Properties

Label 175.2.s.a
Level 175
Weight 2
Character orbit 175.s
Analytic conductor 1.397
Analytic rank 0
Dimension 144
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 175.s (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.39738203537\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) over \(\Q(\zeta_{20})\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144q - 16q^{2} - 20q^{4} - 14q^{7} - 12q^{8} - 20q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 144q - 16q^{2} - 20q^{4} - 14q^{7} - 12q^{8} - 20q^{9} - 12q^{11} - 10q^{14} - 20q^{15} + 12q^{16} - 28q^{18} - 6q^{21} + 16q^{22} - 8q^{23} - 20q^{25} - 70q^{28} + 40q^{30} - 20q^{32} - 40q^{35} - 28q^{36} + 4q^{37} - 60q^{39} - 30q^{42} + 72q^{43} - 20q^{44} - 12q^{46} + 140q^{50} - 32q^{51} - 104q^{53} - 22q^{56} + 120q^{57} - 32q^{58} - 120q^{60} + 48q^{63} + 40q^{64} - 20q^{65} - 16q^{67} + 90q^{70} - 12q^{71} - 64q^{72} + 74q^{77} + 60q^{78} - 20q^{79} - 8q^{81} + 190q^{84} - 12q^{86} + 92q^{88} - 6q^{91} - 20q^{92} - 160q^{93} + 80q^{95} + 162q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −2.43824 + 0.386179i −0.183665 0.0935818i 3.89376 1.26516i 2.22917 + 0.175511i 0.483957 + 0.157247i −2.41722 1.07566i −4.60620 + 2.34698i −1.73838 2.39268i −5.50302 + 0.432920i
13.2 −2.43824 + 0.386179i 0.183665 + 0.0935818i 3.89376 1.26516i −2.22917 0.175511i −0.483957 0.157247i 1.07566 + 2.41722i −4.60620 + 2.34698i −1.73838 2.39268i 5.50302 0.432920i
13.3 −2.26523 + 0.358778i −2.54886 1.29871i 3.10045 1.00740i −0.855724 2.06585i 6.23972 + 2.02741i 0.506256 2.59686i −2.57481 + 1.31193i 3.04670 + 4.19343i 2.67959 + 4.37262i
13.4 −2.26523 + 0.358778i 2.54886 + 1.29871i 3.10045 1.00740i 0.855724 + 2.06585i −6.23972 2.02741i 2.59686 0.506256i −2.57481 + 1.31193i 3.04670 + 4.19343i −2.67959 4.37262i
13.5 −1.23441 + 0.195511i −0.157781 0.0803936i −0.416571 + 0.135352i 0.231923 2.22401i 0.210485 + 0.0683906i 0.122096 + 2.64293i 2.71491 1.38332i −1.74492 2.40168i 0.148530 + 2.79068i
13.6 −1.23441 + 0.195511i 0.157781 + 0.0803936i −0.416571 + 0.135352i −0.231923 + 2.22401i −0.210485 0.0683906i −2.64293 0.122096i 2.71491 1.38332i −1.74492 2.40168i −0.148530 2.79068i
13.7 −1.05758 + 0.167504i −1.81417 0.924365i −0.811695 + 0.263736i −1.08801 + 1.95352i 2.07346 + 0.673709i 2.20364 1.46423i 2.72237 1.38712i 0.673400 + 0.926855i 0.823432 2.24825i
13.8 −1.05758 + 0.167504i 1.81417 + 0.924365i −0.811695 + 0.263736i 1.08801 1.95352i −2.07346 0.673709i 1.46423 2.20364i 2.72237 1.38712i 0.673400 + 0.926855i −0.823432 + 2.24825i
13.9 −0.437303 + 0.0692620i −2.82799 1.44093i −1.71568 + 0.557457i 2.19504 0.426380i 1.33649 + 0.434253i −2.15975 + 1.52822i 1.50065 0.764621i 4.15790 + 5.72286i −0.930365 + 0.338490i
13.10 −0.437303 + 0.0692620i 2.82799 + 1.44093i −1.71568 + 0.557457i −2.19504 + 0.426380i −1.33649 0.434253i −1.52822 + 2.15975i 1.50065 0.764621i 4.15790 + 5.72286i 0.930365 0.338490i
13.11 0.401286 0.0635575i −0.682337 0.347668i −1.74512 + 0.567024i −1.40273 1.74137i −0.295909 0.0961468i −1.74509 1.98864i −1.38827 + 0.707357i −1.41865 1.95260i −0.673572 0.609633i
13.12 0.401286 0.0635575i 0.682337 + 0.347668i −1.74512 + 0.567024i 1.40273 + 1.74137i 0.295909 + 0.0961468i 1.98864 + 1.74509i −1.38827 + 0.707357i −1.41865 1.95260i 0.673572 + 0.609633i
13.13 1.00543 0.159244i −2.24750 1.14516i −0.916583 + 0.297816i −2.16146 + 0.572778i −2.44206 0.793475i 1.00386 + 2.44791i −2.68816 + 1.36968i 1.97652 + 2.72044i −2.08199 + 0.920089i
13.14 1.00543 0.159244i 2.24750 + 1.14516i −0.916583 + 0.297816i 2.16146 0.572778i 2.44206 + 0.793475i −2.44791 1.00386i −2.68816 + 1.36968i 1.97652 + 2.72044i 2.08199 0.920089i
13.15 1.78879 0.283316i −1.39081 0.708653i 1.21738 0.395551i 1.39172 1.75018i −2.68864 0.873591i 2.56523 0.647770i −1.16181 + 0.591970i −0.331192 0.455847i 1.99363 3.52499i
13.16 1.78879 0.283316i 1.39081 + 0.708653i 1.21738 0.395551i −1.39172 + 1.75018i 2.68864 + 0.873591i 0.647770 2.56523i −1.16181 + 0.591970i −0.331192 0.455847i −1.99363 + 3.52499i
13.17 2.28620 0.362099i −0.873949 0.445299i 3.19350 1.03763i 1.80613 + 1.31829i −2.15927 0.701589i −1.94696 + 1.79146i 2.80043 1.42689i −1.19786 1.64871i 4.60653 + 2.35989i
13.18 2.28620 0.362099i 0.873949 + 0.445299i 3.19350 1.03763i −1.80613 1.31829i 2.15927 + 0.701589i −1.79146 + 1.94696i 2.80043 1.42689i −1.19786 1.64871i −4.60653 2.35989i
27.1 −2.43824 0.386179i −0.183665 + 0.0935818i 3.89376 + 1.26516i 2.22917 0.175511i 0.483957 0.157247i −2.41722 + 1.07566i −4.60620 2.34698i −1.73838 + 2.39268i −5.50302 0.432920i
27.2 −2.43824 0.386179i 0.183665 0.0935818i 3.89376 + 1.26516i −2.22917 + 0.175511i −0.483957 + 0.157247i 1.07566 2.41722i −4.60620 2.34698i −1.73838 + 2.39268i 5.50302 + 0.432920i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 167.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
25.f odd 20 1 inner
175.s even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.2.s.a 144
5.b even 2 1 875.2.s.c 144
5.c odd 4 1 875.2.s.a 144
5.c odd 4 1 875.2.s.b 144
7.b odd 2 1 inner 175.2.s.a 144
25.d even 5 1 875.2.s.b 144
25.e even 10 1 875.2.s.a 144
25.f odd 20 1 inner 175.2.s.a 144
25.f odd 20 1 875.2.s.c 144
35.c odd 2 1 875.2.s.c 144
35.f even 4 1 875.2.s.a 144
35.f even 4 1 875.2.s.b 144
175.l odd 10 1 875.2.s.b 144
175.m odd 10 1 875.2.s.a 144
175.s even 20 1 inner 175.2.s.a 144
175.s even 20 1 875.2.s.c 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.s.a 144 1.a even 1 1 trivial
175.2.s.a 144 7.b odd 2 1 inner
175.2.s.a 144 25.f odd 20 1 inner
175.2.s.a 144 175.s even 20 1 inner
875.2.s.a 144 5.c odd 4 1
875.2.s.a 144 25.e even 10 1
875.2.s.a 144 35.f even 4 1
875.2.s.a 144 175.m odd 10 1
875.2.s.b 144 5.c odd 4 1
875.2.s.b 144 25.d even 5 1
875.2.s.b 144 35.f even 4 1
875.2.s.b 144 175.l odd 10 1
875.2.s.c 144 5.b even 2 1
875.2.s.c 144 25.f odd 20 1
875.2.s.c 144 35.c odd 2 1
875.2.s.c 144 175.s even 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(175, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database