Properties

Label 210.2.b.a
Level $210$
Weight $2$
Character orbit 210.b
Analytic conductor $1.677$
Analytic rank $0$
Dimension $4$
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.67685844245\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu + 1 \)
\(\beta_{2}\)\(=\)\( \nu^{3} + 2 \nu \)
\(\beta_{3}\)\(=\)\( -\nu^{2} + \nu - 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + \beta_{1} - 2\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{3} + \beta_{2} - \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
0.618034i
1.61803i
0.618034i
1.61803i
1.00000i −1.61803 + 0.618034i −1.00000 1.00000 0.618034 + 1.61803i 2.61803 0.381966i 1.00000i 2.23607 2.00000i 1.00000i
41.2 1.00000i 0.618034 1.61803i −1.00000 1.00000 −1.61803 0.618034i 0.381966 2.61803i 1.00000i −2.23607 2.00000i 1.00000i
41.3 1.00000i −1.61803 0.618034i −1.00000 1.00000 0.618034 1.61803i 2.61803 + 0.381966i 1.00000i 2.23607 + 2.00000i 1.00000i
41.4 1.00000i 0.618034 + 1.61803i −1.00000 1.00000 −1.61803 + 0.618034i 0.381966 + 2.61803i 1.00000i −2.23607 + 2.00000i 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{2} - 6 T_{17} + 4 \) acting on \(S_{2}^{\mathrm{new}}(210, [\chi])\).