# Properties

 Label 930.2.n.f 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: $$x^{12} - 3 x^{11} + 10 x^{10} - 15 x^{9} + 61 x^{8} - 25 x^{7} + 316 x^{6} + 50 x^{5} + 1336 x^{4} - 1720 x^{3} + 1235 x^{2} - 462 x + 121$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + 10 x^{10} - 15 x^{9} + 61 x^{8} - 25 x^{7} + 316 x^{6} + 50 x^{5} + 1336 x^{4} - 1720 x^{3} + 1235 x^{2} - 462 x + 121$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-2027942507365 \nu^{11} - 6741096197250 \nu^{10} + 8771348618837 \nu^{9} - 72535728446286 \nu^{8} - 17517132180398 \nu^{7} - 615408227668441 \nu^{6} - 860876049003457 \nu^{5} - 3984672827274868 \nu^{4} - 7420702000011733 \nu^{3} - 14341908095525667 \nu^{2} + 7072708206337553 \nu - 1477229912476098$$$$)/ 6988317776409038$$ $$\beta_{3}$$ $$=$$ $$($$$$-134293628406918 \nu^{11} + 425188252801769 \nu^{10} - 1268784225899430 \nu^{9} + 1917919591296563 \nu^{8} - 7394018319912852 \nu^{7} + 3550029164157328 \nu^{6} - 35667296072233237 \nu^{5} + 2754955118692127 \nu^{4} - 135584886451618900 \nu^{3} + 312612762860028023 \nu^{2} - 8091642031761393 \nu - 15756133945716967$$$$)/ 76871495540499418$$ $$\beta_{4}$$ $$=$$ $$($$$$-15260073350092 \nu^{11} + 29181641218473 \nu^{10} - 109621640737688 \nu^{9} + 75584589358371 \nu^{8} - 713395593165066 \nu^{7} - 583970775230722 \nu^{6} - 4827509082611883 \nu^{5} - 5949632017082929 \nu^{4} - 23573729912968530 \nu^{3} + 3498606615917195 \nu^{2} - 519837145022823 \nu - 3740133049967303$$$$)/ 6988317776409038$$ $$\beta_{5}$$ $$=$$ $$($$$$-15433725086754 \nu^{11} + 40585830735791 \nu^{10} - 146367463122124 \nu^{9} + 223757360866823 \nu^{8} - 991662749019128 \nu^{7} + 330398885873348 \nu^{6} - 5395017710926361 \nu^{5} - 1315481216295935 \nu^{4} - 23150104877728406 \nu^{3} + 24914693329265299 \nu^{2} - 10541881812984401 \nu + 32619625852034783$$$$)/ 6988317776409038$$ $$\beta_{6}$$ $$=$$ $$($$$$340012095451573 \nu^{11} - 1187897093205731 \nu^{10} + 3721119007918933 \nu^{9} - 6306019479888163 \nu^{8} + 21572168305488034 \nu^{7} - 16347653911105051 \nu^{6} + 101020143635159126 \nu^{5} - 36101995136152063 \nu^{4} + 388810207335389309 \nu^{3} - 844131833219359390 \nu^{2} + 458399610657781800 \nu - 162803796693877779$$$$)/ 76871495540499418$$ $$\beta_{7}$$ $$=$$ $$($$$$-33528790708398 \nu^{11} + 91949782233819 \nu^{10} - 305918373930930 \nu^{9} + 411412294337441 \nu^{8} - 1894440097124638 \nu^{7} + 331955061958652 \nu^{6} - 10410258827512877 \nu^{5} - 4028400266069701 \nu^{4} - 44017325353458958 \nu^{3} + 48036577117623227 \nu^{2} - 20389233432139957 \nu + 4859347452653617$$$$)/ 6988317776409038$$ $$\beta_{8}$$ $$=$$ $$($$$$-441758859332147 \nu^{11} + 956459880204063 \nu^{10} - 3406140988749461 \nu^{9} + 3261280776741975 \nu^{8} - 22421755181549116 \nu^{7} - 9794869585067343 \nu^{6} - 135944293867413280 \nu^{5} - 136600790069248997 \nu^{4} - 634502238994515103 \nu^{3} + 275634659163244302 \nu^{2} - 17169842981346048 \nu - 20188974742087613$$$$)/ 76871495540499418$$ $$\beta_{9}$$ $$=$$ $$($$$$671468142034590 \nu^{11} - 2125941264008845 \nu^{10} + 6343921129497150 \nu^{9} - 9589597956482815 \nu^{8} + 36970091599564260 \nu^{7} - 17750145820786640 \nu^{6} + 178336480361166185 \nu^{5} - 13774775593460635 \nu^{4} + 677924432258094500 \nu^{3} - 1486192318759640697 \nu^{2} + 40458210158806965 \nu + 78780669728584835$$$$)/ 76871495540499418$$ $$\beta_{10}$$ $$=$$ $$($$$$1265045893001167 \nu^{11} - 4428523508139374 \nu^{10} + 13877760802501913 \nu^{9} - 23565100422588612 \nu^{8} + 80719761458687858 \nu^{7} - 63357386244670403 \nu^{6} + 378209006503488693 \nu^{5} - 135859451182039782 \nu^{4} + 1449278457432522573 \nu^{3} - 3145320278857390257 \nu^{2} + 1639786454689738933 \nu - 606588366342633378$$$$)/ 76871495540499418$$ $$\beta_{11}$$ $$=$$ $$($$$$-1008542670894593 \nu^{11} + 2221788278300577 \nu^{10} - 7880338853403248 \nu^{9} + 7802352181656534 \nu^{8} - 51695687777336817 \nu^{7} - 19938477389819882 \nu^{6} - 312521813195179998 \nu^{5} - 291360942695946612 \nu^{4} - 1453848797788054938 \nu^{3} + 683915303331891880 \nu^{2} - 94302808274772314 \nu - 23125898751850279$$$$)/ 38435747770249709$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} + 5 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} + \beta_{9} - 2 \beta_{8} - \beta_{7} + \beta_{6} + 3 \beta_{3} - 7 \beta_{2} + 1$$ $$\nu^{4}$$ $$=$$ $$8 \beta_{11} - \beta_{10} + \beta_{9} - 27 \beta_{8} - 12 \beta_{7} + 2 \beta_{6} + \beta_{4} + 3 \beta_{3} - 12 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$11 \beta_{11} - 11 \beta_{10} - 21 \beta_{8} - 56 \beta_{7} + 21 \beta_{6} + 2 \beta_{5} + 6 \beta_{4} - 17 \beta_{2} - 11 \beta_{1} - 12$$ $$\nu^{6}$$ $$=$$ $$-50 \beta_{10} - 17 \beta_{9} - 60 \beta_{7} + 169 \beta_{6} + 17 \beta_{5} + 60 \beta_{4} - 57 \beta_{3} - 17 \beta_{2} - 66 \beta_{1} - 40$$ $$\nu^{7}$$ $$=$$ $$-110 \beta_{11} - 143 \beta_{9} + 256 \beta_{8} + 110 \beta_{5} + 379 \beta_{4} - 530 \beta_{3} + 74 \beta_{2} - 74 \beta_{1} - 256$$ $$\nu^{8}$$ $$=$$ $$-596 \beta_{11} + 379 \beta_{10} - 596 \beta_{9} + 1768 \beta_{8} + 693 \beta_{7} - 1219 \beta_{6} + 379 \beta_{5} + 379 \beta_{4} - 2364 \beta_{3} + 890 \beta_{2} - 97 \beta_{1} - 1219$$ $$\nu^{9}$$ $$=$$ $$-1486 \beta_{11} + 1072 \beta_{10} - 1072 \beta_{9} + 4101 \beta_{8} + 4446 \beta_{7} - 2717 \beta_{6} - 1977 \beta_{4} - 3789 \beta_{3} + 4446 \beta_{2} - 905 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-2469 \beta_{11} + 2469 \beta_{10} + 6490 \beta_{8} + 12591 \beta_{7} - 6490 \beta_{6} - 3049 \beta_{5} - 7910 \beta_{4} + 10379 \beta_{2} + 2469 \beta_{1} + 9394$$ $$\nu^{11}$$ $$=$$ $$4681 \beta_{10} + 10379 \beta_{9} + 21807 \beta_{7} - 14366 \beta_{6} - 10379 \beta_{5} - 21807 \beta_{4} + 37810 \beta_{3} + 10379 \beta_{2} + 20173 \beta_{1} + 27431$$

## Character values

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

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{3}$$

## 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
 0.472164 + 0.343047i −1.70225 − 1.23676i 2.53910 + 1.84477i 0.784484 + 2.41439i −0.738812 − 2.27383i 0.145311 + 0.447222i 0.784484 − 2.41439i −0.738812 + 2.27383i 0.145311 − 0.447222i 0.472164 − 0.343047i −1.70225 + 1.23676i 2.53910 − 1.84477i
0.809017 0.587785i 0.809017 + 0.587785i 0.309017 0.951057i 1.00000 1.00000 −1.31116 + 4.03534i −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 0.782218 2.40742i −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 0.837960 2.57897i −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 −4.03161 2.92914i 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 0.545848 + 0.396582i 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 2.67675 + 1.94477i 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 −4.03161 + 2.92914i 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 0.545848 0.396582i 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 2.67675 1.94477i 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 −1.31116 4.03534i −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 0.782218 + 2.40742i −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 0.837960 + 2.57897i −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.f 12
31.d even 5 1 inner 930.2.n.f 12

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

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{3}$$
$3$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{3}$$
$5$ $$( -1 + T )^{12}$$
$7$ $$104976 - 303264 T + 392040 T^{2} - 212490 T^{3} + 106561 T^{4} - 26150 T^{5} + 7459 T^{6} - 875 T^{7} + 306 T^{8} - 25 T^{9} + 15 T^{10} + T^{11} + T^{12}$$
$11$ $$22801 - 81389 T + 145340 T^{2} - 156740 T^{3} + 123091 T^{4} - 68080 T^{5} + 29139 T^{6} - 10325 T^{7} + 3141 T^{8} - 735 T^{9} + 125 T^{10} - 14 T^{11} + T^{12}$$
$13$ $$121 + 66 T + 1215 T^{2} - 1970 T^{3} + 2101 T^{4} - 1100 T^{5} + 1029 T^{6} - 170 T^{7} + 226 T^{8} + 40 T^{9} - 4 T^{11} + T^{12}$$
$17$ $$64883025 - 8941050 T + 30713940 T^{2} + 9367740 T^{3} + 2642386 T^{4} + 543506 T^{5} + 126423 T^{6} + 13735 T^{7} + 1476 T^{8} + 70 T^{9} + 37 T^{10} + 7 T^{11} + T^{12}$$
$19$ $$4000000 - 2600000 T + 2840000 T^{2} - 771000 T^{3} + 338025 T^{4} - 88100 T^{5} + 22565 T^{6} - 2840 T^{7} + 701 T^{8} - 168 T^{9} + 69 T^{10} - 12 T^{11} + T^{12}$$
$23$ $$93025 + 138775 T + 690115 T^{2} - 64805 T^{3} + 257576 T^{4} + 62927 T^{5} + 5872 T^{6} - 775 T^{7} + 1186 T^{8} + 5 T^{9} + 13 T^{10} + 6 T^{11} + T^{12}$$
$29$ $$1229881 - 1319710 T + 700847 T^{2} + 77601 T^{3} + 208249 T^{4} + 127909 T^{5} + 77789 T^{6} + 30482 T^{7} + 8937 T^{8} + 1808 T^{9} + 264 T^{10} + 23 T^{11} + T^{12}$$
$31$ $$887503681 - 973391134 T + 454372332 T^{2} - 104953693 T^{3} + 5783298 T^{4} + 3547702 T^{5} - 1072707 T^{6} + 114442 T^{7} + 6018 T^{8} - 3523 T^{9} + 492 T^{10} - 34 T^{11} + T^{12}$$
$37$ $$( 472249 - 155911 T + 5398 T^{2} + 2617 T^{3} - 198 T^{4} - 11 T^{5} + T^{6} )^{2}$$
$41$ $$5856400 - 20231200 T + 31658440 T^{2} - 26054050 T^{3} + 13106201 T^{4} - 4283450 T^{5} + 1013821 T^{6} - 176255 T^{7} + 24046 T^{8} - 2555 T^{9} + 271 T^{10} - 15 T^{11} + T^{12}$$
$43$ $$22817915136 + 21224576448 T + 7924214160 T^{2} + 283459740 T^{3} + 138783181 T^{4} - 3537865 T^{5} + 2984451 T^{6} + 245885 T^{7} + 43206 T^{8} + 3375 T^{9} + 345 T^{10} + 17 T^{11} + T^{12}$$
$47$ $$4405801 + 4040575 T + 3559517 T^{2} + 1394554 T^{3} + 426374 T^{4} + 52241 T^{5} + 14209 T^{6} + 13378 T^{7} + 7052 T^{8} + 1442 T^{9} + 134 T^{10} + 2 T^{11} + T^{12}$$
$53$ $$8294400 + 10713600 T + 45109440 T^{2} - 10620600 T^{3} - 3902639 T^{4} + 3015609 T^{5} + 2184467 T^{6} + 309267 T^{7} + 41980 T^{8} + 4383 T^{9} + 407 T^{10} + 21 T^{11} + T^{12}$$
$59$ $$5935315681 + 2305374884 T + 962543721 T^{2} + 197536351 T^{3} + 39907837 T^{4} + 5350541 T^{5} + 727181 T^{6} + 68608 T^{7} + 6979 T^{8} + 368 T^{9} + 108 T^{10} + 15 T^{11} + T^{12}$$
$61$ $$( 32400 - 29340 T + 3971 T^{2} + 918 T^{3} - 141 T^{4} - 8 T^{5} + T^{6} )^{2}$$
$67$ $$( 14400 + 19200 T + 8035 T^{2} + 805 T^{3} - 134 T^{4} - 12 T^{5} + T^{6} )^{2}$$
$71$ $$8737936 - 9080832 T + 32691600 T^{2} + 6312750 T^{3} + 711331 T^{4} - 620655 T^{5} + 172091 T^{6} - 42545 T^{7} + 18336 T^{8} - 3495 T^{9} + 385 T^{10} - 23 T^{11} + T^{12}$$
$73$ $$9591460096 - 6968734016 T + 4235125072 T^{2} - 1486105492 T^{3} + 404938953 T^{4} - 86242912 T^{5} + 15220433 T^{6} - 2241132 T^{7} + 269613 T^{8} - 24252 T^{9} + 1497 T^{10} - 56 T^{11} + T^{12}$$
$79$ $$96412761 + 92553894 T + 117164295 T^{2} + 40855785 T^{3} + 11630206 T^{4} + 1744755 T^{5} + 218924 T^{6} + 19935 T^{7} + 5266 T^{8} + 105 T^{9} + 35 T^{10} + 9 T^{11} + T^{12}$$
$83$ $$467856 + 9976824 T + 81717372 T^{2} + 13532838 T^{3} + 30855013 T^{4} + 15855578 T^{5} + 4551178 T^{6} + 750918 T^{7} + 93668 T^{8} + 6558 T^{9} + 427 T^{10} + 24 T^{11} + T^{12}$$
$89$ $$725386076416 + 13650983488 T - 19496860112 T^{2} + 2161113684 T^{3} + 2451593933 T^{4} + 489894301 T^{5} + 60296672 T^{6} + 5150419 T^{7} + 366843 T^{8} + 18571 T^{9} + 968 T^{10} + 27 T^{11} + T^{12}$$
$97$ $$1957531536 - 3842591400 T + 2511909072 T^{2} + 1238338344 T^{3} + 349773409 T^{4} + 25022661 T^{5} + 5990179 T^{6} - 907107 T^{7} + 169022 T^{8} - 14703 T^{9} + 979 T^{10} - 38 T^{11} + T^{12}$$