Properties

Label 10.12.a.a
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)
 
\(f(q)\) \(=\) \( q - 32 q^{2} - 12 q^{3} + 1024 q^{4} + 3125 q^{5} + 384 q^{6} - 14176 q^{7} - 32768 q^{8} - 177003 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 32 q^{2} - 12 q^{3} + 1024 q^{4} + 3125 q^{5} + 384 q^{6} - 14176 q^{7} - 32768 q^{8} - 177003 q^{9} - 100000 q^{10} - 756348 q^{11} - 12288 q^{12} - 905482 q^{13} + 453632 q^{14} - 37500 q^{15} + 1048576 q^{16} + 2803794 q^{17} + 5664096 q^{18} - 5428660 q^{19} + 3200000 q^{20} + 170112 q^{21} + 24203136 q^{22} - 10236672 q^{23} + 393216 q^{24} + 9765625 q^{25} + 28975424 q^{26} + 4249800 q^{27} - 14516224 q^{28} - 197498010 q^{29} + 1200000 q^{30} - 44362288 q^{31} - 33554432 q^{32} + 9076176 q^{33} - 89721408 q^{34} - 44300000 q^{35} - 181251072 q^{36} + 576737054 q^{37} + 173717120 q^{38} + 10865784 q^{39} - 102400000 q^{40} + 930058362 q^{41} - 5443584 q^{42} + 1605598988 q^{43} - 774500352 q^{44} - 553134375 q^{45} + 327573504 q^{46} - 1803684456 q^{47} - 12582912 q^{48} - 1776367767 q^{49} - 312500000 q^{50} - 33645528 q^{51} - 927213568 q^{52} + 1558674798 q^{53} - 135993600 q^{54} - 2363587500 q^{55} + 464519168 q^{56} + 65143920 q^{57} + 6319936320 q^{58} - 9501997020 q^{59} - 38400000 q^{60} + 6736320422 q^{61} + 1419593216 q^{62} + 2509194528 q^{63} + 1073741824 q^{64} - 2829631250 q^{65} - 290437632 q^{66} + 8402906564 q^{67} + 2871085056 q^{68} + 122840064 q^{69} + 1417600000 q^{70} - 4806306168 q^{71} + 5800034304 q^{72} + 7462713338 q^{73} - 18455585728 q^{74} - 117187500 q^{75} - 5558947840 q^{76} + 10721989248 q^{77} - 347705088 q^{78} - 20644540720 q^{79} + 3276800000 q^{80} + 31304552841 q^{81} - 29761867584 q^{82} - 68013349212 q^{83} + 174194688 q^{84} + 8761856250 q^{85} - 51379167616 q^{86} + 2369976120 q^{87} + 24784011264 q^{88} + 69871323210 q^{89} + 17700300000 q^{90} + 12836112832 q^{91} - 10482352128 q^{92} + 532347456 q^{93} + 57717902592 q^{94} - 16964562500 q^{95} + 402653184 q^{96} + 39960952514 q^{97} + 56843768544 q^{98} + 133875865044 q^{99}+O(q^{100}) \) 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 −12.0000 1024.00 3125.00 384.000 −14176.0 −32768.0 −177003. −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.a 1
3.b odd 2 1 90.12.a.g 1
4.b odd 2 1 80.12.a.d 1
5.b even 2 1 50.12.a.d 1
5.c odd 4 2 50.12.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.12.a.a 1 1.a even 1 1 trivial
50.12.a.d 1 5.b even 2 1
50.12.b.c 2 5.c odd 4 2
80.12.a.d 1 4.b odd 2 1
90.12.a.g 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 12 \) 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
$3$ \( T + 12 \) Copy content Toggle raw display
$5$ \( T - 3125 \) Copy content Toggle raw display
$7$ \( T + 14176 \) Copy content Toggle raw display
$11$ \( T + 756348 \) Copy content Toggle raw display
$13$ \( T + 905482 \) Copy content Toggle raw display
$17$ \( T - 2803794 \) Copy content Toggle raw display
$19$ \( T + 5428660 \) Copy content Toggle raw display
$23$ \( T + 10236672 \) Copy content Toggle raw display
$29$ \( T + 197498010 \) Copy content Toggle raw display
$31$ \( T + 44362288 \) Copy content Toggle raw display
$37$ \( T - 576737054 \) Copy content Toggle raw display
$41$ \( T - 930058362 \) Copy content Toggle raw display
$43$ \( T - 1605598988 \) Copy content Toggle raw display
$47$ \( T + 1803684456 \) Copy content Toggle raw display
$53$ \( T - 1558674798 \) Copy content Toggle raw display
$59$ \( T + 9501997020 \) Copy content Toggle raw display
$61$ \( T - 6736320422 \) Copy content Toggle raw display
$67$ \( T - 8402906564 \) Copy content Toggle raw display
$71$ \( T + 4806306168 \) Copy content Toggle raw display
$73$ \( T - 7462713338 \) Copy content Toggle raw display
$79$ \( T + 20644540720 \) Copy content Toggle raw display
$83$ \( T + 68013349212 \) Copy content Toggle raw display
$89$ \( T - 69871323210 \) Copy content Toggle raw display
$97$ \( T - 39960952514 \) Copy content Toggle raw display
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