Properties

Label 605.6.a.a
Level $605$
Weight $6$
Character orbit 605.a
Self dual yes
Analytic conductor $97.032$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 605.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(97.0322109869\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{2} - 4 q^{3} - 28 q^{4} + 25 q^{5} + 8 q^{6} - 192 q^{7} + 120 q^{8} - 227 q^{9} + O(q^{10}) \) \( q - 2 q^{2} - 4 q^{3} - 28 q^{4} + 25 q^{5} + 8 q^{6} - 192 q^{7} + 120 q^{8} - 227 q^{9} - 50 q^{10} + 112 q^{12} - 286 q^{13} + 384 q^{14} - 100 q^{15} + 656 q^{16} + 1678 q^{17} + 454 q^{18} - 1060 q^{19} - 700 q^{20} + 768 q^{21} + 2976 q^{23} - 480 q^{24} + 625 q^{25} + 572 q^{26} + 1880 q^{27} + 5376 q^{28} + 3410 q^{29} + 200 q^{30} - 2448 q^{31} - 5152 q^{32} - 3356 q^{34} - 4800 q^{35} + 6356 q^{36} + 182 q^{37} + 2120 q^{38} + 1144 q^{39} + 3000 q^{40} + 9398 q^{41} - 1536 q^{42} + 1244 q^{43} - 5675 q^{45} - 5952 q^{46} - 12088 q^{47} - 2624 q^{48} + 20057 q^{49} - 1250 q^{50} - 6712 q^{51} + 8008 q^{52} + 23846 q^{53} - 3760 q^{54} - 23040 q^{56} + 4240 q^{57} - 6820 q^{58} - 20020 q^{59} + 2800 q^{60} - 32302 q^{61} + 4896 q^{62} + 43584 q^{63} - 10688 q^{64} - 7150 q^{65} + 60972 q^{67} - 46984 q^{68} - 11904 q^{69} + 9600 q^{70} - 32648 q^{71} - 27240 q^{72} + 38774 q^{73} - 364 q^{74} - 2500 q^{75} + 29680 q^{76} - 2288 q^{78} + 33360 q^{79} + 16400 q^{80} + 47641 q^{81} - 18796 q^{82} - 16716 q^{83} - 21504 q^{84} + 41950 q^{85} - 2488 q^{86} - 13640 q^{87} + 101370 q^{89} + 11350 q^{90} + 54912 q^{91} - 83328 q^{92} + 9792 q^{93} + 24176 q^{94} - 26500 q^{95} + 20608 q^{96} - 119038 q^{97} - 40114 q^{98} + O(q^{100}) \)

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
0
−2.00000 −4.00000 −28.0000 25.0000 8.00000 −192.000 120.000 −227.000 −50.0000
\(n\): 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 605.6.a.a 1
11.b odd 2 1 5.6.a.a 1
33.d even 2 1 45.6.a.b 1
44.c even 2 1 80.6.a.e 1
55.d odd 2 1 25.6.a.a 1
55.e even 4 2 25.6.b.a 2
77.b even 2 1 245.6.a.b 1
88.b odd 2 1 320.6.a.j 1
88.g even 2 1 320.6.a.g 1
132.d odd 2 1 720.6.a.a 1
143.d odd 2 1 845.6.a.b 1
165.d even 2 1 225.6.a.f 1
165.l odd 4 2 225.6.b.e 2
220.g even 2 1 400.6.a.g 1
220.i odd 4 2 400.6.c.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.a.a 1 11.b odd 2 1
25.6.a.a 1 55.d odd 2 1
25.6.b.a 2 55.e even 4 2
45.6.a.b 1 33.d even 2 1
80.6.a.e 1 44.c even 2 1
225.6.a.f 1 165.d even 2 1
225.6.b.e 2 165.l odd 4 2
245.6.a.b 1 77.b even 2 1
320.6.a.g 1 88.g even 2 1
320.6.a.j 1 88.b odd 2 1
400.6.a.g 1 220.g even 2 1
400.6.c.j 2 220.i odd 4 2
605.6.a.a 1 1.a even 1 1 trivial
720.6.a.a 1 132.d odd 2 1
845.6.a.b 1 143.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 2 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(605))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( 4 + T \)
$5$ \( -25 + T \)
$7$ \( 192 + T \)
$11$ \( T \)
$13$ \( 286 + T \)
$17$ \( -1678 + T \)
$19$ \( 1060 + T \)
$23$ \( -2976 + T \)
$29$ \( -3410 + T \)
$31$ \( 2448 + T \)
$37$ \( -182 + T \)
$41$ \( -9398 + T \)
$43$ \( -1244 + T \)
$47$ \( 12088 + T \)
$53$ \( -23846 + T \)
$59$ \( 20020 + T \)
$61$ \( 32302 + T \)
$67$ \( -60972 + T \)
$71$ \( 32648 + T \)
$73$ \( -38774 + T \)
$79$ \( -33360 + T \)
$83$ \( 16716 + T \)
$89$ \( -101370 + T \)
$97$ \( 119038 + T \)
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