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Properties

Label 750.2.m.b
Level $750$
Weight $2$
Character orbit 750.m
Analytic conductor $5.989$
Analytic rank $0$
Dimension $120$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [750,2,Mod(31,750)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

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: \(5.98878015160\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(6\) over \(\Q(\zeta_{25})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{25}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 20 q^{5} - 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 20 q^{5} - 5 q^{7} - 20 q^{10} + 5 q^{11} - 10 q^{13} - 20 q^{14} + 20 q^{15} - 5 q^{17} - 30 q^{18} + 15 q^{19} + 5 q^{20} - 15 q^{23} + 30 q^{24} - 20 q^{25} + 10 q^{26} - 5 q^{29} + 5 q^{30} - 50 q^{31} - 30 q^{32} + 30 q^{34} - 10 q^{35} + 10 q^{37} + 35 q^{38} + 10 q^{39} - 5 q^{40} - 35 q^{41} + 5 q^{43} - 25 q^{44} - 10 q^{46} + 130 q^{47} - 15 q^{49} - 20 q^{50} - 5 q^{51} + 40 q^{52} + 15 q^{53} + 10 q^{55} - 20 q^{57} - 5 q^{58} - 55 q^{59} + 50 q^{61} + 5 q^{62} + 5 q^{63} - 40 q^{65} + 20 q^{66} + 85 q^{67} - 20 q^{68} + 10 q^{69} - 45 q^{70} - 30 q^{71} - 30 q^{74} + 45 q^{75} - 5 q^{76} - 35 q^{77} - 10 q^{78} + 70 q^{79} - 45 q^{82} - 35 q^{83} - 5 q^{84} + 100 q^{85} - 15 q^{86} - 15 q^{87} + 5 q^{88} + 70 q^{89} - 5 q^{90} - 105 q^{91} + 65 q^{92} - 20 q^{94} - 65 q^{95} - 5 q^{97} + 50 q^{98} - 10 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0.0627905 + 0.998027i 0.425779 + 0.904827i −0.992115 + 0.125333i −1.79347 + 1.33546i −0.876307 + 0.481754i 0.974672 − 0.708141i −0.187381 − 0.982287i −0.637424 + 0.770513i −1.44544 − 1.70608i
31.2 0.0627905 + 0.998027i 0.425779 + 0.904827i −0.992115 + 0.125333i −1.48947 − 1.66777i −0.876307 + 0.481754i −1.22348 + 0.888912i −0.187381 − 0.982287i −0.637424 + 0.770513i 1.57096 − 1.59125i
31.3 0.0627905 + 0.998027i 0.425779 + 0.904827i −0.992115 + 0.125333i 0.460463 − 2.18814i −0.876307 + 0.481754i −2.27810 + 1.65513i −0.187381 − 0.982287i −0.637424 + 0.770513i 2.21274 + 0.322160i
31.4 0.0627905 + 0.998027i 0.425779 + 0.904827i −0.992115 + 0.125333i 1.15357 + 1.91554i −0.876307 + 0.481754i 2.74901 − 1.99727i −0.187381 − 0.982287i −0.637424 + 0.770513i −1.83932 + 1.27157i
31.5 0.0627905 + 0.998027i 0.425779 + 0.904827i −0.992115 + 0.125333i 1.87607 + 1.21670i −0.876307 + 0.481754i −3.78150 + 2.74742i −0.187381 − 0.982287i −0.637424 + 0.770513i −1.09650 + 1.94877i
31.6 0.0627905 + 0.998027i 0.425779 + 0.904827i −0.992115 + 0.125333i 1.88480 − 1.20313i −0.876307 + 0.481754i 1.71901 − 1.24893i −0.187381 − 0.982287i −0.637424 + 0.770513i 1.31911 + 1.80554i
61.1 −0.187381 + 0.982287i −0.968583 + 0.248690i −0.929776 − 0.368125i −2.06500 − 0.857776i −0.0627905 − 0.998027i 0.229915 − 0.707607i 0.535827 − 0.844328i 0.876307 − 0.481754i 1.22952 − 1.86769i
61.2 −0.187381 + 0.982287i −0.968583 + 0.248690i −0.929776 − 0.368125i −1.57781 + 1.58446i −0.0627905 − 0.998027i 0.466191 − 1.43479i 0.535827 − 0.844328i 0.876307 − 0.481754i −1.26075 − 1.84676i
61.3 −0.187381 + 0.982287i −0.968583 + 0.248690i −0.929776 − 0.368125i −0.344720 + 2.20934i −0.0627905 − 0.998027i 0.602477 − 1.85423i 0.535827 − 0.844328i 0.876307 − 0.481754i −2.10561 − 0.752603i
61.4 −0.187381 + 0.982287i −0.968583 + 0.248690i −0.929776 − 0.368125i 0.320997 − 2.21291i −0.0627905 − 0.998027i 0.370620 − 1.14065i 0.535827 − 0.844328i 0.876307 − 0.481754i 2.11356 + 0.729969i
61.5 −0.187381 + 0.982287i −0.968583 + 0.248690i −0.929776 − 0.368125i 1.27636 + 1.83600i −0.0627905 − 0.998027i −1.43236 + 4.40835i 0.535827 − 0.844328i 0.876307 − 0.481754i −2.04265 + 0.909725i
61.6 −0.187381 + 0.982287i −0.968583 + 0.248690i −0.929776 − 0.368125i 1.97154 − 1.05501i −0.0627905 − 0.998027i −0.469414 + 1.44471i 0.535827 − 0.844328i 0.876307 − 0.481754i 0.666890 + 2.13431i
91.1 0.728969 + 0.684547i −0.535827 + 0.844328i 0.0627905 + 0.998027i −2.14929 − 0.616880i −0.968583 + 0.248690i −0.0420378 − 0.129379i −0.637424 + 0.770513i −0.425779 − 0.904827i −1.14448 − 1.92098i
91.2 0.728969 + 0.684547i −0.535827 + 0.844328i 0.0627905 + 0.998027i −0.913066 + 2.04115i −0.968583 + 0.248690i 1.33490 + 4.10840i −0.637424 + 0.770513i −0.425779 − 0.904827i −2.06286 + 0.862900i
91.3 0.728969 + 0.684547i −0.535827 + 0.844328i 0.0627905 + 0.998027i −0.820559 − 2.08007i −0.968583 + 0.248690i 0.787886 + 2.42486i −0.637424 + 0.770513i −0.425779 − 0.904827i 0.825743 − 2.07802i
91.4 0.728969 + 0.684547i −0.535827 + 0.844328i 0.0627905 + 0.998027i −0.707391 + 2.12123i −0.968583 + 0.248690i 0.0878963 + 0.270517i −0.637424 + 0.770513i −0.425779 − 0.904827i −1.96774 + 1.06206i
91.5 0.728969 + 0.684547i −0.535827 + 0.844328i 0.0627905 + 0.998027i 1.24227 − 1.85924i −0.968583 + 0.248690i −1.16962 − 3.59972i −0.637424 + 0.770513i −0.425779 − 0.904827i 2.17831 − 0.504932i
91.6 0.728969 + 0.684547i −0.535827 + 0.844328i 0.0627905 + 0.998027i 2.15347 + 0.602149i −0.968583 + 0.248690i −0.426863 − 1.31375i −0.637424 + 0.770513i −0.425779 − 0.904827i 1.15761 + 1.91310i
121.1 0.0627905 − 0.998027i 0.425779 − 0.904827i −0.992115 − 0.125333i −1.79347 − 1.33546i −0.876307 − 0.481754i 0.974672 + 0.708141i −0.187381 + 0.982287i −0.637424 − 0.770513i −1.44544 + 1.70608i
121.2 0.0627905 − 0.998027i 0.425779 − 0.904827i −0.992115 − 0.125333i −1.48947 + 1.66777i −0.876307 − 0.481754i −1.22348 − 0.888912i −0.187381 + 0.982287i −0.637424 − 0.770513i 1.57096 + 1.59125i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.6
Significant digits:
Format:

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.b ✓ 120
125.g even 25 1 inner 750.2.m.b ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
750.2.m.b ✓ 120 1.a even 1 1 trivial
750.2.m.b ✓ 120 125.g even 25 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{120} + 5 T_{7}^{119} + 125 T_{7}^{118} + 665 T_{7}^{117} + 8810 T_{7}^{116} + \cdots + 45\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(750, [\chi])\). Copy content Toggle raw display