Properties

Label 30.8.a.b
Level $30$
Weight $8$
Character orbit 30.a
Self dual yes
Analytic conductor $9.372$
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] = [30,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("30.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(9.37155076452\)
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)
 
\(f(q)\) \(=\) \( q - 8 q^{2} - 27 q^{3} + 64 q^{4} + 125 q^{5} + 216 q^{6} + 416 q^{7} - 512 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} - 27 q^{3} + 64 q^{4} + 125 q^{5} + 216 q^{6} + 416 q^{7} - 512 q^{8} + 729 q^{9} - 1000 q^{10} - 3948 q^{11} - 1728 q^{12} - 7978 q^{13} - 3328 q^{14} - 3375 q^{15} + 4096 q^{16} - 34014 q^{17} - 5832 q^{18} + 3020 q^{19} + 8000 q^{20} - 11232 q^{21} + 31584 q^{22} - 66528 q^{23} + 13824 q^{24} + 15625 q^{25} + 63824 q^{26} - 19683 q^{27} + 26624 q^{28} - 18570 q^{29} + 27000 q^{30} + 208832 q^{31} - 32768 q^{32} + 106596 q^{33} + 272112 q^{34} + 52000 q^{35} + 46656 q^{36} - 301474 q^{37} - 24160 q^{38} + 215406 q^{39} - 64000 q^{40} - 460998 q^{41} + 89856 q^{42} + 343052 q^{43} - 252672 q^{44} + 91125 q^{45} + 532224 q^{46} - 1356264 q^{47} - 110592 q^{48} - 650487 q^{49} - 125000 q^{50} + 918378 q^{51} - 510592 q^{52} + 617982 q^{53} + 157464 q^{54} - 493500 q^{55} - 212992 q^{56} - 81540 q^{57} + 148560 q^{58} + 939540 q^{59} - 216000 q^{60} - 204298 q^{61} - 1670656 q^{62} + 303264 q^{63} + 262144 q^{64} - 997250 q^{65} - 852768 q^{66} - 758524 q^{67} - 2176896 q^{68} + 1796256 q^{69} - 416000 q^{70} + 912072 q^{71} - 373248 q^{72} + 3043322 q^{73} + 2411792 q^{74} - 421875 q^{75} + 193280 q^{76} - 1642368 q^{77} - 1723248 q^{78} + 6010880 q^{79} + 512000 q^{80} + 531441 q^{81} + 3687984 q^{82} + 9723012 q^{83} - 718848 q^{84} - 4251750 q^{85} - 2744416 q^{86} + 501390 q^{87} + 2021376 q^{88} + 7160010 q^{89} - 729000 q^{90} - 3318848 q^{91} - 4257792 q^{92} - 5638464 q^{93} + 10850112 q^{94} + 377500 q^{95} + 884736 q^{96} - 16785214 q^{97} + 5203896 q^{98} - 2878092 q^{99}+O(q^{100}) \) 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
−8.00000 −27.0000 64.0000 125.000 216.000 416.000 −512.000 729.000 −1000.00
\(n\): 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 30.8.a.b 1
3.b odd 2 1 90.8.a.h 1
4.b odd 2 1 240.8.a.m 1
5.b even 2 1 150.8.a.n 1
5.c odd 4 2 150.8.c.g 2
15.d odd 2 1 450.8.a.e 1
15.e even 4 2 450.8.c.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.8.a.b 1 1.a even 1 1 trivial
90.8.a.h 1 3.b odd 2 1
150.8.a.n 1 5.b even 2 1
150.8.c.g 2 5.c odd 4 2
240.8.a.m 1 4.b odd 2 1
450.8.a.e 1 15.d odd 2 1
450.8.c.n 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 416 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(30))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 8 \) Copy content Toggle raw display
$3$ \( T + 27 \) Copy content Toggle raw display
$5$ \( T - 125 \) Copy content Toggle raw display
$7$ \( T - 416 \) Copy content Toggle raw display
$11$ \( T + 3948 \) Copy content Toggle raw display
$13$ \( T + 7978 \) Copy content Toggle raw display
$17$ \( T + 34014 \) Copy content Toggle raw display
$19$ \( T - 3020 \) Copy content Toggle raw display
$23$ \( T + 66528 \) Copy content Toggle raw display
$29$ \( T + 18570 \) Copy content Toggle raw display
$31$ \( T - 208832 \) Copy content Toggle raw display
$37$ \( T + 301474 \) Copy content Toggle raw display
$41$ \( T + 460998 \) Copy content Toggle raw display
$43$ \( T - 343052 \) Copy content Toggle raw display
$47$ \( T + 1356264 \) Copy content Toggle raw display
$53$ \( T - 617982 \) Copy content Toggle raw display
$59$ \( T - 939540 \) Copy content Toggle raw display
$61$ \( T + 204298 \) Copy content Toggle raw display
$67$ \( T + 758524 \) Copy content Toggle raw display
$71$ \( T - 912072 \) Copy content Toggle raw display
$73$ \( T - 3043322 \) Copy content Toggle raw display
$79$ \( T - 6010880 \) Copy content Toggle raw display
$83$ \( T - 9723012 \) Copy content Toggle raw display
$89$ \( T - 7160010 \) Copy content Toggle raw display
$97$ \( T + 16785214 \) Copy content Toggle raw display
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