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Properties

Label 45.21.g.c
Level $45$
Weight $21$
Character orbit 45.g
Analytic conductor $114.081$
Analytic rank $0$
Dimension $40$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,21,Mod(28,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 21, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.28");
 
S:= CuspForms(chi, 21);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 21 \)
Character orbit: \([\chi]\) \(=\) 45.g (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: \(114.081194296\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 205575700 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 205575700 q^{7} + 12601483240 q^{10} - 177887245100 q^{13} - 14966285850740 q^{16} - 120667544894500 q^{22} + 323482147040860 q^{25} + 19\!\cdots\!00 q^{28}+ \cdots - 15\!\cdots\!00 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1 −1421.95 + 1421.95i 0 − 2.99531e6i −9.71738e6 + 969486.i 0 −2.28395e8 + 2.28395e8i 2.76815e9 + 2.76815e9i 0 1.24391e10 − 1.51962e10i
28.2 −1232.14 + 1232.14i 0 − 1.98774e6i 3.03230e6 − 9.28292e6i 0 2.26327e8 − 2.26327e8i 1.15718e9 + 1.15718e9i 0 7.70162e9 + 1.51740e10i
28.3 −1148.57 + 1148.57i 0 − 1.58987e6i 2.57160e6 + 9.42095e6i 0 1.49583e8 − 1.49583e8i 6.21719e8 + 6.21719e8i 0 −1.37743e10 − 7.86700e9i
28.4 −1049.53 + 1049.53i 0 − 1.15445e6i 9.76136e6 − 288449.i 0 −3.15905e8 + 3.15905e8i 1.11120e8 + 1.11120e8i 0 −9.94211e9 + 1.05476e10i
28.5 −790.729 + 790.729i 0 − 201927.i −8.63190e6 − 4.56702e6i 0 −9.57398e7 + 9.57398e7i −6.69469e8 − 6.69469e8i 0 1.04368e10 − 3.21421e9i
28.6 −764.666 + 764.666i 0 − 120852.i −6.95816e6 + 6.85211e6i 0 1.70437e8 − 1.70437e8i −7.09399e8 − 7.09399e8i 0 8.10897e7 − 1.05602e10i
28.7 −492.830 + 492.830i 0 562814.i 1.66607e6 − 9.62246e6i 0 6.52529e7 − 6.52529e7i −7.94141e8 − 7.94141e8i 0 3.92114e9 + 5.56332e9i
28.8 −336.121 + 336.121i 0 822622.i 9.33673e6 + 2.86233e6i 0 −1.25334e7 + 1.25334e7i −6.28948e8 − 6.28948e8i 0 −4.10036e9 + 2.17618e9i
28.9 −219.628 + 219.628i 0 952103.i −2.50873e6 + 9.43789e6i 0 −2.50717e8 + 2.50717e8i −4.39405e8 − 4.39405e8i 0 −1.52184e9 − 2.62381e9i
28.10 −185.168 + 185.168i 0 980001.i 9.62201e6 + 1.66865e6i 0 3.43083e8 − 3.43083e8i −3.75628e8 − 3.75628e8i 0 −2.09067e9 + 1.47271e9i
28.11 185.168 − 185.168i 0 980001.i −9.62201e6 − 1.66865e6i 0 3.43083e8 − 3.43083e8i 3.75628e8 + 3.75628e8i 0 −2.09067e9 + 1.47271e9i
28.12 219.628 − 219.628i 0 952103.i 2.50873e6 − 9.43789e6i 0 −2.50717e8 + 2.50717e8i 4.39405e8 + 4.39405e8i 0 −1.52184e9 − 2.62381e9i
28.13 336.121 − 336.121i 0 822622.i −9.33673e6 − 2.86233e6i 0 −1.25334e7 + 1.25334e7i 6.28948e8 + 6.28948e8i 0 −4.10036e9 + 2.17618e9i
28.14 492.830 − 492.830i 0 562814.i −1.66607e6 + 9.62246e6i 0 6.52529e7 − 6.52529e7i 7.94141e8 + 7.94141e8i 0 3.92114e9 + 5.56332e9i
28.15 764.666 − 764.666i 0 − 120852.i 6.95816e6 − 6.85211e6i 0 1.70437e8 − 1.70437e8i 7.09399e8 + 7.09399e8i 0 8.10897e7 − 1.05602e10i
28.16 790.729 − 790.729i 0 − 201927.i 8.63190e6 + 4.56702e6i 0 −9.57398e7 + 9.57398e7i 6.69469e8 + 6.69469e8i 0 1.04368e10 − 3.21421e9i
28.17 1049.53 − 1049.53i 0 − 1.15445e6i −9.76136e6 + 288449.i 0 −3.15905e8 + 3.15905e8i −1.11120e8 − 1.11120e8i 0 −9.94211e9 + 1.05476e10i
28.18 1148.57 − 1148.57i 0 − 1.58987e6i −2.57160e6 − 9.42095e6i 0 1.49583e8 − 1.49583e8i −6.21719e8 − 6.21719e8i 0 −1.37743e10 − 7.86700e9i
28.19 1232.14 − 1232.14i 0 − 1.98774e6i −3.03230e6 + 9.28292e6i 0 2.26327e8 − 2.26327e8i −1.15718e9 − 1.15718e9i 0 7.70162e9 + 1.51740e10i
28.20 1421.95 − 1421.95i 0 − 2.99531e6i 9.71738e6 − 969486.i 0 −2.28395e8 + 2.28395e8i −2.76815e9 − 2.76815e9i 0 1.24391e10 − 1.51962e10i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.21.g.c ✓ 40
3.b odd 2 1 inner 45.21.g.c ✓ 40
5.c odd 4 1 inner 45.21.g.c ✓ 40
15.e even 4 1 inner 45.21.g.c ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.21.g.c ✓ 40 1.a even 1 1 trivial
45.21.g.c ✓ 40 3.b odd 2 1 inner
45.21.g.c ✓ 40 5.c odd 4 1 inner
45.21.g.c ✓ 40 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 40619323536925 T_{2}^{36} + \cdots + 57\!\cdots\!00 \) acting on \(S_{21}^{\mathrm{new}}(45, [\chi])\). Copy content Toggle raw display