Properties

Label 33.8.a.d
Level $33$
Weight $8$
Character orbit 33.a
Self dual yes
Analytic conductor $10.309$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 33.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.3087058410\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.115512.1
Defining polynomial: \(x^{3} - x^{2} - 70 x - 194\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 + \beta_{1} ) q^{2} -27 q^{3} + ( -5 + 13 \beta_{1} - \beta_{2} ) q^{4} + ( -148 + 14 \beta_{1} - 6 \beta_{2} ) q^{5} + ( -81 - 27 \beta_{1} ) q^{6} + ( 538 + 84 \beta_{1} + 28 \beta_{2} ) q^{7} + ( 1051 + 5 \beta_{1} - 9 \beta_{2} ) q^{8} + 729 q^{9} +O(q^{10})\) \( q + ( 3 + \beta_{1} ) q^{2} -27 q^{3} + ( -5 + 13 \beta_{1} - \beta_{2} ) q^{4} + ( -148 + 14 \beta_{1} - 6 \beta_{2} ) q^{5} + ( -81 - 27 \beta_{1} ) q^{6} + ( 538 + 84 \beta_{1} + 28 \beta_{2} ) q^{7} + ( 1051 + 5 \beta_{1} - 9 \beta_{2} ) q^{8} + 729 q^{9} + ( 960 + 40 \beta_{1} + 10 \beta_{2} ) q^{10} -1331 q^{11} + ( 135 - 351 \beta_{1} + 27 \beta_{2} ) q^{12} + ( 6924 + 466 \beta_{1} - 186 \beta_{2} ) q^{13} + ( 12086 + 1154 \beta_{1} - 196 \beta_{2} ) q^{14} + ( 3996 - 378 \beta_{1} + 162 \beta_{2} ) q^{15} + ( 4075 - 491 \beta_{1} + 159 \beta_{2} ) q^{16} + ( -4846 - 1462 \beta_{1} + 542 \beta_{2} ) q^{17} + ( 2187 + 729 \beta_{1} ) q^{18} + ( 8164 - 2074 \beta_{1} - 526 \beta_{2} ) q^{19} + ( 26704 - 512 \beta_{1} + 688 \beta_{2} ) q^{20} + ( -14526 - 2268 \beta_{1} - 756 \beta_{2} ) q^{21} + ( -3993 - 1331 \beta_{1} ) q^{22} + ( 11698 + 310 \beta_{1} - 2030 \beta_{2} ) q^{23} + ( -28377 - 135 \beta_{1} + 243 \beta_{2} ) q^{24} + ( 9707 - 4596 \beta_{1} + 3044 \beta_{2} ) q^{25} + ( 67944 + 13072 \beta_{1} + 278 \beta_{2} ) q^{26} -19683 q^{27} + ( 92678 + 14442 \beta_{1} - 3954 \beta_{2} ) q^{28} + ( -59954 + 7634 \beta_{1} + 2278 \beta_{2} ) q^{29} + ( -25920 - 1080 \beta_{1} - 270 \beta_{2} ) q^{30} + ( 96296 - 11080 \beta_{1} + 2728 \beta_{2} ) q^{31} + ( -173189 - 2747 \beta_{1} + 1007 \beta_{2} ) q^{32} + 35937 q^{33} + ( -163862 - 23802 \beta_{1} - 706 \beta_{2} ) q^{34} + ( -177624 + 19012 \beta_{1} - 9108 \beta_{2} ) q^{35} + ( -3645 + 9477 \beta_{1} - 729 \beta_{2} ) q^{36} + ( 35854 + 5828 \beta_{1} + 1996 \beta_{2} ) q^{37} + ( -228776 - 8368 \beta_{1} + 4178 \beta_{2} ) q^{38} + ( -186948 - 12582 \beta_{1} + 5022 \beta_{2} ) q^{39} + ( -79120 + 10960 \beta_{1} - 3520 \beta_{2} ) q^{40} + ( -45066 - 4694 \beta_{1} + 2462 \beta_{2} ) q^{41} + ( -326322 - 31158 \beta_{1} + 5292 \beta_{2} ) q^{42} + ( 64512 - 18594 \beta_{1} - 7718 \beta_{2} ) q^{43} + ( 6655 - 17303 \beta_{1} + 1331 \beta_{2} ) q^{44} + ( -107892 + 10206 \beta_{1} - 4374 \beta_{2} ) q^{45} + ( 5474 + 31038 \beta_{1} + 7810 \beta_{2} ) q^{46} + ( -197162 - 65462 \beta_{1} + 1294 \beta_{2} ) q^{47} + ( -110025 + 13257 \beta_{1} - 4293 \beta_{2} ) q^{48} + ( 1487053 + 33152 \beta_{1} - 3584 \beta_{2} ) q^{49} + ( -397415 - 60605 \beta_{1} - 7580 \beta_{2} ) q^{50} + ( 130842 + 39474 \beta_{1} - 14634 \beta_{2} ) q^{51} + ( 816664 + 136792 \beta_{1} + 9624 \beta_{2} ) q^{52} + ( 26348 - 72182 \beta_{1} + 9102 \beta_{2} ) q^{53} + ( -59049 - 19683 \beta_{1} ) q^{54} + ( 196988 - 18634 \beta_{1} + 7986 \beta_{2} ) q^{55} + ( 250886 + 121018 \beta_{1} + 26462 \beta_{2} ) q^{56} + ( -220428 + 55998 \beta_{1} + 14202 \beta_{2} ) q^{57} + ( 763310 - 1838 \beta_{1} - 16746 \beta_{2} ) q^{58} + ( 844256 + 43644 \beta_{1} - 37356 \beta_{2} ) q^{59} + ( -721008 + 13824 \beta_{1} - 18576 \beta_{2} ) q^{60} + ( 2226264 - 115710 \beta_{1} + 7574 \beta_{2} ) q^{61} + ( -886936 - 36328 \beta_{1} + 168 \beta_{2} ) q^{62} + ( 392202 + 61236 \beta_{1} + 20412 \beta_{2} ) q^{63} + ( -1322101 - 145867 \beta_{1} - 21633 \beta_{2} ) q^{64} + ( 1063944 + 18628 \beta_{1} + 26188 \beta_{2} ) q^{65} + ( 107811 + 35937 \beta_{1} ) q^{66} + ( 2383452 + 31184 \beta_{1} - 2480 \beta_{2} ) q^{67} + ( -2607318 - 209098 \beta_{1} - 42750 \beta_{2} ) q^{68} + ( -315846 - 8370 \beta_{1} + 54810 \beta_{2} ) q^{69} + ( 1343040 + 85360 \beta_{1} + 17420 \beta_{2} ) q^{70} + ( 463466 + 315766 \beta_{1} + 40050 \beta_{2} ) q^{71} + ( 766179 + 3645 \beta_{1} - 6561 \beta_{2} ) q^{72} + ( -2143038 - 58412 \beta_{1} - 30404 \beta_{2} ) q^{73} + ( 835826 + 78166 \beta_{1} - 13812 \beta_{2} ) q^{74} + ( -262089 + 124092 \beta_{1} - 82188 \beta_{2} ) q^{75} + ( -2551576 - 80408 \beta_{1} + 58984 \beta_{2} ) q^{76} + ( -716078 - 111804 \beta_{1} - 37268 \beta_{2} ) q^{77} + ( -1834488 - 352944 \beta_{1} - 7506 \beta_{2} ) q^{78} + ( 2291062 - 163544 \beta_{1} + 95960 \beta_{2} ) q^{79} + ( -2518672 + 124176 \beta_{1} - 84944 \beta_{2} ) q^{80} + 531441 q^{81} + ( -591530 - 111702 \beta_{1} - 5154 \beta_{2} ) q^{82} + ( 2168532 - 376356 \beta_{1} - 2892 \beta_{2} ) q^{83} + ( -2502306 - 389934 \beta_{1} + 106758 \beta_{2} ) q^{84} + ( -5515344 + 160552 \beta_{1} - 171208 \beta_{2} ) q^{85} + ( -2173156 - 59684 \beta_{1} + 49466 \beta_{2} ) q^{86} + ( 1618758 - 206118 \beta_{1} - 61506 \beta_{2} ) q^{87} + ( -1398881 - 6655 \beta_{1} + 11979 \beta_{2} ) q^{88} + ( -2947654 + 103240 \beta_{1} + 109592 \beta_{2} ) q^{89} + ( 699840 + 29160 \beta_{1} + 7290 \beta_{2} ) q^{90} + ( 1022216 + 1585244 \beta_{1} + 31028 \beta_{2} ) q^{91} + ( 2307330 + 213694 \beta_{1} + 197562 \beta_{2} ) q^{92} + ( -2599992 + 299160 \beta_{1} - 73656 \beta_{2} ) q^{93} + ( -8012746 - 862134 \beta_{1} + 60286 \beta_{2} ) q^{94} + ( -63656 - 100372 \beta_{1} + 47588 \beta_{2} ) q^{95} + ( 4676103 + 74169 \beta_{1} - 27189 \beta_{2} ) q^{96} + ( -588258 + 148640 \beta_{1} - 47392 \beta_{2} ) q^{97} + ( 8125799 + 1847245 \beta_{1} - 18816 \beta_{2} ) q^{98} -970299 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 9q^{2} - 81q^{3} - 15q^{4} - 444q^{5} - 243q^{6} + 1614q^{7} + 3153q^{8} + 2187q^{9} + O(q^{10}) \) \( 3q + 9q^{2} - 81q^{3} - 15q^{4} - 444q^{5} - 243q^{6} + 1614q^{7} + 3153q^{8} + 2187q^{9} + 2880q^{10} - 3993q^{11} + 405q^{12} + 20772q^{13} + 36258q^{14} + 11988q^{15} + 12225q^{16} - 14538q^{17} + 6561q^{18} + 24492q^{19} + 80112q^{20} - 43578q^{21} - 11979q^{22} + 35094q^{23} - 85131q^{24} + 29121q^{25} + 203832q^{26} - 59049q^{27} + 278034q^{28} - 179862q^{29} - 77760q^{30} + 288888q^{31} - 519567q^{32} + 107811q^{33} - 491586q^{34} - 532872q^{35} - 10935q^{36} + 107562q^{37} - 686328q^{38} - 560844q^{39} - 237360q^{40} - 135198q^{41} - 978966q^{42} + 193536q^{43} + 19965q^{44} - 323676q^{45} + 16422q^{46} - 591486q^{47} - 330075q^{48} + 4461159q^{49} - 1192245q^{50} + 392526q^{51} + 2449992q^{52} + 79044q^{53} - 177147q^{54} + 590964q^{55} + 752658q^{56} - 661284q^{57} + 2289930q^{58} + 2532768q^{59} - 2163024q^{60} + 6678792q^{61} - 2660808q^{62} + 1176606q^{63} - 3966303q^{64} + 3191832q^{65} + 323433q^{66} + 7150356q^{67} - 7821954q^{68} - 947538q^{69} + 4029120q^{70} + 1390398q^{71} + 2298537q^{72} - 6429114q^{73} + 2507478q^{74} - 786267q^{75} - 7654728q^{76} - 2148234q^{77} - 5503464q^{78} + 6873186q^{79} - 7556016q^{80} + 1594323q^{81} - 1774590q^{82} + 6505596q^{83} - 7506918q^{84} - 16546032q^{85} - 6519468q^{86} + 4856274q^{87} - 4196643q^{88} - 8842962q^{89} + 2099520q^{90} + 3066648q^{91} + 6921990q^{92} - 7799976q^{93} - 24038238q^{94} - 190968q^{95} + 14028309q^{96} - 1764774q^{97} + 24377397q^{98} - 2910897q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 70 x - 194\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 6 \nu - 45 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 6 \nu - 92 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 2 \beta_{1} + 2\)\()/6\)
\(\nu^{2}\)\(=\)\(\beta_{2} - \beta_{1} + 47\)

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
−3.66999
9.97132
−5.30133
−6.51124 −27.0000 −85.6037 −22.9029 175.804 −1466.13 1390.83 729.000 149.126
1.2 −2.40077 −27.0000 −122.236 −505.769 64.8207 1401.07 600.759 729.000 1214.23
1.3 17.9120 −27.0000 192.840 84.6717 −483.624 1679.06 1161.42 729.000 1516.64
\(n\): 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 33.8.a.d 3
3.b odd 2 1 99.8.a.e 3
4.b odd 2 1 528.8.a.o 3
11.b odd 2 1 363.8.a.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.a.d 3 1.a even 1 1 trivial
99.8.a.e 3 3.b odd 2 1
363.8.a.e 3 11.b odd 2 1
528.8.a.o 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 9 T_{2}^{2} - 144 T_{2} - 280 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(33))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 9 T + 240 T^{2} - 2584 T^{3} + 30720 T^{4} - 147456 T^{5} + 2097152 T^{6} \)
$3$ \( ( 1 + 27 T )^{3} \)
$5$ \( 1 + 444 T + 201195 T^{2} + 68394200 T^{3} + 15718359375 T^{4} + 2709960937500 T^{5} + 476837158203125 T^{6} \)
$7$ \( 1 - 1614 T + 307233 T^{2} + 790656308 T^{3} + 253019586519 T^{4} - 1094652039578286 T^{5} + 558545864083284007 T^{6} \)
$11$ \( ( 1 + 1331 T )^{3} \)
$13$ \( 1 - 20772 T + 232682259 T^{2} - 1941065209864 T^{3} + 14600466684459903 T^{4} - 81787182283745631108 T^{5} + \)\(24\!\cdots\!13\)\( T^{6} \)
$17$ \( 1 + 14538 T + 412756467 T^{2} + 200146212980 T^{3} + 169369940940948291 T^{4} + \)\(24\!\cdots\!02\)\( T^{5} + \)\(69\!\cdots\!17\)\( T^{6} \)
$19$ \( 1 - 24492 T + 1476867765 T^{2} - 49394356429992 T^{3} + 1320130357373593335 T^{4} - \)\(19\!\cdots\!32\)\( T^{5} + \)\(71\!\cdots\!19\)\( T^{6} \)
$23$ \( 1 - 35094 T + 2262349653 T^{2} - 38515282207028 T^{3} + 7702905668546019891 T^{4} - \)\(40\!\cdots\!46\)\( T^{5} + \)\(39\!\cdots\!23\)\( T^{6} \)
$29$ \( 1 + 179862 T + 40311941247 T^{2} + 6266602437157788 T^{3} + \)\(69\!\cdots\!23\)\( T^{4} + \)\(53\!\cdots\!22\)\( T^{5} + \)\(51\!\cdots\!29\)\( T^{6} \)
$31$ \( 1 - 288888 T + 77083618461 T^{2} - 13983720499847440 T^{3} + \)\(21\!\cdots\!71\)\( T^{4} - \)\(21\!\cdots\!48\)\( T^{5} + \)\(20\!\cdots\!31\)\( T^{6} \)
$37$ \( 1 - 107562 T + 273599913795 T^{2} - 19232280372286428 T^{3} + \)\(25\!\cdots\!35\)\( T^{4} - \)\(96\!\cdots\!18\)\( T^{5} + \)\(85\!\cdots\!37\)\( T^{6} \)
$41$ \( 1 + 135198 T + 575331993483 T^{2} + 51384550416508124 T^{3} + \)\(11\!\cdots\!23\)\( T^{4} + \)\(51\!\cdots\!78\)\( T^{5} + \)\(73\!\cdots\!41\)\( T^{6} \)
$43$ \( 1 - 193536 T + 633525664605 T^{2} - 125885577366905616 T^{3} + \)\(17\!\cdots\!35\)\( T^{4} - \)\(14\!\cdots\!64\)\( T^{5} + \)\(20\!\cdots\!43\)\( T^{6} \)
$47$ \( 1 + 591486 T + 908422267245 T^{2} + 693984302571198916 T^{3} + \)\(46\!\cdots\!35\)\( T^{4} + \)\(15\!\cdots\!34\)\( T^{5} + \)\(13\!\cdots\!47\)\( T^{6} \)
$53$ \( 1 - 79044 T + 2529330228603 T^{2} + 108778384552517688 T^{3} + \)\(29\!\cdots\!11\)\( T^{4} - \)\(10\!\cdots\!36\)\( T^{5} + \)\(16\!\cdots\!53\)\( T^{6} \)
$59$ \( 1 - 2532768 T + 6588293215113 T^{2} - 8761483458577361984 T^{3} + \)\(16\!\cdots\!47\)\( T^{4} - \)\(15\!\cdots\!48\)\( T^{5} + \)\(15\!\cdots\!59\)\( T^{6} \)
$61$ \( 1 - 6678792 T + 21974604832851 T^{2} - 46509029370020353824 T^{3} + \)\(69\!\cdots\!71\)\( T^{4} - \)\(65\!\cdots\!72\)\( T^{5} + \)\(31\!\cdots\!61\)\( T^{6} \)
$67$ \( 1 - 7150356 T + 35053255038225 T^{2} - 99829660769282829176 T^{3} + \)\(21\!\cdots\!75\)\( T^{4} - \)\(26\!\cdots\!24\)\( T^{5} + \)\(22\!\cdots\!67\)\( T^{6} \)
$71$ \( 1 - 1390398 T + 6393443942565 T^{2} - 11581466543393663652 T^{3} + \)\(58\!\cdots\!15\)\( T^{4} - \)\(11\!\cdots\!38\)\( T^{5} + \)\(75\!\cdots\!71\)\( T^{6} \)
$73$ \( 1 + 6429114 T + 44280311135751 T^{2} + \)\(14\!\cdots\!12\)\( T^{3} + \)\(48\!\cdots\!47\)\( T^{4} + \)\(78\!\cdots\!26\)\( T^{5} + \)\(13\!\cdots\!73\)\( T^{6} \)
$79$ \( 1 - 6873186 T + 51506564223993 T^{2} - \)\(26\!\cdots\!60\)\( T^{3} + \)\(98\!\cdots\!87\)\( T^{4} - \)\(25\!\cdots\!66\)\( T^{5} + \)\(70\!\cdots\!79\)\( T^{6} \)
$83$ \( 1 - 6505596 T + 71173099306545 T^{2} - \)\(27\!\cdots\!56\)\( T^{3} + \)\(19\!\cdots\!15\)\( T^{4} - \)\(47\!\cdots\!84\)\( T^{5} + \)\(19\!\cdots\!83\)\( T^{6} \)
$89$ \( 1 + 8842962 T + 131349994025895 T^{2} + \)\(75\!\cdots\!48\)\( T^{3} + \)\(58\!\cdots\!55\)\( T^{4} + \)\(17\!\cdots\!42\)\( T^{5} + \)\(86\!\cdots\!89\)\( T^{6} \)
$97$ \( 1 + 1764774 T + 235749361482351 T^{2} + \)\(28\!\cdots\!08\)\( T^{3} + \)\(19\!\cdots\!63\)\( T^{4} + \)\(11\!\cdots\!06\)\( T^{5} + \)\(52\!\cdots\!97\)\( T^{6} \)
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