# 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 $1$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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