# Properties

 Label 605.2.g.g Level $605$ Weight $2$ Character orbit 605.g Analytic conductor $4.831$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# 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: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.159390625.1 Defining polynomial: $$x^{8} - x^{7} + 6 x^{6} - 11 x^{5} + 21 x^{4} - 5 x^{3} + 10 x^{2} + 25 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 6 x^{6} - 11 x^{5} + 21 x^{4} - 5 x^{3} + 10 x^{2} + 25 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$555 \nu^{7} - 2159 \nu^{6} + 7489 \nu^{5} - 18164 \nu^{4} + 40069 \nu^{3} - 84434 \nu^{2} + 43855 \nu + 375$$$$)/94655$$ $$\beta_{3}$$ $$=$$ $$($$$$-970 \nu^{7} - 1002 \nu^{6} - 6608 \nu^{5} + 9063 \nu^{4} - 14943 \nu^{3} + 27673 \nu^{2} - 68120 \nu + 35160$$$$)/94655$$ $$\beta_{4}$$ $$=$$ $$($$$$-1604 \nu^{7} + 4159 \nu^{6} - 12059 \nu^{5} + 28414 \nu^{4} - 81659 \nu^{3} + 38305 \nu^{2} - 13500 \nu - 13875$$$$)/94655$$ $$\beta_{5}$$ $$=$$ $$($$$$-2052 \nu^{7} + 2252 \nu^{6} - 19912 \nu^{5} + 21007 \nu^{4} - 82042 \nu^{3} + 35785 \nu^{2} - 19395 \nu - 90925$$$$)/94655$$ $$\beta_{6}$$ $$=$$ $$($$$$-2667 \nu^{7} + 6691 \nu^{6} - 17466 \nu^{5} + 50856 \nu^{4} - 82441 \nu^{3} + 72554 \nu^{2} - 4035 \nu - 12035$$$$)/94655$$ $$\beta_{7}$$ $$=$$ $$($$$$4024 \nu^{7} - 1464 \nu^{6} + 21519 \nu^{5} - 26434 \nu^{4} + 59219 \nu^{3} + 22635 \nu^{2} + 54640 \nu + 66675$$$$)/94655$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 3 \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{6} + \beta_{5} - 4 \beta_{4} - \beta_{3} - \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$7 \beta_{7} + 7 \beta_{6} + 2 \beta_{5} + 13 \beta_{3} + 13 \beta_{2} - 7$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{7} - 11 \beta_{6} - 20 \beta_{5} + 20 \beta_{4} - 11 \beta_{2} + 8 \beta_{1} - 12$$ $$\nu^{6}$$ $$=$$ $$-19 \beta_{7} - 19 \beta_{4} - 68 \beta_{3} - 36 \beta_{2} - 24 \beta_{1} + 36$$ $$\nu^{7}$$ $$=$$ $$111 \beta_{7} + 81 \beta_{6} + 111 \beta_{5} - 55 \beta_{4} + 81 \beta_{3} + 148 \beta_{2} - 56 \beta_{1}$$

## Character values

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

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$1$$ $$-\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 1.43801 − 1.04478i −0.628998 + 0.456994i −0.762262 − 2.34600i 0.453245 + 1.39494i 1.43801 + 1.04478i −0.628998 − 0.456994i −0.762262 + 2.34600i 0.453245 − 1.39494i
−1.51774 1.10270i −0.549273 + 1.69049i 0.469548 + 1.44512i −0.809017 + 0.587785i 2.69776 1.96004i −1.31579 4.04959i −0.278565 + 0.857334i −0.128998 0.0937225i 1.87603
81.2 1.82676 + 1.32722i 0.240256 0.739431i 0.957503 + 2.94689i −0.809017 + 0.587785i 1.42027 1.03189i −0.0383089 0.117903i −0.766520 + 2.35911i 1.93801 + 1.40805i −2.25800
251.1 −0.780121 + 2.40097i 1.99563 1.44991i −3.53801 2.57052i 0.309017 + 0.951057i 1.92435 + 5.92254i 1.69369 + 1.23053i 4.84703 3.52157i 0.953245 2.93379i −2.52452
251.2 −0.0288961 + 0.0889332i −1.18661 + 0.862123i 1.61096 + 1.17043i 0.309017 + 0.951057i −0.0423829 0.130441i 3.66042 + 2.65945i −0.301943 + 0.219374i −0.262262 + 0.807160i −0.0935099
366.1 −1.51774 + 1.10270i −0.549273 1.69049i 0.469548 1.44512i −0.809017 0.587785i 2.69776 + 1.96004i −1.31579 + 4.04959i −0.278565 0.857334i −0.128998 + 0.0937225i 1.87603
366.2 1.82676 1.32722i 0.240256 + 0.739431i 0.957503 2.94689i −0.809017 0.587785i 1.42027 + 1.03189i −0.0383089 + 0.117903i −0.766520 2.35911i 1.93801 1.40805i −2.25800
511.1 −0.780121 2.40097i 1.99563 + 1.44991i −3.53801 + 2.57052i 0.309017 0.951057i 1.92435 5.92254i 1.69369 1.23053i 4.84703 + 3.52157i 0.953245 + 2.93379i −2.52452
511.2 −0.0288961 0.0889332i −1.18661 0.862123i 1.61096 1.17043i 0.309017 0.951057i −0.0423829 + 0.130441i 3.66042 2.65945i −0.301943 0.219374i −0.262262 0.807160i −0.0935099
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 511.2 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 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.g.g 8
11.b odd 2 1 605.2.g.j 8
11.c even 5 1 605.2.a.i 4
11.c even 5 1 inner 605.2.g.g 8
11.c even 5 2 605.2.g.n 8
11.d odd 10 2 55.2.g.a 8
11.d odd 10 1 605.2.a.l 4
11.d odd 10 1 605.2.g.j 8
33.f even 10 2 495.2.n.f 8
33.f even 10 1 5445.2.a.bg 4
33.h odd 10 1 5445.2.a.bu 4
44.g even 10 2 880.2.bo.e 8
44.g even 10 1 9680.2.a.cs 4
44.h odd 10 1 9680.2.a.cv 4
55.h odd 10 2 275.2.h.b 8
55.h odd 10 1 3025.2.a.v 4
55.j even 10 1 3025.2.a.be 4
55.l even 20 4 275.2.z.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.g.a 8 11.d odd 10 2
275.2.h.b 8 55.h odd 10 2
275.2.z.b 16 55.l even 20 4
495.2.n.f 8 33.f even 10 2
605.2.a.i 4 11.c even 5 1
605.2.a.l 4 11.d odd 10 1
605.2.g.g 8 1.a even 1 1 trivial
605.2.g.g 8 11.c even 5 1 inner
605.2.g.j 8 11.b odd 2 1
605.2.g.j 8 11.d odd 10 1
605.2.g.n 8 11.c even 5 2
880.2.bo.e 8 44.g even 10 2
3025.2.a.v 4 55.h odd 10 1
3025.2.a.be 4 55.j even 10 1
5445.2.a.bg 4 33.f even 10 1
5445.2.a.bu 4 33.h odd 10 1
9680.2.a.cs 4 44.g even 10 1
9680.2.a.cv 4 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 $$S_{2}^{\mathrm{new}}(605, [\chi])$$:

 $$T_{2}^{8} + \cdots$$ $$T_{3}^{8} - T_{3}^{7} + T_{3}^{6} - T_{3}^{5} + 16 T_{3}^{4} + 25 T_{3}^{3} + 35 T_{3}^{2} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 7 T + 117 T^{2} + 45 T^{3} + 6 T^{4} - 5 T^{5} + 3 T^{6} + T^{7} + T^{8}$$
$3$ $$25 + 35 T^{2} + 25 T^{3} + 16 T^{4} - T^{5} + T^{6} - T^{7} + T^{8}$$
$5$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$7$ $$25 + 100 T + 1515 T^{2} - 1550 T^{3} + 711 T^{4} - 162 T^{5} + 39 T^{6} - 8 T^{7} + T^{8}$$
$11$ $$T^{8}$$
$13$ $$121 - 187 T + 257 T^{2} - 225 T^{3} + 176 T^{4} - 105 T^{5} + 53 T^{6} - 11 T^{7} + T^{8}$$
$17$ $$121 - 473 T + 797 T^{2} - 525 T^{3} + 236 T^{4} - 75 T^{5} + 23 T^{6} - 4 T^{7} + T^{8}$$
$19$ $$625 + 125 T + 1400 T^{2} - 1245 T^{3} + 551 T^{4} - 141 T^{5} + 56 T^{6} - 11 T^{7} + T^{8}$$
$23$ $$( -1669 - 706 T - 54 T^{2} + 9 T^{3} + T^{4} )^{2}$$
$29$ $$3025 + 5775 T + 3140 T^{2} - 3685 T^{3} + 1891 T^{4} - 521 T^{5} + 106 T^{6} - 11 T^{7} + T^{8}$$
$31$ $$10201 - 6969 T + 4857 T^{2} - 1197 T^{3} + 280 T^{4} - 3 T^{5} + 37 T^{6} + 9 T^{7} + T^{8}$$
$37$ $$22801 + 4379 T - 4048 T^{2} - 1503 T^{3} + 2855 T^{4} - 147 T^{5} + 92 T^{6} + T^{7} + T^{8}$$
$41$ $$249001 + 309879 T + 171872 T^{2} + 43897 T^{3} + 8005 T^{4} + 1103 T^{5} + 142 T^{6} + 11 T^{7} + T^{8}$$
$43$ $$( -59 + 191 T + 121 T^{2} + 21 T^{3} + T^{4} )^{2}$$
$47$ $$5041 - 5751 T + 23227 T^{2} + 4497 T^{3} + 2330 T^{4} - 27 T^{5} - 23 T^{6} + T^{7} + T^{8}$$
$53$ $$1 + 46 T + 813 T^{2} + 340 T^{3} + 441 T^{4} + 27 T^{6} - 8 T^{7} + T^{8}$$
$59$ $$9150625 - 3811500 T + 968225 T^{2} - 197890 T^{3} + 36451 T^{4} - 4886 T^{5} + 451 T^{6} - 26 T^{7} + T^{8}$$
$61$ $$3025 - 2750 T + 11290 T^{2} + 3250 T^{3} + 1956 T^{4} + 92 T^{5} - 21 T^{6} - 2 T^{7} + T^{8}$$
$67$ $$( -101 - 238 T - 82 T^{2} + T^{3} + T^{4} )^{2}$$
$71$ $$60824401 - 3938495 T + 130344 T^{2} + 28175 T^{3} + 30501 T^{4} + 4345 T^{5} + 454 T^{6} + 25 T^{7} + T^{8}$$
$73$ $$151321 - 53293 T + 243573 T^{2} + 132229 T^{3} + 34230 T^{4} + 5159 T^{5} + 523 T^{6} + 32 T^{7} + T^{8}$$
$79$ $$4644025 - 2241200 T + 619935 T^{2} - 130605 T^{3} + 25116 T^{4} - 3647 T^{5} + 369 T^{6} - 23 T^{7} + T^{8}$$
$83$ $$841 - 580 T - 709 T^{2} + 1080 T^{3} + 2561 T^{4} + 540 T^{5} + 161 T^{6} + 10 T^{7} + T^{8}$$
$89$ $$( 725 - 400 T - 150 T^{2} + T^{4} )^{2}$$
$97$ $$625 + 750 T + 4525 T^{2} - 2670 T^{3} + 691 T^{4} + 42 T^{5} + 129 T^{6} + 18 T^{7} + T^{8}$$