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Properties

Label 550.5.d.a
Level $550$
Weight $5$
Character orbit 550.d
Analytic conductor $56.853$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [550,5,Mod(351,550)]
 
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([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.351");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 550.d (of order \(2\), degree \(1\), 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: \(56.8534796961\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{553})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 271x^{2} + 272x + 19602 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_1 q^{3} - 8 q^{4} - \beta_{3} q^{6} + 12 \beta_{2} q^{7} - 8 \beta_{2} q^{8} + ( - \beta_1 + 57) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_1 q^{3} - 8 q^{4} - \beta_{3} q^{6} + 12 \beta_{2} q^{7} - 8 \beta_{2} q^{8} + ( - \beta_1 + 57) q^{9} + (3 \beta_{3} - 9 \beta_{2} + 4 \beta_1 - 43) q^{11} - 8 \beta_1 q^{12} + ( - 6 \beta_{3} - 42 \beta_{2}) q^{13} - 96 q^{14} + 64 q^{16} + (12 \beta_{3} - 48 \beta_{2}) q^{17} + (\beta_{3} + 57 \beta_{2}) q^{18} + (6 \beta_{3} - 78 \beta_{2}) q^{19} - 12 \beta_{3} q^{21} + ( - 4 \beta_{3} - 43 \beta_{2} + \cdots + 72) q^{22}+ \cdots + (165 \beta_{3} - 99 \beta_{2} + \cdots - 3003) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 32 q^{4} + 230 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 32 q^{4} + 230 q^{9} - 180 q^{11} + 16 q^{12} - 384 q^{14} + 256 q^{16} + 240 q^{22} + 1566 q^{23} + 1440 q^{26} - 506 q^{27} + 4418 q^{31} + 2302 q^{33} + 1344 q^{34} - 1840 q^{36} + 382 q^{37} + 2400 q^{38} + 192 q^{42} + 1440 q^{44} - 5688 q^{47} - 128 q^{48} + 4996 q^{49} + 8568 q^{53} + 3072 q^{56} + 9504 q^{58} - 3390 q^{59} - 2048 q^{64} + 13152 q^{66} + 8734 q^{67} + 26314 q^{69} + 3522 q^{71} + 2880 q^{77} - 27264 q^{78} - 31096 q^{81} + 19968 q^{82} - 10464 q^{86} - 1920 q^{88} - 8766 q^{89} + 17280 q^{91} - 12528 q^{92} - 20458 q^{93} - 17282 q^{97} - 12562 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 271x^{2} + 272x + 19602 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} - 3\nu^{2} - 824\nu + 132 ) / 561 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{3} - 6\nu^{2} - 526\nu + 264 ) / 561 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} + 558\nu^{2} - 824\nu - 76164 ) / 561 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta _1 + 136 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{3} + 412\beta_{2} - 266\beta _1 + 276 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
351.1
12.2580 − 1.41421i
−11.2580 − 1.41421i
12.2580 + 1.41421i
−11.2580 + 1.41421i
− 2.82843i −12.2580 −8.00000 0 34.6708i − 33.9411i 22.6274i 69.2580 0
351.2 − 2.82843i 11.2580 −8.00000 0 − 31.8424i − 33.9411i 22.6274i 45.7420 0
351.3 2.82843i −12.2580 −8.00000 0 − 34.6708i 33.9411i − 22.6274i 69.2580 0
351.4 2.82843i 11.2580 −8.00000 0 31.8424i 33.9411i − 22.6274i 45.7420 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
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 550.5.d.a 4
5.b even 2 1 22.5.b.a ✓ 4
5.c odd 4 2 550.5.c.a 8
11.b odd 2 1 inner 550.5.d.a 4
15.d odd 2 1 198.5.d.a 4
20.d odd 2 1 176.5.h.e 4
40.e odd 2 1 704.5.h.j 4
40.f even 2 1 704.5.h.i 4
55.d odd 2 1 22.5.b.a ✓ 4
55.e even 4 2 550.5.c.a 8
165.d even 2 1 198.5.d.a 4
220.g even 2 1 176.5.h.e 4
440.c even 2 1 704.5.h.j 4
440.o odd 2 1 704.5.h.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.5.b.a ✓ 4 5.b even 2 1
22.5.b.a ✓ 4 55.d odd 2 1
176.5.h.e 4 20.d odd 2 1
176.5.h.e 4 220.g even 2 1
198.5.d.a 4 15.d odd 2 1
198.5.d.a 4 165.d even 2 1
550.5.c.a 8 5.c odd 4 2
550.5.c.a 8 55.e even 4 2
550.5.d.a 4 1.a even 1 1 trivial
550.5.d.a 4 11.b odd 2 1 inner
704.5.h.i 4 40.f even 2 1
704.5.h.i 4 440.o odd 2 1
704.5.h.j 4 40.e odd 2 1
704.5.h.j 4 440.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} - 138 \) acting on \(S_{5}^{\mathrm{new}}(550, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + T - 138)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1152)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 180 T^{3} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{4} + 112032 T^{2} + 557715456 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 21069103104 \) Copy content Toggle raw display
$19$ \( T^{4} + 169632 T^{2} + 26873856 \) Copy content Toggle raw display
$23$ \( (T^{2} - 783 T - 178666)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 443364212736 \) Copy content Toggle raw display
$31$ \( (T^{2} - 2209 T + 1069366)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 191 T - 1694258)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 6140246114304 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 5615307472896 \) Copy content Toggle raw display
$47$ \( (T^{2} + 2844 T - 534988)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 4284 T + 1048964)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 1695 T - 582538)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 74192451158016 \) Copy content Toggle raw display
$67$ \( (T^{2} - 4367 T + 4736566)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 1761 T - 10612234)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 33350347800576 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 205902754873344 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 39\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{2} + 4383 T - 48856162)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8641 T + 14335486)^{2} \) Copy content Toggle raw display
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