# Properties

 Label 75.3.h.a Level $75$ Weight $3$ Character orbit 75.h Analytic conductor $2.044$ Analytic rank $0$ Dimension $72$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,3,Mod(14,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("75.14");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 75.h (of order $$10$$, degree $$4$$, 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: $$2.04360198270$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$18$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$72 q - 5 q^{3} - 38 q^{4} + 5 q^{6} - 13 q^{9}+O(q^{10})$$ 72 * q - 5 * q^3 - 38 * q^4 + 5 * q^6 - 13 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$72 q - 5 q^{3} - 38 q^{4} + 5 q^{6} - 13 q^{9} - 20 q^{10} - 45 q^{12} - 10 q^{13} - 15 q^{15} + 22 q^{16} - 36 q^{19} + 54 q^{21} - 50 q^{22} - 20 q^{24} - 100 q^{25} + 100 q^{27} + 270 q^{28} - 5 q^{30} - 126 q^{31} + 20 q^{33} + 210 q^{34} - 213 q^{36} + 110 q^{37} - 191 q^{39} + 140 q^{40} - 175 q^{42} - 405 q^{45} - 210 q^{46} + 150 q^{48} - 224 q^{49} - 60 q^{51} - 320 q^{52} + 320 q^{54} - 10 q^{55} - 70 q^{58} + 1190 q^{60} + 294 q^{61} + 795 q^{63} + 362 q^{64} - 470 q^{66} - 260 q^{67} + 335 q^{69} + 1200 q^{70} + 215 q^{72} - 150 q^{73} + 200 q^{75} - 16 q^{76} - 1295 q^{78} - 346 q^{79} + 507 q^{81} - 456 q^{84} - 1450 q^{85} - 430 q^{87} - 1710 q^{88} - 820 q^{90} + 538 q^{91} - 560 q^{94} + 740 q^{96} - 150 q^{97} + 60 q^{99}+O(q^{100})$$ 72 * q - 5 * q^3 - 38 * q^4 + 5 * q^6 - 13 * q^9 - 20 * q^10 - 45 * q^12 - 10 * q^13 - 15 * q^15 + 22 * q^16 - 36 * q^19 + 54 * q^21 - 50 * q^22 - 20 * q^24 - 100 * q^25 + 100 * q^27 + 270 * q^28 - 5 * q^30 - 126 * q^31 + 20 * q^33 + 210 * q^34 - 213 * q^36 + 110 * q^37 - 191 * q^39 + 140 * q^40 - 175 * q^42 - 405 * q^45 - 210 * q^46 + 150 * q^48 - 224 * q^49 - 60 * q^51 - 320 * q^52 + 320 * q^54 - 10 * q^55 - 70 * q^58 + 1190 * q^60 + 294 * q^61 + 795 * q^63 + 362 * q^64 - 470 * q^66 - 260 * q^67 + 335 * q^69 + 1200 * q^70 + 215 * q^72 - 150 * q^73 + 200 * q^75 - 16 * q^76 - 1295 * q^78 - 346 * q^79 + 507 * q^81 - 456 * q^84 - 1450 * q^85 - 430 * q^87 - 1710 * q^88 - 820 * q^90 + 538 * q^91 - 560 * q^94 + 740 * q^96 - 150 * q^97 + 60 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
14.1 −2.91455 + 2.11754i 0.529130 2.95297i 2.77453 8.53912i −2.30619 + 4.43638i 4.71086 + 9.72702i 9.61929i 5.54243 + 17.0578i −8.44004 3.12501i −2.67273 17.8135i
14.2 −2.79240 + 2.02880i −2.73887 + 1.22418i 2.44541 7.52621i −1.27184 4.83554i 5.16441 8.97500i 4.79411i 4.17417 + 12.8468i 6.00278 6.70572i 13.3618 + 10.9224i
14.3 −2.48516 + 1.80557i 2.97017 0.422034i 1.67984 5.17002i 3.94761 3.06861i −6.61931 + 6.41167i 8.51768i 1.36320 + 4.19549i 8.64377 2.50702i −4.26983 + 14.7537i
14.4 −2.05424 + 1.49250i 0.392507 + 2.97421i 0.756307 2.32767i 4.44388 + 2.29172i −5.24530 5.52394i 6.97211i −1.21820 3.74924i −8.69188 + 2.33480i −12.5492 + 1.92472i
14.5 −1.88806 + 1.37175i −2.96199 0.476037i 0.446980 1.37566i −0.465978 + 4.97824i 6.24541 3.16433i 12.5544i −1.84155 5.66769i 8.54678 + 2.82003i −5.94912 10.0384i
14.6 −1.27392 + 0.925557i 0.196179 2.99358i −0.469852 + 1.44606i −2.47338 4.34538i 2.52081 + 3.99515i 3.45468i −2.68623 8.26736i −8.92303 1.17455i 7.17279 + 3.24641i
14.7 −0.949573 + 0.689905i −0.131165 + 2.99713i −0.810348 + 2.49399i −4.82831 1.29902i −1.94319 2.93649i 4.52613i −2.40195 7.39246i −8.96559 0.786239i 5.48103 2.09756i
14.8 −0.680034 + 0.494074i −2.26100 1.97177i −1.01773 + 3.13225i 4.99975 + 0.0496921i 2.51176 + 0.223767i 9.76040i −1.89447 5.83059i 1.22427 + 8.91634i −3.42456 + 2.43645i
14.9 −0.624639 + 0.453826i 2.99063 0.236895i −1.05185 + 3.23727i −2.07515 + 4.54904i −1.76055 + 1.50520i 1.71117i −1.76649 5.43671i 8.88776 1.41693i −0.768257 3.78326i
14.10 0.624639 0.453826i 2.28023 + 1.94950i −1.05185 + 3.23727i 2.07515 4.54904i 2.30905 + 0.182906i 1.71117i 1.76649 + 5.43671i 1.39889 + 8.89062i −0.768257 3.78326i
14.11 0.680034 0.494074i −2.98817 + 0.266209i −1.01773 + 3.13225i −4.99975 0.0496921i −1.90053 + 1.65741i 9.76040i 1.89447 + 5.83059i 8.85827 1.59096i −3.42456 + 2.43645i
14.12 0.949573 0.689905i 1.65555 2.50183i −0.810348 + 2.49399i 4.82831 + 1.29902i −0.153954 3.51784i 4.52613i 2.40195 + 7.39246i −3.51828 8.28382i 5.48103 2.09756i
14.13 1.27392 0.925557i −1.60087 + 2.53717i −0.469852 + 1.44606i 2.47338 + 4.34538i 0.308913 + 4.71384i 3.45468i 2.68623 + 8.26736i −3.87443 8.12335i 7.17279 + 3.24641i
14.14 1.88806 1.37175i −2.67611 1.35589i 0.446980 1.37566i 0.465978 4.97824i −6.91259 + 1.11096i 12.5544i 1.84155 + 5.66769i 5.32311 + 7.25703i −5.94912 10.0384i
14.15 2.05424 1.49250i 2.06574 2.17548i 0.756307 2.32767i −4.44388 2.29172i 0.996648 7.55208i 6.97211i 1.21820 + 3.74924i −0.465411 8.98796i −12.5492 + 1.92472i
14.16 2.48516 1.80557i 2.15485 + 2.08725i 1.67984 5.17002i −3.94761 + 3.06861i 9.12382 + 1.29641i 8.51768i −1.36320 4.19549i 0.286751 + 8.99543i −4.26983 + 14.7537i
14.17 2.79240 2.02880i −1.49624 2.60025i 2.44541 7.52621i 1.27184 + 4.83554i −9.45346 4.22537i 4.79411i −4.17417 12.8468i −4.52255 + 7.78116i 13.3618 + 10.9224i
14.18 2.91455 2.11754i −1.30764 + 2.70002i 2.77453 8.53912i 2.30619 4.43638i 1.90623 + 10.6383i 9.61929i −5.54243 17.0578i −5.58018 7.06128i −2.67273 17.8135i
29.1 −1.16846 + 3.59614i −0.604570 2.93845i −8.33088 6.05274i 1.74586 4.68529i 11.2735 + 1.25934i 3.47419i 19.2645 13.9965i −8.26899 + 3.55300i 14.8090 + 11.7529i
29.2 −1.10327 + 3.39551i −0.907156 + 2.85956i −7.07623 5.14118i 2.46900 + 4.34788i −8.70882 6.23512i 0.132726i 13.7104 9.96116i −7.35413 5.18813i −17.4872 + 3.58664i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 14.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.e even 10 1 inner
75.h odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.3.h.a 72
3.b odd 2 1 inner 75.3.h.a 72
5.b even 2 1 375.3.h.a 72
5.c odd 4 2 375.3.j.b 144
15.d odd 2 1 375.3.h.a 72
15.e even 4 2 375.3.j.b 144
25.d even 5 1 375.3.h.a 72
25.e even 10 1 inner 75.3.h.a 72
25.f odd 20 2 375.3.j.b 144
75.h odd 10 1 inner 75.3.h.a 72
75.j odd 10 1 375.3.h.a 72
75.l even 20 2 375.3.j.b 144

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.h.a 72 1.a even 1 1 trivial
75.3.h.a 72 3.b odd 2 1 inner
75.3.h.a 72 25.e even 10 1 inner
75.3.h.a 72 75.h odd 10 1 inner
375.3.h.a 72 5.b even 2 1
375.3.h.a 72 15.d odd 2 1
375.3.h.a 72 25.d even 5 1
375.3.h.a 72 75.j odd 10 1
375.3.j.b 144 5.c odd 4 2
375.3.j.b 144 15.e even 4 2
375.3.j.b 144 25.f odd 20 2
375.3.j.b 144 75.l even 20 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(75, [\chi])$$.