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Properties

Label 595.2.d.b
Level $595$
Weight $2$
Character orbit 595.d
Analytic conductor $4.751$
Analytic rank $0$
Dimension $26$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [595,2,Mod(169,595)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(595, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("595.169");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 595 = 5 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 595.d (of order \(2\), degree \(1\), 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: \(4.75109892027\)
Analytic rank: \(0\)
Dimension: \(26\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 26 q + 6 q^{3} - 20 q^{4} - 26 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q + 6 q^{3} - 20 q^{4} - 26 q^{7} + 24 q^{9} - 4 q^{10} - 20 q^{12} + 2 q^{15} + 16 q^{16} - q^{17} - 12 q^{19} - 19 q^{20} - 6 q^{21} - 8 q^{22} - 14 q^{23} + 8 q^{26} + 42 q^{27} + 20 q^{28} + q^{30} - 10 q^{34} + 6 q^{36} - 14 q^{37} + 15 q^{40} + 4 q^{45} + 58 q^{48} + 26 q^{49} + 17 q^{50} + 19 q^{51} - 2 q^{55} + 24 q^{57} + 16 q^{58} - 32 q^{59} - 69 q^{60} + 52 q^{62} - 24 q^{63} + 12 q^{64} - 16 q^{65} - 76 q^{66} + 34 q^{68} - 20 q^{69} + 4 q^{70} + 8 q^{73} + 8 q^{75} + 12 q^{76} - 24 q^{78} + 68 q^{80} + 10 q^{81} - 34 q^{82} + 20 q^{84} - 13 q^{85} - 14 q^{86} - 12 q^{89} + 3 q^{90} - 8 q^{92} + 80 q^{94} - 44 q^{95} - 62 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1 − 2.60866i 0.411324 −4.80513 −0.309102 − 2.21460i − 1.07301i −1.00000 7.31764i −2.83081 −5.77715 + 0.806342i
169.2 − 2.58800i 2.82533 −4.69773 2.12895 + 0.683807i − 7.31194i −1.00000 6.98173i 4.98246 1.76969 − 5.50971i
169.3 − 2.34168i −0.830517 −3.48348 2.07222 + 0.840173i 1.94481i −1.00000 3.47383i −2.31024 1.96742 − 4.85249i
169.4 − 1.98429i −1.87309 −1.93742 −2.13090 + 0.677690i 3.71675i −1.00000 − 0.124181i 0.508453 1.34474 + 4.22833i
169.5 − 1.96497i 0.524615 −1.86110 −0.874022 + 2.05818i − 1.03085i −1.00000 − 0.272944i −2.72478 4.04425 + 1.71742i
169.6 − 1.81095i 2.42955 −1.27954 −0.155430 − 2.23066i − 4.39979i −1.00000 − 1.30472i 2.90270 −4.03961 + 0.281476i
169.7 − 1.58730i −2.53772 −0.519524 −0.715646 − 2.11845i 4.02812i −1.00000 − 2.34996i 3.44001 −3.36262 + 1.13595i
169.8 − 1.12750i 3.33864 0.728736 −1.84089 + 1.26931i − 3.76433i −1.00000 − 3.07666i 8.14655 1.43115 + 2.07561i
169.9 − 1.06574i −1.69433 0.864200 2.20924 + 0.345361i 1.80572i −1.00000 − 3.05249i −0.129233 0.368065 − 2.35447i
169.10 − 0.646342i −0.203765 1.58224 −2.12840 − 0.685512i 0.131702i −1.00000 − 2.31535i −2.95848 −0.443076 + 1.37567i
169.11 − 0.606154i 2.02727 1.63258 1.94770 − 1.09839i − 1.22884i −1.00000 − 2.20190i 1.10982 −0.665796 − 1.18061i
169.12 − 0.411841i −2.56057 1.83039 0.647485 + 2.14027i 1.05455i −1.00000 − 1.57751i 3.55650 0.881451 − 0.266661i
169.13 − 0.232868i 1.14326 1.94577 −0.851205 + 2.06772i − 0.266229i −1.00000 − 0.918843i −1.69295 0.481504 + 0.198218i
169.14 0.232868i 1.14326 1.94577 −0.851205 − 2.06772i 0.266229i −1.00000 0.918843i −1.69295 0.481504 − 0.198218i
169.15 0.411841i −2.56057 1.83039 0.647485 − 2.14027i − 1.05455i −1.00000 1.57751i 3.55650 0.881451 + 0.266661i
169.16 0.606154i 2.02727 1.63258 1.94770 + 1.09839i 1.22884i −1.00000 2.20190i 1.10982 −0.665796 + 1.18061i
169.17 0.646342i −0.203765 1.58224 −2.12840 + 0.685512i − 0.131702i −1.00000 2.31535i −2.95848 −0.443076 − 1.37567i
169.18 1.06574i −1.69433 0.864200 2.20924 − 0.345361i − 1.80572i −1.00000 3.05249i −0.129233 0.368065 + 2.35447i
169.19 1.12750i 3.33864 0.728736 −1.84089 − 1.26931i 3.76433i −1.00000 3.07666i 8.14655 1.43115 − 2.07561i
169.20 1.58730i −2.53772 −0.519524 −0.715646 + 2.11845i − 4.02812i −1.00000 2.34996i 3.44001 −3.36262 − 1.13595i
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 169.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 595.2.d.b yes 26
5.b even 2 1 595.2.d.a ✓ 26
17.b even 2 1 595.2.d.a ✓ 26
85.c even 2 1 inner 595.2.d.b yes 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
595.2.d.a ✓ 26 5.b even 2 1
595.2.d.a ✓ 26 17.b even 2 1
595.2.d.b yes 26 1.a even 1 1 trivial
595.2.d.b yes 26 85.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{13} - 3 T_{3}^{12} - 21 T_{3}^{11} + 59 T_{3}^{10} + 166 T_{3}^{9} - 422 T_{3}^{8} - 602 T_{3}^{7} + \cdots - 40 \) acting on \(S_{2}^{\mathrm{new}}(595, [\chi])\). Copy content Toggle raw display