↑

Properties

Label 550.3.c.a
Level $550$
Weight $3$
Character orbit 550.c
Analytic conductor $14.986$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [550,3,Mod(549,550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.549");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 550.c (of order \(2\), degree \(1\), 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: \(14.9864145398\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 22)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + \beta_1 q^{3} + 2 q^{4} - \beta_{2} q^{6} + 6 \beta_{3} q^{7} - 2 \beta_{3} q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + \beta_1 q^{3} + 2 q^{4} - \beta_{2} q^{6} + 6 \beta_{3} q^{7} - 2 \beta_{3} q^{8} + 8 q^{9} + ( - 6 \beta_{2} + 7) q^{11} + 2 \beta_1 q^{12} + 6 \beta_{3} q^{13} - 12 q^{14} + 4 q^{16} - 18 \beta_{3} q^{17} - 8 \beta_{3} q^{18} - 18 \beta_{2} q^{19} + 6 \beta_{2} q^{21} + ( - 7 \beta_{3} + 12 \beta_1) q^{22} + 17 \beta_1 q^{23} - 2 \beta_{2} q^{24} - 12 q^{26} + 17 \beta_1 q^{27} + 12 \beta_{3} q^{28} - 24 \beta_{2} q^{29} + 17 q^{31} - 4 \beta_{3} q^{32} + (6 \beta_{3} + 7 \beta_1) q^{33} + 36 q^{34} + 16 q^{36} - 47 \beta_1 q^{37} + 36 \beta_1 q^{38} + 6 \beta_{2} q^{39} - 6 \beta_{2} q^{41} - 12 \beta_1 q^{42} + 12 \beta_{3} q^{43} + ( - 12 \beta_{2} + 14) q^{44} - 17 \beta_{2} q^{46} + 58 \beta_1 q^{47} + 4 \beta_1 q^{48} + 23 q^{49} - 18 \beta_{2} q^{51} + 12 \beta_{3} q^{52} + 2 \beta_1 q^{53} - 17 \beta_{2} q^{54} - 24 q^{56} + 18 \beta_{3} q^{57} + 48 \beta_1 q^{58} + 55 q^{59} + 60 \beta_{2} q^{61} - 17 \beta_{3} q^{62} + 48 \beta_{3} q^{63} + 8 q^{64} + ( - 7 \beta_{2} - 12) q^{66} - 89 \beta_1 q^{67} - 36 \beta_{3} q^{68} - 17 q^{69} - 7 q^{71} - 16 \beta_{3} q^{72} + 90 \beta_{3} q^{73} + 47 \beta_{2} q^{74} - 36 \beta_{2} q^{76} + (42 \beta_{3} - 72 \beta_1) q^{77} - 12 \beta_1 q^{78} - 24 \beta_{2} q^{79} + 55 q^{81} + 12 \beta_1 q^{82} - 24 \beta_{3} q^{83} + 12 \beta_{2} q^{84} - 24 q^{86} + 24 \beta_{3} q^{87} + ( - 14 \beta_{3} + 24 \beta_1) q^{88} + 97 q^{89} + 72 q^{91} + 34 \beta_1 q^{92} + 17 \beta_1 q^{93} - 58 \beta_{2} q^{94} - 4 \beta_{2} q^{96} + 121 \beta_1 q^{97} - 23 \beta_{3} q^{98} + ( - 48 \beta_{2} + 56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} + 32 q^{9} + 28 q^{11} - 48 q^{14} + 16 q^{16} - 48 q^{26} + 68 q^{31} + 144 q^{34} + 64 q^{36} + 56 q^{44} + 92 q^{49} - 96 q^{56} + 220 q^{59} + 32 q^{64} - 48 q^{66} - 68 q^{69} - 28 q^{71} + 220 q^{81} - 96 q^{86} + 388 q^{89} + 288 q^{91} + 224 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display

Display \(\zeta_{8}^j\) in terms of \(\beta_i\)

\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{8}^j\)

Character values

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

\(n\) \(101\) \(177\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
549.1
0.707107 − 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
−0.707107 − 0.707107i
−1.41421 − 1.00000i 2.00000 0 1.41421i 8.48528 −2.82843 8.00000 0
549.2 −1.41421 1.00000i 2.00000 0 − 1.41421i 8.48528 −2.82843 8.00000 0
549.3 1.41421 − 1.00000i 2.00000 0 − 1.41421i −8.48528 2.82843 8.00000 0
549.4 1.41421 1.00000i 2.00000 0 1.41421i −8.48528 2.82843 8.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.b odd 2 1 inner
55.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 550.3.c.a 4
5.b even 2 1 inner 550.3.c.a 4
5.c odd 4 1 22.3.b.a ✓ 2
5.c odd 4 1 550.3.d.a 2
11.b odd 2 1 inner 550.3.c.a 4
15.e even 4 1 198.3.d.b 2
20.e even 4 1 176.3.h.c 2
35.f even 4 1 1078.3.d.a 2
40.i odd 4 1 704.3.h.d 2
40.k even 4 1 704.3.h.e 2
55.d odd 2 1 inner 550.3.c.a 4
55.e even 4 1 22.3.b.a ✓ 2
55.e even 4 1 550.3.d.a 2
55.k odd 20 4 242.3.d.b 8
55.l even 20 4 242.3.d.b 8
60.l odd 4 1 1584.3.j.d 2
165.l odd 4 1 198.3.d.b 2
220.i odd 4 1 176.3.h.c 2
385.l odd 4 1 1078.3.d.a 2
440.t even 4 1 704.3.h.d 2
440.w odd 4 1 704.3.h.e 2
660.q even 4 1 1584.3.j.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.3.b.a ✓ 2 5.c odd 4 1
22.3.b.a ✓ 2 55.e even 4 1
176.3.h.c 2 20.e even 4 1
176.3.h.c 2 220.i odd 4 1
198.3.d.b 2 15.e even 4 1
198.3.d.b 2 165.l odd 4 1
242.3.d.b 8 55.k odd 20 4
242.3.d.b 8 55.l even 20 4
550.3.c.a 4 1.a even 1 1 trivial
550.3.c.a 4 5.b even 2 1 inner
550.3.c.a 4 11.b odd 2 1 inner
550.3.c.a 4 55.d odd 2 1 inner
550.3.d.a 2 5.c odd 4 1
550.3.d.a 2 55.e even 4 1
704.3.h.d 2 40.i odd 4 1
704.3.h.d 2 440.t even 4 1
704.3.h.e 2 40.k even 4 1
704.3.h.e 2 440.w odd 4 1
1078.3.d.a 2 35.f even 4 1
1078.3.d.a 2 385.l odd 4 1
1584.3.j.d 2 60.l odd 4 1
1584.3.j.d 2 660.q even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(550, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 14 T + 121)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 648)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 648)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 289)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1152)^{2} \) Copy content Toggle raw display
$31$ \( (T - 17)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2209)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 288)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 3364)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$59$ \( (T - 55)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 7200)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 7921)^{2} \) Copy content Toggle raw display
$71$ \( (T + 7)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 16200)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 1152)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 1152)^{2} \) Copy content Toggle raw display
$89$ \( (T - 97)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 14641)^{2} \) Copy content Toggle raw display
show more
show less