# Properties

 Label 100.9.f.b Level $100$ Weight $9$ Character orbit 100.f Analytic conductor $40.738$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [100,9,Mod(57,100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(100, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("100.57");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 100.f (of order $$4$$, degree $$2$$, 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: $$40.7378610061$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4 x^{7} + 8 x^{6} + 22254 x^{5} + 4820745 x^{4} + 50131374 x^{3} + 307615702 x^{2} + \cdots + 2405464244$$ x^8 - 4*x^7 + 8*x^6 + 22254*x^5 + 4820745*x^4 + 50131374*x^3 + 307615702*x^2 - 1770757924*x + 2405464244 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{14}\cdot 5^{4}$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 9 \beta_1 + 9) q^{3} + (\beta_{7} - \beta_{4} - \beta_{3} + \cdots + 254) q^{7}+ \cdots + ( - 4 \beta_{7} - 4 \beta_{6} + \cdots + 2648 \beta_1) q^{9}+O(q^{10})$$ q + (b2 + 9*b1 + 9) * q^3 + (b7 - b4 - b3 - 254*b1 + 254) * q^7 + (-4*b7 - 4*b6 + 3*b5 + 4*b4 + 27*b3 + 27*b2 + 2648*b1) * q^9 $$q + (\beta_{2} + 9 \beta_1 + 9) q^{3} + (\beta_{7} - \beta_{4} - \beta_{3} + \cdots + 254) q^{7}+ \cdots + (16577 \beta_{7} + \cdots - 34340848 \beta_1) q^{99}+O(q^{100})$$ q + (b2 + 9*b1 + 9) * q^3 + (b7 - b4 - b3 - 254*b1 + 254) * q^7 + (-4*b7 - 4*b6 + 3*b5 + 4*b4 + 27*b3 + 27*b2 + 2648*b1) * q^9 + (-b7 + b6 + 9*b4 - 17*b3 + 17*b2 - 44) * q^11 + (19*b6 - 15*b5 - 15*b4 + 98*b2 - 4123*b1 - 4123) * q^13 + (20*b7 + 5*b5 - 25*b4 + 646*b3 + 5614*b1 - 5614) * q^17 + (-b7 - b6 + 12*b5 + b4 + 1198*b3 + 1198*b2 + 28846*b1) * q^19 + (27*b7 - 27*b6 - 48*b4 - 1231*b3 + 1231*b2 + 14199) * q^21 + (-157*b6 + 100*b5 + 100*b4 + 1765*b2 + 83350*b1 + 83350) * q^23 + (-208*b7 - 60*b5 + 268*b4 + 3708*b3 + 197932*b1 - 197932) * q^27 + (233*b7 + 233*b6 - 296*b5 - 233*b4 + 2116*b3 + 2116*b2 + 22670*b1) * q^29 + (-235*b7 + 235*b6 - 65*b4 - 935*b3 + 935*b2 - 396844) * q^31 + (422*b6 - 45*b5 - 45*b4 - 4078*b2 + 116983*b1 + 116983) * q^33 + (357*b7 + 330*b5 - 687*b4 - 11040*b3 + 665250*b1 - 665250) * q^37 + (-915*b7 - 915*b6 + 1200*b5 + 915*b4 - 15465*b3 - 15465*b2 + 935772*b1) * q^39 + (954*b7 - 954*b6 + 749*b4 + 16773*b3 - 16773*b2 - 1281543) * q^41 + (-454*b6 - 1020*b5 - 1020*b4 - 239*b2 + 1292779*b1 + 1292779) * q^43 + (1983*b7 - 1200*b5 - 783*b4 - 37101*b3 + 2394756*b1 - 2394756) * q^47 + (182*b7 + 182*b6 - 1239*b5 - 182*b4 - 12971*b3 - 12971*b2 + 2166976*b1) * q^49 + (-1804*b7 + 1804*b6 - 464*b4 - 3643*b3 + 3643*b2 - 5889014) * q^51 + (1573*b6 + 1760*b5 + 1760*b4 - 42664*b2 + 3029414*b1 + 3029414) * q^53 + (-8862*b7 + 3390*b5 + 5472*b4 + 69800*b3 + 11044740*b1 - 11044740) * q^57 + (4453*b7 + 4453*b6 - 1556*b5 - 4453*b4 - 4234*b3 - 4234*b2 + 3199282*b1) * q^59 + (662*b7 - 662*b6 - 3803*b4 - 131*b3 + 131*b2 - 10314395) * q^61 + (-5873*b6 + 3360*b5 + 3360*b4 + 110901*b2 + 9707894*b1 + 9707894) * q^63 + (9702*b7 - 6720*b5 - 2982*b4 + 48375*b3 + 12596585*b1 - 12596585) * q^67 + (-4271*b7 - 4271*b6 - 1473*b5 + 4271*b4 + 231583*b3 + 231583*b2 + 16663315*b1) * q^69 + (2037*b7 - 2037*b6 + 3847*b4 - 208431*b3 + 208431*b2 - 12307044) * q^71 + (-446*b6 - 6990*b5 - 6990*b4 - 5512*b2 + 11689687*b1 + 11689687) * q^73 + (8676*b7 + 4585*b5 - 13261*b4 + 114006*b3 + 16803459*b1 - 16803459) * q^77 + (-9112*b7 - 9112*b6 + 16884*b5 + 9112*b4 - 95144*b3 - 95144*b2 + 13406084*b1) * q^79 + (2788*b7 - 2788*b6 + 8963*b4 + 307461*b3 - 307461*b2 - 20393764) * q^81 + (30176*b6 - 10680*b5 - 10680*b4 + 158413*b2 + 1345897*b1 + 1345897) * q^83 + (-25154*b7 + 16380*b5 + 8774*b4 - 247112*b3 + 20538422*b1 - 20538422) * q^87 + (5654*b7 + 5654*b6 - 5608*b5 - 5654*b4 - 164912*b3 - 164912*b2 + 4382972*b1) * q^89 + (-15514*b7 + 15514*b6 + 646*b4 - 145703*b3 + 145703*b2 - 16262282) * q^91 + (-15870*b6 + 12945*b5 + 12945*b4 - 711614*b2 + 6095049*b1 + 6095049) * q^93 + (28228*b7 - 49530*b5 + 21302*b4 - 261904*b3 - 22511821*b1 + 22511821) * q^97 + (16577*b7 + 16577*b6 - 53244*b5 - 16577*b4 - 457311*b3 - 457311*b2 - 34340848*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 70 q^{3} + 2030 q^{7}+O(q^{10})$$ 8 * q + 70 * q^3 + 2030 * q^7 $$8 q + 70 q^{3} + 2030 q^{7} - 420 q^{11} - 33180 q^{13} - 43620 q^{17} + 108668 q^{21} + 663270 q^{23} - 1576040 q^{27} - 3178492 q^{31} + 944020 q^{33} - 5344080 q^{37} - 10185252 q^{41} + 10342710 q^{43} - 19232250 q^{47} - 47126684 q^{51} + 24320640 q^{53} - 88218320 q^{57} - 82515684 q^{61} + 77441350 q^{63} - 100675930 q^{67} - 99290076 q^{71} + 93528520 q^{73} - 134199660 q^{77} - 161920268 q^{81} + 10450350 q^{83} - 164801600 q^{87} - 130681068 q^{91} + 50183620 q^{93} + 179570760 q^{97}+O(q^{100})$$ 8 * q + 70 * q^3 + 2030 * q^7 - 420 * q^11 - 33180 * q^13 - 43620 * q^17 + 108668 * q^21 + 663270 * q^23 - 1576040 * q^27 - 3178492 * q^31 + 944020 * q^33 - 5344080 * q^37 - 10185252 * q^41 + 10342710 * q^43 - 19232250 * q^47 - 47126684 * q^51 + 24320640 * q^53 - 88218320 * q^57 - 82515684 * q^61 + 77441350 * q^63 - 100675930 * q^67 - 99290076 * q^71 + 93528520 * q^73 - 134199660 * q^77 - 161920268 * q^81 + 10450350 * q^83 - 164801600 * q^87 - 130681068 * q^91 + 50183620 * q^93 + 179570760 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 8 x^{6} + 22254 x^{5} + 4820745 x^{4} + 50131374 x^{3} + 307615702 x^{2} + \cdots + 2405464244$$ :

 $$\beta_{1}$$ $$=$$ $$( 17543298991401 \nu^{7} - 16868436332617 \nu^{6} - 231983879936421 \nu^{5} + \cdots - 17\!\cdots\!12 ) / 11\!\cdots\!00$$ (17543298991401*v^7 - 16868436332617*v^6 - 231983879936421*v^5 + 392578007792287377*v^4 + 85750328228947671494*v^3 + 1137909215057435119602*v^2 + 7291639495351253703576*v - 17201080872629180854312) / 11186314553142756537500 $$\beta_{2}$$ $$=$$ $$( - 83\!\cdots\!98 \nu^{7} + \cdots + 44\!\cdots\!76 ) / 75\!\cdots\!25$$ (-8397852473178398*v^7 + 9230087829528791*v^6 - 246098412514904642*v^5 - 183559907930723391146*v^4 - 41049105751594389826687*v^3 - 544847636925257658459946*v^2 - 5023198320683908632074548*v + 4411956177038996224861776) / 75507623233713606628125 $$\beta_{3}$$ $$=$$ $$( - 13\!\cdots\!93 \nu^{7} + \cdots + 17\!\cdots\!16 ) / 75\!\cdots\!25$$ (-13457794229710593*v^7 + 59832656428792231*v^6 - 242583434153172547*v^5 - 299911930994464745036*v^4 - 64687778407869036729017*v^3 - 648976954800072987776086*v^2 - 4389568545832306052594368*v + 17013291057978802265646116) / 75507623233713606628125 $$\beta_{4}$$ $$=$$ $$( - 2504810371918 \nu^{7} + 24975279644456 \nu^{6} - 81235255354622 \nu^{5} + \cdots + 40\!\cdots\!41 ) / 95\!\cdots\!75$$ (-2504810371918*v^7 + 24975279644456*v^6 - 81235255354622*v^5 - 56354460270811236*v^4 - 11894038044923293192*v^3 - 52271737565208336936*v^2 + 318311125881297533832*v + 4037594741399928975941) / 9563980143598936875 $$\beta_{5}$$ $$=$$ $$( 23\!\cdots\!61 \nu^{7} + \cdots - 22\!\cdots\!32 ) / 30\!\cdots\!00$$ (231940644953970061*v^7 - 954163199902626837*v^6 + 6258704938764385719*v^5 + 5130017024449071699197*v^4 + 1122694509109269734313534*v^3 + 11566522066568880996738922*v^2 + 102159443599004386265643736*v - 228594529135241255584608232) / 302030492934854426512500 $$\beta_{6}$$ $$=$$ $$( - 35\!\cdots\!87 \nu^{7} + \cdots + 11\!\cdots\!44 ) / 24\!\cdots\!00$$ (-3543448084543187*v^7 + 2876078302200779*v^6 + 70016115531916927*v^5 - 80595227235512868899*v^4 - 17321043270604001637178*v^3 - 229773196423349041163974*v^2 - 1364092156812684985315912*v + 1106190455551748291810244) / 2455532462884995337500 $$\beta_{7}$$ $$=$$ $$( - 47\!\cdots\!21 \nu^{7} + \cdots + 80\!\cdots\!52 ) / 30\!\cdots\!00$$ (-470942764374624521*v^7 + 703186185514027657*v^6 + 6874094100223788941*v^5 - 10435370961422585275617*v^4 - 2299576773112436823005774*v^3 - 28998247451815689605587842*v^2 - 163290456523450804974144696*v + 802896311985118436858684752) / 302030492934854426512500
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} + 4\beta_{2} - 39\beta _1 + 41 ) / 80$$ (b5 + b4 + 4*b2 - 39*b1 + 41) / 80 $$\nu^{2}$$ $$=$$ $$( 21\beta_{7} + 21\beta_{6} - 11\beta_{5} - 20\beta_{4} + 12\beta_{3} + 16\beta_{2} + 44747\beta _1 + 1 ) / 40$$ (21*b7 + 21*b6 - 11*b5 - 20*b4 + 12*b3 + 16*b2 + 44747*b1 + 1) / 40 $$\nu^{3}$$ $$=$$ $$( 165 \beta_{7} + 63 \beta_{6} + 988 \beta_{5} - 1186 \beta_{4} + 8158 \beta_{3} + 42 \beta_{2} + \cdots - 335999 ) / 40$$ (165*b7 + 63*b6 + 988*b5 - 1186*b4 + 8158*b3 + 42*b2 + 470280*b1 - 335999) / 40 $$\nu^{4}$$ $$=$$ $$( 9453 \beta_{7} - 9321 \beta_{6} + 804 \beta_{5} - 7210 \beta_{4} + 14870 \beta_{3} - 8342 \beta_{2} + \cdots - 19644975 ) / 8$$ (9453*b7 - 9321*b6 + 804*b5 - 7210*b4 + 14870*b3 - 8342*b2 + 322512*b1 - 19644975) / 8 $$\nu^{5}$$ $$=$$ $$( 234885 \beta_{7} - 929985 \beta_{6} - 2102712 \beta_{5} - 2281412 \beta_{4} + 290290 \beta_{3} + \cdots - 1744509937 ) / 40$$ (234885*b7 - 929985*b6 - 2102712*b5 - 2281412*b4 + 290290*b3 - 18505738*b2 - 1252937932*b1 - 1744509937) / 40 $$\nu^{6}$$ $$=$$ $$( - 103276119 \beta_{7} - 108859329 \beta_{6} - 2298436 \beta_{5} + 90808300 \beta_{4} + \cdots - 9000406409 ) / 40$$ (-103276119*b7 - 108859329*b6 - 2298436*b5 + 90808300*b4 - 182398438*b3 - 293596074*b2 - 228107898188*b1 - 9000406409) / 40 $$\nu^{7}$$ $$=$$ $$( - 3322505895 \beta_{7} - 744118557 \beta_{6} - 4443569742 \beta_{5} + 7750380034 \beta_{4} + \cdots + 3839292487721 ) / 40$$ (-3322505895*b7 - 744118557*b6 - 4443569742*b5 + 7750380034*b4 - 42489368582*b3 - 1668013018*b2 - 5439489405770*b1 + 3839292487721) / 40

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/100\mathbb{Z}\right)^\times$$.

 $$n$$ $$51$$ $$77$$ $$\chi(n)$$ $$1$$ $$-\beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
57.1
 1.89682 − 0.896816i −29.1758 + 30.1758i 36.4975 − 35.4975i −7.21849 + 8.21849i 1.89682 + 0.896816i −29.1758 − 30.1758i 36.4975 + 35.4975i −7.21849 − 8.21849i
0 −80.1275 + 80.1275i 0 0 0 1045.24 + 1045.24i 0 6279.84i 0
57.2 0 −18.1203 + 18.1203i 0 0 0 190.873 + 190.873i 0 5904.31i 0
57.3 0 29.4437 29.4437i 0 0 0 −1846.83 1846.83i 0 4827.14i 0
57.4 0 103.804 103.804i 0 0 0 1625.71 + 1625.71i 0 14989.6i 0
93.1 0 −80.1275 80.1275i 0 0 0 1045.24 1045.24i 0 6279.84i 0
93.2 0 −18.1203 18.1203i 0 0 0 190.873 190.873i 0 5904.31i 0
93.3 0 29.4437 + 29.4437i 0 0 0 −1846.83 + 1846.83i 0 4827.14i 0
93.4 0 103.804 + 103.804i 0 0 0 1625.71 1625.71i 0 14989.6i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 57.4 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.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.9.f.b 8
5.b even 2 1 20.9.f.a 8
5.c odd 4 1 20.9.f.a 8
5.c odd 4 1 inner 100.9.f.b 8
15.d odd 2 1 180.9.l.a 8
15.e even 4 1 180.9.l.a 8
20.d odd 2 1 80.9.p.d 8
20.e even 4 1 80.9.p.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.9.f.a 8 5.b even 2 1
20.9.f.a 8 5.c odd 4 1
80.9.p.d 8 20.d odd 2 1
80.9.p.d 8 20.e even 4 1
100.9.f.b 8 1.a even 1 1 trivial
100.9.f.b 8 5.c odd 4 1 inner
180.9.l.a 8 15.d odd 2 1
180.9.l.a 8 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 70 T_{3}^{7} + 2450 T_{3}^{6} + 774360 T_{3}^{5} + 259170228 T_{3}^{4} + \cdots + 315086072440896$$ acting on $$S_{9}^{\mathrm{new}}(100, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + \cdots + 315086072440896$$
$5$ $$T^{8}$$
$7$ $$T^{8} + \cdots + 57\!\cdots\!16$$
$11$ $$(T^{4} + \cdots - 41\!\cdots\!00)^{2}$$
$13$ $$T^{8} + \cdots + 40\!\cdots\!96$$
$17$ $$T^{8} + \cdots + 27\!\cdots\!36$$
$19$ $$T^{8} + \cdots + 88\!\cdots\!36$$
$23$ $$T^{8} + \cdots + 41\!\cdots\!36$$
$29$ $$T^{8} + \cdots + 32\!\cdots\!96$$
$31$ $$(T^{4} + \cdots - 29\!\cdots\!24)^{2}$$
$37$ $$T^{8} + \cdots + 11\!\cdots\!16$$
$41$ $$(T^{4} + \cdots - 83\!\cdots\!04)^{2}$$
$43$ $$T^{8} + \cdots + 16\!\cdots\!00$$
$47$ $$T^{8} + \cdots + 26\!\cdots\!76$$
$53$ $$T^{8} + \cdots + 10\!\cdots\!96$$
$59$ $$T^{8} + \cdots + 19\!\cdots\!76$$
$61$ $$(T^{4} + \cdots + 41\!\cdots\!16)^{2}$$
$67$ $$T^{8} + \cdots + 41\!\cdots\!36$$
$71$ $$(T^{4} + \cdots + 17\!\cdots\!56)^{2}$$
$73$ $$T^{8} + \cdots + 29\!\cdots\!56$$
$79$ $$T^{8} + \cdots + 12\!\cdots\!16$$
$83$ $$T^{8} + \cdots + 31\!\cdots\!56$$
$89$ $$T^{8} + \cdots + 14\!\cdots\!36$$
$97$ $$T^{8} + \cdots + 44\!\cdots\!36$$