# Properties

 Label 490.2.g.c Level $490$ Weight $2$ Character orbit 490.g Analytic conductor $3.913$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$490 = 2 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 490.g (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.91266969904$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 10 x^{14} + 61 x^{12} + 266 x^{10} + 852 x^{8} + 1438 x^{6} + 1933 x^{4} + 3038 x^{2} + 2401$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 70) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 10 x^{14} + 61 x^{12} + 266 x^{10} + 852 x^{8} + 1438 x^{6} + 1933 x^{4} + 3038 x^{2} + 2401$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-525491 \nu^{14} - 4061221 \nu^{12} - 20248352 \nu^{10} - 70335258 \nu^{8} - 168482130 \nu^{6} + 151049252 \nu^{4} + 415969049 \nu^{2} - 16581649$$$$)/ 785969800$$ $$\beta_{2}$$ $$=$$ $$($$$$34297 \nu^{14} + 364187 \nu^{12} + 2237304 \nu^{10} + 9493246 \nu^{8} + 30357550 \nu^{6} + 51464796 \nu^{4} + 34590357 \nu^{2} + 53786663$$$$)/34172600$$ $$\beta_{3}$$ $$=$$ $$($$$$88213 \nu^{15} + 1057753 \nu^{13} + 8202336 \nu^{11} + 41107094 \nu^{9} + 157143590 \nu^{7} + 407420164 \nu^{5} + 856246593 \nu^{3} + 1691266857 \nu$$$$)/ 785969800$$ $$\beta_{4}$$ $$=$$ $$($$$$-623003 \nu^{14} - 5366258 \nu^{12} - 30938216 \nu^{10} - 125675214 \nu^{8} - 373946420 \nu^{6} - 445540384 \nu^{4} - 843429583 \nu^{2} - 1076511772$$$$)/ 196492450$$ $$\beta_{5}$$ $$=$$ $$($$$$-4174573 \nu^{15} - 307671 \nu^{14} - 41450603 \nu^{13} - 4646621 \nu^{12} - 238976156 \nu^{11} - 14263312 \nu^{10} - 995989274 \nu^{9} - 55869898 \nu^{8} - 2888475470 \nu^{7} - 109893770 \nu^{6} - 3441488944 \nu^{5} - 128527588 \nu^{4} - 1618097653 \nu^{3} - 220726331 \nu^{2} - 6797535927 \nu - 8000085009$$$$)/ 11003577200$$ $$\beta_{6}$$ $$=$$ $$($$$$4174573 \nu^{15} - 307671 \nu^{14} + 41450603 \nu^{13} - 4646621 \nu^{12} + 238976156 \nu^{11} - 14263312 \nu^{10} + 995989274 \nu^{9} - 55869898 \nu^{8} + 2888475470 \nu^{7} - 109893770 \nu^{6} + 3441488944 \nu^{5} - 128527588 \nu^{4} + 1618097653 \nu^{3} - 220726331 \nu^{2} + 6797535927 \nu - 8000085009$$$$)/ 11003577200$$ $$\beta_{7}$$ $$=$$ $$($$$$-1276339 \nu^{15} - 1594698 \nu^{14} - 12140159 \nu^{13} - 13006098 \nu^{12} - 75497868 \nu^{11} - 68962271 \nu^{10} - 335301337 \nu^{9} - 245531944 \nu^{8} - 1083598885 \nu^{7} - 538978895 \nu^{6} - 1831430982 \nu^{5} + 207676721 \nu^{4} - 2464268024 \nu^{3} + 748752592 \nu^{2} - 1945628006 \nu - 269649107$$$$)/ 2750894300$$ $$\beta_{8}$$ $$=$$ $$($$$$1276339 \nu^{15} - 1594698 \nu^{14} + 12140159 \nu^{13} - 13006098 \nu^{12} + 75497868 \nu^{11} - 68962271 \nu^{10} + 335301337 \nu^{9} - 245531944 \nu^{8} + 1083598885 \nu^{7} - 538978895 \nu^{6} + 1831430982 \nu^{5} + 207676721 \nu^{4} + 2464268024 \nu^{3} + 748752592 \nu^{2} + 1945628006 \nu - 269649107$$$$)/ 2750894300$$ $$\beta_{9}$$ $$=$$ $$($$$$-26590 \nu^{15} - 257521 \nu^{13} - 1484692 \nu^{11} - 6079906 \nu^{9} - 17945290 \nu^{7} - 21381008 \nu^{5} - 8929484 \nu^{3} - 42231189 \nu$$$$)/50016260$$ $$\beta_{10}$$ $$=$$ $$($$$$-124363 \nu^{15} - 151501 \nu^{14} - 1257693 \nu^{13} - 1812391 \nu^{12} - 7185176 \nu^{11} - 10972682 \nu^{10} - 31585764 \nu^{9} - 50405418 \nu^{8} - 99003980 \nu^{7} - 158239340 \nu^{6} - 164843024 \nu^{5} - 262434718 \nu^{4} - 223880973 \nu^{3} - 190752681 \nu^{2} - 517014827 \nu - 459236869$$$$)/ 239208200$$ $$\beta_{11}$$ $$=$$ $$($$$$-124363 \nu^{15} + 151501 \nu^{14} - 1257693 \nu^{13} + 1812391 \nu^{12} - 7185176 \nu^{11} + 10972682 \nu^{10} - 31585764 \nu^{9} + 50405418 \nu^{8} - 99003980 \nu^{7} + 158239340 \nu^{6} - 164843024 \nu^{5} + 262434718 \nu^{4} - 223880973 \nu^{3} + 190752681 \nu^{2} - 517014827 \nu + 459236869$$$$)/ 239208200$$ $$\beta_{12}$$ $$=$$ $$($$$$154017 \nu^{15} + 1553547 \nu^{13} + 9597064 \nu^{11} + 41588666 \nu^{9} + 133651610 \nu^{7} + 226254436 \nu^{5} + 303303017 \nu^{3} + 238292243 \nu$$$$)/ 239208200$$ $$\beta_{13}$$ $$=$$ $$($$$$-357305 \nu^{15} - 472647 \nu^{14} - 2868675 \nu^{13} - 3965353 \nu^{12} - 15737980 \nu^{11} - 22633800 \nu^{10} - 60556930 \nu^{9} - 92234954 \nu^{8} - 165340790 \nu^{7} - 272435506 \nu^{6} - 118396640 \nu^{5} - 324550940 \nu^{4} - 281366705 \nu^{3} - 614398939 \nu^{2} - 477019655 \nu - 781884621$$$$)/ 200065040$$ $$\beta_{14}$$ $$=$$ $$($$$$-357305 \nu^{15} + 472647 \nu^{14} - 2868675 \nu^{13} + 3965353 \nu^{12} - 15737980 \nu^{11} + 22633800 \nu^{10} - 60556930 \nu^{9} + 92234954 \nu^{8} - 165340790 \nu^{7} + 272435506 \nu^{6} - 118396640 \nu^{5} + 324550940 \nu^{4} - 281366705 \nu^{3} + 614398939 \nu^{2} - 477019655 \nu + 781884621$$$$)/ 200065040$$ $$\beta_{15}$$ $$=$$ $$($$$$104513 \nu^{15} + 887273 \nu^{13} + 5023936 \nu^{11} + 20069294 \nu^{9} + 58201830 \nu^{7} + 59771364 \nu^{5} + 109455293 \nu^{3} + 147858697 \nu$$$$)/34172600$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{12} - \beta_{9} + \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{14} + \beta_{13} - \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{2} + \beta_{1} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{12} + 8 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 3 \beta_{6} - 3 \beta_{5}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{14} - 5 \beta_{13} - 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{8} - 2 \beta_{7} - 5 \beta_{6} - 5 \beta_{5} + 8 \beta_{4} + 5 \beta_{2} + 5 \beta_{1} - 3$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-22 \beta_{15} - 14 \beta_{14} - 14 \beta_{13} + 19 \beta_{12} - 3 \beta_{11} - 3 \beta_{10} - 19 \beta_{9} + 3 \beta_{8} - 3 \beta_{7} - 14 \beta_{6} + 14 \beta_{5} - 3 \beta_{3}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-7 \beta_{14} + 7 \beta_{13} + 14 \beta_{8} + 14 \beta_{7} + 5 \beta_{6} + 5 \beta_{5} - 11 \beta_{4} - 22 \beta_{1} + 4$$ $$\nu^{7}$$ $$=$$ $$($$$$94 \beta_{15} + 64 \beta_{14} + 64 \beta_{13} - 87 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} - 87 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 64 \beta_{6} + 64 \beta_{5} - 7 \beta_{3}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$75 \beta_{14} - 75 \beta_{13} + 62 \beta_{11} - 62 \beta_{10} - 62 \beta_{8} - 62 \beta_{7} + 75 \beta_{6} + 75 \beta_{5} + 112 \beta_{4} - 87 \beta_{2} + 87 \beta_{1} + 25$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-112 \beta_{15} - 75 \beta_{14} - 75 \beta_{13} - 112 \beta_{12} + 658 \beta_{9} + 150 \beta_{8} - 150 \beta_{7} + 471 \beta_{6} - 471 \beta_{5}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-337 \beta_{14} + 337 \beta_{13} - 198 \beta_{11} + 198 \beta_{10} - 198 \beta_{8} - 198 \beta_{7} + 337 \beta_{6} + 337 \beta_{5} - 486 \beta_{4} + 273 \beta_{2} + 273 \beta_{1} + 759$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-456 \beta_{15} - 332 \beta_{14} - 332 \beta_{13} + 1215 \beta_{12} + 535 \beta_{11} + 535 \beta_{10} - 1215 \beta_{9} - 535 \beta_{8} + 535 \beta_{7} - 332 \beta_{6} + 332 \beta_{5} + 759 \beta_{3}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$-166 \beta_{14} + 166 \beta_{13} + 332 \beta_{8} + 332 \beta_{7} - 2530 \beta_{6} - 2530 \beta_{5} - 228 \beta_{4} - 456 \beta_{1} - 3575$$ $$\nu^{13}$$ $$=$$ $$($$$$2032 \beta_{15} + 1452 \beta_{14} + 1452 \beta_{13} + 1771 \beta_{12} - 2696 \beta_{11} - 2696 \beta_{10} + 1771 \beta_{9} - 2696 \beta_{8} + 2696 \beta_{7} - 1452 \beta_{6} + 1452 \beta_{5} - 3803 \beta_{3}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$10647 \beta_{14} - 10647 \beta_{13} - 1244 \beta_{11} + 1244 \beta_{10} + 1244 \beta_{8} + 1244 \beta_{7} + 10647 \beta_{6} + 10647 \beta_{5} + 15030 \beta_{4} + 1771 \beta_{2} - 1771 \beta_{1} + 16801$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$-15030 \beta_{15} - 10647 \beta_{14} - 10647 \beta_{13} - 15030 \beta_{12} - 30632 \beta_{9} + 21294 \beta_{8} - 21294 \beta_{7} - 21653 \beta_{6} + 21653 \beta_{5}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/490\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$-1$$ $$-\beta_{9}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 0.587308 + 2.01725i 1.45333 − 1.51725i −1.45333 − 1.51725i −0.587308 + 2.01725i −0.144868 + 1.25092i −1.01089 − 0.750919i 1.01089 − 0.750919i 0.144868 + 1.25092i 0.587308 − 2.01725i 1.45333 + 1.51725i −1.45333 + 1.51725i −0.587308 − 2.01725i −0.144868 − 1.25092i −1.01089 + 0.750919i 1.01089 + 0.750919i 0.144868 − 1.25092i
−0.707107 0.707107i −2.05532 2.05532i 1.00000i −0.830578 + 2.07609i 2.90667i 0 0.707107 0.707107i 5.44871i 2.05532 0.880708i
97.2 −0.707107 0.707107i −0.830578 0.830578i 1.00000i −2.05532 0.880708i 1.17462i 0 0.707107 0.707107i 1.62028i 0.830578 + 2.07609i
97.3 −0.707107 0.707107i 0.830578 + 0.830578i 1.00000i 2.05532 + 0.880708i 1.17462i 0 0.707107 0.707107i 1.62028i −0.830578 2.07609i
97.4 −0.707107 0.707107i 2.05532 + 2.05532i 1.00000i 0.830578 2.07609i 2.90667i 0 0.707107 0.707107i 5.44871i −2.05532 + 0.880708i
97.5 0.707107 + 0.707107i −1.42962 1.42962i 1.00000i −0.204875 2.22666i 2.02179i 0 −0.707107 + 0.707107i 1.08763i 1.42962 1.71936i
97.6 0.707107 + 0.707107i −0.204875 0.204875i 1.00000i −1.42962 1.71936i 0.289737i 0 −0.707107 + 0.707107i 2.91605i 0.204875 2.22666i
97.7 0.707107 + 0.707107i 0.204875 + 0.204875i 1.00000i 1.42962 + 1.71936i 0.289737i 0 −0.707107 + 0.707107i 2.91605i −0.204875 + 2.22666i
97.8 0.707107 + 0.707107i 1.42962 + 1.42962i 1.00000i 0.204875 + 2.22666i 2.02179i 0 −0.707107 + 0.707107i 1.08763i −1.42962 + 1.71936i
293.1 −0.707107 + 0.707107i −2.05532 + 2.05532i 1.00000i −0.830578 2.07609i 2.90667i 0 0.707107 + 0.707107i 5.44871i 2.05532 + 0.880708i
293.2 −0.707107 + 0.707107i −0.830578 + 0.830578i 1.00000i −2.05532 + 0.880708i 1.17462i 0 0.707107 + 0.707107i 1.62028i 0.830578 2.07609i
293.3 −0.707107 + 0.707107i 0.830578 0.830578i 1.00000i 2.05532 0.880708i 1.17462i 0 0.707107 + 0.707107i 1.62028i −0.830578 + 2.07609i
293.4 −0.707107 + 0.707107i 2.05532 2.05532i 1.00000i 0.830578 + 2.07609i 2.90667i 0 0.707107 + 0.707107i 5.44871i −2.05532 0.880708i
293.5 0.707107 0.707107i −1.42962 + 1.42962i 1.00000i −0.204875 + 2.22666i 2.02179i 0 −0.707107 0.707107i 1.08763i 1.42962 + 1.71936i
293.6 0.707107 0.707107i −0.204875 + 0.204875i 1.00000i −1.42962 + 1.71936i 0.289737i 0 −0.707107 0.707107i 2.91605i 0.204875 + 2.22666i
293.7 0.707107 0.707107i 0.204875 0.204875i 1.00000i 1.42962 1.71936i 0.289737i 0 −0.707107 0.707107i 2.91605i −0.204875 2.22666i
293.8 0.707107 0.707107i 1.42962 1.42962i 1.00000i 0.204875 2.22666i 2.02179i 0 −0.707107 0.707107i 1.08763i −1.42962 1.71936i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 293.8 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.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.g.c 16
5.c odd 4 1 inner 490.2.g.c 16
7.b odd 2 1 inner 490.2.g.c 16
7.c even 3 1 70.2.k.a 16
7.c even 3 1 490.2.l.c 16
7.d odd 6 1 70.2.k.a 16
7.d odd 6 1 490.2.l.c 16
21.g even 6 1 630.2.bv.c 16
21.h odd 6 1 630.2.bv.c 16
28.f even 6 1 560.2.ci.c 16
28.g odd 6 1 560.2.ci.c 16
35.f even 4 1 inner 490.2.g.c 16
35.i odd 6 1 350.2.o.c 16
35.j even 6 1 350.2.o.c 16
35.k even 12 1 70.2.k.a 16
35.k even 12 1 350.2.o.c 16
35.k even 12 1 490.2.l.c 16
35.l odd 12 1 70.2.k.a 16
35.l odd 12 1 350.2.o.c 16
35.l odd 12 1 490.2.l.c 16
105.w odd 12 1 630.2.bv.c 16
105.x even 12 1 630.2.bv.c 16
140.w even 12 1 560.2.ci.c 16
140.x odd 12 1 560.2.ci.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.k.a 16 7.c even 3 1
70.2.k.a 16 7.d odd 6 1
70.2.k.a 16 35.k even 12 1
70.2.k.a 16 35.l odd 12 1
350.2.o.c 16 35.i odd 6 1
350.2.o.c 16 35.j even 6 1
350.2.o.c 16 35.k even 12 1
350.2.o.c 16 35.l odd 12 1
490.2.g.c 16 1.a even 1 1 trivial
490.2.g.c 16 5.c odd 4 1 inner
490.2.g.c 16 7.b odd 2 1 inner
490.2.g.c 16 35.f even 4 1 inner
490.2.l.c 16 7.c even 3 1
490.2.l.c 16 7.d odd 6 1
490.2.l.c 16 35.k even 12 1
490.2.l.c 16 35.l odd 12 1
560.2.ci.c 16 28.f even 6 1
560.2.ci.c 16 28.g odd 6 1
560.2.ci.c 16 140.w even 12 1
560.2.ci.c 16 140.x odd 12 1
630.2.bv.c 16 21.g even 6 1
630.2.bv.c 16 21.h odd 6 1
630.2.bv.c 16 105.w odd 12 1
630.2.bv.c 16 105.x even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} + 90 T_{3}^{12} + 1361 T_{3}^{8} + 2280 T_{3}^{4} + 16$$ acting on $$S_{2}^{\mathrm{new}}(490, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{4}$$
$3$ $$16 + 2280 T^{4} + 1361 T^{8} + 90 T^{12} + T^{16}$$
$5$ $$390625 + 187500 T^{2} + 45000 T^{4} + 8100 T^{6} + 1454 T^{8} + 324 T^{10} + 72 T^{12} + 12 T^{14} + T^{16}$$
$7$ $$T^{16}$$
$11$ $$( -62 + 96 T - 13 T^{2} - 6 T^{3} + T^{4} )^{4}$$
$13$ $$16 + 2280 T^{4} + 1361 T^{8} + 90 T^{12} + T^{16}$$
$17$ $$9834496 + 26049024 T^{4} + 1517840 T^{8} + 2664 T^{12} + T^{16}$$
$19$ $$( 10000 - 5800 T^{2} + 1049 T^{4} - 62 T^{6} + T^{8} )^{2}$$
$23$ $$( 16129 + 14732 T + 6728 T^{2} - 3204 T^{3} + 770 T^{4} + 12 T^{5} + 8 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$29$ $$( 329476 + 90948 T^{2} + 7325 T^{4} + 162 T^{6} + T^{8} )^{2}$$
$31$ $$( 16 + 9280 T^{2} + 1592 T^{4} + 80 T^{6} + T^{8} )^{2}$$
$37$ $$( 256 - 1280 T + 3200 T^{2} + 1776 T^{3} + 497 T^{4} - 12 T^{5} + 8 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$41$ $$( 18769 + 31468 T^{2} + 5174 T^{4} + 140 T^{6} + T^{8} )^{2}$$
$43$ $$( 784 + 896 T + 512 T^{2} - 976 T^{3} + 673 T^{4} - 140 T^{5} + 8 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$47$ $$9834496 + 997319808 T^{4} + 3446993 T^{8} + 3378 T^{12} + T^{16}$$
$53$ $$( 204304 + 79552 T + 15488 T^{2} - 33248 T^{3} + 14593 T^{4} - 3100 T^{5} + 392 T^{6} - 28 T^{7} + T^{8} )^{2}$$
$59$ $$( 16384 - 25600 T^{2} + 4496 T^{4} - 152 T^{6} + T^{8} )^{2}$$
$61$ $$( 148996 + 51700 T^{2} + 3989 T^{4} + 110 T^{6} + T^{8} )^{2}$$
$67$ $$( 58564 - 53240 T + 24200 T^{2} + 37444 T^{3} + 18709 T^{4} + 4100 T^{5} + 512 T^{6} + 32 T^{7} + T^{8} )^{2}$$
$71$ $$( -4424 + 1816 T - 190 T^{2} - 4 T^{3} + T^{4} )^{4}$$
$73$ $$1017603875209216 + 1356641323008 T^{4} + 366562064 T^{8} + 34248 T^{12} + T^{16}$$
$79$ $$( 7840000 + 729600 T^{2} + 23264 T^{4} + 288 T^{6} + T^{8} )^{2}$$
$83$ $$3812835757370896 + 5479636353768 T^{4} + 1373332049 T^{8} + 69978 T^{12} + T^{16}$$
$89$ $$( 99856 - 62840 T^{2} + 7241 T^{4} - 190 T^{6} + T^{8} )^{2}$$
$97$ $$( 3111696 + 8136 T^{4} + T^{8} )^{2}$$