# Properties

 Label 605.2.e.a Level $605$ Weight $2$ Character orbit 605.e Analytic conductor $4.831$ Analytic rank $0$ Dimension $20$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 67 x^{16} + 1315 x^{12} + 9193 x^{8} + 16040 x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{7}$$ 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 $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{12} q^{3} + ( -\beta_{4} + \beta_{10} - \beta_{13} + \beta_{14} ) q^{4} + ( \beta_{2} + \beta_{8} ) q^{5} + ( \beta_{3} - \beta_{15} - \beta_{17} ) q^{6} + ( \beta_{3} - \beta_{6} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{7} + ( -\beta_{15} + \beta_{16} - \beta_{17} ) q^{8} + ( \beta_{7} + \beta_{8} + \beta_{10} - \beta_{12} - \beta_{13} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{12} q^{3} + ( -\beta_{4} + \beta_{10} - \beta_{13} + \beta_{14} ) q^{4} + ( \beta_{2} + \beta_{8} ) q^{5} + ( \beta_{3} - \beta_{15} - \beta_{17} ) q^{6} + ( \beta_{3} - \beta_{6} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{7} + ( -\beta_{15} + \beta_{16} - \beta_{17} ) q^{8} + ( \beta_{7} + \beta_{8} + \beta_{10} - \beta_{12} - \beta_{13} ) q^{9} + ( -\beta_{3} + \beta_{9} ) q^{10} + ( -1 + 2 \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{8} + \beta_{10} - 3 \beta_{13} + 2 \beta_{14} ) q^{12} + ( \beta_{1} + \beta_{9} - \beta_{11} + \beta_{15} - \beta_{16} + \beta_{18} - \beta_{19} ) q^{13} + ( \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + \beta_{12} + \beta_{14} ) q^{14} + ( 1 + \beta_{2} + \beta_{4} - \beta_{8} - 2 \beta_{10} + \beta_{13} - \beta_{14} ) q^{15} + ( 1 - \beta_{4} + \beta_{5} + 2 \beta_{12} - 2 \beta_{13} ) q^{16} + ( -\beta_{1} + \beta_{3} + \beta_{15} + \beta_{16} ) q^{17} + ( \beta_{6} + \beta_{9} + \beta_{11} - \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{18} + ( -\beta_{1} + \beta_{3} + \beta_{11} - 2 \beta_{16} + \beta_{17} ) q^{19} + ( 1 - \beta_{2} - \beta_{8} - 2 \beta_{10} + 2 \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{20} + ( 2 \beta_{1} - \beta_{3} + \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{21} + ( \beta_{2} - \beta_{4} - 2 \beta_{7} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{23} + ( -\beta_{1} - \beta_{6} + 2 \beta_{9} + 2 \beta_{11} + \beta_{16} - \beta_{17} + \beta_{18} ) q^{24} + ( \beta_{2} - \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{25} + ( 2 - \beta_{2} - \beta_{7} + \beta_{8} - \beta_{12} + \beta_{13} ) q^{26} + ( -1 - \beta_{4} + \beta_{7} + \beta_{10} ) q^{27} + ( -\beta_{1} - \beta_{9} + \beta_{11} + 2 \beta_{15} - 2 \beta_{16} - \beta_{18} + \beta_{19} ) q^{28} + ( 2 \beta_{1} - \beta_{6} + 2 \beta_{9} - \beta_{11} - \beta_{16} - \beta_{17} + \beta_{18} - 2 \beta_{19} ) q^{29} + ( -\beta_{1} - \beta_{3} - \beta_{9} - 3 \beta_{11} + 3 \beta_{15} + \beta_{16} + \beta_{17} ) q^{30} + ( \beta_{4} - \beta_{5} + \beta_{12} - \beta_{13} ) q^{31} + ( 2 \beta_{1} - 2 \beta_{6} + \beta_{9} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{32} + ( \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} - 2 \beta_{10} - \beta_{13} + \beta_{14} ) q^{34} + ( -3 \beta_{1} + 2 \beta_{6} - 2 \beta_{9} - \beta_{11} - 2 \beta_{15} + \beta_{16} - \beta_{18} + 2 \beta_{19} ) q^{35} + ( -1 + 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{12} - 2 \beta_{13} ) q^{36} + ( -4 + 2 \beta_{2} + \beta_{5} - 2 \beta_{7} + \beta_{8} + 4 \beta_{10} - 2 \beta_{13} + 2 \beta_{14} ) q^{37} + ( -2 - \beta_{2} - \beta_{5} + \beta_{7} + \beta_{8} - 2 \beta_{10} - 4 \beta_{12} - \beta_{13} + \beta_{14} ) q^{38} + ( \beta_{1} + \beta_{6} - 2 \beta_{9} - 2 \beta_{11} + 2 \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{39} + ( \beta_{1} - \beta_{3} + \beta_{6} - \beta_{9} - 2 \beta_{11} + 2 \beta_{15} - \beta_{16} + \beta_{17} - \beta_{19} ) q^{40} + ( \beta_{1} + 3 \beta_{11} + \beta_{15} - 2 \beta_{18} + \beta_{19} ) q^{41} + ( -2 + 2 \beta_{10} + 2 \beta_{13} ) q^{42} + ( -\beta_{1} + \beta_{6} - \beta_{11} + \beta_{15} - \beta_{16} - \beta_{18} + \beta_{19} ) q^{43} + ( -2 + \beta_{2} + 3 \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{10} - \beta_{12} + 4 \beta_{13} - 2 \beta_{14} ) q^{45} + ( \beta_{1} - 4 \beta_{6} + \beta_{11} - \beta_{15} + 4 \beta_{17} + 2 \beta_{19} ) q^{46} + ( 1 + \beta_{2} - 3 \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{10} - 4 \beta_{13} + \beta_{14} ) q^{47} + ( -5 - \beta_{4} + 2 \beta_{5} - \beta_{7} - 2 \beta_{8} - 5 \beta_{10} + 3 \beta_{12} ) q^{48} + ( 3 \beta_{4} + \beta_{5} - 3 \beta_{7} - 3 \beta_{8} + 3 \beta_{10} + \beta_{12} + 3 \beta_{13} - 2 \beta_{14} ) q^{49} + ( \beta_{1} + \beta_{11} - 4 \beta_{15} - 3 \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{50} + ( -\beta_{3} + 2 \beta_{11} + \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{51} + ( 3 \beta_{1} + \beta_{3} + \beta_{6} - \beta_{9} - 2 \beta_{15} - 2 \beta_{16} ) q^{52} + ( -3 - 3 \beta_{10} - 2 \beta_{12} ) q^{53} + ( -2 \beta_{1} + \beta_{3} + 2 \beta_{11} + 2 \beta_{16} + \beta_{17} + \beta_{19} ) q^{54} + ( -2 - \beta_{2} - \beta_{4} + \beta_{5} ) q^{56} + ( 3 \beta_{1} - 2 \beta_{3} + 3 \beta_{6} - 2 \beta_{9} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{57} + ( \beta_{2} - 2 \beta_{5} - \beta_{7} + 2 \beta_{8} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{58} + ( \beta_{4} - 3 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} - 4 \beta_{14} ) q^{59} + ( 5 - 2 \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} - 5 \beta_{10} + 5 \beta_{13} ) q^{60} + ( -2 \beta_{1} - 2 \beta_{3} + \beta_{6} - \beta_{11} - \beta_{15} + \beta_{17} - \beta_{18} ) q^{61} + ( -\beta_{1} - 4 \beta_{3} + 2 \beta_{6} - \beta_{9} + \beta_{15} + \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{62} + ( \beta_{1} + \beta_{9} - 3 \beta_{11} - 3 \beta_{15} + 3 \beta_{16} + \beta_{18} - \beta_{19} ) q^{63} + ( -\beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} - \beta_{12} - \beta_{14} ) q^{64} + ( \beta_{3} - 2 \beta_{6} + 2 \beta_{9} - \beta_{11} - 3 \beta_{15} - \beta_{16} - 2 \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{65} + ( 2 - \beta_{2} + \beta_{4} - 2 \beta_{5} - 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{13} - \beta_{14} ) q^{67} + ( 2 \beta_{1} - \beta_{6} + \beta_{9} - 3 \beta_{11} + \beta_{15} - \beta_{16} - \beta_{17} + 2 \beta_{18} - 2 \beta_{19} ) q^{68} + ( 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{10} + \beta_{12} + \beta_{13} ) q^{69} + ( 6 - 2 \beta_{2} - 3 \beta_{4} + 4 \beta_{5} + 3 \beta_{7} - \beta_{8} - 2 \beta_{10} + 2 \beta_{12} - 5 \beta_{13} + \beta_{14} ) q^{70} + ( 2 + 2 \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{12} + \beta_{13} ) q^{71} + ( -2 \beta_{1} - 3 \beta_{3} + 3 \beta_{15} + 3 \beta_{16} ) q^{72} + ( \beta_{1} - 2 \beta_{6} - \beta_{9} - \beta_{11} + 4 \beta_{17} + \beta_{18} - \beta_{19} ) q^{73} + ( -2 \beta_{1} - 3 \beta_{3} - \beta_{6} + 2 \beta_{9} + 3 \beta_{11} - 2 \beta_{16} - 4 \beta_{17} + \beta_{18} - 2 \beta_{19} ) q^{74} + ( 3 + \beta_{2} + \beta_{5} + 2 \beta_{7} + \beta_{8} + 4 \beta_{10} - 3 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} ) q^{75} + ( \beta_{3} + 3 \beta_{6} + \beta_{11} - 2 \beta_{15} - 4 \beta_{17} - \beta_{18} - \beta_{19} ) q^{76} + ( 6 - 4 \beta_{2} + 4 \beta_{7} + 6 \beta_{10} - 4 \beta_{12} - 4 \beta_{13} + 4 \beta_{14} ) q^{78} + ( -3 \beta_{1} + 3 \beta_{11} + 2 \beta_{16} - \beta_{19} ) q^{79} + ( 2 - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{7} + 4 \beta_{8} + \beta_{10} - 2 \beta_{12} - \beta_{14} ) q^{80} + ( 5 + 2 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{12} + 2 \beta_{13} ) q^{81} + ( -6 + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{8} + 6 \beta_{10} + \beta_{14} ) q^{82} + ( \beta_{11} + 3 \beta_{15} - 3 \beta_{16} + \beta_{17} ) q^{83} + ( -4 \beta_{1} + 2 \beta_{6} - 4 \beta_{9} + 2 \beta_{11} - 2 \beta_{16} + 2 \beta_{17} - 2 \beta_{18} + 4 \beta_{19} ) q^{84} + ( -2 \beta_{1} + 2 \beta_{3} + \beta_{6} - \beta_{11} + \beta_{15} + 2 \beta_{16} - \beta_{17} - \beta_{19} ) q^{85} + ( 6 - 3 \beta_{2} - \beta_{4} + \beta_{5} ) q^{86} + ( 2 \beta_{6} + 3 \beta_{15} + 3 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} - 2 \beta_{19} ) q^{87} + ( -2 \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{12} + 4 \beta_{13} - 2 \beta_{14} ) q^{89} + ( -2 \beta_{1} - \beta_{9} - 3 \beta_{11} - 5 \beta_{16} + 2 \beta_{17} - \beta_{18} - \beta_{19} ) q^{90} + ( 2 - 4 \beta_{2} - \beta_{4} + \beta_{5} + 3 \beta_{7} - 3 \beta_{8} + 3 \beta_{12} - 3 \beta_{13} ) q^{91} + ( -1 - \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - \beta_{7} - 2 \beta_{8} + \beta_{10} + 6 \beta_{13} - \beta_{14} ) q^{92} + ( -7 - 2 \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{8} - 7 \beta_{10} + 2 \beta_{12} ) q^{93} + ( -\beta_{3} - \beta_{6} + 2 \beta_{9} + \beta_{11} + 7 \beta_{16} - 2 \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{94} + ( -3 \beta_{1} + 2 \beta_{3} - 3 \beta_{6} - \beta_{9} + \beta_{11} - \beta_{15} + 3 \beta_{16} + 2 \beta_{17} + 3 \beta_{19} ) q^{95} + ( -2 \beta_{11} + 3 \beta_{15} + 2 \beta_{18} - \beta_{19} ) q^{96} + ( 2 - \beta_{2} + 3 \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} - 2 \beta_{10} + \beta_{13} - \beta_{14} ) q^{97} + ( \beta_{1} - 3 \beta_{6} - 2 \beta_{9} - \beta_{11} + 5 \beta_{15} - 5 \beta_{16} + 3 \beta_{17} + \beta_{18} - \beta_{19} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 4q^{3} + 4q^{5} + O(q^{10})$$ $$20q + 4q^{3} + 4q^{5} - 16q^{12} + 16q^{15} + 12q^{16} + 16q^{20} + 12q^{23} + 16q^{25} + 56q^{26} - 20q^{27} - 16q^{31} - 20q^{36} - 72q^{37} - 32q^{38} - 32q^{42} - 28q^{45} + 16q^{47} - 104q^{48} - 52q^{53} - 32q^{56} + 12q^{58} + 112q^{60} + 28q^{67} + 104q^{70} + 24q^{71} + 64q^{75} + 104q^{78} + 44q^{80} + 100q^{81} - 124q^{82} + 128q^{86} - 16q^{92} - 132q^{93} + 32q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 67 x^{16} + 1315 x^{12} + 9193 x^{8} + 16040 x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$253 \nu^{16} + 17445 \nu^{12} + 371423 \nu^{8} + 3732211 \nu^{4} + 14018812$$$$)/4721416$$ $$\beta_{3}$$ $$=$$ $$($$$$627 \nu^{17} + 94553 \nu^{13} + 3691749 \nu^{9} + 38655603 \nu^{5} + 68818580 \nu$$$$)/9442832$$ $$\beta_{4}$$ $$=$$ $$($$$$29131 \nu^{18} + 2794 \nu^{16} + 1994661 \nu^{14} + 90014 \nu^{12} + 40723037 \nu^{10} - 1440730 \nu^{8} + 298958639 \nu^{6} - 27038662 \nu^{4} + 553705724 \nu^{2} - 17207320$$$$)/18885664$$ $$\beta_{5}$$ $$=$$ $$($$$$29131 \nu^{18} - 2794 \nu^{16} + 1994661 \nu^{14} - 90014 \nu^{12} + 40723037 \nu^{10} + 1440730 \nu^{8} + 298958639 \nu^{6} + 27038662 \nu^{4} + 553705724 \nu^{2} + 17207320$$$$)/18885664$$ $$\beta_{6}$$ $$=$$ $$($$$$-68791 \nu^{19} - 31178 \nu^{17} - 4654673 \nu^{15} - 2028502 \nu^{13} - 92769865 \nu^{11} - 37504502 \nu^{9} - 652386595 \nu^{7} - 239341410 \nu^{5} - 1088277028 \nu^{3} - 352512664 \nu$$$$)/37771328$$ $$\beta_{7}$$ $$=$$ $$($$$$-68791 \nu^{18} - 26646 \nu^{16} - 4654673 \nu^{14} - 1510730 \nu^{12} - 92769865 \nu^{10} - 19766122 \nu^{8} - 652386595 \nu^{6} - 54861310 \nu^{4} - 1088277028 \nu^{2} + 34912152$$$$)/37771328$$ $$\beta_{8}$$ $$=$$ $$($$$$-68791 \nu^{18} + 26646 \nu^{16} - 4654673 \nu^{14} + 1510730 \nu^{12} - 92769865 \nu^{10} + 19766122 \nu^{8} - 652386595 \nu^{6} + 54861310 \nu^{4} - 1088277028 \nu^{2} - 34912152$$$$)/37771328$$ $$\beta_{9}$$ $$=$$ $$($$$$-68791 \nu^{19} + 31178 \nu^{17} - 4654673 \nu^{15} + 2028502 \nu^{13} - 92769865 \nu^{11} + 37504502 \nu^{9} - 652386595 \nu^{7} + 239341410 \nu^{5} - 1088277028 \nu^{3} + 352512664 \nu$$$$)/37771328$$ $$\beta_{10}$$ $$=$$ $$($$$$36359 \nu^{18} + 2437065 \nu^{14} + 47881865 \nu^{10} + 335733979 \nu^{6} + 598127204 \nu^{2}$$$$)/18885664$$ $$\beta_{11}$$ $$=$$ $$($$$$36359 \nu^{19} + 2437065 \nu^{15} + 47881865 \nu^{11} + 335733979 \nu^{7} + 598127204 \nu^{3}$$$$)/18885664$$ $$\beta_{12}$$ $$=$$ $$($$$$-124787 \nu^{18} - 242 \nu^{16} - 8385109 \nu^{14} - 119326 \nu^{12} - 165346749 \nu^{10} - 5897806 \nu^{8} - 1158063823 \nu^{6} - 62382362 \nu^{4} - 1983090404 \nu^{2} - 62676248$$$$)/37771328$$ $$\beta_{13}$$ $$=$$ $$($$$$-124787 \nu^{18} + 242 \nu^{16} - 8385109 \nu^{14} + 119326 \nu^{12} - 165346749 \nu^{10} + 5897806 \nu^{8} - 1158063823 \nu^{6} + 62382362 \nu^{4} - 1983090404 \nu^{2} + 62676248$$$$)/37771328$$ $$\beta_{14}$$ $$=$$ $$($$$$-284679 \nu^{18} + 5830 \nu^{16} - 19018177 \nu^{14} + 299354 \nu^{12} - 371191865 \nu^{10} + 3016346 \nu^{8} - 2574550419 \nu^{6} + 8305038 \nu^{4} - 4426670852 \nu^{2} + 28261608$$$$)/37771328$$ $$\beta_{15}$$ $$=$$ $$($$$$415659 \nu^{19} + 2266 \nu^{17} + 27881629 \nu^{15} + 258886 \nu^{13} + 548401669 \nu^{11} + 8869190 \nu^{9} + 3843935655 \nu^{7} + 92240050 \nu^{5} + 6730336708 \nu^{3} + 212598072 \nu$$$$)/37771328$$ $$\beta_{16}$$ $$=$$ $$($$$$-415659 \nu^{19} + 2266 \nu^{17} - 27881629 \nu^{15} + 258886 \nu^{13} - 548401669 \nu^{11} + 8869190 \nu^{9} - 3843935655 \nu^{7} + 92240050 \nu^{5} - 6730336708 \nu^{3} + 212598072 \nu$$$$)/37771328$$ $$\beta_{17}$$ $$=$$ $$($$$$-270223 \nu^{19} - 18133369 \nu^{15} - 356874209 \nu^{11} - 2500999739 \nu^{7} - 4356713556 \nu^{3}$$$$)/18885664$$ $$\beta_{18}$$ $$=$$ $$($$$$-350169 \nu^{19} - 35226 \nu^{17} - 23449903 \nu^{15} - 2307622 \nu^{13} - 459796767 \nu^{11} - 43447270 \nu^{9} - 3209243037 \nu^{7} - 289613954 \nu^{5} - 5578503780 \nu^{3} - 491828168 \nu$$$$)/18885664$$ $$\beta_{19}$$ $$=$$ $$($$$$1171993 \nu^{19} - 39274 \nu^{17} + 78511871 \nu^{15} - 2586742 \nu^{13} + 1540572087 \nu^{11} - 49390038 \nu^{9} + 10768098957 \nu^{7} - 339886498 \nu^{5} + 18782157644 \nu^{3} - 593372344 \nu$$$$)/37771328$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{14} - \beta_{13} + 3 \beta_{10} - \beta_{4}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{17} + \beta_{16} - \beta_{15} + 4 \beta_{11}$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{13} + 2 \beta_{12} + \beta_{5} - \beta_{4} + 6 \beta_{2} - 13$$ $$\nu^{5}$$ $$=$$ $$\beta_{19} + \beta_{18} + \beta_{17} + 8 \beta_{16} + 8 \beta_{15} + \beta_{9} - 2 \beta_{6} - 8 \beta_{3} - 18 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-37 \beta_{14} + 56 \beta_{13} + 19 \beta_{12} - 65 \beta_{10} - 11 \beta_{8} - 11 \beta_{7} - \beta_{5} + 36 \beta_{4}$$ $$\nu^{7}$$ $$=$$ $$\beta_{19} - \beta_{18} + 76 \beta_{17} - 54 \beta_{16} + 54 \beta_{15} - 100 \beta_{11} - 12 \beta_{9} - 11 \beta_{6} - \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$141 \beta_{13} - 141 \beta_{12} - 11 \beta_{8} + 11 \beta_{7} - 88 \beta_{5} + 88 \beta_{4} - 230 \beta_{2} + 355$$ $$\nu^{9}$$ $$=$$ $$-88 \beta_{19} - 88 \beta_{18} - 88 \beta_{17} - 349 \beta_{16} - 349 \beta_{15} - 99 \beta_{9} + 187 \beta_{6} + 336 \beta_{3} + 475 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$1433 \beta_{14} - 2393 \beta_{13} - 960 \beta_{12} + 2047 \beta_{10} + 624 \beta_{8} + 624 \beta_{7} + 86 \beta_{5} - 1347 \beta_{4}$$ $$\nu^{11}$$ $$=$$ $$-86 \beta_{19} + 86 \beta_{18} - 3439 \beta_{17} + 2221 \beta_{16} - 2221 \beta_{15} + 3308 \beta_{11} + 710 \beta_{9} + 624 \beta_{6} + 86 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-6284 \beta_{13} + 6284 \beta_{12} + 594 \beta_{8} - 594 \beta_{7} + 4179 \beta_{5} - 4179 \beta_{4} + 8938 \beta_{2} - 12201$$ $$\nu^{13}$$ $$=$$ $$4179 \beta_{19} + 4179 \beta_{18} + 4179 \beta_{17} + 14034 \beta_{16} + 14034 \beta_{15} + 4773 \beta_{9} - 8952 \beta_{6} - 13148 \beta_{3} - 15772 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-55793 \beta_{14} + 96106 \beta_{13} + 40313 \beta_{12} - 74189 \beta_{10} - 27165 \beta_{8} - 27165 \beta_{7} - 3887 \beta_{5} + 51906 \beta_{4}$$ $$\nu^{15}$$ $$=$$ $$3887 \beta_{19} - 3887 \beta_{18} + 140306 \beta_{17} - 88332 \beta_{16} + 88332 \beta_{15} - 122208 \beta_{11} - 31052 \beta_{9} - 27165 \beta_{6} - 3887 \beta_{1}$$ $$\nu^{16}$$ $$=$$ $$255803 \beta_{13} - 255803 \beta_{12} - 24809 \beta_{8} + 24809 \beta_{7} - 173714 \beta_{5} + 173714 \beta_{4} - 348490 \beta_{2} + 456487$$ $$\nu^{17}$$ $$=$$ $$-173714 \beta_{19} - 173714 \beta_{18} - 173714 \beta_{17} - 554675 \beta_{16} - 554675 \beta_{15} - 198523 \beta_{9} + 372237 \beta_{6} + 512668 \beta_{3} + 581645 \beta_{1}$$ $$\nu^{18}$$ $$=$$ $$2177741 \beta_{14} - 3791035 \beta_{13} - 1613294 \beta_{12} + 2828363 \beta_{10} + 1100626 \beta_{8} + 1100626 \beta_{7} + 156516 \beta_{5} - 2021225 \beta_{4}$$ $$\nu^{19}$$ $$=$$ $$-156516 \beta_{19} + 156516 \beta_{18} - 5560845 \beta_{17} + 3478003 \beta_{16} - 3478003 \beta_{15} + 4693072 \beta_{11} + 1257142 \beta_{9} + 1100626 \beta_{6} + 156516 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$-\beta_{10}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
362.1
 −1.76854 + 1.76854i −1.35329 + 1.35329i −1.31283 + 1.31283i −0.895288 + 0.895288i −0.125683 + 0.125683i 0.125683 − 0.125683i 0.895288 − 0.895288i 1.31283 − 1.31283i 1.35329 − 1.35329i 1.76854 − 1.76854i −1.76854 − 1.76854i −1.35329 − 1.35329i −1.31283 − 1.31283i −0.895288 − 0.895288i −0.125683 − 0.125683i 0.125683 + 0.125683i 0.895288 + 0.895288i 1.31283 + 1.31283i 1.35329 + 1.35329i 1.76854 + 1.76854i
−1.76854 + 1.76854i 1.85563 1.85563i 4.25546i −2.19774 + 0.412263i 6.56351i 0.260724 0.260724i 3.98886 + 3.98886i 3.88674i 3.15768 4.61588i
362.2 −1.35329 + 1.35329i 0.185315 0.185315i 1.66281i −0.610994 + 2.15097i 0.501571i −3.24854 + 3.24854i −0.456321 0.456321i 2.93132i −2.08404 3.73775i
362.3 −1.31283 + 1.31283i −1.09155 + 1.09155i 1.44706i 2.21808 0.283032i 2.86605i 1.45954 1.45954i −0.725917 0.725917i 0.617036i −2.54040 + 3.28355i
362.4 −0.895288 + 0.895288i −1.60711 + 1.60711i 0.396917i −0.452009 2.18991i 2.87765i 1.81197 1.81197i −2.14593 2.14593i 2.16559i 2.36528 + 1.55592i
362.5 −0.125683 + 0.125683i 1.65771 1.65771i 1.96841i 2.04266 + 0.909702i 0.416692i 3.15683 3.15683i −0.498761 0.498761i 2.49602i −0.371061 + 0.142393i
362.6 0.125683 0.125683i 1.65771 1.65771i 1.96841i 2.04266 + 0.909702i 0.416692i −3.15683 + 3.15683i 0.498761 + 0.498761i 2.49602i 0.371061 0.142393i
362.7 0.895288 0.895288i −1.60711 + 1.60711i 0.396917i −0.452009 2.18991i 2.87765i −1.81197 + 1.81197i 2.14593 + 2.14593i 2.16559i −2.36528 1.55592i
362.8 1.31283 1.31283i −1.09155 + 1.09155i 1.44706i 2.21808 0.283032i 2.86605i −1.45954 + 1.45954i 0.725917 + 0.725917i 0.617036i 2.54040 3.28355i
362.9 1.35329 1.35329i 0.185315 0.185315i 1.66281i −0.610994 + 2.15097i 0.501571i 3.24854 3.24854i 0.456321 + 0.456321i 2.93132i 2.08404 + 3.73775i
362.10 1.76854 1.76854i 1.85563 1.85563i 4.25546i −2.19774 + 0.412263i 6.56351i −0.260724 + 0.260724i −3.98886 3.98886i 3.88674i −3.15768 + 4.61588i
483.1 −1.76854 1.76854i 1.85563 + 1.85563i 4.25546i −2.19774 0.412263i 6.56351i 0.260724 + 0.260724i 3.98886 3.98886i 3.88674i 3.15768 + 4.61588i
483.2 −1.35329 1.35329i 0.185315 + 0.185315i 1.66281i −0.610994 2.15097i 0.501571i −3.24854 3.24854i −0.456321 + 0.456321i 2.93132i −2.08404 + 3.73775i
483.3 −1.31283 1.31283i −1.09155 1.09155i 1.44706i 2.21808 + 0.283032i 2.86605i 1.45954 + 1.45954i −0.725917 + 0.725917i 0.617036i −2.54040 3.28355i
483.4 −0.895288 0.895288i −1.60711 1.60711i 0.396917i −0.452009 + 2.18991i 2.87765i 1.81197 + 1.81197i −2.14593 + 2.14593i 2.16559i 2.36528 1.55592i
483.5 −0.125683 0.125683i 1.65771 + 1.65771i 1.96841i 2.04266 0.909702i 0.416692i 3.15683 + 3.15683i −0.498761 + 0.498761i 2.49602i −0.371061 0.142393i
483.6 0.125683 + 0.125683i 1.65771 + 1.65771i 1.96841i 2.04266 0.909702i 0.416692i −3.15683 3.15683i 0.498761 0.498761i 2.49602i 0.371061 + 0.142393i
483.7 0.895288 + 0.895288i −1.60711 1.60711i 0.396917i −0.452009 + 2.18991i 2.87765i −1.81197 1.81197i 2.14593 2.14593i 2.16559i −2.36528 + 1.55592i
483.8 1.31283 + 1.31283i −1.09155 1.09155i 1.44706i 2.21808 + 0.283032i 2.86605i −1.45954 1.45954i 0.725917 0.725917i 0.617036i 2.54040 + 3.28355i
483.9 1.35329 + 1.35329i 0.185315 + 0.185315i 1.66281i −0.610994 2.15097i 0.501571i 3.24854 + 3.24854i 0.456321 0.456321i 2.93132i 2.08404 3.73775i
483.10 1.76854 + 1.76854i 1.85563 + 1.85563i 4.25546i −2.19774 0.412263i 6.56351i −0.260724 0.260724i −3.98886 + 3.98886i 3.88674i −3.15768 4.61588i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 483.10 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
11.b odd 2 1 inner
55.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.e.a 20
5.c odd 4 1 inner 605.2.e.a 20
11.b odd 2 1 inner 605.2.e.a 20
11.c even 5 4 605.2.m.f 80
11.d odd 10 4 605.2.m.f 80
55.e even 4 1 inner 605.2.e.a 20
55.k odd 20 4 605.2.m.f 80
55.l even 20 4 605.2.m.f 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.e.a 20 1.a even 1 1 trivial
605.2.e.a 20 5.c odd 4 1 inner
605.2.e.a 20 11.b odd 2 1 inner
605.2.e.a 20 55.e even 4 1 inner
605.2.m.f 80 11.c even 5 4
605.2.m.f 80 11.d odd 10 4
605.2.m.f 80 55.k odd 20 4
605.2.m.f 80 55.l even 20 4

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{20} + 67 T_{2}^{16} + 1315 T_{2}^{12} + 9193 T_{2}^{8} + 16040 T_{2}^{4} + 16$$ acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 16040 T^{4} + 9193 T^{8} + 1315 T^{12} + 67 T^{16} + T^{20}$$
$3$ $$( 32 - 160 T + 400 T^{2} + 168 T^{3} + 56 T^{4} - 60 T^{5} + 41 T^{6} + 6 T^{7} + 2 T^{8} - 2 T^{9} + T^{10} )^{2}$$
$5$ $$( 3125 - 1250 T - 250 T^{2} - 150 T^{3} - 35 T^{4} + 128 T^{5} - 7 T^{6} - 6 T^{7} - 2 T^{8} - 2 T^{9} + T^{10} )^{2}$$
$7$ $$2560000 + 138715136 T^{4} + 11506176 T^{8} + 229392 T^{12} + 904 T^{16} + T^{20}$$
$11$ $$T^{20}$$
$13$ $$8540717056 + 1464147968 T^{4} + 41713920 T^{8} + 393984 T^{12} + 1152 T^{16} + T^{20}$$
$17$ $$4096 + 507904 T^{4} + 1036288 T^{8} + 223248 T^{12} + 2344 T^{16} + T^{20}$$
$19$ $$( -247808 + 211968 T^{2} - 38144 T^{4} + 2768 T^{6} - 88 T^{8} + T^{10} )^{2}$$
$23$ $$( 2350112 - 1387520 T + 409600 T^{2} + 76680 T^{3} + 44792 T^{4} - 16092 T^{5} + 2945 T^{6} + 50 T^{7} + 18 T^{8} - 6 T^{9} + T^{10} )^{2}$$
$29$ $$( -46208 + 97344 T^{2} - 34472 T^{4} + 3164 T^{6} - 102 T^{8} + T^{10} )^{2}$$
$31$ $$( 2272 + 304 T - 300 T^{2} - 59 T^{3} + 4 T^{4} + T^{5} )^{4}$$
$37$ $$( 75251912 + 57242488 T + 21771556 T^{2} + 3313400 T^{3} + 472898 T^{4} + 136494 T^{5} + 39957 T^{6} + 6550 T^{7} + 648 T^{8} + 36 T^{9} + T^{10} )^{2}$$
$41$ $$( 40572032 + 10554688 T^{2} + 734824 T^{4} + 20636 T^{6} + 246 T^{8} + T^{10} )^{2}$$
$43$ $$13817456889856 + 916357939200 T^{4} + 6225534976 T^{8} + 9338640 T^{12} + 5224 T^{16} + T^{20}$$
$47$ $$( 307328 - 1229312 T + 2458624 T^{2} - 1607200 T^{3} + 580864 T^{4} - 114096 T^{5} + 11972 T^{6} - 320 T^{7} + 32 T^{8} - 8 T^{9} + T^{10} )^{2}$$
$53$ $$( 11552 - 44384 T + 85264 T^{2} - 42624 T^{3} + 656 T^{4} + 16656 T^{5} + 9800 T^{6} + 2400 T^{7} + 338 T^{8} + 26 T^{9} + T^{10} )^{2}$$
$59$ $$( 2408704 + 3812480 T^{2} + 1555664 T^{4} + 42385 T^{6} + 366 T^{8} + T^{10} )^{2}$$
$61$ $$( 8192 + 952832 T^{2} + 544968 T^{4} + 25452 T^{6} + 294 T^{8} + T^{10} )^{2}$$
$67$ $$( 408608 - 339904 T + 141376 T^{2} + 123576 T^{3} + 70576 T^{4} - 7296 T^{5} + 337 T^{6} + 54 T^{7} + 98 T^{8} - 14 T^{9} + T^{10} )^{2}$$
$71$ $$( -2144 + 1072 T + 192 T^{2} - 83 T^{3} - 6 T^{4} + T^{5} )^{4}$$
$73$ $$1011333821168091136 + 4737986996174848 T^{4} + 2804909474048 T^{8} + 571849344 T^{12} + 44784 T^{16} + T^{20}$$
$79$ $$( -397394432 + 92450816 T^{2} - 4132096 T^{4} + 66468 T^{6} - 436 T^{8} + T^{10} )^{2}$$
$83$ $$20199631360000 + 13044730003456 T^{4} + 37636646912 T^{8} + 31183376 T^{12} + 9768 T^{16} + T^{20}$$
$89$ $$( 3748096 + 1835328 T^{2} + 309776 T^{4} + 19657 T^{6} + 282 T^{8} + T^{10} )^{2}$$
$97$ $$( 248110088 - 17063416 T + 586756 T^{2} - 438104 T^{3} + 365986 T^{4} - 38790 T^{5} + 2189 T^{6} - 226 T^{7} + 128 T^{8} - 16 T^{9} + T^{10} )^{2}$$