Properties

Label 35.6.f.a
Level $35$
Weight $6$
Character orbit 35.f
Analytic conductor $5.613$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [35,6,Mod(13,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.13");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 35.f (of order \(4\), 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 \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 4 q^{2} - 196 q^{7} - 368 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 4 q^{2} - 196 q^{7} - 368 q^{8} + 412 q^{11} + 2960 q^{15} - 7672 q^{16} + 3368 q^{18} - 84 q^{21} + 4604 q^{22} + 2008 q^{23} - 7000 q^{25} + 21280 q^{28} - 5260 q^{30} - 6960 q^{32} - 34860 q^{35} + 9848 q^{36} + 7696 q^{37} + 58660 q^{42} - 50252 q^{43} - 8848 q^{46} + 74180 q^{50} - 49868 q^{51} - 85676 q^{53} - 155848 q^{56} - 84020 q^{57} + 226692 q^{58} + 306160 q^{60} + 88032 q^{63} - 157640 q^{65} - 185604 q^{67} - 183820 q^{70} + 12712 q^{71} + 19784 q^{72} + 219716 q^{77} - 200740 q^{78} + 498268 q^{81} + 78860 q^{85} + 22552 q^{86} - 895072 q^{88} - 242284 q^{91} + 517160 q^{92} + 643580 q^{93} + 573820 q^{95} + 413868 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −7.19923 + 7.19923i −13.8307 + 13.8307i 71.6578i 8.68156 55.2235i 199.140i −83.2487 + 99.3813i 285.505 + 285.505i 139.575i 335.066 + 460.067i
13.2 −7.19923 + 7.19923i 13.8307 13.8307i 71.6578i −8.68156 + 55.2235i 199.140i −99.3813 + 83.2487i 285.505 + 285.505i 139.575i −335.066 460.067i
13.3 −5.71951 + 5.71951i −2.93091 + 2.93091i 33.4255i −55.3132 + 8.08990i 33.5267i 33.7470 125.172i 8.15308 + 8.15308i 225.820i 270.094 362.635i
13.4 −5.71951 + 5.71951i 2.93091 2.93091i 33.4255i 55.3132 8.08990i 33.5267i 125.172 33.7470i 8.15308 + 8.15308i 225.820i −270.094 + 362.635i
13.5 −3.69988 + 3.69988i −18.2186 + 18.2186i 4.62173i −9.39510 + 55.1066i 134.813i 44.6849 + 121.697i −135.496 135.496i 420.835i −169.127 238.649i
13.6 −3.69988 + 3.69988i 18.2186 18.2186i 4.62173i 9.39510 55.1066i 134.813i −121.697 44.6849i −135.496 135.496i 420.835i 169.127 + 238.649i
13.7 −2.47595 + 2.47595i −6.55592 + 6.55592i 19.7394i 52.0184 + 20.4716i 32.4643i −114.937 59.9710i −128.104 128.104i 157.040i −179.482 + 78.1083i
13.8 −2.47595 + 2.47595i 6.55592 6.55592i 19.7394i −52.0184 20.4716i 32.4643i 59.9710 + 114.937i −128.104 128.104i 157.040i 179.482 78.1083i
13.9 −0.295977 + 0.295977i −17.5765 + 17.5765i 31.8248i −16.1154 53.5284i 10.4045i 32.8881 125.401i −18.8906 18.8906i 374.868i 20.6129 + 11.0734i
13.10 −0.295977 + 0.295977i 17.5765 17.5765i 31.8248i 16.1154 + 53.5284i 10.4045i 125.401 32.8881i −18.8906 18.8906i 374.868i −20.6129 11.0734i
13.11 1.35053 1.35053i −1.76991 + 1.76991i 28.3521i 33.8544 44.4846i 4.78065i 33.8584 + 125.142i 81.5075 + 81.5075i 236.735i −14.3563 105.799i
13.12 1.35053 1.35053i 1.76991 1.76991i 28.3521i −33.8544 + 44.4846i 4.78065i −125.142 33.8584i 81.5075 + 81.5075i 236.735i 14.3563 + 105.799i
13.13 4.47623 4.47623i −10.4953 + 10.4953i 8.07328i 34.0119 + 44.3643i 93.9592i 122.053 43.7039i 107.102 + 107.102i 22.6955i 350.830 + 46.3396i
13.14 4.47623 4.47623i 10.4953 10.4953i 8.07328i −34.0119 44.3643i 93.9592i 43.7039 122.053i 107.102 + 107.102i 22.6955i −350.830 46.3396i
13.15 5.10578 5.10578i −14.8441 + 14.8441i 20.1381i −55.8880 1.23744i 151.582i −74.7452 + 105.925i 60.5645 + 60.5645i 197.695i −291.670 + 279.034i
13.16 5.10578 5.10578i 14.8441 14.8441i 20.1381i 55.8880 + 1.23744i 151.582i −105.925 + 74.7452i 60.5645 + 60.5645i 197.695i 291.670 279.034i
13.17 7.45800 7.45800i −9.59993 + 9.59993i 79.2434i 39.6521 39.4045i 143.193i −88.8292 94.4265i −352.341 352.341i 58.6826i 1.84648 589.603i
13.18 7.45800 7.45800i 9.59993 9.59993i 79.2434i −39.6521 + 39.4045i 143.193i 94.4265 + 88.8292i −352.341 352.341i 58.6826i −1.84648 + 589.603i
27.1 −7.19923 7.19923i −13.8307 13.8307i 71.6578i 8.68156 + 55.2235i 199.140i −83.2487 99.3813i 285.505 285.505i 139.575i 335.066 460.067i
27.2 −7.19923 7.19923i 13.8307 + 13.8307i 71.6578i −8.68156 55.2235i 199.140i −99.3813 83.2487i 285.505 285.505i 139.575i −335.066 + 460.067i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.6.f.a 36
5.c odd 4 1 inner 35.6.f.a 36
7.b odd 2 1 inner 35.6.f.a 36
35.f even 4 1 inner 35.6.f.a 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.f.a 36 1.a even 1 1 trivial
35.6.f.a 36 5.c odd 4 1 inner
35.6.f.a 36 7.b odd 2 1 inner
35.6.f.a 36 35.f even 4 1 inner

Hecke kernels

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