Properties

Label 10.16.a.a
Level $10$
Weight $16$
Character orbit 10.a
Self dual yes
Analytic conductor $14.269$
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] = [10,16,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, 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("10.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(14.2693505100\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - 128 q^{2} - 5568 q^{3} + 16384 q^{4} + 78125 q^{5} + 712704 q^{6} + 2564996 q^{7} - 2097152 q^{8} + 16653717 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 128 q^{2} - 5568 q^{3} + 16384 q^{4} + 78125 q^{5} + 712704 q^{6} + 2564996 q^{7} - 2097152 q^{8} + 16653717 q^{9} - 10000000 q^{10} - 81067668 q^{11} - 91226112 q^{12} + 351412022 q^{13} - 328319488 q^{14} - 435000000 q^{15} + 268435456 q^{16} - 2157825054 q^{17} - 2131675776 q^{18} - 5107458100 q^{19} + 1280000000 q^{20} - 14281897728 q^{21} + 10376661504 q^{22} + 11784341052 q^{23} + 11676942336 q^{24} + 6103515625 q^{25} - 44980738816 q^{26} - 12833182080 q^{27} + 42024894464 q^{28} - 20400574890 q^{29} + 55680000000 q^{30} - 123613797688 q^{31} - 34359738368 q^{32} + 451384775424 q^{33} + 276201606912 q^{34} + 200390312500 q^{35} + 272854499328 q^{36} - 22499625394 q^{37} + 653754636800 q^{38} - 1956662138496 q^{39} - 163840000000 q^{40} - 1044060129558 q^{41} + 1828082909184 q^{42} - 2984233999768 q^{43} - 1328212672512 q^{44} + 1301071640625 q^{45} - 1508395654656 q^{46} - 2267362482084 q^{47} - 1494648619008 q^{48} + 1831642970073 q^{49} - 781250000000 q^{50} + 12014769900672 q^{51} + 5757534568448 q^{52} - 8655803512338 q^{53} + 1642647306240 q^{54} - 6333411562500 q^{55} - 5379186491392 q^{56} + 28438326700800 q^{57} + 2611273585920 q^{58} - 25953000142380 q^{59} - 7127040000000 q^{60} + 29809710409622 q^{61} + 15822566104064 q^{62} + 42716717490132 q^{63} + 4398046511104 q^{64} + 27454064218750 q^{65} - 57777251254272 q^{66} - 78103662703144 q^{67} - 35353805684736 q^{68} - 65615210977536 q^{69} - 25649960000000 q^{70} + 67746916371072 q^{71} - 34925375913984 q^{72} + 134520120122282 q^{73} + 2879952050432 q^{74} - 33984375000000 q^{75} - 83680593510400 q^{76} - 207938244149328 q^{77} + 250452753727488 q^{78} + 16723463056640 q^{79} + 20971520000000 q^{80} - 167507478615879 q^{81} + 133639696583424 q^{82} + 80883629455632 q^{83} - 233994612375552 q^{84} - 168580082343750 q^{85} + 381981951970304 q^{86} + 113590400987520 q^{87} + 170011222081536 q^{88} - 523835472467190 q^{89} - 166537170000000 q^{90} + 901370430781912 q^{91} + 193074643795968 q^{92} + 688281625526784 q^{93} + 290222397706752 q^{94} - 399020164062500 q^{95} + 191315023233024 q^{96} + 11\!\cdots\!26 q^{97}+ \cdots - 13\!\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 −5568.00 16384.0 78125.0 712704. 2.56500e6 −2.09715e6 1.66537e7 −1.00000e7
\(n\): 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 10.16.a.a 1
3.b odd 2 1 90.16.a.f 1
4.b odd 2 1 80.16.a.c 1
5.b even 2 1 50.16.a.d 1
5.c odd 4 2 50.16.b.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.16.a.a 1 1.a even 1 1 trivial
50.16.a.d 1 5.b even 2 1
50.16.b.d 2 5.c odd 4 2
80.16.a.c 1 4.b odd 2 1
90.16.a.f 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 5568 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(10))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 128 \) Copy content Toggle raw display
$3$ \( T + 5568 \) Copy content Toggle raw display
$5$ \( T - 78125 \) Copy content Toggle raw display
$7$ \( T - 2564996 \) Copy content Toggle raw display
$11$ \( T + 81067668 \) Copy content Toggle raw display
$13$ \( T - 351412022 \) Copy content Toggle raw display
$17$ \( T + 2157825054 \) Copy content Toggle raw display
$19$ \( T + 5107458100 \) Copy content Toggle raw display
$23$ \( T - 11784341052 \) Copy content Toggle raw display
$29$ \( T + 20400574890 \) Copy content Toggle raw display
$31$ \( T + 123613797688 \) Copy content Toggle raw display
$37$ \( T + 22499625394 \) Copy content Toggle raw display
$41$ \( T + 1044060129558 \) Copy content Toggle raw display
$43$ \( T + 2984233999768 \) Copy content Toggle raw display
$47$ \( T + 2267362482084 \) Copy content Toggle raw display
$53$ \( T + 8655803512338 \) Copy content Toggle raw display
$59$ \( T + 25953000142380 \) Copy content Toggle raw display
$61$ \( T - 29809710409622 \) Copy content Toggle raw display
$67$ \( T + 78103662703144 \) Copy content Toggle raw display
$71$ \( T - 67746916371072 \) Copy content Toggle raw display
$73$ \( T - 134520120122282 \) Copy content Toggle raw display
$79$ \( T - 16723463056640 \) Copy content Toggle raw display
$83$ \( T - 80883629455632 \) Copy content Toggle raw display
$89$ \( T + 523835472467190 \) Copy content Toggle raw display
$97$ \( T - 1186642412683826 \) Copy content Toggle raw display
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