# Properties

 Label 414.2.p.a Level $414$ Weight $2$ Character orbit 414.p Analytic conductor $3.306$ Analytic rank $0$ Dimension $480$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$414 = 2 \cdot 3^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 414.p (of order $$66$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.30580664368$$ Analytic rank: $$0$$ Dimension: $$480$$ Relative dimension: $$24$$ over $$\Q(\zeta_{66})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$480 q - 4 q^{3} - 24 q^{4} + 8 q^{6}+O(q^{10})$$ 480 * q - 4 * q^3 - 24 * q^4 + 8 * q^6 $$\operatorname{Tr}(f)(q) =$$ $$480 q - 4 q^{3} - 24 q^{4} + 8 q^{6} + 4 q^{12} - 44 q^{15} + 24 q^{16} + 28 q^{18} + 66 q^{21} + 6 q^{23} + 4 q^{24} + 12 q^{25} + 86 q^{27} - 48 q^{29} + 44 q^{30} - 12 q^{31} - 44 q^{33} + 24 q^{39} - 60 q^{41} + 12 q^{46} + 48 q^{47} + 8 q^{48} - 78 q^{49} + 24 q^{50} - 30 q^{54} - 24 q^{55} - 66 q^{56} - 66 q^{57} - 72 q^{59} - 22 q^{60} - 110 q^{63} + 48 q^{64} - 462 q^{65} - 88 q^{66} - 70 q^{69} + 12 q^{70} - 52 q^{72} - 264 q^{74} - 76 q^{75} - 12 q^{77} - 58 q^{78} - 96 q^{81} - 198 q^{83} - 22 q^{84} + 178 q^{87} + 6 q^{92} - 20 q^{93} - 12 q^{94} + 12 q^{95} - 4 q^{96} - 66 q^{97} - 132 q^{99}+O(q^{100})$$ 480 * q - 4 * q^3 - 24 * q^4 + 8 * q^6 + 4 * q^12 - 44 * q^15 + 24 * q^16 + 28 * q^18 + 66 * q^21 + 6 * q^23 + 4 * q^24 + 12 * q^25 + 86 * q^27 - 48 * q^29 + 44 * q^30 - 12 * q^31 - 44 * q^33 + 24 * q^39 - 60 * q^41 + 12 * q^46 + 48 * q^47 + 8 * q^48 - 78 * q^49 + 24 * q^50 - 30 * q^54 - 24 * q^55 - 66 * q^56 - 66 * q^57 - 72 * q^59 - 22 * q^60 - 110 * q^63 + 48 * q^64 - 462 * q^65 - 88 * q^66 - 70 * q^69 + 12 * q^70 - 52 * q^72 - 264 * q^74 - 76 * q^75 - 12 * q^77 - 58 * q^78 - 96 * q^81 - 198 * q^83 - 22 * q^84 + 178 * q^87 + 6 * q^92 - 20 * q^93 - 12 * q^94 + 12 * q^95 - 4 * q^96 - 66 * q^97 - 132 * q^99

## 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 −0.971812 + 0.235759i −1.65357 + 0.515460i 0.888835 0.458227i 3.05695 + 2.40402i 1.48544 0.890775i 2.82979 + 2.01508i −0.755750 + 0.654861i 2.46860 1.70470i −3.53755 1.61555i
5.2 −0.971812 + 0.235759i −1.53348 0.805250i 0.888835 0.458227i −2.29731 1.80663i 1.68010 + 0.421018i −1.55923 1.11032i −0.755750 + 0.654861i 1.70315 + 2.46967i 2.65848 + 1.21409i
5.3 −0.971812 + 0.235759i −1.46892 + 0.917760i 0.888835 0.458227i −2.89944 2.28015i 1.21114 1.23820i 3.52216 + 2.50812i −0.755750 + 0.654861i 1.31543 2.69623i 3.35528 + 1.53230i
5.4 −0.971812 + 0.235759i −1.37201 + 1.05716i 0.888835 0.458227i 0.729978 + 0.574061i 1.08410 1.35082i −1.96750 1.40105i −0.755750 + 0.654861i 0.764833 2.90087i −0.844741 0.385780i
5.5 −0.971812 + 0.235759i −1.35343 1.08084i 0.888835 0.458227i −0.165116 0.129849i 1.57010 + 0.731288i 0.0204197 + 0.0145408i −0.755750 + 0.654861i 0.663572 + 2.92569i 0.191075 + 0.0872611i
5.6 −0.971812 + 0.235759i −0.304529 + 1.70507i 0.888835 0.458227i 0.311301 + 0.244810i −0.106041 1.72880i −0.315067 0.224358i −0.755750 + 0.654861i −2.81452 1.03849i −0.360242 0.164517i
5.7 −0.971812 + 0.235759i −0.201012 1.72035i 0.888835 0.458227i 0.914714 + 0.719339i 0.600933 + 1.62446i 3.62325 + 2.58010i −0.755750 + 0.654861i −2.91919 + 0.691621i −1.05852 0.483410i
5.8 −0.971812 + 0.235759i 0.843144 1.51298i 0.888835 0.458227i 0.487696 + 0.383528i −0.462678 + 1.66911i −2.10909 1.50188i −0.755750 + 0.654861i −1.57822 2.55132i −0.564369 0.257739i
5.9 −0.971812 + 0.235759i 0.893014 + 1.48409i 0.888835 0.458227i 1.85702 + 1.46038i −1.21773 1.23172i 1.69311 + 1.20566i −0.755750 + 0.654861i −1.40505 + 2.65063i −2.14898 0.981404i
5.10 −0.971812 + 0.235759i 1.41656 + 0.996675i 0.888835 0.458227i −2.36798 1.86220i −1.61160 0.634614i −1.80599 1.28604i −0.755750 + 0.654861i 1.01328 + 2.82370i 2.74026 + 1.25144i
5.11 −0.971812 + 0.235759i 1.56615 0.739717i 0.888835 0.458227i −1.72561 1.35704i −1.34761 + 1.08810i 2.21726 + 1.57890i −0.755750 + 0.654861i 1.90564 2.31701i 1.99691 + 0.911957i
5.12 −0.971812 + 0.235759i 1.62639 + 0.595693i 0.888835 0.458227i 1.33065 + 1.04644i −1.72099 0.195465i −3.58234 2.55097i −0.755750 + 0.654861i 2.29030 + 1.93766i −1.53985 0.703227i
5.13 0.971812 0.235759i −1.71005 + 0.275194i 0.888835 0.458227i 0.483941 + 0.380576i −1.59697 + 0.670596i −3.88305 2.76511i 0.755750 0.654861i 2.84854 0.941191i 0.560024 + 0.255754i
5.14 0.971812 0.235759i −1.39847 1.02190i 0.888835 0.458227i 2.01366 + 1.58356i −1.59997 0.663397i 1.16801 + 0.831733i 0.755750 0.654861i 0.911423 + 2.85820i 2.33024 + 1.06419i
5.15 0.971812 0.235759i −1.36529 + 1.06582i 0.888835 0.458227i −0.463371 0.364399i −1.07553 + 1.35766i 0.209188 + 0.148962i 0.755750 0.654861i 0.728058 2.91031i −0.536220 0.244883i
5.16 0.971812 0.235759i −0.616489 + 1.61862i 0.888835 0.458227i −3.43182 2.69881i −0.217506 + 1.71834i 0.0555390 + 0.0395491i 0.755750 0.654861i −2.23988 1.99573i −3.97135 1.81366i
5.17 0.971812 0.235759i −0.490425 1.66117i 0.888835 0.458227i −1.22350 0.962173i −0.868236 1.49872i −1.60431 1.14243i 0.755750 0.654861i −2.51897 + 1.62936i −1.41585 0.646599i
5.18 0.971812 0.235759i −0.209960 + 1.71928i 0.888835 0.458227i 2.11401 + 1.66247i 0.201293 + 1.72031i 1.78139 + 1.26852i 0.755750 0.654861i −2.91183 0.721960i 2.44636 + 1.11721i
5.19 0.971812 0.235759i 0.786966 1.54295i 0.888835 0.458227i −0.388042 0.305160i 0.401020 1.68499i 3.61967 + 2.57756i 0.755750 0.654861i −1.76137 2.42849i −0.449048 0.205074i
5.20 0.971812 0.235759i 0.834387 1.51783i 0.888835 0.458227i 2.34927 + 1.84748i 0.453026 1.67176i −1.03925 0.740048i 0.755750 0.654861i −1.60760 2.53291i 2.71861 + 1.24155i
See next 80 embeddings (of 480 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 401.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
23.d odd 22 1 inner
207.o even 66 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 414.2.p.a 480
9.d odd 6 1 inner 414.2.p.a 480
23.d odd 22 1 inner 414.2.p.a 480
207.o even 66 1 inner 414.2.p.a 480

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.2.p.a 480 1.a even 1 1 trivial
414.2.p.a 480 9.d odd 6 1 inner
414.2.p.a 480 23.d odd 22 1 inner
414.2.p.a 480 207.o even 66 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(414, [\chi])$$.