# Properties

 Label 21.4.e.b Level 21 Weight 4 Character orbit 21.e Analytic conductor 1.239 Analytic rank 0 Dimension 6 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.9924270768.1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$-\beta_{1} q^{2}$$ $$+ ( 3 - 3 \beta_{4} ) q^{3}$$ $$+ ( -8 + \beta_{1} + \beta_{2} + 8 \beta_{4} + \beta_{5} ) q^{4}$$ $$+ ( \beta_{1} + \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{5}$$ $$+ 3 \beta_{2} q^{6}$$ $$+ ( -4 - 4 \beta_{1} - 3 \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{7}$$ $$+ ( 10 - 9 \beta_{2} - \beta_{3} ) q^{8}$$ $$-9 \beta_{4} q^{9}$$ $$+O(q^{10})$$ $$q$$ $$-\beta_{1} q^{2}$$ $$+ ( 3 - 3 \beta_{4} ) q^{3}$$ $$+ ( -8 + \beta_{1} + \beta_{2} + 8 \beta_{4} + \beta_{5} ) q^{4}$$ $$+ ( \beta_{1} + \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{5}$$ $$+ 3 \beta_{2} q^{6}$$ $$+ ( -4 - 4 \beta_{1} - 3 \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{7}$$ $$+ ( 10 - 9 \beta_{2} - \beta_{3} ) q^{8}$$ $$-9 \beta_{4} q^{9}$$ $$+ ( 22 + 11 \beta_{1} + 11 \beta_{2} - 22 \beta_{4} - \beta_{5} ) q^{10}$$ $$+ ( -12 - \beta_{1} - \beta_{2} + 12 \beta_{4} - 3 \beta_{5} ) q^{11}$$ $$+ ( 3 \beta_{1} - 3 \beta_{3} + 24 \beta_{4} + 3 \beta_{5} ) q^{12}$$ $$+ ( 19 - 5 \beta_{2} + \beta_{3} ) q^{13}$$ $$+ ( -64 - 6 \beta_{1} + \beta_{2} + 3 \beta_{3} + 22 \beta_{4} + \beta_{5} ) q^{14}$$ $$+ ( -12 - 3 \beta_{2} + 3 \beta_{3} ) q^{15}$$ $$+ ( -19 \beta_{1} + \beta_{3} - 74 \beta_{4} - \beta_{5} ) q^{16}$$ $$+ ( -16 + 16 \beta_{4} + 4 \beta_{5} ) q^{17}$$ $$+ ( 9 \beta_{1} + 9 \beta_{2} ) q^{18}$$ $$+ ( 7 \beta_{1} - \beta_{3} + 65 \beta_{4} + \beta_{5} ) q^{19}$$ $$+ ( 150 + 11 \beta_{2} - 3 \beta_{3} ) q^{20}$$ $$+ ( -3 - 9 \beta_{1} + 3 \beta_{2} + 12 \beta_{4} - 3 \beta_{5} ) q^{21}$$ $$+ ( 2 + 13 \beta_{2} + \beta_{3} ) q^{22}$$ $$+ ( 24 \beta_{1} - 4 \beta_{3} - 80 \beta_{4} + 4 \beta_{5} ) q^{23}$$ $$+ ( 30 - 27 \beta_{1} - 27 \beta_{2} - 30 \beta_{4} - 3 \beta_{5} ) q^{24}$$ $$+ ( -53 - 29 \beta_{1} - 29 \beta_{2} + 53 \beta_{4} + \beta_{5} ) q^{25}$$ $$+ ( -16 \beta_{1} + 5 \beta_{3} - 86 \beta_{4} - 5 \beta_{5} ) q^{26}$$ $$-27 q^{27}$$ $$+ ( -110 + 49 \beta_{1} - 16 \beta_{2} - \beta_{3} + 126 \beta_{4} - \beta_{5} ) q^{28}$$ $$+ ( 26 + 25 \beta_{2} - 5 \beta_{3} ) q^{29}$$ $$+ ( 33 \beta_{1} + 3 \beta_{3} - 66 \beta_{4} - 3 \beta_{5} ) q^{30}$$ $$+ ( 39 + 22 \beta_{1} + 22 \beta_{2} - 39 \beta_{4} - 2 \beta_{5} ) q^{31}$$ $$+ ( -218 + 29 \beta_{1} + 29 \beta_{2} + 218 \beta_{4} + 11 \beta_{5} ) q^{32}$$ $$+ ( -3 \beta_{1} + 9 \beta_{3} + 36 \beta_{4} - 9 \beta_{5} ) q^{33}$$ $$+ ( -24 - 48 \beta_{2} ) q^{34}$$ $$+ ( 100 + 24 \beta_{1} + 47 \beta_{2} - 11 \beta_{3} - 158 \beta_{4} + 4 \beta_{5} ) q^{35}$$ $$+ ( 72 - 9 \beta_{2} - 9 \beta_{3} ) q^{36}$$ $$+ ( -19 \beta_{1} + \beta_{3} - 81 \beta_{4} - \beta_{5} ) q^{37}$$ $$+ ( 106 - 80 \beta_{1} - 80 \beta_{2} - 106 \beta_{4} - 7 \beta_{5} ) q^{38}$$ $$+ ( 57 - 15 \beta_{1} - 15 \beta_{2} - 57 \beta_{4} + 3 \beta_{5} ) q^{39}$$ $$+ ( -75 \beta_{1} - 3 \beta_{3} + 18 \beta_{4} + 3 \beta_{5} ) q^{40}$$ $$+ ( 82 + 2 \beta_{2} + 14 \beta_{3} ) q^{41}$$ $$+ ( -126 + 3 \beta_{1} + 21 \beta_{2} - 3 \beta_{3} + 192 \beta_{4} + 12 \beta_{5} ) q^{42}$$ $$+ ( 143 + 69 \beta_{2} + 3 \beta_{3} ) q^{43}$$ $$+ ( 11 \beta_{1} + 11 \beta_{3} + 298 \beta_{4} - 11 \beta_{5} ) q^{44}$$ $$+ ( -36 - 9 \beta_{1} - 9 \beta_{2} + 36 \beta_{4} + 9 \beta_{5} ) q^{45}$$ $$+ ( 360 + 24 \beta_{1} + 24 \beta_{2} - 360 \beta_{4} - 24 \beta_{5} ) q^{46}$$ $$+ ( 72 \beta_{1} - 28 \beta_{3} + 46 \beta_{4} + 28 \beta_{5} ) q^{47}$$ $$+ ( -222 + 57 \beta_{2} + 3 \beta_{3} ) q^{48}$$ $$+ ( -7 - 46 \beta_{1} - 35 \beta_{2} + 25 \beta_{3} - 95 \beta_{4} - 2 \beta_{5} ) q^{49}$$ $$+ ( -470 - 32 \beta_{2} + 29 \beta_{3} ) q^{50}$$ $$+ ( -12 \beta_{3} + 48 \beta_{4} + 12 \beta_{5} ) q^{51}$$ $$+ ( -74 + 102 \beta_{1} + 102 \beta_{2} + 74 \beta_{4} + 24 \beta_{5} ) q^{52}$$ $$+ ( -154 - 69 \beta_{1} - 69 \beta_{2} + 154 \beta_{4} - 11 \beta_{5} ) q^{53}$$ $$+ 27 \beta_{1} q^{54}$$ $$+ ( -350 - 19 \beta_{2} - 25 \beta_{3} ) q^{55}$$ $$+ ( 454 - 81 \beta_{1} - 111 \beta_{2} + 8 \beta_{3} - 522 \beta_{4} - 33 \beta_{5} ) q^{56}$$ $$+ ( 195 - 21 \beta_{2} - 3 \beta_{3} ) q^{57}$$ $$+ ( -41 \beta_{1} - 25 \beta_{3} + 430 \beta_{4} + 25 \beta_{5} ) q^{58}$$ $$+ ( -358 + 69 \beta_{1} + 69 \beta_{2} + 358 \beta_{4} - 29 \beta_{5} ) q^{59}$$ $$+ ( 450 + 33 \beta_{1} + 33 \beta_{2} - 450 \beta_{4} - 9 \beta_{5} ) q^{60}$$ $$+ ( 100 \beta_{1} + 20 \beta_{3} - 10 \beta_{4} - 20 \beta_{5} ) q^{61}$$ $$+ ( 364 + 33 \beta_{2} - 22 \beta_{3} ) q^{62}$$ $$+ ( 27 + 9 \beta_{1} + 36 \beta_{2} + 9 \beta_{3} + 9 \beta_{4} - 9 \beta_{5} ) q^{63}$$ $$+ ( -194 - 183 \beta_{2} - 21 \beta_{3} ) q^{64}$$ $$+ ( -50 \beta_{1} + 18 \beta_{3} + 174 \beta_{4} - 18 \beta_{5} ) q^{65}$$ $$+ ( 6 + 39 \beta_{1} + 39 \beta_{2} - 6 \beta_{4} + 3 \beta_{5} ) q^{66}$$ $$+ ( 215 + 17 \beta_{1} + 17 \beta_{2} - 215 \beta_{4} + 47 \beta_{5} ) q^{67}$$ $$+ ( -24 \beta_{1} + 16 \beta_{3} - 640 \beta_{4} - 16 \beta_{5} ) q^{68}$$ $$+ ( -240 - 72 \beta_{2} - 12 \beta_{3} ) q^{69}$$ $$+ ( 360 - 7 \beta_{1} + 102 \beta_{2} - 47 \beta_{3} + 434 \beta_{4} + 23 \beta_{5} ) q^{70}$$ $$+ ( 66 - 120 \beta_{2} - 12 \beta_{3} ) q^{71}$$ $$+ ( -81 \beta_{1} + 9 \beta_{3} - 90 \beta_{4} - 9 \beta_{5} ) q^{72}$$ $$+ ( -363 - 101 \beta_{1} - 101 \beta_{2} + 363 \beta_{4} - 23 \beta_{5} ) q^{73}$$ $$+ ( -298 + 108 \beta_{1} + 108 \beta_{2} + 298 \beta_{4} + 19 \beta_{5} ) q^{74}$$ $$+ ( -87 \beta_{1} - 3 \beta_{3} + 159 \beta_{4} + 3 \beta_{5} ) q^{75}$$ $$+ ( -718 + 186 \beta_{2} + 72 \beta_{3} ) q^{76}$$ $$+ ( 410 - 25 \beta_{1} + 24 \beta_{2} - 5 \beta_{3} - 438 \beta_{4} + 45 \beta_{5} ) q^{77}$$ $$+ ( -258 + 48 \beta_{2} + 15 \beta_{3} ) q^{78}$$ $$+ ( 36 \beta_{1} + 48 \beta_{3} - 299 \beta_{4} - 48 \beta_{5} ) q^{79}$$ $$+ ( -18 + 121 \beta_{1} + 121 \beta_{2} + 18 \beta_{4} + 51 \beta_{5} ) q^{80}$$ $$+ ( -81 + 81 \beta_{4} ) q^{81}$$ $$+ ( 32 \beta_{1} - 2 \beta_{3} - 52 \beta_{4} + 2 \beta_{5} ) q^{82}$$ $$+ ( 156 - 51 \beta_{2} + 27 \beta_{3} ) q^{83}$$ $$+ ( 48 - 48 \beta_{1} - 195 \beta_{2} + 3 \beta_{3} + 330 \beta_{4} - 6 \beta_{5} ) q^{84}$$ $$+ ( 624 + 72 \beta_{2} ) q^{85}$$ $$+ ( -50 \beta_{1} - 69 \beta_{3} + 1086 \beta_{4} + 69 \beta_{5} ) q^{86}$$ $$+ ( 78 + 75 \beta_{1} + 75 \beta_{2} - 78 \beta_{4} - 15 \beta_{5} ) q^{87}$$ $$+ ( 258 - 117 \beta_{1} - 117 \beta_{2} - 258 \beta_{4} - 3 \beta_{5} ) q^{88}$$ $$+ ( -170 \beta_{1} + 22 \beta_{3} - 532 \beta_{4} - 22 \beta_{5} ) q^{89}$$ $$+ ( -198 - 99 \beta_{2} + 9 \beta_{3} ) q^{90}$$ $$+ ( 18 - 49 \beta_{1} - 74 \beta_{2} - 23 \beta_{3} - 287 \beta_{4} - 23 \beta_{5} ) q^{91}$$ $$+ ( -112 + 336 \beta_{2} - 56 \beta_{3} ) q^{92}$$ $$+ ( 66 \beta_{1} + 6 \beta_{3} - 117 \beta_{4} - 6 \beta_{5} ) q^{93}$$ $$+ ( 984 - 342 \beta_{1} - 342 \beta_{2} - 984 \beta_{4} - 72 \beta_{5} ) q^{94}$$ $$+ ( 246 + 2 \beta_{1} + 2 \beta_{2} - 246 \beta_{4} - 54 \beta_{5} ) q^{95}$$ $$+ ( 87 \beta_{1} - 33 \beta_{3} + 654 \beta_{4} + 33 \beta_{5} ) q^{96}$$ $$+ ( 24 + 53 \beta_{2} - 49 \beta_{3} ) q^{97}$$ $$+ ( -724 + 313 \beta_{1} + 157 \beta_{2} + 35 \beta_{3} + 26 \beta_{4} + 11 \beta_{5} ) q^{98}$$ $$+ ( 108 + 9 \beta_{2} + 27 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q$$ $$\mathstrut -\mathstrut q^{2}$$ $$\mathstrut +\mathstrut 9q^{3}$$ $$\mathstrut -\mathstrut 25q^{4}$$ $$\mathstrut -\mathstrut 11q^{5}$$ $$\mathstrut -\mathstrut 6q^{6}$$ $$\mathstrut -\mathstrut 13q^{7}$$ $$\mathstrut +\mathstrut 78q^{8}$$ $$\mathstrut -\mathstrut 27q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$6q$$ $$\mathstrut -\mathstrut q^{2}$$ $$\mathstrut +\mathstrut 9q^{3}$$ $$\mathstrut -\mathstrut 25q^{4}$$ $$\mathstrut -\mathstrut 11q^{5}$$ $$\mathstrut -\mathstrut 6q^{6}$$ $$\mathstrut -\mathstrut 13q^{7}$$ $$\mathstrut +\mathstrut 78q^{8}$$ $$\mathstrut -\mathstrut 27q^{9}$$ $$\mathstrut +\mathstrut 55q^{10}$$ $$\mathstrut -\mathstrut 35q^{11}$$ $$\mathstrut +\mathstrut 75q^{12}$$ $$\mathstrut +\mathstrut 124q^{13}$$ $$\mathstrut -\mathstrut 326q^{14}$$ $$\mathstrut -\mathstrut 66q^{15}$$ $$\mathstrut -\mathstrut 241q^{16}$$ $$\mathstrut -\mathstrut 48q^{17}$$ $$\mathstrut -\mathstrut 9q^{18}$$ $$\mathstrut +\mathstrut 202q^{19}$$ $$\mathstrut +\mathstrut 878q^{20}$$ $$\mathstrut +\mathstrut 3q^{21}$$ $$\mathstrut -\mathstrut 14q^{22}$$ $$\mathstrut -\mathstrut 216q^{23}$$ $$\mathstrut +\mathstrut 117q^{24}$$ $$\mathstrut -\mathstrut 130q^{25}$$ $$\mathstrut -\mathstrut 274q^{26}$$ $$\mathstrut -\mathstrut 162q^{27}$$ $$\mathstrut -\mathstrut 201q^{28}$$ $$\mathstrut +\mathstrut 106q^{29}$$ $$\mathstrut -\mathstrut 165q^{30}$$ $$\mathstrut +\mathstrut 95q^{31}$$ $$\mathstrut -\mathstrut 683q^{32}$$ $$\mathstrut +\mathstrut 105q^{33}$$ $$\mathstrut -\mathstrut 48q^{34}$$ $$\mathstrut +\mathstrut 56q^{35}$$ $$\mathstrut +\mathstrut 450q^{36}$$ $$\mathstrut -\mathstrut 262q^{37}$$ $$\mathstrut +\mathstrut 398q^{38}$$ $$\mathstrut +\mathstrut 186q^{39}$$ $$\mathstrut -\mathstrut 21q^{40}$$ $$\mathstrut +\mathstrut 488q^{41}$$ $$\mathstrut -\mathstrut 219q^{42}$$ $$\mathstrut +\mathstrut 720q^{43}$$ $$\mathstrut +\mathstrut 905q^{44}$$ $$\mathstrut -\mathstrut 99q^{45}$$ $$\mathstrut +\mathstrut 1056q^{46}$$ $$\mathstrut +\mathstrut 210q^{47}$$ $$\mathstrut -\mathstrut 1446q^{48}$$ $$\mathstrut -\mathstrut 303q^{49}$$ $$\mathstrut -\mathstrut 2756q^{50}$$ $$\mathstrut +\mathstrut 144q^{51}$$ $$\mathstrut -\mathstrut 324q^{52}$$ $$\mathstrut -\mathstrut 393q^{53}$$ $$\mathstrut +\mathstrut 27q^{54}$$ $$\mathstrut -\mathstrut 2062q^{55}$$ $$\mathstrut +\mathstrut 1299q^{56}$$ $$\mathstrut +\mathstrut 1212q^{57}$$ $$\mathstrut +\mathstrut 1249q^{58}$$ $$\mathstrut -\mathstrut 1143q^{59}$$ $$\mathstrut +\mathstrut 1317q^{60}$$ $$\mathstrut +\mathstrut 70q^{61}$$ $$\mathstrut +\mathstrut 2118q^{62}$$ $$\mathstrut +\mathstrut 126q^{63}$$ $$\mathstrut -\mathstrut 798q^{64}$$ $$\mathstrut +\mathstrut 472q^{65}$$ $$\mathstrut -\mathstrut 21q^{66}$$ $$\mathstrut +\mathstrut 628q^{67}$$ $$\mathstrut -\mathstrut 1944q^{68}$$ $$\mathstrut -\mathstrut 1296q^{69}$$ $$\mathstrut +\mathstrut 3251q^{70}$$ $$\mathstrut +\mathstrut 636q^{71}$$ $$\mathstrut -\mathstrut 351q^{72}$$ $$\mathstrut -\mathstrut 988q^{73}$$ $$\mathstrut -\mathstrut 1002q^{74}$$ $$\mathstrut +\mathstrut 390q^{75}$$ $$\mathstrut -\mathstrut 4680q^{76}$$ $$\mathstrut +\mathstrut 1073q^{77}$$ $$\mathstrut -\mathstrut 1644q^{78}$$ $$\mathstrut -\mathstrut 861q^{79}$$ $$\mathstrut -\mathstrut 175q^{80}$$ $$\mathstrut -\mathstrut 243q^{81}$$ $$\mathstrut -\mathstrut 124q^{82}$$ $$\mathstrut +\mathstrut 1038q^{83}$$ $$\mathstrut +\mathstrut 1620q^{84}$$ $$\mathstrut +\mathstrut 3600q^{85}$$ $$\mathstrut +\mathstrut 3208q^{86}$$ $$\mathstrut +\mathstrut 159q^{87}$$ $$\mathstrut +\mathstrut 891q^{88}$$ $$\mathstrut -\mathstrut 1766q^{89}$$ $$\mathstrut -\mathstrut 990q^{90}$$ $$\mathstrut -\mathstrut 654q^{91}$$ $$\mathstrut -\mathstrut 1344q^{92}$$ $$\mathstrut -\mathstrut 285q^{93}$$ $$\mathstrut +\mathstrut 3294q^{94}$$ $$\mathstrut +\mathstrut 736q^{95}$$ $$\mathstrut +\mathstrut 2049q^{96}$$ $$\mathstrut +\mathstrut 38q^{97}$$ $$\mathstrut -\mathstrut 4267q^{98}$$ $$\mathstrut +\mathstrut 630q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6}\mathstrut -\mathstrut$$ $$x^{5}\mathstrut +\mathstrut$$ $$25$$ $$x^{4}\mathstrut +\mathstrut$$ $$12$$ $$x^{3}\mathstrut +\mathstrut$$ $$582$$ $$x^{2}\mathstrut -\mathstrut$$ $$144$$ $$x\mathstrut +\mathstrut$$ $$36$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - 25 \nu^{4} + 625 \nu^{3} - 582 \nu^{2} + 144 \nu - 3600$$$$)/14406$$ $$\beta_{3}$$ $$=$$ $$($$$$-25 \nu^{5} + 625 \nu^{4} - 1219 \nu^{3} + 14550 \nu^{2} - 3600 \nu + 234060$$$$)/14406$$ $$\beta_{4}$$ $$=$$ $$($$$$100 \nu^{5} - 99 \nu^{4} + 2475 \nu^{3} + 1825 \nu^{2} + 57618 \nu + 150$$$$)/14406$$ $$\beta_{5}$$ $$=$$ $$($$$$-1601 \nu^{5} + 1609 \nu^{4} - 40225 \nu^{3} - 14212 \nu^{2} - 936438 \nu + 231696$$$$)/14406$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$16$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$16$$ $$\nu^{3}$$ $$=$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$25$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$10$$ $$\nu^{4}$$ $$=$$ $$-$$$$25$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$394$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$25$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$43$$ $$\beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-$$$$43$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$538$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$637$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$637$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$538$$

## Character Values

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

 $$n$$ $$8$$ $$10$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$

## 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
 2.65415 + 4.59712i 0.124036 + 0.214837i −2.27818 − 3.94593i 2.65415 − 4.59712i 0.124036 − 0.214837i −2.27818 + 3.94593i
−2.65415 4.59712i 1.50000 2.59808i −10.0890 + 17.4746i −2.78070 4.81631i −15.9249 9.67799 15.7904i 64.6443 −4.50000 7.79423i −14.7608 + 25.5664i
4.2 −0.124036 0.214837i 1.50000 2.59808i 3.96923 6.87491i 6.21730 + 10.7687i −0.744216 −18.4385 + 1.73873i −3.95388 −4.50000 7.79423i 1.54234 2.67141i
4.3 2.27818 + 3.94593i 1.50000 2.59808i −6.38024 + 11.0509i −8.93660 15.4786i 13.6691 2.26047 + 18.3818i −21.6905 −4.50000 7.79423i 40.7184 70.5264i
16.1 −2.65415 + 4.59712i 1.50000 + 2.59808i −10.0890 17.4746i −2.78070 + 4.81631i −15.9249 9.67799 + 15.7904i 64.6443 −4.50000 + 7.79423i −14.7608 25.5664i
16.2 −0.124036 + 0.214837i 1.50000 + 2.59808i 3.96923 + 6.87491i 6.21730 10.7687i −0.744216 −18.4385 1.73873i −3.95388 −4.50000 + 7.79423i 1.54234 + 2.67141i
16.3 2.27818 3.94593i 1.50000 + 2.59808i −6.38024 11.0509i −8.93660 + 15.4786i 13.6691 2.26047 18.3818i −21.6905 −4.50000 + 7.79423i 40.7184 + 70.5264i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 16.3 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 $$T_{2}^{6}$$ $$\mathstrut +\mathstrut T_{2}^{5}$$ $$\mathstrut +\mathstrut 25 T_{2}^{4}$$ $$\mathstrut -\mathstrut 12 T_{2}^{3}$$ $$\mathstrut +\mathstrut 582 T_{2}^{2}$$ $$\mathstrut +\mathstrut 144 T_{2}$$ $$\mathstrut +\mathstrut 36$$ acting on $$S_{4}^{\mathrm{new}}(21, [\chi])$$.