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Properties

Label 35.6.j.a
Level $35$
Weight $6$
Character orbit 35.j
Analytic conductor $5.613$
Analytic rank $0$
Dimension $36$
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] = [35,6,Mod(4,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.4");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 35.j (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: \(5.61343369345\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 254 q^{4} - 12 q^{5} + 72 q^{6} + 832 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 254 q^{4} - 12 q^{5} + 72 q^{6} + 832 q^{9} - 84 q^{10} - 360 q^{11} + 158 q^{14} + 688 q^{15} - 2630 q^{16} + 3316 q^{19} - 1096 q^{20} + 1594 q^{21} + 3300 q^{24} - 404 q^{25} - 13570 q^{26} + 16004 q^{29} + 6482 q^{30} - 14860 q^{31} - 38880 q^{34} - 11216 q^{35} - 39756 q^{36} - 1988 q^{39} + 17098 q^{40} + 111852 q^{41} + 7670 q^{44} + 30876 q^{45} - 4818 q^{46} - 28142 q^{49} + 111876 q^{50} - 52732 q^{51} + 20724 q^{54} - 180520 q^{55} + 178668 q^{56} + 89124 q^{59} + 6156 q^{60} - 23122 q^{61} - 73644 q^{64} + 58748 q^{65} - 286316 q^{66} - 110564 q^{69} - 350750 q^{70} - 124264 q^{71} - 102486 q^{74} - 68328 q^{75} + 417020 q^{76} + 113556 q^{79} + 263952 q^{80} + 225014 q^{81} + 94616 q^{84} + 221556 q^{85} - 427796 q^{86} + 302422 q^{89} - 92524 q^{90} + 508268 q^{91} + 305370 q^{94} - 216808 q^{95} - 169668 q^{96} - 120816 q^{99}+O(q^{100}) \) 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 −9.26620 + 5.34984i −6.62610 − 3.82558i 41.2416 − 71.4326i 27.5440 + 48.6449i 81.8650 102.235 − 79.7187i 540.155i −92.2299 − 159.747i −515.471 − 303.397i
4.2 −8.10526 + 4.67957i 16.9158 + 9.76637i 27.7968 − 48.1454i −7.12267 − 55.4461i −182.810 −26.0548 − 126.997i 220.815i 69.2638 + 119.968i 317.195 + 416.074i
4.3 −7.42641 + 4.28764i −18.5335 − 10.7003i 20.7677 − 35.9707i 2.14959 − 55.8604i 183.516 −115.368 + 59.1374i 81.7681i 107.493 + 186.183i 223.545 + 424.058i
4.4 −6.93682 + 4.00497i 6.61661 + 3.82010i 16.0796 − 27.8508i −50.5682 + 23.8298i −61.1976 −6.10907 + 129.498i 1.27596i −92.3137 − 159.892i 255.345 − 367.827i
4.5 −4.66496 + 2.69332i 15.4088 + 8.89629i −1.49207 + 2.58434i 55.0416 − 9.76832i −95.8422 79.8400 + 102.140i − 188.447i 36.7880 + 63.7187i −230.458 + 193.814i
4.6 −3.93557 + 2.27220i −21.4808 − 12.4020i −5.67417 + 9.82796i −51.2816 + 22.2531i 112.719 125.933 − 30.7878i − 196.993i 186.118 + 322.365i 151.259 − 204.101i
4.7 −3.70291 + 2.13788i −7.17064 − 4.13997i −6.85895 + 11.8801i 43.3948 + 35.2404i 35.4030 −117.071 − 55.6909i − 195.479i −87.2213 − 151.072i −236.027 − 37.7194i
4.8 −2.03171 + 1.17301i 21.8875 + 12.6367i −13.2481 + 22.9464i −30.5927 + 46.7877i −59.2919 −33.3578 − 125.277i − 137.233i 197.874 + 342.729i 7.27310 − 130.944i
4.9 −1.15385 + 0.666174i −2.36481 − 1.36533i −15.1124 + 26.1755i −0.183573 − 55.9014i 3.63818 102.879 − 78.8855i − 82.9052i −117.772 − 203.987i 37.4519 + 64.3794i
4.10 1.15385 − 0.666174i 2.36481 + 1.36533i −15.1124 + 26.1755i −48.3202 − 28.1097i 3.63818 −102.879 + 78.8855i 82.9052i −117.772 − 203.987i −74.4801 − 0.244583i
4.11 2.03171 − 1.17301i −21.8875 − 12.6367i −13.2481 + 22.9464i 55.8157 − 3.10020i −59.2919 33.3578 + 125.277i 137.233i 197.874 + 342.729i 109.764 − 71.7708i
4.12 3.70291 − 2.13788i 7.17064 + 4.13997i −6.85895 + 11.8801i 8.82168 + 55.2012i 35.4030 117.071 + 55.6909i 195.479i −87.2213 − 151.072i 150.679 + 185.546i
4.13 3.93557 − 2.27220i 21.4808 + 12.4020i −5.67417 + 9.82796i 44.9126 − 33.2846i 112.719 −125.933 + 30.7878i 196.993i 186.118 + 322.365i 101.127 − 233.044i
4.14 4.66496 − 2.69332i −15.4088 − 8.89629i −1.49207 + 2.58434i −35.9804 + 42.7833i −95.8422 −79.8400 − 102.140i 188.447i 36.7880 + 63.7187i −52.6184 + 296.489i
4.15 6.93682 − 4.00497i −6.61661 − 3.82010i 16.0796 − 27.8508i 45.9213 − 31.8784i −61.1976 6.10907 − 129.498i − 1.27596i −92.3137 − 159.892i 190.875 − 405.049i
4.16 7.42641 − 4.28764i 18.5335 + 10.7003i 20.7677 − 35.9707i −49.4513 − 26.0686i 183.516 115.368 − 59.1374i − 81.7681i 107.493 + 186.183i −479.018 + 18.4333i
4.17 8.10526 − 4.67957i −16.9158 − 9.76637i 27.7968 − 48.1454i −44.4564 − 33.8915i −182.810 26.0548 + 126.997i − 220.815i 69.2638 + 119.968i −518.928 − 66.6621i
4.18 9.26620 − 5.34984i 6.62610 + 3.82558i 41.2416 − 71.4326i 28.3557 + 48.1763i 81.8650 −102.235 + 79.7187i − 540.155i −92.2299 − 159.747i 520.485 + 294.712i
9.1 −9.26620 − 5.34984i −6.62610 + 3.82558i 41.2416 + 71.4326i 27.5440 − 48.6449i 81.8650 102.235 + 79.7187i − 540.155i −92.2299 + 159.747i −515.471 + 303.397i
9.2 −8.10526 − 4.67957i 16.9158 − 9.76637i 27.7968 + 48.1454i −7.12267 + 55.4461i −182.810 −26.0548 + 126.997i − 220.815i 69.2638 − 119.968i 317.195 − 416.074i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.6.j.a ✓ 36
5.b even 2 1 inner 35.6.j.a ✓ 36
7.c even 3 1 inner 35.6.j.a ✓ 36
7.c even 3 1 245.6.b.f 18
7.d odd 6 1 245.6.b.e 18
35.i odd 6 1 245.6.b.e 18
35.j even 6 1 inner 35.6.j.a ✓ 36
35.j even 6 1 245.6.b.f 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.j.a ✓ 36 1.a even 1 1 trivial
35.6.j.a ✓ 36 5.b even 2 1 inner
35.6.j.a ✓ 36 7.c even 3 1 inner
35.6.j.a ✓ 36 35.j even 6 1 inner
245.6.b.e 18 7.d odd 6 1
245.6.b.e 18 35.i odd 6 1
245.6.b.f 18 7.c even 3 1
245.6.b.f 18 35.j even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(35, [\chi])\).