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Properties

Label 755.2.t.a
Level $755$
Weight $2$
Character orbit 755.t
Analytic conductor $6.029$
Analytic rank $0$
Dimension $520$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [755,2,Mod(81,755)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(755, base_ring=CyclotomicField(50)) chi = DirichletCharacter(H, H._module([0, 8])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("755.81"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 755 = 5 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 755.t (of order \(25\), degree \(20\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [520,0,0,-140,0,-5,0,0,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.02870535261\)
Analytic rank: \(0\)
Dimension: \(520\)
Relative dimension: \(26\) over \(\Q(\zeta_{25})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{25}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 520 q - 140 q^{4} - 5 q^{6} - 5 q^{10} - 15 q^{11} + 20 q^{13} - 15 q^{14} - 160 q^{16} - 45 q^{17} - 105 q^{18} - 5 q^{20} - 70 q^{23} - 40 q^{24} - 15 q^{26} - 45 q^{27} + 145 q^{28} - 45 q^{29} - 30 q^{31}+ \cdots - 140 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 −2.22422 − 1.61599i −3.09353 + 0.390804i 1.71771 + 5.28655i 0.968583 − 0.248690i 7.51224 + 4.12989i 2.08469 − 2.51996i 3.02332 − 9.30482i 6.51146 − 1.67186i −2.55623 − 1.01208i
81.2 −2.08978 − 1.51831i 2.66788 − 0.337032i 1.44386 + 4.44375i 0.968583 − 0.248690i −6.08700 − 3.34636i −1.36695 + 1.65236i 2.13320 − 6.56532i 4.09827 − 1.05226i −2.40171 − 0.950905i
81.3 −2.08567 − 1.51533i 0.866226 − 0.109430i 1.43577 + 4.41885i 0.968583 − 0.248690i −1.97249 − 1.08438i 1.83636 − 2.21977i 2.10816 − 6.48824i −2.16738 + 0.556488i −2.39699 − 0.949037i
81.4 −1.71956 − 1.24933i −2.49865 + 0.315653i 0.778018 + 2.39449i 0.968583 − 0.248690i 4.69094 + 2.57887i −2.05815 + 2.48788i 0.340046 − 1.04655i 3.23788 − 0.831345i −1.97623 − 0.782446i
81.5 −1.44293 − 1.04835i 1.36095 − 0.171928i 0.364974 + 1.12327i 0.968583 − 0.248690i −2.14400 − 1.17867i −1.74706 + 2.11184i −0.451347 + 1.38910i −1.08312 + 0.278098i −1.65831 − 0.656572i
81.6 −1.41879 − 1.03081i −1.22405 + 0.154634i 0.332363 + 1.02291i 0.968583 − 0.248690i 1.89608 + 1.04238i 1.40569 − 1.69918i −0.500988 + 1.54188i −1.43135 + 0.367509i −1.63057 − 0.645589i
81.7 −1.17695 − 0.855104i 2.17237 − 0.274434i 0.0359743 + 0.110717i 0.968583 − 0.248690i −2.79144 − 1.53461i 2.69393 − 3.25640i −0.846775 + 2.60611i 1.73814 − 0.446278i −1.35263 − 0.535544i
81.8 −1.16805 − 0.848641i 0.712877 − 0.0900573i 0.0261243 + 0.0804023i 0.968583 − 0.248690i −0.909105 − 0.499785i −1.65484 + 2.00036i −0.854595 + 2.63017i −2.40567 + 0.617670i −1.34241 − 0.531496i
81.9 −0.875623 − 0.636178i −3.00736 + 0.379917i −0.256040 − 0.788009i 0.968583 − 0.248690i 2.87501 + 1.58055i −1.90051 + 2.29732i −0.946036 + 2.91160i 5.99410 − 1.53902i −1.00632 − 0.398432i
81.10 −0.541044 − 0.393092i −1.74185 + 0.220047i −0.479826 − 1.47675i 0.968583 − 0.248690i 1.02892 + 0.565651i 0.256344 − 0.309867i −0.734213 + 2.25968i 0.0798685 − 0.0205068i −0.621804 − 0.246190i
81.11 −0.517411 − 0.375921i −0.599034 + 0.0756756i −0.491636 − 1.51310i 0.968583 − 0.248690i 0.338395 + 0.186034i 0.226667 − 0.273993i −0.709696 + 2.18422i −2.55263 + 0.655405i −0.594644 − 0.235436i
81.12 −0.170328 − 0.123750i 3.29151 − 0.415814i −0.604337 − 1.85996i 0.968583 − 0.248690i −0.612092 − 0.336501i −1.75291 + 2.11890i −0.257354 + 0.792054i 7.75536 − 1.99124i −0.195752 − 0.0775037i
81.13 0.0345306 + 0.0250880i 0.642883 − 0.0812150i −0.617471 − 1.90038i 0.968583 − 0.248690i 0.0242366 + 0.0133242i −0.304731 + 0.368356i 0.0527340 − 0.162299i −2.49905 + 0.641646i 0.0396849 + 0.0157124i
81.14 0.178764 + 0.129880i 1.79421 − 0.226661i −0.602946 − 1.85568i 0.968583 − 0.248690i 0.350180 + 0.192513i 1.97955 − 2.39286i 0.269794 − 0.830340i 0.262065 − 0.0672867i 0.205448 + 0.0813426i
81.15 0.206137 + 0.149767i −2.89669 + 0.365937i −0.597972 − 1.84037i 0.968583 − 0.248690i −0.651921 − 0.358396i 1.76758 − 2.13663i 0.309838 − 0.953582i 5.35114 − 1.37394i 0.236907 + 0.0937980i
81.16 0.487021 + 0.353841i 0.0105432 − 0.00133192i −0.506048 − 1.55746i 0.968583 − 0.248690i 0.00560605 + 0.00308195i −2.63282 + 3.18253i 0.676687 − 2.08263i −2.90564 + 0.746042i 0.559717 + 0.221608i
81.17 0.715742 + 0.520017i −1.38091 + 0.174450i −0.376165 − 1.15772i 0.968583 − 0.248690i −1.07910 − 0.593238i −2.80902 + 3.39553i 0.879574 − 2.70705i −1.02926 + 0.264268i 0.822578 + 0.325682i
81.18 0.917523 + 0.666620i 2.80175 − 0.353943i −0.220567 − 0.678834i 0.968583 − 0.248690i 2.80662 + 1.54295i 1.05459 − 1.27478i 0.951075 − 2.92711i 4.81878 − 1.23725i 1.05448 + 0.417498i
81.19 1.16911 + 0.849405i −1.88581 + 0.238234i 0.0272846 + 0.0839733i 0.968583 − 0.248690i −2.40707 − 1.32330i −0.00346409 + 0.00418736i 0.853688 − 2.62738i 0.593789 − 0.152459i 1.34361 + 0.531974i
81.20 1.30173 + 0.945763i 0.0655351 − 0.00827901i 0.182001 + 0.560143i 0.968583 − 0.248690i 0.0931390 + 0.0512036i 1.93306 − 2.33667i 0.701588 − 2.15927i −2.90152 + 0.744985i 1.49604 + 0.592322i
See next 80 embeddings (of 520 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.26
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
151.h even 25 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 755.2.t.a ✓ 520
151.h even 25 1 inner 755.2.t.a ✓ 520
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
755.2.t.a ✓ 520 1.a even 1 1 trivial
755.2.t.a ✓ 520 151.h even 25 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{520} + 200 T_{2}^{518} + 20510 T_{2}^{516} + 4 T_{2}^{515} + 1437340 T_{2}^{514} + \cdots + 66\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(755, [\chi])\). Copy content Toggle raw display