# Properties

 Label 245.4.e.a Level $245$ Weight $4$ Character orbit 245.e Analytic conductor $14.455$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Learn more

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: $$1$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q - 3 \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{3} + (\zeta_{6} - 1) q^{4} + 5 \zeta_{6} q^{5} + 6 q^{6} - 21 q^{8} + 23 \zeta_{6} q^{9} +O(q^{10})$$ q - 3*z * q^2 + (2*z - 2) * q^3 + (z - 1) * q^4 + 5*z * q^5 + 6 * q^6 - 21 * q^8 + 23*z * q^9 $$q - 3 \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{3} + (\zeta_{6} - 1) q^{4} + 5 \zeta_{6} q^{5} + 6 q^{6} - 21 q^{8} + 23 \zeta_{6} q^{9} + ( - 15 \zeta_{6} + 15) q^{10} + ( - 45 \zeta_{6} + 45) q^{11} - 2 \zeta_{6} q^{12} - 59 q^{13} - 10 q^{15} + 71 \zeta_{6} q^{16} + (54 \zeta_{6} - 54) q^{17} + ( - 69 \zeta_{6} + 69) q^{18} - 121 \zeta_{6} q^{19} - 5 q^{20} - 135 q^{22} - 69 \zeta_{6} q^{23} + ( - 42 \zeta_{6} + 42) q^{24} + (25 \zeta_{6} - 25) q^{25} + 177 \zeta_{6} q^{26} - 100 q^{27} - 162 q^{29} + 30 \zeta_{6} q^{30} + (88 \zeta_{6} - 88) q^{31} + ( - 45 \zeta_{6} + 45) q^{32} + 90 \zeta_{6} q^{33} + 162 q^{34} - 23 q^{36} + 259 \zeta_{6} q^{37} + (363 \zeta_{6} - 363) q^{38} + ( - 118 \zeta_{6} + 118) q^{39} - 105 \zeta_{6} q^{40} - 195 q^{41} - 286 q^{43} + 45 \zeta_{6} q^{44} + (115 \zeta_{6} - 115) q^{45} + (207 \zeta_{6} - 207) q^{46} + 45 \zeta_{6} q^{47} - 142 q^{48} + 75 q^{50} - 108 \zeta_{6} q^{51} + ( - 59 \zeta_{6} + 59) q^{52} + (597 \zeta_{6} - 597) q^{53} + 300 \zeta_{6} q^{54} + 225 q^{55} + 242 q^{57} + 486 \zeta_{6} q^{58} + (360 \zeta_{6} - 360) q^{59} + ( - 10 \zeta_{6} + 10) q^{60} + 392 \zeta_{6} q^{61} + 264 q^{62} + 433 q^{64} - 295 \zeta_{6} q^{65} + ( - 270 \zeta_{6} + 270) q^{66} + ( - 280 \zeta_{6} + 280) q^{67} - 54 \zeta_{6} q^{68} + 138 q^{69} + 48 q^{71} - 483 \zeta_{6} q^{72} + ( - 668 \zeta_{6} + 668) q^{73} + ( - 777 \zeta_{6} + 777) q^{74} - 50 \zeta_{6} q^{75} + 121 q^{76} - 354 q^{78} - 782 \zeta_{6} q^{79} + (355 \zeta_{6} - 355) q^{80} + (421 \zeta_{6} - 421) q^{81} + 585 \zeta_{6} q^{82} - 768 q^{83} - 270 q^{85} + 858 \zeta_{6} q^{86} + ( - 324 \zeta_{6} + 324) q^{87} + (945 \zeta_{6} - 945) q^{88} - 1194 \zeta_{6} q^{89} + 345 q^{90} + 69 q^{92} - 176 \zeta_{6} q^{93} + ( - 135 \zeta_{6} + 135) q^{94} + ( - 605 \zeta_{6} + 605) q^{95} + 90 \zeta_{6} q^{96} - 902 q^{97} + 1035 q^{99} +O(q^{100})$$ q - 3*z * q^2 + (2*z - 2) * q^3 + (z - 1) * q^4 + 5*z * q^5 + 6 * q^6 - 21 * q^8 + 23*z * q^9 + (-15*z + 15) * q^10 + (-45*z + 45) * q^11 - 2*z * q^12 - 59 * q^13 - 10 * q^15 + 71*z * q^16 + (54*z - 54) * q^17 + (-69*z + 69) * q^18 - 121*z * q^19 - 5 * q^20 - 135 * q^22 - 69*z * q^23 + (-42*z + 42) * q^24 + (25*z - 25) * q^25 + 177*z * q^26 - 100 * q^27 - 162 * q^29 + 30*z * q^30 + (88*z - 88) * q^31 + (-45*z + 45) * q^32 + 90*z * q^33 + 162 * q^34 - 23 * q^36 + 259*z * q^37 + (363*z - 363) * q^38 + (-118*z + 118) * q^39 - 105*z * q^40 - 195 * q^41 - 286 * q^43 + 45*z * q^44 + (115*z - 115) * q^45 + (207*z - 207) * q^46 + 45*z * q^47 - 142 * q^48 + 75 * q^50 - 108*z * q^51 + (-59*z + 59) * q^52 + (597*z - 597) * q^53 + 300*z * q^54 + 225 * q^55 + 242 * q^57 + 486*z * q^58 + (360*z - 360) * q^59 + (-10*z + 10) * q^60 + 392*z * q^61 + 264 * q^62 + 433 * q^64 - 295*z * q^65 + (-270*z + 270) * q^66 + (-280*z + 280) * q^67 - 54*z * q^68 + 138 * q^69 + 48 * q^71 - 483*z * q^72 + (-668*z + 668) * q^73 + (-777*z + 777) * q^74 - 50*z * q^75 + 121 * q^76 - 354 * q^78 - 782*z * q^79 + (355*z - 355) * q^80 + (421*z - 421) * q^81 + 585*z * q^82 - 768 * q^83 - 270 * q^85 + 858*z * q^86 + (-324*z + 324) * q^87 + (945*z - 945) * q^88 - 1194*z * q^89 + 345 * q^90 + 69 * q^92 - 176*z * q^93 + (-135*z + 135) * q^94 + (-605*z + 605) * q^95 + 90*z * q^96 - 902 * q^97 + 1035 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - 2 q^{3} - q^{4} + 5 q^{5} + 12 q^{6} - 42 q^{8} + 23 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 - 2 * q^3 - q^4 + 5 * q^5 + 12 * q^6 - 42 * q^8 + 23 * q^9 $$2 q - 3 q^{2} - 2 q^{3} - q^{4} + 5 q^{5} + 12 q^{6} - 42 q^{8} + 23 q^{9} + 15 q^{10} + 45 q^{11} - 2 q^{12} - 118 q^{13} - 20 q^{15} + 71 q^{16} - 54 q^{17} + 69 q^{18} - 121 q^{19} - 10 q^{20} - 270 q^{22} - 69 q^{23} + 42 q^{24} - 25 q^{25} + 177 q^{26} - 200 q^{27} - 324 q^{29} + 30 q^{30} - 88 q^{31} + 45 q^{32} + 90 q^{33} + 324 q^{34} - 46 q^{36} + 259 q^{37} - 363 q^{38} + 118 q^{39} - 105 q^{40} - 390 q^{41} - 572 q^{43} + 45 q^{44} - 115 q^{45} - 207 q^{46} + 45 q^{47} - 284 q^{48} + 150 q^{50} - 108 q^{51} + 59 q^{52} - 597 q^{53} + 300 q^{54} + 450 q^{55} + 484 q^{57} + 486 q^{58} - 360 q^{59} + 10 q^{60} + 392 q^{61} + 528 q^{62} + 866 q^{64} - 295 q^{65} + 270 q^{66} + 280 q^{67} - 54 q^{68} + 276 q^{69} + 96 q^{71} - 483 q^{72} + 668 q^{73} + 777 q^{74} - 50 q^{75} + 242 q^{76} - 708 q^{78} - 782 q^{79} - 355 q^{80} - 421 q^{81} + 585 q^{82} - 1536 q^{83} - 540 q^{85} + 858 q^{86} + 324 q^{87} - 945 q^{88} - 1194 q^{89} + 690 q^{90} + 138 q^{92} - 176 q^{93} + 135 q^{94} + 605 q^{95} + 90 q^{96} - 1804 q^{97} + 2070 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 - 2 * q^3 - q^4 + 5 * q^5 + 12 * q^6 - 42 * q^8 + 23 * q^9 + 15 * q^10 + 45 * q^11 - 2 * q^12 - 118 * q^13 - 20 * q^15 + 71 * q^16 - 54 * q^17 + 69 * q^18 - 121 * q^19 - 10 * q^20 - 270 * q^22 - 69 * q^23 + 42 * q^24 - 25 * q^25 + 177 * q^26 - 200 * q^27 - 324 * q^29 + 30 * q^30 - 88 * q^31 + 45 * q^32 + 90 * q^33 + 324 * q^34 - 46 * q^36 + 259 * q^37 - 363 * q^38 + 118 * q^39 - 105 * q^40 - 390 * q^41 - 572 * q^43 + 45 * q^44 - 115 * q^45 - 207 * q^46 + 45 * q^47 - 284 * q^48 + 150 * q^50 - 108 * q^51 + 59 * q^52 - 597 * q^53 + 300 * q^54 + 450 * q^55 + 484 * q^57 + 486 * q^58 - 360 * q^59 + 10 * q^60 + 392 * q^61 + 528 * q^62 + 866 * q^64 - 295 * q^65 + 270 * q^66 + 280 * q^67 - 54 * q^68 + 276 * q^69 + 96 * q^71 - 483 * q^72 + 668 * q^73 + 777 * q^74 - 50 * q^75 + 242 * q^76 - 708 * q^78 - 782 * q^79 - 355 * q^80 - 421 * q^81 + 585 * q^82 - 1536 * q^83 - 540 * q^85 + 858 * q^86 + 324 * q^87 - 945 * q^88 - 1194 * q^89 + 690 * q^90 + 138 * q^92 - 176 * q^93 + 135 * q^94 + 605 * q^95 + 90 * q^96 - 1804 * q^97 + 2070 * 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
−1.50000 2.59808i −1.00000 + 1.73205i −0.500000 + 0.866025i 2.50000 + 4.33013i 6.00000 0 −21.0000 11.5000 + 19.9186i 7.50000 12.9904i
226.1 −1.50000 + 2.59808i −1.00000 1.73205i −0.500000 0.866025i 2.50000 4.33013i 6.00000 0 −21.0000 11.5000 19.9186i 7.50000 + 12.9904i
 $$n$$: 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.a 2
7.b odd 2 1 35.4.e.a 2
7.c even 3 1 245.4.a.f 1
7.c even 3 1 inner 245.4.e.a 2
7.d odd 6 1 35.4.e.a 2
7.d odd 6 1 245.4.a.e 1
21.c even 2 1 315.4.j.b 2
21.g even 6 1 315.4.j.b 2
21.g even 6 1 2205.4.a.e 1
21.h odd 6 1 2205.4.a.g 1
28.d even 2 1 560.4.q.b 2
28.f even 6 1 560.4.q.b 2
35.c odd 2 1 175.4.e.b 2
35.f even 4 2 175.4.k.b 4
35.i odd 6 1 175.4.e.b 2
35.i odd 6 1 1225.4.a.b 1
35.j even 6 1 1225.4.a.a 1
35.k even 12 2 175.4.k.b 4

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

 $$T_{2}^{2} + 3T_{2} + 9$$ T2^2 + 3*T2 + 9 $$T_{3}^{2} + 2T_{3} + 4$$ T3^2 + 2*T3 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T + 9$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2} - 5T + 25$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 45T + 2025$$
$13$ $$(T + 59)^{2}$$
$17$ $$T^{2} + 54T + 2916$$
$19$ $$T^{2} + 121T + 14641$$
$23$ $$T^{2} + 69T + 4761$$
$29$ $$(T + 162)^{2}$$
$31$ $$T^{2} + 88T + 7744$$
$37$ $$T^{2} - 259T + 67081$$
$41$ $$(T + 195)^{2}$$
$43$ $$(T + 286)^{2}$$
$47$ $$T^{2} - 45T + 2025$$
$53$ $$T^{2} + 597T + 356409$$
$59$ $$T^{2} + 360T + 129600$$
$61$ $$T^{2} - 392T + 153664$$
$67$ $$T^{2} - 280T + 78400$$
$71$ $$(T - 48)^{2}$$
$73$ $$T^{2} - 668T + 446224$$
$79$ $$T^{2} + 782T + 611524$$
$83$ $$(T + 768)^{2}$$
$89$ $$T^{2} + 1194 T + 1425636$$
$97$ $$(T + 902)^{2}$$
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