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Properties

Label 175.2.h.b
Level $175$
Weight $2$
Character orbit 175.h
Analytic conductor $1.397$
Analytic rank $0$
Dimension $28$
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: \(28\)
Relative dimension: \(7\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 6 q^{2} + 4 q^{3} - 6 q^{4} + 8 q^{5} - 15 q^{6} - 28 q^{7} - 2 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 6 q^{2} + 4 q^{3} - 6 q^{4} + 8 q^{5} - 15 q^{6} - 28 q^{7} - 2 q^{8} - 5 q^{9} - 4 q^{10} - 8 q^{11} - 8 q^{12} - 8 q^{13} - 6 q^{14} - 9 q^{15} + 16 q^{16} + 14 q^{17} - 62 q^{18} + 22 q^{19} + 20 q^{20} - 4 q^{21} + 7 q^{22} + 21 q^{23} + 38 q^{24} + 8 q^{25} - 8 q^{26} - 11 q^{27} + 6 q^{28} - 37 q^{29} + 8 q^{30} - q^{31} - 48 q^{32} + 17 q^{33} - 10 q^{34} - 8 q^{35} - 41 q^{36} + 25 q^{37} + 15 q^{38} + 24 q^{39} + 39 q^{40} + 10 q^{41} + 15 q^{42} - 18 q^{43} + 65 q^{44} + 32 q^{45} + 26 q^{46} + 54 q^{47} + 69 q^{48} + 28 q^{49} - 49 q^{50} + 2 q^{51} - 54 q^{52} - 24 q^{53} - 14 q^{54} - 46 q^{55} + 2 q^{56} - 62 q^{57} + 17 q^{58} - 19 q^{59} - 6 q^{60} - 48 q^{61} - 42 q^{62} + 5 q^{63} - 20 q^{64} + 5 q^{65} + 91 q^{66} + 11 q^{67} - 114 q^{68} + 31 q^{69} + 4 q^{70} + 12 q^{71} + 20 q^{72} - 2 q^{73} + 70 q^{74} - 24 q^{75} - 6 q^{76} + 8 q^{77} + 59 q^{78} - 72 q^{79} - 8 q^{80} - 10 q^{81} - 26 q^{82} - 34 q^{83} + 8 q^{84} + 40 q^{85} - 60 q^{86} + 20 q^{87} + 32 q^{88} + 3 q^{89} - 32 q^{90} + 8 q^{91} + 10 q^{92} - 96 q^{93} + 12 q^{94} + 71 q^{95} + 22 q^{96} + 22 q^{97} + 6 q^{98} + 62 q^{99}+O(q^{100}) \) 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 −1.50493 + 1.09340i 0.710176 − 2.18570i 0.451267 − 1.38886i 1.70433 + 1.44750i 1.32107 + 4.06583i −1.00000 −0.310220 − 0.954759i −1.84587 − 1.34110i −4.14759 − 0.314875i
36.2 −1.28672 + 0.934859i −0.306518 + 0.943364i 0.163661 − 0.503697i −1.08759 + 1.95375i −0.487509 − 1.50040i −1.00000 −0.722670 − 2.22415i 1.63107 + 1.18504i −0.427058 − 3.53068i
36.3 −0.358354 + 0.260360i −0.286146 + 0.880667i −0.557403 + 1.71551i 2.21199 − 0.327291i −0.126748 − 0.390092i −1.00000 −0.520660 − 1.60243i 1.73336 + 1.25936i −0.707461 + 0.693198i
36.4 0.933813 − 0.678455i −0.705967 + 2.17274i −0.206328 + 0.635012i −2.22576 + 0.214471i 0.814867 + 2.50790i −1.00000 0.951525 + 2.92849i −1.79537 − 1.30441i −1.93293 + 1.71035i
36.5 1.18544 − 0.861270i 0.474905 − 1.46161i 0.0454391 − 0.139847i −0.316538 − 2.21355i −0.695869 − 2.14166i −1.00000 0.839012 + 2.58221i 0.516290 + 0.375106i −2.28170 − 2.35140i
36.6 1.59291 − 1.15732i 0.775777 − 2.38760i 0.579947 − 1.78489i 0.239897 + 2.22316i −1.52746 − 4.70105i −1.00000 0.0749920 + 0.230802i −2.67174 − 1.94113i 2.95504 + 3.26366i
36.7 2.05588 − 1.49369i −0.780262 + 2.40140i 1.37752 − 4.23957i 1.47367 + 1.68175i 1.98281 + 6.10246i −1.00000 −1.93001 − 5.93997i −2.73086 − 1.98408i 5.54170 + 1.25629i
71.1 −0.766819 + 2.36003i 1.79437 + 1.30369i −3.36368 − 2.44386i 2.12129 + 0.707196i −4.45270 + 3.23507i −1.00000 4.33179 − 3.14723i 0.593118 + 1.82543i −3.29565 + 4.46401i
71.2 −0.459090 + 1.41293i 2.50953 + 1.82328i −0.167580 − 0.121754i −1.77897 − 1.35472i −3.72827 + 2.70875i −1.00000 −2.15486 + 1.56560i 2.04634 + 6.29798i 2.73083 − 1.89163i
71.3 −0.171793 + 0.528723i 0.0113445 + 0.00824229i 1.36800 + 0.993909i 1.03793 + 1.98058i −0.00630680 + 0.00458216i −1.00000 −1.66003 + 1.20608i −0.926990 − 2.85298i −1.22549 + 0.208527i
71.4 0.00200273 − 0.00616377i −1.94895 − 1.41600i 1.61800 + 1.17555i 1.44966 − 1.70249i −0.0126311 + 0.00917702i −1.00000 0.0209726 − 0.0152375i 0.866313 + 2.66624i −0.00759046 − 0.0123450i
71.5 0.511192 − 1.57329i 1.62203 + 1.17848i −0.595879 − 0.432932i −0.827625 + 2.07727i 2.68325 − 1.94950i −1.00000 1.69090 − 1.22851i 0.315138 + 0.969894i 2.84506 + 2.36397i
71.6 0.537726 − 1.65495i −1.85417 − 1.34713i −0.831674 − 0.604246i −2.14680 − 0.625509i −3.22647 + 2.34417i −1.00000 1.36836 − 0.994170i 0.696130 + 2.14247i −2.18957 + 3.21649i
71.7 0.728747 − 2.24285i −0.0161250 − 0.0117155i −2.88129 − 2.09338i 2.14451 − 0.633300i −0.0380272 + 0.0276284i −1.00000 −2.97909 + 2.16444i −0.926928 − 2.85279i 0.142408 − 5.27134i
106.1 −0.766819 − 2.36003i 1.79437 − 1.30369i −3.36368 + 2.44386i 2.12129 − 0.707196i −4.45270 − 3.23507i −1.00000 4.33179 + 3.14723i 0.593118 − 1.82543i −3.29565 − 4.46401i
106.2 −0.459090 − 1.41293i 2.50953 − 1.82328i −0.167580 + 0.121754i −1.77897 + 1.35472i −3.72827 − 2.70875i −1.00000 −2.15486 − 1.56560i 2.04634 − 6.29798i 2.73083 + 1.89163i
106.3 −0.171793 − 0.528723i 0.0113445 − 0.00824229i 1.36800 − 0.993909i 1.03793 − 1.98058i −0.00630680 − 0.00458216i −1.00000 −1.66003 − 1.20608i −0.926990 + 2.85298i −1.22549 − 0.208527i
106.4 0.00200273 + 0.00616377i −1.94895 + 1.41600i 1.61800 − 1.17555i 1.44966 + 1.70249i −0.0126311 − 0.00917702i −1.00000 0.0209726 + 0.0152375i 0.866313 − 2.66624i −0.00759046 + 0.0123450i
106.5 0.511192 + 1.57329i 1.62203 − 1.17848i −0.595879 + 0.432932i −0.827625 − 2.07727i 2.68325 + 1.94950i −1.00000 1.69090 + 1.22851i 0.315138 − 0.969894i 2.84506 − 2.36397i
106.6 0.537726 + 1.65495i −1.85417 + 1.34713i −0.831674 + 0.604246i −2.14680 + 0.625509i −3.22647 − 2.34417i −1.00000 1.36836 + 0.994170i 0.696130 − 2.14247i −2.18957 − 3.21649i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.7
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.b ✓ 28
5.b even 2 1 875.2.h.b 28
5.c odd 4 2 875.2.n.b 56
25.d even 5 1 inner 175.2.h.b ✓ 28
25.d even 5 1 4375.2.a.g 14
25.e even 10 1 875.2.h.b 28
25.e even 10 1 4375.2.a.h 14
25.f odd 20 2 875.2.n.b 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.h.b ✓ 28 1.a even 1 1 trivial
175.2.h.b ✓ 28 25.d even 5 1 inner
875.2.h.b 28 5.b even 2 1
875.2.h.b 28 25.e even 10 1
875.2.n.b 56 5.c odd 4 2
875.2.n.b 56 25.f odd 20 2
4375.2.a.g 14 25.d even 5 1
4375.2.a.h 14 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} - 6 T_{2}^{27} + 28 T_{2}^{26} - 90 T_{2}^{25} + 267 T_{2}^{24} - 626 T_{2}^{23} + 1436 T_{2}^{22} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display