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Properties

Label 550.2.n.a
Level $550$
Weight $2$
Character orbit 550.n
Analytic conductor $4.392$
Analytic rank $0$
Dimension $120$
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(59,550)]
 
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([7, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.59");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.n (of order \(10\), 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: \(120\)
Relative dimension: \(30\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 30 q^{4} + 8 q^{5} + 4 q^{6} + 20 q^{7} - 124 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 30 q^{4} + 8 q^{5} + 4 q^{6} + 20 q^{7} - 124 q^{9} + 6 q^{10} - 6 q^{11} - 10 q^{12} - 26 q^{15} - 30 q^{16} + 10 q^{17} - 8 q^{20} + 8 q^{21} - 4 q^{24} + 4 q^{25} - 12 q^{29} + 4 q^{30} - 20 q^{31} - 60 q^{33} - 2 q^{35} - 26 q^{36} + 10 q^{37} + 30 q^{38} + 60 q^{39} + 4 q^{40} + 10 q^{41} - 50 q^{42} - 4 q^{44} - 66 q^{45} + 24 q^{46} + 42 q^{49} - 24 q^{50} + 8 q^{51} + 30 q^{53} - 10 q^{54} - 84 q^{55} - 20 q^{57} + 12 q^{59} - 4 q^{60} - 8 q^{61} + 20 q^{62} - 90 q^{63} + 30 q^{64} - 76 q^{65} + 16 q^{66} - 50 q^{67} + 10 q^{68} + 30 q^{69} - 16 q^{70} + 58 q^{71} + 30 q^{73} - 20 q^{74} + 14 q^{75} + 60 q^{77} - 50 q^{78} + 8 q^{79} - 2 q^{80} + 176 q^{81} + 20 q^{82} + 50 q^{83} - 8 q^{84} - 78 q^{85} - 6 q^{86} + 20 q^{88} - 20 q^{89} - 20 q^{90} - 80 q^{93} - 16 q^{94} + 130 q^{95} - 16 q^{96} + 40 q^{98} + 58 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 −0.951057 + 0.309017i − 3.02605i 0.809017 − 0.587785i 2.22655 + 0.206121i 0.935100 + 2.87794i −1.23257 + 0.400486i −0.587785 + 0.809017i −6.15696 −2.18127 + 0.492008i
59.2 −0.951057 + 0.309017i − 2.56344i 0.809017 − 0.587785i 0.244243 − 2.22269i 0.792146 + 2.43798i 0.836205 − 0.271700i −0.587785 + 0.809017i −3.57122 0.454560 + 2.18938i
59.3 −0.951057 + 0.309017i − 2.18066i 0.809017 − 0.587785i 1.46724 + 1.68737i 0.673862 + 2.07393i 3.99097 − 1.29675i −0.587785 + 0.809017i −1.75529 −1.91685 − 1.15138i
59.4 −0.951057 + 0.309017i − 1.97724i 0.809017 − 0.587785i −1.43939 − 1.71119i 0.611000 + 1.88047i −3.00221 + 0.975478i −0.587785 + 0.809017i −0.909473 1.89773 + 1.18264i
59.5 −0.951057 + 0.309017i − 1.16405i 0.809017 − 0.587785i −1.54160 + 1.61971i 0.359712 + 1.10708i 0.544583 − 0.176946i −0.587785 + 0.809017i 1.64498 0.965630 − 2.01682i
59.6 −0.951057 + 0.309017i − 1.12473i 0.809017 − 0.587785i −2.12107 − 0.707862i 0.347560 + 1.06968i 3.43547 − 1.11625i −0.587785 + 0.809017i 1.73499 2.23600 + 0.0177703i
59.7 −0.951057 + 0.309017i − 0.382647i 0.809017 − 0.587785i 2.11331 − 0.730709i 0.118244 + 0.363919i −2.04804 + 0.665447i −0.587785 + 0.809017i 2.85358 −1.78407 + 1.34799i
59.8 −0.951057 + 0.309017i − 0.0821346i 0.809017 − 0.587785i −1.47512 + 1.68048i 0.0253810 + 0.0781146i −2.66936 + 0.867328i −0.587785 + 0.809017i 2.99325 0.883629 − 2.05407i
59.9 −0.951057 + 0.309017i 0.196811i 0.809017 − 0.587785i 1.28752 − 1.82820i −0.0608180 − 0.187179i 4.85072 − 1.57609i −0.587785 + 0.809017i 2.96127 −0.659558 + 2.13658i
59.10 −0.951057 + 0.309017i 0.978458i 0.809017 − 0.587785i 0.925704 + 2.03545i −0.302360 − 0.930569i 1.09543 − 0.355927i −0.587785 + 0.809017i 2.04262 −1.50939 − 1.64977i
59.11 −0.951057 + 0.309017i 1.56202i 0.809017 − 0.587785i −1.99109 − 1.01762i −0.482690 − 1.48557i 0.552852 − 0.179633i −0.587785 + 0.809017i 0.560101 2.20810 + 0.352535i
59.12 −0.951057 + 0.309017i 2.08483i 0.809017 − 0.587785i 0.822015 − 2.07949i −0.644249 − 1.98280i −4.87437 + 1.58378i −0.587785 + 0.809017i −1.34654 −0.139184 + 2.23173i
59.13 −0.951057 + 0.309017i 2.37382i 0.809017 − 0.587785i 0.754216 + 2.10503i −0.733551 − 2.25764i −2.29325 + 0.745123i −0.587785 + 0.809017i −2.63503 −1.36779 − 1.76894i
59.14 −0.951057 + 0.309017i 2.98321i 0.809017 − 0.587785i −2.21982 + 0.269074i −0.921862 − 2.83720i 1.18019 − 0.383468i −0.587785 + 0.809017i −5.89954 2.02803 − 0.941866i
59.15 −0.951057 + 0.309017i 3.14622i 0.809017 − 0.587785i 1.94731 − 1.09909i −0.972236 − 2.99224i 3.25141 − 1.05645i −0.587785 + 0.809017i −6.89872 −1.51236 + 1.64705i
59.16 0.951057 − 0.309017i − 3.30651i 0.809017 − 0.587785i −1.69049 − 1.46364i −1.02177 − 3.14468i 0.520443 − 0.169102i 0.587785 − 0.809017i −7.93301 −2.06004 − 0.869610i
59.17 0.951057 − 0.309017i − 3.04287i 0.809017 − 0.587785i 0.311964 + 2.21420i −0.940298 − 2.89394i 4.21725 − 1.37027i 0.587785 − 0.809017i −6.25905 0.980921 + 2.00943i
59.18 0.951057 − 0.309017i − 2.74724i 0.809017 − 0.587785i 1.39053 − 1.75112i −0.848943 − 2.61278i −4.29448 + 1.39536i 0.587785 − 0.809017i −4.54731 0.781351 − 2.09511i
59.19 0.951057 − 0.309017i − 1.58034i 0.809017 − 0.587785i 1.14254 − 1.92213i −0.488353 − 1.50299i 2.51084 − 0.815823i 0.587785 − 0.809017i 0.502519 0.492649 − 2.18112i
59.20 0.951057 − 0.309017i − 1.38221i 0.809017 − 0.587785i −2.10933 + 0.742112i −0.427126 − 1.31456i 2.05179 − 0.666666i 0.587785 − 0.809017i 1.08950 −1.77677 + 1.35761i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
275.t even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 550.2.n.a ✓ 120
11.c even 5 1 550.2.bb.a yes 120
25.e even 10 1 550.2.bb.a yes 120
275.t even 10 1 inner 550.2.n.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.n.a ✓ 120 1.a even 1 1 trivial
550.2.n.a ✓ 120 275.t even 10 1 inner
550.2.bb.a yes 120 11.c even 5 1
550.2.bb.a yes 120 25.e even 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(550, [\chi])\).