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Properties

Label 950.2.u.g
Level $950$
Weight $2$
Character orbit 950.u
Analytic conductor $7.586$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.u (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 190)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36q + 36q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q + 36q^{9} - 24q^{11} - 12q^{14} + 24q^{21} - 18q^{26} + 12q^{29} - 12q^{31} - 36q^{34} - 36q^{36} - 96q^{39} - 42q^{41} - 6q^{44} + 36q^{46} + 78q^{49} + 84q^{51} + 108q^{54} + 60q^{59} + 96q^{61} + 18q^{64} + 48q^{66} + 60q^{69} + 60q^{71} + 6q^{74} - 42q^{76} - 60q^{79} + 36q^{81} - 12q^{84} + 72q^{86} - 60q^{89} - 120q^{91} - 12q^{94} - 342q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1 −0.342020 + 0.939693i −3.16698 + 0.558424i −0.766044 − 0.642788i 0 0.558424 − 3.16698i 0.0202608 + 0.0116976i 0.866025 − 0.500000i 6.89886 − 2.51098i 0
99.2 −0.342020 + 0.939693i 0.0355948 − 0.00627632i −0.766044 − 0.642788i 0 −0.00627632 + 0.0355948i −1.59124 − 0.918706i 0.866025 − 0.500000i −2.81785 + 1.02561i 0
99.3 −0.342020 + 0.939693i 3.13139 − 0.552148i −0.766044 − 0.642788i 0 −0.552148 + 3.13139i 2.89781 + 1.67305i 0.866025 − 0.500000i 6.68164 − 2.43192i 0
99.4 0.342020 − 0.939693i −3.13139 + 0.552148i −0.766044 − 0.642788i 0 −0.552148 + 3.13139i −2.89781 − 1.67305i −0.866025 + 0.500000i 6.68164 − 2.43192i 0
99.5 0.342020 − 0.939693i −0.0355948 + 0.00627632i −0.766044 − 0.642788i 0 −0.00627632 + 0.0355948i 1.59124 + 0.918706i −0.866025 + 0.500000i −2.81785 + 1.02561i 0
99.6 0.342020 − 0.939693i 3.16698 − 0.558424i −0.766044 − 0.642788i 0 0.558424 − 3.16698i −0.0202608 − 0.0116976i −0.866025 + 0.500000i 6.89886 − 2.51098i 0
149.1 −0.642788 + 0.766044i −0.613893 − 1.68666i −0.173648 − 0.984808i 0 1.68666 + 0.613893i −1.17907 + 0.680736i 0.866025 + 0.500000i −0.169813 + 0.142490i 0
149.2 −0.642788 + 0.766044i −0.197144 − 0.541649i −0.173648 − 0.984808i 0 0.541649 + 0.197144i 4.21251 − 2.43209i 0.866025 + 0.500000i 2.04362 − 1.71480i 0
149.3 −0.642788 + 0.766044i 0.811037 + 2.22831i −0.173648 − 0.984808i 0 −2.22831 − 0.811037i −2.73267 + 1.57771i 0.866025 + 0.500000i −2.00943 + 1.68611i 0
149.4 0.642788 − 0.766044i −0.811037 − 2.22831i −0.173648 − 0.984808i 0 −2.22831 − 0.811037i 2.73267 − 1.57771i −0.866025 − 0.500000i −2.00943 + 1.68611i 0
149.5 0.642788 − 0.766044i 0.197144 + 0.541649i −0.173648 − 0.984808i 0 0.541649 + 0.197144i −4.21251 + 2.43209i −0.866025 − 0.500000i 2.04362 − 1.71480i 0
149.6 0.642788 − 0.766044i 0.613893 + 1.68666i −0.173648 − 0.984808i 0 1.68666 + 0.613893i 1.17907 − 0.680736i −0.866025 − 0.500000i −0.169813 + 0.142490i 0
199.1 −0.984808 + 0.173648i −1.49114 − 1.77707i 0.939693 − 0.342020i 0 1.77707 + 1.49114i −4.25802 − 2.45837i −0.866025 + 0.500000i −0.413538 + 2.34529i 0
199.2 −0.984808 + 0.173648i −0.712963 − 0.849676i 0.939693 − 0.342020i 0 0.849676 + 0.712963i 4.26875 + 2.46456i −0.866025 + 0.500000i 0.307311 − 1.74285i 0
199.3 −0.984808 + 0.173648i 2.20410 + 2.62675i 0.939693 − 0.342020i 0 −2.62675 − 2.20410i 1.61687 + 0.933500i −0.866025 + 0.500000i −1.52078 + 8.62480i 0
199.4 0.984808 − 0.173648i −2.20410 − 2.62675i 0.939693 − 0.342020i 0 −2.62675 − 2.20410i −1.61687 − 0.933500i 0.866025 − 0.500000i −1.52078 + 8.62480i 0
199.5 0.984808 − 0.173648i 0.712963 + 0.849676i 0.939693 − 0.342020i 0 0.849676 + 0.712963i −4.26875 − 2.46456i 0.866025 − 0.500000i 0.307311 − 1.74285i 0
199.6 0.984808 − 0.173648i 1.49114 + 1.77707i 0.939693 − 0.342020i 0 1.77707 + 1.49114i 4.25802 + 2.45837i 0.866025 − 0.500000i −0.413538 + 2.34529i 0
499.1 −0.342020 − 0.939693i −3.16698 − 0.558424i −0.766044 + 0.642788i 0 0.558424 + 3.16698i 0.0202608 − 0.0116976i 0.866025 + 0.500000i 6.89886 + 2.51098i 0
499.2 −0.342020 − 0.939693i 0.0355948 + 0.00627632i −0.766044 + 0.642788i 0 −0.00627632 − 0.0355948i −1.59124 + 0.918706i 0.866025 + 0.500000i −2.81785 − 1.02561i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 899.6
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
19.e even 9 1 inner
95.p even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.u.g 36
5.b even 2 1 inner 950.2.u.g 36
5.c odd 4 1 190.2.k.d ✓ 18
5.c odd 4 1 950.2.l.i 18
19.e even 9 1 inner 950.2.u.g 36
95.p even 18 1 inner 950.2.u.g 36
95.q odd 36 1 190.2.k.d ✓ 18
95.q odd 36 1 950.2.l.i 18
95.q odd 36 1 3610.2.a.bi 9
95.r even 36 1 3610.2.a.bj 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.k.d ✓ 18 5.c odd 4 1
190.2.k.d ✓ 18 95.q odd 36 1
950.2.l.i 18 5.c odd 4 1
950.2.l.i 18 95.q odd 36 1
950.2.u.g 36 1.a even 1 1 trivial
950.2.u.g 36 5.b even 2 1 inner
950.2.u.g 36 19.e even 9 1 inner
950.2.u.g 36 95.p even 18 1 inner
3610.2.a.bi 9 95.q odd 36 1
3610.2.a.bj 9 95.r even 36 1

Hecke kernels

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

\(T_{3}^{36} - \cdots\)
\(32\!\cdots\!84\)\( T_{7}^{18} + \)\(20\!\cdots\!72\)\( T_{7}^{16} - \)\(93\!\cdots\!60\)\( T_{7}^{14} + \)\(32\!\cdots\!32\)\( T_{7}^{12} - \)\(80\!\cdots\!48\)\( T_{7}^{10} + \)\(14\!\cdots\!80\)\( T_{7}^{8} - \)\(16\!\cdots\!12\)\( T_{7}^{6} + \)\(11\!\cdots\!68\)\( T_{7}^{4} - 625528209408 T_{7}^{2} + 342102016 \)">\(T_{7}^{36} - \cdots\)