Properties

Label 33.7.h.a
Level $33$
Weight $7$
Character orbit 33.h
Analytic conductor $7.592$
Analytic rank $0$
Dimension $88$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 33.h (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.59178475946\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 88q + 33q^{3} + 634q^{4} + 109q^{6} + 402q^{7} - 2471q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 88q + 33q^{3} + 634q^{4} + 109q^{6} + 402q^{7} - 2471q^{9} - 1016q^{10} + 7594q^{12} - 1830q^{13} - 6973q^{15} - 15506q^{16} - 291q^{18} - 32466q^{19} + 5434q^{21} + 40940q^{22} + 10011q^{24} + 68152q^{25} + 65385q^{27} - 37986q^{28} + 20928q^{30} + 96222q^{31} - 67159q^{33} - 186548q^{34} - 107169q^{36} - 112854q^{37} - 257545q^{39} + 246958q^{40} - 130324q^{42} + 40328q^{43} + 505478q^{45} - 30630q^{46} + 647882q^{48} - 327700q^{49} + 88197q^{51} - 132630q^{52} - 1639642q^{54} - 216226q^{55} + 174477q^{57} + 1571894q^{58} + 772854q^{60} + 559626q^{61} + 1129495q^{63} + 1406654q^{64} + 34620q^{66} - 3229336q^{67} - 2492844q^{69} - 2066360q^{70} - 2672882q^{72} + 1068798q^{73} - 1207057q^{75} + 913268q^{76} + 6763180q^{78} - 255558q^{79} + 3363409q^{81} - 3580072q^{82} - 2958906q^{84} - 2098978q^{85} + 365046q^{87} + 2251838q^{88} - 813500q^{90} + 7887166q^{91} + 4000307q^{93} + 11260802q^{94} + 4588186q^{96} + 3850038q^{97} - 1274785q^{99} + O(q^{100}) \)

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 −14.6057 + 4.74569i 25.8173 7.90347i 139.029 101.011i −39.7419 12.9129i −339.574 + 237.957i −103.376 + 75.1073i −973.541 + 1339.96i 604.070 408.093i 641.741
5.2 −12.6998 + 4.12642i −26.9979 + 0.338512i 92.4810 67.1914i 73.9941 + 24.0421i 341.471 115.704i 162.344 117.950i −394.901 + 543.535i 728.771 18.2782i −1038.92
5.3 −11.7448 + 3.81612i −9.61413 + 25.2303i 71.6006 52.0208i −165.956 53.9225i 16.6343 333.014i −453.911 + 329.785i −177.861 + 244.804i −544.137 485.135i 2154.90
5.4 −11.2768 + 3.66405i 10.7587 + 24.7639i 61.9636 45.0192i 101.905 + 33.1109i −212.060 239.837i 468.165 340.142i −87.7535 + 120.782i −497.500 + 532.855i −1270.48
5.5 −10.0445 + 3.26364i −1.25477 26.9708i 38.4627 27.9448i 135.541 + 44.0400i 100.627 + 266.812i −206.442 + 149.989i 102.165 140.618i −725.851 + 67.6842i −1505.17
5.6 −9.48700 + 3.08251i −4.13517 26.6815i 28.7242 20.8694i −210.493 68.3933i 121.476 + 240.380i 290.276 210.898i 167.074 229.958i −694.801 + 220.665i 2207.77
5.7 −6.30214 + 2.04769i 25.8420 7.82243i −16.2532 + 11.8086i 66.8441 + 21.7190i −146.842 + 102.214i −126.894 + 92.1938i 327.525 450.800i 606.619 404.295i −465.734
5.8 −5.15978 + 1.67652i 21.4926 + 16.3422i −27.9644 + 20.3173i −104.094 33.8222i −138.295 48.2893i −30.2053 + 21.9454i 314.319 432.623i 194.867 + 702.473i 593.806
5.9 −3.91805 + 1.27305i −26.2247 6.42389i −38.0467 + 27.6425i −40.1977 13.0610i 110.927 8.21623i −125.751 + 91.3637i 268.853 370.045i 646.467 + 336.929i 174.124
5.10 −3.46608 + 1.12620i −11.0612 + 24.6303i −41.0317 + 29.8113i 207.457 + 67.4069i 10.6004 97.8275i −309.326 + 224.739i 245.744 338.237i −484.301 544.880i −794.976
5.11 −0.658923 + 0.214097i −17.8783 + 20.2328i −51.3887 + 37.3361i −87.3837 28.3927i 7.44866 17.1596i 484.811 352.236i 51.9309 71.4767i −89.7314 723.456i 63.6579
5.12 0.658923 0.214097i 13.7178 23.2556i −51.3887 + 37.3361i 87.3837 + 28.3927i 4.06004 18.2606i 484.811 352.236i −51.9309 + 71.4767i −352.643 638.031i 63.6579
5.13 3.46608 1.12620i 20.0067 18.1310i −41.0317 + 29.8113i −207.457 67.4069i 48.9257 85.3748i −309.326 + 224.739i −245.744 + 338.237i 71.5354 725.482i −794.976
5.14 3.91805 1.27305i −14.2134 22.9561i −38.0467 + 27.6425i 40.1977 + 13.0610i −84.9128 71.8486i −125.751 + 91.3637i −268.853 + 370.045i −324.961 + 652.565i 174.124
5.15 5.15978 1.67652i 22.1839 + 15.3907i −27.9644 + 20.3173i 104.094 + 33.8222i 140.267 + 42.2210i −30.2053 + 21.9454i −314.319 + 432.623i 255.253 + 682.852i 593.806
5.16 6.30214 2.04769i 0.546048 + 26.9945i −16.2532 + 11.8086i −66.8441 21.7190i 58.7175 + 169.005i −126.894 + 92.1938i −327.525 + 450.800i −728.404 + 29.4805i −465.734
5.17 9.48700 3.08251i −26.6534 + 4.31225i 28.7242 20.8694i 210.493 + 68.3933i −239.568 + 123.070i 290.276 210.898i −167.074 + 229.958i 691.809 229.872i 2207.77
5.18 10.0445 3.26364i −26.0385 + 7.14109i 38.4627 27.9448i −135.541 44.0400i −238.237 + 156.709i −206.442 + 149.989i −102.165 + 140.618i 627.010 371.887i −1505.17
5.19 11.2768 3.66405i 26.8765 + 2.57968i 61.9636 45.0192i −101.905 33.1109i 312.532 69.3863i 468.165 340.142i 87.7535 120.782i 715.690 + 138.666i −1270.48
5.20 11.7448 3.81612i 21.0245 16.9402i 71.6006 52.0208i 165.956 + 53.9225i 182.283 279.191i −453.911 + 329.785i 177.861 244.804i 155.061 712.318i 2154.90
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.22
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 33.7.h.a 88
3.b odd 2 1 inner 33.7.h.a 88
11.c even 5 1 inner 33.7.h.a 88
33.h odd 10 1 inner 33.7.h.a 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.7.h.a 88 1.a even 1 1 trivial
33.7.h.a 88 3.b odd 2 1 inner
33.7.h.a 88 11.c even 5 1 inner
33.7.h.a 88 33.h odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(33, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database