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Properties

Label 755.2.f.d
Level $755$
Weight $2$
Character orbit 755.f
Analytic conductor $6.029$
Analytic rank $0$
Dimension $28$
CM discriminant -151
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [755,2,Mod(452,755)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(755, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("755.452"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 755 = 5 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 755.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [28,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.02870535261\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 112 q^{16} + 168 q^{36} - 14 q^{38} + 126 q^{58} - 154 q^{68} - 252 q^{81}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
452.1 −1.93553 − 1.93553i 0 5.49259i 0.577204 + 2.16029i 0 0 6.76003 − 6.76003i − 3.00000i 3.06411 − 5.29851i
452.2 −1.82731 − 1.82731i 0 4.67811i 1.45736 − 1.69591i 0 0 4.89374 − 4.89374i − 3.00000i −5.76199 + 0.435914i
452.3 −1.77492 − 1.77492i 0 4.30072i −2.04886 + 0.895640i 0 0 4.08360 − 4.08360i − 3.00000i 5.22627 + 2.04688i
452.4 −1.60059 − 1.60059i 0 3.12376i 0.417270 − 2.19679i 0 0 1.79868 − 1.79868i − 3.00000i −4.18403 + 2.84828i
452.5 −0.812984 − 0.812984i 0 − 0.678113i −1.45736 − 1.69591i 0 0 −2.17726 + 2.17726i − 3.00000i −0.193942 + 2.56356i
452.6 −0.503691 − 0.503691i 0 − 1.49259i −0.577204 + 2.16029i 0 0 −1.75919 + 1.75919i − 3.00000i 1.37885 − 0.797385i
452.7 −0.385987 − 0.385987i 0 − 1.70203i 2.23456 + 0.0820234i 0 0 −1.42893 + 1.42893i − 3.00000i −0.830852 − 0.894172i
452.8 −0.0603659 − 0.0603659i 0 − 1.99271i −1.32910 + 1.79819i 0 0 −0.241024 + 0.241024i − 3.00000i 0.188782 − 0.0283173i
452.9 0.921760 + 0.921760i 0 − 0.300717i 2.04886 + 0.895640i 0 0 2.12071 − 2.12071i − 3.00000i 1.06299 + 2.71412i
452.10 1.19922 + 1.19922i 0 0.876236i −0.417270 − 2.19679i 0 0 1.34764 − 1.34764i − 3.00000i 2.13403 − 3.13482i
452.11 1.29361 + 1.29361i 0 1.34684i −1.97768 − 1.04344i 0 0 0.844932 − 0.844932i − 3.00000i −1.20854 − 3.90815i
452.12 1.52531 + 1.52531i 0 2.65316i 1.97768 − 1.04344i 0 0 −0.996273 + 0.996273i − 3.00000i 4.60816 + 1.42501i
452.13 1.96240 + 1.96240i 0 5.70203i −2.23456 + 0.0820234i 0 0 −7.26486 + 7.26486i − 3.00000i −4.54607 − 4.22414i
452.14 1.99909 + 1.99909i 0 5.99271i 1.32910 + 1.79819i 0 0 −7.98179 + 7.98179i − 3.00000i −0.937760 + 6.25173i
603.1 −1.93553 + 1.93553i 0 − 5.49259i 0.577204 − 2.16029i 0 0 6.76003 + 6.76003i 3.00000i 3.06411 + 5.29851i
603.2 −1.82731 + 1.82731i 0 − 4.67811i 1.45736 + 1.69591i 0 0 4.89374 + 4.89374i 3.00000i −5.76199 − 0.435914i
603.3 −1.77492 + 1.77492i 0 − 4.30072i −2.04886 − 0.895640i 0 0 4.08360 + 4.08360i 3.00000i 5.22627 − 2.04688i
603.4 −1.60059 + 1.60059i 0 − 3.12376i 0.417270 + 2.19679i 0 0 1.79868 + 1.79868i 3.00000i −4.18403 − 2.84828i
603.5 −0.812984 + 0.812984i 0 0.678113i −1.45736 + 1.69591i 0 0 −2.17726 − 2.17726i 3.00000i −0.193942 − 2.56356i
603.6 −0.503691 + 0.503691i 0 1.49259i −0.577204 − 2.16029i 0 0 −1.75919 − 1.75919i 3.00000i 1.37885 + 0.797385i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 452.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
151.b odd 2 1 CM by \(\Q(\sqrt{-151}) \)
5.c odd 4 1 inner
755.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 755.2.f.d ✓ 28
5.c odd 4 1 inner 755.2.f.d ✓ 28
151.b odd 2 1 CM 755.2.f.d ✓ 28
755.f even 4 1 inner 755.2.f.d ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
755.2.f.d ✓ 28 1.a even 1 1 trivial
755.2.f.d ✓ 28 5.c odd 4 1 inner
755.2.f.d ✓ 28 151.b odd 2 1 CM
755.2.f.d ✓ 28 755.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(755, [\chi])\):

\( T_{2}^{28} + 168 T_{2}^{24} - 38 T_{2}^{21} + 10192 T_{2}^{20} - 532 T_{2}^{19} - 1064 T_{2}^{17} + \cdots + 11025 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display