# Properties

 Label 2100.2.bi.m Level 2100 Weight 2 Character orbit 2100.bi Analytic conductor 16.769 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.bi (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 3 x^{15} + 9 x^{14} - 18 x^{13} + 32 x^{12} - 36 x^{11} + 51 x^{9} - 167 x^{8} + 153 x^{7} - 972 x^{5} + 2592 x^{4} - 4374 x^{3} + 6561 x^{2} - 6561 x + 6561$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( 1 - \beta_{7} - \beta_{10} ) q^{7} + ( -\beta_{10} - \beta_{13} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + ( 1 - \beta_{7} - \beta_{10} ) q^{7} + ( -\beta_{10} - \beta_{13} ) q^{9} + ( \beta_{2} - \beta_{5} - \beta_{8} - \beta_{14} + \beta_{15} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{10} ) q^{13} + ( \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{14} - \beta_{15} ) q^{17} + ( -2 + \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{9} + \beta_{10} ) q^{19} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{9} + \beta_{12} + \beta_{13} - \beta_{15} ) q^{21} + ( -\beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{13} ) q^{23} + ( \beta_{1} + \beta_{5} - \beta_{9} - \beta_{14} + \beta_{15} ) q^{27} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} - 2 \beta_{12} - 2 \beta_{13} ) q^{29} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{31} + ( 2 - 3 \beta_{1} - 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} + 3 \beta_{9} - \beta_{10} + 2 \beta_{12} + \beta_{13} ) q^{33} + ( \beta_{1} - 6 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{9} + \beta_{10} ) q^{37} + ( 2 + \beta_{2} - 2 \beta_{5} - 2 \beta_{10} - \beta_{12} ) q^{39} + ( 2 \beta_{3} + 2 \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{41} + ( \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{43} + ( -2 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{11} + 4 \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{47} + ( 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - \beta_{7} - 3 \beta_{9} - 3 \beta_{10} ) q^{49} + ( 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{13} - \beta_{14} ) q^{51} + ( -2 \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} + 2 \beta_{12} - \beta_{14} + \beta_{15} ) q^{53} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{57} + ( \beta_{1} + \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{12} - 4 \beta_{13} + \beta_{14} + \beta_{15} ) q^{59} + ( 2 + 2 \beta_{1} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{9} - \beta_{10} ) q^{61} + ( 2 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{6} + \beta_{9} - 2 \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{15} ) q^{63} + ( 3 + 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{9} - 3 \beta_{10} ) q^{67} + ( -4 \beta_{1} + 3 \beta_{3} - \beta_{5} + \beta_{9} + \beta_{14} - \beta_{15} ) q^{69} + ( -\beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{71} + ( 3 + \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{9} + 3 \beta_{10} ) q^{73} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{11} + 2 \beta_{13} - 3 \beta_{14} + \beta_{15} ) q^{77} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{9} + \beta_{10} ) q^{79} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + 3 \beta_{6} + \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{15} ) q^{81} + ( 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{83} + ( 4 - \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + 4 \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{87} + ( -\beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{11} + 4 \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{89} + ( 1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + 3 \beta_{10} ) q^{91} + ( -\beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} - 5 \beta_{10} + 3 \beta_{13} - \beta_{14} ) q^{93} + ( -1 + \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} ) q^{97} + ( -4 + 4 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 3q^{3} + 6q^{7} - 9q^{9} + O(q^{10})$$ $$16q + 3q^{3} + 6q^{7} - 9q^{9} - 18q^{19} - 11q^{21} - 18q^{31} + 12q^{33} - 6q^{37} + 12q^{39} + 4q^{43} - 18q^{49} - q^{51} + 6q^{57} + 36q^{61} + 19q^{63} + 30q^{67} + 54q^{73} + 7q^{81} + 81q^{87} + 20q^{91} - 34q^{93} - 30q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 3 x^{15} + 9 x^{14} - 18 x^{13} + 32 x^{12} - 36 x^{11} + 51 x^{9} - 167 x^{8} + 153 x^{7} - 972 x^{5} + 2592 x^{4} - 4374 x^{3} + 6561 x^{2} - 6561 x + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{15} + 3 \nu^{14} - 9 \nu^{13} + 18 \nu^{12} - 32 \nu^{11} + 36 \nu^{10} - 51 \nu^{8} + 167 \nu^{7} - 153 \nu^{6} + 972 \nu^{4} - 2592 \nu^{3} + 4374 \nu^{2} - 6561 \nu + 6561$$$$)/2187$$ $$\beta_{3}$$ $$=$$ $$($$$$1153 \nu^{15} + 6454 \nu^{14} - 29220 \nu^{13} + 76824 \nu^{12} - 121504 \nu^{11} + 151832 \nu^{10} - 105324 \nu^{9} - 339726 \nu^{8} + 938494 \nu^{7} - 1296236 \nu^{6} - 263352 \nu^{5} + 3207744 \nu^{4} - 9118332 \nu^{3} + 19216764 \nu^{2} - 24411051 \nu + 20670066$$$$)/609687$$ $$\beta_{4}$$ $$=$$ $$($$$$-5395 \nu^{15} - 4206 \nu^{14} + 45069 \nu^{13} - 149400 \nu^{12} + 258388 \nu^{11} - 359337 \nu^{10} + 407982 \nu^{9} + 437691 \nu^{8} - 1965202 \nu^{7} + 3585486 \nu^{6} - 681303 \nu^{5} - 4226967 \nu^{4} + 15893982 \nu^{3} - 39695913 \nu^{2} + 58844394 \nu - 54877662$$$$)/1829061$$ $$\beta_{5}$$ $$=$$ $$($$$$-5968 \nu^{15} + 46258 \nu^{14} - 149151 \nu^{13} + 304524 \nu^{12} - 438368 \nu^{11} + 430760 \nu^{10} + 72792 \nu^{9} - 1670856 \nu^{8} + 3390626 \nu^{7} - 2590688 \nu^{6} - 4108284 \nu^{5} + 17467020 \nu^{4} - 40658976 \nu^{3} + 70727904 \nu^{2} - 81111456 \nu + 52235766$$$$)/1829061$$ $$\beta_{6}$$ $$=$$ $$($$$$6954 \nu^{15} - 71512 \nu^{14} + 241188 \nu^{13} - 534429 \nu^{12} + 786063 \nu^{11} - 848087 \nu^{10} + 73005 \nu^{9} + 2652570 \nu^{8} - 6035418 \nu^{7} + 5956964 \nu^{6} + 5733957 \nu^{5} - 28159650 \nu^{4} + 69652251 \nu^{3} - 124258779 \nu^{2} + 151849728 \nu - 108434376$$$$)/1829061$$ $$\beta_{7}$$ $$=$$ $$($$$$23263 \nu^{15} + 951 \nu^{14} - 143820 \nu^{13} + 498699 \nu^{12} - 887410 \nu^{11} + 1306899 \nu^{10} - 1588383 \nu^{9} - 1318053 \nu^{8} + 7041979 \nu^{7} - 12576744 \nu^{6} + 4191318 \nu^{5} + 13815873 \nu^{4} - 52760484 \nu^{3} + 135325242 \nu^{2} - 202887261 \nu + 185919057$$$$)/5487183$$ $$\beta_{8}$$ $$=$$ $$($$$$24626 \nu^{15} + 32013 \nu^{14} - 242181 \nu^{13} + 767331 \nu^{12} - 1365137 \nu^{11} + 1838910 \nu^{10} - 2035269 \nu^{9} - 2572404 \nu^{8} + 10069919 \nu^{7} - 17596443 \nu^{6} + 3420909 \nu^{5} + 26531901 \nu^{4} - 85116744 \nu^{3} + 211208067 \nu^{2} - 293889789 \nu + 276613947$$$$)/5487183$$ $$\beta_{9}$$ $$=$$ $$($$$$-9427 \nu^{15} + 56618 \nu^{14} - 170922 \nu^{13} + 329391 \nu^{12} - 457733 \nu^{11} + 431323 \nu^{10} + 217959 \nu^{9} - 1928373 \nu^{8} + 3602156 \nu^{7} - 2213422 \nu^{6} - 5344728 \nu^{5} + 19818666 \nu^{4} - 45346419 \nu^{3} + 76141134 \nu^{2} - 82166319 \nu + 47911338$$$$)/1829061$$ $$\beta_{10}$$ $$=$$ $$($$$$-28354 \nu^{15} + 95439 \nu^{14} - 197100 \nu^{13} + 247392 \nu^{12} - 215912 \nu^{11} - 72792 \nu^{10} + 1366488 \nu^{9} - 2393970 \nu^{8} + 1677584 \nu^{7} + 4108284 \nu^{6} - 11666124 \nu^{5} + 25189920 \nu^{4} - 44623872 \nu^{3} + 41955408 \nu^{2} - 13079718 \nu - 33668865$$$$)/5487183$$ $$\beta_{11}$$ $$=$$ $$($$$$-11554 \nu^{15} + 65950 \nu^{14} - 194877 \nu^{13} + 389808 \nu^{12} - 532664 \nu^{11} + 491762 \nu^{10} + 241278 \nu^{9} - 2127084 \nu^{8} + 4183034 \nu^{7} - 2877419 \nu^{6} - 6075699 \nu^{5} + 23354802 \nu^{4} - 52309665 \nu^{3} + 87490611 \nu^{2} - 100513305 \nu + 60732990$$$$)/1829061$$ $$\beta_{12}$$ $$=$$ $$($$$$35773 \nu^{15} - 50757 \nu^{14} - 13698 \nu^{13} + 367722 \nu^{12} - 769078 \nu^{11} + 1323084 \nu^{10} - 2239380 \nu^{9} + 221379 \nu^{8} + 5622202 \nu^{7} - 15338910 \nu^{6} + 10498437 \nu^{5} - 166968 \nu^{4} - 28293624 \nu^{3} + 107884710 \nu^{2} - 195970509 \nu + 220725162$$$$)/5487183$$ $$\beta_{13}$$ $$=$$ $$($$$$35881 \nu^{15} - 110493 \nu^{14} + 242262 \nu^{13} - 315135 \nu^{12} + 321290 \nu^{11} + 42684 \nu^{10} - 1637460 \nu^{9} + 2777847 \nu^{8} - 2550716 \nu^{7} - 4213662 \nu^{6} + 12817755 \nu^{5} - 32506164 \nu^{4} + 56817612 \nu^{3} - 55368522 \nu^{2} + 29541267 \nu + 33668865$$$$)/5487183$$ $$\beta_{14}$$ $$=$$ $$($$$$-21897 \nu^{15} + 62870 \nu^{14} - 114183 \nu^{13} + 108063 \nu^{12} - 49284 \nu^{11} - 176321 \nu^{10} + 1097229 \nu^{9} - 1456956 \nu^{8} + 476724 \nu^{7} + 4215935 \nu^{6} - 8119533 \nu^{5} + 15176142 \nu^{4} - 25329105 \nu^{3} + 12957975 \nu^{2} + 13861206 \nu - 49687182$$$$)/1829061$$ $$\beta_{15}$$ $$=$$ $$($$$$-8452 \nu^{15} + 21901 \nu^{14} - 40300 \nu^{13} + 29256 \nu^{12} - 302 \nu^{11} - 93712 \nu^{10} + 424168 \nu^{9} - 481167 \nu^{8} + 101459 \nu^{7} + 1822111 \nu^{6} - 3083533 \nu^{5} + 5458719 \nu^{4} - 7566768 \nu^{3} + 2059155 \nu^{2} + 8130132 \nu - 23491053$$$$)/609687$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{12} + \beta_{10} + \beta_{9} + \beta_{6} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} - \beta_{14} - \beta_{7} - \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{14} + 2 \beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + 3 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-\beta_{15} + \beta_{14} - 2 \beta_{13} - \beta_{12} + 2 \beta_{11} - 3 \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6} + \beta_{4} + 4 \beta_{3} + 5 \beta_{2} - \beta_{1} - 3$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} + \beta_{11} - 2 \beta_{9} - \beta_{8} + 5 \beta_{6} + 6 \beta_{5} + \beta_{4} + 4 \beta_{3} + 9 \beta_{2} - 3 \beta_{1} + 8$$ $$\nu^{7}$$ $$=$$ $$5 \beta_{15} + 5 \beta_{14} - 2 \beta_{13} - 4 \beta_{12} + 5 \beta_{11} - 19 \beta_{10} - 7 \beta_{9} + \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - 8 \beta_{4} + 10 \beta_{3} + 5 \beta_{2} + 6 \beta_{1} + 38$$ $$\nu^{8}$$ $$=$$ $$3 \beta_{15} - \beta_{14} + 12 \beta_{12} + 2 \beta_{11} + 32 \beta_{10} - 3 \beta_{9} - \beta_{8} + 13 \beta_{7} + 14 \beta_{6} + 4 \beta_{5} - 15 \beta_{4} - 9 \beta_{3} + 9 \beta_{2} + 39 \beta_{1} - 32$$ $$\nu^{9}$$ $$=$$ $$14 \beta_{15} - 14 \beta_{14} - 29 \beta_{13} + 29 \beta_{12} - 13 \beta_{11} + 30 \beta_{10} + 23 \beta_{9} - 13 \beta_{8} - 11 \beta_{7} + 26 \beta_{6} - 24 \beta_{5} - 26 \beta_{4} + 46 \beta_{3} + 38 \beta_{2} - 32 \beta_{1} - 15$$ $$\nu^{10}$$ $$=$$ $$26 \beta_{15} - 37 \beta_{14} + 28 \beta_{13} - 26 \beta_{11} + 50 \beta_{10} + 39 \beta_{9} + 11 \beta_{8} + 9 \beta_{7} - 50 \beta_{6} - 96 \beta_{5} + 70 \beta_{4} + 25 \beta_{3} + 48 \beta_{2} - 66 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$-50 \beta_{15} + 9 \beta_{14} + 40 \beta_{13} + 20 \beta_{12} + 59 \beta_{11} + 80 \beta_{10} + 98 \beta_{9} - 9 \beta_{8} + 110 \beta_{7} + 12 \beta_{6} - 101 \beta_{5} + 151 \beta_{4} - 49 \beta_{3} - 111 \beta_{2} - 110 \beta_{1} + 80$$ $$\nu^{12}$$ $$=$$ $$12 \beta_{15} + 12 \beta_{14} + 105 \beta_{13} + 105 \beta_{12} + 110 \beta_{11} - 72 \beta_{9} - 110 \beta_{8} - 60 \beta_{7} + 102 \beta_{6} + 122 \beta_{5} + 78 \beta_{4} + 184 \beta_{3} + 212 \beta_{2} + 70 \beta_{1} - 239$$ $$\nu^{13}$$ $$=$$ $$102 \beta_{15} + 162 \beta_{14} + 120 \beta_{13} + 240 \beta_{12} + 102 \beta_{11} - 192 \beta_{10} - 90 \beta_{9} + 60 \beta_{8} - 174 \beta_{7} + 174 \beta_{6} + 63 \beta_{5} + 84 \beta_{4} + 264 \beta_{3} + 102 \beta_{2} - 256 \beta_{1} + 384$$ $$\nu^{14}$$ $$=$$ $$174 \beta_{15} + 174 \beta_{14} - 388 \beta_{12} + 348 \beta_{11} - 91 \beta_{10} - 694 \beta_{9} + 174 \beta_{8} + 138 \beta_{7} + 140 \beta_{6} - 336 \beta_{5} - 486 \beta_{4} + 600 \beta_{3} + 846 \beta_{2} + 330 \beta_{1} + 91$$ $$\nu^{15}$$ $$=$$ $$140 \beta_{15} - 140 \beta_{14} - 408 \beta_{13} + 408 \beta_{12} - 138 \beta_{11} + 2208 \beta_{10} - 636 \beta_{9} - 138 \beta_{8} + 772 \beta_{7} + 930 \beta_{6} + 394 \beta_{5} - 930 \beta_{4} + 565 \beta_{3} + 400 \beta_{2} + 229 \beta_{1} - 1104$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$$.

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$\beta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 −1.66463 − 0.478563i −0.734734 − 1.56849i −0.417865 + 1.68089i −0.404332 − 1.68420i 0.747325 − 1.56253i 0.990987 + 1.42054i 1.25639 + 1.19226i 1.72685 + 0.134063i −1.66463 + 0.478563i −0.734734 + 1.56849i −0.417865 − 1.68089i −0.404332 + 1.68420i 0.747325 + 1.56253i 0.990987 − 1.42054i 1.25639 − 1.19226i 1.72685 − 0.134063i
0 −1.66463 + 0.478563i 0 0 0 1.25338 2.33003i 0 2.54196 1.59326i 0
101.2 0 −0.734734 + 1.56849i 0 0 0 2.64111 + 0.156656i 0 −1.92033 2.30485i 0
101.3 0 −0.417865 1.68089i 0 0 0 1.25338 2.33003i 0 −2.65078 + 1.40477i 0
101.4 0 −0.404332 + 1.68420i 0 0 0 −1.60761 + 2.10133i 0 −2.67303 1.36195i 0
101.5 0 0.747325 + 1.56253i 0 0 0 −0.786875 2.52603i 0 −1.88301 + 2.33544i 0
101.6 0 0.990987 1.42054i 0 0 0 2.64111 + 0.156656i 0 −1.03589 2.81548i 0
101.7 0 1.25639 1.19226i 0 0 0 −1.60761 + 2.10133i 0 0.157032 2.99589i 0
101.8 0 1.72685 0.134063i 0 0 0 −0.786875 2.52603i 0 2.96405 0.463016i 0
1601.1 0 −1.66463 0.478563i 0 0 0 1.25338 + 2.33003i 0 2.54196 + 1.59326i 0
1601.2 0 −0.734734 1.56849i 0 0 0 2.64111 0.156656i 0 −1.92033 + 2.30485i 0
1601.3 0 −0.417865 + 1.68089i 0 0 0 1.25338 + 2.33003i 0 −2.65078 1.40477i 0
1601.4 0 −0.404332 1.68420i 0 0 0 −1.60761 2.10133i 0 −2.67303 + 1.36195i 0
1601.5 0 0.747325 1.56253i 0 0 0 −0.786875 + 2.52603i 0 −1.88301 2.33544i 0
1601.6 0 0.990987 + 1.42054i 0 0 0 2.64111 0.156656i 0 −1.03589 + 2.81548i 0
1601.7 0 1.25639 + 1.19226i 0 0 0 −1.60761 2.10133i 0 0.157032 + 2.99589i 0
1601.8 0 1.72685 + 0.134063i 0 0 0 −0.786875 + 2.52603i 0 2.96405 + 0.463016i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1601.8 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
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.bi.m yes 16
3.b odd 2 1 inner 2100.2.bi.m yes 16
5.b even 2 1 2100.2.bi.l 16
5.c odd 4 2 2100.2.bo.i 32
7.d odd 6 1 inner 2100.2.bi.m yes 16
15.d odd 2 1 2100.2.bi.l 16
15.e even 4 2 2100.2.bo.i 32
21.g even 6 1 inner 2100.2.bi.m yes 16
35.i odd 6 1 2100.2.bi.l 16
35.k even 12 2 2100.2.bo.i 32
105.p even 6 1 2100.2.bi.l 16
105.w odd 12 2 2100.2.bo.i 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.bi.l 16 5.b even 2 1
2100.2.bi.l 16 15.d odd 2 1
2100.2.bi.l 16 35.i odd 6 1
2100.2.bi.l 16 105.p even 6 1
2100.2.bi.m yes 16 1.a even 1 1 trivial
2100.2.bi.m yes 16 3.b odd 2 1 inner
2100.2.bi.m yes 16 7.d odd 6 1 inner
2100.2.bi.m yes 16 21.g even 6 1 inner
2100.2.bo.i 32 5.c odd 4 2
2100.2.bo.i 32 15.e even 4 2
2100.2.bo.i 32 35.k even 12 2
2100.2.bo.i 32 105.w odd 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2100, [\chi])$$:

 $$T_{11}^{16} - \cdots$$ $$T_{13}^{8} + 21 T_{13}^{6} + 71 T_{13}^{4} + 78 T_{13}^{2} + 25$$ $$T_{19}^{8} + \cdots$$ $$T_{37}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 3 T + 9 T^{2} - 18 T^{3} + 32 T^{4} - 36 T^{5} + 51 T^{7} - 167 T^{8} + 153 T^{9} - 972 T^{11} + 2592 T^{12} - 4374 T^{13} + 6561 T^{14} - 6561 T^{15} + 6561 T^{16}$$
$5$ 1
$7$ $$( 1 - 3 T + 9 T^{2} - 39 T^{3} + 95 T^{4} - 273 T^{5} + 441 T^{6} - 1029 T^{7} + 2401 T^{8} )^{2}$$
$11$ $$1 + 14 T^{2} - 69 T^{4} - 2576 T^{6} - 25195 T^{8} - 226506 T^{10} - 1311833 T^{12} + 39166148 T^{14} + 900447201 T^{16} + 4739103908 T^{18} - 19206546953 T^{20} - 401269195866 T^{22} - 5400772006795 T^{24} - 66814805772176 T^{26} - 216551557993749 T^{28} + 5316497670165374 T^{30} + 45949729863572161 T^{32}$$
$13$ $$( 1 - 83 T^{2} + 3165 T^{4} - 73411 T^{6} + 1146521 T^{8} - 12406459 T^{10} + 90395565 T^{12} - 400625147 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$1 - 14 T^{2} + 267 T^{4} + 10052 T^{6} - 224695 T^{8} + 3669918 T^{10} + 2136955 T^{12} - 1194075764 T^{14} + 23802887097 T^{16} - 345087895796 T^{18} + 178480618555 T^{20} + 88582898949342 T^{22} - 1567417818205495 T^{24} + 20264770687313348 T^{26} + 155560137340346187 T^{28} - 2357289571831613006 T^{30} + 48661191875666868481 T^{32}$$
$19$ $$( 1 + 9 T + 64 T^{2} + 333 T^{3} + 1299 T^{4} + 3186 T^{5} - 4123 T^{6} - 75567 T^{7} - 435298 T^{8} - 1435773 T^{9} - 1488403 T^{10} + 21852774 T^{11} + 169286979 T^{12} + 824540967 T^{13} + 3010936384 T^{14} + 8044845651 T^{15} + 16983563041 T^{16} )^{2}$$
$23$ $$1 + 105 T^{2} + 5189 T^{4} + 177324 T^{6} + 5348524 T^{8} + 157563000 T^{10} + 4443717224 T^{12} + 116142079557 T^{14} + 2787310422721 T^{16} + 61439160085653 T^{18} + 1243534271681384 T^{20} + 23324978778507000 T^{22} + 418848184239075244 T^{24} + 7345914674449095276 T^{26} +$$$$11\!\cdots\!69$$$$T^{28} +$$$$12\!\cdots\!45$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$( 1 - 90 T^{2} + 2929 T^{4} - 16011 T^{6} - 944082 T^{8} - 13465251 T^{10} + 2071626049 T^{12} - 53534098890 T^{14} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 + 9 T + 97 T^{2} + 630 T^{3} + 3786 T^{4} + 11934 T^{5} + 10454 T^{6} - 251757 T^{7} - 2144539 T^{8} - 7804467 T^{9} + 10046294 T^{10} + 355525794 T^{11} + 3496450506 T^{12} + 18036365130 T^{13} + 86087857057 T^{14} + 247613526999 T^{15} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 + 3 T + 93 T^{3} - 1331 T^{4} - 10584 T^{5} + 15795 T^{6} + 161037 T^{7} + 1169532 T^{8} + 5958369 T^{9} + 21623355 T^{10} - 536111352 T^{11} - 2494508291 T^{12} + 6448988001 T^{13} + 284795631399 T^{15} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 + 175 T^{2} + 13716 T^{4} + 685412 T^{6} + 28562204 T^{8} + 1152177572 T^{10} + 38758137876 T^{12} + 831268242175 T^{14} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 - T + 150 T^{2} - 110 T^{3} + 9290 T^{4} - 4730 T^{5} + 277350 T^{6} - 79507 T^{7} + 3418801 T^{8} )^{4}$$
$47$ $$1 - 34 T^{2} - 1025 T^{4} + 161680 T^{6} - 9103783 T^{8} + 152237258 T^{10} + 4847483963 T^{12} - 753776826296 T^{14} + 44233333645801 T^{16} - 1665093009287864 T^{18} + 23654175392055803 T^{20} + 1640998185078527882 T^{22} -$$$$21\!\cdots\!63$$$$T^{24} +$$$$85\!\cdots\!20$$$$T^{26} -$$$$11\!\cdots\!25$$$$T^{28} -$$$$87\!\cdots\!46$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 + 262 T^{2} + 35395 T^{4} + 3231368 T^{6} + 221761433 T^{8} + 11879858878 T^{10} + 500403165131 T^{12} + 17219962264736 T^{14} + 683285105805889 T^{16} + 48370874001643424 T^{18} + 3948421666806018011 T^{20} +$$$$26\!\cdots\!62$$$$T^{22} +$$$$13\!\cdots\!13$$$$T^{24} +$$$$56\!\cdots\!32$$$$T^{26} +$$$$17\!\cdots\!95$$$$T^{28} +$$$$36\!\cdots\!78$$$$T^{30} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$1 - 179 T^{2} + 16761 T^{4} - 465136 T^{6} - 46752400 T^{8} + 6296484144 T^{10} - 226286292944 T^{12} - 7423721157083 T^{14} + 1182150698554881 T^{16} - 25841973347805923 T^{18} - 2741992700954200784 T^{20} +$$$$26\!\cdots\!04$$$$T^{22} -$$$$68\!\cdots\!00$$$$T^{24} -$$$$23\!\cdots\!36$$$$T^{26} +$$$$29\!\cdots\!41$$$$T^{28} -$$$$11\!\cdots\!19$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$( 1 - 18 T + 325 T^{2} - 3906 T^{3} + 44196 T^{4} - 437580 T^{5} + 4261022 T^{6} - 36703131 T^{7} + 310887998 T^{8} - 2238890991 T^{9} + 15855262862 T^{10} - 99322345980 T^{11} + 611930788836 T^{12} - 3298993151706 T^{13} + 16744121667325 T^{14} - 56569371048378 T^{15} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 - 15 T - 19 T^{2} + 1578 T^{3} - 3827 T^{4} - 92169 T^{5} + 542666 T^{6} + 1629357 T^{7} - 31878578 T^{8} + 109166919 T^{9} + 2436027674 T^{10} - 27721024947 T^{11} - 77118340067 T^{12} + 2130497418846 T^{13} - 1718709261211 T^{14} - 90910674079845 T^{15} + 406067677556641 T^{16} )^{2}$$
$71$ $$( 1 - 139 T^{2} + 17517 T^{4} - 1749386 T^{6} + 123496010 T^{8} - 8818654826 T^{10} + 445136416077 T^{12} - 17805939465019 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 - 27 T + 513 T^{2} - 7290 T^{3} + 85003 T^{4} - 866997 T^{5} + 8113770 T^{6} - 71702037 T^{7} + 621942924 T^{8} - 5234248701 T^{9} + 43238280330 T^{10} - 337276571949 T^{11} + 2413935679723 T^{12} - 15112691912970 T^{13} + 77634458086257 T^{14} - 298279760015619 T^{15} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 - 195 T^{2} - 582 T^{3} + 18208 T^{4} + 79734 T^{5} - 1345644 T^{6} - 3198963 T^{7} + 109759584 T^{8} - 252718077 T^{9} - 8398164204 T^{10} + 39311971626 T^{11} + 709203074848 T^{12} - 1790846824218 T^{13} - 47402053826595 T^{14} + 1517108809906561 T^{16} )^{2}$$
$83$ $$( 1 + 406 T^{2} + 83877 T^{4} + 11459777 T^{6} + 1116323102 T^{8} + 78946403753 T^{10} + 3980661590517 T^{12} + 132737791587814 T^{14} + 2252292232139041 T^{16} )^{2}$$
$89$ $$1 - 118 T^{2} + 3595 T^{4} + 1013020 T^{6} - 177688255 T^{8} + 15671126798 T^{10} - 78833677741 T^{12} - 122913712595300 T^{14} + 16662250146905929 T^{16} - 973599517467371300 T^{18} - 4946201607742157581 T^{20} +$$$$77\!\cdots\!78$$$$T^{22} -$$$$69\!\cdots\!55$$$$T^{24} +$$$$31\!\cdots\!20$$$$T^{26} +$$$$88\!\cdots\!95$$$$T^{28} -$$$$23\!\cdots\!38$$$$T^{30} +$$$$15\!\cdots\!61$$$$T^{32}$$
$97$ $$( 1 - 402 T^{2} + 92431 T^{4} - 14249709 T^{6} + 1608196287 T^{8} - 134075511981 T^{10} + 8182849972111 T^{12} - 334854745981458 T^{14} + 7837433594376961 T^{16} )^{2}$$