 1 + 1908 T + 74176718 T^{2} + 132308269956 T^{3} + 4808584372417849 T^{4} \)
$41$ \( 1 - 36484 T + 564482438 T^{2} - 4226897637284 T^{3} + 13422659310152401 T^{4} \)
$43$ \( 1 - 9768 T + 237972854 T^{2} - 1435978471224 T^{3} + 21611482313284249 T^{4} \)
$47$ \( 1 - 43742 T + 935775830 T^{2} - 10032009296194 T^{3} + 52599132235830049 T^{4} \)
$53$ \( 1 + 12174 T + 457325722 T^{2} + 5091111931782 T^{3} + 174887470365513049 T^{4} \)
$59$ \( 1 + 2788 T + 1399448534 T^{2} + 1993208945612 T^{3} + 511116753300641401 T^{4} \)
$61$ \( 1 + 25302 T + 1815376826 T^{2} + 21369975607902 T^{3} + 713342911662882601 T^{4} \)
$67$ \( 1 + 40520 T + 1779236982 T^{2} + 54707069335640 T^{3} + 1822837804551761449 T^{4} \)
$71$ \( 1 - 31386 T + 3698331094 T^{2} - 56627542410486 T^{3} + 3255243551009881201 T^{4} \)
$73$ \( 1 + 46780 T + 4437463638 T^{2} + 96978289120540 T^{3} + 4297625829703557649 T^{4} \)
$79$ \( 1 + 16850 T + 5148027246 T^{2} + 51848400323150 T^{3} + 9468276082626847201 T^{4} \)
$83$ \( 1 - 79440 T + 5477266486 T^{2} - 312917388679920 T^{3} + 15516041187205853449 T^{4} \)
$89$ \( 1 + 54204 T + 6019532470 T^{2} + 302678358373596 T^{3} + 31181719929966183601 T^{4} \)
$97$ \( 1 + 241568 T + 30950947518 T^{2} + 2074426611202976 T^{3} + 73742412689492826049 T^{4} \)
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