# Properties

 Label 245.4.e.k Level $245$ Weight $4$ Character orbit 245.e Analytic conductor $14.455$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 245.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.4554679514$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{11})$$ Defining polynomial: $$x^{4} + 11x^{2} + 121$$ x^4 + 11*x^2 + 121 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1 - 1) q^{2} - 5 \beta_{2} q^{3} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{4} + (5 \beta_{2} + 5) q^{5} + ( - 5 \beta_{3} - 5) q^{6} + ( - 2 \beta_{3} + 18) q^{8} + (2 \beta_{2} + 2) q^{9}+O(q^{10})$$ q + (-b2 + b1 - 1) * q^2 - 5*b2 * q^3 + (-2*b3 + 4*b2 - 2*b1) * q^4 + (5*b2 + 5) * q^5 + (-5*b3 - 5) * q^6 + (-2*b3 + 18) * q^8 + (2*b2 + 2) * q^9 $$q + ( - \beta_{2} + \beta_1 - 1) q^{2} - 5 \beta_{2} q^{3} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{4} + (5 \beta_{2} + 5) q^{5} + ( - 5 \beta_{3} - 5) q^{6} + ( - 2 \beta_{3} + 18) q^{8} + (2 \beta_{2} + 2) q^{9} + (5 \beta_{3} - 5 \beta_{2} + 5 \beta_1) q^{10} + (4 \beta_{3} + 33 \beta_{2} + 4 \beta_1) q^{11} + (20 \beta_{2} - 10 \beta_1 + 20) q^{12} + ( - 20 \beta_{3} - 5) q^{13} + 25 q^{15} + (36 \beta_{2} + 36) q^{16} + ( - 20 \beta_{3} - 35 \beta_{2} - 20 \beta_1) q^{17} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{18} + (70 \beta_{2} - 20 \beta_1 + 70) q^{19} + ( - 10 \beta_{3} - 20) q^{20} + (29 \beta_{3} - 11) q^{22} + (8 \beta_{2} - 28 \beta_1 + 8) q^{23} + ( - 10 \beta_{3} - 90 \beta_{2} - 10 \beta_1) q^{24} + 25 \beta_{2} q^{25} + (225 \beta_{2} - 25 \beta_1 + 225) q^{26} + 145 q^{27} + ( - 48 \beta_{3} - 129) q^{29} + ( - 25 \beta_{2} + 25 \beta_1 - 25) q^{30} + ( - 60 \beta_{3} + 10 \beta_{2} - 60 \beta_1) q^{31} + (20 \beta_{3} - 180 \beta_{2} + 20 \beta_1) q^{32} + (165 \beta_{2} + 20 \beta_1 + 165) q^{33} + ( - 15 \beta_{3} + 185) q^{34} + ( - 4 \beta_{3} - 8) q^{36} + ( - 164 \beta_{2} + 44 \beta_1 - 164) q^{37} + (90 \beta_{3} - 290 \beta_{2} + 90 \beta_1) q^{38} + ( - 100 \beta_{3} + 25 \beta_{2} - 100 \beta_1) q^{39} + (90 \beta_{2} + 10 \beta_1 + 90) q^{40} + (100 \beta_{3} - 150) q^{41} + (12 \beta_{3} - 58) q^{43} + ( - 44 \beta_{2} + 50 \beta_1 - 44) q^{44} + 10 \beta_{2} q^{45} + (36 \beta_{3} - 316 \beta_{2} + 36 \beta_1) q^{46} + ( - 15 \beta_{2} + 40 \beta_1 - 15) q^{47} + 180 q^{48} + (25 \beta_{3} + 25) q^{50} + ( - 175 \beta_{2} - 100 \beta_1 - 175) q^{51} + (90 \beta_{3} - 460 \beta_{2} + 90 \beta_1) q^{52} + (120 \beta_{3} + 270 \beta_{2} + 120 \beta_1) q^{53} + ( - 145 \beta_{2} + 145 \beta_1 - 145) q^{54} + (20 \beta_{3} - 165) q^{55} + (100 \beta_{3} + 350) q^{57} + (657 \beta_{2} - 177 \beta_1 + 657) q^{58} + (40 \beta_{3} - 190 \beta_{2} + 40 \beta_1) q^{59} + ( - 50 \beta_{3} + 100 \beta_{2} - 50 \beta_1) q^{60} + (540 \beta_{2} + 60 \beta_1 + 540) q^{61} + (70 \beta_{3} + 670) q^{62} + ( - 200 \beta_{3} - 112) q^{64} + ( - 25 \beta_{2} + 100 \beta_1 - 25) q^{65} + (145 \beta_{3} + 55 \beta_{2} + 145 \beta_1) q^{66} + (96 \beta_{3} + 234 \beta_{2} + 96 \beta_1) q^{67} + ( - 300 \beta_{2} + 10 \beta_1 - 300) q^{68} + (140 \beta_{3} + 40) q^{69} + (64 \beta_{3} - 528) q^{71} + (36 \beta_{2} + 4 \beta_1 + 36) q^{72} + ( - 200 \beta_{3} + 430 \beta_{2} - 200 \beta_1) q^{73} + ( - 208 \beta_{3} + 648 \beta_{2} - 208 \beta_1) q^{74} + (125 \beta_{2} + 125) q^{75} + ( - 220 \beta_{3} - 720) q^{76} + (125 \beta_{3} + 1125) q^{78} + ( - 79 \beta_{2} + 348 \beta_1 - 79) q^{79} + 180 \beta_{2} q^{80} - 671 \beta_{2} q^{81} + ( - 950 \beta_{2} - 50 \beta_1 - 950) q^{82} + (200 \beta_{3} - 20) q^{83} + ( - 100 \beta_{3} + 175) q^{85} + ( - 74 \beta_{2} - 46 \beta_1 - 74) q^{86} + ( - 240 \beta_{3} + 645 \beta_{2} - 240 \beta_1) q^{87} + (138 \beta_{3} + 682 \beta_{2} + 138 \beta_1) q^{88} + ( - 120 \beta_{2} - 380 \beta_1 - 120) q^{89} + (10 \beta_{3} + 10) q^{90} + ( - 128 \beta_{3} - 648) q^{92} + (50 \beta_{2} - 300 \beta_1 + 50) q^{93} + ( - 55 \beta_{3} + 455 \beta_{2} - 55 \beta_1) q^{94} + ( - 100 \beta_{3} + 350 \beta_{2} - 100 \beta_1) q^{95} + ( - 900 \beta_{2} + 100 \beta_1 - 900) q^{96} + ( - 180 \beta_{3} - 815) q^{97} + (8 \beta_{3} - 66) q^{99}+O(q^{100})$$ q + (-b2 + b1 - 1) * q^2 - 5*b2 * q^3 + (-2*b3 + 4*b2 - 2*b1) * q^4 + (5*b2 + 5) * q^5 + (-5*b3 - 5) * q^6 + (-2*b3 + 18) * q^8 + (2*b2 + 2) * q^9 + (5*b3 - 5*b2 + 5*b1) * q^10 + (4*b3 + 33*b2 + 4*b1) * q^11 + (20*b2 - 10*b1 + 20) * q^12 + (-20*b3 - 5) * q^13 + 25 * q^15 + (36*b2 + 36) * q^16 + (-20*b3 - 35*b2 - 20*b1) * q^17 + (2*b3 - 2*b2 + 2*b1) * q^18 + (70*b2 - 20*b1 + 70) * q^19 + (-10*b3 - 20) * q^20 + (29*b3 - 11) * q^22 + (8*b2 - 28*b1 + 8) * q^23 + (-10*b3 - 90*b2 - 10*b1) * q^24 + 25*b2 * q^25 + (225*b2 - 25*b1 + 225) * q^26 + 145 * q^27 + (-48*b3 - 129) * q^29 + (-25*b2 + 25*b1 - 25) * q^30 + (-60*b3 + 10*b2 - 60*b1) * q^31 + (20*b3 - 180*b2 + 20*b1) * q^32 + (165*b2 + 20*b1 + 165) * q^33 + (-15*b3 + 185) * q^34 + (-4*b3 - 8) * q^36 + (-164*b2 + 44*b1 - 164) * q^37 + (90*b3 - 290*b2 + 90*b1) * q^38 + (-100*b3 + 25*b2 - 100*b1) * q^39 + (90*b2 + 10*b1 + 90) * q^40 + (100*b3 - 150) * q^41 + (12*b3 - 58) * q^43 + (-44*b2 + 50*b1 - 44) * q^44 + 10*b2 * q^45 + (36*b3 - 316*b2 + 36*b1) * q^46 + (-15*b2 + 40*b1 - 15) * q^47 + 180 * q^48 + (25*b3 + 25) * q^50 + (-175*b2 - 100*b1 - 175) * q^51 + (90*b3 - 460*b2 + 90*b1) * q^52 + (120*b3 + 270*b2 + 120*b1) * q^53 + (-145*b2 + 145*b1 - 145) * q^54 + (20*b3 - 165) * q^55 + (100*b3 + 350) * q^57 + (657*b2 - 177*b1 + 657) * q^58 + (40*b3 - 190*b2 + 40*b1) * q^59 + (-50*b3 + 100*b2 - 50*b1) * q^60 + (540*b2 + 60*b1 + 540) * q^61 + (70*b3 + 670) * q^62 + (-200*b3 - 112) * q^64 + (-25*b2 + 100*b1 - 25) * q^65 + (145*b3 + 55*b2 + 145*b1) * q^66 + (96*b3 + 234*b2 + 96*b1) * q^67 + (-300*b2 + 10*b1 - 300) * q^68 + (140*b3 + 40) * q^69 + (64*b3 - 528) * q^71 + (36*b2 + 4*b1 + 36) * q^72 + (-200*b3 + 430*b2 - 200*b1) * q^73 + (-208*b3 + 648*b2 - 208*b1) * q^74 + (125*b2 + 125) * q^75 + (-220*b3 - 720) * q^76 + (125*b3 + 1125) * q^78 + (-79*b2 + 348*b1 - 79) * q^79 + 180*b2 * q^80 - 671*b2 * q^81 + (-950*b2 - 50*b1 - 950) * q^82 + (200*b3 - 20) * q^83 + (-100*b3 + 175) * q^85 + (-74*b2 - 46*b1 - 74) * q^86 + (-240*b3 + 645*b2 - 240*b1) * q^87 + (138*b3 + 682*b2 + 138*b1) * q^88 + (-120*b2 - 380*b1 - 120) * q^89 + (10*b3 + 10) * q^90 + (-128*b3 - 648) * q^92 + (50*b2 - 300*b1 + 50) * q^93 + (-55*b3 + 455*b2 - 55*b1) * q^94 + (-100*b3 + 350*b2 - 100*b1) * q^95 + (-900*b2 + 100*b1 - 900) * q^96 + (-180*b3 - 815) * q^97 + (8*b3 - 66) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 10 q^{3} - 8 q^{4} + 10 q^{5} - 20 q^{6} + 72 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 + 10 * q^3 - 8 * q^4 + 10 * q^5 - 20 * q^6 + 72 * q^8 + 4 * q^9 $$4 q - 2 q^{2} + 10 q^{3} - 8 q^{4} + 10 q^{5} - 20 q^{6} + 72 q^{8} + 4 q^{9} + 10 q^{10} - 66 q^{11} + 40 q^{12} - 20 q^{13} + 100 q^{15} + 72 q^{16} + 70 q^{17} + 4 q^{18} + 140 q^{19} - 80 q^{20} - 44 q^{22} + 16 q^{23} + 180 q^{24} - 50 q^{25} + 450 q^{26} + 580 q^{27} - 516 q^{29} - 50 q^{30} - 20 q^{31} + 360 q^{32} + 330 q^{33} + 740 q^{34} - 32 q^{36} - 328 q^{37} + 580 q^{38} - 50 q^{39} + 180 q^{40} - 600 q^{41} - 232 q^{43} - 88 q^{44} - 20 q^{45} + 632 q^{46} - 30 q^{47} + 720 q^{48} + 100 q^{50} - 350 q^{51} + 920 q^{52} - 540 q^{53} - 290 q^{54} - 660 q^{55} + 1400 q^{57} + 1314 q^{58} + 380 q^{59} - 200 q^{60} + 1080 q^{61} + 2680 q^{62} - 448 q^{64} - 50 q^{65} - 110 q^{66} - 468 q^{67} - 600 q^{68} + 160 q^{69} - 2112 q^{71} + 72 q^{72} - 860 q^{73} - 1296 q^{74} + 250 q^{75} - 2880 q^{76} + 4500 q^{78} - 158 q^{79} - 360 q^{80} + 1342 q^{81} - 1900 q^{82} - 80 q^{83} + 700 q^{85} - 148 q^{86} - 1290 q^{87} - 1364 q^{88} - 240 q^{89} + 40 q^{90} - 2592 q^{92} + 100 q^{93} - 910 q^{94} - 700 q^{95} - 1800 q^{96} - 3260 q^{97} - 264 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 + 10 * q^3 - 8 * q^4 + 10 * q^5 - 20 * q^6 + 72 * q^8 + 4 * q^9 + 10 * q^10 - 66 * q^11 + 40 * q^12 - 20 * q^13 + 100 * q^15 + 72 * q^16 + 70 * q^17 + 4 * q^18 + 140 * q^19 - 80 * q^20 - 44 * q^22 + 16 * q^23 + 180 * q^24 - 50 * q^25 + 450 * q^26 + 580 * q^27 - 516 * q^29 - 50 * q^30 - 20 * q^31 + 360 * q^32 + 330 * q^33 + 740 * q^34 - 32 * q^36 - 328 * q^37 + 580 * q^38 - 50 * q^39 + 180 * q^40 - 600 * q^41 - 232 * q^43 - 88 * q^44 - 20 * q^45 + 632 * q^46 - 30 * q^47 + 720 * q^48 + 100 * q^50 - 350 * q^51 + 920 * q^52 - 540 * q^53 - 290 * q^54 - 660 * q^55 + 1400 * q^57 + 1314 * q^58 + 380 * q^59 - 200 * q^60 + 1080 * q^61 + 2680 * q^62 - 448 * q^64 - 50 * q^65 - 110 * q^66 - 468 * q^67 - 600 * q^68 + 160 * q^69 - 2112 * q^71 + 72 * q^72 - 860 * q^73 - 1296 * q^74 + 250 * q^75 - 2880 * q^76 + 4500 * q^78 - 158 * q^79 - 360 * q^80 + 1342 * q^81 - 1900 * q^82 - 80 * q^83 + 700 * q^85 - 148 * q^86 - 1290 * q^87 - 1364 * q^88 - 240 * q^89 + 40 * q^90 - 2592 * q^92 + 100 * q^93 - 910 * q^94 - 700 * q^95 - 1800 * q^96 - 3260 * q^97 - 264 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 11x^{2} + 121$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 11$$ (v^2) / 11 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 11$$ (v^3) / 11
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$11\beta_{2}$$ 11*b2 $$\nu^{3}$$ $$=$$ $$11\beta_{3}$$ 11*b3

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1
 −1.65831 − 2.87228i 1.65831 + 2.87228i −1.65831 + 2.87228i 1.65831 − 2.87228i
−2.15831 3.73831i 2.50000 4.33013i −5.31662 + 9.20866i 2.50000 + 4.33013i −21.5831 0 11.3668 1.00000 + 1.73205i 10.7916 18.6915i
116.2 1.15831 + 2.00626i 2.50000 4.33013i 1.31662 2.28046i 2.50000 + 4.33013i 11.5831 0 24.6332 1.00000 + 1.73205i −5.79156 + 10.0313i
226.1 −2.15831 + 3.73831i 2.50000 + 4.33013i −5.31662 9.20866i 2.50000 4.33013i −21.5831 0 11.3668 1.00000 1.73205i 10.7916 + 18.6915i
226.2 1.15831 2.00626i 2.50000 + 4.33013i 1.31662 + 2.28046i 2.50000 4.33013i 11.5831 0 24.6332 1.00000 1.73205i −5.79156 10.0313i
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.4.e.k 4
7.b odd 2 1 245.4.e.j 4
7.c even 3 1 245.4.a.i 2
7.c even 3 1 inner 245.4.e.k 4
7.d odd 6 1 245.4.a.j yes 2
7.d odd 6 1 245.4.e.j 4
21.g even 6 1 2205.4.a.w 2
21.h odd 6 1 2205.4.a.x 2
35.i odd 6 1 1225.4.a.p 2
35.j even 6 1 1225.4.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.i 2 7.c even 3 1
245.4.a.j yes 2 7.d odd 6 1
245.4.e.j 4 7.b odd 2 1
245.4.e.j 4 7.d odd 6 1
245.4.e.k 4 1.a even 1 1 trivial
245.4.e.k 4 7.c even 3 1 inner
1225.4.a.p 2 35.i odd 6 1
1225.4.a.q 2 35.j even 6 1
2205.4.a.w 2 21.g even 6 1
2205.4.a.x 2 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(245, [\chi])$$:

 $$T_{2}^{4} + 2T_{2}^{3} + 14T_{2}^{2} - 20T_{2} + 100$$ T2^4 + 2*T2^3 + 14*T2^2 - 20*T2 + 100 $$T_{3}^{2} - 5T_{3} + 25$$ T3^2 - 5*T3 + 25

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 14 T^{2} - 20 T + 100$$
$3$ $$(T^{2} - 5 T + 25)^{2}$$
$5$ $$(T^{2} - 5 T + 25)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 66 T^{3} + 3443 T^{2} + \cdots + 833569$$
$13$ $$(T^{2} + 10 T - 4375)^{2}$$
$17$ $$T^{4} - 70 T^{3} + 8075 T^{2} + \cdots + 10080625$$
$19$ $$T^{4} - 140 T^{3} + 19100 T^{2} + \cdots + 250000$$
$23$ $$T^{4} - 16 T^{3} + 8816 T^{2} + \cdots + 73273600$$
$29$ $$(T^{2} + 258 T - 8703)^{2}$$
$31$ $$T^{4} + 20 T^{3} + \cdots + 1560250000$$
$37$ $$T^{4} + 328 T^{3} + \cdots + 31360000$$
$41$ $$(T^{2} + 300 T - 87500)^{2}$$
$43$ $$(T^{2} + 116 T + 1780)^{2}$$
$47$ $$T^{4} + 30 T^{3} + \cdots + 301890625$$
$53$ $$T^{4} + 540 T^{3} + \cdots + 7310250000$$
$59$ $$T^{4} - 380 T^{3} + \cdots + 342250000$$
$61$ $$T^{4} - 1080 T^{3} + \cdots + 63504000000$$
$67$ $$T^{4} + 468 T^{3} + \cdots + 2173424400$$
$71$ $$(T^{2} + 1056 T + 233728)^{2}$$
$73$ $$T^{4} + 860 T^{3} + \cdots + 65076010000$$
$79$ $$T^{4} + 158 T^{3} + \cdots + 1758018765409$$
$83$ $$(T^{2} + 40 T - 439600)^{2}$$
$89$ $$T^{4} + 240 T^{3} + \cdots + 2477476000000$$
$97$ $$(T^{2} + 1630 T + 307825)^{2}$$