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Properties

Label 1205.2.b.c
Level $1205$
Weight $2$
Character orbit 1205.b
Analytic conductor $9.622$
Analytic rank $0$
Dimension $46$
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] = [1205,2,Mod(724,1205)]
 
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([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1205.724");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1205 = 5 \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1205.b (of order \(2\), degree \(1\), 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: \(9.62197344356\)
Analytic rank: \(0\)
Dimension: \(46\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 46 q - 34 q^{4} - 8 q^{5} - 4 q^{6} - 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 46 q - 34 q^{4} - 8 q^{5} - 4 q^{6} - 34 q^{9} - 7 q^{10} - 64 q^{11} + 66 q^{14} + 5 q^{15} + 22 q^{16} - 2 q^{20} - 14 q^{21} + 50 q^{24} + 30 q^{25} - 60 q^{26} + 36 q^{29} + 7 q^{30} - 36 q^{31} - 12 q^{34} + 3 q^{35} - 34 q^{36} + 88 q^{39} - 30 q^{40} - 76 q^{41} + 100 q^{44} - 17 q^{45} - 12 q^{46} - 22 q^{49} - 14 q^{50} - 112 q^{51} + 26 q^{54} - 5 q^{55} - 120 q^{56} + 84 q^{59} - 33 q^{60} - 78 q^{61} - 28 q^{64} + 9 q^{65} - 2 q^{66} + 24 q^{69} + 88 q^{70} - 172 q^{71} + 16 q^{74} + 59 q^{75} - 18 q^{76} + 54 q^{79} + 63 q^{80} - 42 q^{81} - 44 q^{84} + 30 q^{85} - 80 q^{86} + 86 q^{89} + 17 q^{90} - 88 q^{91} - 4 q^{94} + q^{95} - 122 q^{96} + 148 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
724.1 − 2.68673i − 2.58072i −5.21850 1.90521 + 1.17055i −6.93370 1.24945i 8.64724i −3.66013 3.14494 − 5.11877i
724.2 − 2.67602i 0.199289i −5.16108 −2.13713 + 0.657778i 0.533301 2.87374i 8.45912i 2.96028 1.76023 + 5.71900i
724.3 − 2.49572i 1.66439i −4.22863 2.11925 − 0.713293i 4.15385 4.24911i 5.56203i 0.229807 −1.78018 − 5.28905i
724.4 − 2.44838i − 0.992209i −3.99457 0.553587 − 2.16646i −2.42931 − 0.744704i 4.88345i 2.01552 −5.30431 − 1.35539i
724.5 − 2.36215i − 2.05253i −3.57974 −0.618559 + 2.14881i −4.84838 3.07728i 3.73158i −1.21289 5.07581 + 1.46113i
724.6 − 1.95888i − 0.350465i −1.83721 −1.30176 − 1.81808i −0.686519 1.14869i − 0.318879i 2.87717 −3.56141 + 2.54999i
724.7 − 1.92263i − 0.557567i −1.69651 −2.01334 + 0.972856i −1.07200 − 0.976687i − 0.583491i 2.68912 1.87044 + 3.87092i
724.8 − 1.88286i 2.55671i −1.54517 −0.979915 − 2.00992i 4.81392 3.51799i − 0.856390i −3.53675 −3.78439 + 1.84504i
724.9 − 1.84209i 1.69681i −1.39329 1.66065 + 1.49741i 3.12567 1.54055i − 1.11761i 0.120844 2.75836 − 3.05907i
724.10 − 1.83307i 2.40621i −1.36013 −1.46796 + 1.68674i 4.41073 − 0.302396i − 1.17293i −2.78983 3.09191 + 2.69086i
724.11 − 1.70963i 1.38336i −0.922819 −1.53605 − 1.62498i 2.36503 − 5.05937i − 1.84158i 1.08631 −2.77811 + 2.62606i
724.12 − 1.49203i − 3.12646i −0.226139 0.643201 − 2.14156i −4.66475 4.66024i − 2.64665i −6.77474 −3.19527 − 0.959673i
724.13 − 1.45267i − 2.41257i −0.110263 −1.02000 − 1.98987i −3.50468 − 1.25057i − 2.74517i −2.82049 −2.89064 + 1.48173i
724.14 − 1.30874i 1.88426i 0.287190 2.21635 − 0.296297i 2.46601 1.28779i − 2.99335i −0.550432 −0.387776 − 2.90063i
724.15 − 1.09514i 0.223927i 0.800662 −1.50518 + 1.65361i 0.245232 − 2.26365i − 3.06713i 2.94986 1.81094 + 1.64839i
724.16 − 0.930197i − 1.32621i 1.13473 −2.23573 − 0.0386721i −1.23364 − 4.33497i − 2.91592i 1.24116 −0.0359727 + 2.07967i
724.17 − 0.779313i 2.96208i 1.39267 2.14053 − 0.646643i 2.30839 − 0.937135i − 2.64395i −5.77395 −0.503937 − 1.66814i
724.18 − 0.760721i − 1.12938i 1.42130 1.88699 + 1.19969i −0.859145 2.53853i − 2.60266i 1.72450 0.912633 − 1.43547i
724.19 − 0.636788i − 2.24729i 1.59450 −1.29399 + 1.82362i −1.43105 1.14493i − 2.28894i −2.05030 1.16126 + 0.824000i
724.20 − 0.434323i 1.75836i 1.81136 0.971846 − 2.01383i 0.763695 2.56007i − 1.65536i −0.0918227 −0.874653 − 0.422095i
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 724.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1205.2.b.c ✓ 46
5.b even 2 1 inner 1205.2.b.c ✓ 46
5.c odd 4 2 6025.2.a.p 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1205.2.b.c ✓ 46 1.a even 1 1 trivial
1205.2.b.c ✓ 46 5.b even 2 1 inner
6025.2.a.p 46 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{46} + 63 T_{2}^{44} + 1836 T_{2}^{42} + 32871 T_{2}^{40} + 404962 T_{2}^{38} + 3644147 T_{2}^{36} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(1205, [\chi])\). Copy content Toggle raw display