# Properties

 Label 672.2.ch.a Level $672$ Weight $2$ Character orbit 672.ch Analytic conductor $5.366$ Analytic rank $0$ Dimension $992$ CM no Inner twists $8$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$672 = 2^{5} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 672.ch (of order $$24$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.36594701583$$ Analytic rank: $$0$$ Dimension: $$992$$ Relative dimension: $$124$$ over $$\Q(\zeta_{24})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$992q - 4q^{3} - 8q^{4} - 16q^{6} - 16q^{7} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$992q - 4q^{3} - 8q^{4} - 16q^{6} - 16q^{7} - 4q^{9} - 8q^{10} - 4q^{12} - 32q^{13} - 32q^{15} + 24q^{16} - 4q^{18} - 8q^{19} - 8q^{21} - 32q^{22} - 44q^{24} - 8q^{25} - 16q^{27} - 16q^{28} - 36q^{30} - 8q^{33} + 64q^{36} - 8q^{37} - 4q^{39} - 72q^{40} - 68q^{42} - 32q^{43} - 4q^{45} - 8q^{46} - 16q^{48} + 20q^{51} + 16q^{52} - 92q^{54} - 32q^{55} - 16q^{57} + 24q^{58} - 20q^{60} - 8q^{61} - 128q^{64} - 20q^{66} - 72q^{67} - 16q^{69} - 16q^{70} - 4q^{72} - 8q^{73} + 16q^{75} - 32q^{76} + 136q^{78} - 16q^{79} - 48q^{82} + 124q^{84} + 48q^{85} - 4q^{87} - 8q^{88} + 128q^{90} - 64q^{91} - 28q^{93} - 64q^{94} - 72q^{96} - 64q^{97} - 16q^{99} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −1.41421 + 0.00202033i −1.50647 + 0.854722i 1.99999 0.00571435i 0.202706 + 1.53970i 2.12874 1.21180i 0.529096 + 2.59231i −2.82840 + 0.0121219i 1.53890 2.57522i −0.289779 2.17705i
11.2 −1.41371 + 0.0378977i −1.72802 0.118141i 1.99713 0.107152i 0.525322 + 3.99022i 2.44738 + 0.101529i −2.15734 1.53163i −2.81929 + 0.227168i 2.97209 + 0.408300i −0.893871 5.62109i
11.3 −1.41292 + 0.0604964i 0.856964 1.50520i 1.99268 0.170953i 0.139662 + 1.06084i −1.11976 + 2.17856i −1.22300 + 2.34612i −2.80515 + 0.362093i −1.53123 2.57980i −0.261508 1.49043i
11.4 −1.40999 + 0.109279i 1.70442 0.308130i 1.97612 0.308164i −0.270326 2.05333i −2.36954 + 0.620717i −2.18436 1.49283i −2.75262 + 0.650455i 2.81011 1.05037i 0.605541 + 2.86562i
11.5 −1.40871 + 0.124645i 0.277691 + 1.70965i 1.96893 0.351176i −0.255599 1.94146i −0.604284 2.37378i 2.57575 + 0.604593i −2.72988 + 0.740122i −2.84578 + 0.949506i 0.602057 + 2.70310i
11.6 −1.40759 0.136707i −1.57871 + 0.712503i 1.96262 + 0.384854i −0.304006 2.30915i 2.31959 0.787091i 2.28276 1.33754i −2.70996 0.810020i 1.98468 2.24968i 0.112239 + 3.29190i
11.7 −1.40248 + 0.181762i −0.483335 1.66325i 1.93393 0.509836i −0.549309 4.17241i 0.980185 + 2.24482i 1.13563 + 2.38963i −2.61963 + 1.06655i −2.53277 + 1.60781i 1.52878 + 5.75190i
11.8 −1.38980 0.261641i 1.00222 + 1.41264i 1.86309 + 0.727258i 0.0693882 + 0.527056i −1.02328 2.22551i −0.176243 2.63987i −2.39904 1.49820i −0.991114 + 2.83155i 0.0414639 0.750657i
11.9 −1.36685 + 0.362941i −0.0108601 + 1.73202i 1.73655 0.992170i 0.167832 + 1.27481i −0.613775 2.37135i −2.51370 + 0.825424i −2.01350 + 1.98641i −2.99976 0.0376198i −0.692080 1.68156i
11.10 −1.35950 0.389562i −0.721966 1.57441i 1.69648 + 1.05922i 0.456804 + 3.46977i 0.368183 + 2.42166i 2.57029 + 0.627378i −1.89374 2.10089i −1.95753 + 2.27334i 0.730665 4.89511i
11.11 −1.34787 + 0.428079i −1.11026 1.32941i 1.63350 1.15399i −0.0969257 0.736224i 2.06557 + 1.31659i −1.56934 2.13007i −1.70774 + 2.25469i −0.534654 + 2.95197i 0.445805 + 0.950841i
11.12 −1.34744 + 0.429412i 0.649335 1.60573i 1.63121 1.15722i 0.323918 + 2.46040i −0.185422 + 2.44246i 1.39830 2.24605i −1.70104 + 2.25975i −2.15673 2.08531i −1.49299 3.17616i
11.13 −1.33331 + 0.471485i 1.63784 + 0.563452i 1.55540 1.25727i 0.441991 + 3.35725i −2.44940 + 0.0209633i 1.71134 + 2.01775i −1.48105 + 2.40967i 2.36504 + 1.84569i −2.17220 4.26785i
11.14 −1.33152 0.476511i −1.64402 0.545154i 1.54587 + 1.26897i −0.104982 0.797415i 1.92927 + 1.50928i 1.28932 2.31034i −1.45368 2.42628i 2.40561 + 1.79249i −0.240192 + 1.11180i
11.15 −1.32881 0.484014i 1.60040 0.662368i 1.53146 + 1.28632i 0.0620289 + 0.471156i −2.44721 + 0.105546i 2.10333 + 1.60499i −1.41242 2.45052i 2.12254 2.12010i 0.145622 0.656099i
11.16 −1.32769 0.487083i −0.590581 + 1.62826i 1.52550 + 1.29339i −0.520784 3.95575i 1.57720 1.87415i −2.59797 + 0.500553i −1.39540 2.46026i −2.30243 1.92323i −1.23534 + 5.50565i
11.17 −1.30196 0.552168i 0.204575 1.71993i 1.39022 + 1.43781i −0.306137 2.32534i −1.21604 + 2.12632i −1.51022 2.17238i −1.01611 2.63961i −2.91630 0.703709i −0.885400 + 3.19655i
11.18 −1.30144 0.553394i 1.57447 + 0.721834i 1.38751 + 1.44042i 0.510185 + 3.87524i −1.64963 1.81073i −2.48556 + 0.906631i −1.00865 2.64247i 1.95791 + 2.27301i 1.48056 5.32573i
11.19 −1.24175 + 0.676794i −1.72435 + 0.163175i 1.08390 1.68082i −0.337107 2.56058i 2.03078 1.36965i −2.18342 + 1.49422i −0.208366 + 2.82074i 2.94675 0.562741i 2.15159 + 2.95146i
11.20 −1.24127 0.677674i −1.06907 1.36275i 1.08152 + 1.68236i 0.0342867 + 0.260434i 0.403515 + 2.41602i −2.18554 + 1.49111i −0.202366 2.82118i −0.714158 + 2.91376i 0.133930 0.346504i
See next 80 embeddings (of 992 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 611.124 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner
32.h odd 8 1 inner
96.o even 8 1 inner
224.bf odd 24 1 inner
672.ch even 24 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.ch.a 992
3.b odd 2 1 inner 672.2.ch.a 992
7.c even 3 1 inner 672.2.ch.a 992
21.h odd 6 1 inner 672.2.ch.a 992
32.h odd 8 1 inner 672.2.ch.a 992
96.o even 8 1 inner 672.2.ch.a 992
224.bf odd 24 1 inner 672.2.ch.a 992
672.ch even 24 1 inner 672.2.ch.a 992

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.ch.a 992 1.a even 1 1 trivial
672.2.ch.a 992 3.b odd 2 1 inner
672.2.ch.a 992 7.c even 3 1 inner
672.2.ch.a 992 21.h odd 6 1 inner
672.2.ch.a 992 32.h odd 8 1 inner
672.2.ch.a 992 96.o even 8 1 inner
672.2.ch.a 992 224.bf odd 24 1 inner
672.2.ch.a 992 672.ch even 24 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(672, [\chi])$$.