# Properties

 Label 315.3.h.d Level 315 Weight 3 Character orbit 315.h Analytic conductor 8.583 Analytic rank 0 Dimension 12 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$8.58312832735$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{12}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( 4 - \beta_{1} ) q^{4} -\beta_{8} q^{5} + ( -1 - \beta_{8} + \beta_{9} ) q^{7} + ( -1 - \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{5} ) q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( 4 - \beta_{1} ) q^{4} -\beta_{8} q^{5} + ( -1 - \beta_{8} + \beta_{9} ) q^{7} + ( -1 - \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{5} ) q^{8} -\beta_{6} q^{10} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{11} ) q^{11} + ( \beta_{4} + 4 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{13} + ( 4 - 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{10} ) q^{14} + ( 9 - 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{9} - 2 \beta_{10} ) q^{16} + ( -\beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{17} + ( 4 \beta_{4} + 2 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{19} + ( -\beta_{7} - 4 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{20} + ( -9 + 3 \beta_{1} - 4 \beta_{2} + \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{22} + ( 3 + 3 \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{9} + 2 \beta_{10} ) q^{23} -5 q^{25} + ( 6 \beta_{4} + 5 \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{26} + ( 4 + 3 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 9 \beta_{8} + 3 \beta_{9} - 2 \beta_{11} ) q^{28} + ( -5 - 5 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{9} - 2 \beta_{10} ) q^{29} + ( 10 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{31} + ( 20 - 4 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} - 6 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{11} ) q^{32} + ( -6 \beta_{4} - 3 \beta_{7} - 5 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} ) q^{34} + ( -5 + \beta_{1} + \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{35} + ( 5 - 5 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{37} + ( 4 \beta_{4} - 3 \beta_{6} + 3 \beta_{7} + 16 \beta_{8} + \beta_{9} - \beta_{10} + 8 \beta_{11} ) q^{38} + ( -5 \beta_{4} - 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{40} + ( -8 \beta_{4} + 3 \beta_{6} + 3 \beta_{8} + 5 \beta_{11} ) q^{41} + ( 13 + 3 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} ) q^{43} + ( -17 + 7 \beta_{1} - 14 \beta_{2} + 7 \beta_{3} + 4 \beta_{4} + 9 \beta_{5} + \beta_{6} - \beta_{8} + 4 \beta_{9} + 4 \beta_{10} + \beta_{11} ) q^{44} + ( 17 - \beta_{1} + 4 \beta_{3} - 2 \beta_{4} + 8 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{46} + ( -\beta_{6} - 7 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} + 4 \beta_{11} ) q^{47} + ( 8 + \beta_{1} - 10 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - 3 \beta_{7} - 9 \beta_{8} - 2 \beta_{10} + 5 \beta_{11} ) q^{49} -5 \beta_{2} q^{50} + ( 5 \beta_{4} + 6 \beta_{7} + 18 \beta_{8} - 3 \beta_{9} + 3 \beta_{10} + 16 \beta_{11} ) q^{52} + ( -15 + 9 \beta_{1} - 12 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{9} + 2 \beta_{10} ) q^{53} + ( 2 \beta_{6} + 3 \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{55} + ( 35 - 7 \beta_{1} + \beta_{2} + \beta_{3} - 10 \beta_{4} - 9 \beta_{6} - 4 \beta_{7} - 11 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} - 6 \beta_{11} ) q^{56} + ( -35 + \beta_{1} + 4 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{58} + ( -4 \beta_{4} - 7 \beta_{6} + 2 \beta_{7} - 7 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} + 5 \beta_{11} ) q^{59} + ( -6 \beta_{4} - 9 \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{61} + ( 8 \beta_{4} + 14 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} ) q^{62} + ( -4 - 9 \beta_{1} + 28 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - 4 \beta_{6} + 4 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 4 \beta_{11} ) q^{64} + ( 22 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{65} + ( 17 - 9 \beta_{1} + 6 \beta_{2} + 8 \beta_{3} - \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{67} + ( -16 \beta_{4} - 5 \beta_{6} + \beta_{7} - 3 \beta_{9} + 3 \beta_{10} - 12 \beta_{11} ) q^{68} + ( -3 + 3 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} - 5 \beta_{11} ) q^{70} + ( -2 + 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + \beta_{6} - \beta_{8} - 6 \beta_{9} - 6 \beta_{10} + \beta_{11} ) q^{71} + ( -12 \beta_{4} - 10 \beta_{6} + 2 \beta_{7} + 14 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 6 \beta_{11} ) q^{73} + ( -21 - 3 \beta_{1} + 12 \beta_{2} - 9 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{74} + ( 14 \beta_{4} + 18 \beta_{6} + 9 \beta_{7} - 27 \beta_{8} + 3 \beta_{11} ) q^{76} + ( -4 + 2 \beta_{1} + 8 \beta_{3} + 20 \beta_{4} + 6 \beta_{5} - \beta_{6} + 3 \beta_{7} - 10 \beta_{8} + 3 \beta_{9} + 7 \beta_{10} + 8 \beta_{11} ) q^{77} + ( 12 + 10 \beta_{1} - 20 \beta_{2} + 6 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + 4 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} ) q^{79} + ( -10 \beta_{4} - 2 \beta_{6} - 7 \beta_{8} - 10 \beta_{11} ) q^{80} + ( 7 \beta_{4} + 8 \beta_{6} + 8 \beta_{7} + 24 \beta_{8} - 3 \beta_{9} + 3 \beta_{10} - 18 \beta_{11} ) q^{82} + ( -6 \beta_{4} + 12 \beta_{6} - 8 \beta_{8} + 4 \beta_{9} - 4 \beta_{10} ) q^{83} + ( 9 + 3 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{85} + ( -33 - 3 \beta_{1} + 20 \beta_{2} + 5 \beta_{3} - 8 \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{8} - 8 \beta_{9} - 8 \beta_{10} - \beta_{11} ) q^{86} + ( -47 + 25 \beta_{1} - 24 \beta_{2} + 14 \beta_{3} + 5 \beta_{4} + 10 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + 5 \beta_{9} + 5 \beta_{10} - 2 \beta_{11} ) q^{88} + ( -4 \beta_{4} - \beta_{6} - 4 \beta_{7} + 31 \beta_{8} + 4 \beta_{9} - 4 \beta_{10} - 7 \beta_{11} ) q^{89} + ( 7 - 11 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} + 2 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} + 5 \beta_{7} - 21 \beta_{8} - 6 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{91} + ( -29 + 19 \beta_{1} - 4 \beta_{2} - \beta_{3} + 10 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + 10 \beta_{9} + 10 \beta_{10} + 2 \beta_{11} ) q^{92} + ( 6 \beta_{4} + 3 \beta_{7} + 7 \beta_{8} - 8 \beta_{9} + 8 \beta_{10} - 27 \beta_{11} ) q^{94} + ( -3 + 3 \beta_{1} + 12 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 5 \beta_{5} - \beta_{6} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{95} + ( -2 \beta_{4} - 6 \beta_{7} - 4 \beta_{8} - 10 \beta_{9} + 10 \beta_{10} + 4 \beta_{11} ) q^{97} + ( -79 + 11 \beta_{1} + \beta_{2} - \beta_{3} + 12 \beta_{4} + \beta_{5} - 7 \beta_{6} + \beta_{7} - 20 \beta_{8} + 3 \beta_{9} - \beta_{10} - 8 \beta_{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 4q^{2} + 44q^{4} - 8q^{7} - 4q^{8} + O(q^{10})$$ $$12q + 4q^{2} + 44q^{4} - 8q^{7} - 4q^{8} + 16q^{11} + 40q^{14} + 92q^{16} - 88q^{22} + 64q^{23} - 60q^{25} + 88q^{28} - 104q^{29} + 228q^{32} - 60q^{35} + 32q^{37} + 152q^{43} - 192q^{44} + 200q^{46} + 60q^{49} - 20q^{50} - 176q^{53} + 368q^{56} - 400q^{58} - 20q^{64} + 240q^{65} + 168q^{67} - 60q^{70} - 32q^{71} - 184q^{74} - 8q^{77} + 120q^{79} + 120q^{85} - 400q^{86} - 536q^{88} + 24q^{91} - 192q^{92} - 884q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + 21 x^{10} - 26 x^{9} + 295 x^{8} - 372 x^{7} + 1704 x^{6} - 1074 x^{5} + 5613 x^{4} - 5472 x^{3} + 4950 x^{2} - 1638 x + 441$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$1163071964 \nu^{11} + 179172244 \nu^{10} + 20630949354 \nu^{9} + 16671982336 \nu^{8} + 315532034086 \nu^{7} + 213549867516 \nu^{6} + 1552126862414 \nu^{5} + 1781364901824 \nu^{4} + 8663666606754 \nu^{3} + 2969181679896 \nu^{2} - 1028895680526 \nu + 2151653525805$$$$)/ 5604734688861$$ $$\beta_{2}$$ $$=$$ $$($$$$5593516534 \nu^{11} - 2605315656 \nu^{10} + 102990651177 \nu^{9} + 27516350446 \nu^{8} + 1489720495401 \nu^{7} + 348399189483 \nu^{6} + 7205061506889 \nu^{5} + 7155359090394 \nu^{4} + 27659028800376 \nu^{3} + 12643415740656 \nu^{2} - 4393836770841 \nu + 10342332760284$$$$)/ 2402029152369$$ $$\beta_{3}$$ $$=$$ $$($$$$42289552636 \nu^{11} - 136320072447 \nu^{10} + 936265293410 \nu^{9} - 2156880862761 \nu^{8} + 12823809975400 \nu^{7} - 31274532528455 \nu^{6} + 76701806074920 \nu^{5} - 135085839058755 \nu^{4} + 221435259480507 \nu^{3} - 573525290759622 \nu^{2} + 198646121553516 \nu - 203704095121695$$$$)/ 16814204066583$$ $$\beta_{4}$$ $$=$$ $$($$$$15660232032 \nu^{11} - 28994320136 \nu^{10} + 329223217160 \nu^{9} - 365904134124 \nu^{8} + 4653112414112 \nu^{7} - 5194542247732 \nu^{6} + 27112135117560 \nu^{5} - 13714835477540 \nu^{4} + 91463612199264 \nu^{3} - 68365456465596 \nu^{2} + 83456511918192 \nu - 16499782051746$$$$)/ 5604734688861$$ $$\beta_{5}$$ $$=$$ $$($$$$2752740836 \nu^{11} + 4168730751 \nu^{10} + 43893742358 \nu^{9} + 122141383813 \nu^{8} + 668319319750 \nu^{7} + 1718774236945 \nu^{6} + 2607856396482 \nu^{5} + 11901611179383 \nu^{4} + 13113600071907 \nu^{3} + 35493670736670 \nu^{2} - 12304269726462 \nu + 17534758495653$$$$)/ 731052350721$$ $$\beta_{6}$$ $$=$$ $$($$$$-79026253784 \nu^{11} + 146272115177 \nu^{10} - 1671226017089 \nu^{9} + 1834614270589 \nu^{8} - 23655237895132 \nu^{7} + 25973059454509 \nu^{6} - 139385258261016 \nu^{5} + 65772299304051 \nu^{4} - 472028460969018 \nu^{3} + 346171354438782 \nu^{2} - 442224670054734 \nu + 86851646987175$$$$)/ 16814204066583$$ $$\beta_{7}$$ $$=$$ $$($$$$82515469676 \nu^{11} - 145734598445 \nu^{10} + 1733118865151 \nu^{9} - 1784598323581 \nu^{8} + 24601833997390 \nu^{7} - 25332409851961 \nu^{6} + 144041638848258 \nu^{5} - 60428204598579 \nu^{4} + 498019460789280 \nu^{3} - 337263809399094 \nu^{2} + 506394799279488 \nu - 97210890476343$$$$)/ 16814204066583$$ $$\beta_{8}$$ $$=$$ $$($$$$-5888 \nu^{11} + 11357 \nu^{10} - 121362 \nu^{9} + 142983 \nu^{8} - 1697439 \nu^{7} + 2055140 \nu^{6} - 9463719 \nu^{5} + 5419563 \nu^{4} - 30382920 \nu^{3} + 29629158 \nu^{2} - 18996579 \nu + 4306806$$$$)/1130283$$ $$\beta_{9}$$ $$=$$ $$($$$$91626156422 \nu^{11} - 196270735236 \nu^{10} + 1900808184921 \nu^{9} - 2547333298184 \nu^{8} + 26419119629440 \nu^{7} - 36605150224288 \nu^{6} + 148372710389544 \nu^{5} - 101412405605496 \nu^{4} + 467993649961155 \nu^{3} - 534812214191700 \nu^{2} + 375544345114512 \nu - 2426520744774$$$$)/ 16814204066583$$ $$\beta_{10}$$ $$=$$ $$($$$$-161873480282 \nu^{11} + 275300339232 \nu^{10} - 3313587375429 \nu^{9} + 3245165108972 \nu^{8} - 46725482453698 \nu^{7} + 46879197803692 \nu^{6} - 261085625853546 \nu^{5} + 105142657549572 \nu^{4} - 876196609907601 \nu^{3} + 678449825160384 \nu^{2} - 604632734322246 \nu + 207714123345456$$$$)/ 16814204066583$$ $$\beta_{11}$$ $$=$$ $$($$$$722764520 \nu^{11} - 1406324924 \nu^{10} + 14880207305 \nu^{9} - 17611281862 \nu^{8} + 207844732437 \nu^{7} - 253771899971 \nu^{6} + 1154612659597 \nu^{5} - 659018133716 \nu^{4} + 3676528796220 \nu^{3} - 3732659132112 \nu^{2} + 2124183465543 \nu - 498205455783$$$$)/ 72788762193$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} - \beta_{1} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} + \beta_{10} + \beta_{9} + 3 \beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{5} + 8 \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_{1} - 13$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{11} - \beta_{8} + \beta_{6} + \beta_{5} - \beta_{3} - 6 \beta_{2} + 11 \beta_{1} - 9$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-23 \beta_{11} + 15 \beta_{10} + 9 \beta_{9} - 59 \beta_{8} + 10 \beta_{7} - 27 \beta_{6} + 15 \beta_{5} - 67 \beta_{4} + 17 \beta_{3} - 32 \beta_{2} + 14 \beta_{1} - 138$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-96 \beta_{11} - 48 \beta_{10} + 54 \beta_{9} - 16 \beta_{8} - 137 \beta_{7} - 145 \beta_{6} - 21 \beta_{5} - 28 \beta_{4} + 13 \beta_{3} + 110 \beta_{2} - 131 \beta_{1} + 117$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$13 \beta_{11} - 144 \beta_{10} - 144 \beta_{9} - 13 \beta_{8} + 13 \beta_{6} - 213 \beta_{5} - 144 \beta_{4} - 239 \beta_{3} + 476 \beta_{2} - 194 \beta_{1} + 1644$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$1022 \beta_{11} + 762 \beta_{10} - 672 \beta_{9} + 1082 \beta_{8} + 1748 \beta_{7} + 1374 \beta_{6} - 357 \beta_{5} + 646 \beta_{4} + 121 \beta_{3} + 1694 \beta_{2} - 1658 \beta_{1} + 1644$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$5220 \beta_{11} + 1416 \beta_{10} + 2142 \beta_{9} + 13222 \beta_{8} - 793 \beta_{7} + 4423 \beta_{6} + 2952 \beta_{5} + 14308 \beta_{4} + 3194 \beta_{3} - 6878 \beta_{2} + 2765 \beta_{1} - 20703$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$3194 \beta_{11} - 429 \beta_{10} - 429 \beta_{9} - 3194 \beta_{8} + 3194 \beta_{6} + 5598 \beta_{5} - 429 \beta_{4} - 790 \beta_{3} - 24704 \beta_{2} + 21683 \beta_{1} - 24291$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-77615 \beta_{11} + 24843 \beta_{10} + 20103 \beta_{9} - 179477 \beta_{8} + 4867 \beta_{7} - 63342 \beta_{6} + 40491 \beta_{5} - 141469 \beta_{4} + 42071 \beta_{3} - 97718 \beta_{2} + 40079 \beta_{1} - 268665$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$-312327 \beta_{11} - 124221 \beta_{10} + 128205 \beta_{9} - 218377 \beta_{8} - 292652 \beta_{7} - 328747 \beta_{6} - 84483 \beta_{5} - 170899 \beta_{4} - 341 \beta_{3} + 352406 \beta_{2} - 288668 \beta_{1} + 366186$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 1.31896 − 2.28450i 1.31896 + 2.28450i 0.378061 + 0.654821i 0.378061 − 0.654821i −1.74681 − 3.02556i −1.74681 + 3.02556i −1.01714 + 1.76174i −1.01714 − 1.76174i 0.198184 − 0.343264i 0.198184 + 0.343264i 1.86875 + 3.23677i 1.86875 − 3.23677i
−3.50369 0 8.27584 2.23607i 0 −6.69736 2.03600i −14.9812 0 7.83449i
181.2 −3.50369 0 8.27584 2.23607i 0 −6.69736 + 2.03600i −14.9812 0 7.83449i
181.3 −2.91758 0 4.51225 2.23607i 0 6.13981 3.36195i −1.49451 0 6.52390i
181.4 −2.91758 0 4.51225 2.23607i 0 6.13981 + 3.36195i −1.49451 0 6.52390i
181.5 0.112974 0 −3.98724 2.23607i 0 −6.71303 1.98374i −0.902349 0 0.252617i
181.6 0.112974 0 −3.98724 2.23607i 0 −6.71303 + 1.98374i −0.902349 0 0.252617i
181.7 1.71214 0 −1.06857 2.23607i 0 −3.33344 + 6.15534i −8.67811 0 3.82847i
181.8 1.71214 0 −1.06857 2.23607i 0 −3.33344 6.15534i −8.67811 0 3.82847i
181.9 2.79155 0 3.79273 2.23607i 0 4.15782 5.63139i −0.578591 0 6.24209i
181.10 2.79155 0 3.79273 2.23607i 0 4.15782 + 5.63139i −0.578591 0 6.24209i
181.11 3.80460 0 10.4750 2.23607i 0 2.44621 6.55866i 24.6348 0 8.50735i
181.12 3.80460 0 10.4750 2.23607i 0 2.44621 + 6.55866i 24.6348 0 8.50735i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 181.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.3.h.d 12
3.b odd 2 1 105.3.h.a 12
7.b odd 2 1 inner 315.3.h.d 12
12.b even 2 1 1680.3.s.c 12
15.d odd 2 1 525.3.h.d 12
15.e even 4 2 525.3.e.c 24
21.c even 2 1 105.3.h.a 12
84.h odd 2 1 1680.3.s.c 12
105.g even 2 1 525.3.h.d 12
105.k odd 4 2 525.3.e.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.h.a 12 3.b odd 2 1
105.3.h.a 12 21.c even 2 1
315.3.h.d 12 1.a even 1 1 trivial
315.3.h.d 12 7.b odd 2 1 inner
525.3.e.c 24 15.e even 4 2
525.3.e.c 24 105.k odd 4 2
525.3.h.d 12 15.d odd 2 1
525.3.h.d 12 105.g even 2 1
1680.3.s.c 12 12.b even 2 1
1680.3.s.c 12 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 2 T_{2}^{5} - 21 T_{2}^{4} + 40 T_{2}^{3} + 103 T_{2}^{2} - 198 T_{2} + 21$$ acting on $$S_{3}^{\mathrm{new}}(315, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 3 T^{2} + 7 T^{4} - 38 T^{5} + 109 T^{6} - 152 T^{7} + 112 T^{8} + 768 T^{10} - 2048 T^{11} + 4096 T^{12} )^{2}$$
$3$ 1
$5$ $$( 1 + 5 T^{2} )^{6}$$
$7$ $$1 + 8 T + 2 T^{2} + 312 T^{3} + 4255 T^{4} + 13888 T^{5} + 43708 T^{6} + 680512 T^{7} + 10216255 T^{8} + 36706488 T^{9} + 11529602 T^{10} + 2259801992 T^{11} + 13841287201 T^{12}$$
$11$ $$( 1 - 8 T + 198 T^{2} - 1416 T^{3} + 30895 T^{4} - 264944 T^{5} + 5610292 T^{6} - 32058224 T^{7} + 452333695 T^{8} - 2508530376 T^{9} + 42443058438 T^{10} - 207499396808 T^{11} + 3138428376721 T^{12} )^{2}$$
$13$ $$1 - 852 T^{2} + 436674 T^{4} - 159409348 T^{6} + 45042421839 T^{8} - 10253858128680 T^{10} + 1907667001388316 T^{12} - 292860442013229480 T^{14} + 36742487242313615919 T^{16} -$$$$37\!\cdots\!88$$$$T^{18} +$$$$29\!\cdots\!34$$$$T^{20} -$$$$16\!\cdots\!52$$$$T^{22} +$$$$54\!\cdots\!61$$$$T^{24}$$
$17$ $$1 - 2172 T^{2} + 2401794 T^{4} - 1757595148 T^{6} + 941667483759 T^{8} - 387997181982840 T^{10} + 125896935467491356 T^{12} - 32405912636388779640 T^{14} +$$$$65\!\cdots\!19$$$$T^{16} -$$$$10\!\cdots\!28$$$$T^{18} +$$$$11\!\cdots\!14$$$$T^{20} -$$$$88\!\cdots\!72$$$$T^{22} +$$$$33\!\cdots\!21$$$$T^{24}$$
$19$ $$1 - 1404 T^{2} + 1342050 T^{4} - 896154316 T^{6} + 500154418383 T^{8} - 227690692547448 T^{10} + 90028611036772572 T^{12} - 29672878743475970808 T^{14} +$$$$84\!\cdots\!03$$$$T^{16} -$$$$19\!\cdots\!76$$$$T^{18} +$$$$38\!\cdots\!50$$$$T^{20} -$$$$52\!\cdots\!04$$$$T^{22} +$$$$48\!\cdots\!21$$$$T^{24}$$
$23$ $$( 1 - 32 T + 2418 T^{2} - 60144 T^{3} + 2694751 T^{4} - 53735792 T^{5} + 1782680284 T^{6} - 28426233968 T^{7} + 754101814591 T^{8} - 8903470508016 T^{9} + 189355962409458 T^{10} - 1325648358836768 T^{11} + 21914624432020321 T^{12} )^{2}$$
$29$ $$( 1 + 52 T + 3990 T^{2} + 132324 T^{3} + 6311983 T^{4} + 164304136 T^{5} + 6334525012 T^{6} + 138179778376 T^{7} + 4464345648223 T^{8} + 78709401128004 T^{9} + 1995983187714390 T^{10} + 21876776131610452 T^{11} + 353814783205469041 T^{12} )^{2}$$
$31$ $$1 - 6228 T^{2} + 19622274 T^{4} - 42341564932 T^{6} + 69647410996719 T^{8} - 91080823295899560 T^{10} + 96735996796679061276 T^{12} -$$$$84\!\cdots\!60$$$$T^{14} +$$$$59\!\cdots\!79$$$$T^{16} -$$$$33\!\cdots\!52$$$$T^{18} +$$$$14\!\cdots\!94$$$$T^{20} -$$$$41\!\cdots\!28$$$$T^{22} +$$$$62\!\cdots\!21$$$$T^{24}$$
$37$ $$( 1 - 16 T + 4622 T^{2} + 32784 T^{3} + 7136719 T^{4} + 278279680 T^{5} + 7429424740 T^{6} + 380964881920 T^{7} + 13375360417759 T^{8} + 84114774592656 T^{9} + 16234680036022862 T^{10} - 76937349958685584 T^{11} + 6582952005840035281 T^{12} )^{2}$$
$41$ $$1 - 7524 T^{2} + 32464386 T^{4} - 93240551380 T^{6} + 206850565610415 T^{8} - 379921653079011144 T^{10} +$$$$65\!\cdots\!56$$$$T^{12} -$$$$10\!\cdots\!84$$$$T^{14} +$$$$16\!\cdots\!15$$$$T^{16} -$$$$21\!\cdots\!80$$$$T^{18} +$$$$20\!\cdots\!26$$$$T^{20} -$$$$13\!\cdots\!24$$$$T^{22} +$$$$50\!\cdots\!61$$$$T^{24}$$
$43$ $$( 1 - 76 T + 9590 T^{2} - 631788 T^{3} + 42146383 T^{4} - 2188827368 T^{5} + 102971644948 T^{6} - 4047141803432 T^{7} + 144090096346783 T^{8} - 3993761318001612 T^{9} + 112089840662193590 T^{10} - 1642472655809602924 T^{11} + 39959630797262576401 T^{12} )^{2}$$
$47$ $$1 - 6132 T^{2} + 27456354 T^{4} - 92403901348 T^{6} + 277727516070639 T^{8} - 754801362269230440 T^{10} +$$$$17\!\cdots\!96$$$$T^{12} -$$$$36\!\cdots\!40$$$$T^{14} +$$$$66\!\cdots\!79$$$$T^{16} -$$$$10\!\cdots\!68$$$$T^{18} +$$$$15\!\cdots\!34$$$$T^{20} -$$$$16\!\cdots\!32$$$$T^{22} +$$$$13\!\cdots\!81$$$$T^{24}$$
$53$ $$( 1 + 88 T + 10434 T^{2} + 574440 T^{3} + 50272015 T^{4} + 2575931008 T^{5} + 181692199804 T^{6} + 7235790201472 T^{7} + 396670379189215 T^{8} + 12732095606942760 T^{9} + 649617609752140674 T^{10} + 15390097392165148312 T^{11} +$$$$49\!\cdots\!41$$$$T^{12} )^{2}$$
$59$ $$1 - 15948 T^{2} + 137050242 T^{4} - 864152710108 T^{6} + 4419517662043791 T^{8} - 18954945757384385304 T^{10} +$$$$70\!\cdots\!16$$$$T^{12} -$$$$22\!\cdots\!44$$$$T^{14} +$$$$64\!\cdots\!11$$$$T^{16} -$$$$15\!\cdots\!48$$$$T^{18} +$$$$29\!\cdots\!22$$$$T^{20} -$$$$41\!\cdots\!48$$$$T^{22} +$$$$31\!\cdots\!61$$$$T^{24}$$
$61$ $$1 - 13716 T^{2} + 109224834 T^{4} - 648593370628 T^{6} + 3203041286245839 T^{8} - 13994290873734287016 T^{10} +$$$$55\!\cdots\!56$$$$T^{12} -$$$$19\!\cdots\!56$$$$T^{14} +$$$$61\!\cdots\!59$$$$T^{16} -$$$$17\!\cdots\!88$$$$T^{18} +$$$$40\!\cdots\!74$$$$T^{20} -$$$$69\!\cdots\!16$$$$T^{22} +$$$$70\!\cdots\!41$$$$T^{24}$$
$67$ $$( 1 - 84 T + 15366 T^{2} - 763540 T^{3} + 102124911 T^{4} - 4033313976 T^{5} + 514098700788 T^{6} - 18105546438264 T^{7} + 2057931438675231 T^{8} - 69068593121318260 T^{9} + 6239635933335345606 T^{10} -$$$$15\!\cdots\!16$$$$T^{11} +$$$$81\!\cdots\!61$$$$T^{12} )^{2}$$
$71$ $$( 1 + 16 T + 16698 T^{2} + 42336 T^{3} + 131112895 T^{4} - 1041328304 T^{5} + 716701578892 T^{6} - 5249335980464 T^{7} + 3331799062726495 T^{8} + 5423253620079456 T^{9} + 10782792464741717178 T^{10} + 52083896816158099216 T^{11} +$$$$16\!\cdots\!41$$$$T^{12} )^{2}$$
$73$ $$1 - 19284 T^{2} + 216036930 T^{4} - 1973102936452 T^{6} + 14880325491672495 T^{8} - 96746229105564519336 T^{10} +$$$$55\!\cdots\!08$$$$T^{12} -$$$$27\!\cdots\!76$$$$T^{14} +$$$$12\!\cdots\!95$$$$T^{16} -$$$$45\!\cdots\!92$$$$T^{18} +$$$$14\!\cdots\!30$$$$T^{20} -$$$$35\!\cdots\!84$$$$T^{22} +$$$$52\!\cdots\!41$$$$T^{24}$$
$79$ $$( 1 - 60 T + 22242 T^{2} - 662476 T^{3} + 201003183 T^{4} - 1192106616 T^{5} + 1261845815388 T^{6} - 7439937390456 T^{7} + 7829090259107823 T^{8} - 161039605183729996 T^{9} + 33743534149941729762 T^{10} -$$$$56\!\cdots\!60$$$$T^{11} +$$$$59\!\cdots\!41$$$$T^{12} )^{2}$$
$83$ $$1 - 45420 T^{2} + 1066844514 T^{4} - 17098817034364 T^{6} + 206563644037003983 T^{8} -$$$$19\!\cdots\!08$$$$T^{10} +$$$$15\!\cdots\!04$$$$T^{12} -$$$$93\!\cdots\!68$$$$T^{14} +$$$$46\!\cdots\!03$$$$T^{16} -$$$$18\!\cdots\!04$$$$T^{18} +$$$$54\!\cdots\!34$$$$T^{20} -$$$$10\!\cdots\!20$$$$T^{22} +$$$$11\!\cdots\!21$$$$T^{24}$$
$89$ $$1 - 46020 T^{2} + 1126013634 T^{4} - 19221077499316 T^{6} + 252845241317038383 T^{8} -$$$$26\!\cdots\!72$$$$T^{10} +$$$$23\!\cdots\!84$$$$T^{12} -$$$$16\!\cdots\!52$$$$T^{14} +$$$$99\!\cdots\!23$$$$T^{16} -$$$$47\!\cdots\!36$$$$T^{18} +$$$$17\!\cdots\!74$$$$T^{20} -$$$$44\!\cdots\!20$$$$T^{22} +$$$$61\!\cdots\!41$$$$T^{24}$$
$97$ $$1 - 51924 T^{2} + 1396974978 T^{4} - 26352399780484 T^{6} + 389015604150559215 T^{8} -$$$$47\!\cdots\!84$$$$T^{10} +$$$$48\!\cdots\!52$$$$T^{12} -$$$$41\!\cdots\!04$$$$T^{14} +$$$$30\!\cdots\!15$$$$T^{16} -$$$$18\!\cdots\!44$$$$T^{18} +$$$$85\!\cdots\!38$$$$T^{20} -$$$$28\!\cdots\!24$$$$T^{22} +$$$$48\!\cdots\!81$$$$T^{24}$$