Properties

Label 605.2.a.g
Level $605$
Weight $2$
Character orbit 605.a
Self dual yes
Analytic conductor $4.831$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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: \(3\)
Coefficient field: 3.3.404.1
Defining polynomial: \(x^{3} - x^{2} - 5 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 5 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)

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
2.86620
−1.65544
−0.210756
−2.34889 2.86620 3.51730 −1.00000 −6.73240 3.38350 −3.56399 5.21509 2.34889
1.2 −1.39593 −1.65544 −0.0513742 −1.00000 2.31088 −4.70682 2.86358 −0.259511 1.39593
1.3 2.74483 −0.210756 5.53407 −1.00000 −0.578488 2.32331 9.70041 −2.95558 −2.74483
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.a.g 3
3.b odd 2 1 5445.2.a.bd 3
4.b odd 2 1 9680.2.a.bz 3
5.b even 2 1 3025.2.a.u 3
11.b odd 2 1 605.2.a.h yes 3
11.c even 5 4 605.2.g.p 12
11.d odd 10 4 605.2.g.o 12
33.d even 2 1 5445.2.a.bb 3
44.c even 2 1 9680.2.a.cb 3
55.d odd 2 1 3025.2.a.p 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.g 3 1.a even 1 1 trivial
605.2.a.h yes 3 11.b odd 2 1
605.2.g.o 12 11.d odd 10 4
605.2.g.p 12 11.c even 5 4
3025.2.a.p 3 55.d odd 2 1
3025.2.a.u 3 5.b even 2 1
5445.2.a.bb 3 33.d even 2 1
5445.2.a.bd 3 3.b odd 2 1
9680.2.a.bz 3 4.b odd 2 1
9680.2.a.cb 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(605))\):

\( T_{2}^{3} + T_{2}^{2} - 7 T_{2} - 9 \)
\( T_{3}^{3} - T_{3}^{2} - 5 T_{3} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -9 - 7 T + T^{2} + T^{3} \)
$3$ \( -1 - 5 T - T^{2} + T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( 37 - 19 T - T^{2} + T^{3} \)
$11$ \( T^{3} \)
$13$ \( -4 + 4 T + 6 T^{2} + T^{3} \)
$17$ \( -48 - 16 T + 4 T^{2} + T^{3} \)
$19$ \( -36 - 12 T + 4 T^{2} + T^{3} \)
$23$ \( -12 + 4 T + 6 T^{2} + T^{3} \)
$29$ \( -72 - 28 T + 2 T^{2} + T^{3} \)
$31$ \( -36 + 48 T - 14 T^{2} + T^{3} \)
$37$ \( -16 - 24 T - 4 T^{2} + T^{3} \)
$41$ \( -297 - 45 T + 9 T^{2} + T^{3} \)
$43$ \( -63 - 3 T + 7 T^{2} + T^{3} \)
$47$ \( -801 - 29 T + 15 T^{2} + T^{3} \)
$53$ \( -12 + 4 T + 6 T^{2} + T^{3} \)
$59$ \( 348 - 52 T - 10 T^{2} + T^{3} \)
$61$ \( -919 - 161 T + 3 T^{2} + T^{3} \)
$67$ \( -59 + 95 T - 19 T^{2} + T^{3} \)
$71$ \( 12 + 4 T - 6 T^{2} + T^{3} \)
$73$ \( 32 + 16 T - 12 T^{2} + T^{3} \)
$79$ \( -296 - 76 T + 2 T^{2} + T^{3} \)
$83$ \( -324 + 36 T + 18 T^{2} + T^{3} \)
$89$ \( 1719 - 157 T - 11 T^{2} + T^{3} \)
$97$ \( 932 - 228 T + 2 T^{2} + T^{3} \)
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