# Properties

 Label 315.2.p.e Level 315 Weight 2 Character orbit 315.p Analytic conductor 2.515 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.p (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( -\beta_{6} - \beta_{7} + \beta_{13} ) q^{4} + ( -\beta_{1} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{15} ) q^{5} + ( \beta_{7} + \beta_{8} - \beta_{11} - \beta_{15} ) q^{7} + ( -2 + \beta_{4} - \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{8} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( -\beta_{6} - \beta_{7} + \beta_{13} ) q^{4} + ( -\beta_{1} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{15} ) q^{5} + ( \beta_{7} + \beta_{8} - \beta_{11} - \beta_{15} ) q^{7} + ( -2 + \beta_{4} - \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{8} + ( -\beta_{1} - \beta_{3} - \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{15} ) q^{10} + ( 2 + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} + \beta_{11} - \beta_{13} ) q^{11} + ( -\beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{13} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{14} + ( -3 + 2 \beta_{2} + 2 \beta_{4} ) q^{16} + ( -\beta_{1} - \beta_{3} - \beta_{9} - \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{17} + ( -2 \beta_{1} - \beta_{5} + \beta_{6} + \beta_{8} - 3 \beta_{10} - 4 \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{19} + ( \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{20} + ( -1 - 2 \beta_{2} + \beta_{7} ) q^{22} + ( 2 - 2 \beta_{4} + 3 \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{23} + ( -1 - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{13} ) q^{25} + ( -2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{15} ) q^{26} + ( 2 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{13} - \beta_{14} ) q^{28} + ( -\beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} + \beta_{11} - \beta_{13} ) q^{29} + ( -2 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} + 3 \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{31} + ( -2 + \beta_{2} + 4 \beta_{7} + 2 \beta_{8} - 2 \beta_{13} ) q^{32} + ( 2 \beta_{1} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 5 \beta_{10} + 8 \beta_{11} + \beta_{12} - 2 \beta_{13} + 4 \beta_{14} - \beta_{15} ) q^{34} + ( -\beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{35} + ( 3 + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{13} ) q^{37} + ( 3 \beta_{1} - 3 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} + 5 \beta_{10} + 5 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{38} + ( -\beta_{1} + 3 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} - 3 \beta_{9} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{40} + ( 2 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} + 4 \beta_{9} - 2 \beta_{11} + 3 \beta_{12} - \beta_{13} - \beta_{15} ) q^{41} + ( -2 + 2 \beta_{8} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{13} ) q^{43} + ( -\beta_{2} + \beta_{4} - 2 \beta_{5} - 6 \beta_{7} - 2 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{44} + ( 4 - 2 \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{8} ) q^{46} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{47} + ( 2 \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{10} - \beta_{13} + 2 \beta_{14} ) q^{49} + ( 4 + \beta_{2} + 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} - \beta_{13} ) q^{50} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{52} + ( -2 + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} - \beta_{13} ) q^{53} + ( 3 \beta_{9} - 3 \beta_{11} + \beta_{14} - \beta_{15} ) q^{55} + ( -2 - \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{9} - 3 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{15} ) q^{56} + ( 2 - 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{13} ) q^{58} + ( -4 \beta_{1} - 2 \beta_{10} - 4 \beta_{11} + 2 \beta_{12} - 2 \beta_{15} ) q^{59} + ( 4 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 3 \beta_{13} + 4 \beta_{15} ) q^{61} + ( \beta_{1} + \beta_{3} + 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{8} + 3 \beta_{9} - \beta_{10} + 7 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} + 3 \beta_{14} ) q^{62} + ( 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - 3 \beta_{7} - 2 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} - \beta_{13} ) q^{64} + ( -4 - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{13} ) q^{65} + ( -2 - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{13} ) q^{67} + ( -5 \beta_{1} + 5 \beta_{3} + \beta_{9} - 6 \beta_{10} - 6 \beta_{11} + \beta_{12} - \beta_{14} ) q^{68} + ( -3 + 3 \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{6} - 3 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{70} + ( -2 - \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{13} ) q^{71} + ( -4 \beta_{1} + 4 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - 5 \beta_{10} - 5 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{73} + ( -\beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{74} + ( -2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} + 4 \beta_{9} - 9 \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} - 3 \beta_{15} ) q^{76} + ( 2 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{10} - \beta_{11} + \beta_{12} - 3 \beta_{13} + \beta_{14} ) q^{77} + ( -4 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{13} ) q^{79} + ( 3 \beta_{1} + 4 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - \beta_{9} - \beta_{10} - 7 \beta_{11} - 3 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{80} + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} - 3 \beta_{9} - \beta_{10} - \beta_{11} - 3 \beta_{12} + \beta_{13} - 3 \beta_{14} + 4 \beta_{15} ) q^{82} + ( -3 \beta_{5} + 3 \beta_{6} + 3 \beta_{8} - 3 \beta_{10} - 3 \beta_{11} - 6 \beta_{12} + 3 \beta_{13} ) q^{83} + ( 3 + 4 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{13} ) q^{85} + ( -4 + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{8} ) q^{86} + ( -5 + 2 \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{8} - 2 \beta_{10} - 2 \beta_{11} + \beta_{13} ) q^{88} + ( \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{89} + ( -4 - 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{13} - 2 \beta_{15} ) q^{91} + ( 4 - 2 \beta_{2} + 5 \beta_{5} - 5 \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{13} ) q^{92} + ( 4 \beta_{1} - \beta_{5} + \beta_{6} + \beta_{8} - 8 \beta_{11} + 2 \beta_{12} + \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{94} + ( 2 + \beta_{2} - \beta_{4} - 3 \beta_{5} + \beta_{6} - 5 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} - 2 \beta_{11} + 4 \beta_{13} ) q^{95} + ( 4 \beta_{1} + 4 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 3 \beta_{9} + 5 \beta_{10} - \beta_{11} + 3 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} ) q^{97} + ( 6 + \beta_{4} + \beta_{5} + 3 \beta_{6} + 6 \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 8q^{7} - 24q^{8} + O(q^{10})$$ $$16q - 8q^{7} - 24q^{8} + 16q^{11} - 48q^{16} - 16q^{22} + 40q^{23} + 24q^{28} - 48q^{32} + 8q^{35} + 32q^{37} - 16q^{43} + 64q^{46} + 72q^{50} - 24q^{53} - 24q^{56} + 32q^{58} - 40q^{65} - 32q^{67} - 40q^{70} - 64q^{71} + 24q^{77} + 48q^{85} - 64q^{86} - 64q^{88} - 48q^{91} + 40q^{92} + 72q^{95} + 96q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{14} + 6 x^{12} - 12 x^{10} + 33 x^{8} - 48 x^{6} + 96 x^{4} - 256 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$13 \nu^{15} - 168 \nu^{13} + 174 \nu^{11} - 84 \nu^{9} - 3 \nu^{7} + 732 \nu^{5} - 336 \nu^{3} - 5056 \nu$$$$)/7680$$ $$\beta_{2}$$ $$=$$ $$($$$$17 \nu^{14} - 72 \nu^{12} + 6 \nu^{10} - 36 \nu^{8} + 513 \nu^{6} - 1332 \nu^{4} + 3696 \nu^{2} - 4544$$$$)/3840$$ $$\beta_{3}$$ $$=$$ $$($$$$31 \nu^{15} - 96 \nu^{13} + 378 \nu^{11} - 588 \nu^{9} + 1359 \nu^{7} - 2676 \nu^{5} + 4368 \nu^{3} - 2752 \nu$$$$)/7680$$ $$\beta_{4}$$ $$=$$ $$($$$$43 \nu^{14} - 48 \nu^{12} - 126 \nu^{10} + 36 \nu^{8} + 27 \nu^{6} + 492 \nu^{4} + 144 \nu^{2} - 1216$$$$)/3840$$ $$\beta_{5}$$ $$=$$ $$($$$$-29 \nu^{15} + 16 \nu^{14} + 144 \nu^{13} - 336 \nu^{12} + 18 \nu^{11} + 288 \nu^{10} + 132 \nu^{9} - 288 \nu^{8} - 621 \nu^{7} + 3024 \nu^{6} + 204 \nu^{5} - 1296 \nu^{4} - 1392 \nu^{3} - 1152 \nu^{2} + 4928 \nu - 17152$$$$)/7680$$ $$\beta_{6}$$ $$=$$ $$($$$$-29 \nu^{15} + 20 \nu^{14} + 144 \nu^{13} + 18 \nu^{11} + 120 \nu^{10} + 132 \nu^{9} + 240 \nu^{8} - 621 \nu^{7} + 1620 \nu^{6} + 204 \nu^{5} - 2160 \nu^{4} - 1392 \nu^{3} + 960 \nu^{2} + 4928 \nu - 1280$$$$)/7680$$ $$\beta_{7}$$ $$=$$ $$($$$$-43 \nu^{14} + 108 \nu^{12} - 114 \nu^{10} + 324 \nu^{8} - 747 \nu^{6} + 528 \nu^{4} - 3024 \nu^{2} + 6016$$$$)/1920$$ $$\beta_{8}$$ $$=$$ $$($$$$-29 \nu^{15} - 104 \nu^{14} + 144 \nu^{13} + 144 \nu^{12} + 18 \nu^{11} - 432 \nu^{10} + 132 \nu^{9} + 1152 \nu^{8} - 621 \nu^{7} - 936 \nu^{6} + 204 \nu^{5} + 4464 \nu^{4} - 1392 \nu^{3} - 4992 \nu^{2} + 4928 \nu + 5888$$$$)/7680$$ $$\beta_{9}$$ $$=$$ $$($$$$83 \nu^{15} - 48 \nu^{13} + 114 \nu^{11} - 444 \nu^{9} + 387 \nu^{7} - 2868 \nu^{5} + 4944 \nu^{3} - 7616 \nu$$$$)/7680$$ $$\beta_{10}$$ $$=$$ $$($$$$\nu^{15} - \nu^{13} + 2 \nu^{11} - 10 \nu^{9} + 13 \nu^{7} - 13 \nu^{5} + 56 \nu^{3} - 80 \nu$$$$)/64$$ $$\beta_{11}$$ $$=$$ $$($$$$-89 \nu^{15} + 204 \nu^{13} - 102 \nu^{11} + 732 \nu^{9} - 1401 \nu^{7} + 984 \nu^{5} - 4752 \nu^{3} + 9728 \nu$$$$)/3840$$ $$\beta_{12}$$ $$=$$ $$($$$$83 \nu^{15} - 270 \nu^{14} - 48 \nu^{13} + 720 \nu^{12} + 114 \nu^{11} - 660 \nu^{10} - 444 \nu^{9} + 3000 \nu^{8} + 387 \nu^{7} - 5550 \nu^{6} + 972 \nu^{5} + 4920 \nu^{4} + 4944 \nu^{3} - 18720 \nu^{2} - 3776 \nu + 44160$$$$)/7680$$ $$\beta_{13}$$ $$=$$ $$($$$$-29 \nu^{15} - 440 \nu^{14} + 144 \nu^{13} + 960 \nu^{12} + 18 \nu^{11} - 720 \nu^{10} + 132 \nu^{9} + 4320 \nu^{8} - 621 \nu^{7} - 8760 \nu^{6} + 204 \nu^{5} + 6240 \nu^{4} - 1392 \nu^{3} - 34560 \nu^{2} + 4928 \nu + 66560$$$$)/7680$$ $$\beta_{14}$$ $$=$$ $$($$$$122 \nu^{15} - 135 \nu^{14} - 312 \nu^{13} + 360 \nu^{12} + 156 \nu^{11} - 330 \nu^{10} - 1176 \nu^{9} + 1500 \nu^{8} + 2298 \nu^{7} - 2775 \nu^{6} - 2352 \nu^{5} + 2460 \nu^{4} + 10176 \nu^{3} - 9360 \nu^{2} - 18944 \nu + 22080$$$$)/3840$$ $$\beta_{15}$$ $$=$$ $$($$$$337 \nu^{15} - 270 \nu^{14} - 672 \nu^{13} + 720 \nu^{12} + 486 \nu^{11} - 660 \nu^{10} - 2676 \nu^{9} + 3000 \nu^{8} + 5793 \nu^{7} - 5550 \nu^{6} - 4812 \nu^{5} + 4920 \nu^{4} + 21936 \nu^{3} - 18720 \nu^{2} - 46144 \nu + 44160$$$$)/7680$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{15} + \beta_{12} - 2 \beta_{11} - \beta_{10} + \beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} + 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{15} + 2 \beta_{14} - 2 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} - 2 \beta_{8} - 2 \beta_{6} + 2 \beta_{5} + \beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{11} - \beta_{10} + 3 \beta_{8} - 3 \beta_{7} - \beta_{5} - \beta_{4} - 3 \beta_{2} + 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{15} - 2 \beta_{13} + 3 \beta_{12} + 4 \beta_{11} + 3 \beta_{10} - 4 \beta_{9} - 2 \beta_{8} - 2 \beta_{6} + 2 \beta_{5} - \beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-2 \beta_{13} - 4 \beta_{11} - 4 \beta_{10} + 3 \beta_{8} + 3 \beta_{7} + 6 \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{2} + 5$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$7 \beta_{15} - 6 \beta_{14} - \beta_{12} - 3 \beta_{11} - 6 \beta_{10} - 8 \beta_{9} + 5 \beta_{3} - 9 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$6 \beta_{13} - 9 \beta_{11} - 9 \beta_{10} + 5 \beta_{8} - 15 \beta_{7} + 10 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} + 5 \beta_{2} - 7$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$13 \beta_{15} - 12 \beta_{14} - 8 \beta_{13} + 15 \beta_{12} + 2 \beta_{11} - 31 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} - 8 \beta_{6} + 8 \beta_{5} + 11 \beta_{3} - 7 \beta_{1}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$12 \beta_{13} - 8 \beta_{11} - 8 \beta_{10} + \beta_{8} - 45 \beta_{7} + 12 \beta_{6} - 9 \beta_{5} - 21 \beta_{4} - 19 \beta_{2} - 11$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$15 \beta_{15} - 14 \beta_{14} - 6 \beta_{13} + 11 \beta_{12} + 31 \beta_{11} + 12 \beta_{10} - 6 \beta_{8} - 6 \beta_{6} + 6 \beta_{5} + 25 \beta_{3} + 31 \beta_{1}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$-12 \beta_{13} - 9 \beta_{11} - 9 \beta_{10} + 7 \beta_{8} + 13 \beta_{7} + 60 \beta_{6} - 37 \beta_{5} - 17 \beta_{4} - 43 \beta_{2} - 71$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$49 \beta_{15} - 24 \beta_{14} - 6 \beta_{13} - 13 \beta_{12} + 88 \beta_{11} + 63 \beta_{10} - 12 \beta_{9} - 6 \beta_{8} - 6 \beta_{6} + 6 \beta_{5} - 17 \beta_{3} - 23 \beta_{1}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$18 \beta_{13} - 12 \beta_{11} - 12 \beta_{10} - 33 \beta_{8} - 69 \beta_{7} + 90 \beta_{6} - 51 \beta_{5} + 111 \beta_{4} - 75 \beta_{2} - 67$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$47 \beta_{15} - 150 \beta_{14} + 12 \beta_{13} + 79 \beta_{12} - 191 \beta_{11} - 106 \beta_{10} + 120 \beta_{9} + 12 \beta_{8} + 12 \beta_{6} - 12 \beta_{5} - 11 \beta_{3} - 49 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$-\beta_{7}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
118.1
 −0.944649 + 1.05244i 0.944649 − 1.05244i −1.36166 + 0.381939i 1.36166 − 0.381939i −1.40927 + 0.118126i 1.40927 − 0.118126i 0.517174 + 1.31626i −0.517174 − 1.31626i −0.944649 − 1.05244i 0.944649 + 1.05244i −1.36166 − 0.381939i 1.36166 + 0.381939i −1.40927 − 0.118126i 1.40927 + 0.118126i 0.517174 − 1.31626i −0.517174 + 1.31626i
−1.48838 + 1.48838i 0 2.43055i −1.28999 + 1.82645i 0 −1.75993 + 1.97552i 0.640825 + 0.640825i 0 −0.798469 4.63845i
118.2 −1.48838 + 1.48838i 0 2.43055i 1.28999 1.82645i 0 −1.97552 + 1.75993i 0.640825 + 0.640825i 0 0.798469 + 4.63845i
118.3 −0.540143 + 0.540143i 0 1.41649i −1.03649 1.98133i 0 0.614060 2.57351i −1.84539 1.84539i 0 1.63006 + 0.510348i
118.4 −0.540143 + 0.540143i 0 1.41649i 1.03649 + 1.98133i 0 2.57351 0.614060i −1.84539 1.84539i 0 −1.63006 0.510348i
118.5 0.167056 0.167056i 0 1.94418i −2.23450 + 0.0836010i 0 −2.64501 + 0.0627175i 0.658899 + 0.658899i 0 −0.359321 + 0.387253i
118.6 0.167056 0.167056i 0 1.94418i 2.23450 0.0836010i 0 −0.0627175 + 2.64501i 0.658899 + 0.658899i 0 0.359321 0.387253i
118.7 1.86147 1.86147i 0 4.93012i −1.50619 1.65269i 0 1.46123 + 2.20563i −5.45433 5.45433i 0 −5.88016 0.272713i
118.8 1.86147 1.86147i 0 4.93012i 1.50619 + 1.65269i 0 −2.20563 1.46123i −5.45433 5.45433i 0 5.88016 + 0.272713i
307.1 −1.48838 1.48838i 0 2.43055i −1.28999 1.82645i 0 −1.75993 1.97552i 0.640825 0.640825i 0 −0.798469 + 4.63845i
307.2 −1.48838 1.48838i 0 2.43055i 1.28999 + 1.82645i 0 −1.97552 1.75993i 0.640825 0.640825i 0 0.798469 4.63845i
307.3 −0.540143 0.540143i 0 1.41649i −1.03649 + 1.98133i 0 0.614060 + 2.57351i −1.84539 + 1.84539i 0 1.63006 0.510348i
307.4 −0.540143 0.540143i 0 1.41649i 1.03649 1.98133i 0 2.57351 + 0.614060i −1.84539 + 1.84539i 0 −1.63006 + 0.510348i
307.5 0.167056 + 0.167056i 0 1.94418i −2.23450 0.0836010i 0 −2.64501 0.0627175i 0.658899 0.658899i 0 −0.359321 0.387253i
307.6 0.167056 + 0.167056i 0 1.94418i 2.23450 + 0.0836010i 0 −0.0627175 2.64501i 0.658899 0.658899i 0 0.359321 + 0.387253i
307.7 1.86147 + 1.86147i 0 4.93012i −1.50619 + 1.65269i 0 1.46123 2.20563i −5.45433 + 5.45433i 0 −5.88016 + 0.272713i
307.8 1.86147 + 1.86147i 0 4.93012i 1.50619 1.65269i 0 −2.20563 + 1.46123i −5.45433 + 5.45433i 0 5.88016 0.272713i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 307.8 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.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.p.e 16
3.b odd 2 1 105.2.m.a 16
5.c odd 4 1 inner 315.2.p.e 16
7.b odd 2 1 inner 315.2.p.e 16
12.b even 2 1 1680.2.cz.d 16
15.d odd 2 1 525.2.m.b 16
15.e even 4 1 105.2.m.a 16
15.e even 4 1 525.2.m.b 16
21.c even 2 1 105.2.m.a 16
21.g even 6 2 735.2.v.a 32
21.h odd 6 2 735.2.v.a 32
35.f even 4 1 inner 315.2.p.e 16
60.l odd 4 1 1680.2.cz.d 16
84.h odd 2 1 1680.2.cz.d 16
105.g even 2 1 525.2.m.b 16
105.k odd 4 1 105.2.m.a 16
105.k odd 4 1 525.2.m.b 16
105.w odd 12 2 735.2.v.a 32
105.x even 12 2 735.2.v.a 32
420.w even 4 1 1680.2.cz.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.m.a 16 3.b odd 2 1
105.2.m.a 16 15.e even 4 1
105.2.m.a 16 21.c even 2 1
105.2.m.a 16 105.k odd 4 1
315.2.p.e 16 1.a even 1 1 trivial
315.2.p.e 16 5.c odd 4 1 inner
315.2.p.e 16 7.b odd 2 1 inner
315.2.p.e 16 35.f even 4 1 inner
525.2.m.b 16 15.d odd 2 1
525.2.m.b 16 15.e even 4 1
525.2.m.b 16 105.g even 2 1
525.2.m.b 16 105.k odd 4 1
735.2.v.a 32 21.g even 6 2
735.2.v.a 32 21.h odd 6 2
735.2.v.a 32 105.w odd 12 2
735.2.v.a 32 105.x even 12 2
1680.2.cz.d 16 12.b even 2 1
1680.2.cz.d 16 60.l odd 4 1
1680.2.cz.d 16 84.h odd 2 1
1680.2.cz.d 16 420.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$:

 $$T_{2}^{8} + 4 T_{2}^{5} + 34 T_{2}^{4} + 24 T_{2}^{3} + 8 T_{2}^{2} - 4 T_{2} + 1$$ $$T_{17}^{16} + 2160 T_{17}^{12} + 268896 T_{17}^{8} + 9541376 T_{17}^{4} + 100000000$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 4 T^{2} - 6 T^{3} + 9 T^{4} - 12 T^{5} + 16 T^{6} - 16 T^{7} + 16 T^{8} )^{2}( 1 + 2 T + 2 T^{3} + 9 T^{4} + 4 T^{5} + 16 T^{7} + 16 T^{8} )^{2}$$
$3$ 
$5$ $$1 + 28 T^{4} - 256 T^{6} - 26 T^{8} - 6400 T^{10} + 17500 T^{12} + 390625 T^{16}$$
$7$ $$1 + 8 T + 32 T^{2} + 88 T^{3} + 196 T^{4} + 248 T^{5} - 416 T^{6} - 2840 T^{7} - 8634 T^{8} - 19880 T^{9} - 20384 T^{10} + 85064 T^{11} + 470596 T^{12} + 1479016 T^{13} + 3764768 T^{14} + 6588344 T^{15} + 5764801 T^{16}$$
$11$ $$( 1 - 4 T + 32 T^{2} - 68 T^{3} + 402 T^{4} - 748 T^{5} + 3872 T^{6} - 5324 T^{7} + 14641 T^{8} )^{4}$$
$13$ $$1 + 424 T^{4} + 47004 T^{8} - 12160488 T^{12} - 4129271418 T^{16} - 347315697768 T^{20} + 38342606809884 T^{24} + 9878388091931944 T^{28} + 665416609183179841 T^{32}$$
$17$ $$1 + 120 T^{4} + 166556 T^{8} - 9625784 T^{12} + 12237871174 T^{16} - 803955105464 T^{20} + 1161854256343196 T^{24} + 69914668467571320 T^{28} + 48661191875666868481 T^{32}$$
$19$ $$( 1 + 48 T^{2} + 1524 T^{4} + 40016 T^{6} + 818246 T^{8} + 14445776 T^{10} + 198609204 T^{12} + 2258202288 T^{14} + 16983563041 T^{16} )^{2}$$
$23$ $$( 1 - 20 T + 200 T^{2} - 1516 T^{3} + 10388 T^{4} - 65340 T^{5} + 378328 T^{6} - 2076676 T^{7} + 10539814 T^{8} - 47763548 T^{9} + 200135512 T^{10} - 794991780 T^{11} + 2906988308 T^{12} - 9757495988 T^{13} + 29607177800 T^{14} - 68096508940 T^{15} + 78310985281 T^{16} )^{2}$$
$29$ $$( 1 - 184 T^{2} + 15868 T^{4} - 835400 T^{6} + 29324070 T^{8} - 702571400 T^{10} + 11223134908 T^{12} - 109447491064 T^{14} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 - 128 T^{2} + 9396 T^{4} - 463040 T^{6} + 16703398 T^{8} - 444981440 T^{10} + 8677403316 T^{12} - 113600471168 T^{14} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 - 16 T + 128 T^{2} - 944 T^{3} + 8860 T^{4} - 73552 T^{5} + 488320 T^{6} - 3175280 T^{7} + 20212134 T^{8} - 117485360 T^{9} + 668510080 T^{10} - 3725629456 T^{11} + 16605066460 T^{12} - 65460695408 T^{13} + 328412980352 T^{14} - 1518910034128 T^{15} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 - 144 T^{2} + 12956 T^{4} - 823792 T^{6} + 38320198 T^{8} - 1384794352 T^{10} + 36610559516 T^{12} - 684015010704 T^{14} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 + 8 T + 32 T^{2} + 280 T^{3} + 2788 T^{4} + 14232 T^{5} + 63840 T^{6} + 569416 T^{7} + 5017638 T^{8} + 24484888 T^{9} + 118040160 T^{10} + 1131543624 T^{11} + 9531617188 T^{12} + 41162364040 T^{13} + 202283617568 T^{14} + 2174548888856 T^{15} + 11688200277601 T^{16} )^{2}$$
$47$ $$1 + 3784 T^{4} + 1124764 T^{8} + 9138019192 T^{12} + 63538455194182 T^{16} + 44590618628837752 T^{20} + 26782078030828949404 T^{24} +$$$$43\!\cdots\!44$$$$T^{28} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$( 1 + 12 T + 72 T^{2} + 572 T^{3} + 1780 T^{4} - 1436 T^{5} + 18200 T^{6} + 441076 T^{7} + 6254598 T^{8} + 23377028 T^{9} + 51123800 T^{10} - 213787372 T^{11} + 14045056180 T^{12} + 239207821996 T^{13} + 1595834001288 T^{14} + 14096533678044 T^{15} + 62259690411361 T^{16} )^{2}$$
$59$ $$( 1 + 312 T^{2} + 49660 T^{4} + 5040712 T^{6} + 354176614 T^{8} + 17546718472 T^{10} + 601748147260 T^{12} + 13160326495992 T^{14} + 146830437604321 T^{16} )^{2}$$
$61$ $$( 1 - 200 T^{2} + 17532 T^{4} - 857912 T^{6} + 37932838 T^{8} - 3192290552 T^{10} + 242745284412 T^{12} - 10304074872200 T^{14} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 + 16 T + 128 T^{2} + 1424 T^{3} + 22436 T^{4} + 211280 T^{5} + 1522560 T^{6} + 15870032 T^{7} + 163564774 T^{8} + 1063292144 T^{9} + 6834771840 T^{10} + 63545206640 T^{11} + 452110550756 T^{12} + 1922578152368 T^{13} + 11578672917632 T^{14} + 96971385685168 T^{15} + 406067677556641 T^{16} )^{2}$$
$71$ $$( 1 + 16 T + 232 T^{2} + 2376 T^{3} + 21730 T^{4} + 168696 T^{5} + 1169512 T^{6} + 5726576 T^{7} + 25411681 T^{8} )^{4}$$
$73$ $$1 - 15256 T^{4} + 80862300 T^{8} + 94053698264 T^{12} - 2362018367550906 T^{16} + 2670959590242353624 T^{20} +$$$$65\!\cdots\!00$$$$T^{24} -$$$$34\!\cdots\!76$$$$T^{28} +$$$$65\!\cdots\!61$$$$T^{32}$$
$79$ $$( 1 - 312 T^{2} + 58396 T^{4} - 7293320 T^{6} + 672141766 T^{8} - 45517610120 T^{10} + 2274528930076 T^{12} - 75843286122552 T^{14} + 1517108809906561 T^{16} )^{2}$$
$83$ $$1 + 5000 T^{4} + 95818588 T^{8} + 596752860728 T^{12} + 4571727903671302 T^{16} + 28320888822097717688 T^{20} +$$$$21\!\cdots\!08$$$$T^{24} +$$$$53\!\cdots\!00$$$$T^{28} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$( 1 + 576 T^{2} + 155068 T^{4} + 25338304 T^{6} + 2740378246 T^{8} + 200704705984 T^{10} + 9729313827388 T^{12} + 286261223593536 T^{14} + 3936588805702081 T^{16} )^{2}$$
$97$ $$1 - 55064 T^{4} + 1465436892 T^{8} - 24336256217256 T^{12} + 274732504520067270 T^{16} -$$$$21\!\cdots\!36$$$$T^{20} +$$$$11\!\cdots\!12$$$$T^{24} -$$$$38\!\cdots\!24$$$$T^{28} +$$$$61\!\cdots\!21$$$$T^{32}$$
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