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Properties

Label 310.3.x.a
Level $310$
Weight $3$
Character orbit 310.x
Analytic conductor $8.447$
Analytic rank $0$
Dimension $256$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [310,3,Mod(7,310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(310, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([15, 56]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("310.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 310 = 2 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 310.x (of order \(60\), degree \(16\), 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: \(256\)
Relative dimension: \(16\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256 q - 64 q^{2} - 4 q^{3} - 8 q^{5} - 8 q^{6} + 24 q^{7} + 128 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 256 q - 64 q^{2} - 4 q^{3} - 8 q^{5} - 8 q^{6} + 24 q^{7} + 128 q^{8} - 52 q^{10} - 88 q^{11} - 8 q^{12} - 190 q^{15} + 256 q^{16} - 8 q^{17} + 96 q^{18} + 12 q^{20} + 120 q^{21} + 152 q^{22} - 94 q^{23} + 34 q^{25} + 80 q^{27} - 8 q^{28} + 52 q^{30} + 236 q^{31} - 1024 q^{32} + 56 q^{33} - 148 q^{35} - 768 q^{36} - 218 q^{37} + 212 q^{38} - 12 q^{40} - 144 q^{41} - 240 q^{42} - 36 q^{43} - 630 q^{45} + 192 q^{46} + 144 q^{47} - 24 q^{48} + 142 q^{50} + 464 q^{51} + 216 q^{53} - 40 q^{55} - 16 q^{56} - 220 q^{57} + 48 q^{58} - 96 q^{60} + 368 q^{61} - 364 q^{62} + 400 q^{63} + 210 q^{65} - 208 q^{66} + 26 q^{67} + 16 q^{68} - 12 q^{70} + 1288 q^{71} + 192 q^{72} + 408 q^{73} - 884 q^{75} + 184 q^{76} + 268 q^{77} + 1000 q^{78} + 32 q^{80} - 144 q^{81} - 4 q^{82} + 1248 q^{83} - 220 q^{85} - 392 q^{86} - 254 q^{87} + 96 q^{88} + 648 q^{90} + 216 q^{91} - 8 q^{92} - 672 q^{93} - 818 q^{95} + 32 q^{96} + 1090 q^{97} + 320 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.642040 + 1.26007i −3.14933 + 4.84955i −1.17557 − 1.61803i 2.52910 + 4.31320i −4.08879 − 7.08199i −0.534858 − 1.39335i 2.79360 − 0.442463i −9.93918 − 22.3238i −7.05873 + 0.417604i
7.2 −0.642040 + 1.26007i −2.78116 + 4.28261i −1.17557 − 1.61803i 2.15998 − 4.50938i −3.61079 − 6.25407i 1.63272 + 4.25338i 2.79360 − 0.442463i −6.94528 − 15.5994i 4.29536 + 5.61693i
7.3 −0.642040 + 1.26007i −1.99452 + 3.07128i −1.17557 − 1.61803i −3.75831 − 3.29774i −2.58949 − 4.48512i −3.03189 − 7.89833i 2.79360 − 0.442463i −1.79406 − 4.02954i 6.56838 − 2.61847i
7.4 −0.642040 + 1.26007i −1.73171 + 2.66661i −1.17557 − 1.61803i 4.99998 + 0.0141902i −2.24829 − 3.89415i −1.55720 − 4.05666i 2.79360 − 0.442463i −0.451321 − 1.01368i −3.22807 + 6.29123i
7.5 −0.642040 + 1.26007i −1.60549 + 2.47224i −1.17557 − 1.61803i −3.84996 + 3.19027i −2.08442 − 3.61031i −2.00887 − 5.23328i 2.79360 − 0.442463i 0.126258 + 0.283581i −1.54815 − 6.89951i
7.6 −0.642040 + 1.26007i −1.15774 + 1.78277i −1.17557 − 1.61803i −2.91327 + 4.06360i −1.50310 − 2.60345i 3.82473 + 9.96377i 2.79360 − 0.442463i 1.82273 + 4.09393i −3.25000 − 6.27993i
7.7 −0.642040 + 1.26007i −0.863874 + 1.33025i −1.17557 − 1.61803i −0.901922 − 4.91798i −1.12157 − 1.94262i 1.44833 + 3.77302i 2.79360 − 0.442463i 2.63734 + 5.92357i 6.77609 + 2.02105i
7.8 −0.642040 + 1.26007i −0.841864 + 1.29636i −1.17557 − 1.61803i 2.43946 + 4.36452i −1.09299 − 1.89312i 1.06956 + 2.78631i 2.79360 − 0.442463i 2.68882 + 6.03920i −7.06585 + 0.271702i
7.9 −0.642040 + 1.26007i 0.249741 − 0.384567i −1.17557 − 1.61803i 4.99474 − 0.229340i 0.324239 + 0.561599i −4.40590 − 11.4778i 2.79360 − 0.442463i 3.57511 + 8.02982i −2.91783 + 6.44098i
7.10 −0.642040 + 1.26007i 0.684757 − 1.05443i −1.17557 − 1.61803i 4.82978 − 1.29353i 0.889023 + 1.53983i 3.09997 + 8.07569i 2.79360 − 0.442463i 3.01769 + 6.77785i −1.47096 + 6.91638i
7.11 −0.642040 + 1.26007i 1.35426 − 2.08539i −1.17557 − 1.61803i −4.54133 − 2.09197i 1.75825 + 3.04537i 3.24117 + 8.44352i 2.79360 − 0.442463i 1.14583 + 2.57358i 5.55175 − 4.37928i
7.12 −0.642040 + 1.26007i 1.38300 − 2.12964i −1.17557 − 1.61803i −4.96627 + 0.579777i 1.79556 + 3.11000i −2.60177 − 6.77784i 2.79360 − 0.442463i 1.03797 + 2.33132i 2.45798 − 6.63011i
7.13 −0.642040 + 1.26007i 1.39985 − 2.15559i −1.17557 − 1.61803i 0.828904 + 4.93081i 1.81744 + 3.14789i −2.99737 − 7.80840i 2.79360 − 0.442463i 0.973668 + 2.18689i −6.74538 − 2.12130i
7.14 −0.642040 + 1.26007i 2.27094 − 3.49695i −1.17557 − 1.61803i 4.57908 + 2.00798i 2.94838 + 5.10674i 3.60882 + 9.40129i 2.79360 − 0.442463i −3.41083 − 7.66084i −5.47016 + 4.48078i
7.15 −0.642040 + 1.26007i 2.43152 − 3.74422i −1.17557 − 1.61803i 0.914383 − 4.91568i 3.15685 + 5.46783i −1.14329 − 2.97838i 2.79360 − 0.442463i −4.44622 − 9.98638i 5.60705 + 4.30825i
7.16 −0.642040 + 1.26007i 3.10534 − 4.78181i −1.17557 − 1.61803i −2.68295 + 4.21922i 4.03167 + 6.98307i 0.567755 + 1.47905i 2.79360 − 0.442463i −9.56189 − 21.4764i −3.59397 − 6.08961i
103.1 −1.39680 + 0.221232i −3.70978 + 4.58120i 1.90211 − 0.618034i 2.61029 + 4.26455i 4.16832 − 7.21975i −5.43356 + 3.52859i −2.52015 + 1.28408i −5.35369 − 25.1871i −4.58951 − 5.37926i
103.2 −1.39680 + 0.221232i −2.68185 + 3.31180i 1.90211 − 0.618034i −0.725089 − 4.94715i 3.01333 − 5.21924i −2.77293 + 1.80076i −2.52015 + 1.28408i −1.90454 − 8.96017i 2.10727 + 6.74977i
103.3 −1.39680 + 0.221232i −2.48545 + 3.06928i 1.90211 − 0.618034i 4.83678 + 1.26710i 2.79266 − 4.83703i 3.36614 − 2.18600i −2.52015 + 1.28408i −1.37179 − 6.45375i −7.03635 − 0.699837i
103.4 −1.39680 + 0.221232i −2.44976 + 3.02520i 1.90211 − 0.618034i 1.05896 − 4.88657i 2.75256 − 4.76757i 5.96300 − 3.87242i −2.52015 + 1.28408i −1.27932 − 6.01871i −0.398092 + 7.05985i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
31.g even 15 1 inner
155.w odd 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 310.3.x.a ✓ 256
5.c odd 4 1 inner 310.3.x.a ✓ 256
31.g even 15 1 inner 310.3.x.a ✓ 256
155.w odd 60 1 inner 310.3.x.a ✓ 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.3.x.a ✓ 256 1.a even 1 1 trivial
310.3.x.a ✓ 256 5.c odd 4 1 inner
310.3.x.a ✓ 256 31.g even 15 1 inner
310.3.x.a ✓ 256 155.w odd 60 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{256} + 4 T_{3}^{255} + 8 T_{3}^{254} - 88 T_{3}^{253} + 1596 T_{3}^{252} + 5720 T_{3}^{251} + \cdots + 61\!\cdots\!25 \) acting on \(S_{3}^{\mathrm{new}}(310, [\chi])\). Copy content Toggle raw display