# Properties

 Label 33.8.a.c 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: $$\Q(\sqrt{97})$$ Defining polynomial: $$x^{2} - x - 24$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + 3\sqrt{97})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} -27 q^{3} + ( 90 + \beta ) q^{4} + ( -90 - 14 \beta ) q^{5} -27 \beta q^{6} + ( -188 - 42 \beta ) q^{7} + ( 218 - 37 \beta ) q^{8} + 729 q^{9} +O(q^{10})$$ $$q + \beta q^{2} -27 q^{3} + ( 90 + \beta ) q^{4} + ( -90 - 14 \beta ) q^{5} -27 \beta q^{6} + ( -188 - 42 \beta ) q^{7} + ( 218 - 37 \beta ) q^{8} + 729 q^{9} + ( -3052 - 104 \beta ) q^{10} + 1331 q^{11} + ( -2430 - 27 \beta ) q^{12} + ( -6642 + 38 \beta ) q^{13} + ( -9156 - 230 \beta ) q^{14} + ( 2430 + 378 \beta ) q^{15} + ( -19586 + 53 \beta ) q^{16} + ( -5218 + 180 \beta ) q^{17} + 729 \beta q^{18} + ( 5912 + 2372 \beta ) q^{19} + ( -11152 - 1364 \beta ) q^{20} + ( 5076 + 1134 \beta ) q^{21} + 1331 \beta q^{22} + ( -5808 - 2050 \beta ) q^{23} + ( -5886 + 999 \beta ) q^{24} + ( -27297 + 2716 \beta ) q^{25} + ( 8284 - 6604 \beta ) q^{26} -19683 q^{27} + ( -26076 - 4010 \beta ) q^{28} + ( 9938 - 4564 \beta ) q^{29} + ( 82404 + 2808 \beta ) q^{30} + ( -24112 + 184 \beta ) q^{31} + ( -16350 - 14797 \beta ) q^{32} -35937 q^{33} + ( 39240 - 5038 \beta ) q^{34} + ( 145104 + 7000 \beta ) q^{35} + ( 65610 + 729 \beta ) q^{36} + ( 130494 + 13104 \beta ) q^{37} + ( 517096 + 8284 \beta ) q^{38} + ( 179334 - 1026 \beta ) q^{39} + ( 93304 + 796 \beta ) q^{40} + ( 363062 + 29712 \beta ) q^{41} + ( 247212 + 6210 \beta ) q^{42} + ( -848440 - 7216 \beta ) q^{43} + ( 119790 + 1331 \beta ) q^{44} + ( -65610 - 10206 \beta ) q^{45} + ( -446900 - 7858 \beta ) q^{46} + ( -612368 + 42642 \beta ) q^{47} + ( 528822 - 1431 \beta ) q^{48} + ( -403647 + 17556 \beta ) q^{49} + ( 592088 - 24581 \beta ) q^{50} + ( 140886 - 4860 \beta ) q^{51} + ( -589496 - 3184 \beta ) q^{52} + ( -1064818 - 26758 \beta ) q^{53} -19683 \beta q^{54} + ( -119790 - 18634 \beta ) q^{55} + ( 297788 - 646 \beta ) q^{56} + ( -159624 - 64044 \beta ) q^{57} + ( -994952 + 5374 \beta ) q^{58} + ( 425476 + 76380 \beta ) q^{59} + ( 301104 + 36828 \beta ) q^{60} + ( -444666 - 172662 \beta ) q^{61} + ( 40112 - 23928 \beta ) q^{62} + ( -137052 - 30618 \beta ) q^{63} + ( -718738 - 37931 \beta ) q^{64} + ( 481804 + 89036 \beta ) q^{65} -35937 \beta q^{66} + ( -1631172 + 2392 \beta ) q^{67} + ( -430380 + 11162 \beta ) q^{68} + ( 156816 + 55350 \beta ) q^{69} + ( 1526000 + 152104 \beta ) q^{70} + ( -2749544 + 3574 \beta ) q^{71} + ( 158922 - 26973 \beta ) q^{72} + ( -2665950 - 118912 \beta ) q^{73} + ( 2856672 + 143598 \beta ) q^{74} + ( 737019 - 73332 \beta ) q^{75} + ( 1049176 + 221764 \beta ) q^{76} + ( -250228 - 55902 \beta ) q^{77} + ( -223668 + 178308 \beta ) q^{78} + ( -746548 - 43494 \beta ) q^{79} + ( 1600984 + 268692 \beta ) q^{80} + 531441 q^{81} + ( 6477216 + 392774 \beta ) q^{82} + ( 4553116 - 255344 \beta ) q^{83} + ( 704052 + 108270 \beta ) q^{84} + ( -79740 + 54332 \beta ) q^{85} + ( -1573088 - 855656 \beta ) q^{86} + ( -268326 + 123228 \beta ) q^{87} + ( 290158 - 49247 \beta ) q^{88} + ( 3441562 - 72992 \beta ) q^{89} + ( -2224908 - 75816 \beta ) q^{90} + ( 900768 + 270224 \beta ) q^{91} + ( -969620 - 192358 \beta ) q^{92} + ( 651024 - 4968 \beta ) q^{93} + ( 9295956 - 569726 \beta ) q^{94} + ( -7771424 - 329456 \beta ) q^{95} + ( 441450 + 399519 \beta ) q^{96} + ( 5189498 - 481620 \beta ) q^{97} + ( 3827208 - 386091 \beta ) q^{98} + 970299 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - 54q^{3} + 181q^{4} - 194q^{5} - 27q^{6} - 418q^{7} + 399q^{8} + 1458q^{9} + O(q^{10})$$ $$2q + q^{2} - 54q^{3} + 181q^{4} - 194q^{5} - 27q^{6} - 418q^{7} + 399q^{8} + 1458q^{9} - 6208q^{10} + 2662q^{11} - 4887q^{12} - 13246q^{13} - 18542q^{14} + 5238q^{15} - 39119q^{16} - 10256q^{17} + 729q^{18} + 14196q^{19} - 23668q^{20} + 11286q^{21} + 1331q^{22} - 13666q^{23} - 10773q^{24} - 51878q^{25} + 9964q^{26} - 39366q^{27} - 56162q^{28} + 15312q^{29} + 167616q^{30} - 48040q^{31} - 47497q^{32} - 71874q^{33} + 73442q^{34} + 297208q^{35} + 131949q^{36} + 274092q^{37} + 1042476q^{38} + 357642q^{39} + 187404q^{40} + 755836q^{41} + 500634q^{42} - 1704096q^{43} + 240911q^{44} - 141426q^{45} - 901658q^{46} - 1182094q^{47} + 1056213q^{48} - 789738q^{49} + 1159595q^{50} + 276912q^{51} - 1182176q^{52} - 2156394q^{53} - 19683q^{54} - 258214q^{55} + 594930q^{56} - 383292q^{57} - 1984530q^{58} + 927332q^{59} + 639036q^{60} - 1061994q^{61} + 56296q^{62} - 304722q^{63} - 1475407q^{64} + 1052644q^{65} - 35937q^{66} - 3259952q^{67} - 849598q^{68} + 368982q^{69} + 3204104q^{70} - 5495514q^{71} + 290871q^{72} - 5450812q^{73} + 5856942q^{74} + 1400706q^{75} + 2320116q^{76} - 556358q^{77} - 269028q^{78} - 1536590q^{79} + 3470660q^{80} + 1062882q^{81} + 13347206q^{82} + 8850888q^{83} + 1516374q^{84} - 105148q^{85} - 4001832q^{86} - 413424q^{87} + 531069q^{88} + 6810132q^{89} - 4525632q^{90} + 2071760q^{91} - 2131598q^{92} + 1297080q^{93} + 18022186q^{94} - 15872304q^{95} + 1282419q^{96} + 9897376q^{97} + 7268325q^{98} + 1940598q^{99} + O(q^{100})$$

## 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
 −4.42443 5.42443
−14.2733 −27.0000 75.7267 109.826 385.379 411.478 746.112 729.000 −1567.58
1.2 15.2733 −27.0000 105.273 −303.826 −412.379 −829.478 −347.112 729.000 −4640.42
 $$n$$: 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.c 2
3.b odd 2 1 99.8.a.b 2
4.b odd 2 1 528.8.a.h 2
11.b odd 2 1 363.8.a.c 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} - 218$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(33))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 38 T^{2} - 128 T^{3} + 16384 T^{4}$$
$3$ $$( 1 + 27 T )^{2}$$
$5$ $$1 + 194 T + 122882 T^{2} + 15156250 T^{3} + 6103515625 T^{4}$$
$7$ $$1 + 418 T + 1305774 T^{2} + 344240974 T^{3} + 678223072849 T^{4}$$
$11$ $$( 1 - 1331 T )^{2}$$
$13$ $$1 + 13246 T + 169046010 T^{2} + 831166856182 T^{3} + 3937376385699289 T^{4}$$
$17$ $$1 + 10256 T + 839902430 T^{2} + 4208433430288 T^{3} + 168377826559400929 T^{4}$$
$19$ $$1 - 14196 T + 610166774 T^{2} - 12689403206844 T^{3} + 799006685782884121 T^{4}$$
$23$ $$1 + 13666 T + 5939145158 T^{2} + 46530344558702 T^{3} + 11592836324538749809 T^{4}$$
$29$ $$1 - 15312 T + 30012198502 T^{2} - 264130106043408 T^{3} +$$$$29\!\cdots\!81$$$$T^{4}$$
$31$ $$1 + 48040 T + 55594799550 T^{2} + 1321705981892440 T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$
$37$ $$1 - 274092 T + 171168601790 T^{2} - 26020068067138236 T^{3} +$$$$90\!\cdots\!89$$$$T^{4}$$
$41$ $$1 - 755836 T + 339658819958 T^{2} - 147202291353119516 T^{3} +$$$$37\!\cdots\!61$$$$T^{4}$$
$43$ $$1 + 1704096 T + 1258258595846 T^{2} + 463205007912994272 T^{3} +$$$$73\!\cdots\!49$$$$T^{4}$$
$47$ $$1 + 1182094 T + 965730056342 T^{2} + 598876150960589522 T^{3} +$$$$25\!\cdots\!69$$$$T^{4}$$
$53$ $$1 + 2156394 T + 3355666109890 T^{2} + 2533140053677667778 T^{3} +$$$$13\!\cdots\!69$$$$T^{4}$$
$59$ $$1 - 927332 T + 3919039493894 T^{2} - 2307806158720172908 T^{3} +$$$$61\!\cdots\!61$$$$T^{4}$$
$61$ $$1 + 1061994 T + 60938203298 T^{2} + 3337574035397285874 T^{3} +$$$$98\!\cdots\!41$$$$T^{4}$$
$67$ $$1 + 3259952 T + 14776996218054 T^{2} + 19757628919195924496 T^{3} +$$$$36\!\cdots\!29$$$$T^{4}$$
$71$ $$1 + 5495514 T + 25737621036694 T^{2} + 49982360162119957974 T^{3} +$$$$82\!\cdots\!81$$$$T^{4}$$
$73$ $$1 + 5450812 T + 26436565990902 T^{2} + 60217292416676156764 T^{3} +$$$$12\!\cdots\!09$$$$T^{4}$$
$79$ $$1 + 1536590 T + 38585225535486 T^{2} + 29508534509042057810 T^{3} +$$$$36\!\cdots\!81$$$$T^{4}$$
$83$ $$1 - 8850888 T + 59626634719558 T^{2} -$$$$24\!\cdots\!76$$$$T^{3} +$$$$73\!\cdots\!29$$$$T^{4}$$
$89$ $$1 - 6810132 T + 98894344907446 T^{2} -$$$$30\!\cdots\!28$$$$T^{3} +$$$$19\!\cdots\!41$$$$T^{4}$$
$97$ $$1 - 9897376 T + 135461286702270 T^{2} -$$$$79\!\cdots\!88$$$$T^{3} +$$$$65\!\cdots\!69$$$$T^{4}$$