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Properties

Label 352.2.n.a
Level $352$
Weight $2$
Character orbit 352.n
Analytic conductor $2.811$
Analytic rank $0$
Dimension $160$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [352,2,Mod(45,352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(352, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("352.45");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 352 = 2^{5} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 352.n (of order \(8\), degree \(4\), 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: \(2.81073415115\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 160 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 16 q^{10} - 16 q^{14} - 16 q^{20} + 24 q^{24} - 48 q^{27} + 40 q^{28} + 80 q^{30} - 48 q^{31} - 48 q^{35} + 56 q^{36} - 16 q^{38} - 48 q^{39} - 64 q^{40} - 104 q^{48} - 80 q^{50} + 32 q^{51} + 64 q^{54} - 40 q^{56} + 8 q^{58} + 32 q^{59} - 8 q^{60} - 64 q^{61} + 24 q^{62} - 24 q^{64} + 48 q^{68} - 64 q^{69} - 24 q^{70} + 32 q^{71} + 72 q^{72} - 128 q^{78} - 16 q^{80} + 40 q^{82} - 80 q^{83} - 152 q^{84} - 64 q^{86} - 112 q^{87} + 144 q^{90} - 72 q^{92} + 96 q^{94} + 56 q^{96} + 56 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1 −1.40899 − 0.121493i 0.295327 − 0.712982i 1.97048 + 0.342364i −0.912619 + 0.378019i −0.502734 + 0.968701i 0.169918 − 0.169918i −2.73478 − 0.721786i 1.70019 + 1.70019i 1.33179 − 0.421746i
45.2 −1.40312 − 0.176771i −0.786365 + 1.89845i 1.93750 + 0.496064i 1.77711 − 0.736103i 1.43896 − 2.52475i −3.53406 + 3.53406i −2.63086 − 1.03853i −0.864430 − 0.864430i −2.62363 + 0.718701i
45.3 −1.39120 + 0.254107i 1.28218 − 3.09546i 1.87086 − 0.707026i −0.541423 + 0.224265i −0.997190 + 4.63221i 3.42929 − 3.42929i −2.42307 + 1.45901i −5.81657 − 5.81657i 0.696239 − 0.449576i
45.4 −1.38982 + 0.261511i −0.342954 + 0.827965i 1.86322 − 0.726908i −3.84471 + 1.59253i 0.260125 − 1.24041i 0.897811 − 0.897811i −2.39946 + 1.49753i 1.55341 + 1.55341i 4.92700 − 3.21877i
45.5 −1.38771 − 0.272508i 0.0495063 − 0.119519i 1.85148 + 0.756324i 3.36527 − 1.39394i −0.101270 + 0.152367i 2.36229 − 2.36229i −2.36321 − 1.55410i 2.10949 + 2.10949i −5.04988 + 1.01732i
45.6 −1.26223 + 0.637791i 0.851231 − 2.05505i 1.18644 − 1.61008i −1.73443 + 0.718422i 0.236247 + 3.13685i −3.06287 + 3.06287i −0.470671 + 2.78899i −1.37733 − 1.37733i 1.73104 − 2.01302i
45.7 −1.21532 + 0.723188i 0.443524 − 1.07076i 0.953999 − 1.75781i 2.95283 − 1.22310i 0.235339 + 1.62207i −0.544294 + 0.544294i 0.111813 + 2.82622i 1.17150 + 1.17150i −2.70410 + 3.62191i
45.8 −1.21451 − 0.724544i 1.24381 − 3.00281i 0.950072 + 1.75993i 3.33233 − 1.38030i −3.68629 + 2.74576i −2.60523 + 2.60523i 0.121276 − 2.82583i −5.34852 − 5.34852i −5.04724 − 0.738036i
45.9 −1.19859 − 0.750585i 0.612055 − 1.47763i 0.873244 + 1.79929i −2.43306 + 1.00780i −1.84269 + 1.31168i −1.62581 + 1.62581i 0.303858 − 2.81206i 0.312536 + 0.312536i 3.67268 + 0.618270i
45.10 −1.17562 − 0.786075i −0.821814 + 1.98403i 0.764172 + 1.84825i 0.279093 − 0.115604i 2.52574 − 1.68647i 2.72415 − 2.72415i 0.554489 − 2.77354i −1.13970 − 1.13970i −0.418981 − 0.0834812i
45.11 −1.05278 − 0.944275i −0.572262 + 1.38156i 0.216689 + 1.98823i −1.26110 + 0.522366i 1.90704 − 0.914108i −0.780224 + 0.780224i 1.64931 − 2.29778i 0.540088 + 0.540088i 1.82092 + 0.640892i
45.12 −0.808743 + 1.16014i −0.834198 + 2.01393i −0.691870 − 1.87652i −3.17138 + 1.31363i −1.66180 − 2.59654i −2.88679 + 2.88679i 2.73658 + 0.714950i −1.23871 − 1.23871i 1.04083 − 4.74165i
45.13 −0.786776 + 1.17515i −0.181061 + 0.437119i −0.761968 − 1.84916i −0.620270 + 0.256924i −0.371228 − 0.556689i 0.666406 − 0.666406i 2.77255 + 0.559448i 1.96303 + 1.96303i 0.186088 − 0.931053i
45.14 −0.736075 + 1.20756i 0.781137 − 1.88583i −0.916388 − 1.77770i −0.243537 + 0.100876i 1.70227 + 2.33138i 1.03521 − 1.03521i 2.82121 + 0.201931i −0.824865 − 0.824865i 0.0574475 − 0.368338i
45.15 −0.720777 − 1.21675i 0.555561 − 1.34124i −0.960961 + 1.75401i 1.75948 − 0.728800i −2.03239 + 0.290758i 1.87846 − 1.87846i 2.82683 − 0.0950021i 0.631036 + 0.631036i −2.15496 − 1.61554i
45.16 −0.485531 − 1.32825i −1.13323 + 2.73587i −1.52852 + 1.28982i 4.06060 − 1.68195i 4.18415 + 0.176873i −0.347099 + 0.347099i 2.45535 + 1.40402i −4.07943 − 4.07943i −4.20561 − 4.57687i
45.17 −0.290095 − 1.38414i −0.104231 + 0.251637i −1.83169 + 0.803065i −3.83200 + 1.58727i 0.378537 + 0.0712722i 2.68118 − 2.68118i 1.64292 + 2.30235i 2.06886 + 2.06886i 3.30865 + 4.84357i
45.18 −0.187349 + 1.40175i −0.504935 + 1.21902i −1.92980 − 0.525232i 2.41202 − 0.999093i −1.61416 − 0.936174i 3.04703 − 3.04703i 1.09779 − 2.60670i 0.890269 + 0.890269i 0.948588 + 3.56823i
45.19 −0.0941526 − 1.41108i 1.19624 − 2.88798i −1.98227 + 0.265713i −2.77785 + 1.15062i −4.18779 − 1.41607i −1.10504 + 1.10504i 0.561577 + 2.77212i −4.78811 − 4.78811i 1.88516 + 3.81142i
45.20 −0.0645658 + 1.41274i 0.243474 − 0.587798i −1.99166 − 0.182429i 2.40247 − 0.995138i 0.814686 + 0.381917i −3.21448 + 3.21448i 0.386318 − 2.80192i 1.83509 + 1.83509i 1.25075 + 3.45832i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 352.2.n.a ✓ 160
32.g even 8 1 inner 352.2.n.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
352.2.n.a ✓ 160 1.a even 1 1 trivial
352.2.n.a ✓ 160 32.g even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(352, [\chi])\).