Properties

Label 210.2.t.a
Level 210
Weight 2
Character orbit 210.t
Analytic conductor 1.677
Analytic rank 1
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.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.67685844245\)
Analytic rank: \(1\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

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

Character values

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

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(1 - \beta_{2}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
−1.18614 + 1.26217i
1.68614 0.396143i
1.68614 + 0.396143i
−1.18614 1.26217i
−0.500000 0.866025i −0.500000 1.65831i −0.500000 + 0.866025i −1.50000 + 1.65831i −1.18614 + 1.26217i −2.50000 + 0.866025i 1.00000 −2.50000 + 1.65831i 2.18614 + 0.469882i
59.2 −0.500000 0.866025i −0.500000 + 1.65831i −0.500000 + 0.866025i −1.50000 1.65831i 1.68614 0.396143i −2.50000 + 0.866025i 1.00000 −2.50000 1.65831i −0.686141 + 2.12819i
89.1 −0.500000 + 0.866025i −0.500000 1.65831i −0.500000 0.866025i −1.50000 + 1.65831i 1.68614 + 0.396143i −2.50000 0.866025i 1.00000 −2.50000 + 1.65831i −0.686141 2.12819i
89.2 −0.500000 + 0.866025i −0.500000 + 1.65831i −0.500000 0.866025i −1.50000 1.65831i −1.18614 1.26217i −2.50000 0.866025i 1.00000 −2.50000 1.65831i 2.18614 0.469882i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
105.p 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.t.a 4
3.b odd 2 1 210.2.t.c yes 4
5.b even 2 1 210.2.t.d yes 4
5.c odd 4 2 1050.2.s.e 8
7.c even 3 1 1470.2.d.d 4
7.d odd 6 1 210.2.t.b yes 4
7.d odd 6 1 1470.2.d.c 4
15.d odd 2 1 210.2.t.b yes 4
15.e even 4 2 1050.2.s.d 8
21.g even 6 1 210.2.t.d yes 4
21.g even 6 1 1470.2.d.a 4
21.h odd 6 1 1470.2.d.b 4
35.i odd 6 1 210.2.t.c yes 4
35.i odd 6 1 1470.2.d.b 4
35.j even 6 1 1470.2.d.a 4
35.k even 12 2 1050.2.s.d 8
105.o odd 6 1 1470.2.d.c 4
105.p even 6 1 inner 210.2.t.a 4
105.p even 6 1 1470.2.d.d 4
105.w odd 12 2 1050.2.s.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.t.a 4 1.a even 1 1 trivial
210.2.t.a 4 105.p even 6 1 inner
210.2.t.b yes 4 7.d odd 6 1
210.2.t.b yes 4 15.d odd 2 1
210.2.t.c yes 4 3.b odd 2 1
210.2.t.c yes 4 35.i odd 6 1
210.2.t.d yes 4 5.b even 2 1
210.2.t.d yes 4 21.g even 6 1
1050.2.s.d 8 15.e even 4 2
1050.2.s.d 8 35.k even 12 2
1050.2.s.e 8 5.c odd 4 2
1050.2.s.e 8 105.w odd 12 2
1470.2.d.a 4 21.g even 6 1
1470.2.d.a 4 35.j even 6 1
1470.2.d.b 4 21.h odd 6 1
1470.2.d.b 4 35.i odd 6 1
1470.2.d.c 4 7.d odd 6 1
1470.2.d.c 4 105.o odd 6 1
1470.2.d.d 4 7.c even 3 1
1470.2.d.d 4 105.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(210, [\chi])\):

\( T_{11}^{4} + 9 T_{11}^{3} + 31 T_{11}^{2} + 36 T_{11} + 16 \)
\( T_{13} + 2 \)
\( T_{23}^{4} + 3 T_{23}^{3} + 15 T_{23}^{2} - 18 T_{23} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( ( 1 + T + 3 T^{2} )^{2} \)
$5$ \( ( 1 + 3 T + 5 T^{2} )^{2} \)
$7$ \( ( 1 + 5 T + 7 T^{2} )^{2} \)
$11$ \( 1 + 9 T + 53 T^{2} + 234 T^{3} + 852 T^{4} + 2574 T^{5} + 6413 T^{6} + 11979 T^{7} + 14641 T^{8} \)
$13$ \( ( 1 + 2 T + 13 T^{2} )^{4} \)
$17$ \( 1 - 10 T^{2} - 189 T^{4} - 2890 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 3 T - 31 T^{2} - 18 T^{3} + 864 T^{4} - 414 T^{5} - 16399 T^{6} + 36501 T^{7} + 279841 T^{8} \)
$29$ \( ( 1 - 47 T^{2} + 841 T^{4} )^{2} \)
$31$ \( 1 - 9 T + 71 T^{2} - 396 T^{3} + 1812 T^{4} - 12276 T^{5} + 68231 T^{6} - 268119 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 6 T - 10 T^{2} - 132 T^{3} - 441 T^{4} - 4884 T^{5} - 13690 T^{6} + 303918 T^{7} + 1874161 T^{8} \)
$41$ \( ( 1 + 9 T + 94 T^{2} + 369 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( 1 - 49 T^{2} + 660 T^{4} - 90601 T^{6} + 3418801 T^{8} \)
$47$ \( 1 + 18 T + 218 T^{2} + 1980 T^{3} + 14967 T^{4} + 93060 T^{5} + 481562 T^{6} + 1868814 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 3 T - 91 T^{2} + 18 T^{3} + 6714 T^{4} + 954 T^{5} - 255619 T^{6} - 446631 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 9 T + 17 T^{2} + 486 T^{3} - 4164 T^{4} + 28674 T^{5} + 59177 T^{6} - 1848411 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 27 T + 401 T^{2} + 4266 T^{3} + 36066 T^{4} + 260226 T^{5} + 1492121 T^{6} + 6128487 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 9 T + 143 T^{2} - 1044 T^{3} + 10776 T^{4} - 69948 T^{5} + 641927 T^{6} - 2706867 T^{7} + 20151121 T^{8} \)
$71$ \( 1 - 208 T^{2} + 19710 T^{4} - 1048528 T^{6} + 25411681 T^{8} \)
$73$ \( ( 1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( 1 + T - 83 T^{2} - 74 T^{3} + 736 T^{4} - 5846 T^{5} - 518003 T^{6} + 493039 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 190 T^{2} + 18051 T^{4} - 1308910 T^{6} + 47458321 T^{8} \)
$89$ \( 1 + 3 T - 163 T^{2} - 18 T^{3} + 20862 T^{4} - 1602 T^{5} - 1291123 T^{6} + 2114907 T^{7} + 62742241 T^{8} \)
$97$ \( ( 1 + 13 T + 162 T^{2} + 1261 T^{3} + 9409 T^{4} )^{2} \)
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