Properties

Label 345.2.m.d
Level $345$
Weight $2$
Character orbit 345.m
Analytic conductor $2.755$
Analytic rank $0$
Dimension $50$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [345,2,Mod(16,345)] 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("345.16"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(345, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 0, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 345.m (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [50,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.75483886973\)
Analytic rank: \(0\)
Dimension: \(50\)
Relative dimension: \(5\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 50 q + 2 q^{2} + 5 q^{3} - 4 q^{4} - 5 q^{5} + 9 q^{6} + 3 q^{7} + 6 q^{8} - 5 q^{9} + 2 q^{10} + 19 q^{11} + 4 q^{12} + 15 q^{13} + 43 q^{14} + 5 q^{15} - 76 q^{16} - 23 q^{17} + 2 q^{18} - 7 q^{19} - 4 q^{20}+ \cdots + 8 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −1.61428 + 1.86298i −0.841254 + 0.540641i −0.580162 − 4.03511i 0.415415 + 0.909632i 0.350817 − 2.43999i −4.31213 + 1.26616i 4.30638 + 2.76754i 0.415415 − 0.909632i −2.36522 − 0.694493i
16.2 −1.17289 + 1.35359i −0.841254 + 0.540641i −0.171901 − 1.19560i 0.415415 + 0.909632i 0.254894 − 1.77283i 4.60299 − 1.35156i −1.19350 − 0.767014i 0.415415 − 0.909632i −1.71851 − 0.504600i
16.3 −0.0160116 + 0.0184784i −0.841254 + 0.540641i 0.284545 + 1.97905i 0.415415 + 0.909632i 0.00347966 − 0.0242015i −1.07504 + 0.315661i −0.0822637 − 0.0528676i 0.415415 − 0.909632i −0.0234600 − 0.00688847i
16.4 0.935055 − 1.07911i −0.841254 + 0.540641i −0.00552303 − 0.0384135i 0.415415 + 0.909632i −0.203207 + 1.41333i −1.20277 + 0.353165i 2.35578 + 1.51397i 0.415415 − 0.909632i 1.37003 + 0.402277i
16.5 1.68174 − 1.94083i −0.841254 + 0.540641i −0.653948 − 4.54831i 0.415415 + 0.909632i −0.365477 + 2.54195i 3.88186 − 1.13982i −5.60645 − 3.60304i 0.415415 − 0.909632i 2.46406 + 0.723514i
31.1 −0.362097 + 2.51844i −0.415415 − 0.909632i −4.29243 − 1.26037i −0.654861 − 0.755750i 2.44127 − 0.716822i 0.729201 + 0.468629i 2.61454 − 5.72503i −0.654861 + 0.755750i 2.14043 − 1.37557i
31.2 −0.213822 + 1.48716i −0.415415 − 0.909632i −0.246951 − 0.0725115i −0.654861 − 0.755750i 1.44160 − 0.423291i −4.04499 − 2.59956i −1.08765 + 2.38161i −0.654861 + 0.755750i 1.26395 − 0.812290i
31.3 −0.128466 + 0.893501i −0.415415 − 0.909632i 1.13714 + 0.333896i −0.654861 − 0.755750i 0.866124 − 0.254317i 0.379627 + 0.243972i −1.19440 + 2.61538i −0.654861 + 0.755750i 0.759391 − 0.488031i
31.4 0.172988 − 1.20316i −0.415415 − 0.909632i 0.501328 + 0.147203i −0.654861 − 0.755750i −1.16629 + 0.342454i 2.59847 + 1.66993i 1.27373 − 2.78908i −0.654861 + 0.755750i −1.02257 + 0.657164i
31.5 0.258297 − 1.79650i −0.415415 − 0.909632i −1.24169 − 0.364594i −0.654861 − 0.755750i −1.74145 + 0.511336i −3.56097 − 2.28850i 0.532214 − 1.16539i −0.654861 + 0.755750i −1.52685 + 0.981246i
121.1 −1.04936 + 2.29779i 0.959493 + 0.281733i −2.86894 − 3.31094i 0.841254 − 0.540641i −1.65422 + 1.90907i 0.551929 − 3.83875i 5.77092 − 1.69450i 0.841254 + 0.540641i 0.359496 + 2.50035i
121.2 −0.243944 + 0.534163i 0.959493 + 0.281733i 1.08390 + 1.25089i 0.841254 − 0.540641i −0.384553 + 0.443798i −0.281976 + 1.96119i −2.05947 + 0.604716i 0.841254 + 0.540641i 0.0835714 + 0.581252i
121.3 0.0769284 − 0.168450i 0.959493 + 0.281733i 1.28726 + 1.48558i 0.841254 − 0.540641i 0.121270 − 0.139953i 0.595692 − 4.14313i 0.704639 − 0.206901i 0.841254 + 0.540641i −0.0263545 − 0.183300i
121.4 0.663397 − 1.45264i 0.959493 + 0.281733i −0.360336 − 0.415850i 0.841254 − 0.540641i 1.04578 − 1.20689i −0.448840 + 3.12175i 2.22140 − 0.652262i 0.841254 + 0.540641i −0.227270 − 1.58069i
121.5 1.09706 − 2.40223i 0.959493 + 0.281733i −3.25744 − 3.75929i 0.841254 − 0.540641i 1.72941 − 1.99584i 0.314772 − 2.18929i −7.53647 + 2.21291i 0.841254 + 0.540641i −0.375837 − 2.61400i
151.1 −1.61428 − 1.86298i −0.841254 − 0.540641i −0.580162 + 4.03511i 0.415415 − 0.909632i 0.350817 + 2.43999i −4.31213 − 1.26616i 4.30638 − 2.76754i 0.415415 + 0.909632i −2.36522 + 0.694493i
151.2 −1.17289 − 1.35359i −0.841254 − 0.540641i −0.171901 + 1.19560i 0.415415 − 0.909632i 0.254894 + 1.77283i 4.60299 + 1.35156i −1.19350 + 0.767014i 0.415415 + 0.909632i −1.71851 + 0.504600i
151.3 −0.0160116 − 0.0184784i −0.841254 − 0.540641i 0.284545 − 1.97905i 0.415415 − 0.909632i 0.00347966 + 0.0242015i −1.07504 − 0.315661i −0.0822637 + 0.0528676i 0.415415 + 0.909632i −0.0234600 + 0.00688847i
151.4 0.935055 + 1.07911i −0.841254 − 0.540641i −0.00552303 + 0.0384135i 0.415415 − 0.909632i −0.203207 − 1.41333i −1.20277 − 0.353165i 2.35578 − 1.51397i 0.415415 + 0.909632i 1.37003 − 0.402277i
151.5 1.68174 + 1.94083i −0.841254 − 0.540641i −0.653948 + 4.54831i 0.415415 − 0.909632i −0.365477 − 2.54195i 3.88186 + 1.13982i −5.60645 + 3.60304i 0.415415 + 0.909632i 2.46406 − 0.723514i
See all 50 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 16.5
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 345.2.m.d ✓ 50
23.c even 11 1 inner 345.2.m.d ✓ 50
23.c even 11 1 7935.2.a.bu 25
23.d odd 22 1 7935.2.a.bt 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.m.d ✓ 50 1.a even 1 1 trivial
345.2.m.d ✓ 50 23.c even 11 1 inner
7935.2.a.bt 25 23.d odd 22 1
7935.2.a.bu 25 23.c even 11 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{50} - 2 T_{2}^{49} + 9 T_{2}^{48} - 20 T_{2}^{47} + 100 T_{2}^{46} - 195 T_{2}^{45} + 731 T_{2}^{44} + \cdots + 529 \) acting on \(S_{2}^{\mathrm{new}}(345, [\chi])\). Copy content Toggle raw display