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

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

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:

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

Atkin-Lehner signs

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

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

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 32 T^{2} \)
$3$ \( 1 + 4 T + 243 T^{2} \)
$5$ \( 1 - 25 T \)
$7$ \( 1 + 192 T + 16807 T^{2} \)
$11$ 1
$13$ \( 1 + 286 T + 371293 T^{2} \)
$17$ \( 1 - 1678 T + 1419857 T^{2} \)
$19$ \( 1 + 1060 T + 2476099 T^{2} \)
$23$ \( 1 - 2976 T + 6436343 T^{2} \)
$29$ \( 1 - 3410 T + 20511149 T^{2} \)
$31$ \( 1 + 2448 T + 28629151 T^{2} \)
$37$ \( 1 - 182 T + 69343957 T^{2} \)
$41$ \( 1 - 9398 T + 115856201 T^{2} \)
$43$ \( 1 - 1244 T + 147008443 T^{2} \)
$47$ \( 1 + 12088 T + 229345007 T^{2} \)
$53$ \( 1 - 23846 T + 418195493 T^{2} \)
$59$ \( 1 + 20020 T + 714924299 T^{2} \)
$61$ \( 1 + 32302 T + 844596301 T^{2} \)
$67$ \( 1 - 60972 T + 1350125107 T^{2} \)
$71$ \( 1 + 32648 T + 1804229351 T^{2} \)
$73$ \( 1 - 38774 T + 2073071593 T^{2} \)
$79$ \( 1 - 33360 T + 3077056399 T^{2} \)
$83$ \( 1 + 16716 T + 3939040643 T^{2} \)
$89$ \( 1 - 101370 T + 5584059449 T^{2} \)
$97$ \( 1 + 119038 T + 8587340257 T^{2} \)
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