# Properties

 Label 245.2.l.c Level 245 Weight 2 Character orbit 245.l Analytic conductor 1.956 Analytic rank 0 Dimension 8 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 245.l (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.95633484952$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: 8.0.3317760000.2 Defining polynomial: $$x^{8} - 25 x^{4} + 625$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{2} - \beta_{4} ) q^{2} + \beta_{1} q^{3} + \beta_{5} q^{5} + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{6} + ( -2 - 2 \beta_{6} ) q^{8} + 2 \beta_{2} q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{2} - \beta_{4} ) q^{2} + \beta_{1} q^{3} + \beta_{5} q^{5} + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{6} + ( -2 - 2 \beta_{6} ) q^{8} + 2 \beta_{2} q^{9} + ( \beta_{1} - \beta_{7} ) q^{10} + \beta_{4} q^{11} + ( \beta_{1} - \beta_{5} ) q^{13} + 5 \beta_{6} q^{15} + ( -4 + 4 \beta_{4} ) q^{16} + \beta_{7} q^{17} + ( 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} ) q^{18} + ( \beta_{3} - \beta_{5} - \beta_{7} ) q^{19} + ( 1 - \beta_{6} ) q^{22} + ( -2 - 2 \beta_{2} + 2 \beta_{4} ) q^{23} + ( -2 \beta_{1} - 2 \beta_{7} ) q^{24} + ( -5 \beta_{2} + 5 \beta_{6} ) q^{25} + ( -\beta_{3} - \beta_{5} + \beta_{7} ) q^{26} -\beta_{3} q^{27} -3 \beta_{6} q^{29} + ( 5 + 5 \beta_{2} - 5 \beta_{4} ) q^{30} + ( -\beta_{1} + \beta_{7} ) q^{31} + \beta_{5} q^{33} + ( \beta_{1} + \beta_{3} - \beta_{5} ) q^{34} + ( 6 - 6 \beta_{2} - 6 \beta_{4} ) q^{37} -2 \beta_{1} q^{38} + ( 5 \beta_{2} - 5 \beta_{6} ) q^{39} + ( 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{40} + ( -3 \beta_{1} + 3 \beta_{3} + 3 \beta_{5} ) q^{41} + ( -3 - 3 \beta_{6} ) q^{43} + 2 \beta_{7} q^{45} + 4 \beta_{4} q^{46} + ( -3 \beta_{3} + 3 \beta_{7} ) q^{47} + ( -4 \beta_{1} + 4 \beta_{5} ) q^{48} + ( 5 + 5 \beta_{6} ) q^{50} + ( -5 + 5 \beta_{4} ) q^{51} + ( \beta_{2} - \beta_{4} - \beta_{6} ) q^{53} + ( -\beta_{3} + \beta_{5} + \beta_{7} ) q^{54} + ( -\beta_{1} + \beta_{5} ) q^{55} + ( 5 - 5 \beta_{6} ) q^{57} + ( -3 - 3 \beta_{2} + 3 \beta_{4} ) q^{58} + ( 3 \beta_{1} + 3 \beta_{7} ) q^{59} + ( 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{61} + 2 \beta_{3} q^{62} + 8 \beta_{6} q^{64} + 5 \beta_{2} q^{65} + ( \beta_{1} - \beta_{7} ) q^{66} + ( \beta_{2} + \beta_{4} - \beta_{6} ) q^{67} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} ) q^{69} -6 q^{71} + ( 4 - 4 \beta_{2} - 4 \beta_{4} ) q^{72} + ( -12 \beta_{2} + 12 \beta_{6} ) q^{74} + ( -5 \beta_{3} + 5 \beta_{7} ) q^{75} + ( -5 - 5 \beta_{6} ) q^{78} + 13 \beta_{2} q^{79} -4 \beta_{1} q^{80} -11 \beta_{4} q^{81} + ( 6 \beta_{3} - 6 \beta_{7} ) q^{82} + ( 2 \beta_{1} - 2 \beta_{5} ) q^{83} -5 q^{85} + ( -6 + 6 \beta_{4} ) q^{86} -3 \beta_{7} q^{87} + ( 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} ) q^{88} + ( -2 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} ) q^{89} + ( 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} ) q^{90} + ( -5 - 5 \beta_{2} + 5 \beta_{4} ) q^{93} + ( 3 \beta_{1} + 3 \beta_{7} ) q^{94} + ( 5 \beta_{2} + 5 \beta_{4} - 5 \beta_{6} ) q^{95} + \beta_{3} q^{97} + 2 \beta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{2} - 16q^{8} + O(q^{10})$$ $$8q + 4q^{2} - 16q^{8} + 4q^{11} - 16q^{16} - 8q^{18} + 8q^{22} - 8q^{23} + 20q^{30} + 24q^{37} - 24q^{43} + 16q^{46} + 40q^{50} - 20q^{51} - 4q^{53} + 40q^{57} - 12q^{58} + 4q^{67} - 48q^{71} + 16q^{72} - 40q^{78} - 44q^{81} - 40q^{85} - 24q^{86} - 8q^{88} - 20q^{93} + 20q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 25 x^{4} + 625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/5$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/5$$ $$\beta_{4}$$ $$=$$ $$\nu^{4}$$$$/25$$ $$\beta_{5}$$ $$=$$ $$\nu^{5}$$$$/25$$ $$\beta_{6}$$ $$=$$ $$\nu^{6}$$$$/125$$ $$\beta_{7}$$ $$=$$ $$\nu^{7}$$$$/125$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$25 \beta_{4}$$ $$\nu^{5}$$ $$=$$ $$25 \beta_{5}$$ $$\nu^{6}$$ $$=$$ $$125 \beta_{6}$$ $$\nu^{7}$$ $$=$$ $$125 \beta_{7}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/245\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$1 - \beta_{4}$$ $$-\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
68.1
 −2.15988 − 0.578737i 2.15988 + 0.578737i −0.578737 + 2.15988i 0.578737 − 2.15988i −0.578737 − 2.15988i 0.578737 + 2.15988i −2.15988 + 0.578737i 2.15988 − 0.578737i
−0.366025 1.36603i −2.15988 0.578737i 0 −0.578737 2.15988i 3.16228i 0 −2.00000 2.00000i 1.73205 + 1.00000i −2.73861 + 1.58114i
68.2 −0.366025 1.36603i 2.15988 + 0.578737i 0 0.578737 + 2.15988i 3.16228i 0 −2.00000 2.00000i 1.73205 + 1.00000i 2.73861 1.58114i
117.1 1.36603 0.366025i −0.578737 + 2.15988i 0 −2.15988 + 0.578737i 3.16228i 0 −2.00000 + 2.00000i −1.73205 1.00000i −2.73861 + 1.58114i
117.2 1.36603 0.366025i 0.578737 2.15988i 0 2.15988 0.578737i 3.16228i 0 −2.00000 + 2.00000i −1.73205 1.00000i 2.73861 1.58114i
178.1 1.36603 + 0.366025i −0.578737 2.15988i 0 −2.15988 0.578737i 3.16228i 0 −2.00000 2.00000i −1.73205 + 1.00000i −2.73861 1.58114i
178.2 1.36603 + 0.366025i 0.578737 + 2.15988i 0 2.15988 + 0.578737i 3.16228i 0 −2.00000 2.00000i −1.73205 + 1.00000i 2.73861 + 1.58114i
227.1 −0.366025 + 1.36603i −2.15988 + 0.578737i 0 −0.578737 + 2.15988i 3.16228i 0 −2.00000 + 2.00000i 1.73205 1.00000i −2.73861 1.58114i
227.2 −0.366025 + 1.36603i 2.15988 0.578737i 0 0.578737 2.15988i 3.16228i 0 −2.00000 + 2.00000i 1.73205 1.00000i 2.73861 + 1.58114i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 227.2 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.c odd 4 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
35.f even 4 1 inner
35.k even 12 1 inner
35.l odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.l.c 8
5.c odd 4 1 inner 245.2.l.c 8
7.b odd 2 1 inner 245.2.l.c 8
7.c even 3 1 35.2.f.a 4
7.c even 3 1 inner 245.2.l.c 8
7.d odd 6 1 35.2.f.a 4
7.d odd 6 1 inner 245.2.l.c 8
21.g even 6 1 315.2.p.c 4
21.h odd 6 1 315.2.p.c 4
28.f even 6 1 560.2.bj.a 4
28.g odd 6 1 560.2.bj.a 4
35.f even 4 1 inner 245.2.l.c 8
35.i odd 6 1 175.2.f.c 4
35.j even 6 1 175.2.f.c 4
35.k even 12 1 35.2.f.a 4
35.k even 12 1 175.2.f.c 4
35.k even 12 1 inner 245.2.l.c 8
35.l odd 12 1 35.2.f.a 4
35.l odd 12 1 175.2.f.c 4
35.l odd 12 1 inner 245.2.l.c 8
105.w odd 12 1 315.2.p.c 4
105.x even 12 1 315.2.p.c 4
140.w even 12 1 560.2.bj.a 4
140.x odd 12 1 560.2.bj.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.f.a 4 7.c even 3 1
35.2.f.a 4 7.d odd 6 1
35.2.f.a 4 35.k even 12 1
35.2.f.a 4 35.l odd 12 1
175.2.f.c 4 35.i odd 6 1
175.2.f.c 4 35.j even 6 1
175.2.f.c 4 35.k even 12 1
175.2.f.c 4 35.l odd 12 1
245.2.l.c 8 1.a even 1 1 trivial
245.2.l.c 8 5.c odd 4 1 inner
245.2.l.c 8 7.b odd 2 1 inner
245.2.l.c 8 7.c even 3 1 inner
245.2.l.c 8 7.d odd 6 1 inner
245.2.l.c 8 35.f even 4 1 inner
245.2.l.c 8 35.k even 12 1 inner
245.2.l.c 8 35.l odd 12 1 inner
315.2.p.c 4 21.g even 6 1
315.2.p.c 4 21.h odd 6 1
315.2.p.c 4 105.w odd 12 1
315.2.p.c 4 105.x even 12 1
560.2.bj.a 4 28.f even 6 1
560.2.bj.a 4 28.g odd 6 1
560.2.bj.a 4 140.w even 12 1
560.2.bj.a 4 140.x odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 2 T_{2}^{3} + 2 T_{2}^{2} - 4 T_{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(245, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 2 T^{2} )^{4}( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2}$$
$3$ $$1 + 17 T^{4} + 208 T^{8} + 1377 T^{12} + 6561 T^{16}$$
$5$ $$1 - 25 T^{4} + 625 T^{8}$$
$7$ 1
$11$ $$( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} )^{4}$$
$13$ $$( 1 + 103 T^{4} + 28561 T^{8} )^{2}$$
$17$ $$1 - 263 T^{4} - 14352 T^{8} - 21966023 T^{12} + 6975757441 T^{16}$$
$19$ $$( 1 - 28 T^{2} + 423 T^{4} - 10108 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$( 1 + 4 T + 8 T^{2} - 152 T^{3} - 833 T^{4} - 3496 T^{5} + 4232 T^{6} + 48668 T^{7} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 49 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 + 52 T^{2} + 1743 T^{4} + 49972 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 12 T + 72 T^{2} + 24 T^{3} - 1513 T^{4} + 888 T^{5} + 98568 T^{6} - 607836 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 + 8 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 + 6 T + 18 T^{2} + 258 T^{3} + 1849 T^{4} )^{4}$$
$47$ $$1 + 2017 T^{4} - 811392 T^{8} + 9842316577 T^{12} + 23811286661761 T^{16}$$
$53$ $$( 1 + 2 T + 2 T^{2} - 208 T^{3} - 3017 T^{4} - 11024 T^{5} + 5618 T^{6} + 297754 T^{7} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 - 28 T^{2} - 2697 T^{4} - 97468 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 82 T^{2} + 3003 T^{4} + 305122 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 - 2 T + 2 T^{2} + 264 T^{3} - 4753 T^{4} + 17688 T^{5} + 8978 T^{6} - 601526 T^{7} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 + 6 T + 71 T^{2} )^{8}$$
$73$ $$( 1 - 5329 T^{4} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 - 142 T^{2} + 6241 T^{4} )^{2}( 1 + 131 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 7538 T^{4} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 138 T^{2} + 11123 T^{4} - 1093098 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 + 16903 T^{4} + 88529281 T^{8} )^{2}$$