# Properties

 Label 245.6.a.a Level $245$ Weight $6$ Character orbit 245.a Self dual yes Analytic conductor $39.294$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 245.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$39.2940358542$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 8 q^{2} - q^{3} + 32 q^{4} - 25 q^{5} + 8 q^{6} - 242 q^{9}+O(q^{10})$$ q - 8 * q^2 - q^3 + 32 * q^4 - 25 * q^5 + 8 * q^6 - 242 * q^9 $$q - 8 q^{2} - q^{3} + 32 q^{4} - 25 q^{5} + 8 q^{6} - 242 q^{9} + 200 q^{10} - 453 q^{11} - 32 q^{12} + 969 q^{13} + 25 q^{15} - 1024 q^{16} - 1637 q^{17} + 1936 q^{18} + 1550 q^{19} - 800 q^{20} + 3624 q^{22} - 1654 q^{23} + 625 q^{25} - 7752 q^{26} + 485 q^{27} - 4985 q^{29} - 200 q^{30} - 1192 q^{31} + 8192 q^{32} + 453 q^{33} + 13096 q^{34} - 7744 q^{36} - 11018 q^{37} - 12400 q^{38} - 969 q^{39} + 1728 q^{41} - 10814 q^{43} - 14496 q^{44} + 6050 q^{45} + 13232 q^{46} - 26237 q^{47} + 1024 q^{48} - 5000 q^{50} + 1637 q^{51} + 31008 q^{52} + 25936 q^{53} - 3880 q^{54} + 11325 q^{55} - 1550 q^{57} + 39880 q^{58} + 4580 q^{59} + 800 q^{60} + 12488 q^{61} + 9536 q^{62} - 32768 q^{64} - 24225 q^{65} - 3624 q^{66} - 15848 q^{67} - 52384 q^{68} + 1654 q^{69} + 51792 q^{71} - 4846 q^{73} + 88144 q^{74} - 625 q^{75} + 49600 q^{76} + 7752 q^{78} + 62765 q^{79} + 25600 q^{80} + 58321 q^{81} - 13824 q^{82} + 23644 q^{83} + 40925 q^{85} + 86512 q^{86} + 4985 q^{87} + 147300 q^{89} - 48400 q^{90} - 52928 q^{92} + 1192 q^{93} + 209896 q^{94} - 38750 q^{95} - 8192 q^{96} + 8343 q^{97} + 109626 q^{99}+O(q^{100})$$ q - 8 * q^2 - q^3 + 32 * q^4 - 25 * q^5 + 8 * q^6 - 242 * q^9 + 200 * q^10 - 453 * q^11 - 32 * q^12 + 969 * q^13 + 25 * q^15 - 1024 * q^16 - 1637 * q^17 + 1936 * q^18 + 1550 * q^19 - 800 * q^20 + 3624 * q^22 - 1654 * q^23 + 625 * q^25 - 7752 * q^26 + 485 * q^27 - 4985 * q^29 - 200 * q^30 - 1192 * q^31 + 8192 * q^32 + 453 * q^33 + 13096 * q^34 - 7744 * q^36 - 11018 * q^37 - 12400 * q^38 - 969 * q^39 + 1728 * q^41 - 10814 * q^43 - 14496 * q^44 + 6050 * q^45 + 13232 * q^46 - 26237 * q^47 + 1024 * q^48 - 5000 * q^50 + 1637 * q^51 + 31008 * q^52 + 25936 * q^53 - 3880 * q^54 + 11325 * q^55 - 1550 * q^57 + 39880 * q^58 + 4580 * q^59 + 800 * q^60 + 12488 * q^61 + 9536 * q^62 - 32768 * q^64 - 24225 * q^65 - 3624 * q^66 - 15848 * q^67 - 52384 * q^68 + 1654 * q^69 + 51792 * q^71 - 4846 * q^73 + 88144 * q^74 - 625 * q^75 + 49600 * q^76 + 7752 * q^78 + 62765 * q^79 + 25600 * q^80 + 58321 * q^81 - 13824 * q^82 + 23644 * q^83 + 40925 * q^85 + 86512 * q^86 + 4985 * q^87 + 147300 * q^89 - 48400 * q^90 - 52928 * q^92 + 1192 * q^93 + 209896 * q^94 - 38750 * q^95 - 8192 * q^96 + 8343 * q^97 + 109626 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−8.00000 −1.00000 32.0000 −25.0000 8.00000 0 0 −242.000 200.000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.6.a.a 1
7.b odd 2 1 35.6.a.a 1
21.c even 2 1 315.6.a.a 1
28.d even 2 1 560.6.a.c 1
35.c odd 2 1 175.6.a.a 1
35.f even 4 2 175.6.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.a 1 7.b odd 2 1
175.6.a.a 1 35.c odd 2 1
175.6.b.b 2 35.f even 4 2
245.6.a.a 1 1.a even 1 1 trivial
315.6.a.a 1 21.c even 2 1
560.6.a.c 1 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(245))$$:

 $$T_{2} + 8$$ T2 + 8 $$T_{3} + 1$$ T3 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 8$$
$3$ $$T + 1$$
$5$ $$T + 25$$
$7$ $$T$$
$11$ $$T + 453$$
$13$ $$T - 969$$
$17$ $$T + 1637$$
$19$ $$T - 1550$$
$23$ $$T + 1654$$
$29$ $$T + 4985$$
$31$ $$T + 1192$$
$37$ $$T + 11018$$
$41$ $$T - 1728$$
$43$ $$T + 10814$$
$47$ $$T + 26237$$
$53$ $$T - 25936$$
$59$ $$T - 4580$$
$61$ $$T - 12488$$
$67$ $$T + 15848$$
$71$ $$T - 51792$$
$73$ $$T + 4846$$
$79$ $$T - 62765$$
$83$ $$T - 23644$$
$89$ $$T - 147300$$
$97$ $$T - 8343$$