Properties

Label 845.2.b.a
Level $845$
Weight $2$
Character orbit 845.b
Analytic conductor $6.747$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
339.1
1.00000i
1.00000i
1.00000i 2.00000i 1.00000 −2.00000 1.00000i 2.00000 0 3.00000i −1.00000 −1.00000 + 2.00000i
339.2 1.00000i 2.00000i 1.00000 −2.00000 + 1.00000i 2.00000 0 3.00000i −1.00000 −1.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
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.a 2
5.b even 2 1 inner 845.2.b.a 2
5.c odd 4 1 4225.2.a.e 1
5.c odd 4 1 4225.2.a.m 1
13.b even 2 1 845.2.b.b 2
13.c even 3 2 845.2.n.a 4
13.d odd 4 1 65.2.d.a 2
13.d odd 4 1 65.2.d.b yes 2
13.e even 6 2 845.2.n.b 4
13.f odd 12 2 845.2.l.a 4
13.f odd 12 2 845.2.l.b 4
39.f even 4 1 585.2.h.b 2
39.f even 4 1 585.2.h.c 2
52.f even 4 1 1040.2.f.a 2
52.f even 4 1 1040.2.f.b 2
65.d even 2 1 845.2.b.b 2
65.f even 4 1 325.2.c.b 2
65.f even 4 1 325.2.c.e 2
65.g odd 4 1 65.2.d.a 2
65.g odd 4 1 65.2.d.b yes 2
65.h odd 4 1 4225.2.a.h 1
65.h odd 4 1 4225.2.a.k 1
65.k even 4 1 325.2.c.b 2
65.k even 4 1 325.2.c.e 2
65.l even 6 2 845.2.n.b 4
65.n even 6 2 845.2.n.a 4
65.s odd 12 2 845.2.l.a 4
65.s odd 12 2 845.2.l.b 4
195.n even 4 1 585.2.h.b 2
195.n even 4 1 585.2.h.c 2
260.u even 4 1 1040.2.f.a 2
260.u even 4 1 1040.2.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.d.a 2 13.d odd 4 1
65.2.d.a 2 65.g odd 4 1
65.2.d.b yes 2 13.d odd 4 1
65.2.d.b yes 2 65.g odd 4 1
325.2.c.b 2 65.f even 4 1
325.2.c.b 2 65.k even 4 1
325.2.c.e 2 65.f even 4 1
325.2.c.e 2 65.k even 4 1
585.2.h.b 2 39.f even 4 1
585.2.h.b 2 195.n even 4 1
585.2.h.c 2 39.f even 4 1
585.2.h.c 2 195.n even 4 1
845.2.b.a 2 1.a even 1 1 trivial
845.2.b.a 2 5.b even 2 1 inner
845.2.b.b 2 13.b even 2 1
845.2.b.b 2 65.d even 2 1
845.2.l.a 4 13.f odd 12 2
845.2.l.a 4 65.s odd 12 2
845.2.l.b 4 13.f odd 12 2
845.2.l.b 4 65.s odd 12 2
845.2.n.a 4 13.c even 3 2
845.2.n.a 4 65.n even 6 2
845.2.n.b 4 13.e even 6 2
845.2.n.b 4 65.l even 6 2
1040.2.f.a 2 52.f even 4 1
1040.2.f.a 2 260.u even 4 1
1040.2.f.b 2 52.f even 4 1
1040.2.f.b 2 260.u even 4 1
4225.2.a.e 1 5.c odd 4 1
4225.2.a.h 1 65.h odd 4 1
4225.2.a.k 1 65.h odd 4 1
4225.2.a.m 1 5.c odd 4 1

Hecke kernels

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

\( T_{2}^{2} + 1 \)
\( T_{11} - 2 \)

Hecke characteristic polynomials

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