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Properties

Label 536.2.y.b
Level $536$
Weight $2$
Character orbit 536.y
Analytic conductor $4.280$
Analytic rank $0$
Dimension $180$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [536,2,Mod(17,536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(536, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([0, 0, 64]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("536.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 536 = 2^{3} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 536.y (of order \(33\), degree \(20\), 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: \(4.27998154834\)
Analytic rank: \(0\)
Dimension: \(180\)
Relative dimension: \(9\) over \(\Q(\zeta_{33})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 180 q - 4 q^{3} + 2 q^{5} + 4 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 180 q - 4 q^{3} + 2 q^{5} + 4 q^{7} - 18 q^{9} + 3 q^{11} - 2 q^{13} - 35 q^{17} + 2 q^{19} - 121 q^{21} - 21 q^{23} - 24 q^{25} - 52 q^{27} - 6 q^{29} + 12 q^{31} - 39 q^{33} - 57 q^{35} - 48 q^{37} + 56 q^{39} - 13 q^{41} + 7 q^{43} + 3 q^{45} - 25 q^{47} + 15 q^{49} - 8 q^{51} + 24 q^{53} + 120 q^{55} + 30 q^{59} - 63 q^{61} - 30 q^{63} + 17 q^{65} + 109 q^{69} + 13 q^{71} - 11 q^{73} + q^{75} + 88 q^{77} - 115 q^{79} + 234 q^{81} - 41 q^{83} + 48 q^{85} - 61 q^{87} - 10 q^{89} - 24 q^{91} + 35 q^{93} + 131 q^{95} + 17 q^{97} + 44 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −2.68367 + 1.72469i 0 0.195752 − 1.36149i 0 −0.0925136 − 0.129917i 0 2.98129 − 6.52812i 0
17.2 0 −1.74595 + 1.12205i 0 −0.436186 + 3.03374i 0 2.53991 + 3.56680i 0 0.543085 − 1.18919i 0
17.3 0 −1.06123 + 0.682010i 0 −0.270844 + 1.88376i 0 −2.29212 − 3.21883i 0 −0.585177 + 1.28136i 0
17.4 0 −0.978160 + 0.628626i 0 0.349704 − 2.43225i 0 −0.932116 − 1.30897i 0 −0.684617 + 1.49910i 0
17.5 0 −0.0894431 + 0.0574816i 0 0.431508 − 3.00120i 0 2.23822 + 3.14314i 0 −1.24155 + 2.71861i 0
17.6 0 0.290693 − 0.186817i 0 −0.189846 + 1.32041i 0 0.516008 + 0.724632i 0 −1.19664 + 2.62028i 0
17.7 0 1.79375 − 1.15278i 0 −0.428123 + 2.97766i 0 0.702410 + 0.986397i 0 0.642417 − 1.40670i 0
17.8 0 1.79865 − 1.15592i 0 0.0530366 − 0.368878i 0 −1.18757 − 1.66771i 0 0.652748 − 1.42932i 0
17.9 0 1.86450 − 1.19824i 0 0.437313 − 3.04158i 0 −2.06041 − 2.89343i 0 0.794337 − 1.73935i 0
33.1 0 −3.10409 + 0.911444i 0 −2.62750 − 3.03230i 0 2.77028 + 1.42818i 0 6.28090 − 4.03649i 0
33.2 0 −2.84895 + 0.836528i 0 1.43392 + 1.65483i 0 −3.32377 − 1.71352i 0 4.89300 − 3.14454i 0
33.3 0 −1.49097 + 0.437790i 0 0.272083 + 0.314000i 0 −1.20691 − 0.622205i 0 −0.492415 + 0.316456i 0
33.4 0 −0.921098 + 0.270459i 0 −0.336769 − 0.388652i 0 0.322941 + 0.166488i 0 −1.74849 + 1.12368i 0
33.5 0 −0.309124 + 0.0907670i 0 2.80924 + 3.24204i 0 4.20099 + 2.16576i 0 −2.43644 + 1.56581i 0
33.6 0 0.0275006 − 0.00807490i 0 −1.30320 − 1.50397i 0 2.53852 + 1.30870i 0 −2.52307 + 1.62148i 0
33.7 0 1.31936 − 0.387400i 0 −1.99401 − 2.30121i 0 −3.06459 − 1.57990i 0 −0.933118 + 0.599678i 0
33.8 0 1.71500 − 0.503571i 0 2.26848 + 2.61797i 0 −2.56754 − 1.32366i 0 0.163897 − 0.105330i 0
33.9 0 2.74258 − 0.805295i 0 0.132614 + 0.153045i 0 1.66104 + 0.856327i 0 4.34950 − 2.79526i 0
49.1 0 −2.35218 − 1.51166i 0 −0.265354 − 1.84558i 0 4.86574 + 0.464622i 0 2.00141 + 4.38248i 0
49.2 0 −2.30176 − 1.47925i 0 0.600588 + 4.17718i 0 1.94096 + 0.185339i 0 1.86368 + 4.08089i 0
See next 80 embeddings (of 180 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.g even 33 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 536.2.y.b ✓ 180
67.g even 33 1 inner 536.2.y.b ✓ 180
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
536.2.y.b ✓ 180 1.a even 1 1 trivial
536.2.y.b ✓ 180 67.g even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{180} + 4 T_{3}^{179} + 44 T_{3}^{178} + 180 T_{3}^{177} + 1111 T_{3}^{176} + \cdots + 44\!\cdots\!69 \) acting on \(S_{2}^{\mathrm{new}}(536, [\chi])\). Copy content Toggle raw display