# Properties

 Label 315.2.m.b Level 315 Weight 2 Character orbit 315.m Analytic conductor 2.515 Analytic rank 0 Dimension 12 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{8} + \beta_{9} ) q^{2} + ( -\beta_{5} - \beta_{6} + \beta_{8} ) q^{4} + ( -\beta_{1} + \beta_{4} + \beta_{10} - \beta_{11} ) q^{5} -\beta_{3} q^{7} + ( -2 + \beta_{2} - 2 \beta_{6} + \beta_{8} + \beta_{11} ) q^{8} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{8} + \beta_{9} ) q^{2} + ( -\beta_{5} - \beta_{6} + \beta_{8} ) q^{4} + ( -\beta_{1} + \beta_{4} + \beta_{10} - \beta_{11} ) q^{5} -\beta_{3} q^{7} + ( -2 + \beta_{2} - 2 \beta_{6} + \beta_{8} + \beta_{11} ) q^{8} + ( 2 + \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{10} + ( \beta_{2} - \beta_{3} + \beta_{5} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{11} + ( -2 \beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{13} + \beta_{7} q^{14} + ( -1 - 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{16} + ( -1 + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{17} + ( -2 \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{19} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} - 4 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{20} + ( 1 - 4 \beta_{2} + 4 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} - 2 \beta_{10} + \beta_{11} ) q^{22} + ( -2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{11} ) q^{23} + ( -1 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{25} + ( -\beta_{1} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{26} + ( \beta_{1} - \beta_{5} + \beta_{10} ) q^{28} + ( 4 + \beta_{1} - \beta_{3} - \beta_{5} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{29} + ( 2 - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{31} + ( -2 - 5 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{32} + ( \beta_{2} - \beta_{3} - 4 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{9} + 4 \beta_{11} ) q^{34} + ( -1 - \beta_{8} - \beta_{9} - \beta_{11} ) q^{35} + ( -1 + 2 \beta_{5} - \beta_{6} + 2 \beta_{10} ) q^{37} + ( -1 - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{8} - 2 \beta_{10} - 2 \beta_{11} ) q^{38} + ( -3 + 5 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{40} + ( \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{41} + ( 4 + \beta_{5} - 4 \beta_{6} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{43} + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{44} + ( -4 \beta_{1} + 4 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{46} + ( -2 + 4 \beta_{1} + 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} ) q^{47} + \beta_{6} q^{49} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - \beta_{9} - 3 \beta_{11} ) q^{50} + ( 2 - 4 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{10} - 3 \beta_{11} ) q^{52} + ( 2 - 4 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{53} + ( -1 + 4 \beta_{1} - 2 \beta_{3} + \beta_{4} + 2 \beta_{6} - 3 \beta_{7} + 2 \beta_{10} - \beta_{11} ) q^{55} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{11} ) q^{56} + ( -4 + 4 \beta_{1} - 3 \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{8} - \beta_{10} + 2 \beta_{11} ) q^{58} + ( 6 + \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{59} + ( -6 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + \beta_{5} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} ) q^{61} + ( -3 - \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - 4 \beta_{9} - \beta_{10} - \beta_{11} ) q^{62} + ( -4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 5 \beta_{6} - 5 \beta_{8} - 2 \beta_{9} - 4 \beta_{10} - 6 \beta_{11} ) q^{64} + ( -4 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{10} + 3 \beta_{11} ) q^{65} + ( -4 + 4 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{67} + ( -3 + 8 \beta_{3} - \beta_{4} + \beta_{5} - 4 \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} ) q^{68} + ( \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} ) q^{70} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 3 \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{71} + ( -2 + 2 \beta_{1} - \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{73} + ( 8 - \beta_{1} - 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} - 4 \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{74} + ( -2 + 3 \beta_{2} - 3 \beta_{3} - \beta_{5} + 4 \beta_{7} - 4 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} ) q^{76} + ( -1 - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{77} + ( 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{79} + ( -2 + \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + \beta_{4} - 6 \beta_{6} + 4 \beta_{8} - \beta_{10} + 5 \beta_{11} ) q^{80} + ( 4 \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} + 3 \beta_{10} + 3 \beta_{11} ) q^{82} + ( -2 + 2 \beta_{2} + 6 \beta_{3} - 2 \beta_{6} ) q^{83} + ( 1 + 6 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{8} + 2 \beta_{9} - 4 \beta_{10} + 3 \beta_{11} ) q^{85} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{5} + 4 \beta_{6} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{86} + ( -1 + 4 \beta_{1} + \beta_{4} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - 3 \beta_{11} ) q^{88} + ( 2 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{89} + ( -\beta_{1} + \beta_{2} - 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{91} + ( 2 + 4 \beta_{4} + \beta_{5} + 2 \beta_{6} - 4 \beta_{7} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{92} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 8 \beta_{6} - 2 \beta_{9} + 2 \beta_{11} ) q^{94} + ( -4 + 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - 3 \beta_{11} ) q^{95} + ( 2 \beta_{2} + 3 \beta_{4} + 3 \beta_{6} + 3 \beta_{7} + \beta_{8} - 2 \beta_{11} ) q^{97} -\beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 4q^{5} - 24q^{8} + O(q^{10})$$ $$12q + 4q^{5} - 24q^{8} + 16q^{10} - 4q^{13} + 4q^{14} - 20q^{16} - 8q^{17} + 12q^{20} - 8q^{22} - 8q^{23} - 8q^{25} + 32q^{29} - 48q^{32} - 8q^{35} + 4q^{37} - 24q^{38} - 28q^{40} + 40q^{43} + 64q^{44} + 16q^{46} - 24q^{47} + 16q^{50} + 36q^{52} + 40q^{53} - 16q^{55} - 28q^{58} + 80q^{59} - 32q^{61} - 16q^{62} - 48q^{65} - 48q^{67} - 32q^{68} + 8q^{70} - 20q^{73} + 64q^{74} + 16q^{76} - 36q^{80} + 20q^{82} - 24q^{83} + 56q^{89} - 8q^{92} - 56q^{95} + 12q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 18 x^{10} + 107 x^{8} + 240 x^{6} + 151 x^{4} + 30 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$5 \nu^{11} - 9 \nu^{10} + 85 \nu^{9} - 159 \nu^{8} + 450 \nu^{7} - 910 \nu^{6} + 740 \nu^{5} - 1860 \nu^{4} - 115 \nu^{3} - 769 \nu^{2} - 145 \nu - 57$$$$)/40$$ $$\beta_{2}$$ $$=$$ $$($$$$3 \nu^{11} + 7 \nu^{10} + 53 \nu^{9} + 127 \nu^{8} + 300 \nu^{7} + 760 \nu^{6} + 570 \nu^{5} + 1690 \nu^{4} + 53 \nu^{3} + 937 \nu^{2} - 171 \nu + 71$$$$)/40$$ $$\beta_{3}$$ $$=$$ $$($$$$5 \nu^{11} + 9 \nu^{10} + 85 \nu^{9} + 159 \nu^{8} + 450 \nu^{7} + 910 \nu^{6} + 740 \nu^{5} + 1860 \nu^{4} - 115 \nu^{3} + 769 \nu^{2} - 145 \nu + 57$$$$)/40$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{11} + 6 \nu^{10} + 37 \nu^{9} + 106 \nu^{8} + 235 \nu^{7} + 610 \nu^{6} + 625 \nu^{5} + 1280 \nu^{4} + 647 \nu^{3} + 616 \nu^{2} + 176 \nu + 38$$$$)/20$$ $$\beta_{5}$$ $$=$$ $$($$$$-6 \nu^{11} + 7 \nu^{10} - 106 \nu^{9} + 122 \nu^{8} - 605 \nu^{7} + 680 \nu^{6} - 1215 \nu^{5} + 1300 \nu^{4} - 411 \nu^{3} + 327 \nu^{2} + 57 \nu - 24$$$$)/20$$ $$\beta_{6}$$ $$=$$ $$($$$$-8 \nu^{11} - 143 \nu^{9} - 840 \nu^{7} - 1840 \nu^{5} - 1058 \nu^{3} - 139 \nu$$$$)/20$$ $$\beta_{7}$$ $$=$$ $$($$$$-6 \nu^{10} - 106 \nu^{8} - 605 \nu^{6} - 1215 \nu^{4} - 421 \nu^{2} - 3$$$$)/10$$ $$\beta_{8}$$ $$=$$ $$($$$$6 \nu^{11} + 7 \nu^{10} + 106 \nu^{9} + 122 \nu^{8} + 605 \nu^{7} + 680 \nu^{6} + 1215 \nu^{5} + 1300 \nu^{4} + 411 \nu^{3} + 327 \nu^{2} - 57 \nu - 24$$$$)/20$$ $$\beta_{9}$$ $$=$$ $$($$$$-7 \nu^{11} - 6 \nu^{10} - 122 \nu^{9} - 106 \nu^{8} - 680 \nu^{7} - 605 \nu^{6} - 1300 \nu^{5} - 1215 \nu^{4} - 327 \nu^{3} - 411 \nu^{2} + 44 \nu + 17$$$$)/20$$ $$\beta_{10}$$ $$=$$ $$($$$$23 \nu^{11} - 6 \nu^{10} + 408 \nu^{9} - 106 \nu^{8} + 2355 \nu^{7} - 610 \nu^{6} + 4915 \nu^{5} - 1280 \nu^{4} + 2258 \nu^{3} - 616 \nu^{2} + 259 \nu - 38$$$$)/20$$ $$\beta_{11}$$ $$=$$ $$($$$$23 \nu^{11} + 6 \nu^{10} + 408 \nu^{9} + 106 \nu^{8} + 2355 \nu^{7} + 610 \nu^{6} + 4915 \nu^{5} + 1280 \nu^{4} + 2258 \nu^{3} + 616 \nu^{2} + 259 \nu + 38$$$$)/20$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} - \beta_{10} - \beta_{8} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} - \beta_{7} - \beta_{5} + \beta_{3} - \beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{11} + 7 \beta_{10} + 2 \beta_{9} + 5 \beta_{8} + 14 \beta_{6} - 3 \beta_{5} + 12 \beta_{4} + 2 \beta_{2} + 2 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-\beta_{11} + \beta_{10} - 8 \beta_{9} + 10 \beta_{7} + 8 \beta_{5} - 4 \beta_{3} + 8 \beta_{2} - 4 \beta_{1} + 9$$ $$\nu^{5}$$ $$=$$ $$($$$$31 \beta_{11} - 51 \beta_{10} - 30 \beta_{9} - 35 \beta_{8} - 98 \beta_{6} + 5 \beta_{5} - 82 \beta_{4} - 2 \beta_{3} - 30 \beta_{2} - 32 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$15 \beta_{11} - 15 \beta_{10} + 65 \beta_{9} - 89 \beta_{7} - 65 \beta_{5} + 13 \beta_{3} - 65 \beta_{2} + 52 \beta_{1} - 46$$ $$\nu^{7}$$ $$=$$ $$($$$$-217 \beta_{11} + 395 \beta_{10} + 312 \beta_{9} + 265 \beta_{8} + 730 \beta_{6} + 47 \beta_{5} + 612 \beta_{4} + 30 \beta_{3} + 312 \beta_{2} + 342 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$-162 \beta_{11} + 162 \beta_{10} - 542 \beta_{9} - 6 \beta_{8} + 766 \beta_{7} + 536 \beta_{5} - 18 \beta_{3} + 542 \beta_{2} - 524 \beta_{1} + 259$$ $$\nu^{9}$$ $$=$$ $$($$$$1649 \beta_{11} - 3181 \beta_{10} - 2892 \beta_{9} - 2097 \beta_{8} - 5714 \beta_{6} - 795 \beta_{5} - 4830 \beta_{4} - 336 \beta_{3} - 2892 \beta_{2} - 3228 \beta_{1}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$1552 \beta_{11} - 1552 \beta_{10} + 4571 \beta_{9} + 106 \beta_{8} - 6515 \beta_{7} - 4465 \beta_{5} - 253 \beta_{3} - 4571 \beta_{2} + 4824 \beta_{1} - 1620$$ $$\nu^{11}$$ $$=$$ $$($$$$-13177 \beta_{11} + 26207 \beta_{10} + 25570 \beta_{9} + 17065 \beta_{8} + 46206 \beta_{6} + 8505 \beta_{5} + 39384 \beta_{4} + 3316 \beta_{3} + 25570 \beta_{2} + 28886 \beta_{1}$$$$)/2$$

## Character values

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

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$\beta_{6}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1
 2.91021i − 2.01185i 0.203482i 2.15459i − 0.699479i − 0.556948i − 2.91021i 2.01185i − 0.203482i − 2.15459i 0.699479i 0.556948i
−1.45137 1.45137i 0 2.21293i 1.31357 1.80957i 0 −0.707107 + 0.707107i 0.309035 0.309035i 0 −4.53282 + 0.719873i
8.2 −1.28118 1.28118i 0 1.28283i −0.565689 + 2.16333i 0 0.707107 0.707107i −0.918816 + 0.918816i 0 3.49636 2.04686i
8.3 −0.876184 0.876184i 0 0.464602i −0.0251942 + 2.23593i 0 −0.707107 + 0.707107i −2.15945 + 2.15945i 0 1.98116 1.93701i
8.4 0.0241053 + 0.0241053i 0 1.99884i −2.20653 0.362277i 0 0.707107 0.707107i 0.0963933 0.0963933i 0 −0.0444562 0.0619218i
8.5 1.62044 + 1.62044i 0 3.25168i 1.83294 + 1.28075i 0 −0.707107 + 0.707107i −2.02827 + 2.02827i 0 0.894803 + 5.04556i
8.6 1.96418 + 1.96418i 0 5.71600i 1.65089 1.50816i 0 0.707107 0.707107i −7.29890 + 7.29890i 0 6.20495 + 0.280357i
197.1 −1.45137 + 1.45137i 0 2.21293i 1.31357 + 1.80957i 0 −0.707107 0.707107i 0.309035 + 0.309035i 0 −4.53282 0.719873i
197.2 −1.28118 + 1.28118i 0 1.28283i −0.565689 2.16333i 0 0.707107 + 0.707107i −0.918816 0.918816i 0 3.49636 + 2.04686i
197.3 −0.876184 + 0.876184i 0 0.464602i −0.0251942 2.23593i 0 −0.707107 0.707107i −2.15945 2.15945i 0 1.98116 + 1.93701i
197.4 0.0241053 0.0241053i 0 1.99884i −2.20653 + 0.362277i 0 0.707107 + 0.707107i 0.0963933 + 0.0963933i 0 −0.0444562 + 0.0619218i
197.5 1.62044 1.62044i 0 3.25168i 1.83294 1.28075i 0 −0.707107 0.707107i −2.02827 2.02827i 0 0.894803 5.04556i
197.6 1.96418 1.96418i 0 5.71600i 1.65089 + 1.50816i 0 0.707107 + 0.707107i −7.29890 7.29890i 0 6.20495 0.280357i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 197.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e 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.m.b yes 12
3.b odd 2 1 315.2.m.a 12
5.b even 2 1 1575.2.m.d 12
5.c odd 4 1 315.2.m.a 12
5.c odd 4 1 1575.2.m.c 12
15.d odd 2 1 1575.2.m.c 12
15.e even 4 1 inner 315.2.m.b yes 12
15.e even 4 1 1575.2.m.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.m.a 12 3.b odd 2 1
315.2.m.a 12 5.c odd 4 1
315.2.m.b yes 12 1.a even 1 1 trivial
315.2.m.b yes 12 15.e even 4 1 inner
1575.2.m.c 12 5.c odd 4 1
1575.2.m.c 12 15.d odd 2 1
1575.2.m.d 12 5.b even 2 1
1575.2.m.d 12 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 8 T^{3} - 5 T^{4} + 32 T^{6} - 32 T^{7} - 5 T^{8} + 80 T^{9} - 96 T^{10} - 72 T^{11} + 193 T^{12} - 144 T^{13} - 384 T^{14} + 640 T^{15} - 80 T^{16} - 1024 T^{17} + 2048 T^{18} - 1280 T^{20} + 4096 T^{21} + 4096 T^{24}$$
$3$ 1
$5$ $$1 - 4 T + 12 T^{2} - 12 T^{3} - 5 T^{4} + 160 T^{5} - 408 T^{6} + 800 T^{7} - 125 T^{8} - 1500 T^{9} + 7500 T^{10} - 12500 T^{11} + 15625 T^{12}$$
$7$ $$( 1 + T^{4} )^{3}$$
$11$ $$1 - 52 T^{2} + 1458 T^{4} - 30084 T^{6} + 494911 T^{8} - 6805160 T^{10} + 80542716 T^{12} - 823424360 T^{14} + 7245991951 T^{16} - 53295641124 T^{18} + 312535248498 T^{20} - 1348746079252 T^{22} + 3138428376721 T^{24}$$
$13$ $$1 + 4 T + 8 T^{2} + 108 T^{3} + 298 T^{4} + 204 T^{5} + 4264 T^{6} + 17924 T^{7} + 42687 T^{8} + 200904 T^{9} + 1148048 T^{10} + 3595224 T^{11} + 7337420 T^{12} + 46737912 T^{13} + 194020112 T^{14} + 441386088 T^{15} + 1219183407 T^{16} + 6655055732 T^{17} + 20581513576 T^{18} + 12800697468 T^{19} + 243087754858 T^{20} + 1145285932284 T^{21} + 1102867934792 T^{22} + 7168641576148 T^{23} + 23298085122481 T^{24}$$
$17$ $$1 + 8 T + 32 T^{2} + 232 T^{3} + 1238 T^{4} + 3432 T^{5} + 14752 T^{6} + 67976 T^{7} + 254479 T^{8} + 1447664 T^{9} + 6834752 T^{10} + 33133872 T^{11} + 165039860 T^{12} + 563275824 T^{13} + 1975243328 T^{14} + 7112373232 T^{15} + 21254340559 T^{16} + 96516199432 T^{17} + 356077417888 T^{18} + 1408282325736 T^{19} + 8635987711958 T^{20} + 27512387347304 T^{21} + 64511804814368 T^{22} + 274175170461064 T^{23} + 582622237229761 T^{24}$$
$19$ $$1 - 84 T^{2} + 4642 T^{4} - 179332 T^{6} + 5560223 T^{8} - 138383432 T^{10} + 2894445596 T^{12} - 49956418952 T^{14} + 724613821583 T^{16} - 8436831931492 T^{18} + 78837699636322 T^{20} - 515009565655284 T^{22} + 2213314919066161 T^{24}$$
$23$ $$1 + 8 T + 32 T^{2} + 184 T^{3} + 518 T^{4} - 1752 T^{5} - 13664 T^{6} - 145640 T^{7} - 745937 T^{8} + 382288 T^{9} + 8743232 T^{10} + 77776560 T^{11} + 600339476 T^{12} + 1788860880 T^{13} + 4625169728 T^{14} + 4651298096 T^{15} - 208743756017 T^{16} - 937388994520 T^{17} - 2022762387296 T^{18} - 5965254183144 T^{19} + 40565090375558 T^{20} + 331412089709192 T^{21} + 1325648358836768 T^{22} + 7622478063311416 T^{23} + 21914624432020321 T^{24}$$
$29$ $$( 1 - 16 T + 216 T^{2} - 1904 T^{3} + 15483 T^{4} - 98368 T^{5} + 589488 T^{6} - 2852672 T^{7} + 13021203 T^{8} - 46436656 T^{9} + 152772696 T^{10} - 328178384 T^{11} + 594823321 T^{12} )^{2}$$
$31$ $$( 1 + 114 T^{2} - 168 T^{3} + 6583 T^{4} - 10856 T^{5} + 253628 T^{6} - 336536 T^{7} + 6326263 T^{8} - 5004888 T^{9} + 105281394 T^{10} + 887503681 T^{12} )^{2}$$
$37$ $$1 - 4 T + 8 T^{2} - 28 T^{3} + 122 T^{4} - 124 T^{5} - 88 T^{6} + 102780 T^{7} - 827937 T^{8} + 2005016 T^{9} - 4255984 T^{10} + 201457960 T^{11} - 3018818132 T^{12} + 7453944520 T^{13} - 5826442096 T^{14} + 101560075448 T^{15} - 1551687235857 T^{16} + 7127171900460 T^{17} - 225783923992 T^{18} - 11771552764492 T^{19} + 428522493378362 T^{20} - 3638928714262156 T^{21} + 38468674979342792 T^{22} - 711670487117841652 T^{23} + 6582952005840035281 T^{24}$$
$41$ $$1 - 232 T^{2} + 30646 T^{4} - 2820520 T^{6} + 198407727 T^{8} - 11090030832 T^{10} + 503431026356 T^{12} - 18642341828592 T^{14} + 560652817055247 T^{16} - 13397764013825320 T^{18} + 244706018571642166 T^{20} - 3114056959955357032 T^{22} + 22563490300366186081 T^{24}$$
$43$ $$1 - 40 T + 800 T^{2} - 11608 T^{3} + 144630 T^{4} - 1609192 T^{5} + 16036512 T^{6} - 146487000 T^{7} + 1250458847 T^{8} - 9986195856 T^{9} + 74890568512 T^{10} - 532487979120 T^{11} + 3591291202036 T^{12} - 22896983102160 T^{13} + 138472661178688 T^{14} - 793972473922992 T^{15} + 4275069956582447 T^{16} - 21534825789741000 T^{17} + 101372614391645088 T^{18} - 437408334444495544 T^{19} + 1690464406149432630 T^{20} - 5834095039362873544 T^{21} + 17289185850627399200 T^{22} - 37171749578848908280 T^{23} + 39959630797262576401 T^{24}$$
$47$ $$1 + 24 T + 288 T^{2} + 2920 T^{3} + 30374 T^{4} + 283512 T^{5} + 2319776 T^{6} + 18683912 T^{7} + 148359759 T^{8} + 1083670000 T^{9} + 7566143296 T^{10} + 53830728976 T^{11} + 378456487636 T^{12} + 2530044261872 T^{13} + 16713610540864 T^{14} + 112509870410000 T^{15} + 723948297156879 T^{16} + 4285061928427384 T^{17} + 25005365019046304 T^{18} + 143633734128706056 T^{19} + 723244021064328614 T^{20} + 3267860981460079640 T^{21} + 15148550083919054112 T^{22} + 59331821162016295272 T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$
$53$ $$1 - 40 T + 800 T^{2} - 11880 T^{3} + 147222 T^{4} - 1477736 T^{5} + 11899040 T^{6} - 74456936 T^{7} + 249326879 T^{8} + 1557249200 T^{9} - 37151695552 T^{10} + 419908594096 T^{11} - 3501062916044 T^{12} + 22255155487088 T^{13} - 104359112805568 T^{14} + 231838589148400 T^{15} + 1967309001538799 T^{16} - 31137555057789448 T^{17} + 263734619648416160 T^{18} - 1735912940938169032 T^{19} + 9165996141741389142 T^{20} - 39201191470609340040 T^{21} +$$$$13\!\cdots\!00$$$$T^{22} -$$$$37\!\cdots\!80$$$$T^{23} +$$$$49\!\cdots\!41$$$$T^{24}$$
$59$ $$( 1 - 40 T + 986 T^{2} - 16824 T^{3} + 222455 T^{4} - 2325552 T^{5} + 19830604 T^{6} - 137207568 T^{7} + 774365855 T^{8} - 3455296296 T^{9} + 11947717946 T^{10} - 28596971960 T^{11} + 42180533641 T^{12} )^{2}$$
$61$ $$( 1 + 16 T + 254 T^{2} + 2800 T^{3} + 31591 T^{4} + 270400 T^{5} + 2316548 T^{6} + 16494400 T^{7} + 117550111 T^{8} + 635546800 T^{9} + 3516843614 T^{10} + 13513540816 T^{11} + 51520374361 T^{12} )^{2}$$
$67$ $$1 + 48 T + 1152 T^{2} + 18992 T^{3} + 248566 T^{4} + 2816976 T^{5} + 29214848 T^{6} + 287145808 T^{7} + 2747657919 T^{8} + 25947593440 T^{9} + 239514732288 T^{10} + 2122341535712 T^{11} + 17856126120308 T^{12} + 142196882892704 T^{13} + 1075181633240832 T^{14} + 7804076045794720 T^{15} + 55368387192377199 T^{16} + 387682764750601456 T^{17} + 2642727885393245312 T^{18} + 17072879135116363248 T^{19} +$$$$10\!\cdots\!06$$$$T^{20} +$$$$51\!\cdots\!24$$$$T^{21} +$$$$20\!\cdots\!48$$$$T^{22} +$$$$58\!\cdots\!84$$$$T^{23} +$$$$81\!\cdots\!61$$$$T^{24}$$
$71$ $$1 - 404 T^{2} + 75490 T^{4} - 8307268 T^{6} + 571593487 T^{8} - 25299911144 T^{10} + 1141503138780 T^{12} - 127536852076904 T^{14} + 14525151353321647 T^{16} - 1064163389407837828 T^{18} + 48747934073742497890 T^{20} -$$$$13\!\cdots\!04$$$$T^{22} +$$$$16\!\cdots\!41$$$$T^{24}$$
$73$ $$1 + 20 T + 200 T^{2} + 1452 T^{3} + 7482 T^{4} + 35804 T^{5} + 273832 T^{6} + 2000484 T^{7} + 27883951 T^{8} + 188107816 T^{9} - 317778928 T^{10} - 34324729064 T^{11} - 473910414484 T^{12} - 2505705221672 T^{13} - 1693443907312 T^{14} + 73177138256872 T^{15} + 791855160530191 T^{16} + 4147146552651012 T^{17} + 41440153853169448 T^{18} + 395541056577748988 T^{19} + 6033934407551514042 T^{20} + 85481543900405009676 T^{21} +$$$$85\!\cdots\!00$$$$T^{22} +$$$$62\!\cdots\!40$$$$T^{23} +$$$$22\!\cdots\!21$$$$T^{24}$$
$79$ $$1 - 372 T^{2} + 73090 T^{4} - 9878564 T^{6} + 1031952175 T^{8} - 90900376040 T^{10} + 7353042693660 T^{12} - 567309246865640 T^{14} + 40194620804376175 T^{16} - 2401354986961351844 T^{18} +$$$$11\!\cdots\!90$$$$T^{20} -$$$$35\!\cdots\!72$$$$T^{22} +$$$$59\!\cdots\!41$$$$T^{24}$$
$83$ $$1 + 24 T + 288 T^{2} + 2760 T^{3} + 26518 T^{4} + 273208 T^{5} + 2728608 T^{6} + 24159272 T^{7} + 240213375 T^{8} + 2803489520 T^{9} + 29745965888 T^{10} + 283235701200 T^{11} + 2619914616372 T^{12} + 23508563199600 T^{13} + 204919959002432 T^{14} + 1602998862172240 T^{15} + 11400123459243375 T^{16} + 95164354313291896 T^{17} + 892092118297640352 T^{18} + 7413786218774013416 T^{19} + 59726285411863089238 T^{20} +$$$$51\!\cdots\!80$$$$T^{21} +$$$$44\!\cdots\!12$$$$T^{22} +$$$$30\!\cdots\!08$$$$T^{23} +$$$$10\!\cdots\!61$$$$T^{24}$$
$89$ $$( 1 - 28 T + 684 T^{2} - 11188 T^{3} + 161827 T^{4} - 1876496 T^{5} + 19427016 T^{6} - 167008144 T^{7} + 1281831667 T^{8} - 7887193172 T^{9} + 42915692844 T^{10} - 156353664572 T^{11} + 496981290961 T^{12} )^{2}$$
$97$ $$1 - 12 T + 72 T^{2} - 884 T^{3} + 15450 T^{4} - 102628 T^{5} + 509864 T^{6} - 1529948 T^{7} - 62272049 T^{8} + 410787880 T^{9} - 1704538608 T^{10} + 69640900504 T^{11} - 1520682376276 T^{12} + 6755167348888 T^{13} - 16038003762672 T^{14} + 374915006803240 T^{15} - 5512899724366769 T^{16} - 13138184051516636 T^{17} + 424702438321119656 T^{18} - 8292166339419780964 T^{19} +$$$$12\!\cdots\!50$$$$T^{20} -$$$$67\!\cdots\!28$$$$T^{21} +$$$$53\!\cdots\!28$$$$T^{22} -$$$$85\!\cdots\!36$$$$T^{23} +$$$$69\!\cdots\!41$$$$T^{24}$$