# Properties

 Label 33.8.f.a Level $33$ Weight $8$ Character orbit 33.f Analytic conductor $10.309$ Analytic rank $0$ Dimension $104$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,8,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, 8, names="a")

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

chi := DirichletCharacter("33.2");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$8$$ 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: $$10.3087058410$$ Analytic rank: $$0$$ Dimension: $$104$$ Relative dimension: $$26$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$104 q - 42 q^{3} - 1542 q^{4} - 1725 q^{6} - 10 q^{7} + 550 q^{9}+O(q^{10})$$ 104 * q - 42 * q^3 - 1542 * q^4 - 1725 * q^6 - 10 * q^7 + 550 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$104 q - 42 q^{3} - 1542 q^{4} - 1725 q^{6} - 10 q^{7} + 550 q^{9} + 12942 q^{12} - 10 q^{13} - 21762 q^{15} - 40042 q^{16} + 60435 q^{18} + 10460 q^{19} - 218044 q^{22} + 330875 q^{24} + 462276 q^{25} - 294096 q^{27} - 1044930 q^{28} + 243010 q^{30} + 412 q^{31} + 32138 q^{33} + 720380 q^{34} + 166067 q^{36} - 494526 q^{37} + 1075280 q^{39} + 363510 q^{40} - 1090804 q^{42} - 3105552 q^{45} + 706710 q^{46} + 3640214 q^{48} + 3722248 q^{49} + 53970 q^{51} - 3236270 q^{52} + 1380424 q^{55} - 3039180 q^{57} - 1651930 q^{58} - 12139486 q^{60} + 3921710 q^{61} + 459870 q^{63} + 3705878 q^{64} + 15158784 q^{66} - 15866932 q^{67} + 8740364 q^{69} + 14057584 q^{70} - 28129160 q^{72} - 26920980 q^{73} - 16391598 q^{75} + 43716808 q^{78} + 59786450 q^{79} - 17715062 q^{81} + 23865872 q^{82} + 71482920 q^{84} - 2864030 q^{85} - 108977170 q^{88} - 54544280 q^{90} - 56013610 q^{91} - 71700158 q^{93} + 53266890 q^{94} + 34615450 q^{96} + 71130138 q^{97} + 46642084 q^{99}+O(q^{100})$$ 104 * q - 42 * q^3 - 1542 * q^4 - 1725 * q^6 - 10 * q^7 + 550 * q^9 + 12942 * q^12 - 10 * q^13 - 21762 * q^15 - 40042 * q^16 + 60435 * q^18 + 10460 * q^19 - 218044 * q^22 + 330875 * q^24 + 462276 * q^25 - 294096 * q^27 - 1044930 * q^28 + 243010 * q^30 + 412 * q^31 + 32138 * q^33 + 720380 * q^34 + 166067 * q^36 - 494526 * q^37 + 1075280 * q^39 + 363510 * q^40 - 1090804 * q^42 - 3105552 * q^45 + 706710 * q^46 + 3640214 * q^48 + 3722248 * q^49 + 53970 * q^51 - 3236270 * q^52 + 1380424 * q^55 - 3039180 * q^57 - 1651930 * q^58 - 12139486 * q^60 + 3921710 * q^61 + 459870 * q^63 + 3705878 * q^64 + 15158784 * q^66 - 15866932 * q^67 + 8740364 * q^69 + 14057584 * q^70 - 28129160 * q^72 - 26920980 * q^73 - 16391598 * q^75 + 43716808 * q^78 + 59786450 * q^79 - 17715062 * q^81 + 23865872 * q^82 + 71482920 * q^84 - 2864030 * q^85 - 108977170 * q^88 - 54544280 * q^90 - 56013610 * q^91 - 71700158 * q^93 + 53266890 * q^94 + 34615450 * q^96 + 71130138 * q^97 + 46642084 * 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 −6.60852 + 20.3389i −0.462116 + 46.7631i −266.445 193.584i 133.323 43.3192i −948.057 318.434i 864.199 1189.47i 3483.52 2530.93i −2186.57 43.2200i 2997.91i
2.2 −6.54483 + 20.1429i 34.8194 31.2188i −259.349 188.428i −311.568 + 101.235i 400.951 + 905.686i 7.55299 10.3958i 3299.65 2397.34i 237.777 2174.04i 6938.46i
2.3 −5.89289 + 18.1365i −33.9797 32.1307i −190.651 138.516i 349.084 113.424i 782.976 426.928i −252.753 + 347.884i 1660.92 1206.73i 122.233 + 2183.58i 6999.55i
2.4 −5.37068 + 16.5293i −46.7045 + 2.38546i −140.818 102.310i −418.675 + 136.036i 211.405 784.802i 46.5852 64.1191i 647.639 470.537i 2175.62 222.823i 7651.00i
2.5 −5.00631 + 15.4078i 43.8799 + 16.1726i −108.784 79.0360i 190.716 61.9674i −468.861 + 595.129i −557.986 + 768.002i 84.7229 61.5548i 1663.89 + 1419.30i 3248.75i
2.6 −4.09750 + 12.6108i −17.5014 + 43.3670i −38.6886 28.1089i 4.24286 1.37859i −475.181 398.404i −778.147 + 1071.03i −860.103 + 624.902i −1574.40 1517.97i 59.1546i
2.7 −3.82493 + 11.7719i 1.38201 46.7449i −20.3938 14.8170i −22.0489 + 7.16411i 544.992 + 195.065i 49.3778 67.9627i −1029.34 + 747.857i −2183.18 129.204i 286.960i
2.8 −2.99455 + 9.21628i 22.8616 + 40.7964i 27.5817 + 20.0393i −419.491 + 136.301i −444.451 + 88.5320i 451.568 621.530i −1270.78 + 923.277i −1141.69 + 1865.34i 4274.31i
2.9 −2.85991 + 8.80190i 44.0288 15.7627i 34.2599 + 24.8913i 118.636 38.5472i 12.8232 + 432.617i 768.087 1057.18i −1275.45 + 926.669i 1690.07 1388.03i 1154.47i
2.10 −2.74972 + 8.46278i −41.5222 + 21.5152i 39.4965 + 28.6959i 404.353 131.382i −67.9040 410.554i 811.138 1116.44i −1272.91 + 924.821i 1261.19 1786.72i 3783.21i
2.11 −0.997925 + 3.07130i −46.7172 2.12148i 95.1172 + 69.1067i −23.9618 + 7.78566i 53.1360 141.365i −324.042 + 446.006i −641.580 + 466.135i 2178.00 + 198.219i 81.3633i
2.12 −0.334381 + 1.02912i −25.6322 39.1151i 102.607 + 74.5483i −221.683 + 72.0293i 48.8250 13.2993i 142.147 195.649i −223.083 + 162.079i −872.979 + 2005.21i 252.224i
2.13 −0.119980 + 0.369259i 44.9099 13.0422i 103.432 + 75.1479i −475.452 + 154.484i −0.572341 + 18.1482i −851.641 + 1172.18i −80.3650 + 58.3886i 1846.80 1171.44i 194.100i
2.14 0.119980 0.369259i 26.2818 38.6816i 103.432 + 75.1479i 475.452 154.484i −11.1303 14.3458i −851.641 + 1172.18i 80.3650 58.3886i −805.539 2033.24i 194.100i
2.15 0.334381 1.02912i 29.2799 + 36.4649i 102.607 + 74.5483i 221.683 72.0293i 47.3173 17.9393i 142.147 195.649i 223.083 162.079i −472.379 + 2135.38i 252.224i
2.16 0.997925 3.07130i −12.4188 + 45.0863i 95.1172 + 69.1067i 23.9618 7.78566i 126.080 + 83.1345i −324.042 + 446.006i 641.580 466.135i −1878.55 1119.83i 81.3633i
2.17 2.74972 8.46278i −33.2933 + 32.8414i 39.4965 + 28.6959i −404.353 + 131.382i 186.382 + 372.058i 811.138 1116.44i 1272.91 924.821i 29.8835 2186.80i 3783.21i
2.18 2.85991 8.80190i 28.5969 37.0030i 34.2599 + 24.8913i −118.636 + 38.5472i −243.912 357.532i 768.087 1057.18i 1275.45 926.669i −551.438 2116.34i 1154.47i
2.19 2.99455 9.21628i −31.7351 34.3495i 27.5817 + 20.0393i 419.491 136.301i −411.607 + 189.618i 451.568 621.530i 1270.78 923.277i −172.770 + 2180.16i 4274.31i
2.20 3.82493 11.7719i 44.8842 + 13.1306i −20.3938 14.8170i 22.0489 7.16411i 326.251 478.149i 49.3778 67.9627i 1029.34 747.857i 1842.17 + 1178.71i 286.960i
See next 80 embeddings (of 104 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2.26 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 33.8.f.a 104
3.b odd 2 1 inner 33.8.f.a 104
11.d odd 10 1 inner 33.8.f.a 104
33.f even 10 1 inner 33.8.f.a 104

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.f.a 104 1.a even 1 1 trivial
33.8.f.a 104 3.b odd 2 1 inner
33.8.f.a 104 11.d odd 10 1 inner
33.8.f.a 104 33.f even 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(33, [\chi])$$.