Properties

Label 475.2.u.c
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 6 q^{4} - 12 q^{6} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 6 q^{4} - 12 q^{6} + 42 q^{9} - 48 q^{14} + 42 q^{16} + 24 q^{19} + 6 q^{21} - 42 q^{24} - 42 q^{26} + 18 q^{29} + 60 q^{31} - 48 q^{34} - 42 q^{36} - 24 q^{39} - 12 q^{41} + 60 q^{44} + 42 q^{46} + 6 q^{49} + 54 q^{51} - 60 q^{54} - 144 q^{56} - 36 q^{59} + 12 q^{61} + 48 q^{64} - 66 q^{66} - 54 q^{69} + 48 q^{71} + 78 q^{74} + 54 q^{76} - 18 q^{79} + 30 q^{81} + 24 q^{84} - 66 q^{86} + 12 q^{89} - 12 q^{91} + 132 q^{94} - 36 q^{96} - 6 q^{99}+O(q^{100}) \) 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 −0.691434 1.89970i −1.87167 0.330026i −1.59869 + 1.34146i 0 0.667187 + 3.78381i −2.67790 + 1.54609i 0.152212 + 0.0878797i 0.575162 + 0.209342i 0
24.2 −0.575828 1.58207i 3.20261 + 0.564707i −0.639290 + 0.536428i 0 −0.950745 5.39194i −0.474919 + 0.274194i −1.69930 0.981094i 7.11876 + 2.59101i 0
24.3 −0.408399 1.12207i 2.23864 + 0.394733i 0.439845 0.369074i 0 −0.471342 2.67312i 1.90222 1.09825i −2.66196 1.53688i 2.03663 + 0.741274i 0
24.4 0.408399 + 1.12207i −2.23864 0.394733i 0.439845 0.369074i 0 −0.471342 2.67312i −1.90222 + 1.09825i 2.66196 + 1.53688i 2.03663 + 0.741274i 0
24.5 0.575828 + 1.58207i −3.20261 0.564707i −0.639290 + 0.536428i 0 −0.950745 5.39194i 0.474919 0.274194i 1.69930 + 0.981094i 7.11876 + 2.59101i 0
24.6 0.691434 + 1.89970i 1.87167 + 0.330026i −1.59869 + 1.34146i 0 0.667187 + 3.78381i 2.67790 1.54609i −0.152212 0.0878797i 0.575162 + 0.209342i 0
74.1 −2.58240 0.455347i 0.597006 0.711484i 4.58206 + 1.66773i 0 −1.86568 + 1.56549i 3.31824 1.91579i −6.53146 3.77094i 0.371151 + 2.10490i 0
74.2 −1.44009 0.253927i 0.970838 1.15700i 0.130002 + 0.0473169i 0 −1.69189 + 1.41966i 3.52622 2.03586i 2.35759 + 1.36116i 0.124823 + 0.707907i 0
74.3 −0.708527 0.124932i 0.793382 0.945515i −1.39298 0.507004i 0 −0.680257 + 0.570804i −1.11885 + 0.645970i 2.16976 + 1.25271i 0.256400 + 1.45411i 0
74.4 0.708527 + 0.124932i −0.793382 + 0.945515i −1.39298 0.507004i 0 −0.680257 + 0.570804i 1.11885 0.645970i −2.16976 1.25271i 0.256400 + 1.45411i 0
74.5 1.44009 + 0.253927i −0.970838 + 1.15700i 0.130002 + 0.0473169i 0 −1.69189 + 1.41966i −3.52622 + 2.03586i −2.35759 1.36116i 0.124823 + 0.707907i 0
74.6 2.58240 + 0.455347i −0.597006 + 0.711484i 4.58206 + 1.66773i 0 −1.86568 + 1.56549i −3.31824 + 1.91579i 6.53146 + 3.77094i 0.371151 + 2.10490i 0
99.1 −0.691434 + 1.89970i −1.87167 + 0.330026i −1.59869 1.34146i 0 0.667187 3.78381i −2.67790 1.54609i 0.152212 0.0878797i 0.575162 0.209342i 0
99.2 −0.575828 + 1.58207i 3.20261 0.564707i −0.639290 0.536428i 0 −0.950745 + 5.39194i −0.474919 0.274194i −1.69930 + 0.981094i 7.11876 2.59101i 0
99.3 −0.408399 + 1.12207i 2.23864 0.394733i 0.439845 + 0.369074i 0 −0.471342 + 2.67312i 1.90222 + 1.09825i −2.66196 + 1.53688i 2.03663 0.741274i 0
99.4 0.408399 1.12207i −2.23864 + 0.394733i 0.439845 + 0.369074i 0 −0.471342 + 2.67312i −1.90222 1.09825i 2.66196 1.53688i 2.03663 0.741274i 0
99.5 0.575828 1.58207i −3.20261 + 0.564707i −0.639290 0.536428i 0 −0.950745 + 5.39194i 0.474919 + 0.274194i 1.69930 0.981094i 7.11876 2.59101i 0
99.6 0.691434 1.89970i 1.87167 0.330026i −1.59869 1.34146i 0 0.667187 3.78381i 2.67790 + 1.54609i −0.152212 + 0.0878797i 0.575162 0.209342i 0
149.1 −1.16796 + 1.39192i 0.0605732 + 0.166424i −0.226016 1.28180i 0 −0.302396 0.110063i 0.929646 0.536732i −1.09903 0.634528i 2.27411 1.90820i 0
149.2 −1.00948 + 1.20305i −0.781523 2.14722i −0.0809872 0.459301i 0 3.37214 + 1.22736i 3.47568 2.00668i −2.08582 1.20425i −1.70162 + 1.42783i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 424.6
Significant digits:
Format:

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.c 36
5.b even 2 1 inner 475.2.u.c 36
5.c odd 4 1 95.2.k.b 18
5.c odd 4 1 475.2.l.b 18
15.e even 4 1 855.2.bs.b 18
19.e even 9 1 inner 475.2.u.c 36
95.p even 18 1 inner 475.2.u.c 36
95.q odd 36 1 95.2.k.b 18
95.q odd 36 1 475.2.l.b 18
95.q odd 36 1 1805.2.a.t 9
95.q odd 36 1 9025.2.a.ce 9
95.r even 36 1 1805.2.a.u 9
95.r even 36 1 9025.2.a.cd 9
285.bi even 36 1 855.2.bs.b 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.k.b 18 5.c odd 4 1
95.2.k.b 18 95.q odd 36 1
475.2.l.b 18 5.c odd 4 1
475.2.l.b 18 95.q odd 36 1
475.2.u.c 36 1.a even 1 1 trivial
475.2.u.c 36 5.b even 2 1 inner
475.2.u.c 36 19.e even 9 1 inner
475.2.u.c 36 95.p even 18 1 inner
855.2.bs.b 18 15.e even 4 1
855.2.bs.b 18 285.bi even 36 1
1805.2.a.t 9 95.q odd 36 1
1805.2.a.u 9 95.r even 36 1
9025.2.a.cd 9 95.r even 36 1
9025.2.a.ce 9 95.q odd 36 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 3 T_{2}^{34} - 24 T_{2}^{32} - 79 T_{2}^{30} + 669 T_{2}^{28} + 3744 T_{2}^{26} + 16520 T_{2}^{24} + 19014 T_{2}^{22} - 12558 T_{2}^{20} - 202302 T_{2}^{18} - 178248 T_{2}^{16} + 29115 T_{2}^{14} + 2164089 T_{2}^{12} + \cdots + 130321 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display