# Properties

 Label 33.8.e.b Level $33$ Weight $8$ Character orbit 33.e Analytic conductor $10.309$ Analytic rank $0$ Dimension $28$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,8,Mod(4,33)]

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, 2]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("33.4");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 33.e (of order $$5$$, 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: $$28$$ Relative dimension: $$7$$ over $$\Q(\zeta_{5})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 6 q^{2} - 189 q^{3} + 160 q^{4} + 773 q^{5} - 162 q^{6} + 1289 q^{7} + 2956 q^{8} - 5103 q^{9}+O(q^{10})$$ 28 * q - 6 * q^2 - 189 * q^3 + 160 * q^4 + 773 * q^5 - 162 * q^6 + 1289 * q^7 + 2956 * q^8 - 5103 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 6 q^{2} - 189 q^{3} + 160 q^{4} + 773 q^{5} - 162 q^{6} + 1289 q^{7} + 2956 q^{8} - 5103 q^{9} - 10640 q^{10} + 13209 q^{11} + 45630 q^{12} + 13499 q^{13} - 3318 q^{14} - 22734 q^{15} - 113196 q^{16} + 30296 q^{17} + 21141 q^{18} - 6858 q^{19} + 76725 q^{20} - 177012 q^{21} - 48859 q^{22} - 166984 q^{23} + 44442 q^{24} + 99832 q^{25} + 340424 q^{26} - 137781 q^{27} + 657387 q^{28} + 192014 q^{29} - 184410 q^{30} + 396068 q^{31} - 303498 q^{32} + 168453 q^{33} - 1229124 q^{34} - 725859 q^{35} + 116640 q^{36} + 811053 q^{37} + 1548798 q^{38} + 364473 q^{39} - 1427263 q^{40} + 1041497 q^{41} + 300024 q^{42} - 5265288 q^{43} - 5341085 q^{44} + 100602 q^{45} + 1198651 q^{46} + 1667505 q^{47} + 730593 q^{48} + 2968140 q^{49} + 11247432 q^{50} + 340767 q^{51} + 2325763 q^{52} - 1625736 q^{53} - 905418 q^{54} - 8338690 q^{55} - 18201906 q^{56} + 1546479 q^{57} + 14965553 q^{58} - 2587454 q^{59} + 2071575 q^{60} + 5801619 q^{61} + 15631121 q^{62} + 939681 q^{63} - 3732846 q^{64} - 15368174 q^{65} - 235683 q^{66} - 7141262 q^{67} + 17394545 q^{68} - 5063553 q^{69} - 4329200 q^{70} + 3569199 q^{71} + 1199934 q^{72} + 125008 q^{73} - 13691530 q^{74} - 11794221 q^{75} - 19690428 q^{76} - 6038739 q^{77} + 12684438 q^{78} + 17075485 q^{79} + 26606436 q^{80} - 3720087 q^{81} + 19886007 q^{82} - 4645575 q^{83} + 1074519 q^{84} + 12002715 q^{85} - 12296287 q^{86} - 12259512 q^{87} - 50905990 q^{88} + 9601664 q^{89} + 8857350 q^{90} - 14129477 q^{91} - 90776021 q^{92} + 10693836 q^{93} + 1047679 q^{94} + 58935661 q^{95} + 26156304 q^{96} - 22170261 q^{97} + 140532178 q^{98} - 6579954 q^{99}+O(q^{100})$$ 28 * q - 6 * q^2 - 189 * q^3 + 160 * q^4 + 773 * q^5 - 162 * q^6 + 1289 * q^7 + 2956 * q^8 - 5103 * q^9 - 10640 * q^10 + 13209 * q^11 + 45630 * q^12 + 13499 * q^13 - 3318 * q^14 - 22734 * q^15 - 113196 * q^16 + 30296 * q^17 + 21141 * q^18 - 6858 * q^19 + 76725 * q^20 - 177012 * q^21 - 48859 * q^22 - 166984 * q^23 + 44442 * q^24 + 99832 * q^25 + 340424 * q^26 - 137781 * q^27 + 657387 * q^28 + 192014 * q^29 - 184410 * q^30 + 396068 * q^31 - 303498 * q^32 + 168453 * q^33 - 1229124 * q^34 - 725859 * q^35 + 116640 * q^36 + 811053 * q^37 + 1548798 * q^38 + 364473 * q^39 - 1427263 * q^40 + 1041497 * q^41 + 300024 * q^42 - 5265288 * q^43 - 5341085 * q^44 + 100602 * q^45 + 1198651 * q^46 + 1667505 * q^47 + 730593 * q^48 + 2968140 * q^49 + 11247432 * q^50 + 340767 * q^51 + 2325763 * q^52 - 1625736 * q^53 - 905418 * q^54 - 8338690 * q^55 - 18201906 * q^56 + 1546479 * q^57 + 14965553 * q^58 - 2587454 * q^59 + 2071575 * q^60 + 5801619 * q^61 + 15631121 * q^62 + 939681 * q^63 - 3732846 * q^64 - 15368174 * q^65 - 235683 * q^66 - 7141262 * q^67 + 17394545 * q^68 - 5063553 * q^69 - 4329200 * q^70 + 3569199 * q^71 + 1199934 * q^72 + 125008 * q^73 - 13691530 * q^74 - 11794221 * q^75 - 19690428 * q^76 - 6038739 * q^77 + 12684438 * q^78 + 17075485 * q^79 + 26606436 * q^80 - 3720087 * q^81 + 19886007 * q^82 - 4645575 * q^83 + 1074519 * q^84 + 12002715 * q^85 - 12296287 * q^86 - 12259512 * q^87 - 50905990 * q^88 + 9601664 * q^89 + 8857350 * q^90 - 14129477 * q^91 - 90776021 * q^92 + 10693836 * q^93 + 1047679 * q^94 + 58935661 * q^95 + 26156304 * q^96 - 22170261 * q^97 + 140532178 * q^98 - 6579954 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −17.1972 12.4945i 8.34346 25.6785i 100.077 + 308.005i 190.798 138.623i −464.324 + 337.351i −503.289 1548.96i 1286.53 3959.52i −589.773 428.495i −5013.22
4.2 −12.1293 8.81245i 8.34346 25.6785i 29.9064 + 92.0423i −3.10808 + 2.25815i −327.491 + 237.936i 300.631 + 925.246i −144.646 + 445.175i −589.773 428.495i 57.5987
4.3 −3.93160 2.85647i 8.34346 25.6785i −32.2561 99.2742i −221.480 + 160.915i −106.153 + 77.1248i −146.721 451.561i −348.978 + 1074.04i −589.773 428.495i 1330.42
4.4 −3.24396 2.35687i 8.34346 25.6785i −34.5858 106.444i 354.050 257.233i −87.5868 + 63.6355i 116.729 + 359.256i −297.283 + 914.942i −589.773 428.495i −1754.79
4.5 7.82727 + 5.68684i 8.34346 25.6785i −10.6283 32.7104i −100.934 + 73.3327i 211.336 153.545i −177.127 545.140i 485.517 1494.27i −589.773 428.495i −1207.07
4.6 14.6841 + 10.6686i 8.34346 25.6785i 62.2484 + 191.581i 425.412 309.080i 396.469 288.052i −184.854 568.921i −411.912 + 1267.74i −589.773 428.495i 9544.22
4.7 16.9629 + 12.3242i 8.34346 25.6785i 96.2976 + 296.374i −380.493 + 276.444i 457.997 332.754i 39.7826 + 122.438i −1189.75 + 3661.68i −589.773 428.495i −9861.22
16.1 −6.47025 19.9134i −21.8435 15.8702i −251.124 + 182.452i −11.5844 + 35.6531i −174.697 + 537.661i 80.4609 58.4583i 3089.84 + 2244.90i 225.273 + 693.320i 784.928
16.2 −4.06039 12.4966i −21.8435 15.8702i −36.1242 + 26.2458i 158.380 487.442i −109.631 + 337.408i 846.952 615.347i −886.010 643.724i 225.273 + 693.320i −6734.46
16.3 −2.99223 9.20914i −21.8435 15.8702i 27.6994 20.1248i −60.2993 + 185.582i −80.7902 + 248.647i −765.654 + 556.280i −1270.94 923.389i 225.273 + 693.320i 1889.48
16.4 −0.970201 2.98597i −21.8435 15.8702i 95.5794 69.4425i −100.399 + 308.998i −26.1954 + 80.6212i 1340.18 973.701i −625.207 454.240i 225.273 + 693.320i 1020.07
16.5 0.813458 + 2.50357i −21.8435 15.8702i 97.9481 71.1634i 78.7070 242.235i 21.9634 67.5963i −1135.08 + 824.681i 530.435 + 385.384i 225.273 + 693.320i 670.476
16.6 2.75376 + 8.47521i −21.8435 15.8702i 39.3082 28.5591i −42.2440 + 130.014i 74.3516 228.831i 29.1631 21.1883i 1273.10 + 924.960i 225.273 + 693.320i −1218.22
16.7 4.95371 + 15.2460i −21.8435 15.8702i −104.346 + 75.8118i 99.6953 306.831i 133.750 411.641i 803.319 583.645i −12.6937 9.22251i 225.273 + 693.320i 5171.79
25.1 −17.1972 + 12.4945i 8.34346 + 25.6785i 100.077 308.005i 190.798 + 138.623i −464.324 337.351i −503.289 + 1548.96i 1286.53 + 3959.52i −589.773 + 428.495i −5013.22
25.2 −12.1293 + 8.81245i 8.34346 + 25.6785i 29.9064 92.0423i −3.10808 2.25815i −327.491 237.936i 300.631 925.246i −144.646 445.175i −589.773 + 428.495i 57.5987
25.3 −3.93160 + 2.85647i 8.34346 + 25.6785i −32.2561 + 99.2742i −221.480 160.915i −106.153 77.1248i −146.721 + 451.561i −348.978 1074.04i −589.773 + 428.495i 1330.42
25.4 −3.24396 + 2.35687i 8.34346 + 25.6785i −34.5858 + 106.444i 354.050 + 257.233i −87.5868 63.6355i 116.729 359.256i −297.283 914.942i −589.773 + 428.495i −1754.79
25.5 7.82727 5.68684i 8.34346 + 25.6785i −10.6283 + 32.7104i −100.934 73.3327i 211.336 + 153.545i −177.127 + 545.140i 485.517 + 1494.27i −589.773 + 428.495i −1207.07
25.6 14.6841 10.6686i 8.34346 + 25.6785i 62.2484 191.581i 425.412 + 309.080i 396.469 + 288.052i −184.854 + 568.921i −411.912 1267.74i −589.773 + 428.495i 9544.22
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.7 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.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.8.e.b 28
3.b odd 2 1 99.8.f.b 28
11.c even 5 1 inner 33.8.e.b 28
11.c even 5 1 363.8.a.o 14
11.d odd 10 1 363.8.a.r 14
33.h odd 10 1 99.8.f.b 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.e.b 28 1.a even 1 1 trivial
33.8.e.b 28 11.c even 5 1 inner
99.8.f.b 28 3.b odd 2 1
99.8.f.b 28 33.h odd 10 1
363.8.a.o 14 11.c even 5 1
363.8.a.r 14 11.d odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{28} + 6 T_{2}^{27} + 386 T_{2}^{26} - 1216 T_{2}^{25} + 235753 T_{2}^{24} + 1933616 T_{2}^{23} + 138235563 T_{2}^{22} + 251006875 T_{2}^{21} + 44479969025 T_{2}^{20} + \cdots + 51\!\cdots\!00$$ acting on $$S_{8}^{\mathrm{new}}(33, [\chi])$$.