↑

Properties

Label 357.2.u.a
Level $357$
Weight $2$
Character orbit 357.u
Analytic conductor $2.851$
Analytic rank $0$
Dimension $32$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [357,2,Mod(43,357)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("357.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(357, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 357 = 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 357.u (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.85065935216\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{6} - 32 q^{10} + 8 q^{14} - 8 q^{16} - 8 q^{17} - 8 q^{18} + 8 q^{19} - 16 q^{20} - 8 q^{23} - 8 q^{24} + 24 q^{25} - 32 q^{26} + 24 q^{29} + 8 q^{31} - 40 q^{32} + 8 q^{33} - 8 q^{34} - 8 q^{35}+ \cdots + 96 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.83460 + 1.83460i −0.923880 − 0.382683i − 4.73152i −0.623364 + 1.50493i 2.39702 − 0.992879i −0.382683 − 0.923880i 5.01125 + 5.01125i 0.707107 + 0.707107i −1.61733 − 3.90458i
43.2 −1.37093 + 1.37093i 0.923880 + 0.382683i − 1.75892i 0.320309 − 0.773295i −1.79121 + 0.741944i 0.382683 + 0.923880i −0.330500 − 0.330500i 0.707107 + 0.707107i 0.621014 + 1.49926i
43.3 −0.952098 + 0.952098i −0.923880 − 0.382683i 0.187018i 0.256532 − 0.619322i 1.24398 − 0.515272i −0.382683 − 0.923880i −2.08226 − 2.08226i 0.707107 + 0.707107i 0.345412 + 0.833899i
43.4 −0.185935 + 0.185935i 0.923880 + 0.382683i 1.93086i −0.965862 + 2.33180i −0.242936 + 0.100627i 0.382683 + 0.923880i −0.730884 − 0.730884i 0.707107 + 0.707107i −0.253975 − 0.613150i
43.5 0.0534575 − 0.0534575i −0.923880 − 0.382683i 1.99428i −0.219585 + 0.530125i −0.0698456 + 0.0289310i −0.382683 − 0.923880i 0.213525 + 0.213525i 0.707107 + 0.707107i 0.0166007 + 0.0400776i
43.6 0.551860 − 0.551860i 0.923880 + 0.382683i 1.39090i 0.697030 − 1.68278i 0.721040 − 0.298665i 0.382683 + 0.923880i 1.87130 + 1.87130i 0.707107 + 0.707107i −0.543996 − 1.31332i
43.7 0.719572 − 0.719572i −0.923880 − 0.382683i 0.964432i 1.67621 − 4.04672i −0.940166 + 0.389430i −0.382683 − 0.923880i 2.13312 + 2.13312i 0.707107 + 0.707107i −1.70576 − 4.11806i
43.8 1.60447 − 1.60447i 0.923880 + 0.382683i − 3.14862i 0.272946 − 0.658950i 2.09634 − 0.868331i 0.382683 + 0.923880i −1.84292 − 1.84292i 0.707107 + 0.707107i −0.619331 − 1.49520i
127.1 −1.48525 + 1.48525i −0.382683 + 0.923880i − 2.41192i −0.446226 − 0.184833i −0.803810 − 1.94057i 0.923880 − 0.382683i 0.611807 + 0.611807i −0.707107 − 0.707107i 0.937278 − 0.388233i
127.2 −1.32716 + 1.32716i 0.382683 − 0.923880i − 1.52271i 1.27840 + 0.529531i 0.718254 + 1.73402i −0.923880 + 0.382683i −0.633442 − 0.633442i −0.707107 − 0.707107i −2.39941 + 0.993870i
127.3 −1.01985 + 1.01985i 0.382683 − 0.923880i − 0.0802040i −1.03308 − 0.427916i 0.551941 + 1.33250i −0.923880 + 0.382683i −1.95791 − 1.95791i −0.707107 − 0.707107i 1.49000 − 0.617180i
127.4 0.124990 − 0.124990i −0.382683 + 0.923880i 1.96875i 3.70116 + 1.53307i 0.0676443 + 0.163308i 0.923880 − 0.382683i 0.496056 + 0.496056i −0.707107 − 0.707107i 0.654228 − 0.270990i
127.5 0.764267 − 0.764267i −0.382683 + 0.923880i 0.831793i −1.58285 − 0.655638i 0.413618 + 0.998563i 0.923880 − 0.382683i 2.16424 + 2.16424i −0.707107 − 0.707107i −1.71080 + 0.708638i
127.6 0.904145 − 0.904145i 0.382683 − 0.923880i 0.365044i 2.54273 + 1.05323i −0.489320 − 1.18132i −0.923880 + 0.382683i 2.13834 + 2.13834i −0.707107 − 0.707107i 3.25127 − 1.34672i
127.7 1.60878 − 1.60878i 0.382683 − 0.923880i − 3.17634i −2.57128 − 1.06506i −0.870665 − 2.10197i −0.923880 + 0.382683i −1.89248 − 1.89248i −0.707107 − 0.707107i −5.85006 + 2.42317i
127.8 1.84429 − 1.84429i −0.382683 + 0.923880i − 4.80284i −3.30307 − 1.36818i 0.998125 + 2.40969i 0.923880 − 0.382683i −5.16926 − 5.16926i −0.707107 − 0.707107i −8.61515 + 3.56851i
253.1 −1.48525 − 1.48525i −0.382683 − 0.923880i 2.41192i −0.446226 + 0.184833i −0.803810 + 1.94057i 0.923880 + 0.382683i 0.611807 − 0.611807i −0.707107 + 0.707107i 0.937278 + 0.388233i
253.2 −1.32716 − 1.32716i 0.382683 + 0.923880i 1.52271i 1.27840 − 0.529531i 0.718254 − 1.73402i −0.923880 − 0.382683i −0.633442 + 0.633442i −0.707107 + 0.707107i −2.39941 − 0.993870i
253.3 −1.01985 − 1.01985i 0.382683 + 0.923880i 0.0802040i −1.03308 + 0.427916i 0.551941 − 1.33250i −0.923880 − 0.382683i −1.95791 + 1.95791i −0.707107 + 0.707107i 1.49000 + 0.617180i
253.4 0.124990 + 0.124990i −0.382683 − 0.923880i − 1.96875i 3.70116 − 1.53307i 0.0676443 − 0.163308i 0.923880 + 0.382683i 0.496056 − 0.496056i −0.707107 + 0.707107i 0.654228 + 0.270990i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 357.2.u.a ✓ 32
17.d even 8 1 inner 357.2.u.a ✓ 32
17.e odd 16 1 6069.2.a.bj 16
17.e odd 16 1 6069.2.a.bk 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
357.2.u.a ✓ 32 1.a even 1 1 trivial
357.2.u.a ✓ 32 17.d even 8 1 inner
6069.2.a.bj 16 17.e odd 16 1
6069.2.a.bk 16 17.e odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 102 T_{2}^{28} + 8 T_{2}^{27} - 56 T_{2}^{25} + 3605 T_{2}^{24} + 504 T_{2}^{23} + 32 T_{2}^{22} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(357, [\chi])\). Copy content Toggle raw display