# Properties

 Label 403.2.be.c Level 403 Weight 2 Character orbit 403.be Analytic conductor 3.218 Analytic rank 0 Dimension 136 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.be (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$136$$ Relative dimension: $$34$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$136q - 4q^{2} - 24q^{3} + 2q^{5} - 36q^{6} - 2q^{7} - 8q^{8} + 60q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$136q - 4q^{2} - 24q^{3} + 2q^{5} - 36q^{6} - 2q^{7} - 8q^{8} + 60q^{9} - 18q^{11} - 6q^{13} + 4q^{14} - 168q^{16} - 50q^{18} - 22q^{19} + 22q^{20} - 54q^{21} + 84q^{22} + 36q^{24} + 12q^{26} + 20q^{28} + 12q^{31} + 20q^{32} - 12q^{33} + 24q^{34} - 16q^{35} + 30q^{37} - 16q^{39} + 16q^{40} + 2q^{41} - 84q^{42} - 90q^{44} - 4q^{45} - 40q^{47} + 12q^{48} + 4q^{50} + 96q^{52} + 84q^{53} - 132q^{57} + 34q^{59} - 20q^{63} + 66q^{65} - 152q^{66} + 24q^{67} - 128q^{70} - 52q^{71} + 60q^{72} + 48q^{74} + 70q^{76} + 124q^{78} + 168q^{79} + 54q^{80} - 28q^{81} - 90q^{83} - 66q^{84} - 126q^{86} + 108q^{87} - 184q^{93} + 56q^{94} + 240q^{96} + 52q^{97} - 12q^{98} + 6q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
57.1 −1.92370 + 1.92370i 2.94175 + 1.69842i 5.40127i −0.883813 0.236817i −8.92630 + 2.39180i −2.87039 + 0.769118i 6.54303 + 6.54303i 4.26925 + 7.39456i 2.15576 1.24463i
57.2 −1.87740 + 1.87740i −2.43895 1.40813i 5.04929i 1.13969 + 0.305379i 7.22253 1.93527i 0.691082 0.185175i 5.72474 + 5.72474i 2.46566 + 4.27065i −2.71297 + 1.56634i
57.3 −1.75183 + 1.75183i 0.421253 + 0.243210i 4.13779i 0.481245 + 0.128949i −1.16402 + 0.311899i 0.251958 0.0675120i 3.74504 + 3.74504i −1.38170 2.39317i −1.06895 + 0.617161i
57.4 −1.71373 + 1.71373i 0.673401 + 0.388788i 3.87374i 3.38103 + 0.905945i −1.82031 + 0.487750i −0.0747797 + 0.0200372i 3.21109 + 3.21109i −1.19769 2.07446i −7.34672 + 4.24163i
57.5 −1.63991 + 1.63991i −1.62343 0.937289i 3.37862i −3.90548 1.04647i 4.19936 1.12521i −3.18409 + 0.853175i 2.26082 + 2.26082i 0.257021 + 0.445173i 8.12076 4.68852i
57.6 −1.60653 + 1.60653i 1.41629 + 0.817697i 3.16189i −2.93216 0.785669i −3.58898 + 0.961663i 4.32754 1.15956i 1.86662 + 1.86662i −0.162744 0.281882i 5.97281 3.44840i
57.7 −1.31740 + 1.31740i −1.87836 1.08447i 1.47108i 1.82236 + 0.488300i 3.90323 1.04587i −2.35057 + 0.629834i −0.696799 0.696799i 0.852156 + 1.47598i −3.04406 + 1.75749i
57.8 −1.10405 + 1.10405i −1.49319 0.862095i 0.437851i −2.41909 0.648194i 2.60035 0.696763i 2.14125 0.573746i −1.72469 1.72469i −0.0135854 0.0235306i 3.38644 1.95516i
57.9 −1.06684 + 1.06684i −1.24662 0.719735i 0.276298i 3.17907 + 0.851829i 2.09779 0.562100i 4.69129 1.25703i −1.83892 1.83892i −0.463962 0.803606i −4.30033 + 2.48279i
57.10 −1.05928 + 1.05928i 1.91771 + 1.10719i 0.244140i −0.214757 0.0575440i −3.20421 + 0.858565i 2.75305 0.737677i −1.85994 1.85994i 0.951740 + 1.64846i 0.288442 0.166532i
57.11 −0.968661 + 0.968661i 0.615455 + 0.355333i 0.123393i 1.15257 + 0.308831i −0.940364 + 0.251970i −2.39214 + 0.640972i −2.05685 2.05685i −1.24748 2.16069i −1.41560 + 0.817300i
57.12 −0.890053 + 0.890053i 2.76707 + 1.59757i 0.415612i 2.13125 + 0.571067i −3.88476 + 1.04092i 1.08500 0.290724i −2.15002 2.15002i 3.60445 + 6.24308i −2.40520 + 1.38865i
57.13 −0.689883 + 0.689883i −2.73323 1.57803i 1.04812i 0.999543 + 0.267827i 2.97426 0.796952i −0.376277 + 0.100823i −2.10285 2.10285i 3.48036 + 6.02817i −0.874337 + 0.504799i
57.14 −0.472261 + 0.472261i 2.26659 + 1.30862i 1.55394i −3.80869 1.02054i −1.68843 + 0.452414i −3.01874 + 0.808868i −1.67839 1.67839i 1.92496 + 3.33413i 2.28066 1.31674i
57.15 −0.456690 + 0.456690i −0.104092 0.0600978i 1.58287i −2.27787 0.610354i 0.0749841 0.0200919i 0.192857 0.0516759i −1.63626 1.63626i −1.49278 2.58556i 1.31902 0.761539i
57.16 −0.308073 + 0.308073i 1.87353 + 1.08168i 1.81018i 3.38449 + 0.906872i −0.910421 + 0.243947i −3.42053 + 0.916529i −1.17382 1.17382i 0.840073 + 1.45505i −1.32205 + 0.763289i
57.17 −0.0974514 + 0.0974514i 0.799719 + 0.461718i 1.98101i 1.71922 + 0.460663i −0.122929 + 0.0329386i 2.76053 0.739683i −0.387954 0.387954i −1.07363 1.85959i −0.212432 + 0.122648i
57.18 0.0740210 0.0740210i −1.19559 0.690275i 1.98904i 2.69078 + 0.720992i −0.139594 + 0.0374041i −0.0708884 + 0.0189945i 0.295273 + 0.295273i −0.547040 0.947500i 0.252543 0.145806i
57.19 0.0798971 0.0798971i −1.31416 0.758731i 1.98723i 1.61887 + 0.433775i −0.165618 + 0.0443772i −4.54713 + 1.21840i 0.318568 + 0.318568i −0.348656 0.603889i 0.164001 0.0946857i
57.20 0.185549 0.185549i −2.28373 1.31851i 1.93114i −3.12819 0.838196i −0.668395 + 0.179096i −1.29814 + 0.347834i 0.729421 + 0.729421i 1.97696 + 3.42420i −0.735961 + 0.424907i
See next 80 embeddings (of 136 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 398.34 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner
31.e odd 6 1 inner
403.be even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.be.c 136
13.d odd 4 1 inner 403.2.be.c 136
31.e odd 6 1 inner 403.2.be.c 136
403.be even 12 1 inner 403.2.be.c 136

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.be.c 136 1.a even 1 1 trivial
403.2.be.c 136 13.d odd 4 1 inner
403.2.be.c 136 31.e odd 6 1 inner
403.2.be.c 136 403.be even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(403, [\chi])$$:

 $$T_{2}^{68} + \cdots$$ $$T_{7}^{136} + \cdots$$

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database