Properties

Label 210.2.d.a
Level 210
Weight 2
Character orbit 210.d
Analytic conductor 1.677
Analytic rank 0
Dimension 8
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.67685844245\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.10070523904.11
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 10 x^{4} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - \nu^{2} \)\()/18\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 7 \nu \)\()/6\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} + 19 \nu^{2} \)\()/18\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} + 3 \nu^{5} + 9 \nu^{4} + \nu^{3} - \nu^{2} - 3 \nu - 45 \)\()/36\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - 3 \nu^{5} + 9 \nu^{4} - \nu^{3} + \nu^{2} + 3 \nu - 45 \)\()/36\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} - 10 \nu^{3} \)\()/27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(-3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(2 \beta_{6} + 2 \beta_{5} + 5\)
\(\nu^{5}\)\(=\)\(6 \beta_{3} + 7 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{4} + 19 \beta_{2}\)
\(\nu^{7}\)\(=\)\(-3 \beta_{7} + 20 \beta_{6} - 20 \beta_{5} + 20 \beta_{3} + 20 \beta_{2} + 20 \beta_{1}\)

Character values

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

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
−1.68014 + 0.420861i
−1.68014 0.420861i
−0.420861 + 1.68014i
−0.420861 1.68014i
0.420861 + 1.68014i
0.420861 1.68014i
1.68014 + 0.420861i
1.68014 0.420861i
−1.00000 −1.68014 0.420861i 1.00000 −1.08495 + 1.95522i 1.68014 + 0.420861i 0.595188 2.57794i −1.00000 2.64575 + 1.41421i 1.08495 1.95522i
209.2 −1.00000 −1.68014 + 0.420861i 1.00000 −1.08495 1.95522i 1.68014 0.420861i 0.595188 + 2.57794i −1.00000 2.64575 1.41421i 1.08495 + 1.95522i
209.3 −1.00000 −0.420861 1.68014i 1.00000 1.95522 1.08495i 0.420861 + 1.68014i 2.37608 + 1.16372i −1.00000 −2.64575 + 1.41421i −1.95522 + 1.08495i
209.4 −1.00000 −0.420861 + 1.68014i 1.00000 1.95522 + 1.08495i 0.420861 1.68014i 2.37608 1.16372i −1.00000 −2.64575 1.41421i −1.95522 1.08495i
209.5 −1.00000 0.420861 1.68014i 1.00000 −1.95522 1.08495i −0.420861 + 1.68014i −2.37608 1.16372i −1.00000 −2.64575 1.41421i 1.95522 + 1.08495i
209.6 −1.00000 0.420861 + 1.68014i 1.00000 −1.95522 + 1.08495i −0.420861 1.68014i −2.37608 + 1.16372i −1.00000 −2.64575 + 1.41421i 1.95522 1.08495i
209.7 −1.00000 1.68014 0.420861i 1.00000 1.08495 + 1.95522i −1.68014 + 0.420861i −0.595188 + 2.57794i −1.00000 2.64575 1.41421i −1.08495 1.95522i
209.8 −1.00000 1.68014 + 0.420861i 1.00000 1.08495 1.95522i −1.68014 0.420861i −0.595188 2.57794i −1.00000 2.64575 + 1.41421i −1.08495 + 1.95522i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.d.a 8
3.b odd 2 1 210.2.d.b yes 8
4.b odd 2 1 1680.2.k.e 8
5.b even 2 1 210.2.d.b yes 8
5.c odd 4 2 1050.2.b.f 16
7.b odd 2 1 inner 210.2.d.a 8
12.b even 2 1 1680.2.k.f 8
15.d odd 2 1 inner 210.2.d.a 8
15.e even 4 2 1050.2.b.f 16
20.d odd 2 1 1680.2.k.f 8
21.c even 2 1 210.2.d.b yes 8
28.d even 2 1 1680.2.k.e 8
35.c odd 2 1 210.2.d.b yes 8
35.f even 4 2 1050.2.b.f 16
60.h even 2 1 1680.2.k.e 8
84.h odd 2 1 1680.2.k.f 8
105.g even 2 1 inner 210.2.d.a 8
105.k odd 4 2 1050.2.b.f 16
140.c even 2 1 1680.2.k.f 8
420.o odd 2 1 1680.2.k.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.d.a 8 1.a even 1 1 trivial
210.2.d.a 8 7.b odd 2 1 inner
210.2.d.a 8 15.d odd 2 1 inner
210.2.d.a 8 105.g even 2 1 inner
210.2.d.b yes 8 3.b odd 2 1
210.2.d.b yes 8 5.b even 2 1
210.2.d.b yes 8 21.c even 2 1
210.2.d.b yes 8 35.c odd 2 1
1050.2.b.f 16 5.c odd 4 2
1050.2.b.f 16 15.e even 4 2
1050.2.b.f 16 35.f even 4 2
1050.2.b.f 16 105.k odd 4 2
1680.2.k.e 8 4.b odd 2 1
1680.2.k.e 8 28.d even 2 1
1680.2.k.e 8 60.h even 2 1
1680.2.k.e 8 420.o odd 2 1
1680.2.k.f 8 12.b even 2 1
1680.2.k.f 8 20.d odd 2 1
1680.2.k.f 8 84.h odd 2 1
1680.2.k.f 8 140.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{2} - 4 T_{23} - 24 \) acting on \(S_{2}^{\mathrm{new}}(210, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{8} \)
$3$ \( 1 - 10 T^{4} + 81 T^{8} \)
$5$ \( 1 + 22 T^{4} + 625 T^{8} \)
$7$ \( 1 + 4 T^{2} - 10 T^{4} + 196 T^{6} + 2401 T^{8} \)
$11$ \( ( 1 - 6 T + 11 T^{2} )^{4}( 1 + 6 T + 11 T^{2} )^{4} \)
$13$ \( ( 1 + 40 T^{2} + 710 T^{4} + 6760 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 44 T^{2} + 950 T^{4} - 12716 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 24 T^{2} + 838 T^{4} - 8664 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 4 T + 22 T^{2} - 92 T^{3} + 529 T^{4} )^{4} \)
$29$ \( ( 1 - 52 T^{2} + 1350 T^{4} - 43732 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 84 T^{2} + 3574 T^{4} - 80724 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 - 20 T^{2} + 38 T^{4} - 27380 T^{6} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 + 84 T^{2} + 4678 T^{4} + 141204 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 44 T^{2} + 2390 T^{4} - 81356 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 - 108 T^{2} + 6886 T^{4} - 238572 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 4 T - 2 T^{2} - 212 T^{3} + 2809 T^{4} )^{4} \)
$59$ \( ( 1 + 216 T^{2} + 18598 T^{4} + 751896 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 136 T^{2} + 9798 T^{4} - 506056 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 140 T^{2} + 12086 T^{4} - 628460 T^{6} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 - 28 T^{2} + 9270 T^{4} - 141148 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 + 252 T^{2} + 26422 T^{4} + 1342908 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 + 4 T + 134 T^{2} + 316 T^{3} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 - 224 T^{2} + 26294 T^{4} - 1543136 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 60 T^{2} + 14950 T^{4} - 475260 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 + 108 T^{2} + 16246 T^{4} + 1016172 T^{6} + 88529281 T^{8} )^{2} \)
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