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Properties

Label 975.2.ci.a
Level $975$
Weight $2$
Character orbit 975.ci
Analytic conductor $7.785$
Analytic rank $0$
Dimension $560$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(112,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 9, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.112");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.ci (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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(560\)
Relative dimension: \(70\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 560 q + 140 q^{4} - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 560 q + 140 q^{4} - 8 q^{5} - 8 q^{12} + 12 q^{13} - 4 q^{15} - 140 q^{16} + 28 q^{17} - 16 q^{18} - 20 q^{19} - 100 q^{20} + 32 q^{22} - 8 q^{23} + 4 q^{25} + 40 q^{29} - 20 q^{33} + 40 q^{34} - 12 q^{37} + 48 q^{40} - 40 q^{41} + 4 q^{45} + 24 q^{47} + 16 q^{48} + 560 q^{49} - 148 q^{50} + 28 q^{52} - 20 q^{53} + 8 q^{55} - 112 q^{58} + 12 q^{60} - 72 q^{62} + 140 q^{64} - 132 q^{65} - 60 q^{68} + 16 q^{70} - 12 q^{72} + 80 q^{73} + 16 q^{75} + 48 q^{77} - 8 q^{78} + 32 q^{80} + 140 q^{81} + 36 q^{82} + 104 q^{83} - 116 q^{85} + 24 q^{87} - 72 q^{88} + 120 q^{89} + 12 q^{90} + 60 q^{91} + 32 q^{92} - 40 q^{94} - 96 q^{95} - 100 q^{96} - 440 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
112.1 −1.61424 + 2.22181i 0.891007 + 0.453990i −1.71264 − 5.27095i 2.20167 + 0.390708i −2.44698 + 1.24680i 4.38436 9.25186 + 3.00611i 0.587785 + 0.809017i −4.42210 + 4.26099i
112.2 −1.56541 + 2.15460i −0.891007 − 0.453990i −1.57377 − 4.84356i −2.05836 − 0.873583i 2.37296 − 1.20908i 3.52126 7.83375 + 2.54534i 0.587785 + 0.809017i 5.10440 − 3.06743i
112.3 −1.55281 + 2.13726i 0.891007 + 0.453990i −1.53864 − 4.73543i −1.30264 − 1.81745i −2.35386 + 1.19935i −0.552323 7.48508 + 2.43205i 0.587785 + 0.809017i 5.90713 + 0.0380743i
112.4 −1.55165 + 2.13567i −0.891007 − 0.453990i −1.53542 − 4.72552i −2.05232 + 0.887677i 2.35211 − 1.19846i −3.21476 7.45333 + 2.42173i 0.587785 + 0.809017i 1.28871 − 5.76045i
112.5 −1.54018 + 2.11987i −0.891007 − 0.453990i −1.50368 − 4.62784i 1.38843 − 1.75278i 2.33471 − 1.18959i −4.82248 7.14223 + 2.32065i 0.587785 + 0.809017i 1.57724 + 5.64290i
112.6 −1.48216 + 2.04002i −0.891007 − 0.453990i −1.34685 − 4.14517i 0.913151 − 2.04112i 2.24677 − 1.14478i 2.12888 5.65609 + 1.83778i 0.587785 + 0.809017i 2.81048 + 4.88811i
112.7 −1.42588 + 1.96256i 0.891007 + 0.453990i −1.20046 − 3.69462i 1.30729 + 1.81411i −2.16145 + 1.10131i −4.87518 4.34836 + 1.41287i 0.587785 + 0.809017i −5.42433 − 0.0210800i
112.8 −1.42303 + 1.95864i 0.891007 + 0.453990i −1.19320 − 3.67230i −2.23541 + 0.0542456i −2.15713 + 1.09912i 0.0854339 4.28564 + 1.39249i 0.587785 + 0.809017i 3.07482 − 4.45555i
112.9 −1.40741 + 1.93714i 0.891007 + 0.453990i −1.15366 − 3.55059i 0.888824 − 2.05183i −2.13346 + 1.08705i 1.06750 3.94716 + 1.28251i 0.587785 + 0.809017i 2.72373 + 4.60954i
112.10 −1.29820 + 1.78682i −0.891007 − 0.453990i −0.889373 − 2.73721i 2.03035 + 0.936844i 1.96791 − 1.00270i 3.14871 1.84442 + 0.599290i 0.587785 + 0.809017i −4.30978 + 2.41166i
112.11 −1.24145 + 1.70870i 0.891007 + 0.453990i −0.760446 − 2.34041i −0.0203879 + 2.23598i −1.88187 + 0.958861i 2.01158 0.925719 + 0.300784i 0.587785 + 0.809017i −3.79531 − 2.81068i
112.12 −1.19269 + 1.64159i 0.891007 + 0.453990i −0.654291 − 2.01370i 2.22533 + 0.218899i −1.80796 + 0.921201i −1.17717 0.226424 + 0.0735696i 0.587785 + 0.809017i −3.01346 + 3.39200i
112.13 −1.12795 + 1.55250i −0.891007 − 0.453990i −0.519929 − 1.60018i 0.281826 + 2.21824i 1.70983 − 0.871204i −2.79960 −0.579412 − 0.188262i 0.587785 + 0.809017i −3.76169 − 2.06454i
112.14 −1.08120 + 1.48815i −0.891007 − 0.453990i −0.427551 − 1.31587i −1.76898 + 1.36773i 1.63896 − 0.835094i −1.73005 −1.07837 − 0.350385i 0.587785 + 0.809017i −0.122760 − 4.11131i
112.15 −1.04732 + 1.44151i 0.891007 + 0.453990i −0.363035 − 1.11731i 1.62806 − 1.53279i −1.58759 + 0.808920i −3.31432 −1.39837 − 0.454356i 0.587785 + 0.809017i 0.504432 + 3.95216i
112.16 −0.998875 + 1.37483i −0.891007 − 0.453990i −0.274382 − 0.844462i −0.264293 − 2.22039i 1.51417 − 0.771506i −0.157796 −1.79736 − 0.583997i 0.587785 + 0.809017i 3.31667 + 1.85454i
112.17 −0.973429 + 1.33981i −0.891007 − 0.453990i −0.229493 − 0.706308i −1.20322 + 1.88474i 1.47559 − 0.751852i 4.84043 −1.98037 − 0.643461i 0.587785 + 0.809017i −1.35395 − 3.44675i
112.18 −0.939990 + 1.29379i 0.891007 + 0.453990i −0.172266 − 0.530180i −2.06177 + 0.865509i −1.42490 + 0.726025i 3.64446 −2.19401 − 0.712876i 0.587785 + 0.809017i 0.818261 − 3.48106i
112.19 −0.936943 + 1.28959i 0.891007 + 0.453990i −0.167151 − 0.514437i −2.03631 + 0.923823i −1.42029 + 0.723672i −4.62300 −2.21199 − 0.718719i 0.587785 + 0.809017i 0.716552 − 3.49158i
112.20 −0.925914 + 1.27441i −0.891007 − 0.453990i −0.148774 − 0.457879i −1.32907 − 1.79822i 1.40357 − 0.715153i 2.17057 −2.27504 − 0.739206i 0.587785 + 0.809017i 3.52227 − 0.0287869i
See next 80 embeddings (of 560 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 112.70
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
325.be even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.ci.a yes 560
13.d odd 4 1 975.2.bx.a ✓ 560
25.f odd 20 1 975.2.bx.a ✓ 560
325.be even 20 1 inner 975.2.ci.a yes 560
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.bx.a ✓ 560 13.d odd 4 1
975.2.bx.a ✓ 560 25.f odd 20 1
975.2.ci.a yes 560 1.a even 1 1 trivial
975.2.ci.a yes 560 325.be even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(975, [\chi])\).