Properties

Label 845.2.c.c
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: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 2 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 2 \beta_{1}\)

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
1.61803i
0.618034i
0.618034i
1.61803i
1.61803i −2.23607 −0.618034 1.00000i 3.61803i 0.236068i 2.23607i 2.00000 −1.61803
506.2 0.618034i 2.23607 1.61803 1.00000i 1.38197i 4.23607i 2.23607i 2.00000 0.618034
506.3 0.618034i 2.23607 1.61803 1.00000i 1.38197i 4.23607i 2.23607i 2.00000 0.618034
506.4 1.61803i −2.23607 −0.618034 1.00000i 3.61803i 0.236068i 2.23607i 2.00000 −1.61803
\(n\): 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.c 4
13.b even 2 1 inner 845.2.c.c 4
13.c even 3 2 845.2.m.e 8
13.d odd 4 1 845.2.a.b 2
13.d odd 4 1 845.2.a.e 2
13.e even 6 2 845.2.m.e 8
13.f odd 12 2 65.2.e.a 4
13.f odd 12 2 845.2.e.g 4
39.f even 4 1 7605.2.a.ba 2
39.f even 4 1 7605.2.a.bf 2
39.k even 12 2 585.2.j.e 4
52.l even 12 2 1040.2.q.n 4
65.g odd 4 1 4225.2.a.u 2
65.g odd 4 1 4225.2.a.y 2
65.o even 12 2 325.2.o.a 8
65.s odd 12 2 325.2.e.b 4
65.t even 12 2 325.2.o.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.e.a 4 13.f odd 12 2
325.2.e.b 4 65.s odd 12 2
325.2.o.a 8 65.o even 12 2
325.2.o.a 8 65.t even 12 2
585.2.j.e 4 39.k even 12 2
845.2.a.b 2 13.d odd 4 1
845.2.a.e 2 13.d odd 4 1
845.2.c.c 4 1.a even 1 1 trivial
845.2.c.c 4 13.b even 2 1 inner
845.2.e.g 4 13.f odd 12 2
845.2.m.e 8 13.c even 3 2
845.2.m.e 8 13.e even 6 2
1040.2.q.n 4 52.l even 12 2
4225.2.a.u 2 65.g odd 4 1
4225.2.a.y 2 65.g odd 4 1
7605.2.a.ba 2 39.f even 4 1
7605.2.a.bf 2 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T^{2} + T^{4} \)
$3$ \( ( -5 + T^{2} )^{2} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( 1 + 18 T^{2} + T^{4} \)
$11$ \( 1 + 18 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( -19 + 2 T + T^{2} )^{2} \)
$19$ \( 1 + 18 T^{2} + T^{4} \)
$23$ \( ( 31 + 12 T + T^{2} )^{2} \)
$29$ \( ( -11 + 6 T + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( 9 + T^{2} )^{2} \)
$41$ \( 5041 + 178 T^{2} + T^{4} \)
$43$ \( ( 11 - 8 T + T^{2} )^{2} \)
$47$ \( 4096 + 192 T^{2} + T^{4} \)
$53$ \( ( -6 + T )^{4} \)
$59$ \( 81 + 162 T^{2} + T^{4} \)
$61$ \( ( -179 - 2 T + T^{2} )^{2} \)
$67$ \( 841 + 122 T^{2} + T^{4} \)
$71$ \( 121 + 42 T^{2} + T^{4} \)
$73$ \( ( 36 + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( ( 80 + T^{2} )^{2} \)
$89$ \( ( 81 + T^{2} )^{2} \)
$97$ \( 361 + 42 T^{2} + T^{4} \)
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