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Properties

Label 825.2.n.p
Level $825$
Weight $2$
Character orbit 825.n
Analytic conductor $6.588$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,2,Mod(301,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.301");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.n (of order \(5\), degree \(4\), 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: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{5})\)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} - 6 q^{3} - 6 q^{4} + 2 q^{6} + 4 q^{7} + 6 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} - 6 q^{3} - 6 q^{4} + 2 q^{6} + 4 q^{7} + 6 q^{8} - 6 q^{9} + 24 q^{12} + 4 q^{13} + 2 q^{14} - 22 q^{16} + 4 q^{17} + 2 q^{18} + 8 q^{19} - 16 q^{21} - 4 q^{22} + 6 q^{24} - 38 q^{26} - 6 q^{27} + 30 q^{28} - 10 q^{31} - 56 q^{32} + 10 q^{33} + 12 q^{34} - 6 q^{36} + 10 q^{37} + 4 q^{38} + 4 q^{39} + 30 q^{41} - 8 q^{42} - 64 q^{43} + 24 q^{44} + 54 q^{46} - 8 q^{47} - 2 q^{48} + 14 q^{49} + 14 q^{51} + 14 q^{52} + 26 q^{53} - 8 q^{54} + 12 q^{56} + 8 q^{57} + 20 q^{58} - 30 q^{59} + 20 q^{61} - 50 q^{62} + 4 q^{63} - 32 q^{64} + 6 q^{66} + 20 q^{67} - 62 q^{68} - 10 q^{69} - 16 q^{71} + 6 q^{72} - 12 q^{73} + 16 q^{74} - 68 q^{76} - 2 q^{77} + 32 q^{78} + 26 q^{79} - 6 q^{81} + 56 q^{82} + 48 q^{83} - 52 q^{86} + 48 q^{88} - 20 q^{89} - 20 q^{91} + 46 q^{92} - 10 q^{93} - 36 q^{94} + 14 q^{96} - 14 q^{97} - 120 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
301.1 −1.91233 − 1.38939i 0.309017 − 0.951057i 1.10857 + 3.41183i 0 −1.91233 + 1.38939i −0.529727 − 1.63033i 1.15951 − 3.56862i −0.809017 − 0.587785i 0
301.2 −0.634375 − 0.460901i 0.309017 − 0.951057i −0.428031 − 1.31734i 0 −0.634375 + 0.460901i −0.469090 − 1.44371i −0.820252 + 2.52448i −0.809017 − 0.587785i 0
301.3 −0.233654 − 0.169760i 0.309017 − 0.951057i −0.592258 − 1.82278i 0 −0.233654 + 0.169760i 1.13329 + 3.48791i −0.349548 + 1.07580i −0.809017 − 0.587785i 0
301.4 0.649936 + 0.472206i 0.309017 − 0.951057i −0.418596 − 1.28830i 0 0.649936 − 0.472206i 0.157137 + 0.483617i 0.832793 − 2.56307i −0.809017 − 0.587785i 0
301.5 1.54314 + 1.12116i 0.309017 − 0.951057i 0.506258 + 1.55810i 0 1.54314 − 1.12116i −1.37739 − 4.23918i 0.213204 − 0.656175i −0.809017 − 0.587785i 0
301.6 2.20531 + 1.60225i 0.309017 − 0.951057i 1.67816 + 5.16484i 0 2.20531 − 1.60225i −0.150285 − 0.462529i −2.88981 + 8.89393i −0.809017 − 0.587785i 0
526.1 −0.800975 + 2.46515i −0.809017 + 0.587785i −3.81735 − 2.77347i 0 −0.800975 − 2.46515i 2.76550 + 2.00926i 5.70065 − 4.14176i 0.309017 − 0.951057i 0
526.2 −0.650947 + 2.00341i −0.809017 + 0.587785i −1.97188 − 1.43266i 0 −0.650947 − 2.00341i −2.68296 − 1.94929i 0.745386 − 0.541555i 0.309017 − 0.951057i 0
526.3 −0.150893 + 0.464400i −0.809017 + 0.587785i 1.42514 + 1.03542i 0 −0.150893 − 0.464400i 4.13322 + 3.00296i −1.48598 + 1.07962i 0.309017 − 0.951057i 0
526.4 −0.0921069 + 0.283476i −0.809017 + 0.587785i 1.54616 + 1.12335i 0 −0.0921069 − 0.283476i −1.87778 − 1.36429i −0.943133 + 0.685226i 0.309017 − 0.951057i 0
526.5 0.392740 − 1.20873i −0.809017 + 0.587785i 0.311255 + 0.226140i 0 0.392740 + 1.20873i −0.390991 − 0.284071i 2.45199 − 1.78148i 0.309017 − 0.951057i 0
526.6 0.684148 − 2.10559i −0.809017 + 0.587785i −2.34742 − 1.70550i 0 0.684148 + 2.10559i 1.28908 + 0.936570i −1.61482 + 1.17323i 0.309017 − 0.951057i 0
676.1 −0.800975 − 2.46515i −0.809017 − 0.587785i −3.81735 + 2.77347i 0 −0.800975 + 2.46515i 2.76550 − 2.00926i 5.70065 + 4.14176i 0.309017 + 0.951057i 0
676.2 −0.650947 − 2.00341i −0.809017 − 0.587785i −1.97188 + 1.43266i 0 −0.650947 + 2.00341i −2.68296 + 1.94929i 0.745386 + 0.541555i 0.309017 + 0.951057i 0
676.3 −0.150893 − 0.464400i −0.809017 − 0.587785i 1.42514 − 1.03542i 0 −0.150893 + 0.464400i 4.13322 − 3.00296i −1.48598 − 1.07962i 0.309017 + 0.951057i 0
676.4 −0.0921069 − 0.283476i −0.809017 − 0.587785i 1.54616 − 1.12335i 0 −0.0921069 + 0.283476i −1.87778 + 1.36429i −0.943133 − 0.685226i 0.309017 + 0.951057i 0
676.5 0.392740 + 1.20873i −0.809017 − 0.587785i 0.311255 − 0.226140i 0 0.392740 − 1.20873i −0.390991 + 0.284071i 2.45199 + 1.78148i 0.309017 + 0.951057i 0
676.6 0.684148 + 2.10559i −0.809017 − 0.587785i −2.34742 + 1.70550i 0 0.684148 − 2.10559i 1.28908 − 0.936570i −1.61482 − 1.17323i 0.309017 + 0.951057i 0
751.1 −1.91233 + 1.38939i 0.309017 + 0.951057i 1.10857 − 3.41183i 0 −1.91233 − 1.38939i −0.529727 + 1.63033i 1.15951 + 3.56862i −0.809017 + 0.587785i 0
751.2 −0.634375 + 0.460901i 0.309017 + 0.951057i −0.428031 + 1.31734i 0 −0.634375 − 0.460901i −0.469090 + 1.44371i −0.820252 − 2.52448i −0.809017 + 0.587785i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 301.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.n.p 24
5.b even 2 1 825.2.n.o 24
5.c odd 4 2 165.2.s.a ✓ 48
11.c even 5 1 inner 825.2.n.p 24
11.c even 5 1 9075.2.a.dy 12
11.d odd 10 1 9075.2.a.ea 12
15.e even 4 2 495.2.ba.c 48
55.h odd 10 1 9075.2.a.dx 12
55.j even 10 1 825.2.n.o 24
55.j even 10 1 9075.2.a.dz 12
55.k odd 20 2 165.2.s.a ✓ 48
55.k odd 20 2 1815.2.c.j 24
55.l even 20 2 1815.2.c.k 24
165.v even 20 2 495.2.ba.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.s.a ✓ 48 5.c odd 4 2
165.2.s.a ✓ 48 55.k odd 20 2
495.2.ba.c 48 15.e even 4 2
495.2.ba.c 48 165.v even 20 2
825.2.n.o 24 5.b even 2 1
825.2.n.o 24 55.j even 10 1
825.2.n.p 24 1.a even 1 1 trivial
825.2.n.p 24 11.c even 5 1 inner
1815.2.c.j 24 55.k odd 20 2
1815.2.c.k 24 55.l even 20 2
9075.2.a.dx 12 55.h odd 10 1
9075.2.a.dy 12 11.c even 5 1
9075.2.a.dz 12 55.j even 10 1
9075.2.a.ea 12 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{24} - 2 T_{2}^{23} + 11 T_{2}^{22} - 24 T_{2}^{21} + 95 T_{2}^{20} - 116 T_{2}^{19} + 556 T_{2}^{18} + \cdots + 25 \) Copy content Toggle raw display
\( T_{13}^{24} - 4 T_{13}^{23} + 35 T_{13}^{22} - 204 T_{13}^{21} + 1406 T_{13}^{20} - 5008 T_{13}^{19} + \cdots + 445252201 \) Copy content Toggle raw display