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Properties

Label 175.2.h.c
Level $175$
Weight $2$
Character orbit 175.h
Analytic conductor $1.397$
Analytic rank $0$
Dimension $32$
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] = [175,2,Mod(36,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.36");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 175.h (of order \(5\), 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: \(1.39738203537\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 3 q^{2} - 4 q^{3} - 9 q^{4} + q^{5} + 15 q^{6} + 32 q^{7} - 9 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 3 q^{2} - 4 q^{3} - 9 q^{4} + q^{5} + 15 q^{6} + 32 q^{7} - 9 q^{8} - 6 q^{9} - 15 q^{10} + 2 q^{11} - 2 q^{12} - 16 q^{13} - 3 q^{14} - 9 q^{15} - 15 q^{16} - 18 q^{17} + 26 q^{18} - 8 q^{19} - 12 q^{20} - 4 q^{21} + 9 q^{22} - 11 q^{23} - 46 q^{24} + 19 q^{25} - 34 q^{26} - 7 q^{27} - 9 q^{28} + 13 q^{29} - 10 q^{30} + 5 q^{31} + 64 q^{32} - 25 q^{33} + 25 q^{34} + q^{35} + 64 q^{36} + 6 q^{37} - 15 q^{38} - 18 q^{39} + 20 q^{40} - 26 q^{41} + 15 q^{42} - 6 q^{43} - 37 q^{44} - 37 q^{45} - 12 q^{46} + 15 q^{48} + 32 q^{49} + 20 q^{50} - 10 q^{51} - 38 q^{52} - 27 q^{53} + 108 q^{54} + 42 q^{55} - 9 q^{56} + 2 q^{57} + 49 q^{58} + 13 q^{59} + 18 q^{60} + 40 q^{61} - 14 q^{62} - 6 q^{63} + q^{64} + 40 q^{65} - 37 q^{66} - 15 q^{67} + 4 q^{68} - 45 q^{69} - 15 q^{70} - 24 q^{71} - 75 q^{72} - 38 q^{73} - 40 q^{74} - 56 q^{75} - 70 q^{76} + 2 q^{77} + 93 q^{78} + 66 q^{79} - 11 q^{80} - 7 q^{81} - 32 q^{82} - 52 q^{83} - 2 q^{84} + 13 q^{85} + 60 q^{86} + 54 q^{87} - 66 q^{88} + 28 q^{89} - 135 q^{90} - 16 q^{91} - 44 q^{92} + 56 q^{93} - 4 q^{94} - 23 q^{95} - 34 q^{96} - 44 q^{97} - 3 q^{98} - 66 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1 −2.25360 + 1.63734i −0.295516 + 0.909504i 1.77981 − 5.47769i 1.23160 + 1.86632i −0.823190 − 2.53352i 1.00000 3.23625 + 9.96014i 1.68718 + 1.22581i −5.83133 − 2.18940i
36.2 −1.82833 + 1.32836i 0.873597 − 2.68865i 0.960212 − 2.95523i −2.20890 − 0.347494i 1.97427 + 6.07619i 1.00000 0.773301 + 2.37998i −4.03864 − 2.93425i 4.50019 − 2.29888i
36.3 −1.45479 + 1.05697i −0.0282165 + 0.0868416i 0.381203 − 1.17322i 0.905283 − 2.04462i −0.0507396 − 0.156160i 1.00000 −0.425874 − 1.31071i 2.42031 + 1.75846i 0.844097 + 3.93135i
36.4 −0.533169 + 0.387370i −0.987230 + 3.03838i −0.483820 + 1.48905i 1.51052 + 1.64873i −0.650618 − 2.00240i 1.00000 −0.726159 − 2.23489i −5.83009 − 4.23581i −1.44403 − 0.293924i
36.5 0.0920209 − 0.0668571i 0.903271 − 2.77998i −0.614036 + 1.88981i 2.08483 − 0.808386i −0.102742 − 0.316207i 1.00000 0.140141 + 0.431309i −4.48535 − 3.25880i 0.137802 − 0.213774i
36.6 0.361567 − 0.262694i 0.155006 − 0.477060i −0.556311 + 1.71215i −0.702720 + 2.12278i −0.0692756 − 0.213209i 1.00000 0.524840 + 1.61529i 2.22349 + 1.61546i 0.303560 + 0.952127i
36.7 1.19734 − 0.869915i −0.571143 + 1.75780i 0.0588259 − 0.181048i 1.31966 − 1.80513i 0.845285 + 2.60152i 1.00000 0.827621 + 2.54716i −0.336596 − 0.244551i 0.00976061 − 3.30934i
36.8 1.99191 − 1.44721i 0.0682653 − 0.210099i 1.25527 − 3.86332i −2.21322 + 0.318852i −0.168079 − 0.517293i 1.00000 −1.56896 − 4.82877i 2.38757 + 1.73467i −3.94709 + 3.83812i
71.1 −0.795678 + 2.44884i −1.85627 − 1.34866i −3.74570 − 2.72141i 2.18069 − 0.494549i 4.77966 − 3.47263i 1.00000 5.47846 − 3.98034i 0.699813 + 2.15380i −0.524056 + 5.73368i
71.2 −0.369205 + 1.13629i 1.01300 + 0.735989i 0.463180 + 0.336520i 1.95484 − 1.08564i −1.21030 + 0.879338i 1.00000 −2.48657 + 1.80660i −0.442558 − 1.36205i 0.511870 + 2.62209i
71.3 −0.172359 + 0.530468i −2.67034 − 1.94012i 1.36635 + 0.992708i −1.72828 + 1.41882i 1.48943 − 1.08213i 1.00000 −1.66459 + 1.20939i 2.43961 + 7.50836i −0.454750 − 1.16135i
71.4 −0.0723910 + 0.222797i 1.27625 + 0.927248i 1.57364 + 1.14331i −2.02395 + 0.950594i −0.298977 + 0.217219i 1.00000 −0.747688 + 0.543227i −0.158032 − 0.486374i −0.0652733 − 0.519743i
71.5 0.324683 − 0.999270i −1.06367 − 0.772801i 0.724912 + 0.526679i 1.85720 + 1.24531i −1.11759 + 0.811978i 1.00000 2.46172 − 1.78855i −0.392880 − 1.20916i 1.84741 − 1.45151i
71.6 0.380117 − 1.16988i −0.0753841 − 0.0547698i 0.393906 + 0.286190i −1.46965 − 1.68527i −0.0927287 + 0.0673714i 1.00000 2.47485 − 1.79809i −0.924368 − 2.84491i −2.53020 + 1.07871i
71.7 0.788620 − 2.42712i 2.47892 + 1.80104i −3.65097 − 2.65259i −1.30865 − 1.81313i 6.32628 − 4.59631i 1.00000 −5.18811 + 3.76938i 1.97425 + 6.07612i −5.43271 + 1.74639i
71.8 0.843265 − 2.59530i −1.22054 − 0.886773i −4.40646 − 3.20148i −0.889246 + 2.05164i −3.33068 + 2.41988i 1.00000 −7.60923 + 5.52843i −0.223704 − 0.688489i 4.57476 + 4.03794i
106.1 −0.795678 − 2.44884i −1.85627 + 1.34866i −3.74570 + 2.72141i 2.18069 + 0.494549i 4.77966 + 3.47263i 1.00000 5.47846 + 3.98034i 0.699813 − 2.15380i −0.524056 − 5.73368i
106.2 −0.369205 − 1.13629i 1.01300 − 0.735989i 0.463180 − 0.336520i 1.95484 + 1.08564i −1.21030 − 0.879338i 1.00000 −2.48657 − 1.80660i −0.442558 + 1.36205i 0.511870 − 2.62209i
106.3 −0.172359 − 0.530468i −2.67034 + 1.94012i 1.36635 − 0.992708i −1.72828 − 1.41882i 1.48943 + 1.08213i 1.00000 −1.66459 − 1.20939i 2.43961 − 7.50836i −0.454750 + 1.16135i
106.4 −0.0723910 − 0.222797i 1.27625 − 0.927248i 1.57364 − 1.14331i −2.02395 − 0.950594i −0.298977 − 0.217219i 1.00000 −0.747688 − 0.543227i −0.158032 + 0.486374i −0.0652733 + 0.519743i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.2.h.c ✓ 32
5.b even 2 1 875.2.h.c 32
5.c odd 4 2 875.2.n.d 64
25.d even 5 1 inner 175.2.h.c ✓ 32
25.d even 5 1 4375.2.a.l 16
25.e even 10 1 875.2.h.c 32
25.e even 10 1 4375.2.a.i 16
25.f odd 20 2 875.2.n.d 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.h.c ✓ 32 1.a even 1 1 trivial
175.2.h.c ✓ 32 25.d even 5 1 inner
875.2.h.c 32 5.b even 2 1
875.2.h.c 32 25.e even 10 1
875.2.n.d 64 5.c odd 4 2
875.2.n.d 64 25.f odd 20 2
4375.2.a.i 16 25.e even 10 1
4375.2.a.l 16 25.d even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 3 T_{2}^{31} + 17 T_{2}^{30} + 49 T_{2}^{29} + 192 T_{2}^{28} + 393 T_{2}^{27} + 1381 T_{2}^{26} + \cdots + 25 \) acting on \(S_{2}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display