# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,4,Mod(116,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("245.116");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

## 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: no Analytic conductor: $$14.4554679514$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 5) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{3} + (8 \zeta_{6} - 8) q^{4} + 5 \zeta_{6} q^{5} - 8 q^{6} + 23 \zeta_{6} q^{9}+O(q^{10})$$ q + 4*z * q^2 + (2*z - 2) * q^3 + (8*z - 8) * q^4 + 5*z * q^5 - 8 * q^6 + 23*z * q^9 $$q + 4 \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{3} + (8 \zeta_{6} - 8) q^{4} + 5 \zeta_{6} q^{5} - 8 q^{6} + 23 \zeta_{6} q^{9} + (20 \zeta_{6} - 20) q^{10} + (32 \zeta_{6} - 32) q^{11} - 16 \zeta_{6} q^{12} - 38 q^{13} - 10 q^{15} + 64 \zeta_{6} q^{16} + (26 \zeta_{6} - 26) q^{17} + (92 \zeta_{6} - 92) q^{18} - 100 \zeta_{6} q^{19} - 40 q^{20} - 128 q^{22} + 78 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} - 152 \zeta_{6} q^{26} - 100 q^{27} - 50 q^{29} - 40 \zeta_{6} q^{30} + ( - 108 \zeta_{6} + 108) q^{31} + (256 \zeta_{6} - 256) q^{32} - 64 \zeta_{6} q^{33} - 104 q^{34} - 184 q^{36} - 266 \zeta_{6} q^{37} + ( - 400 \zeta_{6} + 400) q^{38} + ( - 76 \zeta_{6} + 76) q^{39} + 22 q^{41} + 442 q^{43} - 256 \zeta_{6} q^{44} + (115 \zeta_{6} - 115) q^{45} + (312 \zeta_{6} - 312) q^{46} + 514 \zeta_{6} q^{47} - 128 q^{48} - 100 q^{50} - 52 \zeta_{6} q^{51} + ( - 304 \zeta_{6} + 304) q^{52} + (2 \zeta_{6} - 2) q^{53} - 400 \zeta_{6} q^{54} - 160 q^{55} + 200 q^{57} - 200 \zeta_{6} q^{58} + (500 \zeta_{6} - 500) q^{59} + ( - 80 \zeta_{6} + 80) q^{60} + 518 \zeta_{6} q^{61} + 432 q^{62} - 512 q^{64} - 190 \zeta_{6} q^{65} + ( - 256 \zeta_{6} + 256) q^{66} + (126 \zeta_{6} - 126) q^{67} - 208 \zeta_{6} q^{68} - 156 q^{69} + 412 q^{71} + ( - 878 \zeta_{6} + 878) q^{73} + ( - 1064 \zeta_{6} + 1064) q^{74} - 50 \zeta_{6} q^{75} + 800 q^{76} + 304 q^{78} - 600 \zeta_{6} q^{79} + (320 \zeta_{6} - 320) q^{80} + (421 \zeta_{6} - 421) q^{81} + 88 \zeta_{6} q^{82} + 282 q^{83} - 130 q^{85} + 1768 \zeta_{6} q^{86} + ( - 100 \zeta_{6} + 100) q^{87} + 150 \zeta_{6} q^{89} - 460 q^{90} - 624 q^{92} + 216 \zeta_{6} q^{93} + (2056 \zeta_{6} - 2056) q^{94} + ( - 500 \zeta_{6} + 500) q^{95} - 512 \zeta_{6} q^{96} + 386 q^{97} - 736 q^{99} +O(q^{100})$$ q + 4*z * q^2 + (2*z - 2) * q^3 + (8*z - 8) * q^4 + 5*z * q^5 - 8 * q^6 + 23*z * q^9 + (20*z - 20) * q^10 + (32*z - 32) * q^11 - 16*z * q^12 - 38 * q^13 - 10 * q^15 + 64*z * q^16 + (26*z - 26) * q^17 + (92*z - 92) * q^18 - 100*z * q^19 - 40 * q^20 - 128 * q^22 + 78*z * q^23 + (25*z - 25) * q^25 - 152*z * q^26 - 100 * q^27 - 50 * q^29 - 40*z * q^30 + (-108*z + 108) * q^31 + (256*z - 256) * q^32 - 64*z * q^33 - 104 * q^34 - 184 * q^36 - 266*z * q^37 + (-400*z + 400) * q^38 + (-76*z + 76) * q^39 + 22 * q^41 + 442 * q^43 - 256*z * q^44 + (115*z - 115) * q^45 + (312*z - 312) * q^46 + 514*z * q^47 - 128 * q^48 - 100 * q^50 - 52*z * q^51 + (-304*z + 304) * q^52 + (2*z - 2) * q^53 - 400*z * q^54 - 160 * q^55 + 200 * q^57 - 200*z * q^58 + (500*z - 500) * q^59 + (-80*z + 80) * q^60 + 518*z * q^61 + 432 * q^62 - 512 * q^64 - 190*z * q^65 + (-256*z + 256) * q^66 + (126*z - 126) * q^67 - 208*z * q^68 - 156 * q^69 + 412 * q^71 + (-878*z + 878) * q^73 + (-1064*z + 1064) * q^74 - 50*z * q^75 + 800 * q^76 + 304 * q^78 - 600*z * q^79 + (320*z - 320) * q^80 + (421*z - 421) * q^81 + 88*z * q^82 + 282 * q^83 - 130 * q^85 + 1768*z * q^86 + (-100*z + 100) * q^87 + 150*z * q^89 - 460 * q^90 - 624 * q^92 + 216*z * q^93 + (2056*z - 2056) * q^94 + (-500*z + 500) * q^95 - 512*z * q^96 + 386 * q^97 - 736 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} - 2 q^{3} - 8 q^{4} + 5 q^{5} - 16 q^{6} + 23 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 - 2 * q^3 - 8 * q^4 + 5 * q^5 - 16 * q^6 + 23 * q^9 $$2 q + 4 q^{2} - 2 q^{3} - 8 q^{4} + 5 q^{5} - 16 q^{6} + 23 q^{9} - 20 q^{10} - 32 q^{11} - 16 q^{12} - 76 q^{13} - 20 q^{15} + 64 q^{16} - 26 q^{17} - 92 q^{18} - 100 q^{19} - 80 q^{20} - 256 q^{22} + 78 q^{23} - 25 q^{25} - 152 q^{26} - 200 q^{27} - 100 q^{29} - 40 q^{30} + 108 q^{31} - 256 q^{32} - 64 q^{33} - 208 q^{34} - 368 q^{36} - 266 q^{37} + 400 q^{38} + 76 q^{39} + 44 q^{41} + 884 q^{43} - 256 q^{44} - 115 q^{45} - 312 q^{46} + 514 q^{47} - 256 q^{48} - 200 q^{50} - 52 q^{51} + 304 q^{52} - 2 q^{53} - 400 q^{54} - 320 q^{55} + 400 q^{57} - 200 q^{58} - 500 q^{59} + 80 q^{60} + 518 q^{61} + 864 q^{62} - 1024 q^{64} - 190 q^{65} + 256 q^{66} - 126 q^{67} - 208 q^{68} - 312 q^{69} + 824 q^{71} + 878 q^{73} + 1064 q^{74} - 50 q^{75} + 1600 q^{76} + 608 q^{78} - 600 q^{79} - 320 q^{80} - 421 q^{81} + 88 q^{82} + 564 q^{83} - 260 q^{85} + 1768 q^{86} + 100 q^{87} + 150 q^{89} - 920 q^{90} - 1248 q^{92} + 216 q^{93} - 2056 q^{94} + 500 q^{95} - 512 q^{96} + 772 q^{97} - 1472 q^{99}+O(q^{100})$$ 2 * q + 4 * q^2 - 2 * q^3 - 8 * q^4 + 5 * q^5 - 16 * q^6 + 23 * q^9 - 20 * q^10 - 32 * q^11 - 16 * q^12 - 76 * q^13 - 20 * q^15 + 64 * q^16 - 26 * q^17 - 92 * q^18 - 100 * q^19 - 80 * q^20 - 256 * q^22 + 78 * q^23 - 25 * q^25 - 152 * q^26 - 200 * q^27 - 100 * q^29 - 40 * q^30 + 108 * q^31 - 256 * q^32 - 64 * q^33 - 208 * q^34 - 368 * q^36 - 266 * q^37 + 400 * q^38 + 76 * q^39 + 44 * q^41 + 884 * q^43 - 256 * q^44 - 115 * q^45 - 312 * q^46 + 514 * q^47 - 256 * q^48 - 200 * q^50 - 52 * q^51 + 304 * q^52 - 2 * q^53 - 400 * q^54 - 320 * q^55 + 400 * q^57 - 200 * q^58 - 500 * q^59 + 80 * q^60 + 518 * q^61 + 864 * q^62 - 1024 * q^64 - 190 * q^65 + 256 * q^66 - 126 * q^67 - 208 * q^68 - 312 * q^69 + 824 * q^71 + 878 * q^73 + 1064 * q^74 - 50 * q^75 + 1600 * q^76 + 608 * q^78 - 600 * q^79 - 320 * q^80 - 421 * q^81 + 88 * q^82 + 564 * q^83 - 260 * q^85 + 1768 * q^86 + 100 * q^87 + 150 * q^89 - 920 * q^90 - 1248 * q^92 + 216 * q^93 - 2056 * q^94 + 500 * q^95 - 512 * q^96 + 772 * q^97 - 1472 * q^99

## Character values

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

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1
 0.5 + 0.866025i 0.5 − 0.866025i
2.00000 + 3.46410i −1.00000 + 1.73205i −4.00000 + 6.92820i 2.50000 + 4.33013i −8.00000 0 0 11.5000 + 19.9186i −10.0000 + 17.3205i
226.1 2.00000 3.46410i −1.00000 1.73205i −4.00000 6.92820i 2.50000 4.33013i −8.00000 0 0 11.5000 19.9186i −10.0000 17.3205i
 $$n$$: 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.f 2
7.b odd 2 1 245.4.e.g 2
7.c even 3 1 5.4.a.a 1
7.c even 3 1 inner 245.4.e.f 2
7.d odd 6 1 245.4.a.a 1
7.d odd 6 1 245.4.e.g 2
21.g even 6 1 2205.4.a.q 1
21.h odd 6 1 45.4.a.d 1
28.g odd 6 1 80.4.a.d 1
35.i odd 6 1 1225.4.a.k 1
35.j even 6 1 25.4.a.c 1
35.l odd 12 2 25.4.b.a 2
56.k odd 6 1 320.4.a.h 1
56.p even 6 1 320.4.a.g 1
63.g even 3 1 405.4.e.l 2
63.h even 3 1 405.4.e.l 2
63.j odd 6 1 405.4.e.c 2
63.n odd 6 1 405.4.e.c 2
77.h odd 6 1 605.4.a.d 1
84.n even 6 1 720.4.a.u 1
91.r even 6 1 845.4.a.b 1
105.o odd 6 1 225.4.a.b 1
105.x even 12 2 225.4.b.c 2
112.u odd 12 2 1280.4.d.l 2
112.w even 12 2 1280.4.d.e 2
119.j even 6 1 1445.4.a.a 1
133.r odd 6 1 1805.4.a.h 1
140.p odd 6 1 400.4.a.m 1
140.w even 12 2 400.4.c.k 2
280.bf even 6 1 1600.4.a.bi 1
280.bi odd 6 1 1600.4.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 7.c even 3 1
25.4.a.c 1 35.j even 6 1
25.4.b.a 2 35.l odd 12 2
45.4.a.d 1 21.h odd 6 1
80.4.a.d 1 28.g odd 6 1
225.4.a.b 1 105.o odd 6 1
225.4.b.c 2 105.x even 12 2
245.4.a.a 1 7.d odd 6 1
245.4.e.f 2 1.a even 1 1 trivial
245.4.e.f 2 7.c even 3 1 inner
245.4.e.g 2 7.b odd 2 1
245.4.e.g 2 7.d odd 6 1
320.4.a.g 1 56.p even 6 1
320.4.a.h 1 56.k odd 6 1
400.4.a.m 1 140.p odd 6 1
400.4.c.k 2 140.w even 12 2
405.4.e.c 2 63.j odd 6 1
405.4.e.c 2 63.n odd 6 1
405.4.e.l 2 63.g even 3 1
405.4.e.l 2 63.h even 3 1
605.4.a.d 1 77.h odd 6 1
720.4.a.u 1 84.n even 6 1
845.4.a.b 1 91.r even 6 1
1225.4.a.k 1 35.i odd 6 1
1280.4.d.e 2 112.w even 12 2
1280.4.d.l 2 112.u odd 12 2
1445.4.a.a 1 119.j even 6 1
1600.4.a.s 1 280.bi odd 6 1
1600.4.a.bi 1 280.bf even 6 1
1805.4.a.h 1 133.r odd 6 1
2205.4.a.q 1 21.g even 6 1

## Hecke kernels

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

 $$T_{2}^{2} - 4T_{2} + 16$$ T2^2 - 4*T2 + 16 $$T_{3}^{2} + 2T_{3} + 4$$ T3^2 + 2*T3 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 4T + 16$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2} - 5T + 25$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 32T + 1024$$
$13$ $$(T + 38)^{2}$$
$17$ $$T^{2} + 26T + 676$$
$19$ $$T^{2} + 100T + 10000$$
$23$ $$T^{2} - 78T + 6084$$
$29$ $$(T + 50)^{2}$$
$31$ $$T^{2} - 108T + 11664$$
$37$ $$T^{2} + 266T + 70756$$
$41$ $$(T - 22)^{2}$$
$43$ $$(T - 442)^{2}$$
$47$ $$T^{2} - 514T + 264196$$
$53$ $$T^{2} + 2T + 4$$
$59$ $$T^{2} + 500T + 250000$$
$61$ $$T^{2} - 518T + 268324$$
$67$ $$T^{2} + 126T + 15876$$
$71$ $$(T - 412)^{2}$$
$73$ $$T^{2} - 878T + 770884$$
$79$ $$T^{2} + 600T + 360000$$
$83$ $$(T - 282)^{2}$$
$89$ $$T^{2} - 150T + 22500$$
$97$ $$(T - 386)^{2}$$