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Properties

Label 21.8.g.b
Level $21$
Weight $8$
Character orbit 21.g
Analytic conductor $6.560$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [21,8,Mod(5,21)]
 
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, 5]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.5");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 21.g (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: \(6.56008553517\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 84 q^{3} + 1150 q^{4} + 2464 q^{7} - 3516 q^{9} - 630 q^{10} - 3864 q^{12} + 27576 q^{15} - 19594 q^{16} + 33498 q^{18} - 47880 q^{19} + 151788 q^{21} + 105132 q^{22} - 211554 q^{24} - 331252 q^{25}+ \cdots + 35642232 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −17.9825 − 10.3822i −36.2119 − 29.5923i 151.580 + 262.545i −171.660 + 297.324i 343.947 + 908.102i −505.231 − 753.847i − 3637.11i 435.597 + 2143.18i 6173.77 − 3564.43i
5.2 −17.0599 − 9.84952i −12.0893 + 45.1758i 130.026 + 225.212i 147.583 − 255.621i 651.201 − 651.618i −45.1728 + 906.368i − 2601.29i −1894.70 − 1092.29i −5035.49 + 2907.24i
5.3 −14.6587 − 8.46320i 44.2597 − 15.1022i 79.2516 + 137.268i −120.388 + 208.518i −776.603 − 153.200i 234.885 + 876.568i − 516.309i 1730.84 − 1336.84i 3529.45 − 2037.73i
5.4 −12.6634 − 7.31122i 9.88299 − 45.7092i 42.9078 + 74.3185i 269.104 − 466.101i −459.342 + 506.577i 626.737 − 656.311i 616.838i −1991.65 − 903.486i −6815.54 + 3934.95i
5.5 −10.0113 − 5.78004i 32.5439 + 33.5841i 2.81783 + 4.88062i −33.4474 + 57.9325i −131.690 − 524.327i −470.343 − 776.093i 1414.54i −68.7843 + 2185.92i 669.705 − 386.655i
5.6 −8.08214 − 4.66623i −31.7364 + 34.3482i −20.4526 − 35.4250i −123.341 + 213.633i 416.775 − 129.518i 658.195 − 624.758i 1576.30i −172.602 − 2180.18i 1993.72 − 1151.08i
5.7 −6.40713 − 3.69916i −45.1116 − 12.3265i −36.6324 − 63.4492i 101.547 − 175.885i 243.439 + 245.852i −682.340 + 598.293i 1489.02i 1883.12 + 1112.13i −1301.25 + 751.279i
5.8 −1.73270 − 1.00038i −9.93003 − 45.6990i −61.9985 − 107.385i −181.259 + 313.950i −28.5104 + 89.1165i 799.270 + 429.779i 504.184i −1989.79 + 907.584i 628.136 − 362.654i
5.9 1.73270 + 1.00038i 34.6114 + 31.4491i −61.9985 − 107.385i 181.259 − 313.950i 28.5104 + 89.1165i 799.270 + 429.779i − 504.184i 208.903 + 2177.00i 628.136 − 362.654i
5.10 6.40713 + 3.69916i −11.8808 + 45.2310i −36.6324 − 63.4492i −101.547 + 175.885i −243.439 + 245.852i −682.340 + 598.293i − 1489.02i −1904.69 − 1074.76i −1301.25 + 751.279i
5.11 8.08214 + 4.66623i −45.6146 + 10.3104i −20.4526 − 35.4250i 123.341 − 213.633i −416.775 − 129.518i 658.195 − 624.758i − 1576.30i 1974.39 − 940.611i 1993.72 − 1151.08i
5.12 10.0113 + 5.78004i −12.8127 − 44.9759i 2.81783 + 4.88062i 33.4474 − 57.9325i 131.690 − 524.327i −470.343 − 776.093i − 1414.54i −1858.67 + 1152.53i 669.705 − 386.655i
5.13 12.6634 + 7.31122i 44.5268 + 14.2957i 42.9078 + 74.3185i −269.104 + 466.101i 459.342 + 506.577i 626.737 − 656.311i − 616.838i 1778.27 + 1273.08i −6815.54 + 3934.95i
5.14 14.6587 + 8.46320i 35.2088 − 30.7789i 79.2516 + 137.268i 120.388 − 208.518i 776.603 − 153.200i 234.885 + 876.568i 516.309i 292.317 − 2167.38i 3529.45 − 2037.73i
5.15 17.0599 + 9.84952i −45.1680 − 12.1182i 130.026 + 225.212i −147.583 + 255.621i −651.201 − 651.618i −45.1728 + 906.368i 2601.29i 1893.30 + 1094.71i −5035.49 + 2907.24i
5.16 17.9825 + 10.3822i 7.52171 + 46.1565i 151.580 + 262.545i 171.660 − 297.324i −343.947 + 908.102i −505.231 − 753.847i 3637.11i −2073.85 + 694.352i 6173.77 − 3564.43i
17.1 −17.9825 + 10.3822i −36.2119 + 29.5923i 151.580 − 262.545i −171.660 − 297.324i 343.947 − 908.102i −505.231 + 753.847i 3637.11i 435.597 − 2143.18i 6173.77 + 3564.43i
17.2 −17.0599 + 9.84952i −12.0893 − 45.1758i 130.026 − 225.212i 147.583 + 255.621i 651.201 + 651.618i −45.1728 − 906.368i 2601.29i −1894.70 + 1092.29i −5035.49 − 2907.24i
17.3 −14.6587 + 8.46320i 44.2597 + 15.1022i 79.2516 − 137.268i −120.388 − 208.518i −776.603 + 153.200i 234.885 − 876.568i 516.309i 1730.84 + 1336.84i 3529.45 + 2037.73i
17.4 −12.6634 + 7.31122i 9.88299 + 45.7092i 42.9078 − 74.3185i 269.104 + 466.101i −459.342 − 506.577i 626.737 + 656.311i − 616.838i −1991.65 + 903.486i −6815.54 − 3934.95i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.8.g.b ✓ 32
3.b odd 2 1 inner 21.8.g.b ✓ 32
7.c even 3 1 147.8.c.b 32
7.d odd 6 1 inner 21.8.g.b ✓ 32
7.d odd 6 1 147.8.c.b 32
21.g even 6 1 inner 21.8.g.b ✓ 32
21.g even 6 1 147.8.c.b 32
21.h odd 6 1 147.8.c.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.8.g.b ✓ 32 1.a even 1 1 trivial
21.8.g.b ✓ 32 3.b odd 2 1 inner
21.8.g.b ✓ 32 7.d odd 6 1 inner
21.8.g.b ✓ 32 21.g even 6 1 inner
147.8.c.b 32 7.c even 3 1
147.8.c.b 32 7.d odd 6 1
147.8.c.b 32 21.g even 6 1
147.8.c.b 32 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 1599 T_{2}^{30} + 1524771 T_{2}^{28} - 958606738 T_{2}^{26} + 447659189388 T_{2}^{24} + \cdots + 68\!\cdots\!96 \) acting on \(S_{8}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display