Properties

Label 30.10.a.e
Level $30$
Weight $10$
Character orbit 30.a
Self dual yes
Analytic conductor $15.451$
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] = [30,10,Mod(1,30)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(30, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("30.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 30.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: \(15.4510750849\)
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)
 
\(f(q)\) \(=\) \( q + 16 q^{2} - 81 q^{3} + 256 q^{4} + 625 q^{5} - 1296 q^{6} - 10336 q^{7} + 4096 q^{8} + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 16 q^{2} - 81 q^{3} + 256 q^{4} + 625 q^{5} - 1296 q^{6} - 10336 q^{7} + 4096 q^{8} + 6561 q^{9} + 10000 q^{10} + 27420 q^{11} - 20736 q^{12} - 169762 q^{13} - 165376 q^{14} - 50625 q^{15} + 65536 q^{16} - 385086 q^{17} + 104976 q^{18} - 637084 q^{19} + 160000 q^{20} + 837216 q^{21} + 438720 q^{22} - 1298400 q^{23} - 331776 q^{24} + 390625 q^{25} - 2716192 q^{26} - 531441 q^{27} - 2646016 q^{28} + 7162974 q^{29} - 810000 q^{30} - 7031872 q^{31} + 1048576 q^{32} - 2221020 q^{33} - 6161376 q^{34} - 6460000 q^{35} + 1679616 q^{36} + 1926038 q^{37} - 10193344 q^{38} + 13750722 q^{39} + 2560000 q^{40} + 8896074 q^{41} + 13395456 q^{42} + 32429444 q^{43} + 7019520 q^{44} + 4100625 q^{45} - 20774400 q^{46} + 17206440 q^{47} - 5308416 q^{48} + 66479289 q^{49} + 6250000 q^{50} + 31191966 q^{51} - 43459072 q^{52} - 20642154 q^{53} - 8503056 q^{54} + 17137500 q^{55} - 42336256 q^{56} + 51603804 q^{57} + 114607584 q^{58} - 63193380 q^{59} - 12960000 q^{60} - 63758050 q^{61} - 112509952 q^{62} - 67814496 q^{63} + 16777216 q^{64} - 106101250 q^{65} - 35536320 q^{66} + 145261964 q^{67} - 98582016 q^{68} + 105170400 q^{69} - 103360000 q^{70} - 367656840 q^{71} + 26873856 q^{72} + 252486218 q^{73} + 30816608 q^{74} - 31640625 q^{75} - 163093504 q^{76} - 283413120 q^{77} + 220011552 q^{78} - 185523712 q^{79} + 40960000 q^{80} + 43046721 q^{81} + 142337184 q^{82} - 467897652 q^{83} + 214327296 q^{84} - 240678750 q^{85} + 518871104 q^{86} - 580200894 q^{87} + 112312320 q^{88} + 579096378 q^{89} + 65610000 q^{90} + 1754660032 q^{91} - 332390400 q^{92} + 569581632 q^{93} + 275303040 q^{94} - 398177500 q^{95} - 84934656 q^{96} - 1314516862 q^{97} + 1063668624 q^{98} + 179902620 q^{99}+O(q^{100}) \) 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
16.0000 −81.0000 256.000 625.000 −1296.00 −10336.0 4096.00 6561.00 10000.0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)

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 30.10.a.e 1
3.b odd 2 1 90.10.a.a 1
4.b odd 2 1 240.10.a.j 1
5.b even 2 1 150.10.a.e 1
5.c odd 4 2 150.10.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.10.a.e 1 1.a even 1 1 trivial
90.10.a.a 1 3.b odd 2 1
150.10.a.e 1 5.b even 2 1
150.10.c.c 2 5.c odd 4 2
240.10.a.j 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 10336 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(30))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 16 \) Copy content Toggle raw display
$3$ \( T + 81 \) Copy content Toggle raw display
$5$ \( T - 625 \) Copy content Toggle raw display
$7$ \( T + 10336 \) Copy content Toggle raw display
$11$ \( T - 27420 \) Copy content Toggle raw display
$13$ \( T + 169762 \) Copy content Toggle raw display
$17$ \( T + 385086 \) Copy content Toggle raw display
$19$ \( T + 637084 \) Copy content Toggle raw display
$23$ \( T + 1298400 \) Copy content Toggle raw display
$29$ \( T - 7162974 \) Copy content Toggle raw display
$31$ \( T + 7031872 \) Copy content Toggle raw display
$37$ \( T - 1926038 \) Copy content Toggle raw display
$41$ \( T - 8896074 \) Copy content Toggle raw display
$43$ \( T - 32429444 \) Copy content Toggle raw display
$47$ \( T - 17206440 \) Copy content Toggle raw display
$53$ \( T + 20642154 \) Copy content Toggle raw display
$59$ \( T + 63193380 \) Copy content Toggle raw display
$61$ \( T + 63758050 \) Copy content Toggle raw display
$67$ \( T - 145261964 \) Copy content Toggle raw display
$71$ \( T + 367656840 \) Copy content Toggle raw display
$73$ \( T - 252486218 \) Copy content Toggle raw display
$79$ \( T + 185523712 \) Copy content Toggle raw display
$83$ \( T + 467897652 \) Copy content Toggle raw display
$89$ \( T - 579096378 \) Copy content Toggle raw display
$97$ \( T + 1314516862 \) Copy content Toggle raw display
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