# 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.

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

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