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Properties

Label 525.3.l.e
Level $525$
Weight $3$
Character orbit 525.l
Analytic conductor $14.305$
Analytic rank $0$
Dimension $24$
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] = [525,3,Mod(43,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.43");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.l (of order \(4\), degree \(2\), 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: \(14.3052138789\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{2} + 24 q^{6} + 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{2} + 24 q^{6} + 48 q^{8} + 48 q^{12} - 64 q^{13} - 184 q^{16} - 24 q^{17} - 24 q^{18} - 8 q^{22} - 8 q^{23} - 80 q^{26} + 96 q^{31} - 56 q^{32} + 72 q^{33} + 168 q^{36} - 8 q^{37} - 56 q^{38} + 320 q^{41} + 112 q^{43} + 320 q^{46} - 64 q^{47} - 192 q^{48} - 192 q^{51} - 96 q^{52} + 72 q^{53} - 336 q^{56} - 48 q^{57} + 512 q^{58} - 496 q^{61} + 776 q^{62} - 192 q^{66} + 192 q^{67} - 568 q^{68} - 144 q^{71} - 144 q^{72} - 224 q^{73} + 416 q^{76} - 112 q^{77} + 216 q^{78} - 216 q^{81} - 352 q^{82} + 32 q^{83} + 240 q^{86} - 384 q^{87} - 216 q^{88} - 1304 q^{92} + 168 q^{96} + 816 q^{97} + 56 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −2.74240 + 2.74240i 1.22474 + 1.22474i − 11.0415i 0 −6.71747 −1.87083 + 1.87083i 19.3105 + 19.3105i 3.00000i 0
43.2 −2.41688 + 2.41688i −1.22474 − 1.22474i − 7.68258i 0 5.92011 1.87083 − 1.87083i 8.90034 + 8.90034i 3.00000i 0
43.3 −2.24469 + 2.24469i −1.22474 − 1.22474i − 6.07726i 0 5.49834 −1.87083 + 1.87083i 4.66280 + 4.66280i 3.00000i 0
43.4 −2.08980 + 2.08980i 1.22474 + 1.22474i − 4.73454i 0 −5.11895 1.87083 − 1.87083i 1.53505 + 1.53505i 3.00000i 0
43.5 −0.992944 + 0.992944i −1.22474 − 1.22474i 2.02813i 0 2.43221 −1.87083 + 1.87083i −5.98559 − 5.98559i 3.00000i 0
43.6 −0.675544 + 0.675544i −1.22474 − 1.22474i 3.08728i 0 1.65474 1.87083 − 1.87083i −4.78777 − 4.78777i 3.00000i 0
43.7 −0.408558 + 0.408558i 1.22474 + 1.22474i 3.66616i 0 −1.00076 1.87083 − 1.87083i −3.13207 − 3.13207i 3.00000i 0
43.8 0.867675 − 0.867675i −1.22474 − 1.22474i 2.49428i 0 −2.12536 1.87083 − 1.87083i 5.63493 + 5.63493i 3.00000i 0
43.9 1.01289 − 1.01289i −1.22474 − 1.22474i 1.94811i 0 −2.48106 −1.87083 + 1.87083i 6.02478 + 6.02478i 3.00000i 0
43.10 1.36784 − 1.36784i 1.22474 + 1.22474i 0.258033i 0 3.35051 −1.87083 + 1.87083i 5.82430 + 5.82430i 3.00000i 0
43.11 1.59930 − 1.59930i 1.22474 + 1.22474i − 1.11554i 0 3.91747 −1.87083 + 1.87083i 4.61313 + 4.61313i 3.00000i 0
43.12 2.72310 − 2.72310i 1.22474 + 1.22474i − 10.8306i 0 6.67022 1.87083 − 1.87083i −18.6004 − 18.6004i 3.00000i 0
232.1 −2.74240 − 2.74240i 1.22474 − 1.22474i 11.0415i 0 −6.71747 −1.87083 − 1.87083i 19.3105 − 19.3105i − 3.00000i 0
232.2 −2.41688 − 2.41688i −1.22474 + 1.22474i 7.68258i 0 5.92011 1.87083 + 1.87083i 8.90034 − 8.90034i − 3.00000i 0
232.3 −2.24469 − 2.24469i −1.22474 + 1.22474i 6.07726i 0 5.49834 −1.87083 − 1.87083i 4.66280 − 4.66280i − 3.00000i 0
232.4 −2.08980 − 2.08980i 1.22474 − 1.22474i 4.73454i 0 −5.11895 1.87083 + 1.87083i 1.53505 − 1.53505i − 3.00000i 0
232.5 −0.992944 − 0.992944i −1.22474 + 1.22474i − 2.02813i 0 2.43221 −1.87083 − 1.87083i −5.98559 + 5.98559i − 3.00000i 0
232.6 −0.675544 − 0.675544i −1.22474 + 1.22474i − 3.08728i 0 1.65474 1.87083 + 1.87083i −4.78777 + 4.78777i − 3.00000i 0
232.7 −0.408558 − 0.408558i 1.22474 − 1.22474i − 3.66616i 0 −1.00076 1.87083 + 1.87083i −3.13207 + 3.13207i − 3.00000i 0
232.8 0.867675 + 0.867675i −1.22474 + 1.22474i − 2.49428i 0 −2.12536 1.87083 + 1.87083i 5.63493 − 5.63493i − 3.00000i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.l.e 24
5.b even 2 1 105.3.l.a ✓ 24
5.c odd 4 1 105.3.l.a ✓ 24
5.c odd 4 1 inner 525.3.l.e 24
15.d odd 2 1 315.3.o.b 24
15.e even 4 1 315.3.o.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.l.a ✓ 24 5.b even 2 1
105.3.l.a ✓ 24 5.c odd 4 1
315.3.o.b 24 15.d odd 2 1
315.3.o.b 24 15.e even 4 1
525.3.l.e 24 1.a even 1 1 trivial
525.3.l.e 24 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 8 T_{2}^{23} + 32 T_{2}^{22} + 48 T_{2}^{21} + 278 T_{2}^{20} + 2056 T_{2}^{19} + \cdots + 8151025 \) acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display