# Properties

 Label 605.2.g.o Level $605$ Weight $2$ Character orbit 605.g Analytic conductor $4.831$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.g (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - x^{11} + 6 x^{10} - 12 x^{9} + 43 x^{8} + 72 x^{7} + 155 x^{6} + 162 x^{5} + 541 x^{4} + 114 x^{3} + 24 x^{2} + 5 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{11} q^{2} + \beta_{8} q^{3} + ( -\beta_{1} - 3 \beta_{4} - \beta_{5} ) q^{4} -\beta_{7} q^{5} + ( 2 \beta_{6} + \beta_{7} ) q^{6} + ( 2 \beta_{1} + \beta_{5} ) q^{7} + ( 4 - 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 4 \beta_{7} + \beta_{8} + 2 \beta_{9} - 4 \beta_{10} + 2 \beta_{11} ) q^{8} + ( \beta_{1} + \beta_{2} - \beta_{6} - \beta_{8} - \beta_{11} ) q^{9} +O(q^{10})$$ $$q -\beta_{11} q^{2} + \beta_{8} q^{3} + ( -\beta_{1} - 3 \beta_{4} - \beta_{5} ) q^{4} -\beta_{7} q^{5} + ( 2 \beta_{6} + \beta_{7} ) q^{6} + ( 2 \beta_{1} + \beta_{5} ) q^{7} + ( 4 - 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 4 \beta_{7} + \beta_{8} + 2 \beta_{9} - 4 \beta_{10} + 2 \beta_{11} ) q^{8} + ( \beta_{1} + \beta_{2} - \beta_{6} - \beta_{8} - \beta_{11} ) q^{9} -\beta_{3} q^{10} + ( 2 - 2 \beta_{2} + \beta_{3} ) q^{12} + ( -\beta_{1} - \beta_{2} + \beta_{6} + \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{13} + ( -3 + \beta_{3} + 3 \beta_{4} + \beta_{5} - 3 \beta_{7} - 3 \beta_{8} - \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{14} + \beta_{1} q^{15} + ( 2 \beta_{6} + 5 \beta_{7} + 4 \beta_{9} ) q^{16} + ( -2 \beta_{6} + 2 \beta_{7} ) q^{17} + ( -3 \beta_{1} - 6 \beta_{4} - \beta_{5} ) q^{18} + ( -2 + \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{19} + ( -\beta_{1} - \beta_{2} + \beta_{6} + \beta_{8} + 3 \beta_{10} - \beta_{11} ) q^{20} + ( -5 - 2 \beta_{3} ) q^{21} + ( -2 + \beta_{2} + \beta_{3} ) q^{23} + ( \beta_{1} + \beta_{2} - \beta_{6} - \beta_{8} - 5 \beta_{10} - \beta_{11} ) q^{24} + ( -1 + \beta_{4} - \beta_{7} + \beta_{10} ) q^{25} + ( \beta_{1} - 4 \beta_{4} + \beta_{5} ) q^{26} + ( 4 \beta_{7} - \beta_{9} ) q^{27} + ( -3 \beta_{6} - 8 \beta_{7} - 2 \beta_{9} ) q^{28} + 2 \beta_{5} q^{29} + ( 1 - \beta_{4} + \beta_{7} - 2 \beta_{8} - \beta_{10} ) q^{30} + ( \beta_{1} + \beta_{2} - \beta_{6} - \beta_{8} - 4 \beta_{10} - \beta_{11} ) q^{31} + ( -10 + 2 \beta_{2} + 5 \beta_{3} ) q^{32} + ( -2 + 4 \beta_{2} + 2 \beta_{3} ) q^{34} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{6} - 2 \beta_{8} + \beta_{11} ) q^{35} + ( 2 - 5 \beta_{3} - 2 \beta_{4} - 5 \beta_{5} + 2 \beta_{7} + 3 \beta_{8} + 5 \beta_{9} - 2 \beta_{10} + 5 \beta_{11} ) q^{36} + ( -2 \beta_{4} - 2 \beta_{5} ) q^{37} + ( -3 \beta_{6} - 6 \beta_{7} - 3 \beta_{9} ) q^{38} + ( 3 \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{39} + ( \beta_{1} - 4 \beta_{4} - 2 \beta_{5} ) q^{40} + ( -3 + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} ) q^{41} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{6} + 2 \beta_{8} + 10 \beta_{10} + 3 \beta_{11} ) q^{42} + ( 2 - 2 \beta_{2} - \beta_{3} ) q^{43} + ( \beta_{2} - \beta_{3} ) q^{45} + ( -\beta_{1} - \beta_{2} + \beta_{6} + \beta_{8} - 4 \beta_{10} + 3 \beta_{11} ) q^{46} + ( 4 + 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} + \beta_{8} - 4 \beta_{9} - 4 \beta_{10} - 4 \beta_{11} ) q^{47} + ( \beta_{1} - 2 \beta_{4} + 2 \beta_{5} ) q^{48} + ( -3 \beta_{6} + 6 \beta_{7} - 3 \beta_{9} ) q^{49} -\beta_{9} q^{50} + ( 6 \beta_{4} - 2 \beta_{5} ) q^{51} + ( -\beta_{3} - \beta_{5} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{52} + ( -\beta_{1} - \beta_{2} + \beta_{6} + \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{53} + ( 5 - \beta_{2} + 3 \beta_{3} ) q^{54} + ( 1 - 2 \beta_{2} - 8 \beta_{3} ) q^{56} + ( -\beta_{1} - \beta_{2} + \beta_{6} + \beta_{8} + 4 \beta_{10} + \beta_{11} ) q^{57} + ( -10 + 2 \beta_{3} + 10 \beta_{4} + 2 \beta_{5} - 10 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 10 \beta_{10} - 2 \beta_{11} ) q^{58} + ( -\beta_{1} - 2 \beta_{4} + 3 \beta_{5} ) q^{59} + ( -2 \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{60} + ( -5 \beta_{6} + 3 \beta_{7} + \beta_{9} ) q^{61} + ( -3 \beta_{1} - 6 \beta_{4} + 3 \beta_{5} ) q^{62} + ( 2 + 3 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{63} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{6} + 3 \beta_{8} - 13 \beta_{10} + 7 \beta_{11} ) q^{64} + ( -2 - \beta_{2} - \beta_{3} ) q^{65} + ( 6 + \beta_{2} + 2 \beta_{3} ) q^{67} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{10} + 4 \beta_{11} ) q^{68} + ( 2 + \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{69} + ( -3 \beta_{1} + 3 \beta_{4} + \beta_{5} ) q^{70} + ( \beta_{6} + 2 \beta_{7} + \beta_{9} ) q^{71} + ( 5 \beta_{6} + 16 \beta_{7} + 5 \beta_{9} ) q^{72} + ( -2 \beta_{1} + 4 \beta_{4} - 2 \beta_{5} ) q^{73} + ( 10 - 4 \beta_{3} - 10 \beta_{4} - 4 \beta_{5} + 10 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} - 10 \beta_{10} + 4 \beta_{11} ) q^{74} + ( \beta_{1} + \beta_{2} - \beta_{6} - \beta_{8} ) q^{75} + ( 8 + \beta_{2} - 7 \beta_{3} ) q^{76} + ( -2 - 5 \beta_{2} - \beta_{3} ) q^{78} + ( 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{6} - 4 \beta_{8} + 2 \beta_{11} ) q^{79} + ( 5 - 4 \beta_{3} - 5 \beta_{4} - 4 \beta_{5} + 5 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} - 5 \beta_{10} + 4 \beta_{11} ) q^{80} + ( -3 \beta_{1} - \beta_{4} - 3 \beta_{5} ) q^{81} + ( 3 \beta_{6} - 12 \beta_{7} - 6 \beta_{9} ) q^{82} + ( 3 \beta_{6} + 6 \beta_{7} + 3 \beta_{9} ) q^{83} + ( 7 \beta_{1} + 7 \beta_{4} - 3 \beta_{5} ) q^{84} + ( 2 - 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} ) q^{85} + ( 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{6} - 3 \beta_{8} + 3 \beta_{10} - 3 \beta_{11} ) q^{86} + ( 2 - 4 \beta_{2} ) q^{87} + ( 3 - 6 \beta_{2} - 4 \beta_{3} ) q^{89} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{6} + 3 \beta_{8} + 6 \beta_{10} - \beta_{11} ) q^{90} + ( -8 - 3 \beta_{3} + 8 \beta_{4} - 3 \beta_{5} - 8 \beta_{7} - 7 \beta_{8} + 3 \beta_{9} + 8 \beta_{10} + 3 \beta_{11} ) q^{91} + ( 3 \beta_{1} + 12 \beta_{4} + 5 \beta_{5} ) q^{92} + ( 7 \beta_{6} + 4 \beta_{7} - \beta_{9} ) q^{93} + ( -2 \beta_{6} - 19 \beta_{7} ) q^{94} + ( -\beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{95} + ( 1 + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{96} + ( 7 \beta_{1} + 7 \beta_{2} - 7 \beta_{6} - 7 \beta_{8} + 2 \beta_{10} + 3 \beta_{11} ) q^{97} + ( 12 + 3 \beta_{2} + 3 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - q^{2} - q^{3} - 9q^{4} + 3q^{5} - 5q^{6} + q^{7} + 9q^{8} - 2q^{9} + O(q^{10})$$ $$12q - q^{2} - q^{3} - 9q^{4} + 3q^{5} - 5q^{6} + q^{7} + 9q^{8} - 2q^{9} - 4q^{10} + 36q^{12} - 6q^{13} - 5q^{14} + q^{15} - 13q^{16} - 4q^{17} - 20q^{18} - 4q^{19} + 9q^{20} - 68q^{21} - 24q^{23} - 17q^{24} - 3q^{25} - 12q^{26} - 13q^{27} + 25q^{28} - 2q^{29} + 5q^{30} - 14q^{31} - 108q^{32} - 32q^{34} - q^{35} - 2q^{36} - 4q^{37} + 18q^{38} + 4q^{39} - 9q^{40} - 9q^{41} + 35q^{42} + 28q^{43} - 8q^{45} - 8q^{46} + 15q^{47} - 7q^{48} - 18q^{49} - q^{50} + 20q^{51} - 2q^{52} + 6q^{53} + 76q^{54} - 12q^{56} + 14q^{57} - 30q^{58} - 10q^{59} + 9q^{60} - 3q^{61} - 24q^{62} + 12q^{63} - 29q^{64} - 24q^{65} + 76q^{67} + 8q^{69} + 5q^{70} - 6q^{71} - 48q^{72} + 12q^{73} + 28q^{74} - q^{75} + 64q^{76} - 8q^{78} - 2q^{79} + 13q^{80} - 3q^{81} + 27q^{82} - 18q^{83} + 31q^{84} + 4q^{85} + 3q^{86} + 40q^{87} + 44q^{89} + 20q^{90} - 20q^{91} + 34q^{92} - 20q^{93} + 59q^{94} + 4q^{95} + 7q^{96} + 2q^{97} + 144q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} + 6 x^{10} - 12 x^{9} + 43 x^{8} + 72 x^{7} + 155 x^{6} + 162 x^{5} + 541 x^{4} + 114 x^{3} + 24 x^{2} + 5 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$12 \nu^{10} + 2713 \nu^{5} + 5190$$$$)/24631$$ $$\beta_{3}$$ $$=$$ $$($$$$43 \nu^{10} + 7669 \nu^{5} - 67611$$$$)/24631$$ $$\beta_{4}$$ $$=$$ $$($$$$43 \nu^{11} + 7669 \nu^{6} - 116873 \nu$$$$)/24631$$ $$\beta_{5}$$ $$=$$ $$($$$$141 \nu^{11} + 25720 \nu^{6} - 320798 \nu$$$$)/24631$$ $$\beta_{6}$$ $$=$$ $$($$$$43 \nu^{11} - 258 \nu^{10} + 516 \nu^{9} - 1849 \nu^{8} + 4573 \nu^{7} - 6665 \nu^{6} - 6966 \nu^{5} - 23263 \nu^{4} - 4902 \nu^{3} - 117905 \nu^{2} - 215 \nu - 43$$$$)/24631$$ $$\beta_{7}$$ $$=$$ $$($$$$-227 \nu^{11} + 1362 \nu^{10} - 2724 \nu^{9} + 9761 \nu^{8} - 24714 \nu^{7} + 35185 \nu^{6} + 36774 \nu^{5} + 122807 \nu^{4} + 25878 \nu^{3} + 559992 \nu^{2} + 1135 \nu + 227$$$$)/24631$$ $$\beta_{8}$$ $$=$$ $$($$$$1092 \nu^{11} - 1092 \nu^{10} + 6521 \nu^{9} - 13104 \nu^{8} + 46956 \nu^{7} + 78624 \nu^{6} + 169260 \nu^{5} + 171948 \nu^{4} + 590772 \nu^{3} + 124488 \nu^{2} + 26208 \nu + 5460$$$$)/24631$$ $$\beta_{9}$$ $$=$$ $$($$$$-638 \nu^{11} + 3828 \nu^{10} - 7656 \nu^{9} + 27434 \nu^{8} - 69569 \nu^{7} + 98890 \nu^{6} + 103356 \nu^{5} + 345158 \nu^{4} + 72732 \nu^{3} + 1537440 \nu^{2} + 3190 \nu + 638$$$$)/24631$$ $$\beta_{10}$$ $$=$$ $$($$$$-5460 \nu^{11} + 6552 \nu^{10} - 33852 \nu^{9} + 72041 \nu^{8} - 247884 \nu^{7} - 346164 \nu^{6} - 767676 \nu^{5} - 715260 \nu^{4} - 2781912 \nu^{3} - 31668 \nu^{2} - 6552 \nu - 1092$$$$)/24631$$ $$\beta_{11}$$ $$=$$ $$($$$$15030 \nu^{11} - 18036 \nu^{10} + 93186 \nu^{9} - 198446 \nu^{8} + 682362 \nu^{7} + 952902 \nu^{6} + 2113218 \nu^{5} + 1968930 \nu^{4} + 7637059 \nu^{3} + 87174 \nu^{2} + 18036 \nu + 3006$$$$)/24631$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{9} + 3 \beta_{7} + \beta_{6}$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} + 4 \beta_{10} + 6 \beta_{8} + 6 \beta_{6} - 6 \beta_{2} - 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$6 \beta_{11} + 19 \beta_{10} + 6 \beta_{9} + 12 \beta_{8} - 19 \beta_{7} - 6 \beta_{5} + 19 \beta_{4} - 6 \beta_{3} - 19$$ $$\nu^{5}$$ $$=$$ $$-12 \beta_{3} + 43 \beta_{2} - 42$$ $$\nu^{6}$$ $$=$$ $$43 \beta_{5} - 141 \beta_{4} - 109 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$109 \beta_{9} - 370 \beta_{7} - 336 \beta_{6}$$ $$\nu^{8}$$ $$=$$ $$-336 \beta_{11} - 1117 \beta_{10} - 924 \beta_{8} - 924 \beta_{6} + 924 \beta_{2} + 924 \beta_{1}$$ $$\nu^{9}$$ $$=$$ $$-924 \beta_{11} - 3108 \beta_{10} - 924 \beta_{9} - 2713 \beta_{8} + 3108 \beta_{7} + 924 \beta_{5} - 3108 \beta_{4} + 924 \beta_{3} + 3108$$ $$\nu^{10}$$ $$=$$ $$2713 \beta_{3} - 7669 \beta_{2} + 9063$$ $$\nu^{11}$$ $$=$$ $$-7669 \beta_{5} + 25720 \beta_{4} + 22158 \beta_{1}$$

## Character values

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

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 −0.885704 − 2.72592i 0.511560 + 1.57442i 0.0651271 + 0.200441i −0.170505 − 0.123879i −1.33928 − 0.973045i 2.31880 + 1.68471i −0.885704 + 2.72592i 0.511560 − 1.57442i 0.0651271 − 0.200441i −0.170505 + 0.123879i −1.33928 + 0.973045i 2.31880 − 1.68471i
−1.90030 1.38065i 0.885704 2.72592i 1.08691 + 3.34515i 0.809017 0.587785i −5.44662 + 3.95720i −1.04556 3.21790i 1.10133 3.38955i −4.21910 3.06535i −2.34889
81.2 −1.12933 0.820508i −0.511560 + 1.57442i −0.0158755 0.0488598i 0.809017 0.587785i 1.86955 1.35830i 1.45449 + 4.47645i −0.884894 + 2.72343i 0.209948 + 0.152536i −1.39593
81.3 2.22061 + 1.61337i −0.0651271 + 0.200441i 1.71012 + 5.26321i 0.809017 0.587785i −0.468007 + 0.340027i −0.717944 2.20960i −2.99759 + 9.22564i 2.39112 + 1.73725i 2.74483
251.1 −0.848198 + 2.61048i 0.170505 0.123879i −4.47716 3.25284i −0.309017 0.951057i 0.178763 + 0.550175i 1.87960 + 1.36561i 7.84779 5.70176i −0.913325 + 2.81093i 2.74483
251.2 0.431367 1.32761i 1.33928 0.973045i 0.0415626 + 0.0301970i −0.309017 0.951057i −0.714103 2.19778i −3.80789 2.76660i 2.31668 1.68317i −0.0801932 + 0.246809i −1.39593
251.3 0.725848 2.23393i −2.31880 + 1.68471i −2.84556 2.06742i −0.309017 0.951057i 2.08042 + 6.40289i 2.73731 + 1.98877i −2.88333 + 2.09486i 1.61155 4.95985i −2.34889
366.1 −1.90030 + 1.38065i 0.885704 + 2.72592i 1.08691 3.34515i 0.809017 + 0.587785i −5.44662 3.95720i −1.04556 + 3.21790i 1.10133 + 3.38955i −4.21910 + 3.06535i −2.34889
366.2 −1.12933 + 0.820508i −0.511560 1.57442i −0.0158755 + 0.0488598i 0.809017 + 0.587785i 1.86955 + 1.35830i 1.45449 4.47645i −0.884894 2.72343i 0.209948 0.152536i −1.39593
366.3 2.22061 1.61337i −0.0651271 0.200441i 1.71012 5.26321i 0.809017 + 0.587785i −0.468007 0.340027i −0.717944 + 2.20960i −2.99759 9.22564i 2.39112 1.73725i 2.74483
511.1 −0.848198 2.61048i 0.170505 + 0.123879i −4.47716 + 3.25284i −0.309017 + 0.951057i 0.178763 0.550175i 1.87960 1.36561i 7.84779 + 5.70176i −0.913325 2.81093i 2.74483
511.2 0.431367 + 1.32761i 1.33928 + 0.973045i 0.0415626 0.0301970i −0.309017 + 0.951057i −0.714103 + 2.19778i −3.80789 + 2.76660i 2.31668 + 1.68317i −0.0801932 0.246809i −1.39593
511.3 0.725848 + 2.23393i −2.31880 1.68471i −2.84556 + 2.06742i −0.309017 + 0.951057i 2.08042 6.40289i 2.73731 1.98877i −2.88333 2.09486i 1.61155 + 4.95985i −2.34889
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 511.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.g.o 12
11.b odd 2 1 605.2.g.p 12
11.c even 5 1 605.2.a.h yes 3
11.c even 5 3 inner 605.2.g.o 12
11.d odd 10 1 605.2.a.g 3
11.d odd 10 3 605.2.g.p 12
33.f even 10 1 5445.2.a.bd 3
33.h odd 10 1 5445.2.a.bb 3
44.g even 10 1 9680.2.a.bz 3
44.h odd 10 1 9680.2.a.cb 3
55.h odd 10 1 3025.2.a.u 3
55.j even 10 1 3025.2.a.p 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.g 3 11.d odd 10 1
605.2.a.h yes 3 11.c even 5 1
605.2.g.o 12 1.a even 1 1 trivial
605.2.g.o 12 11.c even 5 3 inner
605.2.g.p 12 11.b odd 2 1
605.2.g.p 12 11.d odd 10 3
3025.2.a.p 3 55.j even 10 1
3025.2.a.u 3 55.h odd 10 1
5445.2.a.bb 3 33.h odd 10 1
5445.2.a.bd 3 33.f even 10 1
9680.2.a.bz 3 44.g even 10 1
9680.2.a.cb 3 44.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$:

 $$T_{2}^{12} + \cdots$$ $$T_{3}^{12} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$6561 + 5103 T + 4698 T^{2} + 3492 T^{3} + 2671 T^{4} + 656 T^{5} + 419 T^{6} + 102 T^{7} + 53 T^{8} + 6 T^{9} + 8 T^{10} + T^{11} + T^{12}$$
$3$ $$1 - 5 T + 24 T^{2} - 114 T^{3} + 541 T^{4} - 162 T^{5} + 155 T^{6} - 72 T^{7} + 43 T^{8} + 12 T^{9} + 6 T^{10} + T^{11} + T^{12}$$
$5$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{3}$$
$7$ $$1874161 - 962407 T + 544862 T^{2} - 255152 T^{3} + 119739 T^{4} - 22556 T^{5} + 7923 T^{6} - 1442 T^{7} + 345 T^{8} - 2 T^{9} + 20 T^{10} - T^{11} + T^{12}$$
$11$ $$T^{12}$$
$13$ $$256 - 256 T + 640 T^{2} - 1088 T^{3} + 2112 T^{4} + 1952 T^{5} + 1488 T^{6} + 912 T^{7} + 832 T^{8} + 164 T^{9} + 32 T^{10} + 6 T^{11} + T^{12}$$
$17$ $$5308416 + 1769472 T + 1032192 T^{2} + 380928 T^{3} + 176128 T^{4} + 11264 T^{5} + 10496 T^{6} + 768 T^{7} + 896 T^{8} + 144 T^{9} + 32 T^{10} + 4 T^{11} + T^{12}$$
$19$ $$1679616 + 559872 T + 373248 T^{2} + 139968 T^{3} + 72576 T^{4} + 5184 T^{5} + 5904 T^{6} + 528 T^{7} + 688 T^{8} + 124 T^{9} + 28 T^{10} + 4 T^{11} + T^{12}$$
$23$ $$( -12 + 4 T + 6 T^{2} + T^{3} )^{4}$$
$29$ $$26873856 + 10450944 T + 4810752 T^{2} + 1787904 T^{3} + 683776 T^{4} + 83968 T^{5} + 26816 T^{6} + 3264 T^{7} + 848 T^{8} + 48 T^{9} + 32 T^{10} + 2 T^{11} + T^{12}$$
$31$ $$1679616 + 2239488 T + 2332800 T^{2} + 2286144 T^{3} + 2203200 T^{4} + 759456 T^{5} + 219312 T^{6} + 58272 T^{7} + 13504 T^{8} + 1436 T^{9} + 148 T^{10} + 14 T^{11} + T^{12}$$
$37$ $$65536 - 98304 T + 131072 T^{2} - 167936 T^{3} + 212992 T^{4} - 58368 T^{5} + 23808 T^{6} - 7808 T^{7} + 2112 T^{8} + 272 T^{9} + 40 T^{10} + 4 T^{11} + T^{12}$$
$41$ $$7780827681 + 1178913285 T + 414405882 T^{2} + 72315342 T^{3} + 19545219 T^{4} + 599238 T^{5} + 439587 T^{6} + 18954 T^{7} + 14175 T^{8} + 1242 T^{9} + 126 T^{10} + 9 T^{11} + T^{12}$$
$43$ $$( 63 - 3 T - 7 T^{2} + T^{3} )^{4}$$
$47$ $$411651843201 - 14903749629 T + 8248422456 T^{2} - 63806058 T^{3} + 138168493 T^{4} - 19400442 T^{5} + 3210155 T^{6} - 307032 T^{7} + 47011 T^{8} - 3444 T^{9} + 254 T^{10} - 15 T^{11} + T^{12}$$
$53$ $$20736 + 6912 T + 12672 T^{2} + 9408 T^{3} + 10048 T^{4} - 6816 T^{5} + 3536 T^{6} - 1392 T^{7} + 736 T^{8} - 156 T^{9} + 32 T^{10} - 6 T^{11} + T^{12}$$
$59$ $$14666178816 + 2191497984 T + 748907136 T^{2} + 132735552 T^{3} + 35056960 T^{4} + 489824 T^{5} + 699152 T^{6} + 17808 T^{7} + 21344 T^{8} + 1692 T^{9} + 152 T^{10} + 10 T^{11} + T^{12}$$
$61$ $$713283282721 + 124960400999 T + 24220320358 T^{2} + 3874939606 T^{3} + 621943239 T^{4} + 40060742 T^{5} + 4832067 T^{6} + 300546 T^{7} + 24835 T^{8} + 74 T^{9} + 170 T^{10} + 3 T^{11} + T^{12}$$
$67$ $$( -59 + 95 T - 19 T^{2} + T^{3} )^{4}$$
$71$ $$20736 - 6912 T + 12672 T^{2} - 9408 T^{3} + 10048 T^{4} + 6816 T^{5} + 3536 T^{6} + 1392 T^{7} + 736 T^{8} + 156 T^{9} + 32 T^{10} + 6 T^{11} + T^{12}$$
$73$ $$1048576 + 524288 T + 655360 T^{2} + 557056 T^{3} + 540672 T^{4} - 249856 T^{5} + 95232 T^{6} - 29184 T^{7} + 13312 T^{8} - 1312 T^{9} + 128 T^{10} - 12 T^{11} + T^{12}$$
$79$ $$7676563456 + 1971009536 T + 557938688 T^{2} + 130637824 T^{3} + 30653184 T^{4} + 2887168 T^{5} + 507072 T^{6} + 46144 T^{7} + 5520 T^{8} + 16 T^{9} + 80 T^{10} + 2 T^{11} + T^{12}$$
$83$ $$11019960576 - 1224440064 T + 748268928 T^{2} - 185177664 T^{3} + 65924928 T^{4} + 14906592 T^{5} + 2577744 T^{6} + 338256 T^{7} + 59616 T^{8} + 4212 T^{9} + 288 T^{10} + 18 T^{11} + T^{12}$$
$89$ $$( 1719 - 157 T - 11 T^{2} + T^{3} )^{4}$$
$97$ $$754507653376 + 184579125504 T + 43535434880 T^{2} + 9444649408 T^{3} + 2019023680 T^{4} + 113691168 T^{5} + 13346448 T^{6} + 854704 T^{7} + 58464 T^{8} - 1852 T^{9} + 232 T^{10} - 2 T^{11} + T^{12}$$