Properties

Label 33.6.f.a
Level 33
Weight 6
Character orbit 33.f
Analytic conductor 5.293
Analytic rank 0
Dimension 72
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 33.f (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.29266605383\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72q + 18q^{3} - 262q^{4} + 15q^{6} - 10q^{7} + 292q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 72q + 18q^{3} - 262q^{4} + 15q^{6} - 10q^{7} + 292q^{9} + 1854q^{12} - 10q^{13} - 762q^{15} - 10122q^{16} + 4815q^{18} + 4460q^{19} + 4628q^{22} - 805q^{24} + 13708q^{25} + 6108q^{27} - 28130q^{28} - 15470q^{30} + 4340q^{31} - 508q^{33} + 18732q^{34} - 56461q^{36} + 978q^{37} + 23360q^{39} + 69750q^{40} + 60788q^{42} - 31356q^{45} - 52090q^{46} + 4238q^{48} - 58448q^{49} - 178950q^{51} - 14190q^{52} + 86600q^{55} + 266190q^{57} + 137102q^{58} + 284090q^{60} - 77890q^{61} - 120330q^{63} - 379114q^{64} - 323304q^{66} + 42668q^{67} - 271816q^{69} + 87176q^{70} + 343960q^{72} + 116440q^{73} + 326202q^{75} + 155512q^{78} - 350590q^{79} - 208088q^{81} - 606424q^{82} - 220680q^{84} + 665610q^{85} + 1152974q^{88} + 293440q^{90} + 621014q^{91} + 478456q^{93} - 521270q^{94} - 1246430q^{96} - 1030446q^{97} - 590000q^{99} + O(q^{100}) \)

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 −3.08173 + 9.48458i −12.5347 + 9.26724i −54.5716 39.6486i 16.3931 5.32643i −49.2674 147.445i −74.1699 + 102.086i 286.047 207.825i 71.2363 232.324i 171.896i
2.2 −2.95379 + 9.09083i 10.1153 11.8609i −48.0298 34.8957i 91.6682 29.7848i 77.9468 + 126.991i 70.5766 97.1403i 211.641 153.766i −38.3609 239.953i 921.319i
2.3 −2.81967 + 8.67805i 12.9564 + 8.66789i −41.4695 30.1293i −53.2208 + 17.2925i −111.753 + 87.9955i −14.1075 + 19.4173i 142.170 103.293i 92.7354 + 224.609i 510.612i
2.4 −2.32473 + 7.15479i −12.7649 8.94746i −19.8981 14.4568i −28.0945 + 9.12845i 93.6922 70.5298i 146.291 201.352i −45.0663 + 32.7426i 82.8858 + 228.427i 222.231i
2.5 −1.97194 + 6.06902i 4.72506 14.8551i −7.05585 5.12637i −59.6930 + 19.3954i 80.8382 + 57.9699i −97.0552 + 133.585i −120.178 + 87.3143i −198.348 140.382i 400.525i
2.6 −1.36557 + 4.20281i 6.31381 + 14.2526i 10.0898 + 7.33064i 98.1154 31.8796i −68.5228 + 7.07275i −5.71539 + 7.86655i −158.991 + 115.514i −163.272 + 179.976i 455.894i
2.7 −1.12695 + 3.46841i −12.9075 8.74056i 15.1287 + 10.9916i 54.0225 17.5530i 44.8620 34.9182i −123.528 + 170.022i −149.586 + 108.681i 90.2051 + 225.637i 207.153i
2.8 −1.01916 + 3.13665i −8.08821 + 13.3259i 17.0887 + 12.4156i −40.8503 + 13.2731i −33.5556 38.9511i 39.2343 54.0014i −141.742 + 102.981i −112.162 215.566i 141.660i
2.9 −0.489986 + 1.50802i 15.5884 + 0.0191829i 23.8545 + 17.3313i −16.6412 + 5.40705i −7.66704 + 23.4983i 36.5406 50.2938i −78.8739 + 57.3052i 242.999 + 0.598063i 27.7446i
2.10 0.489986 1.50802i 4.79885 14.8314i 23.8545 + 17.3313i 16.6412 5.40705i −20.0147 14.5039i 36.5406 50.2938i 78.8739 57.3052i −196.942 142.348i 27.7446i
2.11 1.01916 3.13665i −15.1731 + 3.57441i 17.0887 + 12.4156i 40.8503 13.2731i −4.25217 + 51.2356i 39.2343 54.0014i 141.742 102.981i 217.447 108.470i 141.660i
2.12 1.12695 3.46841i 4.32414 + 14.9767i 15.1287 + 10.9916i −54.0225 + 17.5530i 56.8185 + 1.88017i −123.528 + 170.022i 149.586 108.681i −205.604 + 129.523i 207.153i
2.13 1.36557 4.20281i −11.6039 10.4091i 10.0898 + 7.33064i −98.1154 + 31.8796i −59.5934 + 34.5547i −5.71539 + 7.86655i 158.991 115.514i 26.3023 + 241.572i 455.894i
2.14 1.97194 6.06902i 15.5882 + 0.0966769i −7.05585 5.12637i 59.6930 19.3954i 31.3257 94.4141i −97.0552 + 133.585i 120.178 87.3143i 242.981 + 3.01403i 400.525i
2.15 2.32473 7.15479i 4.56497 + 14.9051i −19.8981 14.4568i 28.0945 9.12845i 117.255 + 1.98889i 146.291 201.352i 45.0663 32.7426i −201.322 + 136.082i 222.231i
2.16 2.81967 8.67805i −4.23991 15.0008i −41.4695 30.1293i 53.2208 17.2925i −142.133 5.50305i −14.1075 + 19.4173i −142.170 + 103.293i −207.046 + 127.204i 510.612i
2.17 2.95379 9.09083i 14.4062 5.95502i −48.0298 34.8957i −91.6682 + 29.7848i −11.5833 148.554i 70.5766 97.1403i −211.641 + 153.766i 172.075 171.578i 921.319i
2.18 3.08173 9.48458i −12.6871 + 9.05745i −54.5716 39.6486i −16.3931 + 5.32643i 46.8079 + 148.244i −74.1699 + 102.086i −286.047 + 207.825i 78.9251 229.826i 171.896i
8.1 −8.92830 + 6.48679i −5.92302 + 14.4194i 27.7475 85.3981i 33.7792 46.4931i −40.6528 167.162i −11.4606 3.72378i 197.091 + 606.584i −172.836 170.812i 634.222i
8.2 −7.80447 + 5.67028i 14.8651 4.69358i 18.8692 58.0733i −46.3038 + 63.7318i −89.4001 + 120.920i −135.347 43.9769i 86.6348 + 266.635i 198.941 139.541i 759.949i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.18
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 33.6.f.a 72
3.b odd 2 1 inner 33.6.f.a 72
11.d odd 10 1 inner 33.6.f.a 72
33.f even 10 1 inner 33.6.f.a 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.f.a 72 1.a even 1 1 trivial
33.6.f.a 72 3.b odd 2 1 inner
33.6.f.a 72 11.d odd 10 1 inner
33.6.f.a 72 33.f even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(33, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database