Properties

Label 605.2.a.m
Level $605$
Weight $2$
Character orbit 605.a
Self dual yes
Analytic conductor $4.831$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.27433728.1
Defining polynomial: \(x^{6} - 9 x^{4} + 15 x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 9 x^{4} + 15 x^{2} - 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - 8 \nu^{2} + 7 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - 8 \nu^{3} + 9 \nu \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} + 10 \nu^{3} - 19 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(2 \beta_{3} + 8 \beta_{2} + 17\)
\(\nu^{5}\)\(=\)\(8 \beta_{5} + 10 \beta_{4} + 31 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.62383
−1.37268
−0.480901
0.480901
1.37268
2.62383
−2.62383 1.33988 4.88448 1.00000 −3.51561 2.62383 −7.56839 −1.20473 −2.62383
1.2 −1.37268 3.26180 −0.115749 1.00000 −4.47741 1.37268 2.90425 7.63935 −1.37268
1.3 −0.480901 −1.60168 −1.76873 1.00000 0.770249 0.480901 1.81239 −0.434624 −0.480901
1.4 0.480901 −1.60168 −1.76873 1.00000 −0.770249 −0.480901 −1.81239 −0.434624 0.480901
1.5 1.37268 3.26180 −0.115749 1.00000 4.47741 −1.37268 −2.90425 7.63935 1.37268
1.6 2.62383 1.33988 4.88448 1.00000 3.51561 −2.62383 7.56839 −1.20473 2.62383
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.a.m 6
3.b odd 2 1 5445.2.a.bx 6
4.b odd 2 1 9680.2.a.cw 6
5.b even 2 1 3025.2.a.bg 6
11.b odd 2 1 inner 605.2.a.m 6
11.c even 5 4 605.2.g.q 24
11.d odd 10 4 605.2.g.q 24
33.d even 2 1 5445.2.a.bx 6
44.c even 2 1 9680.2.a.cw 6
55.d odd 2 1 3025.2.a.bg 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.m 6 1.a even 1 1 trivial
605.2.a.m 6 11.b odd 2 1 inner
605.2.g.q 24 11.c even 5 4
605.2.g.q 24 11.d odd 10 4
3025.2.a.bg 6 5.b even 2 1
3025.2.a.bg 6 55.d odd 2 1
5445.2.a.bx 6 3.b odd 2 1
5445.2.a.bx 6 33.d even 2 1
9680.2.a.cw 6 4.b odd 2 1
9680.2.a.cw 6 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(605))\):

\( T_{2}^{6} - 9 T_{2}^{4} + 15 T_{2}^{2} - 3 \)
\( T_{3}^{3} - 3 T_{3}^{2} - 3 T_{3} + 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + 15 T^{2} - 9 T^{4} + T^{6} \)
$3$ \( ( 7 - 3 T - 3 T^{2} + T^{3} )^{2} \)
$5$ \( ( -1 + T )^{6} \)
$7$ \( -3 + 15 T^{2} - 9 T^{4} + T^{6} \)
$11$ \( T^{6} \)
$13$ \( -3888 + 1296 T^{2} - 72 T^{4} + T^{6} \)
$17$ \( -768 + 384 T^{2} - 48 T^{4} + T^{6} \)
$19$ \( -48 + 96 T^{2} - 36 T^{4} + T^{6} \)
$23$ \( ( 84 - 12 T - 6 T^{2} + T^{3} )^{2} \)
$29$ \( -6912 + 5184 T^{2} - 144 T^{4} + T^{6} \)
$31$ \( ( -124 - 48 T + T^{3} )^{2} \)
$37$ \( ( -16 - 24 T + T^{3} )^{2} \)
$41$ \( -7203 + 2499 T^{2} - 105 T^{4} + T^{6} \)
$43$ \( -43923 + 4695 T^{2} - 129 T^{4} + T^{6} \)
$47$ \( ( -303 + 141 T - 21 T^{2} + T^{3} )^{2} \)
$53$ \( ( 732 - 84 T - 12 T^{2} + T^{3} )^{2} \)
$59$ \( ( 12 - 12 T + T^{3} )^{2} \)
$61$ \( -1083 + 339 T^{2} - 33 T^{4} + T^{6} \)
$67$ \( ( -97 + 69 T - 15 T^{2} + T^{3} )^{2} \)
$71$ \( ( -12 - 12 T + T^{3} )^{2} \)
$73$ \( -49152 + 6144 T^{2} - 192 T^{4} + T^{6} \)
$79$ \( -101568 + 11184 T^{2} - 276 T^{4} + T^{6} \)
$83$ \( -40368 + 14640 T^{2} - 312 T^{4} + T^{6} \)
$89$ \( ( -147 - 21 T + 15 T^{2} + T^{3} )^{2} \)
$97$ \( ( 76 - 36 T - 12 T^{2} + T^{3} )^{2} \)
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