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Properties

Label 250.3.f.f
Level $250$
Weight $3$
Character orbit 250.f
Analytic conductor $6.812$
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] = [250,3,Mod(7,250)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(250, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([17]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("250.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 250 = 2 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 250.f (of order \(20\), degree \(8\), not 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: \(6.81200660901\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(3\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{2} + 2 q^{3} - 4 q^{6} - 2 q^{7} - 12 q^{8} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{2} + 2 q^{3} - 4 q^{6} - 2 q^{7} - 12 q^{8} - 40 q^{9} - 32 q^{11} - 16 q^{12} + 2 q^{13} - 30 q^{14} + 24 q^{16} - 22 q^{17} - 136 q^{18} + 230 q^{19} + 68 q^{21} - 8 q^{22} + 42 q^{23} + 36 q^{26} - 340 q^{27} + 4 q^{28} - 100 q^{29} - 132 q^{31} + 96 q^{32} - 266 q^{33} - 150 q^{34} - 108 q^{36} + 38 q^{37} + 20 q^{38} + 80 q^{39} + 168 q^{41} - 128 q^{42} - 78 q^{43} + 40 q^{44} + 26 q^{46} + 168 q^{47} + 32 q^{48} + 168 q^{51} + 4 q^{52} + 42 q^{53} + 80 q^{54} - 48 q^{56} + 280 q^{57} - 160 q^{58} - 450 q^{59} - 492 q^{61} + 142 q^{62} + 762 q^{63} + 202 q^{66} + 498 q^{67} - 136 q^{68} + 670 q^{69} - 2 q^{71} - 72 q^{72} + 62 q^{73} - 40 q^{76} - 624 q^{77} + 658 q^{78} + 360 q^{79} - 46 q^{81} + 272 q^{82} - 128 q^{83} - 620 q^{84} - 264 q^{86} + 400 q^{87} - 44 q^{88} - 900 q^{89} + 798 q^{91} - 36 q^{92} + 294 q^{93} + 190 q^{94} + 16 q^{96} - 312 q^{97} + 66 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.26007 + 0.642040i −5.23058 + 0.828443i 1.17557 − 1.61803i 0 6.05902 − 4.40214i 4.28629 + 4.28629i −0.442463 + 2.79360i 18.1132 − 5.88532i 0
7.2 −1.26007 + 0.642040i 1.91941 − 0.304004i 1.17557 − 1.61803i 0 −2.22341 + 1.61540i 6.96775 + 6.96775i −0.442463 + 2.79360i −4.96781 + 1.61414i 0
7.3 −1.26007 + 0.642040i 2.41437 − 0.382399i 1.17557 − 1.61803i 0 −2.79677 + 2.03197i −7.03135 − 7.03135i −0.442463 + 2.79360i −2.87654 + 0.934645i 0
43.1 1.39680 − 0.221232i −2.42324 + 4.75588i 1.90211 − 0.618034i 0 −2.33264 + 7.17912i −6.88294 + 6.88294i 2.52015 − 1.28408i −11.4562 − 15.7681i 0
43.2 1.39680 − 0.221232i −0.350339 + 0.687579i 1.90211 − 0.618034i 0 −0.337240 + 1.03792i 2.38638 − 2.38638i 2.52015 − 1.28408i 4.94004 + 6.79938i 0
43.3 1.39680 − 0.221232i 2.63154 − 5.16469i 1.90211 − 0.618034i 0 2.53315 − 7.79623i −2.21400 + 2.21400i 2.52015 − 1.28408i −14.4589 − 19.9010i 0
93.1 0.221232 − 1.39680i −3.03060 + 1.54417i −1.90211 − 0.618034i 0 1.48643 + 4.57476i −2.71827 + 2.71827i −1.28408 + 2.52015i 1.51000 − 2.07833i 0
93.2 0.221232 − 1.39680i 0.587238 − 0.299213i −1.90211 − 0.618034i 0 −0.288025 − 0.886451i −5.08008 + 5.08008i −1.28408 + 2.52015i −5.03475 + 6.92973i 0
93.3 0.221232 − 1.39680i 4.20343 − 2.14176i −1.90211 − 0.618034i 0 −2.06168 − 6.34519i 8.41873 − 8.41873i −1.28408 + 2.52015i 7.79165 − 10.7243i 0
107.1 0.642040 − 1.26007i −0.574508 + 3.62730i −1.17557 − 1.61803i 0 4.20181 + 3.05279i −8.92599 − 8.92599i −2.79360 + 0.442463i −4.26773 − 1.38667i 0
107.2 0.642040 − 1.26007i 0.0294182 − 0.185739i −1.17557 − 1.61803i 0 −0.215157 − 0.156321i 7.34278 + 7.34278i −2.79360 + 0.442463i 8.52588 + 2.77022i 0
107.3 0.642040 − 1.26007i 0.823858 − 5.20163i −1.17557 − 1.61803i 0 −6.02549 − 4.37778i 2.45070 + 2.45070i −2.79360 + 0.442463i −17.8187 − 5.78966i 0
143.1 −1.26007 − 0.642040i −5.23058 − 0.828443i 1.17557 + 1.61803i 0 6.05902 + 4.40214i 4.28629 − 4.28629i −0.442463 − 2.79360i 18.1132 + 5.88532i 0
143.2 −1.26007 − 0.642040i 1.91941 + 0.304004i 1.17557 + 1.61803i 0 −2.22341 − 1.61540i 6.96775 − 6.96775i −0.442463 − 2.79360i −4.96781 − 1.61414i 0
143.3 −1.26007 − 0.642040i 2.41437 + 0.382399i 1.17557 + 1.61803i 0 −2.79677 − 2.03197i −7.03135 + 7.03135i −0.442463 − 2.79360i −2.87654 − 0.934645i 0
157.1 1.39680 + 0.221232i −2.42324 − 4.75588i 1.90211 + 0.618034i 0 −2.33264 − 7.17912i −6.88294 − 6.88294i 2.52015 + 1.28408i −11.4562 + 15.7681i 0
157.2 1.39680 + 0.221232i −0.350339 − 0.687579i 1.90211 + 0.618034i 0 −0.337240 − 1.03792i 2.38638 + 2.38638i 2.52015 + 1.28408i 4.94004 − 6.79938i 0
157.3 1.39680 + 0.221232i 2.63154 + 5.16469i 1.90211 + 0.618034i 0 2.53315 + 7.79623i −2.21400 − 2.21400i 2.52015 + 1.28408i −14.4589 + 19.9010i 0
207.1 0.221232 + 1.39680i −3.03060 − 1.54417i −1.90211 + 0.618034i 0 1.48643 − 4.57476i −2.71827 − 2.71827i −1.28408 − 2.52015i 1.51000 + 2.07833i 0
207.2 0.221232 + 1.39680i 0.587238 + 0.299213i −1.90211 + 0.618034i 0 −0.288025 + 0.886451i −5.08008 − 5.08008i −1.28408 − 2.52015i −5.03475 − 6.92973i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.f odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 250.3.f.f 24
5.b even 2 1 250.3.f.d 24
5.c odd 4 1 50.3.f.b ✓ 24
5.c odd 4 1 250.3.f.e 24
20.e even 4 1 400.3.bg.b 24
25.d even 5 1 50.3.f.b ✓ 24
25.e even 10 1 250.3.f.e 24
25.f odd 20 1 250.3.f.d 24
25.f odd 20 1 inner 250.3.f.f 24
100.j odd 10 1 400.3.bg.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.3.f.b ✓ 24 5.c odd 4 1
50.3.f.b ✓ 24 25.d even 5 1
250.3.f.d 24 5.b even 2 1
250.3.f.d 24 25.f odd 20 1
250.3.f.e 24 5.c odd 4 1
250.3.f.e 24 25.e even 10 1
250.3.f.f 24 1.a even 1 1 trivial
250.3.f.f 24 25.f odd 20 1 inner
400.3.bg.b 24 20.e even 4 1
400.3.bg.b 24 100.j odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 2 T_{3}^{23} + 22 T_{3}^{22} + 84 T_{3}^{21} - 336 T_{3}^{20} + 3078 T_{3}^{19} + \cdots + 533794816 \) acting on \(S_{3}^{\mathrm{new}}(250, [\chi])\). Copy content Toggle raw display