# Properties

 Label 315.4.d.b Level $315$ Weight $4$ Character orbit 315.d Analytic conductor $18.586$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 315.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.5856016518$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 37 x^{8} + 398 x^{6} + 1149 x^{4} + 1040 x^{2} + 100$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{9}\cdot 5$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} + ( -5 + \beta_{1} + \beta_{5} ) q^{4} + ( 2 + \beta_{1} - \beta_{6} - \beta_{8} ) q^{5} + 7 \beta_{2} q^{7} + ( 1 + \beta_{1} + 7 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 6 \beta_{6} - \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{8} +O(q^{10})$$ $$q + \beta_{6} q^{2} + ( -5 + \beta_{1} + \beta_{5} ) q^{4} + ( 2 + \beta_{1} - \beta_{6} - \beta_{8} ) q^{5} + 7 \beta_{2} q^{7} + ( 1 + \beta_{1} + 7 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 6 \beta_{6} - \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{8} + ( 8 - 4 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} + 4 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{10} + ( -14 - 8 \beta_{1} + 2 \beta_{3} ) q^{11} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 7 \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{13} -7 \beta_{1} q^{14} + ( 32 - 6 \beta_{1} - \beta_{2} + 5 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{16} + ( 1 + \beta_{1} - 19 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 5 \beta_{6} - 9 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{17} + ( -31 + \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 7 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{19} + ( -38 - 5 \beta_{1} - 27 \beta_{2} - \beta_{4} + 5 \beta_{5} + 23 \beta_{6} + 4 \beta_{8} - 5 \beta_{9} ) q^{20} + ( -2 - 2 \beta_{1} - 104 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 8 \beta_{6} - 8 \beta_{7} + 6 \beta_{8} - 2 \beta_{9} ) q^{22} + ( -3 - 3 \beta_{1} - 31 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 9 \beta_{6} + 5 \beta_{7} + 9 \beta_{8} - 5 \beta_{9} ) q^{23} + ( -42 + 3 \beta_{1} + 13 \beta_{2} - 5 \beta_{3} + 9 \beta_{4} - 5 \beta_{5} - 15 \beta_{6} - 5 \beta_{7} + \beta_{8} - 5 \beta_{9} ) q^{25} + ( -85 + 9 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 13 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{26} + ( -35 \beta_{2} + 7 \beta_{6} - 7 \beta_{7} ) q^{28} + ( 74 - 10 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{29} + ( 67 - 27 \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 9 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{31} + ( -2 - 2 \beta_{1} + 14 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 27 \beta_{6} - 10 \beta_{7} + 6 \beta_{8} - 6 \beta_{9} ) q^{32} + ( -23 + 83 \beta_{1} + 11 \beta_{2} - 7 \beta_{3} + 33 \beta_{4} - 17 \beta_{5} + 11 \beta_{6} - 11 \beta_{7} + 11 \beta_{8} - 11 \beta_{9} ) q^{34} + ( 7 \beta_{1} + 14 \beta_{2} + 7 \beta_{4} + 7 \beta_{6} ) q^{35} + ( -2 - 2 \beta_{1} - 84 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 26 \beta_{6} - 8 \beta_{7} + 6 \beta_{8} - 18 \beta_{9} ) q^{37} + ( 3 + 3 \beta_{1} - 17 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 81 \beta_{6} - 17 \beta_{7} - 9 \beta_{8} - \beta_{9} ) q^{38} + ( -211 + 33 \beta_{1} + 15 \beta_{2} - 3 \beta_{3} + 18 \beta_{4} + 12 \beta_{5} - 45 \beta_{6} - 14 \beta_{7} - \beta_{8} - 6 \beta_{9} ) q^{40} + ( -161 - 35 \beta_{1} - 3 \beta_{2} + 17 \beta_{3} - 9 \beta_{4} - 17 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} ) q^{41} + ( 2 + 2 \beta_{1} + 116 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 14 \beta_{6} - 6 \beta_{8} - 14 \beta_{9} ) q^{43} + ( 64 + 114 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} + 36 \beta_{4} - 16 \beta_{5} + 12 \beta_{6} - 12 \beta_{7} + 12 \beta_{8} - 12 \beta_{9} ) q^{44} + ( 123 + 7 \beta_{1} + 3 \beta_{2} - 15 \beta_{3} + 9 \beta_{4} + 13 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} ) q^{46} + ( -6 - 6 \beta_{1} - 136 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} - 6 \beta_{5} + 42 \beta_{6} - 4 \beta_{7} + 18 \beta_{8} + 10 \beta_{9} ) q^{47} -49 q^{49} + ( 239 - 17 \beta_{1} + 35 \beta_{2} - 3 \beta_{3} + 13 \beta_{4} - 23 \beta_{5} - 20 \beta_{6} + 21 \beta_{7} - \beta_{8} + 9 \beta_{9} ) q^{50} + ( 7 + 7 \beta_{1} + 59 \beta_{2} - 7 \beta_{3} + 7 \beta_{4} + 7 \beta_{5} - 129 \beta_{6} - 13 \beta_{7} - 21 \beta_{8} + 19 \beta_{9} ) q^{52} + ( -11 - 11 \beta_{1} + 5 \beta_{2} + 11 \beta_{3} - 11 \beta_{4} - 11 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 33 \beta_{8} - 9 \beta_{9} ) q^{53} + ( -60 - 64 \beta_{1} - 54 \beta_{2} + 6 \beta_{3} - 8 \beta_{4} + 6 \beta_{5} - 46 \beta_{6} + 38 \beta_{7} + 18 \beta_{8} + 2 \beta_{9} ) q^{55} + ( -49 + 42 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} + 21 \beta_{4} - 7 \beta_{5} + 7 \beta_{6} - 7 \beta_{7} + 7 \beta_{8} - 7 \beta_{9} ) q^{56} + ( -8 - 8 \beta_{1} - 82 \beta_{2} + 8 \beta_{3} - 8 \beta_{4} - 8 \beta_{5} + 134 \beta_{6} - 10 \beta_{7} + 24 \beta_{8} - 16 \beta_{9} ) q^{58} + ( 164 - 40 \beta_{1} - 12 \beta_{2} + 28 \beta_{3} - 36 \beta_{4} + 40 \beta_{5} - 12 \beta_{6} + 12 \beta_{7} - 12 \beta_{8} + 12 \beta_{9} ) q^{59} + ( 112 - 20 \beta_{1} + 16 \beta_{2} + 4 \beta_{3} + 48 \beta_{4} + 2 \beta_{5} + 16 \beta_{6} - 16 \beta_{7} + 16 \beta_{8} - 16 \beta_{9} ) q^{61} + ( 5 + 5 \beta_{1} - 393 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} + 29 \beta_{6} - 49 \beta_{7} - 15 \beta_{8} + \beta_{9} ) q^{62} + ( -11 + 59 \beta_{1} + 10 \beta_{2} + 14 \beta_{3} + 30 \beta_{4} - 17 \beta_{5} + 10 \beta_{6} - 10 \beta_{7} + 10 \beta_{8} - 10 \beta_{9} ) q^{64} + ( 337 - \beta_{1} + 77 \beta_{2} + 19 \beta_{3} - 23 \beta_{4} - \beta_{5} + 39 \beta_{6} + 17 \beta_{7} + 11 \beta_{8} + 3 \beta_{9} ) q^{65} + ( 12 + 12 \beta_{1} + 70 \beta_{2} - 12 \beta_{3} + 12 \beta_{4} + 12 \beta_{5} + 52 \beta_{6} - 10 \beta_{7} - 36 \beta_{8} + 24 \beta_{9} ) q^{67} + ( 9 + 9 \beta_{1} + 897 \beta_{2} - 9 \beta_{3} + 9 \beta_{4} + 9 \beta_{5} - 33 \beta_{6} + 89 \beta_{7} - 27 \beta_{8} + 21 \beta_{9} ) q^{68} + ( -84 - 28 \beta_{1} + 56 \beta_{2} + 7 \beta_{3} - 14 \beta_{4} + 7 \beta_{5} - 28 \beta_{6} + 21 \beta_{7} - 7 \beta_{8} + 14 \beta_{9} ) q^{70} + ( -316 + 24 \beta_{1} + 8 \beta_{2} - 42 \beta_{3} + 24 \beta_{4} - 10 \beta_{5} + 8 \beta_{6} - 8 \beta_{7} + 8 \beta_{8} - 8 \beta_{9} ) q^{71} + ( 13 + 13 \beta_{1} + 99 \beta_{2} - 13 \beta_{3} + 13 \beta_{4} + 13 \beta_{5} - 115 \beta_{6} + 11 \beta_{7} - 39 \beta_{8} + 11 \beta_{9} ) q^{73} + ( -266 + 176 \beta_{1} + 28 \beta_{2} - 36 \beta_{3} + 84 \beta_{4} + 18 \beta_{5} + 28 \beta_{6} - 28 \beta_{7} + 28 \beta_{8} - 28 \beta_{9} ) q^{74} + ( 871 + 49 \beta_{1} + 7 \beta_{2} + 21 \beta_{3} + 21 \beta_{4} - 71 \beta_{5} + 7 \beta_{6} - 7 \beta_{7} + 7 \beta_{8} - 7 \beta_{9} ) q^{76} + ( -98 \beta_{2} - 56 \beta_{6} + 14 \beta_{9} ) q^{77} + ( 194 + 82 \beta_{1} - 14 \beta_{2} + 10 \beta_{3} - 42 \beta_{4} - 22 \beta_{5} - 14 \beta_{6} + 14 \beta_{7} - 14 \beta_{8} + 14 \beta_{9} ) q^{79} + ( 393 - 36 \beta_{1} + 70 \beta_{2} - 15 \beta_{3} + 30 \beta_{4} - 20 \beta_{5} - 184 \beta_{6} + 25 \beta_{7} - 9 \beta_{8} - 10 \beta_{9} ) q^{80} + ( -37 - 37 \beta_{1} - 317 \beta_{2} + 37 \beta_{3} - 37 \beta_{4} - 37 \beta_{5} + 23 \beta_{6} - 13 \beta_{7} + 111 \beta_{8} - 49 \beta_{9} ) q^{82} + ( 4 + 4 \beta_{1} + 312 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 44 \beta_{6} + 16 \beta_{7} - 12 \beta_{8} - 4 \beta_{9} ) q^{83} + ( -207 - 205 \beta_{1} - 353 \beta_{2} + 15 \beta_{3} + 21 \beta_{4} - 25 \beta_{5} - 23 \beta_{6} - 55 \beta_{7} - 19 \beta_{8} - 5 \beta_{9} ) q^{85} + ( -206 - 136 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} + 36 \beta_{4} + 6 \beta_{5} + 12 \beta_{6} - 12 \beta_{7} + 12 \beta_{8} - 12 \beta_{9} ) q^{86} + ( -16 - 16 \beta_{1} + 602 \beta_{2} + 16 \beta_{3} - 16 \beta_{4} - 16 \beta_{5} - 102 \beta_{6} + 130 \beta_{7} + 48 \beta_{8} + 32 \beta_{9} ) q^{88} + ( 71 + 57 \beta_{1} - 15 \beta_{2} - 35 \beta_{3} - 45 \beta_{4} + 7 \beta_{5} - 15 \beta_{6} + 15 \beta_{7} - 15 \beta_{8} + 15 \beta_{9} ) q^{89} + ( -7 - 49 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} - 21 \beta_{4} + 7 \beta_{5} - 7 \beta_{6} + 7 \beta_{7} - 7 \beta_{8} + 7 \beta_{9} ) q^{91} + ( 7 + 7 \beta_{1} - 271 \beta_{2} - 7 \beta_{3} + 7 \beta_{4} + 7 \beta_{5} - 75 \beta_{6} + 33 \beta_{7} - 21 \beta_{8} + 3 \beta_{9} ) q^{92} + ( -450 + 280 \beta_{1} - 24 \beta_{3} + 58 \beta_{5} ) q^{94} + ( -153 - 97 \beta_{1} - 77 \beta_{2} - 9 \beta_{3} + 3 \beta_{4} + 41 \beta_{5} + 89 \beta_{6} + 23 \beta_{7} + 27 \beta_{8} - 43 \beta_{9} ) q^{95} + ( 1 + \beta_{1} + 91 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 161 \beta_{6} + 19 \beta_{7} - 3 \beta_{8} + 15 \beta_{9} ) q^{97} -49 \beta_{6} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 54q^{4} + 14q^{5} + O(q^{10})$$ $$10q - 54q^{4} + 14q^{5} + 92q^{10} - 132q^{11} + 14q^{14} + 310q^{16} - 348q^{19} - 366q^{20} - 374q^{25} - 892q^{26} + 740q^{29} + 684q^{31} - 224q^{34} - 2156q^{40} - 1604q^{41} + 580q^{44} + 1280q^{46} - 490q^{49} + 2504q^{50} - 452q^{55} - 462q^{56} + 1408q^{59} + 1300q^{61} - 150q^{64} + 3296q^{65} - 882q^{70} - 2940q^{71} - 2624q^{74} + 8740q^{76} + 1640q^{79} + 4126q^{80} - 1704q^{85} - 1664q^{86} + 572q^{89} - 28q^{91} - 5080q^{94} - 1268q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 37 x^{8} + 398 x^{6} + 1149 x^{4} + 1040 x^{2} + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$16 \nu^{8} + 561 \nu^{6} + 5285 \nu^{4} + 8248 \nu^{2} + 880$$$$)/105$$ $$\beta_{2}$$ $$=$$ $$($$$$57 \nu^{9} + 1999 \nu^{7} + 18816 \nu^{5} + 28813 \nu^{3} + 790 \nu$$$$)/700$$ $$\beta_{3}$$ $$=$$ $$($$$$73 \nu^{8} + 2574 \nu^{6} + 24563 \nu^{4} + 41065 \nu^{2} + 6535$$$$)/105$$ $$\beta_{4}$$ $$=$$ $$($$$$88 \nu^{8} + 3096 \nu^{6} + 29414 \nu^{4} + 48052 \nu^{2} + 210 \nu + 5995$$$$)/105$$ $$\beta_{5}$$ $$=$$ $$($$$$-130 \nu^{8} - 4566 \nu^{6} - 43148 \nu^{4} - 67876 \nu^{2} - 5575$$$$)/105$$ $$\beta_{6}$$ $$=$$ $$($$$$157 \nu^{9} + 5514 \nu^{7} + 52136 \nu^{5} + 82603 \nu^{3} + 7690 \nu$$$$)/525$$ $$\beta_{7}$$ $$=$$ $$($$$$949 \nu^{9} + 33483 \nu^{7} + 320432 \nu^{5} + 548881 \nu^{3} + 134830 \nu$$$$)/2100$$ $$\beta_{8}$$ $$=$$ $$($$$$-1223 \nu^{9} - 660 \nu^{8} - 43161 \nu^{7} - 23220 \nu^{6} - 413224 \nu^{5} - 220080 \nu^{4} - 706907 \nu^{3} - 350940 \nu^{2} - 139610 \nu - 34200$$$$)/2100$$ $$\beta_{9}$$ $$=$$ $$($$$$-1373 \nu^{9} - 48591 \nu^{7} - 468664 \nu^{5} - 838937 \nu^{3} - 228710 \nu$$$$)/2100$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{9} - 3 \beta_{8} + 3 \beta_{7} - 3 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2} + \beta_{1} + 1$$$$)/20$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} + 17 \beta_{1} - 68$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$-26 \beta_{9} + 36 \beta_{8} - 11 \beta_{7} + 16 \beta_{6} - 12 \beta_{5} - 12 \beta_{4} + 12 \beta_{3} + 51 \beta_{2} - 12 \beta_{1} - 12$$$$)/10$$ $$\nu^{4}$$ $$=$$ $$($$$$-53 \beta_{9} + 53 \beta_{8} - 53 \beta_{7} + 53 \beta_{6} - 51 \beta_{5} + 159 \beta_{4} - 129 \beta_{3} + 53 \beta_{2} - 591 \beta_{1} + 2059$$$$)/20$$ $$\nu^{5}$$ $$=$$ $$($$$$933 \beta_{9} - 1323 \beta_{8} + 283 \beta_{7} - 133 \beta_{6} + 441 \beta_{5} + 441 \beta_{4} - 441 \beta_{3} - 3053 \beta_{2} + 441 \beta_{1} + 441$$$$)/20$$ $$\nu^{6}$$ $$=$$ $$($$$$1177 \beta_{9} - 1177 \beta_{8} + 1177 \beta_{7} - 1177 \beta_{6} + 639 \beta_{5} - 3531 \beta_{4} + 2741 \beta_{3} - 1177 \beta_{2} + 9679 \beta_{1} - 35351$$$$)/20$$ $$\nu^{7}$$ $$=$$ $$($$$$-4283 \beta_{9} + 6033 \beta_{8} - 1028 \beta_{7} - 1112 \beta_{6} - 2011 \beta_{5} - 2011 \beta_{4} + 2011 \beta_{3} + 18548 \beta_{2} - 2011 \beta_{1} - 2011$$$$)/5$$ $$\nu^{8}$$ $$=$$ $$($$$$-24793 \beta_{9} + 24793 \beta_{8} - 24793 \beta_{7} + 24793 \beta_{6} - 7621 \beta_{5} + 74379 \beta_{4} - 56589 \beta_{3} + 24793 \beta_{2} - 161551 \beta_{1} + 628389$$$$)/20$$ $$\nu^{9}$$ $$=$$ $$($$$$159539 \beta_{9} - 222969 \beta_{8} + 30934 \beta_{7} + 91881 \beta_{6} + 74323 \beta_{5} + 74323 \beta_{4} - 74323 \beta_{3} - 822694 \beta_{2} + 74323 \beta_{1} + 74323$$$$)/10$$

## Character values

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

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 3.71490i − 1.37042i − 4.40248i 1.35311i 0.329739i − 0.329739i − 1.35311i 4.40248i 1.37042i − 3.71490i
5.18660i 0 −18.9008 4.24321 + 10.3438i 0 7.00000i 56.5383i 0 53.6494 22.0078i
64.2 4.88936i 0 −15.9059 9.63020 + 5.67972i 0 7.00000i 38.6546i 0 27.7702 47.0855i
64.3 3.33774i 0 −3.14050 −10.1427 4.70380i 0 7.00000i 16.2197i 0 −15.7000 + 33.8537i
64.4 2.20666i 0 3.13065 1.50045 11.0792i 0 7.00000i 24.5616i 0 −24.4480 3.31098i
64.5 0.428319i 0 7.81654 1.76884 + 11.0395i 0 7.00000i 6.77452i 0 4.72844 0.757628i
64.6 0.428319i 0 7.81654 1.76884 11.0395i 0 7.00000i 6.77452i 0 4.72844 + 0.757628i
64.7 2.20666i 0 3.13065 1.50045 + 11.0792i 0 7.00000i 24.5616i 0 −24.4480 + 3.31098i
64.8 3.33774i 0 −3.14050 −10.1427 + 4.70380i 0 7.00000i 16.2197i 0 −15.7000 33.8537i
64.9 4.88936i 0 −15.9059 9.63020 5.67972i 0 7.00000i 38.6546i 0 27.7702 + 47.0855i
64.10 5.18660i 0 −18.9008 4.24321 10.3438i 0 7.00000i 56.5383i 0 53.6494 + 22.0078i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 64.10 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.4.d.b 10
3.b odd 2 1 105.4.d.b 10
5.b even 2 1 inner 315.4.d.b 10
5.c odd 4 1 1575.4.a.bo 5
5.c odd 4 1 1575.4.a.bp 5
15.d odd 2 1 105.4.d.b 10
15.e even 4 1 525.4.a.w 5
15.e even 4 1 525.4.a.x 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.d.b 10 3.b odd 2 1
105.4.d.b 10 15.d odd 2 1
315.4.d.b 10 1.a even 1 1 trivial
315.4.d.b 10 5.b even 2 1 inner
525.4.a.w 5 15.e even 4 1
525.4.a.x 5 15.e even 4 1
1575.4.a.bo 5 5.c odd 4 1
1575.4.a.bp 5 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} + 67 T_{2}^{8} + 1523 T_{2}^{6} + 13329 T_{2}^{4} + 37280 T_{2}^{2} + 6400$$ acting on $$S_{4}^{\mathrm{new}}(315, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$6400 + 37280 T^{2} + 13329 T^{4} + 1523 T^{6} + 67 T^{8} + T^{10}$$
$3$ $$T^{10}$$
$5$ $$30517578125 - 3417968750 T + 556640625 T^{2} - 18000000 T^{3} + 353750 T^{4} + 184700 T^{5} + 2830 T^{6} - 1152 T^{7} + 285 T^{8} - 14 T^{9} + T^{10}$$
$7$ $$( 49 + T^{2} )^{5}$$
$11$ $$( 55852416 + 1472448 T - 140456 T^{2} - 2100 T^{3} + 66 T^{4} + T^{5} )^{2}$$
$13$ $$1712383850521600 + 20084730574080 T^{2} + 36081658496 T^{4} + 25849248 T^{6} + 8308 T^{8} + T^{10}$$
$17$ $$53832671581241344 + 2085927144738816 T^{2} + 3681475123712 T^{4} + 694030608 T^{6} + 45544 T^{8} + T^{10}$$
$19$ $$( 784374624 - 40426032 T - 1513744 T^{2} - 3640 T^{3} + 174 T^{4} + T^{5} )^{2}$$
$23$ $$238300185600000000 + 4736362037760000 T^{2} + 5601652211200 T^{4} + 921968016 T^{6} + 52808 T^{8} + T^{10}$$
$29$ $$( 1150048 - 14986160 T - 1029840 T^{2} + 38440 T^{3} - 370 T^{4} + T^{5} )^{2}$$
$31$ $$( 52737095200 - 1618226480 T + 13826640 T^{2} - 6904 T^{3} - 342 T^{4} + T^{5} )^{2}$$
$37$ $$99\!\cdots\!64$$$$+ 28828609951449153536 T^{2} + 1752974284050432 T^{4} + 39259401728 T^{6} + 345664 T^{8} + T^{10}$$
$41$ $$( -531402107648 - 12602553024 T - 74825096 T^{2} + 47852 T^{3} + 802 T^{4} + T^{5} )^{2}$$
$43$ $$17\!\cdots\!00$$$$+ 20629646614603694080 T^{2} + 1345199401488384 T^{4} + 31766978048 T^{6} + 309312 T^{8} + T^{10}$$
$47$ $$43\!\cdots\!00$$$$+$$$$17\!\cdots\!00$$$$T^{2} + 7318958884495360 T^{4} + 98765465856 T^{6} + 536048 T^{8} + T^{10}$$
$53$ $$21\!\cdots\!24$$$$+ 87341928855767905536 T^{2} + 4262799772404352 T^{4} + 73000876448 T^{6} + 491764 T^{8} + T^{10}$$
$59$ $$( -33724261457920 + 47252515840 T + 380481408 T^{2} - 599408 T^{3} - 704 T^{4} + T^{5} )^{2}$$
$61$ $$( 1105143174112 + 47217562320 T + 214899120 T^{2} - 445560 T^{3} - 650 T^{4} + T^{5} )^{2}$$
$67$ $$24\!\cdots\!64$$$$+$$$$46\!\cdots\!40$$$$T^{2} + 20650444285009920 T^{4} + 244281647360 T^{6} + 990640 T^{8} + T^{10}$$
$71$ $$( -237519904000 - 40906747200 T - 376607000 T^{2} + 92060 T^{3} + 1470 T^{4} + T^{5} )^{2}$$
$73$ $$37\!\cdots\!24$$$$+$$$$45\!\cdots\!76$$$$T^{2} + 17951407658069632 T^{4} + 273800955808 T^{6} + 1491284 T^{8} + T^{10}$$
$79$ $$( -43229481181184 + 35855795200 T + 590031680 T^{2} - 728400 T^{3} - 820 T^{4} + T^{5} )^{2}$$
$83$ $$16\!\cdots\!96$$$$+$$$$11\!\cdots\!80$$$$T^{2} + 16188157928407040 T^{4} + 246934425600 T^{6} + 974720 T^{8} + T^{10}$$
$89$ $$( 1125486676224 + 41779572288 T + 206349496 T^{2} - 1347380 T^{3} - 286 T^{4} + T^{5} )^{2}$$
$97$ $$39\!\cdots\!56$$$$+$$$$21\!\cdots\!16$$$$T^{2} + 340713117665820288 T^{4} + 1579332498848 T^{6} + 2344596 T^{8} + T^{10}$$