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Properties

Label 325.2.be.b
Level $325$
Weight $2$
Character orbit 325.be
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(47,325)]
 
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([17, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.be (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.

\(\operatorname{Tr}(f)(q) = \) \( 256 q - 10 q^{2} - 16 q^{3} + 68 q^{4} + 6 q^{5} - 6 q^{6} - 12 q^{7} - 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 256 q - 10 q^{2} - 16 q^{3} + 68 q^{4} + 6 q^{5} - 6 q^{6} - 12 q^{7} - 10 q^{8} - 18 q^{10} - 14 q^{11} - 14 q^{13} - 30 q^{14} + 44 q^{15} - 84 q^{16} - 16 q^{18} + 14 q^{19} + 36 q^{20} - 16 q^{21} - 22 q^{22} + 32 q^{23} - 10 q^{24} - 4 q^{25} - 24 q^{26} - 10 q^{27} - 46 q^{28} - 60 q^{29} + 36 q^{30} + 16 q^{31} - 50 q^{33} - 36 q^{34} - 34 q^{35} - 50 q^{36} - 42 q^{37} + 14 q^{38} - 30 q^{39} - 68 q^{40} + 46 q^{41} - 10 q^{42} - 4 q^{43} - 22 q^{44} - 14 q^{45} - 32 q^{46} + 6 q^{47} + 26 q^{48} + 172 q^{49} + 44 q^{50} + 30 q^{52} - 26 q^{53} + 54 q^{54} - 46 q^{55} - 170 q^{56} + 70 q^{58} - 24 q^{59} + 6 q^{60} + 2 q^{61} + 14 q^{62} - 30 q^{63} + 48 q^{64} + 108 q^{65} + 18 q^{66} + 30 q^{67} - 30 q^{68} + 30 q^{69} - 108 q^{70} + 26 q^{71} - 8 q^{72} - 50 q^{73} + 46 q^{75} + 2 q^{76} - 126 q^{77} - 44 q^{78} - 30 q^{79} - 178 q^{80} + 44 q^{81} - 60 q^{82} - 128 q^{83} - 48 q^{84} + 32 q^{85} - 32 q^{86} - 16 q^{87} + 16 q^{88} + 6 q^{89} - 30 q^{90} - 144 q^{91} - 78 q^{92} + 20 q^{93} + 130 q^{94} + 38 q^{95} + 134 q^{96} + 100 q^{98} - 26 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −2.62413 − 0.852630i −1.61459 + 0.255725i 4.54102 + 3.29925i 0.805851 − 2.08581i 4.45492 + 0.705589i 1.06317 −5.85958 − 8.06502i −0.311678 + 0.101270i −3.89308 + 4.78633i
47.2 −2.59008 − 0.841569i 2.78299 − 0.440782i 4.38226 + 3.18390i 2.15033 + 0.613238i −7.57912 − 1.20041i 0.255189 −5.46944 − 7.52804i 4.69756 − 1.52633i −5.05347 − 3.39799i
47.3 −2.30602 − 0.749272i −0.526270 + 0.0833531i 3.13829 + 2.28010i −0.491939 + 2.18128i 1.27604 + 0.202106i 0.402318 −2.67815 − 3.68616i −2.58316 + 0.839318i 2.76880 − 4.66149i
47.4 −2.22080 − 0.721582i 1.05220 − 0.166652i 2.79325 + 2.02941i −2.23466 − 0.0792204i −2.45698 − 0.389148i 2.04800 −1.99380 − 2.74422i −1.77382 + 0.576348i 4.90558 + 1.78843i
47.5 −2.04252 − 0.663656i −3.04245 + 0.481877i 2.11343 + 1.53550i 1.25880 + 1.84809i 6.53407 + 1.03490i −1.36586 −0.772988 − 1.06393i 6.17112 − 2.00512i −1.34463 − 4.61017i
47.6 −1.95080 − 0.633852i −1.86522 + 0.295422i 1.78580 + 1.29746i −1.95674 − 1.08221i 3.82592 + 0.605966i −3.47717 −0.250019 − 0.344122i 0.538611 − 0.175005i 3.13124 + 3.35145i
47.7 −1.76114 − 0.572228i 0.507026 − 0.0803050i 1.15612 + 0.839972i 2.13985 − 0.648867i −0.938894 − 0.148706i −4.74469 0.621450 + 0.855353i −2.60254 + 0.845618i −4.13987 − 0.0817409i
47.8 −1.61447 − 0.524574i 2.36703 − 0.374901i 0.713307 + 0.518248i 0.146593 − 2.23126i −4.01817 − 0.636415i 4.29791 1.11584 + 1.53583i 2.60911 − 0.847753i −1.40713 + 3.52540i
47.9 −1.59822 − 0.519294i 2.23803 − 0.354469i 0.666612 + 0.484322i 0.171721 + 2.22946i −3.76094 − 0.595674i −1.79716 1.16162 + 1.59884i 2.02996 − 0.659573i 0.883298 − 3.65235i
47.10 −1.18028 − 0.383497i −2.53356 + 0.401276i −0.372036 − 0.270300i −0.0160720 − 2.23601i 3.14420 + 0.497993i 3.91848 1.79436 + 2.46972i 3.40472 − 1.10626i −0.838534 + 2.64529i
47.11 −1.17377 − 0.381380i −0.855420 + 0.135485i −0.385757 − 0.280269i 2.16877 − 0.544469i 1.05573 + 0.167212i 0.683755 1.79675 + 2.47302i −2.13978 + 0.695257i −2.75328 − 0.188045i
47.12 −0.977362 − 0.317564i 0.790062 − 0.125134i −0.763645 − 0.554820i −1.85815 + 1.24389i −0.811915 − 0.128595i 1.41627 1.77825 + 2.44755i −2.24463 + 0.729324i 2.21110 − 0.625651i
47.13 −0.850095 − 0.276213i 3.05610 − 0.484038i −0.971666 − 0.705957i −1.72417 − 1.42381i −2.73167 − 0.432654i −3.84365 1.68179 + 2.31478i 6.25227 − 2.03149i 1.07244 + 1.68661i
47.14 −0.677190 − 0.220032i −1.88874 + 0.299147i −1.20786 − 0.877563i −0.481769 + 2.18355i 1.34486 + 0.213005i 1.61531 1.46191 + 2.01215i 0.624689 − 0.202974i 0.806701 − 1.37267i
47.15 −0.284300 − 0.0923746i 2.36885 − 0.375190i −1.54574 − 1.12305i 2.04793 + 0.897756i −0.708123 − 0.112156i 1.68717 0.687127 + 0.945749i 2.61753 − 0.850487i −0.499298 − 0.444409i
47.16 0.0880390 + 0.0286056i −2.34186 + 0.370914i −1.61110 − 1.17053i 2.23107 + 0.149487i −0.216785 − 0.0343354i −0.730009 −0.217178 − 0.298920i 2.49355 − 0.810204i 0.192145 + 0.0769817i
47.17 0.224407 + 0.0729142i −3.31549 + 0.525122i −1.57299 − 1.14285i −2.15733 + 0.588161i −0.782307 − 0.123905i −1.30040 −0.547042 − 0.752939i 7.86356 − 2.55502i −0.527004 − 0.0253125i
47.18 0.299408 + 0.0972834i 1.37643 − 0.218006i −1.53785 − 1.11732i 1.42600 − 1.72236i 0.433323 + 0.0686317i 1.53295 −0.721837 − 0.993523i −1.00612 + 0.326910i 0.594512 − 0.376962i
47.19 0.342539 + 0.111298i 0.0632286 − 0.0100144i −1.51309 − 1.09932i 0.620957 + 2.14812i 0.0227728 + 0.00360686i −4.86355 −0.819341 − 1.12773i −2.84927 + 0.925785i −0.0263785 + 0.804925i
47.20 0.769643 + 0.250072i −0.849855 + 0.134604i −1.08822 − 0.790638i 0.0218966 + 2.23596i −0.687745 − 0.108928i 4.49056 −1.59116 − 2.19004i −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 47.32
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 325.2.be.b yes 256
13.d odd 4 1 325.2.z.b ✓ 256
25.f odd 20 1 325.2.z.b ✓ 256
325.be even 20 1 inner 325.2.be.b yes 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.z.b ✓ 256 13.d odd 4 1
325.2.z.b ✓ 256 25.f odd 20 1
325.2.be.b yes 256 1.a even 1 1 trivial
325.2.be.b yes 256 325.be even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{256} + 10 T_{2}^{255} - 48 T_{2}^{254} - 810 T_{2}^{253} + 588 T_{2}^{252} + \cdots + 20\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display