# Properties

 Label 245.4.a.k Level $245$ Weight $4$ Character orbit 245.a Self dual yes Analytic conductor $14.455$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,4,Mod(1,245)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("245.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 245.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$14.4554679514$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

 $$f(q)$$ $$=$$ $$q + (\beta + 4) q^{2} + (4 \beta - 1) q^{3} + (8 \beta + 10) q^{4} + 5 q^{5} + (15 \beta + 4) q^{6} + (34 \beta + 24) q^{8} + ( - 8 \beta + 6) q^{9}+O(q^{10})$$ q + (b + 4) * q^2 + (4*b - 1) * q^3 + (8*b + 10) * q^4 + 5 * q^5 + (15*b + 4) * q^6 + (34*b + 24) * q^8 + (-8*b + 6) * q^9 $$q + (\beta + 4) q^{2} + (4 \beta - 1) q^{3} + (8 \beta + 10) q^{4} + 5 q^{5} + (15 \beta + 4) q^{6} + (34 \beta + 24) q^{8} + ( - 8 \beta + 6) q^{9} + (5 \beta + 20) q^{10} + ( - 32 \beta - 7) q^{11} + (32 \beta + 54) q^{12} + ( - 4 \beta - 25) q^{13} + (20 \beta - 5) q^{15} + (96 \beta + 84) q^{16} + ( - 44 \beta + 25) q^{17} + ( - 26 \beta + 8) q^{18} + ( - 44 \beta - 18) q^{19} + (40 \beta + 50) q^{20} + ( - 135 \beta - 92) q^{22} + ( - 68 \beta + 122) q^{23} + (62 \beta + 248) q^{24} + 25 q^{25} + ( - 41 \beta - 108) q^{26} + ( - 76 \beta - 43) q^{27} + (24 \beta - 13) q^{29} + (75 \beta + 20) q^{30} + (180 \beta + 60) q^{31} + (196 \beta + 336) q^{32} + (4 \beta - 249) q^{33} + ( - 151 \beta + 12) q^{34} + ( - 32 \beta - 68) q^{36} + ( - 60 \beta + 282) q^{37} + ( - 194 \beta - 160) q^{38} + ( - 96 \beta - 7) q^{39} + (170 \beta + 120) q^{40} + ( - 124 \beta + 164) q^{41} + (68 \beta - 130) q^{43} + ( - 376 \beta - 582) q^{44} + ( - 40 \beta + 30) q^{45} + ( - 150 \beta + 352) q^{46} + (132 \beta + 175) q^{47} + (240 \beta + 684) q^{48} + (25 \beta + 100) q^{50} + (144 \beta - 377) q^{51} + ( - 240 \beta - 314) q^{52} + (128 \beta - 28) q^{53} + ( - 347 \beta - 324) q^{54} + ( - 160 \beta - 35) q^{55} + ( - 28 \beta - 334) q^{57} + (83 \beta - 4) q^{58} + 616 q^{59} + (160 \beta + 270) q^{60} + (108 \beta - 168) q^{61} + (780 \beta + 600) q^{62} + (352 \beta + 1064) q^{64} + ( - 20 \beta - 125) q^{65} + ( - 233 \beta - 988) q^{66} + ( - 64 \beta - 76) q^{67} + ( - 240 \beta - 454) q^{68} + (556 \beta - 666) q^{69} - 952 q^{71} + (12 \beta - 400) q^{72} + (344 \beta - 338) q^{73} + (42 \beta + 1008) q^{74} + (100 \beta - 25) q^{75} + ( - 584 \beta - 884) q^{76} + ( - 391 \beta - 220) q^{78} + (248 \beta + 507) q^{79} + (480 \beta + 420) q^{80} + (120 \beta - 727) q^{81} + ( - 332 \beta + 408) q^{82} + ( - 600 \beta + 188) q^{83} + ( - 220 \beta + 125) q^{85} + (142 \beta - 384) q^{86} + ( - 76 \beta + 205) q^{87} + ( - 1006 \beta - 2344) q^{88} + ( - 44 \beta + 108) q^{89} + ( - 130 \beta + 40) q^{90} + (296 \beta + 132) q^{92} + (60 \beta + 1380) q^{93} + (703 \beta + 964) q^{94} + ( - 220 \beta - 90) q^{95} + (1148 \beta + 1232) q^{96} + ( - 220 \beta - 1371) q^{97} + ( - 136 \beta + 470) q^{99}+O(q^{100})$$ q + (b + 4) * q^2 + (4*b - 1) * q^3 + (8*b + 10) * q^4 + 5 * q^5 + (15*b + 4) * q^6 + (34*b + 24) * q^8 + (-8*b + 6) * q^9 + (5*b + 20) * q^10 + (-32*b - 7) * q^11 + (32*b + 54) * q^12 + (-4*b - 25) * q^13 + (20*b - 5) * q^15 + (96*b + 84) * q^16 + (-44*b + 25) * q^17 + (-26*b + 8) * q^18 + (-44*b - 18) * q^19 + (40*b + 50) * q^20 + (-135*b - 92) * q^22 + (-68*b + 122) * q^23 + (62*b + 248) * q^24 + 25 * q^25 + (-41*b - 108) * q^26 + (-76*b - 43) * q^27 + (24*b - 13) * q^29 + (75*b + 20) * q^30 + (180*b + 60) * q^31 + (196*b + 336) * q^32 + (4*b - 249) * q^33 + (-151*b + 12) * q^34 + (-32*b - 68) * q^36 + (-60*b + 282) * q^37 + (-194*b - 160) * q^38 + (-96*b - 7) * q^39 + (170*b + 120) * q^40 + (-124*b + 164) * q^41 + (68*b - 130) * q^43 + (-376*b - 582) * q^44 + (-40*b + 30) * q^45 + (-150*b + 352) * q^46 + (132*b + 175) * q^47 + (240*b + 684) * q^48 + (25*b + 100) * q^50 + (144*b - 377) * q^51 + (-240*b - 314) * q^52 + (128*b - 28) * q^53 + (-347*b - 324) * q^54 + (-160*b - 35) * q^55 + (-28*b - 334) * q^57 + (83*b - 4) * q^58 + 616 * q^59 + (160*b + 270) * q^60 + (108*b - 168) * q^61 + (780*b + 600) * q^62 + (352*b + 1064) * q^64 + (-20*b - 125) * q^65 + (-233*b - 988) * q^66 + (-64*b - 76) * q^67 + (-240*b - 454) * q^68 + (556*b - 666) * q^69 - 952 * q^71 + (12*b - 400) * q^72 + (344*b - 338) * q^73 + (42*b + 1008) * q^74 + (100*b - 25) * q^75 + (-584*b - 884) * q^76 + (-391*b - 220) * q^78 + (248*b + 507) * q^79 + (480*b + 420) * q^80 + (120*b - 727) * q^81 + (-332*b + 408) * q^82 + (-600*b + 188) * q^83 + (-220*b + 125) * q^85 + (142*b - 384) * q^86 + (-76*b + 205) * q^87 + (-1006*b - 2344) * q^88 + (-44*b + 108) * q^89 + (-130*b + 40) * q^90 + (296*b + 132) * q^92 + (60*b + 1380) * q^93 + (703*b + 964) * q^94 + (-220*b - 90) * q^95 + (1148*b + 1232) * q^96 + (-220*b - 1371) * q^97 + (-136*b + 470) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{2} - 2 q^{3} + 20 q^{4} + 10 q^{5} + 8 q^{6} + 48 q^{8} + 12 q^{9}+O(q^{10})$$ 2 * q + 8 * q^2 - 2 * q^3 + 20 * q^4 + 10 * q^5 + 8 * q^6 + 48 * q^8 + 12 * q^9 $$2 q + 8 q^{2} - 2 q^{3} + 20 q^{4} + 10 q^{5} + 8 q^{6} + 48 q^{8} + 12 q^{9} + 40 q^{10} - 14 q^{11} + 108 q^{12} - 50 q^{13} - 10 q^{15} + 168 q^{16} + 50 q^{17} + 16 q^{18} - 36 q^{19} + 100 q^{20} - 184 q^{22} + 244 q^{23} + 496 q^{24} + 50 q^{25} - 216 q^{26} - 86 q^{27} - 26 q^{29} + 40 q^{30} + 120 q^{31} + 672 q^{32} - 498 q^{33} + 24 q^{34} - 136 q^{36} + 564 q^{37} - 320 q^{38} - 14 q^{39} + 240 q^{40} + 328 q^{41} - 260 q^{43} - 1164 q^{44} + 60 q^{45} + 704 q^{46} + 350 q^{47} + 1368 q^{48} + 200 q^{50} - 754 q^{51} - 628 q^{52} - 56 q^{53} - 648 q^{54} - 70 q^{55} - 668 q^{57} - 8 q^{58} + 1232 q^{59} + 540 q^{60} - 336 q^{61} + 1200 q^{62} + 2128 q^{64} - 250 q^{65} - 1976 q^{66} - 152 q^{67} - 908 q^{68} - 1332 q^{69} - 1904 q^{71} - 800 q^{72} - 676 q^{73} + 2016 q^{74} - 50 q^{75} - 1768 q^{76} - 440 q^{78} + 1014 q^{79} + 840 q^{80} - 1454 q^{81} + 816 q^{82} + 376 q^{83} + 250 q^{85} - 768 q^{86} + 410 q^{87} - 4688 q^{88} + 216 q^{89} + 80 q^{90} + 264 q^{92} + 2760 q^{93} + 1928 q^{94} - 180 q^{95} + 2464 q^{96} - 2742 q^{97} + 940 q^{99}+O(q^{100})$$ 2 * q + 8 * q^2 - 2 * q^3 + 20 * q^4 + 10 * q^5 + 8 * q^6 + 48 * q^8 + 12 * q^9 + 40 * q^10 - 14 * q^11 + 108 * q^12 - 50 * q^13 - 10 * q^15 + 168 * q^16 + 50 * q^17 + 16 * q^18 - 36 * q^19 + 100 * q^20 - 184 * q^22 + 244 * q^23 + 496 * q^24 + 50 * q^25 - 216 * q^26 - 86 * q^27 - 26 * q^29 + 40 * q^30 + 120 * q^31 + 672 * q^32 - 498 * q^33 + 24 * q^34 - 136 * q^36 + 564 * q^37 - 320 * q^38 - 14 * q^39 + 240 * q^40 + 328 * q^41 - 260 * q^43 - 1164 * q^44 + 60 * q^45 + 704 * q^46 + 350 * q^47 + 1368 * q^48 + 200 * q^50 - 754 * q^51 - 628 * q^52 - 56 * q^53 - 648 * q^54 - 70 * q^55 - 668 * q^57 - 8 * q^58 + 1232 * q^59 + 540 * q^60 - 336 * q^61 + 1200 * q^62 + 2128 * q^64 - 250 * q^65 - 1976 * q^66 - 152 * q^67 - 908 * q^68 - 1332 * q^69 - 1904 * q^71 - 800 * q^72 - 676 * q^73 + 2016 * q^74 - 50 * q^75 - 1768 * q^76 - 440 * q^78 + 1014 * q^79 + 840 * q^80 - 1454 * q^81 + 816 * q^82 + 376 * q^83 + 250 * q^85 - 768 * q^86 + 410 * q^87 - 4688 * q^88 + 216 * q^89 + 80 * q^90 + 264 * q^92 + 2760 * q^93 + 1928 * q^94 - 180 * q^95 + 2464 * q^96 - 2742 * q^97 + 940 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
2.58579 −6.65685 −1.31371 5.00000 −17.2132 0 −24.0833 17.3137 12.9289
1.2 5.41421 4.65685 21.3137 5.00000 25.2132 0 72.0833 −5.31371 27.0711
 $$n$$: 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 245.4.a.k 2
3.b odd 2 1 2205.4.a.u 2
5.b even 2 1 1225.4.a.m 2
7.b odd 2 1 35.4.a.b 2
7.c even 3 2 245.4.e.i 4
7.d odd 6 2 245.4.e.h 4
21.c even 2 1 315.4.a.f 2
28.d even 2 1 560.4.a.r 2
35.c odd 2 1 175.4.a.c 2
35.f even 4 2 175.4.b.c 4
56.e even 2 1 2240.4.a.bo 2
56.h odd 2 1 2240.4.a.bn 2
105.g even 2 1 1575.4.a.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.b 2 7.b odd 2 1
175.4.a.c 2 35.c odd 2 1
175.4.b.c 4 35.f even 4 2
245.4.a.k 2 1.a even 1 1 trivial
245.4.e.h 4 7.d odd 6 2
245.4.e.i 4 7.c even 3 2
315.4.a.f 2 21.c even 2 1
560.4.a.r 2 28.d even 2 1
1225.4.a.m 2 5.b even 2 1
1575.4.a.z 2 105.g even 2 1
2205.4.a.u 2 3.b odd 2 1
2240.4.a.bn 2 56.h odd 2 1
2240.4.a.bo 2 56.e even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(245))$$:

 $$T_{2}^{2} - 8T_{2} + 14$$ T2^2 - 8*T2 + 14 $$T_{3}^{2} + 2T_{3} - 31$$ T3^2 + 2*T3 - 31

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 8T + 14$$
$3$ $$T^{2} + 2T - 31$$
$5$ $$(T - 5)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 14T - 1999$$
$13$ $$T^{2} + 50T + 593$$
$17$ $$T^{2} - 50T - 3247$$
$19$ $$T^{2} + 36T - 3548$$
$23$ $$T^{2} - 244T + 5636$$
$29$ $$T^{2} + 26T - 983$$
$31$ $$T^{2} - 120T - 61200$$
$37$ $$T^{2} - 564T + 72324$$
$41$ $$T^{2} - 328T - 3856$$
$43$ $$T^{2} + 260T + 7652$$
$47$ $$T^{2} - 350T - 4223$$
$53$ $$T^{2} + 56T - 31984$$
$59$ $$(T - 616)^{2}$$
$61$ $$T^{2} + 336T + 4896$$
$67$ $$T^{2} + 152T - 2416$$
$71$ $$(T + 952)^{2}$$
$73$ $$T^{2} + 676T - 122428$$
$79$ $$T^{2} - 1014 T + 134041$$
$83$ $$T^{2} - 376T - 684656$$
$89$ $$T^{2} - 216T + 7792$$
$97$ $$T^{2} + 2742 T + 1782841$$