# Properties

 Label 363.2.f.i Level $363$ Weight $2$ Character orbit 363.f Analytic conductor $2.899$ Analytic rank $0$ Dimension $32$ CM no Inner twists $16$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 363.f (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.89856959337$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 4 q^{3} - 16 q^{4} + 8 q^{9}+O(q^{10})$$ 32 * q + 4 * q^3 - 16 * q^4 + 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 4 q^{3} - 16 q^{4} + 8 q^{9} + 16 q^{12} - 4 q^{15} + 8 q^{16} + 4 q^{27} + 40 q^{31} - 96 q^{34} + 40 q^{36} + 56 q^{37} - 64 q^{42} - 160 q^{45} - 28 q^{48} + 8 q^{49} - 104 q^{58} + 28 q^{60} + 16 q^{64} + 64 q^{67} + 16 q^{70} - 24 q^{75} + 240 q^{78} + 8 q^{81} - 96 q^{82} + 48 q^{91} - 56 q^{93} - 32 q^{97}+O(q^{100})$$ 32 * q + 4 * q^3 - 16 * q^4 + 8 * q^9 + 16 * q^12 - 4 * q^15 + 8 * q^16 + 4 * q^27 + 40 * q^31 - 96 * q^34 + 40 * q^36 + 56 * q^37 - 64 * q^42 - 160 * q^45 - 28 * q^48 + 8 * q^49 - 104 * q^58 + 28 * q^60 + 16 * q^64 + 64 * q^67 + 16 * q^70 - 24 * q^75 + 240 * q^78 + 8 * q^81 - 96 * q^82 + 48 * q^91 - 56 * q^93 - 32 * q^97

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1 −1.93692 1.40726i −1.49697 0.871256i 1.15327 + 3.54939i 0.728450 + 1.00263i 1.67343 + 3.79418i −2.68999 + 0.874032i 1.28144 3.94386i 1.48183 + 2.60848i 2.96713i
161.2 −1.93692 1.40726i 1.72318 + 0.175035i 1.15327 + 3.54939i −0.728450 1.00263i −3.09136 2.76399i 2.68999 0.874032i 1.28144 3.94386i 2.93873 + 0.603233i 2.96713i
161.3 −1.21836 0.885188i −1.43489 + 0.970101i 0.0828009 + 0.254835i −1.71005 2.35368i 2.60693 + 0.0882174i −2.68999 + 0.874032i −0.806046 + 2.48075i 1.11781 2.78397i 4.38134i
161.4 −1.21836 0.885188i 0.590638 + 1.62823i 0.0828009 + 0.254835i 1.71005 + 2.35368i 0.721685 2.50660i 2.68999 0.874032i −0.806046 + 2.48075i −2.30229 + 1.92339i 4.38134i
161.5 1.21836 + 0.885188i −1.43489 + 0.970101i 0.0828009 + 0.254835i −1.71005 2.35368i −2.60693 0.0882174i 2.68999 0.874032i 0.806046 2.48075i 1.11781 2.78397i 4.38134i
161.6 1.21836 + 0.885188i 0.590638 + 1.62823i 0.0828009 + 0.254835i 1.71005 + 2.35368i −0.721685 + 2.50660i −2.68999 + 0.874032i 0.806046 2.48075i −2.30229 + 1.92339i 4.38134i
161.7 1.93692 + 1.40726i −1.49697 0.871256i 1.15327 + 3.54939i 0.728450 + 1.00263i −1.67343 3.79418i 2.68999 0.874032i −1.28144 + 3.94386i 1.48183 + 2.60848i 2.96713i
161.8 1.93692 + 1.40726i 1.72318 + 0.175035i 1.15327 + 3.54939i −0.728450 1.00263i 3.09136 + 2.76399i −2.68999 + 0.874032i −1.28144 + 3.94386i 2.93873 + 0.603233i 2.96713i
215.1 −0.739839 2.27699i −1.29120 + 1.15447i −3.01929 + 2.19364i −1.17866 0.382969i 3.58400 + 2.08594i 1.66251 + 2.28825i 3.35485 + 2.43744i 0.334407 2.98130i 2.96713i
215.2 −0.739839 2.27699i 0.698961 1.58476i −3.01929 + 2.19364i 1.17866 + 0.382969i −4.12560 0.419062i −1.66251 2.28825i 3.35485 + 2.43744i −2.02291 2.21537i 2.96713i
215.3 −0.465371 1.43226i 0.479216 + 1.66444i −0.216775 + 0.157497i 2.76692 + 0.899027i 2.16090 1.46094i 1.66251 + 2.28825i −2.11025 1.53319i −2.54070 + 1.59525i 4.38134i
215.4 −0.465371 1.43226i 1.73106 0.0585784i −0.216775 + 0.157497i −2.76692 0.899027i −0.889484 2.45207i −1.66251 2.28825i −2.11025 1.53319i 2.99314 0.202805i 4.38134i
215.5 0.465371 + 1.43226i 0.479216 + 1.66444i −0.216775 + 0.157497i 2.76692 + 0.899027i −2.16090 + 1.46094i −1.66251 2.28825i 2.11025 + 1.53319i −2.54070 + 1.59525i 4.38134i
215.6 0.465371 + 1.43226i 1.73106 0.0585784i −0.216775 + 0.157497i −2.76692 0.899027i 0.889484 + 2.45207i 1.66251 + 2.28825i 2.11025 + 1.53319i 2.99314 0.202805i 4.38134i
215.7 0.739839 + 2.27699i −1.29120 + 1.15447i −3.01929 + 2.19364i −1.17866 0.382969i −3.58400 2.08594i −1.66251 2.28825i −3.35485 2.43744i 0.334407 2.98130i 2.96713i
215.8 0.739839 + 2.27699i 0.698961 1.58476i −3.01929 + 2.19364i 1.17866 + 0.382969i 4.12560 + 0.419062i 1.66251 + 2.28825i −3.35485 2.43744i −2.02291 2.21537i 2.96713i
233.1 −0.739839 + 2.27699i −1.29120 1.15447i −3.01929 2.19364i −1.17866 + 0.382969i 3.58400 2.08594i 1.66251 2.28825i 3.35485 2.43744i 0.334407 + 2.98130i 2.96713i
233.2 −0.739839 + 2.27699i 0.698961 + 1.58476i −3.01929 2.19364i 1.17866 0.382969i −4.12560 + 0.419062i −1.66251 + 2.28825i 3.35485 2.43744i −2.02291 + 2.21537i 2.96713i
233.3 −0.465371 + 1.43226i 0.479216 1.66444i −0.216775 0.157497i 2.76692 0.899027i 2.16090 + 1.46094i 1.66251 2.28825i −2.11025 + 1.53319i −2.54070 1.59525i 4.38134i
233.4 −0.465371 + 1.43226i 1.73106 + 0.0585784i −0.216775 0.157497i −2.76692 + 0.899027i −0.889484 + 2.45207i −1.66251 + 2.28825i −2.11025 + 1.53319i 2.99314 + 0.202805i 4.38134i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
33.d even 2 1 inner
33.f even 10 3 inner
33.h odd 10 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.f.i 32
3.b odd 2 1 inner 363.2.f.i 32
11.b odd 2 1 inner 363.2.f.i 32
11.c even 5 1 363.2.d.e 8
11.c even 5 3 inner 363.2.f.i 32
11.d odd 10 1 363.2.d.e 8
11.d odd 10 3 inner 363.2.f.i 32
33.d even 2 1 inner 363.2.f.i 32
33.f even 10 1 363.2.d.e 8
33.f even 10 3 inner 363.2.f.i 32
33.h odd 10 1 363.2.d.e 8
33.h odd 10 3 inner 363.2.f.i 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.d.e 8 11.c even 5 1
363.2.d.e 8 11.d odd 10 1
363.2.d.e 8 33.f even 10 1
363.2.d.e 8 33.h odd 10 1
363.2.f.i 32 1.a even 1 1 trivial
363.2.f.i 32 3.b odd 2 1 inner
363.2.f.i 32 11.b odd 2 1 inner
363.2.f.i 32 11.c even 5 3 inner
363.2.f.i 32 11.d odd 10 3 inner
363.2.f.i 32 33.d even 2 1 inner
363.2.f.i 32 33.f even 10 3 inner
363.2.f.i 32 33.h odd 10 3 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(363, [\chi])$$:

 $$T_{2}^{16} + 8T_{2}^{14} + 51T_{2}^{12} + 304T_{2}^{10} + 1769T_{2}^{8} + 3952T_{2}^{6} + 8619T_{2}^{4} + 17576T_{2}^{2} + 28561$$ T2^16 + 8*T2^14 + 51*T2^12 + 304*T2^10 + 1769*T2^8 + 3952*T2^6 + 8619*T2^4 + 17576*T2^2 + 28561 $$T_{5}^{16} - 10T_{5}^{14} + 87T_{5}^{12} - 740T_{5}^{10} + 6269T_{5}^{8} - 9620T_{5}^{6} + 14703T_{5}^{4} - 21970T_{5}^{2} + 28561$$ T5^16 - 10*T5^14 + 87*T5^12 - 740*T5^10 + 6269*T5^8 - 9620*T5^6 + 14703*T5^4 - 21970*T5^2 + 28561 $$T_{7}^{8} - 8T_{7}^{6} + 64T_{7}^{4} - 512T_{7}^{2} + 4096$$ T7^8 - 8*T7^6 + 64*T7^4 - 512*T7^2 + 4096