# Properties

 Label 250.2.e.c Level $250$ Weight $2$ Character orbit 250.e Analytic conductor $1.996$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$250 = 2 \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 250.e (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.99626005053$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + x^{14} - 4 x^{12} - 49 x^{10} + 11 x^{8} + 395 x^{6} + 900 x^{4} + 1125 x^{2} + 625$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$5^{2}$$ Twist minimal: no (minimal twist has level 50) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{9} + \beta_{10} + \beta_{12} - \beta_{13} ) q^{2} + ( -\beta_{9} + \beta_{13} - \beta_{14} ) q^{3} -\beta_{3} q^{4} + ( \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} ) q^{6} + ( \beta_{5} - \beta_{9} - \beta_{11} - \beta_{14} ) q^{7} + \beta_{12} q^{8} + ( -1 - \beta_{2} - \beta_{4} - 3 \beta_{6} - \beta_{7} - \beta_{8} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{9} + \beta_{10} + \beta_{12} - \beta_{13} ) q^{2} + ( -\beta_{9} + \beta_{13} - \beta_{14} ) q^{3} -\beta_{3} q^{4} + ( \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} ) q^{6} + ( \beta_{5} - \beta_{9} - \beta_{11} - \beta_{14} ) q^{7} + \beta_{12} q^{8} + ( -1 - \beta_{2} - \beta_{4} - 3 \beta_{6} - \beta_{7} - \beta_{8} ) q^{9} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{11} + ( -\beta_{9} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{12} + ( -\beta_{5} - 2 \beta_{9} + \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{15} ) q^{13} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{14} + \beta_{6} q^{16} + ( -\beta_{10} - 3 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{17} + ( 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} ) q^{18} + ( -\beta_{1} + 3 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} - \beta_{7} - \beta_{8} ) q^{19} + ( -5 + \beta_{1} - 4 \beta_{3} - 5 \beta_{6} ) q^{21} + ( -\beta_{5} + \beta_{10} + \beta_{14} - \beta_{15} ) q^{22} + ( \beta_{5} + \beta_{9} + \beta_{10} - 2 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{23} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{24} + ( 3 + \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + \beta_{8} ) q^{26} + ( -\beta_{5} + \beta_{9} + \beta_{10} + 5 \beta_{12} - 5 \beta_{13} + \beta_{14} ) q^{27} + ( -\beta_{5} + \beta_{10} + \beta_{14} - \beta_{15} ) q^{28} + ( 2 - \beta_{1} - 3 \beta_{3} + 2 \beta_{6} ) q^{29} + ( -\beta_{1} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{31} -\beta_{9} q^{32} + ( -5 \beta_{10} + \beta_{11} - \beta_{12} + 5 \beta_{13} ) q^{33} + ( -1 + \beta_{2} - \beta_{4} - 3 \beta_{6} + \beta_{8} ) q^{34} + ( -2 - 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{6} - \beta_{7} - \beta_{8} ) q^{36} + ( \beta_{9} + \beta_{12} + 2 \beta_{13} + 2 \beta_{15} ) q^{37} + ( \beta_{5} - 3 \beta_{9} - \beta_{11} - 3 \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{38} + ( -2 + 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} - 5 \beta_{6} + \beta_{7} + \beta_{8} ) q^{39} + ( -\beta_{2} + 3 \beta_{6} + \beta_{7} - \beta_{8} ) q^{41} + ( -5 \beta_{10} + \beta_{11} - \beta_{12} + 5 \beta_{13} ) q^{42} + ( \beta_{5} + 3 \beta_{9} + 4 \beta_{10} - \beta_{11} + 4 \beta_{12} - \beta_{14} ) q^{43} + ( \beta_{1} + \beta_{4} + \beta_{7} ) q^{44} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{7} ) q^{46} + ( \beta_{5} - \beta_{10} - \beta_{14} + \beta_{15} ) q^{47} + ( -\beta_{5} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{48} + ( -2 + \beta_{1} + \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + \beta_{8} ) q^{49} + ( 5 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{8} ) q^{51} + ( 3 \beta_{9} + 3 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{15} ) q^{52} + ( 4 \beta_{9} + 4 \beta_{10} - 4 \beta_{13} + 3 \beta_{14} ) q^{53} + ( 4 + \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{6} + \beta_{7} ) q^{54} + ( \beta_{1} + \beta_{4} + \beta_{7} ) q^{56} + ( -\beta_{5} + 3 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} + 2 \beta_{12} + 3 \beta_{14} ) q^{57} + ( 2 \beta_{10} - \beta_{11} + 5 \beta_{12} - 2 \beta_{13} ) q^{58} + ( 2 \beta_{2} + 2 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} ) q^{59} + ( -2 + 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{61} + ( \beta_{5} - 2 \beta_{9} - \beta_{11} - 2 \beta_{12} - 3 \beta_{13} + 2 \beta_{15} ) q^{62} + ( 2 \beta_{5} + \beta_{9} - 2 \beta_{11} + \beta_{12} - 7 \beta_{13} + \beta_{15} ) q^{63} + ( 1 + \beta_{3} + \beta_{4} + \beta_{6} ) q^{64} + ( -5 - 5 \beta_{4} - \beta_{6} - \beta_{7} ) q^{66} + ( -7 \beta_{12} + 3 \beta_{14} - 3 \beta_{15} ) q^{67} + ( \beta_{5} + 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} ) q^{68} + ( 2 \beta_{1} + 7 \beta_{3} + 13 \beta_{4} + 7 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} ) q^{69} + ( -4 - \beta_{1} - \beta_{2} - 7 \beta_{3} - 4 \beta_{6} - \beta_{7} ) q^{71} + ( \beta_{5} + \beta_{9} - 2 \beta_{10} - \beta_{13} + \beta_{15} ) q^{72} + ( -\beta_{5} - 6 \beta_{9} - 6 \beta_{10} + 3 \beta_{11} - 8 \beta_{12} + 8 \beta_{13} + \beta_{14} - 3 \beta_{15} ) q^{73} + ( -3 + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{74} + ( 1 - \beta_{1} + 3 \beta_{3} + 3 \beta_{4} - \beta_{8} ) q^{76} + ( \beta_{5} - 9 \beta_{9} - 9 \beta_{10} - 5 \beta_{12} + 5 \beta_{13} - \beta_{14} ) q^{77} + ( 2 \beta_{5} + 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} ) q^{78} + ( 2 + 2 \beta_{1} + 4 \beta_{3} + 2 \beta_{6} ) q^{79} + ( \beta_{1} + 6 \beta_{3} + 6 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{81} + ( -2 \beta_{5} - 3 \beta_{9} + \beta_{11} + \beta_{14} ) q^{82} + ( -2 \beta_{10} - 2 \beta_{11} - 7 \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{83} + ( -5 - 5 \beta_{4} - \beta_{6} - \beta_{7} ) q^{84} + ( -3 - \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{6} ) q^{86} + ( -2 \beta_{5} - 5 \beta_{9} + 2 \beta_{11} - 5 \beta_{12} + 11 \beta_{13} - 5 \beta_{15} ) q^{87} + ( -\beta_{5} + \beta_{11} + \beta_{13} ) q^{88} + ( 2 - 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{8} ) q^{89} + ( -2 + 3 \beta_{2} - 2 \beta_{4} - 3 \beta_{6} - 2 \beta_{7} + 3 \beta_{8} ) q^{91} + ( 2 \beta_{10} - 2 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{92} + ( -5 \beta_{5} - 2 \beta_{9} + 7 \beta_{10} + 2 \beta_{11} + 7 \beta_{12} + 2 \beta_{14} ) q^{93} + ( -\beta_{1} - \beta_{4} - \beta_{7} ) q^{94} + ( -1 + \beta_{1} - \beta_{6} ) q^{96} + ( 5 \beta_{9} - 3 \beta_{10} - 5 \beta_{13} ) q^{97} + ( \beta_{5} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{98} + ( 6 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 4q^{4} - 6q^{6} + 2q^{9} + O(q^{10})$$ $$16q + 4q^{4} - 6q^{6} + 2q^{9} + 2q^{11} + 2q^{14} - 4q^{16} - 40q^{19} - 38q^{21} - 4q^{24} + 44q^{26} + 30q^{29} - 18q^{31} + 2q^{34} - 2q^{36} + 24q^{39} - 18q^{41} - 2q^{44} + 14q^{46} + 8q^{49} + 52q^{51} + 50q^{54} - 2q^{56} - 20q^{59} + 12q^{61} + 4q^{64} - 52q^{66} - 86q^{69} - 18q^{71} - 48q^{74} - 20q^{76} + 20q^{79} - 34q^{81} - 52q^{84} - 46q^{86} + 30q^{89} + 2q^{91} + 2q^{94} - 6q^{96} + 84q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + x^{14} - 4 x^{12} - 49 x^{10} + 11 x^{8} + 395 x^{6} + 900 x^{4} + 1125 x^{2} + 625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$948392 \nu^{14} - 2081693 \nu^{12} - 1547103 \nu^{10} - 35207443 \nu^{8} + 136185777 \nu^{6} + 77053830 \nu^{4} + 27131400 \nu^{2} + 181387875$$$$)/ 171780125$$ $$\beta_{2}$$ $$=$$ $$($$$$122987 \nu^{14} + 97172 \nu^{12} - 701513 \nu^{10} - 5799603 \nu^{8} + 3932417 \nu^{6} + 54634025 \nu^{4} + 77779875 \nu^{2} + 52412250$$$$)/15616375$$ $$\beta_{3}$$ $$=$$ $$($$$$1379032 \nu^{14} - 312743 \nu^{12} - 4990978 \nu^{10} - 61900293 \nu^{8} + 94590252 \nu^{6} + 424939915 \nu^{4} + 771306350 \nu^{2} + 563878625$$$$)/ 171780125$$ $$\beta_{4}$$ $$=$$ $$($$$$-281280 \nu^{14} + 69219 \nu^{12} + 939009 \nu^{10} + 12381889 \nu^{8} - 18275316 \nu^{6} - 83450271 \nu^{4} - 152170830 \nu^{2} - 167096475$$$$)/34356025$$ $$\beta_{5}$$ $$=$$ $$($$$$-1157122 \nu^{15} + 1428203 \nu^{13} + 8678488 \nu^{11} + 43919978 \nu^{9} - 137248467 \nu^{7} - 474206065 \nu^{5} + 222668250 \nu^{3} + 1705295750 \nu$$$$)/ 858900625$$ $$\beta_{6}$$ $$=$$ $$($$$$-441862 \nu^{14} + 108648 \nu^{12} + 1544473 \nu^{10} + 20140738 \nu^{8} - 29602957 \nu^{6} - 135351515 \nu^{4} - 246968035 \nu^{2} - 215410900$$$$)/34356025$$ $$\beta_{7}$$ $$=$$ $$($$$$-2817591 \nu^{14} + 2379174 \nu^{12} + 8884104 \nu^{10} + 118381374 \nu^{8} - 262226761 \nu^{6} - 716617430 \nu^{4} - 884417625 \nu^{2} - 860602375$$$$)/ 171780125$$ $$\beta_{8}$$ $$=$$ $$($$$$-626638 \nu^{14} + 144621 \nu^{12} + 2783831 \nu^{10} + 27217946 \nu^{8} - 41997039 \nu^{6} - 210462216 \nu^{4} - 281904075 \nu^{2} - 222107450$$$$)/34356025$$ $$\beta_{9}$$ $$=$$ $$($$$$2293 \nu^{15} + 486 \nu^{13} - 12284 \nu^{11} - 101834 \nu^{9} + 117086 \nu^{7} + 906528 \nu^{5} + 1102430 \nu^{3} + 955700 \nu$$$$)/633875$$ $$\beta_{10}$$ $$=$$ $$($$$$3183877 \nu^{15} - 4006568 \nu^{13} - 6543803 \nu^{11} - 134015793 \nu^{9} + 335534227 \nu^{7} + 577777045 \nu^{5} + 1177699300 \nu^{3} + 1245786250 \nu$$$$)/ 858900625$$ $$\beta_{11}$$ $$=$$ $$($$$$8105189 \nu^{15} + 777904 \nu^{13} - 26062291 \nu^{11} - 376639746 \nu^{9} + 389714419 \nu^{7} + 2537763645 \nu^{5} + 5645322900 \nu^{3} + 5975622625 \nu$$$$)/ 858900625$$ $$\beta_{12}$$ $$=$$ $$($$$$-8189122 \nu^{15} + 3158678 \nu^{13} + 32153713 \nu^{11} + 353467203 \nu^{9} - 594131367 \nu^{7} - 2560462840 \nu^{5} - 3581602500 \nu^{3} - 3331016750 \nu$$$$)/ 858900625$$ $$\beta_{13}$$ $$=$$ $$($$$$8211971 \nu^{15} - 5580699 \nu^{13} - 25580004 \nu^{11} - 355005349 \nu^{9} + 691228436 \nu^{7} + 2139497850 \nu^{5} + 3370777175 \nu^{3} + 3295922875 \nu$$$$)/ 858900625$$ $$\beta_{14}$$ $$=$$ $$($$$$-2000435 \nu^{15} + 181037 \nu^{13} + 8319942 \nu^{11} + 89497307 \nu^{9} - 126320558 \nu^{7} - 670609003 \nu^{5} - 1070064880 \nu^{3} - 651093200 \nu$$$$)/ 171780125$$ $$\beta_{15}$$ $$=$$ $$($$$$-14757833 \nu^{15} + 273347 \nu^{13} + 57022912 \nu^{11} + 679004122 \nu^{9} - 877490958 \nu^{7} - 4930710405 \nu^{5} - 8853033325 \nu^{3} - 6958879375 \nu$$$$)/ 858900625$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{15} + 2 \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} + 2 \beta_{10} + 3 \beta_{9} + 2 \beta_{5}$$$$)/5$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{8} + 2 \beta_{7} + 3 \beta_{6} + 2 \beta_{4} + 11 \beta_{3} - 4 \beta_{2} - \beta_{1} + 4$$$$)/5$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{15} - 3 \beta_{14} - 16 \beta_{13} - 6 \beta_{12} - \beta_{11} + 17 \beta_{10} - 2 \beta_{9} + 2 \beta_{5}$$$$)/5$$ $$\nu^{4}$$ $$=$$ $$($$$$-7 \beta_{8} - 13 \beta_{7} + 3 \beta_{6} + 7 \beta_{4} + 6 \beta_{3} - 19 \beta_{2} - 26 \beta_{1} + 14$$$$)/5$$ $$\nu^{5}$$ $$=$$ $$($$$$-16 \beta_{15} + 22 \beta_{14} - 6 \beta_{13} - 51 \beta_{12} - 6 \beta_{11} - 43 \beta_{10} - 67 \beta_{9} - 8 \beta_{5}$$$$)/5$$ $$\nu^{6}$$ $$=$$ $$($$$$23 \beta_{8} - 23 \beta_{7} + 63 \beta_{6} + 52 \beta_{4} + 156 \beta_{3} + 11 \beta_{2} - 11 \beta_{1} + 144$$$$)/5$$ $$\nu^{7}$$ $$=$$ $$($$$$-31 \beta_{15} - 53 \beta_{14} + 9 \beta_{13} - 31 \beta_{12} - 106 \beta_{11} - 33 \beta_{10} - 62 \beta_{9} + 137 \beta_{5}$$$$)/5$$ $$\nu^{8}$$ $$=$$ $$($$$$-57 \beta_{8} - 18 \beta_{7} + 363 \beta_{6} - 208 \beta_{4} + 381 \beta_{3} - 114 \beta_{2} - 96 \beta_{1} + 39$$$$)/5$$ $$\nu^{9}$$ $$=$$ $$($$$$209 \beta_{15} - 418 \beta_{14} - 271 \beta_{13} - 551 \beta_{12} - 76 \beta_{11} - 133 \beta_{10} - 627 \beta_{9} + 342 \beta_{5}$$$$)/5$$ $$\nu^{10}$$ $$=$$ $$($$$$208 \beta_{8} - 688 \beta_{7} + 208 \beta_{6} - 493 \beta_{4} - 149 \beta_{3} - 344 \beta_{2} - 896 \beta_{1} - 606$$$$)/5$$ $$\nu^{11}$$ $$=$$ $$($$$$-136 \beta_{15} + 77 \beta_{14} + 1584 \beta_{13} - 1661 \beta_{12} - 136 \beta_{11} - 4208 \beta_{10} - 4267 \beta_{9} + 77 \beta_{5}$$$$)/5$$ $$\nu^{12}$$ $$=$$ $$($$$$2208 \beta_{8} - 683 \beta_{7} + 2888 \beta_{6} - 683 \beta_{4} + 3956 \beta_{3} + 2891 \beta_{2} + 1104 \beta_{1} + 1784$$$$)/5$$ $$\nu^{13}$$ $$=$$ $$($$$$1299 \beta_{15} - 5453 \beta_{14} + 5559 \beta_{13} + 2144 \beta_{12} - 3376 \beta_{11} - 6858 \beta_{10} - 2077 \beta_{9} + 6752 \beta_{5}$$$$)/5$$ $$\nu^{14}$$ $$=$$ $$($$$$393 \beta_{8} + 1012 \beta_{7} + 10778 \beta_{6} - 18118 \beta_{4} - 619 \beta_{3} + 1631 \beta_{2} + 2024 \beta_{1} - 18511$$$$)/5$$ $$\nu^{15}$$ $$=$$ $$($$$$17084 \beta_{15} - 22378 \beta_{14} + 5294 \beta_{13} - 8776 \beta_{12} + 5294 \beta_{11} - 17318 \beta_{10} - 19517 \beta_{9} + 8542 \beta_{5}$$$$)/5$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/250\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$\chi(n)$$ $$-\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.917186 + 1.66637i −1.86824 − 0.357358i 1.86824 + 0.357358i −0.917186 − 1.66637i 0.644389 + 0.983224i −0.0566033 − 1.17421i 0.0566033 + 1.17421i −0.644389 − 0.983224i 0.644389 − 0.983224i −0.0566033 + 1.17421i 0.0566033 − 1.17421i −0.644389 + 0.983224i 0.917186 − 1.66637i −1.86824 + 0.357358i 1.86824 − 0.357358i −0.917186 + 1.66637i
−0.587785 + 0.809017i −1.63079 + 0.529876i −0.309017 0.951057i 0 0.529876 1.63079i 2.77447i 0.951057 + 0.309017i −0.0483405 + 0.0351215i 0
49.2 −0.587785 + 0.809017i 2.21858 0.720859i −0.309017 0.951057i 0 −0.720859 + 2.21858i 3.77447i 0.951057 + 0.309017i 1.97539 1.43521i 0
49.3 0.587785 0.809017i −2.21858 + 0.720859i −0.309017 0.951057i 0 −0.720859 + 2.21858i 3.77447i −0.951057 0.309017i 1.97539 1.43521i 0
49.4 0.587785 0.809017i 1.63079 0.529876i −0.309017 0.951057i 0 0.529876 1.63079i 2.77447i −0.951057 0.309017i −0.0483405 + 0.0351215i 0
99.1 −0.951057 + 0.309017i −0.792578 1.09089i 0.809017 0.587785i 0 1.09089 + 0.792578i 0.833366i −0.587785 + 0.809017i 0.365190 1.12394i 0
99.2 −0.951057 + 0.309017i 1.74363 + 2.39991i 0.809017 0.587785i 0 −2.39991 1.74363i 1.83337i −0.587785 + 0.809017i −1.79224 + 5.51595i 0
99.3 0.951057 0.309017i −1.74363 2.39991i 0.809017 0.587785i 0 −2.39991 1.74363i 1.83337i 0.587785 0.809017i −1.79224 + 5.51595i 0
99.4 0.951057 0.309017i 0.792578 + 1.09089i 0.809017 0.587785i 0 1.09089 + 0.792578i 0.833366i 0.587785 0.809017i 0.365190 1.12394i 0
149.1 −0.951057 0.309017i −0.792578 + 1.09089i 0.809017 + 0.587785i 0 1.09089 0.792578i 0.833366i −0.587785 0.809017i 0.365190 + 1.12394i 0
149.2 −0.951057 0.309017i 1.74363 2.39991i 0.809017 + 0.587785i 0 −2.39991 + 1.74363i 1.83337i −0.587785 0.809017i −1.79224 5.51595i 0
149.3 0.951057 + 0.309017i −1.74363 + 2.39991i 0.809017 + 0.587785i 0 −2.39991 + 1.74363i 1.83337i 0.587785 + 0.809017i −1.79224 5.51595i 0
149.4 0.951057 + 0.309017i 0.792578 1.09089i 0.809017 + 0.587785i 0 1.09089 0.792578i 0.833366i 0.587785 + 0.809017i 0.365190 + 1.12394i 0
199.1 −0.587785 0.809017i −1.63079 0.529876i −0.309017 + 0.951057i 0 0.529876 + 1.63079i 2.77447i 0.951057 0.309017i −0.0483405 0.0351215i 0
199.2 −0.587785 0.809017i 2.21858 + 0.720859i −0.309017 + 0.951057i 0 −0.720859 2.21858i 3.77447i 0.951057 0.309017i 1.97539 + 1.43521i 0
199.3 0.587785 + 0.809017i −2.21858 0.720859i −0.309017 + 0.951057i 0 −0.720859 2.21858i 3.77447i −0.951057 + 0.309017i 1.97539 + 1.43521i 0
199.4 0.587785 + 0.809017i 1.63079 + 0.529876i −0.309017 + 0.951057i 0 0.529876 + 1.63079i 2.77447i −0.951057 + 0.309017i −0.0483405 0.0351215i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.4 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
25.d even 5 1 inner
25.e even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 250.2.e.c 16
5.b even 2 1 inner 250.2.e.c 16
5.c odd 4 1 50.2.d.b 8
5.c odd 4 1 250.2.d.d 8
15.e even 4 1 450.2.h.e 8
20.e even 4 1 400.2.u.d 8
25.d even 5 1 inner 250.2.e.c 16
25.d even 5 1 1250.2.b.e 8
25.e even 10 1 inner 250.2.e.c 16
25.e even 10 1 1250.2.b.e 8
25.f odd 20 1 50.2.d.b 8
25.f odd 20 1 250.2.d.d 8
25.f odd 20 1 1250.2.a.f 4
25.f odd 20 1 1250.2.a.l 4
75.l even 20 1 450.2.h.e 8
100.l even 20 1 400.2.u.d 8
100.l even 20 1 10000.2.a.t 4
100.l even 20 1 10000.2.a.x 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.b 8 5.c odd 4 1
50.2.d.b 8 25.f odd 20 1
250.2.d.d 8 5.c odd 4 1
250.2.d.d 8 25.f odd 20 1
250.2.e.c 16 1.a even 1 1 trivial
250.2.e.c 16 5.b even 2 1 inner
250.2.e.c 16 25.d even 5 1 inner
250.2.e.c 16 25.e even 10 1 inner
400.2.u.d 8 20.e even 4 1
400.2.u.d 8 100.l even 20 1
450.2.h.e 8 15.e even 4 1
450.2.h.e 8 75.l even 20 1
1250.2.a.f 4 25.f odd 20 1
1250.2.a.l 4 25.f odd 20 1
1250.2.b.e 8 25.d even 5 1
1250.2.b.e 8 25.e even 10 1
10000.2.a.t 4 100.l even 20 1
10000.2.a.x 4 100.l even 20 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(250, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$3$ $$65536 - 28672 T^{2} + 19968 T^{4} - 12224 T^{6} + 4625 T^{8} - 764 T^{10} + 78 T^{12} - 7 T^{14} + T^{16}$$
$5$ $$T^{16}$$
$7$ $$( 256 + 496 T^{2} + 201 T^{4} + 26 T^{6} + T^{8} )^{2}$$
$11$ $$( 256 - 64 T + 352 T^{2} + 128 T^{3} + 105 T^{4} + 7 T^{5} - 3 T^{6} - T^{7} + T^{8} )^{2}$$
$13$ $$1568239201 - 429750052 T^{2} + 57271058 T^{4} - 4132974 T^{6} + 611375 T^{8} - 30774 T^{10} + 923 T^{12} - 17 T^{14} + T^{16}$$
$17$ $$141158161 - 112192283 T^{2} + 34950648 T^{4} - 330601 T^{6} + 626475 T^{8} - 25061 T^{10} + 808 T^{12} - 23 T^{14} + T^{16}$$
$19$ $$( 6400 + 11200 T + 12000 T^{2} + 8000 T^{3} + 3585 T^{4} + 965 T^{5} + 190 T^{6} + 20 T^{7} + T^{8} )^{2}$$
$23$ $$65536 + 135168 T^{2} + 511488 T^{4} + 344896 T^{6} + 503505 T^{8} - 84269 T^{10} + 5673 T^{12} - 117 T^{14} + T^{16}$$
$29$ $$( 25 + 300 T + 10150 T^{2} - 5700 T^{3} + 1735 T^{4} - 390 T^{5} + 105 T^{6} - 15 T^{7} + T^{8} )^{2}$$
$31$ $$( 1597696 + 257856 T + 56432 T^{2} + 228 T^{3} + 805 T^{4} + 237 T^{5} + 87 T^{6} + 9 T^{7} + T^{8} )^{2}$$
$37$ $$25411681 - 9920688 T^{2} + 16392788 T^{4} - 15259546 T^{6} + 7335030 T^{8} - 131476 T^{10} + 2793 T^{12} - 58 T^{14} + T^{16}$$
$41$ $$( 7921 - 11214 T + 9162 T^{2} - 4232 T^{3} + 1425 T^{4} - 128 T^{5} + 27 T^{6} + 9 T^{7} + T^{8} )^{2}$$
$43$ $$( 30976 + 17824 T^{2} + 2641 T^{4} + 114 T^{6} + T^{8} )^{2}$$
$47$ $$65536 + 49152 T^{2} + 132608 T^{4} - 16176 T^{6} + 9025 T^{8} - 2961 T^{10} + 473 T^{12} - 33 T^{14} + T^{16}$$
$53$ $$4294967296 - 25031606272 T^{2} + 55624204288 T^{4} + 955607296 T^{6} + 59314305 T^{8} + 115916 T^{10} + 11158 T^{12} - 167 T^{14} + T^{16}$$
$59$ $$( 102400 + 281600 T + 320000 T^{2} + 105600 T^{3} + 17360 T^{4} + 1320 T^{5} + 260 T^{6} + 10 T^{7} + T^{8} )^{2}$$
$61$ $$( 2920681 - 984384 T + 570837 T^{2} - 56572 T^{3} + 9180 T^{4} + 332 T^{5} - 18 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$67$ $$794123370496 + 149382909952 T^{2} + 140360683008 T^{4} - 12950447296 T^{6} + 462689105 T^{8} - 1334641 T^{10} + 51753 T^{12} - 353 T^{14} + T^{16}$$
$71$ $$( 4096 + 19456 T + 32512 T^{2} - 21432 T^{3} + 5225 T^{4} + 297 T^{5} + 217 T^{6} + 9 T^{7} + T^{8} )^{2}$$
$73$ $$1380756603136 - 131455864832 T^{2} + 7889246048 T^{4} - 317475324 T^{6} + 43986725 T^{8} + 167361 T^{10} + 47318 T^{12} - 352 T^{14} + T^{16}$$
$79$ $$( 102400 - 25600 T + 25600 T^{2} + 3200 T^{3} - 240 T^{4} - 320 T^{5} + 120 T^{6} - 10 T^{7} + T^{8} )^{2}$$
$83$ $$180227832610816 - 12797040463872 T^{2} + 426901737728 T^{4} - 6695212304 T^{6} + 283380705 T^{8} - 3107189 T^{10} + 19833 T^{12} - 77 T^{14} + T^{16}$$
$89$ $$( 9610000 - 2015000 T + 1025500 T^{2} - 30750 T^{3} + 8525 T^{4} + 750 T^{5} + 70 T^{6} - 15 T^{7} + T^{8} )^{2}$$
$97$ $$( 1 - 199 T^{2} + 15126 T^{4} + 76 T^{6} + T^{8} )^{2}$$