# Properties

 Label 275.6.b.b Level $275$ Weight $6$ Character orbit 275.b Analytic conductor $44.106$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,6,Mod(199,275)]

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([1, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("275.199");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$275 = 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 275.b (of order $$2$$, degree $$1$$, 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: $$44.1055504486$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.11877512256.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 34x^{4} - 154x^{3} + 569x^{2} - 6512x + 17216$$ x^6 - 2*x^5 + 34*x^4 - 154*x^3 + 569*x^2 - 6512*x + 17216 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 11) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} + (\beta_{5} - \beta_{4} + 6 \beta_{3}) q^{3} + ( - 4 \beta_{2} - 6 \beta_1 - 30) q^{4} + ( - 13 \beta_{2} - 10 \beta_1 - 72) q^{6} + ( - 10 \beta_{5} - 30 \beta_{4} - 4 \beta_{3}) q^{7} + ( - 26 \beta_{5} - 94 \beta_{3}) q^{8} + ( - 19 \beta_{2} + 11 \beta_1 + 6) q^{9}+O(q^{10})$$ q + b5 * q^2 + (b5 - b4 + 6*b3) * q^3 + (-4*b2 - 6*b1 - 30) * q^4 + (-13*b2 - 10*b1 - 72) * q^6 + (-10*b5 - 30*b4 - 4*b3) * q^7 + (-26*b5 - 94*b3) * q^8 + (-19*b2 + 11*b1 + 6) * q^9 $$q + \beta_{5} q^{2} + (\beta_{5} - \beta_{4} + 6 \beta_{3}) q^{3} + ( - 4 \beta_{2} - 6 \beta_1 - 30) q^{4} + ( - 13 \beta_{2} - 10 \beta_1 - 72) q^{6} + ( - 10 \beta_{5} - 30 \beta_{4} - 4 \beta_{3}) q^{7} + ( - 26 \beta_{5} - 94 \beta_{3}) q^{8} + ( - 19 \beta_{2} + 11 \beta_1 + 6) q^{9} + 121 q^{11} + ( - 112 \beta_{5} + \cdots - 142 \beta_{3}) q^{12}+ \cdots + ( - 2299 \beta_{2} + 1331 \beta_1 + 726) q^{99}+O(q^{100})$$ q + b5 * q^2 + (b5 - b4 + 6*b3) * q^3 + (-4*b2 - 6*b1 - 30) * q^4 + (-13*b2 - 10*b1 - 72) * q^6 + (-10*b5 - 30*b4 - 4*b3) * q^7 + (-26*b5 - 94*b3) * q^8 + (-19*b2 + 11*b1 + 6) * q^9 + 121 * q^11 + (-112*b5 - 70*b4 - 142*b3) * q^12 + (-70*b5 - 18*b4 + 87*b3) * q^13 + (138*b2 - 60*b1 + 320) * q^14 + (164*b2 - 36*b1 + 652) * q^16 + (132*b5 - 60*b4 - 161*b3) * q^17 + (-48*b5 - 158*b4 - 565*b3) * q^18 + (-60*b2 - 60*b1 - 480) * q^19 + (318*b2 + 362*b1 - 182) * q^21 + 121*b5 * q^22 + (21*b5 - 501*b4 - 344*b3) * q^23 + (526*b2 + 72*b1 + 3940) * q^24 + (160*b2 + 348*b1 + 4160) * q^26 + (57*b5 - 329*b4 + 608*b3) * q^27 + (432*b5 + 108*b4 + 3916*b3) * q^28 + (-182*b2 - 138*b1 + 1096) * q^29 + (101*b2 - 69*b1 - 1389) * q^31 + (404*b5 + 1128*b4 + 1692*b3) * q^32 + (121*b5 - 121*b4 + 726*b3) * q^33 + (-26*b2 - 1032*b1 - 8784) * q^34 + (1188*b2 + 8*b1 + 1588) * q^36 + (937*b5 + 591*b4 - 3151*b3) * q^37 + (-840*b5 - 120*b4 - 1500*b3) * q^38 + (844*b2 + 1036*b1 + 2532) * q^39 + (1378*b2 - 282*b1 + 1904) * q^41 + (1814*b5 + 460*b4 + 7818*b3) * q^42 + (-1190*b5 + 1110*b4 - 4738*b3) * q^43 + (-484*b2 - 726*b1 - 3630) * q^44 + (2107*b2 - 2130*b1 - 6312) * q^46 + (-600*b5 - 1272*b4 + 3296*b3) * q^47 + (2604*b5 + 628*b4 + 9968*b3) * q^48 + (340*b2 + 2220*b1 - 15437) * q^49 + (-1034*b2 - 1102*b1 - 7942) * q^51 + (3256*b5 - 1008*b4 + 6220*b3) * q^52 + (476*b5 + 1620*b4 + 7891*b3) * q^53 + (-457*b2 - 1658*b1 - 6824) * q^54 + (-5468*b2 - 4080*b1 - 15464) * q^56 + (-1560*b5 - 720*b4 - 3120*b3) * q^57 + (92*b5 - 540*b4 - 4682*b3) * q^58 + (3141*b2 + 747*b1 + 1495) * q^59 + (1466*b2 + 4038*b1 + 7508) * q^61 + (-1123*b5 + 882*b4 + 3035*b3) * q^62 + (3332*b5 - 308*b4 - 1980*b3) * q^63 + (-3136*b2 + 936*b1 + 7096) * q^64 + (-1573*b2 - 1210*b1 - 8712) * q^66 + (-6575*b5 - 2721*b4 + 8866*b3) * q^67 + (-6728*b5 + 2052*b4 - 2784*b3) * q^68 + (3421*b2 + 3611*b1 - 10477) * q^69 + (3935*b2 + 2673*b1 + 13985) * q^71 + (4820*b5 + 2040*b4 + 15160*b3) * q^72 + (-5370*b5 + 3522*b4 + 1397*b3) * q^73 + (781*b2 - 3258*b1 - 52184) * q^74 + (4800*b2 + 2640*b1 + 35520) * q^76 + (-1210*b5 - 3630*b4 - 484*b3) * q^77 + (7980*b5 + 920*b4 + 20524*b3) * q^78 + (3902*b2 + 1362*b1 - 41262) * q^79 + (-4600*b2 + 6280*b1 - 26879) * q^81 + (6852*b5 + 9396*b4 + 39430*b3) * q^82 + (-3150*b5 - 11250*b4 - 20238*b3) * q^83 + (-14096*b2 + 2540*b1 - 113692) * q^84 + (10906*b2 + 11580*b1 + 84880) * q^86 + (-1960*b5 - 4296*b4 + 4660*b3) * q^87 + (-3146*b5 - 11374*b3) * q^88 + (-5167*b2 - 4569*b1 + 34085) * q^89 + (-6840*b2 + 10536*b1 - 33032) * q^91 + (-1472*b5 + 5130*b4 + 54378*b3) * q^92 + (-421*b5 + 1709*b4 - 3340*b3) * q^93 + (-376*b2 - 1488*b1 + 24480) * q^94 + (-15404*b2 - 10808*b1 - 29088) * q^96 + (123*b5 + 6429*b4 - 3757*b3) * q^97 + (-9637*b5 - 6840*b4 + 2860*b3) * q^98 + (-2299*b2 + 1331*b1 + 726) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 168 q^{4} - 412 q^{6} + 14 q^{9}+O(q^{10})$$ 6 * q - 168 * q^4 - 412 * q^6 + 14 * q^9 $$6 q - 168 q^{4} - 412 q^{6} + 14 q^{9} + 726 q^{11} + 2040 q^{14} + 3984 q^{16} - 2760 q^{19} - 1816 q^{21} + 23496 q^{24} + 24264 q^{26} + 6852 q^{29} - 8196 q^{31} - 50640 q^{34} + 9512 q^{36} + 13120 q^{39} + 11988 q^{41} - 20328 q^{44} - 33612 q^{46} - 97062 q^{49} - 45448 q^{51} - 37628 q^{54} - 84624 q^{56} + 7476 q^{59} + 36972 q^{61} + 40704 q^{64} - 49852 q^{66} - 70084 q^{69} + 78564 q^{71} - 306588 q^{74} + 207840 q^{76} - 250296 q^{79} - 173834 q^{81} - 687232 q^{84} + 486120 q^{86} + 213648 q^{89} - 219264 q^{91} + 149856 q^{94} - 152912 q^{96} + 1694 q^{99}+O(q^{100})$$ 6 * q - 168 * q^4 - 412 * q^6 + 14 * q^9 + 726 * q^11 + 2040 * q^14 + 3984 * q^16 - 2760 * q^19 - 1816 * q^21 + 23496 * q^24 + 24264 * q^26 + 6852 * q^29 - 8196 * q^31 - 50640 * q^34 + 9512 * q^36 + 13120 * q^39 + 11988 * q^41 - 20328 * q^44 - 33612 * q^46 - 97062 * q^49 - 45448 * q^51 - 37628 * q^54 - 84624 * q^56 + 7476 * q^59 + 36972 * q^61 + 40704 * q^64 - 49852 * q^66 - 70084 * q^69 + 78564 * q^71 - 306588 * q^74 + 207840 * q^76 - 250296 * q^79 - 173834 * q^81 - 687232 * q^84 + 486120 * q^86 + 213648 * q^89 - 219264 * q^91 + 149856 * q^94 - 152912 * q^96 + 1694 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 34x^{4} - 154x^{3} + 569x^{2} - 6512x + 17216$$ :

 $$\beta_{1}$$ $$=$$ $$( -137\nu^{5} + 578\nu^{4} - 4930\nu^{3} + 27490\nu^{2} + 2527\nu + 713224 ) / 138448$$ (-137*v^5 + 578*v^4 - 4930*v^3 + 27490*v^2 + 2527*v + 713224) / 138448 $$\beta_{2}$$ $$=$$ $$( 167\nu^{5} + 306\nu^{4} + 1462\nu^{3} - 30478\nu^{2} - 37945\nu - 804728 ) / 138448$$ (167*v^5 + 306*v^4 + 1462*v^3 - 30478*v^2 - 37945*v - 804728) / 138448 $$\beta_{3}$$ $$=$$ $$( -25\nu^{5} - 58\nu^{4} - 1182\nu^{3} + 454\nu^{2} - 15277\nu + 127832 ) / 20360$$ (-25*v^5 - 58*v^4 - 1182*v^3 + 454*v^2 - 15277*v + 127832) / 20360 $$\beta_{4}$$ $$=$$ $$( -1761\nu^{5} - 3434\nu^{4} - 66402\nu^{3} - 25354\nu^{2} - 683001\nu + 7434160 ) / 346120$$ (-1761*v^5 - 3434*v^4 - 66402*v^3 - 25354*v^2 - 683001*v + 7434160) / 346120 $$\beta_{5}$$ $$=$$ $$( 4207\nu^{5} + 3978\nu^{4} + 157454\nu^{3} - 86742\nu^{2} + 2737847\nu - 19126680 ) / 692240$$ (4207*v^5 + 3978*v^4 + 157454*v^3 - 86742*v^2 + 2737847*v - 19126680) / 692240
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} + \beta _1 + 1 ) / 2$$ (b5 + b4 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( -5\beta_{4} + 15\beta_{3} - 5\beta_{2} + \beta _1 - 21 ) / 2$$ (-5*b4 + 15*b3 - 5*b2 + b1 - 21) / 2 $$\nu^{3}$$ $$=$$ $$( -21\beta_{5} - \beta_{4} - 100\beta_{3} - 24\beta_{2} - 29\beta _1 + 79 ) / 2$$ (-21*b5 - b4 - 100*b3 - 24*b2 - 29*b1 + 79) / 2 $$\nu^{4}$$ $$=$$ $$( -60\beta_{5} + 45\beta_{4} - 495\beta_{3} + 181\beta_{2} + 235\beta _1 + 325 ) / 2$$ (-60*b5 + 45*b4 - 495*b3 + 181*b2 + 235*b1 + 325) / 2 $$\nu^{5}$$ $$=$$ $$( 521\beta_{5} - 759\beta_{4} + 4520\beta_{3} + 624\beta_{2} + 233\beta _1 + 4745 ) / 2$$ (521*b5 - 759*b4 + 4520*b3 + 624*b2 + 233*b1 + 4745) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 0.874840 − 6.07300i 3.64914 − 0.444721i −3.52398 − 4.62828i −3.52398 + 4.62828i 3.64914 + 0.444721i 0.874840 + 6.07300i
10.3963i 20.6466i −76.0833 0 −214.649 164.454i 458.304i −183.283 0
199.2 8.18772i 3.48600i −35.0388 0 −28.5424 145.071i 24.8808i 230.848 0
199.3 2.20859i 16.8394i 27.1221 0 37.1913 225.525i 130.577i −40.5643 0
199.4 2.20859i 16.8394i 27.1221 0 37.1913 225.525i 130.577i −40.5643 0
199.5 8.18772i 3.48600i −35.0388 0 −28.5424 145.071i 24.8808i 230.848 0
199.6 10.3963i 20.6466i −76.0833 0 −214.649 164.454i 458.304i −183.283 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 180T_{2}^{4} + 8100T_{2}^{2} + 35344$$ acting on $$S_{6}^{\mathrm{new}}(275, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 180 T^{4} + \cdots + 35344$$
$3$ $$T^{6} + 722 T^{4} + \cdots + 1468944$$
$5$ $$T^{6}$$
$7$ $$T^{6} + \cdots + 28949220680704$$
$11$ $$(T - 121)^{6}$$
$13$ $$T^{6} + \cdots + 26\!\cdots\!64$$
$17$ $$T^{6} + \cdots + 11\!\cdots\!36$$
$19$ $$(T^{3} + 1380 T^{2} + \cdots - 57024000)^{2}$$
$23$ $$T^{6} + \cdots + 28\!\cdots\!84$$
$29$ $$(T^{3} - 3426 T^{2} + \cdots + 4029189120)^{2}$$
$31$ $$(T^{3} + 4098 T^{2} + \cdots + 1094344400)^{2}$$
$37$ $$T^{6} + \cdots + 29\!\cdots\!56$$
$41$ $$(T^{3} - 5994 T^{2} + \cdots + 201929821568)^{2}$$
$43$ $$T^{6} + \cdots + 59\!\cdots\!36$$
$47$ $$T^{6} + \cdots + 49\!\cdots\!96$$
$53$ $$T^{6} + \cdots + 34\!\cdots\!36$$
$59$ $$(T^{3} + \cdots - 7759637437060)^{2}$$
$61$ $$(T^{3} + \cdots + 15233874751008)^{2}$$
$67$ $$T^{6} + \cdots + 21\!\cdots\!44$$
$71$ $$(T^{3} + \cdots - 1290398551704)^{2}$$
$73$ $$T^{6} + \cdots + 11\!\cdots\!04$$
$79$ $$(T^{3} + \cdots - 1279883216320)^{2}$$
$83$ $$T^{6} + \cdots + 16\!\cdots\!76$$
$89$ $$(T^{3} + \cdots + 90320980174650)^{2}$$
$97$ $$T^{6} + \cdots + 10\!\cdots\!36$$