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Properties

Label 575.2.r.b
Level $575$
Weight $2$
Character orbit 575.r
Analytic conductor $4.591$
Analytic rank $0$
Dimension $200$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [575,2,Mod(7,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([11, 38]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.r (of order \(44\), degree \(20\), 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: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(200\)
Relative dimension: \(10\) over \(\Q(\zeta_{44})\)
Twist minimal: no (minimal twist has level 115)
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 200 q + 18 q^{2} + 14 q^{3} - 36 q^{6} + 22 q^{7} + 26 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 200 q + 18 q^{2} + 14 q^{3} - 36 q^{6} + 22 q^{7} + 26 q^{8} - 44 q^{11} + 6 q^{12} + 26 q^{13} - 52 q^{16} + 22 q^{17} - 58 q^{18} - 44 q^{21} - 22 q^{23} - 28 q^{26} + 26 q^{27} - 66 q^{28} - 40 q^{31} + 46 q^{32} - 12 q^{36} - 66 q^{37} + 22 q^{38} - 8 q^{41} - 198 q^{42} + 22 q^{43} - 76 q^{46} - 52 q^{47} - 18 q^{48} - 44 q^{51} - 158 q^{52} + 22 q^{53} + 88 q^{56} - 66 q^{57} + 58 q^{58} + 44 q^{61} - 38 q^{62} + 22 q^{63} + 132 q^{66} + 22 q^{67} + 132 q^{71} + 28 q^{72} - 34 q^{73} + 132 q^{76} + 10 q^{77} - 22 q^{78} - 48 q^{81} + 50 q^{82} + 22 q^{83} + 46 q^{87} + 110 q^{88} - 50 q^{92} + 36 q^{93} + 148 q^{96} + 88 q^{97} + 50 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.13477 + 0.152682i 0.476413 − 2.19004i 2.55431 − 0.367254i 0 −0.682656 + 4.74797i −1.43707 + 2.63180i −1.21416 + 0.264125i −1.84040 − 0.840481i 0
7.2 −1.82025 + 0.130187i −0.475590 + 2.18625i 1.31673 − 0.189318i 0 0.581073 − 4.04145i 0.696418 − 1.27539i 1.19425 − 0.259794i −1.82462 − 0.833276i 0
7.3 −1.60196 + 0.114574i −0.0509080 + 0.234020i 0.573502 − 0.0824571i 0 0.0547398 − 0.380723i 0.562120 − 1.02945i 2.22942 − 0.484980i 2.67672 + 1.22242i 0
7.4 −0.632652 + 0.0452482i 0.364069 − 1.67360i −1.58144 + 0.227377i 0 −0.154602 + 1.07528i −0.0937908 + 0.171765i 2.22976 − 0.485055i 0.0605071 + 0.0276327i 0
7.5 0.107969 − 0.00772212i −0.390334 + 1.79434i −1.96805 + 0.282962i 0 −0.0282880 + 0.196748i −2.36052 + 4.32296i −0.421846 + 0.0917670i −0.338390 − 0.154538i 0
7.6 0.643856 − 0.0460495i 0.683154 − 3.14041i −1.56721 + 0.225331i 0 0.295239 − 2.05343i 1.03991 − 1.90445i −2.26018 + 0.491672i −6.66656 − 3.04452i 0
7.7 1.05653 − 0.0755644i −0.0308132 + 0.141646i −0.869101 + 0.124958i 0 −0.0218516 + 0.151981i −0.717546 + 1.31409i −2.97883 + 0.648004i 2.70978 + 1.23752i 0
7.8 1.30561 − 0.0933793i −0.136266 + 0.626406i −0.283736 + 0.0407951i 0 −0.119418 + 0.830568i 2.05467 − 3.76286i −2.92471 + 0.636232i 2.35508 + 1.07553i 0
7.9 2.43652 − 0.174263i 0.128656 − 0.591424i 3.92662 − 0.564562i 0 0.210410 − 1.46344i 0.0458364 − 0.0839431i 4.69506 − 1.02135i 2.39567 + 1.09406i 0
7.10 2.48666 − 0.177849i −0.639470 + 2.93959i 4.17220 − 0.599872i 0 −1.06734 + 7.42350i −1.40495 + 2.57297i 5.39608 − 1.17384i −5.50340 − 2.51332i 0
43.1 −2.12277 − 1.58909i 2.61802 − 0.976473i 1.41750 + 4.82756i 0 −7.10917 − 2.08744i 0.214285 + 0.0466148i 2.80906 − 7.53137i 3.63331 − 3.14828i 0
43.2 −1.63237 − 1.22197i −1.28961 + 0.480998i 0.607932 + 2.07043i 0 2.69288 + 0.790700i 2.19188 + 0.476815i 0.112470 − 0.301544i −0.835523 + 0.723985i 0
43.3 −1.35022 − 1.01076i 1.01206 − 0.377480i 0.237992 + 0.810525i 0 −1.74805 − 0.513275i −2.09735 − 0.456251i −0.680931 + 1.82565i −1.38547 + 1.20051i 0
43.4 −0.374452 − 0.280311i 2.71879 − 1.01406i −0.501825 − 1.70906i 0 −1.30231 − 0.382392i 1.99318 + 0.433591i −0.618082 + 1.65714i 4.09625 − 3.54942i 0
43.5 −0.315237 − 0.235983i 0.199411 − 0.0743763i −0.519779 − 1.77020i 0 −0.0804131 − 0.0236114i −2.31779 − 0.504204i −0.529109 + 1.41860i −2.23302 + 1.93492i 0
43.6 −0.108362 − 0.0811187i −1.81672 + 0.677602i −0.558303 − 1.90141i 0 0.251830 + 0.0739439i 2.69644 + 0.586574i −0.188348 + 0.504981i 0.574083 − 0.497445i 0
43.7 1.06769 + 0.799265i −2.27240 + 0.847561i −0.0623212 − 0.212247i 0 −3.10365 − 0.911314i −0.301466 − 0.0655800i 1.03527 − 2.77567i 2.17819 − 1.88741i 0
43.8 1.23839 + 0.927048i 1.46916 − 0.547968i 0.110730 + 0.377111i 0 2.32738 + 0.683382i 4.02961 + 0.876587i 0.868729 − 2.32915i −0.409093 + 0.354481i 0
43.9 1.85329 + 1.38736i −1.54338 + 0.575653i 0.946469 + 3.22338i 0 −3.65898 − 1.07437i −3.89694 − 0.847728i −1.09984 + 2.94878i −0.216589 + 0.187675i 0
43.10 2.06627 + 1.54679i 2.23496 − 0.833596i 1.31345 + 4.47320i 0 5.90743 + 1.73458i −3.10636 − 0.675748i −2.40116 + 6.43777i 2.03290 − 1.76152i 0
See next 80 embeddings (of 200 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.d odd 22 1 inner
115.l even 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.2.r.b 200
5.b even 2 1 115.2.l.a ✓ 200
5.c odd 4 1 115.2.l.a ✓ 200
5.c odd 4 1 inner 575.2.r.b 200
23.d odd 22 1 inner 575.2.r.b 200
115.i odd 22 1 115.2.l.a ✓ 200
115.l even 44 1 115.2.l.a ✓ 200
115.l even 44 1 inner 575.2.r.b 200
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.l.a ✓ 200 5.b even 2 1
115.2.l.a ✓ 200 5.c odd 4 1
115.2.l.a ✓ 200 115.i odd 22 1
115.2.l.a ✓ 200 115.l even 44 1
575.2.r.b 200 1.a even 1 1 trivial
575.2.r.b 200 5.c odd 4 1 inner
575.2.r.b 200 23.d odd 22 1 inner
575.2.r.b 200 115.l even 44 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{200} - 18 T_{2}^{199} + 162 T_{2}^{198} - 986 T_{2}^{197} + 4587 T_{2}^{196} - 17344 T_{2}^{195} + \cdots + 294499921 \) acting on \(S_{2}^{\mathrm{new}}(575, [\chi])\). Copy content Toggle raw display