Properties

Label 210.4.a.b
Level $210$
Weight $4$
Character orbit 210.a
Self dual yes
Analytic conductor $12.390$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [210,4,Mod(1,210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(210, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("210.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 210.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: \(12.3904011012\)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} + 5 q^{5} + 6 q^{6} - 7 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} + 5 q^{5} + 6 q^{6} - 7 q^{7} - 8 q^{8} + 9 q^{9} - 10 q^{10} - 44 q^{11} - 12 q^{12} + 54 q^{13} + 14 q^{14} - 15 q^{15} + 16 q^{16} + 98 q^{17} - 18 q^{18} - 60 q^{19} + 20 q^{20} + 21 q^{21} + 88 q^{22} - 144 q^{23} + 24 q^{24} + 25 q^{25} - 108 q^{26} - 27 q^{27} - 28 q^{28} - 210 q^{29} + 30 q^{30} - 208 q^{31} - 32 q^{32} + 132 q^{33} - 196 q^{34} - 35 q^{35} + 36 q^{36} - 226 q^{37} + 120 q^{38} - 162 q^{39} - 40 q^{40} - 502 q^{41} - 42 q^{42} + 484 q^{43} - 176 q^{44} + 45 q^{45} + 288 q^{46} - 232 q^{47} - 48 q^{48} + 49 q^{49} - 50 q^{50} - 294 q^{51} + 216 q^{52} - 530 q^{53} + 54 q^{54} - 220 q^{55} + 56 q^{56} + 180 q^{57} + 420 q^{58} - 764 q^{59} - 60 q^{60} + 814 q^{61} + 416 q^{62} - 63 q^{63} + 64 q^{64} + 270 q^{65} - 264 q^{66} + 60 q^{67} + 392 q^{68} + 432 q^{69} + 70 q^{70} + 848 q^{71} - 72 q^{72} - 958 q^{73} + 452 q^{74} - 75 q^{75} - 240 q^{76} + 308 q^{77} + 324 q^{78} - 152 q^{79} + 80 q^{80} + 81 q^{81} + 1004 q^{82} + 308 q^{83} + 84 q^{84} + 490 q^{85} - 968 q^{86} + 630 q^{87} + 352 q^{88} - 1094 q^{89} - 90 q^{90} - 378 q^{91} - 576 q^{92} + 624 q^{93} + 464 q^{94} - 300 q^{95} + 96 q^{96} + 554 q^{97} - 98 q^{98} - 396 q^{99}+O(q^{100}) \) 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
−2.00000 −3.00000 4.00000 5.00000 6.00000 −7.00000 −8.00000 9.00000 −10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)

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 210.4.a.b 1
3.b odd 2 1 630.4.a.n 1
4.b odd 2 1 1680.4.a.x 1
5.b even 2 1 1050.4.a.w 1
5.c odd 4 2 1050.4.g.k 2
7.b odd 2 1 1470.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.4.a.b 1 1.a even 1 1 trivial
630.4.a.n 1 3.b odd 2 1
1050.4.a.w 1 5.b even 2 1
1050.4.g.k 2 5.c odd 4 2
1470.4.a.i 1 7.b odd 2 1
1680.4.a.x 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(210))\):

\( T_{11} + 44 \) Copy content Toggle raw display
\( T_{13} - 54 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T + 7 \) Copy content Toggle raw display
$11$ \( T + 44 \) Copy content Toggle raw display
$13$ \( T - 54 \) Copy content Toggle raw display
$17$ \( T - 98 \) Copy content Toggle raw display
$19$ \( T + 60 \) Copy content Toggle raw display
$23$ \( T + 144 \) Copy content Toggle raw display
$29$ \( T + 210 \) Copy content Toggle raw display
$31$ \( T + 208 \) Copy content Toggle raw display
$37$ \( T + 226 \) Copy content Toggle raw display
$41$ \( T + 502 \) Copy content Toggle raw display
$43$ \( T - 484 \) Copy content Toggle raw display
$47$ \( T + 232 \) Copy content Toggle raw display
$53$ \( T + 530 \) Copy content Toggle raw display
$59$ \( T + 764 \) Copy content Toggle raw display
$61$ \( T - 814 \) Copy content Toggle raw display
$67$ \( T - 60 \) Copy content Toggle raw display
$71$ \( T - 848 \) Copy content Toggle raw display
$73$ \( T + 958 \) Copy content Toggle raw display
$79$ \( T + 152 \) Copy content Toggle raw display
$83$ \( T - 308 \) Copy content Toggle raw display
$89$ \( T + 1094 \) Copy content Toggle raw display
$97$ \( T - 554 \) Copy content Toggle raw display
show more
show less