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Properties

Label 770.2.bv.a
Level $770$
Weight $2$
Character orbit 770.bv
Analytic conductor $6.148$
Analytic rank $0$
Dimension $768$
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] = [770,2,Mod(3,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([45, 10, 48]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.bv (of order \(60\), degree \(16\), 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: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(768\)
Relative dimension: \(48\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 768 q + 24 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 768 q + 24 q^{5} + 4 q^{7} + 24 q^{10} + 12 q^{11} - 24 q^{15} - 96 q^{16} + 8 q^{22} + 8 q^{23} + 16 q^{25} - 24 q^{26} - 28 q^{28} - 16 q^{30} + 252 q^{33} - 40 q^{35} + 160 q^{36} - 8 q^{37} - 44 q^{42} + 80 q^{43} + 96 q^{45} - 8 q^{46} - 24 q^{47} + 64 q^{50} - 8 q^{51} + 40 q^{53} + 16 q^{56} + 64 q^{57} - 48 q^{58} - 164 q^{63} - 88 q^{65} + 32 q^{67} - 100 q^{70} + 32 q^{71} + 120 q^{73} - 336 q^{75} - 96 q^{77} - 36 q^{80} - 24 q^{81} + 48 q^{82} - 112 q^{85} - 24 q^{87} + 4 q^{88} - 12 q^{91} - 16 q^{92} - 88 q^{93} - 44 q^{95} + 32 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −0.933580 + 0.358368i −2.69941 + 1.75302i 0.743145 − 0.669131i 2.18170 − 0.490086i 1.89189 − 2.60396i 0.165124 − 2.64059i −0.453990 + 0.891007i 2.99352 − 6.72356i −1.86116 + 1.23939i
3.2 −0.933580 + 0.358368i −2.63721 + 1.71262i 0.743145 − 0.669131i −1.28481 + 1.83010i 1.84830 − 2.54396i 2.42583 + 1.05609i −0.453990 + 0.891007i 2.80157 − 6.29244i 0.543627 − 2.16898i
3.3 −0.933580 + 0.358368i −2.34572 + 1.52333i 0.743145 − 0.669131i 2.10146 + 0.764123i 1.64400 − 2.26278i −0.853423 + 2.50433i −0.453990 + 0.891007i 1.96165 − 4.40594i −2.23572 + 0.0397239i
3.4 −0.933580 + 0.358368i −2.15925 + 1.40223i 0.743145 − 0.669131i −2.23222 + 0.131148i 1.51332 − 2.08290i −2.38311 + 1.14925i −0.453990 + 0.891007i 1.47589 − 3.31491i 2.03696 − 0.922393i
3.5 −0.933580 + 0.358368i −1.55619 + 1.01060i 0.743145 − 0.669131i 0.808226 − 2.08489i 1.09066 − 1.50116i 2.61764 − 0.384675i −0.453990 + 0.891007i 0.180196 − 0.404728i −0.00738565 + 2.23606i
3.6 −0.933580 + 0.358368i −1.50590 + 0.977944i 0.743145 − 0.669131i −0.812068 + 2.08340i 1.05542 − 1.45266i 0.435551 − 2.60965i −0.453990 + 0.891007i 0.0911544 − 0.204736i 0.0115077 − 2.23604i
3.7 −0.933580 + 0.358368i −1.36144 + 0.884131i 0.743145 − 0.669131i 0.639380 − 2.14271i 0.954172 − 1.31330i −2.62915 + 0.295904i −0.453990 + 0.891007i −0.148372 + 0.333249i 0.170965 + 2.22952i
3.8 −0.933580 + 0.358368i −1.35519 + 0.880069i 0.743145 − 0.669131i −1.14900 − 1.91828i 0.949788 − 1.30727i 1.70475 + 2.02332i −0.453990 + 0.891007i −0.158198 + 0.355318i 1.76013 + 1.37911i
3.9 −0.933580 + 0.358368i −0.971197 + 0.630703i 0.743145 − 0.669131i −2.11038 − 0.739133i 0.680667 − 0.936858i −0.542344 − 2.58957i −0.453990 + 0.891007i −0.674772 + 1.51556i 2.23509 − 0.0662510i
3.10 −0.933580 + 0.358368i −0.775531 + 0.503636i 0.743145 − 0.669131i 1.94282 + 1.10700i 0.543533 − 0.748110i 2.37238 − 1.17125i −0.453990 + 0.891007i −0.872411 + 1.95947i −2.21049 − 0.337227i
3.11 −0.933580 + 0.358368i −0.647269 + 0.420342i 0.743145 − 0.669131i 1.72525 + 1.42250i 0.453641 − 0.624383i −2.58746 − 0.552333i −0.453990 + 0.891007i −0.977939 + 2.19649i −2.12044 − 0.709742i
3.12 −0.933580 + 0.358368i −0.152462 + 0.0990097i 0.743145 − 0.669131i −0.0612266 + 2.23523i 0.106853 − 0.147071i 0.216184 + 2.63690i −0.453990 + 0.891007i −1.20677 + 2.71045i −0.743875 − 2.10871i
3.13 −0.933580 + 0.358368i −0.0326591 + 0.0212091i 0.743145 − 0.669131i 2.11379 − 0.729310i 0.0228892 − 0.0315043i 0.772445 + 2.53048i −0.453990 + 0.891007i −1.21959 + 2.73925i −1.71203 + 1.43838i
3.14 −0.933580 + 0.358368i 0.872815 − 0.566813i 0.743145 − 0.669131i −1.81587 + 1.30484i −0.611715 + 0.841954i 2.44283 − 1.01617i −0.453990 + 0.891007i −0.779681 + 1.75119i 1.22765 − 1.86893i
3.15 −0.933580 + 0.358368i 0.874801 − 0.568102i 0.743145 − 0.669131i −0.233574 − 2.22384i −0.613107 + 0.843870i −2.62366 − 0.341186i −0.453990 + 0.891007i −0.777674 + 1.74668i 1.01501 + 1.99242i
3.16 −0.933580 + 0.358368i 0.907325 − 0.589224i 0.743145 − 0.669131i −1.48749 − 1.66954i −0.635902 + 0.875244i −0.0897121 + 2.64423i −0.453990 + 0.891007i −0.744156 + 1.67140i 1.98700 + 1.02558i
3.17 −0.933580 + 0.358368i 1.03126 − 0.669709i 0.743145 − 0.669131i −0.319857 + 2.21307i −0.722763 + 0.994798i −2.37940 − 1.15692i −0.453990 + 0.891007i −0.605220 + 1.35935i −0.494482 − 2.18071i
3.18 −0.933580 + 0.358368i 1.39668 − 0.907018i 0.743145 − 0.669131i 1.40042 − 1.74322i −0.978872 + 1.34730i 1.56547 − 2.13291i −0.453990 + 0.891007i −0.0921626 + 0.207001i −0.682687 + 2.12930i
3.19 −0.933580 + 0.358368i 1.48690 − 0.965607i 0.743145 − 0.669131i −2.23596 + 0.0223949i −1.04210 + 1.43433i 2.64052 + 0.166273i −0.453990 + 0.891007i 0.0582768 − 0.130892i 2.07942 − 0.822202i
3.20 −0.933580 + 0.358368i 1.85269 − 1.20315i 0.743145 − 0.669131i 2.01523 − 0.968944i −1.29847 + 1.78719i −1.60576 + 2.10275i −0.453990 + 0.891007i 0.764685 − 1.71751i −1.53414 + 1.62678i
See next 80 embeddings (of 768 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.d odd 6 1 inner
11.c even 5 1 inner
35.k even 12 1 inner
55.k odd 20 1 inner
77.p odd 30 1 inner
385.bu even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.bv.a ✓ 768
5.c odd 4 1 inner 770.2.bv.a ✓ 768
7.d odd 6 1 inner 770.2.bv.a ✓ 768
11.c even 5 1 inner 770.2.bv.a ✓ 768
35.k even 12 1 inner 770.2.bv.a ✓ 768
55.k odd 20 1 inner 770.2.bv.a ✓ 768
77.p odd 30 1 inner 770.2.bv.a ✓ 768
385.bu even 60 1 inner 770.2.bv.a ✓ 768
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.bv.a ✓ 768 1.a even 1 1 trivial
770.2.bv.a ✓ 768 5.c odd 4 1 inner
770.2.bv.a ✓ 768 7.d odd 6 1 inner
770.2.bv.a ✓ 768 11.c even 5 1 inner
770.2.bv.a ✓ 768 35.k even 12 1 inner
770.2.bv.a ✓ 768 55.k odd 20 1 inner
770.2.bv.a ✓ 768 77.p odd 30 1 inner
770.2.bv.a ✓ 768 385.bu even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(770, [\chi])\).