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Properties

Label 550.2.l.d
Level $550$
Weight $2$
Character orbit 550.l
Analytic conductor $4.392$
Analytic rank $0$
Dimension $48$
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] = [550,2,Mod(181,550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(550, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.l (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: \(4.39177211117\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{2} - 22 q^{3} - 12 q^{4} - q^{5} - 3 q^{6} + 11 q^{7} + 12 q^{8} + 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{2} - 22 q^{3} - 12 q^{4} - q^{5} - 3 q^{6} + 11 q^{7} + 12 q^{8} + 70 q^{9} + 6 q^{10} + 3 q^{11} + 8 q^{12} - 34 q^{13} - q^{14} + 7 q^{15} - 12 q^{16} - 9 q^{17} + 10 q^{18} - 6 q^{19} + 19 q^{20} - 15 q^{21} - 8 q^{22} + 14 q^{23} - 8 q^{24} + 7 q^{25} - 16 q^{26} - 34 q^{27} + q^{28} - q^{29} - 12 q^{30} + 17 q^{31} - 48 q^{32} + 4 q^{34} - 41 q^{35} - 25 q^{36} - 23 q^{37} + q^{38} - 20 q^{39} + 6 q^{40} + 29 q^{41} + 22 q^{43} + 13 q^{44} + 62 q^{45} - 9 q^{46} - 52 q^{47} + 3 q^{48} - 23 q^{49} - 2 q^{50} + 4 q^{51} + q^{52} + 44 q^{53} - 11 q^{54} - 7 q^{55} - 11 q^{56} - 5 q^{57} + q^{58} - 18 q^{59} + 2 q^{60} + 16 q^{61} - 17 q^{62} - 13 q^{63} - 12 q^{64} - 2 q^{65} + 20 q^{67} - 4 q^{68} - 2 q^{69} - 59 q^{70} + 39 q^{71} + 25 q^{72} - 42 q^{73} - 37 q^{74} + 20 q^{75} + 14 q^{76} - 13 q^{77} - 15 q^{78} + 53 q^{79} - 6 q^{80} + 56 q^{81} + 36 q^{82} + 14 q^{83} + 6 q^{85} + 13 q^{86} - 18 q^{87} + 17 q^{88} - 8 q^{89} + 28 q^{90} - 5 q^{91} - 46 q^{92} + 15 q^{93} - 13 q^{94} - 24 q^{95} + 22 q^{96} - 16 q^{97} + 23 q^{98} - 54 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1 0.809017 − 0.587785i −3.29068 0.309017 − 0.951057i 1.79924 − 1.32768i −2.66222 + 1.93421i 0.583682 − 0.424070i −0.309017 − 0.951057i 7.82859 0.675221 − 2.13168i
181.2 0.809017 − 0.587785i −2.99748 0.309017 − 0.951057i −1.28647 + 1.82893i −2.42501 + 1.76187i 3.29479 − 2.39380i −0.309017 − 0.951057i 5.98487 0.0342403 + 2.23581i
181.3 0.809017 − 0.587785i −2.25044 0.309017 − 0.951057i 2.09203 + 0.789573i −1.82065 + 1.32278i −0.374768 + 0.272285i −0.309017 − 0.951057i 2.06450 2.15658 − 0.590884i
181.4 0.809017 − 0.587785i −1.91667 0.309017 − 0.951057i −1.59148 + 1.57073i −1.55062 + 1.12659i −0.266407 + 0.193556i −0.309017 − 0.951057i 0.673631 −0.364288 + 2.20619i
181.5 0.809017 − 0.587785i −1.31124 0.309017 − 0.951057i 1.26051 + 1.84692i −1.06082 + 0.770730i −3.19479 + 2.32115i −0.309017 − 0.951057i −1.28064 2.10537 + 0.753278i
181.6 0.809017 − 0.587785i −0.917576 0.309017 − 0.951057i 0.160491 − 2.23030i −0.742335 + 0.539338i 3.50434 − 2.54605i −0.309017 − 0.951057i −2.15805 −1.18110 − 1.89869i
181.7 0.809017 − 0.587785i 0.0324201 0.309017 − 0.951057i −2.10206 − 0.762462i 0.0262284 − 0.0190561i 2.00370 − 1.45577i −0.309017 − 0.951057i −2.99895 −2.14877 + 0.618714i
181.8 0.809017 − 0.587785i 0.0894816 0.309017 − 0.951057i −2.07325 + 0.837648i 0.0723921 − 0.0525959i −1.99429 + 1.44893i −0.309017 − 0.951057i −2.99199 −1.18493 + 1.89629i
181.9 0.809017 − 0.587785i 1.09243 0.309017 − 0.951057i 0.0135632 + 2.23603i 0.883793 − 0.642113i 3.28932 − 2.38983i −0.309017 − 0.951057i −1.80660 1.32528 + 1.80101i
181.10 0.809017 − 0.587785i 1.51732 0.309017 − 0.951057i 2.19389 + 0.432234i 1.22754 − 0.891861i 1.63049 − 1.18462i −0.309017 − 0.951057i −0.697726 2.02896 − 0.939854i
181.11 0.809017 − 0.587785i 2.70405 0.309017 − 0.951057i 0.608928 − 2.15156i 2.18762 − 1.58940i 0.426185 − 0.309642i −0.309017 − 0.951057i 4.31188 −0.772022 − 2.09857i
181.12 0.809017 − 0.587785i 2.86643 0.309017 − 0.951057i 1.46969 + 1.68523i 2.31899 − 1.68484i −2.23913 + 1.62682i −0.309017 − 0.951057i 5.21640 2.17956 + 0.499517i
191.1 −0.309017 − 0.951057i −3.34776 −0.809017 + 0.587785i 0.814715 − 2.08236i 1.03451 + 3.18391i −1.27708 − 3.93046i 0.809017 + 0.587785i 8.20748 −2.23221 − 0.131355i
191.2 −0.309017 − 0.951057i −3.06362 −0.809017 + 0.587785i −1.19915 + 1.88734i 0.946712 + 2.91368i 0.968704 + 2.98137i 0.809017 + 0.587785i 6.38579 2.16552 + 0.557237i
191.3 −0.309017 − 0.951057i −2.20028 −0.809017 + 0.587785i −1.31045 + 1.81183i 0.679923 + 2.09259i −1.59332 − 4.90374i 0.809017 + 0.587785i 1.84122 2.12810 + 0.686427i
191.4 −0.309017 − 0.951057i −2.11004 −0.809017 + 0.587785i 2.08390 + 0.810786i 0.652038 + 2.00677i 0.0819677 + 0.252271i 0.809017 + 0.587785i 1.45226 0.127144 − 2.23245i
191.5 −0.309017 − 0.951057i −1.84306 −0.809017 + 0.587785i −0.786235 − 2.09328i 0.569536 + 1.75285i 0.678426 + 2.08798i 0.809017 + 0.587785i 0.396864 −1.74787 + 1.39461i
191.6 −0.309017 − 0.951057i −1.56726 −0.809017 + 0.587785i −2.22800 − 0.189804i 0.484312 + 1.49056i 0.543135 + 1.67160i 0.809017 + 0.587785i −0.543680 0.507974 + 2.17760i
191.7 −0.309017 − 0.951057i −0.931600 −0.809017 + 0.587785i 1.98944 + 1.02084i 0.287880 + 0.886004i 0.0302337 + 0.0930496i 0.809017 + 0.587785i −2.13212 0.356104 − 2.20753i
191.8 −0.309017 − 0.951057i −0.220703 −0.809017 + 0.587785i −0.141858 − 2.23156i 0.0682010 + 0.209901i −0.182915 − 0.562954i 0.809017 + 0.587785i −2.95129 −2.07851 + 0.824507i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
275.k even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 550.2.l.d yes 48
11.c even 5 1 550.2.i.d ✓ 48
25.d even 5 1 550.2.i.d ✓ 48
275.k even 5 1 inner 550.2.l.d yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.i.d ✓ 48 11.c even 5 1
550.2.i.d ✓ 48 25.d even 5 1
550.2.l.d yes 48 1.a even 1 1 trivial
550.2.l.d yes 48 275.k even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 11 T_{3}^{23} + 7 T_{3}^{22} - 339 T_{3}^{21} - 1085 T_{3}^{20} + 3444 T_{3}^{19} + \cdots + 1019 \) acting on \(S_{2}^{\mathrm{new}}(550, [\chi])\). Copy content Toggle raw display