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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Newspace parameters

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

Self dual: no
Analytic conductor: \(1.67685844245\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

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

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.

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 \(S_{2}^{\mathrm{new}}(210, [\chi])\):

\( T_{11}^{2} - T_{11} + 1 \)
\( T_{13} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( 1 - 4 T + 7 T^{2} \)
$11$ \( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 7 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 8 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 6 T + 5 T^{2} + 186 T^{3} + 961 T^{4} \)
$37$ \( 1 - 3 T - 28 T^{2} - 111 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 9 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \)
$53$ \( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 14 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 - 13 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \)
$83$ \( ( 1 - 12 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 2 T - 85 T^{2} - 178 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 16 T + 97 T^{2} )^{2} \)
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