Properties

 Label 75.3.j.a Level $75$ Weight $3$ Character orbit 75.j 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(11,75)]

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, 8]))

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

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

chi := DirichletCharacter("75.11");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 75.j (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.

 $$\operatorname{Tr}(f)(q) =$$ $$72 q - q^{3} + 26 q^{4} - 11 q^{6} - 8 q^{7} - 13 q^{9}+O(q^{10})$$ 72 * q - q^3 + 26 * q^4 - 11 * q^6 - 8 * q^7 - 13 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$72 q - q^{3} + 26 q^{4} - 11 q^{6} - 8 q^{7} - 13 q^{9} - 20 q^{10} + 31 q^{12} - 42 q^{13} + 45 q^{15} - 130 q^{16} + 30 q^{18} - 36 q^{19} - 60 q^{21} - 70 q^{22} - 72 q^{24} + 100 q^{25} - 154 q^{27} - 62 q^{28} + 15 q^{30} + 114 q^{31} - 10 q^{33} + 178 q^{34} + 271 q^{36} - 98 q^{37} - 155 q^{39} - 120 q^{40} - 475 q^{42} - 52 q^{43} + 35 q^{45} + 198 q^{46} - 326 q^{48} + 112 q^{49} + 44 q^{51} + 412 q^{52} + 304 q^{54} + 10 q^{55} + 622 q^{57} + 190 q^{58} + 360 q^{60} - 306 q^{61} + 293 q^{63} + 474 q^{64} + 320 q^{66} + 472 q^{67} + 269 q^{69} - 840 q^{70} + 175 q^{72} + 318 q^{73} - 310 q^{75} + 112 q^{76} + 815 q^{78} - 346 q^{79} - 373 q^{81} - 1620 q^{82} - 730 q^{84} - 530 q^{85} - 370 q^{87} - 810 q^{88} - 230 q^{90} - 550 q^{91} - 272 q^{93} - 612 q^{94} - 698 q^{96} + 182 q^{97} - 160 q^{99}+O(q^{100})$$ 72 * q - q^3 + 26 * q^4 - 11 * q^6 - 8 * q^7 - 13 * q^9 - 20 * q^10 + 31 * q^12 - 42 * q^13 + 45 * q^15 - 130 * q^16 + 30 * q^18 - 36 * q^19 - 60 * q^21 - 70 * q^22 - 72 * q^24 + 100 * q^25 - 154 * q^27 - 62 * q^28 + 15 * q^30 + 114 * q^31 - 10 * q^33 + 178 * q^34 + 271 * q^36 - 98 * q^37 - 155 * q^39 - 120 * q^40 - 475 * q^42 - 52 * q^43 + 35 * q^45 + 198 * q^46 - 326 * q^48 + 112 * q^49 + 44 * q^51 + 412 * q^52 + 304 * q^54 + 10 * q^55 + 622 * q^57 + 190 * q^58 + 360 * q^60 - 306 * q^61 + 293 * q^63 + 474 * q^64 + 320 * q^66 + 472 * q^67 + 269 * q^69 - 840 * q^70 + 175 * q^72 + 318 * q^73 - 310 * q^75 + 112 * q^76 + 815 * q^78 - 346 * q^79 - 373 * q^81 - 1620 * q^82 - 730 * q^84 - 530 * q^85 - 370 * q^87 - 810 * q^88 - 230 * q^90 - 550 * q^91 - 272 * q^93 - 612 * q^94 - 698 * q^96 + 182 * q^97 - 160 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −2.26850 3.12232i 0.549348 2.94927i −3.36674 + 10.3618i −3.89421 + 3.13610i −10.4548 + 4.97519i −5.08243 25.3081 8.22311i −8.39643 3.24035i 18.6259 + 5.04472i
11.2 −2.06309 2.83960i 2.44039 + 1.74485i −2.57091 + 7.91245i 4.77936 1.46892i −0.0800814 10.5295i 6.80412 14.4196 4.68521i 2.91102 + 8.51622i −14.0314 10.5409i
11.3 −1.71799 2.36461i −2.67836 + 1.35143i −1.40382 + 4.32053i 2.09404 + 4.54038i 7.79701 + 4.01154i 3.73689 1.50905 0.490320i 5.34726 7.23926i 7.13868 12.7519i
11.4 −1.49360 2.05577i −0.154094 + 2.99604i −0.759266 + 2.33678i −4.43527 2.30832i 6.38932 4.15811i −8.13443 −3.72888 + 1.21159i −8.95251 0.923341i 1.87916 + 12.5656i
11.5 −1.37909 1.89816i −1.85450 2.35814i −0.465043 + 1.43126i 3.05671 3.95684i −1.91859 + 6.77224i −3.37306 −5.56759 + 1.80902i −2.12163 + 8.74635i −11.7262 0.345267i
11.6 −1.06164 1.46122i 2.88836 0.810770i 0.227982 0.701657i −4.39665 2.38107i −4.25111 3.35979i 6.57297 −8.13837 + 2.64432i 7.68530 4.68360i 1.18838 + 8.95230i
11.7 −0.733747 1.00992i 1.70510 2.46833i 0.754522 2.32218i 3.80815 + 3.24006i −3.74392 + 0.0891202i −0.783313 −7.64775 + 2.48490i −3.18527 8.41748i 0.477973 6.22330i
11.8 −0.146095 0.201083i −2.97010 0.422474i 1.21698 3.74547i −3.70242 + 3.36037i 0.348966 + 0.658958i −11.1634 −1.87649 + 0.609710i 8.64303 + 2.50958i 1.21662 + 0.253561i
11.9 −0.0749426 0.103150i 1.19040 + 2.75372i 1.23104 3.78877i −0.601131 + 4.96373i 0.194834 0.329160i 8.18661 −0.968106 + 0.314557i −6.16592 + 6.55603i 0.557058 0.309989i
11.10 0.0749426 + 0.103150i −2.58164 + 1.52811i 1.23104 3.78877i 0.601131 4.96373i −0.351099 0.151775i 8.18661 0.968106 0.314557i 4.32978 7.89006i 0.557058 0.309989i
11.11 0.146095 + 0.201083i 2.65119 + 1.40399i 1.21698 3.74547i 3.70242 3.36037i 0.105007 + 0.738225i −11.1634 1.87649 0.609710i 5.05760 + 7.44451i 1.21662 + 0.253561i
11.12 0.733747 + 1.00992i 0.0713918 2.99915i 0.754522 2.32218i −3.80815 3.24006i 3.08127 2.12852i −0.783313 7.64775 2.48490i −8.98981 0.428229i 0.477973 6.22330i
11.13 1.06164 + 1.46122i −1.86018 2.35367i 0.227982 0.701657i 4.39665 + 2.38107i 1.46439 5.21687i 6.57297 8.13837 2.64432i −2.07948 + 8.75647i 1.18838 + 8.95230i
11.14 1.37909 + 1.89816i 2.88640 0.817724i −0.465043 + 1.43126i −3.05671 + 3.95684i 5.53279 + 4.35114i −3.37306 5.56759 1.80902i 7.66265 4.72056i −11.7262 0.345267i
11.15 1.49360 + 2.05577i −1.63636 + 2.51442i −0.759266 + 2.33678i 4.43527 + 2.30832i −7.61315 + 0.391563i −8.13443 3.72888 1.21159i −3.64463 8.22902i 1.87916 + 12.5656i
11.16 1.71799 + 2.36461i 1.37249 + 2.66763i −1.40382 + 4.32053i −2.09404 4.54038i −3.94999 + 7.82837i 3.73689 −1.50905 + 0.490320i −5.23255 + 7.32260i 7.13868 12.7519i
11.17 2.06309 + 2.83960i −2.99991 0.0228156i −2.57091 + 7.91245i −4.77936 + 1.46892i −6.12430 8.56561i 6.80412 −14.4196 + 4.68521i 8.99896 + 0.136890i −14.0314 10.5409i
11.18 2.26850 + 3.12232i 1.28911 2.70891i −3.36674 + 10.3618i 3.89421 3.13610i 11.3824 2.12015i −5.08243 −25.3081 + 8.22311i −5.67640 6.98416i 18.6259 + 5.04472i
41.1 −2.26850 + 3.12232i 0.549348 + 2.94927i −3.36674 10.3618i −3.89421 3.13610i −10.4548 4.97519i −5.08243 25.3081 + 8.22311i −8.39643 + 3.24035i 18.6259 5.04472i
41.2 −2.06309 + 2.83960i 2.44039 1.74485i −2.57091 7.91245i 4.77936 + 1.46892i −0.0800814 + 10.5295i 6.80412 14.4196 + 4.68521i 2.91102 8.51622i −14.0314 + 10.5409i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.18 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.d even 5 1 inner
75.j 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.j.a 72
3.b odd 2 1 inner 75.3.j.a 72
5.b even 2 1 375.3.j.a 72
5.c odd 4 2 375.3.h.b 144
15.d odd 2 1 375.3.j.a 72
15.e even 4 2 375.3.h.b 144
25.d even 5 1 inner 75.3.j.a 72
25.e even 10 1 375.3.j.a 72
25.f odd 20 2 375.3.h.b 144
75.h odd 10 1 375.3.j.a 72
75.j odd 10 1 inner 75.3.j.a 72
75.l even 20 2 375.3.h.b 144

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

Hecke kernels

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