# Properties

 Label 33.4.a.c Level 33 Weight 4 Character orbit 33.a Self dual yes Analytic conductor 1.947 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 33.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.94706303019$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{97})$$ Defining polynomial: $$x^{2} - x - 24$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{97})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} -3 q^{3} + ( 16 + \beta ) q^{4} + ( -6 - 2 \beta ) q^{5} -3 \beta q^{6} + ( 14 - 4 \beta ) q^{7} + ( 24 + 9 \beta ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + \beta q^{2} -3 q^{3} + ( 16 + \beta ) q^{4} + ( -6 - 2 \beta ) q^{5} -3 \beta q^{6} + ( 14 - 4 \beta ) q^{7} + ( 24 + 9 \beta ) q^{8} + 9 q^{9} + ( -48 - 8 \beta ) q^{10} -11 q^{11} + ( -48 - 3 \beta ) q^{12} + ( 14 + 2 \beta ) q^{13} + ( -96 + 10 \beta ) q^{14} + ( 18 + 6 \beta ) q^{15} + ( 88 + 25 \beta ) q^{16} + ( 60 - 14 \beta ) q^{17} + 9 \beta q^{18} + ( 26 - 2 \beta ) q^{19} + ( -144 - 40 \beta ) q^{20} + ( -42 + 12 \beta ) q^{21} -11 \beta q^{22} + ( 72 - 10 \beta ) q^{23} + ( -72 - 27 \beta ) q^{24} + ( 7 + 28 \beta ) q^{25} + ( 48 + 16 \beta ) q^{26} -27 q^{27} + ( 128 - 54 \beta ) q^{28} + ( -96 - 6 \beta ) q^{29} + ( 144 + 24 \beta ) q^{30} + ( 176 + 8 \beta ) q^{31} + ( 408 + 41 \beta ) q^{32} + 33 q^{33} + ( -336 + 46 \beta ) q^{34} + ( 108 + 4 \beta ) q^{35} + ( 144 + 9 \beta ) q^{36} + ( -190 + 52 \beta ) q^{37} + ( -48 + 24 \beta ) q^{38} + ( -42 - 6 \beta ) q^{39} + ( -576 - 120 \beta ) q^{40} + ( -384 - 14 \beta ) q^{41} + ( 288 - 30 \beta ) q^{42} + ( 206 - 26 \beta ) q^{43} + ( -176 - 11 \beta ) q^{44} + ( -54 - 18 \beta ) q^{45} + ( -240 + 62 \beta ) q^{46} + ( 96 + 74 \beta ) q^{47} + ( -264 - 75 \beta ) q^{48} + ( 237 - 96 \beta ) q^{49} + ( 672 + 35 \beta ) q^{50} + ( -180 + 42 \beta ) q^{51} + ( 272 + 48 \beta ) q^{52} + ( -234 - 54 \beta ) q^{53} -27 \beta q^{54} + ( 66 + 22 \beta ) q^{55} + ( -528 - 6 \beta ) q^{56} + ( -78 + 6 \beta ) q^{57} + ( -144 - 102 \beta ) q^{58} + ( -36 - 100 \beta ) q^{59} + ( 432 + 120 \beta ) q^{60} + ( -406 + 34 \beta ) q^{61} + ( 192 + 184 \beta ) q^{62} + ( 126 - 36 \beta ) q^{63} + ( 280 + 249 \beta ) q^{64} + ( -180 - 44 \beta ) q^{65} + 33 \beta q^{66} + ( -340 - 96 \beta ) q^{67} + ( 624 - 178 \beta ) q^{68} + ( -216 + 30 \beta ) q^{69} + ( 96 + 112 \beta ) q^{70} + ( 288 + 54 \beta ) q^{71} + ( 216 + 81 \beta ) q^{72} + ( 662 - 28 \beta ) q^{73} + ( 1248 - 138 \beta ) q^{74} + ( -21 - 84 \beta ) q^{75} + ( 368 - 8 \beta ) q^{76} + ( -154 + 44 \beta ) q^{77} + ( -144 - 48 \beta ) q^{78} + ( 254 + 144 \beta ) q^{79} + ( -1728 - 376 \beta ) q^{80} + 81 q^{81} + ( -336 - 398 \beta ) q^{82} + ( -240 + 156 \beta ) q^{83} + ( -384 + 162 \beta ) q^{84} + ( 312 - 8 \beta ) q^{85} + ( -624 + 180 \beta ) q^{86} + ( 288 + 18 \beta ) q^{87} + ( -264 - 99 \beta ) q^{88} + ( -414 + 72 \beta ) q^{89} + ( -432 - 72 \beta ) q^{90} + ( 4 - 36 \beta ) q^{91} + ( 912 - 98 \beta ) q^{92} + ( -528 - 24 \beta ) q^{93} + ( 1776 + 170 \beta ) q^{94} + ( -60 - 36 \beta ) q^{95} + ( -1224 - 123 \beta ) q^{96} + ( -322 + 192 \beta ) q^{97} + ( -2304 + 141 \beta ) q^{98} -99 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - 6q^{3} + 33q^{4} - 14q^{5} - 3q^{6} + 24q^{7} + 57q^{8} + 18q^{9} + O(q^{10})$$ $$2q + q^{2} - 6q^{3} + 33q^{4} - 14q^{5} - 3q^{6} + 24q^{7} + 57q^{8} + 18q^{9} - 104q^{10} - 22q^{11} - 99q^{12} + 30q^{13} - 182q^{14} + 42q^{15} + 201q^{16} + 106q^{17} + 9q^{18} + 50q^{19} - 328q^{20} - 72q^{21} - 11q^{22} + 134q^{23} - 171q^{24} + 42q^{25} + 112q^{26} - 54q^{27} + 202q^{28} - 198q^{29} + 312q^{30} + 360q^{31} + 857q^{32} + 66q^{33} - 626q^{34} + 220q^{35} + 297q^{36} - 328q^{37} - 72q^{38} - 90q^{39} - 1272q^{40} - 782q^{41} + 546q^{42} + 386q^{43} - 363q^{44} - 126q^{45} - 418q^{46} + 266q^{47} - 603q^{48} + 378q^{49} + 1379q^{50} - 318q^{51} + 592q^{52} - 522q^{53} - 27q^{54} + 154q^{55} - 1062q^{56} - 150q^{57} - 390q^{58} - 172q^{59} + 984q^{60} - 778q^{61} + 568q^{62} + 216q^{63} + 809q^{64} - 404q^{65} + 33q^{66} - 776q^{67} + 1070q^{68} - 402q^{69} + 304q^{70} + 630q^{71} + 513q^{72} + 1296q^{73} + 2358q^{74} - 126q^{75} + 728q^{76} - 264q^{77} - 336q^{78} + 652q^{79} - 3832q^{80} + 162q^{81} - 1070q^{82} - 324q^{83} - 606q^{84} + 616q^{85} - 1068q^{86} + 594q^{87} - 627q^{88} - 756q^{89} - 936q^{90} - 28q^{91} + 1726q^{92} - 1080q^{93} + 3722q^{94} - 156q^{95} - 2571q^{96} - 452q^{97} - 4467q^{98} - 198q^{99} + O(q^{100})$$

## 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
−4.42443 −3.00000 11.5756 2.84886 13.2733 31.6977 −15.8199 9.00000 −12.6046
1.2 5.42443 −3.00000 21.4244 −16.8489 −16.2733 −7.69772 72.8199 9.00000 −91.3954
 $$n$$: 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 33.4.a.c 2
3.b odd 2 1 99.4.a.f 2
4.b odd 2 1 528.4.a.p 2
5.b even 2 1 825.4.a.l 2
5.c odd 4 2 825.4.c.h 4
7.b odd 2 1 1617.4.a.k 2
8.b even 2 1 2112.4.a.bn 2
8.d odd 2 1 2112.4.a.bg 2
11.b odd 2 1 363.4.a.i 2
12.b even 2 1 1584.4.a.bj 2
15.d odd 2 1 2475.4.a.p 2
33.d even 2 1 1089.4.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.c 2 1.a even 1 1 trivial
99.4.a.f 2 3.b odd 2 1
363.4.a.i 2 11.b odd 2 1
528.4.a.p 2 4.b odd 2 1
825.4.a.l 2 5.b even 2 1
825.4.c.h 4 5.c odd 4 2
1089.4.a.u 2 33.d even 2 1
1584.4.a.bj 2 12.b even 2 1
1617.4.a.k 2 7.b odd 2 1
2112.4.a.bg 2 8.d odd 2 1
2112.4.a.bn 2 8.b even 2 1
2475.4.a.p 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} - 24$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(33))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T - 8 T^{2} - 8 T^{3} + 64 T^{4}$$
$3$ $$( 1 + 3 T )^{2}$$
$5$ $$1 + 14 T + 202 T^{2} + 1750 T^{3} + 15625 T^{4}$$
$7$ $$1 - 24 T + 442 T^{2} - 8232 T^{3} + 117649 T^{4}$$
$11$ $$( 1 + 11 T )^{2}$$
$13$ $$1 - 30 T + 4522 T^{2} - 65910 T^{3} + 4826809 T^{4}$$
$17$ $$1 - 106 T + 7882 T^{2} - 520778 T^{3} + 24137569 T^{4}$$
$19$ $$1 - 50 T + 14246 T^{2} - 342950 T^{3} + 47045881 T^{4}$$
$23$ $$1 - 134 T + 26398 T^{2} - 1630378 T^{3} + 148035889 T^{4}$$
$29$ $$1 + 198 T + 57706 T^{2} + 4829022 T^{3} + 594823321 T^{4}$$
$31$ $$1 - 360 T + 90430 T^{2} - 10724760 T^{3} + 887503681 T^{4}$$
$37$ $$1 + 328 T + 62630 T^{2} + 16614184 T^{3} + 2565726409 T^{4}$$
$41$ $$1 + 782 T + 285970 T^{2} + 53896222 T^{3} + 4750104241 T^{4}$$
$43$ $$1 - 386 T + 179870 T^{2} - 30689702 T^{3} + 6321363049 T^{4}$$
$47$ $$1 - 266 T + 92542 T^{2} - 27616918 T^{3} + 10779215329 T^{4}$$
$53$ $$1 + 522 T + 295162 T^{2} + 77713794 T^{3} + 22164361129 T^{4}$$
$59$ $$1 + 172 T + 175654 T^{2} + 35325188 T^{3} + 42180533641 T^{4}$$
$61$ $$1 + 778 T + 577250 T^{2} + 176591218 T^{3} + 51520374361 T^{4}$$
$67$ $$1 + 776 T + 528582 T^{2} + 233392088 T^{3} + 90458382169 T^{4}$$
$71$ $$1 - 630 T + 744334 T^{2} - 225483930 T^{3} + 128100283921 T^{4}$$
$73$ $$1 - 1296 T + 1178926 T^{2} - 504166032 T^{3} + 151334226289 T^{4}$$
$79$ $$1 - 652 T + 589506 T^{2} - 321461428 T^{3} + 243087455521 T^{4}$$
$83$ $$1 + 324 T + 579670 T^{2} + 185258988 T^{3} + 326940373369 T^{4}$$
$89$ $$1 + 756 T + 1427110 T^{2} + 532956564 T^{3} + 496981290961 T^{4}$$
$97$ $$1 + 452 T + 982470 T^{2} + 412528196 T^{3} + 832972004929 T^{4}$$