# Properties

 Label 33.8.e.a 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 22 q^{2} + 189 q^{3} - 692 q^{4} - 777 q^{5} + 594 q^{6} - 83 q^{7} + 3568 q^{8} - 5103 q^{9}+O(q^{10})$$ 28 * q - 22 * q^2 + 189 * q^3 - 692 * q^4 - 777 * q^5 + 594 * q^6 - 83 * q^7 + 3568 * q^8 - 5103 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 22 q^{2} + 189 q^{3} - 692 q^{4} - 777 q^{5} + 594 q^{6} - 83 q^{7} + 3568 q^{8} - 5103 q^{9} - 1352 q^{10} - 18851 q^{11} - 60966 q^{12} + 14635 q^{13} + 62912 q^{14} - 22896 q^{15} - 36148 q^{16} - 10338 q^{17} + 9477 q^{18} + 90342 q^{19} - 289719 q^{20} + 28836 q^{21} - 202339 q^{22} + 138508 q^{23} + 187704 q^{24} - 174068 q^{25} + 466130 q^{26} + 137781 q^{27} - 640233 q^{28} - 487290 q^{29} - 372546 q^{30} - 388352 q^{31} + 2726322 q^{32} - 265923 q^{33} - 1543708 q^{34} + 742659 q^{35} - 504468 q^{36} + 670209 q^{37} + 2282072 q^{38} - 395145 q^{39} + 4641741 q^{40} - 2721357 q^{41} + 620676 q^{42} - 199784 q^{43} - 5887437 q^{44} - 103518 q^{45} + 3389447 q^{46} + 2112053 q^{47} + 2292111 q^{48} + 3927748 q^{49} - 3663090 q^{50} + 615681 q^{51} - 4343053 q^{52} - 1743854 q^{53} - 354294 q^{54} - 8387710 q^{55} - 6307818 q^{56} + 1223721 q^{57} + 546605 q^{58} - 218820 q^{59} + 7822413 q^{60} - 743157 q^{61} + 3996183 q^{62} - 60507 q^{63} - 21725666 q^{64} + 4895794 q^{65} - 8684037 q^{66} - 3235686 q^{67} + 961671 q^{68} + 4712499 q^{69} + 31155916 q^{70} + 9848757 q^{71} - 5068008 q^{72} - 6667768 q^{73} - 31115192 q^{74} - 2544939 q^{75} - 6535748 q^{76} + 8320585 q^{77} + 20060730 q^{78} - 10121851 q^{79} + 44592526 q^{80} - 3720087 q^{81} + 24927115 q^{82} + 13288897 q^{83} - 17477019 q^{84} - 28050005 q^{85} + 20937351 q^{86} - 22226400 q^{87} + 39687014 q^{88} - 40241460 q^{89} - 9565938 q^{90} + 8733043 q^{91} + 11607651 q^{92} + 10485504 q^{93} - 7877385 q^{94} + 14371097 q^{95} + 2790126 q^{96} + 30518151 q^{97} - 16954218 q^{98} + 11659626 q^{99}+O(q^{100})$$ 28 * q - 22 * q^2 + 189 * q^3 - 692 * q^4 - 777 * q^5 + 594 * q^6 - 83 * q^7 + 3568 * q^8 - 5103 * q^9 - 1352 * q^10 - 18851 * q^11 - 60966 * q^12 + 14635 * q^13 + 62912 * q^14 - 22896 * q^15 - 36148 * q^16 - 10338 * q^17 + 9477 * q^18 + 90342 * q^19 - 289719 * q^20 + 28836 * q^21 - 202339 * q^22 + 138508 * q^23 + 187704 * q^24 - 174068 * q^25 + 466130 * q^26 + 137781 * q^27 - 640233 * q^28 - 487290 * q^29 - 372546 * q^30 - 388352 * q^31 + 2726322 * q^32 - 265923 * q^33 - 1543708 * q^34 + 742659 * q^35 - 504468 * q^36 + 670209 * q^37 + 2282072 * q^38 - 395145 * q^39 + 4641741 * q^40 - 2721357 * q^41 + 620676 * q^42 - 199784 * q^43 - 5887437 * q^44 - 103518 * q^45 + 3389447 * q^46 + 2112053 * q^47 + 2292111 * q^48 + 3927748 * q^49 - 3663090 * q^50 + 615681 * q^51 - 4343053 * q^52 - 1743854 * q^53 - 354294 * q^54 - 8387710 * q^55 - 6307818 * q^56 + 1223721 * q^57 + 546605 * q^58 - 218820 * q^59 + 7822413 * q^60 - 743157 * q^61 + 3996183 * q^62 - 60507 * q^63 - 21725666 * q^64 + 4895794 * q^65 - 8684037 * q^66 - 3235686 * q^67 + 961671 * q^68 + 4712499 * q^69 + 31155916 * q^70 + 9848757 * q^71 - 5068008 * q^72 - 6667768 * q^73 - 31115192 * q^74 - 2544939 * q^75 - 6535748 * q^76 + 8320585 * q^77 + 20060730 * q^78 - 10121851 * q^79 + 44592526 * q^80 - 3720087 * q^81 + 24927115 * q^82 + 13288897 * q^83 - 17477019 * q^84 - 28050005 * q^85 + 20937351 * q^86 - 22226400 * q^87 + 39687014 * q^88 - 40241460 * q^89 - 9565938 * q^90 + 8733043 * q^91 + 11607651 * q^92 + 10485504 * q^93 - 7877385 * q^94 + 14371097 * q^95 + 2790126 * q^96 + 30518151 * q^97 - 16954218 * q^98 + 11659626 * 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 −18.2616 13.2678i −8.34346 + 25.6785i 117.897 + 362.849i −299.083 + 217.296i 493.063 358.231i 233.965 + 720.070i 1768.39 5442.55i −589.773 428.495i 8344.78
4.2 −12.1414 8.82122i −8.34346 + 25.6785i 30.0447 + 92.4681i 158.638 115.257i 327.817 238.173i −372.740 1147.18i −142.714 + 439.227i −589.773 428.495i −2942.80
4.3 −8.43584 6.12900i −8.34346 + 25.6785i −5.95538 18.3288i −186.461 + 135.472i 227.768 165.483i −43.6776 134.426i −474.540 + 1460.49i −589.773 428.495i 2403.26
4.4 −2.14212 1.55634i −8.34346 + 25.6785i −37.3877 115.067i 121.253 88.0954i 57.8373 42.0213i 231.374 + 712.095i −203.727 + 627.009i −589.773 428.495i −396.845
4.5 6.67280 + 4.84807i −8.34346 + 25.6785i −18.5317 57.0347i 269.190 195.578i −180.166 + 130.898i −36.4929 112.314i 479.095 1474.50i −589.773 428.495i 2744.43
4.6 8.90518 + 6.46999i −8.34346 + 25.6785i −2.11276 6.50241i −328.988 + 239.024i −240.440 + 174.690i −385.790 1187.34i 458.645 1411.56i −589.773 428.495i −4476.18
4.7 15.4308 + 11.2111i −8.34346 + 25.6785i 72.8662 + 224.259i 0.205524 0.149322i −416.632 + 302.701i 242.486 + 746.295i −635.379 + 1955.49i −589.773 428.495i 4.84548
16.1 −6.06001 18.6508i 21.8435 + 15.8702i −207.575 + 150.812i −96.8862 + 298.185i 163.620 503.572i 902.102 655.415i 2039.90 + 1482.08i 225.273 + 693.320i 6148.53
16.2 −5.89947 18.1567i 21.8435 + 15.8702i −191.308 + 138.993i 133.778 411.726i 159.286 490.231i −1338.24 + 972.291i 1675.32 + 1217.19i 225.273 + 693.320i −8264.81
16.3 −1.78222 5.48510i 21.8435 + 15.8702i 76.6441 55.6852i −142.765 + 439.384i 48.1199 148.098i −506.310 + 367.856i −1039.27 755.075i 225.273 + 693.320i 2664.50
16.4 −0.907839 2.79404i 21.8435 + 15.8702i 96.5717 70.1634i 61.3322 188.761i 24.5117 75.4391i −198.219 + 144.015i −587.936 427.160i 225.273 + 693.320i −583.086
16.5 2.98016 + 9.17200i 21.8435 + 15.8702i 28.3100 20.5684i 33.3401 102.610i −80.4644 + 247.644i 1247.17 906.120i 1271.70 + 923.944i 225.273 + 693.320i 1040.50
16.6 3.92793 + 12.0889i 21.8435 + 15.8702i −27.1593 + 19.7324i −17.6507 + 54.3232i −106.054 + 326.401i −962.124 + 699.024i 971.059 + 705.516i 225.273 + 693.320i −726.039
16.7 6.71359 + 20.6623i 21.8435 + 15.8702i −278.304 + 202.200i −94.4035 + 290.544i −181.267 + 557.882i 945.004 686.586i −3796.54 2758.35i 225.273 + 693.320i −6637.09
25.1 −18.2616 + 13.2678i −8.34346 25.6785i 117.897 362.849i −299.083 217.296i 493.063 + 358.231i 233.965 720.070i 1768.39 + 5442.55i −589.773 + 428.495i 8344.78
25.2 −12.1414 + 8.82122i −8.34346 25.6785i 30.0447 92.4681i 158.638 + 115.257i 327.817 + 238.173i −372.740 + 1147.18i −142.714 439.227i −589.773 + 428.495i −2942.80
25.3 −8.43584 + 6.12900i −8.34346 25.6785i −5.95538 + 18.3288i −186.461 135.472i 227.768 + 165.483i −43.6776 + 134.426i −474.540 1460.49i −589.773 + 428.495i 2403.26
25.4 −2.14212 + 1.55634i −8.34346 25.6785i −37.3877 + 115.067i 121.253 + 88.0954i 57.8373 + 42.0213i 231.374 712.095i −203.727 627.009i −589.773 + 428.495i −396.845
25.5 6.67280 4.84807i −8.34346 25.6785i −18.5317 + 57.0347i 269.190 + 195.578i −180.166 130.898i −36.4929 + 112.314i 479.095 + 1474.50i −589.773 + 428.495i 2744.43
25.6 8.90518 6.46999i −8.34346 25.6785i −2.11276 + 6.50241i −328.988 239.024i −240.440 174.690i −385.790 + 1187.34i 458.645 + 1411.56i −589.773 + 428.495i −4476.18
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.a 28
3.b odd 2 1 99.8.f.c 28
11.c even 5 1 inner 33.8.e.a 28
11.c even 5 1 363.8.a.q 14
11.d odd 10 1 363.8.a.p 14
33.h odd 10 1 99.8.f.c 28

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{28} + 22 T_{2}^{27} + 1036 T_{2}^{26} + 16944 T_{2}^{25} + 574889 T_{2}^{24} + 6855516 T_{2}^{23} + 234979339 T_{2}^{22} + 2089735763 T_{2}^{21} + 94011380549 T_{2}^{20} + \cdots + 74\!\cdots\!00$$ acting on $$S_{8}^{\mathrm{new}}(33, [\chi])$$.