# Properties

 Label 275.6.a.b Level $275$ Weight $6$ Character orbit 275.a Self dual yes Analytic conductor $44.106$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,6,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, 6, names="a")

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

chi := DirichletCharacter("275.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$275 = 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$44.1055504486$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.54492.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 52x - 38$$ x^3 - x^2 - 52*x - 38 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + (\beta_{2} - \beta_1 - 11) q^{3} + ( - 4 \beta_{2} - 6 \beta_1 + 30) q^{4} + (13 \beta_{2} + 10 \beta_1 - 72) q^{6} + (10 \beta_{2} + 30 \beta_1 - 38) q^{7} + ( - 26 \beta_{2} + 188) q^{8} + ( - 19 \beta_{2} + 11 \beta_1 - 6) q^{9}+O(q^{10})$$ q - b2 * q^2 + (b2 - b1 - 11) * q^3 + (-4*b2 - 6*b1 + 30) * q^4 + (13*b2 + 10*b1 - 72) * q^6 + (10*b2 + 30*b1 - 38) * q^7 + (-26*b2 + 188) * q^8 + (-19*b2 + 11*b1 - 6) * q^9 $$q - \beta_{2} q^{2} + (\beta_{2} - \beta_1 - 11) q^{3} + ( - 4 \beta_{2} - 6 \beta_1 + 30) q^{4} + (13 \beta_{2} + 10 \beta_1 - 72) q^{6} + (10 \beta_{2} + 30 \beta_1 - 38) q^{7} + ( - 26 \beta_{2} + 188) q^{8} + ( - 19 \beta_{2} + 11 \beta_1 - 6) q^{9} + 121 q^{11} + (112 \beta_{2} + 70 \beta_1 - 354) q^{12} + ( - 70 \beta_{2} - 18 \beta_1 - 156) q^{13} + (138 \beta_{2} - 60 \beta_1 - 320) q^{14} + ( - 164 \beta_{2} + 36 \beta_1 + 652) q^{16} + ( - 132 \beta_{2} + 60 \beta_1 - 382) q^{17} + ( - 48 \beta_{2} - 158 \beta_1 + 1288) q^{18} + ( - 60 \beta_{2} - 60 \beta_1 + 480) q^{19} + ( - 318 \beta_{2} - 362 \beta_1 - 182) q^{21} - 121 \beta_{2} q^{22} + (21 \beta_{2} - 501 \beta_1 + 1189) q^{23} + (526 \beta_{2} + 72 \beta_1 - 3940) q^{24} + ( - 160 \beta_{2} - 348 \beta_1 + 4160) q^{26} + ( - 57 \beta_{2} + 329 \beta_1 + 887) q^{27} + (432 \beta_{2} + 108 \beta_1 - 7940) q^{28} + ( - 182 \beta_{2} - 138 \beta_1 - 1096) q^{29} + ( - 101 \beta_{2} + 69 \beta_1 - 1389) q^{31} + ( - 404 \beta_{2} - 1128 \beta_1 + 4512) q^{32} + (121 \beta_{2} - 121 \beta_1 - 1331) q^{33} + ( - 26 \beta_{2} - 1032 \beta_1 + 8784) q^{34} + ( - 1188 \beta_{2} - 8 \beta_1 + 1588) q^{36} + ( - 937 \beta_{2} - 591 \beta_1 - 5711) q^{37} + ( - 840 \beta_{2} - 120 \beta_1 + 3120) q^{38} + (844 \beta_{2} + 1036 \beta_1 - 2532) q^{39} + ( - 1378 \beta_{2} + 282 \beta_1 + 1904) q^{41} + ( - 1814 \beta_{2} - 460 \beta_1 + 16096) q^{42} + ( - 1190 \beta_{2} + 1110 \beta_1 + 8366) q^{43} + ( - 484 \beta_{2} - 726 \beta_1 + 3630) q^{44} + ( - 2107 \beta_{2} + 2130 \beta_1 - 6312) q^{46} + (600 \beta_{2} + 1272 \beta_1 + 5320) q^{47} + (2604 \beta_{2} + 628 \beta_1 - 20564) q^{48} + (340 \beta_{2} + 2220 \beta_1 + 15437) q^{49} + (1034 \beta_{2} + 1102 \beta_1 - 7942) q^{51} + ( - 3256 \beta_{2} + 1008 \beta_1 + 11432) q^{52} + (476 \beta_{2} + 1620 \beta_1 - 17402) q^{53} + ( - 457 \beta_{2} - 1658 \beta_1 + 6824) q^{54} + (5468 \beta_{2} + 4080 \beta_1 - 15464) q^{56} + (1560 \beta_{2} + 720 \beta_1 - 6960) q^{57} + (92 \beta_{2} - 540 \beta_1 + 9904) q^{58} + (3141 \beta_{2} + 747 \beta_1 - 1495) q^{59} + ( - 1466 \beta_{2} - 4038 \beta_1 + 7508) q^{61} + (1123 \beta_{2} - 882 \beta_1 + 6952) q^{62} + (3332 \beta_{2} - 308 \beta_1 + 4268) q^{63} + ( - 3136 \beta_{2} + 936 \beta_1 - 7096) q^{64} + (1573 \beta_{2} + 1210 \beta_1 - 8712) q^{66} + (6575 \beta_{2} + 2721 \beta_1 + 15011) q^{67} + ( - 6728 \beta_{2} + 2052 \beta_1 + 3516) q^{68} + (3421 \beta_{2} + 3611 \beta_1 + 10477) q^{69} + ( - 3935 \beta_{2} - 2673 \beta_1 + 13985) q^{71} + ( - 4820 \beta_{2} - 2040 \beta_1 + 32360) q^{72} + ( - 5370 \beta_{2} + 3522 \beta_1 - 6316) q^{73} + (781 \beta_{2} - 3258 \beta_1 + 52184) q^{74} + ( - 4800 \beta_{2} - 2640 \beta_1 + 35520) q^{76} + (1210 \beta_{2} + 3630 \beta_1 - 4598) q^{77} + (7980 \beta_{2} + 920 \beta_1 - 41968) q^{78} + (3902 \beta_{2} + 1362 \beta_1 + 41262) q^{79} + (4600 \beta_{2} - 6280 \beta_1 - 26879) q^{81} + ( - 6852 \beta_{2} - 9396 \beta_1 + 88256) q^{82} + ( - 3150 \beta_{2} - 11250 \beta_1 + 51726) q^{83} + ( - 14096 \beta_{2} + 2540 \beta_1 + 113692) q^{84} + ( - 10906 \beta_{2} - 11580 \beta_1 + 84880) q^{86} + (1960 \beta_{2} + 4296 \beta_1 + 5024) q^{87} + ( - 3146 \beta_{2} + 22748) q^{88} + ( - 5167 \beta_{2} - 4569 \beta_1 - 34085) q^{89} + (6840 \beta_{2} - 10536 \beta_1 - 33032) q^{91} + (1472 \beta_{2} - 5130 \beta_1 + 113886) q^{92} + ( - 421 \beta_{2} + 1709 \beta_1 + 4971) q^{93} + ( - 376 \beta_{2} - 1488 \beta_1 - 24480) q^{94} + (15404 \beta_{2} + 10808 \beta_1 - 29088) q^{96} + ( - 123 \beta_{2} - 6429 \beta_1 - 1085) q^{97} + ( - 9637 \beta_{2} - 6840 \beta_1 + 1120) q^{98} + ( - 2299 \beta_{2} + 1331 \beta_1 - 726) q^{99}+O(q^{100})$$ q - b2 * q^2 + (b2 - b1 - 11) * q^3 + (-4*b2 - 6*b1 + 30) * q^4 + (13*b2 + 10*b1 - 72) * q^6 + (10*b2 + 30*b1 - 38) * q^7 + (-26*b2 + 188) * q^8 + (-19*b2 + 11*b1 - 6) * q^9 + 121 * q^11 + (112*b2 + 70*b1 - 354) * q^12 + (-70*b2 - 18*b1 - 156) * q^13 + (138*b2 - 60*b1 - 320) * q^14 + (-164*b2 + 36*b1 + 652) * q^16 + (-132*b2 + 60*b1 - 382) * q^17 + (-48*b2 - 158*b1 + 1288) * q^18 + (-60*b2 - 60*b1 + 480) * q^19 + (-318*b2 - 362*b1 - 182) * q^21 - 121*b2 * q^22 + (21*b2 - 501*b1 + 1189) * q^23 + (526*b2 + 72*b1 - 3940) * q^24 + (-160*b2 - 348*b1 + 4160) * q^26 + (-57*b2 + 329*b1 + 887) * q^27 + (432*b2 + 108*b1 - 7940) * q^28 + (-182*b2 - 138*b1 - 1096) * q^29 + (-101*b2 + 69*b1 - 1389) * q^31 + (-404*b2 - 1128*b1 + 4512) * q^32 + (121*b2 - 121*b1 - 1331) * q^33 + (-26*b2 - 1032*b1 + 8784) * q^34 + (-1188*b2 - 8*b1 + 1588) * q^36 + (-937*b2 - 591*b1 - 5711) * q^37 + (-840*b2 - 120*b1 + 3120) * q^38 + (844*b2 + 1036*b1 - 2532) * q^39 + (-1378*b2 + 282*b1 + 1904) * q^41 + (-1814*b2 - 460*b1 + 16096) * q^42 + (-1190*b2 + 1110*b1 + 8366) * q^43 + (-484*b2 - 726*b1 + 3630) * q^44 + (-2107*b2 + 2130*b1 - 6312) * q^46 + (600*b2 + 1272*b1 + 5320) * q^47 + (2604*b2 + 628*b1 - 20564) * q^48 + (340*b2 + 2220*b1 + 15437) * q^49 + (1034*b2 + 1102*b1 - 7942) * q^51 + (-3256*b2 + 1008*b1 + 11432) * q^52 + (476*b2 + 1620*b1 - 17402) * q^53 + (-457*b2 - 1658*b1 + 6824) * q^54 + (5468*b2 + 4080*b1 - 15464) * q^56 + (1560*b2 + 720*b1 - 6960) * q^57 + (92*b2 - 540*b1 + 9904) * q^58 + (3141*b2 + 747*b1 - 1495) * q^59 + (-1466*b2 - 4038*b1 + 7508) * q^61 + (1123*b2 - 882*b1 + 6952) * q^62 + (3332*b2 - 308*b1 + 4268) * q^63 + (-3136*b2 + 936*b1 - 7096) * q^64 + (1573*b2 + 1210*b1 - 8712) * q^66 + (6575*b2 + 2721*b1 + 15011) * q^67 + (-6728*b2 + 2052*b1 + 3516) * q^68 + (3421*b2 + 3611*b1 + 10477) * q^69 + (-3935*b2 - 2673*b1 + 13985) * q^71 + (-4820*b2 - 2040*b1 + 32360) * q^72 + (-5370*b2 + 3522*b1 - 6316) * q^73 + (781*b2 - 3258*b1 + 52184) * q^74 + (-4800*b2 - 2640*b1 + 35520) * q^76 + (1210*b2 + 3630*b1 - 4598) * q^77 + (7980*b2 + 920*b1 - 41968) * q^78 + (3902*b2 + 1362*b1 + 41262) * q^79 + (4600*b2 - 6280*b1 - 26879) * q^81 + (-6852*b2 - 9396*b1 + 88256) * q^82 + (-3150*b2 - 11250*b1 + 51726) * q^83 + (-14096*b2 + 2540*b1 + 113692) * q^84 + (-10906*b2 - 11580*b1 + 84880) * q^86 + (1960*b2 + 4296*b1 + 5024) * q^87 + (-3146*b2 + 22748) * q^88 + (-5167*b2 - 4569*b1 - 34085) * q^89 + (6840*b2 - 10536*b1 - 33032) * q^91 + (1472*b2 - 5130*b1 + 113886) * q^92 + (-421*b2 + 1709*b1 + 4971) * q^93 + (-376*b2 - 1488*b1 - 24480) * q^94 + (15404*b2 + 10808*b1 - 29088) * q^96 + (-123*b2 - 6429*b1 - 1085) * q^97 + (-9637*b2 - 6840*b1 + 1120) * q^98 + (-2299*b2 + 1331*b1 - 726) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 34 q^{3} + 84 q^{4} - 206 q^{6} - 84 q^{7} + 564 q^{8} - 7 q^{9}+O(q^{10})$$ 3 * q - 34 * q^3 + 84 * q^4 - 206 * q^6 - 84 * q^7 + 564 * q^8 - 7 * q^9 $$3 q - 34 q^{3} + 84 q^{4} - 206 q^{6} - 84 q^{7} + 564 q^{8} - 7 q^{9} + 363 q^{11} - 992 q^{12} - 486 q^{13} - 1020 q^{14} + 1992 q^{16} - 1086 q^{17} + 3706 q^{18} + 1380 q^{19} - 908 q^{21} + 3066 q^{23} - 11748 q^{24} + 12132 q^{26} + 2990 q^{27} - 23712 q^{28} - 3426 q^{29} - 4098 q^{31} + 12408 q^{32} - 4114 q^{33} + 25320 q^{34} + 4756 q^{36} - 17724 q^{37} + 9240 q^{38} - 6560 q^{39} + 5994 q^{41} + 47828 q^{42} + 26208 q^{43} + 10164 q^{44} - 16806 q^{46} + 17232 q^{47} - 61064 q^{48} + 48531 q^{49} - 22724 q^{51} + 35304 q^{52} - 50586 q^{53} + 18814 q^{54} - 42312 q^{56} - 20160 q^{57} + 29172 q^{58} - 3738 q^{59} + 18486 q^{61} + 19974 q^{62} + 12496 q^{63} - 20352 q^{64} - 24926 q^{66} + 47754 q^{67} + 12600 q^{68} + 35042 q^{69} + 39282 q^{71} + 95040 q^{72} - 15426 q^{73} + 153294 q^{74} + 103920 q^{76} - 10164 q^{77} - 124984 q^{78} + 125148 q^{79} - 86917 q^{81} + 255372 q^{82} + 143928 q^{83} + 343616 q^{84} + 243060 q^{86} + 19368 q^{87} + 68244 q^{88} - 106824 q^{89} - 109632 q^{91} + 336528 q^{92} + 16622 q^{93} - 74928 q^{94} - 76456 q^{96} - 9684 q^{97} - 3480 q^{98} - 847 q^{99}+O(q^{100})$$ 3 * q - 34 * q^3 + 84 * q^4 - 206 * q^6 - 84 * q^7 + 564 * q^8 - 7 * q^9 + 363 * q^11 - 992 * q^12 - 486 * q^13 - 1020 * q^14 + 1992 * q^16 - 1086 * q^17 + 3706 * q^18 + 1380 * q^19 - 908 * q^21 + 3066 * q^23 - 11748 * q^24 + 12132 * q^26 + 2990 * q^27 - 23712 * q^28 - 3426 * q^29 - 4098 * q^31 + 12408 * q^32 - 4114 * q^33 + 25320 * q^34 + 4756 * q^36 - 17724 * q^37 + 9240 * q^38 - 6560 * q^39 + 5994 * q^41 + 47828 * q^42 + 26208 * q^43 + 10164 * q^44 - 16806 * q^46 + 17232 * q^47 - 61064 * q^48 + 48531 * q^49 - 22724 * q^51 + 35304 * q^52 - 50586 * q^53 + 18814 * q^54 - 42312 * q^56 - 20160 * q^57 + 29172 * q^58 - 3738 * q^59 + 18486 * q^61 + 19974 * q^62 + 12496 * q^63 - 20352 * q^64 - 24926 * q^66 + 47754 * q^67 + 12600 * q^68 + 35042 * q^69 + 39282 * q^71 + 95040 * q^72 - 15426 * q^73 + 153294 * q^74 + 103920 * q^76 - 10164 * q^77 - 124984 * q^78 + 125148 * q^79 - 86917 * q^81 + 255372 * q^82 + 143928 * q^83 + 343616 * q^84 + 243060 * q^86 + 19368 * q^87 + 68244 * q^88 - 106824 * q^89 - 109632 * q^91 + 336528 * q^92 + 16622 * q^93 - 74928 * q^94 - 76456 * q^96 - 9684 * q^97 - 3480 * q^98 - 847 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 52x - 38$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} - 3\nu - 34 ) / 3$$ (v^2 - 3*v - 34) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2} + 3\beta _1 + 34$$ 3*b2 + 3*b1 + 34

## 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
 −6.29828 8.04796 −0.749680
−8.18772 3.48600 35.0388 0 −28.5424 −145.071 −24.8808 −230.848 0
1.2 −2.20859 −16.8394 −27.1221 0 37.1913 225.525 130.577 40.5643 0
1.3 10.3963 −20.6466 76.0833 0 −214.649 −164.454 458.304 183.283 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 275.6.a.b 3
5.b even 2 1 11.6.a.b 3
5.c odd 4 2 275.6.b.b 6
15.d odd 2 1 99.6.a.g 3
20.d odd 2 1 176.6.a.i 3
35.c odd 2 1 539.6.a.e 3
40.e odd 2 1 704.6.a.t 3
40.f even 2 1 704.6.a.q 3
55.d odd 2 1 121.6.a.d 3
165.d even 2 1 1089.6.a.r 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.6.a.b 3 5.b even 2 1
99.6.a.g 3 15.d odd 2 1
121.6.a.d 3 55.d odd 2 1
176.6.a.i 3 20.d odd 2 1
275.6.a.b 3 1.a even 1 1 trivial
275.6.b.b 6 5.c odd 4 2
539.6.a.e 3 35.c odd 2 1
704.6.a.q 3 40.f even 2 1
704.6.a.t 3 40.e odd 2 1
1089.6.a.r 3 165.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 90T_{2} - 188$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(275))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 90T - 188$$
$3$ $$T^{3} + 34 T^{2} + \cdots - 1212$$
$5$ $$T^{3}$$
$7$ $$T^{3} + 84 T^{2} + \cdots - 5380448$$
$11$ $$(T - 121)^{3}$$
$13$ $$T^{3} + 486 T^{2} + \cdots - 164136608$$
$17$ $$T^{3} + 1086 T^{2} + \cdots - 331752056$$
$19$ $$T^{3} - 1380 T^{2} + \cdots + 57024000$$
$23$ $$T^{3} + \cdots + 17004325928$$
$29$ $$T^{3} + \cdots - 4029189120$$
$31$ $$T^{3} + \cdots + 1094344400$$
$37$ $$T^{3} + \cdots - 541788167034$$
$41$ $$T^{3} + \cdots + 201929821568$$
$43$ $$T^{3} + \cdots + 2443875098544$$
$47$ $$T^{3} + \cdots + 70174939136$$
$53$ $$T^{3} + \cdots + 1850911309656$$
$59$ $$T^{3} + \cdots + 7759637437060$$
$61$ $$T^{3} + \cdots + 15233874751008$$
$67$ $$T^{3} + \cdots + 147288561330212$$
$71$ $$T^{3} + \cdots - 1290398551704$$
$73$ $$T^{3} + \cdots + 34539701265952$$
$79$ $$T^{3} + \cdots + 1279883216320$$
$83$ $$T^{3} + \cdots + 411597824719824$$
$89$ $$T^{3} + \cdots - 90320980174650$$
$97$ $$T^{3} + \cdots + 10221902527106$$