Properties

Label 21.4.c.b
Level 21
Weight 4
Character orbit 21.c
Analytic conductor 1.239
Analytic rank 0
Dimension 4
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 21.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.23904011012\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-6}, \sqrt{-17})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q\) \( -\beta_{2} q^{2} \) \( + \beta_{3} q^{3} \) \( -9 q^{4} \) \( + ( -\beta_{1} - 2 \beta_{3} ) q^{5} \) \( + ( 9 \beta_{1} + \beta_{3} ) q^{6} \) \( + ( 7 - 7 \beta_{1} ) q^{7} \) \( + \beta_{2} q^{8} \) \( + ( 24 + 3 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{2} q^{2} \) \( + \beta_{3} q^{3} \) \( -9 q^{4} \) \( + ( -\beta_{1} - 2 \beta_{3} ) q^{5} \) \( + ( 9 \beta_{1} + \beta_{3} ) q^{6} \) \( + ( 7 - 7 \beta_{1} ) q^{7} \) \( + \beta_{2} q^{8} \) \( + ( 24 + 3 \beta_{2} ) q^{9} \) \( -17 \beta_{1} q^{10} \) \( + 8 \beta_{2} q^{11} \) \( -9 \beta_{3} q^{12} \) \( + 23 \beta_{1} q^{13} \) \( + ( 7 \beta_{1} - 7 \beta_{2} + 14 \beta_{3} ) q^{14} \) \( + ( -51 - 3 \beta_{2} ) q^{15} \) \( -55 q^{16} \) \( + ( -6 \beta_{1} - 12 \beta_{3} ) q^{17} \) \( + ( 51 - 24 \beta_{2} ) q^{18} \) \( -15 \beta_{1} q^{19} \) \( + ( 9 \beta_{1} + 18 \beta_{3} ) q^{20} \) \( + ( -21 + 21 \beta_{2} + 7 \beta_{3} ) q^{21} \) \( + 136 q^{22} \) \( -22 \beta_{2} q^{23} \) \( + ( -9 \beta_{1} - \beta_{3} ) q^{24} \) \( -23 q^{25} \) \( + ( -23 \beta_{1} - 46 \beta_{3} ) q^{26} \) \( + ( -27 \beta_{1} + 21 \beta_{3} ) q^{27} \) \( + ( -63 + 63 \beta_{1} ) q^{28} \) \( + 14 \beta_{2} q^{29} \) \( + ( -51 + 51 \beta_{2} ) q^{30} \) \( + 104 \beta_{1} q^{31} \) \( + 63 \beta_{2} q^{32} \) \( + ( -72 \beta_{1} - 8 \beta_{3} ) q^{33} \) \( -102 \beta_{1} q^{34} \) \( + ( -7 \beta_{1} - 42 \beta_{2} - 14 \beta_{3} ) q^{35} \) \( + ( -216 - 27 \beta_{2} ) q^{36} \) \( + 230 q^{37} \) \( + ( 15 \beta_{1} + 30 \beta_{3} ) q^{38} \) \( + ( 69 - 69 \beta_{2} ) q^{39} \) \( + 17 \beta_{1} q^{40} \) \( + ( 14 \beta_{1} + 28 \beta_{3} ) q^{41} \) \( + ( 357 + 63 \beta_{1} + 21 \beta_{2} + 7 \beta_{3} ) q^{42} \) \( + 44 q^{43} \) \( -72 \beta_{2} q^{44} \) \( + ( 27 \beta_{1} - 48 \beta_{3} ) q^{45} \) \( -374 q^{46} \) \( + ( 34 \beta_{1} + 68 \beta_{3} ) q^{47} \) \( -55 \beta_{3} q^{48} \) \( + ( -245 - 98 \beta_{1} ) q^{49} \) \( + 23 \beta_{2} q^{50} \) \( + ( -306 - 18 \beta_{2} ) q^{51} \) \( -207 \beta_{1} q^{52} \) \( + 50 \beta_{2} q^{53} \) \( + ( 216 \beta_{1} + 75 \beta_{3} ) q^{54} \) \( + 136 \beta_{1} q^{55} \) \( + ( -7 \beta_{1} + 7 \beta_{2} - 14 \beta_{3} ) q^{56} \) \( + ( -45 + 45 \beta_{2} ) q^{57} \) \( + 238 q^{58} \) \( + ( -13 \beta_{1} - 26 \beta_{3} ) q^{59} \) \( + ( 459 + 27 \beta_{2} ) q^{60} \) \( -29 \beta_{1} q^{61} \) \( + ( -104 \beta_{1} - 208 \beta_{3} ) q^{62} \) \( + ( 168 - 189 \beta_{1} + 21 \beta_{2} - 42 \beta_{3} ) q^{63} \) \( + 631 q^{64} \) \( + 138 \beta_{2} q^{65} \) \( + 136 \beta_{3} q^{66} \) \( -64 q^{67} \) \( + ( 54 \beta_{1} + 108 \beta_{3} ) q^{68} \) \( + ( 198 \beta_{1} + 22 \beta_{3} ) q^{69} \) \( + ( -714 - 119 \beta_{1} ) q^{70} \) \( -112 \beta_{2} q^{71} \) \( + ( -51 + 24 \beta_{2} ) q^{72} \) \( -36 \beta_{1} q^{73} \) \( -230 \beta_{2} q^{74} \) \( -23 \beta_{3} q^{75} \) \( + 135 \beta_{1} q^{76} \) \( + ( -56 \beta_{1} + 56 \beta_{2} - 112 \beta_{3} ) q^{77} \) \( + ( -1173 - 69 \beta_{2} ) q^{78} \) \( -442 q^{79} \) \( + ( 55 \beta_{1} + 110 \beta_{3} ) q^{80} \) \( + ( 423 + 144 \beta_{2} ) q^{81} \) \( + 238 \beta_{1} q^{82} \) \( + ( 49 \beta_{1} + 98 \beta_{3} ) q^{83} \) \( + ( 189 - 189 \beta_{2} - 63 \beta_{3} ) q^{84} \) \( + 612 q^{85} \) \( -44 \beta_{2} q^{86} \) \( + ( -126 \beta_{1} - 14 \beta_{3} ) q^{87} \) \( -136 q^{88} \) \( + ( 48 \beta_{1} + 96 \beta_{3} ) q^{89} \) \( + ( -459 \beta_{1} - 102 \beta_{3} ) q^{90} \) \( + ( 966 + 161 \beta_{1} ) q^{91} \) \( + 198 \beta_{2} q^{92} \) \( + ( 312 - 312 \beta_{2} ) q^{93} \) \( + 578 \beta_{1} q^{94} \) \( -90 \beta_{2} q^{95} \) \( + ( -567 \beta_{1} - 63 \beta_{3} ) q^{96} \) \( -446 \beta_{1} q^{97} \) \( + ( 98 \beta_{1} + 245 \beta_{2} + 196 \beta_{3} ) q^{98} \) \( + ( -408 + 192 \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 36q^{4} \) \(\mathstrut +\mathstrut 28q^{7} \) \(\mathstrut +\mathstrut 96q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 36q^{4} \) \(\mathstrut +\mathstrut 28q^{7} \) \(\mathstrut +\mathstrut 96q^{9} \) \(\mathstrut -\mathstrut 204q^{15} \) \(\mathstrut -\mathstrut 220q^{16} \) \(\mathstrut +\mathstrut 204q^{18} \) \(\mathstrut -\mathstrut 84q^{21} \) \(\mathstrut +\mathstrut 544q^{22} \) \(\mathstrut -\mathstrut 92q^{25} \) \(\mathstrut -\mathstrut 252q^{28} \) \(\mathstrut -\mathstrut 204q^{30} \) \(\mathstrut -\mathstrut 864q^{36} \) \(\mathstrut +\mathstrut 920q^{37} \) \(\mathstrut +\mathstrut 276q^{39} \) \(\mathstrut +\mathstrut 1428q^{42} \) \(\mathstrut +\mathstrut 176q^{43} \) \(\mathstrut -\mathstrut 1496q^{46} \) \(\mathstrut -\mathstrut 980q^{49} \) \(\mathstrut -\mathstrut 1224q^{51} \) \(\mathstrut -\mathstrut 180q^{57} \) \(\mathstrut +\mathstrut 952q^{58} \) \(\mathstrut +\mathstrut 1836q^{60} \) \(\mathstrut +\mathstrut 672q^{63} \) \(\mathstrut +\mathstrut 2524q^{64} \) \(\mathstrut -\mathstrut 256q^{67} \) \(\mathstrut -\mathstrut 2856q^{70} \) \(\mathstrut -\mathstrut 204q^{72} \) \(\mathstrut -\mathstrut 4692q^{78} \) \(\mathstrut -\mathstrut 1768q^{79} \) \(\mathstrut +\mathstrut 1692q^{81} \) \(\mathstrut +\mathstrut 756q^{84} \) \(\mathstrut +\mathstrut 2448q^{85} \) \(\mathstrut -\mathstrut 544q^{88} \) \(\mathstrut +\mathstrut 3864q^{91} \) \(\mathstrut +\mathstrut 1248q^{93} \) \(\mathstrut -\mathstrut 1632q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(46\) \(x^{2}\mathstrut +\mathstrut \) \(121\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} - 35 \nu \)\()/22\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 57 \nu \)\()/22\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 11 \nu^{2} + 35 \nu + 253 \)\()/44\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)
\(\nu^{2}\)\(=\)\(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut -\mathstrut \) \(23\)
\(\nu^{3}\)\(=\)\(-\)\(35\) \(\beta_{2}\mathstrut -\mathstrut \) \(57\) \(\beta_{1}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1
6.57260i
1.67362i
6.57260i
1.67362i
4.12311i −5.04975 1.22474i −9.00000 10.0995 −5.04975 + 20.8207i 7.00000 17.1464i 4.12311i 24.0000 + 12.3693i 41.6413i
20.2 4.12311i 5.04975 + 1.22474i −9.00000 −10.0995 5.04975 20.8207i 7.00000 + 17.1464i 4.12311i 24.0000 + 12.3693i 41.6413i
20.3 4.12311i −5.04975 + 1.22474i −9.00000 10.0995 −5.04975 20.8207i 7.00000 + 17.1464i 4.12311i 24.0000 12.3693i 41.6413i
20.4 4.12311i 5.04975 1.22474i −9.00000 −10.0995 5.04975 + 20.8207i 7.00000 17.1464i 4.12311i 24.0000 12.3693i 41.6413i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
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Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
7.b Odd 1 yes
21.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut +\mathstrut 17 \) acting on \(S_{4}^{\mathrm{new}}(21, [\chi])\).