# Properties

 Label 403.2.cd.a Level 403 Weight 2 Character orbit 403.cd Analytic conductor 3.218 Analytic rank 0 Dimension 576 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.cd (of order $$60$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$576$$ Relative dimension: $$36$$ over $$\Q(\zeta_{60})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$576q - 12q^{2} - 28q^{3} - 8q^{5} + 6q^{6} - 16q^{7} - 46q^{8} - 104q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$576q - 12q^{2} - 28q^{3} - 8q^{5} + 6q^{6} - 16q^{7} - 46q^{8} - 104q^{9} - 18q^{11} - 24q^{13} - 44q^{14} + 10q^{15} + 120q^{16} - 10q^{18} - 16q^{19} - 32q^{20} - 28q^{21} - 28q^{24} - 72q^{26} - 100q^{27} + 38q^{28} - 40q^{29} - 66q^{31} - 124q^{32} + 52q^{33} + 76q^{34} - 32q^{35} - 36q^{37} + 10q^{39} - 52q^{40} + 2q^{41} + 8q^{42} + 12q^{44} + 30q^{45} + 20q^{46} - 20q^{47} - 40q^{48} + 50q^{52} - 84q^{53} - 80q^{54} - 28q^{55} - 126q^{57} + 70q^{58} - 74q^{59} - 50q^{60} + 212q^{63} + 84q^{65} + 136q^{66} - 20q^{67} - 60q^{68} + 18q^{70} - 62q^{71} - 54q^{72} - 32q^{73} - 112q^{74} - 64q^{76} + 6q^{78} - 56q^{79} + 60q^{80} - 168q^{81} + 238q^{83} - 206q^{84} - 30q^{85} - 158q^{86} + 76q^{87} + 140q^{89} - 50q^{91} + 70q^{93} - 48q^{94} + 8q^{96} - 92q^{97} + 50q^{98} - 156q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
21.1 −0.438402 2.76796i −2.21379 + 0.232679i −5.56730 + 1.80893i 0.467617 1.74517i 1.61458 + 6.02569i −2.46919 + 1.60351i 4.90317 + 9.62301i 1.91230 0.406472i −5.03557 0.529260i
21.2 −0.420851 2.65715i 0.222716 0.0234084i −4.98120 + 1.61849i −0.667546 + 2.49132i −0.155930 0.581938i 0.821932 0.533769i 3.95420 + 7.76055i −2.88539 + 0.613308i 6.90073 + 0.725296i
21.3 −0.375607 2.37149i 2.23445 0.234850i −3.58075 + 1.16346i 0.997640 3.72324i −1.39622 5.21075i 0.525344 0.341162i 1.92397 + 3.77601i 2.00316 0.425785i −9.20434 0.967415i
21.4 −0.342253 2.16090i 1.06144 0.111561i −2.65023 + 0.861112i −0.672966 + 2.51154i −0.604352 2.25547i −3.40576 + 2.21173i 0.781311 + 1.53341i −1.82024 + 0.386904i 5.65752 + 0.594629i
21.5 −0.339072 2.14081i −2.52783 + 0.265685i −2.56601 + 0.833746i −1.03003 + 3.84414i 1.42590 + 5.32152i 2.45967 1.59733i 0.686907 + 1.34813i 3.38488 0.719478i 8.57885 + 0.901674i
21.6 −0.336779 2.12634i 2.51608 0.264450i −2.50579 + 0.814180i −0.295700 + 1.10357i −1.40967 5.26097i 0.467445 0.303562i 0.620378 + 1.21756i 3.32626 0.707019i 2.44614 + 0.257100i
21.7 −0.331707 2.09431i −1.13363 + 0.119149i −2.37401 + 0.771361i 0.0970019 0.362016i 0.625567 + 2.33465i 0.682237 0.443050i 0.477646 + 0.937433i −1.66353 + 0.353594i −0.790351 0.0830693i
21.8 −0.310808 1.96236i 0.486514 0.0511347i −1.85216 + 0.601802i 0.642313 2.39714i −0.251557 0.938825i −3.43497 + 2.23070i −0.0473792 0.0929870i −2.70036 + 0.573980i −4.90370 0.515400i
21.9 −0.302034 1.90697i −2.73963 + 0.287946i −1.64319 + 0.533905i 0.850977 3.17589i 1.37656 + 5.13741i 3.33174 2.16366i −0.238636 0.468349i 4.48819 0.953995i −6.31334 0.663559i
21.10 −0.226478 1.42992i 2.63019 0.276444i −0.0912802 + 0.0296587i −0.411870 + 1.53712i −0.990974 3.69837i 2.42932 1.57762i −1.25145 2.45610i 3.90704 0.830467i 2.29124 + 0.240819i
21.11 −0.222256 1.40327i −3.26546 + 0.343213i −0.0176593 + 0.00573785i 0.213111 0.795340i 1.20739 + 4.50604i −2.75126 + 1.78669i −1.27805 2.50831i 7.61098 1.61776i −1.16344 0.122283i
21.12 −0.184795 1.16675i −0.0935769 + 0.00983533i 0.574955 0.186814i −0.0228269 + 0.0851910i 0.0287679 + 0.107363i 2.60956 1.69467i −1.39681 2.74139i −2.92578 + 0.621894i 0.103615 + 0.0108904i
21.13 −0.183398 1.15793i −1.43456 + 0.150779i 0.594952 0.193312i −0.478261 + 1.78489i 0.437686 + 1.63347i −1.94871 + 1.26551i −1.39744 2.74262i −0.899204 + 0.191132i 2.15449 + 0.226446i
21.14 −0.159860 1.00931i 1.15285 0.121170i 0.908952 0.295336i 0.515241 1.92291i −0.306593 1.14422i 0.690634 0.448503i −1.37125 2.69124i −1.62005 + 0.344353i −2.02318 0.212645i
21.15 −0.138227 0.872731i −1.52885 + 0.160688i 1.15956 0.376764i −0.666091 + 2.48588i 0.351566 + 1.31206i −0.721464 + 0.468524i −1.29140 2.53451i −0.622893 + 0.132400i 2.26158 + 0.237702i
21.16 −0.0719155 0.454056i 2.88221 0.302933i 1.70112 0.552727i 0.524460 1.95731i −0.344824 1.28690i −3.88603 + 2.52362i −0.790719 1.55187i 5.28094 1.12250i −0.926446 0.0973734i
21.17 −0.0567540 0.358331i 2.32789 0.244671i 1.77693 0.577361i −0.959878 + 3.58231i −0.219790 0.820268i −2.32142 + 1.50755i −0.637148 1.25047i 2.42476 0.515398i 1.33813 + 0.140643i
21.18 −0.0348740 0.220186i −1.73304 + 0.182150i 1.85485 0.602676i 0.939504 3.50628i 0.100545 + 0.375238i 3.04041 1.97447i −0.399803 0.784658i 0.0357975 0.00760900i −0.804796 0.0845874i
21.19 −0.0103892 0.0655950i 1.19020 0.125096i 1.89792 0.616671i 0.380352 1.41949i −0.0205710 0.0767719i 0.375204 0.243660i −0.120470 0.236436i −1.53350 + 0.325956i −0.0970632 0.0102018i
21.20 0.0347043 + 0.219114i −2.53826 + 0.266782i 1.85531 0.602826i 0.413236 1.54222i −0.146544 0.546910i −1.87187 + 1.21561i 0.397906 + 0.780935i 3.43713 0.730585i 0.352263 + 0.0370243i
See next 80 embeddings (of 576 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 385.36 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner
31.h odd 30 1 inner
403.cd even 60 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.cd.a 576
13.d odd 4 1 inner 403.2.cd.a 576
31.h odd 30 1 inner 403.2.cd.a 576
403.cd even 60 1 inner 403.2.cd.a 576

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.cd.a 576 1.a even 1 1 trivial
403.2.cd.a 576 13.d odd 4 1 inner
403.2.cd.a 576 31.h odd 30 1 inner
403.2.cd.a 576 403.cd even 60 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(403, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database