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Properties

Label 153.2.r.a
Level $153$
Weight $2$
Character orbit 153.r
Analytic conductor $1.222$
Analytic rank $0$
Dimension $128$
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] = [153,2,Mod(25,153)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(153, base_ring=CyclotomicField(24))
 
chi = DirichletCharacter(H, H._module([16, 15]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("153.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 153.r (of order \(24\), degree \(8\), 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: \(1.22171115093\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(16\) over \(\Q(\zeta_{24})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 4 q^{2} - 8 q^{3} - 4 q^{5} - 8 q^{6} - 4 q^{7} - 32 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 4 q^{2} - 8 q^{3} - 4 q^{5} - 8 q^{6} - 4 q^{7} - 32 q^{8} - 20 q^{9} - 16 q^{10} - 44 q^{12} - 4 q^{14} - 4 q^{15} + 32 q^{16} - 16 q^{17} - 16 q^{19} - 36 q^{20} - 4 q^{22} + 8 q^{23} + 28 q^{24} - 4 q^{25} - 32 q^{26} - 32 q^{27} - 48 q^{28} - 4 q^{29} - 4 q^{31} + 28 q^{32} - 32 q^{33} + 20 q^{34} + 96 q^{35} - 44 q^{36} - 16 q^{37} - 28 q^{39} + 12 q^{40} + 20 q^{41} + 48 q^{42} - 16 q^{43} + 16 q^{44} - 4 q^{45} + 32 q^{46} - 24 q^{48} - 4 q^{49} - 32 q^{50} - 40 q^{51} - 72 q^{52} + 16 q^{53} + 88 q^{54} - 76 q^{56} + 64 q^{57} - 4 q^{58} + 56 q^{59} - 8 q^{60} - 4 q^{61} - 16 q^{62} + 84 q^{63} - 52 q^{65} + 176 q^{66} - 8 q^{67} + 12 q^{68} - 112 q^{69} - 12 q^{70} + 48 q^{71} - 16 q^{73} + 52 q^{74} - 16 q^{75} - 4 q^{76} + 8 q^{77} - 72 q^{78} - 4 q^{79} - 8 q^{80} - 136 q^{82} + 44 q^{83} - 56 q^{84} - 52 q^{85} - 128 q^{86} - 52 q^{87} + 200 q^{90} - 24 q^{91} + 76 q^{92} - 4 q^{93} - 20 q^{94} - 28 q^{95} + 180 q^{96} + 32 q^{97} + 44 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −0.692433 − 2.58420i 1.38163 − 1.04456i −4.46656 + 2.57877i −0.381054 − 2.89439i −3.65603 − 2.84711i −0.322275 + 2.44792i 5.97331 + 5.97331i 0.817792 − 2.88638i −7.21583 + 2.98889i
25.2 −0.620919 − 2.31730i −0.314680 + 1.70323i −3.25229 + 1.87771i 0.221301 + 1.68095i 4.14227 − 0.328356i −0.520754 + 3.95552i 2.97786 + 2.97786i −2.80195 − 1.07194i 3.75785 − 1.55655i
25.3 −0.507788 − 1.89509i 1.59887 + 0.666048i −1.60147 + 0.924611i 0.267635 + 2.03289i 0.450336 − 3.36821i 0.492209 − 3.73870i −0.209174 − 0.209174i 2.11276 + 2.12985i 3.71662 − 1.53947i
25.4 −0.433302 − 1.61711i −1.05547 − 1.37331i −0.695232 + 0.401392i −0.252374 − 1.91697i −1.76346 + 2.30186i −0.100213 + 0.761197i −1.41727 − 1.41727i −0.771980 + 2.89897i −2.99059 + 1.23874i
25.5 −0.414826 − 1.54815i −1.42086 + 0.990533i −0.492643 + 0.284428i −0.138734 − 1.05379i 2.12291 + 1.78881i 0.285047 − 2.16514i −1.62195 − 1.62195i 1.03769 − 2.81482i −1.57388 + 0.651922i
25.6 −0.210675 − 0.786249i 0.934231 + 1.45850i 1.15825 − 0.668715i −0.490990 − 3.72944i 0.949922 − 1.04181i −0.183997 + 1.39760i −1.92094 − 1.92094i −1.25443 + 2.72515i −2.82883 + 1.17174i
25.7 −0.167615 − 0.625547i 1.67849 − 0.427404i 1.36884 − 0.790298i 0.122411 + 0.929804i −0.548702 − 0.978335i −0.404753 + 3.07440i −1.63967 − 1.63967i 2.63465 − 1.43479i 0.561119 − 0.232423i
25.8 −0.135259 − 0.504794i −1.73173 + 0.0331966i 1.49553 − 0.863444i 0.559597 + 4.25056i 0.250990 + 0.869679i −0.0693900 + 0.527070i −1.37722 − 1.37722i 2.99780 − 0.114975i 2.06997 − 0.857409i
25.9 0.0203196 + 0.0758337i −0.0241731 − 1.73188i 1.72671 − 0.996918i 0.123272 + 0.936344i 0.130844 − 0.0370242i 0.0673665 − 0.511699i 0.221714 + 0.221714i −2.99883 + 0.0837300i −0.0685016 + 0.0283743i
25.10 0.163921 + 0.611762i 0.365298 + 1.69309i 1.38467 − 0.799438i 0.222007 + 1.68631i −0.975889 + 0.501009i 0.492780 − 3.74303i 1.61172 + 1.61172i −2.73311 + 1.23697i −0.995230 + 0.412238i
25.11 0.272551 + 1.01717i −0.852920 + 1.50749i 0.771695 − 0.445538i −0.0439997 − 0.334211i −1.76584 − 0.456699i −0.560031 + 4.25386i 2.15276 + 2.15276i −1.54506 − 2.57154i 0.327958 − 0.135845i
25.12 0.300955 + 1.12318i −1.62103 − 0.610144i 0.561091 − 0.323946i −0.333892 − 2.53616i 0.197446 − 2.00433i 0.433348 − 3.29160i 2.17716 + 2.17716i 2.25545 + 1.97812i 2.74808 − 1.13829i
25.13 0.440272 + 1.64312i 0.884865 − 1.48896i −0.773943 + 0.446836i −0.398874 − 3.02975i 2.83612 + 0.798388i −0.359108 + 2.72769i 1.33074 + 1.33074i −1.43403 − 2.63507i 4.80262 − 1.98931i
25.14 0.577377 + 2.15480i 1.21948 − 1.22999i −2.57775 + 1.48827i 0.437040 + 3.31965i 3.35448 + 1.91756i 0.458076 − 3.47944i −1.54040 − 1.54040i −0.0257533 − 2.99989i −6.90085 + 2.85843i
25.15 0.621435 + 2.31923i 1.28808 + 1.15795i −3.26058 + 1.88250i −0.258179 − 1.96107i −1.88509 + 3.70694i 0.0106414 − 0.0808292i −2.99660 − 2.99660i 0.318306 + 2.98307i 4.38772 − 1.81745i
25.16 0.634374 + 2.36752i −1.49813 + 0.869259i −3.47066 + 2.00379i 0.103653 + 0.787320i −3.00836 − 2.99541i 0.0398722 − 0.302859i −3.47941 − 3.47941i 1.48878 − 2.60452i −1.79824 + 0.744855i
43.1 −0.608047 − 2.26926i 1.32892 − 1.11084i −3.04777 + 1.75963i 3.21696 − 0.423520i −3.32883 − 2.34023i −0.646459 − 0.0851079i 2.52383 + 2.52383i 0.532075 − 2.95244i −2.91714 − 7.04259i
43.2 −0.542746 − 2.02556i −0.544703 − 1.64417i −2.07625 + 1.19872i −1.85569 + 0.244306i −3.03472 + 1.99569i 0.270414 + 0.0356006i 0.589341 + 0.589341i −2.40660 + 1.79117i 1.50202 + 3.62620i
43.3 −0.540865 − 2.01854i −0.507581 + 1.65601i −2.04991 + 1.18351i 0.974701 − 0.128322i 3.61725 + 0.128894i 5.02505 + 0.661561i 0.542348 + 0.542348i −2.48472 − 1.68112i −0.786204 − 1.89807i
43.4 −0.325994 − 1.21663i 1.60404 + 0.653486i 0.358141 − 0.206773i −0.410190 + 0.0540025i 0.272139 − 2.16456i 0.472446 + 0.0621988i −2.14958 − 2.14958i 2.14591 + 2.09644i 0.199420 + 0.481444i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
17.d even 8 1 inner
153.r even 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 153.2.r.a ✓ 128
3.b odd 2 1 459.2.v.a 128
9.c even 3 1 inner 153.2.r.a ✓ 128
9.d odd 6 1 459.2.v.a 128
17.d even 8 1 inner 153.2.r.a ✓ 128
51.g odd 8 1 459.2.v.a 128
153.q odd 24 1 459.2.v.a 128
153.r even 24 1 inner 153.2.r.a ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.2.r.a ✓ 128 1.a even 1 1 trivial
153.2.r.a ✓ 128 9.c even 3 1 inner
153.2.r.a ✓ 128 17.d even 8 1 inner
153.2.r.a ✓ 128 153.r even 24 1 inner
459.2.v.a 128 3.b odd 2 1
459.2.v.a 128 9.d odd 6 1
459.2.v.a 128 51.g odd 8 1
459.2.v.a 128 153.q odd 24 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(153, [\chi])\).