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Properties

Label 875.2.h.e
Level $875$
Weight $2$
Character orbit 875.h
Analytic conductor $6.987$
Analytic rank $0$
Dimension $56$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [875,2,Mod(176,875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(875, 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("875.176");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 875 = 5^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 875.h (of order \(5\), degree \(4\), 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: \(6.98691017686\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{5})\)
Twist minimal: no (minimal twist has level 175)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{2} + 4 q^{3} - 12 q^{4} + 56 q^{7} + 12 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{2} + 4 q^{3} - 12 q^{4} + 56 q^{7} + 12 q^{8} - 6 q^{9} + 8 q^{11} + 12 q^{12} + 8 q^{13} + 4 q^{14} - 32 q^{16} + 20 q^{17} - 48 q^{18} - 12 q^{19} + 4 q^{21} + 8 q^{22} + 16 q^{23} + 28 q^{24} + 12 q^{26} + 16 q^{27} - 12 q^{28} + 2 q^{29} + 12 q^{31} - 112 q^{32} + 14 q^{33} - 14 q^{36} + 16 q^{37} + 20 q^{38} + 4 q^{39} + 4 q^{41} - 32 q^{43} - 22 q^{44} - 4 q^{46} + 18 q^{47} + 48 q^{48} + 56 q^{49} - 44 q^{51} - 16 q^{52} + 20 q^{53} - 54 q^{54} + 12 q^{56} - 152 q^{57} - 32 q^{58} + 6 q^{59} - 4 q^{61} - 18 q^{62} - 6 q^{63} - 24 q^{64} - 74 q^{66} + 32 q^{67} - 124 q^{68} + 78 q^{69} - 8 q^{71} + 100 q^{72} + 48 q^{73} - 60 q^{74} + 52 q^{76} + 8 q^{77} - 124 q^{78} - 72 q^{81} - 44 q^{82} - 10 q^{83} + 12 q^{84} - 20 q^{86} - 26 q^{87} + 88 q^{88} - 38 q^{89} + 8 q^{91} + 96 q^{92} - 96 q^{93} - 88 q^{94} - 28 q^{96} + 90 q^{97} + 4 q^{98} - 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
176.1 −1.93486 + 1.40576i −0.850362 + 2.61714i 1.14949 − 3.53776i 0 −2.03374 − 6.25920i 1.00000 1.27104 + 3.91185i −3.69928 − 2.68768i 0
176.2 −1.82369 + 1.32499i 0.135168 − 0.416003i 0.952210 − 2.93060i 0 0.304695 + 0.937754i 1.00000 0.753300 + 2.31842i 2.27226 + 1.65090i 0
176.3 −1.21819 + 0.885063i 0.167360 − 0.515081i 0.0826041 − 0.254229i 0 0.252004 + 0.775589i 1.00000 −0.806229 − 2.48132i 2.18975 + 1.59095i 0
176.4 −0.771080 + 0.560222i 0.337689 − 1.03930i −0.337319 + 1.03816i 0 0.321853 + 0.990563i 1.00000 −0.910553 − 2.80240i 1.46094 + 1.06144i 0
176.5 −0.627580 + 0.455964i −0.536677 + 1.65172i −0.432080 + 1.32981i 0 −0.416318 − 1.28129i 1.00000 −0.814607 − 2.50710i −0.0131148 − 0.00952845i 0
176.6 −0.317350 + 0.230568i 0.860550 − 2.64850i −0.570485 + 1.75577i 0 0.337565 + 1.03892i 1.00000 −0.466216 − 1.43486i −3.84697 − 2.79498i 0
176.7 0.160894 − 0.116896i −0.477493 + 1.46957i −0.605812 + 1.86450i 0 0.0949619 + 0.292263i 1.00000 0.243394 + 0.749089i 0.495409 + 0.359936i 0
176.8 0.347070 − 0.252161i −0.297691 + 0.916200i −0.561162 + 1.72708i 0 0.127710 + 0.393052i 1.00000 0.505878 + 1.55693i 1.67625 + 1.21787i 0
176.9 0.875013 − 0.635734i −0.987513 + 3.03925i −0.256544 + 0.789562i 0 1.06807 + 3.28718i 1.00000 0.945922 + 2.91125i −5.83482 − 4.23925i 0
176.10 0.959091 − 0.696821i 0.536033 − 1.64974i −0.183737 + 0.565483i 0 −0.635468 − 1.95577i 1.00000 0.950501 + 2.92534i −0.00725735 − 0.00527277i 0
176.11 1.52910 − 1.11095i 0.561715 − 1.72878i 0.485887 − 1.49541i 0 −1.06168 − 3.26751i 1.00000 0.249767 + 0.768705i −0.246106 − 0.178807i 0
176.12 1.72620 − 1.25416i −0.523784 + 1.61204i 0.788824 − 2.42775i 0 1.11760 + 3.43962i 1.00000 −0.364416 − 1.12156i 0.102725 + 0.0746342i 0
176.13 2.10536 − 1.52964i 0.533289 − 1.64129i 1.47474 − 4.53877i 0 −1.38782 − 4.27126i 1.00000 −2.22946 − 6.86157i 0.0176014 + 0.0127882i 0
176.14 2.22607 − 1.61734i −0.694351 + 2.13699i 1.72159 − 5.29851i 0 1.91056 + 5.88011i 1.00000 −3.03652 − 9.34546i −1.65757 − 1.20429i 0
351.1 −0.825507 + 2.54065i −0.340724 − 0.247551i −4.15540 − 3.01908i 0 0.910210 − 0.661306i 1.00000 6.77832 − 4.92473i −0.872239 − 2.68448i 0
351.2 −0.793475 + 2.44207i 2.59187 + 1.88310i −3.71604 − 2.69986i 0 −6.65525 + 4.83532i 1.00000 5.38714 − 3.91399i 2.24466 + 6.90836i 0
351.3 −0.648406 + 1.99559i 0.616687 + 0.448050i −1.94391 − 1.41233i 0 −1.29399 + 0.940136i 1.00000 0.683774 − 0.496791i −0.747496 − 2.30056i 0
351.4 −0.591863 + 1.82157i −2.32470 − 1.68900i −1.34977 − 0.980668i 0 4.45253 − 3.23495i 1.00000 −0.513801 + 0.373298i 1.62449 + 4.99966i 0
351.5 −0.388594 + 1.19597i −0.515546 − 0.374566i 0.338697 + 0.246078i 0 0.648308 − 0.471023i 1.00000 −2.46062 + 1.78775i −0.801563 − 2.46696i 0
351.6 −0.317588 + 0.977435i 2.32169 + 1.68681i 0.763516 + 0.554727i 0 −2.38608 + 1.73359i 1.00000 −2.44761 + 1.77829i 1.61787 + 4.97930i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 176.14
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 875.2.h.e 56
5.b even 2 1 875.2.h.d 56
5.c odd 4 1 175.2.n.a ✓ 56
5.c odd 4 1 875.2.n.c 56
25.d even 5 1 inner 875.2.h.e 56
25.d even 5 1 4375.2.a.o 28
25.e even 10 1 875.2.h.d 56
25.e even 10 1 4375.2.a.p 28
25.f odd 20 1 175.2.n.a ✓ 56
25.f odd 20 1 875.2.n.c 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.n.a ✓ 56 5.c odd 4 1
175.2.n.a ✓ 56 25.f odd 20 1
875.2.h.d 56 5.b even 2 1
875.2.h.d 56 25.e even 10 1
875.2.h.e 56 1.a even 1 1 trivial
875.2.h.e 56 25.d even 5 1 inner
875.2.n.c 56 5.c odd 4 1
875.2.n.c 56 25.f odd 20 1
4375.2.a.o 28 25.d even 5 1
4375.2.a.p 28 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} - 4 T_{2}^{55} + 28 T_{2}^{54} - 100 T_{2}^{53} + 462 T_{2}^{52} - 1336 T_{2}^{51} + \cdots + 42025 \) acting on \(S_{2}^{\mathrm{new}}(875, [\chi])\). Copy content Toggle raw display