Properties

Label 475.2.u.b
Level $475$
Weight $2$
Character orbit 475.u
Analytic conductor $3.793$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.u (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 95)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 6 q^{4} - 12 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 6 q^{4} - 12 q^{6} - 6 q^{9} + 24 q^{14} - 6 q^{16} - 42 q^{21} + 30 q^{24} + 6 q^{26} - 30 q^{29} - 36 q^{31} + 24 q^{34} + 150 q^{36} - 72 q^{39} - 60 q^{41} - 84 q^{44} + 18 q^{46} - 18 q^{49} - 90 q^{51} + 132 q^{54} - 36 q^{59} - 60 q^{61} - 72 q^{64} + 78 q^{66} - 30 q^{69} - 24 q^{71} + 30 q^{74} - 66 q^{76} + 102 q^{79} + 54 q^{81} - 96 q^{84} + 126 q^{86} + 108 q^{89} + 60 q^{91} - 60 q^{94} - 132 q^{96} + 186 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1 −0.883478 2.42733i 0.243956 + 0.0430161i −3.57933 + 3.00342i 0 −0.111115 0.630167i −0.347830 + 0.200820i 5.97847 + 3.45167i −2.76141 1.00507i 0
24.2 −0.429906 1.18116i −2.96649 0.523072i 0.321776 0.270002i 0 0.657481 + 3.72876i 3.22501 1.86196i −2.63437 1.52095i 5.70738 + 2.07732i 0
24.3 −0.105338 0.289414i −1.61894 0.285463i 1.45942 1.22460i 0 0.0879194 + 0.498616i −0.0772459 + 0.0445979i −1.04160 0.601369i −0.279590 0.101762i 0
24.4 0.105338 + 0.289414i 1.61894 + 0.285463i 1.45942 1.22460i 0 0.0879194 + 0.498616i 0.0772459 0.0445979i 1.04160 + 0.601369i −0.279590 0.101762i 0
24.5 0.429906 + 1.18116i 2.96649 + 0.523072i 0.321776 0.270002i 0 0.657481 + 3.72876i −3.22501 + 1.86196i 2.63437 + 1.52095i 5.70738 + 2.07732i 0
24.6 0.883478 + 2.42733i −0.243956 0.0430161i −3.57933 + 3.00342i 0 −0.111115 0.630167i 0.347830 0.200820i −5.97847 3.45167i −2.76141 1.00507i 0
74.1 −2.18897 0.385975i 0.666572 0.794389i 2.76323 + 1.00573i 0 −1.76572 + 1.48162i −1.75377 + 1.01254i −1.81055 1.04532i 0.334208 + 1.89539i 0
74.2 −2.09998 0.370282i −1.43367 + 1.70859i 2.39340 + 0.871127i 0 3.64334 3.05712i −1.28659 + 0.742812i −1.01015 0.583208i −0.342900 1.94468i 0
74.3 −0.207778 0.0366369i −0.0513559 + 0.0612035i −1.83756 0.668816i 0 0.0129129 0.0108352i 1.46118 0.843614i 0.722735 + 0.417271i 0.519836 + 2.94814i 0
74.4 0.207778 + 0.0366369i 0.0513559 0.0612035i −1.83756 0.668816i 0 0.0129129 0.0108352i −1.46118 + 0.843614i −0.722735 0.417271i 0.519836 + 2.94814i 0
74.5 2.09998 + 0.370282i 1.43367 1.70859i 2.39340 + 0.871127i 0 3.64334 3.05712i 1.28659 0.742812i 1.01015 + 0.583208i −0.342900 1.94468i 0
74.6 2.18897 + 0.385975i −0.666572 + 0.794389i 2.76323 + 1.00573i 0 −1.76572 + 1.48162i 1.75377 1.01254i 1.81055 + 1.04532i 0.334208 + 1.89539i 0
99.1 −0.883478 + 2.42733i 0.243956 0.0430161i −3.57933 3.00342i 0 −0.111115 + 0.630167i −0.347830 0.200820i 5.97847 3.45167i −2.76141 + 1.00507i 0
99.2 −0.429906 + 1.18116i −2.96649 + 0.523072i 0.321776 + 0.270002i 0 0.657481 3.72876i 3.22501 + 1.86196i −2.63437 + 1.52095i 5.70738 2.07732i 0
99.3 −0.105338 + 0.289414i −1.61894 + 0.285463i 1.45942 + 1.22460i 0 0.0879194 0.498616i −0.0772459 0.0445979i −1.04160 + 0.601369i −0.279590 + 0.101762i 0
99.4 0.105338 0.289414i 1.61894 0.285463i 1.45942 + 1.22460i 0 0.0879194 0.498616i 0.0772459 + 0.0445979i 1.04160 0.601369i −0.279590 + 0.101762i 0
99.5 0.429906 1.18116i 2.96649 0.523072i 0.321776 + 0.270002i 0 0.657481 3.72876i −3.22501 1.86196i 2.63437 1.52095i 5.70738 2.07732i 0
99.6 0.883478 2.42733i −0.243956 + 0.0430161i −3.57933 3.00342i 0 −0.111115 + 0.630167i 0.347830 + 0.200820i −5.97847 + 3.45167i −2.76141 + 1.00507i 0
149.1 −1.47196 + 1.75422i 1.13116 + 3.10785i −0.563307 3.19467i 0 −7.11687 2.59033i −2.54534 + 1.46955i 2.46697 + 1.42431i −6.08105 + 5.10261i 0
149.2 −0.443593 + 0.528654i −0.237653 0.652945i 0.264596 + 1.50060i 0 0.450603 + 0.164006i −2.02186 + 1.16732i −2.10598 1.21589i 1.92827 1.61801i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 424.6
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 475.2.u.b 36
5.b even 2 1 inner 475.2.u.b 36
5.c odd 4 1 95.2.k.a 18
5.c odd 4 1 475.2.l.c 18
15.e even 4 1 855.2.bs.c 18
19.e even 9 1 inner 475.2.u.b 36
95.p even 18 1 inner 475.2.u.b 36
95.q odd 36 1 95.2.k.a 18
95.q odd 36 1 475.2.l.c 18
95.q odd 36 1 1805.2.a.v 9
95.q odd 36 1 9025.2.a.cc 9
95.r even 36 1 1805.2.a.s 9
95.r even 36 1 9025.2.a.cf 9
285.bi even 36 1 855.2.bs.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.k.a 18 5.c odd 4 1
95.2.k.a 18 95.q odd 36 1
475.2.l.c 18 5.c odd 4 1
475.2.l.c 18 95.q odd 36 1
475.2.u.b 36 1.a even 1 1 trivial
475.2.u.b 36 5.b even 2 1 inner
475.2.u.b 36 19.e even 9 1 inner
475.2.u.b 36 95.p even 18 1 inner
855.2.bs.c 18 15.e even 4 1
855.2.bs.c 18 285.bi even 36 1
1805.2.a.s 9 95.r even 36 1
1805.2.a.v 9 95.q odd 36 1
9025.2.a.cc 9 95.q odd 36 1
9025.2.a.cf 9 95.r even 36 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 3 T_{2}^{34} - 12 T_{2}^{32} - 175 T_{2}^{30} + 1281 T_{2}^{28} - 2784 T_{2}^{26} + 39776 T_{2}^{24} - 183258 T_{2}^{22} - 49458 T_{2}^{20} + 648786 T_{2}^{18} + 1796508 T_{2}^{16} + 868779 T_{2}^{14} + 719613 T_{2}^{12} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display