# Properties

 Label 33.4.a.b Level $33$ Weight $4$ Character orbit 33.a Self dual yes Analytic conductor $1.947$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,4,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, 4, names="a")

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

chi := DirichletCharacter("33.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$1.94706303019$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - 3 q^{3} - 7 q^{4} - 4 q^{5} + 3 q^{6} - 26 q^{7} + 15 q^{8} + 9 q^{9}+O(q^{10})$$ q - q^2 - 3 * q^3 - 7 * q^4 - 4 * q^5 + 3 * q^6 - 26 * q^7 + 15 * q^8 + 9 * q^9 $$q - q^{2} - 3 q^{3} - 7 q^{4} - 4 q^{5} + 3 q^{6} - 26 q^{7} + 15 q^{8} + 9 q^{9} + 4 q^{10} + 11 q^{11} + 21 q^{12} - 32 q^{13} + 26 q^{14} + 12 q^{15} + 41 q^{16} + 74 q^{17} - 9 q^{18} - 60 q^{19} + 28 q^{20} + 78 q^{21} - 11 q^{22} - 182 q^{23} - 45 q^{24} - 109 q^{25} + 32 q^{26} - 27 q^{27} + 182 q^{28} - 90 q^{29} - 12 q^{30} - 8 q^{31} - 161 q^{32} - 33 q^{33} - 74 q^{34} + 104 q^{35} - 63 q^{36} - 66 q^{37} + 60 q^{38} + 96 q^{39} - 60 q^{40} + 422 q^{41} - 78 q^{42} + 408 q^{43} - 77 q^{44} - 36 q^{45} + 182 q^{46} - 506 q^{47} - 123 q^{48} + 333 q^{49} + 109 q^{50} - 222 q^{51} + 224 q^{52} + 348 q^{53} + 27 q^{54} - 44 q^{55} - 390 q^{56} + 180 q^{57} + 90 q^{58} - 200 q^{59} - 84 q^{60} + 132 q^{61} + 8 q^{62} - 234 q^{63} - 167 q^{64} + 128 q^{65} + 33 q^{66} - 1036 q^{67} - 518 q^{68} + 546 q^{69} - 104 q^{70} + 762 q^{71} + 135 q^{72} - 542 q^{73} + 66 q^{74} + 327 q^{75} + 420 q^{76} - 286 q^{77} - 96 q^{78} - 550 q^{79} - 164 q^{80} + 81 q^{81} - 422 q^{82} - 132 q^{83} - 546 q^{84} - 296 q^{85} - 408 q^{86} + 270 q^{87} + 165 q^{88} + 570 q^{89} + 36 q^{90} + 832 q^{91} + 1274 q^{92} + 24 q^{93} + 506 q^{94} + 240 q^{95} + 483 q^{96} + 14 q^{97} - 333 q^{98} + 99 q^{99}+O(q^{100})$$ q - q^2 - 3 * q^3 - 7 * q^4 - 4 * q^5 + 3 * q^6 - 26 * q^7 + 15 * q^8 + 9 * q^9 + 4 * q^10 + 11 * q^11 + 21 * q^12 - 32 * q^13 + 26 * q^14 + 12 * q^15 + 41 * q^16 + 74 * q^17 - 9 * q^18 - 60 * q^19 + 28 * q^20 + 78 * q^21 - 11 * q^22 - 182 * q^23 - 45 * q^24 - 109 * q^25 + 32 * q^26 - 27 * q^27 + 182 * q^28 - 90 * q^29 - 12 * q^30 - 8 * q^31 - 161 * q^32 - 33 * q^33 - 74 * q^34 + 104 * q^35 - 63 * q^36 - 66 * q^37 + 60 * q^38 + 96 * q^39 - 60 * q^40 + 422 * q^41 - 78 * q^42 + 408 * q^43 - 77 * q^44 - 36 * q^45 + 182 * q^46 - 506 * q^47 - 123 * q^48 + 333 * q^49 + 109 * q^50 - 222 * q^51 + 224 * q^52 + 348 * q^53 + 27 * q^54 - 44 * q^55 - 390 * q^56 + 180 * q^57 + 90 * q^58 - 200 * q^59 - 84 * q^60 + 132 * q^61 + 8 * q^62 - 234 * q^63 - 167 * q^64 + 128 * q^65 + 33 * q^66 - 1036 * q^67 - 518 * q^68 + 546 * q^69 - 104 * q^70 + 762 * q^71 + 135 * q^72 - 542 * q^73 + 66 * q^74 + 327 * q^75 + 420 * q^76 - 286 * q^77 - 96 * q^78 - 550 * q^79 - 164 * q^80 + 81 * q^81 - 422 * q^82 - 132 * q^83 - 546 * q^84 - 296 * q^85 - 408 * q^86 + 270 * q^87 + 165 * q^88 + 570 * q^89 + 36 * q^90 + 832 * q^91 + 1274 * q^92 + 24 * q^93 + 506 * q^94 + 240 * q^95 + 483 * q^96 + 14 * q^97 - 333 * q^98 + 99 * 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
 0
−1.00000 −3.00000 −7.00000 −4.00000 3.00000 −26.0000 15.0000 9.00000 4.00000
 $$n$$: 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.b 1
3.b odd 2 1 99.4.a.a 1
4.b odd 2 1 528.4.a.h 1
5.b even 2 1 825.4.a.f 1
5.c odd 4 2 825.4.c.f 2
7.b odd 2 1 1617.4.a.d 1
8.b even 2 1 2112.4.a.u 1
8.d odd 2 1 2112.4.a.h 1
11.b odd 2 1 363.4.a.d 1
12.b even 2 1 1584.4.a.l 1
15.d odd 2 1 2475.4.a.e 1
33.d even 2 1 1089.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.b 1 1.a even 1 1 trivial
99.4.a.a 1 3.b odd 2 1
363.4.a.d 1 11.b odd 2 1
528.4.a.h 1 4.b odd 2 1
825.4.a.f 1 5.b even 2 1
825.4.c.f 2 5.c odd 4 2
1089.4.a.e 1 33.d even 2 1
1584.4.a.l 1 12.b even 2 1
1617.4.a.d 1 7.b odd 2 1
2112.4.a.h 1 8.d odd 2 1
2112.4.a.u 1 8.b even 2 1
2475.4.a.e 1 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(33))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 3$$
$5$ $$T + 4$$
$7$ $$T + 26$$
$11$ $$T - 11$$
$13$ $$T + 32$$
$17$ $$T - 74$$
$19$ $$T + 60$$
$23$ $$T + 182$$
$29$ $$T + 90$$
$31$ $$T + 8$$
$37$ $$T + 66$$
$41$ $$T - 422$$
$43$ $$T - 408$$
$47$ $$T + 506$$
$53$ $$T - 348$$
$59$ $$T + 200$$
$61$ $$T - 132$$
$67$ $$T + 1036$$
$71$ $$T - 762$$
$73$ $$T + 542$$
$79$ $$T + 550$$
$83$ $$T + 132$$
$89$ $$T - 570$$
$97$ $$T - 14$$