# Properties

 Label 504.2.bt.a Level 504 Weight 2 Character orbit 504.bt Analytic conductor 4.024 Analytic rank 0 Dimension 184 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.bt (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$184$$ Relative dimension: $$92$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$184q - 2q^{3} - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$184q - 2q^{3} - 2q^{4} - 2q^{6} - 2q^{9} - 6q^{10} - 6q^{11} - 8q^{12} + 12q^{14} - 2q^{16} + 2q^{18} - 4q^{19} - 6q^{20} + 2q^{22} - 8q^{24} - 74q^{25} - 6q^{26} - 8q^{27} + 3q^{30} - 14q^{33} - 4q^{34} + 30q^{35} - 38q^{36} + 39q^{38} + 6q^{40} - 12q^{41} - 20q^{42} - 4q^{43} + 9q^{44} - 6q^{46} - 5q^{48} - 2q^{49} - 21q^{50} - 34q^{51} + 9q^{52} + 47q^{54} - 24q^{56} + 4q^{57} - 3q^{58} - 11q^{60} - 8q^{64} - 26q^{66} - 4q^{67} - 42q^{68} - 3q^{70} + 52q^{72} - 4q^{73} + 27q^{74} + 30q^{75} + 2q^{76} - 29q^{78} + 87q^{80} + 14q^{81} - 4q^{82} - 72q^{83} - 59q^{84} - 27q^{86} - 7q^{88} - 24q^{89} - 49q^{90} - 36q^{91} - 36q^{92} - 18q^{94} + 23q^{96} - 4q^{97} + 57q^{98} + 10q^{99} + O(q^{100})$$

## 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 −1.41347 + 0.0459458i 1.18561 1.26267i 1.99578 0.129886i −0.120967 + 0.209521i −1.61780 + 1.83922i 2.21818 + 1.44212i −2.81500 + 0.275287i −0.188680 2.99406i 0.161356 0.301709i
11.2 −1.41299 + 0.0589246i −0.617940 1.61807i 1.99306 0.166519i −1.47387 + 2.55282i 0.968485 + 2.24990i −2.64405 + 0.0949942i −2.80635 + 0.352729i −2.23630 + 1.99974i 1.93213 3.69394i
11.3 −1.40622 + 0.150130i −1.62573 + 0.597489i 1.95492 0.422232i 1.01528 1.75852i 2.19644 1.08427i −2.61403 0.408451i −2.68567 + 0.887244i 2.28601 1.94271i −1.16371 + 2.62530i
11.4 −1.40502 + 0.160980i 0.138508 + 1.72650i 1.94817 0.452362i 0.152023 0.263311i −0.472540 2.40348i 2.13879 1.55742i −2.66440 + 0.949196i −2.96163 + 0.478270i −0.171207 + 0.394430i
11.5 −1.40362 + 0.172746i −1.32668 + 1.11351i 1.94032 0.484940i −1.78605 + 3.09352i 1.66981 1.79213i 1.43753 2.22115i −2.63970 + 1.01585i 0.520180 2.95456i 1.97254 4.65068i
11.6 −1.39667 0.222043i 0.996010 + 1.41703i 1.90139 + 0.620242i 2.02288 3.50373i −1.07646 2.20028i −1.02043 + 2.44105i −2.51791 1.28847i −1.01593 + 2.82274i −3.60328 + 4.44440i
11.7 −1.38769 0.272632i 1.73194 + 0.0199803i 1.85134 + 0.756656i 0.662921 1.14821i −2.39793 0.499908i 0.655004 2.56339i −2.36279 1.55474i 2.99920 + 0.0692093i −1.23297 + 1.41263i
11.8 −1.33582 + 0.464308i −0.893431 1.48384i 1.56884 1.24047i 0.431911 0.748092i 1.88242 + 1.56732i −0.623850 + 2.57115i −1.51972 + 2.38546i −1.40356 + 2.65142i −0.229611 + 1.19986i
11.9 −1.33451 0.468067i −1.72686 0.133969i 1.56183 + 1.24928i −0.658616 + 1.14076i 2.24181 + 0.987070i 1.31090 + 2.29816i −1.49952 2.39821i 2.96410 + 0.462692i 1.41288 1.21407i
11.10 −1.33426 0.468774i −1.54596 0.781037i 1.56050 + 1.25093i 1.27843 2.21431i 1.69658 + 1.76681i 0.102649 2.64376i −1.49571 2.40059i 1.77996 + 2.41490i −2.74377 + 2.35517i
11.11 −1.32352 + 0.498286i 1.29060 + 1.15514i 1.50342 1.31898i −1.45478 + 2.51976i −2.28373 0.885761i 0.649643 + 2.56475i −1.33258 + 2.49484i 0.331314 + 2.98165i 0.669877 4.05985i
11.12 −1.31497 0.520448i 0.732856 1.56937i 1.45827 + 1.36874i 1.20934 2.09463i −1.78046 + 1.68225i −2.58361 + 0.570073i −1.20521 2.55880i −1.92584 2.30025i −2.68038 + 2.12497i
11.13 −1.28291 0.595101i 0.671530 + 1.59657i 1.29171 + 1.52692i −1.33380 + 2.31020i 0.0886094 2.44789i −2.19987 1.46989i −0.748476 2.72760i −2.09809 + 2.14430i 3.08594 2.17004i
11.14 −1.25253 + 0.656631i 0.406083 1.68377i 1.13767 1.64490i 1.72880 2.99437i 0.596986 + 2.37563i −0.995123 2.45148i −0.344876 + 2.80732i −2.67019 1.36751i −0.199181 + 4.88573i
11.15 −1.22535 0.706063i −0.326067 1.70108i 1.00295 + 1.73034i −1.27390 + 2.20646i −0.801525 + 2.31464i 2.38039 1.15488i −0.00723210 2.82842i −2.78736 + 1.10933i 3.11887 1.80422i
11.16 −1.21905 + 0.716886i −0.0776217 + 1.73031i 0.972150 1.74783i 0.664172 1.15038i −1.14581 2.16498i −2.29630 1.31415i 0.0679007 + 2.82761i −2.98795 0.268619i 0.0150339 + 1.87850i
11.17 −1.21438 + 0.724765i 1.22201 1.22748i 0.949431 1.76028i −1.73881 + 3.01171i −0.594345 + 2.37629i −0.680594 2.55672i 0.122820 + 2.82576i −0.0134011 2.99997i −0.0712072 4.91759i
11.18 −1.20074 + 0.747147i −1.53340 0.805409i 0.883544 1.79425i −0.00857237 + 0.0148478i 2.44297 0.178589i 2.28903 1.32678i 0.279666 + 2.81457i 1.70263 + 2.47003i −0.000800297 0.0242331i
11.19 −1.13389 0.845156i −0.862894 + 1.50180i 0.571423 + 1.91663i 1.48267 2.56806i 2.24769 0.973603i 2.51854 0.810517i 0.971921 2.65619i −1.51083 2.59179i −3.85161 + 1.65882i
11.20 −1.08666 + 0.905084i 1.73140 0.0474789i 0.361645 1.96703i 1.91888 3.32360i −1.83846 + 1.61866i 2.42950 + 1.04762i 1.38734 + 2.46481i 2.99549 0.164410i 0.922971 + 5.34836i
See next 80 embeddings (of 184 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 275.92 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.d odd 2 1 inner
63.j odd 6 1 inner
504.bt even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.bt.a 184
7.c even 3 1 504.2.cy.a yes 184
8.d odd 2 1 inner 504.2.bt.a 184
9.d odd 6 1 504.2.cy.a yes 184
56.k odd 6 1 504.2.cy.a yes 184
63.j odd 6 1 inner 504.2.bt.a 184
72.l even 6 1 504.2.cy.a yes 184
504.bt even 6 1 inner 504.2.bt.a 184

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bt.a 184 1.a even 1 1 trivial
504.2.bt.a 184 8.d odd 2 1 inner
504.2.bt.a 184 63.j odd 6 1 inner
504.2.bt.a 184 504.bt even 6 1 inner
504.2.cy.a yes 184 7.c even 3 1
504.2.cy.a yes 184 9.d odd 6 1
504.2.cy.a yes 184 56.k odd 6 1
504.2.cy.a yes 184 72.l even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database