# Properties

 Label 33.6.a.c Level $33$ Weight $6$ Character orbit 33.a Self dual yes Analytic conductor $5.293$ Analytic rank $1$ 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: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -2 - \beta ) q^{2} -9 q^{3} + ( 16 + 5 \beta ) q^{4} + ( 24 + 10 \beta ) q^{5} + ( 18 + 9 \beta ) q^{6} + ( -138 - 10 \beta ) q^{7} + ( -188 + \beta ) q^{8} + 81 q^{9} +O(q^{10})$$ $$q + ( -2 - \beta ) q^{2} -9 q^{3} + ( 16 + 5 \beta ) q^{4} + ( 24 + 10 \beta ) q^{5} + ( 18 + 9 \beta ) q^{6} + ( -138 - 10 \beta ) q^{7} + ( -188 + \beta ) q^{8} + 81 q^{9} + ( -488 - 54 \beta ) q^{10} -121 q^{11} + ( -144 - 45 \beta ) q^{12} + ( -64 - 38 \beta ) q^{13} + ( 716 + 168 \beta ) q^{14} + ( -216 - 90 \beta ) q^{15} + ( -180 + 25 \beta ) q^{16} + ( -370 - 60 \beta ) q^{17} + ( -162 - 81 \beta ) q^{18} + ( -744 + 12 \beta ) q^{19} + ( 2584 + 330 \beta ) q^{20} + ( 1242 + 90 \beta ) q^{21} + ( 242 + 121 \beta ) q^{22} + ( -1502 - 366 \beta ) q^{23} + ( 1692 - 9 \beta ) q^{24} + ( 1851 + 580 \beta ) q^{25} + ( 1800 + 178 \beta ) q^{26} -729 q^{27} + ( -4408 - 900 \beta ) q^{28} + ( 3506 - 412 \beta ) q^{29} + ( 4392 + 486 \beta ) q^{30} + ( -3592 - 344 \beta ) q^{31} + ( 5276 + 73 \beta ) q^{32} + 1089 q^{33} + ( 3380 + 550 \beta ) q^{34} + ( -7712 - 1720 \beta ) q^{35} + ( 1296 + 405 \beta ) q^{36} + ( -15026 + 136 \beta ) q^{37} + ( 960 + 708 \beta ) q^{38} + ( 576 + 342 \beta ) q^{39} + ( -4072 - 1846 \beta ) q^{40} + ( -3246 + 712 \beta ) q^{41} + ( -6444 - 1512 \beta ) q^{42} + ( -9076 + 1496 \beta ) q^{43} + ( -1936 - 605 \beta ) q^{44} + ( 1944 + 810 \beta ) q^{45} + ( 19108 + 2600 \beta ) q^{46} + ( 2662 + 2526 \beta ) q^{47} + ( 1620 - 225 \beta ) q^{48} + ( 6637 + 2860 \beta ) q^{49} + ( -29222 - 3591 \beta ) q^{50} + ( 3330 + 540 \beta ) q^{51} + ( -9384 - 1118 \beta ) q^{52} + ( 8192 - 2206 \beta ) q^{53} + ( 1458 + 729 \beta ) q^{54} + ( -2904 - 1210 \beta ) q^{55} + ( 25504 + 1732 \beta ) q^{56} + ( 6696 - 108 \beta ) q^{57} + ( 11116 - 2270 \beta ) q^{58} + ( 7912 + 1476 \beta ) q^{59} + ( -23256 - 2970 \beta ) q^{60} + ( -308 - 2330 \beta ) q^{61} + ( 22320 + 4624 \beta ) q^{62} + ( -11178 - 810 \beta ) q^{63} + ( -8004 - 6295 \beta ) q^{64} + ( -18256 - 1932 \beta ) q^{65} + ( -2178 - 1089 \beta ) q^{66} + ( 17268 - 3200 \beta ) q^{67} + ( -19120 - 3110 \beta ) q^{68} + ( 13518 + 3294 \beta ) q^{69} + ( 91104 + 12872 \beta ) q^{70} + ( -18454 + 3098 \beta ) q^{71} + ( -15228 + 81 \beta ) q^{72} + ( 34090 - 7536 \beta ) q^{73} + ( 24068 + 14618 \beta ) q^{74} + ( -16659 - 5220 \beta ) q^{75} + ( -9264 - 3468 \beta ) q^{76} + ( 16698 + 1210 \beta ) q^{77} + ( -16200 - 1602 \beta ) q^{78} + ( -3806 + 9482 \beta ) q^{79} + ( 6680 - 950 \beta ) q^{80} + 6561 q^{81} + ( -24836 + 1110 \beta ) q^{82} + ( -27852 - 2592 \beta ) q^{83} + ( 39672 + 8100 \beta ) q^{84} + ( -35280 - 5740 \beta ) q^{85} + ( -47672 + 4588 \beta ) q^{86} + ( -31554 + 3708 \beta ) q^{87} + ( 22748 - 121 \beta ) q^{88} + ( 53722 - 15056 \beta ) q^{89} + ( -39528 - 4374 \beta ) q^{90} + ( 25552 + 6264 \beta ) q^{91} + ( -104552 - 15196 \beta ) q^{92} + ( 32328 + 3096 \beta ) q^{93} + ( -116468 - 10240 \beta ) q^{94} + ( -12576 - 7032 \beta ) q^{95} + ( -47484 - 657 \beta ) q^{96} + ( -7090 + 21300 \beta ) q^{97} + ( -139114 - 15217 \beta ) q^{98} -9801 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 5q^{2} - 18q^{3} + 37q^{4} + 58q^{5} + 45q^{6} - 286q^{7} - 375q^{8} + 162q^{9} + O(q^{10})$$ $$2q - 5q^{2} - 18q^{3} + 37q^{4} + 58q^{5} + 45q^{6} - 286q^{7} - 375q^{8} + 162q^{9} - 1030q^{10} - 242q^{11} - 333q^{12} - 166q^{13} + 1600q^{14} - 522q^{15} - 335q^{16} - 800q^{17} - 405q^{18} - 1476q^{19} + 5498q^{20} + 2574q^{21} + 605q^{22} - 3370q^{23} + 3375q^{24} + 4282q^{25} + 3778q^{26} - 1458q^{27} - 9716q^{28} + 6600q^{29} + 9270q^{30} - 7528q^{31} + 10625q^{32} + 2178q^{33} + 7310q^{34} - 17144q^{35} + 2997q^{36} - 29916q^{37} + 2628q^{38} + 1494q^{39} - 9990q^{40} - 5780q^{41} - 14400q^{42} - 16656q^{43} - 4477q^{44} + 4698q^{45} + 40816q^{46} + 7850q^{47} + 3015q^{48} + 16134q^{49} - 62035q^{50} + 7200q^{51} - 19886q^{52} + 14178q^{53} + 3645q^{54} - 7018q^{55} + 52740q^{56} + 13284q^{57} + 19962q^{58} + 17300q^{59} - 49482q^{60} - 2946q^{61} + 49264q^{62} - 23166q^{63} - 22303q^{64} - 38444q^{65} - 5445q^{66} + 31336q^{67} - 41350q^{68} + 30330q^{69} + 195080q^{70} - 33810q^{71} - 30375q^{72} + 60644q^{73} + 62754q^{74} - 38538q^{75} - 21996q^{76} + 34606q^{77} - 34002q^{78} + 1870q^{79} + 12410q^{80} + 13122q^{81} - 48562q^{82} - 58296q^{83} + 87444q^{84} - 76300q^{85} - 90756q^{86} - 59400q^{87} + 45375q^{88} + 92388q^{89} - 83430q^{90} + 57368q^{91} - 224300q^{92} + 67752q^{93} - 243176q^{94} - 32184q^{95} - 95625q^{96} + 7120q^{97} - 293445q^{98} - 19602q^{99} + O(q^{100})$$

## 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
 7.15207 −6.15207
−9.15207 −9.00000 51.7603 95.5207 82.3686 −209.521 −180.848 81.0000 −874.212
1.2 4.15207 −9.00000 −14.7603 −37.5207 −37.3686 −76.4793 −194.152 81.0000 −155.788
 $$n$$: 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.c 2
3.b odd 2 1 99.6.a.f 2
4.b odd 2 1 528.6.a.s 2
5.b even 2 1 825.6.a.e 2
11.b odd 2 1 363.6.a.j 2
33.d even 2 1 1089.6.a.j 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 5 T_{2} - 38$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(33))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 5 T + 26 T^{2} + 160 T^{3} + 1024 T^{4}$$
$3$ $$( 1 + 9 T )^{2}$$
$5$ $$1 - 58 T + 2666 T^{2} - 181250 T^{3} + 9765625 T^{4}$$
$7$ $$1 + 286 T + 49638 T^{2} + 4806802 T^{3} + 282475249 T^{4}$$
$11$ $$( 1 + 121 T )^{2}$$
$13$ $$1 + 166 T + 685578 T^{2} + 61634638 T^{3} + 137858491849 T^{4}$$
$17$ $$1 + 800 T + 2840414 T^{2} + 1135885600 T^{3} + 2015993900449 T^{4}$$
$19$ $$1 + 1476 T + 5490470 T^{2} + 3654722124 T^{3} + 6131066257801 T^{4}$$
$23$ $$1 + 3370 T + 9784358 T^{2} + 21690475910 T^{3} + 41426511213649 T^{4}$$
$29$ $$1 - 6600 T + 44401126 T^{2} - 135373583400 T^{3} + 420707233300201 T^{4}$$
$31$ $$1 + 7528 T + 66189630 T^{2} + 215520248728 T^{3} + 819628286980801 T^{4}$$
$37$ $$1 + 29916 T + 361611230 T^{2} + 2074493817612 T^{3} + 4808584372417849 T^{4}$$
$41$ $$1 + 5780 T + 217632230 T^{2} + 669648841780 T^{3} + 13422659310152401 T^{4}$$
$43$ $$1 + 16656 T + 264340262 T^{2} + 2448572626608 T^{3} + 21611482313284249 T^{4}$$
$47$ $$1 - 7850 T + 191750726 T^{2} - 1800358304950 T^{3} + 52599132235830049 T^{4}$$
$53$ $$1 - 14178 T + 671305114 T^{2} - 5929175699754 T^{3} + 174887470365513049 T^{4}$$
$59$ $$1 - 17300 T + 1408269110 T^{2} - 12368190372700 T^{3} + 511116753300641401 T^{4}$$
$61$ $$1 + 2946 T + 1451133506 T^{2} + 2488180702746 T^{3} + 713342911662882601 T^{4}$$
$67$ $$1 - 31336 T + 2492616438 T^{2} - 42307520352952 T^{3} + 1822837804551761449 T^{4}$$
$71$ $$1 + 33810 T + 3469543750 T^{2} + 61000994357310 T^{3} + 3255243551009881201 T^{4}$$
$73$ $$1 - 60644 T + 2552552022 T^{2} - 125719353685892 T^{3} + 4297625829703557649 T^{4}$$
$79$ $$1 - 1870 T + 2176543686 T^{2} - 5754095466130 T^{3} + 9468276082626847201 T^{4}$$
$83$ $$1 + 58296 T + 8430395158 T^{2} + 229630313324328 T^{3} + 15516041187205853449 T^{4}$$
$89$ $$1 - 92388 T + 3271275766 T^{2} - 515900084374212 T^{3} + 31181719929966183601 T^{4}$$
$97$ $$1 - 7120 T - 2888428386 T^{2} - 61141862629840 T^{3} + 73742412689492826049 T^{4}$$