Properties

Label 50.18.b.f
Level $50$
Weight $18$
Character orbit 50.b
Analytic conductor $91.611$
Analytic rank $0$
Dimension $4$
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] = [50,18,Mod(49,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.49");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 50.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: \(91.6110436723\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{2941})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1471x^{2} + 540225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 10)
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.

\(f(q)\) \(=\) \( q - 128 \beta_1 q^{2} + (\beta_{2} + 4407 \beta_1) q^{3} - 65536 q^{4} + (128 \beta_{3} + 2256384) q^{6} + ( - 117 \beta_{2} - 6921049 \beta_1) q^{7} + 8388608 \beta_1 q^{8} + ( - 8814 \beta_{3} - 117948033) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 128 \beta_1 q^{2} + (\beta_{2} + 4407 \beta_1) q^{3} - 65536 q^{4} + (128 \beta_{3} + 2256384) q^{6} + ( - 117 \beta_{2} - 6921049 \beta_1) q^{7} + 8388608 \beta_1 q^{8} + ( - 8814 \beta_{3} - 117948033) q^{9} + ( - 45441 \beta_{3} + 32257632) q^{11} + ( - 65536 \beta_{2} - 288817152 \beta_1) q^{12} + ( - 74988 \beta_{2} + 723959617 \beta_1) q^{13} + ( - 14976 \beta_{3} - 3543577088) q^{14} + 4294967296 q^{16} + ( - 1394748 \beta_{2} - 395053149 \beta_1) q^{17} + (4512768 \beta_{2} + 15097348224 \beta_1) q^{18} + (3580956 \beta_{3} - 22106856380) q^{19} + (7436668 \beta_{3} + 141824238972) q^{21} + (23265792 \beta_{2} - 4128976896 \beta_1) q^{22} + ( - 23366889 \beta_{2} + 121887445707 \beta_1) q^{23} + ( - 8388608 \beta_{3} - 147874381824) q^{24} + ( - 9598464 \beta_{3} + 370667323904) q^{26} + ( - 144181062 \beta_{2} - 1443781985490 \beta_1) q^{27} + (7667712 \beta_{2} + 453577867264 \beta_1) q^{28} + (1294812 \beta_{3} - 1993657431750) q^{29} + (24213483 \beta_{3} - 2746130669668) q^{31} - 549755813888 \beta_1 q^{32} + ( - 768776316 \beta_{2} - 7555618721376 \beta_1) q^{33} + ( - 178527744 \beta_{3} - 202267212288) q^{34} + (577634304 \beta_{3} + 7729842290688) q^{36} + (503271432 \beta_{2} + 15678821909471 \beta_1) q^{37} + ( - 1833449472 \beta_{2} + 2829677616640 \beta_1) q^{38} + ( - 393487501 \beta_{3} - 58872947676) q^{39} + (2512768842 \beta_{3} + 11705738638662) q^{41} + ( - 3807574016 \beta_{2} - 18153502588416 \beta_1) q^{42} + ( - 2880330687 \beta_{2} - 31214230797773 \beta_1) q^{43} + (2978021376 \beta_{3} - 2114036170752) q^{44} + ( - 2990961792 \beta_{3} + \cdots + 62406372201984) q^{46}+ \cdots + (5075357799105 \beta_{3} + \cdots + 26\!\cdots\!44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 262144 q^{4} + 9025536 q^{6} - 471792132 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 262144 q^{4} + 9025536 q^{6} - 471792132 q^{9} + 129030528 q^{11} - 14174308352 q^{14} + 17179869184 q^{16} - 88427425520 q^{19} + 567296955888 q^{21} - 591497527296 q^{24} + 1482669295616 q^{26} - 7974629727000 q^{29} - 10984522678672 q^{31} - 809068849152 q^{34} + 30919369162752 q^{36} - 235491790704 q^{39} + 46822954554648 q^{41} - 8456144683008 q^{44} + 249625488807936 q^{46} + 154831593772812 q^{49} + 972946158829488 q^{51} - 29\!\cdots\!20 q^{54}+ \cdots + 10\!\cdots\!76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 1471x^{2} + 540225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} + 1472\nu ) / 735 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\nu^{3} + 35296\nu ) / 49 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 960\nu^{2} + 706080 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 120\beta_1 ) / 480 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 706080 ) / 960 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -46\beta_{2} + 16545\beta_1 ) / 30 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
27.6155i
26.6155i
26.6155i
27.6155i
256.000i 4201.44i −65536.0 0 −1.07557e6 1.23193e7i 1.67772e7i 1.11488e8 0
49.2 256.000i 21829.4i −65536.0 0 5.58834e6 1.53649e7i 1.67772e7i −3.47384e8 0
49.3 256.000i 21829.4i −65536.0 0 5.58834e6 1.53649e7i 1.67772e7i −3.47384e8 0
49.4 256.000i 4201.44i −65536.0 0 −1.07557e6 1.23193e7i 1.67772e7i 1.11488e8 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 50.18.b.f 4
5.b even 2 1 inner 50.18.b.f 4
5.c odd 4 1 10.18.a.d 2
5.c odd 4 1 50.18.a.c 2
15.e even 4 1 90.18.a.j 2
20.e even 4 1 80.18.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.18.a.d 2 5.c odd 4 1
50.18.a.c 2 5.c odd 4 1
50.18.b.f 4 1.a even 1 1 trivial
50.18.b.f 4 5.b even 2 1 inner
80.18.a.b 2 20.e even 4 1
90.18.a.j 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 494176392T_{3}^{2} + 8411641958720016 \) acting on \(S_{18}^{\mathrm{new}}(50, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 65536)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 494176392 T^{2} + \cdots + 84\!\cdots\!16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 387845231088008 T^{2} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{2} - 64515264 T - 13\!\cdots\!76)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{2} + 44213712760 T - 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{2} + 3987314863500 T + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 5492261339336 T + 71\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 88\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{2} - 23411477277324 T - 41\!\cdots\!56)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 62\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 55\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{2} + 902179170360600 T - 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots - 87\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 69\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{2} + 67588560434136 T - 91\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 81\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 54\!\cdots\!56 \) Copy content Toggle raw display
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