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Properties

Label 403.2.c.b
Level 403
Weight 2
Character orbit 403.c
Analytic conductor 3.218
Analytic rank 0
Dimension 32
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 403 = 13 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 403.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.21797120146\)
Analytic rank: \(0\)
Dimension: \(32\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 32q - 4q^{3} - 36q^{4} + 20q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q - 4q^{3} - 36q^{4} + 20q^{9} + 4q^{10} - 16q^{12} + 10q^{13} - 16q^{14} + 28q^{16} - 8q^{17} - 16q^{22} - 8q^{23} + 4q^{25} + 18q^{26} + 20q^{27} - 16q^{29} + 40q^{30} - 4q^{35} - 44q^{36} + 12q^{38} + 4q^{39} + 28q^{40} + 28q^{42} - 32q^{43} - 64q^{49} - 64q^{52} - 12q^{53} + 44q^{55} + 8q^{56} + 16q^{61} + 8q^{62} - 76q^{64} - 66q^{65} - 68q^{66} + 64q^{68} + 20q^{69} + 16q^{74} - 32q^{77} - 20q^{78} + 64q^{79} - 16q^{81} + 12q^{82} - 72q^{87} + 80q^{88} + 68q^{90} + 22q^{91} + 28q^{92} + 88q^{94} + 4q^{95} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
311.1 − 2.71381i 0.265781 −5.36475 0.954130i − 0.721280i 4.28452i 9.13130i −2.92936 2.58933
311.2 − 2.67507i −1.67043 −5.15599 − 2.86564i 4.46851i − 5.21058i 8.44248i −0.209666 −7.66579
311.3 − 2.45635i 2.80039 −4.03363 3.48535i − 6.87871i − 3.54003i 4.99531i 4.84216 8.56123
311.4 − 2.44691i 2.35527 −3.98738 − 2.67528i − 5.76313i 0.955539i 4.86294i 2.54728 −6.54617
311.5 − 2.21678i −2.59178 −2.91410 1.00571i 5.74540i 1.82096i 2.02635i 3.71732 2.22943
311.6 − 1.99449i −1.58843 −1.97798 − 2.66789i 3.16810i 1.73927i − 0.0439183i −0.476895 −5.32107
311.7 − 1.88981i −0.327950 −1.57140 1.75790i 0.619765i − 0.652358i − 0.809976i −2.89245 3.32211
311.8 − 1.48947i 2.99747 −0.218531 0.692562i − 4.46466i 1.15260i − 2.65345i 5.98485 1.03155
311.9 − 1.44935i 0.0382575 −0.100611 3.85977i − 0.0554484i 1.66255i − 2.75288i −2.99854 5.59415
311.10 − 1.44015i −2.79139 −0.0740342 2.61757i 4.02002i − 4.99660i − 2.77368i 4.79185 3.76969
311.11 − 1.40506i 1.52886 0.0258025 − 1.01618i − 2.14814i − 2.90958i − 2.84638i −0.662589 −1.42779
311.12 − 1.01245i −1.05136 0.974950 − 3.24438i 1.06445i − 1.79572i − 3.01198i −1.89463 −3.28477
311.13 − 0.937316i 1.68575 1.12144 0.290760i − 1.58008i 3.85549i − 2.92577i −0.158249 0.272534
311.14 − 0.624841i −2.78388 1.60957 − 0.619834i 1.73948i 2.35216i − 2.25541i 4.74999 −0.387298
311.15 − 0.475674i 0.345654 1.77373 − 0.161594i − 0.164419i − 4.07996i − 1.79507i −2.88052 −0.0768661
311.16 − 0.327248i −1.21221 1.89291 − 2.01769i 0.396693i 0.699230i − 1.27395i −1.53055 −0.660284
311.17 0.327248i −1.21221 1.89291 2.01769i − 0.396693i − 0.699230i 1.27395i −1.53055 −0.660284
311.18 0.475674i 0.345654 1.77373 0.161594i 0.164419i 4.07996i 1.79507i −2.88052 −0.0768661
311.19 0.624841i −2.78388 1.60957 0.619834i − 1.73948i − 2.35216i 2.25541i 4.74999 −0.387298
311.20 0.937316i 1.68575 1.12144 − 0.290760i 1.58008i − 3.85549i 2.92577i −0.158249 0.272534
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 311.32
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.c.b ✓ 32
13.b even 2 1 inner 403.2.c.b ✓ 32
13.d odd 4 1 5239.2.a.k 16
13.d odd 4 1 5239.2.a.l 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.c.b ✓ 32 1.a even 1 1 trivial
403.2.c.b ✓ 32 13.b even 2 1 inner
5239.2.a.k 16 13.d odd 4 1
5239.2.a.l 16 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{32} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(403, [\chi])\).