Properties

Label 210.4.g.a
Level $210$
Weight $4$
Character orbit 210.g
Analytic conductor $12.390$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 210.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.3904011012\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q + 2 \beta_{1} q^{2} + 3 \beta_{1} q^{3} -4 q^{4} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{5} -6 q^{6} + 7 \beta_{1} q^{7} -8 \beta_{1} q^{8} -9 q^{9} +O(q^{10})\) \( q + 2 \beta_{1} q^{2} + 3 \beta_{1} q^{3} -4 q^{4} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{5} -6 q^{6} + 7 \beta_{1} q^{7} -8 \beta_{1} q^{8} -9 q^{9} + ( 4 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{10} + ( 6 - 4 \beta_{3} ) q^{11} -12 \beta_{1} q^{12} + ( 50 \beta_{1} - 8 \beta_{2} ) q^{13} -14 q^{14} + ( 6 - 3 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{15} + 16 q^{16} + ( -40 \beta_{1} - 12 \beta_{2} ) q^{17} -18 \beta_{1} q^{18} + ( 2 - 2 \beta_{3} ) q^{19} + ( 4 + 8 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{20} -21 q^{21} + ( 12 \beta_{1} - 8 \beta_{2} ) q^{22} + ( 32 \beta_{1} - 26 \beta_{2} ) q^{23} + 24 q^{24} + ( 69 - 92 \beta_{1} + 6 \beta_{2} + 8 \beta_{3} ) q^{25} + ( -100 + 16 \beta_{3} ) q^{26} -27 \beta_{1} q^{27} -28 \beta_{1} q^{28} + ( -10 + 2 \beta_{3} ) q^{29} + ( 6 + 12 \beta_{1} - 6 \beta_{2} + 12 \beta_{3} ) q^{30} + ( -146 - 4 \beta_{3} ) q^{31} + 32 \beta_{1} q^{32} + ( 18 \beta_{1} - 12 \beta_{2} ) q^{33} + ( 80 + 24 \beta_{3} ) q^{34} + ( 14 - 7 \beta_{1} - 14 \beta_{2} - 7 \beta_{3} ) q^{35} + 36 q^{36} + ( 136 \beta_{1} - 30 \beta_{2} ) q^{37} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{38} + ( -150 + 24 \beta_{3} ) q^{39} + ( -16 + 8 \beta_{1} + 16 \beta_{2} + 8 \beta_{3} ) q^{40} + ( -66 + 16 \beta_{3} ) q^{41} -42 \beta_{1} q^{42} + ( 16 \beta_{1} + 70 \beta_{2} ) q^{43} + ( -24 + 16 \beta_{3} ) q^{44} + ( 9 + 18 \beta_{1} - 9 \beta_{2} + 18 \beta_{3} ) q^{45} + ( -64 + 52 \beta_{3} ) q^{46} + ( -284 \beta_{1} - 34 \beta_{2} ) q^{47} + 48 \beta_{1} q^{48} -49 q^{49} + ( 184 + 138 \beta_{1} + 16 \beta_{2} - 12 \beta_{3} ) q^{50} + ( 120 + 36 \beta_{3} ) q^{51} + ( -200 \beta_{1} + 32 \beta_{2} ) q^{52} + ( 174 \beta_{1} - 94 \beta_{2} ) q^{53} + 54 q^{54} + ( 186 - 108 \beta_{1} + 14 \beta_{2} - 8 \beta_{3} ) q^{55} + 56 q^{56} + ( 6 \beta_{1} - 6 \beta_{2} ) q^{57} + ( -20 \beta_{1} + 4 \beta_{2} ) q^{58} + ( -112 - 128 \beta_{3} ) q^{59} + ( -24 + 12 \beta_{1} + 24 \beta_{2} + 12 \beta_{3} ) q^{60} + ( -6 - 82 \beta_{3} ) q^{61} + ( -292 \beta_{1} - 8 \beta_{2} ) q^{62} -63 \beta_{1} q^{63} -64 q^{64} + ( 292 + 334 \beta_{1} - 92 \beta_{2} - 66 \beta_{3} ) q^{65} + ( -36 + 24 \beta_{3} ) q^{66} + ( -460 \beta_{1} + 78 \beta_{2} ) q^{67} + ( 160 \beta_{1} + 48 \beta_{2} ) q^{68} + ( -96 + 78 \beta_{3} ) q^{69} + ( 14 + 28 \beta_{1} - 14 \beta_{2} + 28 \beta_{3} ) q^{70} + ( -78 - 98 \beta_{3} ) q^{71} + 72 \beta_{1} q^{72} + ( 166 \beta_{1} - 28 \beta_{2} ) q^{73} + ( -272 + 60 \beta_{3} ) q^{74} + ( 276 + 207 \beta_{1} + 24 \beta_{2} - 18 \beta_{3} ) q^{75} + ( -8 + 8 \beta_{3} ) q^{76} + ( 42 \beta_{1} - 28 \beta_{2} ) q^{77} + ( -300 \beta_{1} + 48 \beta_{2} ) q^{78} + ( -188 + 16 \beta_{3} ) q^{79} + ( -16 - 32 \beta_{1} + 16 \beta_{2} - 32 \beta_{3} ) q^{80} + 81 q^{81} + ( -132 \beta_{1} + 32 \beta_{2} ) q^{82} + ( 1008 \beta_{1} + 52 \beta_{2} ) q^{83} + 84 q^{84} + ( 208 + 616 \beta_{1} + 92 \beta_{2} + 16 \beta_{3} ) q^{85} + ( -32 - 140 \beta_{3} ) q^{86} + ( -30 \beta_{1} + 6 \beta_{2} ) q^{87} + ( -48 \beta_{1} + 32 \beta_{2} ) q^{88} + ( -274 - 228 \beta_{3} ) q^{89} + ( -36 + 18 \beta_{1} + 36 \beta_{2} + 18 \beta_{3} ) q^{90} + ( -350 + 56 \beta_{3} ) q^{91} + ( -128 \beta_{1} + 104 \beta_{2} ) q^{92} + ( -438 \beta_{1} - 12 \beta_{2} ) q^{93} + ( 568 + 68 \beta_{3} ) q^{94} + ( 94 - 52 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{95} -96 q^{96} + ( -810 \beta_{1} + 80 \beta_{2} ) q^{97} -98 \beta_{1} q^{98} + ( -54 + 36 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 16q^{4} - 4q^{5} - 24q^{6} - 36q^{9} + O(q^{10}) \) \( 4q - 16q^{4} - 4q^{5} - 24q^{6} - 36q^{9} + 16q^{10} + 24q^{11} - 56q^{14} + 24q^{15} + 64q^{16} + 8q^{19} + 16q^{20} - 84q^{21} + 96q^{24} + 276q^{25} - 400q^{26} - 40q^{29} + 24q^{30} - 584q^{31} + 320q^{34} + 56q^{35} + 144q^{36} - 600q^{39} - 64q^{40} - 264q^{41} - 96q^{44} + 36q^{45} - 256q^{46} - 196q^{49} + 736q^{50} + 480q^{51} + 216q^{54} + 744q^{55} + 224q^{56} - 448q^{59} - 96q^{60} - 24q^{61} - 256q^{64} + 1168q^{65} - 144q^{66} - 384q^{69} + 56q^{70} - 312q^{71} - 1088q^{74} + 1104q^{75} - 32q^{76} - 752q^{79} - 64q^{80} + 324q^{81} + 336q^{84} + 832q^{85} - 128q^{86} - 1096q^{89} - 144q^{90} - 1400q^{91} + 2272q^{94} + 376q^{95} - 384q^{96} - 216q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{3} + 6 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{3} + 6 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\(3 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{3} + 3 \beta_{2}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1
1.22474 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
−1.22474 1.22474i
2.00000i 3.00000i −4.00000 −10.7980 2.89898i −6.00000 7.00000i 8.00000i −9.00000 −5.79796 + 21.5959i
169.2 2.00000i 3.00000i −4.00000 8.79796 + 6.89898i −6.00000 7.00000i 8.00000i −9.00000 13.7980 17.5959i
169.3 2.00000i 3.00000i −4.00000 −10.7980 + 2.89898i −6.00000 7.00000i 8.00000i −9.00000 −5.79796 21.5959i
169.4 2.00000i 3.00000i −4.00000 8.79796 6.89898i −6.00000 7.00000i 8.00000i −9.00000 13.7980 + 17.5959i
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.4.g.a 4
3.b odd 2 1 630.4.g.e 4
5.b even 2 1 inner 210.4.g.a 4
5.c odd 4 1 1050.4.a.bc 2
5.c odd 4 1 1050.4.a.bg 2
15.d odd 2 1 630.4.g.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.4.g.a 4 1.a even 1 1 trivial
210.4.g.a 4 5.b even 2 1 inner
630.4.g.e 4 3.b odd 2 1
630.4.g.e 4 15.d odd 2 1
1050.4.a.bc 2 5.c odd 4 1
1050.4.a.bg 2 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} - 12 T_{11} - 348 \) acting on \(S_{4}^{\mathrm{new}}(210, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + T^{2} )^{2} \)
$3$ \( ( 9 + T^{2} )^{2} \)
$5$ \( 15625 + 500 T - 130 T^{2} + 4 T^{3} + T^{4} \)
$7$ \( ( 49 + T^{2} )^{2} \)
$11$ \( ( -348 - 12 T + T^{2} )^{2} \)
$13$ \( 929296 + 8072 T^{2} + T^{4} \)
$17$ \( 3444736 + 10112 T^{2} + T^{4} \)
$19$ \( ( -92 - 4 T + T^{2} )^{2} \)
$23$ \( 231040000 + 34496 T^{2} + T^{4} \)
$29$ \( ( 4 + 20 T + T^{2} )^{2} \)
$31$ \( ( 20932 + 292 T + T^{2} )^{2} \)
$37$ \( 9634816 + 80192 T^{2} + T^{4} \)
$41$ \( ( -1788 + 132 T + T^{2} )^{2} \)
$43$ \( 13769614336 + 235712 T^{2} + T^{4} \)
$47$ \( 2799679744 + 216800 T^{2} + T^{4} \)
$53$ \( 33046876944 + 484680 T^{2} + T^{4} \)
$59$ \( ( -380672 + 224 T + T^{2} )^{2} \)
$61$ \( ( -161340 + 12 T + T^{2} )^{2} \)
$67$ \( 4301261056 + 715232 T^{2} + T^{4} \)
$71$ \( ( -224412 + 156 T + T^{2} )^{2} \)
$73$ \( 76387600 + 92744 T^{2} + T^{4} \)
$79$ \( ( 29200 + 376 T + T^{2} )^{2} \)
$83$ \( 904720564224 + 2161920 T^{2} + T^{4} \)
$89$ \( ( -1172540 + 548 T + T^{2} )^{2} \)
$97$ \( 252506250000 + 1619400 T^{2} + T^{4} \)
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