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Properties

Label 69.6.c.b
Level $69$
Weight $6$
Character orbit 69.c
Analytic conductor $11.066$
Analytic rank $0$
Dimension $32$
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] = [69,6,Mod(68,69)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("69.68");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 69.c (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: \(11.0664835671\)
Analytic rank: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 408 q^{4} - 528 q^{6} - 444 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 408 q^{4} - 528 q^{6} - 444 q^{9} - 2484 q^{12} + 520 q^{13} + 4936 q^{16} + 7188 q^{18} + 18660 q^{24} + 36032 q^{25} - 22032 q^{27} + 6544 q^{31} - 33912 q^{36} - 63912 q^{39} + 54328 q^{46} + 88284 q^{48} - 207664 q^{49} + 46296 q^{52} - 38628 q^{54} - 139296 q^{55} - 184144 q^{58} + 486584 q^{64} - 113580 q^{69} + 37176 q^{70} - 15504 q^{72} - 93896 q^{73} + 249840 q^{75} + 368028 q^{78} - 339372 q^{81} - 23512 q^{82} + 259584 q^{85} + 509928 q^{87} + 82740 q^{93} - 562000 q^{94} + 1404 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1 − 9.90483i 13.3502 − 8.04819i −66.1056 −91.0951 −79.7159 − 132.231i 106.321i 337.810i 113.453 − 214.889i 902.281i
68.2 − 9.90483i 13.3502 − 8.04819i −66.1056 91.0951 −79.7159 − 132.231i − 106.321i 337.810i 113.453 − 214.889i − 902.281i
68.3 − 9.59549i −8.65480 − 12.9651i −60.0734 −42.3345 −124.407 + 83.0470i − 238.780i 269.378i −93.1890 + 224.421i 406.221i
68.4 − 9.59549i −8.65480 − 12.9651i −60.0734 42.3345 −124.407 + 83.0470i 238.780i 269.378i −93.1890 + 224.421i − 406.221i
68.5 − 8.75944i 10.6432 + 11.3896i −44.7279 −43.1688 99.7663 − 93.2285i − 151.444i 111.489i −16.4449 + 242.443i 378.135i
68.6 − 8.75944i 10.6432 + 11.3896i −44.7279 43.1688 99.7663 − 93.2285i 151.444i 111.489i −16.4449 + 242.443i − 378.135i
68.7 − 7.06404i −15.4896 + 1.75248i −17.9007 −82.3579 12.3796 + 109.419i 96.3127i − 99.5980i 236.858 − 54.2905i 581.779i
68.8 − 7.06404i −15.4896 + 1.75248i −17.9007 82.3579 12.3796 + 109.419i − 96.3127i − 99.5980i 236.858 − 54.2905i − 581.779i
68.9 − 5.41836i 3.03549 − 15.2901i 2.64141 −41.1824 −82.8470 − 16.4474i 67.8449i − 187.700i −224.572 − 92.8256i 223.141i
68.10 − 5.41836i 3.03549 − 15.2901i 2.64141 41.1824 −82.8470 − 16.4474i − 67.8449i − 187.700i −224.572 − 92.8256i − 223.141i
68.11 − 2.78145i −1.49341 + 15.5168i 24.2635 −76.0392 43.1591 + 4.15386i − 165.966i − 156.494i −238.539 − 46.3458i 211.500i
68.12 − 2.78145i −1.49341 + 15.5168i 24.2635 76.0392 43.1591 + 4.15386i 165.966i − 156.494i −238.539 − 46.3458i − 211.500i
68.13 − 1.57119i −12.8969 − 8.75610i 29.5314 −37.1969 −13.7575 + 20.2635i − 122.242i − 96.6776i 89.6613 + 225.854i 58.4435i
68.14 − 1.57119i −12.8969 − 8.75610i 29.5314 37.1969 −13.7575 + 20.2635i 122.242i − 96.6776i 89.6613 + 225.854i − 58.4435i
68.15 − 1.27619i 11.5059 + 10.5173i 30.3713 −80.0597 13.4221 − 14.6838i 196.848i − 79.5978i 21.7727 + 242.023i 102.171i
68.16 − 1.27619i 11.5059 + 10.5173i 30.3713 80.0597 13.4221 − 14.6838i − 196.848i − 79.5978i 21.7727 + 242.023i − 102.171i
68.17 1.27619i 11.5059 − 10.5173i 30.3713 −80.0597 13.4221 + 14.6838i − 196.848i 79.5978i 21.7727 − 242.023i − 102.171i
68.18 1.27619i 11.5059 − 10.5173i 30.3713 80.0597 13.4221 + 14.6838i 196.848i 79.5978i 21.7727 − 242.023i 102.171i
68.19 1.57119i −12.8969 + 8.75610i 29.5314 −37.1969 −13.7575 − 20.2635i 122.242i 96.6776i 89.6613 − 225.854i − 58.4435i
68.20 1.57119i −12.8969 + 8.75610i 29.5314 37.1969 −13.7575 − 20.2635i − 122.242i 96.6776i 89.6613 − 225.854i 58.4435i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.32
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.6.c.b ✓ 32
3.b odd 2 1 inner 69.6.c.b ✓ 32
23.b odd 2 1 inner 69.6.c.b ✓ 32
69.c even 2 1 inner 69.6.c.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.6.c.b ✓ 32 1.a even 1 1 trivial
69.6.c.b ✓ 32 3.b odd 2 1 inner
69.6.c.b ✓ 32 23.b odd 2 1 inner
69.6.c.b ✓ 32 69.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 358 T_{2}^{14} + 50377 T_{2}^{12} + 3516244 T_{2}^{10} + 126194032 T_{2}^{8} + \cdots + 31583309824 \) acting on \(S_{6}^{\mathrm{new}}(69, [\chi])\). Copy content Toggle raw display