# Properties

 Label 33.4.a.d 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: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 3 q^{3} + \beta q^{4} + ( 10 - 4 \beta ) q^{5} + 3 \beta q^{6} + ( 2 - 2 \beta ) q^{7} + ( 8 - 7 \beta ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + \beta q^{2} + 3 q^{3} + \beta q^{4} + ( 10 - 4 \beta ) q^{5} + 3 \beta q^{6} + ( 2 - 2 \beta ) q^{7} + ( 8 - 7 \beta ) q^{8} + 9 q^{9} + ( -32 + 6 \beta ) q^{10} + 11 q^{11} + 3 \beta q^{12} + ( -42 + 8 \beta ) q^{13} -16 q^{14} + ( 30 - 12 \beta ) q^{15} + ( -56 - 7 \beta ) q^{16} + ( -28 + 30 \beta ) q^{17} + 9 \beta q^{18} + ( -18 - 18 \beta ) q^{19} + ( -32 + 6 \beta ) q^{20} + ( 6 - 6 \beta ) q^{21} + 11 \beta q^{22} + 112 q^{23} + ( 24 - 21 \beta ) q^{24} + ( 103 - 64 \beta ) q^{25} + ( 64 - 34 \beta ) q^{26} + 27 q^{27} -16 q^{28} + ( 88 + 46 \beta ) q^{29} + ( -96 + 18 \beta ) q^{30} + ( -72 + 104 \beta ) q^{31} + ( -120 - 7 \beta ) q^{32} + 33 q^{33} + ( 240 + 2 \beta ) q^{34} + ( 84 - 20 \beta ) q^{35} + 9 \beta q^{36} + ( -46 + 44 \beta ) q^{37} + ( -144 - 36 \beta ) q^{38} + ( -126 + 24 \beta ) q^{39} + ( 304 - 74 \beta ) q^{40} + ( -248 + 2 \beta ) q^{41} -48 q^{42} + ( 10 - 86 \beta ) q^{43} + 11 \beta q^{44} + ( 90 - 36 \beta ) q^{45} + 112 \beta q^{46} + ( -8 - 48 \beta ) q^{47} + ( -168 - 21 \beta ) q^{48} + ( -307 - 4 \beta ) q^{49} + ( -512 + 39 \beta ) q^{50} + ( -84 + 90 \beta ) q^{51} + ( 64 - 34 \beta ) q^{52} + ( 22 - 128 \beta ) q^{53} + 27 \beta q^{54} + ( 110 - 44 \beta ) q^{55} + ( 128 - 16 \beta ) q^{56} + ( -54 - 54 \beta ) q^{57} + ( 368 + 134 \beta ) q^{58} + 196 q^{59} + ( -96 + 18 \beta ) q^{60} + ( -526 - 52 \beta ) q^{61} + ( 832 + 32 \beta ) q^{62} + ( 18 - 18 \beta ) q^{63} + ( 392 - 71 \beta ) q^{64} + ( -676 + 216 \beta ) q^{65} + 33 \beta q^{66} + ( 388 + 152 \beta ) q^{67} + ( 240 + 2 \beta ) q^{68} + 336 q^{69} + ( -160 + 64 \beta ) q^{70} + ( 136 + 184 \beta ) q^{71} + ( 72 - 63 \beta ) q^{72} + ( -170 - 252 \beta ) q^{73} + ( 352 - 2 \beta ) q^{74} + ( 309 - 192 \beta ) q^{75} + ( -144 - 36 \beta ) q^{76} + ( 22 - 22 \beta ) q^{77} + ( 192 - 102 \beta ) q^{78} + ( -78 - 74 \beta ) q^{79} + ( -336 + 182 \beta ) q^{80} + 81 q^{81} + ( 16 - 246 \beta ) q^{82} + ( 336 - 324 \beta ) q^{83} -48 q^{84} + ( -1240 + 292 \beta ) q^{85} + ( -688 - 76 \beta ) q^{86} + ( 264 + 138 \beta ) q^{87} + ( 88 - 77 \beta ) q^{88} + ( 482 + 8 \beta ) q^{89} + ( -288 + 54 \beta ) q^{90} + ( -212 + 84 \beta ) q^{91} + 112 \beta q^{92} + ( -216 + 312 \beta ) q^{93} + ( -384 - 56 \beta ) q^{94} + ( 396 - 36 \beta ) q^{95} + ( -360 - 21 \beta ) q^{96} + ( -802 + 420 \beta ) q^{97} + ( -32 - 311 \beta ) q^{98} + 99 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + 6q^{3} + q^{4} + 16q^{5} + 3q^{6} + 2q^{7} + 9q^{8} + 18q^{9} + O(q^{10})$$ $$2q + q^{2} + 6q^{3} + q^{4} + 16q^{5} + 3q^{6} + 2q^{7} + 9q^{8} + 18q^{9} - 58q^{10} + 22q^{11} + 3q^{12} - 76q^{13} - 32q^{14} + 48q^{15} - 119q^{16} - 26q^{17} + 9q^{18} - 54q^{19} - 58q^{20} + 6q^{21} + 11q^{22} + 224q^{23} + 27q^{24} + 142q^{25} + 94q^{26} + 54q^{27} - 32q^{28} + 222q^{29} - 174q^{30} - 40q^{31} - 247q^{32} + 66q^{33} + 482q^{34} + 148q^{35} + 9q^{36} - 48q^{37} - 324q^{38} - 228q^{39} + 534q^{40} - 494q^{41} - 96q^{42} - 66q^{43} + 11q^{44} + 144q^{45} + 112q^{46} - 64q^{47} - 357q^{48} - 618q^{49} - 985q^{50} - 78q^{51} + 94q^{52} - 84q^{53} + 27q^{54} + 176q^{55} + 240q^{56} - 162q^{57} + 870q^{58} + 392q^{59} - 174q^{60} - 1104q^{61} + 1696q^{62} + 18q^{63} + 713q^{64} - 1136q^{65} + 33q^{66} + 928q^{67} + 482q^{68} + 672q^{69} - 256q^{70} + 456q^{71} + 81q^{72} - 592q^{73} + 702q^{74} + 426q^{75} - 324q^{76} + 22q^{77} + 282q^{78} - 230q^{79} - 490q^{80} + 162q^{81} - 214q^{82} + 348q^{83} - 96q^{84} - 2188q^{85} - 1452q^{86} + 666q^{87} + 99q^{88} + 972q^{89} - 522q^{90} - 340q^{91} + 112q^{92} - 120q^{93} - 824q^{94} + 756q^{95} - 741q^{96} - 1184q^{97} - 375q^{98} + 198q^{99} + O(q^{100})$$

## 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
 −2.37228 3.37228
−2.37228 3.00000 −2.37228 19.4891 −7.11684 6.74456 24.6060 9.00000 −46.2337
1.2 3.37228 3.00000 3.37228 −3.48913 10.1168 −4.74456 −15.6060 9.00000 −11.7663
 $$n$$: 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.d 2
3.b odd 2 1 99.4.a.e 2
4.b odd 2 1 528.4.a.o 2
5.b even 2 1 825.4.a.k 2
5.c odd 4 2 825.4.c.i 4
7.b odd 2 1 1617.4.a.j 2
8.b even 2 1 2112.4.a.ba 2
8.d odd 2 1 2112.4.a.bh 2
11.b odd 2 1 363.4.a.j 2
12.b even 2 1 1584.4.a.x 2
15.d odd 2 1 2475.4.a.o 2
33.d even 2 1 1089.4.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.d 2 1.a even 1 1 trivial
99.4.a.e 2 3.b odd 2 1
363.4.a.j 2 11.b odd 2 1
528.4.a.o 2 4.b odd 2 1
825.4.a.k 2 5.b even 2 1
825.4.c.i 4 5.c odd 4 2
1089.4.a.t 2 33.d even 2 1
1584.4.a.x 2 12.b even 2 1
1617.4.a.j 2 7.b odd 2 1
2112.4.a.ba 2 8.b even 2 1
2112.4.a.bh 2 8.d odd 2 1
2475.4.a.o 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} - 8$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(33))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 8 T^{2} - 8 T^{3} + 64 T^{4}$$
$3$ $$( 1 - 3 T )^{2}$$
$5$ $$1 - 16 T + 182 T^{2} - 2000 T^{3} + 15625 T^{4}$$
$7$ $$1 - 2 T + 654 T^{2} - 686 T^{3} + 117649 T^{4}$$
$11$ $$( 1 - 11 T )^{2}$$
$13$ $$1 + 76 T + 5310 T^{2} + 166972 T^{3} + 4826809 T^{4}$$
$17$ $$1 + 26 T + 2570 T^{2} + 127738 T^{3} + 24137569 T^{4}$$
$19$ $$1 + 54 T + 11774 T^{2} + 370386 T^{3} + 47045881 T^{4}$$
$23$ $$( 1 - 112 T + 12167 T^{2} )^{2}$$
$29$ $$1 - 222 T + 43642 T^{2} - 5414358 T^{3} + 594823321 T^{4}$$
$31$ $$1 + 40 T - 29250 T^{2} + 1191640 T^{3} + 887503681 T^{4}$$
$37$ $$1 + 48 T + 85910 T^{2} + 2431344 T^{3} + 2565726409 T^{4}$$
$41$ $$1 + 494 T + 198818 T^{2} + 34046974 T^{3} + 4750104241 T^{4}$$
$43$ $$1 + 66 T + 99086 T^{2} + 5247462 T^{3} + 6321363049 T^{4}$$
$47$ $$1 + 64 T + 189662 T^{2} + 6644672 T^{3} + 10779215329 T^{4}$$
$53$ $$1 + 84 T + 164350 T^{2} + 12505668 T^{3} + 22164361129 T^{4}$$
$59$ $$( 1 - 196 T + 205379 T^{2} )^{2}$$
$61$ $$1 + 1104 T + 736358 T^{2} + 250587024 T^{3} + 51520374361 T^{4}$$
$67$ $$1 - 928 T + 626214 T^{2} - 279108064 T^{3} + 90458382169 T^{4}$$
$71$ $$1 - 456 T + 488494 T^{2} - 163207416 T^{3} + 128100283921 T^{4}$$
$73$ $$1 + 592 T + 341742 T^{2} + 230298064 T^{3} + 151334226289 T^{4}$$
$79$ $$1 + 230 T + 954126 T^{2} + 113398970 T^{3} + 243087455521 T^{4}$$
$83$ $$1 - 348 T + 307798 T^{2} - 198981876 T^{3} + 326940373369 T^{4}$$
$89$ $$1 - 972 T + 1645606 T^{2} - 685229868 T^{3} + 496981290961 T^{4}$$
$97$ $$1 + 1184 T + 720510 T^{2} + 1080604832 T^{3} + 832972004929 T^{4}$$