Properties

Label 950.2.bb.d
Level $950$
Weight $2$
Character orbit 950.bb
Analytic conductor $7.586$
Analytic rank $0$
Dimension $96$
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.bb (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(8\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96q - 12q^{6} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 96q - 12q^{6} + 12q^{26} - 72q^{31} + 12q^{36} - 24q^{41} + 12q^{51} - 72q^{61} - 36q^{66} + 168q^{71} - 12q^{76} + 12q^{81} - 48q^{86} - 72q^{91} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1 −0.573576 + 0.819152i −0.282278 + 3.22645i −0.342020 0.939693i 0 −2.48104 2.08184i −0.0261014 0.0974119i 0.965926 + 0.258819i −7.37587 1.30056i 0
143.2 −0.573576 + 0.819152i −0.0476128 + 0.544217i −0.342020 0.939693i 0 −0.418487 0.351152i −0.123323 0.460247i 0.965926 + 0.258819i 2.66052 + 0.469121i 0
143.3 −0.573576 + 0.819152i 0.134247 1.53445i −0.342020 0.939693i 0 1.17995 + 0.990096i −1.26063 4.70472i 0.965926 + 0.258819i 0.617895 + 0.108952i 0
143.4 −0.573576 + 0.819152i 0.225912 2.58219i −0.342020 0.939693i 0 1.98563 + 1.66614i 0.690517 + 2.57704i 0.965926 + 0.258819i −3.66223 0.645749i 0
143.5 0.573576 0.819152i −0.225912 + 2.58219i −0.342020 0.939693i 0 1.98563 + 1.66614i −0.690517 2.57704i −0.965926 0.258819i −3.66223 0.645749i 0
143.6 0.573576 0.819152i −0.134247 + 1.53445i −0.342020 0.939693i 0 1.17995 + 0.990096i 1.26063 + 4.70472i −0.965926 0.258819i 0.617895 + 0.108952i 0
143.7 0.573576 0.819152i 0.0476128 0.544217i −0.342020 0.939693i 0 −0.418487 0.351152i 0.123323 + 0.460247i −0.965926 0.258819i 2.66052 + 0.469121i 0
143.8 0.573576 0.819152i 0.282278 3.22645i −0.342020 0.939693i 0 −2.48104 2.08184i 0.0261014 + 0.0974119i −0.965926 0.258819i −7.37587 1.30056i 0
193.1 −0.0871557 0.996195i −1.07287 + 2.30077i −0.984808 + 0.173648i 0 2.38552 + 0.868257i 3.67829 + 0.985594i 0.258819 + 0.965926i −2.21412 2.63869i 0
193.2 −0.0871557 0.996195i −0.0142429 + 0.0305440i −0.984808 + 0.173648i 0 0.0316692 + 0.0115266i −2.42021 0.648494i 0.258819 + 0.965926i 1.92763 + 2.29726i 0
193.3 −0.0871557 0.996195i 0.368687 0.790651i −0.984808 + 0.173648i 0 −0.819776 0.298374i 1.54311 + 0.413475i 0.258819 + 0.965926i 1.43916 + 1.71513i 0
193.4 −0.0871557 0.996195i 1.36591 2.92920i −0.984808 + 0.173648i 0 −3.03710 1.10542i 2.44739 + 0.655775i 0.258819 + 0.965926i −4.78616 5.70392i 0
193.5 0.0871557 + 0.996195i −1.36591 + 2.92920i −0.984808 + 0.173648i 0 −3.03710 1.10542i −2.44739 0.655775i −0.258819 0.965926i −4.78616 5.70392i 0
193.6 0.0871557 + 0.996195i −0.368687 + 0.790651i −0.984808 + 0.173648i 0 −0.819776 0.298374i −1.54311 0.413475i −0.258819 0.965926i 1.43916 + 1.71513i 0
193.7 0.0871557 + 0.996195i 0.0142429 0.0305440i −0.984808 + 0.173648i 0 0.0316692 + 0.0115266i 2.42021 + 0.648494i −0.258819 0.965926i 1.92763 + 2.29726i 0
193.8 0.0871557 + 0.996195i 1.07287 2.30077i −0.984808 + 0.173648i 0 2.38552 + 0.868257i −3.67829 0.985594i −0.258819 0.965926i −2.21412 2.63869i 0
243.1 −0.422618 0.906308i −2.62162 1.83568i −0.642788 + 0.766044i 0 −0.555745 + 3.15179i −0.894036 3.33659i 0.965926 + 0.258819i 2.47712 + 6.80584i 0
243.2 −0.422618 0.906308i −1.05824 0.740990i −0.642788 + 0.766044i 0 −0.224332 + 1.27225i 1.24089 + 4.63106i 0.965926 + 0.258819i −0.455247 1.25078i 0
243.3 −0.422618 0.906308i 0.451001 + 0.315794i −0.642788 + 0.766044i 0 0.0956055 0.542206i −0.410083 1.53045i 0.965926 + 0.258819i −0.922385 2.53423i 0
243.4 −0.422618 0.906308i 1.68936 + 1.18290i −0.642788 + 0.766044i 0 0.358120 2.03100i −0.500613 1.86831i 0.965926 + 0.258819i 0.428624 + 1.17763i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 907.8
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.f odd 18 1 inner
95.o odd 18 1 inner
95.r even 36 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.bb.d 96
5.b even 2 1 inner 950.2.bb.d 96
5.c odd 4 2 inner 950.2.bb.d 96
19.f odd 18 1 inner 950.2.bb.d 96
95.o odd 18 1 inner 950.2.bb.d 96
95.r even 36 2 inner 950.2.bb.d 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.bb.d 96 1.a even 1 1 trivial
950.2.bb.d 96 5.b even 2 1 inner
950.2.bb.d 96 5.c odd 4 2 inner
950.2.bb.d 96 19.f odd 18 1 inner
950.2.bb.d 96 95.o odd 18 1 inner
950.2.bb.d 96 95.r even 36 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(16\!\cdots\!60\)\( T_{3}^{72} - \)\(78\!\cdots\!33\)\( T_{3}^{68} + \)\(17\!\cdots\!38\)\( T_{3}^{64} - \)\(11\!\cdots\!83\)\( T_{3}^{60} + \)\(24\!\cdots\!81\)\( T_{3}^{56} + \)\(84\!\cdots\!87\)\( T_{3}^{52} + \)\(12\!\cdots\!20\)\( T_{3}^{48} - \)\(11\!\cdots\!20\)\( T_{3}^{44} - \)\(66\!\cdots\!12\)\( T_{3}^{40} + \)\(11\!\cdots\!01\)\( T_{3}^{36} + \)\(46\!\cdots\!74\)\( T_{3}^{32} + \)\(11\!\cdots\!49\)\( T_{3}^{28} + \)\(14\!\cdots\!21\)\( T_{3}^{24} - \)\(50\!\cdots\!08\)\( T_{3}^{20} - \)\(10\!\cdots\!24\)\( T_{3}^{16} - \)\(35\!\cdots\!80\)\( T_{3}^{12} + \)\(10\!\cdots\!96\)\( T_{3}^{8} + \)\(45\!\cdots\!92\)\( T_{3}^{4} + 16777216 \)">\(T_{3}^{96} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\).