↑

Properties

Label 44.4.g.a
Level $44$
Weight $4$
Character orbit 44.g
Analytic conductor $2.596$
Analytic rank $0$
Dimension $64$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [44,4,Mod(7,44)] 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("44.7"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(44, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 7])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 44 = 2^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 44.g (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.59608404025\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 5 q^{2} + 7 q^{4} - 6 q^{5} - 5 q^{6} - 5 q^{8} + 122 q^{9} - 38 q^{12} - 10 q^{13} + 66 q^{14} - 73 q^{16} - 10 q^{17} - 230 q^{18} - 180 q^{20} - 747 q^{22} - 455 q^{24} - 374 q^{25} + 330 q^{26}+ \cdots + 3904 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.82727 + 0.0808108i −8.33696 − 2.70884i 7.98694 − 0.456948i 2.66384 − 1.93540i 23.7898 + 6.98492i 9.36317 + 28.8169i −22.5443 + 1.93735i 40.3236 + 29.2968i −7.37501 + 5.68716i
7.2 −2.61226 + 1.08447i −0.385577 − 0.125282i 5.64784 − 5.66586i −12.5983 + 9.15323i 1.14309 − 0.0908796i −7.49067 − 23.0539i −8.60916 + 20.9256i −21.7105 − 15.7736i 22.9837 − 37.5732i
7.3 −2.52462 − 1.27526i 9.06350 + 2.94491i 4.74744 + 6.43908i −11.2635 + 8.18338i −19.1264 − 18.9931i 4.41947 + 13.6017i −3.77402 − 22.3105i 51.6311 + 37.5122i 38.8719 − 6.29616i
7.4 −2.47419 − 1.37055i −0.359909 − 0.116941i 4.24321 + 6.78198i 8.74268 − 6.35193i 0.730208 + 0.782607i −8.16533 − 25.1303i −1.20348 − 22.5954i −21.7276 − 15.7860i −30.3366 + 3.73362i
7.5 −2.25286 + 1.71015i 5.41384 + 1.75906i 2.15076 − 7.70547i 9.36097 − 6.80114i −15.2049 + 5.29557i 4.13125 + 12.7147i 8.33217 + 21.0375i 4.37190 + 3.17637i −9.45796 + 31.3307i
7.6 −1.36627 − 2.47655i −3.16826 − 1.02943i −4.26660 + 6.76728i −7.20176 + 5.23238i 1.77927 + 9.25284i 4.94145 + 15.2082i 22.5889 + 1.32051i −12.8653 − 9.34719i 22.7978 + 10.6867i
7.7 −0.930280 + 2.67106i −5.41384 − 1.75906i −6.26916 − 4.96967i 9.36097 − 6.80114i 9.73495 − 12.8243i −4.13125 − 12.7147i 19.1064 − 12.1221i 4.37190 + 3.17637i 9.45796 + 31.3307i
7.8 −0.237151 − 2.81847i 5.16645 + 1.67868i −7.88752 + 1.33681i 12.4335 − 9.03344i 3.50608 − 14.9596i −0.106317 − 0.327210i 5.63828 + 21.9137i 2.03079 + 1.47545i −28.4091 − 32.9010i
7.9 −0.224162 + 2.81953i 0.385577 + 0.125282i −7.89950 − 1.26406i −12.5983 + 9.15323i −0.439667 + 1.05906i 7.49067 + 23.0539i 5.33483 − 21.9895i −21.7105 − 15.7736i −22.9837 − 37.5732i
7.10 0.796820 + 2.71387i 8.33696 + 2.70884i −6.73016 + 4.32493i 2.66384 − 1.93540i −0.708387 + 24.7839i −9.36317 − 28.8169i −17.1000 − 14.8186i 40.3236 + 29.2968i 7.37501 + 5.68716i
7.11 1.14487 − 2.58636i −5.25156 − 1.70634i −5.37855 − 5.92210i −3.44642 + 2.50397i −10.4256 + 11.6289i −3.17779 − 9.78024i −21.4744 + 7.13085i 2.82388 + 2.05167i 2.53048 + 11.7804i
7.12 1.99299 + 2.00698i −9.06350 − 2.94491i −0.0559633 + 7.99980i −11.2635 + 8.18338i −12.1531 − 24.0595i −4.41947 − 13.6017i −16.1670 + 15.8312i 51.6311 + 37.5122i −38.8719 − 6.29616i
7.13 2.06803 + 1.92957i 0.359909 + 0.116941i 0.553518 + 7.98083i 8.74268 − 6.35193i 0.518657 + 0.936308i 8.16533 + 25.1303i −14.2549 + 17.5727i −21.7276 − 15.7860i 30.3366 + 3.73362i
7.14 2.10599 − 1.88807i 5.25156 + 1.70634i 0.870417 − 7.95251i −3.44642 + 2.50397i 14.2814 − 6.32176i 3.17779 + 9.78024i −13.1818 − 18.3913i 2.82388 + 2.05167i −2.53048 + 11.7804i
7.15 2.75381 − 0.645410i −5.16645 − 1.67868i 7.16689 − 3.55467i 12.4335 − 9.03344i −15.3108 − 1.28828i 0.106317 + 0.327210i 17.4420 − 14.4144i 2.03079 + 1.47545i 28.4091 − 32.9010i
7.16 2.77754 + 0.534106i 3.16826 + 1.02943i 7.42946 + 2.96700i −7.20176 + 5.23238i 8.25015 + 4.55147i −4.94145 − 15.2082i 19.0509 + 12.2091i −12.8653 − 9.34719i −22.7978 + 10.6867i
19.1 −2.82727 − 0.0808108i −8.33696 + 2.70884i 7.98694 + 0.456948i 2.66384 + 1.93540i 23.7898 − 6.98492i 9.36317 − 28.8169i −22.5443 − 1.93735i 40.3236 − 29.2968i −7.37501 − 5.68716i
19.2 −2.61226 − 1.08447i −0.385577 + 0.125282i 5.64784 + 5.66586i −12.5983 − 9.15323i 1.14309 + 0.0908796i −7.49067 + 23.0539i −8.60916 − 20.9256i −21.7105 + 15.7736i 22.9837 + 37.5732i
19.3 −2.52462 + 1.27526i 9.06350 − 2.94491i 4.74744 − 6.43908i −11.2635 − 8.18338i −19.1264 + 18.9931i 4.41947 − 13.6017i −3.77402 + 22.3105i 51.6311 − 37.5122i 38.8719 + 6.29616i
19.4 −2.47419 + 1.37055i −0.359909 + 0.116941i 4.24321 − 6.78198i 8.74268 + 6.35193i 0.730208 − 0.782607i −8.16533 + 25.1303i −1.20348 + 22.5954i −21.7276 + 15.7860i −30.3366 − 3.73362i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.d odd 10 1 inner
44.g even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 44.4.g.a ✓ 64
4.b odd 2 1 inner 44.4.g.a ✓ 64
11.d odd 10 1 inner 44.4.g.a ✓ 64
44.g even 10 1 inner 44.4.g.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.4.g.a ✓ 64 1.a even 1 1 trivial
44.4.g.a ✓ 64 4.b odd 2 1 inner
44.4.g.a ✓ 64 11.d odd 10 1 inner
44.4.g.a ✓ 64 44.g even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(44, [\chi])\).