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Properties

Label 975.2.m.b
Level $975$
Weight $2$
Character orbit 975.m
Analytic conductor $7.785$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [975,2,Mod(443,975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("975.443"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(975, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,0,0,12,0,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 12 q^{6} - 32 q^{9} + 80 q^{14} - 8 q^{16} + 4 q^{21} + 16 q^{24} + 24 q^{31} - 16 q^{36} + 4 q^{39} - 16 q^{44} - 32 q^{46} + 40 q^{51} + 112 q^{54} - 72 q^{59} - 32 q^{66} - 40 q^{69} - 256 q^{74}+ \cdots - 136 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
443.1 −1.87465 − 1.87465i −1.20401 + 1.24513i 5.02865i 0 4.59130 − 0.0770814i −3.00921 + 3.00921i 5.67767 − 5.67767i −0.100703 − 2.99831i 0
443.2 −1.63741 − 1.63741i 1.07007 + 1.36197i 3.36220i 0 0.477964 − 3.98223i −1.97947 + 1.97947i 2.23048 − 2.23048i −0.709918 + 2.91479i 0
443.3 −1.51844 − 1.51844i −0.145999 − 1.72589i 2.61134i 0 −2.39897 + 2.84235i 0.988598 − 0.988598i 0.928279 − 0.928279i −2.95737 + 0.503955i 0
443.4 −1.47851 − 1.47851i −1.61522 − 0.625353i 2.37199i 0 1.46353 + 3.31271i −2.32423 + 2.32423i 0.549992 − 0.549992i 2.21787 + 2.02017i 0
443.5 −1.08900 − 1.08900i 0.390653 − 1.68742i 0.371844i 0 −2.26302 + 1.41218i −2.02596 + 2.02596i −1.77306 + 1.77306i −2.69478 − 1.31839i 0
443.6 −0.727164 − 0.727164i 0.502116 + 1.65767i − 0.942464i 0 0.840280 − 1.57052i 1.99337 − 1.99337i −2.13965 + 2.13965i −2.49576 + 1.66469i 0
443.7 −0.572109 − 0.572109i −1.49934 + 0.867167i − 1.34538i 0 1.35390 + 0.361672i −0.327985 + 0.327985i −1.91392 + 1.91392i 1.49604 − 2.60036i 0
443.8 −0.520491 − 0.520491i 0.349729 − 1.69638i − 1.45818i 0 −1.06498 + 0.700917i 3.38546 − 3.38546i −1.79995 + 1.79995i −2.75538 − 1.18654i 0
443.9 0.520491 + 0.520491i −0.349729 + 1.69638i − 1.45818i 0 −1.06498 + 0.700917i −3.38546 + 3.38546i 1.79995 − 1.79995i −2.75538 − 1.18654i 0
443.10 0.572109 + 0.572109i 1.49934 − 0.867167i − 1.34538i 0 1.35390 + 0.361672i 0.327985 − 0.327985i 1.91392 − 1.91392i 1.49604 − 2.60036i 0
443.11 0.727164 + 0.727164i −0.502116 − 1.65767i − 0.942464i 0 0.840280 − 1.57052i −1.99337 + 1.99337i 2.13965 − 2.13965i −2.49576 + 1.66469i 0
443.12 1.08900 + 1.08900i −0.390653 + 1.68742i 0.371844i 0 −2.26302 + 1.41218i 2.02596 − 2.02596i 1.77306 − 1.77306i −2.69478 − 1.31839i 0
443.13 1.47851 + 1.47851i 1.61522 + 0.625353i 2.37199i 0 1.46353 + 3.31271i 2.32423 − 2.32423i −0.549992 + 0.549992i 2.21787 + 2.02017i 0
443.14 1.51844 + 1.51844i 0.145999 + 1.72589i 2.61134i 0 −2.39897 + 2.84235i −0.988598 + 0.988598i −0.928279 + 0.928279i −2.95737 + 0.503955i 0
443.15 1.63741 + 1.63741i −1.07007 − 1.36197i 3.36220i 0 0.477964 − 3.98223i 1.97947 − 1.97947i −2.23048 + 2.23048i −0.709918 + 2.91479i 0
443.16 1.87465 + 1.87465i 1.20401 − 1.24513i 5.02865i 0 4.59130 − 0.0770814i 3.00921 − 3.00921i −5.67767 + 5.67767i −0.100703 − 2.99831i 0
482.1 −1.87465 + 1.87465i −1.20401 − 1.24513i − 5.02865i 0 4.59130 + 0.0770814i −3.00921 − 3.00921i 5.67767 + 5.67767i −0.100703 + 2.99831i 0
482.2 −1.63741 + 1.63741i 1.07007 − 1.36197i − 3.36220i 0 0.477964 + 3.98223i −1.97947 − 1.97947i 2.23048 + 2.23048i −0.709918 − 2.91479i 0
482.3 −1.51844 + 1.51844i −0.145999 + 1.72589i − 2.61134i 0 −2.39897 − 2.84235i 0.988598 + 0.988598i 0.928279 + 0.928279i −2.95737 − 0.503955i 0
482.4 −1.47851 + 1.47851i −1.61522 + 0.625353i − 2.37199i 0 1.46353 − 3.31271i −2.32423 − 2.32423i 0.549992 + 0.549992i 2.21787 − 2.02017i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 443.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.m.b ✓ 32
3.b odd 2 1 975.2.m.c yes 32
5.b even 2 1 inner 975.2.m.b ✓ 32
5.c odd 4 2 975.2.m.c yes 32
15.d odd 2 1 975.2.m.c yes 32
15.e even 4 2 inner 975.2.m.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.m.b ✓ 32 1.a even 1 1 trivial
975.2.m.b ✓ 32 5.b even 2 1 inner
975.2.m.b ✓ 32 15.e even 4 2 inner
975.2.m.c yes 32 3.b odd 2 1
975.2.m.c yes 32 5.c odd 4 2
975.2.m.c yes 32 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{32} + 126 T_{2}^{28} + 5879 T_{2}^{24} + 127668 T_{2}^{20} + 1299639 T_{2}^{16} + 5343510 T_{2}^{12} + \cdots + 456976 \) Copy content Toggle raw display
\( T_{29}^{8} - 100T_{29}^{6} + 72T_{29}^{5} + 2086T_{29}^{4} + 432T_{29}^{3} - 8964T_{29}^{2} - 6552T_{29} + 1521 \) Copy content Toggle raw display