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Properties

Label 345.3.e.a
Level $345$
Weight $3$
Character orbit 345.e
Analytic conductor $9.401$
Analytic rank $0$
Dimension $60$
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(116,345)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(345, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("345.116");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 345.e (of order \(2\), degree \(1\), 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: \(60\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 8 q^{3} - 128 q^{4} - 22 q^{6} + 24 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 8 q^{3} - 128 q^{4} - 22 q^{6} + 24 q^{7} + 20 q^{9} - 38 q^{12} - 64 q^{13} + 20 q^{15} + 288 q^{16} + 90 q^{18} - 24 q^{19} - 28 q^{21} + 120 q^{22} - 36 q^{24} - 300 q^{25} - 88 q^{27} - 248 q^{28} + 80 q^{30} + 152 q^{31} - 64 q^{33} - 24 q^{34} - 126 q^{36} + 76 q^{39} + 176 q^{42} - 32 q^{43} + 326 q^{48} + 524 q^{49} - 20 q^{51} + 188 q^{52} - 436 q^{54} - 112 q^{57} - 180 q^{58} - 20 q^{60} - 224 q^{61} + 384 q^{63} - 116 q^{64} - 564 q^{66} - 520 q^{67} - 120 q^{70} + 60 q^{72} + 464 q^{73} - 40 q^{75} + 176 q^{76} + 422 q^{78} - 24 q^{79} + 316 q^{81} - 148 q^{82} + 264 q^{84} - 120 q^{85} - 452 q^{87} - 392 q^{88} + 180 q^{90} + 568 q^{91} + 292 q^{93} - 364 q^{94} - 378 q^{96} + 464 q^{97} - 488 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1 − 3.80864i −2.47278 − 1.69863i −10.5057 − 2.23607i −6.46948 + 9.41792i 6.62236 24.7779i 3.22929 + 8.40069i −8.51637
116.2 − 3.80643i 2.07450 + 2.16713i −10.4889 2.23607i 8.24902 − 7.89645i −1.30788 24.6996i −0.392883 + 8.99142i 8.51144
116.3 − 3.75623i 2.11684 − 2.12579i −10.1093 2.23607i −7.98496 − 7.95134i −5.33453 22.9478i −0.0379757 − 8.99992i 8.39918
116.4 − 3.70300i 2.99127 + 0.228735i −9.71222 − 2.23607i 0.847006 − 11.0767i 9.67807 21.1524i 8.89536 + 1.36841i −8.28016
116.5 − 3.48797i 0.956684 − 2.84337i −8.16593 − 2.23607i −9.91759 − 3.33688i −5.18659 14.5306i −7.16951 − 5.44041i −7.79934
116.6 − 3.47425i −0.554823 + 2.94825i −8.07038 − 2.23607i 10.2429 + 1.92759i 7.87612 14.1415i −8.38434 − 3.27151i −7.76865
116.7 − 3.37743i −2.50617 − 1.64898i −7.40701 2.23607i −5.56930 + 8.46440i −7.76992 11.5069i 3.56176 + 8.26522i 7.55216
116.8 − 3.19498i −2.97156 + 0.412070i −6.20788 2.23607i 1.31655 + 9.49408i 10.6589 7.05411i 8.66040 − 2.44899i 7.14419
116.9 − 3.15979i −0.349953 + 2.97952i −5.98430 2.23607i 9.41466 + 1.10578i 4.33882 6.26996i −8.75507 − 2.08539i 7.06551
116.10 − 3.08103i 2.48088 − 1.68678i −5.49274 2.23607i −5.19702 − 7.64367i 12.0310 4.59916i 3.30955 − 8.36940i 6.88939
116.11 − 3.03382i 1.37829 + 2.66464i −5.20409 − 2.23607i 8.08406 − 4.18149i −9.37880 3.65300i −5.20064 + 7.34530i −6.78384
116.12 − 2.80702i −2.81389 + 1.04021i −3.87937 − 2.23607i 2.91989 + 7.89864i −7.40993 − 0.338608i 6.83593 − 5.85407i −6.27669
116.13 − 2.56459i 2.98355 − 0.313734i −2.57714 2.23607i −0.804600 − 7.65160i −13.0908 − 3.64905i 8.80314 − 1.87208i 5.73461
116.14 − 2.52617i −0.428982 − 2.96917i −2.38154 2.23607i −7.50064 + 1.08368i −1.96180 − 4.08850i −8.63195 + 2.54744i 5.64869
116.15 − 2.39841i −2.99994 + 0.0185638i −1.75237 − 2.23607i 0.0445236 + 7.19509i 4.01397 − 5.39074i 8.99931 − 0.111381i −5.36301
116.16 − 2.20057i −1.81400 − 2.38944i −0.842496 − 2.23607i −5.25812 + 3.99182i 4.30011 − 6.94830i −2.41883 + 8.66887i −4.92062
116.17 − 2.09750i 2.57694 − 1.53603i −0.399513 − 2.23607i −3.22184 − 5.40513i −0.584859 − 7.55203i 4.28119 − 7.91653i −4.69016
116.18 − 1.88378i 1.96199 + 2.26949i 0.451382 − 2.23607i 4.27521 − 3.69596i 10.7298 − 8.38541i −1.30117 + 8.90545i −4.21225
116.19 − 1.81497i 2.76305 + 1.16858i 0.705875 2.23607i 2.12094 − 5.01485i 3.12281 − 8.54103i 6.26885 + 6.45767i 4.05840
116.20 − 1.55425i −0.698242 − 2.91761i 1.58429 − 2.23607i −4.53471 + 1.08525i −12.6446 − 8.67941i −8.02492 + 4.07440i −3.47542
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.60
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 345.3.e.a ✓ 60
3.b odd 2 1 inner 345.3.e.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.3.e.a ✓ 60 1.a even 1 1 trivial
345.3.e.a ✓ 60 3.b odd 2 1 inner

Hecke kernels

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