Properties

Label 50.16.a.b
Level $50$
Weight $16$
Character orbit 50.a
Self dual yes
Analytic conductor $71.347$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [50,16,Mod(1,50)]
 
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([0]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)

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: yes
Analytic conductor: \(71.3467525500\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 128 q^{2} - 6252 q^{3} + 16384 q^{4} - 800256 q^{6} - 56 q^{7} + 2097152 q^{8} + 24738597 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 128 q^{2} - 6252 q^{3} + 16384 q^{4} - 800256 q^{6} - 56 q^{7} + 2097152 q^{8} + 24738597 q^{9} - 95889948 q^{11} - 102432768 q^{12} + 59782138 q^{13} - 7168 q^{14} + 268435456 q^{16} + 1355814414 q^{17} + 3166540416 q^{18} + 3783593180 q^{19} + 350112 q^{21} - 12273913344 q^{22} + 11608845528 q^{23} - 13111394304 q^{24} + 7652113664 q^{26} - 64956341880 q^{27} - 917504 q^{28} - 28959105930 q^{29} + 253685353952 q^{31} + 34359738368 q^{32} + 599503954896 q^{33} + 173544244992 q^{34} + 405317173248 q^{36} - 817641294446 q^{37} + 484299927040 q^{38} - 373757926776 q^{39} - 682333284198 q^{41} + 44814336 q^{42} - 366945604292 q^{43} - 1571060908032 q^{44} + 1485932227584 q^{46} - 695741581776 q^{47} - 1678258470912 q^{48} - 4747561506807 q^{49} - 8476551716328 q^{51} + 979470548992 q^{52} - 12993372468702 q^{53} - 8314411760640 q^{54} - 117440512 q^{56} - 23655024561360 q^{57} - 3706765559040 q^{58} + 9209035340340 q^{59} - 42338641200298 q^{61} + 32471725305856 q^{62} - 1385361432 q^{63} + 4398046511104 q^{64} + 76736506226688 q^{66} - 30029787950636 q^{67} + 22213663358976 q^{68} - 72578502241056 q^{69} + 115328696975352 q^{71} + 51880598175744 q^{72} - 43787346432122 q^{73} - 104658085689088 q^{74} + 61990390661120 q^{76} + 5369837088 q^{77} - 47841014627328 q^{78} + 79603813043120 q^{79} + 51135221770281 q^{81} - 87338660377344 q^{82} + 3417068864868 q^{83} + 5736235008 q^{84} - 46969037349376 q^{86} + 181052330274360 q^{87} - 201095796228096 q^{88} - 377306179184790 q^{89} - 3347799728 q^{91} + 190199325130752 q^{92} - 15\!\cdots\!04 q^{93}+ \cdots - 23\!\cdots\!56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
128.000 −6252.00 16384.0 0 −800256. −56.0000 2.09715e6 2.47386e7 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.16.a.b 1
5.b even 2 1 2.16.a.a 1
5.c odd 4 2 50.16.b.a 2
15.d odd 2 1 18.16.a.e 1
20.d odd 2 1 16.16.a.a 1
35.c odd 2 1 98.16.a.a 1
35.i odd 6 2 98.16.c.d 2
35.j even 6 2 98.16.c.a 2
40.e odd 2 1 64.16.a.k 1
40.f even 2 1 64.16.a.a 1
60.h even 2 1 144.16.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.16.a.a 1 5.b even 2 1
16.16.a.a 1 20.d odd 2 1
18.16.a.e 1 15.d odd 2 1
50.16.a.b 1 1.a even 1 1 trivial
50.16.b.a 2 5.c odd 4 2
64.16.a.a 1 40.f even 2 1
64.16.a.k 1 40.e odd 2 1
98.16.a.a 1 35.c odd 2 1
98.16.c.a 2 35.j even 6 2
98.16.c.d 2 35.i odd 6 2
144.16.a.d 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 6252 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(50))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 128 \) Copy content Toggle raw display
$3$ \( T + 6252 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 56 \) Copy content Toggle raw display
$11$ \( T + 95889948 \) Copy content Toggle raw display
$13$ \( T - 59782138 \) Copy content Toggle raw display
$17$ \( T - 1355814414 \) Copy content Toggle raw display
$19$ \( T - 3783593180 \) Copy content Toggle raw display
$23$ \( T - 11608845528 \) Copy content Toggle raw display
$29$ \( T + 28959105930 \) Copy content Toggle raw display
$31$ \( T - 253685353952 \) Copy content Toggle raw display
$37$ \( T + 817641294446 \) Copy content Toggle raw display
$41$ \( T + 682333284198 \) Copy content Toggle raw display
$43$ \( T + 366945604292 \) Copy content Toggle raw display
$47$ \( T + 695741581776 \) Copy content Toggle raw display
$53$ \( T + 12993372468702 \) Copy content Toggle raw display
$59$ \( T - 9209035340340 \) Copy content Toggle raw display
$61$ \( T + 42338641200298 \) Copy content Toggle raw display
$67$ \( T + 30029787950636 \) Copy content Toggle raw display
$71$ \( T - 115328696975352 \) Copy content Toggle raw display
$73$ \( T + 43787346432122 \) Copy content Toggle raw display
$79$ \( T - 79603813043120 \) Copy content Toggle raw display
$83$ \( T - 3417068864868 \) Copy content Toggle raw display
$89$ \( T + 377306179184790 \) Copy content Toggle raw display
$97$ \( T - 166982186657374 \) Copy content Toggle raw display
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