# Properties

 Label 33.4.a.a 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)

 $$f(q)$$ $$=$$ $$q - 5 q^{2} + 3 q^{3} + 17 q^{4} - 14 q^{5} - 15 q^{6} - 32 q^{7} - 45 q^{8} + 9 q^{9}+O(q^{10})$$ q - 5 * q^2 + 3 * q^3 + 17 * q^4 - 14 * q^5 - 15 * q^6 - 32 * q^7 - 45 * q^8 + 9 * q^9 $$q - 5 q^{2} + 3 q^{3} + 17 q^{4} - 14 q^{5} - 15 q^{6} - 32 q^{7} - 45 q^{8} + 9 q^{9} + 70 q^{10} - 11 q^{11} + 51 q^{12} - 38 q^{13} + 160 q^{14} - 42 q^{15} + 89 q^{16} - 2 q^{17} - 45 q^{18} + 72 q^{19} - 238 q^{20} - 96 q^{21} + 55 q^{22} + 68 q^{23} - 135 q^{24} + 71 q^{25} + 190 q^{26} + 27 q^{27} - 544 q^{28} - 54 q^{29} + 210 q^{30} - 152 q^{31} - 85 q^{32} - 33 q^{33} + 10 q^{34} + 448 q^{35} + 153 q^{36} + 174 q^{37} - 360 q^{38} - 114 q^{39} + 630 q^{40} + 94 q^{41} + 480 q^{42} - 528 q^{43} - 187 q^{44} - 126 q^{45} - 340 q^{46} - 340 q^{47} + 267 q^{48} + 681 q^{49} - 355 q^{50} - 6 q^{51} - 646 q^{52} - 438 q^{53} - 135 q^{54} + 154 q^{55} + 1440 q^{56} + 216 q^{57} + 270 q^{58} + 20 q^{59} - 714 q^{60} + 570 q^{61} + 760 q^{62} - 288 q^{63} - 287 q^{64} + 532 q^{65} + 165 q^{66} - 460 q^{67} - 34 q^{68} + 204 q^{69} - 2240 q^{70} - 1092 q^{71} - 405 q^{72} + 562 q^{73} - 870 q^{74} + 213 q^{75} + 1224 q^{76} + 352 q^{77} + 570 q^{78} - 16 q^{79} - 1246 q^{80} + 81 q^{81} - 470 q^{82} + 372 q^{83} - 1632 q^{84} + 28 q^{85} + 2640 q^{86} - 162 q^{87} + 495 q^{88} - 966 q^{89} + 630 q^{90} + 1216 q^{91} + 1156 q^{92} - 456 q^{93} + 1700 q^{94} - 1008 q^{95} - 255 q^{96} - 526 q^{97} - 3405 q^{98} - 99 q^{99}+O(q^{100})$$ q - 5 * q^2 + 3 * q^3 + 17 * q^4 - 14 * q^5 - 15 * q^6 - 32 * q^7 - 45 * q^8 + 9 * q^9 + 70 * q^10 - 11 * q^11 + 51 * q^12 - 38 * q^13 + 160 * q^14 - 42 * q^15 + 89 * q^16 - 2 * q^17 - 45 * q^18 + 72 * q^19 - 238 * q^20 - 96 * q^21 + 55 * q^22 + 68 * q^23 - 135 * q^24 + 71 * q^25 + 190 * q^26 + 27 * q^27 - 544 * q^28 - 54 * q^29 + 210 * q^30 - 152 * q^31 - 85 * q^32 - 33 * q^33 + 10 * q^34 + 448 * q^35 + 153 * q^36 + 174 * q^37 - 360 * q^38 - 114 * q^39 + 630 * q^40 + 94 * q^41 + 480 * q^42 - 528 * q^43 - 187 * q^44 - 126 * q^45 - 340 * q^46 - 340 * q^47 + 267 * q^48 + 681 * q^49 - 355 * q^50 - 6 * q^51 - 646 * q^52 - 438 * q^53 - 135 * q^54 + 154 * q^55 + 1440 * q^56 + 216 * q^57 + 270 * q^58 + 20 * q^59 - 714 * q^60 + 570 * q^61 + 760 * q^62 - 288 * q^63 - 287 * q^64 + 532 * q^65 + 165 * q^66 - 460 * q^67 - 34 * q^68 + 204 * q^69 - 2240 * q^70 - 1092 * q^71 - 405 * q^72 + 562 * q^73 - 870 * q^74 + 213 * q^75 + 1224 * q^76 + 352 * q^77 + 570 * q^78 - 16 * q^79 - 1246 * q^80 + 81 * q^81 - 470 * q^82 + 372 * q^83 - 1632 * q^84 + 28 * q^85 + 2640 * q^86 - 162 * q^87 + 495 * q^88 - 966 * q^89 + 630 * q^90 + 1216 * q^91 + 1156 * q^92 - 456 * q^93 + 1700 * q^94 - 1008 * q^95 - 255 * q^96 - 526 * q^97 - 3405 * 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
−5.00000 3.00000 17.0000 −14.0000 −15.0000 −32.0000 −45.0000 9.00000 70.0000
 $$n$$: 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.a 1
3.b odd 2 1 99.4.a.b 1
4.b odd 2 1 528.4.a.a 1
5.b even 2 1 825.4.a.i 1
5.c odd 4 2 825.4.c.a 2
7.b odd 2 1 1617.4.a.a 1
8.b even 2 1 2112.4.a.l 1
8.d odd 2 1 2112.4.a.y 1
11.b odd 2 1 363.4.a.h 1
12.b even 2 1 1584.4.a.t 1
15.d odd 2 1 2475.4.a.b 1
33.d even 2 1 1089.4.a.a 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 5$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(33))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 5$$
$3$ $$T - 3$$
$5$ $$T + 14$$
$7$ $$T + 32$$
$11$ $$T + 11$$
$13$ $$T + 38$$
$17$ $$T + 2$$
$19$ $$T - 72$$
$23$ $$T - 68$$
$29$ $$T + 54$$
$31$ $$T + 152$$
$37$ $$T - 174$$
$41$ $$T - 94$$
$43$ $$T + 528$$
$47$ $$T + 340$$
$53$ $$T + 438$$
$59$ $$T - 20$$
$61$ $$T - 570$$
$67$ $$T + 460$$
$71$ $$T + 1092$$
$73$ $$T - 562$$
$79$ $$T + 16$$
$83$ $$T - 372$$
$89$ $$T + 966$$
$97$ $$T + 526$$