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Properties

Label 353.2.k.a
Level $353$
Weight $2$
Character orbit 353.k
Analytic conductor $2.819$
Analytic rank $0$
Dimension $2320$
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] = [353,2,Mod(9,353)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(353, base_ring=CyclotomicField(176))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("353.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 353 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 353.k (of order \(176\), degree \(80\), 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: \(2.81871919135\)
Analytic rank: \(0\)
Dimension: \(2320\)
Relative dimension: \(29\) over \(\Q(\zeta_{176})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{176}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2320 q - 80 q^{2} - 80 q^{3} - 88 q^{4} - 80 q^{5} - 80 q^{6} - 80 q^{7} - 96 q^{8} - 88 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 2320 q - 80 q^{2} - 80 q^{3} - 88 q^{4} - 80 q^{5} - 80 q^{6} - 80 q^{7} - 96 q^{8} - 88 q^{9} - 112 q^{10} - 96 q^{11} - 160 q^{12} - 80 q^{13} - 80 q^{14} - 80 q^{15} + 160 q^{16} - 80 q^{17} - 24 q^{18} - 80 q^{19} - 96 q^{20} - 112 q^{21} - 40 q^{22} - 104 q^{23} - 120 q^{24} - 56 q^{25} + 160 q^{26} - 104 q^{27} - 48 q^{28} - 80 q^{29} - 80 q^{30} - 80 q^{31} - 96 q^{32} + 48 q^{33} - 88 q^{34} - 88 q^{35} - 168 q^{36} - 80 q^{37} - 40 q^{38} - 128 q^{39} - 128 q^{40} - 24 q^{41} + 256 q^{43} - 200 q^{44} + 72 q^{45} - 48 q^{46} - 192 q^{47} - 104 q^{48} - 64 q^{49} - 16 q^{50} + 56 q^{51} - 112 q^{52} - 80 q^{53} - 136 q^{54} - 16 q^{55} - 32 q^{56} - 112 q^{57} - 216 q^{58} - 80 q^{59} + 1120 q^{60} - 80 q^{61} - 48 q^{62} - 80 q^{63} - 88 q^{64} - 8 q^{65} - 56 q^{66} - 24 q^{67} + 280 q^{68} - 296 q^{69} - 96 q^{70} - 104 q^{71} - 112 q^{72} - 96 q^{73} - 88 q^{74} - 80 q^{75} - 208 q^{76} + 224 q^{77} - 112 q^{78} - 104 q^{79} - 72 q^{80} - 64 q^{81} - 104 q^{82} + 16 q^{83} - 96 q^{84} + 888 q^{85} - 208 q^{86} + 16 q^{87} - 88 q^{88} - 16 q^{89} + 120 q^{90} - 112 q^{91} - 112 q^{92} + 24 q^{93} - 144 q^{94} - 48 q^{95} + 32 q^{96} - 40 q^{97} - 128 q^{98} - 80 q^{99}+O(q^{100}) \) 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 −2.65496 + 0.577552i −3.02711 − 0.0540395i 4.89599 − 2.23592i 1.73689 − 1.67596i 8.06807 − 1.60484i 0.971899 − 4.88607i −7.35708 + 5.50744i 6.16239 + 0.220090i −3.64342 + 5.45276i
9.2 −2.42807 + 0.528195i 2.99379 + 0.0534447i 3.79729 − 1.73417i 1.78657 − 1.72390i −7.29737 + 1.45154i 0.167849 − 0.843835i −4.32566 + 3.23815i 5.96182 + 0.212927i −3.42737 + 5.12942i
9.3 −2.39102 + 0.520136i −1.43195 − 0.0255630i 3.62719 − 1.65648i −2.59855 + 2.50740i 3.43712 − 0.683686i −0.133849 + 0.672906i −3.89336 + 2.91453i −0.948262 − 0.0338673i 4.90901 − 7.34685i
9.4 −2.36315 + 0.514071i −0.496028 − 0.00885501i 3.50093 − 1.59882i 1.48881 − 1.43659i 1.17674 − 0.234068i −0.739797 + 3.71921i −3.57922 + 2.67937i −2.75212 − 0.0982923i −2.77977 + 4.16022i
9.5 −2.34887 + 0.510966i 1.23566 + 0.0220589i 3.43686 − 1.56956i −1.65139 + 1.59346i −2.91369 + 0.579569i 0.420291 − 2.11295i −3.42206 + 2.56173i −1.47171 − 0.0525623i 3.06469 − 4.58664i
9.6 −1.82978 + 0.398045i 2.41683 + 0.0431448i 1.37041 − 0.625844i −1.45804 + 1.40689i −4.43945 + 0.883061i −0.653728 + 3.28651i 0.739718 − 0.553746i 2.84110 + 0.101470i 2.10789 − 3.15468i
9.7 −1.63152 + 0.354916i −1.92813 − 0.0344207i 0.716631 − 0.327275i −0.869089 + 0.838604i 3.15800 − 0.628165i 0.242057 − 1.21690i 1.62025 − 1.21290i 0.718405 + 0.0256579i 1.12030 − 1.67665i
9.8 −1.61113 + 0.350479i −0.283910 − 0.00506833i 0.653633 − 0.298504i 2.58952 − 2.49869i 0.459192 − 0.0913390i 0.246937 − 1.24144i 1.69141 − 1.26617i −2.91751 − 0.104199i −3.29632 + 4.93329i
9.9 −1.57046 + 0.341633i −3.35844 − 0.0599544i 0.530373 − 0.242213i 0.391006 − 0.377291i 5.29479 − 1.05320i −0.871981 + 4.38375i 1.82306 − 1.36473i 8.27745 + 0.295630i −0.485165 + 0.726101i
9.10 −1.15376 + 0.250986i 0.914985 + 0.0163342i −0.551085 + 0.251672i 0.0353179 − 0.0340791i −1.05978 + 0.210803i 0.0107260 − 0.0539232i 2.46313 − 1.84388i −2.16116 − 0.0771860i −0.0321952 + 0.0481835i
9.11 −0.950060 + 0.206673i 2.72803 + 0.0487004i −0.959365 + 0.438127i 0.298723 − 0.288244i −2.60186 + 0.517542i 0.262827 − 1.32132i 2.37760 − 1.77985i 4.44169 + 0.158636i −0.224232 + 0.335587i
9.12 −0.487960 + 0.106149i 1.16890 + 0.0208671i −1.59243 + 0.727237i −2.83329 + 2.73390i −0.572593 + 0.113896i 0.914341 − 4.59670i 1.49938 − 1.12242i −1.63219 − 0.0582938i 1.09233 − 1.63478i
9.13 −0.373907 + 0.0813386i −1.81205 − 0.0323485i −1.68607 + 0.770004i −0.817942 + 0.789251i 0.680171 − 0.135294i −0.195001 + 0.980334i 1.18046 − 0.883683i 0.284403 + 0.0101575i 0.241638 − 0.361637i
9.14 −0.275816 + 0.0600002i 0.271836 + 0.00485278i −1.74679 + 0.797732i 0.223565 − 0.215722i −0.0752680 + 0.0149717i −0.535516 + 2.69222i 0.885862 − 0.663148i −2.92422 − 0.104439i −0.0487194 + 0.0729137i
9.15 0.0140002 − 0.00304556i 2.35085 + 0.0419670i −1.81908 + 0.830745i 2.97181 − 2.86757i 0.0330403 − 0.00657212i −0.703865 + 3.53857i −0.0458772 + 0.0343432i 2.52665 + 0.0902395i 0.0328727 − 0.0491975i
9.16 0.0350567 − 0.00762611i −2.28801 − 0.0408452i −1.81809 + 0.830295i 1.12453 − 1.08509i −0.0805216 + 0.0160167i 0.504556 − 2.53658i −0.114846 + 0.0859724i 2.23524 + 0.0798316i 0.0311474 − 0.0466154i
9.17 0.448373 − 0.0975376i −2.99706 − 0.0535030i −1.62774 + 0.743364i −2.35830 + 2.27557i −1.34902 + 0.268336i 0.471922 − 2.37251i −1.39200 + 1.04204i 5.98140 + 0.213626i −0.835442 + 1.25033i
9.18 0.519581 − 0.113028i 0.164769 + 0.00294143i −1.56208 + 0.713376i 1.60165 − 1.54547i 0.0859433 − 0.0170952i 0.827108 − 4.15816i −1.58234 + 1.18453i −2.97095 − 0.106108i 0.657507 − 0.984029i
9.19 0.560168 − 0.121857i 2.86228 + 0.0510970i −1.52032 + 0.694309i −2.58102 + 2.49048i 1.60958 − 0.320166i −0.655357 + 3.29470i −1.68488 + 1.26129i 5.19194 + 0.185431i −1.14232 + 1.70960i
9.20 1.02847 − 0.223729i 3.07378 + 0.0548726i −0.811577 + 0.370635i 0.505511 − 0.487779i 3.17355 − 0.631258i 0.604216 − 3.03760i −2.43693 + 1.82426i 6.44700 + 0.230255i 0.410771 − 0.614762i
See next 80 embeddings (of 2320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.29
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
353.k even 176 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 353.2.k.a ✓ 2320
353.k even 176 1 inner 353.2.k.a ✓ 2320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
353.2.k.a ✓ 2320 1.a even 1 1 trivial
353.2.k.a ✓ 2320 353.k even 176 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(353, [\chi])\).