# Properties

 Label 930.2.n.d Level $930$ Weight $2$ Character orbit 930.n Analytic conductor $7.426$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2 x^{11} + 13 x^{10} - 9 x^{9} + 60 x^{8} + x^{7} + 263 x^{6} + 823 x^{5} + 1931 x^{4} + 1170 x^{3} + 1340 x^{2} + 1000 x + 400$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$5$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{3} + \beta_{5} - \beta_{6} ) q^{2} + \beta_{5} q^{3} + \beta_{3} q^{4} - q^{5} - q^{6} + ( -\beta_{1} - \beta_{4} - \beta_{7} ) q^{7} + \beta_{6} q^{8} + \beta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{3} + \beta_{5} - \beta_{6} ) q^{2} + \beta_{5} q^{3} + \beta_{3} q^{4} - q^{5} - q^{6} + ( -\beta_{1} - \beta_{4} - \beta_{7} ) q^{7} + \beta_{6} q^{8} + \beta_{6} q^{9} + ( 1 + \beta_{3} - \beta_{5} + \beta_{6} ) q^{10} + ( 1 + \beta_{3} - \beta_{5} + \beta_{8} ) q^{11} + ( 1 + \beta_{3} - \beta_{5} + \beta_{6} ) q^{12} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{13} + ( \beta_{2} + \beta_{7} ) q^{14} -\beta_{5} q^{15} -\beta_{5} q^{16} + ( \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{17} -\beta_{5} q^{18} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{10} ) q^{19} -\beta_{3} q^{20} + ( \beta_{1} + \beta_{2} + \beta_{7} ) q^{21} + ( -\beta_{3} + \beta_{5} + \beta_{11} ) q^{22} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{23} -\beta_{3} q^{24} + q^{25} + ( 2 - 2 \beta_{2} + \beta_{3} + \beta_{6} + \beta_{9} ) q^{26} -\beta_{3} q^{27} + ( \beta_{1} + \beta_{7} ) q^{28} + ( \beta_{1} + \beta_{2} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{29} + q^{30} + ( 1 - \beta_{2} + \beta_{5} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{31} + q^{32} + ( 1 + \beta_{3} - \beta_{10} ) q^{33} + ( 2 + \beta_{1} + 2 \beta_{6} + \beta_{7} - \beta_{9} + \beta_{11} ) q^{34} + ( \beta_{1} + \beta_{4} + \beta_{7} ) q^{35} + q^{36} + ( -2 + 3 \beta_{3} + 3 \beta_{6} ) q^{37} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{7} - \beta_{8} ) q^{38} + ( \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{11} ) q^{39} -\beta_{6} q^{40} + ( -\beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{41} + ( \beta_{1} + \beta_{4} + \beta_{7} ) q^{42} + ( 4 - \beta_{1} - \beta_{2} + 4 \beta_{3} - 5 \beta_{5} + 5 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{43} + ( -1 - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{44} -\beta_{6} q^{45} + ( -1 + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{46} + ( -2 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{10} ) q^{47} -\beta_{6} q^{48} + ( 3 + 3 \beta_{1} + \beta_{2} + \beta_{4} + 3 \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{10} ) q^{49} + ( -1 - \beta_{3} + \beta_{5} - \beta_{6} ) q^{50} + ( 2 + \beta_{1} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{51} + ( -2 - 2 \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{10} ) q^{52} + ( 3 \beta_{2} + \beta_{4} + 3 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{53} -\beta_{6} q^{54} + ( -1 - \beta_{3} + \beta_{5} - \beta_{8} ) q^{55} + ( \beta_{1} + \beta_{4} ) q^{56} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{9} ) q^{57} + ( -\beta_{1} + \beta_{4} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{58} + ( \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{10} ) q^{59} + ( -1 - \beta_{3} + \beta_{5} - \beta_{6} ) q^{60} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{8} - 3 \beta_{9} + \beta_{11} ) q^{61} + ( -2 - \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{62} + ( \beta_{1} + \beta_{4} ) q^{63} + ( -1 - \beta_{3} + \beta_{5} - \beta_{6} ) q^{64} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{65} + ( -1 - \beta_{3} + \beta_{5} - \beta_{8} ) q^{66} + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{4} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} ) q^{67} + ( -2 + \beta_{1} - 2 \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{68} + ( -1 - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - 2 \beta_{10} - 2 \beta_{11} ) q^{69} + ( -\beta_{2} - \beta_{7} ) q^{70} + ( 3 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{71} + ( -1 - \beta_{3} + \beta_{5} - \beta_{6} ) q^{72} + ( 4 - 3 \beta_{1} + 7 \beta_{3} - \beta_{4} - 4 \beta_{5} - \beta_{7} + \beta_{8} ) q^{73} + ( 2 + 2 \beta_{3} - 5 \beta_{5} + 5 \beta_{6} ) q^{74} + \beta_{5} q^{75} + ( \beta_{2} + 2 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{11} ) q^{76} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{10} ) q^{77} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{78} + ( -2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{7} + 2 \beta_{11} ) q^{79} + \beta_{5} q^{80} + ( -1 - \beta_{3} + \beta_{5} - \beta_{6} ) q^{81} + ( 1 - \beta_{1} - \beta_{5} - \beta_{8} + \beta_{10} + \beta_{11} ) q^{82} + ( -3 - 3 \beta_{3} + 3 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{83} + ( -\beta_{2} - \beta_{7} ) q^{84} + ( -\beta_{2} - 2 \beta_{3} + 2 \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{85} + ( 1 - \beta_{1} - 4 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} ) q^{86} + ( \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{8} - \beta_{11} ) q^{87} + ( 1 + \beta_{3} + \beta_{6} - \beta_{9} ) q^{88} + ( -2 + 3 \beta_{1} + \beta_{3} + 2 \beta_{5} + \beta_{8} ) q^{89} + \beta_{5} q^{90} + ( 2 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 5 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{91} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - 2 \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{92} + ( -\beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{93} + ( 3 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{8} + \beta_{11} ) q^{94} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{10} ) q^{95} + \beta_{5} q^{96} + ( -3 + 2 \beta_{1} - 3 \beta_{3} + 3 \beta_{5} + 2 \beta_{8} + 3 \beta_{10} + 3 \beta_{11} ) q^{97} + ( -3 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{6} + \beta_{8} + \beta_{11} ) q^{98} + ( 1 + \beta_{3} + \beta_{6} - \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 3q^{2} + 3q^{3} - 3q^{4} - 12q^{5} - 12q^{6} - q^{7} - 3q^{8} - 3q^{9} + O(q^{10})$$ $$12q - 3q^{2} + 3q^{3} - 3q^{4} - 12q^{5} - 12q^{6} - q^{7} - 3q^{8} - 3q^{9} + 3q^{10} + 6q^{11} + 3q^{12} - 6q^{13} - q^{14} - 3q^{15} - 3q^{16} - 13q^{17} - 3q^{18} - 10q^{19} + 3q^{20} + q^{21} + 6q^{22} + 8q^{23} + 3q^{24} + 12q^{25} + 14q^{26} + 3q^{27} - q^{28} + 7q^{29} + 12q^{30} + 16q^{31} + 12q^{32} + 9q^{33} + 17q^{34} + q^{35} + 12q^{36} - 42q^{37} + q^{39} + 3q^{40} - 3q^{41} + q^{42} + 5q^{43} - 9q^{44} + 3q^{45} - 2q^{46} - 16q^{47} + 3q^{48} + 28q^{49} - 3q^{50} + 13q^{51} - 6q^{52} - q^{53} + 3q^{54} - 6q^{55} + 4q^{56} - 20q^{57} - 3q^{58} - 3q^{59} - 3q^{60} + 8q^{61} - 14q^{62} + 4q^{63} - 3q^{64} + 6q^{65} - 6q^{66} + 24q^{67} - 8q^{68} - 8q^{69} + q^{70} - 19q^{71} - 3q^{72} + 10q^{73} - 12q^{74} + 3q^{75} + 6q^{78} + 3q^{79} + 3q^{80} - 3q^{81} + 7q^{82} - 36q^{83} + q^{84} + 13q^{85} + 20q^{86} + 8q^{87} + 6q^{88} - 15q^{89} + 3q^{90} + 51q^{91} - 12q^{92} - q^{93} + 24q^{94} + 10q^{95} + 3q^{96} - 14q^{97} - 12q^{98} + 6q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + 13 x^{10} - 9 x^{9} + 60 x^{8} + x^{7} + 263 x^{6} + 823 x^{5} + 1931 x^{4} + 1170 x^{3} + 1340 x^{2} + 1000 x + 400$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-17849017864 \nu^{11} + 17625053431 \nu^{10} - 178529954095 \nu^{9} - 111496966669 \nu^{8} - 495858189935 \nu^{7} - 1902177059364 \nu^{6} + 216123116006 \nu^{5} - 22850390835465 \nu^{4} - 38781999340004 \nu^{3} - 33360822706280 \nu^{2} - 14600309174840 \nu + 60463522673300$$$$)/ 107519805038390$$ $$\beta_{3}$$ $$=$$ $$($$$$23361760035 \nu^{11} - 106837550849 \nu^{10} + 512829808605 \nu^{9} - 1139133730021 \nu^{8} + 2950126373569 \nu^{7} - 3751671881315 \nu^{6} + 10498141078366 \nu^{5} + 3863106684695 \nu^{4} + 20833220372194 \nu^{3} - 23111477672266 \nu^{2} + 148807987112405 \nu - 15010929677480$$$$)/ 107519805038390$$ $$\beta_{4}$$ $$=$$ $$($$$$-146654331477 \nu^{11} + 214983815436 \nu^{10} - 1782233600297 \nu^{9} + 626624859189 \nu^{8} - 9624980905700 \nu^{7} - 637596034947 \nu^{6} - 48748704399099 \nu^{5} - 127545198552457 \nu^{4} - 382232489020961 \nu^{3} - 302595918088550 \nu^{2} - 315224424729940 \nu - 205909846055800$$$$)/ 215039610076780$$ $$\beta_{5}$$ $$=$$ $$($$$$259180521843 \nu^{11} - 820941472449 \nu^{10} + 4037834562557 \nu^{9} - 6329673984486 \nu^{8} + 19051303779369 \nu^{7} - 17907584046427 \nu^{6} + 72430523469356 \nu^{5} + 134261867661482 \nu^{4} + 278674930944664 \nu^{3} - 257444750971091 \nu^{2} + 212990793099520 \nu + 53049709415200$$$$)/ 215039610076780$$ $$\beta_{6}$$ $$=$$ $$($$$$-604635226733 \nu^{11} + 1137874382010 \nu^{10} - 7789757733805 \nu^{9} + 4727597224217 \nu^{8} - 36724101470656 \nu^{7} - 2588067986473 \nu^{6} - 166627772868235 \nu^{5} - 496750299137235 \nu^{4} - 1258952186163283 \nu^{3} - 862551212637626 \nu^{2} - 943654494647340 \nu - 663036463432360$$$$)/ 430079220153560$$ $$\beta_{7}$$ $$=$$ $$($$$$302580428763 \nu^{11} - 668487778598 \nu^{10} + 3997049287899 \nu^{9} - 3500472468789 \nu^{8} + 18166764568270 \nu^{7} - 4266046224647 \nu^{6} + 79043701815307 \nu^{5} + 221802656734169 \nu^{4} + 560685961527401 \nu^{3} + 134311106170100 \nu^{2} + 206130812427800 \nu + 103672208737200$$$$)/ 215039610076780$$ $$\beta_{8}$$ $$=$$ $$($$$$377873811249 \nu^{11} - 1246157058016 \nu^{10} + 6108799410656 \nu^{9} - 10496150942691 \nu^{8} + 30636118502149 \nu^{7} - 34700982692066 \nu^{6} + 115791065278799 \nu^{5} + 168010412077976 \nu^{4} + 381784730133684 \nu^{3} - 359469623547681 \nu^{2} + 87053037871525 \nu - 228519186690980$$$$)/ 107519805038390$$ $$\beta_{9}$$ $$=$$ $$($$$$-811947191771 \nu^{11} + 1070796378400 \nu^{10} - 8730178441851 \nu^{9} - 1669598790561 \nu^{8} - 32876644690520 \nu^{7} - 47642147786361 \nu^{6} - 156547132073865 \nu^{5} - 848523389812191 \nu^{4} - 1801907179914651 \nu^{3} - 1493427761388610 \nu^{2} - 583577757140760 \nu - 797530905591380$$$$)/ 215039610076780$$ $$\beta_{10}$$ $$=$$ $$($$$$2196856365657 \nu^{11} - 6036163855492 \nu^{10} + 32819025256117 \nu^{9} - 42245615072839 \nu^{8} + 156459753762298 \nu^{7} - 91084996607083 \nu^{6} + 604977651733493 \nu^{5} + 1468271773696657 \nu^{4} + 3005207185931801 \nu^{3} + 578163383466308 \nu^{2} + 3110122517771760 \nu + 1778835879579520$$$$)/ 430079220153560$$ $$\beta_{11}$$ $$=$$ $$($$$$-2487542474027 \nu^{11} + 6234274597186 \nu^{10} - 33979177074555 \nu^{9} + 35569652132971 \nu^{8} - 145106089008796 \nu^{7} + 42032352335873 \nu^{6} - 567823275403729 \nu^{5} - 1830786108310665 \nu^{4} - 3452889698168529 \nu^{3} - 226203310184226 \nu^{2} - 1085086519421180 \nu - 1579157825751160$$$$)/ 430079220153560$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{11} + \beta_{9} + \beta_{7} - \beta_{6} - 4 \beta_{5} + \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{10} + \beta_{9} - \beta_{8} + 7 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - \beta_{2} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$11 \beta_{11} + 9 \beta_{10} - 9 \beta_{9} + 12 \beta_{7} + 28 \beta_{6} + 11 \beta_{5} + 4 \beta_{4} - 11 \beta_{3} + 12 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$13 \beta_{11} - 27 \beta_{9} + 13 \beta_{8} + 43 \beta_{6} + 14 \beta_{4} + 43 \beta_{3} + 71 \beta_{2} + 14 \beta_{1} + 21$$ $$\nu^{6}$$ $$=$$ $$-84 \beta_{11} - 84 \beta_{10} + 28 \beta_{8} - 127 \beta_{7} - 114 \beta_{5} - 127 \beta_{4} + 342 \beta_{3} - 52 \beta_{1} + 114$$ $$\nu^{7}$$ $$=$$ $$-314 \beta_{11} - 155 \beta_{10} + 314 \beta_{9} - 155 \beta_{8} - 491 \beta_{7} - 499 \beta_{6} + 109 \beta_{5} - 665 \beta_{4} - 665 \beta_{2} - 491 \beta_{1} - 499$$ $$\nu^{8}$$ $$=$$ $$329 \beta_{10} + 824 \beta_{9} - 824 \beta_{8} - 267 \beta_{7} - 1983 \beta_{6} + 1983 \beta_{5} - 3185 \beta_{3} - 1304 \beta_{2} - 1304 \beta_{1} - 3185$$ $$\nu^{9}$$ $$=$$ $$3494 \beta_{11} + 1861 \beta_{10} - 1861 \beta_{9} + 2088 \beta_{7} - 93 \beta_{6} + 5429 \beta_{5} + 6466 \beta_{4} - 5429 \beta_{3} + 2088 \beta_{2}$$ $$\nu^{10}$$ $$=$$ $$8327 \beta_{11} - 12048 \beta_{9} + 8327 \beta_{8} + 12957 \beta_{6} + 13301 \beta_{4} + 12957 \beta_{3} + 12779 \beta_{2} + 13301 \beta_{1} + 30924$$ $$\nu^{11}$$ $$=$$ $$-21106 \beta_{11} - 21106 \beta_{10} + 17022 \beta_{8} - 24483 \beta_{7} - 57656 \beta_{5} - 24483 \beta_{4} + 65148 \beta_{3} + 40117 \beta_{1} + 57656$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$$.

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
481.1
 −0.781164 + 2.40418i 0.270443 − 0.832338i 1.01072 − 3.11068i 2.30431 + 1.67418i −0.479361 − 0.348276i −1.32494 − 0.962628i 2.30431 − 1.67418i −0.479361 + 0.348276i −1.32494 + 0.962628i −0.781164 − 2.40418i 0.270443 + 0.832338i 1.01072 + 3.11068i
−0.809017 + 0.587785i 0.809017 + 0.587785i 0.309017 0.951057i −1.00000 −1.00000 −0.482786 + 1.48586i 0.309017 + 0.951057i 0.309017 + 0.951057i 0.809017 0.587785i
481.2 −0.809017 + 0.587785i 0.809017 + 0.587785i 0.309017 0.951057i −1.00000 −1.00000 0.167143 0.514413i 0.309017 + 0.951057i 0.309017 + 0.951057i 0.809017 0.587785i
481.3 −0.809017 + 0.587785i 0.809017 + 0.587785i 0.309017 0.951057i −1.00000 −1.00000 0.624660 1.92251i 0.309017 + 0.951057i 0.309017 + 0.951057i 0.809017 0.587785i
721.1 0.309017 0.951057i −0.309017 0.951057i −0.809017 0.587785i −1.00000 −1.00000 −3.72844 2.70887i −0.809017 + 0.587785i −0.809017 + 0.587785i −0.309017 + 0.951057i
721.2 0.309017 0.951057i −0.309017 0.951057i −0.809017 0.587785i −1.00000 −1.00000 0.775623 + 0.563523i −0.809017 + 0.587785i −0.809017 + 0.587785i −0.309017 + 0.951057i
721.3 0.309017 0.951057i −0.309017 0.951057i −0.809017 0.587785i −1.00000 −1.00000 2.14380 + 1.55756i −0.809017 + 0.587785i −0.809017 + 0.587785i −0.309017 + 0.951057i
841.1 0.309017 + 0.951057i −0.309017 + 0.951057i −0.809017 + 0.587785i −1.00000 −1.00000 −3.72844 + 2.70887i −0.809017 0.587785i −0.809017 0.587785i −0.309017 0.951057i
841.2 0.309017 + 0.951057i −0.309017 + 0.951057i −0.809017 + 0.587785i −1.00000 −1.00000 0.775623 0.563523i −0.809017 0.587785i −0.809017 0.587785i −0.309017 0.951057i
841.3 0.309017 + 0.951057i −0.309017 + 0.951057i −0.809017 + 0.587785i −1.00000 −1.00000 2.14380 1.55756i −0.809017 0.587785i −0.809017 0.587785i −0.309017 0.951057i
901.1 −0.809017 0.587785i 0.809017 0.587785i 0.309017 + 0.951057i −1.00000 −1.00000 −0.482786 1.48586i 0.309017 0.951057i 0.309017 0.951057i 0.809017 + 0.587785i
901.2 −0.809017 0.587785i 0.809017 0.587785i 0.309017 + 0.951057i −1.00000 −1.00000 0.167143 + 0.514413i 0.309017 0.951057i 0.309017 0.951057i 0.809017 + 0.587785i
901.3 −0.809017 0.587785i 0.809017 0.587785i 0.309017 + 0.951057i −1.00000 −1.00000 0.624660 + 1.92251i 0.309017 0.951057i 0.309017 0.951057i 0.809017 + 0.587785i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 901.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.n.d 12
31.d even 5 1 inner 930.2.n.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.n.d 12 1.a even 1 1 trivial
930.2.n.d 12 31.d even 5 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{12} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{3}$$
$3$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{3}$$
$5$ $$( 1 + T )^{12}$$
$7$ $$400 - 1200 T + 2960 T^{2} - 3850 T^{3} + 3421 T^{4} - 2116 T^{5} + 1417 T^{6} - 543 T^{7} + 190 T^{8} - 17 T^{9} - 3 T^{10} + T^{11} + T^{12}$$
$11$ $$24025 - 44175 T + 38200 T^{2} - 13300 T^{3} + 16015 T^{4} - 3260 T^{5} + 1765 T^{6} - 175 T^{7} + 181 T^{8} - 71 T^{9} + 31 T^{10} - 6 T^{11} + T^{12}$$
$13$ $$1079521 - 2098780 T + 14820553 T^{2} + 6280122 T^{3} + 1841889 T^{4} + 274132 T^{5} + 41001 T^{6} + 4654 T^{7} + 1872 T^{8} + 214 T^{9} + 36 T^{10} + 6 T^{11} + T^{12}$$
$17$ $$13830961 + 6813208 T + 2765786 T^{2} + 915050 T^{3} + 444690 T^{4} + 58058 T^{5} + 24429 T^{6} + 6113 T^{7} + 2850 T^{8} + 710 T^{9} + 131 T^{10} + 13 T^{11} + T^{12}$$
$19$ $$160000 - 840000 T + 13006000 T^{2} + 4917500 T^{3} + 3576025 T^{4} + 443550 T^{5} + 97125 T^{6} + 20450 T^{7} + 2315 T^{8} - 110 T^{9} + 25 T^{10} + 10 T^{11} + T^{12}$$
$23$ $$257442025 - 210109275 T + 179059575 T^{2} - 55699925 T^{3} + 9329740 T^{4} - 600535 T^{5} + 149300 T^{6} - 22715 T^{7} + 6706 T^{8} - 787 T^{9} + 99 T^{10} - 8 T^{11} + T^{12}$$
$29$ $$74149321 - 62498638 T + 59446413 T^{2} - 20556439 T^{3} + 5424493 T^{4} - 925241 T^{5} + 151487 T^{6} - 16794 T^{7} + 2853 T^{8} - 116 T^{9} + 28 T^{10} - 7 T^{11} + T^{12}$$
$31$ $$887503681 - 458066416 T + 94199142 T^{2} - 10635387 T^{3} + 257548 T^{4} + 464628 T^{5} - 142027 T^{6} + 14988 T^{7} + 268 T^{8} - 357 T^{9} + 102 T^{10} - 16 T^{11} + T^{12}$$
$37$ $$( 1 + 7 T + T^{2} )^{6}$$
$41$ $$144400 - 372400 T + 472800 T^{2} - 251450 T^{3} + 233065 T^{4} - 13030 T^{5} + 63515 T^{6} + 5305 T^{7} + 3436 T^{8} - 133 T^{9} - 21 T^{10} + 3 T^{11} + T^{12}$$
$43$ $$281702656 + 186168128 T + 325205264 T^{2} + 604628 T^{3} + 2466197 T^{4} - 704013 T^{5} + 199479 T^{6} + 4579 T^{7} + 5594 T^{8} - 549 T^{9} + 57 T^{10} - 5 T^{11} + T^{12}$$
$47$ $$398161 + 1710641 T + 3681137 T^{2} + 4897922 T^{3} + 4425128 T^{4} + 2665897 T^{5} + 1006843 T^{6} + 184562 T^{7} + 27128 T^{8} + 2722 T^{9} + 242 T^{10} + 16 T^{11} + T^{12}$$
$53$ $$10736275456 + 7250633216 T + 2378774272 T^{2} + 333866672 T^{3} + 309849193 T^{4} + 1081627 T^{5} + 1544713 T^{6} + 4907 T^{7} + 21608 T^{8} + 537 T^{9} + 197 T^{10} + T^{11} + T^{12}$$
$59$ $$17181025 + 3316000 T + 95865 T^{2} + 80075 T^{3} + 309051 T^{4} + 103413 T^{5} + 38883 T^{6} + 9436 T^{7} + 2405 T^{8} + 346 T^{9} + 68 T^{10} + 3 T^{11} + T^{12}$$
$61$ $$( -944 - 9784 T + 4205 T^{2} + 980 T^{3} - 195 T^{4} - 4 T^{5} + T^{6} )^{2}$$
$67$ $$( 9920 - 26200 T + 3495 T^{2} + 1225 T^{3} - 124 T^{4} - 12 T^{5} + T^{6} )^{2}$$
$71$ $$379314576 + 664910640 T + 424522728 T^{2} - 78005442 T^{3} + 25369279 T^{4} - 3179197 T^{5} + 3640131 T^{6} + 433561 T^{7} + 72732 T^{8} + 5011 T^{9} + 361 T^{10} + 19 T^{11} + T^{12}$$
$73$ $$5895782656 + 3749209152 T + 868311344 T^{2} - 146528108 T^{3} + 121341657 T^{4} - 15928782 T^{5} + 2397669 T^{6} - 86794 T^{7} + 5659 T^{8} - 106 T^{9} + 217 T^{10} - 10 T^{11} + T^{12}$$
$79$ $$12091421521 - 2989839590 T + 899760097 T^{2} - 59149261 T^{3} + 9137484 T^{4} - 2197889 T^{5} + 1063334 T^{6} - 64697 T^{7} + 16402 T^{8} - 1863 T^{9} + 169 T^{10} - 3 T^{11} + T^{12}$$
$83$ $$157275696400 - 18016629400 T - 448874460 T^{2} + 835919770 T^{3} + 471260341 T^{4} + 99772334 T^{5} + 15679922 T^{6} + 1761142 T^{7} + 169540 T^{8} + 12698 T^{9} + 807 T^{10} + 36 T^{11} + T^{12}$$
$89$ $$304711936 + 238169664 T + 369549376 T^{2} + 44532296 T^{3} + 2555257 T^{4} + 164911 T^{5} + 487426 T^{6} + 134753 T^{7} + 25459 T^{8} + 2473 T^{9} + 208 T^{10} + 15 T^{11} + T^{12}$$
$97$ $$29659728400 - 10851582200 T + 3783199800 T^{2} - 1080221500 T^{3} + 267651365 T^{4} - 38675505 T^{5} + 4343155 T^{6} - 197655 T^{7} + 8466 T^{8} - 221 T^{9} + 201 T^{10} + 14 T^{11} + T^{12}$$