Properties

Label 845.2.c.b
Level $845$
Weight $2$
Character orbit 845.c
Analytic conductor $6.747$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{2} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{3} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{4} -\zeta_{8}^{2} q^{5} + ( -\zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{6} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{7} + ( -\zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{8} - q^{9} +O(q^{10})\) \( q + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{2} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{3} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{4} -\zeta_{8}^{2} q^{5} + ( -\zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{6} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{7} + ( -\zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{8} - q^{9} + ( 1 + \zeta_{8} - \zeta_{8}^{3} ) q^{10} + ( \zeta_{8} + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{11} + ( 4 + \zeta_{8} - \zeta_{8}^{3} ) q^{12} + ( -6 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{14} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{15} + 3 q^{16} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{17} + ( -\zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{18} + ( \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{19} + ( 2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{20} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{21} + ( -4 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{22} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{23} + ( 3 \zeta_{8} + 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{24} - q^{25} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{27} + ( -6 \zeta_{8} - 10 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{28} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{29} + ( -2 - \zeta_{8} + \zeta_{8}^{3} ) q^{30} + ( 3 \zeta_{8} - 6 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{31} + ( \zeta_{8} - 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{32} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{33} -2 \zeta_{8}^{2} q^{34} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{35} + ( 1 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{36} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{37} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{38} + ( -3 - \zeta_{8} + \zeta_{8}^{3} ) q^{40} + ( -2 \zeta_{8} + 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{41} + ( 8 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{42} + ( 4 + 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{43} + ( -5 \zeta_{8} - 6 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{44} + \zeta_{8}^{2} q^{45} + ( -\zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{46} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{47} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{48} + ( -5 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{49} + ( -\zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{50} + ( 4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{51} + ( -6 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{53} + ( 4 \zeta_{8} + 8 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{54} + ( 2 + \zeta_{8} - \zeta_{8}^{3} ) q^{55} + ( 10 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{56} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{57} + ( -4 \zeta_{8} - 8 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{58} + ( -3 \zeta_{8} + 6 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{59} + ( -\zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{60} -8 q^{61} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{62} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{63} + ( 7 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{64} + ( 6 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{66} + 2 \zeta_{8}^{2} q^{67} + ( 6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{68} + 2 q^{69} + ( 4 \zeta_{8} + 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{70} + ( -7 \zeta_{8} - 2 \zeta_{8}^{2} - 7 \zeta_{8}^{3} ) q^{71} + ( \zeta_{8} + 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{72} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{73} + ( 12 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{74} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{75} + ( 3 \zeta_{8} - 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{76} + ( -8 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{77} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{79} -3 \zeta_{8}^{2} q^{80} -5 q^{81} + ( -2 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{82} + ( -2 \zeta_{8} + 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{83} + ( 10 \zeta_{8} + 12 \zeta_{8}^{2} + 10 \zeta_{8}^{3} ) q^{84} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{85} + ( 9 \zeta_{8} + 14 \zeta_{8}^{2} + 9 \zeta_{8}^{3} ) q^{86} + 8 q^{87} + ( 8 + 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{88} + 6 \zeta_{8}^{2} q^{89} + ( -1 - \zeta_{8} + \zeta_{8}^{3} ) q^{90} + ( 4 + \zeta_{8} - \zeta_{8}^{3} ) q^{92} + ( 6 \zeta_{8} - 6 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{93} + ( 6 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{94} + ( -2 + \zeta_{8} - \zeta_{8}^{3} ) q^{95} + ( 3 \zeta_{8} - 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{96} + ( 4 \zeta_{8} + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{97} + ( -13 \zeta_{8} - 21 \zeta_{8}^{2} - 13 \zeta_{8}^{3} ) q^{98} + ( -\zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} - 4q^{9} + 4q^{10} + 16q^{12} - 24q^{14} + 12q^{16} + 8q^{17} - 16q^{22} - 4q^{25} - 8q^{30} + 8q^{35} + 4q^{36} - 12q^{40} + 32q^{42} + 16q^{43} - 20q^{49} + 16q^{51} - 24q^{53} + 8q^{55} + 40q^{56} - 32q^{61} + 28q^{64} + 24q^{66} + 24q^{68} + 8q^{69} + 48q^{74} - 32q^{77} - 20q^{81} - 8q^{82} + 32q^{87} + 32q^{88} - 4q^{90} + 16q^{92} + 24q^{94} - 8q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
506.1
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
2.41421i −1.41421 −3.82843 1.00000i 3.41421i 4.82843i 4.41421i −1.00000 2.41421
506.2 0.414214i 1.41421 1.82843 1.00000i 0.585786i 0.828427i 1.58579i −1.00000 −0.414214
506.3 0.414214i 1.41421 1.82843 1.00000i 0.585786i 0.828427i 1.58579i −1.00000 −0.414214
506.4 2.41421i −1.41421 −3.82843 1.00000i 3.41421i 4.82843i 4.41421i −1.00000 2.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.c.b 4
13.b even 2 1 inner 845.2.c.b 4
13.c even 3 2 845.2.m.f 8
13.d odd 4 1 65.2.a.b 2
13.d odd 4 1 845.2.a.g 2
13.e even 6 2 845.2.m.f 8
13.f odd 12 2 845.2.e.c 4
13.f odd 12 2 845.2.e.h 4
39.f even 4 1 585.2.a.m 2
39.f even 4 1 7605.2.a.x 2
52.f even 4 1 1040.2.a.j 2
65.f even 4 1 325.2.b.f 4
65.g odd 4 1 325.2.a.i 2
65.g odd 4 1 4225.2.a.r 2
65.k even 4 1 325.2.b.f 4
91.i even 4 1 3185.2.a.j 2
104.j odd 4 1 4160.2.a.bf 2
104.m even 4 1 4160.2.a.z 2
143.g even 4 1 7865.2.a.j 2
156.l odd 4 1 9360.2.a.cd 2
195.j odd 4 1 2925.2.c.r 4
195.n even 4 1 2925.2.a.u 2
195.u odd 4 1 2925.2.c.r 4
260.u even 4 1 5200.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.b 2 13.d odd 4 1
325.2.a.i 2 65.g odd 4 1
325.2.b.f 4 65.f even 4 1
325.2.b.f 4 65.k even 4 1
585.2.a.m 2 39.f even 4 1
845.2.a.g 2 13.d odd 4 1
845.2.c.b 4 1.a even 1 1 trivial
845.2.c.b 4 13.b even 2 1 inner
845.2.e.c 4 13.f odd 12 2
845.2.e.h 4 13.f odd 12 2
845.2.m.f 8 13.c even 3 2
845.2.m.f 8 13.e even 6 2
1040.2.a.j 2 52.f even 4 1
2925.2.a.u 2 195.n even 4 1
2925.2.c.r 4 195.j odd 4 1
2925.2.c.r 4 195.u odd 4 1
3185.2.a.j 2 91.i even 4 1
4160.2.a.z 2 104.m even 4 1
4160.2.a.bf 2 104.j odd 4 1
4225.2.a.r 2 65.g odd 4 1
5200.2.a.bu 2 260.u even 4 1
7605.2.a.x 2 39.f even 4 1
7865.2.a.j 2 143.g even 4 1
9360.2.a.cd 2 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 6 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T^{2} + T^{4} \)
$3$ \( ( -2 + T^{2} )^{2} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( 16 + 24 T^{2} + T^{4} \)
$11$ \( 4 + 12 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( -4 - 4 T + T^{2} )^{2} \)
$19$ \( 4 + 12 T^{2} + T^{4} \)
$23$ \( ( -2 + T^{2} )^{2} \)
$29$ \( ( -32 + T^{2} )^{2} \)
$31$ \( 324 + 108 T^{2} + T^{4} \)
$37$ \( ( 72 + T^{2} )^{2} \)
$41$ \( 784 + 88 T^{2} + T^{4} \)
$43$ \( ( -34 - 8 T + T^{2} )^{2} \)
$47$ \( 16 + 24 T^{2} + T^{4} \)
$53$ \( ( -36 + 12 T + T^{2} )^{2} \)
$59$ \( 324 + 108 T^{2} + T^{4} \)
$61$ \( ( 8 + T )^{4} \)
$67$ \( ( 4 + T^{2} )^{2} \)
$71$ \( 8836 + 204 T^{2} + T^{4} \)
$73$ \( ( 72 + T^{2} )^{2} \)
$79$ \( ( -72 + T^{2} )^{2} \)
$83$ \( 784 + 88 T^{2} + T^{4} \)
$89$ \( ( 36 + T^{2} )^{2} \)
$97$ \( 784 + 72 T^{2} + T^{4} \)
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