Properties

Label 2548.2.u.d
Level 25482548
Weight 22
Character orbit 2548.u
Analytic conductor 20.34620.346
Analytic rank 00
Dimension 1818
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2548,2,Mod(589,2548)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2548, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2548.589");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 2548=227213 2548 = 2^{2} \cdot 7^{2} \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 2548.u (of order 66, degree 22, 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: 20.345882435020.3458824350
Analytic rank: 00
Dimension: 1818
Relative dimension: 99 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q[x]/(x18)\mathbb{Q}[x]/(x^{18} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x18x17+17x166x15+188x1449x13+1116x12x11+4649x10++16 x^{18} - x^{17} + 17 x^{16} - 6 x^{15} + 188 x^{14} - 49 x^{13} + 1116 x^{12} - x^{11} + 4649 x^{10} + \cdots + 16 Copy content Toggle raw display
Coefficient ring: Z[a1,,a9]\Z[a_1, \ldots, a_{9}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 364)
Sato-Tate group: SU(2)[C6]\mathrm{SU}(2)[C_{6}]

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,,β171,\beta_1,\ldots,\beta_{17} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == qβ1q3+(β11+β10)q5+(β8β4)q9+(β17β16++β7)q11+(β13β12+β10++1)q13++(β17+3β15+β14++2)q99+O(q100) q - \beta_1 q^{3} + (\beta_{11} + \beta_{10}) q^{5} + (\beta_{8} - \beta_{4}) q^{9} + (\beta_{17} - \beta_{16} + \cdots + \beta_{7}) q^{11} + ( - \beta_{13} - \beta_{12} + \beta_{10} + \cdots + 1) q^{13}+ \cdots + (\beta_{17} + 3 \beta_{15} + \beta_{14} + \cdots + 2) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 18qq36q9+6q11+4q13+6q15+10q17+21q196q2310q25+20q27+2q29+12q3318q3725q39+9q4114q43+30q454q51++18q97+O(q100) 18 q - q^{3} - 6 q^{9} + 6 q^{11} + 4 q^{13} + 6 q^{15} + 10 q^{17} + 21 q^{19} - 6 q^{23} - 10 q^{25} + 20 q^{27} + 2 q^{29} + 12 q^{33} - 18 q^{37} - 25 q^{39} + 9 q^{41} - 14 q^{43} + 30 q^{45} - 4 q^{51}+ \cdots + 18 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x18x17+17x166x15+188x1449x13+1116x12x11+4649x10++16 x^{18} - x^{17} + 17 x^{16} - 6 x^{15} + 188 x^{14} - 49 x^{13} + 1116 x^{12} - x^{11} + 4649 x^{10} + \cdots + 16 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (51 ⁣ ⁣75ν17++19 ⁣ ⁣64)/47 ⁣ ⁣71 ( - 51\!\cdots\!75 \nu^{17} + \cdots + 19\!\cdots\!64 ) / 47\!\cdots\!71 Copy content Toggle raw display
β3\beta_{3}== (13 ⁣ ⁣80ν17++78 ⁣ ⁣24)/47 ⁣ ⁣71 ( 13\!\cdots\!80 \nu^{17} + \cdots + 78\!\cdots\!24 ) / 47\!\cdots\!71 Copy content Toggle raw display
β4\beta_{4}== (19 ⁣ ⁣81ν17++18 ⁣ ⁣68)/18 ⁣ ⁣84 ( - 19\!\cdots\!81 \nu^{17} + \cdots + 18\!\cdots\!68 ) / 18\!\cdots\!84 Copy content Toggle raw display
β5\beta_{5}== (16 ⁣ ⁣85ν17+11 ⁣ ⁣92)/14 ⁣ ⁣13 ( - 16\!\cdots\!85 \nu^{17} + \cdots - 11\!\cdots\!92 ) / 14\!\cdots\!13 Copy content Toggle raw display
β6\beta_{6}== (20 ⁣ ⁣81ν17+22 ⁣ ⁣88)/14 ⁣ ⁣13 ( 20\!\cdots\!81 \nu^{17} + \cdots - 22\!\cdots\!88 ) / 14\!\cdots\!13 Copy content Toggle raw display
β7\beta_{7}== (80 ⁣ ⁣19ν17+28 ⁣ ⁣54)/28 ⁣ ⁣26 ( - 80\!\cdots\!19 \nu^{17} + \cdots - 28\!\cdots\!54 ) / 28\!\cdots\!26 Copy content Toggle raw display
β8\beta_{8}== (19 ⁣ ⁣81ν17++18 ⁣ ⁣68)/47 ⁣ ⁣71 ( - 19\!\cdots\!81 \nu^{17} + \cdots + 18\!\cdots\!68 ) / 47\!\cdots\!71 Copy content Toggle raw display
β9\beta_{9}== (12 ⁣ ⁣15ν17++31 ⁣ ⁣14)/28 ⁣ ⁣26 ( - 12\!\cdots\!15 \nu^{17} + \cdots + 31\!\cdots\!14 ) / 28\!\cdots\!26 Copy content Toggle raw display
β10\beta_{10}== (27 ⁣ ⁣49ν17++22 ⁣ ⁣44)/56 ⁣ ⁣52 ( 27\!\cdots\!49 \nu^{17} + \cdots + 22\!\cdots\!44 ) / 56\!\cdots\!52 Copy content Toggle raw display
β11\beta_{11}== (28 ⁣ ⁣73ν17+42 ⁣ ⁣72)/56 ⁣ ⁣52 ( 28\!\cdots\!73 \nu^{17} + \cdots - 42\!\cdots\!72 ) / 56\!\cdots\!52 Copy content Toggle raw display
β12\beta_{12}== (13 ⁣ ⁣18ν17++36 ⁣ ⁣22)/14 ⁣ ⁣13 ( 13\!\cdots\!18 \nu^{17} + \cdots + 36\!\cdots\!22 ) / 14\!\cdots\!13 Copy content Toggle raw display
β13\beta_{13}== (34 ⁣ ⁣17ν17+15 ⁣ ⁣64)/18 ⁣ ⁣84 ( 34\!\cdots\!17 \nu^{17} + \cdots - 15\!\cdots\!64 ) / 18\!\cdots\!84 Copy content Toggle raw display
β14\beta_{14}== (26 ⁣ ⁣98ν17+43 ⁣ ⁣55)/14 ⁣ ⁣13 ( - 26\!\cdots\!98 \nu^{17} + \cdots - 43\!\cdots\!55 ) / 14\!\cdots\!13 Copy content Toggle raw display
β15\beta_{15}== (10 ⁣ ⁣55ν17++52 ⁣ ⁣40)/56 ⁣ ⁣52 ( - 10\!\cdots\!55 \nu^{17} + \cdots + 52\!\cdots\!40 ) / 56\!\cdots\!52 Copy content Toggle raw display
β16\beta_{16}== (58 ⁣ ⁣98ν17+28 ⁣ ⁣11)/14 ⁣ ⁣13 ( 58\!\cdots\!98 \nu^{17} + \cdots - 28\!\cdots\!11 ) / 14\!\cdots\!13 Copy content Toggle raw display
β17\beta_{17}== (14 ⁣ ⁣53ν17++17 ⁣ ⁣40)/28 ⁣ ⁣26 ( - 14\!\cdots\!53 \nu^{17} + \cdots + 17\!\cdots\!40 ) / 28\!\cdots\!26 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β84β4 \beta_{8} - 4\beta_{4} Copy content Toggle raw display
ν3\nu^{3}== β17β16β11β10β9+β7β6β55β3+β22 -\beta_{17} - \beta_{16} - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - 5\beta_{3} + \beta_{2} - 2 Copy content Toggle raw display
ν4\nu^{4}== β172β162β15β14β132β12β11+27 - \beta_{17} - 2 \beta_{16} - 2 \beta_{15} - \beta_{14} - \beta_{13} - 2 \beta_{12} - \beta_{11} + \cdots - 27 Copy content Toggle raw display
ν5\nu^{5}== 2β172β162β1512β146β1310β12+32β1 2 \beta_{17} - 2 \beta_{16} - 2 \beta_{15} - 12 \beta_{14} - 6 \beta_{13} - 10 \beta_{12} + \cdots - 32 \beta_1 Copy content Toggle raw display
ν6\nu^{6}== 29β17+15β16+12β1514β14+26β13+14β12++207 29 \beta_{17} + 15 \beta_{16} + 12 \beta_{15} - 14 \beta_{14} + 26 \beta_{13} + 14 \beta_{12} + \cdots + 207 Copy content Toggle raw display
ν7\nu^{7}== 89β17+122β16+63β15+89β14+89β13+122β12++294 89 \beta_{17} + 122 \beta_{16} + 63 \beta_{15} + 89 \beta_{14} + 89 \beta_{13} + 122 \beta_{12} + \cdots + 294 Copy content Toggle raw display
ν8\nu^{8}== 157β17+157β16+173β15+327β14125β13+170β12++209β1 - 157 \beta_{17} + 157 \beta_{16} + 173 \beta_{15} + 327 \beta_{14} - 125 \beta_{13} + 170 \beta_{12} + \cdots + 209 \beta_1 Copy content Toggle raw display
ν9\nu^{9}== 1193β17789β16343β15+404β14747β13404β12+3143 - 1193 \beta_{17} - 789 \beta_{16} - 343 \beta_{15} + 404 \beta_{14} - 747 \beta_{13} - 404 \beta_{12} + \cdots - 3143 Copy content Toggle raw display
ν10\nu^{10}== 1771β173399β163008β151771β141589β13+14725 - 1771 \beta_{17} - 3399 \beta_{16} - 3008 \beta_{15} - 1771 \beta_{14} - 1589 \beta_{13} + \cdots - 14725 Copy content Toggle raw display
ν11\nu^{11}== 4427β174427β164544β1511550β14+1002β13+14581β1 4427 \beta_{17} - 4427 \beta_{16} - 4544 \beta_{15} - 11550 \beta_{14} + 1002 \beta_{13} + \cdots - 14581 \beta_1 Copy content Toggle raw display
ν12\nu^{12}== 34133β17+17845β16+11667β1516288β14+27955β13++132000 34133 \beta_{17} + 17845 \beta_{16} + 11667 \beta_{15} - 16288 \beta_{14} + 27955 \beta_{13} + \cdots + 132000 Copy content Toggle raw display
ν13\nu^{13}== 65591β17+111597β16+83765β15+65591β14+63794β13++327660 65591 \beta_{17} + 111597 \beta_{16} + 83765 \beta_{15} + 65591 \beta_{14} + 63794 \beta_{13} + \cdots + 327660 Copy content Toggle raw display
ν14\nu^{14}== 160099β17+160099β16+177531β15+337065β14113959β13++274743β1 - 160099 \beta_{17} + 160099 \beta_{16} + 177531 \beta_{15} + 337065 \beta_{14} - 113959 \beta_{13} + \cdots + 274743 \beta_1 Copy content Toggle raw display
ν15\nu^{15}== 1078369β17613705β16354497β15+464664β14819161β13+3260085 - 1078369 \beta_{17} - 613705 \beta_{16} - 354497 \beta_{15} + 464664 \beta_{14} - 819161 \beta_{13} + \cdots - 3260085 Copy content Toggle raw display
ν16\nu^{16}== 1739546β173299254β162823663β151739546β141576975β13+11361824 - 1739546 \beta_{17} - 3299254 \beta_{16} - 2823663 \beta_{15} - 1739546 \beta_{14} - 1576975 \beta_{13} + \cdots - 11361824 Copy content Toggle raw display
ν17\nu^{17}== 4616962β174616962β164884083β1510425525β14+10358511β1 4616962 \beta_{17} - 4616962 \beta_{16} - 4884083 \beta_{15} - 10425525 \beta_{14} + \cdots - 10358511 \beta_1 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/2548Z)×\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times.

nn 197197 885885 12751275
χ(n)\chi(n) β4\beta_{4} 11 11

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
589.1
1.55551 2.69422i
1.19055 2.06209i
0.893365 1.54735i
0.141496 0.245078i
0.118765 0.205706i
−0.302937 + 0.524702i
−0.591009 + 1.02366i
−1.23985 + 2.14749i
−1.26588 + 2.19257i
1.55551 + 2.69422i
1.19055 + 2.06209i
0.893365 + 1.54735i
0.141496 + 0.245078i
0.118765 + 0.205706i
−0.302937 0.524702i
−0.591009 1.02366i
−1.23985 2.14749i
−1.26588 2.19257i
0 −1.55551 + 2.69422i 0 1.19594i 0 0 0 −3.33921 5.78368i 0
589.2 0 −1.19055 + 2.06209i 0 1.91620i 0 0 0 −1.33482 2.31198i 0
589.3 0 −0.893365 + 1.54735i 0 1.41239i 0 0 0 −0.0962010 0.166625i 0
589.4 0 −0.141496 + 0.245078i 0 0.818894i 0 0 0 1.45996 + 2.52872i 0
589.5 0 −0.118765 + 0.205706i 0 2.57768i 0 0 0 1.47179 + 2.54921i 0
589.6 0 0.302937 0.524702i 0 4.02670i 0 0 0 1.31646 + 2.28017i 0
589.7 0 0.591009 1.02366i 0 3.88123i 0 0 0 0.801416 + 1.38809i 0
589.8 0 1.23985 2.14749i 0 0.559885i 0 0 0 −1.57447 2.72707i 0
589.9 0 1.26588 2.19257i 0 1.99909i 0 0 0 −1.70492 2.95301i 0
1765.1 0 −1.55551 2.69422i 0 1.19594i 0 0 0 −3.33921 + 5.78368i 0
1765.2 0 −1.19055 2.06209i 0 1.91620i 0 0 0 −1.33482 + 2.31198i 0
1765.3 0 −0.893365 1.54735i 0 1.41239i 0 0 0 −0.0962010 + 0.166625i 0
1765.4 0 −0.141496 0.245078i 0 0.818894i 0 0 0 1.45996 2.52872i 0
1765.5 0 −0.118765 0.205706i 0 2.57768i 0 0 0 1.47179 2.54921i 0
1765.6 0 0.302937 + 0.524702i 0 4.02670i 0 0 0 1.31646 2.28017i 0
1765.7 0 0.591009 + 1.02366i 0 3.88123i 0 0 0 0.801416 1.38809i 0
1765.8 0 1.23985 + 2.14749i 0 0.559885i 0 0 0 −1.57447 + 2.72707i 0
1765.9 0 1.26588 + 2.19257i 0 1.99909i 0 0 0 −1.70492 + 2.95301i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 589.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2548.2.u.d 18
7.b odd 2 1 2548.2.u.e 18
7.c even 3 1 2548.2.bb.f 18
7.c even 3 1 2548.2.bq.f 18
7.d odd 6 1 364.2.bb.a 18
7.d odd 6 1 364.2.bq.a yes 18
13.e even 6 1 inner 2548.2.u.d 18
21.g even 6 1 3276.2.fe.i 18
21.g even 6 1 3276.2.hi.i 18
91.k even 6 1 2548.2.bq.f 18
91.l odd 6 1 364.2.bq.a yes 18
91.p odd 6 1 364.2.bb.a 18
91.t odd 6 1 2548.2.u.e 18
91.u even 6 1 2548.2.bb.f 18
273.y even 6 1 3276.2.hi.i 18
273.br even 6 1 3276.2.fe.i 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.bb.a 18 7.d odd 6 1
364.2.bb.a 18 91.p odd 6 1
364.2.bq.a yes 18 7.d odd 6 1
364.2.bq.a yes 18 91.l odd 6 1
2548.2.u.d 18 1.a even 1 1 trivial
2548.2.u.d 18 13.e even 6 1 inner
2548.2.u.e 18 7.b odd 2 1
2548.2.u.e 18 91.t odd 6 1
2548.2.bb.f 18 7.c even 3 1
2548.2.bb.f 18 91.u even 6 1
2548.2.bq.f 18 7.c even 3 1
2548.2.bq.f 18 91.k even 6 1
3276.2.fe.i 18 21.g even 6 1
3276.2.fe.i 18 273.br even 6 1
3276.2.hi.i 18 21.g even 6 1
3276.2.hi.i 18 273.y even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T318+T317+17T316+6T315+188T314+49T313+1116T312++16 T_{3}^{18} + T_{3}^{17} + 17 T_{3}^{16} + 6 T_{3}^{15} + 188 T_{3}^{14} + 49 T_{3}^{13} + 1116 T_{3}^{12} + \cdots + 16 acting on S2new(2548,[χ])S_{2}^{\mathrm{new}}(2548, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T18 T^{18} Copy content Toggle raw display
33 T18+T17++16 T^{18} + T^{17} + \cdots + 16 Copy content Toggle raw display
55 T18+50T16++14283 T^{18} + 50 T^{16} + \cdots + 14283 Copy content Toggle raw display
77 T18 T^{18} Copy content Toggle raw display
1111 T186T17++1581228 T^{18} - 6 T^{17} + \cdots + 1581228 Copy content Toggle raw display
1313 T18++10604499373 T^{18} + \cdots + 10604499373 Copy content Toggle raw display
1717 T18++14846935104 T^{18} + \cdots + 14846935104 Copy content Toggle raw display
1919 T1821T17++2988012 T^{18} - 21 T^{17} + \cdots + 2988012 Copy content Toggle raw display
2323 T18++2496801024 T^{18} + \cdots + 2496801024 Copy content Toggle raw display
2929 T182T17++4100625 T^{18} - 2 T^{17} + \cdots + 4100625 Copy content Toggle raw display
3131 T18++10703377083 T^{18} + \cdots + 10703377083 Copy content Toggle raw display
3737 T18++1104538032 T^{18} + \cdots + 1104538032 Copy content Toggle raw display
4141 T18++3793327443 T^{18} + \cdots + 3793327443 Copy content Toggle raw display
4343 T18++1883413140625 T^{18} + \cdots + 1883413140625 Copy content Toggle raw display
4747 T18++14333519021787 T^{18} + \cdots + 14333519021787 Copy content Toggle raw display
5353 (T913T8+1053081)2 (T^{9} - 13 T^{8} + \cdots - 1053081)^{2} Copy content Toggle raw display
5959 T18++501089328 T^{18} + \cdots + 501089328 Copy content Toggle raw display
6161 T18++1673300836 T^{18} + \cdots + 1673300836 Copy content Toggle raw display
6767 T18++26266034700 T^{18} + \cdots + 26266034700 Copy content Toggle raw display
7171 T18++44122876875 T^{18} + \cdots + 44122876875 Copy content Toggle raw display
7373 T18++3223781883 T^{18} + \cdots + 3223781883 Copy content Toggle raw display
7979 (T914T8++2307735)2 (T^{9} - 14 T^{8} + \cdots + 2307735)^{2} Copy content Toggle raw display
8383 T18++93 ⁣ ⁣00 T^{18} + \cdots + 93\!\cdots\!00 Copy content Toggle raw display
8989 T18++465414910128 T^{18} + \cdots + 465414910128 Copy content Toggle raw display
9797 T18++938674322067 T^{18} + \cdots + 938674322067 Copy content Toggle raw display
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