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Properties

Label 77.4.l.b
Level $77$
Weight $4$
Character orbit 77.l
Analytic conductor $4.543$
Analytic rank $0$
Dimension $80$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [77,4,Mod(6,77)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("77.6"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(77, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 9])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 77.l (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.54314707044\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(20\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 10 q^{2} + 86 q^{4} - 15 q^{7} - 250 q^{8} + 162 q^{9} - 62 q^{11} + 128 q^{14} + 198 q^{15} - 18 q^{16} - 460 q^{18} - 874 q^{22} - 380 q^{23} + 1430 q^{25} + 860 q^{28} + 690 q^{29} - 280 q^{30}+ \cdots + 6512 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −5.00802 + 1.62721i −5.21947 + 7.18399i 15.9604 − 11.5959i −11.2359 − 3.65078i 14.4494 − 44.4707i −17.6075 − 5.74250i −36.2999 + 49.9625i −16.0233 − 49.3148i 62.2104
6.2 −5.00802 + 1.62721i 5.21947 − 7.18399i 15.9604 − 11.5959i 11.2359 + 3.65078i −14.4494 + 44.4707i −0.0204298 − 18.5202i −36.2999 + 49.9625i −16.0233 − 49.3148i −62.2104
6.3 −4.13371 + 1.34312i −0.700278 + 0.963850i 8.81145 − 6.40189i −1.85018 − 0.601161i 1.60018 − 4.92484i 13.0769 − 13.1147i −7.38724 + 10.1677i 7.90484 + 24.3286i 8.45556
6.4 −4.13371 + 1.34312i 0.700278 − 0.963850i 8.81145 − 6.40189i 1.85018 + 0.601161i −1.60018 + 4.92484i −16.5138 + 8.38418i −7.38724 + 10.1677i 7.90484 + 24.3286i −8.45556
6.5 −3.80616 + 1.23670i −3.60538 + 4.96237i 6.48532 − 4.71186i 20.4395 + 6.64118i 7.58569 − 23.3464i 13.4238 + 12.7594i −0.0383256 + 0.0527506i −3.28297 − 10.1039i −86.0090
6.6 −3.80616 + 1.23670i 3.60538 − 4.96237i 6.48532 − 4.71186i −20.4395 − 6.64118i −7.58569 + 23.3464i 7.98674 + 16.7096i −0.0383256 + 0.0527506i −3.28297 − 10.1039i 86.0090
6.7 −2.04005 + 0.662854i −1.71361 + 2.35859i −2.74969 + 1.99777i −9.85761 − 3.20293i 1.93247 − 5.94752i 8.06357 − 16.6727i 14.3719 − 19.7812i 5.71699 + 17.5951i 22.2331
6.8 −2.04005 + 0.662854i 1.71361 − 2.35859i −2.74969 + 1.99777i 9.85761 + 3.20293i −1.93247 + 5.94752i −18.3485 + 2.51676i 14.3719 − 19.7812i 5.71699 + 17.5951i −22.2331
6.9 −1.73562 + 0.563937i −4.70761 + 6.47947i −3.77778 + 2.74472i −7.62325 − 2.47694i 4.51661 − 13.9007i 4.07512 + 18.0664i 13.5903 − 18.7055i −11.4785 − 35.3272i 14.6279
6.10 −1.73562 + 0.563937i 4.70761 − 6.47947i −3.77778 + 2.74472i 7.62325 + 2.47694i −4.51661 + 13.9007i 15.9229 + 9.45848i 13.5903 − 18.7055i −11.4785 − 35.3272i −14.6279
6.11 −0.415018 + 0.134848i −4.31258 + 5.93575i −6.31808 + 4.59035i 10.0880 + 3.27779i 0.989377 − 3.04499i −11.5610 − 14.4687i 4.05508 − 5.58135i −8.29139 − 25.5183i −4.62871
6.12 −0.415018 + 0.134848i 4.31258 − 5.93575i −6.31808 + 4.59035i −10.0880 − 3.27779i −0.989377 + 3.04499i −10.1880 − 15.4662i 4.05508 − 5.58135i −8.29139 − 25.5183i 4.62871
6.13 1.58825 − 0.516052i −0.166675 + 0.229408i −4.21592 + 3.06305i −15.5125 − 5.04030i −0.146334 + 0.450369i −17.6464 + 5.62193i −12.9679 + 17.8488i 8.31861 + 25.6021i −27.2387
6.14 1.58825 − 0.516052i 0.166675 − 0.229408i −4.21592 + 3.06305i 15.5125 + 5.04030i 0.146334 − 0.450369i 10.7998 − 15.0454i −12.9679 + 17.8488i 8.31861 + 25.6021i 27.2387
6.15 2.96168 − 0.962307i −4.44588 + 6.11923i 1.37336 − 0.997804i −19.7976 − 6.43262i −7.27868 + 22.4015i 11.1839 − 14.7621i −11.5361 + 15.8781i −9.33563 − 28.7321i −64.8242
6.16 2.96168 − 0.962307i 4.44588 − 6.11923i 1.37336 − 0.997804i 19.7976 + 6.43262i 7.27868 − 22.4015i −17.4956 + 6.07481i −11.5361 + 15.8781i −9.33563 − 28.7321i 64.8242
6.17 3.04571 − 0.989610i −4.30054 + 5.91918i 1.82486 − 1.32584i 6.11040 + 1.98539i −7.24049 + 22.2839i −15.7286 + 9.77811i −10.8129 + 14.8826i −8.19864 − 25.2328i 20.5752
6.18 3.04571 − 0.989610i 4.30054 − 5.91918i 1.82486 − 1.32584i −6.11040 − 1.98539i 7.24049 − 22.2839i 14.1599 − 11.9372i −10.8129 + 14.8826i −8.19864 − 25.2328i −20.5752
6.19 4.37984 − 1.42310i −3.13063 + 4.30894i 10.6857 − 7.76359i 7.36182 + 2.39200i −7.57961 + 23.3277i 17.6038 + 5.75366i 14.0981 − 19.4043i −0.422668 − 1.30084i 35.6476
6.20 4.37984 − 1.42310i 3.13063 − 4.30894i 10.6857 − 7.76359i −7.36182 − 2.39200i 7.57961 − 23.3277i 0.0321680 + 18.5202i 14.0981 − 19.4043i −0.422668 − 1.30084i −35.6476
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.20
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.d odd 10 1 inner
77.l even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.4.l.b ✓ 80
7.b odd 2 1 inner 77.4.l.b ✓ 80
11.d odd 10 1 inner 77.4.l.b ✓ 80
77.l even 10 1 inner 77.4.l.b ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.l.b ✓ 80 1.a even 1 1 trivial
77.4.l.b ✓ 80 7.b odd 2 1 inner
77.4.l.b ✓ 80 11.d odd 10 1 inner
77.4.l.b ✓ 80 77.l even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 5 T_{2}^{39} - 49 T_{2}^{38} - 165 T_{2}^{37} + 2157 T_{2}^{36} + 4240 T_{2}^{35} + \cdots + 22\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(77, [\chi])\). Copy content Toggle raw display