# 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

# Related objects

## 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 36q^{4} + 28q^{7} + 96q^{9} + O(q^{10})$$ $$4q - 36q^{4} + 28q^{7} + 96q^{9} - 204q^{15} - 220q^{16} + 204q^{18} - 84q^{21} + 544q^{22} - 92q^{25} - 252q^{28} - 204q^{30} - 864q^{36} + 920q^{37} + 276q^{39} + 1428q^{42} + 176q^{43} - 1496q^{46} - 980q^{49} - 1224q^{51} - 180q^{57} + 952q^{58} + 1836q^{60} + 672q^{63} + 2524q^{64} - 256q^{67} - 2856q^{70} - 204q^{72} - 4692q^{78} - 1768q^{79} + 1692q^{81} + 756q^{84} + 2448q^{85} - 544q^{88} + 3864q^{91} + 1248q^{93} - 1632q^{99} + O(q^{100})$$

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

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

## 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: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$T_{2}^{2} + 17$$ acting on $$S_{4}^{\mathrm{new}}(21, [\chi])$$.