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Properties

Label 950.2.q.f
Level $950$
Weight $2$
Character orbit 950.q
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.q (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32q + 4q^{6} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q + 4q^{6} + 72q^{11} + 16q^{16} + 60q^{21} + 8q^{26} - 28q^{36} - 84q^{41} - 84q^{51} - 52q^{61} - 24q^{71} + 16q^{76} + 64q^{81} - 36q^{86} - 84q^{91} + 8q^{96} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −0.965926 − 0.258819i −0.824123 + 3.07567i 0.866025 + 0.500000i 0 1.59208 − 2.75757i −3.29328 − 3.29328i −0.707107 − 0.707107i −6.18248 − 3.56945i 0
107.2 −0.965926 − 0.258819i −0.313773 + 1.17102i 0.866025 + 0.500000i 0 0.606164 − 1.04991i 1.88504 + 1.88504i −0.707107 − 0.707107i 1.32525 + 0.765131i 0
107.3 −0.965926 − 0.258819i 0.206366 − 0.770169i 0.866025 + 0.500000i 0 −0.398669 + 0.690515i −0.349095 − 0.349095i −0.707107 − 0.707107i 2.04750 + 1.18213i 0
107.4 −0.965926 − 0.258819i 0.672711 − 2.51059i 0.866025 + 0.500000i 0 −1.29958 + 2.25093i −0.692149 − 0.692149i −0.707107 − 0.707107i −3.25245 − 1.87780i 0
107.5 0.965926 + 0.258819i −0.672711 + 2.51059i 0.866025 + 0.500000i 0 −1.29958 + 2.25093i 0.692149 + 0.692149i 0.707107 + 0.707107i −3.25245 − 1.87780i 0
107.6 0.965926 + 0.258819i −0.206366 + 0.770169i 0.866025 + 0.500000i 0 −0.398669 + 0.690515i 0.349095 + 0.349095i 0.707107 + 0.707107i 2.04750 + 1.18213i 0
107.7 0.965926 + 0.258819i 0.313773 − 1.17102i 0.866025 + 0.500000i 0 0.606164 − 1.04991i −1.88504 − 1.88504i 0.707107 + 0.707107i 1.32525 + 0.765131i 0
107.8 0.965926 + 0.258819i 0.824123 − 3.07567i 0.866025 + 0.500000i 0 1.59208 − 2.75757i 3.29328 + 3.29328i 0.707107 + 0.707107i −6.18248 − 3.56945i 0
293.1 −0.965926 + 0.258819i −0.824123 − 3.07567i 0.866025 − 0.500000i 0 1.59208 + 2.75757i −3.29328 + 3.29328i −0.707107 + 0.707107i −6.18248 + 3.56945i 0
293.2 −0.965926 + 0.258819i −0.313773 − 1.17102i 0.866025 − 0.500000i 0 0.606164 + 1.04991i 1.88504 − 1.88504i −0.707107 + 0.707107i 1.32525 − 0.765131i 0
293.3 −0.965926 + 0.258819i 0.206366 + 0.770169i 0.866025 − 0.500000i 0 −0.398669 − 0.690515i −0.349095 + 0.349095i −0.707107 + 0.707107i 2.04750 − 1.18213i 0
293.4 −0.965926 + 0.258819i 0.672711 + 2.51059i 0.866025 − 0.500000i 0 −1.29958 − 2.25093i −0.692149 + 0.692149i −0.707107 + 0.707107i −3.25245 + 1.87780i 0
293.5 0.965926 − 0.258819i −0.672711 − 2.51059i 0.866025 − 0.500000i 0 −1.29958 − 2.25093i 0.692149 − 0.692149i 0.707107 − 0.707107i −3.25245 + 1.87780i 0
293.6 0.965926 − 0.258819i −0.206366 − 0.770169i 0.866025 − 0.500000i 0 −0.398669 − 0.690515i 0.349095 − 0.349095i 0.707107 − 0.707107i 2.04750 − 1.18213i 0
293.7 0.965926 − 0.258819i 0.313773 + 1.17102i 0.866025 − 0.500000i 0 0.606164 + 1.04991i −1.88504 + 1.88504i 0.707107 − 0.707107i 1.32525 − 0.765131i 0
293.8 0.965926 − 0.258819i 0.824123 + 3.07567i 0.866025 − 0.500000i 0 1.59208 + 2.75757i 3.29328 − 3.29328i 0.707107 − 0.707107i −6.18248 + 3.56945i 0
407.1 −0.258819 − 0.965926i −3.07567 + 0.824123i −0.866025 + 0.500000i 0 1.59208 + 2.75757i −3.29328 − 3.29328i 0.707107 + 0.707107i 6.18248 − 3.56945i 0
407.2 −0.258819 − 0.965926i −1.17102 + 0.313773i −0.866025 + 0.500000i 0 0.606164 + 1.04991i 1.88504 + 1.88504i 0.707107 + 0.707107i −1.32525 + 0.765131i 0
407.3 −0.258819 − 0.965926i 0.770169 − 0.206366i −0.866025 + 0.500000i 0 −0.398669 − 0.690515i −0.349095 − 0.349095i 0.707107 + 0.707107i −2.04750 + 1.18213i 0
407.4 −0.258819 − 0.965926i 2.51059 − 0.672711i −0.866025 + 0.500000i 0 −1.29958 − 2.25093i −0.692149 − 0.692149i 0.707107 + 0.707107i 3.25245 − 1.87780i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 943.8
Significant digits:
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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.d odd 6 1 inner
95.h odd 6 1 inner
95.l even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.q.f ✓ 32
5.b even 2 1 inner 950.2.q.f ✓ 32
5.c odd 4 2 inner 950.2.q.f ✓ 32
19.d odd 6 1 inner 950.2.q.f ✓ 32
95.h odd 6 1 inner 950.2.q.f ✓ 32
95.l even 12 2 inner 950.2.q.f ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.q.f ✓ 32 1.a even 1 1 trivial
950.2.q.f ✓ 32 5.b even 2 1 inner
950.2.q.f ✓ 32 5.c odd 4 2 inner
950.2.q.f ✓ 32 19.d odd 6 1 inner
950.2.q.f ✓ 32 95.h odd 6 1 inner
950.2.q.f ✓ 32 95.l even 12 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\(T_{3}^{32} - \cdots\)
\( T_{7}^{16} + 522 T_{7}^{12} + 24273 T_{7}^{8} + 23256 T_{7}^{4} + 1296 \)