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Properties

Label 75.4.i.a
Level $75$
Weight $4$
Character orbit 75.i
Analytic conductor $4.425$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,4,Mod(4,75)]
 
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([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.i (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: \(4.42514325043\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 72 q^{4} - 6 q^{5} - 12 q^{6} + 210 q^{8} + 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 72 q^{4} - 6 q^{5} - 12 q^{6} + 210 q^{8} + 144 q^{9} - 14 q^{10} - 84 q^{11} - 132 q^{14} - 54 q^{15} - 460 q^{16} + 320 q^{17} + 188 q^{19} - 404 q^{20} - 84 q^{21} + 290 q^{22} - 160 q^{23} - 576 q^{24} - 496 q^{25} + 348 q^{26} - 360 q^{28} - 604 q^{29} + 564 q^{30} - 786 q^{31} + 480 q^{33} + 584 q^{34} + 1160 q^{35} - 648 q^{36} + 1030 q^{37} + 2490 q^{38} + 312 q^{39} + 3072 q^{40} + 884 q^{41} - 1020 q^{42} - 1694 q^{44} + 54 q^{45} - 1436 q^{46} - 1720 q^{47} - 5220 q^{49} - 3644 q^{50} + 1632 q^{51} - 3720 q^{52} - 5170 q^{53} + 108 q^{54} + 1484 q^{55} - 2700 q^{56} + 5030 q^{58} + 1392 q^{59} - 2646 q^{60} - 240 q^{61} + 8830 q^{62} - 180 q^{63} + 6618 q^{64} + 4582 q^{65} + 744 q^{66} - 4980 q^{67} - 72 q^{69} - 3180 q^{70} - 1432 q^{71} + 1890 q^{72} + 840 q^{73} + 2908 q^{74} + 576 q^{75} + 8832 q^{76} + 1120 q^{77} - 320 q^{79} - 444 q^{80} - 1296 q^{81} + 9380 q^{83} + 4392 q^{84} - 602 q^{85} + 3996 q^{86} + 2280 q^{87} + 2030 q^{88} - 1242 q^{89} - 144 q^{90} + 1484 q^{91} - 11230 q^{92} - 4018 q^{94} - 9884 q^{95} - 4344 q^{96} - 3870 q^{97} - 7020 q^{98} - 504 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −5.15598 − 1.67528i 1.76336 − 2.42705i 17.3054 + 12.5731i −10.7764 − 2.97815i −13.1578 + 9.55971i − 14.0951i −42.6704 − 58.7308i −2.78115 − 8.55951i 50.5736 + 33.4087i
4.2 −4.17173 − 1.35548i −1.76336 + 2.42705i 9.09391 + 6.60711i 3.86783 + 10.4900i 10.6461 − 7.73482i − 28.2575i −8.35540 − 11.5002i −2.78115 − 8.55951i −1.91660 − 49.0042i
4.3 −4.02273 − 1.30706i −1.76336 + 2.42705i 8.00179 + 5.81364i −11.0080 − 1.95576i 10.2658 − 7.45855i 16.6083i −4.70077 − 6.47005i −2.78115 − 8.55951i 41.7257 + 22.2556i
4.4 −3.37780 − 1.09751i 1.76336 − 2.42705i 3.73287 + 2.71209i 3.23678 + 10.7016i −8.61999 + 6.26279i 3.89279i 7.06844 + 9.72887i −2.78115 − 8.55951i 0.811917 − 39.7001i
4.5 −3.23774 − 1.05200i 1.76336 − 2.42705i 2.90409 + 2.10995i 8.47742 − 7.28926i −8.26255 + 6.00310i − 3.93904i 8.82524 + 12.1469i −2.78115 − 8.55951i −35.1160 + 14.6824i
4.6 −2.11566 − 0.687420i −1.76336 + 2.42705i −2.46866 − 1.79358i 4.55739 − 10.2093i 5.39907 − 3.92265i 13.2560i 14.4503 + 19.8891i −2.78115 − 8.55951i −16.6600 + 18.4666i
4.7 −1.21421 − 0.394522i 1.76336 − 2.42705i −5.15347 − 3.74422i −11.0564 − 1.65995i −3.09861 + 2.25127i 32.4814i 10.7836 + 14.8424i −2.78115 − 8.55951i 12.7700 + 6.37753i
4.8 −0.596564 − 0.193835i −1.76336 + 2.42705i −6.15382 − 4.47101i 11.0878 + 1.43544i 1.52240 − 1.10609i − 11.1579i 5.75408 + 7.91981i −2.78115 − 8.55951i −6.33635 − 3.00554i
4.9 0.623209 + 0.202493i 1.76336 − 2.42705i −6.12475 − 4.44989i −7.34487 + 8.42929i 1.59040 − 1.15549i − 35.0184i −5.99724 − 8.25449i −2.78115 − 8.55951i −6.28425 + 3.76592i
4.10 1.45424 + 0.472511i −1.76336 + 2.42705i −4.58059 − 3.32799i −8.39792 − 7.38071i −3.71115 + 2.69631i − 11.0201i −12.2789 − 16.9005i −2.78115 − 8.55951i −8.72513 − 14.7014i
4.11 1.62249 + 0.527178i 1.76336 − 2.42705i −4.11759 − 2.99160i 3.27805 − 10.6890i 4.14051 − 3.00826i 2.02294i −13.1256 − 18.0659i −2.78115 − 8.55951i 10.9536 − 15.6146i
4.12 2.02545 + 0.658107i −1.76336 + 2.42705i −2.80281 − 2.03636i −1.80424 + 11.0338i −5.16884 + 3.75538i 25.6982i −14.3512 − 19.7527i −2.78115 − 8.55951i −10.9158 + 21.1610i
4.13 3.89397 + 1.26523i 1.76336 − 2.42705i 7.09004 + 5.15122i 5.95095 + 9.46500i 9.93722 − 7.21981i 3.23603i 1.83811 + 2.52994i −2.78115 − 8.55951i 11.1974 + 44.3857i
4.14 4.37032 + 1.42000i −1.76336 + 2.42705i 10.6112 + 7.70947i 7.28855 − 8.47803i −11.1529 + 8.10302i 33.1983i 13.8187 + 19.0198i −2.78115 − 8.55951i 43.8922 − 26.7020i
4.15 4.94396 + 1.60639i 1.76336 − 2.42705i 15.3901 + 11.1815i −5.32210 − 9.83236i 12.6167 − 9.16660i − 2.85806i 33.6817 + 46.3589i −2.78115 − 8.55951i −10.5177 − 57.1601i
4.16 4.95879 + 1.61121i −1.76336 + 2.42705i 15.5215 + 11.2770i −5.83325 + 9.53799i −12.6546 + 9.19411i − 24.6028i 34.2806 + 47.1832i −2.78115 − 8.55951i −44.2936 + 37.8983i
19.1 −5.15598 + 1.67528i 1.76336 + 2.42705i 17.3054 − 12.5731i −10.7764 + 2.97815i −13.1578 − 9.55971i 14.0951i −42.6704 + 58.7308i −2.78115 + 8.55951i 50.5736 − 33.4087i
19.2 −4.17173 + 1.35548i −1.76336 − 2.42705i 9.09391 − 6.60711i 3.86783 − 10.4900i 10.6461 + 7.73482i 28.2575i −8.35540 + 11.5002i −2.78115 + 8.55951i −1.91660 + 49.0042i
19.3 −4.02273 + 1.30706i −1.76336 − 2.42705i 8.00179 − 5.81364i −11.0080 + 1.95576i 10.2658 + 7.45855i − 16.6083i −4.70077 + 6.47005i −2.78115 + 8.55951i 41.7257 − 22.2556i
19.4 −3.37780 + 1.09751i 1.76336 + 2.42705i 3.73287 − 2.71209i 3.23678 − 10.7016i −8.61999 − 6.26279i − 3.89279i 7.06844 − 9.72887i −2.78115 + 8.55951i 0.811917 + 39.7001i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.4.i.a ✓ 64
3.b odd 2 1 225.4.m.c 64
25.e even 10 1 inner 75.4.i.a ✓ 64
25.f odd 20 1 1875.4.a.m 32
25.f odd 20 1 1875.4.a.n 32
75.h odd 10 1 225.4.m.c 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.i.a ✓ 64 1.a even 1 1 trivial
75.4.i.a ✓ 64 25.e even 10 1 inner
225.4.m.c 64 3.b odd 2 1
225.4.m.c 64 75.h odd 10 1
1875.4.a.m 32 25.f odd 20 1
1875.4.a.n 32 25.f odd 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(75, [\chi])\).