Properties

Label 385.2.o.b
Level $385$
Weight $2$
Character orbit 385.o
Analytic conductor $3.074$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.07424047782\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72q - 28q^{4} - 60q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 72q - 28q^{4} - 60q^{9} - 4q^{11} - 20q^{14} - 4q^{15} - 20q^{16} - 48q^{25} + 24q^{26} - 36q^{31} + 168q^{36} + 36q^{44} + 30q^{45} + 28q^{49} + 84q^{56} - 84q^{59} - 80q^{64} + 60q^{66} - 162q^{70} - 32q^{71} - 108q^{75} + 30q^{80} + 92q^{81} - 36q^{86} - 72q^{89} + 80q^{91} + 120q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
54.1 −1.30077 + 2.25300i −0.878341 1.52133i −2.38400 4.12920i −1.24339 + 1.85849i 4.57007 2.30616 + 1.29678i 7.20103 −0.0429672 + 0.0744215i −2.56981 5.21881i
54.2 −1.30077 + 2.25300i 0.878341 + 1.52133i −2.38400 4.12920i 0.987807 2.00605i −4.57007 2.30616 + 1.29678i 7.20103 −0.0429672 + 0.0744215i 3.23472 + 4.83493i
54.3 −1.15617 + 2.00254i −1.18194 2.04718i −1.67346 2.89851i −2.07840 0.824775i 5.46608 −2.64218 + 0.137378i 3.11452 −1.29395 + 2.24119i 4.05463 3.20851i
54.4 −1.15617 + 2.00254i 1.18194 + 2.04718i −1.67346 2.89851i −1.75348 1.38756i −5.46608 −2.64218 + 0.137378i 3.11452 −1.29395 + 2.24119i 4.80597 1.90716i
54.5 −1.14254 + 1.97894i −1.52900 2.64830i −1.61081 2.79001i 2.14435 0.633864i 6.98779 1.82663 1.91400i 2.79153 −3.17567 + 5.50043i −1.19563 + 4.96776i
54.6 −1.14254 + 1.97894i 1.52900 + 2.64830i −1.61081 2.79001i 0.523230 + 2.17399i −6.98779 1.82663 1.91400i 2.79153 −3.17567 + 5.50043i −4.90002 1.44844i
54.7 −0.901007 + 1.56059i −0.984554 1.70530i −0.623628 1.08016i 1.31996 1.80491i 3.54836 −0.686066 + 2.55525i −1.35646 −0.438692 + 0.759836i 1.62742 + 3.68616i
54.8 −0.901007 + 1.56059i 0.984554 + 1.70530i −0.623628 1.08016i −0.903114 + 2.04558i −3.54836 −0.686066 + 2.55525i −1.35646 −0.438692 + 0.759836i −2.37860 3.25247i
54.9 −0.736827 + 1.27622i −0.269187 0.466245i −0.0858291 0.148660i −1.48744 + 1.66958i 0.793377 −0.997019 2.45070i −2.69434 1.35508 2.34706i −1.03477 3.12850i
54.10 −0.736827 + 1.27622i 0.269187 + 0.466245i −0.0858291 0.148660i 0.702179 2.12296i −0.793377 −0.997019 2.45070i −2.69434 1.35508 2.34706i 2.19198 + 2.46039i
54.11 −0.559049 + 0.968301i −1.09998 1.90522i 0.374928 + 0.649395i 1.09488 + 1.94968i 2.45977 0.753941 + 2.53605i −3.07461 −0.919918 + 1.59334i −2.49997 0.0297884i
54.12 −0.559049 + 0.968301i 1.09998 + 1.90522i 0.374928 + 0.649395i 2.23591 0.0266420i −2.45977 0.753941 + 2.53605i −3.07461 −0.919918 + 1.59334i −1.22419 + 2.17993i
54.13 −0.408979 + 0.708373i −1.44107 2.49601i 0.665472 + 1.15263i −1.82905 + 1.28630i 2.35748 2.08705 1.62611i −2.72457 −2.65339 + 4.59580i −0.163134 1.82172i
54.14 −0.408979 + 0.708373i 1.44107 + 2.49601i 0.665472 + 1.15263i 0.199440 2.22716i −2.35748 2.08705 1.62611i −2.72457 −2.65339 + 4.59580i 1.49609 + 1.05214i
54.15 −0.249992 + 0.433000i −1.24153 2.15040i 0.875008 + 1.51556i −0.407192 2.19868i 1.24149 −2.20302 1.46517i −1.87495 −1.58280 + 2.74149i 1.05382 + 0.373339i
54.16 −0.249992 + 0.433000i 1.24153 + 2.15040i 0.875008 + 1.51556i −2.10771 + 0.746701i −1.24149 −2.20302 1.46517i −1.87495 −1.58280 + 2.74149i 0.203590 1.09931i
54.17 −0.137269 + 0.237757i −0.351914 0.609533i 0.962314 + 1.66678i 2.12913 0.683216i 0.193228 2.52423 0.792646i −1.07746 1.25231 2.16907i −0.129825 + 0.600002i
54.18 −0.137269 + 0.237757i 0.351914 + 0.609533i 0.962314 + 1.66678i 0.472885 + 2.18549i −0.193228 2.52423 0.792646i −1.07746 1.25231 2.16907i −0.584530 0.187569i
54.19 0.137269 0.237757i −0.351914 0.609533i 0.962314 + 1.66678i 2.12913 0.683216i −0.193228 −2.52423 + 0.792646i 1.07746 1.25231 2.16907i 0.129825 0.600002i
54.20 0.137269 0.237757i 0.351914 + 0.609533i 0.962314 + 1.66678i 0.472885 + 2.18549i 0.193228 −2.52423 + 0.792646i 1.07746 1.25231 2.16907i 0.584530 + 0.187569i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 164.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
11.b odd 2 1 inner
35.i odd 6 1 inner
55.d odd 2 1 inner
77.i even 6 1 inner
385.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 385.2.o.b 72
5.b even 2 1 inner 385.2.o.b 72
7.d odd 6 1 inner 385.2.o.b 72
11.b odd 2 1 inner 385.2.o.b 72
35.i odd 6 1 inner 385.2.o.b 72
55.d odd 2 1 inner 385.2.o.b 72
77.i even 6 1 inner 385.2.o.b 72
385.o even 6 1 inner 385.2.o.b 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.o.b 72 1.a even 1 1 trivial
385.2.o.b 72 5.b even 2 1 inner
385.2.o.b 72 7.d odd 6 1 inner
385.2.o.b 72 11.b odd 2 1 inner
385.2.o.b 72 35.i odd 6 1 inner
385.2.o.b 72 55.d odd 2 1 inner
385.2.o.b 72 77.i even 6 1 inner
385.2.o.b 72 385.o even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{36} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(385, [\chi])\).