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Properties

Label 349.2.i.a
Level $349$
Weight $2$
Character orbit 349.i
Analytic conductor $2.787$
Analytic rank $0$
Dimension $1568$
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] = [349,2,Mod(9,349)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(349, base_ring=CyclotomicField(174))
 
chi = DirichletCharacter(H, H._module([26]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("349.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 349 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 349.i (of order \(87\), degree \(56\), 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: \(2.78677903054\)
Analytic rank: \(0\)
Dimension: \(1568\)
Relative dimension: \(28\) over \(\Q(\zeta_{87})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{87}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1568 q - 58 q^{2} - 58 q^{3} - 28 q^{4} - 62 q^{5} - 72 q^{6} - 60 q^{7} + 23 q^{8} - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1568 q - 58 q^{2} - 58 q^{3} - 28 q^{4} - 62 q^{5} - 72 q^{6} - 60 q^{7} + 23 q^{8} - 30 q^{9} - 56 q^{10} - 56 q^{11} - 69 q^{12} - 56 q^{13} - 60 q^{14} - 67 q^{15} - 24 q^{16} - 58 q^{17} + 40 q^{18} - 169 q^{19} - 72 q^{20} + 46 q^{21} - 51 q^{22} - 47 q^{23} + q^{24} - 52 q^{25} + 11 q^{26} - 28 q^{27} - 6 q^{28} - 66 q^{29} - 37 q^{30} - 10 q^{31} - 52 q^{32} - 70 q^{33} - 44 q^{34} - 42 q^{35} - 124 q^{36} - 159 q^{37} - 118 q^{38} - 32 q^{39} - 82 q^{40} - 403 q^{42} - 35 q^{43} + 78 q^{44} - 76 q^{45} - 63 q^{46} - 32 q^{47} + 22 q^{48} - 32 q^{49} + 308 q^{50} + 44 q^{51} + 255 q^{52} - 46 q^{53} - 51 q^{54} + 48 q^{55} - 39 q^{56} - 83 q^{57} - 46 q^{58} - 42 q^{59} - 742 q^{60} + 190 q^{61} - 31 q^{62} - 89 q^{63} + 149 q^{64} - 130 q^{65} + 385 q^{66} - 24 q^{67} - q^{68} + 48 q^{69} - 6 q^{70} - 48 q^{71} + 243 q^{72} - 81 q^{73} + 75 q^{74} - 32 q^{75} - 67 q^{76} - 48 q^{77} - 547 q^{78} - 106 q^{79} + 148 q^{80} + 418 q^{81} - 50 q^{82} - 304 q^{83} + 338 q^{84} + 90 q^{85} - 46 q^{86} + 102 q^{87} - 116 q^{88} + 340 q^{89} - 126 q^{90} - 112 q^{91} - 106 q^{92} - q^{93} - 91 q^{94} + 120 q^{95} - 914 q^{96} - 58 q^{97} + 316 q^{98} - 131 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −2.46125 − 1.24848i 0.387351 − 0.146288i 3.31766 + 4.53187i 1.25689 − 0.324652i −1.13601 − 0.123548i 4.59338 + 0.165940i −1.61470 − 9.84922i −2.12241 + 1.86979i −3.49886 − 0.770156i
9.2 −2.17302 − 1.10227i 1.67917 − 0.634157i 2.32562 + 3.17675i −1.08355 + 0.279878i −4.34788 − 0.472861i −4.21470 − 0.152260i −0.763578 − 4.65762i 0.166393 − 0.146588i 2.66309 + 0.586190i
9.3 −2.14572 − 1.08842i −2.78217 + 1.05072i 2.23804 + 3.05713i 2.44600 − 0.631793i 7.11339 + 0.773627i −1.34778 − 0.0486897i −0.696275 − 4.24709i 4.38540 − 3.86344i −5.93608 − 1.30663i
9.4 −2.00904 − 1.01909i −1.03329 + 0.390235i 1.81630 + 2.48103i 0.540716 − 0.139665i 2.47361 + 0.269021i −0.925700 − 0.0334418i −0.391715 − 2.38936i −1.33564 + 1.17667i −1.22865 − 0.270447i
9.5 −1.95105 − 0.989675i 2.11329 − 0.798109i 1.64572 + 2.24803i −2.39358 + 0.618254i −4.91299 − 0.534320i 2.11340 + 0.0763486i −0.278204 − 1.69697i 1.57796 − 1.39014i 5.28186 + 1.16263i
9.6 −1.86652 − 0.946796i 2.80322 − 1.05867i 1.40605 + 1.92064i 2.94974 − 0.761907i −6.23459 − 0.678053i −0.347829 − 0.0125657i −0.128769 − 0.785453i 4.48620 − 3.95224i −6.22710 − 1.37069i
9.7 −1.76537 − 0.895491i −1.48767 + 0.561835i 1.13323 + 1.54797i −3.26214 + 0.842600i 3.12940 + 0.340343i 1.24531 + 0.0449880i 0.0261182 + 0.159314i −0.353559 + 0.311477i 6.51343 + 1.43371i
9.8 −1.18360 − 0.600385i −0.682614 + 0.257797i −0.140957 − 0.192545i 3.36183 − 0.868350i 0.962720 + 0.104702i 1.83035 + 0.0661230i 0.480660 + 2.93189i −1.85155 + 1.63117i −4.50041 − 0.990615i
9.9 −1.05413 − 0.534709i 0.535376 − 0.202191i −0.356137 − 0.486477i 1.37934 − 0.356279i −0.672467 − 0.0731352i −3.69432 − 0.133461i 0.497739 + 3.03607i −2.00530 + 1.76663i −1.64451 − 0.361983i
9.10 −0.818321 − 0.415096i −2.39483 + 0.904438i −0.684062 − 0.934416i −1.74670 + 0.451166i 2.33517 + 0.253965i −4.80765 − 0.173681i 0.468806 + 2.85959i 2.66617 − 2.34884i 1.61663 + 0.355848i
9.11 −0.737293 − 0.373994i 0.783668 − 0.295962i −0.777676 − 1.06229i −2.65376 + 0.685457i −0.688481 − 0.0748768i 1.75900 + 0.0635456i 0.443583 + 2.70573i −1.72451 + 1.51925i 2.21296 + 0.487108i
9.12 −0.736757 − 0.373722i 2.21410 − 0.836181i −0.778263 − 1.06309i 0.343155 − 0.0886359i −1.94375 − 0.211396i 5.07513 + 0.183344i 0.443393 + 2.70458i 1.95198 − 1.71965i −0.285947 − 0.0629418i
9.13 −0.379345 − 0.192424i 2.17990 − 0.823266i −1.07453 − 1.46779i 2.33094 − 0.602075i −0.985352 − 0.107163i −2.42172 − 0.0874869i 0.262811 + 1.60307i 1.82315 − 1.60615i −1.00009 − 0.220136i
9.14 −0.0280982 − 0.0142529i −2.20107 + 0.831260i −1.18082 − 1.61298i 0.943566 − 0.243720i 0.0736939 + 0.00801469i 0.689625 + 0.0249134i 0.0203836 + 0.124335i 1.90266 − 1.67620i −0.0299862 − 0.00660047i
9.15 0.0699482 + 0.0354814i −0.760409 + 0.287178i −1.17777 − 1.60881i −2.29345 + 0.592390i −0.0633787 − 0.00689285i 2.08428 + 0.0752967i −0.0506779 − 0.309122i −1.75530 + 1.54638i −0.181442 − 0.0399383i
9.16 0.200938 + 0.101926i −1.14709 + 0.433214i −1.15142 − 1.57282i 0.757667 − 0.195703i −0.274651 − 0.0298701i 2.54552 + 0.0919592i −0.143955 − 0.878084i −1.12290 + 0.989249i 0.172191 + 0.0379022i
9.17 0.436267 + 0.221298i 1.26243 − 0.476772i −1.04005 − 1.42069i −3.45502 + 0.892420i 0.656265 + 0.0713731i −3.40184 − 0.122895i −0.297627 − 1.81544i −0.884633 + 0.779341i −1.70480 − 0.375255i
9.18 0.581003 + 0.294716i 2.59259 − 0.979122i −0.930699 − 1.27132i 0.258546 − 0.0667815i 1.79486 + 0.195203i −0.0188644 0.000681494i −0.376856 − 2.29872i 3.51178 − 3.09380i 0.169897 + 0.0373972i
9.19 1.07325 + 0.544407i −2.77045 + 1.04630i −0.325928 − 0.445212i 3.94990 − 1.02025i −3.54299 − 0.385323i −2.35651 − 0.0851310i −0.496810 − 3.03041i 4.32963 − 3.81430i 4.79464 + 1.05538i
9.20 1.18924 + 0.603247i 0.788193 − 0.297671i −0.131017 − 0.178967i 3.38595 − 0.874580i 1.11692 + 0.121472i −0.168277 − 0.00607917i −0.479320 − 2.92372i −1.71841 + 1.51388i 4.55430 + 1.00248i
See next 80 embeddings (of 1568 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
349.i even 87 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 349.2.i.a ✓ 1568
349.i even 87 1 inner 349.2.i.a ✓ 1568
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
349.2.i.a ✓ 1568 1.a even 1 1 trivial
349.2.i.a ✓ 1568 349.i even 87 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(349, [\chi])\).