# Properties

 Label 450.2.l.d Level $450$ Weight $2$ Character orbit 450.l Analytic conductor $3.593$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 450.l (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.59326809096$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -\beta_{11} q^{2} + ( 1 + \beta_{3} + \beta_{4} + \beta_{5} ) q^{4} + ( -\beta_{8} - \beta_{9} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{5} + ( 1 + 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{7} + \beta_{12} q^{8} +O(q^{10})$$ $$q -\beta_{11} q^{2} + ( 1 + \beta_{3} + \beta_{4} + \beta_{5} ) q^{4} + ( -\beta_{8} - \beta_{9} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{5} + ( 1 + 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{7} + \beta_{12} q^{8} + ( 1 + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{10} ) q^{10} + ( -\beta_{9} + \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{11} + ( -3 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{10} ) q^{13} + ( -\beta_{1} + \beta_{8} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{14} + \beta_{3} q^{16} + ( -\beta_{1} + \beta_{8} + 2 \beta_{11} + 3 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{17} + ( -2 - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{10} ) q^{19} + ( -2 \beta_{1} - 2 \beta_{8} - \beta_{9} - \beta_{13} + \beta_{15} ) q^{20} + ( -1 + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{22} + ( -2 \beta_{1} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} - 2 \beta_{13} - 2 \beta_{15} ) q^{23} + ( 2 + 2 \beta_{2} - 3 \beta_{3} - \beta_{6} + 2 \beta_{10} ) q^{25} + ( 2 \beta_{1} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{14} - 2 \beta_{15} ) q^{26} + ( \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{28} + ( 2 \beta_{1} + \beta_{8} + \beta_{9} - 4 \beta_{11} + 4 \beta_{12} + 3 \beta_{13} + 3 \beta_{14} ) q^{29} + ( 1 - 3 \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{10} ) q^{31} -\beta_{8} q^{32} + ( -1 + \beta_{2} + 3 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{10} ) q^{34} + ( -3 \beta_{1} + \beta_{8} - 2 \beta_{9} + 3 \beta_{11} - \beta_{12} - 3 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{35} + ( -5 - \beta_{2} - 5 \beta_{3} - 8 \beta_{4} - 5 \beta_{5} - \beta_{6} - \beta_{10} ) q^{37} + ( 4 \beta_{1} + 2 \beta_{8} + 2 \beta_{13} - 2 \beta_{15} ) q^{38} + ( 1 + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} + \beta_{10} ) q^{40} + ( -3 \beta_{8} - \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{15} ) q^{41} + ( -2 - 4 \beta_{3} + 4 \beta_{7} ) q^{43} + ( -\beta_{1} - 2 \beta_{8} + \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{44} + ( 6 + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{10} ) q^{46} + ( -4 \beta_{8} - 4 \beta_{9} - 6 \beta_{11} - 6 \beta_{12} ) q^{47} + ( 4 + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} + \beta_{7} - \beta_{10} ) q^{49} + ( -2 \beta_{1} + 2 \beta_{8} - \beta_{9} - \beta_{11} + \beta_{14} + 2 \beta_{15} ) q^{50} + ( -2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{10} ) q^{52} + ( \beta_{1} + 6 \beta_{8} + 5 \beta_{9} + 4 \beta_{11} + 4 \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{53} + ( -1 + 4 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{10} ) q^{55} + ( 2 \beta_{9} + \beta_{11} + \beta_{12} + \beta_{14} + \beta_{15} ) q^{56} + ( -2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{7} + \beta_{10} ) q^{58} + ( 5 \beta_{8} + 5 \beta_{9} + 4 \beta_{11} + 2 \beta_{12} + 3 \beta_{13} + 3 \beta_{15} ) q^{59} + ( 2 + 2 \beta_{2} - \beta_{3} - 7 \beta_{4} + 3 \beta_{6} + 4 \beta_{7} + 3 \beta_{10} ) q^{61} + ( -2 \beta_{1} + 2 \beta_{8} + \beta_{9} + 3 \beta_{11} - \beta_{13} - 6 \beta_{14} - 2 \beta_{15} ) q^{62} -\beta_{4} q^{64} + ( \beta_{1} - 5 \beta_{8} - 3 \beta_{11} - 7 \beta_{12} - 2 \beta_{15} ) q^{65} + ( -8 - 8 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} - 6 \beta_{10} ) q^{67} + ( \beta_{1} - 2 \beta_{8} + 2 \beta_{9} + \beta_{11} + \beta_{14} + \beta_{15} ) q^{68} + ( 3 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{70} + ( 4 \beta_{1} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} - 2 \beta_{12} ) q^{71} + ( 6 \beta_{2} + \beta_{3} - 3 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} + 6 \beta_{7} + 3 \beta_{10} ) q^{73} + ( \beta_{1} - 4 \beta_{8} - 4 \beta_{9} - 5 \beta_{11} - 8 \beta_{12} + \beta_{14} - \beta_{15} ) q^{74} + ( -4 - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{7} - 2 \beta_{10} ) q^{76} + ( \beta_{1} + \beta_{8} + \beta_{9} + 5 \beta_{11} - \beta_{14} ) q^{77} + ( -5 - 4 \beta_{2} - \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - 8 \beta_{10} ) q^{79} + ( -\beta_{8} - \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{80} + ( 3 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{10} ) q^{82} + ( -2 \beta_{1} - 2 \beta_{8} - 3 \beta_{11} - 5 \beta_{12} + 4 \beta_{13} + \beta_{14} - \beta_{15} ) q^{83} + ( -1 + 7 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} + 3 \beta_{6} + 3 \beta_{7} + \beta_{10} ) q^{85} + ( 4 \beta_{1} + 8 \beta_{8} + 4 \beta_{9} + 2 \beta_{11} + 4 \beta_{12} + 4 \beta_{13} - 4 \beta_{15} ) q^{86} + ( \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{10} ) q^{88} + ( 2 \beta_{1} + 8 \beta_{8} + 4 \beta_{9} + 3 \beta_{11} + 4 \beta_{12} + 5 \beta_{13} - 3 \beta_{14} - 5 \beta_{15} ) q^{89} + ( 2 - 6 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} + 2 \beta_{10} ) q^{91} + ( -4 \beta_{1} - 2 \beta_{8} + 2 \beta_{12} - 2 \beta_{14} + 2 \beta_{15} ) q^{92} + ( 2 + 2 \beta_{4} + 6 \beta_{5} ) q^{94} + ( 2 \beta_{1} - 6 \beta_{8} + 2 \beta_{11} + 2 \beta_{13} - 2 \beta_{14} - 4 \beta_{15} ) q^{95} + ( -7 - 7 \beta_{2} + \beta_{4} - \beta_{5} - 7 \beta_{6} - 7 \beta_{7} ) q^{97} + ( 2 \beta_{1} + 2 \beta_{8} + 3 \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 4q^{4} + O(q^{10})$$ $$16q + 4q^{4} - 4q^{16} - 16q^{19} - 20q^{22} + 20q^{25} - 10q^{28} + 6q^{31} - 26q^{34} + 10q^{37} + 20q^{46} + 28q^{49} - 20q^{55} + 32q^{61} + 4q^{64} - 40q^{67} - 30q^{70} - 24q^{76} - 36q^{79} - 70q^{85} + 10q^{88} + 52q^{91} - 70q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-1807 \nu^{14} - 3112 \nu^{12} + 10523 \nu^{10} + 91863 \nu^{8} + 793 \nu^{6} - 961270 \nu^{4} - 1623925 \nu^{2} - 1433125$$$$)/633875$$ $$\beta_{3}$$ $$=$$ $$($$$$1379032 \nu^{14} - 312743 \nu^{12} - 4990978 \nu^{10} - 61900293 \nu^{8} + 94590252 \nu^{6} + 424939915 \nu^{4} + 771306350 \nu^{2} + 563878625$$$$)/ 171780125$$ $$\beta_{4}$$ $$=$$ $$($$$$-281280 \nu^{14} + 69219 \nu^{12} + 939009 \nu^{10} + 12381889 \nu^{8} - 18275316 \nu^{6} - 83450271 \nu^{4} - 152170830 \nu^{2} - 167096475$$$$)/34356025$$ $$\beta_{5}$$ $$=$$ $$($$$$-441862 \nu^{14} + 108648 \nu^{12} + 1544473 \nu^{10} + 20140738 \nu^{8} - 29602957 \nu^{6} - 135351515 \nu^{4} - 246968035 \nu^{2} - 215410900$$$$)/34356025$$ $$\beta_{6}$$ $$=$$ $$($$$$-454006 \nu^{14} + 189729 \nu^{12} + 1810849 \nu^{10} + 19437794 \nu^{8} - 33302741 \nu^{6} - 142368195 \nu^{4} - 198805085 \nu^{2} - 184475875$$$$)/34356025$$ $$\beta_{7}$$ $$=$$ $$($$$$2758534 \nu^{14} - 1453576 \nu^{12} - 9476246 \nu^{10} - 120179351 \nu^{8} + 220846139 \nu^{6} + 803999345 \nu^{4} + 1188508900 \nu^{2} + 1026496375$$$$)/ 171780125$$ $$\beta_{8}$$ $$=$$ $$($$$$-2293 \nu^{15} - 486 \nu^{13} + 12284 \nu^{11} + 101834 \nu^{9} - 117086 \nu^{7} - 906528 \nu^{5} - 1102430 \nu^{3} - 955700 \nu$$$$)/633875$$ $$\beta_{9}$$ $$=$$ $$($$$$-3183877 \nu^{15} + 4006568 \nu^{13} + 6543803 \nu^{11} + 134015793 \nu^{9} - 335534227 \nu^{7} - 577777045 \nu^{5} - 1177699300 \nu^{3} - 1245786250 \nu$$$$)/ 858900625$$ $$\beta_{10}$$ $$=$$ $$($$$$-4538397 \nu^{14} + 2417483 \nu^{12} + 16184468 \nu^{10} + 196094683 \nu^{8} - 355909112 \nu^{6} - 1301216635 \nu^{4} - 1924774325 \nu^{2} - 1833539875$$$$)/ 171780125$$ $$\beta_{11}$$ $$=$$ $$($$$$8189122 \nu^{15} - 3158678 \nu^{13} - 32153713 \nu^{11} - 353467203 \nu^{9} + 594131367 \nu^{7} + 2560462840 \nu^{5} + 3581602500 \nu^{3} + 3331016750 \nu$$$$)/ 858900625$$ $$\beta_{12}$$ $$=$$ $$($$$$8211971 \nu^{15} - 5580699 \nu^{13} - 25580004 \nu^{11} - 355005349 \nu^{9} + 691228436 \nu^{7} + 2139497850 \nu^{5} + 3370777175 \nu^{3} + 3295922875 \nu$$$$)/ 858900625$$ $$\beta_{13}$$ $$=$$ $$($$$$9262311 \nu^{15} - 650299 \nu^{13} - 34740779 \nu^{11} - 420559724 \nu^{9} + 526962886 \nu^{7} + 3011969710 \nu^{5} + 5422654650 \nu^{3} + 5129227500 \nu$$$$)/ 858900625$$ $$\beta_{14}$$ $$=$$ $$($$$$-441862 \nu^{15} + 108648 \nu^{13} + 1544473 \nu^{11} + 20140738 \nu^{9} - 29602957 \nu^{7} - 135351515 \nu^{5} - 246968035 \nu^{3} - 215410900 \nu$$$$)/34356025$$ $$\beta_{15}$$ $$=$$ $$($$$$723142 \nu^{15} - 177867 \nu^{13} - 2483482 \nu^{11} - 32522627 \nu^{9} + 47878273 \nu^{7} + 218801786 \nu^{5} + 399138865 \nu^{3} + 382507375 \nu$$$$)/34356025$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{10} - \beta_{6} + 2 \beta_{3} + \beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} - \beta_{13} - 4 \beta_{12} + \beta_{11} - 4 \beta_{9} - \beta_{8} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{10} + 5 \beta_{7} - 4 \beta_{6} - 2 \beta_{5} - \beta_{4} - 4 \beta_{3} + \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$-4 \beta_{15} - 6 \beta_{14} + 13 \beta_{11} + 13 \beta_{9} + 19 \beta_{8} + 4 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$7 \beta_{7} + 7 \beta_{6} + 15 \beta_{5} + 8 \beta_{4} + 29 \beta_{3} - 2 \beta_{2} + 29$$ $$\nu^{7}$$ $$=$$ $$23 \beta_{15} - 38 \beta_{13} - 15 \beta_{12} - 4 \beta_{9} - 15 \beta_{8}$$ $$\nu^{8}$$ $$=$$ $$27 \beta_{10} + 15 \beta_{7} - 27 \beta_{6} + 57 \beta_{5} - 53 \beta_{4} + 57 \beta_{3} + 15 \beta_{2}$$ $$\nu^{9}$$ $$=$$ $$95 \beta_{15} + 95 \beta_{14} - 57 \beta_{13} - 96 \beta_{12} + 57 \beta_{11} - 57 \beta_{9} - 57 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$152 \beta_{10} + 248 \beta_{7} - 209 \beta_{4} - 209 \beta_{3} - 190$$ $$\nu^{11}$$ $$=$$ $$-39 \beta_{14} - 39 \beta_{13} + 305 \beta_{12} + 344 \beta_{11} + 857 \beta_{9} + 857 \beta_{8} + 19 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-494 \beta_{10} + 799 \beta_{6} + 935 \beta_{5} + 1012 \beta_{3} - 494 \beta_{2} + 441$$ $$\nu^{13}$$ $$=$$ $$1506 \beta_{15} + 935 \beta_{14} - 1506 \beta_{13} + 281 \beta_{12} - 1104 \beta_{11} + 281 \beta_{9} - 1506 \beta_{8} - 571 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-450 \beta_{10} - 450 \beta_{7} + 281 \beta_{6} + 2358 \beta_{5} - 3376 \beta_{4} + 281 \beta_{3} - 169 \beta_{2} - 3545$$ $$\nu^{15}$$ $$=$$ $$3826 \beta_{15} + 6184 \beta_{14} - 1012 \beta_{11} - 1012 \beta_{9} - 1631 \beta_{8} - 3826 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$1 + \beta_{3} + \beta_{4} + \beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 1.86824 − 0.357358i −0.917186 + 1.66637i 0.917186 − 1.66637i −1.86824 + 0.357358i −0.0566033 − 1.17421i 0.644389 + 0.983224i −0.644389 − 0.983224i 0.0566033 + 1.17421i −0.0566033 + 1.17421i 0.644389 − 0.983224i −0.644389 + 0.983224i 0.0566033 − 1.17421i 1.86824 + 0.357358i −0.917186 − 1.66637i 0.917186 + 1.66637i −1.86824 − 0.357358i
−0.951057 + 0.309017i 0 0.809017 0.587785i −1.95894 1.07822i 0 3.03582i −0.587785 + 0.809017i 0 2.19625 + 0.420099i
19.2 −0.951057 + 0.309017i 0 0.809017 0.587785i 0.420099 + 2.19625i 0 0.407162i −0.587785 + 0.809017i 0 −1.07822 1.95894i
19.3 0.951057 0.309017i 0 0.809017 0.587785i −0.420099 2.19625i 0 0.407162i 0.587785 0.809017i 0 −1.07822 1.95894i
19.4 0.951057 0.309017i 0 0.809017 0.587785i 1.95894 + 1.07822i 0 3.03582i 0.587785 0.809017i 0 2.19625 + 0.420099i
109.1 −0.587785 + 0.809017i 0 −0.309017 0.951057i −1.87020 + 1.22570i 0 3.26086i 0.951057 + 0.309017i 0 0.107666 2.23347i
109.2 −0.587785 + 0.809017i 0 −0.309017 0.951057i 2.23347 0.107666i 0 0.992398i 0.951057 + 0.309017i 0 −1.22570 + 1.87020i
109.3 0.587785 0.809017i 0 −0.309017 0.951057i −2.23347 + 0.107666i 0 0.992398i −0.951057 0.309017i 0 −1.22570 + 1.87020i
109.4 0.587785 0.809017i 0 −0.309017 0.951057i 1.87020 1.22570i 0 3.26086i −0.951057 0.309017i 0 0.107666 2.23347i
289.1 −0.587785 0.809017i 0 −0.309017 + 0.951057i −1.87020 1.22570i 0 3.26086i 0.951057 0.309017i 0 0.107666 + 2.23347i
289.2 −0.587785 0.809017i 0 −0.309017 + 0.951057i 2.23347 + 0.107666i 0 0.992398i 0.951057 0.309017i 0 −1.22570 1.87020i
289.3 0.587785 + 0.809017i 0 −0.309017 + 0.951057i −2.23347 0.107666i 0 0.992398i −0.951057 + 0.309017i 0 −1.22570 1.87020i
289.4 0.587785 + 0.809017i 0 −0.309017 + 0.951057i 1.87020 + 1.22570i 0 3.26086i −0.951057 + 0.309017i 0 0.107666 + 2.23347i
379.1 −0.951057 0.309017i 0 0.809017 + 0.587785i −1.95894 + 1.07822i 0 3.03582i −0.587785 0.809017i 0 2.19625 0.420099i
379.2 −0.951057 0.309017i 0 0.809017 + 0.587785i 0.420099 2.19625i 0 0.407162i −0.587785 0.809017i 0 −1.07822 + 1.95894i
379.3 0.951057 + 0.309017i 0 0.809017 + 0.587785i −0.420099 + 2.19625i 0 0.407162i 0.587785 + 0.809017i 0 −1.07822 + 1.95894i
379.4 0.951057 + 0.309017i 0 0.809017 + 0.587785i 1.95894 1.07822i 0 3.03582i 0.587785 + 0.809017i 0 2.19625 0.420099i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 379.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.e even 10 1 inner
75.h odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.l.d 16
3.b odd 2 1 inner 450.2.l.d 16
25.e even 10 1 inner 450.2.l.d 16
75.h odd 10 1 inner 450.2.l.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.l.d 16 1.a even 1 1 trivial
450.2.l.d 16 3.b odd 2 1 inner
450.2.l.d 16 25.e even 10 1 inner
450.2.l.d 16 75.h odd 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + 21 T_{7}^{6} + 121 T_{7}^{4} + 116 T_{7}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(450, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$3$ $$T^{16}$$
$5$ $$390625 - 156250 T^{2} + 21875 T^{4} + 2500 T^{6} - 1475 T^{8} + 100 T^{10} + 35 T^{12} - 10 T^{14} + T^{16}$$
$7$ $$( 16 + 116 T^{2} + 121 T^{4} + 21 T^{6} + T^{8} )^{2}$$
$11$ $$3748096 + 642752 T^{2} + 339408 T^{4} + 111724 T^{6} + 20825 T^{8} + 2104 T^{10} + 198 T^{12} + 17 T^{14} + T^{16}$$
$13$ $$( 16 + 200 T + 844 T^{2} + 1190 T^{3} + 631 T^{4} + 85 T^{5} + 14 T^{6} + T^{8} )^{2}$$
$17$ $$3748096 + 1486848 T^{2} + 3994208 T^{4} - 1226604 T^{6} + 171205 T^{8} - 8859 T^{10} + 2078 T^{12} - 72 T^{14} + T^{16}$$
$19$ $$( 30976 - 26752 T + 9408 T^{2} + 576 T^{3} + 1200 T^{4} + 176 T^{5} + 48 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$23$ $$10485760000 - 13762560000 T^{2} + 6955008000 T^{4} - 111104000 T^{6} + 6278400 T^{8} - 190400 T^{10} + 4320 T^{12} + T^{16}$$
$29$ $$65536 - 1568768 T^{2} + 103797248 T^{4} + 84939904 T^{6} + 26937305 T^{8} + 612569 T^{10} + 6698 T^{12} + 32 T^{14} + T^{16}$$
$31$ $$( 5180176 + 277672 T + 175828 T^{2} + 4074 T^{3} + 2825 T^{4} + 24 T^{5} + 28 T^{6} - 3 T^{7} + T^{8} )^{2}$$
$37$ $$( 633616 + 453720 T + 91256 T^{2} + 2180 T^{3} + 2021 T^{4} + 880 T^{5} + 36 T^{6} - 5 T^{7} + T^{8} )^{2}$$
$41$ $$160000 + 800000 T^{2} + 1508000 T^{4} - 377500 T^{6} + 1118525 T^{8} - 18875 T^{10} + 3770 T^{12} + 100 T^{14} + T^{16}$$
$43$ $$( 246016 + 84736 T^{2} + 7776 T^{4} + 176 T^{6} + T^{8} )^{2}$$
$47$ $$( 160000 + 8000 T^{2} + 2400 T^{4} - 80 T^{6} + T^{8} )^{2}$$
$53$ $$104060401 - 59849267 T^{2} + 18822688 T^{4} - 4098149 T^{6} + 1832175 T^{8} - 135949 T^{10} + 4008 T^{12} + 13 T^{14} + T^{16}$$
$59$ $$9250941239296 + 1554930532352 T^{2} + 107277163008 T^{4} + 1751596304 T^{6} + 46401705 T^{8} + 919784 T^{10} + 12578 T^{12} - 3 T^{14} + T^{16}$$
$61$ $$( 495616 - 1109504 T + 1005632 T^{2} - 188472 T^{3} + 33905 T^{4} - 2973 T^{5} + 242 T^{6} - 16 T^{7} + T^{8} )^{2}$$
$67$ $$( 25563136 - 7685120 T + 1256704 T^{2} + 92480 T^{3} - 16464 T^{4} - 1880 T^{5} + 64 T^{6} + 20 T^{7} + T^{8} )^{2}$$
$71$ $$60523872256 + 18295717888 T^{2} + 2484236288 T^{4} + 117354496 T^{6} + 31642880 T^{8} + 862976 T^{10} + 9968 T^{12} + 8 T^{14} + T^{16}$$
$73$ $$( 19360000 - 4400000 T + 988000 T^{2} + 382500 T^{3} + 18025 T^{4} - 2125 T^{5} - 180 T^{6} + T^{8} )^{2}$$
$79$ $$( 20214016 - 4729792 T + 1643248 T^{2} - 100204 T^{3} + 4405 T^{4} + 61 T^{5} + 188 T^{6} + 18 T^{7} + T^{8} )^{2}$$
$83$ $$7052454973378816 - 284752953232128 T^{2} + 5512956319968 T^{4} - 58725935596 T^{6} + 917467905 T^{8} - 8203036 T^{10} + 41658 T^{12} - 63 T^{14} + T^{16}$$
$89$ $$26639462656 - 10134407872 T^{2} + 6789364128 T^{4} - 951715904 T^{6} + 265788905 T^{8} + 7827376 T^{10} + 95658 T^{12} + 473 T^{14} + T^{16}$$
$97$ $$( 12313081 + 9912925 T + 5729896 T^{2} - 249655 T^{3} - 71229 T^{4} - 1355 T^{5} + 406 T^{6} + 35 T^{7} + T^{8} )^{2}$$