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Properties

Label 310.3.u.b
Level $310$
Weight $3$
Character orbit 310.u
Analytic conductor $8.447$
Analytic rank $0$
Dimension $80$
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(11,310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(310, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 23]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("310.11");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 310 = 2 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 310.u (of order \(30\), degree \(8\), 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: \(80\)
Relative dimension: \(10\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 6 q^{3} - 40 q^{4} + 16 q^{7} - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 6 q^{3} - 40 q^{4} + 16 q^{7} - 44 q^{9} + 24 q^{11} - 12 q^{12} - 48 q^{13} + 24 q^{14} - 80 q^{16} - 40 q^{17} - 56 q^{18} - 102 q^{19} + 100 q^{20} - 40 q^{21} + 4 q^{22} + 140 q^{23} - 200 q^{25} + 48 q^{26} + 30 q^{27} + 72 q^{28} - 60 q^{29} + 40 q^{31} - 54 q^{33} - 16 q^{34} - 60 q^{35} + 192 q^{36} - 96 q^{37} - 188 q^{38} + 298 q^{39} - 72 q^{41} + 80 q^{42} - 116 q^{43} - 52 q^{44} - 140 q^{45} + 120 q^{46} + 136 q^{47} - 24 q^{48} + 794 q^{49} + 16 q^{51} + 24 q^{52} + 170 q^{53} - 240 q^{54} - 50 q^{55} + 8 q^{56} - 60 q^{57} - 26 q^{59} - 176 q^{62} - 476 q^{63} - 160 q^{64} - 180 q^{65} + 432 q^{66} + 76 q^{67} - 240 q^{68} - 1204 q^{69} + 20 q^{70} - 140 q^{71} + 128 q^{72} + 218 q^{73} - 144 q^{74} + 30 q^{75} + 336 q^{76} + 980 q^{77} + 660 q^{78} + 914 q^{79} - 200 q^{80} - 606 q^{81} - 504 q^{82} + 860 q^{83} + 300 q^{84} - 130 q^{85} + 832 q^{86} + 484 q^{87} - 72 q^{88} + 200 q^{89} + 240 q^{90} + 90 q^{91} + 1130 q^{93} + 8 q^{94} - 120 q^{95} - 96 q^{97} - 256 q^{98} - 2526 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −0.437016 − 1.34500i −4.08416 + 3.67739i −1.61803 + 1.17557i −1.11803 − 1.93649i 6.73092 + 3.88610i −0.614228 − 5.84399i 2.28825 + 1.66251i 2.21637 − 21.0874i −2.11598 + 2.35003i
11.2 −0.437016 − 1.34500i −1.86669 + 1.68077i −1.61803 + 1.17557i −1.11803 − 1.93649i 3.07641 + 1.77616i 0.494079 + 4.70085i 2.28825 + 1.66251i −0.281232 + 2.67575i −2.11598 + 2.35003i
11.3 −0.437016 − 1.34500i 0.924651 − 0.832559i −1.61803 + 1.17557i −1.11803 − 1.93649i −1.52388 − 0.879811i 0.393783 + 3.74660i 2.28825 + 1.66251i −0.778932 + 7.41104i −2.11598 + 2.35003i
11.4 −0.437016 − 1.34500i 2.07035 − 1.86415i −1.61803 + 1.17557i −1.11803 − 1.93649i −3.41206 − 1.96995i −0.136380 − 1.29757i 2.28825 + 1.66251i −0.129467 + 1.23179i −2.11598 + 2.35003i
11.5 −0.437016 − 1.34500i 4.24301 − 3.82042i −1.61803 + 1.17557i −1.11803 − 1.93649i −6.99271 − 4.03725i −1.09800 − 10.4468i 2.28825 + 1.66251i 2.46674 − 23.4694i −2.11598 + 2.35003i
11.6 0.437016 + 1.34500i −3.07807 + 2.77151i −1.61803 + 1.17557i −1.11803 − 1.93649i −5.07283 − 2.92880i −0.478138 − 4.54918i −2.28825 − 1.66251i 0.852509 − 8.11108i 2.11598 − 2.35003i
11.7 0.437016 + 1.34500i −0.623498 + 0.561400i −1.61803 + 1.17557i −1.11803 − 1.93649i −1.02756 − 0.593262i 0.951406 + 9.05203i −2.28825 − 1.66251i −0.867176 + 8.25063i 2.11598 − 2.35003i
11.8 0.437016 + 1.34500i 0.591263 − 0.532376i −1.61803 + 1.17557i −1.11803 − 1.93649i 0.974435 + 0.562591i −0.303545 − 2.88804i −2.28825 − 1.66251i −0.874588 + 8.32115i 2.11598 − 2.35003i
11.9 0.437016 + 1.34500i 0.664670 − 0.598471i −1.61803 + 1.17557i −1.11803 − 1.93649i 1.09541 + 0.632437i −1.19076 − 11.3293i −2.28825 − 1.66251i −0.857138 + 8.15513i 2.11598 − 2.35003i
11.10 0.437016 + 1.34500i 3.73280 − 3.36103i −1.61803 + 1.17557i −1.11803 − 1.93649i 6.15187 + 3.55178i −0.00152406 − 0.0145005i −2.28825 − 1.66251i 1.69653 − 16.1414i 2.11598 − 2.35003i
21.1 −1.14412 − 0.831254i −3.83177 + 0.402736i 0.618034 + 1.90211i 1.11803 + 1.93649i 4.71879 + 2.72440i −12.8674 − 2.73504i 0.874032 − 2.68999i 5.71696 − 1.21518i 0.330548 − 3.14495i
21.2 −1.14412 − 0.831254i −3.41869 + 0.359319i 0.618034 + 1.90211i 1.11803 + 1.93649i 4.21009 + 2.43070i 4.86026 + 1.03308i 0.874032 − 2.68999i 2.75503 − 0.585601i 0.330548 − 3.14495i
21.3 −1.14412 − 0.831254i 0.708936 − 0.0745122i 0.618034 + 1.90211i 1.11803 + 1.93649i −0.873048 − 0.504055i −1.66862 − 0.354675i 0.874032 − 2.68999i −8.30629 + 1.76556i 0.330548 − 3.14495i
21.4 −1.14412 − 0.831254i 1.03656 − 0.108947i 0.618034 + 1.90211i 1.11803 + 1.93649i −1.27651 − 0.736996i −1.79397 − 0.381321i 0.874032 − 2.68999i −7.74074 + 1.64535i 0.330548 − 3.14495i
21.5 −1.14412 − 0.831254i 3.78241 − 0.397547i 0.618034 + 1.90211i 1.11803 + 1.93649i −4.65800 − 2.68930i 12.7733 + 2.71504i 0.874032 − 2.68999i 5.34525 − 1.13617i 0.330548 − 3.14495i
21.6 1.14412 + 0.831254i −5.79769 + 0.609362i 0.618034 + 1.90211i 1.11803 + 1.93649i −7.13980 − 4.12217i 11.0007 + 2.33826i −0.874032 + 2.68999i 24.4386 − 5.19458i −0.330548 + 3.14495i
21.7 1.14412 + 0.831254i −2.12020 + 0.222842i 0.618034 + 1.90211i 1.11803 + 1.93649i −2.61101 − 1.50747i −3.22866 − 0.686272i −0.874032 + 2.68999i −4.35774 + 0.926266i −0.330548 + 3.14495i
21.8 1.14412 + 0.831254i 0.135010 − 0.0141901i 0.618034 + 1.90211i 1.11803 + 1.93649i 0.166263 + 0.0959921i −9.36908 − 1.99146i −0.874032 + 2.68999i −8.78530 + 1.86737i −0.330548 + 3.14495i
21.9 1.14412 + 0.831254i 1.42333 − 0.149598i 0.618034 + 1.90211i 1.11803 + 1.93649i 1.75281 + 1.01199i 7.38203 + 1.56910i −0.874032 + 2.68999i −6.79985 + 1.44535i −0.330548 + 3.14495i
21.10 1.14412 + 0.831254i 4.63699 − 0.487368i 0.618034 + 1.90211i 1.11803 + 1.93649i 5.71042 + 3.29691i 0.930943 + 0.197878i −0.874032 + 2.68999i 12.4608 − 2.64864i −0.330548 + 3.14495i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.h odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 310.3.u.b ✓ 80
31.h odd 30 1 inner 310.3.u.b ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.3.u.b ✓ 80 1.a even 1 1 trivial
310.3.u.b ✓ 80 31.h odd 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{80} + 6 T_{3}^{79} + 85 T_{3}^{78} + 428 T_{3}^{77} + 3164 T_{3}^{76} + 18800 T_{3}^{75} + \cdots + 11\!\cdots\!61 \) acting on \(S_{3}^{\mathrm{new}}(310, [\chi])\). Copy content Toggle raw display