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Properties

Label 750.2.m.c
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
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 20 q^{5} - 5 q^{7} + 20 q^{10} + 10 q^{13} + 20 q^{14} + 15 q^{17} + 30 q^{18} - 5 q^{19} - 5 q^{21} + 10 q^{22} - 15 q^{23} - 30 q^{24} - 20 q^{25} - 10 q^{26} + 5 q^{28} - 5 q^{29} + 5 q^{30} + 55 q^{31} + 30 q^{32} + 10 q^{33} + 25 q^{34} - 10 q^{35} - 5 q^{37} - 35 q^{38} - 10 q^{39} - 5 q^{40} + 55 q^{41} + 5 q^{42} + 5 q^{43} + 15 q^{44} - 5 q^{45} - 10 q^{46} + 10 q^{47} - 25 q^{49} + 20 q^{50} + 15 q^{51} - 40 q^{52} + 30 q^{53} - 10 q^{55} - 5 q^{56} + 60 q^{57} + 5 q^{58} + 20 q^{59} - 20 q^{60} - 60 q^{61} + 25 q^{62} + 5 q^{63} + 15 q^{65} - 25 q^{66} + 95 q^{67} + 60 q^{68} - 10 q^{69} + 40 q^{70} - 90 q^{71} - 30 q^{74} - 55 q^{75} + 15 q^{76} + 95 q^{77} - 5 q^{78} - 90 q^{79} - 5 q^{80} - 45 q^{82} + 95 q^{83} - 5 q^{84} + 115 q^{85} - 15 q^{86} + 10 q^{87} + 65 q^{89} + 5 q^{90} - 85 q^{91} - 65 q^{92} - 30 q^{93} - 10 q^{94} - 55 q^{95} - 35 q^{98} + 10 q^{99}+O(q^{100}) \) 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 −2.23585 + 0.0310538i 0.876307 − 0.481754i 2.95579 − 2.14750i 0.187381 + 0.982287i −0.637424 + 0.770513i 0.171383 + 2.22949i
31.2 −0.0627905 − 0.998027i 0.425779 + 0.904827i −0.992115 + 0.125333i −1.18727 − 1.89484i 0.876307 − 0.481754i −0.762228 + 0.553791i 0.187381 + 0.982287i −0.637424 + 0.770513i −1.81655 + 1.30390i
31.3 −0.0627905 − 0.998027i 0.425779 + 0.904827i −0.992115 + 0.125333i −0.425058 + 2.19530i 0.876307 − 0.481754i −3.26211 + 2.37006i 0.187381 + 0.982287i −0.637424 + 0.770513i 2.21765 + 0.286376i
31.4 −0.0627905 − 0.998027i 0.425779 + 0.904827i −0.992115 + 0.125333i 1.03434 − 1.98246i 0.876307 − 0.481754i −3.43331 + 2.49445i 0.187381 + 0.982287i −0.637424 + 0.770513i −2.04349 − 0.907816i
31.5 −0.0627905 − 0.998027i 0.425779 + 0.904827i −0.992115 + 0.125333i 1.47222 + 1.68302i 0.876307 − 0.481754i 3.13516 − 2.27783i 0.187381 + 0.982287i −0.637424 + 0.770513i 1.58726 − 1.57499i
31.6 −0.0627905 − 0.998027i 0.425779 + 0.904827i −0.992115 + 0.125333i 2.08569 + 0.806166i 0.876307 − 0.481754i −0.776870 + 0.564429i 0.187381 + 0.982287i −0.637424 + 0.770513i 0.673614 − 2.13219i
61.1 0.187381 − 0.982287i −0.968583 + 0.248690i −0.929776 − 0.368125i −2.18926 − 0.455140i 0.0627905 + 0.998027i −0.678564 + 2.08841i −0.535827 + 0.844328i 0.876307 − 0.481754i −0.857304 + 2.06519i
61.2 0.187381 − 0.982287i −0.968583 + 0.248690i −0.929776 − 0.368125i −1.23586 + 1.86351i 0.0627905 + 0.998027i 0.110092 − 0.338830i −0.535827 + 0.844328i 0.876307 − 0.481754i 1.59892 + 1.56315i
61.3 0.187381 − 0.982287i −0.968583 + 0.248690i −0.929776 − 0.368125i −1.03501 − 1.98211i 0.0627905 + 0.998027i 1.53574 − 4.72653i −0.535827 + 0.844328i 0.876307 − 0.481754i −2.14094 + 0.645264i
61.4 0.187381 − 0.982287i −0.968583 + 0.248690i −0.929776 − 0.368125i 1.50692 + 1.65202i 0.0627905 + 0.998027i 0.459838 − 1.41523i −0.535827 + 0.844328i 0.876307 − 0.481754i 1.90513 − 1.17067i
61.5 0.187381 − 0.982287i −0.968583 + 0.248690i −0.929776 − 0.368125i 1.67563 − 1.48063i 0.0627905 + 0.998027i −0.697030 + 2.14524i −0.535827 + 0.844328i 0.876307 − 0.481754i −1.14042 − 1.92339i
61.6 0.187381 − 0.982287i −0.968583 + 0.248690i −0.929776 − 0.368125i 2.11026 + 0.739458i 0.0627905 + 0.998027i −1.29381 + 3.98193i −0.535827 + 0.844328i 0.876307 − 0.481754i 1.12178 − 1.93432i
91.1 −0.728969 − 0.684547i −0.535827 + 0.844328i 0.0627905 + 0.998027i −1.58297 − 1.57930i 0.968583 − 0.248690i 0.654948 + 2.01572i 0.637424 − 0.770513i −0.425779 − 0.904827i 0.0728305 + 2.23488i
91.2 −0.728969 − 0.684547i −0.535827 + 0.844328i 0.0627905 + 0.998027i −1.52290 + 1.63731i 0.968583 − 0.248690i −0.492764 − 1.51657i 0.637424 − 0.770513i −0.425779 − 0.904827i 2.23096 − 0.151051i
91.3 −0.728969 − 0.684547i −0.535827 + 0.844328i 0.0627905 + 0.998027i 0.530678 + 2.17218i 0.968583 − 0.248690i 0.441871 + 1.35994i 0.637424 − 0.770513i −0.425779 − 0.904827i 1.10011 − 1.94673i
91.4 −0.728969 − 0.684547i −0.535827 + 0.844328i 0.0627905 + 0.998027i 0.595540 − 2.15530i 0.968583 − 0.248690i −0.577883 − 1.77854i 0.637424 − 0.770513i −0.425779 − 0.904827i −1.90954 + 1.16347i
91.5 −0.728969 − 0.684547i −0.535827 + 0.844328i 0.0627905 + 0.998027i 1.97321 − 1.05187i 0.968583 − 0.248690i 1.27023 + 3.90938i 0.637424 − 0.770513i −0.425779 − 0.904827i −2.15847 − 0.583972i
91.6 −0.728969 − 0.684547i −0.535827 + 0.844328i 0.0627905 + 0.998027i 2.08356 + 0.811657i 0.968583 − 0.248690i −0.330294 − 1.01654i 0.637424 − 0.770513i −0.425779 − 0.904827i −0.963230 − 2.01797i
121.1 −0.0627905 + 0.998027i 0.425779 − 0.904827i −0.992115 − 0.125333i −2.23585 − 0.0310538i 0.876307 + 0.481754i 2.95579 + 2.14750i 0.187381 − 0.982287i −0.637424 − 0.770513i 0.171383 − 2.22949i
121.2 −0.0627905 + 0.998027i 0.425779 − 0.904827i −0.992115 − 0.125333i −1.18727 + 1.89484i 0.876307 + 0.481754i −0.762228 − 0.553791i 0.187381 − 0.982287i −0.637424 − 0.770513i −1.81655 − 1.30390i
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.c ✓ 120
125.g even 25 1 inner 750.2.m.c ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
750.2.m.c ✓ 120 1.a even 1 1 trivial
750.2.m.c ✓ 120 125.g even 25 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{120} + 5 T_{7}^{119} + 130 T_{7}^{118} + 650 T_{7}^{117} + 9440 T_{7}^{116} + \cdots + 29\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(750, [\chi])\). Copy content Toggle raw display