# Properties

 Label 136.3.q.a Level $136$ Weight $3$ Character orbit 136.q Analytic conductor $3.706$ Analytic rank $0$ Dimension $272$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$136 = 2^{3} \cdot 17$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 136.q (of order $$16$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.70573159530$$ Analytic rank: $$0$$ Dimension: $$272$$ Relative dimension: $$34$$ over $$\Q(\zeta_{16})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$272 q - 8 q^{2} - 8 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{8} - 16 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$272 q - 8 q^{2} - 8 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{8} - 16 q^{9} - 8 q^{10} - 8 q^{12} - 8 q^{14} - 16 q^{15} - 16 q^{17} - 16 q^{18} - 8 q^{20} - 8 q^{22} - 16 q^{23} + 136 q^{24} - 16 q^{25} - 232 q^{26} + 232 q^{28} - 200 q^{30} - 16 q^{31} + 32 q^{32} + 56 q^{34} - 200 q^{36} + 192 q^{38} - 16 q^{39} - 456 q^{40} - 16 q^{41} + 424 q^{42} - 360 q^{44} + 200 q^{46} - 16 q^{47} - 80 q^{48} - 16 q^{49} - 16 q^{52} - 456 q^{54} - 16 q^{55} - 448 q^{56} - 336 q^{57} - 512 q^{58} - 736 q^{60} - 288 q^{62} - 16 q^{63} - 152 q^{64} + 144 q^{65} - 80 q^{66} + 80 q^{68} + 288 q^{70} - 16 q^{71} + 512 q^{72} + 464 q^{73} + 328 q^{74} + 496 q^{76} + 1136 q^{78} - 16 q^{79} + 520 q^{80} - 464 q^{81} + 592 q^{82} + 1184 q^{86} - 16 q^{87} - 840 q^{88} - 16 q^{89} + 1512 q^{90} - 1016 q^{92} + 664 q^{94} - 16 q^{95} - 768 q^{96} - 16 q^{97} + 400 q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −1.99615 0.123996i −2.49131 + 1.66464i 3.96925 + 0.495030i −5.28331 1.05092i 5.17944 3.01396i 0.430227 + 2.16290i −7.86185 1.48033i −0.00856088 + 0.0206678i 10.4160 + 2.75290i
5.2 −1.95431 0.425075i 4.73781 3.16571i 3.63862 + 1.66145i −5.99545 1.19257i −10.6048 + 4.17283i −1.47940 7.43743i −6.40474 4.79367i 8.98103 21.6821i 11.2100 + 4.87916i
5.3 −1.93868 + 0.491460i 1.98785 1.32824i 3.51693 1.90556i 8.85748 + 1.76186i −3.20102 + 3.55198i −1.81827 9.14104i −5.88169 + 5.42271i −1.25682 + 3.03423i −18.0377 + 0.937417i
5.4 −1.86396 0.725012i 2.06162 1.37753i 2.94871 + 2.70279i 4.63458 + 0.921876i −4.84150 + 1.07296i 2.38099 + 11.9700i −3.53674 7.17576i −1.09147 + 2.63504i −7.97032 5.07847i
5.5 −1.83427 0.797164i −2.62580 + 1.75450i 2.72906 + 2.92442i 3.72340 + 0.740630i 6.21503 1.12503i −0.824370 4.14439i −2.67457 7.53967i 0.372384 0.899013i −6.23929 4.32667i
5.6 −1.82287 + 0.822889i 0.416587 0.278355i 2.64571 3.00004i −4.26113 0.847591i −0.530329 + 0.850210i −0.301354 1.51501i −2.35408 + 7.64580i −3.34809 + 8.08300i 8.46495 1.96139i
5.7 −1.75423 + 0.960563i −3.89540 + 2.60282i 2.15464 3.37009i 6.75022 + 1.34270i 4.33324 8.30772i 2.12232 + 10.6696i −0.542538 + 7.98158i 4.95529 11.9631i −13.1312 + 4.12860i
5.8 −1.56990 + 1.23912i 3.79009 2.53245i 0.929162 3.89059i −0.274813 0.0546637i −2.81203 + 8.67207i 1.94983 + 9.80246i 3.36221 + 7.25917i 4.50728 10.8815i 0.499163 0.254710i
5.9 −1.39201 1.43608i 1.32480 0.885206i −0.124624 + 3.99806i 0.988766 + 0.196678i −3.11536 0.670305i −1.85512 9.32633i 5.91499 5.38636i −2.47263 + 5.96947i −1.09393 1.69372i
5.10 −1.20172 1.59871i 1.03356 0.690603i −1.11173 + 3.84240i −6.72566 1.33782i −2.34613 0.822447i 1.20217 + 6.04371i 7.47886 2.84018i −2.85284 + 6.88735i 5.94360 + 12.3600i
5.11 −1.18540 + 1.61084i −2.34076 + 1.56405i −1.18963 3.81900i 0.703425 + 0.139920i 0.255314 5.62462i −1.92692 9.68727i 7.56201 + 2.61075i −0.411234 + 0.992807i −1.05923 + 0.967245i
5.12 −0.962010 1.75344i −4.61315 + 3.08241i −2.14907 + 3.37365i −4.96080 0.986765i 9.84269 + 5.12355i 0.206012 + 1.03569i 7.98290 + 0.522780i 8.33574 20.1243i 3.04211 + 9.64772i
5.13 −0.809891 + 1.82868i 0.402825 0.269159i −2.68815 2.96207i 0.164157 + 0.0326529i 0.165962 + 0.954627i 1.23928 + 6.23029i 7.59379 2.51683i −3.35433 + 8.09807i −0.192661 + 0.273746i
5.14 −0.559622 1.92011i 4.61315 3.08241i −3.37365 + 2.14907i 4.96080 + 0.986765i −8.50018 7.13277i 0.206012 + 1.03569i 6.01442 + 5.27510i 8.33574 20.1243i −0.881479 10.0775i
5.15 −0.301940 + 1.97708i 2.58349 1.72623i −3.81766 1.19392i −9.15038 1.82012i 2.63284 + 5.62898i −1.42493 7.16362i 3.51317 7.18732i 0.250398 0.604514i 6.36139 17.5414i
5.16 −0.295486 + 1.97805i 3.57309 2.38746i −3.82538 1.16897i 5.68249 + 1.13032i 3.66673 + 7.77323i −1.07342 5.39645i 3.44263 7.22138i 3.62286 8.74635i −3.91492 + 10.9063i
5.17 −0.280710 1.98020i −1.03356 + 0.690603i −3.84240 + 1.11173i 6.72566 + 1.33782i 1.65767 + 1.85280i 1.20217 + 6.04371i 3.28004 + 7.29667i −2.85284 + 6.88735i 0.761189 13.6937i
5.18 −0.0311599 1.99976i −1.32480 + 0.885206i −3.99806 + 0.124624i −0.988766 0.196678i 1.81148 + 2.62170i −1.85512 9.32633i 0.373797 + 7.99126i −2.47263 + 5.96947i −0.362498 + 1.98342i
5.19 0.108110 + 1.99708i −3.69899 + 2.47159i −3.97662 + 0.431807i −1.02401 0.203689i −5.33584 7.11996i 0.149919 + 0.753693i −1.29226 7.89494i 4.12963 9.96982i 0.296076 2.06705i
5.20 0.506868 + 1.93471i −0.155374 + 0.103818i −3.48617 + 1.96128i 6.64253 + 1.32128i −0.279611 0.247982i 0.946911 + 4.76044i −5.56153 5.75060i −3.43079 + 8.28265i 0.810599 + 13.5211i
See next 80 embeddings (of 272 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 133.34 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
17.e odd 16 1 inner
136.q odd 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.3.q.a 272
8.b even 2 1 inner 136.3.q.a 272
17.e odd 16 1 inner 136.3.q.a 272
136.q odd 16 1 inner 136.3.q.a 272

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.3.q.a 272 1.a even 1 1 trivial
136.3.q.a 272 8.b even 2 1 inner
136.3.q.a 272 17.e odd 16 1 inner
136.3.q.a 272 136.q odd 16 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(136, [\chi])$$.