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Properties

Label 21.9.h.b
Level $21$
Weight $9$
Character orbit 21.h
Analytic conductor $8.555$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [21,9,Mod(2,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.2");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\), degree \(2\), 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: \(8.55495081128\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 80 q^{3} + 2302 q^{4} - 1028 q^{6} - 3206 q^{7} - 286 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 80 q^{3} + 2302 q^{4} - 1028 q^{6} - 3206 q^{7} - 286 q^{9} + 4862 q^{10} + 61828 q^{12} + 47312 q^{13} - 132676 q^{15} + 13558 q^{16} + 127516 q^{18} + 273416 q^{19} - 213080 q^{21} - 428116 q^{22} - 238362 q^{24} + 1520644 q^{25} - 951136 q^{27} - 2265942 q^{28} - 55820 q^{30} - 552182 q^{31} + 2140034 q^{33} + 5992464 q^{34} - 11269292 q^{36} + 2961940 q^{37} - 6244780 q^{39} - 11393286 q^{40} + 12825218 q^{42} + 30770112 q^{43} + 8604500 q^{45} + 2834700 q^{46} - 17753720 q^{48} - 488922 q^{49} - 8525928 q^{51} - 11542416 q^{52} + 36951838 q^{54} - 17482820 q^{55} - 98494480 q^{57} + 26032034 q^{58} - 38452490 q^{60} - 83468788 q^{61} + 114737140 q^{63} + 127817964 q^{64} + 72740152 q^{66} + 114922616 q^{67} - 26793528 q^{69} - 228954530 q^{70} - 38169168 q^{72} - 165056480 q^{73} + 171335056 q^{75} + 14012088 q^{76} - 277758680 q^{78} + 140842074 q^{79} - 17381110 q^{81} - 109600652 q^{82} + 242375000 q^{84} + 208839936 q^{85} + 89390378 q^{87} + 96328590 q^{88} + 229037236 q^{90} - 13959372 q^{91} + 93228660 q^{93} - 109682268 q^{94} - 23819614 q^{96} - 662841404 q^{97} - 330294424 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −24.2165 − 13.9814i 9.01105 − 80.4972i 262.960 + 455.461i −807.622 − 466.281i −1343.68 + 1823.38i −2054.96 + 1241.75i − 7547.75i −6398.60 − 1450.73i 13038.5 + 22583.4i
2.2 −23.6192 − 13.6365i 55.5579 + 58.9433i 243.911 + 422.466i −131.892 − 76.1477i −508.450 − 2149.81i 2.47410 − 2401.00i − 6322.49i −387.632 + 6549.54i 2076.78 + 3597.09i
2.3 −22.6560 − 13.0804i −68.4744 + 43.2695i 214.195 + 370.997i 74.7674 + 43.1670i 2117.34 − 84.6381i 708.863 + 2293.97i − 4509.88i 2816.50 − 5925.71i −1129.29 − 1955.98i
2.4 −19.8804 − 11.4779i −45.1440 − 67.2534i 135.487 + 234.670i 961.540 + 555.145i 125.550 + 1855.18i 1449.34 − 1914.21i − 343.721i −2485.04 + 6072.18i −12743.9 − 22073.0i
2.5 −15.3286 − 8.84999i 80.5361 − 8.65683i 28.6448 + 49.6143i 543.476 + 313.776i −1311.12 − 580.046i −1152.87 + 2106.11i 3517.17i 6411.12 − 1394.37i −5553.83 − 9619.51i
2.6 −10.2436 − 5.91415i −80.9415 − 3.07787i −58.0457 − 100.538i −466.837 − 269.529i 810.930 + 510.228i −1725.86 − 1669.20i 4401.21i 6542.05 + 498.256i 3188.06 + 5521.89i
2.7 −9.39142 − 5.42214i 1.17874 + 80.9914i −69.2009 − 119.859i −487.734 − 281.593i 428.077 − 767.016i 2288.33 + 726.860i 4277.00i −6558.22 + 190.936i 3053.67 + 5289.12i
2.8 −9.35049 − 5.39851i 57.0876 − 57.4631i −69.7122 − 120.745i −328.020 − 189.382i −844.012 + 229.120i 2016.47 − 1303.33i 4269.40i −43.0100 − 6560.86i 2044.76 + 3541.63i
2.9 −4.78453 − 2.76235i −9.65985 + 80.4219i −112.739 − 195.269i 865.089 + 499.459i 268.371 − 358.097i −2333.29 − 566.186i 2660.02i −6374.37 − 1553.73i −2759.36 − 4779.35i
2.10 4.78453 + 2.76235i 74.4774 + 31.8453i −112.739 − 195.269i −865.089 − 499.459i 268.371 + 358.097i −2333.29 − 566.186i − 2660.02i 4532.75 + 4743.51i −2759.36 − 4779.35i
2.11 9.35049 + 5.39851i −78.3083 + 20.7078i −69.7122 − 120.745i 328.020 + 189.382i −844.012 − 229.120i 2016.47 − 1303.33i − 4269.40i 5703.38 − 3243.18i 2044.76 + 3541.63i
2.12 9.39142 + 5.42214i 69.5513 + 41.5165i −69.2009 − 119.859i 487.734 + 281.593i 428.077 + 767.016i 2288.33 + 726.860i − 4277.00i 3113.76 + 5775.05i 3053.67 + 5289.12i
2.13 10.2436 + 5.91415i 37.8052 − 71.6363i −58.0457 − 100.538i 466.837 + 269.529i 810.930 − 510.228i −1725.86 − 1669.20i − 4401.21i −3702.53 − 5416.46i 3188.06 + 5521.89i
2.14 15.3286 + 8.84999i −47.7651 + 65.4179i 28.6448 + 49.6143i −543.476 − 313.776i −1311.12 + 580.046i −1152.87 + 2106.11i − 3517.17i −1998.00 − 6249.38i −5553.83 − 9619.51i
2.15 19.8804 + 11.4779i −35.6711 − 72.7226i 135.487 + 234.670i −961.540 − 555.145i 125.550 − 1855.18i 1449.34 − 1914.21i 343.721i −4016.14 + 5188.19i −12743.9 − 22073.0i
2.16 22.6560 + 13.0804i 71.7097 − 37.6658i 214.195 + 370.997i −74.7674 − 43.1670i 2117.34 + 84.6381i 708.863 + 2293.97i 4509.88i 3723.57 − 5402.01i −1129.29 − 1955.98i
2.17 23.6192 + 13.6365i 23.2675 + 77.5862i 243.911 + 422.466i 131.892 + 76.1477i −508.450 + 2149.81i 2.47410 − 2401.00i 6322.49i −5478.25 + 3610.47i 2076.78 + 3597.09i
2.18 24.2165 + 13.9814i −74.2182 − 32.4448i 262.960 + 455.461i 807.622 + 466.281i −1343.68 − 1823.38i −2054.96 + 1241.75i 7547.75i 4455.67 + 4815.99i 13038.5 + 22583.4i
11.1 −24.2165 + 13.9814i 9.01105 + 80.4972i 262.960 − 455.461i −807.622 + 466.281i −1343.68 − 1823.38i −2054.96 − 1241.75i 7547.75i −6398.60 + 1450.73i 13038.5 − 22583.4i
11.2 −23.6192 + 13.6365i 55.5579 − 58.9433i 243.911 − 422.466i −131.892 + 76.1477i −508.450 + 2149.81i 2.47410 + 2401.00i 6322.49i −387.632 − 6549.54i 2076.78 − 3597.09i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.9.h.b ✓ 36
3.b odd 2 1 inner 21.9.h.b ✓ 36
7.c even 3 1 inner 21.9.h.b ✓ 36
21.h odd 6 1 inner 21.9.h.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.9.h.b ✓ 36 1.a even 1 1 trivial
21.9.h.b ✓ 36 3.b odd 2 1 inner
21.9.h.b ✓ 36 7.c even 3 1 inner
21.9.h.b ✓ 36 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 3455 T_{2}^{34} + 6996931 T_{2}^{32} - 9498102458 T_{2}^{30} + 9641771581840 T_{2}^{28} + \cdots + 14\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display