# Properties

 Label 245.6.b.a Level 245 Weight 6 Character orbit 245.b Analytic conductor 39.294 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 245.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$39.2940358542$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ Defining polynomial: $$x^{2} - x + 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 5) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} -3 \beta q^{3} -12 q^{4} + ( 45 + 5 \beta ) q^{5} -132 q^{6} -20 \beta q^{8} -153 q^{9} +O(q^{10})$$ $$q -\beta q^{2} -3 \beta q^{3} -12 q^{4} + ( 45 + 5 \beta ) q^{5} -132 q^{6} -20 \beta q^{8} -153 q^{9} + ( 220 - 45 \beta ) q^{10} + 252 q^{11} + 36 \beta q^{12} -18 \beta q^{13} + ( 660 - 135 \beta ) q^{15} -1264 q^{16} -104 \beta q^{17} + 153 \beta q^{18} + 220 q^{19} + ( -540 - 60 \beta ) q^{20} -252 \beta q^{22} -367 \beta q^{23} -2640 q^{24} + ( 925 + 450 \beta ) q^{25} -792 q^{26} -270 \beta q^{27} -6930 q^{29} + ( -5940 - 660 \beta ) q^{30} -6752 q^{31} + 624 \beta q^{32} -756 \beta q^{33} -4576 q^{34} + 1836 q^{36} -2106 \beta q^{37} -220 \beta q^{38} -2376 q^{39} + ( 4400 - 900 \beta ) q^{40} + 198 q^{41} + 63 \beta q^{43} -3024 q^{44} + ( -6885 - 765 \beta ) q^{45} -16148 q^{46} -1589 \beta q^{47} + 3792 \beta q^{48} + ( 19800 - 925 \beta ) q^{50} -13728 q^{51} + 216 \beta q^{52} + 878 \beta q^{53} -11880 q^{54} + ( 11340 + 1260 \beta ) q^{55} -660 \beta q^{57} + 6930 \beta q^{58} + 24660 q^{59} + ( -7920 + 1620 \beta ) q^{60} + 5698 q^{61} + 6752 \beta q^{62} -12992 q^{64} + ( 3960 - 810 \beta ) q^{65} -33264 q^{66} + 6579 \beta q^{67} + 1248 \beta q^{68} -48444 q^{69} + 53352 q^{71} + 3060 \beta q^{72} + 10692 \beta q^{73} -92664 q^{74} + ( 59400 - 2775 \beta ) q^{75} -2640 q^{76} + 2376 \beta q^{78} + 51920 q^{79} + ( -56880 - 6320 \beta ) q^{80} -72819 q^{81} -198 \beta q^{82} -9323 \beta q^{83} + ( 22880 - 4680 \beta ) q^{85} + 2772 q^{86} + 20790 \beta q^{87} -5040 \beta q^{88} + 9990 q^{89} + ( -33660 + 6885 \beta ) q^{90} + 4404 \beta q^{92} + 20256 \beta q^{93} -69916 q^{94} + ( 9900 + 1100 \beta ) q^{95} + 82368 q^{96} -15264 \beta q^{97} -38556 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 24q^{4} + 90q^{5} - 264q^{6} - 306q^{9} + O(q^{10})$$ $$2q - 24q^{4} + 90q^{5} - 264q^{6} - 306q^{9} + 440q^{10} + 504q^{11} + 1320q^{15} - 2528q^{16} + 440q^{19} - 1080q^{20} - 5280q^{24} + 1850q^{25} - 1584q^{26} - 13860q^{29} - 11880q^{30} - 13504q^{31} - 9152q^{34} + 3672q^{36} - 4752q^{39} + 8800q^{40} + 396q^{41} - 6048q^{44} - 13770q^{45} - 32296q^{46} + 39600q^{50} - 27456q^{51} - 23760q^{54} + 22680q^{55} + 49320q^{59} - 15840q^{60} + 11396q^{61} - 25984q^{64} + 7920q^{65} - 66528q^{66} - 96888q^{69} + 106704q^{71} - 185328q^{74} + 118800q^{75} - 5280q^{76} + 103840q^{79} - 113760q^{80} - 145638q^{81} + 45760q^{85} + 5544q^{86} + 19980q^{89} - 67320q^{90} - 139832q^{94} + 19800q^{95} + 164736q^{96} - 77112q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 0.5 + 1.65831i 0.5 − 1.65831i
6.63325i 19.8997i −12.0000 45.0000 + 33.1662i −132.000 0 132.665i −153.000 220.000 298.496i
99.2 6.63325i 19.8997i −12.0000 45.0000 33.1662i −132.000 0 132.665i −153.000 220.000 + 298.496i
 $$n$$: e.g. 2-40 or 990-1000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.6.b.a 2
5.b even 2 1 inner 245.6.b.a 2
7.b odd 2 1 5.6.b.a 2
21.c even 2 1 45.6.b.b 2
28.d even 2 1 80.6.c.a 2
35.c odd 2 1 5.6.b.a 2
35.f even 4 2 25.6.a.c 2
56.e even 2 1 320.6.c.g 2
56.h odd 2 1 320.6.c.f 2
84.h odd 2 1 720.6.f.f 2
105.g even 2 1 45.6.b.b 2
105.k odd 4 2 225.6.a.n 2
140.c even 2 1 80.6.c.a 2
140.j odd 4 2 400.6.a.t 2
280.c odd 2 1 320.6.c.f 2
280.n even 2 1 320.6.c.g 2
420.o odd 2 1 720.6.f.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.b.a 2 7.b odd 2 1
5.6.b.a 2 35.c odd 2 1
25.6.a.c 2 35.f even 4 2
45.6.b.b 2 21.c even 2 1
45.6.b.b 2 105.g even 2 1
80.6.c.a 2 28.d even 2 1
80.6.c.a 2 140.c even 2 1
225.6.a.n 2 105.k odd 4 2
245.6.b.a 2 1.a even 1 1 trivial
245.6.b.a 2 5.b even 2 1 inner
320.6.c.f 2 56.h odd 2 1
320.6.c.f 2 280.c odd 2 1
320.6.c.g 2 56.e even 2 1
320.6.c.g 2 280.n even 2 1
400.6.a.t 2 140.j odd 4 2
720.6.f.f 2 84.h odd 2 1
720.6.f.f 2 420.o odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} + 44$$ $$T_{19} - 220$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 20 T^{2} + 1024 T^{4}$$
$3$ $$( 1 - 24 T + 243 T^{2} )( 1 + 24 T + 243 T^{2} )$$
$5$ $$1 - 90 T + 3125 T^{2}$$
$7$ 1
$11$ $$( 1 - 252 T + 161051 T^{2} )^{2}$$
$13$ $$1 - 728330 T^{2} + 137858491849 T^{4}$$
$17$ $$1 - 2363810 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 - 220 T + 2476099 T^{2} )^{2}$$
$23$ $$1 - 6946370 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 + 6930 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 + 6752 T + 28629151 T^{2} )^{2}$$
$37$ $$1 + 56462470 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 - 198 T + 115856201 T^{2} )^{2}$$
$43$ $$1 - 293842250 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 - 347593490 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 - 802472090 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 - 24660 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 - 5698 T + 844596301 T^{2} )^{2}$$
$67$ $$1 - 795787610 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 - 53352 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 + 883886830 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 - 51920 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 - 4053674810 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 - 9990 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 - 6923133890 T^{2} + 73742412689492826049 T^{4}$$