Properties

Label 33.4.f.a
Level 33
Weight 4
Character orbit 33.f
Analytic conductor 1.947
Analytic rank 0
Dimension 40
CM no
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.94706303019\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40q - 3q^{3} - 38q^{4} + 45q^{6} - 10q^{7} - 65q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q - 3q^{3} - 38q^{4} + 45q^{6} - 10q^{7} - 65q^{9} - 90q^{12} - 10q^{13} + 33q^{15} + 310q^{16} + 225q^{18} - 460q^{19} - 340q^{22} - 565q^{24} - 604q^{25} - 435q^{27} + 1190q^{28} + 910q^{30} + 840q^{31} + 1208q^{33} - 188q^{34} + 1991q^{36} + 126q^{37} - 1075q^{39} - 90q^{40} - 3340q^{42} - 1662q^{45} + 430q^{46} - 346q^{48} + 376q^{49} - 210q^{51} - 4270q^{52} - 546q^{55} + 1800q^{57} - 4582q^{58} + 674q^{60} + 650q^{61} + 3945q^{63} + 7238q^{64} + 3504q^{66} + 4556q^{67} + 3860q^{69} + 2964q^{70} - 1640q^{72} + 3860q^{73} - 6048q^{75} - 7640q^{78} - 3550q^{79} - 2453q^{81} - 5812q^{82} - 7080q^{84} - 8230q^{85} - 9298q^{88} + 9220q^{90} - 6766q^{91} + 5659q^{93} + 3530q^{94} + 14890q^{96} + 8004q^{97} + 955q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −1.66959 + 5.13848i −2.33142 4.64375i −17.1443 12.4560i −7.03009 + 2.28422i 27.7543 4.22679i −8.32735 + 11.4616i 57.6606 41.8928i −16.1289 + 21.6531i 39.9377i
2.2 −1.27640 + 3.92834i 0.664082 + 5.15354i −7.33056 5.32596i 0.616497 0.200312i −21.0925 3.96922i 8.50755 11.7096i 3.54571 2.57611i −26.1180 + 6.84475i 2.67749i
2.3 −0.882408 + 2.71577i 4.90539 1.71383i −0.124638 0.0905550i 4.91761 1.59783i 0.325809 + 14.8342i −9.45167 + 13.0091i −18.1255 + 13.1689i 21.1256 16.8139i 14.7651i
2.4 −0.448331 + 1.37982i −5.12333 + 0.866882i 4.76923 + 3.46505i −14.6309 + 4.75385i 1.10081 7.45793i −7.94038 + 10.9290i −16.3093 + 11.8494i 25.4970 8.88265i 22.3193i
2.5 −0.325060 + 1.00043i −1.99228 4.79904i 5.57694 + 4.05188i 11.6880 3.79766i 5.44873 0.433161i 13.1668 18.1225i −12.6746 + 9.20865i −19.0617 + 19.1221i 12.9275i
2.6 0.325060 1.00043i 3.94851 + 3.37776i 5.57694 + 4.05188i −11.6880 + 3.79766i 4.66272 2.85225i 13.1668 18.1225i 12.6746 9.20865i 4.18154 + 26.6742i 12.9275i
2.7 0.448331 1.37982i −2.40765 + 4.60470i 4.76923 + 3.46505i 14.6309 4.75385i 5.27423 + 5.38655i −7.94038 + 10.9290i 16.3093 11.8494i −15.4064 22.1730i 22.3193i
2.8 0.882408 2.71577i 3.14579 4.13570i −0.124638 0.0905550i −4.91761 + 1.59783i −8.45574 12.1926i −9.45167 + 13.0091i 18.1255 13.1689i −7.20798 26.0201i 14.7651i
2.9 1.27640 3.92834i −4.69610 2.22411i −7.33056 5.32596i −0.616497 + 0.200312i −14.7312 + 15.6090i 8.50755 11.7096i −3.54571 + 2.57611i 17.1067 + 20.8893i 2.67749i
2.10 1.66959 5.13848i 3.69602 + 3.65232i −17.1443 12.4560i 7.03009 2.28422i 24.9382 12.8940i −8.32735 + 11.4616i −57.6606 + 41.8928i 0.321176 + 26.9981i 39.9377i
8.1 −3.83133 + 2.78363i 2.59912 4.49940i 4.45840 13.7215i 2.11757 2.91459i 2.56657 + 24.4737i 28.5187 + 9.26628i 9.40652 + 28.9503i −13.4892 23.3889i 17.0613i
8.2 −3.30778 + 2.40324i 2.27962 + 4.66940i 2.69368 8.29030i −3.98805 + 5.48909i −18.7622 9.96686i −17.8876 5.81204i 0.905818 + 2.78782i −16.6067 + 21.2889i 27.7409i
8.3 −2.97363 + 2.16047i −5.19431 + 0.138465i 1.70270 5.24038i 10.1182 13.9265i 15.1468 11.6339i −16.0632 5.21926i −2.82814 8.70411i 26.9617 1.43845i 63.2724i
8.4 −1.29832 + 0.943283i −2.93846 4.28549i −1.67629 + 5.15909i −10.7269 + 14.7643i 7.85748 + 2.79213i −4.11498 1.33704i −6.65743 20.4895i −9.73090 + 25.1855i 29.2872i
8.5 −0.315290 + 0.229071i 5.06463 + 1.16169i −2.42520 + 7.46400i 2.50207 3.44380i −1.86294 + 0.793892i 11.0922 + 3.60409i −1.90859 5.87403i 24.3009 + 11.7671i 1.65895i
8.6 0.315290 0.229071i −3.41455 + 3.91674i −2.42520 + 7.46400i −2.50207 + 3.44380i −0.179357 + 2.01708i 11.0922 + 3.60409i 1.90859 + 5.87403i −3.68176 26.7478i 1.65895i
8.7 1.29832 0.943283i −0.141685 5.19422i −1.67629 + 5.15909i 10.7269 14.7643i −5.08357 6.61010i −4.11498 1.33704i 6.65743 + 20.4895i −26.9599 + 1.47188i 29.2872i
8.8 2.97363 2.16047i 4.28367 2.94112i 1.70270 5.24038i −10.1182 + 13.9265i 6.38385 18.0005i −16.0632 5.21926i 2.82814 + 8.70411i 9.69966 25.1976i 63.2724i
8.9 3.30778 2.40324i 0.900354 + 5.11755i 2.69368 8.29030i 3.98805 5.48909i 15.2769 + 14.7640i −17.8876 5.81204i −0.905818 2.78782i −25.3787 + 9.21522i 27.7409i
8.10 3.83133 2.78363i −4.74741 2.11237i 4.45840 13.7215i −2.11757 + 2.91459i −24.0689 + 5.12183i 28.5187 + 9.26628i −9.40652 28.9503i 18.0758 + 20.0566i 17.0613i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.4.f.a 40
3.b odd 2 1 inner 33.4.f.a 40
11.c even 5 1 363.4.d.d 40
11.d odd 10 1 inner 33.4.f.a 40
11.d odd 10 1 363.4.d.d 40
33.f even 10 1 inner 33.4.f.a 40
33.f even 10 1 363.4.d.d 40
33.h odd 10 1 363.4.d.d 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.f.a 40 1.a even 1 1 trivial
33.4.f.a 40 3.b odd 2 1 inner
33.4.f.a 40 11.d odd 10 1 inner
33.4.f.a 40 33.f even 10 1 inner
363.4.d.d 40 11.c even 5 1
363.4.d.d 40 11.d odd 10 1
363.4.d.d 40 33.f even 10 1
363.4.d.d 40 33.h odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(33, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database