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Properties

Label 760.2.cj.a
Level $760$
Weight $2$
Character orbit 760.cj
Analytic conductor $6.069$
Analytic rank $0$
Dimension $696$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [760,2,Mod(149,760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(760, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 9, 9, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("760.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.cj (of order \(18\), degree \(6\), 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: \(6.06863055362\)
Analytic rank: \(0\)
Dimension: \(696\)
Relative dimension: \(116\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 696 q - 6 q^{4} - 18 q^{6} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 696 q - 6 q^{4} - 18 q^{6} - 24 q^{9} - 15 q^{10} - 42 q^{14} - 12 q^{15} - 6 q^{16} - 42 q^{20} - 36 q^{24} - 12 q^{25} + 18 q^{26} + 18 q^{30} - 84 q^{31} + 12 q^{34} + 30 q^{36} - 48 q^{39} - 18 q^{40} - 24 q^{41} - 36 q^{44} - 6 q^{46} + 264 q^{49} + 6 q^{50} - 60 q^{54} - 42 q^{55} - 108 q^{56} - 30 q^{60} - 30 q^{64} - 6 q^{65} - 48 q^{66} - 63 q^{70} - 24 q^{71} - 126 q^{74} - 48 q^{76} - 24 q^{79} + 45 q^{80} + 12 q^{81} - 126 q^{84} - 6 q^{86} - 24 q^{89} - 66 q^{90} + 36 q^{94} + 48 q^{95} - 72 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1 −1.41361 + 0.0412957i 1.86911 − 0.680300i 1.99659 − 0.116752i 1.57330 + 1.58894i −2.61410 + 1.03887i −3.83962 + 2.21681i −2.81758 + 0.247492i 0.732625 − 0.614746i −2.28965 − 2.18117i
149.2 −1.41340 + 0.0480814i −0.853547 + 0.310666i 1.99538 − 0.135916i 2.15311 + 0.603411i 1.19146 − 0.480133i 2.16960 − 1.25262i −2.81372 + 0.288044i −1.66610 + 1.39803i −3.07221 − 0.749333i
149.3 −1.41143 + 0.0886226i −1.94489 + 0.707883i 1.98429 − 0.250170i 1.10315 + 1.94501i 2.68235 − 1.17149i 1.82545 − 1.05393i −2.77853 + 0.528951i 0.983380 − 0.825154i −1.72939 − 2.64749i
149.4 −1.41091 − 0.0966802i 2.39983 − 0.873468i 1.98131 + 0.272813i −2.17683 + 0.511279i −3.47038 + 1.00036i 0.0619134 − 0.0357457i −2.76906 − 0.576467i 2.69812 − 2.26399i 3.12073 − 0.510910i
149.5 −1.40436 + 0.166634i −1.64940 + 0.600333i 1.94447 − 0.468028i 0.944226 − 2.02693i 2.21632 − 1.11793i −3.02392 + 1.74586i −2.65275 + 0.981294i 0.0619929 − 0.0520182i −0.988282 + 3.00388i
149.6 −1.40153 − 0.189013i −0.518967 + 0.188888i 1.92855 + 0.529812i −2.11164 + 0.735512i 0.763047 − 0.166641i −3.01707 + 1.74191i −2.60277 − 1.10707i −2.06449 + 1.73231i 3.09854 − 0.631712i
149.7 −1.39742 − 0.217322i 1.09597 − 0.398899i 1.90554 + 0.607378i 2.09970 − 0.768942i −1.61821 + 0.319250i 0.483147 − 0.278945i −2.53084 − 1.26288i −1.25611 + 1.05400i −3.10126 + 0.618221i
149.8 −1.39435 + 0.236198i 0.181085 − 0.0659096i 1.88842 − 0.658684i −1.20584 − 1.88307i −0.236928 + 0.134673i 3.57472 − 2.06386i −2.47754 + 1.36448i −2.26969 + 1.90449i 2.12613 + 2.34085i
149.9 −1.38086 + 0.305315i −2.47932 + 0.902398i 1.81357 − 0.843195i −2.16660 + 0.553023i 3.14808 − 2.00306i 1.32061 − 0.762456i −2.24685 + 1.71805i 3.03456 − 2.54630i 2.82294 − 1.42514i
149.10 −1.37589 − 0.327012i 2.63491 − 0.959028i 1.78613 + 0.899863i −0.872850 − 2.05867i −3.93895 + 0.457867i 2.05129 − 1.18431i −2.16324 − 1.82219i 3.72487 − 3.12554i 0.527731 + 3.11793i
149.11 −1.32515 − 0.493943i −1.32943 + 0.483874i 1.51204 + 1.30910i 0.0572752 − 2.23533i 2.00070 + 0.0154588i −0.0930200 + 0.0537051i −1.35706 − 2.48161i −0.764873 + 0.641805i −1.18003 + 2.93386i
149.12 −1.31248 + 0.526680i 0.442441 − 0.161036i 1.44522 − 1.38252i −1.10537 − 1.94375i −0.495882 + 0.444381i −0.597304 + 0.344854i −1.16868 + 2.57569i −2.12831 + 1.78587i 2.47451 + 1.96895i
149.13 −1.30920 + 0.534788i 3.07500 − 1.11921i 1.42800 − 1.40029i 1.78750 − 1.34344i −3.42725 + 3.10974i 0.114581 − 0.0661535i −1.12068 + 2.59694i 5.90487 − 4.95477i −1.62174 + 2.71477i
149.14 −1.29814 − 0.561099i −2.30483 + 0.838890i 1.37034 + 1.45677i −0.917555 + 2.03914i 3.46270 + 0.204241i −1.38219 + 0.798010i −0.961497 − 2.65999i 2.31038 − 1.93864i 2.33527 − 2.13225i
149.15 −1.29455 + 0.569339i −0.246229 + 0.0896200i 1.35171 − 1.47407i −0.226534 + 2.22456i 0.267731 − 0.256205i −2.61757 + 1.51125i −0.910599 + 2.67784i −2.24554 + 1.88423i −0.973272 − 3.00878i
149.16 −1.29213 + 0.574799i 2.09534 − 0.762642i 1.33921 − 1.48543i 0.924482 + 2.03601i −2.26909 + 2.18983i 3.33797 − 1.92718i −0.876613 + 2.68915i 1.51070 − 1.26763i −2.36485 − 2.09940i
149.17 −1.27278 + 0.616457i 0.939590 − 0.341983i 1.23996 − 1.56923i −1.29806 + 1.82072i −0.985078 + 1.01449i 0.641202 − 0.370198i −0.610837 + 2.76168i −1.53226 + 1.28572i 0.529754 − 3.11759i
149.18 −1.24486 − 0.671057i 1.93130 − 0.702935i 1.09937 + 1.67075i −0.0703711 − 2.23496i −2.87591 − 0.420953i −4.29143 + 2.47766i −0.247394 − 2.81759i 0.937664 − 0.786793i −1.41218 + 2.82944i
149.19 −1.23832 − 0.683058i 0.426270 − 0.155150i 1.06686 + 1.69169i 0.820306 + 2.08017i −0.633834 − 0.0990427i −1.16648 + 0.673468i −0.165596 − 2.82358i −2.14050 + 1.79609i 0.405076 − 3.13623i
149.20 −1.22504 − 0.706589i −3.12152 + 1.13614i 1.00147 + 1.73120i 2.11063 − 0.738394i 4.62678 + 0.813809i 3.34997 − 1.93410i −0.00358954 − 2.82842i 6.15494 − 5.16461i −3.10736 − 0.586785i
See next 80 embeddings (of 696 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.116
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
19.e even 9 1 inner
40.f even 2 1 inner
95.p even 18 1 inner
152.t even 18 1 inner
760.cj even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.cj.a ✓ 696
5.b even 2 1 inner 760.2.cj.a ✓ 696
8.b even 2 1 inner 760.2.cj.a ✓ 696
19.e even 9 1 inner 760.2.cj.a ✓ 696
40.f even 2 1 inner 760.2.cj.a ✓ 696
95.p even 18 1 inner 760.2.cj.a ✓ 696
152.t even 18 1 inner 760.2.cj.a ✓ 696
760.cj even 18 1 inner 760.2.cj.a ✓ 696
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.cj.a ✓ 696 1.a even 1 1 trivial
760.2.cj.a ✓ 696 5.b even 2 1 inner
760.2.cj.a ✓ 696 8.b even 2 1 inner
760.2.cj.a ✓ 696 19.e even 9 1 inner
760.2.cj.a ✓ 696 40.f even 2 1 inner
760.2.cj.a ✓ 696 95.p even 18 1 inner
760.2.cj.a ✓ 696 152.t even 18 1 inner
760.2.cj.a ✓ 696 760.cj even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(760, [\chi])\).