# 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

## Newspace parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 33.e (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28q - 6q^{2} - 189q^{3} + 160q^{4} + 773q^{5} - 162q^{6} + 1289q^{7} + 2956q^{8} - 5103q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q - 6q^{2} - 189q^{3} + 160q^{4} + 773q^{5} - 162q^{6} + 1289q^{7} + 2956q^{8} - 5103q^{9} - 10640q^{10} + 13209q^{11} + 45630q^{12} + 13499q^{13} - 3318q^{14} - 22734q^{15} - 113196q^{16} + 30296q^{17} + 21141q^{18} - 6858q^{19} + 76725q^{20} - 177012q^{21} - 48859q^{22} - 166984q^{23} + 44442q^{24} + 99832q^{25} + 340424q^{26} - 137781q^{27} + 657387q^{28} + 192014q^{29} - 184410q^{30} + 396068q^{31} - 303498q^{32} + 168453q^{33} - 1229124q^{34} - 725859q^{35} + 116640q^{36} + 811053q^{37} + 1548798q^{38} + 364473q^{39} - 1427263q^{40} + 1041497q^{41} + 300024q^{42} - 5265288q^{43} - 5341085q^{44} + 100602q^{45} + 1198651q^{46} + 1667505q^{47} + 730593q^{48} + 2968140q^{49} + 11247432q^{50} + 340767q^{51} + 2325763q^{52} - 1625736q^{53} - 905418q^{54} - 8338690q^{55} - 18201906q^{56} + 1546479q^{57} + 14965553q^{58} - 2587454q^{59} + 2071575q^{60} + 5801619q^{61} + 15631121q^{62} + 939681q^{63} - 3732846q^{64} - 15368174q^{65} - 235683q^{66} - 7141262q^{67} + 17394545q^{68} - 5063553q^{69} - 4329200q^{70} + 3569199q^{71} + 1199934q^{72} + 125008q^{73} - 13691530q^{74} - 11794221q^{75} - 19690428q^{76} - 6038739q^{77} + 12684438q^{78} + 17075485q^{79} + 26606436q^{80} - 3720087q^{81} + 19886007q^{82} - 4645575q^{83} + 1074519q^{84} + 12002715q^{85} - 12296287q^{86} - 12259512q^{87} - 50905990q^{88} + 9601664q^{89} + 8857350q^{90} - 14129477q^{91} - 90776021q^{92} + 10693836q^{93} + 1047679q^{94} + 58935661q^{95} + 26156304q^{96} - 22170261q^{97} + 140532178q^{98} - 6579954q^{99} + O(q^{100})$$

## 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.

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 31.7 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 $$T_{2}^{28} + \cdots$$ acting on $$S_{8}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database