# Properties

 Label 245.3.h.b Level $245$ Weight $3$ Character orbit 245.h Analytic conductor $6.676$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 245.h (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.67576647683$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 5x^{2} + 25$$ x^4 - 5*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} - 2 \beta_1 q^{3} + ( - 3 \beta_{2} + 3) q^{4} + ( - \beta_{3} + \beta_1) q^{5} - 2 \beta_{3} q^{6} + 7 q^{8} + 11 \beta_{2} q^{9}+O(q^{10})$$ q + b2 * q^2 - 2*b1 * q^3 + (-3*b2 + 3) * q^4 + (-b3 + b1) * q^5 - 2*b3 * q^6 + 7 * q^8 + 11*b2 * q^9 $$q + \beta_{2} q^{2} - 2 \beta_1 q^{3} + ( - 3 \beta_{2} + 3) q^{4} + ( - \beta_{3} + \beta_1) q^{5} - 2 \beta_{3} q^{6} + 7 q^{8} + 11 \beta_{2} q^{9} + \beta_1 q^{10} + (2 \beta_{2} - 2) q^{11} + (6 \beta_{3} - 6 \beta_1) q^{12} - 6 \beta_{3} q^{13} - 10 q^{15} - 5 \beta_{2} q^{16} - 12 \beta_1 q^{17} + (11 \beta_{2} - 11) q^{18} + (6 \beta_{3} - 6 \beta_1) q^{19} - 3 \beta_{3} q^{20} - 2 q^{22} - 26 \beta_{2} q^{23} - 14 \beta_1 q^{24} + ( - 5 \beta_{2} + 5) q^{25} + ( - 6 \beta_{3} + 6 \beta_1) q^{26} - 4 \beta_{3} q^{27} - 22 q^{29} - 10 \beta_{2} q^{30} + 24 \beta_1 q^{31} + ( - 33 \beta_{2} + 33) q^{32} + ( - 4 \beta_{3} + 4 \beta_1) q^{33} - 12 \beta_{3} q^{34} + 33 q^{36} - 14 \beta_{2} q^{37} - 6 \beta_1 q^{38} + (60 \beta_{2} - 60) q^{39} + ( - 7 \beta_{3} + 7 \beta_1) q^{40} - 12 \beta_{3} q^{41} - 34 q^{43} + 6 \beta_{2} q^{44} + 11 \beta_1 q^{45} + ( - 26 \beta_{2} + 26) q^{46} + (12 \beta_{3} - 12 \beta_1) q^{47} + 10 \beta_{3} q^{48} + 5 q^{50} + 120 \beta_{2} q^{51} - 18 \beta_1 q^{52} + ( - 34 \beta_{2} + 34) q^{53} + ( - 4 \beta_{3} + 4 \beta_1) q^{54} + 2 \beta_{3} q^{55} + 60 q^{57} - 22 \beta_{2} q^{58} + 18 \beta_1 q^{59} + (30 \beta_{2} - 30) q^{60} + ( - 42 \beta_{3} + 42 \beta_1) q^{61} + 24 \beta_{3} q^{62} + 13 q^{64} - 30 \beta_{2} q^{65} + 4 \beta_1 q^{66} + (14 \beta_{2} - 14) q^{67} + (36 \beta_{3} - 36 \beta_1) q^{68} + 52 \beta_{3} q^{69} + 62 q^{71} + 77 \beta_{2} q^{72} + 24 \beta_1 q^{73} + ( - 14 \beta_{2} + 14) q^{74} + (10 \beta_{3} - 10 \beta_1) q^{75} + 18 \beta_{3} q^{76} - 60 q^{78} - 38 \beta_{2} q^{79} - 5 \beta_1 q^{80} + ( - 59 \beta_{2} + 59) q^{81} + ( - 12 \beta_{3} + 12 \beta_1) q^{82} + 18 \beta_{3} q^{83} - 60 q^{85} - 34 \beta_{2} q^{86} + 44 \beta_1 q^{87} + (14 \beta_{2} - 14) q^{88} + (12 \beta_{3} - 12 \beta_1) q^{89} + 11 \beta_{3} q^{90} - 78 q^{92} - 240 \beta_{2} q^{93} - 12 \beta_1 q^{94} + (30 \beta_{2} - 30) q^{95} + (66 \beta_{3} - 66 \beta_1) q^{96} - 12 \beta_{3} q^{97} - 22 q^{99}+O(q^{100})$$ q + b2 * q^2 - 2*b1 * q^3 + (-3*b2 + 3) * q^4 + (-b3 + b1) * q^5 - 2*b3 * q^6 + 7 * q^8 + 11*b2 * q^9 + b1 * q^10 + (2*b2 - 2) * q^11 + (6*b3 - 6*b1) * q^12 - 6*b3 * q^13 - 10 * q^15 - 5*b2 * q^16 - 12*b1 * q^17 + (11*b2 - 11) * q^18 + (6*b3 - 6*b1) * q^19 - 3*b3 * q^20 - 2 * q^22 - 26*b2 * q^23 - 14*b1 * q^24 + (-5*b2 + 5) * q^25 + (-6*b3 + 6*b1) * q^26 - 4*b3 * q^27 - 22 * q^29 - 10*b2 * q^30 + 24*b1 * q^31 + (-33*b2 + 33) * q^32 + (-4*b3 + 4*b1) * q^33 - 12*b3 * q^34 + 33 * q^36 - 14*b2 * q^37 - 6*b1 * q^38 + (60*b2 - 60) * q^39 + (-7*b3 + 7*b1) * q^40 - 12*b3 * q^41 - 34 * q^43 + 6*b2 * q^44 + 11*b1 * q^45 + (-26*b2 + 26) * q^46 + (12*b3 - 12*b1) * q^47 + 10*b3 * q^48 + 5 * q^50 + 120*b2 * q^51 - 18*b1 * q^52 + (-34*b2 + 34) * q^53 + (-4*b3 + 4*b1) * q^54 + 2*b3 * q^55 + 60 * q^57 - 22*b2 * q^58 + 18*b1 * q^59 + (30*b2 - 30) * q^60 + (-42*b3 + 42*b1) * q^61 + 24*b3 * q^62 + 13 * q^64 - 30*b2 * q^65 + 4*b1 * q^66 + (14*b2 - 14) * q^67 + (36*b3 - 36*b1) * q^68 + 52*b3 * q^69 + 62 * q^71 + 77*b2 * q^72 + 24*b1 * q^73 + (-14*b2 + 14) * q^74 + (10*b3 - 10*b1) * q^75 + 18*b3 * q^76 - 60 * q^78 - 38*b2 * q^79 - 5*b1 * q^80 + (-59*b2 + 59) * q^81 + (-12*b3 + 12*b1) * q^82 + 18*b3 * q^83 - 60 * q^85 - 34*b2 * q^86 + 44*b1 * q^87 + (14*b2 - 14) * q^88 + (12*b3 - 12*b1) * q^89 + 11*b3 * q^90 - 78 * q^92 - 240*b2 * q^93 - 12*b1 * q^94 + (30*b2 - 30) * q^95 + (66*b3 - 66*b1) * q^96 - 12*b3 * q^97 - 22 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 6 q^{4} + 28 q^{8} + 22 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 + 6 * q^4 + 28 * q^8 + 22 * q^9 $$4 q + 2 q^{2} + 6 q^{4} + 28 q^{8} + 22 q^{9} - 4 q^{11} - 40 q^{15} - 10 q^{16} - 22 q^{18} - 8 q^{22} - 52 q^{23} + 10 q^{25} - 88 q^{29} - 20 q^{30} + 66 q^{32} + 132 q^{36} - 28 q^{37} - 120 q^{39} - 136 q^{43} + 12 q^{44} + 52 q^{46} + 20 q^{50} + 240 q^{51} + 68 q^{53} + 240 q^{57} - 44 q^{58} - 60 q^{60} + 52 q^{64} - 60 q^{65} - 28 q^{67} + 248 q^{71} + 154 q^{72} + 28 q^{74} - 240 q^{78} - 76 q^{79} + 118 q^{81} - 240 q^{85} - 68 q^{86} - 28 q^{88} - 312 q^{92} - 480 q^{93} - 60 q^{95} - 88 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 + 6 * q^4 + 28 * q^8 + 22 * q^9 - 4 * q^11 - 40 * q^15 - 10 * q^16 - 22 * q^18 - 8 * q^22 - 52 * q^23 + 10 * q^25 - 88 * q^29 - 20 * q^30 + 66 * q^32 + 132 * q^36 - 28 * q^37 - 120 * q^39 - 136 * q^43 + 12 * q^44 + 52 * q^46 + 20 * q^50 + 240 * q^51 + 68 * q^53 + 240 * q^57 - 44 * q^58 - 60 * q^60 + 52 * q^64 - 60 * q^65 - 28 * q^67 + 248 * q^71 + 154 * q^72 + 28 * q^74 - 240 * q^78 - 76 * q^79 + 118 * q^81 - 240 * q^85 - 68 * q^86 - 28 * q^88 - 312 * q^92 - 480 * q^93 - 60 * q^95 - 88 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 5$$ (v^2) / 5 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 5$$ (v^3) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$5\beta_{2}$$ 5*b2 $$\nu^{3}$$ $$=$$ $$5\beta_{3}$$ 5*b3

## Character values

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

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 1.93649 + 1.11803i −1.93649 − 1.11803i 1.93649 − 1.11803i −1.93649 + 1.11803i
0.500000 + 0.866025i −3.87298 2.23607i 1.50000 2.59808i 1.93649 1.11803i 4.47214i 0 7.00000 5.50000 + 9.52628i 1.93649 + 1.11803i
31.2 0.500000 + 0.866025i 3.87298 + 2.23607i 1.50000 2.59808i −1.93649 + 1.11803i 4.47214i 0 7.00000 5.50000 + 9.52628i −1.93649 1.11803i
166.1 0.500000 0.866025i −3.87298 + 2.23607i 1.50000 + 2.59808i 1.93649 + 1.11803i 4.47214i 0 7.00000 5.50000 9.52628i 1.93649 1.11803i
166.2 0.500000 0.866025i 3.87298 2.23607i 1.50000 + 2.59808i −1.93649 1.11803i 4.47214i 0 7.00000 5.50000 9.52628i −1.93649 + 1.11803i
 $$n$$: 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
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.3.h.b 4
7.b odd 2 1 inner 245.3.h.b 4
7.c even 3 1 35.3.d.a 2
7.c even 3 1 inner 245.3.h.b 4
7.d odd 6 1 35.3.d.a 2
7.d odd 6 1 inner 245.3.h.b 4
21.g even 6 1 315.3.h.b 2
21.h odd 6 1 315.3.h.b 2
28.f even 6 1 560.3.f.a 2
28.g odd 6 1 560.3.f.a 2
35.i odd 6 1 175.3.d.f 2
35.j even 6 1 175.3.d.f 2
35.k even 12 2 175.3.c.d 4
35.l odd 12 2 175.3.c.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.a 2 7.c even 3 1
35.3.d.a 2 7.d odd 6 1
175.3.c.d 4 35.k even 12 2
175.3.c.d 4 35.l odd 12 2
175.3.d.f 2 35.i odd 6 1
175.3.d.f 2 35.j even 6 1
245.3.h.b 4 1.a even 1 1 trivial
245.3.h.b 4 7.b odd 2 1 inner
245.3.h.b 4 7.c even 3 1 inner
245.3.h.b 4 7.d odd 6 1 inner
315.3.h.b 2 21.g even 6 1
315.3.h.b 2 21.h odd 6 1
560.3.f.a 2 28.f even 6 1
560.3.f.a 2 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} + 1$$ acting on $$S_{3}^{\mathrm{new}}(245, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$T^{4} - 20T^{2} + 400$$
$5$ $$T^{4} - 5T^{2} + 25$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 2 T + 4)^{2}$$
$13$ $$(T^{2} + 180)^{2}$$
$17$ $$T^{4} - 720 T^{2} + 518400$$
$19$ $$T^{4} - 180 T^{2} + 32400$$
$23$ $$(T^{2} + 26 T + 676)^{2}$$
$29$ $$(T + 22)^{4}$$
$31$ $$T^{4} - 2880 T^{2} + \cdots + 8294400$$
$37$ $$(T^{2} + 14 T + 196)^{2}$$
$41$ $$(T^{2} + 720)^{2}$$
$43$ $$(T + 34)^{4}$$
$47$ $$T^{4} - 720 T^{2} + 518400$$
$53$ $$(T^{2} - 34 T + 1156)^{2}$$
$59$ $$T^{4} - 1620 T^{2} + \cdots + 2624400$$
$61$ $$T^{4} - 8820 T^{2} + \cdots + 77792400$$
$67$ $$(T^{2} + 14 T + 196)^{2}$$
$71$ $$(T - 62)^{4}$$
$73$ $$T^{4} - 2880 T^{2} + \cdots + 8294400$$
$79$ $$(T^{2} + 38 T + 1444)^{2}$$
$83$ $$(T^{2} + 1620)^{2}$$
$89$ $$T^{4} - 720 T^{2} + 518400$$
$97$ $$(T^{2} + 720)^{2}$$