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Properties

Label 750.2.m.d
Level $750$
Weight $2$
Character orbit 750.m
Analytic conductor $5.989$
Analytic rank $0$
Dimension $140$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [750,2,Mod(31,750)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(750, base_ring=CyclotomicField(50)) chi = DirichletCharacter(H, H._module([0, 24])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("750.31"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 750.m (of order \(25\), degree \(20\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [140,0,0,0,-20,0,5,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.98878015160\)
Analytic rank: \(0\)
Dimension: \(140\)
Relative dimension: \(7\) over \(\Q(\zeta_{25})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{25}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 140 q - 20 q^{5} + 5 q^{7} + 20 q^{10} + 5 q^{11} - 10 q^{13} + 20 q^{14} - 5 q^{17} + 35 q^{18} - 5 q^{19} - 5 q^{23} + 35 q^{24} - 40 q^{25} - 10 q^{26} + 25 q^{29} + 5 q^{30} + 30 q^{31} + 35 q^{32}+ \cdots - 10 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −0.0627905 − 0.998027i −0.425779 − 0.904827i −0.992115 + 0.125333i −2.16028 + 0.577210i −0.876307 + 0.481754i 2.54596 − 1.84975i 0.187381 + 0.982287i −0.637424 + 0.770513i 0.711716 + 2.11978i
31.2 −0.0627905 − 0.998027i −0.425779 − 0.904827i −0.992115 + 0.125333i −1.80451 − 1.32050i −0.876307 + 0.481754i −1.41226 + 1.02607i 0.187381 + 0.982287i −0.637424 + 0.770513i −1.20459 + 1.88387i
31.3 −0.0627905 − 0.998027i −0.425779 − 0.904827i −0.992115 + 0.125333i −0.984802 + 2.00753i −0.876307 + 0.481754i −0.654086 + 0.475221i 0.187381 + 0.982287i −0.637424 + 0.770513i 2.06540 + 0.856805i
31.4 −0.0627905 − 0.998027i −0.425779 − 0.904827i −0.992115 + 0.125333i 0.377671 − 2.20394i −0.876307 + 0.481754i 0.579424 − 0.420976i 0.187381 + 0.982287i −0.637424 + 0.770513i −2.22331 − 0.238540i
31.5 −0.0627905 − 0.998027i −0.425779 − 0.904827i −0.992115 + 0.125333i 0.438702 + 2.19261i −0.876307 + 0.481754i 2.44269 − 1.77472i 0.187381 + 0.982287i −0.637424 + 0.770513i 2.16074 − 0.575512i
31.6 −0.0627905 − 0.998027i −0.425779 − 0.904827i −0.992115 + 0.125333i 1.64002 − 1.51998i −0.876307 + 0.481754i −3.47694 + 2.52615i 0.187381 + 0.982287i −0.637424 + 0.770513i −1.61996 − 1.54134i
31.7 −0.0627905 − 0.998027i −0.425779 − 0.904827i −0.992115 + 0.125333i 2.15939 + 0.580559i −0.876307 + 0.481754i −0.247132 + 0.179552i 0.187381 + 0.982287i −0.637424 + 0.770513i 0.443825 − 2.19158i
61.1 0.187381 − 0.982287i 0.968583 − 0.248690i −0.929776 − 0.368125i −2.23432 + 0.0883165i −0.0627905 − 0.998027i 0.931547 − 2.86701i −0.535827 + 0.844328i 0.876307 − 0.481754i −0.331918 + 2.21130i
61.2 0.187381 − 0.982287i 0.968583 − 0.248690i −0.929776 − 0.368125i −1.92109 − 1.14429i −0.0627905 − 0.998027i −0.982764 + 3.02464i −0.535827 + 0.844328i 0.876307 − 0.481754i −1.48400 + 1.67264i
61.3 0.187381 − 0.982287i 0.968583 − 0.248690i −0.929776 − 0.368125i −1.54173 + 1.61959i −0.0627905 − 0.998027i 0.541008 − 1.66505i −0.535827 + 0.844328i 0.876307 − 0.481754i 1.30201 + 1.81790i
61.4 0.187381 − 0.982287i 0.968583 − 0.248690i −0.929776 − 0.368125i −0.422838 + 2.19572i −0.0627905 − 0.998027i −0.666032 + 2.04983i −0.535827 + 0.844328i 0.876307 − 0.481754i 2.07760 + 0.826787i
61.5 0.187381 − 0.982287i 0.968583 − 0.248690i −0.929776 − 0.368125i 0.451587 − 2.18999i −0.0627905 − 0.998027i −0.473017 + 1.45580i −0.535827 + 0.844328i 0.876307 − 0.481754i −2.06658 − 0.853952i
61.6 0.187381 − 0.982287i 0.968583 − 0.248690i −0.929776 − 0.368125i 1.98800 + 1.02364i −0.0627905 − 0.998027i 1.12925 − 3.47548i −0.535827 + 0.844328i 0.876307 − 0.481754i 1.37802 − 1.76098i
61.7 0.187381 − 0.982287i 0.968583 − 0.248690i −0.929776 − 0.368125i 2.23035 + 0.159814i −0.0627905 − 0.998027i −1.33060 + 4.09516i −0.535827 + 0.844328i 0.876307 − 0.481754i 0.574909 − 2.16090i
91.1 −0.728969 − 0.684547i 0.535827 − 0.844328i 0.0627905 + 0.998027i −2.20812 + 0.352409i −0.968583 + 0.248690i −0.260934 − 0.803073i 0.637424 − 0.770513i −0.425779 − 0.904827i 1.85089 + 1.25467i
91.2 −0.728969 − 0.684547i 0.535827 − 0.844328i 0.0627905 + 0.998027i −1.61863 − 1.54274i −0.968583 + 0.248690i 1.03821 + 3.19530i 0.637424 − 0.770513i −0.425779 − 0.904827i 0.123852 + 2.23264i
91.3 −0.728969 − 0.684547i 0.535827 − 0.844328i 0.0627905 + 0.998027i −1.57126 + 1.59096i −0.968583 + 0.248690i 0.0482591 + 0.148526i 0.637424 − 0.770513i −0.425779 − 0.904827i 2.23448 − 0.0841620i
91.4 −0.728969 − 0.684547i 0.535827 − 0.844328i 0.0627905 + 0.998027i −1.13246 − 1.92809i −0.968583 + 0.248690i −1.48776 − 4.57885i 0.637424 − 0.770513i −0.425779 − 0.904827i −0.494339 + 2.18074i
91.5 −0.728969 − 0.684547i 0.535827 − 0.844328i 0.0627905 + 0.998027i 0.231671 + 2.22403i −0.968583 + 0.248690i −0.975717 − 3.00295i 0.637424 − 0.770513i −0.425779 − 0.904827i 1.35358 − 1.77984i
91.6 −0.728969 − 0.684547i 0.535827 − 0.844328i 0.0627905 + 0.998027i 1.69474 + 1.45871i −0.968583 + 0.248690i 1.39889 + 4.30534i 0.637424 − 0.770513i −0.425779 − 0.904827i −0.236856 − 2.22349i
See next 80 embeddings (of 140 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.7
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
125.g even 25 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 750.2.m.d ✓ 140
125.g even 25 1 inner 750.2.m.d ✓ 140
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
750.2.m.d ✓ 140 1.a even 1 1 trivial
750.2.m.d ✓ 140 125.g even 25 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{140} - 5 T_{7}^{139} + 165 T_{7}^{138} - 755 T_{7}^{137} + 14990 T_{7}^{136} + \cdots + 13\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(750, [\chi])\). Copy content Toggle raw display