# Properties

 Label 11.4.c.a Level 11 Weight 4 Character orbit 11.c Analytic conductor 0.649 Analytic rank 0 Dimension 8 CM No Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$11$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 11.c (of order $$5$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.649021010063$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.29283765625.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{2}$$ $$+ ( -2 - 2 \beta_{2} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{3}$$ $$+ ( 3 - 2 \beta_{1} + 5 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{4}$$ $$+ ( -2 \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{5}$$ $$+ ( -2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} + 13 \beta_{6} - 2 \beta_{7} ) q^{6}$$ $$+ ( -6 + 3 \beta_{1} + 3 \beta_{5} - 6 \beta_{6} + 5 \beta_{7} ) q^{7}$$ $$+ ( -7 + 6 \beta_{1} - 7 \beta_{2} + 6 \beta_{3} - 26 \beta_{4} + 3 \beta_{5} - 26 \beta_{6} ) q^{8}$$ $$+ ( 3 \beta_{1} - 14 \beta_{2} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{2}$$ $$+ ( -2 - 2 \beta_{2} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{3}$$ $$+ ( 3 - 2 \beta_{1} + 5 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{4}$$ $$+ ( -2 \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{5}$$ $$+ ( -2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} + 13 \beta_{6} - 2 \beta_{7} ) q^{6}$$ $$+ ( -6 + 3 \beta_{1} + 3 \beta_{5} - 6 \beta_{6} + 5 \beta_{7} ) q^{7}$$ $$+ ( -7 + 6 \beta_{1} - 7 \beta_{2} + 6 \beta_{3} - 26 \beta_{4} + 3 \beta_{5} - 26 \beta_{6} ) q^{8}$$ $$+ ( 3 \beta_{1} - 14 \beta_{2} ) q^{9}$$ $$+ ( 16 + 28 \beta_{2} - 6 \beta_{3} + 28 \beta_{6} ) q^{10}$$ $$+ ( 25 - 4 \beta_{1} + 24 \beta_{2} - 7 \beta_{3} + 36 \beta_{4} - 9 \beta_{5} + 19 \beta_{6} - 10 \beta_{7} ) q^{11}$$ $$+ ( 15 - 24 \beta_{2} + 9 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} - 24 \beta_{6} + 5 \beta_{7} ) q^{12}$$ $$+ ( -7 - 4 \beta_{1} - \beta_{2} + 7 \beta_{3} - 14 \beta_{4} + 7 \beta_{7} ) q^{13}$$ $$+ ( -50 - 4 \beta_{1} - 50 \beta_{2} - 4 \beta_{3} - 22 \beta_{4} + 4 \beta_{5} - 22 \beta_{6} ) q^{14}$$ $$+ ( -25 + 2 \beta_{1} - 12 \beta_{4} + 2 \beta_{5} - 25 \beta_{6} - \beta_{7} ) q^{15}$$ $$+ ( -5 \beta_{1} + 75 \beta_{2} - 11 \beta_{3} + 80 \beta_{4} - 11 \beta_{5} + 50 \beta_{6} - 5 \beta_{7} ) q^{16}$$ $$+ ( 7 \beta_{1} - 18 \beta_{2} + 9 \beta_{3} - 25 \beta_{4} + 9 \beta_{5} + 45 \beta_{6} + 7 \beta_{7} ) q^{17}$$ $$+ ( -14 - 3 \beta_{1} + 13 \beta_{4} - 3 \beta_{5} - 14 \beta_{6} - 14 \beta_{7} ) q^{18}$$ $$+ ( -19 - 12 \beta_{1} - 19 \beta_{2} - 12 \beta_{3} - 58 \beta_{4} - 6 \beta_{5} - 58 \beta_{6} ) q^{19}$$ $$+ ( 48 - 42 \beta_{2} + 10 \beta_{3} + 38 \beta_{4} + 10 \beta_{7} ) q^{20}$$ $$+ ( 67 + 39 \beta_{2} + 7 \beta_{3} + 9 \beta_{4} - 9 \beta_{5} + 39 \beta_{6} - 9 \beta_{7} ) q^{21}$$ $$+ ( 82 + 19 \beta_{1} + 7 \beta_{2} + 25 \beta_{3} - 6 \beta_{4} + 29 \beta_{5} - 38 \beta_{6} + 31 \beta_{7} ) q^{22}$$ $$+ ( -4 + 26 \beta_{2} - 36 \beta_{3} + 10 \beta_{4} - 10 \beta_{5} + 26 \beta_{6} - 10 \beta_{7} ) q^{23}$$ $$+ ( -92 + 4 \beta_{1} - 13 \beta_{2} - 25 \beta_{3} - 67 \beta_{4} - 25 \beta_{7} ) q^{24}$$ $$+ ( -2 + 19 \beta_{1} - 2 \beta_{2} + 19 \beta_{3} + 30 \beta_{4} - 12 \beta_{5} + 30 \beta_{6} ) q^{25}$$ $$+ ( -50 + 4 \beta_{1} - 72 \beta_{4} + 4 \beta_{5} - 50 \beta_{6} - 8 \beta_{7} ) q^{26}$$ $$+ ( 38 \beta_{1} + 38 \beta_{3} - 38 \beta_{4} + 38 \beta_{5} - 55 \beta_{6} + 38 \beta_{7} ) q^{27}$$ $$+ ( -6 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} + 12 \beta_{4} - 6 \beta_{5} + 12 \beta_{6} - 6 \beta_{7} ) q^{28}$$ $$+ ( 12 - 31 \beta_{1} + 104 \beta_{4} - 31 \beta_{5} + 12 \beta_{6} + \beta_{7} ) q^{29}$$ $$+ ( 22 - 28 \beta_{1} + 22 \beta_{2} - 28 \beta_{3} + 78 \beta_{4} - 18 \beta_{5} + 78 \beta_{6} ) q^{30}$$ $$+ ( -89 - 40 \beta_{1} - 45 \beta_{2} - 41 \beta_{3} - 48 \beta_{4} - 41 \beta_{7} ) q^{31}$$ $$+ ( 39 - 27 \beta_{2} + 43 \beta_{3} - 38 \beta_{4} + 38 \beta_{5} - 27 \beta_{6} + 38 \beta_{7} ) q^{32}$$ $$+ ( -25 - 29 \beta_{1} - 68 \beta_{2} - 4 \beta_{3} + 52 \beta_{4} - 2 \beta_{5} + 25 \beta_{6} - \beta_{7} ) q^{33}$$ $$+ ( -18 - 9 \beta_{2} + 34 \beta_{3} + 27 \beta_{4} - 27 \beta_{5} - 9 \beta_{6} - 27 \beta_{7} ) q^{34}$$ $$+ ( 85 + 3 \beta_{1} + 132 \beta_{2} + 4 \beta_{3} + 81 \beta_{4} + 4 \beta_{7} ) q^{35}$$ $$+ ( 15 - 25 \beta_{1} + 15 \beta_{2} - 25 \beta_{3} + 3 \beta_{4} - 33 \beta_{5} + 3 \beta_{6} ) q^{36}$$ $$+ ( 8 + 35 \beta_{1} + 10 \beta_{4} + 35 \beta_{5} + 8 \beta_{6} + 17 \beta_{7} ) q^{37}$$ $$+ ( -7 \beta_{1} - 11 \beta_{2} - 40 \beta_{3} - 4 \beta_{4} - 40 \beta_{5} + 101 \beta_{6} - 7 \beta_{7} ) q^{38}$$ $$+ ( 5 \beta_{1} - 49 \beta_{2} + 5 \beta_{3} - 54 \beta_{4} + 5 \beta_{5} - 87 \beta_{6} + 5 \beta_{7} ) q^{39}$$ $$+ ( -4 + 58 \beta_{1} - 218 \beta_{4} + 58 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{40}$$ $$+ ( -26 + 62 \beta_{1} - 26 \beta_{2} + 62 \beta_{3} - 66 \beta_{4} + 47 \beta_{5} - 66 \beta_{6} ) q^{41}$$ $$+ ( 92 + 58 \beta_{1} + 172 \beta_{2} + 32 \beta_{3} + 60 \beta_{4} + 32 \beta_{7} ) q^{42}$$ $$+ ( -177 - 168 \beta_{2} - 39 \beta_{3} + 31 \beta_{4} - 31 \beta_{5} - 168 \beta_{6} - 31 \beta_{7} ) q^{43}$$ $$+ ( -225 + 14 \beta_{1} + 81 \beta_{2} - 69 \beta_{3} - 192 \beta_{4} - 40 \beta_{5} - 39 \beta_{6} - 42 \beta_{7} ) q^{44}$$ $$+ ( 96 + 95 \beta_{2} + 11 \beta_{3} - 54 \beta_{4} + 54 \beta_{5} + 95 \beta_{6} + 54 \beta_{7} ) q^{45}$$ $$+ ( 146 - 14 \beta_{1} - 264 \beta_{2} + 62 \beta_{3} + 84 \beta_{4} + 62 \beta_{7} ) q^{46}$$ $$+ ( 241 - 9 \beta_{1} + 241 \beta_{2} - 9 \beta_{3} + 161 \beta_{4} + 28 \beta_{5} + 161 \beta_{6} ) q^{47}$$ $$+ ( 204 - 81 \beta_{1} + 219 \beta_{4} - 81 \beta_{5} + 204 \beta_{6} - 20 \beta_{7} ) q^{48}$$ $$+ ( -112 \beta_{1} - 285 \beta_{2} - 43 \beta_{3} - 173 \beta_{4} - 43 \beta_{5} - 165 \beta_{6} - 112 \beta_{7} ) q^{49}$$ $$+ ( -21 \beta_{1} + 109 \beta_{2} + 23 \beta_{3} + 130 \beta_{4} + 23 \beta_{5} - 200 \beta_{6} - 21 \beta_{7} ) q^{50}$$ $$+ ( -54 + 20 \beta_{1} - 47 \beta_{4} + 20 \beta_{5} - 54 \beta_{6} + 56 \beta_{7} ) q^{51}$$ $$+ ( 144 - 6 \beta_{1} + 144 \beta_{2} - 6 \beta_{3} + 366 \beta_{4} - 8 \beta_{5} + 366 \beta_{6} ) q^{52}$$ $$+ ( -197 + 2 \beta_{1} + 95 \beta_{2} + 41 \beta_{3} - 238 \beta_{4} + 41 \beta_{7} ) q^{53}$$ $$+ ( -397 + 55 \beta_{2} - 93 \beta_{3} + 38 \beta_{4} - 38 \beta_{5} + 55 \beta_{6} - 38 \beta_{7} ) q^{54}$$ $$+ ( -140 - 4 \beta_{1} - 119 \beta_{2} + 59 \beta_{3} + 47 \beta_{4} - 53 \beta_{5} + 162 \beta_{6} - 10 \beta_{7} ) q^{55}$$ $$+ ( -334 - 242 \beta_{2} - 14 \beta_{3} - 44 \beta_{4} + 44 \beta_{5} - 242 \beta_{6} + 44 \beta_{7} ) q^{56}$$ $$+ ( 30 + 25 \beta_{1} + 92 \beta_{2} - 27 \beta_{3} + 57 \beta_{4} - 27 \beta_{7} ) q^{57}$$ $$+ ( -114 + 44 \beta_{1} - 114 \beta_{2} + 44 \beta_{3} - 510 \beta_{4} + 168 \beta_{5} - 510 \beta_{6} ) q^{58}$$ $$+ ( 169 - 32 \beta_{1} + 136 \beta_{4} - 32 \beta_{5} + 169 \beta_{6} + 2 \beta_{7} ) q^{59}$$ $$+ ( 42 \beta_{1} - 186 \beta_{2} + 100 \beta_{3} - 228 \beta_{4} + 100 \beta_{5} - 270 \beta_{6} + 42 \beta_{7} ) q^{60}$$ $$+ ( 42 \beta_{1} + 189 \beta_{2} - 105 \beta_{3} + 147 \beta_{4} - 105 \beta_{5} + 84 \beta_{6} + 42 \beta_{7} ) q^{61}$$ $$+ ( 502 - 90 \beta_{1} + 190 \beta_{4} - 90 \beta_{5} + 502 \beta_{6} - 4 \beta_{7} ) q^{62}$$ $$+ ( 19 + 33 \beta_{1} + 19 \beta_{2} + 33 \beta_{3} + 16 \beta_{4} - 19 \beta_{5} + 16 \beta_{6} ) q^{63}$$ $$+ ( 165 + 37 \beta_{1} + 311 \beta_{2} - 22 \beta_{3} + 187 \beta_{4} - 22 \beta_{7} ) q^{64}$$ $$+ ( 127 - 156 \beta_{2} - 44 \beta_{3} + 13 \beta_{4} - 13 \beta_{5} - 156 \beta_{6} - 13 \beta_{7} ) q^{65}$$ $$+ ( -60 + 3 \beta_{1} - 128 \beta_{2} + 30 \beta_{3} - 401 \beta_{4} + 81 \beta_{5} - 204 \beta_{6} - 64 \beta_{7} ) q^{66}$$ $$+ ( 204 + 309 \beta_{2} + 46 \beta_{3} + 63 \beta_{4} - 63 \beta_{5} + 309 \beta_{6} - 63 \beta_{7} ) q^{67}$$ $$+ ( 99 + 11 \beta_{1} + 36 \beta_{2} - 59 \beta_{3} + 158 \beta_{4} - 59 \beta_{7} ) q^{68}$$ $$+ ( 242 - 36 \beta_{1} + 242 \beta_{2} - 36 \beta_{3} + 384 \beta_{4} - 66 \beta_{5} + 384 \beta_{6} ) q^{69}$$ $$+ ( -74 + 86 \beta_{1} - 128 \beta_{4} + 86 \beta_{5} - 74 \beta_{6} + 128 \beta_{7} ) q^{70}$$ $$+ ( 120 \beta_{1} - 253 \beta_{2} + 43 \beta_{3} - 373 \beta_{4} + 43 \beta_{5} - 10 \beta_{6} + 120 \beta_{7} ) q^{71}$$ $$+ ( -72 \beta_{1} - 104 \beta_{2} - 27 \beta_{3} - 32 \beta_{4} - 27 \beta_{5} - 283 \beta_{6} - 72 \beta_{7} ) q^{72}$$ $$+ ( -480 + 44 \beta_{1} - 293 \beta_{4} + 44 \beta_{5} - 480 \beta_{6} - 51 \beta_{7} ) q^{73}$$ $$+ ( -180 - 10 \beta_{1} - 180 \beta_{2} - 10 \beta_{3} + 100 \beta_{4} - 26 \beta_{5} + 100 \beta_{6} ) q^{74}$$ $$+ ( -107 - 29 \beta_{1} - 275 \beta_{2} + 13 \beta_{3} - 120 \beta_{4} + 13 \beta_{7} ) q^{75}$$ $$+ ( 12 - 75 \beta_{2} + 222 \beta_{3} - 125 \beta_{4} + 125 \beta_{5} - 75 \beta_{6} + 125 \beta_{7} ) q^{76}$$ $$+ ( -336 + 19 \beta_{1} - 4 \beta_{2} - 52 \beta_{3} + 82 \beta_{4} + 95 \beta_{5} - 302 \beta_{6} + 97 \beta_{7} ) q^{77}$$ $$+ ( -132 + 136 \beta_{2} - 92 \beta_{3} + 54 \beta_{4} - 54 \beta_{5} + 136 \beta_{6} - 54 \beta_{7} ) q^{78}$$ $$+ ( -81 - 128 \beta_{1} + 143 \beta_{2} - 13 \beta_{3} - 68 \beta_{4} - 13 \beta_{7} ) q^{79}$$ $$+ ( -78 + 14 \beta_{1} - 78 \beta_{2} + 14 \beta_{3} + 286 \beta_{4} - 262 \beta_{5} + 286 \beta_{6} ) q^{80}$$ $$+ ( 100 \beta_{4} - 174 \beta_{7} ) q^{81}$$ $$+ ( -88 \beta_{1} + 664 \beta_{2} - 175 \beta_{3} + 752 \beta_{4} - 175 \beta_{5} + 595 \beta_{6} - 88 \beta_{7} ) q^{82}$$ $$+ ( -14 \beta_{1} - 283 \beta_{2} + 194 \beta_{3} - 269 \beta_{4} + 194 \beta_{5} + 235 \beta_{6} - 14 \beta_{7} ) q^{83}$$ $$+ ( 12 - 6 \beta_{1} + 66 \beta_{4} - 6 \beta_{5} + 12 \beta_{6} + 12 \beta_{7} ) q^{84}$$ $$+ ( 279 - 197 \beta_{1} + 279 \beta_{2} - 197 \beta_{3} - 27 \beta_{4} + 10 \beta_{5} - 27 \beta_{6} ) q^{85}$$ $$+ ( 120 - 208 \beta_{1} - 81 \beta_{2} - 129 \beta_{3} + 249 \beta_{4} - 129 \beta_{7} ) q^{86}$$ $$+ ( 195 - 303 \beta_{2} + 137 \beta_{3} - 43 \beta_{4} + 43 \beta_{5} - 303 \beta_{6} + 43 \beta_{7} ) q^{87}$$ $$+ ( 935 - 33 \beta_{1} + 165 \beta_{2} + 44 \beta_{3} + 561 \beta_{4} - 154 \beta_{5} + 462 \beta_{6} + 198 \beta_{7} ) q^{88}$$ $$+ ( 281 - 428 \beta_{2} + 57 \beta_{3} - 25 \beta_{4} + 25 \beta_{5} - 428 \beta_{6} + 25 \beta_{7} ) q^{89}$$ $$+ ( -392 + 150 \beta_{1} - 386 \beta_{2} + 84 \beta_{3} - 476 \beta_{4} + 84 \beta_{7} ) q^{90}$$ $$+ ( 44 + 8 \beta_{1} + 44 \beta_{2} + 8 \beta_{3} + 21 \beta_{4} - 21 \beta_{5} + 21 \beta_{6} ) q^{91}$$ $$+ ( -906 + 142 \beta_{1} - 1292 \beta_{4} + 142 \beta_{5} - 906 \beta_{6} - 118 \beta_{7} ) q^{92}$$ $$+ ( 85 \beta_{1} + 547 \beta_{2} - 45 \beta_{3} + 462 \beta_{4} - 45 \beta_{5} + 97 \beta_{6} + 85 \beta_{7} ) q^{93}$$ $$+ ( 250 \beta_{1} - 162 \beta_{2} + 142 \beta_{3} - 412 \beta_{4} + 142 \beta_{5} + 10 \beta_{6} + 250 \beta_{7} ) q^{94}$$ $$+ ( -113 + 60 \beta_{1} + 458 \beta_{4} + 60 \beta_{5} - 113 \beta_{6} - 157 \beta_{7} ) q^{95}$$ $$+ ( -123 + 65 \beta_{1} - 123 \beta_{2} + 65 \beta_{3} - 519 \beta_{4} + 109 \beta_{5} - 519 \beta_{6} ) q^{96}$$ $$+ ( -471 + 144 \beta_{1} - 594 \beta_{2} - 50 \beta_{3} - 421 \beta_{4} - 50 \beta_{7} ) q^{97}$$ $$+ ( 843 + 1071 \beta_{2} - 191 \beta_{3} + 242 \beta_{4} - 242 \beta_{5} + 1071 \beta_{6} - 242 \beta_{7} ) q^{98}$$ $$+ ( 368 + 41 \beta_{1} + 95 \beta_{2} - 96 \beta_{3} + 203 \beta_{4} + 95 \beta_{5} + 391 \beta_{6} - 13 \beta_{7} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q$$ $$\mathstrut -\mathstrut 7q^{2}$$ $$\mathstrut -\mathstrut 3q^{3}$$ $$\mathstrut +\mathstrut 3q^{4}$$ $$\mathstrut -\mathstrut 7q^{5}$$ $$\mathstrut -\mathstrut 29q^{6}$$ $$\mathstrut -\mathstrut 35q^{7}$$ $$\mathstrut +\mathstrut 47q^{8}$$ $$\mathstrut +\mathstrut 31q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$8q$$ $$\mathstrut -\mathstrut 7q^{2}$$ $$\mathstrut -\mathstrut 3q^{3}$$ $$\mathstrut +\mathstrut 3q^{4}$$ $$\mathstrut -\mathstrut 7q^{5}$$ $$\mathstrut -\mathstrut 29q^{6}$$ $$\mathstrut -\mathstrut 35q^{7}$$ $$\mathstrut +\mathstrut 47q^{8}$$ $$\mathstrut +\mathstrut 31q^{9}$$ $$\mathstrut +\mathstrut 40q^{10}$$ $$\mathstrut +\mathstrut 67q^{11}$$ $$\mathstrut +\mathstrut 190q^{12}$$ $$\mathstrut -\mathstrut 65q^{13}$$ $$\mathstrut -\mathstrut 196q^{14}$$ $$\mathstrut -\mathstrut 121q^{15}$$ $$\mathstrut -\mathstrut 377q^{16}$$ $$\mathstrut -\mathstrut 31q^{17}$$ $$\mathstrut -\mathstrut 102q^{18}$$ $$\mathstrut +\mathstrut 148q^{19}$$ $$\mathstrut +\mathstrut 342q^{20}$$ $$\mathstrut +\mathstrut 334q^{21}$$ $$\mathstrut +\mathstrut 647q^{22}$$ $$\mathstrut -\mathstrut 12q^{23}$$ $$\mathstrut -\mathstrut 447q^{24}$$ $$\mathstrut -\mathstrut 201q^{25}$$ $$\mathstrut -\mathstrut 140q^{26}$$ $$\mathstrut +\mathstrut 72q^{27}$$ $$\mathstrut -\mathstrut 42q^{28}$$ $$\mathstrut -\mathstrut 199q^{29}$$ $$\mathstrut -\mathstrut 114q^{30}$$ $$\mathstrut -\mathstrut 361q^{31}$$ $$\mathstrut +\mathstrut 324q^{32}$$ $$\mathstrut -\mathstrut 232q^{33}$$ $$\mathstrut -\mathstrut 298q^{34}$$ $$\mathstrut +\mathstrut 237q^{35}$$ $$\mathstrut +\mathstrut 120q^{36}$$ $$\mathstrut +\mathstrut 81q^{37}$$ $$\mathstrut -\mathstrut 52q^{38}$$ $$\mathstrut +\mathstrut 365q^{39}$$ $$\mathstrut +\mathstrut 532q^{40}$$ $$\mathstrut -\mathstrut 31q^{41}$$ $$\mathstrut +\mathstrut 170q^{42}$$ $$\mathstrut -\mathstrut 650q^{43}$$ $$\mathstrut -\mathstrut 1208q^{44}$$ $$\mathstrut +\mathstrut 452q^{45}$$ $$\mathstrut +\mathstrut 1204q^{46}$$ $$\mathstrut +\mathstrut 857q^{47}$$ $$\mathstrut +\mathstrut 644q^{48}$$ $$\mathstrut +\mathstrut 1375q^{49}$$ $$\mathstrut -\mathstrut 147q^{50}$$ $$\mathstrut -\mathstrut 246q^{51}$$ $$\mathstrut -\mathstrut 590q^{52}$$ $$\mathstrut -\mathstrut 1493q^{53}$$ $$\mathstrut -\mathstrut 3100q^{54}$$ $$\mathstrut -\mathstrut 1583q^{55}$$ $$\mathstrut -\mathstrut 1560q^{56}$$ $$\mathstrut +\mathstrut 102q^{57}$$ $$\mathstrut +\mathstrut 1392q^{58}$$ $$\mathstrut +\mathstrut 676q^{59}$$ $$\mathstrut +\mathstrut 1068q^{60}$$ $$\mathstrut -\mathstrut 525q^{61}$$ $$\mathstrut +\mathstrut 2456q^{62}$$ $$\mathstrut -\mathstrut 68q^{63}$$ $$\mathstrut +\mathstrut 471q^{64}$$ $$\mathstrut +\mathstrut 1790q^{65}$$ $$\mathstrut +\mathstrut 1014q^{66}$$ $$\mathstrut +\mathstrut 86q^{67}$$ $$\mathstrut +\mathstrut 710q^{68}$$ $$\mathstrut -\mathstrut 42q^{69}$$ $$\mathstrut -\mathstrut 144q^{70}$$ $$\mathstrut +\mathstrut 1143q^{71}$$ $$\mathstrut +\mathstrut 919q^{72}$$ $$\mathstrut -\mathstrut 2155q^{73}$$ $$\mathstrut -\mathstrut 1476q^{74}$$ $$\mathstrut -\mathstrut 160q^{75}$$ $$\mathstrut -\mathstrut 242q^{76}$$ $$\mathstrut -\mathstrut 2015q^{77}$$ $$\mathstrut -\mathstrut 1340q^{78}$$ $$\mathstrut -\mathstrut 861q^{79}$$ $$\mathstrut -\mathstrut 1916q^{80}$$ $$\mathstrut -\mathstrut 26q^{81}$$ $$\mathstrut -\mathstrut 3497q^{82}$$ $$\mathstrut +\mathstrut 52q^{83}$$ $$\mathstrut -\mathstrut 84q^{84}$$ $$\mathstrut +\mathstrut 2383q^{85}$$ $$\mathstrut +\mathstrut 1061q^{86}$$ $$\mathstrut +\mathstrut 2310q^{87}$$ $$\mathstrut +\mathstrut 4543q^{88}$$ $$\mathstrut +\mathstrut 3782q^{89}$$ $$\mathstrut -\mathstrut 1682q^{90}$$ $$\mathstrut +\mathstrut 135q^{91}$$ $$\mathstrut -\mathstrut 2450q^{92}$$ $$\mathstrut -\mathstrut 2077q^{93}$$ $$\mathstrut +\mathstrut 702q^{94}$$ $$\mathstrut -\mathstrut 1317q^{95}$$ $$\mathstrut +\mathstrut 1252q^{96}$$ $$\mathstrut -\mathstrut 1344q^{97}$$ $$\mathstrut +\mathstrut 2740q^{98}$$ $$\mathstrut +\mathstrut 2099q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8}\mathstrut -\mathstrut$$ $$x^{7}\mathstrut +\mathstrut$$ $$10$$ $$x^{6}\mathstrut -\mathstrut$$ $$19$$ $$x^{5}\mathstrut +\mathstrut$$ $$109$$ $$x^{4}\mathstrut +\mathstrut$$ $$171$$ $$x^{3}\mathstrut +\mathstrut$$ $$810$$ $$x^{2}\mathstrut +\mathstrut$$ $$729$$ $$x\mathstrut +\mathstrut$$ $$6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 280 \nu$$$$)/981$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 171$$$$)/109$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 1261 \nu^{2}$$$$)/8829$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - 10 \nu^{6} + 19 \nu^{5} - 109 \nu^{4} + 1090 \nu^{3} - 810 \nu^{2} - 729 \nu - 6561$$$$)/8829$$ $$\beta_{6}$$ $$=$$ $$($$$$-19 \nu^{7} + 19 \nu^{6} - 190 \nu^{5} + 1090 \nu^{4} - 2071 \nu^{3} - 3249 \nu^{2} - 15390 \nu - 13851$$$$)/79461$$ $$\beta_{7}$$ $$=$$ $$($$$$10 \nu^{7} + 3781 \nu^{2}$$$$)/8829$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-$$$$\beta_{7}\mathstrut +\mathstrut$$ $$10$$ $$\beta_{4}$$ $$\nu^{3}$$ $$=$$ $$9$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$10$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$9$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$9$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$9$$ $$\nu^{4}$$ $$=$$ $$19$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$90$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$19$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$19$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$19$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$19$$ $$\beta_{1}$$ $$\nu^{5}$$ $$=$$ $$109$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$171$$ $$\nu^{6}$$ $$=$$ $$981$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$280$$ $$\beta_{1}$$ $$\nu^{7}$$ $$=$$ $$1261$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$3781$$ $$\beta_{4}$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/11\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 −2.05602 + 1.49379i 2.86504 − 2.08157i −2.05602 − 1.49379i 2.86504 + 2.08157i −1.09435 − 3.36805i 0.785330 + 2.41700i −1.09435 + 3.36805i 0.785330 − 2.41700i
−4.17405 + 3.03263i 1.40336 + 4.31911i 5.75376 17.7083i 6.98613 + 5.07572i −18.9560 13.7723i 0.513765 1.58121i 16.9313 + 52.1091i 5.15818 3.74763i −44.5532
3.2 0.747004 0.542730i −0.476313 1.46594i −2.20868 + 6.79761i −7.05908 5.12872i −1.15142 0.836554i 0.239524 0.737179i 4.32202 + 13.3018i 19.9214 14.4737i −8.05666
4.1 −4.17405 3.03263i 1.40336 4.31911i 5.75376 + 17.7083i 6.98613 5.07572i −18.9560 + 13.7723i 0.513765 + 1.58121i 16.9313 52.1091i 5.15818 + 3.74763i −44.5532
4.2 0.747004 + 0.542730i −0.476313 + 1.46594i −2.20868 6.79761i −7.05908 + 5.12872i −1.15142 + 0.836554i 0.239524 + 0.737179i 4.32202 13.3018i 19.9214 + 14.4737i −8.05666
5.1 −0.976313 3.00478i 1.24700 + 0.906001i −1.60339 + 1.16493i −5.33576 + 16.4218i 1.50487 4.63152i 7.73807 5.62204i −15.3824 11.1760i −7.60928 23.4190i 54.5532
5.2 0.903364 + 2.78027i −3.67405 2.66936i −0.441690 + 0.320907i 1.90871 5.87440i 4.10252 12.6263i −25.9914 + 18.8838i 17.6291 + 12.8083i −1.97025 6.06380i 18.0567
9.1 −0.976313 + 3.00478i 1.24700 0.906001i −1.60339 1.16493i −5.33576 16.4218i 1.50487 + 4.63152i 7.73807 + 5.62204i −15.3824 + 11.1760i −7.60928 + 23.4190i 54.5532
9.2 0.903364 2.78027i −3.67405 + 2.66936i −0.441690 0.320907i 1.90871 + 5.87440i 4.10252 + 12.6263i −25.9914 18.8838i 17.6291 12.8083i −1.97025 + 6.06380i 18.0567
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 9.2 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
11.c Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{4}^{\mathrm{new}}(11, [\chi])$$.