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Properties

Label 349.3.l.a
Level $349$
Weight $3$
Character orbit 349.l
Analytic conductor $9.510$
Analytic rank $0$
Dimension $6384$
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] = [349,3,Mod(2,349)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(349, base_ring=CyclotomicField(348))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("349.2");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 349 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 349.l (of order \(348\), degree \(112\), 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: \(9.50956122617\)
Analytic rank: \(0\)
Dimension: \(6384\)
Relative dimension: \(57\) over \(\Q(\zeta_{348})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{348}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 6384 q - 110 q^{2} - 104 q^{3} - 122 q^{4} - 116 q^{5} - 78 q^{6} - 138 q^{7} - 104 q^{8} - 438 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 6384 q - 110 q^{2} - 104 q^{3} - 122 q^{4} - 116 q^{5} - 78 q^{6} - 138 q^{7} - 104 q^{8} - 438 q^{9} - 174 q^{10} - 102 q^{11} - 78 q^{12} - 140 q^{13} - 144 q^{14} - 78 q^{15} - 550 q^{16} - 116 q^{17} + 242 q^{18} - 34 q^{19} - 56 q^{20} - 1084 q^{21} - 74 q^{22} - 216 q^{23} - 26 q^{24} - 502 q^{25} - 152 q^{26} - 116 q^{27} + 224 q^{28} - 284 q^{29} - 212 q^{30} - 108 q^{31} - 168 q^{32} - 192 q^{33} + 102 q^{34} + 326 q^{35} - 116 q^{36} - 116 q^{37} - 108 q^{38} + 2 q^{39} - 88 q^{40} - 628 q^{41} - 290 q^{42} + 68 q^{43} + 1810 q^{44} - 116 q^{45} + 84 q^{46} + 198 q^{47} - 116 q^{48} + 64 q^{49} - 288 q^{50} - 228 q^{51} + 3436 q^{52} + 122 q^{53} - 296 q^{54} - 2758 q^{55} + 904 q^{56} + 568 q^{57} + 644 q^{58} - 166 q^{59} - 116 q^{60} - 4394 q^{61} - 424 q^{62} - 282 q^{63} + 3538 q^{64} - 60 q^{65} - 372 q^{66} + 148 q^{67} + 86 q^{68} + 2436 q^{69} - 116 q^{70} + 10 q^{71} - 4438 q^{72} - 14 q^{73} - 472 q^{74} - 116 q^{75} - 794 q^{76} - 286 q^{77} + 352 q^{78} - 196 q^{79} - 116 q^{80} - 2466 q^{81} - 260 q^{82} - 38 q^{83} + 2816 q^{84} - 460 q^{85} + 1508 q^{86} - 172 q^{87} - 44 q^{88} - 4580 q^{89} - 238 q^{90} - 758 q^{91} - 116 q^{92} + 658 q^{93} - 10 q^{94} - 782 q^{95} - 12 q^{96} + 278 q^{97} - 510 q^{98} - 136 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −3.88872 − 0.0351066i 5.28293 + 1.26328i 11.1216 + 0.200823i 6.37990 − 2.02213i −20.4995 − 5.09799i −3.92151 − 2.71084i −27.6918 − 0.750152i 18.2870 + 9.27616i −24.8806 + 7.63953i
2.2 −3.85150 − 0.0347706i −1.10346 − 0.263863i 10.8335 + 0.195621i 2.79479 − 0.885820i 4.24078 + 1.05463i 6.01116 + 4.15537i −26.3173 − 0.712917i −6.87843 − 3.48911i −10.7949 + 3.31456i
2.3 −3.78589 − 0.0341783i 0.120211 + 0.0287453i 10.3325 + 0.186574i −4.14197 + 1.31281i −0.454122 − 0.112935i −10.6286 − 7.34731i −23.9726 − 0.649401i −8.01280 − 4.06452i 15.7259 − 4.82860i
2.4 −3.53833 − 0.0319434i −5.24221 − 1.25354i 8.51944 + 0.153836i 4.15233 − 1.31610i 18.5086 + 4.60289i 3.69603 + 2.55498i −15.9910 − 0.433185i 17.8830 + 9.07119i −14.7344 + 4.52415i
2.5 −3.51034 − 0.0316907i −4.80721 − 1.14952i 8.32216 + 0.150274i −8.95167 + 2.83726i 16.8385 + 4.18755i 0.0842867 + 0.0582654i −15.1721 − 0.411001i 13.7614 + 6.98053i 31.5133 − 9.67609i
2.6 −3.45961 − 0.0312327i 3.94972 + 0.944473i 7.96857 + 0.143889i −6.49705 + 2.05927i −13.6350 − 3.39087i 3.87222 + 2.67678i −13.7297 − 0.371928i 6.68182 + 3.38937i 22.5416 − 6.92133i
2.7 −3.21222 − 0.0289993i −0.632500 − 0.151246i 6.31819 + 0.114088i −3.69963 + 1.17261i 2.02734 + 0.504178i 1.82051 + 1.25847i −7.44742 − 0.201745i −7.64924 − 3.88010i 11.9181 − 3.65941i
2.8 −3.16986 − 0.0286169i −2.78073 − 0.664940i 6.04787 + 0.109207i 7.47507 − 2.36925i 8.79551 + 2.18734i −10.7373 − 7.42243i −6.49247 − 0.175877i −0.736105 − 0.373392i −23.7627 + 7.29629i
2.9 −3.07240 − 0.0277370i 2.15753 + 0.515918i 5.43952 + 0.0982219i 3.56025 − 1.12843i −6.61449 − 1.64495i 1.52195 + 1.05209i −4.42406 − 0.119845i −3.63765 − 1.84521i −10.9698 + 3.36825i
2.10 −2.98111 − 0.0269129i 2.28436 + 0.546247i 4.88692 + 0.0882435i 5.16497 − 1.63706i −6.79523 − 1.68990i −2.17863 − 1.50603i −2.64550 − 0.0716648i −3.10649 − 1.57578i −15.4414 + 4.74123i
2.11 −2.77595 − 0.0250607i −3.91415 − 0.935967i 3.70590 + 0.0669178i −0.226277 + 0.0717195i 10.8420 + 2.69629i −3.11980 − 2.15664i 0.814457 + 0.0220631i 6.41809 + 3.25560i 0.629931 − 0.193419i
2.12 −2.65901 − 0.0240050i 4.53782 + 1.08510i 3.07039 + 0.0554423i −3.42197 + 1.08461i −12.0400 − 2.99422i −5.09830 − 3.52433i 2.46970 + 0.0669025i 11.3879 + 5.77656i 9.12508 − 2.80183i
2.13 −2.63878 − 0.0238224i −1.10809 − 0.264972i 2.96322 + 0.0535071i −4.82261 + 1.52854i 2.91770 + 0.725598i 8.61751 + 5.95708i 2.73366 + 0.0740531i −6.86876 − 3.48420i 12.7622 − 3.91860i
2.14 −2.28235 − 0.0206046i 4.32377 + 1.03392i 1.20935 + 0.0218374i 3.59333 − 1.13892i −9.84707 − 2.44886i 11.2269 + 7.76087i 6.36671 + 0.172470i 9.59961 + 4.86944i −8.22470 + 2.52537i
2.15 −2.19204 − 0.0197893i −2.82773 − 0.676179i 0.805302 + 0.0145414i 7.60504 − 2.41045i 6.18512 + 1.53817i 9.20658 + 6.36429i 7.00034 + 0.189634i −0.487575 − 0.247324i −16.7183 + 5.13330i
2.16 −2.02072 − 0.0182427i −0.0681003 − 0.0162844i 0.0836425 + 0.00151034i −6.65975 + 2.11083i 0.137315 + 0.0341487i −8.57453 − 5.92737i 7.91127 + 0.214311i −8.02205 − 4.06921i 13.4960 − 4.14392i
2.17 −1.91513 − 0.0172894i 2.36607 + 0.565785i −0.331942 − 0.00599390i 0.913038 − 0.289391i −4.52154 − 1.12446i −7.26767 − 5.02397i 8.29361 + 0.224668i −2.74824 − 1.39405i −1.75359 + 0.538434i
2.18 −1.89519 − 0.0171094i −3.21901 − 0.769744i −0.407892 − 0.00736535i −3.35579 + 1.06363i 6.08747 + 1.51389i −0.0748365 − 0.0517327i 8.35120 + 0.226228i 1.74311 + 0.884197i 6.37807 − 1.95837i
2.19 −1.32802 − 0.0119892i −0.432264 − 0.103365i −2.23584 − 0.0403728i 6.91530 − 2.19183i 0.572818 + 0.142453i −2.50761 − 1.73345i 8.27914 + 0.224276i −7.85026 − 3.98207i −9.20996 + 2.82789i
2.20 −1.20859 − 0.0109109i −1.76201 − 0.421339i −2.53879 − 0.0458431i 2.91289 − 0.923251i 2.12494 + 0.528450i −0.375371 − 0.259485i 7.90061 + 0.214022i −5.09927 − 2.58662i −3.53055 + 1.08405i
See next 80 embeddings (of 6384 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.57
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
349.l odd 348 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 349.3.l.a ✓ 6384
349.l odd 348 1 inner 349.3.l.a ✓ 6384
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
349.3.l.a ✓ 6384 1.a even 1 1 trivial
349.3.l.a ✓ 6384 349.l odd 348 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(349, [\chi])\).