↑

Properties

Label 550.2.h.l
Level $550$
Weight $2$
Character orbit 550.h
Analytic conductor $4.392$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: 8.0.682515625.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} + 2x^{5} + 19x^{4} + 28x^{3} + 100x^{2} + 88x + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
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.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{3} + \beta_{2} - 1) q^{2} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots + \beta_1) q^{3}+ \cdots + (\beta_{7} - \beta_{6} + \beta_{4} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - \beta_{3} + \beta_{2} - 1) q^{2} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots + \beta_1) q^{3}+ \cdots + (2 \beta_{7} + 4 \beta_{6} + \beta_{5} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 4 q^{3} - 2 q^{4} - q^{6} + q^{7} - 2 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 4 q^{3} - 2 q^{4} - q^{6} + q^{7} - 2 q^{8} - 6 q^{9} + 2 q^{11} - 6 q^{12} - q^{13} + q^{14} - 2 q^{16} + 12 q^{17} - q^{18} - 7 q^{19} + 32 q^{21} - 8 q^{22} - 26 q^{23} - q^{24} - 6 q^{26} + q^{27} - 4 q^{28} - 22 q^{29} - 26 q^{31} + 8 q^{32} - 19 q^{33} + 2 q^{34} - q^{36} + 8 q^{37} + 3 q^{38} - 48 q^{39} - 15 q^{41} + 2 q^{42} - 38 q^{43} + 7 q^{44} - 6 q^{46} - 6 q^{47} + 4 q^{48} + 27 q^{49} - 5 q^{51} - 6 q^{52} + 4 q^{53} - 24 q^{54} + 6 q^{56} - 47 q^{57} - 22 q^{58} + 39 q^{59} - 20 q^{61} + 14 q^{62} + 57 q^{63} - 2 q^{64} + 36 q^{66} + 26 q^{67} + 12 q^{68} - 10 q^{69} + 2 q^{71} - 6 q^{72} - 8 q^{73} + 8 q^{74} + 8 q^{76} - 16 q^{77} + 12 q^{78} + 14 q^{79} - 31 q^{81} + 15 q^{82} + 21 q^{83} - 18 q^{84} + 17 q^{86} - 20 q^{87} + 7 q^{88} + 32 q^{89} + 53 q^{91} + 19 q^{92} + 38 q^{93} + 9 q^{94} + 4 q^{96} + 31 q^{97} + 22 q^{98} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} + 2x^{5} + 19x^{4} + 28x^{3} + 100x^{2} + 88x + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 528 \nu^{7} + 2098 \nu^{6} - 15725 \nu^{5} + 33439 \nu^{4} + 71401 \nu^{3} - 332708 \nu^{2} + \cdots + 440220 ) / 1168519 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5794 \nu^{7} - 9973 \nu^{6} - 30517 \nu^{5} + 195125 \nu^{4} - 61888 \nu^{3} + 104068 \nu^{2} + \cdots + 528473 ) / 1168519 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7409 \nu^{7} - 59487 \nu^{6} + 183537 \nu^{5} - 171974 \nu^{4} - 58164 \nu^{3} - 77439 \nu^{2} + \cdots - 701074 ) / 1168519 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8817 \nu^{7} + 16927 \nu^{6} - 106264 \nu^{5} + 200474 \nu^{4} + 521745 \nu^{3} + 380907 \nu^{2} + \cdots + 2809884 ) / 1168519 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11971 \nu^{7} + 3536 \nu^{6} + 58156 \nu^{5} - 228404 \nu^{4} - 102852 \nu^{3} - 979996 \nu^{2} + \cdots - 2305776 ) / 1168519 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13790 \nu^{7} + 57068 \nu^{6} - 113608 \nu^{5} + 65418 \nu^{4} - 266949 \nu^{3} + 6060 \nu^{2} + \cdots + 665808 ) / 1168519 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{3} - 5\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 6\beta_{6} + 6\beta_{5} + 2\beta_{4} + 4\beta_{3} - 10\beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{7} + 10\beta_{6} + 13\beta_{5} + 13\beta_{4} + 14\beta_{3} - 13\beta_{2} - 10\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 43\beta_{7} + 25\beta_{5} + 49\beta_{4} + 18\beta_{2} - 25\beta _1 - 62 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 97\beta_{7} - 92\beta_{6} + 92\beta_{4} - 97\beta_{3} + 221\beta_{2} - 44\beta _1 - 221 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -449\beta_{6} - 260\beta_{5} - 412\beta_{3} + 896\beta_{2} - 412 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(177\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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
−1.20316 + 0.874145i
2.51217 − 1.82520i
−0.390899 + 1.20306i
0.581882 − 1.79085i
−1.20316 − 0.874145i
2.51217 + 1.82520i
−0.390899 − 1.20306i
0.581882 + 1.79085i
−0.809017 + 0.587785i −0.768582 − 2.36545i 0.309017 − 0.951057i 0 2.01217 + 1.46193i 0.133479 − 0.410805i 0.309017 + 0.951057i −2.57760 + 1.87274i 0
201.2 −0.809017 + 0.587785i 0.650548 + 2.00218i 0.309017 − 0.951057i 0 −1.70316 − 1.23742i 0.675538 − 2.07909i 0.309017 + 0.951057i −1.15847 + 0.841677i 0
251.1 0.309017 − 0.951057i −0.214371 + 0.155750i −0.809017 − 0.587785i 0 0.0818824 + 0.252008i −3.48828 − 2.53438i −0.809017 + 0.587785i −0.905354 + 2.78639i 0
251.2 0.309017 − 0.951057i 2.33240 − 1.69459i −0.809017 − 0.587785i 0 −0.890899 − 2.74191i 3.17926 + 2.30987i −0.809017 + 0.587785i 1.64142 − 5.05178i 0
301.1 −0.809017 − 0.587785i −0.768582 + 2.36545i 0.309017 + 0.951057i 0 2.01217 − 1.46193i 0.133479 + 0.410805i 0.309017 − 0.951057i −2.57760 − 1.87274i 0
301.2 −0.809017 − 0.587785i 0.650548 − 2.00218i 0.309017 + 0.951057i 0 −1.70316 + 1.23742i 0.675538 + 2.07909i 0.309017 − 0.951057i −1.15847 − 0.841677i 0
401.1 0.309017 + 0.951057i −0.214371 − 0.155750i −0.809017 + 0.587785i 0 0.0818824 − 0.252008i −3.48828 + 2.53438i −0.809017 − 0.587785i −0.905354 − 2.78639i 0
401.2 0.309017 + 0.951057i 2.33240 + 1.69459i −0.809017 + 0.587785i 0 −0.890899 + 2.74191i 3.17926 − 2.30987i −0.809017 − 0.587785i 1.64142 + 5.05178i 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.l 8
5.b even 2 1 110.2.g.c ✓ 8
5.c odd 4 2 550.2.ba.f 16
11.c even 5 1 inner 550.2.h.l 8
11.c even 5 1 6050.2.a.dh 4
11.d odd 10 1 6050.2.a.cy 4
15.d odd 2 1 990.2.n.j 8
20.d odd 2 1 880.2.bo.g 8
55.h odd 10 1 1210.2.a.v 4
55.j even 10 1 110.2.g.c ✓ 8
55.j even 10 1 1210.2.a.u 4
55.k odd 20 2 550.2.ba.f 16
165.o odd 10 1 990.2.n.j 8
220.n odd 10 1 880.2.bo.g 8
220.n odd 10 1 9680.2.a.cj 4
220.o even 10 1 9680.2.a.ci 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.g.c ✓ 8 5.b even 2 1
110.2.g.c ✓ 8 55.j even 10 1
550.2.h.l 8 1.a even 1 1 trivial
550.2.h.l 8 11.c even 5 1 inner
550.2.ba.f 16 5.c odd 4 2
550.2.ba.f 16 55.k odd 20 2
880.2.bo.g 8 20.d odd 2 1
880.2.bo.g 8 220.n odd 10 1
990.2.n.j 8 15.d odd 2 1
990.2.n.j 8 165.o odd 10 1
1210.2.a.u 4 55.j even 10 1
1210.2.a.v 4 55.h odd 10 1
6050.2.a.cy 4 11.d odd 10 1
6050.2.a.dh 4 11.c even 5 1
9680.2.a.ci 4 220.o even 10 1
9680.2.a.cj 4 220.n odd 10 1

Hecke kernels

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

\( T_{3}^{8} - 4T_{3}^{7} + 14T_{3}^{6} - 33T_{3}^{5} + 89T_{3}^{4} - 96T_{3}^{3} + 176T_{3}^{2} + 88T_{3} + 16 \) Copy content Toggle raw display
\( T_{7}^{8} - T_{7}^{7} - 6T_{7}^{6} + 8T_{7}^{5} + 249T_{7}^{4} - 504T_{7}^{3} + 1536T_{7}^{2} - 448T_{7} + 256 \) 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
$3$ \( T^{8} - 4 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - T^{7} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{3} + 11 T^{2} + \cdots + 121)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + T^{7} + \cdots + 30976 \) Copy content Toggle raw display
$17$ \( T^{8} - 12 T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$19$ \( T^{8} + 7 T^{7} + \cdots + 22201 \) Copy content Toggle raw display
$23$ \( (T^{4} + 13 T^{3} + \cdots - 484)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 22 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$31$ \( T^{8} + 26 T^{7} + \cdots + 1290496 \) Copy content Toggle raw display
$37$ \( T^{8} - 8 T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$41$ \( T^{8} + 15 T^{7} + \cdots + 73441 \) Copy content Toggle raw display
$43$ \( (T^{4} + 19 T^{3} + \cdots - 44)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 6 T^{7} + \cdots + 59536 \) Copy content Toggle raw display
$53$ \( T^{8} - 4 T^{7} + \cdots + 2298256 \) Copy content Toggle raw display
$59$ \( T^{8} - 39 T^{7} + \cdots + 31933801 \) Copy content Toggle raw display
$61$ \( T^{8} + 20 T^{7} + \cdots + 7929856 \) Copy content Toggle raw display
$67$ \( (T^{4} - 13 T^{3} + \cdots - 604)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 2 T^{7} + \cdots + 28047616 \) Copy content Toggle raw display
$73$ \( T^{8} + 8 T^{7} + \cdots + 16613776 \) Copy content Toggle raw display
$79$ \( T^{8} - 14 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$83$ \( T^{8} - 21 T^{7} + \cdots + 1752976 \) Copy content Toggle raw display
$89$ \( (T^{4} - 16 T^{3} + 37 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 31 T^{7} + \cdots + 5683456 \) Copy content Toggle raw display
show more
show less