# Properties

 Label 36.3.f.b Level $36$ Weight $3$ Character orbit 36.f Analytic conductor $0.981$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [36,3,Mod(7,36)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(36, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4]))

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

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

chi := DirichletCharacter("36.7");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 36.f (of order $$6$$, degree $$2$$, 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: $$0.980928951697$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - 3 \zeta_{6} q^{3} + 4 q^{4} + (4 \zeta_{6} - 4) q^{5} - 6 \zeta_{6} q^{6} + (2 \zeta_{6} - 4) q^{7} + 8 q^{8} + (9 \zeta_{6} - 9) q^{9} +O(q^{10})$$ q + 2 * q^2 - 3*z * q^3 + 4 * q^4 + (4*z - 4) * q^5 - 6*z * q^6 + (2*z - 4) * q^7 + 8 * q^8 + (9*z - 9) * q^9 $$q + 2 q^{2} - 3 \zeta_{6} q^{3} + 4 q^{4} + (4 \zeta_{6} - 4) q^{5} - 6 \zeta_{6} q^{6} + (2 \zeta_{6} - 4) q^{7} + 8 q^{8} + (9 \zeta_{6} - 9) q^{9} + (8 \zeta_{6} - 8) q^{10} + (7 \zeta_{6} - 14) q^{11} - 12 \zeta_{6} q^{12} + ( - 22 \zeta_{6} + 22) q^{13} + (4 \zeta_{6} - 8) q^{14} + 12 q^{15} + 16 q^{16} - 11 q^{17} + (18 \zeta_{6} - 18) q^{18} + ( - 18 \zeta_{6} + 9) q^{19} + (16 \zeta_{6} - 16) q^{20} + (6 \zeta_{6} + 6) q^{21} + (14 \zeta_{6} - 28) q^{22} + (14 \zeta_{6} + 14) q^{23} - 24 \zeta_{6} q^{24} + 9 \zeta_{6} q^{25} + ( - 44 \zeta_{6} + 44) q^{26} + 27 q^{27} + (8 \zeta_{6} - 16) q^{28} - 34 \zeta_{6} q^{29} + 24 q^{30} + (4 \zeta_{6} + 4) q^{31} + 32 q^{32} + (21 \zeta_{6} + 21) q^{33} - 22 q^{34} + ( - 16 \zeta_{6} + 8) q^{35} + (36 \zeta_{6} - 36) q^{36} - 16 q^{37} + ( - 36 \zeta_{6} + 18) q^{38} - 66 q^{39} + (32 \zeta_{6} - 32) q^{40} + (13 \zeta_{6} - 13) q^{41} + (12 \zeta_{6} + 12) q^{42} + (29 \zeta_{6} - 58) q^{43} + (28 \zeta_{6} - 56) q^{44} - 36 \zeta_{6} q^{45} + (28 \zeta_{6} + 28) q^{46} + ( - 2 \zeta_{6} + 4) q^{47} - 48 \zeta_{6} q^{48} + (37 \zeta_{6} - 37) q^{49} + 18 \zeta_{6} q^{50} + 33 \zeta_{6} q^{51} + ( - 88 \zeta_{6} + 88) q^{52} + 52 q^{53} + 54 q^{54} + ( - 56 \zeta_{6} + 28) q^{55} + (16 \zeta_{6} - 32) q^{56} + (27 \zeta_{6} - 54) q^{57} - 68 \zeta_{6} q^{58} + ( - 31 \zeta_{6} - 31) q^{59} + 48 q^{60} + 16 \zeta_{6} q^{61} + (8 \zeta_{6} + 8) q^{62} + ( - 36 \zeta_{6} + 18) q^{63} + 64 q^{64} + 88 \zeta_{6} q^{65} + (42 \zeta_{6} + 42) q^{66} + (67 \zeta_{6} + 67) q^{67} - 44 q^{68} + ( - 84 \zeta_{6} + 42) q^{69} + ( - 32 \zeta_{6} + 16) q^{70} + (72 \zeta_{6} - 72) q^{72} - 25 q^{73} - 32 q^{74} + ( - 27 \zeta_{6} + 27) q^{75} + ( - 72 \zeta_{6} + 36) q^{76} + ( - 42 \zeta_{6} + 42) q^{77} - 132 q^{78} + ( - 16 \zeta_{6} + 32) q^{79} + (64 \zeta_{6} - 64) q^{80} - 81 \zeta_{6} q^{81} + (26 \zeta_{6} - 26) q^{82} + ( - 20 \zeta_{6} + 40) q^{83} + (24 \zeta_{6} + 24) q^{84} + ( - 44 \zeta_{6} + 44) q^{85} + (58 \zeta_{6} - 116) q^{86} + (102 \zeta_{6} - 102) q^{87} + (56 \zeta_{6} - 112) q^{88} - 2 q^{89} - 72 \zeta_{6} q^{90} + (88 \zeta_{6} - 44) q^{91} + (56 \zeta_{6} + 56) q^{92} + ( - 24 \zeta_{6} + 12) q^{93} + ( - 4 \zeta_{6} + 8) q^{94} + (36 \zeta_{6} + 36) q^{95} - 96 \zeta_{6} q^{96} + 43 \zeta_{6} q^{97} + (74 \zeta_{6} - 74) q^{98} + ( - 126 \zeta_{6} + 63) q^{99} +O(q^{100})$$ q + 2 * q^2 - 3*z * q^3 + 4 * q^4 + (4*z - 4) * q^5 - 6*z * q^6 + (2*z - 4) * q^7 + 8 * q^8 + (9*z - 9) * q^9 + (8*z - 8) * q^10 + (7*z - 14) * q^11 - 12*z * q^12 + (-22*z + 22) * q^13 + (4*z - 8) * q^14 + 12 * q^15 + 16 * q^16 - 11 * q^17 + (18*z - 18) * q^18 + (-18*z + 9) * q^19 + (16*z - 16) * q^20 + (6*z + 6) * q^21 + (14*z - 28) * q^22 + (14*z + 14) * q^23 - 24*z * q^24 + 9*z * q^25 + (-44*z + 44) * q^26 + 27 * q^27 + (8*z - 16) * q^28 - 34*z * q^29 + 24 * q^30 + (4*z + 4) * q^31 + 32 * q^32 + (21*z + 21) * q^33 - 22 * q^34 + (-16*z + 8) * q^35 + (36*z - 36) * q^36 - 16 * q^37 + (-36*z + 18) * q^38 - 66 * q^39 + (32*z - 32) * q^40 + (13*z - 13) * q^41 + (12*z + 12) * q^42 + (29*z - 58) * q^43 + (28*z - 56) * q^44 - 36*z * q^45 + (28*z + 28) * q^46 + (-2*z + 4) * q^47 - 48*z * q^48 + (37*z - 37) * q^49 + 18*z * q^50 + 33*z * q^51 + (-88*z + 88) * q^52 + 52 * q^53 + 54 * q^54 + (-56*z + 28) * q^55 + (16*z - 32) * q^56 + (27*z - 54) * q^57 - 68*z * q^58 + (-31*z - 31) * q^59 + 48 * q^60 + 16*z * q^61 + (8*z + 8) * q^62 + (-36*z + 18) * q^63 + 64 * q^64 + 88*z * q^65 + (42*z + 42) * q^66 + (67*z + 67) * q^67 - 44 * q^68 + (-84*z + 42) * q^69 + (-32*z + 16) * q^70 + (72*z - 72) * q^72 - 25 * q^73 - 32 * q^74 + (-27*z + 27) * q^75 + (-72*z + 36) * q^76 + (-42*z + 42) * q^77 - 132 * q^78 + (-16*z + 32) * q^79 + (64*z - 64) * q^80 - 81*z * q^81 + (26*z - 26) * q^82 + (-20*z + 40) * q^83 + (24*z + 24) * q^84 + (-44*z + 44) * q^85 + (58*z - 116) * q^86 + (102*z - 102) * q^87 + (56*z - 112) * q^88 - 2 * q^89 - 72*z * q^90 + (88*z - 44) * q^91 + (56*z + 56) * q^92 + (-24*z + 12) * q^93 + (-4*z + 8) * q^94 + (36*z + 36) * q^95 - 96*z * q^96 + 43*z * q^97 + (74*z - 74) * q^98 + (-126*z + 63) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} - 3 q^{3} + 8 q^{4} - 4 q^{5} - 6 q^{6} - 6 q^{7} + 16 q^{8} - 9 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 - 3 * q^3 + 8 * q^4 - 4 * q^5 - 6 * q^6 - 6 * q^7 + 16 * q^8 - 9 * q^9 $$2 q + 4 q^{2} - 3 q^{3} + 8 q^{4} - 4 q^{5} - 6 q^{6} - 6 q^{7} + 16 q^{8} - 9 q^{9} - 8 q^{10} - 21 q^{11} - 12 q^{12} + 22 q^{13} - 12 q^{14} + 24 q^{15} + 32 q^{16} - 22 q^{17} - 18 q^{18} - 16 q^{20} + 18 q^{21} - 42 q^{22} + 42 q^{23} - 24 q^{24} + 9 q^{25} + 44 q^{26} + 54 q^{27} - 24 q^{28} - 34 q^{29} + 48 q^{30} + 12 q^{31} + 64 q^{32} + 63 q^{33} - 44 q^{34} - 36 q^{36} - 32 q^{37} - 132 q^{39} - 32 q^{40} - 13 q^{41} + 36 q^{42} - 87 q^{43} - 84 q^{44} - 36 q^{45} + 84 q^{46} + 6 q^{47} - 48 q^{48} - 37 q^{49} + 18 q^{50} + 33 q^{51} + 88 q^{52} + 104 q^{53} + 108 q^{54} - 48 q^{56} - 81 q^{57} - 68 q^{58} - 93 q^{59} + 96 q^{60} + 16 q^{61} + 24 q^{62} + 128 q^{64} + 88 q^{65} + 126 q^{66} + 201 q^{67} - 88 q^{68} - 72 q^{72} - 50 q^{73} - 64 q^{74} + 27 q^{75} + 42 q^{77} - 264 q^{78} + 48 q^{79} - 64 q^{80} - 81 q^{81} - 26 q^{82} + 60 q^{83} + 72 q^{84} + 44 q^{85} - 174 q^{86} - 102 q^{87} - 168 q^{88} - 4 q^{89} - 72 q^{90} + 168 q^{92} + 12 q^{94} + 108 q^{95} - 96 q^{96} + 43 q^{97} - 74 q^{98}+O(q^{100})$$ 2 * q + 4 * q^2 - 3 * q^3 + 8 * q^4 - 4 * q^5 - 6 * q^6 - 6 * q^7 + 16 * q^8 - 9 * q^9 - 8 * q^10 - 21 * q^11 - 12 * q^12 + 22 * q^13 - 12 * q^14 + 24 * q^15 + 32 * q^16 - 22 * q^17 - 18 * q^18 - 16 * q^20 + 18 * q^21 - 42 * q^22 + 42 * q^23 - 24 * q^24 + 9 * q^25 + 44 * q^26 + 54 * q^27 - 24 * q^28 - 34 * q^29 + 48 * q^30 + 12 * q^31 + 64 * q^32 + 63 * q^33 - 44 * q^34 - 36 * q^36 - 32 * q^37 - 132 * q^39 - 32 * q^40 - 13 * q^41 + 36 * q^42 - 87 * q^43 - 84 * q^44 - 36 * q^45 + 84 * q^46 + 6 * q^47 - 48 * q^48 - 37 * q^49 + 18 * q^50 + 33 * q^51 + 88 * q^52 + 104 * q^53 + 108 * q^54 - 48 * q^56 - 81 * q^57 - 68 * q^58 - 93 * q^59 + 96 * q^60 + 16 * q^61 + 24 * q^62 + 128 * q^64 + 88 * q^65 + 126 * q^66 + 201 * q^67 - 88 * q^68 - 72 * q^72 - 50 * q^73 - 64 * q^74 + 27 * q^75 + 42 * q^77 - 264 * q^78 + 48 * q^79 - 64 * q^80 - 81 * q^81 - 26 * q^82 + 60 * q^83 + 72 * q^84 + 44 * q^85 - 174 * q^86 - 102 * q^87 - 168 * q^88 - 4 * q^89 - 72 * q^90 + 168 * q^92 + 12 * q^94 + 108 * q^95 - 96 * q^96 + 43 * q^97 - 74 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/36\mathbb{Z}\right)^\times$$.

 $$n$$ $$19$$ $$29$$ $$\chi(n)$$ $$-1$$ $$-1 + \zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.5 − 0.866025i 0.5 + 0.866025i
2.00000 −1.50000 + 2.59808i 4.00000 −2.00000 3.46410i −3.00000 + 5.19615i −3.00000 1.73205i 8.00000 −4.50000 7.79423i −4.00000 6.92820i
31.1 2.00000 −1.50000 2.59808i 4.00000 −2.00000 + 3.46410i −3.00000 5.19615i −3.00000 + 1.73205i 8.00000 −4.50000 + 7.79423i −4.00000 + 6.92820i
 $$n$$: 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
36.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.3.f.b yes 2
3.b odd 2 1 108.3.f.a 2
4.b odd 2 1 36.3.f.a 2
8.b even 2 1 576.3.o.b 2
8.d odd 2 1 576.3.o.a 2
9.c even 3 1 36.3.f.a 2
9.c even 3 1 324.3.d.b 2
9.d odd 6 1 108.3.f.b 2
9.d odd 6 1 324.3.d.c 2
12.b even 2 1 108.3.f.b 2
24.f even 2 1 1728.3.o.b 2
24.h odd 2 1 1728.3.o.a 2
36.f odd 6 1 inner 36.3.f.b yes 2
36.f odd 6 1 324.3.d.b 2
36.h even 6 1 108.3.f.a 2
36.h even 6 1 324.3.d.c 2
72.j odd 6 1 1728.3.o.b 2
72.l even 6 1 1728.3.o.a 2
72.n even 6 1 576.3.o.a 2
72.p odd 6 1 576.3.o.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.f.a 2 4.b odd 2 1
36.3.f.a 2 9.c even 3 1
36.3.f.b yes 2 1.a even 1 1 trivial
36.3.f.b yes 2 36.f odd 6 1 inner
108.3.f.a 2 3.b odd 2 1
108.3.f.a 2 36.h even 6 1
108.3.f.b 2 9.d odd 6 1
108.3.f.b 2 12.b even 2 1
324.3.d.b 2 9.c even 3 1
324.3.d.b 2 36.f odd 6 1
324.3.d.c 2 9.d odd 6 1
324.3.d.c 2 36.h even 6 1
576.3.o.a 2 8.d odd 2 1
576.3.o.a 2 72.n even 6 1
576.3.o.b 2 8.b even 2 1
576.3.o.b 2 72.p odd 6 1
1728.3.o.a 2 24.h odd 2 1
1728.3.o.a 2 72.l even 6 1
1728.3.o.b 2 24.f even 2 1
1728.3.o.b 2 72.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(36, [\chi])$$:

 $$T_{5}^{2} + 4T_{5} + 16$$ T5^2 + 4*T5 + 16 $$T_{7}^{2} + 6T_{7} + 12$$ T7^2 + 6*T7 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 2)^{2}$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$T^{2} + 4T + 16$$
$7$ $$T^{2} + 6T + 12$$
$11$ $$T^{2} + 21T + 147$$
$13$ $$T^{2} - 22T + 484$$
$17$ $$(T + 11)^{2}$$
$19$ $$T^{2} + 243$$
$23$ $$T^{2} - 42T + 588$$
$29$ $$T^{2} + 34T + 1156$$
$31$ $$T^{2} - 12T + 48$$
$37$ $$(T + 16)^{2}$$
$41$ $$T^{2} + 13T + 169$$
$43$ $$T^{2} + 87T + 2523$$
$47$ $$T^{2} - 6T + 12$$
$53$ $$(T - 52)^{2}$$
$59$ $$T^{2} + 93T + 2883$$
$61$ $$T^{2} - 16T + 256$$
$67$ $$T^{2} - 201T + 13467$$
$71$ $$T^{2}$$
$73$ $$(T + 25)^{2}$$
$79$ $$T^{2} - 48T + 768$$
$83$ $$T^{2} - 60T + 1200$$
$89$ $$(T + 2)^{2}$$
$97$ $$T^{2} - 43T + 1849$$