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 $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,9,Mod(23,33)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("33.23");

S:= CuspForms(chi, 9);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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.

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

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 $$S_{9}^{\mathrm{new}}(33, [\chi])$$.