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Properties

Label 9025.2.a.cr
Level $9025$
Weight $2$
Character orbit 9025.a
Self dual yes
Analytic conductor $72.065$
Analytic rank $0$
Dimension $21$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9025,2,Mod(1,9025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9025, 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("9025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.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: \(72.0649878242\)
Analytic rank: \(0\)
Dimension: \(21\)
Twist minimal: no (minimal twist has level 475)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 6 q^{2} + 9 q^{3} + 24 q^{4} + 18 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q + 6 q^{2} + 9 q^{3} + 24 q^{4} + 18 q^{8} + 24 q^{9} + 24 q^{12} + 12 q^{13} - 6 q^{14} + 30 q^{16} - 9 q^{17} + 18 q^{18} + 12 q^{22} - 6 q^{23} + 18 q^{24} + 18 q^{26} + 30 q^{27} - 15 q^{28} - 3 q^{29} - 6 q^{31} + 57 q^{32} + 45 q^{33} - 3 q^{34} + 60 q^{36} + 24 q^{37} - 30 q^{39} + 12 q^{41} + 18 q^{42} + 18 q^{43} + 24 q^{44} - 15 q^{46} + 18 q^{47} + 84 q^{48} + 33 q^{49} - 12 q^{51} + 36 q^{52} + 42 q^{53} - 18 q^{56} + 12 q^{58} + 6 q^{59} - 36 q^{61} - 3 q^{62} - 6 q^{63} - 21 q^{66} + 24 q^{67} - 78 q^{68} + 15 q^{69} + 12 q^{71} + 87 q^{72} + 18 q^{73} - 45 q^{74} - 9 q^{77} - 60 q^{78} + 3 q^{79} + 21 q^{81} + 42 q^{82} + 36 q^{84} + 30 q^{86} - 6 q^{87} + 60 q^{88} + 3 q^{89} - 3 q^{91} + 60 q^{92} + 12 q^{93} + 18 q^{94} + 111 q^{96} + 12 q^{97} + 105 q^{98} + 84 q^{99}+O(q^{100}) \) 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 −2.31644 2.09061 3.36590 0 −4.84277 2.98971 −3.16403 1.37064 0
1.2 −2.28640 −1.01181 3.22763 0 2.31341 0.320153 −2.80684 −1.97624 0
1.3 −2.24024 1.26498 3.01866 0 −2.83386 −1.37484 −2.28203 −1.39981 0
1.4 −2.15200 3.25314 2.63110 0 −7.00075 −2.62239 −1.35813 7.58291 0
1.5 −1.75619 −2.00195 1.08421 0 3.51581 0.476786 1.60831 1.00782 0
1.6 −0.947979 −2.35813 −1.10134 0 2.23545 −2.44908 2.94000 2.56075 0
1.7 −0.617674 −0.0232442 −1.61848 0 0.0143573 4.67118 2.23504 −2.99946 0
1.8 −0.558603 0.670247 −1.68796 0 −0.374402 −2.55490 2.06011 −2.55077 0
1.9 −0.471157 1.07129 −1.77801 0 −0.504745 2.74411 1.78004 −1.85234 0
1.10 0.0440787 3.34883 −1.99806 0 0.147612 −1.48290 −0.176229 8.21469 0
1.11 0.369763 2.43235 −1.86328 0 0.899393 2.99691 −1.42849 2.91634 0
1.12 0.587076 −2.63527 −1.65534 0 −1.54710 1.99298 −2.14596 3.94466 0
1.13 1.17921 1.37815 −0.609457 0 1.62514 −4.21997 −3.07711 −1.10069 0
1.14 1.27496 0.195863 −0.374490 0 0.249717 −1.39855 −3.02737 −2.96164 0
1.15 1.56420 −0.289164 0.446737 0 −0.452312 −4.01995 −2.42962 −2.91638 0
1.16 1.89059 −2.61121 1.57432 0 −4.93672 0.975485 −0.804792 3.81841 0
1.17 1.95063 −0.963432 1.80494 0 −1.87930 5.24962 −0.380482 −2.07180 0
1.18 2.46757 2.83450 4.08889 0 6.99432 4.66735 5.15448 5.03438 0
1.19 2.59627 1.65081 4.74062 0 4.28594 −2.00548 7.11538 −0.274832 0
1.20 2.67570 −2.23729 5.15937 0 −5.98632 −3.57682 8.45352 2.00548 0
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(19\) \(-1\)

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 9025.2.a.cr 21
5.b even 2 1 9025.2.a.cp 21
19.b odd 2 1 9025.2.a.cq 21
19.f odd 18 2 475.2.l.d ✓ 42
95.d odd 2 1 9025.2.a.cs 21
95.o odd 18 2 475.2.l.e yes 42
95.r even 36 4 475.2.u.d 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.l.d ✓ 42 19.f odd 18 2
475.2.l.e yes 42 95.o odd 18 2
475.2.u.d 84 95.r even 36 4
9025.2.a.cp 21 5.b even 2 1
9025.2.a.cq 21 19.b odd 2 1
9025.2.a.cr 21 1.a even 1 1 trivial
9025.2.a.cs 21 95.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):

\( T_{2}^{21} - 6 T_{2}^{20} - 15 T_{2}^{19} + 148 T_{2}^{18} + 3 T_{2}^{17} - 1485 T_{2}^{16} + 1258 T_{2}^{15} + \cdots + 27 \) Copy content Toggle raw display
\( T_{3}^{21} - 9 T_{3}^{20} - 3 T_{3}^{19} + 242 T_{3}^{18} - 444 T_{3}^{17} - 2439 T_{3}^{16} + 7552 T_{3}^{15} + \cdots + 89 \) Copy content Toggle raw display
\( T_{7}^{21} - 90 T_{7}^{19} - 59 T_{7}^{18} + 3339 T_{7}^{17} + 4161 T_{7}^{16} - 65180 T_{7}^{15} + \cdots + 6558193 \) Copy content Toggle raw display
\( T_{11}^{21} - 114 T_{11}^{19} + 34 T_{11}^{18} + 5262 T_{11}^{17} - 3396 T_{11}^{16} - 127360 T_{11}^{15} + \cdots + 58182921 \) Copy content Toggle raw display
\( T_{29}^{21} + 3 T_{29}^{20} - 282 T_{29}^{19} - 648 T_{29}^{18} + 33453 T_{29}^{17} + 56463 T_{29}^{16} + \cdots - 13449210273 \) Copy content Toggle raw display