Properties

Label 1205.2.b.c
Level $1205$
Weight $2$
Character orbit 1205.b
Analytic conductor $9.622$
Analytic rank $0$
Dimension $46$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1205 = 5 \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1205.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.62197344356\)
Analytic rank: \(0\)
Dimension: \(46\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 46q - 34q^{4} - 8q^{5} - 4q^{6} - 34q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 46q - 34q^{4} - 8q^{5} - 4q^{6} - 34q^{9} - 7q^{10} - 64q^{11} + 66q^{14} + 5q^{15} + 22q^{16} - 2q^{20} - 14q^{21} + 50q^{24} + 30q^{25} - 60q^{26} + 36q^{29} + 7q^{30} - 36q^{31} - 12q^{34} + 3q^{35} - 34q^{36} + 88q^{39} - 30q^{40} - 76q^{41} + 100q^{44} - 17q^{45} - 12q^{46} - 22q^{49} - 14q^{50} - 112q^{51} + 26q^{54} - 5q^{55} - 120q^{56} + 84q^{59} - 33q^{60} - 78q^{61} - 28q^{64} + 9q^{65} - 2q^{66} + 24q^{69} + 88q^{70} - 172q^{71} + 16q^{74} + 59q^{75} - 18q^{76} + 54q^{79} + 63q^{80} - 42q^{81} - 44q^{84} + 30q^{85} - 80q^{86} + 86q^{89} + 17q^{90} - 88q^{91} - 4q^{94} + q^{95} - 122q^{96} + 148q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
724.1 2.68673i 2.58072i −5.21850 1.90521 + 1.17055i −6.93370 1.24945i 8.64724i −3.66013 3.14494 5.11877i
724.2 2.67602i 0.199289i −5.16108 −2.13713 + 0.657778i 0.533301 2.87374i 8.45912i 2.96028 1.76023 + 5.71900i
724.3 2.49572i 1.66439i −4.22863 2.11925 0.713293i 4.15385 4.24911i 5.56203i 0.229807 −1.78018 5.28905i
724.4 2.44838i 0.992209i −3.99457 0.553587 2.16646i −2.42931 0.744704i 4.88345i 2.01552 −5.30431 1.35539i
724.5 2.36215i 2.05253i −3.57974 −0.618559 + 2.14881i −4.84838 3.07728i 3.73158i −1.21289 5.07581 + 1.46113i
724.6 1.95888i 0.350465i −1.83721 −1.30176 1.81808i −0.686519 1.14869i 0.318879i 2.87717 −3.56141 + 2.54999i
724.7 1.92263i 0.557567i −1.69651 −2.01334 + 0.972856i −1.07200 0.976687i 0.583491i 2.68912 1.87044 + 3.87092i
724.8 1.88286i 2.55671i −1.54517 −0.979915 2.00992i 4.81392 3.51799i 0.856390i −3.53675 −3.78439 + 1.84504i
724.9 1.84209i 1.69681i −1.39329 1.66065 + 1.49741i 3.12567 1.54055i 1.11761i 0.120844 2.75836 3.05907i
724.10 1.83307i 2.40621i −1.36013 −1.46796 + 1.68674i 4.41073 0.302396i 1.17293i −2.78983 3.09191 + 2.69086i
724.11 1.70963i 1.38336i −0.922819 −1.53605 1.62498i 2.36503 5.05937i 1.84158i 1.08631 −2.77811 + 2.62606i
724.12 1.49203i 3.12646i −0.226139 0.643201 2.14156i −4.66475 4.66024i 2.64665i −6.77474 −3.19527 0.959673i
724.13 1.45267i 2.41257i −0.110263 −1.02000 1.98987i −3.50468 1.25057i 2.74517i −2.82049 −2.89064 + 1.48173i
724.14 1.30874i 1.88426i 0.287190 2.21635 0.296297i 2.46601 1.28779i 2.99335i −0.550432 −0.387776 2.90063i
724.15 1.09514i 0.223927i 0.800662 −1.50518 + 1.65361i 0.245232 2.26365i 3.06713i 2.94986 1.81094 + 1.64839i
724.16 0.930197i 1.32621i 1.13473 −2.23573 0.0386721i −1.23364 4.33497i 2.91592i 1.24116 −0.0359727 + 2.07967i
724.17 0.779313i 2.96208i 1.39267 2.14053 0.646643i 2.30839 0.937135i 2.64395i −5.77395 −0.503937 1.66814i
724.18 0.760721i 1.12938i 1.42130 1.88699 + 1.19969i −0.859145 2.53853i 2.60266i 1.72450 0.912633 1.43547i
724.19 0.636788i 2.24729i 1.59450 −1.29399 + 1.82362i −1.43105 1.14493i 2.28894i −2.05030 1.16126 + 0.824000i
724.20 0.434323i 1.75836i 1.81136 0.971846 2.01383i 0.763695 2.56007i 1.65536i −0.0918227 −0.874653 0.422095i
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 724.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1205.2.b.c 46
5.b even 2 1 inner 1205.2.b.c 46
5.c odd 4 2 6025.2.a.p 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1205.2.b.c 46 1.a even 1 1 trivial
1205.2.b.c 46 5.b even 2 1 inner
6025.2.a.p 46 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{46} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(1205, [\chi])\).