# Properties

 Label 33.5.g.a Level $33$ Weight $5$ Character orbit 33.g Analytic conductor $3.411$ Analytic rank $0$ Dimension $32$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,5,Mod(7,33)]

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([0, 7]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("33.7");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 33.g (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: $$3.41120878177$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 76 q^{4} + 36 q^{5} + 150 q^{7} + 480 q^{8} - 216 q^{9}+O(q^{10})$$ 32 * q + 76 * q^4 + 36 * q^5 + 150 * q^7 + 480 * q^8 - 216 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 76 q^{4} + 36 q^{5} + 150 q^{7} + 480 q^{8} - 216 q^{9} - 246 q^{11} + 360 q^{12} - 510 q^{13} - 1290 q^{14} - 468 q^{15} - 232 q^{16} + 2490 q^{17} + 810 q^{18} - 582 q^{20} - 510 q^{22} - 2196 q^{23} - 3510 q^{24} - 370 q^{25} - 5226 q^{26} + 4310 q^{28} + 960 q^{29} + 3780 q^{30} + 1658 q^{31} - 1008 q^{33} - 2320 q^{34} + 1920 q^{35} + 2052 q^{36} + 1374 q^{37} + 12054 q^{38} + 11070 q^{40} + 9360 q^{41} - 2844 q^{42} - 4350 q^{44} + 972 q^{45} - 12950 q^{46} - 3450 q^{47} + 4464 q^{48} - 11838 q^{49} - 11550 q^{50} + 5580 q^{51} - 19250 q^{52} - 2790 q^{53} + 12356 q^{55} - 5604 q^{56} - 6300 q^{57} + 9486 q^{58} + 2682 q^{59} - 19548 q^{60} - 17190 q^{61} - 39360 q^{62} - 4050 q^{63} + 16248 q^{64} + 2520 q^{66} + 2796 q^{67} + 68160 q^{68} + 4014 q^{69} + 18188 q^{70} + 132 q^{71} - 12150 q^{72} - 21790 q^{73} - 2130 q^{74} + 12168 q^{75} + 4542 q^{77} + 53640 q^{78} + 12270 q^{79} + 32346 q^{80} - 5832 q^{81} + 29442 q^{82} + 35430 q^{83} + 28620 q^{84} - 11990 q^{85} - 49416 q^{86} + 1176 q^{88} - 38748 q^{89} - 10260 q^{90} - 51858 q^{91} - 25590 q^{92} - 32616 q^{93} - 34510 q^{94} - 71670 q^{95} - 49950 q^{96} + 30306 q^{97} - 13932 q^{99}+O(q^{100})$$ 32 * q + 76 * q^4 + 36 * q^5 + 150 * q^7 + 480 * q^8 - 216 * q^9 - 246 * q^11 + 360 * q^12 - 510 * q^13 - 1290 * q^14 - 468 * q^15 - 232 * q^16 + 2490 * q^17 + 810 * q^18 - 582 * q^20 - 510 * q^22 - 2196 * q^23 - 3510 * q^24 - 370 * q^25 - 5226 * q^26 + 4310 * q^28 + 960 * q^29 + 3780 * q^30 + 1658 * q^31 - 1008 * q^33 - 2320 * q^34 + 1920 * q^35 + 2052 * q^36 + 1374 * q^37 + 12054 * q^38 + 11070 * q^40 + 9360 * q^41 - 2844 * q^42 - 4350 * q^44 + 972 * q^45 - 12950 * q^46 - 3450 * q^47 + 4464 * q^48 - 11838 * q^49 - 11550 * q^50 + 5580 * q^51 - 19250 * q^52 - 2790 * q^53 + 12356 * q^55 - 5604 * q^56 - 6300 * q^57 + 9486 * q^58 + 2682 * q^59 - 19548 * q^60 - 17190 * q^61 - 39360 * q^62 - 4050 * q^63 + 16248 * q^64 + 2520 * q^66 + 2796 * q^67 + 68160 * q^68 + 4014 * q^69 + 18188 * q^70 + 132 * q^71 - 12150 * q^72 - 21790 * q^73 - 2130 * q^74 + 12168 * q^75 + 4542 * q^77 + 53640 * q^78 + 12270 * q^79 + 32346 * q^80 - 5832 * q^81 + 29442 * q^82 + 35430 * q^83 + 28620 * q^84 - 11990 * q^85 - 49416 * q^86 + 1176 * q^88 - 38748 * q^89 - 10260 * q^90 - 51858 * q^91 - 25590 * q^92 - 32616 * q^93 - 34510 * q^94 - 71670 * q^95 - 49950 * q^96 + 30306 * q^97 - 13932 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −4.37135 + 6.01665i −1.60570 + 4.94183i −12.1471 37.3849i −24.8968 + 18.0886i −22.7142 31.2634i 40.6127 13.1959i 164.863 + 53.5673i −21.8435 15.8702i 228.867i
7.2 −4.20830 + 5.79223i 1.60570 4.94183i −10.8958 33.5339i 33.2604 24.1651i 21.8670 + 30.0973i −32.2093 + 10.4654i 131.142 + 42.6108i −21.8435 15.8702i 294.346i
7.3 −2.44632 + 3.36708i 1.60570 4.94183i −0.408428 1.25701i −26.0050 + 18.8937i 12.7115 + 17.4958i −15.8191 + 5.13994i −58.1002 18.8779i −21.8435 15.8702i 133.781i
7.4 −1.95593 + 2.69210i −1.60570 + 4.94183i 1.52251 + 4.68579i 6.63359 4.81958i −10.1633 13.9886i −71.1959 + 23.1329i −66.2286 21.5190i −21.8435 15.8702i 27.2851i
7.5 −0.0886329 + 0.121993i 1.60570 4.94183i 4.93725 + 15.1953i 11.7622 8.54577i 0.460550 + 0.633892i 66.7007 21.6724i −4.58589 1.49005i −21.8435 15.8702i 2.19234i
7.6 1.01888 1.40237i −1.60570 + 4.94183i 4.01574 + 12.3592i −32.0081 + 23.2552i 5.29428 + 7.28695i 27.3116 8.87408i 47.8012 + 15.5316i −21.8435 15.8702i 68.5817i
7.7 1.95429 2.68985i −1.60570 + 4.94183i 1.52822 + 4.70338i 35.4362 25.7459i 10.1548 + 13.9769i 8.04612 2.61434i 66.2318 + 21.5200i −21.8435 15.8702i 145.633i
7.8 3.38915 4.66477i 1.60570 4.94183i −5.32944 16.4023i 4.81744 3.50008i −17.6106 24.2389i −13.8978 + 4.51567i −6.83522 2.22090i −21.8435 15.8702i 34.3346i
13.1 −7.29601 2.37062i 4.20378 3.05422i 34.6676 + 25.1875i 2.18122 + 6.71310i −37.9112 + 12.3181i 28.6722 39.4639i −121.078 166.650i 8.34346 25.6785i 54.1497i
13.2 −4.49285 1.45982i −4.20378 + 3.05422i 5.11038 + 3.71291i 5.35863 + 16.4922i 23.3455 7.58543i 48.2339 66.3883i 26.8877 + 37.0078i 8.34346 25.6785i 81.9194i
13.3 −0.173259 0.0562952i 4.20378 3.05422i −12.9174 9.38506i −6.40673 19.7179i −0.900279 + 0.292519i 12.6762 17.4473i 3.42300 + 4.71136i 8.34346 25.6785i 3.77697i
13.4 0.619915 + 0.201423i −4.20378 + 3.05422i −12.6005 9.15483i 12.5302 + 38.5641i −3.22117 + 1.04662i −44.8315 + 61.7053i −12.0973 16.6506i 8.34346 25.6785i 26.4303i
13.5 0.743344 + 0.241527i −4.20378 + 3.05422i −12.4500 9.04549i −9.71709 29.9061i −3.86253 + 1.25501i 16.8776 23.2301i −14.4205 19.8482i 8.34346 25.6785i 24.5775i
13.6 4.40557 + 1.43146i 4.20378 3.05422i 4.41572 + 3.20821i 12.7557 + 39.2579i 22.8920 7.43807i 37.7759 51.9941i −28.7033 39.5067i 8.34346 25.6785i 191.213i
13.7 6.41780 + 2.08527i 4.20378 3.05422i 23.8955 + 17.3611i −9.80980 30.1915i 33.3478 10.8354i −46.3989 + 63.8626i 53.6911 + 73.8995i 8.34346 25.6785i 214.219i
13.8 6.48369 + 2.10668i −4.20378 + 3.05422i 24.6559 + 17.9136i 2.10789 + 6.48741i −33.6903 + 10.9466i 12.4454 17.1296i 58.0090 + 79.8425i 8.34346 25.6785i 46.5030i
19.1 −4.37135 6.01665i −1.60570 4.94183i −12.1471 + 37.3849i −24.8968 18.0886i −22.7142 + 31.2634i 40.6127 + 13.1959i 164.863 53.5673i −21.8435 + 15.8702i 228.867i
19.2 −4.20830 5.79223i 1.60570 + 4.94183i −10.8958 + 33.5339i 33.2604 + 24.1651i 21.8670 30.0973i −32.2093 10.4654i 131.142 42.6108i −21.8435 + 15.8702i 294.346i
19.3 −2.44632 3.36708i 1.60570 + 4.94183i −0.408428 + 1.25701i −26.0050 18.8937i 12.7115 17.4958i −15.8191 5.13994i −58.1002 + 18.8779i −21.8435 + 15.8702i 133.781i
19.4 −1.95593 2.69210i −1.60570 4.94183i 1.52251 4.68579i 6.63359 + 4.81958i −10.1633 + 13.9886i −71.1959 23.1329i −66.2286 + 21.5190i −21.8435 + 15.8702i 27.2851i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.5.g.a 32
3.b odd 2 1 99.5.k.c 32
11.c even 5 1 363.5.c.e 32
11.d odd 10 1 inner 33.5.g.a 32
11.d odd 10 1 363.5.c.e 32
33.f even 10 1 99.5.k.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.5.g.a 32 1.a even 1 1 trivial
33.5.g.a 32 11.d odd 10 1 inner
99.5.k.c 32 3.b odd 2 1
99.5.k.c 32 33.f even 10 1
363.5.c.e 32 11.c even 5 1
363.5.c.e 32 11.d odd 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(33, [\chi])$$.