# Properties

 Label 33.6.a.d Level 33 Weight 6 Character orbit 33.a Self dual yes Analytic conductor 5.293 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 33.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.29266605383$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{313})$$ Defining polynomial: $$x^{2} - x - 78$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{313})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} -9 q^{3} + ( 46 + \beta ) q^{4} + ( -24 + 10 \beta ) q^{5} -9 \beta q^{6} + ( -10 + 2 \beta ) q^{7} + ( 78 + 15 \beta ) q^{8} + 81 q^{9} +O(q^{10})$$ $$q + \beta q^{2} -9 q^{3} + ( 46 + \beta ) q^{4} + ( -24 + 10 \beta ) q^{5} -9 \beta q^{6} + ( -10 + 2 \beta ) q^{7} + ( 78 + 15 \beta ) q^{8} + 81 q^{9} + ( 780 - 14 \beta ) q^{10} + 121 q^{11} + ( -414 - 9 \beta ) q^{12} + ( 20 - 106 \beta ) q^{13} + ( 156 - 8 \beta ) q^{14} + ( 216 - 90 \beta ) q^{15} + ( -302 + 61 \beta ) q^{16} + ( -462 + 4 \beta ) q^{17} + 81 \beta q^{18} + ( -1468 + 4 \beta ) q^{19} + ( -324 + 446 \beta ) q^{20} + ( 90 - 18 \beta ) q^{21} + 121 \beta q^{22} + ( 2610 + 26 \beta ) q^{23} + ( -702 - 135 \beta ) q^{24} + ( 5251 - 380 \beta ) q^{25} + ( -8268 - 86 \beta ) q^{26} -729 q^{27} + ( -304 + 84 \beta ) q^{28} + ( -6234 - 132 \beta ) q^{29} + ( -7020 + 126 \beta ) q^{30} + ( 4664 + 608 \beta ) q^{31} + ( 2262 - 721 \beta ) q^{32} -1089 q^{33} + ( 312 - 458 \beta ) q^{34} + ( 1800 - 128 \beta ) q^{35} + ( 3726 + 81 \beta ) q^{36} + ( 3158 - 320 \beta ) q^{37} + ( 312 - 1464 \beta ) q^{38} + ( -180 + 954 \beta ) q^{39} + ( 9828 + 570 \beta ) q^{40} + ( 12486 - 728 \beta ) q^{41} + ( -1404 + 72 \beta ) q^{42} + ( 9560 + 1240 \beta ) q^{43} + ( 5566 + 121 \beta ) q^{44} + ( -1944 + 810 \beta ) q^{45} + ( 2028 + 2636 \beta ) q^{46} + ( -2514 - 778 \beta ) q^{47} + ( 2718 - 549 \beta ) q^{48} + ( -16395 - 36 \beta ) q^{49} + ( -29640 + 4871 \beta ) q^{50} + ( 4158 - 36 \beta ) q^{51} + ( -7348 - 4962 \beta ) q^{52} + ( 20088 + 594 \beta ) q^{53} -729 \beta q^{54} + ( -2904 + 1210 \beta ) q^{55} + ( 1560 + 36 \beta ) q^{56} + ( 13212 - 36 \beta ) q^{57} + ( -10296 - 6366 \beta ) q^{58} + ( 10944 - 3676 \beta ) q^{59} + ( 2916 - 4014 \beta ) q^{60} + ( -7072 + 2746 \beta ) q^{61} + ( 47424 + 5272 \beta ) q^{62} + ( -810 + 162 \beta ) q^{63} + ( -46574 - 411 \beta ) q^{64} + ( -83160 + 1684 \beta ) q^{65} -1089 \beta q^{66} + ( 32300 + 768 \beta ) q^{67} + ( -20940 - 274 \beta ) q^{68} + ( -23490 - 234 \beta ) q^{69} + ( -9984 + 1672 \beta ) q^{70} + ( 32274 - 3102 \beta ) q^{71} + ( 6318 + 1215 \beta ) q^{72} + ( 26546 + 320 \beta ) q^{73} + ( -24960 + 2838 \beta ) q^{74} + ( -47259 + 3420 \beta ) q^{75} + ( -67216 - 1280 \beta ) q^{76} + ( -1210 + 242 \beta ) q^{77} + ( 74412 + 774 \beta ) q^{78} + ( 9626 - 2130 \beta ) q^{79} + ( 54828 - 3874 \beta ) q^{80} + 6561 q^{81} + ( -56784 + 11758 \beta ) q^{82} + ( -5388 - 3528 \beta ) q^{83} + ( 2736 - 756 \beta ) q^{84} + ( 14208 - 4676 \beta ) q^{85} + ( 96720 + 10800 \beta ) q^{86} + ( 56106 + 1188 \beta ) q^{87} + ( 9438 + 1815 \beta ) q^{88} + ( -30582 + 3024 \beta ) q^{89} + ( 63180 - 1134 \beta ) q^{90} + ( -16736 + 888 \beta ) q^{91} + ( 122088 + 3832 \beta ) q^{92} + ( -41976 - 5472 \beta ) q^{93} + ( -60684 - 3292 \beta ) q^{94} + ( 38352 - 14736 \beta ) q^{95} + ( -20358 + 6489 \beta ) q^{96} + ( -92074 + 1092 \beta ) q^{97} + ( -2808 - 16431 \beta ) q^{98} + 9801 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - 18q^{3} + 93q^{4} - 38q^{5} - 9q^{6} - 18q^{7} + 171q^{8} + 162q^{9} + O(q^{10})$$ $$2q + q^{2} - 18q^{3} + 93q^{4} - 38q^{5} - 9q^{6} - 18q^{7} + 171q^{8} + 162q^{9} + 1546q^{10} + 242q^{11} - 837q^{12} - 66q^{13} + 304q^{14} + 342q^{15} - 543q^{16} - 920q^{17} + 81q^{18} - 2932q^{19} - 202q^{20} + 162q^{21} + 121q^{22} + 5246q^{23} - 1539q^{24} + 10122q^{25} - 16622q^{26} - 1458q^{27} - 524q^{28} - 12600q^{29} - 13914q^{30} + 9936q^{31} + 3803q^{32} - 2178q^{33} + 166q^{34} + 3472q^{35} + 7533q^{36} + 5996q^{37} - 840q^{38} + 594q^{39} + 20226q^{40} + 24244q^{41} - 2736q^{42} + 20360q^{43} + 11253q^{44} - 3078q^{45} + 6692q^{46} - 5806q^{47} + 4887q^{48} - 32826q^{49} - 54409q^{50} + 8280q^{51} - 19658q^{52} + 40770q^{53} - 729q^{54} - 4598q^{55} + 3156q^{56} + 26388q^{57} - 26958q^{58} + 18212q^{59} + 1818q^{60} - 11398q^{61} + 100120q^{62} - 1458q^{63} - 93559q^{64} - 164636q^{65} - 1089q^{66} + 65368q^{67} - 42154q^{68} - 47214q^{69} - 18296q^{70} + 61446q^{71} + 13851q^{72} + 53412q^{73} - 47082q^{74} - 91098q^{75} - 135712q^{76} - 2178q^{77} + 149598q^{78} + 17122q^{79} + 105782q^{80} + 13122q^{81} - 101810q^{82} - 14304q^{83} + 4716q^{84} + 23740q^{85} + 204240q^{86} + 113400q^{87} + 20691q^{88} - 58140q^{89} + 125226q^{90} - 32584q^{91} + 248008q^{92} - 89424q^{93} - 124660q^{94} + 61968q^{95} - 34227q^{96} - 183056q^{97} - 22047q^{98} + 19602q^{99} + O(q^{100})$$

## 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
 −8.34590 9.34590
−8.34590 −9.00000 37.6541 −107.459 75.1131 −26.6918 −47.1885 81.0000 896.843
1.2 9.34590 −9.00000 55.3459 69.4590 −84.1131 8.69181 218.189 81.0000 649.157
 $$n$$: 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 33.6.a.d 2
3.b odd 2 1 99.6.a.e 2
4.b odd 2 1 528.6.a.q 2
5.b even 2 1 825.6.a.d 2
11.b odd 2 1 363.6.a.g 2
33.d even 2 1 1089.6.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.d 2 1.a even 1 1 trivial
99.6.a.e 2 3.b odd 2 1
363.6.a.g 2 11.b odd 2 1
528.6.a.q 2 4.b odd 2 1
825.6.a.d 2 5.b even 2 1
1089.6.a.o 2 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} - 78$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(33))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T - 14 T^{2} - 32 T^{3} + 1024 T^{4}$$
$3$ $$( 1 + 9 T )^{2}$$
$5$ $$1 + 38 T - 1214 T^{2} + 118750 T^{3} + 9765625 T^{4}$$
$7$ $$1 + 18 T + 33382 T^{2} + 302526 T^{3} + 282475249 T^{4}$$
$11$ $$( 1 - 121 T )^{2}$$
$13$ $$1 + 66 T - 135542 T^{2} + 24505338 T^{3} + 137858491849 T^{4}$$
$17$ $$1 + 920 T + 3050062 T^{2} + 1306268440 T^{3} + 2015993900449 T^{4}$$
$19$ $$1 + 2932 T + 7100102 T^{2} + 7259922268 T^{3} + 6131066257801 T^{4}$$
$23$ $$1 - 5246 T + 19699918 T^{2} - 33765055378 T^{3} + 41426511213649 T^{4}$$
$29$ $$1 + 12600 T + 79348870 T^{2} + 258440477400 T^{3} + 420707233300201 T^{4}$$
$31$ $$1 - 9936 T + 53013118 T^{2} - 284459244336 T^{3} + 819628286980801 T^{4}$$
$37$ $$1 - 5996 T + 139663118 T^{2} - 415786366172 T^{3} + 4808584372417849 T^{4}$$
$41$ $$1 - 24244 T + 337184038 T^{2} - 2808817737044 T^{3} + 13422659310152401 T^{4}$$
$43$ $$1 - 20360 T + 277332086 T^{2} - 2993091899480 T^{3} + 21611482313284249 T^{4}$$
$47$ $$1 + 5806 T + 419753950 T^{2} + 1331577110642 T^{3} + 52599132235830049 T^{4}$$
$53$ $$1 - 40770 T + 1224329794 T^{2} - 17049830249610 T^{3} + 174887470365513049 T^{4}$$
$59$ $$1 - 18212 T + 455377462 T^{2} - 13020201333388 T^{3} + 511116753300641401 T^{4}$$
$61$ $$1 + 11398 T + 1131625826 T^{2} + 9626708638798 T^{3} + 713342911662882601 T^{4}$$
$67$ $$1 - 65368 T + 3722340342 T^{2} - 88254977994376 T^{3} + 1822837804551761449 T^{4}$$
$71$ $$1 - 61446 T + 3799408318 T^{2} - 110862676701546 T^{3} + 3255243551009881201 T^{4}$$
$73$ $$1 - 53412 T + 4851340822 T^{2} - 110726899925316 T^{3} + 4297625829703557649 T^{4}$$
$79$ $$1 - 17122 T + 5872391094 T^{2} - 52685359663678 T^{3} + 9468276082626847201 T^{4}$$
$83$ $$1 + 14304 T + 6955271542 T^{2} + 56344037357472 T^{3} + 15516041187205853449 T^{4}$$
$89$ $$1 + 58140 T + 11297620726 T^{2} + 324657216364860 T^{3} + 31181719929966183601 T^{4}$$
$97$ $$1 + 183056 T + 25458744990 T^{2} + 1571964158085392 T^{3} + 73742412689492826049 T^{4}$$