Properties

 Label 845.2.d.c Level $845$ Weight $2$ Character orbit 845.d Analytic conductor $6.747$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.d (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.49787136.1 Defining polynomial: $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 5 \nu^{5} + 15 \nu^{3} + 42 \nu$$$$)/20$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{6} + 5 \nu^{4} + 15 \nu^{2} + 8$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{7} - 5 \nu^{5} + 5 \nu^{3} - 16 \nu$$$$)/40$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} - 3 \nu^{4} - \nu^{2} - 8$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - 3 \nu^{5} - 5 \nu^{3} - 4 \nu$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{6} + 5 \nu^{4} + 15 \nu^{2} + 44$$$$)/20$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{5} - 3 \nu^{3} - 6 \nu$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + \beta_{4} + 2 \beta_{2} - 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} - \beta_{5} + 5 \beta_{3}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{6} - 3 \beta_{4} + \beta_{2} - 2$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{7} - 6 \beta_{5} - 6 \beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$5 \beta_{6} - 5 \beta_{2} - 9$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-10 \beta_{7} + 3 \beta_{5} - 3 \beta_{3} - 7 \beta_{1}$$$$)/2$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
844.1
 −0.228425 + 1.39564i −0.228425 − 1.39564i −1.09445 − 0.895644i −1.09445 + 0.895644i 1.09445 − 0.895644i 1.09445 + 0.895644i 0.228425 + 1.39564i 0.228425 − 1.39564i
−2.18890 1.00000i 2.79129 −0.456850 2.18890i 2.18890i −1.73205 −1.73205 2.00000 1.00000 + 4.79129i
844.2 −2.18890 1.00000i 2.79129 −0.456850 + 2.18890i 2.18890i −1.73205 −1.73205 2.00000 1.00000 4.79129i
844.3 −0.456850 1.00000i −1.79129 −2.18890 0.456850i 0.456850i 1.73205 1.73205 2.00000 1.00000 + 0.208712i
844.4 −0.456850 1.00000i −1.79129 −2.18890 + 0.456850i 0.456850i 1.73205 1.73205 2.00000 1.00000 0.208712i
844.5 0.456850 1.00000i −1.79129 2.18890 + 0.456850i 0.456850i −1.73205 −1.73205 2.00000 1.00000 + 0.208712i
844.6 0.456850 1.00000i −1.79129 2.18890 0.456850i 0.456850i −1.73205 −1.73205 2.00000 1.00000 0.208712i
844.7 2.18890 1.00000i 2.79129 0.456850 + 2.18890i 2.18890i 1.73205 1.73205 2.00000 1.00000 + 4.79129i
844.8 2.18890 1.00000i 2.79129 0.456850 2.18890i 2.18890i 1.73205 1.73205 2.00000 1.00000 4.79129i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 844.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.b even 2 1 inner
65.d 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.d.c 8
5.b even 2 1 inner 845.2.d.c 8
13.b even 2 1 inner 845.2.d.c 8
13.c even 3 1 65.2.l.a 8
13.c even 3 1 845.2.l.c 8
13.d odd 4 2 845.2.b.f 8
13.e even 6 1 65.2.l.a 8
13.e even 6 1 845.2.l.c 8
13.f odd 12 2 845.2.n.c 8
13.f odd 12 2 845.2.n.d 8
39.h odd 6 1 585.2.bf.a 8
39.i odd 6 1 585.2.bf.a 8
52.i odd 6 1 1040.2.df.b 8
52.j odd 6 1 1040.2.df.b 8
65.d even 2 1 inner 845.2.d.c 8
65.f even 4 1 4225.2.a.bj 4
65.f even 4 1 4225.2.a.bk 4
65.g odd 4 2 845.2.b.f 8
65.k even 4 1 4225.2.a.bj 4
65.k even 4 1 4225.2.a.bk 4
65.l even 6 1 65.2.l.a 8
65.l even 6 1 845.2.l.c 8
65.n even 6 1 65.2.l.a 8
65.n even 6 1 845.2.l.c 8
65.q odd 12 1 325.2.n.b 4
65.q odd 12 1 325.2.n.c 4
65.r odd 12 1 325.2.n.b 4
65.r odd 12 1 325.2.n.c 4
65.s odd 12 2 845.2.n.c 8
65.s odd 12 2 845.2.n.d 8
195.x odd 6 1 585.2.bf.a 8
195.y odd 6 1 585.2.bf.a 8
260.v odd 6 1 1040.2.df.b 8
260.w odd 6 1 1040.2.df.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.l.a 8 13.c even 3 1
65.2.l.a 8 13.e even 6 1
65.2.l.a 8 65.l even 6 1
65.2.l.a 8 65.n even 6 1
325.2.n.b 4 65.q odd 12 1
325.2.n.b 4 65.r odd 12 1
325.2.n.c 4 65.q odd 12 1
325.2.n.c 4 65.r odd 12 1
585.2.bf.a 8 39.h odd 6 1
585.2.bf.a 8 39.i odd 6 1
585.2.bf.a 8 195.x odd 6 1
585.2.bf.a 8 195.y odd 6 1
845.2.b.f 8 13.d odd 4 2
845.2.b.f 8 65.g odd 4 2
845.2.d.c 8 1.a even 1 1 trivial
845.2.d.c 8 5.b even 2 1 inner
845.2.d.c 8 13.b even 2 1 inner
845.2.d.c 8 65.d even 2 1 inner
845.2.l.c 8 13.c even 3 1
845.2.l.c 8 13.e even 6 1
845.2.l.c 8 65.l even 6 1
845.2.l.c 8 65.n even 6 1
845.2.n.c 8 13.f odd 12 2
845.2.n.c 8 65.s odd 12 2
845.2.n.d 8 13.f odd 12 2
845.2.n.d 8 65.s odd 12 2
1040.2.df.b 8 52.i odd 6 1
1040.2.df.b 8 52.j odd 6 1
1040.2.df.b 8 260.v odd 6 1
1040.2.df.b 8 260.w odd 6 1
4225.2.a.bj 4 65.f even 4 1
4225.2.a.bj 4 65.k even 4 1
4225.2.a.bk 4 65.f even 4 1
4225.2.a.bk 4 65.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 5 T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(845, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 5 T^{2} + T^{4} )^{2}$$
$3$ $$( 1 + T^{2} )^{4}$$
$5$ $$625 - 34 T^{4} + T^{8}$$
$7$ $$( -3 + T^{2} )^{4}$$
$11$ $$( 7 + T^{2} )^{4}$$
$13$ $$T^{8}$$
$17$ $$( 21 + T^{2} )^{4}$$
$19$ $$( 3 + T^{2} )^{4}$$
$23$ $$( 21 + T^{2} )^{4}$$
$29$ $$( -21 + T^{2} )^{4}$$
$31$ $$( 3600 + 132 T^{2} + T^{4} )^{2}$$
$37$ $$( -63 + T^{2} )^{4}$$
$41$ $$( 7 + T^{2} )^{4}$$
$43$ $$( 225 + 114 T^{2} + T^{4} )^{2}$$
$47$ $$( 256 - 80 T^{2} + T^{4} )^{2}$$
$53$ $$( 144 + 60 T^{2} + T^{4} )^{2}$$
$59$ $$( 2209 + 206 T^{2} + T^{4} )^{2}$$
$61$ $$( 15 + 12 T + T^{2} )^{4}$$
$67$ $$( 225 - 222 T^{2} + T^{4} )^{2}$$
$71$ $$( 625 + 62 T^{2} + T^{4} )^{2}$$
$73$ $$T^{8}$$
$79$ $$( -6 + T )^{8}$$
$83$ $$( 4624 - 164 T^{2} + T^{4} )^{2}$$
$89$ $$( 1681 + 110 T^{2} + T^{4} )^{2}$$
$97$ $$( 2601 - 150 T^{2} + T^{4} )^{2}$$