# Properties

 Label 245.4.e.h Level $245$ Weight $4$ Character orbit 245.e Analytic conductor $14.455$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ 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 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + 4 \beta_{2} - \beta_1) q^{2} + ( - \beta_{2} - 4 \beta_1 - 1) q^{3} + ( - 10 \beta_{2} + 8 \beta_1 - 10) q^{4} - 5 \beta_{2} q^{5} + ( - 15 \beta_{3} - 4) q^{6} + (34 \beta_{3} + 24) q^{8} + (8 \beta_{3} + 6 \beta_{2} + 8 \beta_1) q^{9}+O(q^{10})$$ q + (-b3 + 4*b2 - b1) * q^2 + (-b2 - 4*b1 - 1) * q^3 + (-10*b2 + 8*b1 - 10) * q^4 - 5*b2 * q^5 + (-15*b3 - 4) * q^6 + (34*b3 + 24) * q^8 + (8*b3 + 6*b2 + 8*b1) * q^9 $$q + ( - \beta_{3} + 4 \beta_{2} - \beta_1) q^{2} + ( - \beta_{2} - 4 \beta_1 - 1) q^{3} + ( - 10 \beta_{2} + 8 \beta_1 - 10) q^{4} - 5 \beta_{2} q^{5} + ( - 15 \beta_{3} - 4) q^{6} + (34 \beta_{3} + 24) q^{8} + (8 \beta_{3} + 6 \beta_{2} + 8 \beta_1) q^{9} + (20 \beta_{2} - 5 \beta_1 + 20) q^{10} + (7 \beta_{2} - 32 \beta_1 + 7) q^{11} + (32 \beta_{3} - 54 \beta_{2} + 32 \beta_1) q^{12} + (4 \beta_{3} + 25) q^{13} + (20 \beta_{3} - 5) q^{15} + ( - 96 \beta_{3} + 84 \beta_{2} - 96 \beta_1) q^{16} + (25 \beta_{2} + 44 \beta_1 + 25) q^{17} + ( - 8 \beta_{2} - 26 \beta_1 - 8) q^{18} + ( - 44 \beta_{3} + 18 \beta_{2} - 44 \beta_1) q^{19} + ( - 40 \beta_{3} - 50) q^{20} + ( - 135 \beta_{3} - 92) q^{22} + (68 \beta_{3} + 122 \beta_{2} + 68 \beta_1) q^{23} + (248 \beta_{2} - 62 \beta_1 + 248) q^{24} + ( - 25 \beta_{2} - 25) q^{25} + ( - 41 \beta_{3} + 108 \beta_{2} - 41 \beta_1) q^{26} + (76 \beta_{3} + 43) q^{27} + (24 \beta_{3} - 13) q^{29} + ( - 75 \beta_{3} + 20 \beta_{2} - 75 \beta_1) q^{30} + (60 \beta_{2} - 180 \beta_1 + 60) q^{31} + ( - 336 \beta_{2} + 196 \beta_1 - 336) q^{32} + (4 \beta_{3} + 249 \beta_{2} + 4 \beta_1) q^{33} + (151 \beta_{3} - 12) q^{34} + ( - 32 \beta_{3} - 68) q^{36} + (60 \beta_{3} + 282 \beta_{2} + 60 \beta_1) q^{37} + ( - 160 \beta_{2} + 194 \beta_1 - 160) q^{38} + (7 \beta_{2} - 96 \beta_1 + 7) q^{39} + (170 \beta_{3} - 120 \beta_{2} + 170 \beta_1) q^{40} + (124 \beta_{3} - 164) q^{41} + (68 \beta_{3} - 130) q^{43} + (376 \beta_{3} - 582 \beta_{2} + 376 \beta_1) q^{44} + (30 \beta_{2} + 40 \beta_1 + 30) q^{45} + ( - 352 \beta_{2} - 150 \beta_1 - 352) q^{46} + (132 \beta_{3} - 175 \beta_{2} + 132 \beta_1) q^{47} + ( - 240 \beta_{3} - 684) q^{48} + (25 \beta_{3} + 100) q^{50} + ( - 144 \beta_{3} - 377 \beta_{2} - 144 \beta_1) q^{51} + ( - 314 \beta_{2} + 240 \beta_1 - 314) q^{52} + (28 \beta_{2} + 128 \beta_1 + 28) q^{53} + ( - 347 \beta_{3} + 324 \beta_{2} - 347 \beta_1) q^{54} + (160 \beta_{3} + 35) q^{55} + ( - 28 \beta_{3} - 334) q^{57} + ( - 83 \beta_{3} - 4 \beta_{2} - 83 \beta_1) q^{58} + (616 \beta_{2} + 616) q^{59} + ( - 270 \beta_{2} + 160 \beta_1 - 270) q^{60} + (108 \beta_{3} + 168 \beta_{2} + 108 \beta_1) q^{61} + ( - 780 \beta_{3} - 600) q^{62} + (352 \beta_{3} + 1064) q^{64} + (20 \beta_{3} - 125 \beta_{2} + 20 \beta_1) q^{65} + ( - 988 \beta_{2} + 233 \beta_1 - 988) q^{66} + (76 \beta_{2} - 64 \beta_1 + 76) q^{67} + ( - 240 \beta_{3} + 454 \beta_{2} - 240 \beta_1) q^{68} + ( - 556 \beta_{3} + 666) q^{69} - 952 q^{71} + ( - 12 \beta_{3} - 400 \beta_{2} - 12 \beta_1) q^{72} + ( - 338 \beta_{2} - 344 \beta_1 - 338) q^{73} + ( - 1008 \beta_{2} + 42 \beta_1 - 1008) q^{74} + (100 \beta_{3} + 25 \beta_{2} + 100 \beta_1) q^{75} + (584 \beta_{3} + 884) q^{76} + ( - 391 \beta_{3} - 220) q^{78} + ( - 248 \beta_{3} + 507 \beta_{2} - 248 \beta_1) q^{79} + (420 \beta_{2} - 480 \beta_1 + 420) q^{80} + (727 \beta_{2} + 120 \beta_1 + 727) q^{81} + ( - 332 \beta_{3} - 408 \beta_{2} - 332 \beta_1) q^{82} + (600 \beta_{3} - 188) q^{83} + ( - 220 \beta_{3} + 125) q^{85} + ( - 142 \beta_{3} - 384 \beta_{2} - 142 \beta_1) q^{86} + (205 \beta_{2} + 76 \beta_1 + 205) q^{87} + (2344 \beta_{2} - 1006 \beta_1 + 2344) q^{88} + ( - 44 \beta_{3} - 108 \beta_{2} - 44 \beta_1) q^{89} + (130 \beta_{3} - 40) q^{90} + (296 \beta_{3} + 132) q^{92} + ( - 60 \beta_{3} + 1380 \beta_{2} - 60 \beta_1) q^{93} + (964 \beta_{2} - 703 \beta_1 + 964) q^{94} + (90 \beta_{2} - 220 \beta_1 + 90) q^{95} + (1148 \beta_{3} - 1232 \beta_{2} + 1148 \beta_1) q^{96} + (220 \beta_{3} + 1371) q^{97} + ( - 136 \beta_{3} + 470) q^{99}+O(q^{100})$$ q + (-b3 + 4*b2 - b1) * q^2 + (-b2 - 4*b1 - 1) * q^3 + (-10*b2 + 8*b1 - 10) * q^4 - 5*b2 * q^5 + (-15*b3 - 4) * q^6 + (34*b3 + 24) * q^8 + (8*b3 + 6*b2 + 8*b1) * q^9 + (20*b2 - 5*b1 + 20) * q^10 + (7*b2 - 32*b1 + 7) * q^11 + (32*b3 - 54*b2 + 32*b1) * q^12 + (4*b3 + 25) * q^13 + (20*b3 - 5) * q^15 + (-96*b3 + 84*b2 - 96*b1) * q^16 + (25*b2 + 44*b1 + 25) * q^17 + (-8*b2 - 26*b1 - 8) * q^18 + (-44*b3 + 18*b2 - 44*b1) * q^19 + (-40*b3 - 50) * q^20 + (-135*b3 - 92) * q^22 + (68*b3 + 122*b2 + 68*b1) * q^23 + (248*b2 - 62*b1 + 248) * q^24 + (-25*b2 - 25) * q^25 + (-41*b3 + 108*b2 - 41*b1) * q^26 + (76*b3 + 43) * q^27 + (24*b3 - 13) * q^29 + (-75*b3 + 20*b2 - 75*b1) * q^30 + (60*b2 - 180*b1 + 60) * q^31 + (-336*b2 + 196*b1 - 336) * q^32 + (4*b3 + 249*b2 + 4*b1) * q^33 + (151*b3 - 12) * q^34 + (-32*b3 - 68) * q^36 + (60*b3 + 282*b2 + 60*b1) * q^37 + (-160*b2 + 194*b1 - 160) * q^38 + (7*b2 - 96*b1 + 7) * q^39 + (170*b3 - 120*b2 + 170*b1) * q^40 + (124*b3 - 164) * q^41 + (68*b3 - 130) * q^43 + (376*b3 - 582*b2 + 376*b1) * q^44 + (30*b2 + 40*b1 + 30) * q^45 + (-352*b2 - 150*b1 - 352) * q^46 + (132*b3 - 175*b2 + 132*b1) * q^47 + (-240*b3 - 684) * q^48 + (25*b3 + 100) * q^50 + (-144*b3 - 377*b2 - 144*b1) * q^51 + (-314*b2 + 240*b1 - 314) * q^52 + (28*b2 + 128*b1 + 28) * q^53 + (-347*b3 + 324*b2 - 347*b1) * q^54 + (160*b3 + 35) * q^55 + (-28*b3 - 334) * q^57 + (-83*b3 - 4*b2 - 83*b1) * q^58 + (616*b2 + 616) * q^59 + (-270*b2 + 160*b1 - 270) * q^60 + (108*b3 + 168*b2 + 108*b1) * q^61 + (-780*b3 - 600) * q^62 + (352*b3 + 1064) * q^64 + (20*b3 - 125*b2 + 20*b1) * q^65 + (-988*b2 + 233*b1 - 988) * q^66 + (76*b2 - 64*b1 + 76) * q^67 + (-240*b3 + 454*b2 - 240*b1) * q^68 + (-556*b3 + 666) * q^69 - 952 * q^71 + (-12*b3 - 400*b2 - 12*b1) * q^72 + (-338*b2 - 344*b1 - 338) * q^73 + (-1008*b2 + 42*b1 - 1008) * q^74 + (100*b3 + 25*b2 + 100*b1) * q^75 + (584*b3 + 884) * q^76 + (-391*b3 - 220) * q^78 + (-248*b3 + 507*b2 - 248*b1) * q^79 + (420*b2 - 480*b1 + 420) * q^80 + (727*b2 + 120*b1 + 727) * q^81 + (-332*b3 - 408*b2 - 332*b1) * q^82 + (600*b3 - 188) * q^83 + (-220*b3 + 125) * q^85 + (-142*b3 - 384*b2 - 142*b1) * q^86 + (205*b2 + 76*b1 + 205) * q^87 + (2344*b2 - 1006*b1 + 2344) * q^88 + (-44*b3 - 108*b2 - 44*b1) * q^89 + (130*b3 - 40) * q^90 + (296*b3 + 132) * q^92 + (-60*b3 + 1380*b2 - 60*b1) * q^93 + (964*b2 - 703*b1 + 964) * q^94 + (90*b2 - 220*b1 + 90) * q^95 + (1148*b3 - 1232*b2 + 1148*b1) * q^96 + (220*b3 + 1371) * q^97 + (-136*b3 + 470) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{2} - 2 q^{3} - 20 q^{4} + 10 q^{5} - 16 q^{6} + 96 q^{8} - 12 q^{9}+O(q^{10})$$ 4 * q - 8 * q^2 - 2 * q^3 - 20 * q^4 + 10 * q^5 - 16 * q^6 + 96 * q^8 - 12 * q^9 $$4 q - 8 q^{2} - 2 q^{3} - 20 q^{4} + 10 q^{5} - 16 q^{6} + 96 q^{8} - 12 q^{9} + 40 q^{10} + 14 q^{11} + 108 q^{12} + 100 q^{13} - 20 q^{15} - 168 q^{16} + 50 q^{17} - 16 q^{18} - 36 q^{19} - 200 q^{20} - 368 q^{22} - 244 q^{23} + 496 q^{24} - 50 q^{25} - 216 q^{26} + 172 q^{27} - 52 q^{29} - 40 q^{30} + 120 q^{31} - 672 q^{32} - 498 q^{33} - 48 q^{34} - 272 q^{36} - 564 q^{37} - 320 q^{38} + 14 q^{39} + 240 q^{40} - 656 q^{41} - 520 q^{43} + 1164 q^{44} + 60 q^{45} - 704 q^{46} + 350 q^{47} - 2736 q^{48} + 400 q^{50} + 754 q^{51} - 628 q^{52} + 56 q^{53} - 648 q^{54} + 140 q^{55} - 1336 q^{57} + 8 q^{58} + 1232 q^{59} - 540 q^{60} - 336 q^{61} - 2400 q^{62} + 4256 q^{64} + 250 q^{65} - 1976 q^{66} + 152 q^{67} - 908 q^{68} + 2664 q^{69} - 3808 q^{71} + 800 q^{72} - 676 q^{73} - 2016 q^{74} - 50 q^{75} + 3536 q^{76} - 880 q^{78} - 1014 q^{79} + 840 q^{80} + 1454 q^{81} + 816 q^{82} - 752 q^{83} + 500 q^{85} + 768 q^{86} + 410 q^{87} + 4688 q^{88} + 216 q^{89} - 160 q^{90} + 528 q^{92} - 2760 q^{93} + 1928 q^{94} + 180 q^{95} + 2464 q^{96} + 5484 q^{97} + 1880 q^{99}+O(q^{100})$$ 4 * q - 8 * q^2 - 2 * q^3 - 20 * q^4 + 10 * q^5 - 16 * q^6 + 96 * q^8 - 12 * q^9 + 40 * q^10 + 14 * q^11 + 108 * q^12 + 100 * q^13 - 20 * q^15 - 168 * q^16 + 50 * q^17 - 16 * q^18 - 36 * q^19 - 200 * q^20 - 368 * q^22 - 244 * q^23 + 496 * q^24 - 50 * q^25 - 216 * q^26 + 172 * q^27 - 52 * q^29 - 40 * q^30 + 120 * q^31 - 672 * q^32 - 498 * q^33 - 48 * q^34 - 272 * q^36 - 564 * q^37 - 320 * q^38 + 14 * q^39 + 240 * q^40 - 656 * q^41 - 520 * q^43 + 1164 * q^44 + 60 * q^45 - 704 * q^46 + 350 * q^47 - 2736 * q^48 + 400 * q^50 + 754 * q^51 - 628 * q^52 + 56 * q^53 - 648 * q^54 + 140 * q^55 - 1336 * q^57 + 8 * q^58 + 1232 * q^59 - 540 * q^60 - 336 * q^61 - 2400 * q^62 + 4256 * q^64 + 250 * q^65 - 1976 * q^66 + 152 * q^67 - 908 * q^68 + 2664 * q^69 - 3808 * q^71 + 800 * q^72 - 676 * q^73 - 2016 * q^74 - 50 * q^75 + 3536 * q^76 - 880 * q^78 - 1014 * q^79 + 840 * q^80 + 1454 * q^81 + 816 * q^82 - 752 * q^83 + 500 * q^85 + 768 * q^86 + 410 * q^87 + 4688 * q^88 + 216 * q^89 - 160 * q^90 + 528 * q^92 - 2760 * q^93 + 1928 * q^94 + 180 * q^95 + 2464 * q^96 + 5484 * q^97 + 1880 * 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.

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
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−2.70711 4.68885i 2.32843 4.03295i −10.6569 + 18.4582i 2.50000 + 4.33013i −25.2132 0 72.0833 2.65685 + 4.60181i 13.5355 23.4442i
116.2 −1.29289 2.23936i −3.32843 + 5.76500i 0.656854 1.13770i 2.50000 + 4.33013i 17.2132 0 −24.0833 −8.65685 14.9941i 6.46447 11.1968i
226.1 −2.70711 + 4.68885i 2.32843 + 4.03295i −10.6569 18.4582i 2.50000 4.33013i −25.2132 0 72.0833 2.65685 4.60181i 13.5355 + 23.4442i
226.2 −1.29289 + 2.23936i −3.32843 5.76500i 0.656854 + 1.13770i 2.50000 4.33013i 17.2132 0 −24.0833 −8.65685 + 14.9941i 6.46447 + 11.1968i
 $$n$$: 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.h 4
7.b odd 2 1 245.4.e.i 4
7.c even 3 1 35.4.a.b 2
7.c even 3 1 inner 245.4.e.h 4
7.d odd 6 1 245.4.a.k 2
7.d odd 6 1 245.4.e.i 4
21.g even 6 1 2205.4.a.u 2
21.h odd 6 1 315.4.a.f 2
28.g odd 6 1 560.4.a.r 2
35.i odd 6 1 1225.4.a.m 2
35.j even 6 1 175.4.a.c 2
35.l odd 12 2 175.4.b.c 4
56.k odd 6 1 2240.4.a.bo 2
56.p even 6 1 2240.4.a.bn 2
105.o odd 6 1 1575.4.a.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.b 2 7.c even 3 1
175.4.a.c 2 35.j even 6 1
175.4.b.c 4 35.l odd 12 2
245.4.a.k 2 7.d odd 6 1
245.4.e.h 4 1.a even 1 1 trivial
245.4.e.h 4 7.c even 3 1 inner
245.4.e.i 4 7.b odd 2 1
245.4.e.i 4 7.d odd 6 1
315.4.a.f 2 21.h odd 6 1
560.4.a.r 2 28.g odd 6 1
1225.4.a.m 2 35.i odd 6 1
1575.4.a.z 2 105.o odd 6 1
2205.4.a.u 2 21.g even 6 1
2240.4.a.bn 2 56.p even 6 1
2240.4.a.bo 2 56.k odd 6 1

## Hecke kernels

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

 $$T_{2}^{4} + 8T_{2}^{3} + 50T_{2}^{2} + 112T_{2} + 196$$ T2^4 + 8*T2^3 + 50*T2^2 + 112*T2 + 196 $$T_{3}^{4} + 2T_{3}^{3} + 35T_{3}^{2} - 62T_{3} + 961$$ T3^4 + 2*T3^3 + 35*T3^2 - 62*T3 + 961

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 8 T^{3} + \cdots + 196$$
$3$ $$T^{4} + 2 T^{3} + \cdots + 961$$
$5$ $$(T^{2} - 5 T + 25)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 14 T^{3} + \cdots + 3996001$$
$13$ $$(T^{2} - 50 T + 593)^{2}$$
$17$ $$T^{4} - 50 T^{3} + \cdots + 10543009$$
$19$ $$T^{4} + 36 T^{3} + \cdots + 12588304$$
$23$ $$T^{4} + 244 T^{3} + \cdots + 31764496$$
$29$ $$(T^{2} + 26 T - 983)^{2}$$
$31$ $$T^{4} + \cdots + 3745440000$$
$37$ $$T^{4} + \cdots + 5230760976$$
$41$ $$(T^{2} + 328 T - 3856)^{2}$$
$43$ $$(T^{2} + 260 T + 7652)^{2}$$
$47$ $$T^{4} - 350 T^{3} + \cdots + 17833729$$
$53$ $$T^{4} + \cdots + 1022976256$$
$59$ $$(T^{2} - 616 T + 379456)^{2}$$
$61$ $$T^{4} + 336 T^{3} + \cdots + 23970816$$
$67$ $$T^{4} - 152 T^{3} + \cdots + 5837056$$
$71$ $$(T + 952)^{4}$$
$73$ $$T^{4} + \cdots + 14988615184$$
$79$ $$T^{4} + \cdots + 17966989681$$
$83$ $$(T^{2} + 376 T - 684656)^{2}$$
$89$ $$T^{4} - 216 T^{3} + \cdots + 60715264$$
$97$ $$(T^{2} - 2742 T + 1782841)^{2}$$