Properties

Label 10.12.a.c
Level $10$
Weight $12$
Character orbit 10.a
Self dual yes
Analytic conductor $7.683$
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] = [10,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
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: \(7.68343180560\)
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)
 
\(f(q)\) \(=\) \( q + 32 q^{2} - 318 q^{3} + 1024 q^{4} - 3125 q^{5} - 10176 q^{6} - 70714 q^{7} + 32768 q^{8} - 76023 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 32 q^{2} - 318 q^{3} + 1024 q^{4} - 3125 q^{5} - 10176 q^{6} - 70714 q^{7} + 32768 q^{8} - 76023 q^{9} - 100000 q^{10} + 238272 q^{11} - 325632 q^{12} - 2097478 q^{13} - 2262848 q^{14} + 993750 q^{15} + 1048576 q^{16} + 5955546 q^{17} - 2432736 q^{18} + 10210820 q^{19} - 3200000 q^{20} + 22487052 q^{21} + 7624704 q^{22} - 3535758 q^{23} - 10420224 q^{24} + 9765625 q^{25} - 67119296 q^{26} + 80508060 q^{27} - 72411136 q^{28} - 139304850 q^{29} + 31800000 q^{30} - 101002348 q^{31} + 33554432 q^{32} - 75770496 q^{33} + 190577472 q^{34} + 220981250 q^{35} - 77847552 q^{36} - 524913814 q^{37} + 326746240 q^{38} + 666998004 q^{39} - 102400000 q^{40} + 284590422 q^{41} + 719585664 q^{42} - 1253635078 q^{43} + 243990528 q^{44} + 237571875 q^{45} - 113144256 q^{46} - 216106434 q^{47} - 333447168 q^{48} + 3023143053 q^{49} + 312500000 q^{50} - 1893863628 q^{51} - 2147817472 q^{52} - 4881275358 q^{53} + 2576257920 q^{54} - 744600000 q^{55} - 2317156352 q^{56} - 3247040760 q^{57} - 4457755200 q^{58} + 8692473300 q^{59} + 1017600000 q^{60} + 3296491802 q^{61} - 3232075136 q^{62} + 5375890422 q^{63} + 1073741824 q^{64} + 6554618750 q^{65} - 2424655872 q^{66} + 18275027966 q^{67} + 6098479104 q^{68} + 1124371044 q^{69} + 7071400000 q^{70} - 13287447588 q^{71} - 2491121664 q^{72} - 32505250798 q^{73} - 16797242048 q^{74} - 3105468750 q^{75} + 10455879680 q^{76} - 16849166208 q^{77} + 21343936128 q^{78} + 9297455960 q^{79} - 3276800000 q^{80} - 12134316699 q^{81} + 9106893504 q^{82} - 22741484838 q^{83} + 23026741248 q^{84} - 18611081250 q^{85} - 40116322496 q^{86} + 44298942300 q^{87} + 7807696896 q^{88} - 93378882390 q^{89} + 7602300000 q^{90} + 148321059292 q^{91} - 3620616192 q^{92} + 32118746664 q^{93} - 6915405888 q^{94} - 31908812500 q^{95} - 10670309376 q^{96} - 5811134014 q^{97} + 96740577696 q^{98} - 18114152256 q^{99}+O(q^{100}) \) 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
32.0000 −318.000 1024.00 −3125.00 −10176.0 −70714.0 32768.0 −76023.0 −100000.
\(n\): 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 10.12.a.c 1
3.b odd 2 1 90.12.a.b 1
4.b odd 2 1 80.12.a.e 1
5.b even 2 1 50.12.a.b 1
5.c odd 4 2 50.12.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.12.a.c 1 1.a even 1 1 trivial
50.12.a.b 1 5.b even 2 1
50.12.b.b 2 5.c odd 4 2
80.12.a.e 1 4.b odd 2 1
90.12.a.b 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 318 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(10))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 32 \) Copy content Toggle raw display
$3$ \( T + 318 \) Copy content Toggle raw display
$5$ \( T + 3125 \) Copy content Toggle raw display
$7$ \( T + 70714 \) Copy content Toggle raw display
$11$ \( T - 238272 \) Copy content Toggle raw display
$13$ \( T + 2097478 \) Copy content Toggle raw display
$17$ \( T - 5955546 \) Copy content Toggle raw display
$19$ \( T - 10210820 \) Copy content Toggle raw display
$23$ \( T + 3535758 \) Copy content Toggle raw display
$29$ \( T + 139304850 \) Copy content Toggle raw display
$31$ \( T + 101002348 \) Copy content Toggle raw display
$37$ \( T + 524913814 \) Copy content Toggle raw display
$41$ \( T - 284590422 \) Copy content Toggle raw display
$43$ \( T + 1253635078 \) Copy content Toggle raw display
$47$ \( T + 216106434 \) Copy content Toggle raw display
$53$ \( T + 4881275358 \) Copy content Toggle raw display
$59$ \( T - 8692473300 \) Copy content Toggle raw display
$61$ \( T - 3296491802 \) Copy content Toggle raw display
$67$ \( T - 18275027966 \) Copy content Toggle raw display
$71$ \( T + 13287447588 \) Copy content Toggle raw display
$73$ \( T + 32505250798 \) Copy content Toggle raw display
$79$ \( T - 9297455960 \) Copy content Toggle raw display
$83$ \( T + 22741484838 \) Copy content Toggle raw display
$89$ \( T + 93378882390 \) Copy content Toggle raw display
$97$ \( T + 5811134014 \) Copy content Toggle raw display
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