Properties

Label 136.3.p.c
Level $136$
Weight $3$
Character orbit 136.p
Analytic conductor $3.706$
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] = [136,3,Mod(19,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 4, 7]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 136.p (of order \(8\), degree \(4\), 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: \(3.70573159530\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 4 q^{2} - 8 q^{3} - 28 q^{6} - 4 q^{8} - 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 4 q^{2} - 8 q^{3} - 28 q^{6} - 4 q^{8} - 64 q^{9} - 4 q^{10} + 48 q^{11} + 64 q^{12} - 20 q^{14} + 120 q^{16} - 8 q^{17} - 152 q^{18} + 120 q^{19} - 68 q^{20} - 28 q^{22} - 280 q^{24} - 56 q^{25} + 92 q^{26} + 64 q^{27} - 188 q^{28} - 124 q^{32} - 16 q^{33} - 36 q^{34} - 16 q^{35} - 68 q^{36} + 304 q^{40} + 72 q^{41} - 16 q^{42} - 64 q^{43} + 236 q^{44} + 116 q^{46} + 12 q^{48} - 8 q^{49} + 752 q^{50} + 312 q^{51} - 136 q^{52} - 220 q^{54} - 196 q^{56} + 536 q^{57} - 480 q^{58} - 336 q^{59} - 312 q^{60} - 364 q^{62} - 288 q^{65} - 868 q^{66} - 16 q^{67} + 364 q^{68} + 528 q^{70} - 248 q^{73} + 332 q^{74} - 208 q^{75} - 40 q^{76} + 784 q^{78} + 152 q^{80} + 712 q^{82} - 1600 q^{83} + 112 q^{84} - 1152 q^{86} + 360 q^{88} - 452 q^{90} - 400 q^{91} + 716 q^{92} - 88 q^{94} + 948 q^{96} - 384 q^{97} + 1200 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.99989 0.0206161i 2.82466 1.17001i 3.99915 + 0.0824601i −6.65544 + 2.75677i −5.67315 + 2.28167i −9.71464 4.02394i −7.99617 0.247358i 0.245829 0.245829i 13.3670 5.37604i
19.2 −1.97912 0.288256i 1.21149 0.501817i 3.83382 + 1.14099i 6.29726 2.60841i −2.54234 + 0.643934i 2.88787 + 1.19620i −7.25868 3.36327i −5.14807 + 5.14807i −13.2149 + 3.34713i
19.3 −1.93870 + 0.491361i −4.34673 + 1.80048i 3.51713 1.90520i 0.750539 0.310883i 7.54234 5.62640i −11.2238 4.64906i −5.88252 + 5.42180i 9.28842 9.28842i −1.30232 + 0.971496i
19.4 −1.87130 + 0.705866i −2.10387 + 0.871452i 3.00351 2.64177i −1.18140 + 0.489353i 3.32184 3.11580i 7.56608 + 3.13397i −3.75572 + 7.06361i −2.69711 + 2.69711i 1.86534 1.74964i
19.5 −1.79251 0.887078i −0.958919 + 0.397197i 2.42619 + 3.18019i 0.484158 0.200545i 2.07122 + 0.138655i −3.06598 1.26997i −1.52789 7.85274i −5.60220 + 5.60220i −1.04576 0.0700070i
19.6 −1.75586 + 0.957582i 4.91989 2.03789i 2.16607 3.36276i 4.41938 1.83057i −6.68719 + 8.28944i 0.225959 + 0.0935954i −0.583198 + 7.97871i 13.6884 13.6884i −6.00688 + 7.44614i
19.7 −1.68630 1.07536i −3.95884 + 1.63981i 1.68722 + 3.62675i −7.46648 + 3.09272i 8.43917 + 1.49196i 2.68777 + 1.11331i 1.05489 7.93015i 6.61949 6.61949i 15.9165 + 2.81387i
19.8 −1.66343 1.11041i 4.47944 1.85545i 1.53399 + 3.69417i −4.45144 + 1.84385i −9.51154 1.88761i 12.7752 + 5.29167i 1.55035 7.84834i 10.2588 10.2588i 9.45207 + 1.87580i
19.9 −1.51389 + 1.30695i 1.83762 0.761166i 0.583738 3.95718i −4.01021 + 1.66108i −1.78715 + 3.55401i 4.10587 + 1.70071i 4.28814 + 6.75366i −3.56650 + 3.56650i 3.90006 7.75587i
19.10 −1.11041 1.66343i 4.47944 1.85545i −1.53399 + 3.69417i 4.45144 1.84385i −8.06041 5.39093i −12.7752 5.29167i 7.84834 1.55035i 10.2588 10.2588i −8.01001 5.35723i
19.11 −1.07536 1.68630i −3.95884 + 1.63981i −1.68722 + 3.62675i 7.46648 3.09272i 7.02237 + 4.91242i −2.68777 1.11331i 7.93015 1.05489i 6.61949 6.61949i −13.2444 9.26495i
19.12 −0.987360 + 1.73929i −3.13949 + 1.30042i −2.05024 3.43461i 7.34303 3.04158i 0.838003 6.74445i 4.30870 + 1.78472i 7.99809 0.174765i 1.80133 1.80133i −1.96003 + 15.7748i
19.13 −0.934433 + 1.76829i 0.813063 0.336782i −2.25367 3.30469i −1.46394 + 0.606383i −0.164227 + 1.75243i −7.48231 3.09928i 7.94954 0.897117i −5.81631 + 5.81631i 0.295694 3.15529i
19.14 −0.887078 1.79251i −0.958919 + 0.397197i −2.42619 + 3.18019i −0.484158 + 0.200545i 1.56262 + 1.36653i 3.06598 + 1.26997i 7.85274 + 1.52789i −5.60220 + 5.60220i 0.788964 + 0.689959i
19.15 −0.610091 + 1.90468i −4.33611 + 1.79608i −3.25558 2.32405i −5.90990 + 2.44796i −0.775521 9.35466i 1.71003 + 0.708317i 6.41276 4.78294i 9.21202 9.21202i −1.05700 12.7499i
19.16 −0.288256 1.97912i 1.21149 0.501817i −3.83382 + 1.14099i −6.29726 + 2.60841i −1.34237 2.25303i −2.88787 1.19620i 3.36327 + 7.25868i −5.14807 + 5.14807i 6.97758 + 11.7111i
19.17 −0.170951 + 1.99268i 3.48144 1.44206i −3.94155 0.681301i 3.91706 1.62250i 2.27841 + 7.18392i 1.50905 + 0.625068i 2.03143 7.73778i 3.67692 3.67692i 2.56350 + 8.08282i
19.18 −0.0206161 1.99989i 2.82466 1.17001i −3.99915 + 0.0824601i 6.65544 2.75677i −2.39814 5.62490i 9.71464 + 4.02394i 0.247358 + 7.99617i 0.245829 0.245829i −5.65046 13.2533i
19.19 0.449222 + 1.94890i 1.11155 0.460421i −3.59640 + 1.75098i −7.74186 + 3.20678i 1.39665 + 1.95947i 9.91980 + 4.10892i −5.02805 6.22243i −5.34040 + 5.34040i −9.72751 13.6475i
19.20 0.463170 + 1.94563i −1.79304 + 0.742700i −3.57095 + 1.80232i 0.987744 0.409137i −2.27550 3.14459i −8.55035 3.54167i −5.16059 6.11296i −3.70059 + 3.70059i 1.25352 + 1.73228i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
17.d even 8 1 inner
136.p odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.3.p.c 128
8.d odd 2 1 inner 136.3.p.c 128
17.d even 8 1 inner 136.3.p.c 128
136.p odd 8 1 inner 136.3.p.c 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.3.p.c 128 1.a even 1 1 trivial
136.3.p.c 128 8.d odd 2 1 inner
136.3.p.c 128 17.d even 8 1 inner
136.3.p.c 128 136.p odd 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} + 4 T_{3}^{63} + 24 T_{3}^{62} + 40 T_{3}^{61} + 128 T_{3}^{60} + 248 T_{3}^{59} - 5424 T_{3}^{58} - 1792 T_{3}^{57} + 992596 T_{3}^{56} + 5091712 T_{3}^{55} + 22737104 T_{3}^{54} + 29693376 T_{3}^{53} + \cdots + 25\!\cdots\!24 \) acting on \(S_{3}^{\mathrm{new}}(136, [\chi])\). Copy content Toggle raw display