Properties

Label 325.2.z.b
Level $325$
Weight $2$
Character orbit 325.z
Analytic conductor $2.595$
Analytic rank $0$
Dimension $256$
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(8,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.z (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: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 256 q - 2 q^{2} - 16 q^{3} - 68 q^{4} - 10 q^{5} + 4 q^{6} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 256 q - 2 q^{2} - 16 q^{3} - 68 q^{4} - 10 q^{5} + 4 q^{6} + 6 q^{8} + 18 q^{10} - 4 q^{11} - 10 q^{13} - 30 q^{14} - 40 q^{15} - 84 q^{16} + 16 q^{19} + 30 q^{20} + 14 q^{21} - 22 q^{22} - 32 q^{23} + 10 q^{24} + 4 q^{25} - 24 q^{26} - 10 q^{27} + 40 q^{28} - 60 q^{29} - 36 q^{30} - 14 q^{31} + 80 q^{32} + 62 q^{33} - 14 q^{34} - 34 q^{35} + 50 q^{36} + 80 q^{37} - 14 q^{38} - 20 q^{39} - 68 q^{40} - 64 q^{41} - 10 q^{42} + 4 q^{43} + 12 q^{44} - 52 q^{45} + 18 q^{46} - 70 q^{47} + 26 q^{48} - 172 q^{49} + 106 q^{50} - 164 q^{52} - 26 q^{53} + 86 q^{54} - 46 q^{55} + 170 q^{56} - 20 q^{57} - 90 q^{58} - 6 q^{59} - 74 q^{60} + 2 q^{61} - 14 q^{62} + 98 q^{63} - 48 q^{64} - 26 q^{65} + 18 q^{66} - 4 q^{67} - 30 q^{68} - 30 q^{69} - 38 q^{70} - 54 q^{71} + 10 q^{72} + 112 q^{73} - 46 q^{75} + 2 q^{76} + 126 q^{77} + 120 q^{78} - 30 q^{79} + 176 q^{80} + 44 q^{81} + 60 q^{82} - 110 q^{83} + 38 q^{84} + 18 q^{85} - 22 q^{86} - 16 q^{87} - 16 q^{88} - 86 q^{89} + 30 q^{90} + 156 q^{91} - 78 q^{92} + 130 q^{94} - 38 q^{95} - 136 q^{96} + 10 q^{97} - 10 q^{98} + 26 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −0.852630 2.62413i −1.61459 0.255725i −4.54102 + 3.29925i −2.08581 + 0.805851i 0.705589 + 4.45492i 1.06317i 8.06502 + 5.85958i −0.311678 0.101270i 3.89308 + 4.78633i
8.2 −0.841569 2.59008i 2.78299 + 0.440782i −4.38226 + 3.18390i 0.613238 + 2.15033i −1.20041 7.57912i 0.255189i 7.52804 + 5.46944i 4.69756 + 1.52633i 5.05347 3.39799i
8.3 −0.749272 2.30602i −0.526270 0.0833531i −3.13829 + 2.28010i 2.18128 0.491939i 0.202106 + 1.27604i 0.402318i 3.68616 + 2.67815i −2.58316 0.839318i −2.76880 4.66149i
8.4 −0.721582 2.22080i 1.05220 + 0.166652i −2.79325 + 2.02941i −0.0792204 2.23466i −0.389148 2.45698i 2.04800i 2.74422 + 1.99380i −1.77382 0.576348i −4.90558 + 1.78843i
8.5 −0.663656 2.04252i −3.04245 0.481877i −2.11343 + 1.53550i 1.84809 + 1.25880i 1.03490 + 6.53407i 1.36586i 1.06393 + 0.772988i 6.17112 + 2.00512i 1.34463 4.61017i
8.6 −0.633852 1.95080i −1.86522 0.295422i −1.78580 + 1.29746i −1.08221 1.95674i 0.605966 + 3.82592i 3.47717i 0.344122 + 0.250019i 0.538611 + 0.175005i −3.13124 + 3.35145i
8.7 −0.572228 1.76114i 0.507026 + 0.0803050i −1.15612 + 0.839972i −0.648867 + 2.13985i −0.148706 0.938894i 4.74469i −0.855353 0.621450i −2.60254 0.845618i 4.13987 0.0817409i
8.8 −0.524574 1.61447i 2.36703 + 0.374901i −0.713307 + 0.518248i −2.23126 + 0.146593i −0.636415 4.01817i 4.29791i −1.53583 1.11584i 2.60911 + 0.847753i 1.40713 + 3.52540i
8.9 −0.519294 1.59822i 2.23803 + 0.354469i −0.666612 + 0.484322i 2.22946 + 0.171721i −0.595674 3.76094i 1.79716i −1.59884 1.16162i 2.02996 + 0.659573i −0.883298 3.65235i
8.10 −0.383497 1.18028i −2.53356 0.401276i 0.372036 0.270300i −2.23601 0.0160720i 0.497993 + 3.14420i 3.91848i −2.46972 1.79436i 3.40472 + 1.10626i 0.838534 + 2.64529i
8.11 −0.381380 1.17377i −0.855420 0.135485i 0.385757 0.280269i −0.544469 + 2.16877i 0.167212 + 1.05573i 0.683755i −2.47302 1.79675i −2.13978 0.695257i 2.75328 0.188045i
8.12 −0.317564 0.977362i 0.790062 + 0.125134i 0.763645 0.554820i 1.24389 1.85815i −0.128595 0.811915i 1.41627i −2.44755 1.77825i −2.24463 0.729324i −2.21110 0.625651i
8.13 −0.276213 0.850095i 3.05610 + 0.484038i 0.971666 0.705957i −1.42381 1.72417i −0.432654 2.73167i 3.84365i −2.31478 1.68179i 6.25227 + 2.03149i −1.07244 + 1.68661i
8.14 −0.220032 0.677190i −1.88874 0.299147i 1.20786 0.877563i 2.18355 0.481769i 0.213005 + 1.34486i 1.61531i −2.01215 1.46191i 0.624689 + 0.202974i −0.806701 1.37267i
8.15 −0.0923746 0.284300i 2.36885 + 0.375190i 1.54574 1.12305i 0.897756 + 2.04793i −0.112156 0.708123i 1.68717i −0.945749 0.687127i 2.61753 + 0.850487i 0.499298 0.444409i
8.16 0.0286056 + 0.0880390i −2.34186 0.370914i 1.61110 1.17053i 0.149487 + 2.23107i −0.0343354 0.216785i 0.730009i 0.298920 + 0.217178i 2.49355 + 0.810204i −0.192145 + 0.0769817i
8.17 0.0729142 + 0.224407i −3.31549 0.525122i 1.57299 1.14285i 0.588161 2.15733i −0.123905 0.782307i 1.30040i 0.752939 + 0.547042i 7.86356 + 2.55502i 0.527004 0.0253125i
8.18 0.0972834 + 0.299408i 1.37643 + 0.218006i 1.53785 1.11732i −1.72236 + 1.42600i 0.0686317 + 0.433323i 1.53295i 0.993523 + 0.721837i −1.00612 0.326910i −0.594512 0.376962i
8.19 0.111298 + 0.342539i 0.0632286 + 0.0100144i 1.51309 1.09932i 2.14812 + 0.620957i 0.00360686 + 0.0227728i 4.86355i 1.12773 + 0.819341i −2.84927 0.925785i 0.0263785 + 0.804925i
8.20 0.250072 + 0.769643i −0.849855 0.134604i 1.08822 0.790638i 2.23596 + 0.0218966i −0.108928 0.687745i 4.49056i 2.19004 + 1.59116i −2.14903 0.698264i 0.542299 + 1.72637i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.32
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.z.b 256
13.d odd 4 1 325.2.be.b yes 256
25.f odd 20 1 325.2.be.b yes 256
325.z even 20 1 inner 325.2.z.b 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.z.b 256 1.a even 1 1 trivial
325.2.z.b 256 325.z even 20 1 inner
325.2.be.b yes 256 13.d odd 4 1
325.2.be.b yes 256 25.f odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{256} + 2 T_{2}^{255} + 100 T_{2}^{254} + 198 T_{2}^{253} + 5236 T_{2}^{252} + 10168 T_{2}^{251} + 190978 T_{2}^{250} + 361444 T_{2}^{249} + 5449809 T_{2}^{248} + 10010684 T_{2}^{247} + 129515090 T_{2}^{246} + \cdots + 20\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display