Properties

Label 33.5.h.a
Level $33$
Weight $5$
Character orbit 33.h
Analytic conductor $3.411$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 33.h (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.41120878177\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56q + 90q^{4} - 83q^{6} - 86q^{7} + 232q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 56q + 90q^{4} - 83q^{6} - 86q^{7} + 232q^{9} + 136q^{10} - 710q^{12} + 94q^{13} - 406q^{15} + 334q^{16} + 1149q^{18} + 958q^{19} - 728q^{21} - 1300q^{22} + 795q^{24} - 1740q^{25} - 1026q^{27} - 4962q^{28} - 744q^{30} - 130q^{31} - 2875q^{33} + 4844q^{34} + 11631q^{36} + 2802q^{37} + 8696q^{39} - 626q^{40} + 8804q^{42} - 980q^{43} - 17356q^{45} + 8698q^{46} - 17854q^{48} + 7516q^{49} + 1461q^{51} + 11818q^{52} - 9082q^{54} - 4916q^{55} - 2661q^{57} - 38922q^{58} - 28194q^{60} - 15150q^{61} - 488q^{63} - 30690q^{64} + 20100q^{66} + 40544q^{67} + 25662q^{69} + 15720q^{70} + 72430q^{72} - 20554q^{73} + 28841q^{75} + 70996q^{76} - 39716q^{78} + 46458q^{79} - 24284q^{81} + 26200q^{82} - 8346q^{84} - 15494q^{85} - 66228q^{87} - 78050q^{88} - 43484q^{90} - 49854q^{91} - 30184q^{93} - 119550q^{94} + 50098q^{96} - 64476q^{97} + 52640q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −7.21606 + 2.34464i −5.43034 7.17715i 33.6300 24.4336i −10.4402 3.39224i 56.0135 + 39.0586i −22.5552 + 16.3873i −114.032 + 156.951i −22.0229 + 77.9487i 83.2911
5.2 −6.11716 + 1.98759i 5.92004 + 6.77888i 20.5249 14.9122i 26.2596 + 8.53227i −49.6874 29.7009i −49.1578 + 35.7152i −35.4248 + 48.7581i −10.9063 + 80.2624i −177.593
5.3 −4.82086 + 1.56639i 8.27412 3.54103i 7.84286 5.69817i −20.8854 6.78609i −34.3418 + 30.0314i 52.5734 38.1968i 18.7876 25.8588i 55.9222 58.5978i 111.316
5.4 −4.59281 + 1.49229i −6.55843 + 6.16336i 5.92268 4.30308i −5.60370 1.82075i 20.9241 38.0942i 15.1996 11.0431i 24.6359 33.9084i 5.02599 80.8439i 28.4538
5.5 −2.36002 + 0.766817i −7.05112 5.59301i −7.96259 + 5.78516i 40.2655 + 13.0831i 20.9296 + 7.79270i 50.3841 36.6062i 37.6929 51.8798i 18.4365 + 78.8739i −105.060
5.6 −1.90239 + 0.618124i 1.59869 8.85687i −9.70726 + 7.05274i −8.40855 2.73210i 2.43332 + 17.8374i −61.9407 + 45.0026i 32.9194 45.3097i −75.8884 28.3188i 17.6851
5.7 −0.493178 + 0.160243i 3.56251 + 8.26490i −12.7267 + 9.24651i −30.1183 9.78604i −3.08135 3.50520i 4.18763 3.04249i 9.67167 13.3119i −55.6170 + 58.8876i 16.4219
5.8 0.493178 0.160243i 8.96126 + 0.834156i −12.7267 + 9.24651i 30.1183 + 9.78604i 4.55317 1.02459i 4.18763 3.04249i −9.67167 + 13.3119i 79.6084 + 14.9502i 16.4219
5.9 1.90239 0.618124i −7.92936 + 4.25737i −9.70726 + 7.05274i 8.40855 + 2.73210i −12.4532 + 13.0005i −61.9407 + 45.0026i −32.9194 + 45.3097i 44.7496 67.5164i 17.6851
5.10 2.36002 0.766817i −7.49818 4.97767i −7.96259 + 5.78516i −40.2655 13.0831i −21.5128 5.99767i 50.3841 36.6062i −37.6929 + 51.8798i 31.4455 + 74.6470i −105.060
5.11 4.59281 1.49229i 3.83504 8.14202i 5.92268 4.30308i 5.60370 + 1.82075i 5.46331 43.1177i 15.1996 11.0431i −24.6359 + 33.9084i −51.5850 62.4499i 28.4538
5.12 4.82086 1.56639i −0.810876 + 8.96340i 7.84286 5.69817i 20.8854 + 6.78609i 10.1311 + 44.4815i 52.5734 38.1968i −18.7876 + 25.8588i −79.6850 14.5364i 111.316
5.13 6.11716 1.98759i 8.27649 + 3.53550i 20.5249 14.9122i −26.2596 8.53227i 57.6557 + 5.17701i −49.1578 + 35.7152i 35.4248 48.7581i 56.0004 + 58.5231i −177.593
5.14 7.21606 2.34464i −8.50394 2.94670i 33.6300 24.4336i 10.4402 + 3.39224i −68.2739 1.32487i −22.5552 + 16.3873i 114.032 156.951i 63.6340 + 50.1171i 83.2911
14.1 −4.15672 5.72123i 8.19184 + 3.72744i −10.5099 + 32.3462i −25.2509 + 34.7549i −12.7256 62.3613i −8.14613 + 25.0712i 121.136 39.3593i 53.2124 + 61.0691i 303.802
14.2 −3.94112 5.42449i −8.99437 + 0.318385i −8.94835 + 27.5402i 2.90630 4.00017i 37.1750 + 47.5350i −0.401365 + 1.23527i 82.6281 26.8475i 80.7973 5.72735i −33.1529
14.3 −3.20461 4.41077i 6.23157 6.49365i −4.24110 + 13.0528i 23.1325 31.8392i −48.6118 6.67639i −13.4280 + 41.3270i −11.7990 + 3.83372i −3.33506 80.9313i −214.566
14.4 −2.72011 3.74391i 1.23934 + 8.91426i −1.67360 + 5.15080i 15.2071 20.9308i 30.0031 28.8877i 20.6582 63.5795i −46.5832 + 15.1358i −77.9281 + 22.0955i −119.728
14.5 −1.95750 2.69427i −3.92492 8.09908i 1.51699 4.66881i −11.1246 + 15.3116i −14.1381 + 26.4288i 0.0531312 0.163521i −66.2255 + 21.5180i −50.1900 + 63.5764i 63.0301
14.6 −0.966669 1.33051i −3.49528 + 8.29355i 4.10848 12.6446i −16.0026 + 22.0258i 14.4134 3.36663i −23.7358 + 73.0514i −45.8209 + 14.8881i −56.5660 57.9766i 44.7746
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
33.h 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.h.a 56
3.b odd 2 1 inner 33.5.h.a 56
11.c even 5 1 inner 33.5.h.a 56
33.h odd 10 1 inner 33.5.h.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.5.h.a 56 1.a even 1 1 trivial
33.5.h.a 56 3.b odd 2 1 inner
33.5.h.a 56 11.c even 5 1 inner
33.5.h.a 56 33.h odd 10 1 inner

Hecke kernels

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

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database