Properties

Label 1205.2.a.e
Level $1205$
Weight $2$
Character orbit 1205.a
Self dual yes
Analytic conductor $9.622$
Analytic rank $0$
Dimension $25$
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] = [1205,2,Mod(1,1205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1205, 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("1205.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1205 = 5 \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1205.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: \(9.62197344356\)
Analytic rank: \(0\)
Dimension: \(25\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 25 q + 6 q^{2} + 15 q^{3} + 32 q^{4} + 25 q^{5} - q^{6} + 19 q^{7} + 15 q^{8} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 25 q + 6 q^{2} + 15 q^{3} + 32 q^{4} + 25 q^{5} - q^{6} + 19 q^{7} + 15 q^{8} + 32 q^{9} + 6 q^{10} + 2 q^{11} + 20 q^{12} + 14 q^{13} - 5 q^{14} + 15 q^{15} + 38 q^{16} + 7 q^{17} + 9 q^{18} + 30 q^{19} + 32 q^{20} + q^{21} + q^{22} + 43 q^{23} - 6 q^{24} + 25 q^{25} - 22 q^{26} + 42 q^{27} + 32 q^{28} - 4 q^{29} - q^{30} + 14 q^{31} + 26 q^{32} + 4 q^{33} + 7 q^{34} + 19 q^{35} + 15 q^{36} + 16 q^{37} + 14 q^{38} - 21 q^{39} + 15 q^{40} - q^{41} - 25 q^{42} + 35 q^{43} - 52 q^{44} + 32 q^{45} - 27 q^{46} + 50 q^{47} + 26 q^{48} + 46 q^{49} + 6 q^{50} - 7 q^{51} + 3 q^{52} + 4 q^{53} - 31 q^{54} + 2 q^{55} - 51 q^{56} + 2 q^{58} + 6 q^{59} + 20 q^{60} + 19 q^{61} + 28 q^{63} + 49 q^{64} + 14 q^{65} - 27 q^{66} + 65 q^{67} - 25 q^{68} + 2 q^{69} - 5 q^{70} - 34 q^{71} - 10 q^{72} + 8 q^{73} - 42 q^{74} + 15 q^{75} + 71 q^{76} + q^{77} - 59 q^{78} - 12 q^{79} + 38 q^{80} + 29 q^{81} + 11 q^{82} + 41 q^{83} - 10 q^{84} + 7 q^{85} - 13 q^{86} + 40 q^{87} - 52 q^{88} - 24 q^{89} + 9 q^{90} + 46 q^{91} + 85 q^{92} - 30 q^{93} + 14 q^{94} + 30 q^{95} - 30 q^{96} + 9 q^{97} - 64 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.73903 2.66987 5.50229 1.00000 −7.31286 4.66369 −9.59289 4.12822 −2.73903
1.2 −2.51976 −0.981670 4.34921 1.00000 2.47358 3.43036 −5.91946 −2.03632 −2.51976
1.3 −2.18227 2.05137 2.76228 1.00000 −4.47663 −1.39304 −1.66351 1.20812 −2.18227
1.4 −2.14165 −1.43070 2.58667 1.00000 3.06405 −0.0457027 −1.25644 −0.953111 −2.14165
1.5 −1.83164 0.379920 1.35491 1.00000 −0.695877 −4.92875 1.18157 −2.85566 −1.83164
1.6 −1.58885 2.11200 0.524458 1.00000 −3.35566 3.95616 2.34442 1.46053 −1.58885
1.7 −1.25748 −2.42991 −0.418732 1.00000 3.05557 3.44014 3.04152 2.90444 −1.25748
1.8 −1.16007 2.40515 −0.654229 1.00000 −2.79015 1.39900 3.07910 2.78474 −1.16007
1.9 −1.11717 3.26371 −0.751931 1.00000 −3.64612 −3.89846 3.07438 7.65181 −1.11717
1.10 −0.566303 −0.484852 −1.67930 1.00000 0.274573 −0.0113879 2.08360 −2.76492 −0.566303
1.11 −0.0694400 −2.28670 −1.99518 1.00000 0.158789 0.976003 0.277425 2.22902 −0.0694400
1.12 0.300307 0.482693 −1.90982 1.00000 0.144956 5.09571 −1.17415 −2.76701 0.300307
1.13 0.362712 3.41577 −1.86844 1.00000 1.23894 3.34046 −1.40313 8.66752 0.362712
1.14 0.707449 1.14949 −1.49952 1.00000 0.813204 1.94091 −2.47573 −1.67868 0.707449
1.15 0.782724 −1.54504 −1.38734 1.00000 −1.20934 −2.90223 −2.65136 −0.612841 0.782724
1.16 1.05446 2.79403 −0.888118 1.00000 2.94619 −1.63707 −3.04540 4.80663 1.05446
1.17 1.31571 −0.792486 −0.268903 1.00000 −1.04268 −1.49796 −2.98522 −2.37197 1.31571
1.18 1.91059 2.10440 1.65034 1.00000 4.02063 2.16427 −0.668063 1.42849 1.91059
1.19 1.96718 2.44636 1.86979 1.00000 4.81242 2.28114 −0.256155 2.98466 1.96718
1.20 2.03980 −3.04377 2.16080 1.00000 −6.20869 3.73609 0.327997 6.26453 2.03980
See all 25 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.25
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(241\) \(1\)

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 1205.2.a.e 25
5.b even 2 1 6025.2.a.j 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1205.2.a.e 25 1.a even 1 1 trivial
6025.2.a.j 25 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{25} - 6 T_{2}^{24} - 23 T_{2}^{23} + 197 T_{2}^{22} + 127 T_{2}^{21} - 2741 T_{2}^{20} + 1209 T_{2}^{19} + 21093 T_{2}^{18} - 21526 T_{2}^{17} - 98096 T_{2}^{16} + 140217 T_{2}^{15} + 281530 T_{2}^{14} - 511289 T_{2}^{13} + \cdots + 409 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1205))\). Copy content Toggle raw display