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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 $4$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 5 q^{3} - 38 q^{4} + 5 q^{6} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 5 q^{3} - 38 q^{4} + 5 q^{6} - 13 q^{9} - 20 q^{10} - 45 q^{12} - 10 q^{13} - 15 q^{15} + 22 q^{16} - 36 q^{19} + 54 q^{21} - 50 q^{22} - 20 q^{24} - 100 q^{25} + 100 q^{27} + 270 q^{28} - 5 q^{30} - 126 q^{31} + 20 q^{33} + 210 q^{34} - 213 q^{36} + 110 q^{37} - 191 q^{39} + 140 q^{40} - 175 q^{42} - 405 q^{45} - 210 q^{46} + 150 q^{48} - 224 q^{49} - 60 q^{51} - 320 q^{52} + 320 q^{54} - 10 q^{55} - 70 q^{58} + 1190 q^{60} + 294 q^{61} + 795 q^{63} + 362 q^{64} - 470 q^{66} - 260 q^{67} + 335 q^{69} + 1200 q^{70} + 215 q^{72} - 150 q^{73} + 200 q^{75} - 16 q^{76} - 1295 q^{78} - 346 q^{79} + 507 q^{81} - 456 q^{84} - 1450 q^{85} - 430 q^{87} - 1710 q^{88} - 820 q^{90} + 538 q^{91} - 560 q^{94} + 740 q^{96} - 150 q^{97} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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:

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 \(S_{3}^{\mathrm{new}}(75, [\chi])\).