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Properties

Label 960.2.cf.a
Level $960$
Weight $2$
Character orbit 960.cf
Analytic conductor $7.666$
Analytic rank $0$
Dimension $1504$
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] = [960,2,Mod(173,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 7, 8, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.173");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.cf (of order \(16\), degree \(8\), 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: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(1504\)
Relative dimension: \(188\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1504 q - 8 q^{3} - 16 q^{6} - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1504 q - 8 q^{3} - 16 q^{6} - 16 q^{7} - 16 q^{10} - 8 q^{12} - 16 q^{13} - 8 q^{15} - 32 q^{16} - 8 q^{18} - 16 q^{21} - 48 q^{22} - 16 q^{25} - 8 q^{27} - 16 q^{28} + 56 q^{30} + 64 q^{34} - 16 q^{36} - 16 q^{37} + 48 q^{40} - 8 q^{42} - 16 q^{43} - 8 q^{45} - 32 q^{46} - 112 q^{48} - 16 q^{51} - 16 q^{52} - 16 q^{55} - 8 q^{57} + 48 q^{58} + 48 q^{60} - 32 q^{61} - 16 q^{63} - 16 q^{66} - 16 q^{67} - 16 q^{70} - 128 q^{72} - 16 q^{73} - 8 q^{75} - 144 q^{76} - 160 q^{78} - 32 q^{79} - 16 q^{81} - 96 q^{82} + 64 q^{85} + 104 q^{87} - 16 q^{88} - 8 q^{90} - 32 q^{91} + 16 q^{93} + 64 q^{94} - 16 q^{96} + 128 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
173.1 −1.41421 + 0.00479026i −1.07979 − 1.35427i 1.99995 − 0.0135488i 1.88433 − 1.20386i 1.53354 + 1.91004i 0.369171 + 0.152916i −2.82828 + 0.0287411i −0.668095 + 2.92466i −2.65907 + 1.71154i
173.2 −1.41420 + 0.00691885i 1.38736 − 1.03694i 1.99990 − 0.0195692i 0.991059 − 2.00445i −1.95482 + 1.47604i −2.74614 − 1.13749i −2.82812 + 0.0415118i 0.849513 − 2.87721i −1.38768 + 2.84154i
173.3 −1.41412 + 0.0166267i 1.73196 + 0.0176105i 1.99945 − 0.0470242i −2.04148 + 0.912335i −2.44949 + 0.00389359i −3.45721 − 1.43202i −2.82667 + 0.0997420i 2.99938 + 0.0610013i 2.87172 − 1.32409i
173.4 −1.41399 + 0.0252352i −0.578904 + 1.63244i 1.99873 − 0.0713647i −1.52302 + 1.63719i 0.777369 − 2.32286i 2.18503 + 0.905069i −2.82437 + 0.151347i −2.32974 − 1.89006i 2.11222 − 2.35340i
173.5 −1.41339 + 0.0481896i 0.643428 + 1.60810i 1.99536 − 0.136222i −2.06568 + 0.856126i −0.986911 − 2.24188i 0.502886 + 0.208302i −2.81366 + 0.288690i −2.17200 + 2.06940i 2.87836 − 1.30959i
173.6 −1.41336 − 0.0490580i 1.20565 + 1.24355i 1.99519 + 0.138673i 2.23316 − 0.114074i −1.64302 − 1.81673i 1.44707 + 0.599395i −2.81312 − 0.293876i −0.0928134 + 2.99856i −3.16186 + 0.0516734i
173.7 −1.41250 − 0.0696551i 1.66390 − 0.481076i 1.99030 + 0.196775i 0.386106 + 2.20248i −2.38376 + 0.563619i 4.35662 + 1.80457i −2.79758 − 0.416579i 2.53713 − 1.60092i −0.391959 − 3.13789i
173.8 −1.41041 + 0.103679i −0.319744 − 1.70228i 1.97850 − 0.292459i −0.600616 + 2.15389i 0.627460 + 2.36776i −3.35976 − 1.39166i −2.76017 + 0.617615i −2.79553 + 1.08859i 0.623800 − 3.10014i
173.9 −1.40299 − 0.177819i 0.195966 − 1.72093i 1.93676 + 0.498958i 1.54622 + 1.61530i −0.580953 + 2.37960i 0.142370 + 0.0589716i −2.62853 − 1.04443i −2.92319 − 0.674488i −1.88210 − 2.54120i
173.10 −1.39774 + 0.215210i −0.720335 + 1.57516i 1.90737 − 0.601617i 1.66077 + 1.49728i 0.667853 − 2.35669i −4.63029 − 1.91793i −2.53654 + 1.25139i −1.96224 − 2.26928i −2.64356 − 1.73540i
173.11 −1.39756 − 0.216415i −1.46767 − 0.919752i 1.90633 + 0.604904i −1.74237 − 1.40147i 1.85210 + 1.60303i −3.09754 − 1.28305i −2.53329 − 1.25794i 1.30811 + 2.69979i 2.13177 + 2.33572i
173.12 −1.38439 + 0.288903i −1.28211 + 1.16456i 1.83307 − 0.799909i −1.33279 − 1.79546i 1.43849 − 1.98261i −0.829798 − 0.343713i −2.30659 + 1.63697i 0.287589 − 2.98618i 2.36382 + 2.10056i
173.13 −1.37858 − 0.315461i 0.134368 − 1.72683i 1.80097 + 0.869776i −0.762833 − 2.10192i −0.729985 + 2.33819i 2.49329 + 1.03275i −2.20840 − 1.76719i −2.96389 − 0.464063i 0.388552 + 3.13832i
173.14 −1.37724 + 0.321282i 0.897190 − 1.48157i 1.79356 − 0.884961i −2.23040 + 0.159050i −0.759641 + 2.32872i 1.64420 + 0.681050i −2.18583 + 1.79504i −1.39010 − 2.65850i 3.02069 − 0.935637i
173.15 −1.37148 − 0.345036i −1.22564 + 1.22385i 1.76190 + 0.946417i 0.0691490 + 2.23500i 2.10321 − 1.25559i 0.948932 + 0.393060i −2.08986 − 1.90591i 0.00439449 − 3.00000i 0.676318 − 3.08911i
173.16 −1.37073 + 0.347985i −1.71798 + 0.220299i 1.75781 − 0.953988i 1.01112 − 1.99440i 2.27823 − 0.899804i 2.97501 + 1.23229i −2.07752 + 1.91935i 2.90294 − 0.756941i −0.691956 + 3.08564i
173.17 −1.35292 + 0.411842i 0.699595 + 1.58448i 1.66077 − 1.11438i 1.16416 − 1.90912i −1.59905 − 1.85554i −3.58023 − 1.48298i −1.78794 + 2.19163i −2.02113 + 2.21698i −0.788753 + 3.06233i
173.18 −1.34557 − 0.435249i −0.333590 + 1.69962i 1.62112 + 1.17132i 0.790264 − 2.09177i 1.18863 − 2.14177i 2.93804 + 1.21698i −1.67151 − 2.28168i −2.77744 − 1.13395i −1.97379 + 2.47066i
173.19 −1.33646 + 0.462462i −1.73111 + 0.0569926i 1.57226 − 1.23613i 1.63761 + 1.52257i 2.28721 − 0.876742i 0.882848 + 0.365688i −1.52960 + 2.37914i 2.99350 − 0.197321i −2.89274 − 1.27753i
173.20 −1.33552 − 0.465170i 1.59757 + 0.669151i 1.56723 + 1.24249i −0.318258 − 2.21330i −1.82232 − 1.63681i 2.88068 + 1.19322i −1.51511 − 2.38840i 2.10447 + 2.13803i −0.604522 + 3.10396i
See next 80 embeddings (of 1504 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 173.188
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
320.bi odd 16 1 inner
960.cf even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.2.cf.a ✓ 1504
3.b odd 2 1 inner 960.2.cf.a ✓ 1504
5.c odd 4 1 960.2.cr.a yes 1504
15.e even 4 1 960.2.cr.a yes 1504
64.i even 16 1 960.2.cr.a yes 1504
192.q odd 16 1 960.2.cr.a yes 1504
320.bi odd 16 1 inner 960.2.cf.a ✓ 1504
960.cf even 16 1 inner 960.2.cf.a ✓ 1504
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.cf.a ✓ 1504 1.a even 1 1 trivial
960.2.cf.a ✓ 1504 3.b odd 2 1 inner
960.2.cf.a ✓ 1504 320.bi odd 16 1 inner
960.2.cf.a ✓ 1504 960.cf even 16 1 inner
960.2.cr.a yes 1504 5.c odd 4 1
960.2.cr.a yes 1504 15.e even 4 1
960.2.cr.a yes 1504 64.i even 16 1
960.2.cr.a yes 1504 192.q odd 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(960, [\chi])\).