Properties

Label 845.6.a.b
Level 845
Weight 6
Character orbit 845.a
Self dual yes
Analytic conductor 135.524
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 845.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(135.524327742\)
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

\(f(q)\) \(=\) \( q - 2q^{2} - 4q^{3} - 28q^{4} - 25q^{5} + 8q^{6} - 192q^{7} + 120q^{8} - 227q^{9} + O(q^{10}) \) \( q - 2q^{2} - 4q^{3} - 28q^{4} - 25q^{5} + 8q^{6} - 192q^{7} + 120q^{8} - 227q^{9} + 50q^{10} + 148q^{11} + 112q^{12} + 384q^{14} + 100q^{15} + 656q^{16} - 1678q^{17} + 454q^{18} - 1060q^{19} + 700q^{20} + 768q^{21} - 296q^{22} + 2976q^{23} - 480q^{24} + 625q^{25} + 1880q^{27} + 5376q^{28} - 3410q^{29} - 200q^{30} + 2448q^{31} - 5152q^{32} - 592q^{33} + 3356q^{34} + 4800q^{35} + 6356q^{36} - 182q^{37} + 2120q^{38} - 3000q^{40} + 9398q^{41} - 1536q^{42} - 1244q^{43} - 4144q^{44} + 5675q^{45} - 5952q^{46} + 12088q^{47} - 2624q^{48} + 20057q^{49} - 1250q^{50} + 6712q^{51} + 23846q^{53} - 3760q^{54} - 3700q^{55} - 23040q^{56} + 4240q^{57} + 6820q^{58} + 20020q^{59} - 2800q^{60} + 32302q^{61} - 4896q^{62} + 43584q^{63} - 10688q^{64} + 1184q^{66} - 60972q^{67} + 46984q^{68} - 11904q^{69} - 9600q^{70} + 32648q^{71} - 27240q^{72} + 38774q^{73} + 364q^{74} - 2500q^{75} + 29680q^{76} - 28416q^{77} - 33360q^{79} - 16400q^{80} + 47641q^{81} - 18796q^{82} - 16716q^{83} - 21504q^{84} + 41950q^{85} + 2488q^{86} + 13640q^{87} + 17760q^{88} - 101370q^{89} - 11350q^{90} - 83328q^{92} - 9792q^{93} - 24176q^{94} + 26500q^{95} + 20608q^{96} + 119038q^{97} - 40114q^{98} - 33596q^{99} + O(q^{100}) \)

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 845.6.a.b 1
13.b even 2 1 5.6.a.a 1
39.d odd 2 1 45.6.a.b 1
52.b odd 2 1 80.6.a.e 1
65.d even 2 1 25.6.a.a 1
65.h odd 4 2 25.6.b.a 2
91.b odd 2 1 245.6.a.b 1
104.e even 2 1 320.6.a.j 1
104.h odd 2 1 320.6.a.g 1
143.d odd 2 1 605.6.a.a 1
156.h even 2 1 720.6.a.a 1
195.e odd 2 1 225.6.a.f 1
195.s even 4 2 225.6.b.e 2
260.g odd 2 1 400.6.a.g 1
260.p even 4 2 400.6.c.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.a.a 1 13.b even 2 1
25.6.a.a 1 65.d even 2 1
25.6.b.a 2 65.h odd 4 2
45.6.a.b 1 39.d odd 2 1
80.6.a.e 1 52.b odd 2 1
225.6.a.f 1 195.e odd 2 1
225.6.b.e 2 195.s even 4 2
245.6.a.b 1 91.b odd 2 1
320.6.a.g 1 104.h odd 2 1
320.6.a.j 1 104.e even 2 1
400.6.a.g 1 260.g odd 2 1
400.6.c.j 2 260.p even 4 2
605.6.a.a 1 143.d odd 2 1
720.6.a.a 1 156.h even 2 1
845.6.a.b 1 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(13\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 2 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(845))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 32 T^{2} \)
$3$ \( 1 + 4 T + 243 T^{2} \)
$5$ \( 1 + 25 T \)
$7$ \( 1 + 192 T + 16807 T^{2} \)
$11$ \( 1 - 148 T + 161051 T^{2} \)
$13$ 1
$17$ \( 1 + 1678 T + 1419857 T^{2} \)
$19$ \( 1 + 1060 T + 2476099 T^{2} \)
$23$ \( 1 - 2976 T + 6436343 T^{2} \)
$29$ \( 1 + 3410 T + 20511149 T^{2} \)
$31$ \( 1 - 2448 T + 28629151 T^{2} \)
$37$ \( 1 + 182 T + 69343957 T^{2} \)
$41$ \( 1 - 9398 T + 115856201 T^{2} \)
$43$ \( 1 + 1244 T + 147008443 T^{2} \)
$47$ \( 1 - 12088 T + 229345007 T^{2} \)
$53$ \( 1 - 23846 T + 418195493 T^{2} \)
$59$ \( 1 - 20020 T + 714924299 T^{2} \)
$61$ \( 1 - 32302 T + 844596301 T^{2} \)
$67$ \( 1 + 60972 T + 1350125107 T^{2} \)
$71$ \( 1 - 32648 T + 1804229351 T^{2} \)
$73$ \( 1 - 38774 T + 2073071593 T^{2} \)
$79$ \( 1 + 33360 T + 3077056399 T^{2} \)
$83$ \( 1 + 16716 T + 3939040643 T^{2} \)
$89$ \( 1 + 101370 T + 5584059449 T^{2} \)
$97$ \( 1 - 119038 T + 8587340257 T^{2} \)
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