# Properties

 Label 403.2.l.c Level 403 Weight 2 Character orbit 403.l Analytic conductor 3.218 Analytic rank 0 Dimension 68 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.l (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$68$$ Relative dimension: $$34$$ over $$\Q(\zeta_{6})$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$68q - 6q^{3} - 76q^{4} - 40q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$68q - 6q^{3} - 76q^{4} - 40q^{9} + 8q^{10} - 10q^{12} - 3q^{13} + 10q^{14} + 84q^{16} + 6q^{17} + 4q^{22} - 44q^{23} + 30q^{25} - 3q^{26} - 12q^{27} + 48q^{29} - 4q^{30} - 48q^{35} + 40q^{36} + 60q^{38} - 14q^{39} + 20q^{40} - 10q^{42} - 12q^{43} + 32q^{48} + 58q^{49} + 20q^{51} - 27q^{52} + 8q^{53} - 36q^{55} - 50q^{56} - 12q^{61} - 74q^{62} - 15q^{65} + 164q^{66} + 4q^{68} - 34q^{69} - 4q^{74} + 20q^{75} - 200q^{77} - 58q^{78} - 80q^{79} - 82q^{81} - 66q^{82} + 52q^{87} + 16q^{88} - 14q^{90} - 70q^{91} + 108q^{92} - 4q^{94} + 76q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1 2.78624i −0.733304 1.27012i −5.76313 0.320386 + 0.184975i −3.53886 + 2.04316i −1.80035 + 1.03943i 10.4850i 0.424529 0.735307i 0.515384 0.892671i
25.2 2.59015i 0.881022 + 1.52597i −4.70886 −1.89705 1.09526i 3.95250 2.28198i 3.93715 2.27311i 7.01635i −0.0523990 + 0.0907577i −2.83688 + 4.91363i
25.3 2.46210i 0.199198 + 0.345021i −4.06196 3.00007 + 1.73209i 0.849477 0.490446i −1.28546 + 0.742161i 5.07675i 1.42064 2.46062i 4.26458 7.38647i
25.4 2.42820i 1.37274 + 2.37765i −3.89617 −2.09477 1.20941i 5.77342 3.33328i −4.33186 + 2.50100i 4.60429i −2.26881 + 3.92970i −2.93671 + 5.08652i
25.5 2.22973i −0.613568 1.06273i −2.97169 −2.33602 1.34870i −2.36960 + 1.36809i −1.24184 + 0.716977i 2.16660i 0.747068 1.29396i −3.00724 + 5.20869i
25.6 2.06821i −0.348471 0.603570i −2.27750 1.11091 + 0.641386i −1.24831 + 0.720713i 2.98944 1.72596i 0.573935i 1.25714 2.17742i 1.32652 2.29760i
25.7 2.00957i −1.67472 2.90070i −2.03837 −0.151987 0.0877499i −5.82915 + 3.36546i −2.39273 + 1.38144i 0.0771108i −4.10936 + 7.11762i −0.176340 + 0.305429i
25.8 1.94813i 1.62865 + 2.82091i −1.79519 2.42854 + 1.40212i 5.49549 3.17282i 0.340725 0.196718i 0.398987i −3.80502 + 6.59049i 2.73150 4.73109i
25.9 1.84455i −1.26208 2.18598i −1.40238 3.26306 + 1.88393i −4.03216 + 2.32797i 1.26392 0.729726i 1.10234i −1.68567 + 2.91967i 3.47501 6.01889i
25.10 1.46972i 0.869265 + 1.50561i −0.160067 0.337316 + 0.194750i 2.21282 1.27757i 1.34672 0.777530i 2.70418i −0.0112448 + 0.0194765i 0.286227 0.495759i
25.11 1.25574i −1.30537 2.26097i 0.423126 −3.09634 1.78767i −2.83918 + 1.63920i 4.10494 2.36999i 3.04281i −1.90798 + 3.30473i −2.24485 + 3.88819i
25.12 0.921357i −0.264605 0.458309i 1.15110 −2.89414 1.67093i −0.422266 + 0.243795i −0.336375 + 0.194206i 2.90329i 1.35997 2.35553i −1.53952 + 2.66653i
25.13 0.909522i 0.193979 + 0.335982i 1.17277 3.26453 + 1.88478i 0.305583 0.176429i −3.57672 + 2.06502i 2.88570i 1.42474 2.46773i 1.71425 2.96916i
25.14 0.768888i −0.723761 1.25359i 1.40881 0.354729 + 0.204803i −0.963871 + 0.556491i −0.855619 + 0.493992i 2.62099i 0.452340 0.783476i 0.157470 0.272747i
25.15 0.216546i 1.35890 + 2.35368i 1.95311 0.933057 + 0.538701i 0.509681 0.294264i −2.50436 + 1.44590i 0.856030i −2.19321 + 3.79876i 0.116654 0.202050i
25.16 0.177509i 0.311192 + 0.539000i 1.96849 1.82613 + 1.05431i 0.0956771 0.0552392i 1.99646 1.15265i 0.704441i 1.30632 2.26261i 0.187150 0.324153i
25.17 0.0456796i −1.38907 2.40594i 1.99791 −1.25901 0.726890i −0.109902 + 0.0634522i −3.37083 + 1.94615i 0.182623i −2.35904 + 4.08597i −0.0332040 + 0.0575110i
25.18 0.0456796i −1.38907 2.40594i 1.99791 1.25901 + 0.726890i 0.109902 0.0634522i 3.37083 1.94615i 0.182623i −2.35904 + 4.08597i −0.0332040 + 0.0575110i
25.19 0.177509i 0.311192 + 0.539000i 1.96849 −1.82613 1.05431i −0.0956771 + 0.0552392i −1.99646 + 1.15265i 0.704441i 1.30632 2.26261i 0.187150 0.324153i
25.20 0.216546i 1.35890 + 2.35368i 1.95311 −0.933057 0.538701i −0.509681 + 0.294264i 2.50436 1.44590i 0.856030i −2.19321 + 3.79876i 0.116654 0.202050i
See all 68 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 129.34 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.b even 2 1 inner
31.c even 3 1 inner
403.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.l.c 68
13.b even 2 1 inner 403.2.l.c 68
31.c even 3 1 inner 403.2.l.c 68
403.l even 6 1 inner 403.2.l.c 68

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.l.c 68 1.a even 1 1 trivial
403.2.l.c 68 13.b even 2 1 inner
403.2.l.c 68 31.c even 3 1 inner
403.2.l.c 68 403.l even 6 1 inner

## Hecke kernels

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

 $$T_{2}^{34} + \cdots$$ $$T_{5}^{68} - \cdots$$

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database