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Properties

Label 775.2.ca.a
Level $775$
Weight $2$
Character orbit 775.ca
Analytic conductor $6.188$
Analytic rank $0$
Dimension $48$
CM discriminant -31
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [775,2,Mod(92,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([13, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.92");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.ca (of order \(20\), degree \(8\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{8} + 48 q^{16} - 54 q^{28} + 12 q^{32} + 24 q^{35} - 72 q^{36} - 264 q^{38} - 36 q^{40} - 48 q^{47} + 450 q^{64} + 72 q^{67} - 6 q^{70} - 18 q^{72} + 84 q^{80} + 108 q^{81} + 42 q^{82} - 216 q^{90} - 96 q^{95} - 114 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
92.1 −1.27609 + 2.50447i 0 −3.46839 − 4.77383i −2.20011 + 0.399410i 0 −3.74153 − 3.74153i 10.8295 − 1.71522i −2.85317 − 0.927051i 1.80723 − 6.01979i
92.2 −0.649672 + 1.27505i 0 −0.0281154 − 0.0386976i 2.17235 − 0.529985i 0 −2.75999 − 2.75999i −2.75921 + 0.437015i −2.85317 − 0.927051i −0.735558 + 3.11418i
92.3 −0.634377 + 1.24503i 0 0.0278931 + 0.0383915i −1.54516 − 1.61632i 0 3.56788 + 3.56788i −2.82576 + 0.447556i −2.85317 − 0.927051i 2.99259 − 0.898418i
92.4 0.514203 − 1.00918i 0 0.421532 + 0.580189i 1.44595 + 1.70564i 0 1.84422 + 1.84422i 3.03964 − 0.481431i −2.85317 − 0.927051i 2.46481 − 0.582179i
92.5 0.761889 − 1.49529i 0 −0.479851 − 0.660458i 0.754154 − 2.10505i 0 1.89731 + 1.89731i 1.96192 − 0.310737i −2.85317 − 0.927051i −2.57309 − 2.73150i
92.6 1.28405 − 2.52009i 0 −3.52649 − 4.85380i −0.627196 + 2.14631i 0 −0.807888 − 0.807888i −11.1731 + 1.76964i −2.85317 − 0.927051i 4.60353 + 4.33655i
123.1 −0.421738 + 2.66275i 0 −5.01026 − 1.62793i −1.84744 − 1.25974i 0 −2.48080 + 2.48080i 3.99993 − 7.85030i 1.76336 − 2.42705i 4.13352 − 4.38800i
123.2 −0.359741 + 2.27131i 0 −3.12733 − 1.01613i 1.76898 + 1.36774i 0 −3.72424 + 3.72424i 1.34497 − 2.63965i 1.76336 − 2.42705i −3.74294 + 3.52587i
123.3 −0.0432228 + 0.272898i 0 1.82951 + 0.594443i −2.06899 + 0.848110i 0 1.54982 − 1.54982i −0.492174 + 0.965946i 1.76336 − 2.42705i −0.142020 − 0.601280i
123.4 0.0949675 − 0.599601i 0 1.55161 + 0.504149i −0.167248 + 2.22980i 0 3.66614 − 3.66614i 1.00085 − 1.96429i 1.76336 − 2.42705i 1.32111 + 0.312041i
123.5 0.326771 − 2.06315i 0 −2.24769 − 0.730319i 2.01469 − 0.970061i 0 −1.18535 + 1.18535i −0.344585 + 0.676286i 1.76336 − 2.42705i −1.34304 − 4.47360i
123.6 0.402963 − 2.54421i 0 −4.40851 − 1.43241i 0.300009 − 2.21585i 0 2.17442 − 2.17442i −3.08194 + 6.04865i 1.76336 − 2.42705i −5.51670 − 1.65619i
247.1 −2.30926 + 1.17663i 0 2.77267 − 3.81626i −1.44595 + 1.70564i 0 −3.25559 − 3.25559i −1.10164 + 6.95547i 2.85317 − 0.927051i 1.33218 − 5.64013i
247.2 −2.19113 + 1.11643i 0 2.37903 − 3.27446i 1.54516 − 1.61632i 0 −1.12705 − 1.12705i −0.787650 + 4.97302i 2.85317 − 0.927051i −1.58112 + 5.26663i
247.3 0.0173309 − 0.00883053i 0 −1.17535 + 1.61773i 0.627196 + 2.14631i 0 3.65340 + 3.65340i −0.0121700 + 0.0768385i 2.85317 − 0.927051i 0.0298229 + 0.0316589i
247.4 0.280657 − 0.143002i 0 −1.11725 + 1.53777i 2.20011 + 0.399410i 0 0.0306497 + 0.0306497i −0.192211 + 1.21357i 2.85317 − 0.927051i 0.674591 − 0.202522i
247.5 2.02861 − 1.03363i 0 1.87129 − 2.57561i −0.754154 − 2.10505i 0 3.22494 + 3.22494i 0.421566 − 2.66166i 2.85317 − 0.927051i −3.70572 − 3.49081i
247.6 2.17379 − 1.10760i 0 2.32303 − 3.19737i −2.17235 − 0.529985i 0 −2.52635 − 2.52635i 0.745053 − 4.70408i 2.85317 − 0.927051i −5.30926 + 1.25403i
278.1 −1.27609 − 2.50447i 0 −3.46839 + 4.77383i −2.20011 − 0.399410i 0 −3.74153 + 3.74153i 10.8295 + 1.71522i −2.85317 + 0.927051i 1.80723 + 6.01979i
278.2 −0.649672 − 1.27505i 0 −0.0281154 + 0.0386976i 2.17235 + 0.529985i 0 −2.75999 + 2.75999i −2.75921 − 0.437015i −2.85317 + 0.927051i −0.735558 − 3.11418i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 92.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)
25.f odd 20 1 inner
775.ca even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.ca.a ✓ 48
25.f odd 20 1 inner 775.2.ca.a ✓ 48
31.b odd 2 1 CM 775.2.ca.a ✓ 48
775.ca even 20 1 inner 775.2.ca.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.ca.a ✓ 48 1.a even 1 1 trivial
775.2.ca.a ✓ 48 25.f odd 20 1 inner
775.2.ca.a ✓ 48 31.b odd 2 1 CM
775.2.ca.a ✓ 48 775.ca even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 2 T_{2}^{45} - 72 T_{2}^{44} - 12 T_{2}^{43} - 73 T_{2}^{42} - 144 T_{2}^{41} + \cdots + 801025 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display