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Properties

Label 375.3.j.b
Level $375$
Weight $3$
Character orbit 375.j
Analytic conductor $10.218$
Analytic rank $0$
Dimension $144$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [375,3,Mod(26,375)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(375, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 4])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("375.26"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 375.j (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [144] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.2180099135\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(36\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 76 q^{4} + 10 q^{6} + 26 q^{9} + 44 q^{16} + 72 q^{19} + 108 q^{21} + 40 q^{24} - 252 q^{31} - 420 q^{34} - 426 q^{36} + 382 q^{39} - 420 q^{46} + 448 q^{49} - 120 q^{51} - 640 q^{54} + 588 q^{61}+ \cdots - 120 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −3.59614 + 1.16846i −2.93845 − 0.604570i 8.33088 − 6.05274i 0 11.2735 − 1.25934i 3.47419 −13.9965 + 19.2645i 8.26899 + 3.55300i 0
26.2 −3.59614 + 1.16846i 0.333051 − 2.98146i 8.33088 − 6.05274i 0 2.28600 + 11.1109i −3.47419 −13.9965 + 19.2645i −8.77815 − 1.98595i 0
26.3 −3.39551 + 1.10327i −1.74641 + 2.43927i 7.07623 − 5.14118i 0 3.23878 − 10.2093i 0.132726 −9.96116 + 13.7104i −2.90011 − 8.51994i 0
26.4 −3.39551 + 1.10327i 2.85956 − 0.907156i 7.07623 − 5.14118i 0 −8.70882 + 6.23512i −0.132726 −9.96116 + 13.7104i 7.35413 − 5.18813i 0
26.5 −2.91717 + 0.947845i −0.724906 + 2.91110i 4.37538 − 3.17890i 0 −0.644600 − 9.17926i −13.1629 −2.53897 + 3.49459i −7.94902 − 4.22055i 0
26.6 −2.91717 + 0.947845i 2.99263 + 0.210153i 4.37538 − 3.17890i 0 −8.92919 + 2.22350i 13.1629 −2.53897 + 3.49459i 8.91167 + 1.25782i 0
26.7 −2.35931 + 0.766585i −2.98708 − 0.278125i 1.74261 − 1.26608i 0 7.26065 − 1.63367i −4.33341 2.69174 − 3.70486i 8.84529 + 1.66157i 0
26.8 −2.35931 + 0.766585i 0.658545 − 2.92683i 1.74261 − 1.26608i 0 0.689952 + 7.41011i 4.33341 2.69174 − 3.70486i −8.13264 − 3.85490i 0
26.9 −2.16649 + 0.703935i 0.536292 + 2.95168i 0.962078 − 0.698991i 0 −3.23966 − 6.01726i 7.40536 3.76357 − 5.18010i −8.42478 + 3.16592i 0
26.10 −2.16649 + 0.703935i 2.64149 + 1.42216i 0.962078 − 0.698991i 0 −6.72386 − 1.22166i −7.40536 3.76357 − 5.18010i 4.95491 + 7.51325i 0
26.11 −1.81520 + 0.589794i −2.32377 − 1.89739i −0.288978 + 0.209955i 0 5.33718 + 2.07358i −4.41289 4.88814 − 6.72795i 1.79986 + 8.81819i 0
26.12 −1.81520 + 0.589794i −1.08643 − 2.79637i −0.288978 + 0.209955i 0 3.62137 + 4.43519i 4.41289 4.88814 − 6.72795i −6.63932 + 6.07614i 0
26.13 −1.05391 + 0.342438i −1.51263 + 2.59074i −2.24260 + 1.62934i 0 0.707019 − 3.24840i 8.38289 4.41098 − 6.07119i −4.42388 − 7.83768i 0
26.14 −1.05391 + 0.342438i 2.93137 − 0.638017i −2.24260 + 1.62934i 0 −2.87093 + 1.67623i −8.38289 4.41098 − 6.07119i 8.18587 − 3.74053i 0
26.15 −0.697111 + 0.226505i −2.55471 + 1.57273i −2.80141 + 2.03534i 0 1.42468 − 1.67502i 0.520138 3.21523 − 4.42538i 4.05306 − 8.03571i 0
26.16 −0.697111 + 0.226505i 2.28520 − 1.94367i −2.80141 + 2.03534i 0 −1.15279 + 1.87256i −0.520138 3.21523 − 4.42538i 1.44428 − 8.88336i 0
26.17 −0.612148 + 0.198899i 0.680442 + 2.92181i −2.90090 + 2.10763i 0 −0.997676 − 1.65324i −8.00714 2.86989 − 3.95006i −8.07400 + 3.97625i 0
26.18 −0.612148 + 0.198899i 2.56854 + 1.55003i −2.90090 + 2.10763i 0 −1.88063 − 0.437966i 8.00714 2.86989 − 3.95006i 4.19482 + 7.96263i 0
26.19 0.612148 − 0.198899i −2.56854 − 1.55003i −2.90090 + 2.10763i 0 −1.88063 − 0.437966i −8.00714 −2.86989 + 3.95006i 4.19482 + 7.96263i 0
26.20 0.612148 − 0.198899i −0.680442 − 2.92181i −2.90090 + 2.10763i 0 −0.997676 − 1.65324i 8.00714 −2.86989 + 3.95006i −8.07400 + 3.97625i 0
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner
75.h odd 10 1 inner
75.j odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 375.3.j.b 144
3.b odd 2 1 inner 375.3.j.b 144
5.b even 2 1 inner 375.3.j.b 144
5.c odd 4 1 75.3.h.a ✓ 72
5.c odd 4 1 375.3.h.a 72
15.d odd 2 1 inner 375.3.j.b 144
15.e even 4 1 75.3.h.a ✓ 72
15.e even 4 1 375.3.h.a 72
25.d even 5 1 inner 375.3.j.b 144
25.e even 10 1 inner 375.3.j.b 144
25.f odd 20 1 75.3.h.a ✓ 72
25.f odd 20 1 375.3.h.a 72
75.h odd 10 1 inner 375.3.j.b 144
75.j odd 10 1 inner 375.3.j.b 144
75.l even 20 1 75.3.h.a ✓ 72
75.l even 20 1 375.3.h.a 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.h.a ✓ 72 5.c odd 4 1
75.3.h.a ✓ 72 15.e even 4 1
75.3.h.a ✓ 72 25.f odd 20 1
75.3.h.a ✓ 72 75.l even 20 1
375.3.h.a 72 5.c odd 4 1
375.3.h.a 72 15.e even 4 1
375.3.h.a 72 25.f odd 20 1
375.3.h.a 72 75.l even 20 1
375.3.j.b 144 1.a even 1 1 trivial
375.3.j.b 144 3.b odd 2 1 inner
375.3.j.b 144 5.b even 2 1 inner
375.3.j.b 144 15.d odd 2 1 inner
375.3.j.b 144 25.d even 5 1 inner
375.3.j.b 144 25.e even 10 1 inner
375.3.j.b 144 75.h odd 10 1 inner
375.3.j.b 144 75.j odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} - 55 T_{2}^{70} + 1735 T_{2}^{68} - 41465 T_{2}^{66} + 847535 T_{2}^{64} + \cdots + 16\!\cdots\!25 \) acting on \(S_{3}^{\mathrm{new}}(375, [\chi])\). Copy content Toggle raw display