# Properties

 Label 414.3.h.a Level $414$ Weight $3$ Character orbit 414.h Analytic conductor $11.281$ Analytic rank $0$ Dimension $96$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$414 = 2 \cdot 3^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 414.h (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.2806829445$$ Analytic rank: $$0$$ Dimension: $$96$$ Relative dimension: $$48$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$96 q + 4 q^{3} - 96 q^{4} + 16 q^{6} + 36 q^{9}+O(q^{10})$$ 96 * q + 4 * q^3 - 96 * q^4 + 16 * q^6 + 36 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$96 q + 4 q^{3} - 96 q^{4} + 16 q^{6} + 36 q^{9} + 8 q^{12} - 192 q^{16} + 16 q^{18} + 6 q^{23} - 16 q^{24} + 228 q^{25} + 96 q^{26} - 20 q^{27} + 12 q^{29} + 60 q^{31} - 144 q^{36} + 12 q^{39} - 312 q^{41} - 24 q^{46} + 240 q^{47} - 32 q^{48} + 384 q^{49} + 96 q^{50} - 112 q^{54} + 264 q^{55} + 288 q^{59} + 144 q^{62} + 768 q^{64} - 286 q^{69} + 120 q^{70} - 696 q^{71} - 160 q^{72} - 56 q^{75} - 84 q^{77} - 296 q^{78} - 212 q^{81} + 512 q^{87} + 12 q^{92} - 220 q^{93} + 168 q^{94} - 456 q^{95} - 32 q^{96} - 288 q^{98}+O(q^{100})$$ 96 * q + 4 * q^3 - 96 * q^4 + 16 * q^6 + 36 * q^9 + 8 * q^12 - 192 * q^16 + 16 * q^18 + 6 * q^23 - 16 * q^24 + 228 * q^25 + 96 * q^26 - 20 * q^27 + 12 * q^29 + 60 * q^31 - 144 * q^36 + 12 * q^39 - 312 * q^41 - 24 * q^46 + 240 * q^47 - 32 * q^48 + 384 * q^49 + 96 * q^50 - 112 * q^54 + 264 * q^55 + 288 * q^59 + 144 * q^62 + 768 * q^64 - 286 * q^69 + 120 * q^70 - 696 * q^71 - 160 * q^72 - 56 * q^75 - 84 * q^77 - 296 * q^78 - 212 * q^81 + 512 * q^87 + 12 * q^92 - 220 * q^93 + 168 * q^94 - 456 * q^95 - 32 * q^96 - 288 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
229.1 −0.707107 1.22474i −2.89884 0.772474i −1.00000 + 1.73205i −5.56174 3.21107i 1.10371 + 4.09656i 7.31386 4.22266i 2.82843 7.80657 + 4.47856i 9.08228i
229.2 −0.707107 1.22474i −2.89884 0.772474i −1.00000 + 1.73205i 5.56174 + 3.21107i 1.10371 + 4.09656i −7.31386 + 4.22266i 2.82843 7.80657 + 4.47856i 9.08228i
229.3 −0.707107 1.22474i −2.69891 + 1.30993i −1.00000 + 1.73205i −3.12737 1.80559i 3.51274 + 2.37921i 4.42508 2.55482i 2.82843 5.56819 7.07073i 5.10698i
229.4 −0.707107 1.22474i −2.69891 + 1.30993i −1.00000 + 1.73205i 3.12737 + 1.80559i 3.51274 + 2.37921i −4.42508 + 2.55482i 2.82843 5.56819 7.07073i 5.10698i
229.5 −0.707107 1.22474i −2.61963 1.46204i −1.00000 + 1.73205i −5.22993 3.01950i 0.0617296 + 4.24219i −11.0196 + 6.36214i 2.82843 4.72488 + 7.65999i 8.54043i
229.6 −0.707107 1.22474i −2.61963 1.46204i −1.00000 + 1.73205i 5.22993 + 3.01950i 0.0617296 + 4.24219i 11.0196 6.36214i 2.82843 4.72488 + 7.65999i 8.54043i
229.7 −0.707107 1.22474i −1.45535 + 2.62335i −1.00000 + 1.73205i −6.63175 3.82884i 4.24202 0.0725559i −9.09135 + 5.24889i 2.82843 −4.76392 7.63578i 10.8296i
229.8 −0.707107 1.22474i −1.45535 + 2.62335i −1.00000 + 1.73205i 6.63175 + 3.82884i 4.24202 0.0725559i 9.09135 5.24889i 2.82843 −4.76392 7.63578i 10.8296i
229.9 −0.707107 1.22474i −1.38507 2.66112i −1.00000 + 1.73205i −2.02474 1.16898i −2.27981 + 3.57806i −2.31483 + 1.33647i 2.82843 −5.16316 + 7.37169i 3.30639i
229.10 −0.707107 1.22474i −1.38507 2.66112i −1.00000 + 1.73205i 2.02474 + 1.16898i −2.27981 + 3.57806i 2.31483 1.33647i 2.82843 −5.16316 + 7.37169i 3.30639i
229.11 −0.707107 1.22474i −0.276911 + 2.98719i −1.00000 + 1.73205i −3.00967 1.73763i 3.85435 1.77312i 8.08407 4.66734i 2.82843 −8.84664 1.65437i 4.91476i
229.12 −0.707107 1.22474i −0.276911 + 2.98719i −1.00000 + 1.73205i 3.00967 + 1.73763i 3.85435 1.77312i −8.08407 + 4.66734i 2.82843 −8.84664 1.65437i 4.91476i
229.13 −0.707107 1.22474i 0.170504 2.99515i −1.00000 + 1.73205i −8.47335 4.89209i −3.78886 + 1.90907i 8.68607 5.01491i 2.82843 −8.94186 1.02137i 13.8369i
229.14 −0.707107 1.22474i 0.170504 2.99515i −1.00000 + 1.73205i 8.47335 + 4.89209i −3.78886 + 1.90907i −8.68607 + 5.01491i 2.82843 −8.94186 1.02137i 13.8369i
229.15 −0.707107 1.22474i 1.39491 + 2.65598i −1.00000 + 1.73205i −3.95291 2.28222i 2.26655 3.58647i 2.69328 1.55496i 2.82843 −5.10845 + 7.40971i 6.45508i
229.16 −0.707107 1.22474i 1.39491 + 2.65598i −1.00000 + 1.73205i 3.95291 + 2.28222i 2.26655 3.58647i −2.69328 + 1.55496i 2.82843 −5.10845 + 7.40971i 6.45508i
229.17 −0.707107 1.22474i 1.61253 2.52977i −1.00000 + 1.73205i −0.849237 0.490307i −4.23856 0.186126i 3.58678 2.07083i 2.82843 −3.79946 8.15868i 1.38680i
229.18 −0.707107 1.22474i 1.61253 2.52977i −1.00000 + 1.73205i 0.849237 + 0.490307i −4.23856 0.186126i −3.58678 + 2.07083i 2.82843 −3.79946 8.15868i 1.38680i
229.19 −0.707107 1.22474i 2.13720 + 2.10532i −1.00000 + 1.73205i −4.98011 2.87527i 1.06725 4.10621i −1.58371 + 0.914355i 2.82843 0.135264 + 8.99898i 8.13249i
229.20 −0.707107 1.22474i 2.13720 + 2.10532i −1.00000 + 1.73205i 4.98011 + 2.87527i 1.06725 4.10621i 1.58371 0.914355i 2.82843 0.135264 + 8.99898i 8.13249i
See all 96 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 367.48 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.c even 3 1 inner
23.b odd 2 1 inner
207.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 414.3.h.a 96
3.b odd 2 1 1242.3.h.a 96
9.c even 3 1 inner 414.3.h.a 96
9.d odd 6 1 1242.3.h.a 96
23.b odd 2 1 inner 414.3.h.a 96
69.c even 2 1 1242.3.h.a 96
207.f odd 6 1 inner 414.3.h.a 96
207.g even 6 1 1242.3.h.a 96

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.3.h.a 96 1.a even 1 1 trivial
414.3.h.a 96 9.c even 3 1 inner
414.3.h.a 96 23.b odd 2 1 inner
414.3.h.a 96 207.f odd 6 1 inner
1242.3.h.a 96 3.b odd 2 1
1242.3.h.a 96 9.d odd 6 1
1242.3.h.a 96 69.c even 2 1
1242.3.h.a 96 207.g even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(414, [\chi])$$.