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Properties

Label 175.2.n.a
Level $175$
Weight $2$
Character orbit 175.n
Analytic conductor $1.397$
Analytic rank $0$
Dimension $56$
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(29,175)]
 
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([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 175.n (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: \(1.39738203537\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 12 q^{4} - 6 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 12 q^{4} - 6 q^{5} + 6 q^{9} - 4 q^{10} + 8 q^{11} - 40 q^{12} - 4 q^{14} - 18 q^{15} - 32 q^{16} + 12 q^{19} + 12 q^{20} + 4 q^{21} - 30 q^{22} + 10 q^{23} - 28 q^{24} + 4 q^{25} + 12 q^{26} + 30 q^{27} - 2 q^{29} + 28 q^{30} + 12 q^{31} + 20 q^{33} + 2 q^{35} - 14 q^{36} - 70 q^{37} - 70 q^{38} - 4 q^{39} - 30 q^{40} + 4 q^{41} + 50 q^{42} + 22 q^{44} - 52 q^{45} - 4 q^{46} - 10 q^{47} + 30 q^{48} - 56 q^{49} - 54 q^{50} - 44 q^{51} - 20 q^{53} + 54 q^{54} - 2 q^{55} + 12 q^{56} + 10 q^{58} - 6 q^{59} + 16 q^{60} - 4 q^{61} + 50 q^{62} + 20 q^{63} + 24 q^{64} - 18 q^{65} - 74 q^{66} + 10 q^{67} - 78 q^{69} + 8 q^{70} - 8 q^{71} + 140 q^{72} + 40 q^{73} + 60 q^{74} - 8 q^{75} + 52 q^{76} - 20 q^{77} - 90 q^{78} + 124 q^{80} - 72 q^{81} - 30 q^{83} - 12 q^{84} + 96 q^{85} - 20 q^{86} + 30 q^{87} + 140 q^{88} + 38 q^{89} - 8 q^{90} + 8 q^{91} + 80 q^{92} + 88 q^{94} - 70 q^{95} - 28 q^{96} - 30 q^{97} + 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −2.54065 − 0.825507i −0.247551 + 0.340724i 4.15540 + 3.01908i −1.54620 − 1.61532i 0.910210 − 0.661306i 1.00000i −4.92473 − 6.77832i 0.872239 + 2.68448i 2.59489 + 5.38036i
29.2 −2.05165 − 0.666622i 0.505071 − 0.695171i 2.14685 + 1.55978i 2.21022 − 0.338990i −1.49965 + 1.08956i − 1.00000i −0.828832 − 1.14079i 0.698885 + 2.15095i −4.76059 − 0.777894i
29.3 −1.82157 − 0.591863i −1.68900 + 2.32470i 1.34977 + 0.980668i 2.12640 − 0.691667i 4.45253 − 3.23495i 1.00000i 0.373298 + 0.513801i −1.62449 − 4.99966i −4.28276 + 0.00137701i
29.4 −1.51762 − 0.493105i −1.52541 + 2.09954i 0.441990 + 0.321125i −2.23549 + 0.0510350i 3.35029 − 2.43413i − 1.00000i 1.36346 + 1.87664i −1.15416 − 3.55215i 3.41779 + 1.02488i
29.5 −0.977435 − 0.317588i 1.68681 − 2.32169i −0.763516 − 0.554727i 0.852197 − 2.06731i −2.38608 + 1.73359i 1.00000i 1.77829 + 2.44761i −1.61787 − 4.97930i −1.48952 + 1.75001i
29.6 −0.798956 − 0.259597i −0.142668 + 0.196366i −1.04709 − 0.760758i 1.61677 + 1.54469i 0.164962 − 0.119852i 1.00000i 1.62666 + 2.23890i 0.908846 + 2.79714i −0.890733 − 1.65384i
29.7 −0.387846 − 0.126019i −0.531527 + 0.731584i −1.48349 − 1.07782i 0.0211801 + 2.23597i 0.298344 − 0.216760i − 1.00000i 0.918945 + 1.26482i 0.674357 + 2.07546i 0.273560 − 0.869881i
29.8 0.225611 + 0.0733055i 1.21485 − 1.67209i −1.57251 − 1.14249i −2.18117 − 0.492433i 0.396656 − 0.288188i − 1.00000i −0.549895 − 0.756865i −0.392990 − 1.20950i −0.455999 − 0.270990i
29.9 1.03051 + 0.334833i −1.58321 + 2.17910i −0.668196 − 0.485473i −2.00240 + 0.995194i −2.36115 + 1.71547i 1.00000i −1.79981 − 2.47723i −1.31487 − 4.04676i −2.39671 + 0.355089i
29.10 1.09195 + 0.354796i 1.51811 − 2.08949i −0.551558 − 0.400730i 0.904522 + 2.04495i 2.39904 − 1.74301i 1.00000i −1.80982 − 2.49101i −1.13429 − 3.49098i 0.262151 + 2.55391i
29.11 1.19597 + 0.388594i 0.374566 − 0.515546i −0.338697 − 0.246078i 1.94508 − 1.10303i 0.648308 − 0.471023i − 1.00000i −1.78775 − 2.46062i 0.801563 + 2.46696i 2.75488 − 0.563341i
29.12 1.99559 + 0.648406i −0.448050 + 0.616687i 1.94391 + 1.41233i −0.794894 + 2.09001i −1.29399 + 0.940136i − 1.00000i 0.496791 + 0.683774i 0.747496 + 2.30056i −2.94146 + 3.65539i
29.13 2.11404 + 0.686892i 0.515049 − 0.708904i 2.37929 + 1.72866i −1.55449 − 1.60734i 1.57577 − 1.14486i 1.00000i 1.22941 + 1.69214i 0.689782 + 2.12293i −2.18219 − 4.46575i
29.14 2.44207 + 0.793475i −1.88310 + 2.59187i 3.71604 + 2.69986i 0.256301 − 2.22133i −6.65525 + 4.83532i − 1.00000i 3.91399 + 5.38714i −2.24466 − 6.90836i 2.38847 − 5.22127i
64.1 −1.61734 − 2.22607i −2.13699 − 0.694351i −1.72159 + 5.29851i −2.23415 + 0.0925149i 1.91056 + 5.88011i − 1.00000i 9.34546 − 3.03652i 1.65757 + 1.20429i 3.81932 + 4.82376i
64.2 −1.40576 − 1.93486i 2.61714 + 0.850362i −1.14949 + 3.53776i −0.290501 + 2.21712i −2.03374 − 6.25920i 1.00000i 3.91185 − 1.27104i 3.69928 + 2.68768i 4.69818 − 2.55465i
64.3 −1.11095 − 1.52910i 1.72878 + 0.561715i −0.485887 + 1.49541i −1.28703 − 1.82854i −1.06168 − 3.26751i − 1.00000i −0.768705 + 0.249767i 0.246106 + 0.178807i −1.36619 + 3.99942i
64.4 −0.635734 − 0.875013i −3.03925 − 0.987513i 0.256544 − 0.789562i 0.574707 − 2.16095i 1.06807 + 3.28718i − 1.00000i −2.91125 + 0.945922i 5.83482 + 4.23925i −2.25622 + 0.870914i
64.5 −0.560222 − 0.771080i −1.03930 − 0.337689i 0.337319 − 1.03816i −2.23402 − 0.0956332i 0.321853 + 0.990563i 1.00000i −2.80240 + 0.910553i −1.46094 − 1.06144i 1.17781 + 1.77618i
64.6 −0.455964 − 0.627580i 1.65172 + 0.536677i 0.432080 − 1.32981i 0.928918 − 2.03399i −0.416318 − 1.28129i 1.00000i −2.50710 + 0.814607i 0.0131148 + 0.00952845i −1.70004 + 0.344454i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.14
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 175.2.n.a ✓ 56
5.b even 2 1 875.2.n.c 56
5.c odd 4 1 875.2.h.d 56
5.c odd 4 1 875.2.h.e 56
25.d even 5 1 875.2.n.c 56
25.e even 10 1 inner 175.2.n.a ✓ 56
25.f odd 20 1 875.2.h.d 56
25.f odd 20 1 875.2.h.e 56
25.f odd 20 1 4375.2.a.o 28
25.f odd 20 1 4375.2.a.p 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.n.a ✓ 56 1.a even 1 1 trivial
175.2.n.a ✓ 56 25.e even 10 1 inner
875.2.h.d 56 5.c odd 4 1
875.2.h.d 56 25.f odd 20 1
875.2.h.e 56 5.c odd 4 1
875.2.h.e 56 25.f odd 20 1
875.2.n.c 56 5.b even 2 1
875.2.n.c 56 25.d even 5 1
4375.2.a.o 28 25.f odd 20 1
4375.2.a.p 28 25.f odd 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(175, [\chi])\).