# Properties

 Label 21.5.b.a Level $21$ Weight $5$ Character orbit 21.b Analytic conductor $2.171$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 21.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17076922476$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 82 x^{6} + 2017 x^{4} + 13020 x^{2} + 756$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 3^{3}\cdot 7^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{3} q^{3} + ( -5 + \beta_{2} ) q^{4} -\beta_{6} q^{5} + ( -5 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{6} + \beta_{4} q^{7} + ( -1 - \beta_{1} + 2 \beta_{3} - \beta_{5} + \beta_{7} ) q^{8} + ( 9 + 5 \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{3} q^{3} + ( -5 + \beta_{2} ) q^{4} -\beta_{6} q^{5} + ( -5 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{6} + \beta_{4} q^{7} + ( -1 - \beta_{1} + 2 \beta_{3} - \beta_{5} + \beta_{7} ) q^{8} + ( 9 + 5 \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{9} + ( 1 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{10} + ( 1 + 2 \beta_{1} + 2 \beta_{3} + \beta_{5} - 2 \beta_{6} - 5 \beta_{7} ) q^{11} + ( 13 - 9 \beta_{1} - \beta_{2} + 2 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{12} + ( 57 + \beta_{1} - 10 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} + \beta_{5} ) q^{13} + ( 1 - 3 \beta_{1} - 4 \beta_{3} + \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{14} + ( 11 - 17 \beta_{1} - 7 \beta_{2} - 2 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{15} + ( -61 + 2 \beta_{1} + 9 \beta_{2} + 6 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} ) q^{16} + ( -5 - 19 \beta_{1} + 9 \beta_{3} - 5 \beta_{5} + 11 \beta_{6} + 6 \beta_{7} ) q^{17} + ( -96 + 28 \beta_{1} + 13 \beta_{2} + 6 \beta_{3} - 10 \beta_{4} - \beta_{5} - 8 \beta_{6} - 5 \beta_{7} ) q^{18} + ( -55 + 7 \beta_{1} + 10 \beta_{2} + 21 \beta_{3} - 2 \beta_{4} + 7 \beta_{5} ) q^{19} + ( 5 + 7 \beta_{1} - 23 \beta_{3} + 5 \beta_{5} + 2 \beta_{6} + 8 \beta_{7} ) q^{20} + ( 16 + 15 \beta_{1} - 7 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{21} + ( 2 - 16 \beta_{1} + 8 \beta_{2} - 48 \beta_{3} + 12 \beta_{4} - 16 \beta_{5} ) q^{22} + ( 7 - 46 \beta_{1} - 16 \beta_{3} + 7 \beta_{5} - 22 \beta_{6} - 5 \beta_{7} ) q^{23} + ( 138 + 24 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 9 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} ) q^{24} + ( 137 - 2 \beta_{1} - 8 \beta_{2} - 6 \beta_{3} - 12 \beta_{4} - 2 \beta_{5} ) q^{25} + ( 17 + 165 \beta_{1} - 41 \beta_{3} + 17 \beta_{5} + 12 \beta_{6} - 10 \beta_{7} ) q^{26} + ( -243 - 25 \beta_{1} + 32 \beta_{2} - 6 \beta_{3} + 10 \beta_{4} + 13 \beta_{5} + 8 \beta_{6} - 4 \beta_{7} ) q^{27} + ( 42 + 7 \beta_{1} + 7 \beta_{2} + 21 \beta_{3} - 4 \beta_{4} + 7 \beta_{5} ) q^{28} + ( -30 + 54 \beta_{1} + 92 \beta_{3} - 30 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{29} + ( 308 + 93 \beta_{1} - 29 \beta_{2} + 8 \beta_{4} + 7 \beta_{5} + 10 \beta_{6} + 7 \beta_{7} ) q^{30} + ( -330 - 16 \beta_{1} - 18 \beta_{2} - 48 \beta_{3} - 14 \beta_{4} - 16 \beta_{5} ) q^{31} + ( -19 - 185 \beta_{1} + 40 \beta_{3} - 19 \beta_{5} + 17 \beta_{7} ) q^{32} + ( 191 - 215 \beta_{1} - 16 \beta_{2} - 15 \beta_{3} + 34 \beta_{4} + 21 \beta_{5} + 11 \beta_{6} + 2 \beta_{7} ) q^{33} + ( 368 + 24 \beta_{1} - 28 \beta_{2} + 72 \beta_{3} - 26 \beta_{4} + 24 \beta_{5} ) q^{34} + ( 6 - 67 \beta_{1} - 17 \beta_{3} + 6 \beta_{5} - 10 \beta_{6} - \beta_{7} ) q^{35} + ( -411 - 167 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + 14 \beta_{4} - 31 \beta_{5} + 4 \beta_{6} - 11 \beta_{7} ) q^{36} + ( -332 + 18 \beta_{1} + 6 \beta_{2} + 54 \beta_{3} - 44 \beta_{4} + 18 \beta_{5} ) q^{37} + ( -5 - 127 \beta_{1} + 49 \beta_{3} - 5 \beta_{5} - 48 \beta_{6} - 34 \beta_{7} ) q^{38} + ( -304 + 165 \beta_{1} - 29 \beta_{2} - 32 \beta_{3} + 62 \beta_{4} - 11 \beta_{5} + 28 \beta_{6} + 25 \beta_{7} ) q^{39} + ( -253 + 15 \beta_{1} - 4 \beta_{2} + 45 \beta_{3} + 10 \beta_{4} + 15 \beta_{5} ) q^{40} + ( 23 + 209 \beta_{1} - 23 \beta_{3} + 23 \beta_{5} - 23 \beta_{6} - 46 \beta_{7} ) q^{41} + ( -307 + 109 \beta_{1} + 14 \beta_{2} + 3 \beta_{3} - 5 \beta_{4} + 15 \beta_{5} - 4 \beta_{6} - 13 \beta_{7} ) q^{42} + ( 592 - 30 \beta_{1} + 26 \beta_{2} - 90 \beta_{3} - 80 \beta_{4} - 30 \beta_{5} ) q^{43} + ( 4 - 194 \beta_{1} - 48 \beta_{3} + 4 \beta_{5} + 100 \beta_{6} + 36 \beta_{7} ) q^{44} + ( 1194 + 146 \beta_{1} + 26 \beta_{2} + 12 \beta_{3} + 34 \beta_{4} - 20 \beta_{5} + 11 \beta_{6} - 10 \beta_{7} ) q^{45} + ( 982 - 30 \beta_{1} - 50 \beta_{2} - 90 \beta_{3} + 56 \beta_{4} - 30 \beta_{5} ) q^{46} + ( 2 + 368 \beta_{1} - 68 \beta_{3} + 2 \beta_{5} - 24 \beta_{6} + 62 \beta_{7} ) q^{47} + ( -359 - 51 \beta_{1} - 31 \beta_{2} + 32 \beta_{3} - 29 \beta_{4} + 20 \beta_{5} - 52 \beta_{6} + 5 \beta_{7} ) q^{48} + 343 q^{49} + ( -6 + 257 \beta_{1} + 26 \beta_{3} - 6 \beta_{5} - 24 \beta_{6} - 8 \beta_{7} ) q^{50} + ( 561 + 360 \beta_{1} + 78 \beta_{2} + 18 \beta_{3} + 12 \beta_{4} + 15 \beta_{5} - 66 \beta_{6} - 21 \beta_{7} ) q^{51} + ( -2601 + 15 \beta_{1} + 124 \beta_{2} + 45 \beta_{3} - 34 \beta_{4} + 15 \beta_{5} ) q^{52} + ( 30 - 266 \beta_{1} - 120 \beta_{3} + 30 \beta_{5} + 80 \beta_{6} + 30 \beta_{7} ) q^{53} + ( 576 - 632 \beta_{1} + 31 \beta_{2} + 51 \beta_{3} - 73 \beta_{4} - 13 \beta_{5} + \beta_{6} + 13 \beta_{7} ) q^{54} + ( -54 - 16 \beta_{1} + 170 \beta_{2} - 48 \beta_{3} - 22 \beta_{4} - 16 \beta_{5} ) q^{55} + ( 12 - 36 \beta_{1} - 13 \beta_{3} + 12 \beta_{5} - 6 \beta_{6} - 23 \beta_{7} ) q^{56} + ( -1792 - 225 \beta_{1} + 25 \beta_{2} + 34 \beta_{3} - 10 \beta_{4} - 5 \beta_{5} - 80 \beta_{6} + 7 \beta_{7} ) q^{57} + ( -876 - 34 \beta_{1} - 90 \beta_{2} - 102 \beta_{3} + 176 \beta_{4} - 34 \beta_{5} ) q^{58} + ( -21 - 147 \beta_{1} + 17 \beta_{3} - 21 \beta_{5} + 48 \beta_{6} + 46 \beta_{7} ) q^{59} + ( -1736 + 419 \beta_{1} + \beta_{2} + 24 \beta_{3} - 100 \beta_{4} - 21 \beta_{5} + 28 \beta_{6} - 17 \beta_{7} ) q^{60} + ( 191 - \beta_{1} - 138 \beta_{2} - 3 \beta_{3} - 22 \beta_{4} - \beta_{5} ) q^{61} + ( -12 - 168 \beta_{1} - 28 \beta_{3} - 12 \beta_{5} + 54 \beta_{6} + 64 \beta_{7} ) q^{62} + ( -780 + 121 \beta_{1} + 7 \beta_{2} + 12 \beta_{3} + 5 \beta_{4} - 31 \beta_{5} + 40 \beta_{6} + 25 \beta_{7} ) q^{63} + ( 2891 + 64 \beta_{1} - 153 \beta_{2} + 192 \beta_{3} + 144 \beta_{4} + 64 \beta_{5} ) q^{64} + ( 6 - 386 \beta_{1} + 80 \beta_{3} + 6 \beta_{5} - 98 \beta_{6} - 98 \beta_{7} ) q^{65} + ( 4616 + 348 \beta_{1} - 140 \beta_{2} - 54 \beta_{3} - 124 \beta_{4} + 88 \beta_{5} + 52 \beta_{6} - 32 \beta_{7} ) q^{66} + ( 1142 + 80 \beta_{1} + 186 \beta_{2} + 240 \beta_{3} - 4 \beta_{4} + 80 \beta_{5} ) q^{67} + ( -54 + 622 \beta_{1} + 264 \beta_{3} - 54 \beta_{5} - 46 \beta_{6} - 102 \beta_{7} ) q^{68} + ( -615 - 411 \beta_{1} - 228 \beta_{2} - 15 \beta_{3} + 12 \beta_{4} - 81 \beta_{5} + 33 \beta_{6} - 30 \beta_{7} ) q^{69} + ( 1379 - 7 \beta_{1} - 56 \beta_{2} - 21 \beta_{3} + 6 \beta_{4} - 7 \beta_{5} ) q^{70} + ( 89 + 56 \beta_{1} - 264 \beta_{3} + 89 \beta_{5} + 66 \beta_{6} - 3 \beta_{7} ) q^{71} + ( 1791 - 120 \beta_{1} + 15 \beta_{2} - 144 \beta_{3} - 78 \beta_{4} - 60 \beta_{5} + 24 \beta_{6} + 42 \beta_{7} ) q^{72} + ( -3868 - 6 \beta_{1} - 260 \beta_{2} - 18 \beta_{3} + 180 \beta_{4} - 6 \beta_{5} ) q^{73} + ( -32 - 164 \beta_{1} + 242 \beta_{3} - 32 \beta_{5} - 240 \beta_{6} - 146 \beta_{7} ) q^{74} + ( 244 - 78 \beta_{1} + 86 \beta_{2} - 103 \beta_{3} + 16 \beta_{4} - 34 \beta_{5} + 56 \beta_{6} - 22 \beta_{7} ) q^{75} + ( 2215 - 43 \beta_{1} - 54 \beta_{2} - 129 \beta_{3} + 258 \beta_{4} - 43 \beta_{5} ) q^{76} + ( -67 - 289 \beta_{1} + 135 \beta_{3} - 67 \beta_{5} + 37 \beta_{6} + 66 \beta_{7} ) q^{77} + ( -3764 - 79 \beta_{1} + 181 \beta_{2} - 30 \beta_{3} - 136 \beta_{4} + 183 \beta_{5} + 226 \beta_{6} + 73 \beta_{7} ) q^{78} + ( -2912 + 8 \beta_{1} + 24 \beta_{2} + 24 \beta_{3} - 276 \beta_{4} + 8 \beta_{5} ) q^{79} + ( 109 - 33 \beta_{1} - 371 \beta_{3} + 109 \beta_{5} - 28 \beta_{6} + 44 \beta_{7} ) q^{80} + ( -2079 - 70 \beta_{1} - 94 \beta_{2} + 138 \beta_{3} - 80 \beta_{4} + 166 \beta_{5} - 172 \beta_{6} - 76 \beta_{7} ) q^{81} + ( -4090 - 138 \beta_{1} + 324 \beta_{2} - 414 \beta_{3} + 46 \beta_{4} - 138 \beta_{5} ) q^{82} + ( -171 + 397 \beta_{1} + 445 \beta_{3} - 171 \beta_{5} + 20 \beta_{6} + 68 \beta_{7} ) q^{83} + ( -1863 - 228 \beta_{1} + 42 \beta_{2} - 52 \beta_{3} + 3 \beta_{4} - 15 \beta_{5} - 66 \beta_{6} + 6 \beta_{7} ) q^{84} + ( 3922 + 110 \beta_{1} + 50 \beta_{2} + 330 \beta_{3} + 124 \beta_{4} + 110 \beta_{5} ) q^{85} + ( -136 + 340 \beta_{1} + 282 \beta_{3} - 136 \beta_{5} - 60 \beta_{6} + 126 \beta_{7} ) q^{86} + ( 6194 - 578 \beta_{1} + 188 \beta_{2} - 6 \beta_{3} + 118 \beta_{4} - 66 \beta_{5} - 100 \beta_{6} + 170 \beta_{7} ) q^{87} + ( 3614 - 44 \beta_{1} + 118 \beta_{2} - 132 \beta_{3} - 304 \beta_{4} - 44 \beta_{5} ) q^{88} + ( 1 - 529 \beta_{1} + 67 \beta_{3} + \beta_{5} + 139 \beta_{6} - 70 \beta_{7} ) q^{89} + ( -3099 + 703 \beta_{1} + 136 \beta_{2} - 201 \beta_{3} + 38 \beta_{4} - 31 \beta_{5} + 160 \beta_{6} + 118 \beta_{7} ) q^{90} + ( 1547 - 77 \beta_{1} - 28 \beta_{2} - 231 \beta_{3} + 40 \beta_{4} - 77 \beta_{5} ) q^{91} + ( 188 + 498 \beta_{1} - 670 \beta_{3} + 188 \beta_{5} - 4 \beta_{6} + 106 \beta_{7} ) q^{92} + ( 3862 + 192 \beta_{1} + 68 \beta_{2} + 382 \beta_{3} - 20 \beta_{4} - 16 \beta_{5} + 182 \beta_{6} - 58 \beta_{7} ) q^{93} + ( -8340 + 164 \beta_{1} + 268 \beta_{2} + 492 \beta_{3} - 40 \beta_{4} + 164 \beta_{5} ) q^{94} + ( 178 + 806 \beta_{1} - 532 \beta_{3} + 178 \beta_{5} - 32 \beta_{6} - 2 \beta_{7} ) q^{95} + ( 3608 + 544 \beta_{1} - 109 \beta_{2} + 51 \beta_{3} + 127 \beta_{4} - 159 \beta_{5} - 280 \beta_{6} - 109 \beta_{7} ) q^{96} + ( 3662 - 160 \beta_{1} - 136 \beta_{2} - 480 \beta_{3} - 328 \beta_{4} - 160 \beta_{5} ) q^{97} + 343 \beta_{1} q^{98} + ( -3747 + 1220 \beta_{1} - 394 \beta_{2} - 150 \beta_{3} + 328 \beta_{4} + 103 \beta_{5} - 202 \beta_{6} - 187 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{3} - 36q^{4} - 34q^{6} + 64q^{9} + O(q^{10})$$ $$8q - 2q^{3} - 36q^{4} - 34q^{6} + 64q^{9} - 4q^{10} + 98q^{12} + 420q^{13} + 76q^{15} - 444q^{16} - 712q^{18} - 372q^{19} + 98q^{21} - 16q^{22} + 1146q^{24} + 1056q^{25} - 1862q^{27} + 392q^{28} + 2348q^{30} - 2776q^{31} + 1396q^{33} + 2928q^{34} - 3268q^{36} - 2560q^{37} - 2540q^{39} - 1980q^{40} - 2450q^{42} + 4720q^{43} + 9700q^{45} + 7536q^{46} - 2962q^{48} + 2744q^{49} + 4764q^{51} - 20252q^{52} + 4886q^{54} + 184q^{55} - 14144q^{57} - 7504q^{58} - 13828q^{60} + 972q^{61} - 6076q^{63} + 22772q^{64} + 36020q^{66} + 10200q^{67} - 5760q^{69} + 10780q^{70} + 14304q^{72} - 32008q^{73} + 2114q^{75} + 17332q^{76} - 29668q^{78} - 23168q^{79} - 17216q^{81} - 31976q^{82} - 14798q^{84} + 32016q^{85} + 50764q^{87} + 29208q^{88} - 24352q^{90} + 11956q^{91} + 31848q^{93} - 64992q^{94} + 28630q^{96} + 28112q^{97} - 32432q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 82 x^{6} + 2017 x^{4} + 13020 x^{2} + 756$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 21$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} + 4 \nu^{5} - 44 \nu^{4} + 188 \nu^{3} - 285 \nu^{2} + 1560 \nu + 198$$$$)/48$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + 48 \nu^{4} + 441 \nu^{2} + 322$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} - 4 \nu^{5} - 44 \nu^{4} - 188 \nu^{3} - 285 \nu^{2} - 1576 \nu + 182$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} + 64 \nu^{5} + 1177 \nu^{3} + 6114 \nu$$$$)/48$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{6} - 20 \nu^{5} - 44 \nu^{4} - 892 \nu^{3} - 285 \nu^{2} - 6264 \nu + 198$$$$)/48$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 21$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{5} + 2 \beta_{3} - 33 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} + 4 \beta_{4} + 6 \beta_{3} - 39 \beta_{2} + 2 \beta_{1} + 691$$ $$\nu^{5}$$ $$=$$ $$-47 \beta_{7} + 45 \beta_{5} - 88 \beta_{3} + 1159 \beta_{1} + 45$$ $$\nu^{6}$$ $$=$$ $$-96 \beta_{5} - 176 \beta_{4} - 288 \beta_{3} + 1431 \beta_{2} - 96 \beta_{1} - 24229$$ $$\nu^{7}$$ $$=$$ $$1831 \beta_{7} + 48 \beta_{6} - 1703 \beta_{5} + 3278 \beta_{3} - 41449 \beta_{1} - 1703$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1
 − 6.02741i − 5.97075i − 3.15624i − 0.242064i 0.242064i 3.15624i 5.97075i 6.02741i
6.02741i −8.92230 + 1.18010i −20.3296 15.7540i 7.11292 + 53.7783i −18.5203 26.0965i 78.2147 21.0583i −94.9556
8.2 5.97075i 5.29264 7.27928i −19.6499 22.9681i −43.4628 31.6011i 18.5203 21.7927i −24.9759 77.0533i 137.137
8.3 3.15624i 7.09942 + 5.53157i 6.03813 16.3361i 17.4590 22.4075i −18.5203 69.5577i 19.8035 + 78.5418i −51.5608
8.4 0.242064i −4.46977 + 7.81161i 15.9414 30.4863i 1.89091 + 1.08197i 18.5203 7.73187i −41.0424 69.8321i 7.37963
8.5 0.242064i −4.46977 7.81161i 15.9414 30.4863i 1.89091 1.08197i 18.5203 7.73187i −41.0424 + 69.8321i 7.37963
8.6 3.15624i 7.09942 5.53157i 6.03813 16.3361i 17.4590 + 22.4075i −18.5203 69.5577i 19.8035 78.5418i −51.5608
8.7 5.97075i 5.29264 + 7.27928i −19.6499 22.9681i −43.4628 + 31.6011i 18.5203 21.7927i −24.9759 + 77.0533i 137.137
8.8 6.02741i −8.92230 1.18010i −20.3296 15.7540i 7.11292 53.7783i −18.5203 26.0965i 78.2147 + 21.0583i −94.9556
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 8.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.5.b.a 8
3.b odd 2 1 inner 21.5.b.a 8
4.b odd 2 1 336.5.d.b 8
7.b odd 2 1 147.5.b.e 8
7.c even 3 2 147.5.h.e 16
7.d odd 6 2 147.5.h.c 16
12.b even 2 1 336.5.d.b 8
21.c even 2 1 147.5.b.e 8
21.g even 6 2 147.5.h.c 16
21.h odd 6 2 147.5.h.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.5.b.a 8 1.a even 1 1 trivial
21.5.b.a 8 3.b odd 2 1 inner
147.5.b.e 8 7.b odd 2 1
147.5.b.e 8 21.c even 2 1
147.5.h.c 16 7.d odd 6 2
147.5.h.c 16 21.g even 6 2
147.5.h.e 16 7.c even 3 2
147.5.h.e 16 21.h odd 6 2
336.5.d.b 8 4.b odd 2 1
336.5.d.b 8 12.b even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$756 + 13020 T^{2} + 2017 T^{4} + 82 T^{6} + T^{8}$$
$3$ $$43046721 + 1062882 T - 196830 T^{2} + 45198 T^{3} + 5994 T^{4} + 558 T^{5} - 30 T^{6} + 2 T^{7} + T^{8}$$
$5$ $$32473916496 + 349027728 T^{2} + 1306936 T^{4} + 1972 T^{6} + T^{8}$$
$7$ $$( -343 + T^{2} )^{4}$$
$11$ $$107333491578594624 + 29564999292192 T^{2} + 2662526836 T^{4} + 89596 T^{6} + T^{8}$$
$13$ $$( 137696008 + 8164128 T - 50954 T^{2} - 210 T^{3} + T^{4} )^{2}$$
$17$ $$3367331079306820416 + 681767060496480 T^{2} + 28993077684 T^{4} + 319812 T^{6} + T^{8}$$
$19$ $$( 5829791872 - 9364416 T - 178502 T^{2} + 186 T^{3} + T^{4} )^{2}$$
$23$ $$26\!\cdots\!16$$$$+ 19272018742800768 T^{2} + 335291481540 T^{4} + 1126788 T^{6} + T^{8}$$
$29$ $$12\!\cdots\!44$$$$+ 5385753101577510144 T^{2} + 8254967116816 T^{4} + 5004664 T^{6} + T^{8}$$
$31$ $$( 74477967264 - 377920464 T - 301172 T^{2} + 1388 T^{3} + T^{4} )^{2}$$
$37$ $$( -319722971328 - 1867503648 T - 1920764 T^{2} + 1280 T^{3} + T^{4} )^{2}$$
$41$ $$25\!\cdots\!96$$$$+ 6015668960189441952 T^{2} + 21100860583348 T^{4} + 10616116 T^{6} + T^{8}$$
$43$ $$( 117096670336 - 702886496 T - 4617372 T^{2} - 2360 T^{3} + T^{4} )^{2}$$
$47$ $$34\!\cdots\!96$$$$+$$$$14\!\cdots\!08$$$$T^{2} + 375494394157632 T^{4} + 33700560 T^{6} + T^{8}$$
$53$ $$15\!\cdots\!96$$$$+$$$$35\!\cdots\!40$$$$T^{2} + 178583957915472 T^{4} + 26879352 T^{6} + T^{8}$$
$59$ $$86\!\cdots\!84$$$$+ 7496402363091258624 T^{2} + 17499432670164 T^{4} + 10355208 T^{6} + T^{8}$$
$61$ $$( 13501614097288 + 7213168944 T - 10406090 T^{2} - 486 T^{3} + T^{4} )^{2}$$
$67$ $$( 16869420316288 + 72161588832 T - 24163004 T^{2} - 5100 T^{3} + T^{4} )^{2}$$
$71$ $$62\!\cdots\!44$$$$+$$$$40\!\cdots\!96$$$$T^{2} + 815263906771188 T^{4} + 51778188 T^{6} + T^{8}$$
$73$ $$( -737570944880496 - 191818436976 T + 48874024 T^{2} + 16004 T^{3} + T^{4} )^{2}$$
$79$ $$( 301006877602048 - 199933316096 T - 1730784 T^{2} + 11584 T^{3} + T^{4} )^{2}$$
$83$ $$66\!\cdots\!64$$$$+$$$$32\!\cdots\!92$$$$T^{2} + 4599564869824596 T^{4} + 138447288 T^{6} + T^{8}$$
$89$ $$39\!\cdots\!56$$$$+$$$$24\!\cdots\!72$$$$T^{2} + 3014292955806196 T^{4} + 100361812 T^{6} + T^{8}$$
$97$ $$( -2352511907880048 + 1080834544608 T - 78456008 T^{2} - 14056 T^{3} + T^{4} )^{2}$$