# Properties

 Label 245.4.e.l 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: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{2} + ( - \beta_{2} - 3 \beta_1 - 1) q^{3} + ( - 3 \beta_{2} + 6 \beta_1 - 3) q^{4} - 5 \beta_{2} q^{5} + (8 \beta_{3} + 3) q^{6} + ( - 13 \beta_{3} + 3) q^{8} + (6 \beta_{3} - 8 \beta_{2} + 6 \beta_1) q^{9}+O(q^{10})$$ q + (b3 - 3*b2 + b1) * q^2 + (-b2 - 3*b1 - 1) * q^3 + (-3*b2 + 6*b1 - 3) * q^4 - 5*b2 * q^5 + (8*b3 + 3) * q^6 + (-13*b3 + 3) * q^8 + (6*b3 - 8*b2 + 6*b1) * q^9 $$q + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{2} + ( - \beta_{2} - 3 \beta_1 - 1) q^{3} + ( - 3 \beta_{2} + 6 \beta_1 - 3) q^{4} - 5 \beta_{2} q^{5} + (8 \beta_{3} + 3) q^{6} + ( - 13 \beta_{3} + 3) q^{8} + (6 \beta_{3} - 8 \beta_{2} + 6 \beta_1) q^{9} + ( - 15 \beta_{2} + 5 \beta_1 - 15) q^{10} + ( - 14 \beta_{2} - 10 \beta_1 - 14) q^{11} + (3 \beta_{3} - 33 \beta_{2} + 3 \beta_1) q^{12} + (10 \beta_{3} + 18) q^{13} + (15 \beta_{3} - 5) q^{15} + (12 \beta_{3} - 7 \beta_{2} + 12 \beta_1) q^{16} + ( - 38 \beta_{2} + 54 \beta_1 - 38) q^{17} + ( - 36 \beta_{2} + 26 \beta_1 - 36) q^{18} + ( - 26 \beta_{3} - 80 \beta_{2} - 26 \beta_1) q^{19} + ( - 30 \beta_{3} - 15) q^{20} + (16 \beta_{3} - 22) q^{22} + ( - 117 \beta_{3} - 11 \beta_{2} - 117 \beta_1) q^{23} + ( - 81 \beta_{2} - 22 \beta_1 - 81) q^{24} + ( - 25 \beta_{2} - 25) q^{25} + (48 \beta_{3} - 74 \beta_{2} + 48 \beta_1) q^{26} + (99 \beta_{3} + 1) q^{27} + (60 \beta_{3} - 125) q^{29} + (40 \beta_{3} - 15 \beta_{2} + 40 \beta_1) q^{30} + ( - 66 \beta_{2} - 100 \beta_1 - 66) q^{31} + ( - 21 \beta_{2} + 147 \beta_1 - 21) q^{32} + (52 \beta_{3} + 74 \beta_{2} + 52 \beta_1) q^{33} + ( - 200 \beta_{3} - 222) q^{34} + ( - 66 \beta_{3} - 96) q^{36} + (136 \beta_{3} - 208 \beta_{2} + 136 \beta_1) q^{37} + ( - 188 \beta_{2} + 2 \beta_1 - 188) q^{38} + (42 \beta_{2} - 44 \beta_1 + 42) q^{39} + ( - 65 \beta_{3} - 15 \beta_{2} - 65 \beta_1) q^{40} + (30 \beta_{3} + 53) q^{41} + ( - 5 \beta_{3} - 333) q^{43} + ( - 54 \beta_{3} - 78 \beta_{2} - 54 \beta_1) q^{44} + ( - 40 \beta_{2} + 30 \beta_1 - 40) q^{45} + (201 \beta_{2} - 340 \beta_1 + 201) q^{46} + (64 \beta_{3} + 98 \beta_{2} + 64 \beta_1) q^{47} + (9 \beta_{3} + 65) q^{48} + ( - 25 \beta_{3} - 75) q^{50} + (60 \beta_{3} - 286 \beta_{2} + 60 \beta_1) q^{51} + ( - 174 \beta_{2} + 138 \beta_1 - 174) q^{52} + (476 \beta_{2} - 142 \beta_1 + 476) q^{53} + (298 \beta_{3} - 201 \beta_{2} + 298 \beta_1) q^{54} + (50 \beta_{3} - 70) q^{55} + (266 \beta_{3} - 236) q^{57} + (55 \beta_{3} + 255 \beta_{2} + 55 \beta_1) q^{58} + (420 \beta_{2} - 266 \beta_1 + 420) q^{59} + ( - 165 \beta_{2} + 15 \beta_1 - 165) q^{60} + ( - 570 \beta_{3} - 49 \beta_{2} - 570 \beta_1) q^{61} + (234 \beta_{3} + 2) q^{62} + ( - 366 \beta_{3} - 301) q^{64} + (50 \beta_{3} - 90 \beta_{2} + 50 \beta_1) q^{65} + (118 \beta_{2} + 82 \beta_1 + 118) q^{66} + (643 \beta_{2} - 69 \beta_1 + 643) q^{67} + ( - 390 \beta_{3} + 762 \beta_{2} - 390 \beta_1) q^{68} + (150 \beta_{3} - 713) q^{69} + ( - 350 \beta_{3} + 532) q^{71} + ( - 86 \beta_{3} + 132 \beta_{2} - 86 \beta_1) q^{72} + ( - 86 \beta_{2} - 118 \beta_1 - 86) q^{73} + ( - 896 \beta_{2} + 616 \beta_1 - 896) q^{74} + (75 \beta_{3} + 25 \beta_{2} + 75 \beta_1) q^{75} + ( - 402 \beta_{3} + 72) q^{76} + (174 \beta_{3} + 214) q^{78} + ( - 214 \beta_{3} - 620 \beta_{2} - 214 \beta_1) q^{79} + ( - 35 \beta_{2} + 60 \beta_1 - 35) q^{80} + (377 \beta_{2} + 258 \beta_1 + 377) q^{81} + (143 \beta_{3} - 219 \beta_{2} + 143 \beta_1) q^{82} + ( - 5 \beta_{3} + 953) q^{83} + ( - 270 \beta_{3} - 190) q^{85} + ( - 348 \beta_{3} + 1009 \beta_{2} - 348 \beta_1) q^{86} + (485 \beta_{2} + 435 \beta_1 + 485) q^{87} + ( - 302 \beta_{2} - 212 \beta_1 - 302) q^{88} + (324 \beta_{3} - 325 \beta_{2} + 324 \beta_1) q^{89} + ( - 130 \beta_{3} - 180) q^{90} + (285 \beta_{3} + 1371) q^{92} + (298 \beta_{3} + 666 \beta_{2} + 298 \beta_1) q^{93} + (166 \beta_{2} + 94 \beta_1 + 166) q^{94} + ( - 400 \beta_{2} - 130 \beta_1 - 400) q^{95} + ( - 84 \beta_{3} - 861 \beta_{2} - 84 \beta_1) q^{96} + ( - 500 \beta_{3} + 314) q^{97} + ( - 4 \beta_{3} + 8) q^{99}+O(q^{100})$$ q + (b3 - 3*b2 + b1) * q^2 + (-b2 - 3*b1 - 1) * q^3 + (-3*b2 + 6*b1 - 3) * q^4 - 5*b2 * q^5 + (8*b3 + 3) * q^6 + (-13*b3 + 3) * q^8 + (6*b3 - 8*b2 + 6*b1) * q^9 + (-15*b2 + 5*b1 - 15) * q^10 + (-14*b2 - 10*b1 - 14) * q^11 + (3*b3 - 33*b2 + 3*b1) * q^12 + (10*b3 + 18) * q^13 + (15*b3 - 5) * q^15 + (12*b3 - 7*b2 + 12*b1) * q^16 + (-38*b2 + 54*b1 - 38) * q^17 + (-36*b2 + 26*b1 - 36) * q^18 + (-26*b3 - 80*b2 - 26*b1) * q^19 + (-30*b3 - 15) * q^20 + (16*b3 - 22) * q^22 + (-117*b3 - 11*b2 - 117*b1) * q^23 + (-81*b2 - 22*b1 - 81) * q^24 + (-25*b2 - 25) * q^25 + (48*b3 - 74*b2 + 48*b1) * q^26 + (99*b3 + 1) * q^27 + (60*b3 - 125) * q^29 + (40*b3 - 15*b2 + 40*b1) * q^30 + (-66*b2 - 100*b1 - 66) * q^31 + (-21*b2 + 147*b1 - 21) * q^32 + (52*b3 + 74*b2 + 52*b1) * q^33 + (-200*b3 - 222) * q^34 + (-66*b3 - 96) * q^36 + (136*b3 - 208*b2 + 136*b1) * q^37 + (-188*b2 + 2*b1 - 188) * q^38 + (42*b2 - 44*b1 + 42) * q^39 + (-65*b3 - 15*b2 - 65*b1) * q^40 + (30*b3 + 53) * q^41 + (-5*b3 - 333) * q^43 + (-54*b3 - 78*b2 - 54*b1) * q^44 + (-40*b2 + 30*b1 - 40) * q^45 + (201*b2 - 340*b1 + 201) * q^46 + (64*b3 + 98*b2 + 64*b1) * q^47 + (9*b3 + 65) * q^48 + (-25*b3 - 75) * q^50 + (60*b3 - 286*b2 + 60*b1) * q^51 + (-174*b2 + 138*b1 - 174) * q^52 + (476*b2 - 142*b1 + 476) * q^53 + (298*b3 - 201*b2 + 298*b1) * q^54 + (50*b3 - 70) * q^55 + (266*b3 - 236) * q^57 + (55*b3 + 255*b2 + 55*b1) * q^58 + (420*b2 - 266*b1 + 420) * q^59 + (-165*b2 + 15*b1 - 165) * q^60 + (-570*b3 - 49*b2 - 570*b1) * q^61 + (234*b3 + 2) * q^62 + (-366*b3 - 301) * q^64 + (50*b3 - 90*b2 + 50*b1) * q^65 + (118*b2 + 82*b1 + 118) * q^66 + (643*b2 - 69*b1 + 643) * q^67 + (-390*b3 + 762*b2 - 390*b1) * q^68 + (150*b3 - 713) * q^69 + (-350*b3 + 532) * q^71 + (-86*b3 + 132*b2 - 86*b1) * q^72 + (-86*b2 - 118*b1 - 86) * q^73 + (-896*b2 + 616*b1 - 896) * q^74 + (75*b3 + 25*b2 + 75*b1) * q^75 + (-402*b3 + 72) * q^76 + (174*b3 + 214) * q^78 + (-214*b3 - 620*b2 - 214*b1) * q^79 + (-35*b2 + 60*b1 - 35) * q^80 + (377*b2 + 258*b1 + 377) * q^81 + (143*b3 - 219*b2 + 143*b1) * q^82 + (-5*b3 + 953) * q^83 + (-270*b3 - 190) * q^85 + (-348*b3 + 1009*b2 - 348*b1) * q^86 + (485*b2 + 435*b1 + 485) * q^87 + (-302*b2 - 212*b1 - 302) * q^88 + (324*b3 - 325*b2 + 324*b1) * q^89 + (-130*b3 - 180) * q^90 + (285*b3 + 1371) * q^92 + (298*b3 + 666*b2 + 298*b1) * q^93 + (166*b2 + 94*b1 + 166) * q^94 + (-400*b2 - 130*b1 - 400) * q^95 + (-84*b3 - 861*b2 - 84*b1) * q^96 + (-500*b3 + 314) * q^97 + (-4*b3 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{2} - 2 q^{3} - 6 q^{4} + 10 q^{5} + 12 q^{6} + 12 q^{8} + 16 q^{9}+O(q^{10})$$ 4 * q + 6 * q^2 - 2 * q^3 - 6 * q^4 + 10 * q^5 + 12 * q^6 + 12 * q^8 + 16 * q^9 $$4 q + 6 q^{2} - 2 q^{3} - 6 q^{4} + 10 q^{5} + 12 q^{6} + 12 q^{8} + 16 q^{9} - 30 q^{10} - 28 q^{11} + 66 q^{12} + 72 q^{13} - 20 q^{15} + 14 q^{16} - 76 q^{17} - 72 q^{18} + 160 q^{19} - 60 q^{20} - 88 q^{22} + 22 q^{23} - 162 q^{24} - 50 q^{25} + 148 q^{26} + 4 q^{27} - 500 q^{29} + 30 q^{30} - 132 q^{31} - 42 q^{32} - 148 q^{33} - 888 q^{34} - 384 q^{36} + 416 q^{37} - 376 q^{38} + 84 q^{39} + 30 q^{40} + 212 q^{41} - 1332 q^{43} + 156 q^{44} - 80 q^{45} + 402 q^{46} - 196 q^{47} + 260 q^{48} - 300 q^{50} + 572 q^{51} - 348 q^{52} + 952 q^{53} + 402 q^{54} - 280 q^{55} - 944 q^{57} - 510 q^{58} + 840 q^{59} - 330 q^{60} + 98 q^{61} + 8 q^{62} - 1204 q^{64} + 180 q^{65} + 236 q^{66} + 1286 q^{67} - 1524 q^{68} - 2852 q^{69} + 2128 q^{71} - 264 q^{72} - 172 q^{73} - 1792 q^{74} - 50 q^{75} + 288 q^{76} + 856 q^{78} + 1240 q^{79} - 70 q^{80} + 754 q^{81} + 438 q^{82} + 3812 q^{83} - 760 q^{85} - 2018 q^{86} + 970 q^{87} - 604 q^{88} + 650 q^{89} - 720 q^{90} + 5484 q^{92} - 1332 q^{93} + 332 q^{94} - 800 q^{95} + 1722 q^{96} + 1256 q^{97} + 32 q^{99}+O(q^{100})$$ 4 * q + 6 * q^2 - 2 * q^3 - 6 * q^4 + 10 * q^5 + 12 * q^6 + 12 * q^8 + 16 * q^9 - 30 * q^10 - 28 * q^11 + 66 * q^12 + 72 * q^13 - 20 * q^15 + 14 * q^16 - 76 * q^17 - 72 * q^18 + 160 * q^19 - 60 * q^20 - 88 * q^22 + 22 * q^23 - 162 * q^24 - 50 * q^25 + 148 * q^26 + 4 * q^27 - 500 * q^29 + 30 * q^30 - 132 * q^31 - 42 * q^32 - 148 * q^33 - 888 * q^34 - 384 * q^36 + 416 * q^37 - 376 * q^38 + 84 * q^39 + 30 * q^40 + 212 * q^41 - 1332 * q^43 + 156 * q^44 - 80 * q^45 + 402 * q^46 - 196 * q^47 + 260 * q^48 - 300 * q^50 + 572 * q^51 - 348 * q^52 + 952 * q^53 + 402 * q^54 - 280 * q^55 - 944 * q^57 - 510 * q^58 + 840 * q^59 - 330 * q^60 + 98 * q^61 + 8 * q^62 - 1204 * q^64 + 180 * q^65 + 236 * q^66 + 1286 * q^67 - 1524 * q^68 - 2852 * q^69 + 2128 * q^71 - 264 * q^72 - 172 * q^73 - 1792 * q^74 - 50 * q^75 + 288 * q^76 + 856 * q^78 + 1240 * q^79 - 70 * q^80 + 754 * q^81 + 438 * q^82 + 3812 * q^83 - 760 * q^85 - 2018 * q^86 + 970 * q^87 - 604 * q^88 + 650 * q^89 - 720 * q^90 + 5484 * q^92 - 1332 * q^93 + 332 * q^94 - 800 * q^95 + 1722 * q^96 + 1256 * q^97 + 32 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1
 0.707107 − 1.22474i −0.707107 + 1.22474i 0.707107 + 1.22474i −0.707107 − 1.22474i
0.792893 + 1.37333i −2.62132 + 4.54026i 2.74264 4.75039i 2.50000 + 4.33013i −8.31371 0 21.3848 −0.242641 0.420266i −3.96447 + 6.86666i
116.2 2.20711 + 3.82282i 1.62132 2.80821i −5.74264 + 9.94655i 2.50000 + 4.33013i 14.3137 0 −15.3848 8.24264 + 14.2767i −11.0355 + 19.1141i
226.1 0.792893 1.37333i −2.62132 4.54026i 2.74264 + 4.75039i 2.50000 4.33013i −8.31371 0 21.3848 −0.242641 + 0.420266i −3.96447 6.86666i
226.2 2.20711 3.82282i 1.62132 + 2.80821i −5.74264 9.94655i 2.50000 4.33013i 14.3137 0 −15.3848 8.24264 14.2767i −11.0355 19.1141i
 $$n$$: 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.l 4
7.b odd 2 1 35.4.e.b 4
7.c even 3 1 245.4.a.h 2
7.c even 3 1 inner 245.4.e.l 4
7.d odd 6 1 35.4.e.b 4
7.d odd 6 1 245.4.a.g 2
21.c even 2 1 315.4.j.c 4
21.g even 6 1 315.4.j.c 4
21.g even 6 1 2205.4.a.bf 2
21.h odd 6 1 2205.4.a.bg 2
28.d even 2 1 560.4.q.i 4
28.f even 6 1 560.4.q.i 4
35.c odd 2 1 175.4.e.c 4
35.f even 4 2 175.4.k.c 8
35.i odd 6 1 175.4.e.c 4
35.i odd 6 1 1225.4.a.x 2
35.j even 6 1 1225.4.a.v 2
35.k even 12 2 175.4.k.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.b 4 7.b odd 2 1
35.4.e.b 4 7.d odd 6 1
175.4.e.c 4 35.c odd 2 1
175.4.e.c 4 35.i odd 6 1
175.4.k.c 8 35.f even 4 2
175.4.k.c 8 35.k even 12 2
245.4.a.g 2 7.d odd 6 1
245.4.a.h 2 7.c even 3 1
245.4.e.l 4 1.a even 1 1 trivial
245.4.e.l 4 7.c even 3 1 inner
315.4.j.c 4 21.c even 2 1
315.4.j.c 4 21.g even 6 1
560.4.q.i 4 28.d even 2 1
560.4.q.i 4 28.f even 6 1
1225.4.a.v 2 35.j even 6 1
1225.4.a.x 2 35.i odd 6 1
2205.4.a.bf 2 21.g even 6 1
2205.4.a.bg 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 $$S_{4}^{\mathrm{new}}(245, [\chi])$$:

 $$T_{2}^{4} - 6T_{2}^{3} + 29T_{2}^{2} - 42T_{2} + 49$$ T2^4 - 6*T2^3 + 29*T2^2 - 42*T2 + 49 $$T_{3}^{4} + 2T_{3}^{3} + 21T_{3}^{2} - 34T_{3} + 289$$ T3^4 + 2*T3^3 + 21*T3^2 - 34*T3 + 289

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 6 T^{3} + 29 T^{2} - 42 T + 49$$
$3$ $$T^{4} + 2 T^{3} + 21 T^{2} - 34 T + 289$$
$5$ $$(T^{2} - 5 T + 25)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 28 T^{3} + 788 T^{2} + \cdots + 16$$
$13$ $$(T^{2} - 36 T + 124)^{2}$$
$17$ $$T^{4} + 76 T^{3} + 10164 T^{2} + \cdots + 19254544$$
$19$ $$T^{4} - 160 T^{3} + \cdots + 25482304$$
$23$ $$T^{4} - 22 T^{3} + \cdots + 742944049$$
$29$ $$(T^{2} + 250 T + 8425)^{2}$$
$31$ $$T^{4} + 132 T^{3} + \cdots + 244734736$$
$37$ $$T^{4} - 416 T^{3} + \cdots + 39337984$$
$41$ $$(T^{2} - 106 T + 1009)^{2}$$
$43$ $$(T^{2} + 666 T + 110839)^{2}$$
$47$ $$T^{4} + 196 T^{3} + 37004 T^{2} + \cdots + 1993744$$
$53$ $$T^{4} - 952 T^{3} + \cdots + 34688317504$$
$59$ $$T^{4} - 840 T^{3} + \cdots + 1217172544$$
$61$ $$T^{4} - 98 T^{3} + \cdots + 419125465201$$
$67$ $$T^{4} - 1286 T^{3} + \cdots + 163157021329$$
$71$ $$(T^{2} - 1064 T + 38024)^{2}$$
$73$ $$T^{4} + 172 T^{3} + \cdots + 418284304$$
$79$ $$T^{4} - 1240 T^{3} + \cdots + 85736524864$$
$83$ $$(T^{2} - 1906 T + 908159)^{2}$$
$89$ $$T^{4} - 650 T^{3} + \cdots + 10884122929$$
$97$ $$(T^{2} - 628 T - 401404)^{2}$$