Properties

Label 33.9.b.a
Level 33
Weight 9
Character orbit 33.b
Analytic conductor 13.443
Analytic rank 0
Dimension 26
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 33.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.4434941320\)
Analytic rank: \(0\)
Dimension: \(26\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 26q - 35q^{3} - 2596q^{4} - 3746q^{6} + 7156q^{7} + 9011q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 26q - 35q^{3} - 2596q^{4} - 3746q^{6} + 7156q^{7} + 9011q^{9} - 31836q^{10} - 28900q^{12} - 131624q^{13} + 71041q^{15} + 311972q^{16} - 675394q^{18} + 134608q^{19} + 490306q^{21} - 59088q^{24} - 2324740q^{25} + 2011426q^{27} - 1996688q^{28} - 324146q^{30} + 964738q^{31} - 512435q^{33} + 9219648q^{34} - 6887660q^{36} - 5721542q^{37} - 5782712q^{39} + 8363496q^{40} + 10350076q^{42} + 4260820q^{43} + 6595181q^{45} - 39680292q^{46} + 22674164q^{48} + 20017254q^{49} - 7985018q^{51} + 48711952q^{52} - 18774176q^{54} - 4304454q^{55} + 15476796q^{57} - 16060008q^{58} - 26730016q^{60} + 58840q^{61} - 42877282q^{63} - 57365836q^{64} - 10395110q^{66} - 63186734q^{67} + 100738079q^{69} + 50969160q^{70} + 48890880q^{72} - 21879656q^{73} - 122009926q^{75} - 119289368q^{76} - 123440744q^{78} + 117211444q^{79} + 138667019q^{81} + 121755480q^{82} - 28504816q^{84} + 28880604q^{85} + 14774868q^{87} + 90481380q^{88} + 189044834q^{90} + 192183008q^{91} - 113622071q^{93} - 453996696q^{94} + 11988416q^{96} + 24314206q^{97} - 57905155q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 30.7079i 79.3894 16.0724i −686.976 657.051i −493.550 2437.88i −1255.62 13234.4i 6044.35 2551.96i −20176.7
23.2 27.8469i −68.6229 + 43.0336i −519.450 436.996i 1198.35 + 1910.94i −547.160 7336.27i 2857.21 5906.19i 12169.0
23.3 27.3024i −36.6798 72.2191i −489.423 131.340i −1971.76 + 1001.45i −44.0163 6373.03i −3870.19 + 5297.96i 3585.90
23.4 23.5789i −17.0295 + 79.1896i −299.967 958.765i 1867.21 + 401.538i 3892.73 1036.68i −5980.99 2697.12i −22606.7
23.5 22.5549i 76.8411 25.6211i −252.723 1149.93i −577.881 1733.14i 3915.62 73.9026i 5248.12 3937.51i 25936.5
23.6 17.8320i 18.7312 78.8045i −61.9787 794.590i −1405.24 334.014i 2438.07 3459.78i −5859.29 2952.20i −14169.1
23.7 16.2016i −80.7607 6.22143i −6.49112 809.269i −100.797 + 1308.45i −3530.96 4042.44i 6483.59 + 1004.89i −13111.4
23.8 14.0184i 36.8732 72.1205i 59.4833 519.567i −1011.02 516.905i −3946.73 4422.58i −3841.73 5318.63i 7283.52
23.9 12.0893i −77.6992 22.8875i 109.849 288.931i −276.694 + 939.329i 3024.09 4422.86i 5513.32 + 3556.68i 3492.97
23.10 9.43364i 80.9807 + 1.77018i 167.006 654.316i 16.6993 763.942i −1003.87 3990.49i 6554.73 + 286.701i −6172.58
23.11 8.17075i −40.5699 + 70.1076i 189.239 920.235i 572.832 + 331.487i −1093.89 3637.93i −3269.16 5688.52i 7519.00
23.12 4.28405i 49.1853 + 64.3569i 237.647 147.746i 275.708 210.712i 2560.99 2114.81i −1722.62 + 6330.82i 632.952
23.13 0.463681i −38.1388 + 71.4593i 255.785 649.991i 33.1343 + 17.6842i −831.270 237.305i −3651.86 5450.75i −301.389
23.14 0.463681i −38.1388 71.4593i 255.785 649.991i 33.1343 17.6842i −831.270 237.305i −3651.86 + 5450.75i −301.389
23.15 4.28405i 49.1853 64.3569i 237.647 147.746i 275.708 + 210.712i 2560.99 2114.81i −1722.62 6330.82i 632.952
23.16 8.17075i −40.5699 70.1076i 189.239 920.235i 572.832 331.487i −1093.89 3637.93i −3269.16 + 5688.52i 7519.00
23.17 9.43364i 80.9807 1.77018i 167.006 654.316i 16.6993 + 763.942i −1003.87 3990.49i 6554.73 286.701i −6172.58
23.18 12.0893i −77.6992 + 22.8875i 109.849 288.931i −276.694 939.329i 3024.09 4422.86i 5513.32 3556.68i 3492.97
23.19 14.0184i 36.8732 + 72.1205i 59.4833 519.567i −1011.02 + 516.905i −3946.73 4422.58i −3841.73 + 5318.63i 7283.52
23.20 16.2016i −80.7607 + 6.22143i −6.49112 809.269i −100.797 1308.45i −3530.96 4042.44i 6483.59 1004.89i −13111.4
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.9.b.a 26
3.b odd 2 1 inner 33.9.b.a 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.9.b.a 26 1.a even 1 1 trivial
33.9.b.a 26 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(33, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database