# 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,6,Mod(1,33)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("33.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$5.29266605383$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{313})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 78$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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

$p$ $F_p(T)$
$2$ $$T^{2} - T - 78$$
$3$ $$(T + 9)^{2}$$
$5$ $$T^{2} + 38T - 7464$$
$7$ $$T^{2} + 18T - 232$$
$11$ $$(T - 121)^{2}$$
$13$ $$T^{2} + 66T - 878128$$
$17$ $$T^{2} + 920T + 210348$$
$19$ $$T^{2} + 2932 T + 2147904$$
$23$ $$T^{2} - 5246 T + 6827232$$
$29$ $$T^{2} + 12600 T + 38326572$$
$31$ $$T^{2} - 9936 T - 4245184$$
$37$ $$T^{2} - 5996 T + 975204$$
$41$ $$T^{2} - 24244 T + 105471636$$
$43$ $$T^{2} - 20360 T - 16684800$$
$47$ $$T^{2} + 5806 T - 38936064$$
$53$ $$T^{2} - 40770 T + 387938808$$
$59$ $$T^{2} - 18212 T - 974471136$$
$61$ $$T^{2} + 11398 T - 557566776$$
$67$ $$T^{2} - 65368 T + 1022090128$$
$71$ $$T^{2} - 61446 T + 190949616$$
$73$ $$T^{2} - 53412 T + 705197636$$
$79$ $$T^{2} - 17122 T - 281721704$$
$83$ $$T^{2} + 14304 T - 922809744$$
$89$ $$T^{2} + 58140 T + 129501828$$
$97$ $$T^{2} + 183056 T + 8284064476$$