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Properties

Label 345.3.k.a
Level $345$
Weight $3$
Character orbit 345.k
Analytic conductor $9.401$
Analytic rank $0$
Dimension $88$
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] = [345,3,Mod(208,345)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(345, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("345.208");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 345.k (of order \(4\), degree \(2\), 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: \(9.40056912043\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(44\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 8 q^{2} + 16 q^{5} + 16 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 8 q^{2} + 16 q^{5} + 16 q^{7} - 24 q^{8} - 8 q^{10} - 48 q^{12} + 56 q^{13} - 48 q^{15} - 256 q^{16} + 24 q^{17} + 24 q^{18} + 112 q^{20} + 48 q^{21} + 48 q^{22} - 168 q^{25} - 304 q^{26} - 312 q^{28} + 144 q^{30} - 88 q^{31} - 128 q^{32} + 144 q^{33} + 168 q^{35} + 528 q^{36} + 320 q^{38} - 416 q^{40} + 264 q^{41} + 96 q^{42} - 128 q^{43} - 72 q^{45} - 184 q^{47} + 192 q^{48} + 120 q^{50} + 32 q^{52} - 288 q^{53} + 216 q^{55} + 288 q^{56} - 48 q^{57} - 328 q^{58} - 272 q^{61} - 128 q^{62} + 48 q^{63} + 32 q^{65} + 144 q^{66} + 592 q^{67} + 656 q^{68} - 152 q^{70} + 552 q^{71} + 72 q^{72} - 192 q^{73} + 144 q^{75} - 464 q^{76} - 88 q^{77} - 336 q^{78} + 224 q^{80} - 792 q^{81} + 568 q^{82} + 288 q^{83} + 296 q^{85} - 352 q^{86} - 144 q^{87} - 312 q^{88} + 192 q^{90} - 528 q^{91} - 896 q^{95} - 632 q^{97} - 192 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
208.1 −2.75965 + 2.75965i −1.22474 − 1.22474i − 11.2314i −2.51860 + 4.31934i 6.75974 2.92739 − 2.92739i 19.9560 + 19.9560i 3.00000i −4.96941 − 18.8703i
208.2 −2.52653 + 2.52653i 1.22474 + 1.22474i − 8.76668i −3.57999 − 3.49051i −6.18870 −0.352714 + 0.352714i 12.0432 + 12.0432i 3.00000i 17.8638 − 0.226053i
208.3 −2.49894 + 2.49894i −1.22474 − 1.22474i − 8.48937i 2.88241 − 4.08555i 6.12112 6.54086 − 6.54086i 11.2187 + 11.2187i 3.00000i 3.00659 + 17.4125i
208.4 −2.46101 + 2.46101i 1.22474 + 1.22474i − 8.11314i −3.33404 + 3.72615i −6.02822 4.22537 − 4.22537i 10.1225 + 10.1225i 3.00000i −0.964971 − 17.3752i
208.5 −2.16185 + 2.16185i −1.22474 − 1.22474i − 5.34715i 4.99765 + 0.153299i 5.29542 −0.602215 + 0.602215i 2.91233 + 2.91233i 3.00000i −11.1356 + 10.4727i
208.6 −2.11115 + 2.11115i −1.22474 − 1.22474i − 4.91389i 2.94727 + 4.03901i 5.17124 −4.11858 + 4.11858i 1.92936 + 1.92936i 3.00000i −14.7491 − 2.30481i
208.7 −2.09859 + 2.09859i 1.22474 + 1.22474i − 4.80819i 2.70072 + 4.20786i −5.14048 6.97301 − 6.97301i 1.69605 + 1.69605i 3.00000i −14.4983 − 3.16288i
208.8 −2.09516 + 2.09516i −1.22474 − 1.22474i − 4.77943i −4.96308 − 0.606477i 5.13208 −3.35978 + 3.35978i 1.63304 + 1.63304i 3.00000i 11.6691 − 9.12781i
208.9 −2.09213 + 2.09213i 1.22474 + 1.22474i − 4.75401i −3.10949 + 3.91549i −5.12465 −9.30568 + 9.30568i 1.57748 + 1.57748i 3.00000i −1.68625 − 14.6972i
208.10 −1.49620 + 1.49620i 1.22474 + 1.22474i − 0.477210i −2.30935 − 4.43474i −3.66492 0.396141 − 0.396141i −5.27079 − 5.27079i 3.00000i 10.0905 + 3.17999i
208.11 −1.41610 + 1.41610i 1.22474 + 1.22474i − 0.0106838i 4.99512 − 0.220789i −3.46872 5.18852 − 5.18852i −5.64927 − 5.64927i 3.00000i −6.76094 + 7.38626i
208.12 −1.38782 + 1.38782i −1.22474 − 1.22474i 0.147924i −1.39429 − 4.80166i 3.39945 3.45675 − 3.45675i −5.75656 − 5.75656i 3.00000i 8.59885 + 4.72881i
208.13 −1.30981 + 1.30981i −1.22474 − 1.22474i 0.568803i 1.59586 − 4.73848i 3.20836 −9.52475 + 9.52475i −5.98426 − 5.98426i 3.00000i 4.11623 + 8.29678i
208.14 −1.30862 + 1.30862i −1.22474 − 1.22474i 0.575036i 1.60139 + 4.73662i 3.20545 8.75050 − 8.75050i −5.98698 − 5.98698i 3.00000i −8.29404 − 4.10281i
208.15 −1.19890 + 1.19890i 1.22474 + 1.22474i 1.12527i 4.50752 + 2.16386i −2.93670 −8.99074 + 8.99074i −6.14469 − 6.14469i 3.00000i −7.99832 + 2.80981i
208.16 −1.06187 + 1.06187i 1.22474 + 1.22474i 1.74487i −4.42546 + 2.32709i −2.60103 3.30764 − 3.30764i −6.10030 − 6.10030i 3.00000i 2.22819 − 7.17031i
208.17 −0.573440 + 0.573440i −1.22474 − 1.22474i 3.34233i −4.16459 + 2.76698i 1.40464 −5.25117 + 5.25117i −4.21039 − 4.21039i 3.00000i 0.801447 − 3.97484i
208.18 −0.458207 + 0.458207i 1.22474 + 1.22474i 3.58009i 2.56859 − 4.28979i −1.12237 −2.98825 + 2.98825i −3.47325 − 3.47325i 3.00000i 0.788667 + 3.14256i
208.19 −0.347591 + 0.347591i 1.22474 + 1.22474i 3.75836i −0.342492 + 4.98826i −0.851421 −0.00701469 + 0.00701469i −2.69674 − 2.69674i 3.00000i −1.61483 − 1.85292i
208.20 −0.241915 + 0.241915i −1.22474 − 1.22474i 3.88295i 2.03436 + 4.56742i 0.592568 −2.21557 + 2.21557i −1.90701 − 1.90701i 3.00000i −1.59707 − 0.612786i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 208.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 345.3.k.a ✓ 88
5.c odd 4 1 inner 345.3.k.a ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.3.k.a ✓ 88 1.a even 1 1 trivial
345.3.k.a ✓ 88 5.c odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(345, [\chi])\).