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Properties

Label 755.2.t.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 520 q - 140 q^{4} - 5 q^{6} + 5 q^{10} + 5 q^{11} - 20 q^{13} - 15 q^{14} - 160 q^{16} + 25 q^{17} + 25 q^{18} + 5 q^{20} - 30 q^{23} + 20 q^{24} - 15 q^{26} - 45 q^{27} - 145 q^{28} - 25 q^{29} - 30 q^{31}+ \cdots - 60 q^{99}+O(q^{100}) \) 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 −2.23996 − 1.62742i −0.643868 + 0.0813395i 1.75087 + 5.38861i −0.968583 + 0.248690i 1.57461 + 0.865650i −0.657273 + 0.794506i 3.13652 − 9.65321i −2.49780 + 0.641326i 2.57431 + 1.01924i
81.2 −1.90794 − 1.38620i 2.32456 − 0.293660i 1.10065 + 3.38744i −0.968583 + 0.248690i −4.84218 − 2.66201i −2.16369 + 2.61546i 1.13817 − 3.50291i 2.41159 − 0.619190i 2.19273 + 0.868163i
81.3 −1.83638 − 1.33421i 0.599758 − 0.0757671i 0.974140 + 2.99809i −0.968583 + 0.248690i −1.20247 − 0.661064i 0.638098 − 0.771329i 0.808319 − 2.48775i −2.55178 + 0.655186i 2.11049 + 0.835601i
81.4 −1.81803 − 1.32087i −2.79557 + 0.353163i 0.942477 + 2.90065i −0.968583 + 0.248690i 5.54890 + 3.05053i −0.886606 + 1.07172i 0.729089 − 2.24391i 4.78475 − 1.22851i 2.08940 + 0.827250i
81.5 −1.71499 − 1.24601i −1.66264 + 0.210040i 0.770601 + 2.37167i −0.968583 + 0.248690i 3.11311 + 1.71145i 1.67912 − 2.02970i 0.323418 − 0.995377i −0.185507 + 0.0476301i 1.97098 + 0.780365i
81.6 −1.42663 − 1.03650i 3.05364 − 0.385765i 0.342887 + 1.05530i −0.968583 + 0.248690i −4.75625 − 2.61477i −0.101084 + 0.122190i −0.485197 + 1.49328i 6.27016 − 1.60990i 1.63957 + 0.649153i
81.7 −1.24763 − 0.906459i 0.831980 − 0.105104i 0.116889 + 0.359747i −0.968583 + 0.248690i −1.13328 − 0.623025i 1.48165 − 1.79100i −0.772847 + 2.37858i −2.22461 + 0.571182i 1.43386 + 0.567707i
81.8 −1.11268 − 0.808411i −1.08882 + 0.137550i −0.0335003 − 0.103103i −0.968583 + 0.248690i 1.32271 + 0.727164i −2.47240 + 2.98862i −0.896088 + 2.75788i −1.73914 + 0.446536i 1.27877 + 0.506301i
81.9 −0.904703 − 0.657305i −3.15965 + 0.399157i −0.231597 − 0.712782i −0.968583 + 0.248690i 3.12092 + 1.71574i 2.20408 − 2.66427i −0.950120 + 2.92417i 6.91834 − 1.77633i 1.03974 + 0.411664i
81.10 −0.599053 − 0.435238i −1.29331 + 0.163384i −0.448601 − 1.38065i −0.968583 + 0.248690i 0.845875 + 0.465024i 3.11383 − 3.76397i −0.789812 + 2.43079i −1.25978 + 0.323457i 0.688472 + 0.272585i
81.11 −0.586771 − 0.426314i 0.987311 − 0.124726i −0.455478 − 1.40182i −0.968583 + 0.248690i −0.632498 − 0.347719i −2.02683 + 2.45002i −0.778606 + 2.39630i −1.94652 + 0.499782i 0.674356 + 0.266997i
81.12 −0.505909 − 0.367564i 2.82871 − 0.357350i −0.497194 − 1.53021i −0.968583 + 0.248690i −1.56242 − 0.858948i 1.60392 − 1.93880i −0.697394 + 2.14636i 4.96818 − 1.27561i 0.581424 + 0.230202i
81.13 −0.146807 − 0.106662i 2.03426 − 0.256987i −0.607858 − 1.87080i −0.968583 + 0.248690i −0.326055 − 0.179250i −2.82772 + 3.41813i −0.222455 + 0.684646i 1.16644 − 0.299490i 0.168721 + 0.0668012i
81.14 −0.0974170 − 0.0707776i −2.56738 + 0.324336i −0.613553 − 1.88832i −0.968583 + 0.248690i 0.273063 + 0.150117i −1.89586 + 2.29170i −0.148300 + 0.456422i 3.58052 − 0.919320i 0.111958 + 0.0443274i
81.15 0.0212278 + 0.0154229i −2.24151 + 0.283169i −0.617821 − 1.90146i −0.968583 + 0.248690i −0.0519497 − 0.0285596i −0.144839 + 0.175080i 0.0324277 − 0.0998021i 2.03843 − 0.523381i −0.0243965 − 0.00965924i
81.16 0.510763 + 0.371091i 0.519377 − 0.0656126i −0.494864 − 1.52303i −0.968583 + 0.248690i 0.289627 + 0.159224i 3.03916 − 3.67371i 0.702614 − 2.16242i −2.64030 + 0.677914i −0.587003 − 0.232411i
81.17 0.661339 + 0.480491i −0.389156 + 0.0491618i −0.411536 − 1.26658i −0.968583 + 0.248690i −0.280986 − 0.154473i −0.260453 + 0.314833i 0.841633 − 2.59028i −2.75672 + 0.707806i −0.760055 − 0.300927i
81.18 0.741540 + 0.538761i 2.53779 − 0.320598i −0.358415 − 1.10309i −0.968583 + 0.248690i 2.05460 + 1.12953i −0.157771 + 0.190712i 0.895007 − 2.75455i 3.43185 − 0.881149i −0.852228 − 0.337421i
81.19 0.884143 + 0.642367i −0.234372 + 0.0296081i −0.248961 − 0.766224i −0.968583 + 0.248690i −0.226238 − 0.124375i −0.926357 + 1.11977i 0.947505 − 2.91612i −2.85170 + 0.732191i −1.01612 − 0.402309i
81.20 1.21456 + 0.882430i 2.58366 − 0.326392i 0.0784417 + 0.241419i −0.968583 + 0.248690i 3.42603 + 1.88348i 1.11754 − 1.35087i 0.810079 − 2.49317i 3.66303 − 0.940505i −1.39586 − 0.552658i
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.b ✓ 520
151.h even 25 1 inner 755.2.t.b ✓ 520
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
755.2.t.b ✓ 520 1.a even 1 1 trivial
755.2.t.b ✓ 520 151.h even 25 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{520} + 200 T_{2}^{518} + 20510 T_{2}^{516} + 1437340 T_{2}^{514} - 35 T_{2}^{513} + \cdots + 23\!\cdots\!25 \) acting on \(S_{2}^{\mathrm{new}}(755, [\chi])\). Copy content Toggle raw display