# Properties

 Label 33.3.c.a Level $33$ Weight $3$ Character orbit 33.c Analytic conductor $0.899$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,3,Mod(10,33)]

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, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("33.10");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 33.c (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$0.899184872389$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.39744.5 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} + 12x^{2} + 4x + 22$$ x^4 - 2*x^3 + 12*x^2 + 4*x + 22 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} - \beta_1 q^{3} + ( - 2 \beta_1 - 5) q^{4} + (3 \beta_1 + 1) q^{5} + (\beta_{3} - \beta_{2}) q^{6} + \beta_{2} q^{7} + (3 \beta_{3} - 2 \beta_{2}) q^{8} + 3 q^{9}+O(q^{10})$$ q - b3 * q^2 - b1 * q^3 + (-2*b1 - 5) * q^4 + (3*b1 + 1) * q^5 + (b3 - b2) * q^6 + b2 * q^7 + (3*b3 - 2*b2) * q^8 + 3 * q^9 $$q - \beta_{3} q^{2} - \beta_1 q^{3} + ( - 2 \beta_1 - 5) q^{4} + (3 \beta_1 + 1) q^{5} + (\beta_{3} - \beta_{2}) q^{6} + \beta_{2} q^{7} + (3 \beta_{3} - 2 \beta_{2}) q^{8} + 3 q^{9} + ( - 4 \beta_{3} + 3 \beta_{2}) q^{10} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 5) q^{11} + (5 \beta_1 + 6) q^{12} + (6 \beta_{3} - \beta_{2}) q^{13} + ( - 7 \beta_1 + 3) q^{14} + ( - \beta_1 - 9) q^{15} + (12 \beta_1 + 1) q^{16} + (2 \beta_{3} - 2 \beta_{2}) q^{17} - 3 \beta_{3} q^{18} + ( - 2 \beta_{3} - 4 \beta_{2}) q^{19} + ( - 17 \beta_1 - 23) q^{20} + (2 \beta_{3} + \beta_{2}) q^{21} + ( - 6 \beta_{3} + \beta_{2} - 16 \beta_1 - 3) q^{22} + (5 \beta_1 - 23) q^{23} + ( - 7 \beta_{3} + \beta_{2}) q^{24} + (6 \beta_1 + 3) q^{25} + (19 \beta_1 + 51) q^{26} - 3 \beta_1 q^{27} + (4 \beta_{3} - 3 \beta_{2}) q^{28} + ( - 8 \beta_{3} + 4 \beta_{2}) q^{29} + (10 \beta_{3} - \beta_{2}) q^{30} + ( - 18 \beta_1 + 20) q^{31} + ( - \beta_{3} + 4 \beta_{2}) q^{32} + (5 \beta_{3} + \beta_{2} - 5 \beta_1 - 3) q^{33} + (18 \beta_1 + 12) q^{34} + ( - 6 \beta_{3} - 2 \beta_{2}) q^{35} + ( - 6 \beta_1 - 15) q^{36} + (2 \beta_1 - 16) q^{37} + (24 \beta_1 - 30) q^{38} + ( - 8 \beta_{3} + 5 \beta_{2}) q^{39} + (24 \beta_{3} - 5 \beta_{2}) q^{40} + (4 \beta_{3} + 6 \beta_{2}) q^{41} + ( - 3 \beta_1 + 21) q^{42} + ( - 2 \beta_{3} - 12 \beta_{2}) q^{43} + (15 \beta_{3} - 8 \beta_{2} - 15 \beta_1 - 31) q^{44} + (9 \beta_1 + 3) q^{45} + (18 \beta_{3} + 5 \beta_{2}) q^{46} + ( - 3 \beta_1 + 25) q^{47} + ( - \beta_1 - 36) q^{48} + (10 \beta_1 + 25) q^{49} + ( - 9 \beta_{3} + 6 \beta_{2}) q^{50} - 6 \beta_{3} q^{51} + ( - 46 \beta_{3} + 15 \beta_{2}) q^{52} + ( - 11 \beta_1 + 7) q^{53} + (3 \beta_{3} - 3 \beta_{2}) q^{54} + ( - 16 \beta_{3} - \beta_{2} + 16 \beta_1 + 14) q^{55} + (\beta_1 + 39) q^{56} + ( - 6 \beta_{3} - 6 \beta_{2}) q^{57} + ( - 44 \beta_1 - 60) q^{58} + ( - 42 \beta_1 + 10) q^{59} + (23 \beta_1 + 51) q^{60} + (2 \beta_{3} + 11 \beta_{2}) q^{61} + ( - 2 \beta_{3} - 18 \beta_{2}) q^{62} + 3 \beta_{2} q^{63} + (18 \beta_1 + 7) q^{64} + (30 \beta_{3} - 16 \beta_{2}) q^{65} + (8 \beta_{3} - 5 \beta_{2} + 3 \beta_1 + 48) q^{66} - 34 q^{67} + ( - 22 \beta_{3} + 10 \beta_{2}) q^{68} + (23 \beta_1 - 15) q^{69} + (2 \beta_1 - 60) q^{70} + (\beta_1 - 71) q^{71} + (9 \beta_{3} - 6 \beta_{2}) q^{72} + ( - 4 \beta_{3} + 10 \beta_{2}) q^{73} + (14 \beta_{3} + 2 \beta_{2}) q^{74} + ( - 3 \beta_1 - 18) q^{75} + ( - 2 \beta_{3} + 8 \beta_{2}) q^{76} + ( - 2 \beta_{3} + 4 \beta_{2} + 13 \beta_1 - 45) q^{77} + ( - 51 \beta_1 - 57) q^{78} + (24 \beta_{3} - 5 \beta_{2}) q^{79} + (15 \beta_1 + 109) q^{80} + 9 q^{81} + ( - 34 \beta_1 + 54) q^{82} + ( - 16 \beta_{3} + 6 \beta_{2}) q^{83} + ( - 10 \beta_{3} + \beta_{2}) q^{84} + (20 \beta_{3} - 2 \beta_{2}) q^{85} + (80 \beta_1 - 54) q^{86} + (16 \beta_{3} - 4 \beta_{2}) q^{87} + (22 \beta_{3} - 11 \beta_{2} + 22 \beta_1 + 99) q^{88} + (18 \beta_1 + 76) q^{89} + ( - 12 \beta_{3} + 9 \beta_{2}) q^{90} + (32 \beta_1 + 6) q^{91} + (21 \beta_1 + 85) q^{92} + ( - 20 \beta_1 + 54) q^{93} + ( - 22 \beta_{3} - 3 \beta_{2}) q^{94} + (16 \beta_{3} + 14 \beta_{2}) q^{95} + (9 \beta_{3} + 3 \beta_{2}) q^{96} + ( - 42 \beta_1 - 94) q^{97} + ( - 35 \beta_{3} + 10 \beta_{2}) q^{98} + ( - 3 \beta_{3} + 6 \beta_{2} + 3 \beta_1 + 15) q^{99}+O(q^{100})$$ q - b3 * q^2 - b1 * q^3 + (-2*b1 - 5) * q^4 + (3*b1 + 1) * q^5 + (b3 - b2) * q^6 + b2 * q^7 + (3*b3 - 2*b2) * q^8 + 3 * q^9 + (-4*b3 + 3*b2) * q^10 + (-b3 + 2*b2 + b1 + 5) * q^11 + (5*b1 + 6) * q^12 + (6*b3 - b2) * q^13 + (-7*b1 + 3) * q^14 + (-b1 - 9) * q^15 + (12*b1 + 1) * q^16 + (2*b3 - 2*b2) * q^17 - 3*b3 * q^18 + (-2*b3 - 4*b2) * q^19 + (-17*b1 - 23) * q^20 + (2*b3 + b2) * q^21 + (-6*b3 + b2 - 16*b1 - 3) * q^22 + (5*b1 - 23) * q^23 + (-7*b3 + b2) * q^24 + (6*b1 + 3) * q^25 + (19*b1 + 51) * q^26 - 3*b1 * q^27 + (4*b3 - 3*b2) * q^28 + (-8*b3 + 4*b2) * q^29 + (10*b3 - b2) * q^30 + (-18*b1 + 20) * q^31 + (-b3 + 4*b2) * q^32 + (5*b3 + b2 - 5*b1 - 3) * q^33 + (18*b1 + 12) * q^34 + (-6*b3 - 2*b2) * q^35 + (-6*b1 - 15) * q^36 + (2*b1 - 16) * q^37 + (24*b1 - 30) * q^38 + (-8*b3 + 5*b2) * q^39 + (24*b3 - 5*b2) * q^40 + (4*b3 + 6*b2) * q^41 + (-3*b1 + 21) * q^42 + (-2*b3 - 12*b2) * q^43 + (15*b3 - 8*b2 - 15*b1 - 31) * q^44 + (9*b1 + 3) * q^45 + (18*b3 + 5*b2) * q^46 + (-3*b1 + 25) * q^47 + (-b1 - 36) * q^48 + (10*b1 + 25) * q^49 + (-9*b3 + 6*b2) * q^50 - 6*b3 * q^51 + (-46*b3 + 15*b2) * q^52 + (-11*b1 + 7) * q^53 + (3*b3 - 3*b2) * q^54 + (-16*b3 - b2 + 16*b1 + 14) * q^55 + (b1 + 39) * q^56 + (-6*b3 - 6*b2) * q^57 + (-44*b1 - 60) * q^58 + (-42*b1 + 10) * q^59 + (23*b1 + 51) * q^60 + (2*b3 + 11*b2) * q^61 + (-2*b3 - 18*b2) * q^62 + 3*b2 * q^63 + (18*b1 + 7) * q^64 + (30*b3 - 16*b2) * q^65 + (8*b3 - 5*b2 + 3*b1 + 48) * q^66 - 34 * q^67 + (-22*b3 + 10*b2) * q^68 + (23*b1 - 15) * q^69 + (2*b1 - 60) * q^70 + (b1 - 71) * q^71 + (9*b3 - 6*b2) * q^72 + (-4*b3 + 10*b2) * q^73 + (14*b3 + 2*b2) * q^74 + (-3*b1 - 18) * q^75 + (-2*b3 + 8*b2) * q^76 + (-2*b3 + 4*b2 + 13*b1 - 45) * q^77 + (-51*b1 - 57) * q^78 + (24*b3 - 5*b2) * q^79 + (15*b1 + 109) * q^80 + 9 * q^81 + (-34*b1 + 54) * q^82 + (-16*b3 + 6*b2) * q^83 + (-10*b3 + b2) * q^84 + (20*b3 - 2*b2) * q^85 + (80*b1 - 54) * q^86 + (16*b3 - 4*b2) * q^87 + (22*b3 - 11*b2 + 22*b1 + 99) * q^88 + (18*b1 + 76) * q^89 + (-12*b3 + 9*b2) * q^90 + (32*b1 + 6) * q^91 + (21*b1 + 85) * q^92 + (-20*b1 + 54) * q^93 + (-22*b3 - 3*b2) * q^94 + (16*b3 + 14*b2) * q^95 + (9*b3 + 3*b2) * q^96 + (-42*b1 - 94) * q^97 + (-35*b3 + 10*b2) * q^98 + (-3*b3 + 6*b2 + 3*b1 + 15) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 20 q^{4} + 4 q^{5} + 12 q^{9}+O(q^{10})$$ 4 * q - 20 * q^4 + 4 * q^5 + 12 * q^9 $$4 q - 20 q^{4} + 4 q^{5} + 12 q^{9} + 20 q^{11} + 24 q^{12} + 12 q^{14} - 36 q^{15} + 4 q^{16} - 92 q^{20} - 12 q^{22} - 92 q^{23} + 12 q^{25} + 204 q^{26} + 80 q^{31} - 12 q^{33} + 48 q^{34} - 60 q^{36} - 64 q^{37} - 120 q^{38} + 84 q^{42} - 124 q^{44} + 12 q^{45} + 100 q^{47} - 144 q^{48} + 100 q^{49} + 28 q^{53} + 56 q^{55} + 156 q^{56} - 240 q^{58} + 40 q^{59} + 204 q^{60} + 28 q^{64} + 192 q^{66} - 136 q^{67} - 60 q^{69} - 240 q^{70} - 284 q^{71} - 72 q^{75} - 180 q^{77} - 228 q^{78} + 436 q^{80} + 36 q^{81} + 216 q^{82} - 216 q^{86} + 396 q^{88} + 304 q^{89} + 24 q^{91} + 340 q^{92} + 216 q^{93} - 376 q^{97} + 60 q^{99}+O(q^{100})$$ 4 * q - 20 * q^4 + 4 * q^5 + 12 * q^9 + 20 * q^11 + 24 * q^12 + 12 * q^14 - 36 * q^15 + 4 * q^16 - 92 * q^20 - 12 * q^22 - 92 * q^23 + 12 * q^25 + 204 * q^26 + 80 * q^31 - 12 * q^33 + 48 * q^34 - 60 * q^36 - 64 * q^37 - 120 * q^38 + 84 * q^42 - 124 * q^44 + 12 * q^45 + 100 * q^47 - 144 * q^48 + 100 * q^49 + 28 * q^53 + 56 * q^55 + 156 * q^56 - 240 * q^58 + 40 * q^59 + 204 * q^60 + 28 * q^64 + 192 * q^66 - 136 * q^67 - 60 * q^69 - 240 * q^70 - 284 * q^71 - 72 * q^75 - 180 * q^77 - 228 * q^78 + 436 * q^80 + 36 * q^81 + 216 * q^82 - 216 * q^86 + 396 * q^88 + 304 * q^89 + 24 * q^91 + 340 * q^92 + 216 * q^93 - 376 * q^97 + 60 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} + 12x^{2} + 4x + 22$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + \nu^{2} + 2\nu + 23 ) / 13$$ (v^3 + v^2 + 2*v + 23) / 13 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + \nu^{2} + 28\nu + 10 ) / 13$$ (v^3 + v^2 + 28*v + 10) / 13 $$\beta_{3}$$ $$=$$ $$( -3\nu^{3} + 10\nu^{2} - 32\nu + 9 ) / 13$$ (-3*v^3 + 10*v^2 - 32*v + 9) / 13
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta _1 + 1 ) / 2$$ (b2 - b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 2\beta _1 - 5$$ b3 + b2 + 2*b1 - 5 $$\nu^{3}$$ $$=$$ $$-\beta_{3} - 2\beta_{2} + 12\beta _1 - 19$$ -b3 - 2*b2 + 12*b1 - 19

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 −0.366025 − 1.29224i 1.36603 + 3.21405i 1.36603 − 3.21405i −0.366025 + 1.29224i
3.53045i −1.73205 −8.46410 6.19615 6.11492i 2.58447i 15.7603i 3.00000 21.8752i
10.2 2.35285i 1.73205 −1.53590 −4.19615 4.07525i 6.42810i 5.79766i 3.00000 9.87291i
10.3 2.35285i 1.73205 −1.53590 −4.19615 4.07525i 6.42810i 5.79766i 3.00000 9.87291i
10.4 3.53045i −1.73205 −8.46410 6.19615 6.11492i 2.58447i 15.7603i 3.00000 21.8752i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.3.c.a 4
3.b odd 2 1 99.3.c.b 4
4.b odd 2 1 528.3.j.c 4
5.b even 2 1 825.3.b.a 4
5.c odd 4 2 825.3.h.a 8
8.b even 2 1 2112.3.j.a 4
8.d odd 2 1 2112.3.j.d 4
11.b odd 2 1 inner 33.3.c.a 4
11.c even 5 4 363.3.g.e 16
11.d odd 10 4 363.3.g.e 16
12.b even 2 1 1584.3.j.f 4
33.d even 2 1 99.3.c.b 4
44.c even 2 1 528.3.j.c 4
55.d odd 2 1 825.3.b.a 4
55.e even 4 2 825.3.h.a 8
88.b odd 2 1 2112.3.j.a 4
88.g even 2 1 2112.3.j.d 4
132.d odd 2 1 1584.3.j.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.c.a 4 1.a even 1 1 trivial
33.3.c.a 4 11.b odd 2 1 inner
99.3.c.b 4 3.b odd 2 1
99.3.c.b 4 33.d even 2 1
363.3.g.e 16 11.c even 5 4
363.3.g.e 16 11.d odd 10 4
528.3.j.c 4 4.b odd 2 1
528.3.j.c 4 44.c even 2 1
825.3.b.a 4 5.b even 2 1
825.3.b.a 4 55.d odd 2 1
825.3.h.a 8 5.c odd 4 2
825.3.h.a 8 55.e even 4 2
1584.3.j.f 4 12.b even 2 1
1584.3.j.f 4 132.d odd 2 1
2112.3.j.a 4 8.b even 2 1
2112.3.j.a 4 88.b odd 2 1
2112.3.j.d 4 8.d odd 2 1
2112.3.j.d 4 88.g even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 18T^{2} + 69$$
$3$ $$(T^{2} - 3)^{2}$$
$5$ $$(T^{2} - 2 T - 26)^{2}$$
$7$ $$T^{4} + 48T^{2} + 276$$
$11$ $$T^{4} - 20 T^{3} + 330 T^{2} + \cdots + 14641$$
$13$ $$T^{4} + 624 T^{2} + 33396$$
$17$ $$T^{4} + 216T^{2} + 9936$$
$19$ $$T^{4} + 936T^{2} + 9936$$
$23$ $$(T^{2} + 46 T + 454)^{2}$$
$29$ $$T^{4} + 1536 T^{2} + 70656$$
$31$ $$(T^{2} - 40 T - 572)^{2}$$
$37$ $$(T^{2} + 32 T + 244)^{2}$$
$41$ $$T^{4} + 2304 T^{2} + 4416$$
$43$ $$T^{4} + 7272 T^{2} + \cdots + 3843024$$
$47$ $$(T^{2} - 50 T + 598)^{2}$$
$53$ $$(T^{2} - 14 T - 314)^{2}$$
$59$ $$(T^{2} - 20 T - 5192)^{2}$$
$61$ $$T^{4} + 6144 T^{2} + \cdots + 2596884$$
$67$ $$(T + 34)^{4}$$
$71$ $$(T^{2} + 142 T + 5038)^{2}$$
$73$ $$T^{4} + 4608 T^{2} + \cdots + 4809024$$
$79$ $$T^{4} + 10128 T^{2} + \cdots + 5643924$$
$83$ $$T^{4} + 5184 T^{2} + 4416$$
$89$ $$(T^{2} - 152 T + 4804)^{2}$$
$97$ $$(T^{2} + 188 T + 3544)^{2}$$