Properties

Label 845.2.b.e
Level $845$
Weight $2$
Character orbit 845.b
Analytic conductor $6.747$
Analytic rank $0$
Dimension $6$
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.49843600.1
Defining polynomial: \(x^{6} + 8 x^{4} + 10 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( \beta_{1} - \beta_{5} ) q^{3} + ( -1 + \beta_{2} ) q^{4} + ( 1 + \beta_{3} ) q^{5} + ( -2 + \beta_{2} ) q^{6} + ( \beta_{1} + \beta_{5} ) q^{7} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{8} + ( -1 - \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( \beta_{1} - \beta_{5} ) q^{3} + ( -1 + \beta_{2} ) q^{4} + ( 1 + \beta_{3} ) q^{5} + ( -2 + \beta_{2} ) q^{6} + ( \beta_{1} + \beta_{5} ) q^{7} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{8} + ( -1 - \beta_{3} - \beta_{4} ) q^{9} + ( -1 + \beta_{2} - \beta_{4} ) q^{10} + ( \beta_{3} + \beta_{4} ) q^{11} + ( -1 - 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{12} + ( -4 + \beta_{2} ) q^{14} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{5} ) q^{15} + ( 3 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{16} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{17} + ( -1 - \beta_{3} + \beta_{4} ) q^{18} + ( 2 - \beta_{3} - \beta_{4} ) q^{19} + ( -1 - 3 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{20} + ( 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{21} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{22} + ( -\beta_{1} + \beta_{5} ) q^{23} + ( 6 - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{24} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{25} + ( \beta_{1} + \beta_{5} ) q^{27} + ( -1 - 4 \beta_{1} - \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{28} -3 q^{29} + ( -2 - 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{30} + ( 2 - 2 \beta_{2} ) q^{31} + ( 2 \beta_{1} + \beta_{5} ) q^{32} + ( \beta_{1} - 3 \beta_{5} ) q^{33} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{34} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{35} + ( 1 - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{36} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{37} + ( -1 + 3 \beta_{1} - \beta_{3} + \beta_{4} ) q^{38} + ( 7 - 3 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{40} + ( 3 - 2 \beta_{2} ) q^{41} + ( -1 - 5 \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{42} + ( \beta_{1} - 3 \beta_{5} ) q^{43} + ( \beta_{2} + \beta_{3} + \beta_{4} ) q^{44} + ( -4 - 2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{45} + ( 2 - \beta_{2} ) q^{46} + ( -2 + 6 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{47} + ( 1 + 5 \beta_{1} + \beta_{3} - \beta_{4} - 4 \beta_{5} ) q^{48} + ( -1 - \beta_{3} - \beta_{4} ) q^{49} + ( -6 - 3 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{50} + ( 2 + \beta_{3} + \beta_{4} ) q^{51} + ( -1 + 2 \beta_{1} - \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{53} + ( -4 + \beta_{2} ) q^{54} + ( 3 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{55} + ( 4 - 4 \beta_{2} + \beta_{3} + \beta_{4} ) q^{56} + ( \beta_{1} + \beta_{5} ) q^{57} -3 \beta_{1} q^{58} + ( -2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{59} + ( 8 - \beta_{1} - 3 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} ) q^{60} + ( -1 - 2 \beta_{3} - 2 \beta_{4} ) q^{61} + ( 2 + 6 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{62} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{63} + ( -1 + 2 \beta_{3} + 2 \beta_{4} ) q^{64} + \beta_{2} q^{66} + ( -3 \beta_{1} - \beta_{5} ) q^{67} + ( 2 \beta_{1} + \beta_{5} ) q^{68} + ( 4 + \beta_{3} + \beta_{4} ) q^{69} + ( -4 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{70} + ( -2 - \beta_{3} - \beta_{4} ) q^{71} + ( -1 + 6 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{72} + ( 1 - 6 \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{73} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{74} + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} ) q^{75} + ( -2 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{76} + ( 2 - 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} ) q^{77} + ( -8 - 2 \beta_{2} ) q^{79} + ( 6 + 5 \beta_{1} + \beta_{3} ) q^{80} + ( -3 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{81} + ( 2 + 7 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{82} + ( 2 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{83} + ( 16 - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{84} + ( 5 - 3 \beta_{1} - 3 \beta_{4} + \beta_{5} ) q^{85} + \beta_{2} q^{86} + ( -3 \beta_{1} + 3 \beta_{5} ) q^{87} + ( 2 - 5 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{88} + ( 2 + 4 \beta_{2} - \beta_{3} - \beta_{4} ) q^{89} + ( 6 - 2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{90} + ( 1 + 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{92} + ( 2 + 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{93} + ( -10 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{94} + ( -1 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{95} + ( -2 + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{96} + ( 2 + 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{97} + ( -1 - \beta_{3} + \beta_{4} ) q^{98} + ( -8 - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 4q^{4} + 3q^{5} - 10q^{6} - 6q^{9} + O(q^{10}) \) \( 6q - 4q^{4} + 3q^{5} - 10q^{6} - 6q^{9} - 7q^{10} - 22q^{14} - 4q^{15} + 16q^{16} + 12q^{19} - q^{20} + 4q^{21} + 32q^{24} - q^{25} - 18q^{29} - 4q^{30} + 8q^{31} - 8q^{34} - 10q^{35} + 2q^{36} + 35q^{40} + 14q^{41} + 2q^{44} - 29q^{45} + 10q^{46} - 6q^{49} - 31q^{50} + 12q^{51} - 22q^{54} + 26q^{55} + 16q^{56} - 4q^{59} + 48q^{60} - 6q^{61} - 6q^{64} + 2q^{66} + 24q^{69} - 10q^{70} - 12q^{71} - 8q^{74} - 2q^{75} - 10q^{76} - 52q^{79} + 33q^{80} - 14q^{81} + 90q^{84} + 21q^{85} + 2q^{86} + 20q^{89} + 31q^{90} - 56q^{94} - 20q^{95} - 6q^{96} - 52q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 8 x^{4} + 10 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 3 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + \nu^{4} + 7 \nu^{3} + 7 \nu^{2} + 5 \nu + 3 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 7 \nu^{3} + 7 \nu^{2} - 5 \nu + 5 \)\()/2\)
\(\beta_{5}\)\(=\)\( \nu^{5} + 8 \nu^{3} + 10 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} - \beta_{3} - 5 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} - 7 \beta_{2} + 17\)
\(\nu^{5}\)\(=\)\(-7 \beta_{5} - 8 \beta_{4} + 8 \beta_{3} + 30 \beta_{1} + 8\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
339.1
2.54574i
1.18733i
0.330837i
0.330837i
1.18733i
2.54574i
2.54574i 2.15293i −4.48079 0.817544 2.08125i −5.48079 2.93855i 6.31544i −1.63509 −5.29833 2.08125i
339.2 1.18733i 0.345110i 0.590239 −1.44045 + 1.71029i −0.409761 2.02956i 3.07548i 2.88090 2.03069 + 1.71029i
339.3 0.330837i 2.69180i 1.89055 2.12291 0.702335i 0.890547 3.35348i 1.28714i −4.24581 −0.232358 0.702335i
339.4 0.330837i 2.69180i 1.89055 2.12291 + 0.702335i 0.890547 3.35348i 1.28714i −4.24581 −0.232358 + 0.702335i
339.5 1.18733i 0.345110i 0.590239 −1.44045 1.71029i −0.409761 2.02956i 3.07548i 2.88090 2.03069 1.71029i
339.6 2.54574i 2.15293i −4.48079 0.817544 + 2.08125i −5.48079 2.93855i 6.31544i −1.63509 −5.29833 + 2.08125i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 339.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.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.b.e 6
5.b even 2 1 inner 845.2.b.e 6
5.c odd 4 2 4225.2.a.bq 6
13.b even 2 1 845.2.b.d 6
13.c even 3 2 845.2.n.e 12
13.d odd 4 2 845.2.d.d 12
13.e even 6 2 65.2.n.a 12
13.f odd 12 4 845.2.l.f 24
39.h odd 6 2 585.2.bs.a 12
52.i odd 6 2 1040.2.dh.a 12
65.d even 2 1 845.2.b.d 6
65.g odd 4 2 845.2.d.d 12
65.h odd 4 2 4225.2.a.br 6
65.l even 6 2 65.2.n.a 12
65.n even 6 2 845.2.n.e 12
65.r odd 12 4 325.2.e.e 12
65.s odd 12 4 845.2.l.f 24
195.y odd 6 2 585.2.bs.a 12
260.w odd 6 2 1040.2.dh.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.n.a 12 13.e even 6 2
65.2.n.a 12 65.l even 6 2
325.2.e.e 12 65.r odd 12 4
585.2.bs.a 12 39.h odd 6 2
585.2.bs.a 12 195.y odd 6 2
845.2.b.d 6 13.b even 2 1
845.2.b.d 6 65.d even 2 1
845.2.b.e 6 1.a even 1 1 trivial
845.2.b.e 6 5.b even 2 1 inner
845.2.d.d 12 13.d odd 4 2
845.2.d.d 12 65.g odd 4 2
845.2.l.f 24 13.f odd 12 4
845.2.l.f 24 65.s odd 12 4
845.2.n.e 12 13.c even 3 2
845.2.n.e 12 65.n even 6 2
1040.2.dh.a 12 52.i odd 6 2
1040.2.dh.a 12 260.w odd 6 2
4225.2.a.bq 6 5.c odd 4 2
4225.2.a.br 6 65.h odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\):

\( T_{2}^{6} + 8 T_{2}^{4} + 10 T_{2}^{2} + 1 \)
\( T_{11}^{3} - 13 T_{11} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 10 T^{2} + 8 T^{4} + T^{6} \)
$3$ \( 4 + 35 T^{2} + 12 T^{4} + T^{6} \)
$5$ \( 125 - 75 T + 25 T^{2} - 10 T^{3} + 5 T^{4} - 3 T^{5} + T^{6} \)
$7$ \( 400 + 179 T^{2} + 24 T^{4} + T^{6} \)
$11$ \( ( 8 - 13 T + T^{3} )^{2} \)
$13$ \( T^{6} \)
$17$ \( 169 + 163 T^{2} + 35 T^{4} + T^{6} \)
$19$ \( ( 10 - T - 6 T^{2} + T^{3} )^{2} \)
$23$ \( 4 + 35 T^{2} + 12 T^{4} + T^{6} \)
$29$ \( ( 3 + T )^{6} \)
$31$ \( ( -40 - 40 T - 4 T^{2} + T^{3} )^{2} \)
$37$ \( 169 + 163 T^{2} + 35 T^{4} + T^{6} \)
$41$ \( ( -5 - 29 T - 7 T^{2} + T^{3} )^{2} \)
$43$ \( 256 + 283 T^{2} + 80 T^{4} + T^{6} \)
$47$ \( 270400 + 14640 T^{2} + 236 T^{4} + T^{6} \)
$53$ \( 400 + 1040 T^{2} + 171 T^{4} + T^{6} \)
$59$ \( ( -136 - 55 T + 2 T^{2} + T^{3} )^{2} \)
$61$ \( ( -115 - 49 T + 3 T^{2} + T^{3} )^{2} \)
$67$ \( 20164 + 2603 T^{2} + 100 T^{4} + T^{6} \)
$71$ \( ( -26 - T + 6 T^{2} + T^{3} )^{2} \)
$73$ \( 250000 + 13900 T^{2} + 215 T^{4} + T^{6} \)
$79$ \( ( 160 + 180 T + 26 T^{2} + T^{3} )^{2} \)
$83$ \( 640000 + 23600 T^{2} + 276 T^{4} + T^{6} \)
$89$ \( ( 1586 - 157 T - 10 T^{2} + T^{3} )^{2} \)
$97$ \( 204304 + 14363 T^{2} + 280 T^{4} + T^{6} \)
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