# Properties

 Label 75.22.b.b Level $75$ Weight $22$ Character orbit 75.b Analytic conductor $209.608$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,22,Mod(49,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("75.49");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 75.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: $$209.608008215$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 1728 i q^{2} + 59049 i q^{3} - 888832 q^{4} - 102036672 q^{6} + 538429808 i q^{7} + 2087976960 i q^{8} - 3486784401 q^{9}+O(q^{10})$$ q + 1728*i * q^2 + 59049*i * q^3 - 888832 * q^4 - 102036672 * q^6 + 538429808*i * q^7 + 2087976960*i * q^8 - 3486784401 * q^9 $$q + 1728 i q^{2} + 59049 i q^{3} - 888832 q^{4} - 102036672 q^{6} + 538429808 i q^{7} + 2087976960 i q^{8} - 3486784401 q^{9} - 64113040188 q^{11} - 52484640768 i q^{12} + 130980107986 i q^{13} - 930406708224 q^{14} - 5472039993344 q^{16} + 8242029723618 i q^{17} - 6025163444928 i q^{18} - 13492101753020 q^{19} - 31793741732592 q^{21} - 110787333444864 i q^{22} + 233184825844776 i q^{23} - 123292951511040 q^{24} - 226333626599808 q^{26} - 205891132094649 i q^{27} - 478573643104256 i q^{28} + 20\!\cdots\!70 q^{29} + \cdots + 22\!\cdots\!88 q^{99} +O(q^{100})$$ q + 1728*i * q^2 + 59049*i * q^3 - 888832 * q^4 - 102036672 * q^6 + 538429808*i * q^7 + 2087976960*i * q^8 - 3486784401 * q^9 - 64113040188 * q^11 - 52484640768*i * q^12 + 130980107986*i * q^13 - 930406708224 * q^14 - 5472039993344 * q^16 + 8242029723618*i * q^17 - 6025163444928*i * q^18 - 13492101753020 * q^19 - 31793741732592 * q^21 - 110787333444864*i * q^22 + 233184825844776*i * q^23 - 123292951511040 * q^24 - 226333626599808 * q^26 - 205891132094649*i * q^27 - 478573643104256*i * q^28 + 2024562031123770 * q^29 - 6869194988701768 * q^31 - 5076880050880512*i * q^32 - 3785810910061212*i * q^33 - 14242227362411904 * q^34 + 3099165552709632 * q^36 + 3443998107027638*i * q^37 - 23314351829218560*i * q^38 - 7734244396465314 * q^39 - 21842403084625158 * q^41 - 54939585713918976*i * q^42 + 71792816814133756*i * q^43 + 56985721736380416 * q^44 - 402943379059772928 * q^46 + 283544719418655648*i * q^47 - 323118489566969856*i * q^48 + 268639205940367143 * q^49 - 486683613149919282 * q^51 - 116419311341412352*i * q^52 + 2172285419049898146*i * q^53 + 355779876259553472 * q^54 - 1124229033681223680 * q^56 - 796695116414077980*i * q^57 + 3498443189781874560*i * q^58 - 1534831476719068260 * q^59 + 4311589520797626062 * q^61 - 11869968940476655104*i * q^62 - 1877388655567825008*i * q^63 - 2702850888199831552 * q^64 + 6541881252585774336 * q^66 + 9243910904037307868*i * q^67 - 7325779763302834176*i * q^68 - 13769330781308178024 * q^69 - 20387361256404760728 * q^71 - 7280325493775400960*i * q^72 - 16617754439328636074*i * q^73 - 5951228728943758464 * q^74 + 11992211785340272640 * q^76 - 34520371918721123904*i * q^77 - 13364774317092062592*i * q^78 - 67940304745507627880 * q^79 + 12157665459056928801 * q^81 - 37743672530232273024*i * q^82 - 39503732340682314684*i * q^83 + 28259295051663212544 * q^84 - 124057987454823130368 * q^86 + 119548363375827494730*i * q^87 - 133866550748098068480*i * q^88 - 41611676186839694490 * q^89 - 70523594394721246688 * q^91 - 207262135125263941632*i * q^92 - 405619094887850698632*i * q^93 - 489965275155436959744 * q^94 + 299784690124443353088 * q^96 + 57181473208903260098*i * q^97 + 464208547864954423104*i * q^98 + 223548348428204507388 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 1777664 q^{4} - 204073344 q^{6} - 6973568802 q^{9}+O(q^{10})$$ 2 * q - 1777664 * q^4 - 204073344 * q^6 - 6973568802 * q^9 $$2 q - 1777664 q^{4} - 204073344 q^{6} - 6973568802 q^{9} - 128226080376 q^{11} - 1860813416448 q^{14} - 10944079986688 q^{16} - 26984203506040 q^{19} - 63587483465184 q^{21} - 246585903022080 q^{24} - 452667253199616 q^{26} + 40\!\cdots\!40 q^{29}+ \cdots + 44\!\cdots\!76 q^{99}+O(q^{100})$$ 2 * q - 1777664 * q^4 - 204073344 * q^6 - 6973568802 * q^9 - 128226080376 * q^11 - 1860813416448 * q^14 - 10944079986688 * q^16 - 26984203506040 * q^19 - 63587483465184 * q^21 - 246585903022080 * q^24 - 452667253199616 * q^26 + 4049124062247540 * q^29 - 13738389977403536 * q^31 - 28484454724823808 * q^34 + 6198331105419264 * q^36 - 15468488792930628 * q^39 - 43684806169250316 * q^41 + 113971443472760832 * q^44 - 805886758119545856 * q^46 + 537278411880734286 * q^49 - 973367226299838564 * q^51 + 711559752519106944 * q^54 - 2248458067362447360 * q^56 - 3069662953438136520 * q^59 + 8623179041595252124 * q^61 - 5405701776399663104 * q^64 + 13083762505171548672 * q^66 - 27538661562616356048 * q^69 - 40774722512809521456 * q^71 - 11902457457887516928 * q^74 + 23984423570680545280 * q^76 - 135880609491015255760 * q^79 + 24315330918113857602 * q^81 + 56518590103326425088 * q^84 - 248115974909646260736 * q^86 - 83223352373679388980 * q^89 - 141047188789442493376 * q^91 - 979930550310873919488 * q^94 + 599569380248886706176 * q^96 + 447096696856409014776 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
1728.00i 59049.0i −888832. 0 −1.02037e8 5.38430e8i 2.08798e9i −3.48678e9 0
49.2 1728.00i 59049.0i −888832. 0 −1.02037e8 5.38430e8i 2.08798e9i −3.48678e9 0
 $$n$$: 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
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 75.22.b.b 2
5.b even 2 1 inner 75.22.b.b 2
5.c odd 4 1 3.22.a.b 1
5.c odd 4 1 75.22.a.a 1
15.e even 4 1 9.22.a.a 1
20.e even 4 1 48.22.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.22.a.b 1 5.c odd 4 1
9.22.a.a 1 15.e even 4 1
48.22.a.d 1 20.e even 4 1
75.22.a.a 1 5.c odd 4 1
75.22.b.b 2 1.a even 1 1 trivial
75.22.b.b 2 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2985984$$ acting on $$S_{22}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2985984$$
$3$ $$T^{2} + 3486784401$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 28\!\cdots\!64$$
$11$ $$(T + 64113040188)^{2}$$
$13$ $$T^{2} + 17\!\cdots\!96$$
$17$ $$T^{2} + 67\!\cdots\!24$$
$19$ $$(T + 13492101753020)^{2}$$
$23$ $$T^{2} + 54\!\cdots\!76$$
$29$ $$(T - 20\!\cdots\!70)^{2}$$
$31$ $$(T + 68\!\cdots\!68)^{2}$$
$37$ $$T^{2} + 11\!\cdots\!44$$
$41$ $$(T + 21\!\cdots\!58)^{2}$$
$43$ $$T^{2} + 51\!\cdots\!36$$
$47$ $$T^{2} + 80\!\cdots\!04$$
$53$ $$T^{2} + 47\!\cdots\!16$$
$59$ $$(T + 15\!\cdots\!60)^{2}$$
$61$ $$(T - 43\!\cdots\!62)^{2}$$
$67$ $$T^{2} + 85\!\cdots\!24$$
$71$ $$(T + 20\!\cdots\!28)^{2}$$
$73$ $$T^{2} + 27\!\cdots\!76$$
$79$ $$(T + 67\!\cdots\!80)^{2}$$
$83$ $$T^{2} + 15\!\cdots\!56$$
$89$ $$(T + 41\!\cdots\!90)^{2}$$
$97$ $$T^{2} + 32\!\cdots\!04$$