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Properties

Label 310.3.m.b
Level $310$
Weight $3$
Character orbit 310.m
Analytic conductor $8.447$
Analytic rank $0$
Dimension $48$
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] = [310,3,Mod(91,310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(310, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("310.91");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 310 = 2 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 310.m (of order \(10\), degree \(4\), 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: \(8.44688819517\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 24 q^{4} + 6 q^{7} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 24 q^{4} + 6 q^{7} + 28 q^{9} + 10 q^{11} + 30 q^{13} - 12 q^{14} - 48 q^{16} + 70 q^{17} - 24 q^{18} - 32 q^{19} + 120 q^{20} - 40 q^{21} + 40 q^{22} + 70 q^{23} + 240 q^{25} - 270 q^{27} + 12 q^{28} + 50 q^{29} + 156 q^{31} + 106 q^{33} - 120 q^{34} - 30 q^{35} - 384 q^{36} - 76 q^{38} - 48 q^{39} - 352 q^{41} - 160 q^{42} + 410 q^{43} + 80 q^{44} - 220 q^{45} - 40 q^{46} + 64 q^{47} - 226 q^{49} + 300 q^{51} - 60 q^{52} + 120 q^{53} + 120 q^{54} + 90 q^{55} + 16 q^{56} + 80 q^{58} + 140 q^{59} + 200 q^{62} + 148 q^{63} - 96 q^{64} - 130 q^{65} - 376 q^{66} - 604 q^{67} - 276 q^{69} + 20 q^{70} - 154 q^{71} - 48 q^{72} + 30 q^{73} + 520 q^{74} + 36 q^{76} + 330 q^{77} + 92 q^{78} + 50 q^{79} - 240 q^{80} - 260 q^{81} + 72 q^{82} - 160 q^{83} + 320 q^{84} + 290 q^{85} + 80 q^{86} + 288 q^{87} + 90 q^{89} - 120 q^{90} + 680 q^{91} + 392 q^{93} - 144 q^{94} - 60 q^{95} - 6 q^{97} - 80 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1 −1.14412 − 0.831254i −2.50014 − 3.44115i 0.618034 + 1.90211i 2.23607 6.01535i −2.45831 − 7.56591i 0.874032 − 2.68999i −2.80965 + 8.64720i −2.55834 − 1.85874i
91.2 −1.14412 − 0.831254i −1.80304 − 2.48167i 0.618034 + 1.90211i 2.23607 4.33812i 1.02897 + 3.16685i 0.874032 − 2.68999i −0.126587 + 0.389595i −2.55834 − 1.85874i
91.3 −1.14412 − 0.831254i −1.39445 − 1.91929i 0.618034 + 1.90211i 2.23607 3.35505i 3.79988 + 11.6948i 0.874032 − 2.68999i 1.04195 − 3.20679i −2.55834 − 1.85874i
91.4 −1.14412 − 0.831254i 0.981325 + 1.35068i 0.618034 + 1.90211i 2.23607 − 2.36107i 0.396989 + 1.22181i 0.874032 − 2.68999i 1.91982 − 5.90860i −2.55834 − 1.85874i
91.5 −1.14412 − 0.831254i 1.35045 + 1.85874i 0.618034 + 1.90211i 2.23607 − 3.24919i −3.26889 − 10.0606i 0.874032 − 2.68999i 1.14997 − 3.53924i −2.55834 − 1.85874i
91.6 −1.14412 − 0.831254i 3.36585 + 4.63270i 0.618034 + 1.90211i 2.23607 − 8.09826i −0.695780 − 2.14139i 0.874032 − 2.68999i −7.35178 + 22.6265i −2.55834 − 1.85874i
91.7 1.14412 + 0.831254i −2.89917 − 3.99036i 0.618034 + 1.90211i 2.23607 − 6.97541i 3.21926 + 9.90786i −0.874032 + 2.68999i −4.73666 + 14.5779i 2.55834 + 1.85874i
91.8 1.14412 + 0.831254i −2.69254 − 3.70596i 0.618034 + 1.90211i 2.23607 − 6.47826i −3.40106 − 10.4674i −0.874032 + 2.68999i −3.70323 + 11.3974i 2.55834 + 1.85874i
91.9 1.14412 + 0.831254i −0.343554 − 0.472862i 0.618034 + 1.90211i 2.23607 − 0.826593i −1.97712 − 6.08496i −0.874032 + 2.68999i 2.67558 − 8.23460i 2.55834 + 1.85874i
91.10 1.14412 + 0.831254i 0.443176 + 0.609979i 0.618034 + 1.90211i 2.23607 1.06628i 1.87820 + 5.78050i −0.874032 + 2.68999i 2.60548 − 8.01885i 2.55834 + 1.85874i
91.11 1.14412 + 0.831254i 2.31351 + 3.18427i 0.618034 + 1.90211i 2.23607 5.56631i 2.44245 + 7.51709i −0.874032 + 2.68999i −2.00610 + 6.17415i 2.55834 + 1.85874i
91.12 1.14412 + 0.831254i 3.17858 + 4.37493i 0.618034 + 1.90211i 2.23607 7.64767i −2.81869 − 8.67503i −0.874032 + 2.68999i −6.25555 + 19.2526i 2.55834 + 1.85874i
151.1 −0.437016 − 1.34500i −5.50622 − 1.78908i −1.61803 + 1.17557i −2.23607 8.18770i −1.44507 + 1.04990i 2.28825 + 1.66251i 19.8365 + 14.4120i 0.977198 + 3.00750i
151.2 −0.437016 − 1.34500i −2.24839 − 0.730547i −1.61803 + 1.17557i −2.23607 3.34334i 9.21847 − 6.69761i 2.28825 + 1.66251i −2.75958 − 2.00495i 0.977198 + 3.00750i
151.3 −0.437016 − 1.34500i −2.12242 − 0.689615i −1.61803 + 1.17557i −2.23607 3.15602i −3.87635 + 2.81633i 2.28825 + 1.66251i −3.25207 − 2.36277i 0.977198 + 3.00750i
151.4 −0.437016 − 1.34500i 2.33407 + 0.758384i −1.61803 + 1.17557i −2.23607 − 3.47074i 3.65101 − 2.65262i 2.28825 + 1.66251i −2.40844 − 1.74983i 0.977198 + 3.00750i
151.5 −0.437016 − 1.34500i 3.32023 + 1.07881i −1.61803 + 1.17557i −2.23607 − 4.93715i −8.88019 + 6.45184i 2.28825 + 1.66251i 2.57893 + 1.87370i 0.977198 + 3.00750i
151.6 −0.437016 − 1.34500i 4.22273 + 1.37205i −1.61803 + 1.17557i −2.23607 − 6.27917i 5.61041 − 4.07620i 2.28825 + 1.66251i 8.66781 + 6.29753i 0.977198 + 3.00750i
151.7 0.437016 + 1.34500i −4.64274 − 1.50852i −1.61803 + 1.17557i −2.23607 − 6.90372i 5.97000 − 4.33746i −2.28825 − 1.66251i 11.9983 + 8.71725i −0.977198 − 3.00750i
151.8 0.437016 + 1.34500i −3.10963 − 1.01038i −1.61803 + 1.17557i −2.23607 − 4.62400i −5.75112 + 4.17843i −2.28825 − 1.66251i 1.36779 + 0.993758i −0.977198 − 3.00750i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.f odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 310.3.m.b ✓ 48
31.f odd 10 1 inner 310.3.m.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.3.m.b ✓ 48 1.a even 1 1 trivial
310.3.m.b ✓ 48 31.f odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 68 T_{3}^{46} + 90 T_{3}^{45} + 3322 T_{3}^{44} - 6120 T_{3}^{43} - 134373 T_{3}^{42} + \cdots + 90\!\cdots\!61 \) acting on \(S_{3}^{\mathrm{new}}(310, [\chi])\). Copy content Toggle raw display