Properties

Label 550.6.b.f
Level $550$
Weight $6$
Character orbit 550.b
Analytic conductor $88.211$
Analytic rank $0$
Dimension $2$
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] = [550,6,Mod(199,550)]
 
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, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.199");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 550.b (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: \(88.2111008971\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) 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 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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 i q^{2} + i q^{3} - 16 q^{4} - 4 q^{6} + 166 i q^{7} - 64 i q^{8} + 242 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 i q^{2} + i q^{3} - 16 q^{4} - 4 q^{6} + 166 i q^{7} - 64 i q^{8} + 242 q^{9} - 121 q^{11} - 16 i q^{12} + 692 i q^{13} - 664 q^{14} + 256 q^{16} + 738 i q^{17} + 968 i q^{18} - 1424 q^{19} - 166 q^{21} - 484 i q^{22} - 1779 i q^{23} + 64 q^{24} - 2768 q^{26} + 485 i q^{27} - 2656 i q^{28} + 2064 q^{29} + 6245 q^{31} + 1024 i q^{32} - 121 i q^{33} - 2952 q^{34} - 3872 q^{36} + 14785 i q^{37} - 5696 i q^{38} - 692 q^{39} + 5304 q^{41} - 664 i q^{42} + 17798 i q^{43} + 1936 q^{44} + 7116 q^{46} + 17184 i q^{47} + 256 i q^{48} - 10749 q^{49} - 738 q^{51} - 11072 i q^{52} - 30726 i q^{53} - 1940 q^{54} + 10624 q^{56} - 1424 i q^{57} + 8256 i q^{58} + 34989 q^{59} - 45940 q^{61} + 24980 i q^{62} + 40172 i q^{63} - 4096 q^{64} + 484 q^{66} - 25343 i q^{67} - 11808 i q^{68} + 1779 q^{69} + 13311 q^{71} - 15488 i q^{72} - 53260 i q^{73} - 59140 q^{74} + 22784 q^{76} - 20086 i q^{77} - 2768 i q^{78} - 77234 q^{79} + 58321 q^{81} + 21216 i q^{82} + 55014 i q^{83} + 2656 q^{84} - 71192 q^{86} + 2064 i q^{87} + 7744 i q^{88} - 125415 q^{89} - 114872 q^{91} + 28464 i q^{92} + 6245 i q^{93} - 68736 q^{94} - 1024 q^{96} + 88807 i q^{97} - 42996 i q^{98} - 29282 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4} - 8 q^{6} + 484 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} - 8 q^{6} + 484 q^{9} - 242 q^{11} - 1328 q^{14} + 512 q^{16} - 2848 q^{19} - 332 q^{21} + 128 q^{24} - 5536 q^{26} + 4128 q^{29} + 12490 q^{31} - 5904 q^{34} - 7744 q^{36} - 1384 q^{39} + 10608 q^{41} + 3872 q^{44} + 14232 q^{46} - 21498 q^{49} - 1476 q^{51} - 3880 q^{54} + 21248 q^{56} + 69978 q^{59} - 91880 q^{61} - 8192 q^{64} + 968 q^{66} + 3558 q^{69} + 26622 q^{71} - 118280 q^{74} + 45568 q^{76} - 154468 q^{79} + 116642 q^{81} + 5312 q^{84} - 142384 q^{86} - 250830 q^{89} - 229744 q^{91} - 137472 q^{94} - 2048 q^{96} - 58564 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
4.00000i 1.00000i −16.0000 0 −4.00000 166.000i 64.0000i 242.000 0
199.2 4.00000i 1.00000i −16.0000 0 −4.00000 166.000i 64.0000i 242.000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 550.6.b.f 2
5.b even 2 1 inner 550.6.b.f 2
5.c odd 4 1 22.6.a.b 1
5.c odd 4 1 550.6.a.f 1
15.e even 4 1 198.6.a.i 1
20.e even 4 1 176.6.a.b 1
35.f even 4 1 1078.6.a.a 1
40.i odd 4 1 704.6.a.e 1
40.k even 4 1 704.6.a.f 1
55.e even 4 1 242.6.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.a.b 1 5.c odd 4 1
176.6.a.b 1 20.e even 4 1
198.6.a.i 1 15.e even 4 1
242.6.a.d 1 55.e even 4 1
550.6.a.f 1 5.c odd 4 1
550.6.b.f 2 1.a even 1 1 trivial
550.6.b.f 2 5.b even 2 1 inner
704.6.a.e 1 40.i odd 4 1
704.6.a.f 1 40.k even 4 1
1078.6.a.a 1 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(550, [\chi])\):

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 27556 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 27556 \) Copy content Toggle raw display
$11$ \( (T + 121)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 478864 \) Copy content Toggle raw display
$17$ \( T^{2} + 544644 \) Copy content Toggle raw display
$19$ \( (T + 1424)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3164841 \) Copy content Toggle raw display
$29$ \( (T - 2064)^{2} \) Copy content Toggle raw display
$31$ \( (T - 6245)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 218596225 \) Copy content Toggle raw display
$41$ \( (T - 5304)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 316768804 \) Copy content Toggle raw display
$47$ \( T^{2} + 295289856 \) Copy content Toggle raw display
$53$ \( T^{2} + 944087076 \) Copy content Toggle raw display
$59$ \( (T - 34989)^{2} \) Copy content Toggle raw display
$61$ \( (T + 45940)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 642267649 \) Copy content Toggle raw display
$71$ \( (T - 13311)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2836627600 \) Copy content Toggle raw display
$79$ \( (T + 77234)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 3026540196 \) Copy content Toggle raw display
$89$ \( (T + 125415)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 7886683249 \) Copy content Toggle raw display
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