↑

Properties

Label 310.3.r.a
Level $310$
Weight $3$
Character orbit 310.r
Analytic conductor $8.447$
Analytic rank $0$
Dimension $128$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [310,3,Mod(33,310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(310, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([15, 16]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("310.33");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 310 = 2 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 310.r (of order \(20\), 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: \(128\)
Relative dimension: \(16\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 32 q^{2} + 4 q^{3} + 8 q^{5} + 8 q^{6} - 12 q^{7} + 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 32 q^{2} + 4 q^{3} + 8 q^{5} + 8 q^{6} - 12 q^{7} + 64 q^{8} + 34 q^{10} + 64 q^{11} + 8 q^{12} + 10 q^{15} + 128 q^{16} - 16 q^{17} - 96 q^{18} - 48 q^{20} - 120 q^{21} + 64 q^{22} + 94 q^{23} + 68 q^{25} - 14 q^{27} - 16 q^{28} + 20 q^{30} + 160 q^{31} - 512 q^{32} + 76 q^{33} + 40 q^{35} + 768 q^{36} - 628 q^{37} - 116 q^{38} - 24 q^{40} + 204 q^{41} - 120 q^{42} - 12 q^{43} + 372 q^{45} - 192 q^{46} - 144 q^{47} + 24 q^{48} - 64 q^{50} + 400 q^{51} - 84 q^{53} - 404 q^{55} - 32 q^{56} + 124 q^{57} - 72 q^{58} - 120 q^{60} + 352 q^{61} + 280 q^{62} - 400 q^{63} - 426 q^{65} - 188 q^{66} + 4 q^{67} + 32 q^{68} - 36 q^{70} + 560 q^{71} - 192 q^{72} - 144 q^{73} - 190 q^{75} - 232 q^{76} - 244 q^{77} + 380 q^{78} - 32 q^{80} + 48 q^{81} + 64 q^{82} + 282 q^{83} + 364 q^{85} + 296 q^{86} - 316 q^{87} + 192 q^{88} - 144 q^{90} + 504 q^{91} + 8 q^{92} - 382 q^{95} - 32 q^{96} - 298 q^{97} - 1448 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1 −1.39680 + 0.221232i −5.47277 − 0.866801i 1.90211 − 0.618034i −1.71383 − 4.69711i 7.83614 2.83187 + 1.44291i −2.52015 + 1.28408i 20.6403 + 6.70645i 3.43302 + 6.18178i
33.2 −1.39680 + 0.221232i −4.91209 − 0.777998i 1.90211 − 0.618034i 4.11343 + 2.84248i 7.03333 −7.44490 − 3.79337i −2.52015 + 1.28408i 14.9638 + 4.86204i −6.37449 − 3.06037i
33.3 −1.39680 + 0.221232i −3.69079 − 0.584563i 1.90211 − 0.618034i −4.18348 + 2.73834i 5.28462 9.00285 + 4.58718i −2.52015 + 1.28408i 4.72068 + 1.53384i 5.23769 − 4.75043i
33.4 −1.39680 + 0.221232i −3.44483 − 0.545607i 1.90211 − 0.618034i 4.82373 − 1.31592i 4.93244 9.12983 + 4.65188i −2.52015 + 1.28408i 3.00962 + 0.977886i −6.44667 + 2.90524i
33.5 −1.39680 + 0.221232i −3.02630 − 0.479319i 1.90211 − 0.618034i −3.96124 + 3.05099i 4.33318 −5.00369 − 2.54951i −2.52015 + 1.28408i 0.369231 + 0.119970i 4.85810 − 5.13798i
33.6 −1.39680 + 0.221232i −2.28952 − 0.362624i 1.90211 − 0.618034i −0.0220877 − 4.99995i 3.27823 −1.88026 − 0.958038i −2.52015 + 1.28408i −3.44912 − 1.12069i 1.13700 + 6.97906i
33.7 −1.39680 + 0.221232i 0.00838011 + 0.00132728i 1.90211 − 0.618034i −4.33748 − 2.48723i −0.0119990 −7.36725 − 3.75380i −2.52015 + 1.28408i −8.55944 − 2.78113i 6.60885 + 2.51458i
33.8 −1.39680 + 0.221232i 0.150754 + 0.0238771i 1.90211 − 0.618034i 2.68006 + 4.22105i −0.215856 10.0109 + 5.10079i −2.52015 + 1.28408i −8.53735 − 2.77395i −4.67735 − 5.30306i
33.9 −1.39680 + 0.221232i 0.593250 + 0.0939615i 1.90211 − 0.618034i 0.824983 + 4.93147i −0.849440 −3.53497 − 1.80115i −2.52015 + 1.28408i −8.21639 − 2.66967i −2.24334 − 6.70578i
33.10 −1.39680 + 0.221232i 0.770886 + 0.122096i 1.90211 − 0.618034i 4.75049 − 1.55976i −1.10379 −8.54482 − 4.35380i −2.52015 + 1.28408i −7.98015 − 2.59291i −6.29043 + 3.22963i
33.11 −1.39680 + 0.221232i 1.84197 + 0.291739i 1.90211 − 0.618034i −4.82467 − 1.31246i −2.63741 3.49502 + 1.78080i −2.52015 + 1.28408i −5.25177 − 1.70640i 7.02947 + 0.765883i
33.12 −1.39680 + 0.221232i 2.00224 + 0.317124i 1.90211 − 0.618034i 0.577812 − 4.96650i −2.86689 2.96858 + 1.51256i −2.52015 + 1.28408i −4.65111 − 1.51124i 0.291659 + 7.06505i
33.13 −1.39680 + 0.221232i 4.15228 + 0.657657i 1.90211 − 0.618034i 1.70001 − 4.70212i −5.94541 7.52283 + 3.83307i −2.52015 + 1.28408i 8.24941 + 2.68040i −1.33432 + 6.94403i
33.14 −1.39680 + 0.221232i 4.21394 + 0.667422i 1.90211 − 0.618034i 4.93143 + 0.825205i −6.03369 −5.49948 − 2.80212i −2.52015 + 1.28408i 8.75230 + 2.84380i −7.07080 − 0.0616586i
33.15 −1.39680 + 0.221232i 4.59311 + 0.727478i 1.90211 − 0.618034i −2.56058 + 4.29458i −6.57661 3.75867 + 1.91514i −2.52015 + 1.28408i 12.0080 + 3.90163i 2.62653 − 6.56516i
33.16 −1.39680 + 0.221232i 5.37274 + 0.850958i 1.90211 − 0.618034i −4.03465 − 2.95324i −7.69291 −11.0027 − 5.60615i −2.52015 + 1.28408i 19.5827 + 6.36280i 6.28896 + 3.23250i
47.1 −1.39680 − 0.221232i −5.47277 + 0.866801i 1.90211 + 0.618034i −1.71383 + 4.69711i 7.83614 2.83187 − 1.44291i −2.52015 − 1.28408i 20.6403 − 6.70645i 3.43302 − 6.18178i
47.2 −1.39680 − 0.221232i −4.91209 + 0.777998i 1.90211 + 0.618034i 4.11343 − 2.84248i 7.03333 −7.44490 + 3.79337i −2.52015 − 1.28408i 14.9638 − 4.86204i −6.37449 + 3.06037i
47.3 −1.39680 − 0.221232i −3.69079 + 0.584563i 1.90211 + 0.618034i −4.18348 − 2.73834i 5.28462 9.00285 − 4.58718i −2.52015 − 1.28408i 4.72068 − 1.53384i 5.23769 + 4.75043i
47.4 −1.39680 − 0.221232i −3.44483 + 0.545607i 1.90211 + 0.618034i 4.82373 + 1.31592i 4.93244 9.12983 − 4.65188i −2.52015 − 1.28408i 3.00962 − 0.977886i −6.44667 − 2.90524i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
31.d even 5 1 inner
155.s odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 310.3.r.a ✓ 128
5.c odd 4 1 inner 310.3.r.a ✓ 128
31.d even 5 1 inner 310.3.r.a ✓ 128
155.s odd 20 1 inner 310.3.r.a ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.3.r.a ✓ 128 1.a even 1 1 trivial
310.3.r.a ✓ 128 5.c odd 4 1 inner
310.3.r.a ✓ 128 31.d even 5 1 inner
310.3.r.a ✓ 128 155.s odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{128} - 4 T_{3}^{127} + 8 T_{3}^{126} - 42 T_{3}^{125} - 1820 T_{3}^{124} + 7832 T_{3}^{123} + \cdots + 73\!\cdots\!41 \) acting on \(S_{3}^{\mathrm{new}}(310, [\chi])\). Copy content Toggle raw display