↑

Properties

Label 7203.2.a.m
Level $7203$
Weight $2$
Character orbit 7203.a
Self dual yes
Analytic conductor $57.516$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7203,2,Mod(1,7203)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7203, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7203.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7203 = 3 \cdot 7^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7203.a (trivial)

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: yes
Analytic conductor: \(57.5162445759\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: no (minimal twist has level 147)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 6 q^{2} - 30 q^{3} + 30 q^{4} - 2 q^{5} - 6 q^{6} + 24 q^{8} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 6 q^{2} - 30 q^{3} + 30 q^{4} - 2 q^{5} - 6 q^{6} + 24 q^{8} + 30 q^{9} - 20 q^{10} + 18 q^{11} - 30 q^{12} - 13 q^{13} + 2 q^{15} + 38 q^{16} - 2 q^{17} + 6 q^{18} - 29 q^{19} - 10 q^{20} + 36 q^{22} + 32 q^{23} - 24 q^{24} + 50 q^{25} + 15 q^{26} - 30 q^{27} + 44 q^{29} + 20 q^{30} - 39 q^{31} + 64 q^{32} - 18 q^{33} - 24 q^{34} + 30 q^{36} + 59 q^{37} + 20 q^{38} + 13 q^{39} - 56 q^{40} + 7 q^{41} + 43 q^{43} + 58 q^{44} - 2 q^{45} + 26 q^{46} - 12 q^{47} - 38 q^{48} - 88 q^{50} + 2 q^{51} + 85 q^{52} - 5 q^{53} - 6 q^{54} - 64 q^{55} + 29 q^{57} - 67 q^{58} + 29 q^{59} + 10 q^{60} + 48 q^{61} - 4 q^{62} + 58 q^{64} - 40 q^{65} - 36 q^{66} + 55 q^{67} - 22 q^{68} - 32 q^{69} + 30 q^{71} + 24 q^{72} - 32 q^{73} - 77 q^{74} - 50 q^{75} + 65 q^{76} - 15 q^{78} + 51 q^{79} - 5 q^{80} + 30 q^{81} - 20 q^{82} + 20 q^{83} - 8 q^{85} + 86 q^{86} - 44 q^{87} + 158 q^{88} + 12 q^{89} - 20 q^{90} + 154 q^{92} + 39 q^{93} - 60 q^{94} + 76 q^{95} - 64 q^{96} + 84 q^{97} + 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.57322 −1.00000 4.62145 3.91327 2.57322 0 −6.74556 1.00000 −10.0697
1.2 −2.37436 −1.00000 3.63761 4.01931 2.37436 0 −3.88827 1.00000 −9.54331
1.3 −2.25450 −1.00000 3.08275 −3.70267 2.25450 0 −2.44106 1.00000 8.34765
1.4 −2.18642 −1.00000 2.78041 −0.690192 2.18642 0 −1.70630 1.00000 1.50905
1.5 −1.94613 −1.00000 1.78743 −3.28946 1.94613 0 0.413695 1.00000 6.40172
1.6 −1.87872 −1.00000 1.52958 −1.84344 1.87872 0 0.883793 1.00000 3.46331
1.7 −1.55460 −1.00000 0.416791 3.66912 1.55460 0 2.46126 1.00000 −5.70403
1.8 −1.35024 −1.00000 −0.176841 0.0991132 1.35024 0 2.93927 1.00000 −0.133827
1.9 −0.953851 −1.00000 −1.09017 3.04928 0.953851 0 2.94756 1.00000 −2.90856
1.10 −0.883366 −1.00000 −1.21966 −1.10254 0.883366 0 2.84414 1.00000 0.973945
1.11 −0.685341 −1.00000 −1.53031 3.23719 0.685341 0 2.41946 1.00000 −2.21858
1.12 −0.593081 −1.00000 −1.64825 −2.68780 0.593081 0 2.16371 1.00000 1.59408
1.13 −0.310370 −1.00000 −1.90367 −4.19120 0.310370 0 1.21158 1.00000 1.30082
1.14 −0.216894 −1.00000 −1.95296 0.832138 0.216894 0 0.857373 1.00000 −0.180486
1.15 0.0545765 −1.00000 −1.99702 3.13263 −0.0545765 0 −0.218143 1.00000 0.170968
1.16 0.230260 −1.00000 −1.94698 −1.19031 −0.230260 0 −0.908833 1.00000 −0.274081
1.17 0.397778 −1.00000 −1.84177 1.00801 −0.397778 0 −1.52817 1.00000 0.400965
1.18 0.870359 −1.00000 −1.24248 −3.33563 −0.870359 0 −2.82212 1.00000 −2.90319
1.19 1.26731 −1.00000 −0.393937 −1.29612 −1.26731 0 −3.03385 1.00000 −1.64258
1.20 1.34024 −1.00000 −0.203751 3.60834 −1.34024 0 −2.95356 1.00000 4.83605
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7203.2.a.m 30
7.b odd 2 1 7203.2.a.n 30
49.h odd 42 2 147.2.m.b ✓ 60
147.o even 42 2 441.2.bb.e 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.m.b ✓ 60 49.h odd 42 2
441.2.bb.e 60 147.o even 42 2
7203.2.a.m 30 1.a even 1 1 trivial
7203.2.a.n 30 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7203))\):

\( T_{2}^{30} - 6 T_{2}^{29} - 27 T_{2}^{28} + 218 T_{2}^{27} + 238 T_{2}^{26} - 3476 T_{2}^{25} + \cdots + 64 \) Copy content Toggle raw display
\( T_{5}^{30} + 2 T_{5}^{29} - 98 T_{5}^{28} - 204 T_{5}^{27} + 4220 T_{5}^{26} + 9176 T_{5}^{25} + \cdots - 11105856 \) Copy content Toggle raw display