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.

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

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:

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 \(S_{2}^{\mathrm{new}}(363, [\chi])\):

\( T_{2}^{16} + 8T_{2}^{14} + 51T_{2}^{12} + 304T_{2}^{10} + 1769T_{2}^{8} + 3952T_{2}^{6} + 8619T_{2}^{4} + 17576T_{2}^{2} + 28561 \) Copy content Toggle raw display
\( T_{5}^{16} - 10T_{5}^{14} + 87T_{5}^{12} - 740T_{5}^{10} + 6269T_{5}^{8} - 9620T_{5}^{6} + 14703T_{5}^{4} - 21970T_{5}^{2} + 28561 \) Copy content Toggle raw display
\( T_{7}^{8} - 8T_{7}^{6} + 64T_{7}^{4} - 512T_{7}^{2} + 4096 \) Copy content Toggle raw display