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Properties

Label 175.4.x.a
Level $175$
Weight $4$
Character orbit 175.x
Analytic conductor $10.325$
Analytic rank $0$
Dimension $928$
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(3,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([21, 10]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.3");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.x (of order \(60\), degree \(16\), 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: \(928\)
Relative dimension: \(58\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 928 q - 8 q^{2} - 24 q^{3} - 10 q^{4} - 24 q^{7} + 84 q^{8} - 10 q^{9} - 96 q^{10} - 6 q^{11} - 72 q^{12} - 20 q^{14} - 368 q^{15} - 1670 q^{16} + 120 q^{17} - 14 q^{18} - 30 q^{19} - 12 q^{21} - 880 q^{22}+ \cdots - 19478 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −2.98654 + 4.59886i −1.16114 + 3.02487i −8.97624 − 20.1610i 9.73544 + 5.49738i −10.4432 − 14.3738i 7.57257 + 16.9014i 76.1974 + 12.0685i 12.2633 + 11.0419i −54.3569 + 28.3538i
3.2 −2.93952 + 4.52647i 2.08359 − 5.42794i −8.59424 − 19.3030i 5.69869 − 9.61899i 18.4447 + 25.3869i −18.5027 − 0.805399i 69.9913 + 11.0855i −5.05630 − 4.55272i 26.7886 + 54.0702i
3.3 −2.78034 + 4.28135i 0.882639 − 2.29935i −7.34575 − 16.4988i −10.5665 + 3.65353i 7.39030 + 10.1719i −6.43718 − 17.3656i 50.7243 + 8.03394i 15.5569 + 14.0075i 13.7365 − 55.3971i
3.4 −2.76823 + 4.26271i −3.25777 + 8.48679i −7.25365 − 16.2920i −10.0021 + 4.99589i −27.1584 − 37.3803i 14.5748 − 11.4270i 49.3668 + 7.81893i −41.3476 − 37.2295i 6.39202 − 56.4656i
3.5 −2.71831 + 4.18582i 3.28166 − 8.54901i −6.87804 − 15.4483i −6.26616 + 9.25933i 26.8641 + 36.9752i 12.3275 + 13.8214i 43.9240 + 6.95687i −42.2513 − 38.0433i −21.7246 − 51.3987i
3.6 −2.62864 + 4.04775i 0.173128 − 0.451014i −6.22064 − 13.9718i −6.75712 − 8.90737i 1.37050 + 1.88633i 16.2876 + 8.81554i 34.7703 + 5.50708i 19.8915 + 17.9104i 53.8168 − 3.93686i
3.7 −2.46147 + 3.79033i −1.42743 + 3.71858i −5.05389 − 11.3512i 2.23871 + 10.9539i −10.5811 − 14.5636i −18.1267 − 3.79783i 19.7544 + 3.12879i 8.27464 + 7.45052i −47.0295 − 18.4773i
3.8 −2.44758 + 3.76895i −2.50628 + 6.52907i −4.96040 − 11.1412i −4.07235 − 10.4123i −18.4734 − 25.4265i −15.2656 + 10.4863i 18.6227 + 2.94955i −16.2825 − 14.6608i 49.2108 + 10.1365i
3.9 −2.38969 + 3.67979i 1.74074 − 4.53478i −4.57638 − 10.2787i 10.9198 + 2.39967i 12.5272 + 17.2422i 14.0149 − 12.1072i 14.0907 + 2.23174i 2.53089 + 2.27882i −34.9251 + 34.4481i
3.10 −2.26197 + 3.48313i −1.55172 + 4.04237i −3.76181 − 8.44916i 8.63834 − 7.09782i −10.5702 − 14.5486i 3.60941 − 18.1651i 5.12243 + 0.811313i 6.13196 + 5.52124i 5.18297 + 46.1436i
3.11 −1.98207 + 3.05213i 0.456252 − 1.18858i −2.13297 − 4.79073i −9.54840 + 5.81619i 2.72336 + 3.74839i −7.21114 + 17.0587i −9.90591 − 1.56894i 18.8604 + 16.9819i 1.17388 − 40.6711i
3.12 −1.97283 + 3.03788i 2.51318 − 6.54707i −2.08281 − 4.67806i 9.50586 + 5.88546i 14.9312 + 20.5510i −16.0333 + 9.27000i −10.3009 − 1.63150i −16.4831 − 14.8414i −36.6327 + 17.2667i
3.13 −1.96169 + 3.02073i 3.41104 − 8.88607i −2.02272 − 4.54311i −3.76740 − 10.5265i 20.1510 + 27.7355i 11.4301 − 14.5724i −10.7683 − 1.70553i −47.2621 − 42.5550i 39.1881 + 9.26933i
3.14 −1.78858 + 2.75417i −3.22364 + 8.39787i −1.13255 − 2.54375i 11.1404 − 0.944160i −17.3634 − 23.8987i 13.6862 + 12.4775i −16.9168 − 2.67935i −40.0675 − 36.0769i −17.3251 + 32.3713i
3.15 −1.67008 + 2.57170i 1.53287 − 3.99327i −0.570578 − 1.28154i 4.62112 − 10.1806i 7.70948 + 10.6112i 5.95919 + 17.5353i −19.9805 − 3.16461i 6.46837 + 5.82415i 18.4639 + 28.8866i
3.16 −1.55621 + 2.39635i 1.14091 − 2.97218i −0.0668230 − 0.150087i −9.96831 − 5.06288i 5.34689 + 7.35936i −12.1399 − 13.9866i −22.1135 − 3.50243i 12.5327 + 11.2845i 27.6452 − 16.0087i
3.17 −1.55612 + 2.39621i −0.0369693 + 0.0963083i −0.0664387 − 0.149224i 1.07427 + 11.1286i −0.173247 − 0.238454i 18.4311 − 1.81536i −22.1149 − 3.50265i 20.0570 + 18.0594i −28.3382 − 14.7433i
3.18 −1.25359 + 1.93036i −1.53819 + 4.00713i 1.09910 + 2.46861i −10.5630 − 3.66383i −5.80693 − 7.99256i 13.9174 − 12.2190i −24.3299 − 3.85348i 6.37387 + 5.73906i 20.3141 − 15.7974i
3.19 −1.25167 + 1.92741i −3.02258 + 7.87409i 1.10568 + 2.48340i 2.53086 + 10.8901i −11.3933 − 15.6815i −8.19908 − 16.6065i −24.3295 − 3.85341i −32.8005 − 29.5337i −24.1575 − 8.75286i
3.20 −1.13801 + 1.75238i −2.76584 + 7.20527i 1.47812 + 3.31992i −10.3242 + 4.29074i −9.47881 − 13.0465i −2.38757 + 18.3657i −24.0099 − 3.80279i −24.2011 − 21.7908i 4.23004 − 22.9748i
See next 80 embeddings (of 928 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
25.f odd 20 1 inner
175.x even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.4.x.a ✓ 928
7.d odd 6 1 inner 175.4.x.a ✓ 928
25.f odd 20 1 inner 175.4.x.a ✓ 928
175.x even 60 1 inner 175.4.x.a ✓ 928
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.4.x.a ✓ 928 1.a even 1 1 trivial
175.4.x.a ✓ 928 7.d odd 6 1 inner
175.4.x.a ✓ 928 25.f odd 20 1 inner
175.4.x.a ✓ 928 175.x even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(175, [\chi])\).