# Properties

 Label 66.4.a.c Level $66$ Weight $4$ Character orbit 66.a Self dual yes Analytic conductor $3.894$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [66,4,Mod(1,66)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(66, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("66.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$66 = 2 \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 66.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: $$3.89412606038$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{97})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 24$$ x^2 - x - 24 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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 $$\beta = \sqrt{97}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( - \beta + 5) q^{5} + 6 q^{6} + (3 \beta - 1) q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10})$$ q + 2 * q^2 + 3 * q^3 + 4 * q^4 + (-b + 5) * q^5 + 6 * q^6 + (3*b - 1) * q^7 + 8 * q^8 + 9 * q^9 $$q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( - \beta + 5) q^{5} + 6 q^{6} + (3 \beta - 1) q^{7} + 8 q^{8} + 9 q^{9} + ( - 2 \beta + 10) q^{10} - 11 q^{11} + 12 q^{12} + ( - 7 \beta + 7) q^{13} + (6 \beta - 2) q^{14} + ( - 3 \beta + 15) q^{15} + 16 q^{16} + (2 \beta - 40) q^{17} + 18 q^{18} + (10 \beta - 30) q^{19} + ( - 4 \beta + 20) q^{20} + (9 \beta - 3) q^{21} - 22 q^{22} + ( - 5 \beta - 101) q^{23} + 24 q^{24} + ( - 10 \beta - 3) q^{25} + ( - 14 \beta + 14) q^{26} + 27 q^{27} + (12 \beta - 4) q^{28} + (6 \beta - 168) q^{29} + ( - 6 \beta + 30) q^{30} + ( - 16 \beta + 64) q^{31} + 32 q^{32} - 33 q^{33} + (4 \beta - 80) q^{34} + (16 \beta - 296) q^{35} + 36 q^{36} + (32 \beta + 94) q^{37} + (20 \beta - 60) q^{38} + ( - 21 \beta + 21) q^{39} + ( - 8 \beta + 40) q^{40} + ( - 24 \beta - 66) q^{41} + (18 \beta - 6) q^{42} + ( - 8 \beta + 240) q^{43} - 44 q^{44} + ( - 9 \beta + 45) q^{45} + ( - 10 \beta - 202) q^{46} + ( - 3 \beta - 195) q^{47} + 48 q^{48} + ( - 6 \beta + 531) q^{49} + ( - 20 \beta - 6) q^{50} + (6 \beta - 120) q^{51} + ( - 28 \beta + 28) q^{52} + (35 \beta + 305) q^{53} + 54 q^{54} + (11 \beta - 55) q^{55} + (24 \beta - 8) q^{56} + (30 \beta - 90) q^{57} + (12 \beta - 336) q^{58} + ( - 18 \beta + 186) q^{59} + ( - 12 \beta + 60) q^{60} + (7 \beta + 525) q^{61} + ( - 32 \beta + 128) q^{62} + (27 \beta - 9) q^{63} + 64 q^{64} + ( - 42 \beta + 714) q^{65} - 66 q^{66} + (16 \beta + 204) q^{67} + (8 \beta - 160) q^{68} + ( - 15 \beta - 303) q^{69} + (32 \beta - 592) q^{70} + (15 \beta + 471) q^{71} + 72 q^{72} + ( - 64 \beta + 370) q^{73} + (64 \beta + 188) q^{74} + ( - 30 \beta - 9) q^{75} + (40 \beta - 120) q^{76} + ( - 33 \beta + 11) q^{77} + ( - 42 \beta + 42) q^{78} + ( - 27 \beta - 823) q^{79} + ( - 16 \beta + 80) q^{80} + 81 q^{81} + ( - 48 \beta - 132) q^{82} + (52 \beta - 176) q^{83} + (36 \beta - 12) q^{84} + (50 \beta - 394) q^{85} + ( - 16 \beta + 480) q^{86} + (18 \beta - 504) q^{87} - 88 q^{88} + (16 \beta + 1018) q^{89} + ( - 18 \beta + 90) q^{90} + (28 \beta - 2044) q^{91} + ( - 20 \beta - 404) q^{92} + ( - 48 \beta + 192) q^{93} + ( - 6 \beta - 390) q^{94} + (80 \beta - 1120) q^{95} + 96 q^{96} + (70 \beta + 168) q^{97} + ( - 12 \beta + 1062) q^{98} - 99 q^{99}+O(q^{100})$$ q + 2 * q^2 + 3 * q^3 + 4 * q^4 + (-b + 5) * q^5 + 6 * q^6 + (3*b - 1) * q^7 + 8 * q^8 + 9 * q^9 + (-2*b + 10) * q^10 - 11 * q^11 + 12 * q^12 + (-7*b + 7) * q^13 + (6*b - 2) * q^14 + (-3*b + 15) * q^15 + 16 * q^16 + (2*b - 40) * q^17 + 18 * q^18 + (10*b - 30) * q^19 + (-4*b + 20) * q^20 + (9*b - 3) * q^21 - 22 * q^22 + (-5*b - 101) * q^23 + 24 * q^24 + (-10*b - 3) * q^25 + (-14*b + 14) * q^26 + 27 * q^27 + (12*b - 4) * q^28 + (6*b - 168) * q^29 + (-6*b + 30) * q^30 + (-16*b + 64) * q^31 + 32 * q^32 - 33 * q^33 + (4*b - 80) * q^34 + (16*b - 296) * q^35 + 36 * q^36 + (32*b + 94) * q^37 + (20*b - 60) * q^38 + (-21*b + 21) * q^39 + (-8*b + 40) * q^40 + (-24*b - 66) * q^41 + (18*b - 6) * q^42 + (-8*b + 240) * q^43 - 44 * q^44 + (-9*b + 45) * q^45 + (-10*b - 202) * q^46 + (-3*b - 195) * q^47 + 48 * q^48 + (-6*b + 531) * q^49 + (-20*b - 6) * q^50 + (6*b - 120) * q^51 + (-28*b + 28) * q^52 + (35*b + 305) * q^53 + 54 * q^54 + (11*b - 55) * q^55 + (24*b - 8) * q^56 + (30*b - 90) * q^57 + (12*b - 336) * q^58 + (-18*b + 186) * q^59 + (-12*b + 60) * q^60 + (7*b + 525) * q^61 + (-32*b + 128) * q^62 + (27*b - 9) * q^63 + 64 * q^64 + (-42*b + 714) * q^65 - 66 * q^66 + (16*b + 204) * q^67 + (8*b - 160) * q^68 + (-15*b - 303) * q^69 + (32*b - 592) * q^70 + (15*b + 471) * q^71 + 72 * q^72 + (-64*b + 370) * q^73 + (64*b + 188) * q^74 + (-30*b - 9) * q^75 + (40*b - 120) * q^76 + (-33*b + 11) * q^77 + (-42*b + 42) * q^78 + (-27*b - 823) * q^79 + (-16*b + 80) * q^80 + 81 * q^81 + (-48*b - 132) * q^82 + (52*b - 176) * q^83 + (36*b - 12) * q^84 + (50*b - 394) * q^85 + (-16*b + 480) * q^86 + (18*b - 504) * q^87 - 88 * q^88 + (16*b + 1018) * q^89 + (-18*b + 90) * q^90 + (28*b - 2044) * q^91 + (-20*b - 404) * q^92 + (-48*b + 192) * q^93 + (-6*b - 390) * q^94 + (80*b - 1120) * q^95 + 96 * q^96 + (70*b + 168) * q^97 + (-12*b + 1062) * q^98 - 99 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 10 q^{5} + 12 q^{6} - 2 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 10 * q^5 + 12 * q^6 - 2 * q^7 + 16 * q^8 + 18 * q^9 $$2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 10 q^{5} + 12 q^{6} - 2 q^{7} + 16 q^{8} + 18 q^{9} + 20 q^{10} - 22 q^{11} + 24 q^{12} + 14 q^{13} - 4 q^{14} + 30 q^{15} + 32 q^{16} - 80 q^{17} + 36 q^{18} - 60 q^{19} + 40 q^{20} - 6 q^{21} - 44 q^{22} - 202 q^{23} + 48 q^{24} - 6 q^{25} + 28 q^{26} + 54 q^{27} - 8 q^{28} - 336 q^{29} + 60 q^{30} + 128 q^{31} + 64 q^{32} - 66 q^{33} - 160 q^{34} - 592 q^{35} + 72 q^{36} + 188 q^{37} - 120 q^{38} + 42 q^{39} + 80 q^{40} - 132 q^{41} - 12 q^{42} + 480 q^{43} - 88 q^{44} + 90 q^{45} - 404 q^{46} - 390 q^{47} + 96 q^{48} + 1062 q^{49} - 12 q^{50} - 240 q^{51} + 56 q^{52} + 610 q^{53} + 108 q^{54} - 110 q^{55} - 16 q^{56} - 180 q^{57} - 672 q^{58} + 372 q^{59} + 120 q^{60} + 1050 q^{61} + 256 q^{62} - 18 q^{63} + 128 q^{64} + 1428 q^{65} - 132 q^{66} + 408 q^{67} - 320 q^{68} - 606 q^{69} - 1184 q^{70} + 942 q^{71} + 144 q^{72} + 740 q^{73} + 376 q^{74} - 18 q^{75} - 240 q^{76} + 22 q^{77} + 84 q^{78} - 1646 q^{79} + 160 q^{80} + 162 q^{81} - 264 q^{82} - 352 q^{83} - 24 q^{84} - 788 q^{85} + 960 q^{86} - 1008 q^{87} - 176 q^{88} + 2036 q^{89} + 180 q^{90} - 4088 q^{91} - 808 q^{92} + 384 q^{93} - 780 q^{94} - 2240 q^{95} + 192 q^{96} + 336 q^{97} + 2124 q^{98} - 198 q^{99}+O(q^{100})$$ 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 10 * q^5 + 12 * q^6 - 2 * q^7 + 16 * q^8 + 18 * q^9 + 20 * q^10 - 22 * q^11 + 24 * q^12 + 14 * q^13 - 4 * q^14 + 30 * q^15 + 32 * q^16 - 80 * q^17 + 36 * q^18 - 60 * q^19 + 40 * q^20 - 6 * q^21 - 44 * q^22 - 202 * q^23 + 48 * q^24 - 6 * q^25 + 28 * q^26 + 54 * q^27 - 8 * q^28 - 336 * q^29 + 60 * q^30 + 128 * q^31 + 64 * q^32 - 66 * q^33 - 160 * q^34 - 592 * q^35 + 72 * q^36 + 188 * q^37 - 120 * q^38 + 42 * q^39 + 80 * q^40 - 132 * q^41 - 12 * q^42 + 480 * q^43 - 88 * q^44 + 90 * q^45 - 404 * q^46 - 390 * q^47 + 96 * q^48 + 1062 * q^49 - 12 * q^50 - 240 * q^51 + 56 * q^52 + 610 * q^53 + 108 * q^54 - 110 * q^55 - 16 * q^56 - 180 * q^57 - 672 * q^58 + 372 * q^59 + 120 * q^60 + 1050 * q^61 + 256 * q^62 - 18 * q^63 + 128 * q^64 + 1428 * q^65 - 132 * q^66 + 408 * q^67 - 320 * q^68 - 606 * q^69 - 1184 * q^70 + 942 * q^71 + 144 * q^72 + 740 * q^73 + 376 * q^74 - 18 * q^75 - 240 * q^76 + 22 * q^77 + 84 * q^78 - 1646 * q^79 + 160 * q^80 + 162 * q^81 - 264 * q^82 - 352 * q^83 - 24 * q^84 - 788 * q^85 + 960 * q^86 - 1008 * q^87 - 176 * q^88 + 2036 * q^89 + 180 * q^90 - 4088 * q^91 - 808 * q^92 + 384 * q^93 - 780 * q^94 - 2240 * q^95 + 192 * q^96 + 336 * q^97 + 2124 * q^98 - 198 * 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
 5.42443 −4.42443
2.00000 3.00000 4.00000 −4.84886 6.00000 28.5466 8.00000 9.00000 −9.69772
1.2 2.00000 3.00000 4.00000 14.8489 6.00000 −30.5466 8.00000 9.00000 29.6977
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$11$$ $$1$$

## 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 66.4.a.c 2
3.b odd 2 1 198.4.a.h 2
4.b odd 2 1 528.4.a.n 2
5.b even 2 1 1650.4.a.s 2
5.c odd 4 2 1650.4.c.u 4
8.b even 2 1 2112.4.a.bb 2
8.d odd 2 1 2112.4.a.bi 2
11.b odd 2 1 726.4.a.o 2
12.b even 2 1 1584.4.a.ba 2
33.d even 2 1 2178.4.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.4.a.c 2 1.a even 1 1 trivial
198.4.a.h 2 3.b odd 2 1
528.4.a.n 2 4.b odd 2 1
726.4.a.o 2 11.b odd 2 1
1584.4.a.ba 2 12.b even 2 1
1650.4.a.s 2 5.b even 2 1
1650.4.c.u 4 5.c odd 4 2
2112.4.a.bb 2 8.b even 2 1
2112.4.a.bi 2 8.d odd 2 1
2178.4.a.bf 2 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 10T_{5} - 72$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(66))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 2)^{2}$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2} - 10T - 72$$
$7$ $$T^{2} + 2T - 872$$
$11$ $$(T + 11)^{2}$$
$13$ $$T^{2} - 14T - 4704$$
$17$ $$T^{2} + 80T + 1212$$
$19$ $$T^{2} + 60T - 8800$$
$23$ $$T^{2} + 202T + 7776$$
$29$ $$T^{2} + 336T + 24732$$
$31$ $$T^{2} - 128T - 20736$$
$37$ $$T^{2} - 188T - 90492$$
$41$ $$T^{2} + 132T - 51516$$
$43$ $$T^{2} - 480T + 51392$$
$47$ $$T^{2} + 390T + 37152$$
$53$ $$T^{2} - 610T - 25800$$
$59$ $$T^{2} - 372T + 3168$$
$61$ $$T^{2} - 1050 T + 270872$$
$67$ $$T^{2} - 408T + 16784$$
$71$ $$T^{2} - 942T + 200016$$
$73$ $$T^{2} - 740T - 260412$$
$79$ $$T^{2} + 1646 T + 606616$$
$83$ $$T^{2} + 352T - 231312$$
$89$ $$T^{2} - 2036 T + 1011492$$
$97$ $$T^{2} - 336T - 447076$$
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