Properties

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

Related objects

Downloads

Learn more about

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: \(\Z[a_1, a_2, a_3]\)
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.

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

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 \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2} + \beta_{3} + \beta_{6}\)

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
0.453245 1.39494i
−0.762262 + 2.34600i
−0.628998 + 0.456994i
1.43801 1.04478i
0.453245 + 1.39494i
−0.762262 2.34600i
−0.628998 0.456994i
1.43801 + 1.04478i
0.0756511 + 0.0549637i 0.453245 1.39494i −0.615332 1.89380i −0.809017 + 0.587785i 0.110960 0.0806171i −1.39815 4.30308i 0.115332 0.354955i 0.686611 + 0.498852i −0.0935099
81.2 2.04238 + 1.48388i −0.762262 + 2.34600i 1.35140 + 4.15918i −0.809017 + 0.587785i −5.03801 + 3.66033i −0.646930 1.99105i −1.85140 + 5.69802i −2.49563 1.81318i −2.52452
251.1 −0.697759 + 2.14748i −0.628998 + 0.456994i −2.50678 1.82128i 0.309017 + 0.951057i −0.542497 1.66963i 0.100294 + 0.0728678i 2.00678 1.45801i −0.740256 + 2.27827i −2.25800
251.2 0.579725 1.78421i 1.43801 1.04478i −1.22929 0.893133i 0.309017 + 0.951057i −1.03045 3.17141i 3.44479 + 2.50279i 0.729292 0.529862i 0.0492728 0.151646i 1.87603
366.1 0.0756511 0.0549637i 0.453245 + 1.39494i −0.615332 + 1.89380i −0.809017 0.587785i 0.110960 + 0.0806171i −1.39815 + 4.30308i 0.115332 + 0.354955i 0.686611 0.498852i −0.0935099
366.2 2.04238 1.48388i −0.762262 2.34600i 1.35140 4.15918i −0.809017 0.587785i −5.03801 3.66033i −0.646930 + 1.99105i −1.85140 5.69802i −2.49563 + 1.81318i −2.52452
511.1 −0.697759 2.14748i −0.628998 0.456994i −2.50678 + 1.82128i 0.309017 0.951057i −0.542497 + 1.66963i 0.100294 0.0728678i 2.00678 + 1.45801i −0.740256 2.27827i −2.25800
511.2 0.579725 + 1.78421i 1.43801 + 1.04478i −1.22929 + 0.893133i 0.309017 0.951057i −1.03045 + 3.17141i 3.44479 2.50279i 0.729292 + 0.529862i 0.0492728 + 0.151646i 1.87603
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 511.2
Significant digits:
Format:

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

\(T_{2}^{8} - \cdots\)
\(T_{3}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 18 T + 127 T^{2} - 90 T^{3} + 71 T^{4} - 30 T^{5} + 13 T^{6} - 4 T^{7} + T^{8} \)
$3$ \( 25 + 25 T + 10 T^{2} - 5 T^{3} + 21 T^{4} - 11 T^{5} + 6 T^{6} - T^{7} + T^{8} \)
$5$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$7$ \( 25 - 325 T + 1615 T^{2} + 15 T^{3} + 356 T^{4} - 87 T^{5} + 19 T^{6} - 3 T^{7} + T^{8} \)
$11$ \( T^{8} \)
$13$ \( 121 + 88 T + 37 T^{2} - 20 T^{3} + 21 T^{4} + 20 T^{5} + 33 T^{6} + 4 T^{7} + T^{8} \)
$17$ \( 121 + 187 T + 522 T^{2} + 185 T^{3} + 81 T^{4} - 5 T^{5} - 2 T^{6} + T^{7} + T^{8} \)
$19$ \( 625 - 1750 T + 2025 T^{2} - 545 T^{3} + 426 T^{4} - 151 T^{5} + 31 T^{6} - T^{7} + T^{8} \)
$23$ \( ( -1669 - 706 T - 54 T^{2} + 9 T^{3} + T^{4} )^{2} \)
$29$ \( 3025 - 550 T + 6715 T^{2} + 5825 T^{3} + 2856 T^{4} + 869 T^{5} + 171 T^{6} + 19 T^{7} + T^{8} \)
$31$ \( 10201 + 6666 T + 2837 T^{2} + 78 T^{3} + 55 T^{4} - 78 T^{5} + 57 T^{6} - 6 T^{7} + T^{8} \)
$37$ \( 22801 - 151 T + 13317 T^{2} + 1747 T^{3} + 2880 T^{4} + 203 T^{5} - 23 T^{6} - 4 T^{7} + T^{8} \)
$41$ \( 249001 - 119261 T + 99517 T^{2} - 12353 T^{3} + 2130 T^{4} + 103 T^{5} - 3 T^{6} - 4 T^{7} + T^{8} \)
$43$ \( ( -59 + 191 T + 121 T^{2} + 21 T^{3} + T^{4} )^{2} \)
$47$ \( 5041 + 15194 T + 15417 T^{2} - 10428 T^{3} + 3105 T^{4} - 352 T^{5} + 87 T^{6} - 4 T^{7} + T^{8} \)
$53$ \( 1 - 19 T + 823 T^{2} - 1275 T^{3} + 866 T^{4} - 255 T^{5} + 37 T^{6} - 3 T^{7} + T^{8} \)
$59$ \( 9150625 + 196625 T + 211975 T^{2} - 17065 T^{3} + 4926 T^{4} + 869 T^{5} + 201 T^{6} + 19 T^{7} + T^{8} \)
$61$ \( 3025 + 7700 T + 6065 T^{2} - 6060 T^{3} + 2336 T^{4} - 288 T^{5} + 74 T^{6} - 2 T^{7} + T^{8} \)
$67$ \( ( -101 - 238 T - 82 T^{2} + T^{3} + T^{4} )^{2} \)
$71$ \( 60824401 - 13765235 T + 3366929 T^{2} - 718575 T^{3} + 113726 T^{4} - 11955 T^{5} + 869 T^{6} - 40 T^{7} + T^{8} \)
$73$ \( 151321 + 285137 T + 183278 T^{2} - 74121 T^{3} + 21755 T^{4} - 3441 T^{5} + 368 T^{6} - 23 T^{7} + T^{8} \)
$79$ \( 4644025 + 226275 T + 156610 T^{2} - 10875 T^{3} + 2981 T^{4} + 483 T^{5} + 154 T^{6} + 17 T^{7} + T^{8} \)
$83$ \( 841 - 145 T + 2481 T^{2} - 1745 T^{3} + 2886 T^{4} - 1285 T^{5} + 271 T^{6} - 25 T^{7} + T^{8} \)
$89$ \( ( 725 - 400 T - 150 T^{2} + T^{4} )^{2} \)
$97$ \( 625 - 3000 T + 5775 T^{2} - 1770 T^{3} + 1591 T^{4} - 738 T^{5} + 179 T^{6} - 12 T^{7} + T^{8} \)
show more
show less