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Properties

Label 9025.2.a.cq
Level $9025$
Weight $2$
Character orbit 9025.a
Self dual yes
Analytic conductor $72.065$
Analytic rank $1$
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: \(1\)
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 −2.74664 −2.94073 5.54404 0 8.07713 −1.37939 −9.73421 5.64789 0
1.2 −2.67570 2.23729 5.15937 0 −5.98632 −3.57682 −8.45352 2.00548 0
1.3 −2.59627 −1.65081 4.74062 0 4.28594 −2.00548 −7.11538 −0.274832 0
1.4 −2.46757 −2.83450 4.08889 0 6.99432 4.66735 −5.15448 5.03438 0
1.5 −1.95063 0.963432 1.80494 0 −1.87930 5.24962 0.380482 −2.07180 0
1.6 −1.89059 2.61121 1.57432 0 −4.93672 0.975485 0.804792 3.81841 0
1.7 −1.56420 0.289164 0.446737 0 −0.452312 −4.01995 2.42962 −2.91638 0
1.8 −1.27496 −0.195863 −0.374490 0 0.249717 −1.39855 3.02737 −2.96164 0
1.9 −1.17921 −1.37815 −0.609457 0 1.62514 −4.21997 3.07711 −1.10069 0
1.10 −0.587076 2.63527 −1.65534 0 −1.54710 1.99298 2.14596 3.94466 0
1.11 −0.369763 −2.43235 −1.86328 0 0.899393 2.99691 1.42849 2.91634 0
1.12 −0.0440787 −3.34883 −1.99806 0 0.147612 −1.48290 0.176229 8.21469 0
1.13 0.471157 −1.07129 −1.77801 0 −0.504745 2.74411 −1.78004 −1.85234 0
1.14 0.558603 −0.670247 −1.68796 0 −0.374402 −2.55490 −2.06011 −2.55077 0
1.15 0.617674 0.0232442 −1.61848 0 0.0143573 4.67118 −2.23504 −2.99946 0
1.16 0.947979 2.35813 −1.10134 0 2.23545 −2.44908 −2.94000 2.56075 0
1.17 1.75619 2.00195 1.08421 0 3.51581 0.476786 −1.60831 1.00782 0
1.18 2.15200 −3.25314 2.63110 0 −7.00075 −2.62239 1.35813 7.58291 0
1.19 2.24024 −1.26498 3.01866 0 −2.83386 −1.37484 2.28203 −1.39981 0
1.20 2.28640 1.01181 3.22763 0 2.31341 0.320153 2.80684 −1.97624 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.cq 21
5.b even 2 1 9025.2.a.cs 21
19.b odd 2 1 9025.2.a.cr 21
19.e even 9 2 475.2.l.d ✓ 42
95.d odd 2 1 9025.2.a.cp 21
95.p even 18 2 475.2.l.e yes 42
95.q odd 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.e even 9 2
475.2.l.e yes 42 95.p even 18 2
475.2.u.d 84 95.q odd 36 4
9025.2.a.cp 21 95.d odd 2 1
9025.2.a.cq 21 1.a even 1 1 trivial
9025.2.a.cr 21 19.b odd 2 1
9025.2.a.cs 21 5.b even 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