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Properties

Label 8025.2.a.bp
Level $8025$
Weight $2$
Character orbit 8025.a
Self dual yes
Analytic conductor $64.080$
Analytic rank $1$
Dimension $22$
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] = [8025,2,Mod(1,8025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8025 = 3 \cdot 5^{2} \cdot 107 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8025.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: \(64.0799476221\)
Analytic rank: \(1\)
Dimension: \(22\)
Twist minimal: no (minimal twist has level 1605)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 3 q^{2} - 22 q^{3} + 13 q^{4} - 3 q^{6} + 2 q^{7} + 9 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 3 q^{2} - 22 q^{3} + 13 q^{4} - 3 q^{6} + 2 q^{7} + 9 q^{8} + 22 q^{9} - 16 q^{11} - 13 q^{12} - 20 q^{14} - q^{16} + 20 q^{17} + 3 q^{18} - 16 q^{19} - 2 q^{21} + 4 q^{22} + 14 q^{23} - 9 q^{24} - 16 q^{26} - 22 q^{27} - q^{28} - 24 q^{29} - 42 q^{31} + 11 q^{32} + 16 q^{33} - 14 q^{34} + 13 q^{36} + 2 q^{37} + 21 q^{38} - 26 q^{41} + 20 q^{42} + 2 q^{43} - 24 q^{44} - 16 q^{46} + 26 q^{47} + q^{48} - 10 q^{49} - 20 q^{51} + 6 q^{52} + 22 q^{53} - 3 q^{54} - 42 q^{56} + 16 q^{57} - 69 q^{58} - 34 q^{59} - 16 q^{61} + 34 q^{62} + 2 q^{63} - 39 q^{64} - 4 q^{66} + 6 q^{68} - 14 q^{69} - 76 q^{71} + 9 q^{72} - 14 q^{73} - 12 q^{74} - 48 q^{76} + 54 q^{77} + 16 q^{78} - 72 q^{79} + 22 q^{81} + 2 q^{82} + 28 q^{83} + q^{84} - 22 q^{86} + 24 q^{87} - 19 q^{88} - 22 q^{89} - 58 q^{91} + 34 q^{92} + 42 q^{93} - 8 q^{94} - 11 q^{96} + 10 q^{97} - 3 q^{98} - 16 q^{99}+O(q^{100}) \) 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 −2.27471 −1.00000 3.17431 0 2.27471 −0.883209 −2.67121 1.00000 0
1.2 −2.24431 −1.00000 3.03694 0 2.24431 2.48180 −2.32721 1.00000 0
1.3 −2.17402 −1.00000 2.72637 0 2.17402 4.66914 −1.57914 1.00000 0
1.4 −2.08676 −1.00000 2.35456 0 2.08676 −0.375749 −0.739880 1.00000 0
1.5 −1.27694 −1.00000 −0.369419 0 1.27694 −0.700732 3.02561 1.00000 0
1.6 −1.09634 −1.00000 −0.798029 0 1.09634 1.73554 3.06760 1.00000 0
1.7 −0.861682 −1.00000 −1.25750 0 0.861682 0.233165 2.80693 1.00000 0
1.8 −0.861326 −1.00000 −1.25812 0 0.861326 1.26306 2.80630 1.00000 0
1.9 −0.811713 −1.00000 −1.34112 0 0.811713 −3.87908 2.71203 1.00000 0
1.10 −0.492378 −1.00000 −1.75756 0 0.492378 −2.62877 1.85014 1.00000 0
1.11 0.145947 −1.00000 −1.97870 0 −0.145947 4.33134 −0.580678 1.00000 0
1.12 0.363804 −1.00000 −1.86765 0 −0.363804 −0.0273194 −1.40706 1.00000 0
1.13 0.736346 −1.00000 −1.45779 0 −0.736346 −4.38760 −2.54613 1.00000 0
1.14 0.822726 −1.00000 −1.32312 0 −0.822726 1.32511 −2.73402 1.00000 0
1.15 0.844776 −1.00000 −1.28635 0 −0.844776 3.24504 −2.77623 1.00000 0
1.16 1.30136 −1.00000 −0.306455 0 −1.30136 0.670302 −3.00153 1.00000 0
1.17 1.59994 −1.00000 0.559801 0 −1.59994 −2.78827 −2.30423 1.00000 0
1.18 1.89104 −1.00000 1.57603 0 −1.89104 3.50248 −0.801740 1.00000 0
1.19 2.13388 −1.00000 2.55344 0 −2.13388 −3.51369 1.18098 1.00000 0
1.20 2.25096 −1.00000 3.06684 0 −2.25096 0.523586 2.40142 1.00000 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(107\) \(-1\)

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 8025.2.a.bp 22
5.b even 2 1 8025.2.a.bo 22
5.c odd 4 2 1605.2.b.d ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1605.2.b.d ✓ 44 5.c odd 4 2
8025.2.a.bo 22 5.b even 2 1
8025.2.a.bp 22 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8025))\):

\( T_{2}^{22} - 3 T_{2}^{21} - 24 T_{2}^{20} + 74 T_{2}^{19} + 239 T_{2}^{18} - 763 T_{2}^{17} - 1281 T_{2}^{16} + \cdots + 32 \) Copy content Toggle raw display
\( T_{7}^{22} - 2 T_{7}^{21} - 70 T_{7}^{20} + 126 T_{7}^{19} + 1978 T_{7}^{18} - 3190 T_{7}^{17} + \cdots - 608 \) Copy content Toggle raw display
\( T_{11}^{22} + 16 T_{11}^{21} + 33 T_{11}^{20} - 744 T_{11}^{19} - 4172 T_{11}^{18} + 8562 T_{11}^{17} + \cdots - 141824 \) Copy content Toggle raw display
\( T_{13}^{22} - 139 T_{13}^{20} + 10 T_{13}^{19} + 7879 T_{13}^{18} - 1690 T_{13}^{17} - 236417 T_{13}^{16} + \cdots + 113745920 \) Copy content Toggle raw display