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Properties

Label 550.2.h.k
Level $550$
Weight $2$
Character orbit 550.h
Analytic conductor $4.392$
Analytic rank $0$
Dimension $8$
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(201,550)]
 
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([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.201");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.h (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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.159390625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + (\beta_{7} - \beta_{6} + \beta_{5} + \cdots + 1) q^{3}+ \cdots + ( - \beta_{6} + \beta_{5} + 4 \beta_{3} + \cdots - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + (\beta_{7} - \beta_{6} + \beta_{5} + \cdots + 1) q^{3}+ \cdots + (5 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} + \cdots - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 4 q^{3} - 2 q^{4} - q^{6} - 2 q^{7} - 2 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 4 q^{3} - 2 q^{4} - q^{6} - 2 q^{7} - 2 q^{8} - 24 q^{9} + 5 q^{11} - 6 q^{12} - 4 q^{13} - 2 q^{14} - 2 q^{16} - 18 q^{17} + 11 q^{18} + 17 q^{19} - 40 q^{21} - 5 q^{22} + 4 q^{23} - q^{24} + 6 q^{26} + 19 q^{27} + 8 q^{28} + 2 q^{29} - 2 q^{31} + 8 q^{32} + 23 q^{33} + 2 q^{34} + 11 q^{36} + 8 q^{37} - 18 q^{38} + 24 q^{39} + 6 q^{41} + 20 q^{42} + 16 q^{43} + 10 q^{44} - 6 q^{46} + 12 q^{47} + 4 q^{48} - 6 q^{49} - 17 q^{51} + 6 q^{52} - 8 q^{53} - 6 q^{54} - 12 q^{56} + 19 q^{57} + 2 q^{58} - 12 q^{59} - 2 q^{61} - 22 q^{62} - 48 q^{63} - 2 q^{64} - 27 q^{66} - 34 q^{67} - 18 q^{68} + 20 q^{69} + 20 q^{71} - 24 q^{72} - 26 q^{73} + 8 q^{74} + 2 q^{76} + 26 q^{77} - 36 q^{78} + 14 q^{79} + 35 q^{81} - 9 q^{82} - 9 q^{83} + 26 q^{86} - 56 q^{87} - 5 q^{88} + 2 q^{89} - 52 q^{91} + 4 q^{92} + 32 q^{93} - 18 q^{94} + 4 q^{96} - 26 q^{97} + 64 q^{98} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 555\nu^{7} - 2159\nu^{6} + 7489\nu^{5} - 18164\nu^{4} + 40069\nu^{3} - 84434\nu^{2} + 43855\nu + 375 ) / 94655 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -970\nu^{7} - 1002\nu^{6} - 6608\nu^{5} + 9063\nu^{4} - 14943\nu^{3} + 27673\nu^{2} - 68120\nu + 35160 ) / 94655 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -1604\nu^{7} + 4159\nu^{6} - 12059\nu^{5} + 28414\nu^{4} - 81659\nu^{3} + 38305\nu^{2} - 13500\nu - 13875 ) / 94655 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2052\nu^{7} + 2252\nu^{6} - 19912\nu^{5} + 21007\nu^{4} - 82042\nu^{3} + 35785\nu^{2} - 19395\nu - 90925 ) / 94655 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2667\nu^{7} + 6691\nu^{6} - 17466\nu^{5} + 50856\nu^{4} - 82441\nu^{3} + 72554\nu^{2} - 4035\nu - 12035 ) / 94655 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4024\nu^{7} - 1464\nu^{6} + 21519\nu^{5} - 26434\nu^{4} + 59219\nu^{3} + 22635\nu^{2} + 54640\nu + 66675 ) / 94655 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 3\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{6} + \beta_{5} - 4\beta_{4} - \beta_{3} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{7} + 7\beta_{6} + 2\beta_{5} + 13\beta_{3} + 13\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{7} - 11\beta_{6} - 20\beta_{5} + 20\beta_{4} - 11\beta_{2} + 8\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -19\beta_{7} - 19\beta_{4} - 68\beta_{3} - 36\beta_{2} - 24\beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 111\beta_{7} + 81\beta_{6} + 111\beta_{5} - 55\beta_{4} + 81\beta_{3} + 148\beta_{2} - 56\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/550\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(-\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1
0.453245 − 1.39494i
−0.762262 + 2.34600i
−0.628998 − 0.456994i
1.43801 + 1.04478i
0.453245 + 1.39494i
−0.762262 − 2.34600i
−0.628998 + 0.456994i
1.43801 − 1.04478i
−0.809017 + 0.587785i −1.04238 − 3.20812i 0.309017 − 0.951057i 0 2.72899 + 1.98273i −0.0577923 + 0.177866i 0.309017 + 0.951057i −6.77845 + 4.92483i 0
201.2 −0.809017 + 0.587785i 0.924349 + 2.84485i 0.309017 − 0.951057i 0 −2.41998 − 1.75822i −1.56024 + 4.80193i 0.309017 + 0.951057i −4.81172 + 3.49592i 0
251.1 0.309017 − 0.951057i 0.420275 − 0.305348i −0.809017 − 0.587785i 0 −0.160531 − 0.494063i 3.65351 + 2.65443i −0.809017 + 0.587785i −0.843657 + 2.59651i 0
251.2 0.309017 − 0.951057i 1.69776 − 1.23349i −0.809017 − 0.587785i 0 −0.648486 − 1.99584i −3.03548 − 2.20541i −0.809017 + 0.587785i 0.433827 − 1.33518i 0
301.1 −0.809017 − 0.587785i −1.04238 + 3.20812i 0.309017 + 0.951057i 0 2.72899 − 1.98273i −0.0577923 − 0.177866i 0.309017 − 0.951057i −6.77845 − 4.92483i 0
301.2 −0.809017 − 0.587785i 0.924349 − 2.84485i 0.309017 + 0.951057i 0 −2.41998 + 1.75822i −1.56024 − 4.80193i 0.309017 − 0.951057i −4.81172 − 3.49592i 0
401.1 0.309017 + 0.951057i 0.420275 + 0.305348i −0.809017 + 0.587785i 0 −0.160531 + 0.494063i 3.65351 − 2.65443i −0.809017 − 0.587785i −0.843657 − 2.59651i 0
401.2 0.309017 + 0.951057i 1.69776 + 1.23349i −0.809017 + 0.587785i 0 −0.648486 + 1.99584i −3.03548 + 2.20541i −0.809017 − 0.587785i 0.433827 + 1.33518i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 201.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c 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.h.k ✓ 8
5.b even 2 1 550.2.h.m yes 8
5.c odd 4 2 550.2.ba.g 16
11.c even 5 1 inner 550.2.h.k ✓ 8
11.c even 5 1 6050.2.a.dg 4
11.d odd 10 1 6050.2.a.cz 4
55.h odd 10 1 6050.2.a.dn 4
55.j even 10 1 550.2.h.m yes 8
55.j even 10 1 6050.2.a.df 4
55.k odd 20 2 550.2.ba.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.h.k ✓ 8 1.a even 1 1 trivial
550.2.h.k ✓ 8 11.c even 5 1 inner
550.2.h.m yes 8 5.b even 2 1
550.2.h.m yes 8 55.j even 10 1
550.2.ba.g 16 5.c odd 4 2
550.2.ba.g 16 55.k odd 20 2
6050.2.a.cz 4 11.d odd 10 1
6050.2.a.df 4 55.j even 10 1
6050.2.a.dg 4 11.c even 5 1
6050.2.a.dn 4 55.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(550, [\chi])\):

\( T_{3}^{8} - 4T_{3}^{7} + 23T_{3}^{6} - 75T_{3}^{5} + 236T_{3}^{4} - 525T_{3}^{3} + 797T_{3}^{2} - 473T_{3} + 121 \) Copy content Toggle raw display
\( T_{7}^{8} + 2T_{7}^{7} + 12T_{7}^{6} - 40T_{7}^{5} + 96T_{7}^{4} + 1440T_{7}^{3} + 7488T_{7}^{2} + 896T_{7} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{8} - 5 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + \cdots + 6400 \) Copy content Toggle raw display
$17$ \( T^{8} + 18 T^{7} + \cdots + 21025 \) Copy content Toggle raw display
$19$ \( T^{8} - 17 T^{7} + \cdots + 21025 \) Copy content Toggle raw display
$23$ \( (T^{4} - 2 T^{3} - 48 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 2 T^{7} + \cdots + 6400 \) Copy content Toggle raw display
$31$ \( T^{8} + 2 T^{7} + \cdots + 6400 \) Copy content Toggle raw display
$37$ \( T^{8} - 8 T^{7} + \cdots + 30976 \) Copy content Toggle raw display
$41$ \( T^{8} - 6 T^{7} + \cdots + 10201 \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} + \cdots + 955)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 12 T^{7} + \cdots + 774400 \) Copy content Toggle raw display
$53$ \( T^{8} + 8 T^{7} + \cdots + 3748096 \) Copy content Toggle raw display
$59$ \( T^{8} + 12 T^{7} + \cdots + 21025 \) Copy content Toggle raw display
$61$ \( T^{8} + 2 T^{7} + \cdots + 6400 \) Copy content Toggle raw display
$67$ \( (T^{4} + 17 T^{3} + \cdots - 2011)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 20 T^{7} + \cdots + 30976 \) Copy content Toggle raw display
$73$ \( T^{8} + 26 T^{7} + \cdots + 203401 \) Copy content Toggle raw display
$79$ \( T^{8} - 14 T^{7} + \cdots + 160000 \) Copy content Toggle raw display
$83$ \( T^{8} + 9 T^{7} + \cdots + 3025 \) Copy content Toggle raw display
$89$ \( (T^{4} - T^{3} - 236 T^{2} + \cdots + 2245)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 13 T^{3} + \cdots + 361)^{2} \) Copy content Toggle raw display
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