Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.l (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{2} - \beta_1 q^{3} + \beta_{2} q^{4} + (\beta_{3} - 1) q^{5} + (\beta_{3} - \beta_1) q^{6} + 3 q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{2} - \beta_1 q^{3} + \beta_{2} q^{4} + (\beta_{3} - 1) q^{5} + (\beta_{3} - \beta_1) q^{6} + 3 q^{8} + \beta_{2} q^{9} + (\beta_{2} + \beta_1 - 1) q^{10} - \beta_1 q^{11} - \beta_{3} q^{12} + ( - 4 \beta_{2} + \beta_1 + 4) q^{15} + ( - \beta_{2} + 1) q^{16} + q^{18} + (3 \beta_{3} - 3 \beta_1) q^{19} + (\beta_{3} - \beta_{2} - \beta_1) q^{20} + (\beta_{3} - \beta_1) q^{22} - 3 \beta_1 q^{23} - 3 \beta_1 q^{24} + ( - 2 \beta_{3} - 3) q^{25} + 2 \beta_{3} q^{27} + (6 \beta_{2} - 6) q^{29} + ( - \beta_{3} - 4 \beta_{2} + \beta_1) q^{30} + 3 \beta_{3} q^{31} + 5 \beta_{2} q^{32} + 4 \beta_{2} q^{33} + (\beta_{2} - 1) q^{36} + (6 \beta_{2} - 6) q^{37} + 3 \beta_{3} q^{38} + (3 \beta_{3} - 3) q^{40} + 4 \beta_1 q^{41} + (3 \beta_{3} - 3 \beta_1) q^{43} - \beta_{3} q^{44} + (\beta_{3} - \beta_{2} - \beta_1) q^{45} + (3 \beta_{3} - 3 \beta_1) q^{46} - 8 q^{47} + (\beta_{3} - \beta_1) q^{48} + ( - 7 \beta_{2} + 7) q^{49} + (3 \beta_{2} - 2 \beta_1 - 3) q^{50} - 6 \beta_{3} q^{53} + 2 \beta_1 q^{54} + ( - 4 \beta_{2} + \beta_1 + 4) q^{55} + 12 q^{57} + 6 \beta_{2} q^{58} + ( - \beta_{3} + \beta_1) q^{59} + (\beta_{3} + 4) q^{60} - 6 \beta_{2} q^{61} + 3 \beta_1 q^{62} + 7 q^{64} + 4 q^{66} + (12 \beta_{2} - 12) q^{67} + 12 \beta_{2} q^{69} + (\beta_{3} - \beta_1) q^{71} + 3 \beta_{2} q^{72} - 6 q^{73} + 6 \beta_{2} q^{74} + (8 \beta_{2} + 3 \beta_1 - 8) q^{75} - 3 \beta_1 q^{76} + (\beta_{2} + \beta_1 - 1) q^{80} + ( - 11 \beta_{2} + 11) q^{81} + ( - 4 \beta_{3} + 4 \beta_1) q^{82} - 4 q^{83} + 3 \beta_{3} q^{86} + ( - 6 \beta_{3} + 6 \beta_1) q^{87} - 3 \beta_1 q^{88} + 4 \beta_1 q^{89} + (\beta_{3} - 1) q^{90} - 3 \beta_{3} q^{92} + ( - 12 \beta_{2} + 12) q^{93} + (8 \beta_{2} - 8) q^{94} + ( - 3 \beta_{3} - 12 \beta_{2} + 3 \beta_1) q^{95} - 5 \beta_{3} q^{96} + 6 \beta_{2} q^{97} - 7 \beta_{2} q^{98} - \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 12 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 12 q^{8} + 2 q^{9} - 2 q^{10} + 8 q^{15} + 2 q^{16} + 4 q^{18} - 2 q^{20} - 12 q^{25} - 12 q^{29} - 8 q^{30} + 10 q^{32} + 8 q^{33} - 2 q^{36} - 12 q^{37} - 12 q^{40} - 2 q^{45} - 32 q^{47} + 14 q^{49} - 6 q^{50} + 8 q^{55} + 48 q^{57} + 12 q^{58} + 16 q^{60} - 12 q^{61} + 28 q^{64} + 16 q^{66} - 24 q^{67} + 24 q^{69} + 6 q^{72} - 24 q^{73} + 12 q^{74} - 16 q^{75} - 2 q^{80} + 22 q^{81} - 16 q^{83} - 4 q^{90} + 24 q^{93} - 16 q^{94} - 24 q^{95} + 12 q^{97} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(171\) \(677\)
\(\chi(n)\) \(\beta_{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
654.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.500000 0.866025i −1.73205 1.00000i 0.500000 + 0.866025i −1.00000 + 2.00000i −1.73205 + 1.00000i 0 3.00000 0.500000 + 0.866025i 1.23205 + 1.86603i
654.2 0.500000 0.866025i 1.73205 + 1.00000i 0.500000 + 0.866025i −1.00000 2.00000i 1.73205 1.00000i 0 3.00000 0.500000 + 0.866025i −2.23205 0.133975i
699.1 0.500000 + 0.866025i −1.73205 + 1.00000i 0.500000 0.866025i −1.00000 2.00000i −1.73205 1.00000i 0 3.00000 0.500000 0.866025i 1.23205 1.86603i
699.2 0.500000 + 0.866025i 1.73205 1.00000i 0.500000 0.866025i −1.00000 + 2.00000i 1.73205 + 1.00000i 0 3.00000 0.500000 0.866025i −2.23205 + 0.133975i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
65.d even 2 1 inner
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.l.b 4
5.b even 2 1 845.2.l.a 4
13.b even 2 1 845.2.l.a 4
13.c even 3 1 65.2.d.a 2
13.c even 3 1 inner 845.2.l.b 4
13.d odd 4 1 845.2.n.a 4
13.d odd 4 1 845.2.n.b 4
13.e even 6 1 65.2.d.b yes 2
13.e even 6 1 845.2.l.a 4
13.f odd 12 1 845.2.b.a 2
13.f odd 12 1 845.2.b.b 2
13.f odd 12 1 845.2.n.a 4
13.f odd 12 1 845.2.n.b 4
39.h odd 6 1 585.2.h.b 2
39.i odd 6 1 585.2.h.c 2
52.i odd 6 1 1040.2.f.b 2
52.j odd 6 1 1040.2.f.a 2
65.d even 2 1 inner 845.2.l.b 4
65.g odd 4 1 845.2.n.a 4
65.g odd 4 1 845.2.n.b 4
65.l even 6 1 65.2.d.a 2
65.l even 6 1 inner 845.2.l.b 4
65.n even 6 1 65.2.d.b yes 2
65.n even 6 1 845.2.l.a 4
65.o even 12 1 4225.2.a.e 1
65.o even 12 1 4225.2.a.h 1
65.q odd 12 1 325.2.c.b 2
65.q odd 12 1 325.2.c.e 2
65.r odd 12 1 325.2.c.b 2
65.r odd 12 1 325.2.c.e 2
65.s odd 12 1 845.2.b.a 2
65.s odd 12 1 845.2.b.b 2
65.s odd 12 1 845.2.n.a 4
65.s odd 12 1 845.2.n.b 4
65.t even 12 1 4225.2.a.k 1
65.t even 12 1 4225.2.a.m 1
195.x odd 6 1 585.2.h.b 2
195.y odd 6 1 585.2.h.c 2
260.v odd 6 1 1040.2.f.b 2
260.w odd 6 1 1040.2.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.d.a 2 13.c even 3 1
65.2.d.a 2 65.l even 6 1
65.2.d.b yes 2 13.e even 6 1
65.2.d.b yes 2 65.n even 6 1
325.2.c.b 2 65.q odd 12 1
325.2.c.b 2 65.r odd 12 1
325.2.c.e 2 65.q odd 12 1
325.2.c.e 2 65.r odd 12 1
585.2.h.b 2 39.h odd 6 1
585.2.h.b 2 195.x odd 6 1
585.2.h.c 2 39.i odd 6 1
585.2.h.c 2 195.y odd 6 1
845.2.b.a 2 13.f odd 12 1
845.2.b.a 2 65.s odd 12 1
845.2.b.b 2 13.f odd 12 1
845.2.b.b 2 65.s odd 12 1
845.2.l.a 4 5.b even 2 1
845.2.l.a 4 13.b even 2 1
845.2.l.a 4 13.e even 6 1
845.2.l.a 4 65.n even 6 1
845.2.l.b 4 1.a even 1 1 trivial
845.2.l.b 4 13.c even 3 1 inner
845.2.l.b 4 65.d even 2 1 inner
845.2.l.b 4 65.l even 6 1 inner
845.2.n.a 4 13.d odd 4 1
845.2.n.a 4 13.f odd 12 1
845.2.n.a 4 65.g odd 4 1
845.2.n.a 4 65.s odd 12 1
845.2.n.b 4 13.d odd 4 1
845.2.n.b 4 13.f odd 12 1
845.2.n.b 4 65.g odd 4 1
845.2.n.b 4 65.s odd 12 1
1040.2.f.a 2 52.j odd 6 1
1040.2.f.a 2 260.w odd 6 1
1040.2.f.b 2 52.i odd 6 1
1040.2.f.b 2 260.v odd 6 1
4225.2.a.e 1 65.o even 12 1
4225.2.a.h 1 65.o even 12 1
4225.2.a.k 1 65.t even 12 1
4225.2.a.m 1 65.t even 12 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$23$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$43$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$47$ \( (T + 8)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 12 T + 144)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$73$ \( (T + 6)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T + 4)^{4} \) Copy content Toggle raw display
$89$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$97$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
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