# Properties

 Label 33.8.a.b Level 33 Weight 8 Character orbit 33.a Self dual yes Analytic conductor 10.309 Analytic rank 1 Dimension 2 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{177})$$ Defining polynomial: $$x^{2} - x - 44$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{177})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -9 - \beta ) q^{2} + 27 q^{3} + ( -3 + 19 \beta ) q^{4} + ( 2 - 38 \beta ) q^{5} + ( -243 - 27 \beta ) q^{6} + ( -160 + 154 \beta ) q^{7} + ( 343 - 59 \beta ) q^{8} + 729 q^{9} +O(q^{10})$$ $$q + ( -9 - \beta ) q^{2} + 27 q^{3} + ( -3 + 19 \beta ) q^{4} + ( 2 - 38 \beta ) q^{5} + ( -243 - 27 \beta ) q^{6} + ( -160 + 154 \beta ) q^{7} + ( 343 - 59 \beta ) q^{8} + 729 q^{9} + ( 1654 + 378 \beta ) q^{10} -1331 q^{11} + ( -81 + 513 \beta ) q^{12} + ( -6774 + 878 \beta ) q^{13} + ( -5336 - 1380 \beta ) q^{14} + ( 54 - 1026 \beta ) q^{15} + ( -107 - 2185 \beta ) q^{16} + ( -30674 - 996 \beta ) q^{17} + ( -6561 - 729 \beta ) q^{18} + ( -19416 + 852 \beta ) q^{19} + ( -31774 - 570 \beta ) q^{20} + ( -4320 + 4158 \beta ) q^{21} + ( 11979 + 1331 \beta ) q^{22} + ( -48508 + 6330 \beta ) q^{23} + ( 9261 - 1593 \beta ) q^{24} + ( -14585 + 1292 \beta ) q^{25} + ( 22334 - 2006 \beta ) q^{26} + 19683 q^{27} + ( 129224 - 576 \beta ) q^{28} + ( 21818 - 29644 \beta ) q^{29} + ( 44658 + 10206 \beta ) q^{30} + ( 131304 - 17608 \beta ) q^{31} + ( 53199 + 29509 \beta ) q^{32} -35937 q^{33} + ( 319890 + 40634 \beta ) q^{34} + ( -257808 + 536 \beta ) q^{35} + ( -2187 + 13851 \beta ) q^{36} + ( 177662 - 27472 \beta ) q^{37} + ( 137256 + 10896 \beta ) q^{38} + ( -182898 + 23706 \beta ) q^{39} + ( 99334 - 10910 \beta ) q^{40} + ( -94018 - 87896 \beta ) q^{41} + ( -144072 - 37260 \beta ) q^{42} + ( -157520 + 70936 \beta ) q^{43} + ( 3993 - 25289 \beta ) q^{44} + ( 1458 - 27702 \beta ) q^{45} + ( 158052 - 14792 \beta ) q^{46} + ( 231020 + 74886 \beta ) q^{47} + ( -2889 - 58995 \beta ) q^{48} + ( 245561 - 25564 \beta ) q^{49} + ( 74417 + 1665 \beta ) q^{50} + ( -828198 - 26892 \beta ) q^{51} + ( 754330 - 114658 \beta ) q^{52} + ( -960358 + 98834 \beta ) q^{53} + ( -177147 - 19683 \beta ) q^{54} + ( -2662 + 50578 \beta ) q^{55} + ( -454664 + 53176 \beta ) q^{56} + ( -524232 + 23004 \beta ) q^{57} + ( 1107974 + 274622 \beta ) q^{58} + ( 1156244 + 189540 \beta ) q^{59} + ( -857898 - 15390 \beta ) q^{60} + ( -994582 - 201934 \beta ) q^{61} + ( -406984 + 44776 \beta ) q^{62} + ( -116640 + 112266 \beta ) q^{63} + ( -1763491 - 68609 \beta ) q^{64} + ( -1481564 + 225804 \beta ) q^{65} + ( 323433 + 35937 \beta ) q^{66} + ( 1069444 - 464104 \beta ) q^{67} + ( -740634 - 598742 \beta ) q^{68} + ( -1309716 + 170910 \beta ) q^{69} + ( 2296688 + 252448 \beta ) q^{70} + ( -165988 - 36334 \beta ) q^{71} + ( 250047 - 43011 \beta ) q^{72} + ( -1449806 - 436992 \beta ) q^{73} + ( -390190 + 97058 \beta ) q^{74} + ( -393795 + 34884 \beta ) q^{75} + ( 770520 - 355272 \beta ) q^{76} + ( 212960 - 204974 \beta ) q^{77} + ( 603018 - 54162 \beta ) q^{78} + ( -1195632 + 708646 \beta ) q^{79} + ( 3653106 + 82726 \beta ) q^{80} + 531441 q^{81} + ( 4713586 + 972978 \beta ) q^{82} + ( -3957564 - 461376 \beta ) q^{83} + ( 3489048 - 15552 \beta ) q^{84} + ( 1603964 + 1201468 \beta ) q^{85} + ( -1703504 - 551840 \beta ) q^{86} + ( 589086 - 800388 \beta ) q^{87} + ( -456533 + 78529 \beta ) q^{88} + ( 4061818 + 903568 \beta ) q^{89} + ( 1205766 + 275562 \beta ) q^{90} + ( 7033168 - 1048464 \beta ) q^{91} + ( 5437404 - 820372 \beta ) q^{92} + ( 3545208 - 475416 \beta ) q^{93} + ( -5374164 - 979880 \beta ) q^{94} + ( -1463376 + 707136 \beta ) q^{95} + ( 1436373 + 796743 \beta ) q^{96} + ( -8353150 + 2748 \beta ) q^{97} + ( -1085233 + 10079 \beta ) q^{98} -970299 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 19q^{2} + 54q^{3} + 13q^{4} - 34q^{5} - 513q^{6} - 166q^{7} + 627q^{8} + 1458q^{9} + O(q^{10})$$ $$2q - 19q^{2} + 54q^{3} + 13q^{4} - 34q^{5} - 513q^{6} - 166q^{7} + 627q^{8} + 1458q^{9} + 3686q^{10} - 2662q^{11} + 351q^{12} - 12670q^{13} - 12052q^{14} - 918q^{15} - 2399q^{16} - 62344q^{17} - 13851q^{18} - 37980q^{19} - 64118q^{20} - 4482q^{21} + 25289q^{22} - 90686q^{23} + 16929q^{24} - 27878q^{25} + 42662q^{26} + 39366q^{27} + 257872q^{28} + 13992q^{29} + 99522q^{30} + 245000q^{31} + 135907q^{32} - 71874q^{33} + 680414q^{34} - 515080q^{35} + 9477q^{36} + 327852q^{37} + 285408q^{38} - 342090q^{39} + 187758q^{40} - 275932q^{41} - 325404q^{42} - 244104q^{43} - 17303q^{44} - 24786q^{45} + 301312q^{46} + 536926q^{47} - 64773q^{48} + 465558q^{49} + 150499q^{50} - 1683288q^{51} + 1394002q^{52} - 1821882q^{53} - 373977q^{54} + 45254q^{55} - 856152q^{56} - 1025460q^{57} + 2490570q^{58} + 2502028q^{59} - 1731186q^{60} - 2191098q^{61} - 769192q^{62} - 121014q^{63} - 3595591q^{64} - 2737324q^{65} + 682803q^{66} + 1674784q^{67} - 2080010q^{68} - 2448522q^{69} + 4845824q^{70} - 368310q^{71} + 457083q^{72} - 3336604q^{73} - 683322q^{74} - 752706q^{75} + 1185768q^{76} + 220946q^{77} + 1151874q^{78} - 1682618q^{79} + 7388938q^{80} + 1062882q^{81} + 10400150q^{82} - 8376504q^{83} + 6962544q^{84} + 4409396q^{85} - 3958848q^{86} + 377784q^{87} - 834537q^{88} + 9027204q^{89} + 2687094q^{90} + 13017872q^{91} + 10054436q^{92} + 6615000q^{93} - 11728208q^{94} - 2219616q^{95} + 3669489q^{96} - 16703552q^{97} - 2160387q^{98} - 1940598q^{99} + O(q^{100})$$

## 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
 7.15207 −6.15207
−16.1521 27.0000 132.889 −269.779 −436.106 941.418 −78.9720 729.000 4357.48
1.2 −2.84793 27.0000 −119.889 235.779 −76.8942 −1107.42 705.972 729.000 −671.481
 $$n$$: 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.b 2
3.b odd 2 1 99.8.a.d 2
4.b odd 2 1 528.8.a.f 2
11.b odd 2 1 363.8.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.a.b 2 1.a even 1 1 trivial
99.8.a.d 2 3.b odd 2 1
363.8.a.d 2 11.b odd 2 1
528.8.a.f 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 19 T_{2} + 46$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(33))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 19 T + 302 T^{2} + 2432 T^{3} + 16384 T^{4}$$
$3$ $$( 1 - 27 T )^{2}$$
$5$ $$1 + 34 T + 92642 T^{2} + 2656250 T^{3} + 6103515625 T^{4}$$
$7$ $$1 + 166 T + 604542 T^{2} + 136708138 T^{3} + 678223072849 T^{4}$$
$11$ $$( 1 + 1331 T )^{2}$$
$13$ $$1 + 12670 T + 131517642 T^{2} + 795023710390 T^{3} + 3937376385699289 T^{4}$$
$17$ $$1 + 62344 T + 1748474222 T^{2} + 25582154229512 T^{3} + 168377826559400929 T^{4}$$
$19$ $$1 + 37980 T + 2116242326 T^{2} + 33949248647220 T^{3} + 799006685782884121 T^{4}$$
$23$ $$1 + 90686 T + 7092589718 T^{2} + 308770000486642 T^{3} + 11592836324538749809 T^{4}$$
$29$ $$1 - 13992 T - 4336731434 T^{2} - 241360269315528 T^{3} +$$$$29\!\cdots\!81$$$$T^{4}$$
$31$ $$1 - 245000 T + 56312134590 T^{2} - 6740590457195000 T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$
$37$ $$1 - 327852 T + 183339535550 T^{2} - 31123605781808316 T^{3} +$$$$90\!\cdots\!89$$$$T^{4}$$
$41$ $$1 + 275932 T + 66680638310 T^{2} + 53738936300532092 T^{3} +$$$$37\!\cdots\!61$$$$T^{4}$$
$43$ $$1 + 244104 T + 335871625670 T^{2} + 66352010245663128 T^{3} +$$$$73\!\cdots\!49$$$$T^{4}$$
$47$ $$1 - 536926 T + 837168473222 T^{2} - 272019125577716738 T^{3} +$$$$25\!\cdots\!69$$$$T^{4}$$
$53$ $$1 + 1821882 T + 2746994724802 T^{2} + 2140185080868513234 T^{3} +$$$$13\!\cdots\!69$$$$T^{4}$$
$59$ $$1 - 2502028 T + 4952639534534 T^{2} - 6226675697258712932 T^{3} +$$$$61\!\cdots\!61$$$$T^{4}$$
$61$ $$1 + 2191098 T + 5681315972690 T^{2} + 6886057542519941058 T^{3} +$$$$98\!\cdots\!41$$$$T^{4}$$
$67$ $$1 - 1674784 T + 3291529437702 T^{2} - 10150382825209275232 T^{3} +$$$$36\!\cdots\!29$$$$T^{4}$$
$71$ $$1 + 368310 T + 18165736320454 T^{2} + 3349823705536989210 T^{3} +$$$$82\!\cdots\!81$$$$T^{4}$$
$73$ $$1 + 3336604 T + 16427959744566 T^{2} + 36860794088413126588 T^{3} +$$$$12\!\cdots\!09$$$$T^{4}$$
$79$ $$1 + 1682618 T + 16894191271566 T^{2} + 32312842930472884262 T^{3} +$$$$36\!\cdots\!81$$$$T^{4}$$
$83$ $$1 + 8376504 T + 62394156052870 T^{2} +$$$$22\!\cdots\!08$$$$T^{3} +$$$$73\!\cdots\!29$$$$T^{4}$$
$89$ $$1 - 9027204 T + 72708018275350 T^{2} -$$$$39\!\cdots\!16$$$$T^{3} +$$$$19\!\cdots\!41$$$$T^{4}$$
$97$ $$1 + 16703552 T + 231348397156350 T^{2} +$$$$13\!\cdots\!76$$$$T^{3} +$$$$65\!\cdots\!69$$$$T^{4}$$