# Properties

 Label 75.22.a.a Level $75$ Weight $22$ Character orbit 75.a Self dual yes Analytic conductor $209.608$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,22,Mod(1,75)]

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, 0]))

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

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 75.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: $$209.608008215$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3) 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)

 $$f(q)$$ $$=$$ $$q - 1728 q^{2} + 59049 q^{3} + 888832 q^{4} - 102036672 q^{6} - 538429808 q^{7} + 2087976960 q^{8} + 3486784401 q^{9}+O(q^{10})$$ q - 1728 * q^2 + 59049 * q^3 + 888832 * q^4 - 102036672 * q^6 - 538429808 * q^7 + 2087976960 * q^8 + 3486784401 * q^9 $$q - 1728 q^{2} + 59049 q^{3} + 888832 q^{4} - 102036672 q^{6} - 538429808 q^{7} + 2087976960 q^{8} + 3486784401 q^{9} - 64113040188 q^{11} + 52484640768 q^{12} + 130980107986 q^{13} + 930406708224 q^{14} - 5472039993344 q^{16} - 8242029723618 q^{17} - 6025163444928 q^{18} + 13492101753020 q^{19} - 31793741732592 q^{21} + 110787333444864 q^{22} + 233184825844776 q^{23} + 123292951511040 q^{24} - 226333626599808 q^{26} + 205891132094649 q^{27} - 478573643104256 q^{28} - 20\!\cdots\!70 q^{29}+ \cdots - 22\!\cdots\!88 q^{99}+O(q^{100})$$ q - 1728 * q^2 + 59049 * q^3 + 888832 * q^4 - 102036672 * q^6 - 538429808 * q^7 + 2087976960 * q^8 + 3486784401 * q^9 - 64113040188 * q^11 + 52484640768 * q^12 + 130980107986 * q^13 + 930406708224 * q^14 - 5472039993344 * q^16 - 8242029723618 * q^17 - 6025163444928 * q^18 + 13492101753020 * q^19 - 31793741732592 * q^21 + 110787333444864 * q^22 + 233184825844776 * q^23 + 123292951511040 * q^24 - 226333626599808 * q^26 + 205891132094649 * q^27 - 478573643104256 * q^28 - 2024562031123770 * q^29 - 6869194988701768 * q^31 + 5076880050880512 * q^32 - 3785810910061212 * q^33 + 14242227362411904 * q^34 + 3099165552709632 * q^36 - 3443998107027638 * q^37 - 23314351829218560 * q^38 + 7734244396465314 * q^39 - 21842403084625158 * q^41 + 54939585713918976 * q^42 + 71792816814133756 * q^43 - 56985721736380416 * q^44 - 402943379059772928 * q^46 - 283544719418655648 * q^47 - 323118489566969856 * q^48 - 268639205940367143 * q^49 - 486683613149919282 * q^51 + 116419311341412352 * q^52 + 2172285419049898146 * q^53 - 355779876259553472 * q^54 - 1124229033681223680 * q^56 + 796695116414077980 * q^57 + 3498443189781874560 * q^58 + 1534831476719068260 * q^59 + 4311589520797626062 * q^61 + 11869968940476655104 * q^62 - 1877388655567825008 * q^63 + 2702850888199831552 * q^64 + 6541881252585774336 * q^66 - 9243910904037307868 * q^67 - 7325779763302834176 * q^68 + 13769330781308178024 * q^69 - 20387361256404760728 * q^71 + 7280325493775400960 * q^72 - 16617754439328636074 * q^73 + 5951228728943758464 * q^74 + 11992211785340272640 * q^76 + 34520371918721123904 * q^77 - 13364774317092062592 * q^78 + 67940304745507627880 * q^79 + 12157665459056928801 * q^81 + 37743672530232273024 * q^82 - 39503732340682314684 * q^83 - 28259295051663212544 * q^84 - 124057987454823130368 * q^86 - 119548363375827494730 * q^87 - 133866550748098068480 * q^88 + 41611676186839694490 * q^89 - 70523594394721246688 * q^91 + 207262135125263941632 * q^92 - 405619094887850698632 * q^93 + 489965275155436959744 * q^94 + 299784690124443353088 * q^96 - 57181473208903260098 * q^97 + 464208547864954423104 * q^98 - 223548348428204507388 * 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
 0
−1728.00 59049.0 888832. 0 −1.02037e8 −5.38430e8 2.08798e9 3.48678e9 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
$$3$$ $$-1$$
$$5$$ $$1$$

## 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 75.22.a.a 1
5.b even 2 1 3.22.a.b 1
5.c odd 4 2 75.22.b.b 2
15.d odd 2 1 9.22.a.a 1
20.d odd 2 1 48.22.a.d 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1728$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(75))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1728$$
$3$ $$T - 59049$$
$5$ $$T$$
$7$ $$T + 538429808$$
$11$ $$T + 64113040188$$
$13$ $$T - 130980107986$$
$17$ $$T + 8242029723618$$
$19$ $$T - 13492101753020$$
$23$ $$T - 233184825844776$$
$29$ $$T + 2024562031123770$$
$31$ $$T + 6869194988701768$$
$37$ $$T + 3443998107027638$$
$41$ $$T + 21\!\cdots\!58$$
$43$ $$T - 71\!\cdots\!56$$
$47$ $$T + 28\!\cdots\!48$$
$53$ $$T - 21\!\cdots\!46$$
$59$ $$T - 15\!\cdots\!60$$
$61$ $$T - 43\!\cdots\!62$$
$67$ $$T + 92\!\cdots\!68$$
$71$ $$T + 20\!\cdots\!28$$
$73$ $$T + 16\!\cdots\!74$$
$79$ $$T - 67\!\cdots\!80$$
$83$ $$T + 39\!\cdots\!84$$
$89$ $$T - 41\!\cdots\!90$$
$97$ $$T + 57\!\cdots\!98$$
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