Properties

Label 845.2.c.e
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: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

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
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
1.73205i −0.732051 −1.00000 1.00000i 1.26795i 2.00000i 1.73205i −2.46410 1.73205
506.2 1.73205i 2.73205 −1.00000 1.00000i 4.73205i 2.00000i 1.73205i 4.46410 −1.73205
506.3 1.73205i −0.732051 −1.00000 1.00000i 1.26795i 2.00000i 1.73205i −2.46410 1.73205
506.4 1.73205i 2.73205 −1.00000 1.00000i 4.73205i 2.00000i 1.73205i 4.46410 −1.73205
\(n\): 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.e 4
13.b even 2 1 inner 845.2.c.e 4
13.c even 3 1 845.2.m.a 4
13.c even 3 1 845.2.m.c 4
13.d odd 4 1 65.2.a.c 2
13.d odd 4 1 845.2.a.d 2
13.e even 6 1 845.2.m.a 4
13.e even 6 1 845.2.m.c 4
13.f odd 12 2 845.2.e.e 4
13.f odd 12 2 845.2.e.f 4
39.f even 4 1 585.2.a.k 2
39.f even 4 1 7605.2.a.be 2
52.f even 4 1 1040.2.a.h 2
65.f even 4 1 325.2.b.e 4
65.g odd 4 1 325.2.a.g 2
65.g odd 4 1 4225.2.a.w 2
65.k even 4 1 325.2.b.e 4
91.i even 4 1 3185.2.a.k 2
104.j odd 4 1 4160.2.a.y 2
104.m even 4 1 4160.2.a.bj 2
143.g even 4 1 7865.2.a.h 2
156.l odd 4 1 9360.2.a.cm 2
195.j odd 4 1 2925.2.c.v 4
195.n even 4 1 2925.2.a.z 2
195.u odd 4 1 2925.2.c.v 4
260.u even 4 1 5200.2.a.ca 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.c 2 13.d odd 4 1
325.2.a.g 2 65.g odd 4 1
325.2.b.e 4 65.f even 4 1
325.2.b.e 4 65.k even 4 1
585.2.a.k 2 39.f even 4 1
845.2.a.d 2 13.d odd 4 1
845.2.c.e 4 1.a even 1 1 trivial
845.2.c.e 4 13.b even 2 1 inner
845.2.e.e 4 13.f odd 12 2
845.2.e.f 4 13.f odd 12 2
845.2.m.a 4 13.c even 3 1
845.2.m.a 4 13.e even 6 1
845.2.m.c 4 13.c even 3 1
845.2.m.c 4 13.e even 6 1
1040.2.a.h 2 52.f even 4 1
2925.2.a.z 2 195.n even 4 1
2925.2.c.v 4 195.j odd 4 1
2925.2.c.v 4 195.u odd 4 1
3185.2.a.k 2 91.i even 4 1
4160.2.a.y 2 104.j odd 4 1
4160.2.a.bj 2 104.m even 4 1
4225.2.a.w 2 65.g odd 4 1
5200.2.a.ca 2 260.u even 4 1
7605.2.a.be 2 39.f even 4 1
7865.2.a.h 2 143.g even 4 1
9360.2.a.cm 2 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 3 + T^{2} )^{2} \)
$3$ \( ( -2 - 2 T + T^{2} )^{2} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( ( 4 + T^{2} )^{2} \)
$11$ \( 36 + 24 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( -12 + T^{2} )^{2} \)
$19$ \( 676 + 56 T^{2} + T^{4} \)
$23$ \( ( 6 + 6 T + T^{2} )^{2} \)
$29$ \( ( 24 + 12 T + T^{2} )^{2} \)
$31$ \( 4 + 104 T^{2} + T^{4} \)
$37$ \( ( 16 + T^{2} )^{2} \)
$41$ \( ( 12 + T^{2} )^{2} \)
$43$ \( ( -2 + 10 T + T^{2} )^{2} \)
$47$ \( ( 36 + T^{2} )^{2} \)
$53$ \( ( -108 + T^{2} )^{2} \)
$59$ \( 19044 + 312 T^{2} + T^{4} \)
$61$ \( ( -104 - 4 T + T^{2} )^{2} \)
$67$ \( 8464 + 248 T^{2} + T^{4} \)
$71$ \( 36 + 24 T^{2} + T^{4} \)
$73$ \( ( 16 + T^{2} )^{2} \)
$79$ \( ( -104 - 4 T + T^{2} )^{2} \)
$83$ \( ( 36 + T^{2} )^{2} \)
$89$ \( 144 + 168 T^{2} + T^{4} \)
$97$ \( ( 4 + T^{2} )^{2} \)
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