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Properties

Label 345.2.t.a
Level $345$
Weight $2$
Character orbit 345.t
Analytic conductor $2.755$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [345,2,Mod(4,345)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(345, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 11, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("345.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 345.t (of order \(22\), degree \(10\), 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: \(2.75483886973\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(24\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 28 q^{4} - 4 q^{5} - 4 q^{6} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 28 q^{4} - 4 q^{5} - 4 q^{6} + 24 q^{9} + 8 q^{14} + 4 q^{15} - 36 q^{16} - 8 q^{19} - 92 q^{20} + 8 q^{21} + 12 q^{24} - 44 q^{25} + 32 q^{26} - 12 q^{29} + 8 q^{30} - 20 q^{31} - 54 q^{34} + 36 q^{35} - 28 q^{36} - 36 q^{39} - 24 q^{40} - 28 q^{41} - 224 q^{44} + 4 q^{45} - 8 q^{46} + 52 q^{49} - 48 q^{50} - 18 q^{54} - 22 q^{55} - 56 q^{59} - 36 q^{60} + 44 q^{61} + 12 q^{64} - 16 q^{65} + 8 q^{66} + 8 q^{69} - 152 q^{70} + 4 q^{71} + 88 q^{74} - 8 q^{75} - 100 q^{76} - 84 q^{79} + 122 q^{80} - 24 q^{81} - 56 q^{84} + 186 q^{85} - 392 q^{86} + 48 q^{89} - 304 q^{91} + 80 q^{94} + 190 q^{95} + 16 q^{96} + 44 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.50929 + 1.14595i −0.281733 − 0.959493i 3.67359 − 4.23955i 2.18488 + 0.475730i 1.80648 + 2.08479i 4.53926 − 0.652647i −2.80541 + 9.55436i −0.841254 + 0.540641i −6.02764 + 1.31002i
4.2 −2.31175 + 1.05574i 0.281733 + 0.959493i 2.91987 − 3.36970i −1.68154 − 1.47392i −1.66427 − 1.92067i 2.16722 − 0.311599i −1.76046 + 5.99559i −0.841254 + 0.540641i 5.44336 + 1.63206i
4.3 −2.19367 + 1.00182i −0.281733 − 0.959493i 2.49885 − 2.88382i −2.20845 + 0.350356i 1.57927 + 1.82257i −1.24858 + 0.179518i −1.23374 + 4.20172i −0.841254 + 0.540641i 4.49363 − 2.98103i
4.4 −2.08503 + 0.952203i 0.281733 + 0.959493i 2.13095 − 2.45925i 0.236368 + 2.22354i −1.50105 − 1.73231i 0.207303 − 0.0298056i −0.809843 + 2.75807i −0.841254 + 0.540641i −2.61010 − 4.41109i
4.5 −1.89471 + 0.865286i −0.281733 − 0.959493i 1.53149 − 1.76743i 0.959621 − 2.01968i 1.36404 + 1.57418i −3.38318 + 0.486427i −0.198730 + 0.676813i −0.841254 + 0.540641i −0.0706006 + 4.65707i
4.6 −1.53032 + 0.698876i 0.281733 + 0.959493i 0.543744 − 0.627515i −2.23426 − 0.0897803i −1.10171 − 1.27144i −4.10020 + 0.589520i 0.554398 − 1.88811i −0.841254 + 0.540641i 3.48190 − 1.42408i
4.7 −1.19202 + 0.544376i −0.281733 − 0.959493i −0.185160 + 0.213686i 1.95154 + 1.09156i 0.858156 + 0.990364i −0.0307032 + 0.00441445i 0.842776 − 2.87023i −0.841254 + 0.540641i −2.92049 − 0.238790i
4.8 −0.940856 + 0.429675i −0.281733 − 0.959493i −0.609131 + 0.702975i −0.524248 + 2.17374i 0.677340 + 0.781692i −1.21306 + 0.174411i 0.853861 − 2.90798i −0.841254 + 0.540641i −0.440761 − 2.27044i
4.9 −0.932222 + 0.425732i 0.281733 + 0.959493i −0.621931 + 0.717746i −0.0661665 − 2.23509i −0.671124 − 0.774518i 3.06052 − 0.440036i 0.851669 − 2.90052i −0.841254 + 0.540641i 1.01323 + 2.05543i
4.10 −0.853361 + 0.389717i 0.281733 + 0.959493i −0.733376 + 0.846361i 1.71593 − 1.43374i −0.614350 − 0.708998i −3.82750 + 0.550312i 0.824602 − 2.80834i −0.841254 + 0.540641i −0.905553 + 1.89222i
4.11 −0.737618 + 0.336859i −0.281733 − 0.959493i −0.879115 + 1.01455i −1.87316 − 1.22118i 0.531025 + 0.612835i 2.28235 − 0.328152i 0.763602 − 2.60059i −0.841254 + 0.540641i 1.79304 + 0.269776i
4.12 −0.00990606 + 0.00452394i 0.281733 + 0.959493i −1.30964 + 1.51141i −0.715854 + 2.11838i −0.00713155 − 0.00823025i −0.520522 + 0.0748397i 0.0122721 − 0.0417950i −0.841254 + 0.540641i −0.00249216 − 0.0242233i
4.13 0.00990606 − 0.00452394i −0.281733 − 0.959493i −1.30964 + 1.51141i 1.62957 − 1.53117i −0.00713155 − 0.00823025i 0.520522 − 0.0748397i −0.0122721 + 0.0417950i −0.841254 + 0.540641i 0.00921571 − 0.0225400i
4.14 0.737618 − 0.336859i 0.281733 + 0.959493i −0.879115 + 1.01455i −1.88896 − 1.19659i 0.531025 + 0.612835i −2.28235 + 0.328152i −0.763602 + 2.60059i −0.841254 + 0.540641i −1.79641 − 0.246310i
4.15 0.853361 − 0.389717i −0.281733 − 0.959493i −0.733376 + 0.846361i −0.591351 + 2.15646i −0.614350 − 0.708998i 3.82750 − 0.550312i −0.824602 + 2.80834i −0.841254 + 0.540641i 0.335772 + 2.07070i
4.16 0.932222 − 0.425732i −0.281733 − 0.959493i −0.621931 + 0.717746i −2.06059 + 0.868302i −0.671124 − 0.774518i −3.06052 + 0.440036i −0.851669 + 2.90052i −0.841254 + 0.540641i −1.55127 + 1.68671i
4.17 0.940856 − 0.429675i 0.281733 + 0.959493i −0.609131 + 0.702975i 1.75953 − 1.37988i 0.677340 + 0.781692i 1.21306 − 0.174411i −0.853861 + 2.90798i −0.841254 + 0.540641i 1.06256 − 2.05429i
4.18 1.19202 − 0.544376i 0.281733 + 0.959493i −0.185160 + 0.213686i 1.80362 + 1.32173i 0.858156 + 0.990364i 0.0307032 − 0.00441445i −0.842776 + 2.87023i −0.841254 + 0.540641i 2.86946 + 0.593679i
4.19 1.53032 − 0.698876i −0.281733 − 0.959493i 0.543744 − 0.627515i −1.00981 − 1.99506i −1.10171 − 1.27144i 4.10020 − 0.589520i −0.554398 + 1.88811i −0.841254 + 0.540641i −2.93964 − 2.34736i
4.20 1.89471 − 0.865286i 0.281733 + 0.959493i 1.53149 − 1.76743i −1.43853 + 1.71191i 1.36404 + 1.57418i 3.38318 − 0.486427i 0.198730 − 0.676813i −0.841254 + 0.540641i −1.24431 + 4.48831i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.c even 11 1 inner
115.j even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 345.2.t.a ✓ 240
5.b even 2 1 inner 345.2.t.a ✓ 240
23.c even 11 1 inner 345.2.t.a ✓ 240
115.j even 22 1 inner 345.2.t.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.t.a ✓ 240 1.a even 1 1 trivial
345.2.t.a ✓ 240 5.b even 2 1 inner
345.2.t.a ✓ 240 23.c even 11 1 inner
345.2.t.a ✓ 240 115.j even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(345, [\chi])\).