Properties

Label 210.2.i.a
Level $210$
Weight $2$
Character orbit 210.i
Analytic conductor $1.677$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [210,2,Mod(121,210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(210, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("210.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 210.i (of order \(3\), degree \(2\), minimal)

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: no
Analytic conductor: \(1.67685844245\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + q^{6} + ( - 2 \zeta_{6} + 3) q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + q^{6} + ( - 2 \zeta_{6} + 3) q^{7} + q^{8} - \zeta_{6} q^{9} + (\zeta_{6} - 1) q^{10} + ( - \zeta_{6} + 1) q^{11} - \zeta_{6} q^{12} + 7 q^{13} + ( - \zeta_{6} - 2) q^{14} + q^{15} - \zeta_{6} q^{16} + ( - 4 \zeta_{6} + 4) q^{17} + (\zeta_{6} - 1) q^{18} - \zeta_{6} q^{19} + q^{20} + (3 \zeta_{6} - 1) q^{21} - q^{22} - \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{24} + (\zeta_{6} - 1) q^{25} - 7 \zeta_{6} q^{26} + q^{27} + (3 \zeta_{6} - 1) q^{28} - 8 q^{29} - \zeta_{6} q^{30} + (6 \zeta_{6} - 6) q^{31} + (\zeta_{6} - 1) q^{32} + \zeta_{6} q^{33} - 4 q^{34} + ( - \zeta_{6} - 2) q^{35} + q^{36} + 3 \zeta_{6} q^{37} + (\zeta_{6} - 1) q^{38} + (7 \zeta_{6} - 7) q^{39} - \zeta_{6} q^{40} + 9 q^{41} + ( - 2 \zeta_{6} + 3) q^{42} - 4 q^{43} + \zeta_{6} q^{44} + (\zeta_{6} - 1) q^{45} + (\zeta_{6} - 1) q^{46} + 3 \zeta_{6} q^{47} + q^{48} + ( - 8 \zeta_{6} + 5) q^{49} + q^{50} + 4 \zeta_{6} q^{51} + (7 \zeta_{6} - 7) q^{52} + ( - \zeta_{6} + 1) q^{53} - \zeta_{6} q^{54} - q^{55} + ( - 2 \zeta_{6} + 3) q^{56} + q^{57} + 8 \zeta_{6} q^{58} + (12 \zeta_{6} - 12) q^{59} + (\zeta_{6} - 1) q^{60} + 4 \zeta_{6} q^{61} + 6 q^{62} + ( - \zeta_{6} - 2) q^{63} + q^{64} - 7 \zeta_{6} q^{65} + ( - \zeta_{6} + 1) q^{66} + (12 \zeta_{6} - 12) q^{67} + 4 \zeta_{6} q^{68} + q^{69} + (3 \zeta_{6} - 1) q^{70} - 14 q^{71} - \zeta_{6} q^{72} + ( - 14 \zeta_{6} + 14) q^{73} + ( - 3 \zeta_{6} + 3) q^{74} - \zeta_{6} q^{75} + q^{76} + ( - 3 \zeta_{6} + 1) q^{77} + 7 q^{78} - 4 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{80} + (\zeta_{6} - 1) q^{81} - 9 \zeta_{6} q^{82} + 12 q^{83} + ( - \zeta_{6} - 2) q^{84} - 4 q^{85} + 4 \zeta_{6} q^{86} + ( - 8 \zeta_{6} + 8) q^{87} + ( - \zeta_{6} + 1) q^{88} + 2 \zeta_{6} q^{89} + q^{90} + ( - 14 \zeta_{6} + 21) q^{91} + q^{92} - 6 \zeta_{6} q^{93} + ( - 3 \zeta_{6} + 3) q^{94} + (\zeta_{6} - 1) q^{95} - \zeta_{6} q^{96} - 16 q^{97} + (3 \zeta_{6} - 8) q^{98} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{3} - q^{4} - q^{5} + 2 q^{6} + 4 q^{7} + 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{3} - q^{4} - q^{5} + 2 q^{6} + 4 q^{7} + 2 q^{8} - q^{9} - q^{10} + q^{11} - q^{12} + 14 q^{13} - 5 q^{14} + 2 q^{15} - q^{16} + 4 q^{17} - q^{18} - q^{19} + 2 q^{20} + q^{21} - 2 q^{22} - q^{23} - q^{24} - q^{25} - 7 q^{26} + 2 q^{27} + q^{28} - 16 q^{29} - q^{30} - 6 q^{31} - q^{32} + q^{33} - 8 q^{34} - 5 q^{35} + 2 q^{36} + 3 q^{37} - q^{38} - 7 q^{39} - q^{40} + 18 q^{41} + 4 q^{42} - 8 q^{43} + q^{44} - q^{45} - q^{46} + 3 q^{47} + 2 q^{48} + 2 q^{49} + 2 q^{50} + 4 q^{51} - 7 q^{52} + q^{53} - q^{54} - 2 q^{55} + 4 q^{56} + 2 q^{57} + 8 q^{58} - 12 q^{59} - q^{60} + 4 q^{61} + 12 q^{62} - 5 q^{63} + 2 q^{64} - 7 q^{65} + q^{66} - 12 q^{67} + 4 q^{68} + 2 q^{69} + q^{70} - 28 q^{71} - q^{72} + 14 q^{73} + 3 q^{74} - q^{75} + 2 q^{76} - q^{77} + 14 q^{78} - 4 q^{79} - q^{80} - q^{81} - 9 q^{82} + 24 q^{83} - 5 q^{84} - 8 q^{85} + 4 q^{86} + 8 q^{87} + q^{88} + 2 q^{89} + 2 q^{90} + 28 q^{91} + 2 q^{92} - 6 q^{93} + 3 q^{94} - q^{95} - q^{96} - 32 q^{97} - 13 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 1.00000 2.00000 + 1.73205i 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
151.1 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 1.00000 2.00000 1.73205i 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.i.a 2
3.b odd 2 1 630.2.k.h 2
4.b odd 2 1 1680.2.bg.k 2
5.b even 2 1 1050.2.i.s 2
5.c odd 4 2 1050.2.o.j 4
7.b odd 2 1 1470.2.i.i 2
7.c even 3 1 inner 210.2.i.a 2
7.c even 3 1 1470.2.a.r 1
7.d odd 6 1 1470.2.a.k 1
7.d odd 6 1 1470.2.i.i 2
21.g even 6 1 4410.2.a.q 1
21.h odd 6 1 630.2.k.h 2
21.h odd 6 1 4410.2.a.g 1
28.g odd 6 1 1680.2.bg.k 2
35.i odd 6 1 7350.2.a.ba 1
35.j even 6 1 1050.2.i.s 2
35.j even 6 1 7350.2.a.j 1
35.l odd 12 2 1050.2.o.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.a 2 1.a even 1 1 trivial
210.2.i.a 2 7.c even 3 1 inner
630.2.k.h 2 3.b odd 2 1
630.2.k.h 2 21.h odd 6 1
1050.2.i.s 2 5.b even 2 1
1050.2.i.s 2 35.j even 6 1
1050.2.o.j 4 5.c odd 4 2
1050.2.o.j 4 35.l odd 12 2
1470.2.a.k 1 7.d odd 6 1
1470.2.a.r 1 7.c even 3 1
1470.2.i.i 2 7.b odd 2 1
1470.2.i.i 2 7.d odd 6 1
1680.2.bg.k 2 4.b odd 2 1
1680.2.bg.k 2 28.g odd 6 1
4410.2.a.g 1 21.h odd 6 1
4410.2.a.q 1 21.g even 6 1
7350.2.a.j 1 35.j even 6 1
7350.2.a.ba 1 35.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(210, [\chi])\):

\( T_{11}^{2} - T_{11} + 1 \) Copy content Toggle raw display
\( T_{13} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( (T - 7)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$19$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$29$ \( (T + 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$37$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$41$ \( (T - 9)^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$53$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$71$ \( (T + 14)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$97$ \( (T + 16)^{2} \) Copy content Toggle raw display
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