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Properties

Label 325.2.r.a
Level $325$
Weight $2$
Character orbit 325.r
Analytic conductor $2.595$
Analytic rank $0$
Dimension $120$
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] = [325,2,Mod(14,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.14");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.r (of order \(10\), degree \(4\), 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: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 30 q^{4} - 10 q^{5} + 4 q^{6} - 30 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 30 q^{4} - 10 q^{5} + 4 q^{6} - 30 q^{8} + 26 q^{9} - 2 q^{10} + 2 q^{11} - 40 q^{12} - 12 q^{14} - 16 q^{15} - 42 q^{16} + 10 q^{17} - 8 q^{19} + 20 q^{20} + 20 q^{21} + 40 q^{22} - 30 q^{23} + 20 q^{24} + 28 q^{25} - 24 q^{26} + 60 q^{28} + 6 q^{29} - 48 q^{30} + 12 q^{31} - 40 q^{34} + 2 q^{35} - 74 q^{36} - 40 q^{38} - 8 q^{39} - 6 q^{40} + 16 q^{41} + 80 q^{42} + 6 q^{44} + 84 q^{45} - 16 q^{46} + 40 q^{47} - 100 q^{48} - 164 q^{49} - 86 q^{50} + 12 q^{51} - 10 q^{53} - 16 q^{54} + 56 q^{55} - 36 q^{56} + 70 q^{58} + 6 q^{59} + 22 q^{60} - 20 q^{61} + 30 q^{62} - 50 q^{63} + 72 q^{64} + 2 q^{65} - 50 q^{66} - 50 q^{67} - 70 q^{69} - 60 q^{70} - 34 q^{71} - 10 q^{72} + 40 q^{73} - 72 q^{74} + 24 q^{75} - 20 q^{76} + 40 q^{77} - 12 q^{79} + 92 q^{80} - 52 q^{81} + 10 q^{83} + 96 q^{84} + 2 q^{85} - 100 q^{87} - 130 q^{88} - 16 q^{89} - 10 q^{90} + 8 q^{91} + 140 q^{92} + 40 q^{94} + 14 q^{95} + 110 q^{97} + 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −1.65366 − 2.27606i 0.396751 + 0.128912i −1.82785 + 5.62555i −0.653630 + 2.13840i −0.362678 − 1.11621i − 2.94877i 10.4754 − 3.40366i −2.28626 − 1.66106i 5.94802 − 2.04848i
14.2 −1.42928 − 1.96723i −2.78589 − 0.905192i −1.20914 + 3.72134i −1.74984 − 1.39214i 2.20110 + 6.77427i 0.758221i 4.42370 − 1.43735i 4.51478 + 3.28018i −0.237655 + 5.43210i
14.3 −1.42602 − 1.96275i 2.55567 + 0.830389i −1.20081 + 3.69571i −2.15463 − 0.597959i −2.01460 − 6.20029i 4.84079i 4.35143 − 1.41387i 3.41488 + 2.48105i 1.89891 + 5.08170i
14.4 −1.38556 − 1.90706i 2.98194 + 0.968892i −1.09907 + 3.38260i 1.88989 − 1.19512i −2.28393 − 7.02921i − 4.07845i 3.48988 − 1.13393i 5.52617 + 4.01500i −4.89773 − 1.94823i
14.5 −1.18538 − 1.63153i −1.65852 − 0.538887i −0.638748 + 1.96586i −1.48003 + 1.67616i 1.08677 + 3.34472i 1.87805i 0.128565 − 0.0417732i 0.0332481 + 0.0241561i 4.48911 + 0.427829i
14.6 −0.994837 − 1.36928i 1.47689 + 0.479869i −0.267181 + 0.822298i 1.60351 + 1.55845i −0.812187 − 2.49965i − 0.578984i −1.82761 + 0.593825i −0.476136 − 0.345933i 0.538714 − 3.74605i
14.7 −0.968348 − 1.33282i 1.19994 + 0.389884i −0.220668 + 0.679145i −2.15055 − 0.612468i −0.642315 − 1.97684i − 1.78988i −2.01478 + 0.654642i −1.13921 − 0.827682i 1.26618 + 3.45937i
14.8 −0.959327 − 1.32040i −1.97858 − 0.642881i −0.205115 + 0.631280i 0.993075 + 2.00345i 1.04925 + 3.22926i − 4.53640i −2.07413 + 0.673927i 1.07444 + 0.780630i 1.69267 − 3.23322i
14.9 −0.945265 − 1.30105i −1.98308 − 0.644341i −0.181160 + 0.557553i 1.16099 − 1.91105i 1.03622 + 3.18915i − 3.25557i −2.16230 + 0.702572i 1.09038 + 0.792205i −3.58380 + 0.295938i
14.10 −0.658396 − 0.906204i 0.534463 + 0.173658i 0.230313 − 0.708831i 0.00877619 − 2.23605i −0.194519 − 0.598668i − 1.04428i −2.92460 + 0.950259i −2.17156 − 1.57773i −2.03210 + 1.46425i
14.11 −0.469137 − 0.645711i −0.243145 − 0.0790027i 0.421180 − 1.29626i −1.58576 + 1.57651i 0.0630555 + 0.194065i 3.03723i −2.55276 + 0.829442i −2.37417 − 1.72494i 1.76191 + 0.284345i
14.12 −0.451412 − 0.621315i 2.25005 + 0.731085i 0.435774 − 1.34118i 1.24697 − 1.85609i −0.561464 − 1.72801i 3.76375i −2.49080 + 0.809312i 2.10118 + 1.52659i −1.71611 + 0.0630960i
14.13 −0.413451 − 0.569066i −3.19205 − 1.03716i 0.465139 − 1.43155i 2.02946 + 0.938764i 0.729542 + 2.24530i 4.13676i −2.34491 + 0.761909i 6.68641 + 4.85796i −0.304865 − 1.54303i
14.14 −0.299733 − 0.412547i 2.90973 + 0.945430i 0.537679 − 1.65480i −0.192288 + 2.22778i −0.482109 − 1.48378i 1.67815i −1.81380 + 0.589340i 5.14566 + 3.73854i 0.976702 − 0.588413i
14.15 −0.0856250 − 0.117853i −2.43507 − 0.791203i 0.611476 − 1.88193i −1.86969 − 1.22648i 0.115258 + 0.354727i − 0.518253i −0.551237 + 0.179108i 2.87652 + 2.08991i 0.0155487 + 0.325366i
14.16 0.0781258 + 0.107531i −1.10539 − 0.359163i 0.612575 − 1.88531i 1.94032 − 1.11137i −0.0477383 − 0.146924i − 0.641073i 0.503408 − 0.163567i −1.33416 − 0.969326i 0.271096 + 0.121818i
14.17 0.0798204 + 0.109863i 0.863517 + 0.280574i 0.612335 − 1.88457i −1.53225 + 1.62856i 0.0381015 + 0.117264i − 4.87000i 0.514227 − 0.167082i −1.76011 − 1.27879i −0.301224 − 0.0383466i
14.18 0.0837177 + 0.115228i 1.75853 + 0.571381i 0.611765 − 1.88282i 2.11153 + 0.735823i 0.0813814 + 0.250466i − 1.97939i 0.539085 − 0.175159i 0.338904 + 0.246228i 0.0919855 + 0.304908i
14.19 0.304149 + 0.418625i 0.111612 + 0.0362650i 0.535293 − 1.64746i −1.60219 − 1.55981i 0.0187653 + 0.0577537i 1.87565i 1.83673 − 0.596789i −2.41591 − 1.75526i 0.165670 − 1.14513i
14.20 0.651038 + 0.896077i 0.0288467 + 0.00937285i 0.238930 − 0.735352i 0.868869 + 2.06036i 0.0103815 + 0.0319509i 4.30104i 2.92129 − 0.949184i −2.42631 − 1.76282i −1.28057 + 2.11994i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.r.a ✓ 120
25.e even 10 1 inner 325.2.r.a ✓ 120
25.f odd 20 1 8125.2.a.o 60
25.f odd 20 1 8125.2.a.p 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.r.a ✓ 120 1.a even 1 1 trivial
325.2.r.a ✓ 120 25.e even 10 1 inner
8125.2.a.o 60 25.f odd 20 1
8125.2.a.p 60 25.f odd 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(325, [\chi])\).