# Properties

 Label 33.6.f.a Level $33$ Weight $6$ Character orbit 33.f Analytic conductor $5.293$ Analytic rank $0$ Dimension $72$ CM no Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,6,Mod(2,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("33.2");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 33.f (of order $$10$$, degree $$4$$, 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: $$5.29266605383$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$18$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$72 q + 18 q^{3} - 262 q^{4} + 15 q^{6} - 10 q^{7} + 292 q^{9}+O(q^{10})$$ 72 * q + 18 * q^3 - 262 * q^4 + 15 * q^6 - 10 * q^7 + 292 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$72 q + 18 q^{3} - 262 q^{4} + 15 q^{6} - 10 q^{7} + 292 q^{9} + 1854 q^{12} - 10 q^{13} - 762 q^{15} - 10122 q^{16} + 4815 q^{18} + 4460 q^{19} + 4628 q^{22} - 805 q^{24} + 13708 q^{25} + 6108 q^{27} - 28130 q^{28} - 15470 q^{30} + 4340 q^{31} - 508 q^{33} + 18732 q^{34} - 56461 q^{36} + 978 q^{37} + 23360 q^{39} + 69750 q^{40} + 60788 q^{42} - 31356 q^{45} - 52090 q^{46} + 4238 q^{48} - 58448 q^{49} - 178950 q^{51} - 14190 q^{52} + 86600 q^{55} + 266190 q^{57} + 137102 q^{58} + 284090 q^{60} - 77890 q^{61} - 120330 q^{63} - 379114 q^{64} - 323304 q^{66} + 42668 q^{67} - 271816 q^{69} + 87176 q^{70} + 343960 q^{72} + 116440 q^{73} + 326202 q^{75} + 155512 q^{78} - 350590 q^{79} - 208088 q^{81} - 606424 q^{82} - 220680 q^{84} + 665610 q^{85} + 1152974 q^{88} + 293440 q^{90} + 621014 q^{91} + 478456 q^{93} - 521270 q^{94} - 1246430 q^{96} - 1030446 q^{97} - 590000 q^{99}+O(q^{100})$$ 72 * q + 18 * q^3 - 262 * q^4 + 15 * q^6 - 10 * q^7 + 292 * q^9 + 1854 * q^12 - 10 * q^13 - 762 * q^15 - 10122 * q^16 + 4815 * q^18 + 4460 * q^19 + 4628 * q^22 - 805 * q^24 + 13708 * q^25 + 6108 * q^27 - 28130 * q^28 - 15470 * q^30 + 4340 * q^31 - 508 * q^33 + 18732 * q^34 - 56461 * q^36 + 978 * q^37 + 23360 * q^39 + 69750 * q^40 + 60788 * q^42 - 31356 * q^45 - 52090 * q^46 + 4238 * q^48 - 58448 * q^49 - 178950 * q^51 - 14190 * q^52 + 86600 * q^55 + 266190 * q^57 + 137102 * q^58 + 284090 * q^60 - 77890 * q^61 - 120330 * q^63 - 379114 * q^64 - 323304 * q^66 + 42668 * q^67 - 271816 * q^69 + 87176 * q^70 + 343960 * q^72 + 116440 * q^73 + 326202 * q^75 + 155512 * q^78 - 350590 * q^79 - 208088 * q^81 - 606424 * q^82 - 220680 * q^84 + 665610 * q^85 + 1152974 * q^88 + 293440 * q^90 + 621014 * q^91 + 478456 * q^93 - 521270 * q^94 - 1246430 * q^96 - 1030446 * q^97 - 590000 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −3.08173 + 9.48458i −12.5347 + 9.26724i −54.5716 39.6486i 16.3931 5.32643i −49.2674 147.445i −74.1699 + 102.086i 286.047 207.825i 71.2363 232.324i 171.896i
2.2 −2.95379 + 9.09083i 10.1153 11.8609i −48.0298 34.8957i 91.6682 29.7848i 77.9468 + 126.991i 70.5766 97.1403i 211.641 153.766i −38.3609 239.953i 921.319i
2.3 −2.81967 + 8.67805i 12.9564 + 8.66789i −41.4695 30.1293i −53.2208 + 17.2925i −111.753 + 87.9955i −14.1075 + 19.4173i 142.170 103.293i 92.7354 + 224.609i 510.612i
2.4 −2.32473 + 7.15479i −12.7649 8.94746i −19.8981 14.4568i −28.0945 + 9.12845i 93.6922 70.5298i 146.291 201.352i −45.0663 + 32.7426i 82.8858 + 228.427i 222.231i
2.5 −1.97194 + 6.06902i 4.72506 14.8551i −7.05585 5.12637i −59.6930 + 19.3954i 80.8382 + 57.9699i −97.0552 + 133.585i −120.178 + 87.3143i −198.348 140.382i 400.525i
2.6 −1.36557 + 4.20281i 6.31381 + 14.2526i 10.0898 + 7.33064i 98.1154 31.8796i −68.5228 + 7.07275i −5.71539 + 7.86655i −158.991 + 115.514i −163.272 + 179.976i 455.894i
2.7 −1.12695 + 3.46841i −12.9075 8.74056i 15.1287 + 10.9916i 54.0225 17.5530i 44.8620 34.9182i −123.528 + 170.022i −149.586 + 108.681i 90.2051 + 225.637i 207.153i
2.8 −1.01916 + 3.13665i −8.08821 + 13.3259i 17.0887 + 12.4156i −40.8503 + 13.2731i −33.5556 38.9511i 39.2343 54.0014i −141.742 + 102.981i −112.162 215.566i 141.660i
2.9 −0.489986 + 1.50802i 15.5884 + 0.0191829i 23.8545 + 17.3313i −16.6412 + 5.40705i −7.66704 + 23.4983i 36.5406 50.2938i −78.8739 + 57.3052i 242.999 + 0.598063i 27.7446i
2.10 0.489986 1.50802i 4.79885 14.8314i 23.8545 + 17.3313i 16.6412 5.40705i −20.0147 14.5039i 36.5406 50.2938i 78.8739 57.3052i −196.942 142.348i 27.7446i
2.11 1.01916 3.13665i −15.1731 + 3.57441i 17.0887 + 12.4156i 40.8503 13.2731i −4.25217 + 51.2356i 39.2343 54.0014i 141.742 102.981i 217.447 108.470i 141.660i
2.12 1.12695 3.46841i 4.32414 + 14.9767i 15.1287 + 10.9916i −54.0225 + 17.5530i 56.8185 + 1.88017i −123.528 + 170.022i 149.586 108.681i −205.604 + 129.523i 207.153i
2.13 1.36557 4.20281i −11.6039 10.4091i 10.0898 + 7.33064i −98.1154 + 31.8796i −59.5934 + 34.5547i −5.71539 + 7.86655i 158.991 115.514i 26.3023 + 241.572i 455.894i
2.14 1.97194 6.06902i 15.5882 + 0.0966769i −7.05585 5.12637i 59.6930 19.3954i 31.3257 94.4141i −97.0552 + 133.585i 120.178 87.3143i 242.981 + 3.01403i 400.525i
2.15 2.32473 7.15479i 4.56497 + 14.9051i −19.8981 14.4568i 28.0945 9.12845i 117.255 + 1.98889i 146.291 201.352i 45.0663 32.7426i −201.322 + 136.082i 222.231i
2.16 2.81967 8.67805i −4.23991 15.0008i −41.4695 30.1293i 53.2208 17.2925i −142.133 5.50305i −14.1075 + 19.4173i −142.170 + 103.293i −207.046 + 127.204i 510.612i
2.17 2.95379 9.09083i 14.4062 5.95502i −48.0298 34.8957i −91.6682 + 29.7848i −11.5833 148.554i 70.5766 97.1403i −211.641 + 153.766i 172.075 171.578i 921.319i
2.18 3.08173 9.48458i −12.6871 + 9.05745i −54.5716 39.6486i −16.3931 + 5.32643i 46.8079 + 148.244i −74.1699 + 102.086i −286.047 + 207.825i 78.9251 229.826i 171.896i
8.1 −8.92830 + 6.48679i −5.92302 + 14.4194i 27.7475 85.3981i 33.7792 46.4931i −40.6528 167.162i −11.4606 3.72378i 197.091 + 606.584i −172.836 170.812i 634.222i
8.2 −7.80447 + 5.67028i 14.8651 4.69358i 18.8692 58.0733i −46.3038 + 63.7318i −89.4001 + 120.920i −135.347 43.9769i 86.6348 + 266.635i 198.941 139.541i 759.949i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.6.f.a 72
3.b odd 2 1 inner 33.6.f.a 72
11.d odd 10 1 inner 33.6.f.a 72
33.f even 10 1 inner 33.6.f.a 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.f.a 72 1.a even 1 1 trivial
33.6.f.a 72 3.b odd 2 1 inner
33.6.f.a 72 11.d odd 10 1 inner
33.6.f.a 72 33.f even 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(33, [\chi])$$.