# Properties

 Label 245.4.a.h Level $245$ Weight $4$ Character orbit 245.a Self dual yes Analytic conductor $14.455$ Analytic rank $1$ 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: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta - 3) q^{2} + (3 \beta + 1) q^{3} + ( - 6 \beta + 3) q^{4} - 5 q^{5} + ( - 8 \beta + 3) q^{6} + (13 \beta + 3) q^{8} + (6 \beta - 8) q^{9}+O(q^{10})$$ q + (b - 3) * q^2 + (3*b + 1) * q^3 + (-6*b + 3) * q^4 - 5 * q^5 + (-8*b + 3) * q^6 + (13*b + 3) * q^8 + (6*b - 8) * q^9 $$q + (\beta - 3) q^{2} + (3 \beta + 1) q^{3} + ( - 6 \beta + 3) q^{4} - 5 q^{5} + ( - 8 \beta + 3) q^{6} + (13 \beta + 3) q^{8} + (6 \beta - 8) q^{9} + ( - 5 \beta + 15) q^{10} + (10 \beta + 14) q^{11} + (3 \beta - 33) q^{12} + ( - 10 \beta + 18) q^{13} + ( - 15 \beta - 5) q^{15} + (12 \beta - 7) q^{16} + ( - 54 \beta + 38) q^{17} + ( - 26 \beta + 36) q^{18} + ( - 26 \beta - 80) q^{19} + (30 \beta - 15) q^{20} + ( - 16 \beta - 22) q^{22} + ( - 117 \beta - 11) q^{23} + (22 \beta + 81) q^{24} + 25 q^{25} + (48 \beta - 74) q^{26} + ( - 99 \beta + 1) q^{27} + ( - 60 \beta - 125) q^{29} + (40 \beta - 15) q^{30} + (100 \beta + 66) q^{31} + ( - 147 \beta + 21) q^{32} + (52 \beta + 74) q^{33} + (200 \beta - 222) q^{34} + (66 \beta - 96) q^{36} + (136 \beta - 208) q^{37} + ( - 2 \beta + 188) q^{38} + (44 \beta - 42) q^{39} + ( - 65 \beta - 15) q^{40} + ( - 30 \beta + 53) q^{41} + (5 \beta - 333) q^{43} + ( - 54 \beta - 78) q^{44} + ( - 30 \beta + 40) q^{45} + (340 \beta - 201) q^{46} + (64 \beta + 98) q^{47} + ( - 9 \beta + 65) q^{48} + (25 \beta - 75) q^{50} + (60 \beta - 286) q^{51} + ( - 138 \beta + 174) q^{52} + (142 \beta - 476) q^{53} + (298 \beta - 201) q^{54} + ( - 50 \beta - 70) q^{55} + ( - 266 \beta - 236) q^{57} + (55 \beta + 255) q^{58} + (266 \beta - 420) q^{59} + ( - 15 \beta + 165) q^{60} + ( - 570 \beta - 49) q^{61} + ( - 234 \beta + 2) q^{62} + (366 \beta - 301) q^{64} + (50 \beta - 90) q^{65} + ( - 82 \beta - 118) q^{66} + (69 \beta - 643) q^{67} + ( - 390 \beta + 762) q^{68} + ( - 150 \beta - 713) q^{69} + (350 \beta + 532) q^{71} + ( - 86 \beta + 132) q^{72} + (118 \beta + 86) q^{73} + ( - 616 \beta + 896) q^{74} + (75 \beta + 25) q^{75} + (402 \beta + 72) q^{76} + ( - 174 \beta + 214) q^{78} + ( - 214 \beta - 620) q^{79} + ( - 60 \beta + 35) q^{80} + ( - 258 \beta - 377) q^{81} + (143 \beta - 219) q^{82} + (5 \beta + 953) q^{83} + (270 \beta - 190) q^{85} + ( - 348 \beta + 1009) q^{86} + ( - 435 \beta - 485) q^{87} + (212 \beta + 302) q^{88} + (324 \beta - 325) q^{89} + (130 \beta - 180) q^{90} + ( - 285 \beta + 1371) q^{92} + (298 \beta + 666) q^{93} + ( - 94 \beta - 166) q^{94} + (130 \beta + 400) q^{95} + ( - 84 \beta - 861) q^{96} + (500 \beta + 314) q^{97} + (4 \beta + 8) q^{99}+O(q^{100})$$ q + (b - 3) * q^2 + (3*b + 1) * q^3 + (-6*b + 3) * q^4 - 5 * q^5 + (-8*b + 3) * q^6 + (13*b + 3) * q^8 + (6*b - 8) * q^9 + (-5*b + 15) * q^10 + (10*b + 14) * q^11 + (3*b - 33) * q^12 + (-10*b + 18) * q^13 + (-15*b - 5) * q^15 + (12*b - 7) * q^16 + (-54*b + 38) * q^17 + (-26*b + 36) * q^18 + (-26*b - 80) * q^19 + (30*b - 15) * q^20 + (-16*b - 22) * q^22 + (-117*b - 11) * q^23 + (22*b + 81) * q^24 + 25 * q^25 + (48*b - 74) * q^26 + (-99*b + 1) * q^27 + (-60*b - 125) * q^29 + (40*b - 15) * q^30 + (100*b + 66) * q^31 + (-147*b + 21) * q^32 + (52*b + 74) * q^33 + (200*b - 222) * q^34 + (66*b - 96) * q^36 + (136*b - 208) * q^37 + (-2*b + 188) * q^38 + (44*b - 42) * q^39 + (-65*b - 15) * q^40 + (-30*b + 53) * q^41 + (5*b - 333) * q^43 + (-54*b - 78) * q^44 + (-30*b + 40) * q^45 + (340*b - 201) * q^46 + (64*b + 98) * q^47 + (-9*b + 65) * q^48 + (25*b - 75) * q^50 + (60*b - 286) * q^51 + (-138*b + 174) * q^52 + (142*b - 476) * q^53 + (298*b - 201) * q^54 + (-50*b - 70) * q^55 + (-266*b - 236) * q^57 + (55*b + 255) * q^58 + (266*b - 420) * q^59 + (-15*b + 165) * q^60 + (-570*b - 49) * q^61 + (-234*b + 2) * q^62 + (366*b - 301) * q^64 + (50*b - 90) * q^65 + (-82*b - 118) * q^66 + (69*b - 643) * q^67 + (-390*b + 762) * q^68 + (-150*b - 713) * q^69 + (350*b + 532) * q^71 + (-86*b + 132) * q^72 + (118*b + 86) * q^73 + (-616*b + 896) * q^74 + (75*b + 25) * q^75 + (402*b + 72) * q^76 + (-174*b + 214) * q^78 + (-214*b - 620) * q^79 + (-60*b + 35) * q^80 + (-258*b - 377) * q^81 + (143*b - 219) * q^82 + (5*b + 953) * q^83 + (270*b - 190) * q^85 + (-348*b + 1009) * q^86 + (-435*b - 485) * q^87 + (212*b + 302) * q^88 + (324*b - 325) * q^89 + (130*b - 180) * q^90 + (-285*b + 1371) * q^92 + (298*b + 666) * q^93 + (-94*b - 166) * q^94 + (130*b + 400) * q^95 + (-84*b - 861) * q^96 + (500*b + 314) * q^97 + (4*b + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 10 q^{5} + 6 q^{6} + 6 q^{8} - 16 q^{9}+O(q^{10})$$ 2 * q - 6 * q^2 + 2 * q^3 + 6 * q^4 - 10 * q^5 + 6 * q^6 + 6 * q^8 - 16 * q^9 $$2 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 10 q^{5} + 6 q^{6} + 6 q^{8} - 16 q^{9} + 30 q^{10} + 28 q^{11} - 66 q^{12} + 36 q^{13} - 10 q^{15} - 14 q^{16} + 76 q^{17} + 72 q^{18} - 160 q^{19} - 30 q^{20} - 44 q^{22} - 22 q^{23} + 162 q^{24} + 50 q^{25} - 148 q^{26} + 2 q^{27} - 250 q^{29} - 30 q^{30} + 132 q^{31} + 42 q^{32} + 148 q^{33} - 444 q^{34} - 192 q^{36} - 416 q^{37} + 376 q^{38} - 84 q^{39} - 30 q^{40} + 106 q^{41} - 666 q^{43} - 156 q^{44} + 80 q^{45} - 402 q^{46} + 196 q^{47} + 130 q^{48} - 150 q^{50} - 572 q^{51} + 348 q^{52} - 952 q^{53} - 402 q^{54} - 140 q^{55} - 472 q^{57} + 510 q^{58} - 840 q^{59} + 330 q^{60} - 98 q^{61} + 4 q^{62} - 602 q^{64} - 180 q^{65} - 236 q^{66} - 1286 q^{67} + 1524 q^{68} - 1426 q^{69} + 1064 q^{71} + 264 q^{72} + 172 q^{73} + 1792 q^{74} + 50 q^{75} + 144 q^{76} + 428 q^{78} - 1240 q^{79} + 70 q^{80} - 754 q^{81} - 438 q^{82} + 1906 q^{83} - 380 q^{85} + 2018 q^{86} - 970 q^{87} + 604 q^{88} - 650 q^{89} - 360 q^{90} + 2742 q^{92} + 1332 q^{93} - 332 q^{94} + 800 q^{95} - 1722 q^{96} + 628 q^{97} + 16 q^{99}+O(q^{100})$$ 2 * q - 6 * q^2 + 2 * q^3 + 6 * q^4 - 10 * q^5 + 6 * q^6 + 6 * q^8 - 16 * q^9 + 30 * q^10 + 28 * q^11 - 66 * q^12 + 36 * q^13 - 10 * q^15 - 14 * q^16 + 76 * q^17 + 72 * q^18 - 160 * q^19 - 30 * q^20 - 44 * q^22 - 22 * q^23 + 162 * q^24 + 50 * q^25 - 148 * q^26 + 2 * q^27 - 250 * q^29 - 30 * q^30 + 132 * q^31 + 42 * q^32 + 148 * q^33 - 444 * q^34 - 192 * q^36 - 416 * q^37 + 376 * q^38 - 84 * q^39 - 30 * q^40 + 106 * q^41 - 666 * q^43 - 156 * q^44 + 80 * q^45 - 402 * q^46 + 196 * q^47 + 130 * q^48 - 150 * q^50 - 572 * q^51 + 348 * q^52 - 952 * q^53 - 402 * q^54 - 140 * q^55 - 472 * q^57 + 510 * q^58 - 840 * q^59 + 330 * q^60 - 98 * q^61 + 4 * q^62 - 602 * q^64 - 180 * q^65 - 236 * q^66 - 1286 * q^67 + 1524 * q^68 - 1426 * q^69 + 1064 * q^71 + 264 * q^72 + 172 * q^73 + 1792 * q^74 + 50 * q^75 + 144 * q^76 + 428 * q^78 - 1240 * q^79 + 70 * q^80 - 754 * q^81 - 438 * q^82 + 1906 * q^83 - 380 * q^85 + 2018 * q^86 - 970 * q^87 + 604 * q^88 - 650 * q^89 - 360 * q^90 + 2742 * q^92 + 1332 * q^93 - 332 * q^94 + 800 * q^95 - 1722 * q^96 + 628 * q^97 + 16 * 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
−4.41421 −3.24264 11.4853 −5.00000 14.3137 0 −15.3848 −16.4853 22.0711
1.2 −1.58579 5.24264 −5.48528 −5.00000 −8.31371 0 21.3848 0.485281 7.92893
 $$n$$: 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.h 2
3.b odd 2 1 2205.4.a.bg 2
5.b even 2 1 1225.4.a.v 2
7.b odd 2 1 245.4.a.g 2
7.c even 3 2 245.4.e.l 4
7.d odd 6 2 35.4.e.b 4
21.c even 2 1 2205.4.a.bf 2
21.g even 6 2 315.4.j.c 4
28.f even 6 2 560.4.q.i 4
35.c odd 2 1 1225.4.a.x 2
35.i odd 6 2 175.4.e.c 4
35.k even 12 4 175.4.k.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.b 4 7.d odd 6 2
175.4.e.c 4 35.i odd 6 2
175.4.k.c 8 35.k even 12 4
245.4.a.g 2 7.b odd 2 1
245.4.a.h 2 1.a even 1 1 trivial
245.4.e.l 4 7.c even 3 2
315.4.j.c 4 21.g even 6 2
560.4.q.i 4 28.f even 6 2
1225.4.a.v 2 5.b even 2 1
1225.4.a.x 2 35.c odd 2 1
2205.4.a.bf 2 21.c even 2 1
2205.4.a.bg 2 3.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} + 6T_{2} + 7$$ T2^2 + 6*T2 + 7 $$T_{3}^{2} - 2T_{3} - 17$$ T3^2 - 2*T3 - 17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 6T + 7$$
$3$ $$T^{2} - 2T - 17$$
$5$ $$(T + 5)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 28T - 4$$
$13$ $$T^{2} - 36T + 124$$
$17$ $$T^{2} - 76T - 4388$$
$19$ $$T^{2} + 160T + 5048$$
$23$ $$T^{2} + 22T - 27257$$
$29$ $$T^{2} + 250T + 8425$$
$31$ $$T^{2} - 132T - 15644$$
$37$ $$T^{2} + 416T + 6272$$
$41$ $$T^{2} - 106T + 1009$$
$43$ $$T^{2} + 666T + 110839$$
$47$ $$T^{2} - 196T + 1412$$
$53$ $$T^{2} + 952T + 186248$$
$59$ $$T^{2} + 840T + 34888$$
$61$ $$T^{2} + 98T - 647399$$
$67$ $$T^{2} + 1286 T + 403927$$
$71$ $$T^{2} - 1064T + 38024$$
$73$ $$T^{2} - 172T - 20452$$
$79$ $$T^{2} + 1240 T + 292808$$
$83$ $$T^{2} - 1906 T + 908159$$
$89$ $$T^{2} + 650T - 104327$$
$97$ $$T^{2} - 628T - 401404$$