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Properties

Label 350.3.w.b
Level $350$
Weight $3$
Character orbit 350.w
Analytic conductor $9.537$
Analytic rank $0$
Dimension $320$
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] = [350,3,Mod(23,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([33, 20]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.23");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 350.w (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: \(9.53680925261\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 320 q + 40 q^{2} + 2 q^{5} + 2 q^{7} + 160 q^{8} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q + 40 q^{2} + 2 q^{5} + 2 q^{7} + 160 q^{8} + 40 q^{9} + 16 q^{10} + 16 q^{11} + 30 q^{14} + 20 q^{15} - 160 q^{16} + 94 q^{17} - 464 q^{18} + 40 q^{19} - 16 q^{20} + 68 q^{21} - 32 q^{22} - 16 q^{23} + 66 q^{25} + 144 q^{27} - 8 q^{28} + 200 q^{29} - 98 q^{30} + 84 q^{31} + 640 q^{32} - 178 q^{33} - 200 q^{35} - 384 q^{36} - 186 q^{37} - 16 q^{38} + 80 q^{39} + 4 q^{40} + 128 q^{41} - 136 q^{42} + 100 q^{43} - 40 q^{44} - 174 q^{45} + 48 q^{46} - 110 q^{47} + 20 q^{50} + 56 q^{51} - 262 q^{53} + 192 q^{55} - 48 q^{56} - 244 q^{57} + 20 q^{58} - 120 q^{59} + 64 q^{60} + 216 q^{61} + 552 q^{62} + 388 q^{63} + 130 q^{65} - 16 q^{66} + 270 q^{67} - 88 q^{68} + 340 q^{69} + 62 q^{70} - 340 q^{71} + 272 q^{72} + 134 q^{73} - 220 q^{75} + 64 q^{76} - 1058 q^{77} - 468 q^{78} + 80 q^{79} + 72 q^{80} - 608 q^{81} + 256 q^{82} - 736 q^{83} - 220 q^{84} + 176 q^{85} - 40 q^{86} - 54 q^{87} + 148 q^{88} + 510 q^{89} - 1312 q^{90} + 42 q^{91} + 544 q^{92} + 66 q^{93} + 236 q^{95} + 1024 q^{97} + 212 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −1.09905 − 0.889993i −4.51865 − 2.93445i 0.415823 + 1.95630i 4.64788 − 1.84314i 2.35459 + 7.24668i −6.65042 − 2.18446i 1.28408 − 2.52015i 8.14661 + 18.2976i −6.74864 − 2.11088i
23.2 −1.09905 − 0.889993i −3.75708 − 2.43988i 0.415823 + 1.95630i 0.542682 + 4.97046i 1.95775 + 6.02532i −2.39094 + 6.57901i 1.28408 − 2.52015i 4.50202 + 10.1117i 3.82724 − 5.94577i
23.3 −1.09905 − 0.889993i −3.53426 − 2.29518i 0.415823 + 1.95630i −4.07275 + 2.90047i 1.84164 + 5.66799i 6.56041 − 2.44151i 1.28408 − 2.52015i 3.56254 + 8.00161i 7.05755 + 0.436959i
23.4 −1.09905 − 0.889993i −2.80325 − 1.82045i 0.415823 + 1.95630i −2.34150 − 4.41785i 1.46072 + 4.49564i −1.81873 − 6.75960i 1.28408 − 2.52015i 0.883533 + 1.98445i −1.35842 + 6.93936i
23.5 −1.09905 − 0.889993i −2.80305 − 1.82032i 0.415823 + 1.95630i 1.95222 + 4.60313i 1.46062 + 4.49532i 4.44510 − 5.40750i 1.28408 − 2.52015i 0.882883 + 1.98299i 1.95117 − 6.79654i
23.6 −1.09905 − 0.889993i −2.78790 − 1.81049i 0.415823 + 1.95630i −4.99948 + 0.0721797i 1.45273 + 4.47103i −5.90229 + 3.76338i 1.28408 − 2.52015i 0.833917 + 1.87301i 5.55892 + 4.37017i
23.7 −1.09905 − 0.889993i −2.16468 − 1.40576i 0.415823 + 1.95630i 3.05544 − 3.95781i 1.12798 + 3.47156i 0.353012 + 6.99109i 1.28408 − 2.52015i −0.950940 − 2.13585i −6.88051 + 1.63051i
23.8 −1.09905 − 0.889993i −1.45769 − 0.946636i 0.415823 + 1.95630i 2.30440 − 4.43731i 0.759577 + 2.33774i 6.52871 − 2.52506i 1.28408 − 2.52015i −2.43188 − 5.46210i −6.48183 + 2.82593i
23.9 −1.09905 − 0.889993i −0.414765 − 0.269352i 0.415823 + 1.95630i 0.523415 + 4.97253i 0.216127 + 0.665169i −5.25248 − 4.62725i 1.28408 − 2.52015i −3.56115 − 7.99847i 3.85026 − 5.93090i
23.10 −1.09905 − 0.889993i −0.187835 − 0.121981i 0.415823 + 1.95630i −4.92559 − 0.859398i 0.0978773 + 0.301235i 5.76409 + 3.97181i 1.28408 − 2.52015i −3.64023 − 8.17608i 4.64861 + 5.32826i
23.11 −1.09905 − 0.889993i −0.0727315 − 0.0472324i 0.415823 + 1.95630i 4.40858 + 2.35889i 0.0378991 + 0.116641i 4.55135 + 5.31838i 1.28408 − 2.52015i −3.65757 − 8.21504i −2.74586 − 6.51615i
23.12 −1.09905 − 0.889993i 0.685415 + 0.445113i 0.415823 + 1.95630i 4.61653 + 1.92033i −0.357157 − 1.09922i −6.99208 − 0.332875i 1.28408 − 2.52015i −3.38896 − 7.61173i −3.36472 − 6.21922i
23.13 −1.09905 − 0.889993i 1.52793 + 0.992250i 0.415823 + 1.95630i −1.80465 − 4.66297i −0.796177 − 2.45038i −6.23029 − 3.19116i 1.28408 − 2.52015i −2.31062 − 5.18973i −2.16661 + 6.73096i
23.14 −1.09905 − 0.889993i 1.79928 + 1.16846i 0.415823 + 1.95630i −2.99903 − 4.00072i −0.937571 − 2.88555i −2.19726 + 6.64621i 1.28408 − 2.52015i −1.78854 − 4.01713i −0.264530 + 7.06612i
23.15 −1.09905 − 0.889993i 1.87023 + 1.21454i 0.415823 + 1.95630i 3.46044 − 3.60907i −0.974544 − 2.99934i 5.10775 − 4.78653i 1.28408 − 2.52015i −1.63798 − 3.67895i −7.01524 + 0.886777i
23.16 −1.09905 − 0.889993i 2.08216 + 1.35217i 0.415823 + 1.95630i −4.24058 + 2.64905i −1.08498 − 3.33922i 1.05933 − 6.91938i 1.28408 − 2.52015i −1.15359 − 2.59101i 7.01825 + 0.862656i
23.17 −1.09905 − 0.889993i 3.49368 + 2.26882i 0.415823 + 1.95630i 3.67476 + 3.39060i −1.82049 − 5.60290i 6.99999 + 0.0123205i 1.28408 − 2.52015i 3.39761 + 7.63116i −1.02114 − 6.99695i
23.18 −1.09905 − 0.889993i 3.70913 + 2.40873i 0.415823 + 1.95630i −1.92479 + 4.61467i −1.93276 − 5.94842i −4.25893 + 5.55531i 1.28408 − 2.52015i 4.29498 + 9.64668i 6.22247 − 3.35870i
23.19 −1.09905 − 0.889993i 4.58979 + 2.98064i 0.415823 + 1.95630i 3.23359 − 3.81365i −2.39165 − 7.36076i −4.73311 − 5.15730i 1.28408 − 2.52015i 8.52128 + 19.1391i −6.94800 + 1.31352i
23.20 −1.09905 − 0.889993i 4.74429 + 3.08098i 0.415823 + 1.95630i −4.24513 − 2.64176i −2.47216 − 7.60854i 6.96558 + 0.693339i 1.28408 − 2.52015i 9.35524 + 21.0122i 2.31446 + 6.68156i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
25.f odd 20 1 inner
175.w odd 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.3.w.b ✓ 320
7.c even 3 1 inner 350.3.w.b ✓ 320
25.f odd 20 1 inner 350.3.w.b ✓ 320
175.w odd 60 1 inner 350.3.w.b ✓ 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.3.w.b ✓ 320 1.a even 1 1 trivial
350.3.w.b ✓ 320 7.c even 3 1 inner
350.3.w.b ✓ 320 25.f odd 20 1 inner
350.3.w.b ✓ 320 175.w odd 60 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{320} - 20 T_{3}^{318} - 48 T_{3}^{317} + 2890 T_{3}^{316} + 2240 T_{3}^{315} + \cdots + 22\!\cdots\!25 \) acting on \(S_{3}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display