Properties

 Label 21.4.e.a Level 21 Weight 4 Character orbit 21.e Analytic conductor 1.239 Analytic rank 0 Dimension 2 CM No Inner twists 2

Related objects

Newspace parameters

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

Newform invariants

 Self dual: No Analytic conductor: $$1.23904011012$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 3 - 3 \zeta_{6} ) q^{2}$$ $$-3 \zeta_{6} q^{3}$$ $$-\zeta_{6} q^{4}$$ $$+ ( 3 - 3 \zeta_{6} ) q^{5}$$ $$-9 q^{6}$$ $$+ ( -14 + 21 \zeta_{6} ) q^{7}$$ $$+ 21 q^{8}$$ $$+ ( -9 + 9 \zeta_{6} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 3 - 3 \zeta_{6} ) q^{2}$$ $$-3 \zeta_{6} q^{3}$$ $$-\zeta_{6} q^{4}$$ $$+ ( 3 - 3 \zeta_{6} ) q^{5}$$ $$-9 q^{6}$$ $$+ ( -14 + 21 \zeta_{6} ) q^{7}$$ $$+ 21 q^{8}$$ $$+ ( -9 + 9 \zeta_{6} ) q^{9}$$ $$-9 \zeta_{6} q^{10}$$ $$+ 15 \zeta_{6} q^{11}$$ $$+ ( -3 + 3 \zeta_{6} ) q^{12}$$ $$-64 q^{13}$$ $$+ ( 21 + 42 \zeta_{6} ) q^{14}$$ $$-9 q^{15}$$ $$+ ( 71 - 71 \zeta_{6} ) q^{16}$$ $$-84 \zeta_{6} q^{17}$$ $$+ 27 \zeta_{6} q^{18}$$ $$+ ( 16 - 16 \zeta_{6} ) q^{19}$$ $$-3 q^{20}$$ $$+ ( 63 - 21 \zeta_{6} ) q^{21}$$ $$+ 45 q^{22}$$ $$+ ( 84 - 84 \zeta_{6} ) q^{23}$$ $$-63 \zeta_{6} q^{24}$$ $$+ 116 \zeta_{6} q^{25}$$ $$+ ( -192 + 192 \zeta_{6} ) q^{26}$$ $$+ 27 q^{27}$$ $$+ ( 21 - 7 \zeta_{6} ) q^{28}$$ $$-297 q^{29}$$ $$+ ( -27 + 27 \zeta_{6} ) q^{30}$$ $$+ 253 \zeta_{6} q^{31}$$ $$-45 \zeta_{6} q^{32}$$ $$+ ( 45 - 45 \zeta_{6} ) q^{33}$$ $$-252 q^{34}$$ $$+ ( 21 + 42 \zeta_{6} ) q^{35}$$ $$+ 9 q^{36}$$ $$+ ( 316 - 316 \zeta_{6} ) q^{37}$$ $$-48 \zeta_{6} q^{38}$$ $$+ 192 \zeta_{6} q^{39}$$ $$+ ( 63 - 63 \zeta_{6} ) q^{40}$$ $$+ 360 q^{41}$$ $$+ ( 126 - 189 \zeta_{6} ) q^{42}$$ $$+ 26 q^{43}$$ $$+ ( 15 - 15 \zeta_{6} ) q^{44}$$ $$+ 27 \zeta_{6} q^{45}$$ $$-252 \zeta_{6} q^{46}$$ $$+ ( 30 - 30 \zeta_{6} ) q^{47}$$ $$-213 q^{48}$$ $$+ ( -245 - 147 \zeta_{6} ) q^{49}$$ $$+ 348 q^{50}$$ $$+ ( -252 + 252 \zeta_{6} ) q^{51}$$ $$+ 64 \zeta_{6} q^{52}$$ $$-363 \zeta_{6} q^{53}$$ $$+ ( 81 - 81 \zeta_{6} ) q^{54}$$ $$+ 45 q^{55}$$ $$+ ( -294 + 441 \zeta_{6} ) q^{56}$$ $$-48 q^{57}$$ $$+ ( -891 + 891 \zeta_{6} ) q^{58}$$ $$+ 15 \zeta_{6} q^{59}$$ $$+ 9 \zeta_{6} q^{60}$$ $$+ ( 118 - 118 \zeta_{6} ) q^{61}$$ $$+ 759 q^{62}$$ $$+ ( -63 - 126 \zeta_{6} ) q^{63}$$ $$+ 433 q^{64}$$ $$+ ( -192 + 192 \zeta_{6} ) q^{65}$$ $$-135 \zeta_{6} q^{66}$$ $$+ 370 \zeta_{6} q^{67}$$ $$+ ( -84 + 84 \zeta_{6} ) q^{68}$$ $$-252 q^{69}$$ $$+ ( 189 - 63 \zeta_{6} ) q^{70}$$ $$-342 q^{71}$$ $$+ ( -189 + 189 \zeta_{6} ) q^{72}$$ $$-362 \zeta_{6} q^{73}$$ $$-948 \zeta_{6} q^{74}$$ $$+ ( 348 - 348 \zeta_{6} ) q^{75}$$ $$-16 q^{76}$$ $$+ ( -315 + 105 \zeta_{6} ) q^{77}$$ $$+ 576 q^{78}$$ $$+ ( -467 + 467 \zeta_{6} ) q^{79}$$ $$-213 \zeta_{6} q^{80}$$ $$-81 \zeta_{6} q^{81}$$ $$+ ( 1080 - 1080 \zeta_{6} ) q^{82}$$ $$+ 477 q^{83}$$ $$+ ( -21 - 42 \zeta_{6} ) q^{84}$$ $$-252 q^{85}$$ $$+ ( 78 - 78 \zeta_{6} ) q^{86}$$ $$+ 891 \zeta_{6} q^{87}$$ $$+ 315 \zeta_{6} q^{88}$$ $$+ ( -906 + 906 \zeta_{6} ) q^{89}$$ $$+ 81 q^{90}$$ $$+ ( 896 - 1344 \zeta_{6} ) q^{91}$$ $$-84 q^{92}$$ $$+ ( 759 - 759 \zeta_{6} ) q^{93}$$ $$-90 \zeta_{6} q^{94}$$ $$-48 \zeta_{6} q^{95}$$ $$+ ( -135 + 135 \zeta_{6} ) q^{96}$$ $$+ 503 q^{97}$$ $$+ ( -1176 + 735 \zeta_{6} ) q^{98}$$ $$-135 q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut +\mathstrut 3q^{2}$$ $$\mathstrut -\mathstrut 3q^{3}$$ $$\mathstrut -\mathstrut q^{4}$$ $$\mathstrut +\mathstrut 3q^{5}$$ $$\mathstrut -\mathstrut 18q^{6}$$ $$\mathstrut -\mathstrut 7q^{7}$$ $$\mathstrut +\mathstrut 42q^{8}$$ $$\mathstrut -\mathstrut 9q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut +\mathstrut 3q^{2}$$ $$\mathstrut -\mathstrut 3q^{3}$$ $$\mathstrut -\mathstrut q^{4}$$ $$\mathstrut +\mathstrut 3q^{5}$$ $$\mathstrut -\mathstrut 18q^{6}$$ $$\mathstrut -\mathstrut 7q^{7}$$ $$\mathstrut +\mathstrut 42q^{8}$$ $$\mathstrut -\mathstrut 9q^{9}$$ $$\mathstrut -\mathstrut 9q^{10}$$ $$\mathstrut +\mathstrut 15q^{11}$$ $$\mathstrut -\mathstrut 3q^{12}$$ $$\mathstrut -\mathstrut 128q^{13}$$ $$\mathstrut +\mathstrut 84q^{14}$$ $$\mathstrut -\mathstrut 18q^{15}$$ $$\mathstrut +\mathstrut 71q^{16}$$ $$\mathstrut -\mathstrut 84q^{17}$$ $$\mathstrut +\mathstrut 27q^{18}$$ $$\mathstrut +\mathstrut 16q^{19}$$ $$\mathstrut -\mathstrut 6q^{20}$$ $$\mathstrut +\mathstrut 105q^{21}$$ $$\mathstrut +\mathstrut 90q^{22}$$ $$\mathstrut +\mathstrut 84q^{23}$$ $$\mathstrut -\mathstrut 63q^{24}$$ $$\mathstrut +\mathstrut 116q^{25}$$ $$\mathstrut -\mathstrut 192q^{26}$$ $$\mathstrut +\mathstrut 54q^{27}$$ $$\mathstrut +\mathstrut 35q^{28}$$ $$\mathstrut -\mathstrut 594q^{29}$$ $$\mathstrut -\mathstrut 27q^{30}$$ $$\mathstrut +\mathstrut 253q^{31}$$ $$\mathstrut -\mathstrut 45q^{32}$$ $$\mathstrut +\mathstrut 45q^{33}$$ $$\mathstrut -\mathstrut 504q^{34}$$ $$\mathstrut +\mathstrut 84q^{35}$$ $$\mathstrut +\mathstrut 18q^{36}$$ $$\mathstrut +\mathstrut 316q^{37}$$ $$\mathstrut -\mathstrut 48q^{38}$$ $$\mathstrut +\mathstrut 192q^{39}$$ $$\mathstrut +\mathstrut 63q^{40}$$ $$\mathstrut +\mathstrut 720q^{41}$$ $$\mathstrut +\mathstrut 63q^{42}$$ $$\mathstrut +\mathstrut 52q^{43}$$ $$\mathstrut +\mathstrut 15q^{44}$$ $$\mathstrut +\mathstrut 27q^{45}$$ $$\mathstrut -\mathstrut 252q^{46}$$ $$\mathstrut +\mathstrut 30q^{47}$$ $$\mathstrut -\mathstrut 426q^{48}$$ $$\mathstrut -\mathstrut 637q^{49}$$ $$\mathstrut +\mathstrut 696q^{50}$$ $$\mathstrut -\mathstrut 252q^{51}$$ $$\mathstrut +\mathstrut 64q^{52}$$ $$\mathstrut -\mathstrut 363q^{53}$$ $$\mathstrut +\mathstrut 81q^{54}$$ $$\mathstrut +\mathstrut 90q^{55}$$ $$\mathstrut -\mathstrut 147q^{56}$$ $$\mathstrut -\mathstrut 96q^{57}$$ $$\mathstrut -\mathstrut 891q^{58}$$ $$\mathstrut +\mathstrut 15q^{59}$$ $$\mathstrut +\mathstrut 9q^{60}$$ $$\mathstrut +\mathstrut 118q^{61}$$ $$\mathstrut +\mathstrut 1518q^{62}$$ $$\mathstrut -\mathstrut 252q^{63}$$ $$\mathstrut +\mathstrut 866q^{64}$$ $$\mathstrut -\mathstrut 192q^{65}$$ $$\mathstrut -\mathstrut 135q^{66}$$ $$\mathstrut +\mathstrut 370q^{67}$$ $$\mathstrut -\mathstrut 84q^{68}$$ $$\mathstrut -\mathstrut 504q^{69}$$ $$\mathstrut +\mathstrut 315q^{70}$$ $$\mathstrut -\mathstrut 684q^{71}$$ $$\mathstrut -\mathstrut 189q^{72}$$ $$\mathstrut -\mathstrut 362q^{73}$$ $$\mathstrut -\mathstrut 948q^{74}$$ $$\mathstrut +\mathstrut 348q^{75}$$ $$\mathstrut -\mathstrut 32q^{76}$$ $$\mathstrut -\mathstrut 525q^{77}$$ $$\mathstrut +\mathstrut 1152q^{78}$$ $$\mathstrut -\mathstrut 467q^{79}$$ $$\mathstrut -\mathstrut 213q^{80}$$ $$\mathstrut -\mathstrut 81q^{81}$$ $$\mathstrut +\mathstrut 1080q^{82}$$ $$\mathstrut +\mathstrut 954q^{83}$$ $$\mathstrut -\mathstrut 84q^{84}$$ $$\mathstrut -\mathstrut 504q^{85}$$ $$\mathstrut +\mathstrut 78q^{86}$$ $$\mathstrut +\mathstrut 891q^{87}$$ $$\mathstrut +\mathstrut 315q^{88}$$ $$\mathstrut -\mathstrut 906q^{89}$$ $$\mathstrut +\mathstrut 162q^{90}$$ $$\mathstrut +\mathstrut 448q^{91}$$ $$\mathstrut -\mathstrut 168q^{92}$$ $$\mathstrut +\mathstrut 759q^{93}$$ $$\mathstrut -\mathstrut 90q^{94}$$ $$\mathstrut -\mathstrut 48q^{95}$$ $$\mathstrut -\mathstrut 135q^{96}$$ $$\mathstrut +\mathstrut 1006q^{97}$$ $$\mathstrut -\mathstrut 1617q^{98}$$ $$\mathstrut -\mathstrut 270q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Character Values

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

 $$n$$ $$8$$ $$10$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{6}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.50000 + 2.59808i −1.50000 + 2.59808i −0.500000 + 0.866025i 1.50000 + 2.59808i −9.00000 −3.50000 18.1865i 21.0000 −4.50000 7.79423i −4.50000 + 7.79423i
16.1 1.50000 2.59808i −1.50000 2.59808i −0.500000 0.866025i 1.50000 2.59808i −9.00000 −3.50000 + 18.1865i 21.0000 −4.50000 + 7.79423i −4.50000 7.79423i
 $$n$$: 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
7.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{2}$$ $$\mathstrut -\mathstrut 3 T_{2}$$ $$\mathstrut +\mathstrut 9$$ acting on $$S_{4}^{\mathrm{new}}(21, [\chi])$$.