Properties

Label 950.2.f.d
Level $950$
Weight $2$
Character orbit 950.f
Analytic conductor $7.586$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32q - 8q^{6} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q - 8q^{6} - 24q^{11} - 32q^{16} + 32q^{26} + 56q^{36} - 64q^{61} + 72q^{66} + 4q^{76} - 32q^{81} + 8q^{96} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
493.1 −0.707107 0.707107i −2.10096 + 2.10096i 1.00000i 0 2.97120 −0.978377 + 0.978377i 0.707107 0.707107i 5.82806i 0
493.2 −0.707107 0.707107i −2.10096 + 2.10096i 1.00000i 0 2.97120 0.978377 0.978377i 0.707107 0.707107i 5.82806i 0
493.3 −0.707107 0.707107i −0.267657 + 0.267657i 1.00000i 0 0.378524 −0.709827 + 0.709827i 0.707107 0.707107i 2.85672i 0
493.4 −0.707107 0.707107i −0.267657 + 0.267657i 1.00000i 0 0.378524 0.709827 0.709827i 0.707107 0.707107i 2.85672i 0
493.5 −0.707107 0.707107i 1.16084 1.16084i 1.00000i 0 −1.64167 −2.19318 + 2.19318i 0.707107 0.707107i 0.304919i 0
493.6 −0.707107 0.707107i 1.16084 1.16084i 1.00000i 0 −1.64167 2.19318 2.19318i 0.707107 0.707107i 0.304919i 0
493.7 −0.707107 0.707107i 1.91489 1.91489i 1.00000i 0 −2.70806 −2.95447 + 2.95447i 0.707107 0.707107i 4.33358i 0
493.8 −0.707107 0.707107i 1.91489 1.91489i 1.00000i 0 −2.70806 2.95447 2.95447i 0.707107 0.707107i 4.33358i 0
493.9 0.707107 + 0.707107i −1.91489 + 1.91489i 1.00000i 0 −2.70806 −2.95447 + 2.95447i −0.707107 + 0.707107i 4.33358i 0
493.10 0.707107 + 0.707107i −1.91489 + 1.91489i 1.00000i 0 −2.70806 2.95447 2.95447i −0.707107 + 0.707107i 4.33358i 0
493.11 0.707107 + 0.707107i −1.16084 + 1.16084i 1.00000i 0 −1.64167 −2.19318 + 2.19318i −0.707107 + 0.707107i 0.304919i 0
493.12 0.707107 + 0.707107i −1.16084 + 1.16084i 1.00000i 0 −1.64167 2.19318 2.19318i −0.707107 + 0.707107i 0.304919i 0
493.13 0.707107 + 0.707107i 0.267657 0.267657i 1.00000i 0 0.378524 −0.709827 + 0.709827i −0.707107 + 0.707107i 2.85672i 0
493.14 0.707107 + 0.707107i 0.267657 0.267657i 1.00000i 0 0.378524 0.709827 0.709827i −0.707107 + 0.707107i 2.85672i 0
493.15 0.707107 + 0.707107i 2.10096 2.10096i 1.00000i 0 2.97120 −0.978377 + 0.978377i −0.707107 + 0.707107i 5.82806i 0
493.16 0.707107 + 0.707107i 2.10096 2.10096i 1.00000i 0 2.97120 0.978377 0.978377i −0.707107 + 0.707107i 5.82806i 0
607.1 −0.707107 + 0.707107i −2.10096 2.10096i 1.00000i 0 2.97120 −0.978377 0.978377i 0.707107 + 0.707107i 5.82806i 0
607.2 −0.707107 + 0.707107i −2.10096 2.10096i 1.00000i 0 2.97120 0.978377 + 0.978377i 0.707107 + 0.707107i 5.82806i 0
607.3 −0.707107 + 0.707107i −0.267657 0.267657i 1.00000i 0 0.378524 −0.709827 0.709827i 0.707107 + 0.707107i 2.85672i 0
607.4 −0.707107 + 0.707107i −0.267657 0.267657i 1.00000i 0 0.378524 0.709827 + 0.709827i 0.707107 + 0.707107i 2.85672i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 607.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
19.b odd 2 1 inner
95.d odd 2 1 inner
95.g even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.f.d 32
5.b even 2 1 inner 950.2.f.d 32
5.c odd 4 2 inner 950.2.f.d 32
19.b odd 2 1 inner 950.2.f.d 32
95.d odd 2 1 inner 950.2.f.d 32
95.g even 4 2 inner 950.2.f.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.f.d 32 1.a even 1 1 trivial
950.2.f.d 32 5.b even 2 1 inner
950.2.f.d 32 5.c odd 4 2 inner
950.2.f.d 32 19.b odd 2 1 inner
950.2.f.d 32 95.d odd 2 1 inner
950.2.f.d 32 95.g even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 139 T_{3}^{12} + 5151 T_{3}^{8} + 30550 T_{3}^{4} + 625 \) acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\).