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Properties

Label 115.4.j.a
Level $115$
Weight $4$
Character orbit 115.j
Analytic conductor $6.785$
Analytic rank $0$
Dimension $340$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,4,Mod(4,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 115.j (of order \(22\), degree \(10\), 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: \(6.78521965066\)
Analytic rank: \(0\)
Dimension: \(340\)
Relative dimension: \(34\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 340 q + 122 q^{4} - 25 q^{5} - 46 q^{6} + 288 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 340 q + 122 q^{4} - 25 q^{5} - 46 q^{6} + 288 q^{9} - 25 q^{10} - 14 q^{11} - 258 q^{14} + 363 q^{15} - 462 q^{16} - 166 q^{19} - 535 q^{20} + 10 q^{21} - 152 q^{24} - 497 q^{25} + 122 q^{26} - 78 q^{29} + 1779 q^{30} + 58 q^{31} + 1628 q^{34} - 127 q^{35} - 1886 q^{36} + 1046 q^{39} - 661 q^{40} - 290 q^{41} - 3910 q^{44} + 1324 q^{45} - 206 q^{46} - 726 q^{49} - 131 q^{50} + 1082 q^{51} + 3914 q^{54} + 1337 q^{55} - 3530 q^{56} + 4668 q^{59} + 945 q^{60} - 594 q^{61} - 4346 q^{64} - 5487 q^{65} + 7444 q^{66} - 6950 q^{69} - 6974 q^{70} + 5438 q^{71} - 4954 q^{74} - 7913 q^{75} + 17316 q^{76} - 2142 q^{79} + 428 q^{80} - 3792 q^{81} + 13278 q^{84} + 9743 q^{85} - 12530 q^{86} - 1218 q^{89} + 15918 q^{90} - 15516 q^{91} + 2494 q^{94} + 95 q^{95} - 44100 q^{96} - 10066 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −4.93951 + 2.25580i 2.15470 + 7.33824i 14.0712 − 16.2391i 7.60207 − 8.19808i −27.1968 − 31.3867i −3.85374 + 0.554083i −20.6339 + 70.2727i −26.4932 + 17.0262i −19.0573 + 57.6432i
4.2 −4.74255 + 2.16585i −0.951207 − 3.23951i 12.5620 − 14.4973i −8.81948 + 6.87144i 11.5274 + 13.3034i 17.7098 − 2.54629i −16.4259 + 55.9415i 13.1242 − 8.43440i 26.9443 − 51.6898i
4.3 −4.57634 + 2.08995i −1.90357 − 6.48296i 11.3361 − 13.0826i 10.7939 − 2.91399i 22.2604 + 25.6899i 6.56594 − 0.944040i −13.1970 + 44.9449i −15.6914 + 10.0842i −43.3066 + 35.8941i
4.4 −4.26424 + 1.94741i −0.504685 − 1.71880i 9.15245 − 10.5625i −8.32418 − 7.46378i 5.49931 + 6.34654i −30.2094 + 4.34345i −7.89290 + 26.8808i 20.0143 − 12.8624i 50.0314 + 15.6167i
4.5 −4.05855 + 1.85348i 0.705697 + 2.40338i 7.79759 − 8.99890i 7.16321 + 8.58419i −7.31873 − 8.44627i 6.68222 − 0.960759i −4.91150 + 16.7270i 17.4356 − 11.2052i −44.9829 − 21.5626i
4.6 −3.66922 + 1.67567i 1.43026 + 4.87101i 5.41638 − 6.25083i −9.79971 − 5.38198i −13.4102 − 15.4762i 22.0763 − 3.17409i −0.308014 + 1.04900i 1.03271 − 0.663679i 44.9757 + 3.32653i
4.7 −3.60562 + 1.64663i 2.19909 + 7.48942i 5.05023 − 5.82828i −4.03696 + 10.4261i −20.2614 − 23.3829i −15.3985 + 2.21397i 0.321717 − 1.09567i −28.5415 + 18.3425i −2.61215 − 44.2399i
4.8 −3.14604 + 1.43675i −2.78164 − 9.47338i 2.59445 − 2.99416i −6.86167 + 8.82709i 22.3620 + 25.8072i −20.6381 + 2.96731i 3.93477 − 13.4006i −59.2936 + 38.1057i 8.90478 − 37.6289i
4.9 −2.76142 + 1.26110i −1.90107 − 6.47446i 0.796197 − 0.918860i −0.923165 − 11.1422i 13.4146 + 15.4813i 14.2011 − 2.04181i 5.80231 − 19.7609i −15.5907 + 10.0196i 16.6006 + 29.6040i
4.10 −2.57569 + 1.17628i −0.733725 − 2.49884i 0.0116440 − 0.0134379i 5.73082 + 9.59988i 4.82917 + 5.57316i −1.07107 + 0.153997i 6.36778 − 21.6867i 17.0080 − 10.9304i −26.0529 − 17.9852i
4.11 −2.53535 + 1.15786i 0.328991 + 1.12044i −0.151507 + 0.174848i 9.36420 − 6.10833i −2.13142 − 2.45979i −27.9526 + 4.01898i 6.46370 − 22.0133i 21.5667 − 13.8601i −16.6690 + 26.3292i
4.12 −2.01658 + 0.920943i 1.25855 + 4.28623i −2.02041 + 2.33168i 9.30656 − 6.19580i −6.48534 − 7.48449i 28.4900 − 4.09624i 6.92362 − 23.5797i 5.92605 − 3.80844i −13.0615 + 21.0652i
4.13 −1.47551 + 0.673845i 2.53930 + 8.64808i −3.51581 + 4.05746i −5.00373 − 9.99814i −9.57424 − 11.0493i −16.5891 + 2.38515i 6.10952 − 20.8071i −45.6273 + 29.3229i 14.1203 + 11.3807i
4.14 −1.19431 + 0.545423i −0.846860 − 2.88414i −4.11000 + 4.74319i −10.8354 − 2.75587i 2.58449 + 2.98266i 9.64830 − 1.38722i 5.28079 − 17.9847i 15.1128 − 9.71238i 14.4439 − 2.61849i
4.15 −1.05762 + 0.482997i 0.0756092 + 0.257501i −4.35362 + 5.02435i −9.24453 + 6.28798i −0.204338 − 0.235818i −10.0278 + 1.44178i 4.79825 − 16.3413i 22.6533 − 14.5584i 6.74008 − 11.1154i
4.16 −0.355756 + 0.162468i −1.93102 − 6.57646i −5.13872 + 5.93040i 8.88702 + 6.78387i 1.75544 + 2.02588i 14.3430 − 2.06222i 1.74611 − 5.94670i −16.8072 + 10.8013i −4.26377 − 0.969542i
4.17 −0.354528 + 0.161907i 2.06990 + 7.04943i −5.13941 + 5.93120i −3.60282 + 10.5839i −1.87519 − 2.16409i 29.9575 − 4.30723i 1.74020 − 5.92657i −22.6962 + 14.5859i −0.436316 − 4.33562i
4.18 0.354528 − 0.161907i −2.06990 − 7.04943i −5.13941 + 5.93120i 8.13082 − 7.67397i −1.87519 − 2.16409i −29.9575 + 4.30723i −1.74020 + 5.92657i −22.6962 + 14.5859i 1.64013 − 4.03708i
4.19 0.355756 − 0.162468i 1.93102 + 6.57646i −5.13872 + 5.93040i 9.86263 + 5.26580i 1.75544 + 2.02588i −14.3430 + 2.06222i −1.74611 + 5.94670i −16.8072 + 10.8013i 4.36421 + 0.270973i
4.20 1.05762 − 0.482997i −0.0756092 − 0.257501i −4.35362 + 5.02435i 1.87944 − 11.0212i −0.204338 − 0.235818i 10.0278 − 1.44178i −4.79825 + 16.3413i 22.6533 − 14.5584i −3.33550 − 12.5640i
See next 80 embeddings (of 340 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.c even 11 1 inner
115.j even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.4.j.a ✓ 340
5.b even 2 1 inner 115.4.j.a ✓ 340
23.c even 11 1 inner 115.4.j.a ✓ 340
115.j even 22 1 inner 115.4.j.a ✓ 340
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.j.a ✓ 340 1.a even 1 1 trivial
115.4.j.a ✓ 340 5.b even 2 1 inner
115.4.j.a ✓ 340 23.c even 11 1 inner
115.4.j.a ✓ 340 115.j even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(115, [\chi])\).