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Properties

Label 105.4.s.a
Level $105$
Weight $4$
Character orbit 105.s
Analytic conductor $6.195$
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] = [105,4,Mod(26,105)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(105, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("105.26");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.s (of order \(6\), degree \(2\), 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: \(6.19520055060\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{3} + 64 q^{4} - 80 q^{5} - 28 q^{6} + 46 q^{7} + 100 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{3} + 64 q^{4} - 80 q^{5} - 28 q^{6} + 46 q^{7} + 100 q^{9} + 36 q^{11} + 246 q^{12} + 18 q^{14} + 20 q^{15} - 376 q^{16} - 72 q^{17} - 442 q^{18} - 198 q^{19} - 640 q^{20} - 218 q^{21} + 204 q^{22} + 72 q^{23} - 50 q^{24} - 400 q^{25} - 312 q^{26} + 508 q^{27} + 350 q^{28} + 40 q^{30} + 510 q^{31} + 810 q^{32} + 290 q^{33} - 70 q^{35} - 612 q^{36} - 658 q^{37} - 192 q^{38} - 648 q^{39} - 1404 q^{41} + 1892 q^{42} + 332 q^{43} + 2034 q^{44} - 490 q^{45} - 468 q^{46} + 408 q^{47} + 2810 q^{48} + 980 q^{49} - 888 q^{51} + 3378 q^{52} + 1152 q^{53} + 2714 q^{54} - 3354 q^{56} - 816 q^{57} - 1080 q^{58} - 48 q^{59} - 420 q^{60} - 1662 q^{61} - 2076 q^{62} + 874 q^{63} - 1952 q^{64} + 870 q^{65} - 1892 q^{66} - 1298 q^{67} + 1182 q^{68} + 2450 q^{69} - 450 q^{70} - 2708 q^{72} + 378 q^{73} + 2898 q^{74} - 50 q^{75} - 3528 q^{77} - 1896 q^{78} - 326 q^{79} - 1880 q^{80} - 3308 q^{81} - 2916 q^{82} - 1536 q^{83} + 1380 q^{84} + 720 q^{85} + 5202 q^{86} - 1090 q^{87} + 1668 q^{88} - 1590 q^{89} + 910 q^{90} + 2082 q^{91} - 4950 q^{93} - 1152 q^{94} + 990 q^{95} + 7416 q^{96} - 7830 q^{98} + 3128 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −4.57212 − 2.63971i 4.83065 − 1.91438i 9.93617 + 17.2100i −2.50000 + 4.33013i −27.1397 − 3.99877i −17.5731 + 5.84690i − 62.6791i 19.6703 − 18.4954i 22.8606 − 13.1986i
26.2 −4.56345 − 2.63471i −4.97118 − 1.51242i 9.88336 + 17.1185i −2.50000 + 4.33013i 18.7009 + 19.9994i 12.9395 − 13.2503i − 62.0037i 22.4252 + 15.0370i 22.8172 − 13.1735i
26.3 −3.31734 − 1.91526i 4.42611 + 2.72205i 3.33648 + 5.77895i −2.50000 + 4.33013i −9.46943 − 17.5071i 8.94735 − 16.2156i 5.08329i 12.1808 + 24.0962i 16.5867 − 9.57632i
26.4 −3.06378 − 1.76887i −0.433556 − 5.17803i 2.25781 + 3.91065i −2.50000 + 4.33013i −7.83096 + 16.6312i 5.77785 + 17.5959i 12.3268i −26.6241 + 4.48993i 15.3189 − 8.84436i
26.5 −2.51429 − 1.45163i −2.26232 + 4.67781i 0.214449 + 0.371437i −2.50000 + 4.33013i 12.4786 − 8.47733i 17.9959 + 4.37591i 21.9809i −16.7638 − 21.1654i 12.5715 − 7.25814i
26.6 −1.35336 − 0.781362i −5.18197 + 0.383668i −2.77895 − 4.81328i −2.50000 + 4.33013i 7.31285 + 3.52975i −17.7664 + 5.23007i 21.1872i 26.7056 − 3.97631i 6.76679 − 3.90681i
26.7 −1.20225 − 0.694119i 3.08234 − 4.18320i −3.03640 − 5.25919i −2.50000 + 4.33013i −6.60938 + 2.88975i −11.1211 − 14.8095i 19.5364i −7.99841 − 25.7881i 6.01125 − 3.47060i
26.8 −0.133140 − 0.0768685i 2.47104 + 4.57099i −3.98818 − 6.90773i −2.50000 + 4.33013i 0.0223714 − 0.798528i −15.0457 + 10.7995i 2.45616i −14.7880 + 22.5902i 0.665701 − 0.384343i
26.9 0.155491 + 0.0897728i 5.17065 + 0.514148i −3.98388 − 6.90029i −2.50000 + 4.33013i 0.757834 + 0.544130i 18.1774 + 3.54688i − 2.86694i 26.4713 + 5.31696i −0.777456 + 0.448864i
26.10 0.815639 + 0.470909i −4.77022 − 2.06035i −3.55649 − 6.16002i −2.50000 + 4.33013i −2.92054 − 3.92684i 13.9941 + 12.1312i − 14.2337i 18.5099 + 19.6566i −4.07820 + 2.35455i
26.11 1.48940 + 0.859905i −4.11977 + 3.16663i −2.52113 − 4.36672i −2.50000 + 4.33013i −8.85898 + 1.17376i 2.28179 − 18.3792i − 22.4302i 6.94494 − 26.0915i −7.44700 + 4.29953i
26.12 2.43022 + 1.40309i −2.49879 − 4.55588i −0.0626967 − 0.108594i −2.50000 + 4.33013i 0.319698 − 14.5778i −12.1846 − 13.9476i − 22.8013i −14.5121 + 22.7684i −12.1511 + 7.01543i
26.13 3.53626 + 2.04166i −2.57442 + 4.51357i 4.33676 + 7.51148i −2.50000 + 4.33013i −18.3190 + 10.7051i 0.627588 + 18.5096i 2.75017i −13.7447 − 23.2397i −17.6813 + 10.2083i
26.14 3.64936 + 2.10696i 4.80130 − 1.98684i 4.87855 + 8.44990i −2.50000 + 4.33013i 21.7079 + 2.86543i 16.7999 − 7.79504i 7.40430i 19.1049 − 19.0788i −18.2468 + 10.5348i
26.15 3.95258 + 2.28202i 3.88244 + 3.45350i 6.41528 + 11.1116i −2.50000 + 4.33013i 7.46472 + 22.5101i −15.3726 − 10.3287i 22.0469i 3.14674 + 26.8160i −19.7629 + 11.4101i
26.16 4.69077 + 2.70822i −2.85230 − 4.34332i 10.6689 + 18.4790i −2.50000 + 4.33013i −1.61684 − 28.0981i 14.5222 + 11.4937i 72.2429i −10.7288 + 24.7769i −23.4538 + 13.5411i
101.1 −4.57212 + 2.63971i 4.83065 + 1.91438i 9.93617 − 17.2100i −2.50000 − 4.33013i −27.1397 + 3.99877i −17.5731 − 5.84690i 62.6791i 19.6703 + 18.4954i 22.8606 + 13.1986i
101.2 −4.56345 + 2.63471i −4.97118 + 1.51242i 9.88336 − 17.1185i −2.50000 − 4.33013i 18.7009 − 19.9994i 12.9395 + 13.2503i 62.0037i 22.4252 − 15.0370i 22.8172 + 13.1735i
101.3 −3.31734 + 1.91526i 4.42611 − 2.72205i 3.33648 − 5.77895i −2.50000 − 4.33013i −9.46943 + 17.5071i 8.94735 + 16.2156i − 5.08329i 12.1808 − 24.0962i 16.5867 + 9.57632i
101.4 −3.06378 + 1.76887i −0.433556 + 5.17803i 2.25781 − 3.91065i −2.50000 − 4.33013i −7.83096 − 16.6312i 5.77785 − 17.5959i − 12.3268i −26.6241 − 4.48993i 15.3189 + 8.84436i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.4.s.a ✓ 32
3.b odd 2 1 105.4.s.b yes 32
7.d odd 6 1 105.4.s.b yes 32
21.g even 6 1 inner 105.4.s.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.s.a ✓ 32 1.a even 1 1 trivial
105.4.s.a ✓ 32 21.g even 6 1 inner
105.4.s.b yes 32 3.b odd 2 1
105.4.s.b yes 32 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 96 T_{2}^{30} + 5598 T_{2}^{28} - 162 T_{2}^{27} - 211216 T_{2}^{26} + 9126 T_{2}^{25} + \cdots + 16057958400 \) acting on \(S_{4}^{\mathrm{new}}(105, [\chi])\). Copy content Toggle raw display