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Properties

Label 175.4.k.e
Level $175$
Weight $4$
Character orbit 175.k
Analytic conductor $10.325$
Analytic rank $0$
Dimension $32$
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] = [175,4,Mod(74,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.74");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.k (of order \(6\), degree \(2\), not 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: \(10.3253342510\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 54 q^{4} + 16 q^{6} + 84 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 54 q^{4} + 16 q^{6} + 84 q^{9} + 130 q^{14} - 382 q^{16} - 84 q^{19} + 604 q^{21} - 584 q^{24} - 46 q^{26} - 1152 q^{29} - 36 q^{31} + 1952 q^{34} - 2844 q^{36} + 44 q^{39} + 672 q^{41} - 1760 q^{44} + 196 q^{46} + 2228 q^{49} - 1256 q^{51} + 4326 q^{54} + 6102 q^{56} + 1116 q^{59} + 2992 q^{61} - 9744 q^{64} + 1364 q^{66} - 3288 q^{69} - 2896 q^{71} + 470 q^{74} - 6036 q^{76} - 1680 q^{79} - 4784 q^{81} + 10542 q^{84} - 3632 q^{86} + 5356 q^{89} + 4092 q^{91} + 7250 q^{94} + 13202 q^{96} - 16344 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1 −4.73715 + 2.73499i 3.69486 + 2.13323i 10.9604 − 18.9839i 0 −23.3375 17.0044 − 7.33838i 76.1465i −4.39867 − 7.61872i 0
74.2 −4.32202 + 2.49532i −0.123328 − 0.0712035i 8.45325 − 14.6415i 0 0.710702 −15.6400 − 9.91913i 44.4491i −13.4899 − 23.3651i 0
74.3 −3.23834 + 1.86966i −4.94996 − 2.85786i 2.99122 − 5.18095i 0 21.3729 −18.4439 − 1.68048i − 7.54426i 2.83475 + 4.90994i 0
74.4 −2.52961 + 1.46047i −8.70206 − 5.02414i 0.265967 − 0.460668i 0 29.3505 −2.32782 + 18.3734i − 21.8138i 36.9839 + 64.0580i 0
74.5 −2.34941 + 1.35643i 1.29309 + 0.746568i −0.320193 + 0.554591i 0 −4.05067 18.5163 − 0.381254i − 23.4402i −12.3853 − 21.4519i 0
74.6 −1.91472 + 1.10546i 5.01658 + 2.89632i −1.55590 + 2.69490i 0 −12.8071 −4.81410 + 17.8836i − 24.5674i 3.27735 + 5.67654i 0
74.7 −0.810327 + 0.467843i 1.08497 + 0.626409i −3.56225 + 6.16999i 0 −1.17224 10.2860 − 15.4013i − 14.1518i −12.7152 − 22.0234i 0
74.8 −0.633465 + 0.365731i 7.18259 + 4.14687i −3.73248 + 6.46485i 0 −6.06655 −17.3007 − 6.60948i − 11.3120i 20.8930 + 36.1878i 0
74.9 0.633465 − 0.365731i −7.18259 − 4.14687i −3.73248 + 6.46485i 0 −6.06655 17.3007 + 6.60948i 11.3120i 20.8930 + 36.1878i 0
74.10 0.810327 − 0.467843i −1.08497 − 0.626409i −3.56225 + 6.16999i 0 −1.17224 −10.2860 + 15.4013i 14.1518i −12.7152 − 22.0234i 0
74.11 1.91472 − 1.10546i −5.01658 − 2.89632i −1.55590 + 2.69490i 0 −12.8071 4.81410 − 17.8836i 24.5674i 3.27735 + 5.67654i 0
74.12 2.34941 − 1.35643i −1.29309 − 0.746568i −0.320193 + 0.554591i 0 −4.05067 −18.5163 + 0.381254i 23.4402i −12.3853 − 21.4519i 0
74.13 2.52961 − 1.46047i 8.70206 + 5.02414i 0.265967 − 0.460668i 0 29.3505 2.32782 − 18.3734i 21.8138i 36.9839 + 64.0580i 0
74.14 3.23834 − 1.86966i 4.94996 + 2.85786i 2.99122 − 5.18095i 0 21.3729 18.4439 + 1.68048i 7.54426i 2.83475 + 4.90994i 0
74.15 4.32202 − 2.49532i 0.123328 + 0.0712035i 8.45325 − 14.6415i 0 0.710702 15.6400 + 9.91913i − 44.4491i −13.4899 − 23.3651i 0
74.16 4.73715 − 2.73499i −3.69486 − 2.13323i 10.9604 − 18.9839i 0 −23.3375 −17.0044 + 7.33838i − 76.1465i −4.39867 − 7.61872i 0
149.1 −4.73715 − 2.73499i 3.69486 − 2.13323i 10.9604 + 18.9839i 0 −23.3375 17.0044 + 7.33838i − 76.1465i −4.39867 + 7.61872i 0
149.2 −4.32202 − 2.49532i −0.123328 + 0.0712035i 8.45325 + 14.6415i 0 0.710702 −15.6400 + 9.91913i − 44.4491i −13.4899 + 23.3651i 0
149.3 −3.23834 − 1.86966i −4.94996 + 2.85786i 2.99122 + 5.18095i 0 21.3729 −18.4439 + 1.68048i 7.54426i 2.83475 − 4.90994i 0
149.4 −2.52961 − 1.46047i −8.70206 + 5.02414i 0.265967 + 0.460668i 0 29.3505 −2.32782 − 18.3734i 21.8138i 36.9839 − 64.0580i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.4.k.e 32
5.b even 2 1 inner 175.4.k.e 32
5.c odd 4 1 175.4.e.e ✓ 16
5.c odd 4 1 175.4.e.f yes 16
7.c even 3 1 inner 175.4.k.e 32
35.j even 6 1 inner 175.4.k.e 32
35.k even 12 1 1225.4.a.bk 8
35.k even 12 1 1225.4.a.bo 8
35.l odd 12 1 175.4.e.e ✓ 16
35.l odd 12 1 175.4.e.f yes 16
35.l odd 12 1 1225.4.a.bl 8
35.l odd 12 1 1225.4.a.bn 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.4.e.e ✓ 16 5.c odd 4 1
175.4.e.e ✓ 16 35.l odd 12 1
175.4.e.f yes 16 5.c odd 4 1
175.4.e.f yes 16 35.l odd 12 1
175.4.k.e 32 1.a even 1 1 trivial
175.4.k.e 32 5.b even 2 1 inner
175.4.k.e 32 7.c even 3 1 inner
175.4.k.e 32 35.j even 6 1 inner
1225.4.a.bk 8 35.k even 12 1
1225.4.a.bl 8 35.l odd 12 1
1225.4.a.bn 8 35.l odd 12 1
1225.4.a.bo 8 35.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 91 T_{2}^{30} + 5072 T_{2}^{28} - 179623 T_{2}^{26} + 4659123 T_{2}^{24} + \cdots + 2244531326976 \) acting on \(S_{4}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display