# Properties

 Label 2100.2.bo.h Level 2100 Weight 2 Character orbit 2100.bo Analytic conductor 16.769 Analytic rank 0 Dimension 20 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.bo (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 3 x^{18} - 16 x^{16} + 87 x^{14} + 91 x^{12} - 1104 x^{10} + 819 x^{8} + 7047 x^{6} - 11664 x^{4} - 19683 x^{2} + 59049$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 420) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( \beta_{1} - \beta_{2} - \beta_{18} ) q^{7} + ( -\beta_{7} + \beta_{8} + \beta_{11} + \beta_{12} - \beta_{15} + \beta_{16} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( \beta_{1} - \beta_{2} - \beta_{18} ) q^{7} + ( -\beta_{7} + \beta_{8} + \beta_{11} + \beta_{12} - \beta_{15} + \beta_{16} ) q^{9} + ( -\beta_{8} - \beta_{12} ) q^{11} + ( \beta_{1} - 2 \beta_{2} - \beta_{4} + 2 \beta_{6} - 2 \beta_{10} + \beta_{13} + \beta_{17} - \beta_{18} ) q^{13} + ( -\beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{10} + \beta_{13} - 2 \beta_{17} ) q^{17} + ( -\beta_{3} + \beta_{9} - \beta_{14} - \beta_{16} ) q^{19} + ( -\beta_{9} + \beta_{12} - \beta_{14} ) q^{21} + ( -\beta_{2} + 2 \beta_{5} + \beta_{6} + \beta_{10} - \beta_{18} - \beta_{19} ) q^{23} + ( -\beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{10} - \beta_{13} - \beta_{17} ) q^{27} + ( -2 - \beta_{3} + 2 \beta_{7} - \beta_{9} + \beta_{11} + 2 \beta_{12} - \beta_{15} + \beta_{16} ) q^{29} + ( 1 + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{12} - \beta_{14} + \beta_{16} ) q^{31} + ( \beta_{1} - 3 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{10} + 2 \beta_{13} - 2 \beta_{18} + 2 \beta_{19} ) q^{33} + ( -\beta_{1} - 3 \beta_{2} + \beta_{6} - 2 \beta_{10} - \beta_{13} - \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{37} + ( \beta_{3} - 3 \beta_{8} - \beta_{12} + 2 \beta_{14} + 2 \beta_{15} + \beta_{16} ) q^{39} + ( -2 + 3 \beta_{3} - 2 \beta_{9} + \beta_{14} + 3 \beta_{16} ) q^{41} + ( -3 \beta_{5} - \beta_{6} - \beta_{10} - 3 \beta_{18} + 3 \beta_{19} ) q^{43} + ( 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{4} + 2 \beta_{6} - 4 \beta_{10} - 4 \beta_{18} + 3 \beta_{19} ) q^{47} + ( 1 + 2 \beta_{3} + \beta_{7} + \beta_{8} - 2 \beta_{12} + \beta_{14} - \beta_{15} + \beta_{16} ) q^{49} + ( -6 + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} - \beta_{14} + \beta_{16} ) q^{51} + ( 2 \beta_{1} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{13} - 2 \beta_{18} + 2 \beta_{19} ) q^{53} + ( 2 \beta_{2} - \beta_{6} + \beta_{10} + 2 \beta_{18} + \beta_{19} ) q^{57} + ( 2 \beta_{3} + \beta_{8} - 4 \beta_{11} - \beta_{12} - 2 \beta_{16} ) q^{59} + ( 6 - \beta_{3} - 3 \beta_{7} + \beta_{9} - \beta_{14} - \beta_{16} ) q^{61} + ( -\beta_{1} - \beta_{2} + 4 \beta_{4} - \beta_{6} + \beta_{13} - \beta_{18} ) q^{63} + ( 3 \beta_{1} - \beta_{2} - 3 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{10} + 4 \beta_{13} - \beta_{17} ) q^{67} + ( -3 \beta_{3} + \beta_{7} - \beta_{8} + 4 \beta_{11} + 2 \beta_{12} - 2 \beta_{14} - 2 \beta_{15} ) q^{69} + ( 1 + \beta_{7} - 2 \beta_{9} - \beta_{11} - 3 \beta_{12} + 2 \beta_{14} + 2 \beta_{15} ) q^{71} + ( -3 \beta_{1} - \beta_{2} - 3 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{10} - \beta_{13} - 4 \beta_{17} - 3 \beta_{18} + 2 \beta_{19} ) q^{73} + ( 2 \beta_{2} - \beta_{4} + 3 \beta_{5} + 2 \beta_{10} - 2 \beta_{13} + \beta_{17} + 3 \beta_{18} - 2 \beta_{19} ) q^{77} + ( 2 + \beta_{3} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{11} - 4 \beta_{12} + 3 \beta_{14} - 3 \beta_{16} ) q^{79} + ( 6 - 3 \beta_{3} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{14} ) q^{81} + ( -4 \beta_{1} - 2 \beta_{5} + 4 \beta_{13} - 5 \beta_{18} + 2 \beta_{19} ) q^{83} + ( \beta_{2} + \beta_{4} + \beta_{5} + \beta_{10} + \beta_{13} + 2 \beta_{17} - 3 \beta_{18} ) q^{87} + ( -2 \beta_{3} - 6 \beta_{7} - 2 \beta_{9} - \beta_{14} + \beta_{15} + \beta_{16} ) q^{89} + ( 2 - 5 \beta_{3} - 3 \beta_{7} + 2 \beta_{9} - \beta_{12} - 3 \beta_{14} - 3 \beta_{16} ) q^{91} + ( -6 \beta_{2} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{10} + 5 \beta_{13} + \beta_{17} - 4 \beta_{18} + 3 \beta_{19} ) q^{93} + ( -\beta_{1} - 2 \beta_{4} + \beta_{6} - \beta_{10} - \beta_{13} + 2 \beta_{17} ) q^{97} + ( -3 + \beta_{3} - 3 \beta_{7} - \beta_{9} - \beta_{11} + \beta_{12} - \beta_{14} + 3 \beta_{15} + \beta_{16} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 6q^{9} + O(q^{10})$$ $$20q + 6q^{9} - 12q^{11} - 6q^{19} + 20q^{21} + 30q^{31} - 30q^{39} - 16q^{41} + 26q^{49} - 88q^{51} + 84q^{61} + 28q^{69} - 2q^{79} + 82q^{81} - 56q^{89} - 22q^{91} - 72q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 3 x^{18} - 16 x^{16} + 87 x^{14} + 91 x^{12} - 1104 x^{10} + 819 x^{8} + 7047 x^{6} - 11664 x^{4} - 19683 x^{2} + 59049$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-29 \nu^{19} - 966 \nu^{17} + 3218 \nu^{15} + 6063 \nu^{13} - 4664 \nu^{11} + 13224 \nu^{9} - 164367 \nu^{7} - 18954 \nu^{5} + 1981422 \nu^{3} - 2119203 \nu$$$$)/5038848$$ $$\beta_{3}$$ $$=$$ $$($$$$-47 \nu^{18} + 654 \nu^{16} - 1354 \nu^{14} + 1797 \nu^{12} + 18808 \nu^{10} - 57192 \nu^{8} - 63765 \nu^{6} + 300834 \nu^{4} - 1217430 \nu^{2} + 1673055$$$$)/1679616$$ $$\beta_{4}$$ $$=$$ $$($$$$83 \nu^{19} + 930 \nu^{17} - 86 \nu^{15} - 29625 \nu^{13} + 64280 \nu^{11} + 292920 \nu^{9} - 840303 \nu^{7} - 966978 \nu^{5} + 6617862 \nu^{3} - 1541835 \nu$$$$)/5038848$$ $$\beta_{5}$$ $$=$$ $$($$$$19 \nu^{19} + 222 \nu^{17} - 466 \nu^{15} - 1677 \nu^{13} + 13744 \nu^{11} - 528 \nu^{9} - 97623 \nu^{7} + 99306 \nu^{5} + 477738 \nu^{3} - 448335 \nu$$$$)/839808$$ $$\beta_{6}$$ $$=$$ $$($$$$-31 \nu^{19} + 48 \nu^{17} - 908 \nu^{15} - 3921 \nu^{13} + 26636 \nu^{11} + 1212 \nu^{9} - 275409 \nu^{7} + 262764 \nu^{5} + 997272 \nu^{3} - 2512863 \nu$$$$)/1259712$$ $$\beta_{7}$$ $$=$$ $$($$$$-131 \nu^{18} - 138 \nu^{16} + 4094 \nu^{14} - 6303 \nu^{12} - 42728 \nu^{10} + 100920 \nu^{8} + 172431 \nu^{6} - 842886 \nu^{4} - 10206 \nu^{2} + 2224179$$$$)/1679616$$ $$\beta_{8}$$ $$=$$ $$($$$$-53 \nu^{18} - 366 \nu^{16} + 3098 \nu^{14} + 63 \nu^{12} - 42344 \nu^{10} + 74808 \nu^{8} + 293625 \nu^{6} - 905202 \nu^{4} - 166698 \nu^{2} + 3759453$$$$)/559872$$ $$\beta_{9}$$ $$=$$ $$($$$$-22 \nu^{18} + 111 \nu^{16} + 865 \nu^{14} - 2391 \nu^{12} - 10642 \nu^{10} + 43206 \nu^{8} + 65628 \nu^{6} - 339309 \nu^{4} + 6561 \nu^{2} + 1581201$$$$)/209952$$ $$\beta_{10}$$ $$=$$ $$($$$$49 \nu^{19} + 672 \nu^{17} - 2188 \nu^{15} - 6897 \nu^{13} + 32620 \nu^{11} + 30396 \nu^{9} - 228033 \nu^{7} + 48924 \nu^{5} + 437400 \nu^{3} + 426465 \nu$$$$)/1259712$$ $$\beta_{11}$$ $$=$$ $$($$$$-37 \nu^{18} + 138 \nu^{16} + 322 \nu^{14} - 2841 \nu^{12} + 3464 \nu^{10} + 18600 \nu^{8} - 43047 \nu^{6} - 66906 \nu^{4} + 251262 \nu^{2} - 37179$$$$)/279936$$ $$\beta_{12}$$ $$=$$ $$($$$$221 \nu^{18} - 1194 \nu^{16} - 4130 \nu^{14} + 32097 \nu^{12} - 39208 \nu^{10} - 303240 \nu^{8} + 714735 \nu^{6} + 720090 \nu^{4} - 5259006 \nu^{2} + 4323699$$$$)/1679616$$ $$\beta_{13}$$ $$=$$ $$($$$$-\nu^{19} + 3 \nu^{17} + 16 \nu^{15} - 87 \nu^{13} - 91 \nu^{11} + 1104 \nu^{9} - 819 \nu^{7} - 7047 \nu^{5} + 11664 \nu^{3} + 19683 \nu$$$$)/19683$$ $$\beta_{14}$$ $$=$$ $$($$$$38 \nu^{18} - 33 \nu^{16} - 527 \nu^{14} + 309 \nu^{12} + 3134 \nu^{10} - 1290 \nu^{8} - 15696 \nu^{6} - 30861 \nu^{4} + 169857 \nu^{2} + 137781$$$$)/209952$$ $$\beta_{15}$$ $$=$$ $$($$$$59 \nu^{18} - 246 \nu^{16} - 62 \nu^{14} + 4455 \nu^{12} - 9976 \nu^{10} - 20952 \nu^{8} + 150201 \nu^{6} - 142938 \nu^{4} - 557442 \nu^{2} + 1502469$$$$)/279936$$ $$\beta_{16}$$ $$=$$ $$($$$$383 \nu^{18} - 150 \nu^{16} - 3374 \nu^{14} + 5187 \nu^{12} + 42872 \nu^{10} - 57576 \nu^{8} - 263691 \nu^{6} + 696438 \nu^{4} + 2576286 \nu^{2} - 4139991$$$$)/1679616$$ $$\beta_{17}$$ $$=$$ $$($$$$131 \nu^{19} + 138 \nu^{17} - 4094 \nu^{15} + 6303 \nu^{13} + 42728 \nu^{11} - 100920 \nu^{9} - 172431 \nu^{7} + 842886 \nu^{5} + 10206 \nu^{3} - 2224179 \nu$$$$)/1679616$$ $$\beta_{18}$$ $$=$$ $$($$$$127 \nu^{19} - 48 \nu^{17} - 4084 \nu^{15} + 3777 \nu^{13} + 48628 \nu^{11} - 119868 \nu^{9} - 339759 \nu^{7} + 1085076 \nu^{5} + 868968 \nu^{3} - 5570289 \nu$$$$)/1259712$$ $$\beta_{19}$$ $$=$$ $$($$$$65 \nu^{19} - 118 \nu^{17} - 902 \nu^{15} + 3073 \nu^{13} + 6224 \nu^{11} - 33136 \nu^{9} - 4269 \nu^{7} + 144846 \nu^{5} - 37746 \nu^{3} - 277749 \nu$$$$)/279936$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{16} - \beta_{15} + \beta_{12} + \beta_{11} + \beta_{8} - \beta_{7}$$ $$\nu^{3}$$ $$=$$ $$\beta_{17} + \beta_{13} - 2 \beta_{10} + \beta_{5} + \beta_{4} - 2 \beta_{2} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{14} - \beta_{11} + \beta_{9} - \beta_{8} - 2 \beta_{7} - 3 \beta_{3} + 6$$ $$\nu^{5}$$ $$=$$ $$-3 \beta_{19} + 2 \beta_{18} + 2 \beta_{17} - 5 \beta_{13} - 4 \beta_{6} - 3 \beta_{5} + 4 \beta_{4} + 2 \beta_{2} + 6 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{16} + \beta_{15} - 5 \beta_{14} + 4 \beta_{12} + 9 \beta_{11} + 3 \beta_{9} - 12 \beta_{7} - 10 \beta_{3} - 3$$ $$\nu^{7}$$ $$=$$ $$\beta_{19} - 12 \beta_{18} + 12 \beta_{17} - 10 \beta_{13} - 20 \beta_{10} + 12 \beta_{6} + 9 \beta_{5} + 14 \beta_{4} - 30 \beta_{2} + \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-19 \beta_{16} + 8 \beta_{15} - 9 \beta_{14} - 21 \beta_{12} - 30 \beta_{11} + 7 \beta_{9} - 5 \beta_{8} - 37 \beta_{7} - 16 \beta_{3} + 12$$ $$\nu^{9}$$ $$=$$ $$-19 \beta_{19} - 2 \beta_{18} + 37 \beta_{17} - 25 \beta_{13} + 16 \beta_{10} - 26 \beta_{6} - 46 \beta_{5} + 29 \beta_{4} + 42 \beta_{2} - 9 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-32 \beta_{16} + 65 \beta_{15} - 29 \beta_{14} - 48 \beta_{12} + 26 \beta_{11} - 6 \beta_{9} - 19 \beta_{8} - 68 \beta_{7} - 31 \beta_{3} + 12$$ $$\nu^{11}$$ $$=$$ $$34 \beta_{19} - 144 \beta_{18} + 68 \beta_{17} - 101 \beta_{13} - 84 \beta_{10} + 168 \beta_{6} + 98 \beta_{5} + 40 \beta_{4} - 124 \beta_{2} - 36 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-28 \beta_{16} - 58 \beta_{15} - 10 \beta_{14} - 120 \beta_{12} - 242 \beta_{11} - 42 \beta_{9} + 176 \beta_{8} - 296 \beta_{7} + 236 \beta_{3} - 183$$ $$\nu^{13}$$ $$=$$ $$-70 \beta_{19} + 20 \beta_{18} + 296 \beta_{17} + 268 \beta_{13} + 184 \beta_{10} - 268 \beta_{6} - 118 \beta_{5} - 164 \beta_{4} + 628 \beta_{2} - 303 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$137 \beta_{16} + 493 \beta_{15} - 330 \beta_{14} - 403 \beta_{12} - 37 \beta_{11} - 162 \beta_{9} - 211 \beta_{8} + 703 \beta_{7} + 336 \beta_{3} + 312$$ $$\nu^{15}$$ $$=$$ $$42 \beta_{19} - 228 \beta_{18} - 703 \beta_{17} - 907 \beta_{13} + 302 \beta_{10} + 96 \beta_{6} + 995 \beta_{5} - 295 \beta_{4} + 878 \beta_{2} - 91 \beta_{1}$$ $$\nu^{16}$$ $$=$$ $$1824 \beta_{16} - 1326 \beta_{15} - 232 \beta_{14} + 1056 \beta_{12} + 193 \beta_{11} + 491 \beta_{9} + 979 \beta_{8} - 94 \beta_{7} + 2541 \beta_{3} - 3606$$ $$\nu^{17}$$ $$=$$ $$-849 \beta_{19} + 1834 \beta_{18} + 94 \beta_{17} + 3011 \beta_{13} + 1716 \beta_{10} - 4100 \beta_{6} + 2103 \beta_{5} - 2308 \beta_{4} + 3358 \beta_{2} - 2550 \beta_{1}$$ $$\nu^{18}$$ $$=$$ $$2062 \beta_{16} + 5951 \beta_{15} - 775 \beta_{14} - 3268 \beta_{12} - 3105 \beta_{11} + 1305 \beta_{9} - 7392 \beta_{8} + 14316 \beta_{7} + 394 \beta_{3} + 2109$$ $$\nu^{19}$$ $$=$$ $$467 \beta_{19} + 6744 \beta_{18} - 14316 \beta_{17} - 11798 \beta_{13} + 12332 \beta_{10} - 13080 \beta_{6} - 1773 \beta_{5} + 3010 \beta_{4} + 14286 \beta_{2} - 1159 \beta_{1}$$

## Character values

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

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$\beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1349.1
 1.70278 + 0.317079i 1.64368 + 0.546177i 1.56233 − 0.747749i 0.368412 + 1.69242i 0.268793 − 1.71107i −0.268793 + 1.71107i −0.368412 − 1.69242i −1.56233 + 0.747749i −1.64368 − 0.546177i −1.70278 − 0.317079i 1.70278 − 0.317079i 1.64368 − 0.546177i 1.56233 + 0.747749i 0.368412 − 1.69242i 0.268793 + 1.71107i −0.268793 − 1.71107i −0.368412 + 1.69242i −1.56233 − 0.747749i −1.64368 + 0.546177i −1.70278 + 0.317079i
0 −1.70278 0.317079i 0 0 0 1.99797 + 1.73439i 0 2.79892 + 1.07983i 0
1349.2 0 −1.64368 0.546177i 0 0 0 2.41433 1.08214i 0 2.40338 + 1.79548i 0
1349.3 0 −1.56233 + 0.747749i 0 0 0 −2.60608 0.456468i 0 1.88174 2.33646i 0
1349.4 0 −0.368412 1.69242i 0 0 0 −1.93884 + 1.80025i 0 −2.72854 + 1.24701i 0
1349.5 0 −0.268793 + 1.71107i 0 0 0 0.615143 2.57325i 0 −2.85550 0.919845i 0
1349.6 0 0.268793 1.71107i 0 0 0 −0.615143 + 2.57325i 0 −2.85550 0.919845i 0
1349.7 0 0.368412 + 1.69242i 0 0 0 1.93884 1.80025i 0 −2.72854 + 1.24701i 0
1349.8 0 1.56233 0.747749i 0 0 0 2.60608 + 0.456468i 0 1.88174 2.33646i 0
1349.9 0 1.64368 + 0.546177i 0 0 0 −2.41433 + 1.08214i 0 2.40338 + 1.79548i 0
1349.10 0 1.70278 + 0.317079i 0 0 0 −1.99797 1.73439i 0 2.79892 + 1.07983i 0
1949.1 0 −1.70278 + 0.317079i 0 0 0 1.99797 1.73439i 0 2.79892 1.07983i 0
1949.2 0 −1.64368 + 0.546177i 0 0 0 2.41433 + 1.08214i 0 2.40338 1.79548i 0
1949.3 0 −1.56233 0.747749i 0 0 0 −2.60608 + 0.456468i 0 1.88174 + 2.33646i 0
1949.4 0 −0.368412 + 1.69242i 0 0 0 −1.93884 1.80025i 0 −2.72854 1.24701i 0
1949.5 0 −0.268793 1.71107i 0 0 0 0.615143 + 2.57325i 0 −2.85550 + 0.919845i 0
1949.6 0 0.268793 + 1.71107i 0 0 0 −0.615143 2.57325i 0 −2.85550 + 0.919845i 0
1949.7 0 0.368412 1.69242i 0 0 0 1.93884 + 1.80025i 0 −2.72854 1.24701i 0
1949.8 0 1.56233 + 0.747749i 0 0 0 2.60608 0.456468i 0 1.88174 + 2.33646i 0
1949.9 0 1.64368 0.546177i 0 0 0 −2.41433 1.08214i 0 2.40338 1.79548i 0
1949.10 0 1.70278 0.317079i 0 0 0 −1.99797 + 1.73439i 0 2.79892 1.07983i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1949.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
21.g even 6 1 inner
105.p 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.bo.h 20
3.b odd 2 1 2100.2.bo.g 20
5.b even 2 1 inner 2100.2.bo.h 20
5.c odd 4 1 420.2.bh.a 10
5.c odd 4 1 2100.2.bi.k 10
7.d odd 6 1 2100.2.bo.g 20
15.d odd 2 1 2100.2.bo.g 20
15.e even 4 1 420.2.bh.b yes 10
15.e even 4 1 2100.2.bi.j 10
21.g even 6 1 inner 2100.2.bo.h 20
35.i odd 6 1 2100.2.bo.g 20
35.k even 12 1 420.2.bh.b yes 10
35.k even 12 1 2100.2.bi.j 10
35.k even 12 1 2940.2.d.a 10
35.l odd 12 1 2940.2.d.b 10
105.p even 6 1 inner 2100.2.bo.h 20
105.w odd 12 1 420.2.bh.a 10
105.w odd 12 1 2100.2.bi.k 10
105.w odd 12 1 2940.2.d.b 10
105.x even 12 1 2940.2.d.a 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bh.a 10 5.c odd 4 1
420.2.bh.a 10 105.w odd 12 1
420.2.bh.b yes 10 15.e even 4 1
420.2.bh.b yes 10 35.k even 12 1
2100.2.bi.j 10 15.e even 4 1
2100.2.bi.j 10 35.k even 12 1
2100.2.bi.k 10 5.c odd 4 1
2100.2.bi.k 10 105.w odd 12 1
2100.2.bo.g 20 3.b odd 2 1
2100.2.bo.g 20 7.d odd 6 1
2100.2.bo.g 20 15.d odd 2 1
2100.2.bo.g 20 35.i odd 6 1
2100.2.bo.h 20 1.a even 1 1 trivial
2100.2.bo.h 20 5.b even 2 1 inner
2100.2.bo.h 20 21.g even 6 1 inner
2100.2.bo.h 20 105.p even 6 1 inner
2940.2.d.a 10 35.k even 12 1
2940.2.d.a 10 105.x even 12 1
2940.2.d.b 10 35.l odd 12 1
2940.2.d.b 10 105.w odd 12 1

## Hecke kernels

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

 $$T_{11}^{10} + \cdots$$ $$T_{13}^{10} - 81 T_{13}^{8} + 2183 T_{13}^{6} - 22251 T_{13}^{4} + 65704 T_{13}^{2} - 3888$$ $$T_{19}^{10} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 3 T^{2} - 16 T^{4} + 87 T^{6} + 91 T^{8} - 1104 T^{10} + 819 T^{12} + 7047 T^{14} - 11664 T^{16} - 19683 T^{18} + 59049 T^{20}$$
$5$ 1
$7$ $$1 - 13 T^{2} + 119 T^{4} - 562 T^{6} + 565 T^{8} + 10473 T^{10} + 27685 T^{12} - 1349362 T^{14} + 14000231 T^{16} - 74942413 T^{18} + 282475249 T^{20}$$
$11$ $$( 1 + 6 T + 51 T^{2} + 234 T^{3} + 1195 T^{4} + 3666 T^{5} + 12622 T^{6} + 19434 T^{7} + 25645 T^{8} - 184704 T^{9} - 509783 T^{10} - 2031744 T^{11} + 3103045 T^{12} + 25866654 T^{13} + 184798702 T^{14} + 590412966 T^{15} + 2117015395 T^{16} + 4559998014 T^{17} + 10932302931 T^{18} + 14147686146 T^{19} + 25937424601 T^{20} )^{2}$$
$13$ $$( 1 + 49 T^{2} + 1364 T^{4} + 28371 T^{6} + 474775 T^{8} + 6687888 T^{10} + 80236975 T^{12} + 810304131 T^{14} + 6583767476 T^{16} + 39970805329 T^{18} + 137858491849 T^{20} )^{2}$$
$17$ $$1 + 66 T^{2} + 2267 T^{4} + 48026 T^{6} + 572883 T^{8} - 801368 T^{10} - 179842402 T^{12} - 3509885976 T^{14} - 18012284507 T^{16} + 684073009458 T^{18} + 19619688493209 T^{20} + 197697099733362 T^{22} - 1504404014309147 T^{24} - 84720114927832344 T^{26} - 1254536973958813282 T^{28} - 1615553000015014232 T^{30} +$$$$33\!\cdots\!63$$$$T^{32} +$$$$80\!\cdots\!54$$$$T^{34} +$$$$11\!\cdots\!27$$$$T^{36} +$$$$92\!\cdots\!94$$$$T^{38} +$$$$40\!\cdots\!01$$$$T^{40}$$
$19$ $$( 1 + 3 T + 83 T^{2} + 240 T^{3} + 3800 T^{4} + 10380 T^{5} + 124845 T^{6} + 315315 T^{7} + 3213415 T^{8} + 7312140 T^{9} + 67225536 T^{10} + 138930660 T^{11} + 1160042815 T^{12} + 2162745585 T^{13} + 16269925245 T^{14} + 25701907620 T^{15} + 178774347800 T^{16} + 214529217360 T^{17} + 1409635732403 T^{18} + 968063093337 T^{19} + 6131066257801 T^{20} )^{2}$$
$23$ $$1 - 100 T^{2} + 4454 T^{4} - 135804 T^{6} + 3628499 T^{8} - 75944748 T^{10} + 808939708 T^{12} + 10318464096 T^{14} - 898818094003 T^{16} + 35402708232120 T^{18} - 973535494901246 T^{20} + 18728032654791480 T^{22} - 251526154243893523 T^{24} + 1527503005565941344 T^{26} + 63348865566404437948 T^{28} -$$$$31\!\cdots\!52$$$$T^{30} +$$$$79\!\cdots\!79$$$$T^{32} -$$$$15\!\cdots\!36$$$$T^{34} +$$$$27\!\cdots\!94$$$$T^{36} -$$$$32\!\cdots\!00$$$$T^{38} +$$$$17\!\cdots\!01$$$$T^{40}$$
$29$ $$( 1 - 148 T^{2} + 11438 T^{4} - 619594 T^{6} + 25499257 T^{8} - 826380936 T^{10} + 21444875137 T^{12} - 438227063914 T^{14} + 6803589145598 T^{16} - 74036469118228 T^{18} + 420707233300201 T^{20} )^{2}$$
$31$ $$( 1 - 15 T + 167 T^{2} - 1380 T^{3} + 9668 T^{4} - 61404 T^{5} + 363093 T^{6} - 2129475 T^{7} + 12161347 T^{8} - 68552376 T^{9} + 387742776 T^{10} - 2125123656 T^{11} + 11687054467 T^{12} - 63439189725 T^{13} + 335324010453 T^{14} - 1757944388004 T^{15} + 8580385587908 T^{16} - 37967407473180 T^{17} + 142432803252647 T^{18} - 396594332410065 T^{19} + 819628286980801 T^{20} )^{2}$$
$37$ $$1 + 235 T^{2} + 27377 T^{4} + 2212926 T^{6} + 144837806 T^{8} + 8250414270 T^{10} + 421139833759 T^{12} + 19588270538109 T^{14} + 844751235316463 T^{16} + 34241451973802580 T^{18} + 1307081650323006964 T^{20} + 46876547752135732020 T^{22} +$$$$15\!\cdots\!43$$$$T^{24} +$$$$50\!\cdots\!81$$$$T^{26} +$$$$14\!\cdots\!39$$$$T^{28} +$$$$39\!\cdots\!30$$$$T^{30} +$$$$95\!\cdots\!86$$$$T^{32} +$$$$19\!\cdots\!14$$$$T^{34} +$$$$33\!\cdots\!57$$$$T^{36} +$$$$39\!\cdots\!15$$$$T^{38} +$$$$23\!\cdots\!01$$$$T^{40}$$
$41$ $$( 1 + 4 T + 90 T^{2} + 592 T^{3} + 3749 T^{4} + 36434 T^{5} + 153709 T^{6} + 995152 T^{7} + 6202890 T^{8} + 11303044 T^{9} + 115856201 T^{10} )^{4}$$
$43$ $$( 1 - 177 T^{2} + 15651 T^{4} - 980958 T^{6} + 51911037 T^{8} - 2406825463 T^{10} + 95983507413 T^{12} - 3353700191358 T^{14} + 98935653079899 T^{16} - 2068811449135377 T^{18} + 21611482313284249 T^{20} )^{2}$$
$47$ $$1 + 98 T^{2} + 1463 T^{4} - 172214 T^{6} - 7589337 T^{8} + 17111036 T^{10} + 16661205666 T^{12} + 813009774836 T^{14} - 550471722003 T^{16} - 837271397783898 T^{18} - 19354757954951431 T^{20} - 1849532517704630682 T^{22} - 2686126402895321043 T^{24} +$$$$87\!\cdots\!44$$$$T^{26} +$$$$39\!\cdots\!26$$$$T^{28} +$$$$90\!\cdots\!64$$$$T^{30} -$$$$88\!\cdots\!17$$$$T^{32} -$$$$44\!\cdots\!66$$$$T^{34} +$$$$82\!\cdots\!23$$$$T^{36} +$$$$12\!\cdots\!22$$$$T^{38} +$$$$27\!\cdots\!01$$$$T^{40}$$
$53$ $$1 - 98 T^{2} + 3839 T^{4} + 145862 T^{6} - 27406857 T^{8} + 1377609796 T^{10} - 37452311886 T^{12} - 261577435028 T^{14} + 89930621026509 T^{16} - 7083814035902694 T^{18} + 497135157539747921 T^{20} - 19898433626850667446 T^{22} +$$$$70\!\cdots\!29$$$$T^{24} -$$$$57\!\cdots\!12$$$$T^{26} -$$$$23\!\cdots\!46$$$$T^{28} +$$$$24\!\cdots\!04$$$$T^{30} -$$$$13\!\cdots\!37$$$$T^{32} +$$$$20\!\cdots\!78$$$$T^{34} +$$$$14\!\cdots\!19$$$$T^{36} -$$$$10\!\cdots\!22$$$$T^{38} +$$$$30\!\cdots\!01$$$$T^{40}$$
$59$ $$( 1 - 135 T^{2} + 756 T^{3} + 7351 T^{4} - 84450 T^{5} - 41974 T^{6} + 3597210 T^{7} - 8726459 T^{8} - 56020614 T^{9} + 414268763 T^{10} - 3305216226 T^{11} - 30376803779 T^{12} + 738791392590 T^{13} - 508614110614 T^{14} - 60375357050550 T^{15} + 310069102794991 T^{16} + 1881420522523164 T^{17} - 19822109076583335 T^{18} + 511116753300641401 T^{20} )^{2}$$
$61$ $$( 1 - 42 T + 1112 T^{2} - 22008 T^{3} + 361073 T^{4} - 5062200 T^{5} + 62481246 T^{6} - 687242394 T^{7} + 6821292673 T^{8} - 61325603712 T^{9} + 502052393310 T^{10} - 3740861826432 T^{11} + 25382030036233 T^{12} - 155990965832514 T^{13} + 865105397597886 T^{14} - 4275515394922200 T^{15} + 18602616131649353 T^{16} - 69165484335150168 T^{17} + 213178532052976472 T^{18} - 491154135899033922 T^{19} + 713342911662882601 T^{20} )^{2}$$
$67$ $$1 + 237 T^{2} + 33122 T^{4} + 3625607 T^{6} + 334131306 T^{8} + 26888076115 T^{10} + 2089931081048 T^{12} + 163796249955747 T^{14} + 12948927080804797 T^{16} + 981044928724302918 T^{18} + 68872931546114565948 T^{20} +$$$$44\!\cdots\!02$$$$T^{22} +$$$$26\!\cdots\!37$$$$T^{24} +$$$$14\!\cdots\!43$$$$T^{26} +$$$$84\!\cdots\!68$$$$T^{28} +$$$$49\!\cdots\!35$$$$T^{30} +$$$$27\!\cdots\!66$$$$T^{32} +$$$$13\!\cdots\!03$$$$T^{34} +$$$$54\!\cdots\!82$$$$T^{36} +$$$$17\!\cdots\!33$$$$T^{38} +$$$$33\!\cdots\!01$$$$T^{40}$$
$71$ $$( 1 - 422 T^{2} + 92093 T^{4} - 13410012 T^{6} + 1429352290 T^{8} - 115763220252 T^{10} + 7205364893890 T^{12} - 340770947150172 T^{14} + 11797139447136653 T^{16} - 272507990185711142 T^{18} + 3255243551009881201 T^{20} )^{2}$$
$73$ $$1 - 221 T^{2} + 29033 T^{4} - 2168298 T^{6} + 83474342 T^{8} + 2057167494 T^{10} - 407779306505 T^{12} + 6444321727917 T^{14} + 3050137018094543 T^{16} - 468394461607084380 T^{18} + 40020648370576768996 T^{20} -$$$$24\!\cdots\!20$$$$T^{22} +$$$$86\!\cdots\!63$$$$T^{24} +$$$$97\!\cdots\!13$$$$T^{26} -$$$$32\!\cdots\!05$$$$T^{28} +$$$$88\!\cdots\!06$$$$T^{30} +$$$$19\!\cdots\!82$$$$T^{32} -$$$$26\!\cdots\!82$$$$T^{34} +$$$$18\!\cdots\!13$$$$T^{36} -$$$$76\!\cdots\!49$$$$T^{38} +$$$$18\!\cdots\!01$$$$T^{40}$$
$79$ $$( 1 + T - 201 T^{2} - 740 T^{3} + 18128 T^{4} + 100900 T^{5} - 805347 T^{6} - 5115611 T^{7} + 21370651 T^{8} + 77003688 T^{9} - 699661200 T^{10} + 6083291352 T^{11} + 133374232891 T^{12} - 2522195731829 T^{13} - 31368330883107 T^{14} + 310474990659100 T^{15} + 4406689393684688 T^{16} - 14210892649757660 T^{17} - 304938870791218761 T^{18} + 119851595982618319 T^{19} + 9468276082626847201 T^{20} )^{2}$$
$83$ $$( 1 - 56 T^{2} + 4922 T^{4} - 896022 T^{6} + 72023497 T^{8} - 6972835512 T^{10} + 496169870833 T^{12} - 42523699699062 T^{14} + 1609200517722218 T^{16} - 126128364999786296 T^{18} + 15516041187205853449 T^{20} )^{2}$$
$89$ $$( 1 + 28 T + 118 T^{2} - 2240 T^{3} + 12999 T^{4} + 635638 T^{5} + 2053984 T^{6} - 15180164 T^{7} + 276922069 T^{8} + 3387791550 T^{9} + 10662822610 T^{10} + 301513447950 T^{11} + 2193499708549 T^{12} - 10701545034916 T^{13} + 128871559138144 T^{14} + 3549440380043462 T^{15} + 6460259801202039 T^{16} - 99078190165984960 T^{17} + 464517479072845558 T^{18} + 9809979303809585852 T^{19} + 31181719929966183601 T^{20} )^{2}$$
$97$ $$( 1 + 758 T^{2} + 269933 T^{4} + 59888792 T^{6} + 9235543570 T^{8} + 1040392525668 T^{10} + 86897229450130 T^{12} + 5301911695718552 T^{14} + 224846632206499757 T^{16} + 5940774664537736438 T^{18} + 73742412689492826049 T^{20} )^{2}$$