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Properties

Label 165.6.c.a
Level $165$
Weight $6$
Character orbit 165.c
Analytic conductor $26.463$
Analytic rank $0$
Dimension $26$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [165,6,Mod(34,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.34");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 165.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: \(26.4633302691\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 458 q^{4} - 98 q^{5} - 54 q^{6} - 2106 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q - 458 q^{4} - 98 q^{5} - 54 q^{6} - 2106 q^{9} - 1944 q^{10} - 3146 q^{11} + 5516 q^{14} + 8282 q^{16} + 620 q^{19} + 1594 q^{20} - 5112 q^{21} + 6642 q^{24} + 10778 q^{25} + 4560 q^{26} + 9696 q^{29} - 9378 q^{30} - 48 q^{31} - 11036 q^{34} + 17748 q^{35} + 37098 q^{36} + 6588 q^{39} + 46308 q^{40} - 30240 q^{41} + 55418 q^{44} + 7938 q^{45} - 40560 q^{46} - 163674 q^{49} + 81164 q^{50} - 37008 q^{51} + 4374 q^{54} + 11858 q^{55} - 393204 q^{56} + 131184 q^{59} + 127260 q^{60} - 104084 q^{61} - 371954 q^{64} + 43108 q^{65} + 6534 q^{66} + 65016 q^{69} + 228364 q^{70} - 155640 q^{71} + 480056 q^{74} + 74232 q^{75} - 542348 q^{76} + 224516 q^{79} + 428642 q^{80} + 170586 q^{81} + 142128 q^{84} - 124796 q^{85} - 638500 q^{86} + 591620 q^{89} + 157464 q^{90} - 48400 q^{91} + 716288 q^{94} - 35124 q^{95} - 685998 q^{96} + 254826 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 − 11.2895i − 9.00000i −95.4537 55.2796 − 8.31644i −101.606 156.135i 716.362i −81.0000 −93.8888 − 624.081i
34.2 − 10.3606i 9.00000i −75.3416 −51.1216 − 22.6182i 93.2452 248.805i 449.044i −81.0000 −234.338 + 529.649i
34.3 − 9.81979i − 9.00000i −64.4282 −26.0681 − 49.4515i −88.3781 − 193.324i 318.438i −81.0000 −485.603 + 255.984i
34.4 − 8.30687i 9.00000i −37.0041 41.8926 + 37.0137i 74.7618 54.6353i 41.5686i −81.0000 307.468 − 347.996i
34.5 − 7.80204i 9.00000i −28.8719 −42.3840 + 36.4500i 70.2184 − 85.0125i − 24.4055i −81.0000 284.384 + 330.682i
34.6 − 7.38043i − 9.00000i −22.4707 −15.8201 − 53.6164i −66.4239 191.451i − 70.3301i −81.0000 −395.712 + 116.759i
34.7 − 6.98295i 9.00000i −16.7616 16.7841 − 53.3225i 62.8465 45.4926i − 106.409i −81.0000 −372.349 − 117.203i
34.8 − 5.95684i − 9.00000i −3.48390 −55.6004 − 5.79634i −53.6115 − 79.6742i − 169.866i −81.0000 −34.5279 + 331.202i
34.9 − 3.87461i − 9.00000i 16.9874 −20.4262 + 52.0362i −34.8714 − 134.363i − 189.807i −81.0000 201.620 + 79.1434i
34.10 − 3.81085i − 9.00000i 17.4775 54.9847 + 10.0835i −34.2976 89.4064i − 188.551i −81.0000 38.4268 − 209.538i
34.11 − 3.62063i 9.00000i 18.8910 −7.10978 − 55.4477i 32.5857 − 187.526i − 184.258i −81.0000 −200.756 + 25.7419i
34.12 − 1.48806i 9.00000i 29.7857 −54.4419 + 12.6919i 13.3926 − 7.94259i − 91.9409i −81.0000 18.8863 + 81.0128i
34.13 − 0.570908i 9.00000i 31.6741 55.0310 − 9.82803i 5.13817 245.179i − 36.3520i −81.0000 −5.61090 − 31.4176i
34.14 0.570908i − 9.00000i 31.6741 55.0310 + 9.82803i 5.13817 − 245.179i 36.3520i −81.0000 −5.61090 + 31.4176i
34.15 1.48806i − 9.00000i 29.7857 −54.4419 − 12.6919i 13.3926 7.94259i 91.9409i −81.0000 18.8863 − 81.0128i
34.16 3.62063i − 9.00000i 18.8910 −7.10978 + 55.4477i 32.5857 187.526i 184.258i −81.0000 −200.756 − 25.7419i
34.17 3.81085i 9.00000i 17.4775 54.9847 − 10.0835i −34.2976 − 89.4064i 188.551i −81.0000 38.4268 + 209.538i
34.18 3.87461i 9.00000i 16.9874 −20.4262 − 52.0362i −34.8714 134.363i 189.807i −81.0000 201.620 − 79.1434i
34.19 5.95684i 9.00000i −3.48390 −55.6004 + 5.79634i −53.6115 79.6742i 169.866i −81.0000 −34.5279 − 331.202i
34.20 6.98295i − 9.00000i −16.7616 16.7841 + 53.3225i 62.8465 − 45.4926i 106.409i −81.0000 −372.349 + 117.203i
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.26
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.6.c.a ✓ 26
5.b even 2 1 inner 165.6.c.a ✓ 26
5.c odd 4 1 825.6.a.w 13
5.c odd 4 1 825.6.a.x 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.c.a ✓ 26 1.a even 1 1 trivial
165.6.c.a ✓ 26 5.b even 2 1 inner
825.6.a.w 13 5.c odd 4 1
825.6.a.x 13 5.c odd 4 1