# Properties

 Label 33.6.a.a Level $33$ Weight $6$ Character orbit 33.a Self dual yes Analytic conductor $5.293$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,6,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, 6, names="a")

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

chi := DirichletCharacter("33.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$5.29266605383$$ Analytic rank: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} - 9 q^{3} - 28 q^{4} + 46 q^{5} + 18 q^{6} + 148 q^{7} + 120 q^{8} + 81 q^{9}+O(q^{10})$$ q - 2 * q^2 - 9 * q^3 - 28 * q^4 + 46 * q^5 + 18 * q^6 + 148 * q^7 + 120 * q^8 + 81 * q^9 $$q - 2 q^{2} - 9 q^{3} - 28 q^{4} + 46 q^{5} + 18 q^{6} + 148 q^{7} + 120 q^{8} + 81 q^{9} - 92 q^{10} + 121 q^{11} + 252 q^{12} + 574 q^{13} - 296 q^{14} - 414 q^{15} + 656 q^{16} - 722 q^{17} - 162 q^{18} + 2160 q^{19} - 1288 q^{20} - 1332 q^{21} - 242 q^{22} - 2536 q^{23} - 1080 q^{24} - 1009 q^{25} - 1148 q^{26} - 729 q^{27} - 4144 q^{28} + 4650 q^{29} + 828 q^{30} + 5032 q^{31} - 5152 q^{32} - 1089 q^{33} + 1444 q^{34} + 6808 q^{35} - 2268 q^{36} + 8118 q^{37} - 4320 q^{38} - 5166 q^{39} + 5520 q^{40} - 5138 q^{41} + 2664 q^{42} + 8304 q^{43} - 3388 q^{44} + 3726 q^{45} + 5072 q^{46} + 24728 q^{47} - 5904 q^{48} + 5097 q^{49} + 2018 q^{50} + 6498 q^{51} - 16072 q^{52} - 28746 q^{53} + 1458 q^{54} + 5566 q^{55} + 17760 q^{56} - 19440 q^{57} - 9300 q^{58} - 5860 q^{59} + 11592 q^{60} - 53658 q^{61} - 10064 q^{62} + 11988 q^{63} - 10688 q^{64} + 26404 q^{65} + 2178 q^{66} + 30908 q^{67} + 20216 q^{68} + 22824 q^{69} - 13616 q^{70} - 69648 q^{71} + 9720 q^{72} - 18446 q^{73} - 16236 q^{74} + 9081 q^{75} - 60480 q^{76} + 17908 q^{77} + 10332 q^{78} - 25300 q^{79} + 30176 q^{80} + 6561 q^{81} + 10276 q^{82} - 17556 q^{83} + 37296 q^{84} - 33212 q^{85} - 16608 q^{86} - 41850 q^{87} + 14520 q^{88} + 132570 q^{89} - 7452 q^{90} + 84952 q^{91} + 71008 q^{92} - 45288 q^{93} - 49456 q^{94} + 99360 q^{95} + 46368 q^{96} + 70658 q^{97} - 10194 q^{98} + 9801 q^{99}+O(q^{100})$$ q - 2 * q^2 - 9 * q^3 - 28 * q^4 + 46 * q^5 + 18 * q^6 + 148 * q^7 + 120 * q^8 + 81 * q^9 - 92 * q^10 + 121 * q^11 + 252 * q^12 + 574 * q^13 - 296 * q^14 - 414 * q^15 + 656 * q^16 - 722 * q^17 - 162 * q^18 + 2160 * q^19 - 1288 * q^20 - 1332 * q^21 - 242 * q^22 - 2536 * q^23 - 1080 * q^24 - 1009 * q^25 - 1148 * q^26 - 729 * q^27 - 4144 * q^28 + 4650 * q^29 + 828 * q^30 + 5032 * q^31 - 5152 * q^32 - 1089 * q^33 + 1444 * q^34 + 6808 * q^35 - 2268 * q^36 + 8118 * q^37 - 4320 * q^38 - 5166 * q^39 + 5520 * q^40 - 5138 * q^41 + 2664 * q^42 + 8304 * q^43 - 3388 * q^44 + 3726 * q^45 + 5072 * q^46 + 24728 * q^47 - 5904 * q^48 + 5097 * q^49 + 2018 * q^50 + 6498 * q^51 - 16072 * q^52 - 28746 * q^53 + 1458 * q^54 + 5566 * q^55 + 17760 * q^56 - 19440 * q^57 - 9300 * q^58 - 5860 * q^59 + 11592 * q^60 - 53658 * q^61 - 10064 * q^62 + 11988 * q^63 - 10688 * q^64 + 26404 * q^65 + 2178 * q^66 + 30908 * q^67 + 20216 * q^68 + 22824 * q^69 - 13616 * q^70 - 69648 * q^71 + 9720 * q^72 - 18446 * q^73 - 16236 * q^74 + 9081 * q^75 - 60480 * q^76 + 17908 * q^77 + 10332 * q^78 - 25300 * q^79 + 30176 * q^80 + 6561 * q^81 + 10276 * q^82 - 17556 * q^83 + 37296 * q^84 - 33212 * q^85 - 16608 * q^86 - 41850 * q^87 + 14520 * q^88 + 132570 * q^89 - 7452 * q^90 + 84952 * q^91 + 71008 * q^92 - 45288 * q^93 - 49456 * q^94 + 99360 * q^95 + 46368 * q^96 + 70658 * q^97 - 10194 * q^98 + 9801 * 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
−2.00000 −9.00000 −28.0000 46.0000 18.0000 148.000 120.000 81.0000 −92.0000
 $$n$$: 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 33.6.a.a 1
3.b odd 2 1 99.6.a.b 1
4.b odd 2 1 528.6.a.i 1
5.b even 2 1 825.6.a.b 1
11.b odd 2 1 363.6.a.c 1
33.d even 2 1 1089.6.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.a 1 1.a even 1 1 trivial
99.6.a.b 1 3.b odd 2 1
363.6.a.c 1 11.b odd 2 1
528.6.a.i 1 4.b odd 2 1
825.6.a.b 1 5.b even 2 1
1089.6.a.d 1 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 2$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(33))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T + 9$$
$5$ $$T - 46$$
$7$ $$T - 148$$
$11$ $$T - 121$$
$13$ $$T - 574$$
$17$ $$T + 722$$
$19$ $$T - 2160$$
$23$ $$T + 2536$$
$29$ $$T - 4650$$
$31$ $$T - 5032$$
$37$ $$T - 8118$$
$41$ $$T + 5138$$
$43$ $$T - 8304$$
$47$ $$T - 24728$$
$53$ $$T + 28746$$
$59$ $$T + 5860$$
$61$ $$T + 53658$$
$67$ $$T - 30908$$
$71$ $$T + 69648$$
$73$ $$T + 18446$$
$79$ $$T + 25300$$
$83$ $$T + 17556$$
$89$ $$T - 132570$$
$97$ $$T - 70658$$
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