# Properties

 Label 275.4.a.b Level $275$ Weight $4$ Character orbit 275.a Self dual yes Analytic conductor $16.226$ 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] = [275,4,Mod(1,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, 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("275.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$275 = 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 275.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: $$16.2255252516$$ 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: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} + ( - 4 \beta + 1) q^{3} + ( - 2 \beta - 4) q^{4} + (5 \beta - 13) q^{6} + ( - 4 \beta - 10) q^{7} + ( - 10 \beta + 6) q^{8} + ( - 8 \beta + 22) q^{9}+O(q^{10})$$ q + (b - 1) * q^2 + (-4*b + 1) * q^3 + (-2*b - 4) * q^4 + (5*b - 13) * q^6 + (-4*b - 10) * q^7 + (-10*b + 6) * q^8 + (-8*b + 22) * q^9 $$q + (\beta - 1) q^{2} + ( - 4 \beta + 1) q^{3} + ( - 2 \beta - 4) q^{4} + (5 \beta - 13) q^{6} + ( - 4 \beta - 10) q^{7} + ( - 10 \beta + 6) q^{8} + ( - 8 \beta + 22) q^{9} - 11 q^{11} + (14 \beta + 20) q^{12} + ( - 20 \beta - 40) q^{13} + ( - 6 \beta - 2) q^{14} + (32 \beta - 4) q^{16} + (12 \beta + 62) q^{17} + (30 \beta - 46) q^{18} + ( - 60 \beta + 36) q^{19} + (36 \beta + 38) q^{21} + ( - 11 \beta + 11) q^{22} + ( - 36 \beta + 49) q^{23} + ( - 34 \beta + 126) q^{24} + ( - 20 \beta - 20) q^{26} + (12 \beta + 91) q^{27} + (36 \beta + 64) q^{28} + (56 \beta + 72) q^{29} + ( - 28 \beta - 17) q^{31} + (44 \beta + 52) q^{32} + (44 \beta - 11) q^{33} + (50 \beta - 26) q^{34} + ( - 12 \beta - 40) q^{36} + ( - 8 \beta - 27) q^{37} + (96 \beta - 216) q^{38} + (140 \beta + 200) q^{39} + (4 \beta + 268) q^{41} + (2 \beta + 70) q^{42} + ( - 16 \beta + 30) q^{43} + (22 \beta + 44) q^{44} + (85 \beta - 157) q^{46} + ( - 120 \beta + 136) q^{47} + (48 \beta - 388) q^{48} + (80 \beta - 195) q^{49} + ( - 236 \beta - 82) q^{51} + (160 \beta + 280) q^{52} + ( - 56 \beta + 246) q^{53} + (79 \beta - 55) q^{54} + (76 \beta + 60) q^{56} + ( - 204 \beta + 756) q^{57} + (16 \beta + 96) q^{58} + (132 \beta + 317) q^{59} + ( - 184 \beta + 420) q^{61} + (11 \beta - 67) q^{62} + ( - 8 \beta - 124) q^{63} + ( - 248 \beta + 112) q^{64} + ( - 55 \beta + 143) q^{66} + ( - 20 \beta - 377) q^{67} + ( - 172 \beta - 320) q^{68} + ( - 232 \beta + 481) q^{69} + ( - 76 \beta - 339) q^{71} + ( - 268 \beta + 372) q^{72} + ( - 468 \beta + 200) q^{73} + ( - 19 \beta + 3) q^{74} + (168 \beta + 216) q^{76} + (44 \beta + 110) q^{77} + (60 \beta + 220) q^{78} + ( - 656 \beta + 158) q^{79} + ( - 136 \beta - 647) q^{81} + (264 \beta - 256) q^{82} + (120 \beta - 234) q^{83} + ( - 220 \beta - 368) q^{84} + (46 \beta - 78) q^{86} + ( - 232 \beta - 600) q^{87} + (110 \beta - 66) q^{88} + (328 \beta - 921) q^{89} + (360 \beta + 640) q^{91} + (46 \beta + 20) q^{92} + (40 \beta + 319) q^{93} + (256 \beta - 496) q^{94} + ( - 164 \beta - 476) q^{96} + (144 \beta - 1097) q^{97} + ( - 275 \beta + 435) q^{98} + (88 \beta - 242) q^{99}+O(q^{100})$$ q + (b - 1) * q^2 + (-4*b + 1) * q^3 + (-2*b - 4) * q^4 + (5*b - 13) * q^6 + (-4*b - 10) * q^7 + (-10*b + 6) * q^8 + (-8*b + 22) * q^9 - 11 * q^11 + (14*b + 20) * q^12 + (-20*b - 40) * q^13 + (-6*b - 2) * q^14 + (32*b - 4) * q^16 + (12*b + 62) * q^17 + (30*b - 46) * q^18 + (-60*b + 36) * q^19 + (36*b + 38) * q^21 + (-11*b + 11) * q^22 + (-36*b + 49) * q^23 + (-34*b + 126) * q^24 + (-20*b - 20) * q^26 + (12*b + 91) * q^27 + (36*b + 64) * q^28 + (56*b + 72) * q^29 + (-28*b - 17) * q^31 + (44*b + 52) * q^32 + (44*b - 11) * q^33 + (50*b - 26) * q^34 + (-12*b - 40) * q^36 + (-8*b - 27) * q^37 + (96*b - 216) * q^38 + (140*b + 200) * q^39 + (4*b + 268) * q^41 + (2*b + 70) * q^42 + (-16*b + 30) * q^43 + (22*b + 44) * q^44 + (85*b - 157) * q^46 + (-120*b + 136) * q^47 + (48*b - 388) * q^48 + (80*b - 195) * q^49 + (-236*b - 82) * q^51 + (160*b + 280) * q^52 + (-56*b + 246) * q^53 + (79*b - 55) * q^54 + (76*b + 60) * q^56 + (-204*b + 756) * q^57 + (16*b + 96) * q^58 + (132*b + 317) * q^59 + (-184*b + 420) * q^61 + (11*b - 67) * q^62 + (-8*b - 124) * q^63 + (-248*b + 112) * q^64 + (-55*b + 143) * q^66 + (-20*b - 377) * q^67 + (-172*b - 320) * q^68 + (-232*b + 481) * q^69 + (-76*b - 339) * q^71 + (-268*b + 372) * q^72 + (-468*b + 200) * q^73 + (-19*b + 3) * q^74 + (168*b + 216) * q^76 + (44*b + 110) * q^77 + (60*b + 220) * q^78 + (-656*b + 158) * q^79 + (-136*b - 647) * q^81 + (264*b - 256) * q^82 + (120*b - 234) * q^83 + (-220*b - 368) * q^84 + (46*b - 78) * q^86 + (-232*b - 600) * q^87 + (110*b - 66) * q^88 + (328*b - 921) * q^89 + (360*b + 640) * q^91 + (46*b + 20) * q^92 + (40*b + 319) * q^93 + (256*b - 496) * q^94 + (-164*b - 476) * q^96 + (144*b - 1097) * q^97 + (-275*b + 435) * q^98 + (88*b - 242) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} - 8 q^{4} - 26 q^{6} - 20 q^{7} + 12 q^{8} + 44 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 - 8 * q^4 - 26 * q^6 - 20 * q^7 + 12 * q^8 + 44 * q^9 $$2 q - 2 q^{2} + 2 q^{3} - 8 q^{4} - 26 q^{6} - 20 q^{7} + 12 q^{8} + 44 q^{9} - 22 q^{11} + 40 q^{12} - 80 q^{13} - 4 q^{14} - 8 q^{16} + 124 q^{17} - 92 q^{18} + 72 q^{19} + 76 q^{21} + 22 q^{22} + 98 q^{23} + 252 q^{24} - 40 q^{26} + 182 q^{27} + 128 q^{28} + 144 q^{29} - 34 q^{31} + 104 q^{32} - 22 q^{33} - 52 q^{34} - 80 q^{36} - 54 q^{37} - 432 q^{38} + 400 q^{39} + 536 q^{41} + 140 q^{42} + 60 q^{43} + 88 q^{44} - 314 q^{46} + 272 q^{47} - 776 q^{48} - 390 q^{49} - 164 q^{51} + 560 q^{52} + 492 q^{53} - 110 q^{54} + 120 q^{56} + 1512 q^{57} + 192 q^{58} + 634 q^{59} + 840 q^{61} - 134 q^{62} - 248 q^{63} + 224 q^{64} + 286 q^{66} - 754 q^{67} - 640 q^{68} + 962 q^{69} - 678 q^{71} + 744 q^{72} + 400 q^{73} + 6 q^{74} + 432 q^{76} + 220 q^{77} + 440 q^{78} + 316 q^{79} - 1294 q^{81} - 512 q^{82} - 468 q^{83} - 736 q^{84} - 156 q^{86} - 1200 q^{87} - 132 q^{88} - 1842 q^{89} + 1280 q^{91} + 40 q^{92} + 638 q^{93} - 992 q^{94} - 952 q^{96} - 2194 q^{97} + 870 q^{98} - 484 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 - 8 * q^4 - 26 * q^6 - 20 * q^7 + 12 * q^8 + 44 * q^9 - 22 * q^11 + 40 * q^12 - 80 * q^13 - 4 * q^14 - 8 * q^16 + 124 * q^17 - 92 * q^18 + 72 * q^19 + 76 * q^21 + 22 * q^22 + 98 * q^23 + 252 * q^24 - 40 * q^26 + 182 * q^27 + 128 * q^28 + 144 * q^29 - 34 * q^31 + 104 * q^32 - 22 * q^33 - 52 * q^34 - 80 * q^36 - 54 * q^37 - 432 * q^38 + 400 * q^39 + 536 * q^41 + 140 * q^42 + 60 * q^43 + 88 * q^44 - 314 * q^46 + 272 * q^47 - 776 * q^48 - 390 * q^49 - 164 * q^51 + 560 * q^52 + 492 * q^53 - 110 * q^54 + 120 * q^56 + 1512 * q^57 + 192 * q^58 + 634 * q^59 + 840 * q^61 - 134 * q^62 - 248 * q^63 + 224 * q^64 + 286 * q^66 - 754 * q^67 - 640 * q^68 + 962 * q^69 - 678 * q^71 + 744 * q^72 + 400 * q^73 + 6 * q^74 + 432 * q^76 + 220 * q^77 + 440 * q^78 + 316 * q^79 - 1294 * q^81 - 512 * q^82 - 468 * q^83 - 736 * q^84 - 156 * q^86 - 1200 * q^87 - 132 * q^88 - 1842 * q^89 + 1280 * q^91 + 40 * q^92 + 638 * q^93 - 992 * q^94 - 952 * q^96 - 2194 * q^97 + 870 * q^98 - 484 * 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
 −1.73205 1.73205
−2.73205 7.92820 −0.535898 0 −21.6603 −3.07180 23.3205 35.8564 0
1.2 0.732051 −5.92820 −7.46410 0 −4.33975 −16.9282 −11.3205 8.14359 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$+1$$
$$11$$ $$+1$$

## 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 275.4.a.b 2
3.b odd 2 1 2475.4.a.q 2
5.b even 2 1 11.4.a.a 2
5.c odd 4 2 275.4.b.c 4
15.d odd 2 1 99.4.a.c 2
20.d odd 2 1 176.4.a.i 2
35.c odd 2 1 539.4.a.e 2
40.e odd 2 1 704.4.a.n 2
40.f even 2 1 704.4.a.p 2
55.d odd 2 1 121.4.a.c 2
55.h odd 10 4 121.4.c.f 8
55.j even 10 4 121.4.c.c 8
60.h even 2 1 1584.4.a.bc 2
65.d even 2 1 1859.4.a.a 2
165.d even 2 1 1089.4.a.v 2
220.g even 2 1 1936.4.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 5.b even 2 1
99.4.a.c 2 15.d odd 2 1
121.4.a.c 2 55.d odd 2 1
121.4.c.c 8 55.j even 10 4
121.4.c.f 8 55.h odd 10 4
176.4.a.i 2 20.d odd 2 1
275.4.a.b 2 1.a even 1 1 trivial
275.4.b.c 4 5.c odd 4 2
539.4.a.e 2 35.c odd 2 1
704.4.a.n 2 40.e odd 2 1
704.4.a.p 2 40.f even 2 1
1089.4.a.v 2 165.d even 2 1
1584.4.a.bc 2 60.h even 2 1
1859.4.a.a 2 65.d even 2 1
1936.4.a.w 2 220.g even 2 1
2475.4.a.q 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2T_{2} - 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(275))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 2$$
$3$ $$T^{2} - 2T - 47$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 20T + 52$$
$11$ $$(T + 11)^{2}$$
$13$ $$T^{2} + 80T + 400$$
$17$ $$T^{2} - 124T + 3412$$
$19$ $$T^{2} - 72T - 9504$$
$23$ $$T^{2} - 98T - 1487$$
$29$ $$T^{2} - 144T - 4224$$
$31$ $$T^{2} + 34T - 2063$$
$37$ $$T^{2} + 54T + 537$$
$41$ $$T^{2} - 536T + 71776$$
$43$ $$T^{2} - 60T + 132$$
$47$ $$T^{2} - 272T - 24704$$
$53$ $$T^{2} - 492T + 51108$$
$59$ $$T^{2} - 634T + 48217$$
$61$ $$T^{2} - 840T + 74832$$
$67$ $$T^{2} + 754T + 140929$$
$71$ $$T^{2} + 678T + 97593$$
$73$ $$T^{2} - 400T - 617072$$
$79$ $$T^{2} - 316 T - 1266044$$
$83$ $$T^{2} + 468T + 11556$$
$89$ $$T^{2} + 1842 T + 525489$$
$97$ $$T^{2} + 2194 T + 1141201$$