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Properties

Label 345.2.m.b
Level $345$
Weight $2$
Character orbit 345.m
Analytic conductor $2.755$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [345,2,Mod(16,345)]
 
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, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//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);
 
Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 345.m (of order \(11\), 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: \(30\)
Relative dimension: \(3\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 6 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{5} - 5 q^{6} + 7 q^{7} - 4 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 6 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{5} - 5 q^{6} + 7 q^{7} - 4 q^{8} - 3 q^{9} + 6 q^{10} + 5 q^{11} + 2 q^{12} + 5 q^{13} - 35 q^{14} - 3 q^{15} - 30 q^{16} - 25 q^{17} + 6 q^{18} + 5 q^{19} + 2 q^{20} - 4 q^{21} + 14 q^{22} + 10 q^{23} - 48 q^{24} - 3 q^{25} + 14 q^{26} - 3 q^{27} + 46 q^{28} + 26 q^{29} + 6 q^{30} + 2 q^{31} - 15 q^{32} + 16 q^{33} - 57 q^{34} - 4 q^{35} - 9 q^{36} + 23 q^{37} - 91 q^{38} + 5 q^{39} - 15 q^{40} - 4 q^{41} + 9 q^{42} - q^{43} + 45 q^{44} + 30 q^{45} + 24 q^{46} - 66 q^{47} - 63 q^{48} + 34 q^{49} - 5 q^{50} - 3 q^{51} + 61 q^{52} + 18 q^{53} - 5 q^{54} - 6 q^{55} + 8 q^{56} - 6 q^{57} + 16 q^{58} - 8 q^{59} - 9 q^{60} - 23 q^{61} + 89 q^{62} - 4 q^{63} + 42 q^{64} + 16 q^{65} + 36 q^{66} - 46 q^{67} + 78 q^{68} - q^{69} - 2 q^{70} + 31 q^{71} - 4 q^{72} - 59 q^{73} + 85 q^{74} - 3 q^{75} - 2 q^{76} - 76 q^{77} + 25 q^{78} - 73 q^{79} - 63 q^{80} - 3 q^{81} - 18 q^{82} - 15 q^{83} + 46 q^{84} + 8 q^{85} - 36 q^{86} + 37 q^{87} - 8 q^{88} - 34 q^{89} + 6 q^{90} + 40 q^{91} - 71 q^{92} - 20 q^{93} - 25 q^{94} + 5 q^{95} - 4 q^{96} - 18 q^{97} + 18 q^{98} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −0.284528 + 0.328363i 0.841254 − 0.540641i 0.257764 + 1.79279i 0.415415 + 0.909632i −0.0618339 + 0.430064i 4.08245 − 1.19872i −1.39305 − 0.895261i 0.415415 − 0.909632i −0.416887 − 0.122409i
16.2 0.677241 − 0.781578i 0.841254 − 0.540641i 0.132421 + 0.921009i 0.415415 + 0.909632i 0.147179 − 1.02365i −1.03146 + 0.302863i 2.54953 + 1.63848i 0.415415 − 0.909632i 0.992284 + 0.291361i
16.3 1.83242 − 2.11473i 0.841254 − 0.540641i −0.829675 − 5.77052i 0.415415 + 0.909632i 0.398224 − 2.76971i −1.70229 + 0.499837i −9.01543 − 5.79386i 0.415415 − 0.909632i 2.68484 + 0.788341i
31.1 −0.244703 + 1.70195i 0.415415 + 0.909632i −0.917753 − 0.269477i −0.654861 − 0.755750i −1.64980 + 0.484424i 0.105585 + 0.0678552i −0.745357 + 1.63210i −0.654861 + 0.755750i 1.44649 − 0.929603i
31.2 0.0765926 − 0.532713i 0.415415 + 0.909632i 1.64107 + 0.481861i −0.654861 − 0.755750i 0.516391 − 0.151626i 1.59341 + 1.02402i 0.829533 − 1.81642i −0.654861 + 0.755750i −0.452755 + 0.290968i
31.3 0.297879 − 2.07180i 0.415415 + 0.909632i −2.28462 − 0.670824i −0.654861 − 0.755750i 2.00831 − 0.589694i −2.36896 − 1.52244i −0.331340 + 0.725533i −0.654861 + 0.755750i −1.76083 + 1.13162i
121.1 −0.610201 + 1.33615i −0.959493 − 0.281733i −0.103240 − 0.119145i 0.841254 − 0.540641i 0.961921 − 1.11012i −0.00784623 + 0.0545718i −2.59660 + 0.762429i 0.841254 + 0.540641i 0.209045 + 1.45394i
121.2 0.296607 − 0.649478i −0.959493 − 0.281733i 0.975876 + 1.12622i 0.841254 − 0.540641i −0.467571 + 0.539606i −0.101384 + 0.705139i 2.39106 − 0.702080i 0.841254 + 0.540641i −0.101613 − 0.706733i
121.3 0.824018 − 1.80435i −0.959493 − 0.281733i −1.26694 − 1.46213i 0.841254 − 0.540641i −1.29898 + 1.49911i 0.468021 − 3.25516i 0.124331 − 0.0365068i 0.841254 + 0.540641i −0.282296 − 1.96341i
151.1 −0.284528 − 0.328363i 0.841254 + 0.540641i 0.257764 − 1.79279i 0.415415 − 0.909632i −0.0618339 − 0.430064i 4.08245 + 1.19872i −1.39305 + 0.895261i 0.415415 + 0.909632i −0.416887 + 0.122409i
151.2 0.677241 + 0.781578i 0.841254 + 0.540641i 0.132421 − 0.921009i 0.415415 − 0.909632i 0.147179 + 1.02365i −1.03146 − 0.302863i 2.54953 − 1.63848i 0.415415 + 0.909632i 0.992284 − 0.291361i
151.3 1.83242 + 2.11473i 0.841254 + 0.540641i −0.829675 + 5.77052i 0.415415 − 0.909632i 0.398224 + 2.76971i −1.70229 − 0.499837i −9.01543 + 5.79386i 0.415415 + 0.909632i 2.68484 − 0.788341i
196.1 −2.11954 + 1.36215i −0.142315 + 0.989821i 1.80619 − 3.95501i −0.959493 + 0.281733i −1.04664 − 2.29183i 2.22316 + 2.56567i 0.841878 + 5.85539i −0.959493 − 0.281733i 1.64993 − 1.90412i
196.2 −1.32537 + 0.851761i −0.142315 + 0.989821i 0.200268 − 0.438525i −0.959493 + 0.281733i −0.654472 − 1.43309i 0.0793365 + 0.0915592i −0.340334 − 2.36707i −0.959493 − 0.281733i 1.03171 − 1.19066i
196.3 1.30291 − 0.837330i −0.142315 + 0.989821i 0.165625 − 0.362667i −0.959493 + 0.281733i 0.643383 + 1.40881i 1.30797 + 1.50948i 0.352949 + 2.45481i −0.959493 − 0.281733i −1.01423 + 1.17048i
211.1 −0.610201 − 1.33615i −0.959493 + 0.281733i −0.103240 + 0.119145i 0.841254 + 0.540641i 0.961921 + 1.11012i −0.00784623 − 0.0545718i −2.59660 − 0.762429i 0.841254 − 0.540641i 0.209045 − 1.45394i
211.2 0.296607 + 0.649478i −0.959493 + 0.281733i 0.975876 − 1.12622i 0.841254 + 0.540641i −0.467571 − 0.539606i −0.101384 − 0.705139i 2.39106 + 0.702080i 0.841254 − 0.540641i −0.101613 + 0.706733i
211.3 0.824018 + 1.80435i −0.959493 + 0.281733i −1.26694 + 1.46213i 0.841254 + 0.540641i −1.29898 − 1.49911i 0.468021 + 3.25516i 0.124331 + 0.0365068i 0.841254 − 0.540641i −0.282296 + 1.96341i
256.1 −0.244703 − 1.70195i 0.415415 − 0.909632i −0.917753 + 0.269477i −0.654861 + 0.755750i −1.64980 − 0.484424i 0.105585 − 0.0678552i −0.745357 − 1.63210i −0.654861 − 0.755750i 1.44649 + 0.929603i
256.2 0.0765926 + 0.532713i 0.415415 − 0.909632i 1.64107 − 0.481861i −0.654861 + 0.755750i 0.516391 + 0.151626i 1.59341 − 1.02402i 0.829533 + 1.81642i −0.654861 − 0.755750i −0.452755 − 0.290968i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.3
Significant digits:
Format:

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.b ✓ 30
23.c even 11 1 inner 345.2.m.b ✓ 30
23.c even 11 1 7935.2.a.bo 15
23.d odd 22 1 7935.2.a.bn 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.m.b ✓ 30 1.a even 1 1 trivial
345.2.m.b ✓ 30 23.c even 11 1 inner
7935.2.a.bn 15 23.d odd 22 1
7935.2.a.bo 15 23.c even 11 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} - 6 T_{2}^{29} + 20 T_{2}^{28} - 40 T_{2}^{27} + 71 T_{2}^{26} - 183 T_{2}^{25} + 635 T_{2}^{24} + \cdots + 529 \) acting on \(S_{2}^{\mathrm{new}}(345, [\chi])\). Copy content Toggle raw display