# Properties

 Label 1470.2.n.l Level $1470$ Weight $2$ Character orbit 1470.n Analytic conductor $11.738$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1470.n (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.7380090971$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 18 x^{14} + 227 x^{12} - 1394 x^{10} + 6177 x^{8} - 14768 x^{6} + 24768 x^{4} - 11264 x^{2} + 4096$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + \beta_{5} q^{3} + ( 1 + \beta_{3} ) q^{4} -\beta_{8} q^{5} - q^{6} + ( -\beta_{2} + \beta_{5} ) q^{8} -\beta_{3} q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + \beta_{5} q^{3} + ( 1 + \beta_{3} ) q^{4} -\beta_{8} q^{5} - q^{6} + ( -\beta_{2} + \beta_{5} ) q^{8} -\beta_{3} q^{9} + \beta_{13} q^{10} + ( -\beta_{5} - \beta_{6} - \beta_{13} + \beta_{14} ) q^{11} + \beta_{2} q^{12} + ( -2 \beta_{4} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{13} + \beta_{12} q^{15} + \beta_{3} q^{16} + ( -\beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} ) q^{17} -\beta_{5} q^{18} + ( -2 \beta_{1} + \beta_{2} + 4 \beta_{3} + 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{19} -\beta_{10} q^{20} + ( \beta_{4} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{22} + ( \beta_{2} - 2 \beta_{4} - \beta_{7} - \beta_{8} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{23} + ( -1 - \beta_{3} ) q^{24} + ( 2 \beta_{6} - \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{14} ) q^{25} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{6} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{26} + ( -\beta_{2} + \beta_{5} ) q^{27} + ( -\beta_{1} - \beta_{2} + \beta_{5} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{15} ) q^{29} + \beta_{8} q^{30} + ( 4 + 4 \beta_{3} + \beta_{5} + \beta_{6} + \beta_{13} - \beta_{14} ) q^{31} + \beta_{5} q^{32} + ( -\beta_{4} - \beta_{7} - \beta_{8} ) q^{33} + ( \beta_{1} - \beta_{2} + \beta_{5} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{15} ) q^{34} + q^{36} + ( 2 \beta_{2} + 3 \beta_{4} - \beta_{7} - \beta_{8} ) q^{37} + ( 3 \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} + \beta_{14} ) q^{38} + ( -\beta_{5} - 2 \beta_{6} - \beta_{13} + \beta_{14} ) q^{39} + ( \beta_{12} + \beta_{13} ) q^{40} + ( -4 - 2 \beta_{1} - \beta_{10} + \beta_{11} ) q^{41} + ( -\beta_{2} - \beta_{4} + \beta_{5} + \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{15} ) q^{43} + ( \beta_{1} - \beta_{2} - \beta_{6} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{44} + ( -\beta_{8} + \beta_{10} ) q^{45} + ( \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{46} + ( 3 \beta_{2} - \beta_{4} - 2 \beta_{7} - 2 \beta_{8} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{47} + ( \beta_{2} - \beta_{5} ) q^{48} + ( -1 + 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{9} + \beta_{11} - \beta_{12} ) q^{50} + ( -\beta_{1} + \beta_{2} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{51} + ( \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{52} + ( -5 \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} + \beta_{14} ) q^{53} + \beta_{3} q^{54} + ( -3 \beta_{1} + 5 \beta_{2} + 2 \beta_{4} - 5 \beta_{5} - 2 \beta_{9} + \beta_{12} - \beta_{15} ) q^{55} + ( 3 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{15} ) q^{57} + ( -2 \beta_{2} + \beta_{4} - \beta_{7} - \beta_{8} + 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{58} + ( -4 - 4 \beta_{3} - \beta_{5} + 4 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{59} -\beta_{13} q^{60} + ( 3 \beta_{7} - 3 \beta_{8} ) q^{61} + ( -4 \beta_{2} - \beta_{4} + 4 \beta_{5} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{62} - q^{64} + ( -2 \beta_{1} + 3 \beta_{3} - 4 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{14} - \beta_{15} ) q^{65} + ( \beta_{5} + \beta_{6} + \beta_{13} - \beta_{14} ) q^{66} + ( -\beta_{5} + \beta_{9} + \beta_{13} + \beta_{14} ) q^{67} + ( -\beta_{2} - \beta_{4} - \beta_{7} - \beta_{8} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{68} + ( -2 \beta_{1} + \beta_{2} - \beta_{5} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{15} ) q^{69} + ( -8 - 2 \beta_{1} - \beta_{2} + \beta_{5} - 3 \beta_{10} + 3 \beta_{11} - \beta_{12} + \beta_{15} ) q^{71} -\beta_{2} q^{72} + ( -3 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - \beta_{13} - \beta_{14} ) q^{73} + ( -2 - 2 \beta_{3} + \beta_{5} - 3 \beta_{6} + \beta_{13} - \beta_{14} ) q^{74} + ( -2 \beta_{1} - \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{12} + \beta_{13} ) q^{75} + ( -4 - 2 \beta_{1} + \beta_{2} - \beta_{5} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{15} ) q^{76} + ( 2 \beta_{4} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{78} + ( 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{79} + ( \beta_{8} - \beta_{10} ) q^{80} + ( -1 - \beta_{3} ) q^{81} + ( 3 \beta_{2} + 2 \beta_{4} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{82} + ( 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{9} + 2 \beta_{12} + 2 \beta_{15} ) q^{83} + ( 3 + 4 \beta_{1} + 5 \beta_{2} + \beta_{4} - 5 \beta_{5} - \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} ) q^{85} + ( -\beta_{1} - 2 \beta_{2} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{86} + ( 2 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{13} - 2 \beta_{14} ) q^{87} + ( -\beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{88} + ( -4 \beta_{1} + 12 \beta_{3} + 4 \beta_{6} + \beta_{7} - \beta_{8} ) q^{89} -\beta_{12} q^{90} + ( \beta_{2} - 2 \beta_{4} - \beta_{5} + 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{15} ) q^{92} + ( 4 \beta_{2} + \beta_{4} + \beta_{7} + \beta_{8} ) q^{93} + ( -4 - 4 \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} + 2 \beta_{13} - 2 \beta_{14} ) q^{94} + ( -4 - 4 \beta_{3} + 5 \beta_{5} + 6 \beta_{6} + \beta_{7} + 3 \beta_{8} - 3 \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{95} -\beta_{3} q^{96} + ( -5 \beta_{2} + 4 \beta_{4} + 5 \beta_{5} - 4 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{15} ) q^{97} + ( -\beta_{1} + \beta_{2} - \beta_{5} + \beta_{12} - \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 8q^{4} + 4q^{5} - 16q^{6} + 8q^{9} + O(q^{10})$$ $$16q + 8q^{4} + 4q^{5} - 16q^{6} + 8q^{9} - 8q^{16} - 24q^{19} + 8q^{20} - 8q^{24} - 4q^{25} + 32q^{29} - 4q^{30} + 32q^{31} + 16q^{34} + 16q^{36} - 48q^{41} - 4q^{45} - 8q^{46} - 8q^{50} - 8q^{51} - 8q^{54} - 40q^{59} + 24q^{61} - 16q^{64} - 28q^{65} + 16q^{69} - 80q^{71} - 16q^{74} + 4q^{75} - 48q^{76} - 16q^{79} + 4q^{80} - 8q^{81} + 56q^{85} - 8q^{86} - 88q^{89} - 24q^{94} - 24q^{95} + 8q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 18 x^{14} + 227 x^{12} - 1394 x^{10} + 6177 x^{8} - 14768 x^{6} + 24768 x^{4} - 11264 x^{2} + 4096$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$19 \nu^{14} - 326 \nu^{12} + 3705 \nu^{10} - 18886 \nu^{8} + 51859 \nu^{6} - 85120 \nu^{4} + 38912 \nu^{2} - 716288$$$$)/496640$$ $$\beta_{2}$$ $$=$$ $$($$$$3301 \nu^{15} - 53930 \nu^{13} + 643695 \nu^{11} - 3281194 \nu^{9} + 12013925 \nu^{7} - 14788480 \nu^{5} + 6760448 \nu^{3} + 46691840 \nu$$$$)/33771520$$ $$\beta_{3}$$ $$=$$ $$($$$$269 \nu^{14} - 4686 \nu^{12} + 58575 \nu^{10} - 344566 \nu^{8} + 1506549 \nu^{6} - 3327060 \nu^{4} + 5963712 \nu^{2} - 2710528$$$$)/2110720$$ $$\beta_{4}$$ $$=$$ $$($$$$-2031 \nu^{15} + 32894 \nu^{13} - 396045 \nu^{11} + 2018814 \nu^{9} - 7668591 \nu^{7} + 9098880 \nu^{5} - 4159488 \nu^{3} - 33238528 \nu$$$$)/16885760$$ $$\beta_{5}$$ $$=$$ $$($$$$-4893 \nu^{15} + 93154 \nu^{13} - 1194855 \nu^{11} + 7811442 \nu^{9} - 35273581 \nu^{7} + 89641160 \nu^{5} - 143948224 \nu^{3} + 65518592 \nu$$$$)/33771520$$ $$\beta_{6}$$ $$=$$ $$($$$$-733 \nu^{14} + 12142 \nu^{12} - 151775 \nu^{10} + 854742 \nu^{8} - 3903653 \nu^{6} + 8620820 \nu^{4} - 16501984 \nu^{2} + 1554176$$$$)/4221440$$ $$\beta_{7}$$ $$=$$ $$($$$$1385 \nu^{15} + 5482 \nu^{14} - 21810 \nu^{13} - 111332 \nu^{12} + 270075 \nu^{11} + 1452510 \nu^{10} - 1376690 \nu^{9} - 10109828 \nu^{8} + 5905385 \nu^{7} + 46442378 \nu^{6} - 6204800 \nu^{5} - 126049360 \nu^{4} + 2836480 \nu^{3} + 192477056 \nu^{2} + 57592320 \nu - 87668736$$$$)/33771520$$ $$\beta_{8}$$ $$=$$ $$($$$$1385 \nu^{15} - 5482 \nu^{14} - 21810 \nu^{13} + 111332 \nu^{12} + 270075 \nu^{11} - 1452510 \nu^{10} - 1376690 \nu^{9} + 10109828 \nu^{8} + 5905385 \nu^{7} - 46442378 \nu^{6} - 6204800 \nu^{5} + 126049360 \nu^{4} + 2836480 \nu^{3} - 192477056 \nu^{2} + 57592320 \nu + 87668736$$$$)/33771520$$ $$\beta_{9}$$ $$=$$ $$($$$$3715 \nu^{15} - 69038 \nu^{13} + 878105 \nu^{11} - 5601470 \nu^{9} + 25102547 \nu^{7} - 63272840 \nu^{5} + 101725760 \nu^{3} - 46285824 \nu$$$$)/16885760$$ $$\beta_{10}$$ $$=$$ $$($$$$4339 \nu^{15} + 3301 \nu^{14} - 75250 \nu^{13} - 53930 \nu^{12} + 940625 \nu^{11} + 643695 \nu^{10} - 5519626 \nu^{9} - 3281194 \nu^{8} + 24192875 \nu^{7} + 12013925 \nu^{6} - 53427500 \nu^{5} - 14788480 \nu^{4} + 85946592 \nu^{3} + 6760448 \nu^{2} - 9632000 \nu + 29806080$$$$)/16885760$$ $$\beta_{11}$$ $$=$$ $$($$$$4339 \nu^{15} - 3301 \nu^{14} - 75250 \nu^{13} + 53930 \nu^{12} + 940625 \nu^{11} - 643695 \nu^{10} - 5519626 \nu^{9} + 3281194 \nu^{8} + 24192875 \nu^{7} - 12013925 \nu^{6} - 53427500 \nu^{5} + 14788480 \nu^{4} + 85946592 \nu^{3} - 6760448 \nu^{2} - 9632000 \nu - 29806080$$$$)/16885760$$ $$\beta_{12}$$ $$=$$ $$($$$$6491 \nu^{15} - 761 \nu^{14} - 112738 \nu^{13} + 11858 \nu^{12} + 1409225 \nu^{11} - 148395 \nu^{10} - 8276154 \nu^{9} + 756434 \nu^{8} + 36245267 \nu^{7} - 3323257 \nu^{6} - 80043980 \nu^{5} + 3409280 \nu^{4} + 133656288 \nu^{3} - 1558528 \nu^{2} - 14430464 \nu - 19785216$$$$)/16885760$$ $$\beta_{13}$$ $$=$$ $$($$$$-11597 \nu^{15} + 6596 \nu^{14} + 203666 \nu^{13} - 113560 \nu^{12} - 2548375 \nu^{11} + 1419500 \nu^{10} + 15175618 \nu^{9} - 8295864 \nu^{8} - 66585149 \nu^{7} + 36509540 \nu^{6} + 153883160 \nu^{5} - 80627600 \nu^{4} - 264476096 \nu^{3} + 122123648 \nu^{2} + 120224768 \nu - 14535680$$$$)/33771520$$ $$\beta_{14}$$ $$=$$ $$($$$$-485 \nu^{15} - 194 \nu^{14} + 8730 \nu^{13} + 3340 \nu^{12} - 110095 \nu^{11} - 41750 \nu^{10} + 676090 \nu^{9} + 243996 \nu^{8} - 2995845 \nu^{7} - 1073810 \nu^{6} + 7162480 \nu^{5} + 2371400 \nu^{4} - 12012480 \nu^{3} - 3591872 \nu^{2} + 5463040 \nu + 427520$$$$)/993280$$ $$\beta_{15}$$ $$=$$ $$($$$$10588 \nu^{15} + 761 \nu^{14} - 186280 \nu^{13} - 11858 \nu^{12} + 2328500 \nu^{11} + 148395 \nu^{10} - 13822472 \nu^{9} - 756434 \nu^{8} + 59889020 \nu^{7} + 3323257 \nu^{6} - 132258800 \nu^{5} - 3409280 \nu^{4} + 209010624 \nu^{3} + 1558528 \nu^{2} - 23843840 \nu + 19785216$$$$)/16885760$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{8} - \beta_{7} - \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{14} + \beta_{13} + \beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} + 4 \beta_{6} + \beta_{5} + 8 \beta_{3} + 8$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{15} + 5 \beta_{12} - 9 \beta_{11} - 9 \beta_{10} + 8 \beta_{9} + 13 \beta_{5} - 8 \beta_{4} - 13 \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-17 \beta_{15} - 17 \beta_{14} + 17 \beta_{13} + 17 \beta_{12} + 9 \beta_{8} - 9 \beta_{7} + 44 \beta_{6} + 56 \beta_{3} + 17 \beta_{2} - 44 \beta_{1}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-37 \beta_{14} - 37 \beta_{13} - 81 \beta_{11} - 81 \beta_{10} + 112 \beta_{9} + 81 \beta_{8} + 81 \beta_{7} + 141 \beta_{5}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-201 \beta_{15} + 201 \beta_{12} - 89 \beta_{11} + 89 \beta_{10} - 201 \beta_{5} + 201 \beta_{2} - 436 \beta_{1} - 472$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-325 \beta_{15} - 325 \beta_{14} - 325 \beta_{13} - 325 \beta_{12} + 761 \beta_{8} + 761 \beta_{7} + 1240 \beta_{4} + 1485 \beta_{2}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$2145 \beta_{14} - 2145 \beta_{13} - 905 \beta_{11} + 905 \beta_{10} - 905 \beta_{8} + 905 \beta_{7} - 4284 \beta_{6} - 2145 \beta_{5} - 4344 \beta_{3} - 4344$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-3077 \beta_{15} - 3077 \beta_{12} + 7361 \beta_{11} + 7361 \beta_{10} - 12864 \beta_{9} - 15277 \beta_{5} + 12864 \beta_{4} + 15277 \beta_{2}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$22041 \beta_{15} + 22041 \beta_{14} - 22041 \beta_{13} - 22041 \beta_{12} - 9177 \beta_{8} + 9177 \beta_{7} - 42308 \beta_{6} - 41752 \beta_{3} - 22041 \beta_{2} + 42308 \beta_{1}$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$30053 \beta_{14} + 30053 \beta_{13} + 72361 \beta_{11} + 72361 \beta_{10} - 130472 \beta_{9} - 72361 \beta_{8} - 72361 \beta_{7} - 154925 \beta_{5}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$222961 \beta_{15} - 222961 \beta_{12} + 92489 \beta_{11} - 92489 \beta_{10} + 222961 \beta_{5} - 222961 \beta_{2} + 419916 \beta_{1} + 409656$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$297317 \beta_{15} + 297317 \beta_{14} + 297317 \beta_{13} + 297317 \beta_{12} - 717233 \beta_{8} - 717233 \beta_{7} - 1311760 \beta_{4} - 1559117 \beta_{2}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$-2239977 \beta_{14} + 2239977 \beta_{13} + 928217 \beta_{11} - 928217 \beta_{10} + 928217 \beta_{8} - 928217 \beta_{7} + 4180692 \beta_{6} + 2239977 \beta_{5} + 4058200 \beta_{3} + 4058200$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$2957317 \beta_{15} + 2957317 \beta_{12} - 7138009 \beta_{11} - 7138009 \beta_{10} + 13140600 \beta_{9} + 15630093 \beta_{5} - 13140600 \beta_{4} - 15630093 \beta_{2}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times$$.

 $$n$$ $$491$$ $$1081$$ $$1177$$ $$\chi(n)$$ $$1$$ $$\beta_{3}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.599547 + 0.346149i −2.73710 − 1.58027i −1.46557 − 0.846149i 1.87108 + 1.08027i −1.87108 − 1.08027i −0.599547 − 0.346149i 1.46557 + 0.846149i 2.73710 + 1.58027i 0.599547 − 0.346149i −2.73710 + 1.58027i −1.46557 + 0.846149i 1.87108 − 1.08027i −1.87108 + 1.08027i −0.599547 + 0.346149i 1.46557 − 0.846149i 2.73710 − 1.58027i
−0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −1.98078 + 1.03755i −1.00000 0 1.00000i 0.500000 0.866025i 2.23418 + 0.0918501i
79.2 −0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −0.506647 2.17791i −1.00000 0 1.00000i 0.500000 0.866025i −0.650187 + 2.13945i
79.3 −0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 1.51486 1.64475i −1.00000 0 1.00000i 0.500000 0.866025i −2.13428 + 0.666969i
79.4 −0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 1.97257 + 1.05307i −1.00000 0 1.00000i 0.500000 0.866025i −1.18176 1.89827i
79.5 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −1.89827 1.18176i −1.00000 0 1.00000i 0.500000 0.866025i −1.05307 1.97257i
79.6 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0.0918501 + 2.23418i −1.00000 0 1.00000i 0.500000 0.866025i −1.03755 + 1.98078i
79.7 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0.666969 2.13428i −1.00000 0 1.00000i 0.500000 0.866025i 1.64475 1.51486i
79.8 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 2.13945 0.650187i −1.00000 0 1.00000i 0.500000 0.866025i 2.17791 + 0.506647i
949.1 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i −1.98078 1.03755i −1.00000 0 1.00000i 0.500000 + 0.866025i 2.23418 0.0918501i
949.2 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i −0.506647 + 2.17791i −1.00000 0 1.00000i 0.500000 + 0.866025i −0.650187 2.13945i
949.3 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i 1.51486 + 1.64475i −1.00000 0 1.00000i 0.500000 + 0.866025i −2.13428 0.666969i
949.4 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i 1.97257 1.05307i −1.00000 0 1.00000i 0.500000 + 0.866025i −1.18176 + 1.89827i
949.5 0.866025 0.500000i −0.866025 0.500000i 0.500000 0.866025i −1.89827 + 1.18176i −1.00000 0 1.00000i 0.500000 + 0.866025i −1.05307 + 1.97257i
949.6 0.866025 0.500000i −0.866025 0.500000i 0.500000 0.866025i 0.0918501 2.23418i −1.00000 0 1.00000i 0.500000 + 0.866025i −1.03755 1.98078i
949.7 0.866025 0.500000i −0.866025 0.500000i 0.500000 0.866025i 0.666969 + 2.13428i −1.00000 0 1.00000i 0.500000 + 0.866025i 1.64475 + 1.51486i
949.8 0.866025 0.500000i −0.866025 0.500000i 0.500000 0.866025i 2.13945 + 0.650187i −1.00000 0 1.00000i 0.500000 + 0.866025i 2.17791 0.506647i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 949.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.n.l 16
5.b even 2 1 inner 1470.2.n.l 16
7.b odd 2 1 1470.2.n.k 16
7.c even 3 1 1470.2.g.j 8
7.c even 3 1 inner 1470.2.n.l 16
7.d odd 6 1 1470.2.g.k yes 8
7.d odd 6 1 1470.2.n.k 16
35.c odd 2 1 1470.2.n.k 16
35.i odd 6 1 1470.2.g.k yes 8
35.i odd 6 1 1470.2.n.k 16
35.j even 6 1 1470.2.g.j 8
35.j even 6 1 inner 1470.2.n.l 16
35.k even 12 1 7350.2.a.dr 4
35.k even 12 1 7350.2.a.du 4
35.l odd 12 1 7350.2.a.ds 4
35.l odd 12 1 7350.2.a.dt 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.g.j 8 7.c even 3 1
1470.2.g.j 8 35.j even 6 1
1470.2.g.k yes 8 7.d odd 6 1
1470.2.g.k yes 8 35.i odd 6 1
1470.2.n.k 16 7.b odd 2 1
1470.2.n.k 16 7.d odd 6 1
1470.2.n.k 16 35.c odd 2 1
1470.2.n.k 16 35.i odd 6 1
1470.2.n.l 16 1.a even 1 1 trivial
1470.2.n.l 16 5.b even 2 1 inner
1470.2.n.l 16 7.c even 3 1 inner
1470.2.n.l 16 35.j even 6 1 inner
7350.2.a.dr 4 35.k even 12 1
7350.2.a.ds 4 35.l odd 12 1
7350.2.a.dt 4 35.l odd 12 1
7350.2.a.du 4 35.k even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1470, [\chi])$$:

 $$T_{11}^{8} + 18 T_{11}^{6} - 32 T_{11}^{5} + 292 T_{11}^{4} - 288 T_{11}^{3} + 832 T_{11}^{2} + 512 T_{11} + 1024$$ $$T_{17}^{16} - \cdots$$ $$T_{19}^{8} + \cdots$$ $$T_{31}^{8} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{4}$$
$3$ $$( 1 - T^{2} + T^{4} )^{4}$$
$5$ $$390625 - 312500 T + 156250 T^{2} - 50000 T^{3} + 11250 T^{4} - 3500 T^{5} + 5600 T^{6} - 4340 T^{7} + 2399 T^{8} - 868 T^{9} + 224 T^{10} - 28 T^{11} + 18 T^{12} - 16 T^{13} + 10 T^{14} - 4 T^{15} + T^{16}$$
$7$ $$T^{16}$$
$11$ $$( 1024 + 512 T + 832 T^{2} - 288 T^{3} + 292 T^{4} - 32 T^{5} + 18 T^{6} + T^{8} )^{2}$$
$13$ $$( 256 + 1472 T^{2} + 644 T^{4} + 52 T^{6} + T^{8} )^{2}$$
$17$ $$236421376 - 123008000 T^{2} + 43457664 T^{4} - 8227840 T^{6} + 1129520 T^{8} - 90880 T^{10} + 5064 T^{12} - 80 T^{14} + T^{16}$$
$19$ $$( 200704 + 100352 T + 55552 T^{2} + 8064 T^{3} + 3280 T^{4} + 592 T^{5} + 132 T^{6} + 12 T^{7} + T^{8} )^{2}$$
$23$ $$268435456 - 150994944 T^{2} + 59506688 T^{4} - 11419648 T^{6} + 1581312 T^{8} - 118144 T^{10} + 6192 T^{12} - 88 T^{14} + T^{16}$$
$29$ $$( 1568 + 336 T - 66 T^{2} - 8 T^{3} + T^{4} )^{4}$$
$31$ $$( 4096 - 8192 T + 11392 T^{2} - 7936 T^{3} + 3972 T^{4} - 992 T^{5} + 178 T^{6} - 16 T^{7} + T^{8} )^{2}$$
$37$ $$1600000000 - 1248000000 T^{2} + 810080000 T^{4} - 115580800 T^{6} + 12021456 T^{8} - 542032 T^{10} + 17820 T^{12} - 148 T^{14} + T^{16}$$
$41$ $$( -376 - 120 T + 26 T^{2} + 12 T^{3} + T^{4} )^{4}$$
$43$ $$( 541696 + 297344 T^{2} + 16516 T^{4} + 236 T^{6} + T^{8} )^{2}$$
$47$ $$1099511627776 - 210453397504 T^{2} + 28600958976 T^{4} - 1808023552 T^{6} + 82107408 T^{8} - 1871152 T^{10} + 30476 T^{12} - 204 T^{14} + T^{16}$$
$53$ $$16777216 - 48234496 T^{2} + 116588544 T^{4} - 61988864 T^{6} + 26902784 T^{8} - 968576 T^{10} + 28464 T^{12} - 184 T^{14} + T^{16}$$
$59$ $$( 9048064 + 2213888 T + 698112 T^{2} + 82048 T^{3} + 20432 T^{4} + 2512 T^{5} + 348 T^{6} + 20 T^{7} + T^{8} )^{2}$$
$61$ $$( 419904 - 979776 T + 2204496 T^{2} - 206064 T^{3} + 34668 T^{4} - 1512 T^{5} + 270 T^{6} - 12 T^{7} + T^{8} )^{2}$$
$67$ $$1048576 - 1179648 T^{2} + 929792 T^{4} - 356864 T^{6} + 98832 T^{8} - 14768 T^{10} + 1548 T^{12} - 44 T^{14} + T^{16}$$
$71$ $$( 2944 - 1472 T - 36 T^{2} + 20 T^{3} + T^{4} )^{4}$$
$73$ $$3782742016 - 15046338560 T^{2} + 58633164544 T^{4} - 4796194432 T^{6} + 313247952 T^{8} - 5756144 T^{10} + 80092 T^{12} - 316 T^{14} + T^{16}$$
$79$ $$( 12845056 + 6422528 T + 2580480 T^{2} + 372736 T^{3} + 48896 T^{4} + 2176 T^{5} + 240 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$83$ $$( 16384 + 26624 T^{2} + 6976 T^{4} + 240 T^{6} + T^{8} )^{2}$$
$89$ $$( 25482304 + 18617024 T + 10279760 T^{2} + 1982480 T^{3} + 265644 T^{4} + 21576 T^{5} + 1278 T^{6} + 44 T^{7} + T^{8} )^{2}$$
$97$ $$( 107661376 + 4479328 T^{2} + 66804 T^{4} + 428 T^{6} + T^{8} )^{2}$$