Properties

Label 210.2.n.a
Level 210
Weight 2
Character orbit 210.n
Analytic conductor 1.677
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 210.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.67685844245\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −0.133975 + 2.23205i −1.00000 1.73205 2.00000i 1.00000i 0.500000 0.866025i 1.23205 1.86603i
79.2 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −1.86603 + 1.23205i −1.00000 −1.73205 + 2.00000i 1.00000i 0.500000 0.866025i −2.23205 + 0.133975i
109.1 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i −0.133975 2.23205i −1.00000 1.73205 + 2.00000i 1.00000i 0.500000 + 0.866025i 1.23205 + 1.86603i
109.2 0.866025 0.500000i −0.866025 0.500000i 0.500000 0.866025i −1.86603 1.23205i −1.00000 −1.73205 2.00000i 1.00000i 0.500000 + 0.866025i −2.23205 0.133975i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.n.a 4
3.b odd 2 1 630.2.u.c 4
4.b odd 2 1 1680.2.di.a 4
5.b even 2 1 inner 210.2.n.a 4
5.c odd 4 1 1050.2.i.f 2
5.c odd 4 1 1050.2.i.o 2
7.b odd 2 1 1470.2.n.i 4
7.c even 3 1 inner 210.2.n.a 4
7.c even 3 1 1470.2.g.f 2
7.d odd 6 1 1470.2.g.a 2
7.d odd 6 1 1470.2.n.i 4
15.d odd 2 1 630.2.u.c 4
20.d odd 2 1 1680.2.di.a 4
21.h odd 6 1 630.2.u.c 4
28.g odd 6 1 1680.2.di.a 4
35.c odd 2 1 1470.2.n.i 4
35.i odd 6 1 1470.2.g.a 2
35.i odd 6 1 1470.2.n.i 4
35.j even 6 1 inner 210.2.n.a 4
35.j even 6 1 1470.2.g.f 2
35.k even 12 1 7350.2.a.b 1
35.k even 12 1 7350.2.a.ch 1
35.l odd 12 1 1050.2.i.f 2
35.l odd 12 1 1050.2.i.o 2
35.l odd 12 1 7350.2.a.t 1
35.l odd 12 1 7350.2.a.bn 1
105.o odd 6 1 630.2.u.c 4
140.p odd 6 1 1680.2.di.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.a 4 1.a even 1 1 trivial
210.2.n.a 4 5.b even 2 1 inner
210.2.n.a 4 7.c even 3 1 inner
210.2.n.a 4 35.j even 6 1 inner
630.2.u.c 4 3.b odd 2 1
630.2.u.c 4 15.d odd 2 1
630.2.u.c 4 21.h odd 6 1
630.2.u.c 4 105.o odd 6 1
1050.2.i.f 2 5.c odd 4 1
1050.2.i.f 2 35.l odd 12 1
1050.2.i.o 2 5.c odd 4 1
1050.2.i.o 2 35.l odd 12 1
1470.2.g.a 2 7.d odd 6 1
1470.2.g.a 2 35.i odd 6 1
1470.2.g.f 2 7.c even 3 1
1470.2.g.f 2 35.j even 6 1
1470.2.n.i 4 7.b odd 2 1
1470.2.n.i 4 7.d odd 6 1
1470.2.n.i 4 35.c odd 2 1
1470.2.n.i 4 35.i odd 6 1
1680.2.di.a 4 4.b odd 2 1
1680.2.di.a 4 20.d odd 2 1
1680.2.di.a 4 28.g odd 6 1
1680.2.di.a 4 140.p odd 6 1
7350.2.a.b 1 35.k even 12 1
7350.2.a.t 1 35.l odd 12 1
7350.2.a.bn 1 35.l odd 12 1
7350.2.a.ch 1 35.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} - 5 T_{11} + 25 \) acting on \(S_{2}^{\mathrm{new}}(210, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \)
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 25 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} ) \)
$19$ \( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + T + 19 T^{2} )^{2} \)
$23$ \( 1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 29 T^{2} )^{4} \)
$31$ \( ( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} )^{2} \)
$37$ \( 1 + 49 T^{2} + 1032 T^{4} + 67081 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 + 9 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 + 14 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 - 75 T^{2} + 3416 T^{4} - 165675 T^{6} + 4879681 T^{8} \)
$53$ \( 1 + 105 T^{2} + 8216 T^{4} + 294945 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 + 98 T^{2} + 5115 T^{4} + 439922 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 + 2 T + 71 T^{2} )^{4} \)
$73$ \( 1 + 130 T^{2} + 11571 T^{4} + 692770 T^{6} + 28398241 T^{8} \)
$79$ \( ( 1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 66 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 10 T + 11 T^{2} - 890 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 18 T + 97 T^{2} )^{2}( 1 + 18 T + 97 T^{2} )^{2} \)
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