Properties

Label 2548.2.l.n
Level 25482548
Weight 22
Character orbit 2548.l
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(373,2548)]
 
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, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2548.373");
 
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.l (of order 33, degree 22, not 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(ζ3)\Q(\zeta_{3})
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+19x1616x15+244x14181x13+1600x12607x11++1296 x^{18} - x^{17} + 19 x^{16} - 16 x^{15} + 244 x^{14} - 181 x^{13} + 1600 x^{12} - 607 x^{11} + \cdots + 1296 Copy content Toggle raw display
Coefficient ring: Z[a1,,a25]\Z[a_1, \ldots, a_{25}]
Coefficient ring index: 34 3^{4}
Twist minimal: no (minimal twist has level 364)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

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) == qβ2q3+β9q5+(β3+1)q9+β6q11β12q13+(β17β14β13+2)q15+(β14β13+β6)q17++(β17+β15+β2)q99+O(q100) q - \beta_{2} q^{3} + \beta_{9} q^{5} + (\beta_{3} + 1) q^{9} + \beta_{6} q^{11} - \beta_{12} q^{13} + (\beta_{17} - \beta_{14} - \beta_{13} + \cdots - 2) q^{15} + ( - \beta_{14} - \beta_{13} + \cdots - \beta_{6}) q^{17}+ \cdots + ( - \beta_{17} + \beta_{15} + \cdots - \beta_{2}) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 18q2q3+20q94q114q1310q15+6q172q19+6q235q25+4q27+2q297q3128q338q37+17q397q412q43+10q45+8q99+O(q100) 18 q - 2 q^{3} + 20 q^{9} - 4 q^{11} - 4 q^{13} - 10 q^{15} + 6 q^{17} - 2 q^{19} + 6 q^{23} - 5 q^{25} + 4 q^{27} + 2 q^{29} - 7 q^{31} - 28 q^{33} - 8 q^{37} + 17 q^{39} - 7 q^{41} - 2 q^{43} + 10 q^{45}+ \cdots - 8 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x18x17+19x1616x15+244x14181x13+1600x12607x11++1296 x^{18} - x^{17} + 19 x^{16} - 16 x^{15} + 244 x^{14} - 181 x^{13} + 1600 x^{12} - 607 x^{11} + \cdots + 1296 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (22 ⁣ ⁣28ν17+92 ⁣ ⁣92)/34 ⁣ ⁣13 ( 22\!\cdots\!28 \nu^{17} + \cdots - 92\!\cdots\!92 ) / 34\!\cdots\!13 Copy content Toggle raw display
β3\beta_{3}== (18 ⁣ ⁣85ν17+13 ⁣ ⁣40)/34 ⁣ ⁣13 ( 18\!\cdots\!85 \nu^{17} + \cdots - 13\!\cdots\!40 ) / 34\!\cdots\!13 Copy content Toggle raw display
β4\beta_{4}== (70 ⁣ ⁣47ν17++48 ⁣ ⁣71)/52 ⁣ ⁣95 ( - 70\!\cdots\!47 \nu^{17} + \cdots + 48\!\cdots\!71 ) / 52\!\cdots\!95 Copy content Toggle raw display
β5\beta_{5}== (28 ⁣ ⁣33ν17++16 ⁣ ⁣76)/13 ⁣ ⁣52 ( 28\!\cdots\!33 \nu^{17} + \cdots + 16\!\cdots\!76 ) / 13\!\cdots\!52 Copy content Toggle raw display
β6\beta_{6}== (11 ⁣ ⁣57ν17++42 ⁣ ⁣16)/52 ⁣ ⁣95 ( - 11\!\cdots\!57 \nu^{17} + \cdots + 42\!\cdots\!16 ) / 52\!\cdots\!95 Copy content Toggle raw display
β7\beta_{7}== (39 ⁣ ⁣91ν17++48 ⁣ ⁣68)/52 ⁣ ⁣95 ( - 39\!\cdots\!91 \nu^{17} + \cdots + 48\!\cdots\!68 ) / 52\!\cdots\!95 Copy content Toggle raw display
β8\beta_{8}== (28 ⁣ ⁣33ν17+16 ⁣ ⁣76)/34 ⁣ ⁣13 ( - 28\!\cdots\!33 \nu^{17} + \cdots - 16\!\cdots\!76 ) / 34\!\cdots\!13 Copy content Toggle raw display
β9\beta_{9}== (19 ⁣ ⁣35ν17+57 ⁣ ⁣44)/20 ⁣ ⁣80 ( 19\!\cdots\!35 \nu^{17} + \cdots - 57\!\cdots\!44 ) / 20\!\cdots\!80 Copy content Toggle raw display
β10\beta_{10}== (11 ⁣ ⁣64ν17++15 ⁣ ⁣28)/10 ⁣ ⁣39 ( - 11\!\cdots\!64 \nu^{17} + \cdots + 15\!\cdots\!28 ) / 10\!\cdots\!39 Copy content Toggle raw display
β11\beta_{11}== (23 ⁣ ⁣02ν17++22 ⁣ ⁣43)/17 ⁣ ⁣65 ( 23\!\cdots\!02 \nu^{17} + \cdots + 22\!\cdots\!43 ) / 17\!\cdots\!65 Copy content Toggle raw display
β12\beta_{12}== (30 ⁣ ⁣71ν17+13 ⁣ ⁣59)/17 ⁣ ⁣65 ( 30\!\cdots\!71 \nu^{17} + \cdots - 13\!\cdots\!59 ) / 17\!\cdots\!65 Copy content Toggle raw display
β13\beta_{13}== (20 ⁣ ⁣63ν17+47 ⁣ ⁣40)/52 ⁣ ⁣95 ( - 20\!\cdots\!63 \nu^{17} + \cdots - 47\!\cdots\!40 ) / 52\!\cdots\!95 Copy content Toggle raw display
β14\beta_{14}== (22 ⁣ ⁣90ν17++44 ⁣ ⁣92)/52 ⁣ ⁣95 ( 22\!\cdots\!90 \nu^{17} + \cdots + 44\!\cdots\!92 ) / 52\!\cdots\!95 Copy content Toggle raw display
β15\beta_{15}== (33 ⁣ ⁣51ν17++70 ⁣ ⁣16)/69 ⁣ ⁣60 ( 33\!\cdots\!51 \nu^{17} + \cdots + 70\!\cdots\!16 ) / 69\!\cdots\!60 Copy content Toggle raw display
β16\beta_{16}== (52 ⁣ ⁣39ν17++16 ⁣ ⁣56)/10 ⁣ ⁣39 ( 52\!\cdots\!39 \nu^{17} + \cdots + 16\!\cdots\!56 ) / 10\!\cdots\!39 Copy content Toggle raw display
β17\beta_{17}== (36 ⁣ ⁣27ν17++36 ⁣ ⁣08)/69 ⁣ ⁣60 ( 36\!\cdots\!27 \nu^{17} + \cdots + 36\!\cdots\!08 ) / 69\!\cdots\!60 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β84β5 -\beta_{8} - 4\beta_{5} Copy content Toggle raw display
ν3\nu^{3}== β17β15β10+2β76β21 \beta_{17} - \beta_{15} - \beta_{10} + 2\beta_{7} - 6\beta_{2} - 1 Copy content Toggle raw display
ν4\nu^{4}== β17+β15+β14+2β13β112β9+10β8+28 \beta_{17} + \beta_{15} + \beta_{14} + 2 \beta_{13} - \beta_{11} - 2 \beta_{9} + 10 \beta_{8} + \cdots - 28 Copy content Toggle raw display
ν5\nu^{5}== 2β17+12β16+11β1525β145β1311β12+43β1 - 2 \beta_{17} + 12 \beta_{16} + 11 \beta_{15} - 25 \beta_{14} - 5 \beta_{13} - 11 \beta_{12} + \cdots - 43 \beta_1 Copy content Toggle raw display
ν6\nu^{6}== 14β12+14β116β10+21β727β630β4++230 - 14 \beta_{12} + 14 \beta_{11} - 6 \beta_{10} + 21 \beta_{7} - 27 \beta_{6} - 30 \beta_{4} + \cdots + 230 Copy content Toggle raw display
ν7\nu^{7}== 106β17123β16+35β15+263β14+85β13+141β12++16 - 106 \beta_{17} - 123 \beta_{16} + 35 \beta_{15} + 263 \beta_{14} + 85 \beta_{13} + 141 \beta_{12} + \cdots + 16 Copy content Toggle raw display
ν8\nu^{8}== 157β17+114β16146β15302β14298β13+146β12+254β1 - 157 \beta_{17} + 114 \beta_{16} - 146 \beta_{15} - 302 \beta_{14} - 298 \beta_{13} + 146 \beta_{12} + \cdots - 254 \beta_1 Copy content Toggle raw display
ν9\nu^{9}== 1449β171449β15448β12+448β111221β10+2661β7++277 1449 \beta_{17} - 1449 \beta_{15} - 448 \beta_{12} + 448 \beta_{11} - 1221 \beta_{10} + 2661 \beta_{7} + \cdots + 277 Copy content Toggle raw display
ν10\nu^{10}== 1366β171557β16+1660β15+3757β14+3143β13+294β12+18622 1366 \beta_{17} - 1557 \beta_{16} + 1660 \beta_{15} + 3757 \beta_{14} + 3143 \beta_{13} + 294 \beta_{12} + \cdots - 18622 Copy content Toggle raw display
ν11\nu^{11}== 5123β17+12081β16+9425β1526677β1412044β13+27139β1 - 5123 \beta_{17} + 12081 \beta_{16} + 9425 \beta_{15} - 26677 \beta_{14} - 12044 \beta_{13} + \cdots - 27139 \beta_1 Copy content Toggle raw display
ν12\nu^{12}== 5202β175202β1517252β12+17252β1118831β10++175508 5202 \beta_{17} - 5202 \beta_{15} - 17252 \beta_{12} + 17252 \beta_{11} - 18831 \beta_{10} + \cdots + 175508 Copy content Toggle raw display
ν13\nu^{13}== 88768β17120024β16+55706β15+267329β14+129439β13+108695 - 88768 \beta_{17} - 120024 \beta_{16} + 55706 \beta_{15} + 267329 \beta_{14} + 129439 \beta_{13} + \cdots - 108695 Copy content Toggle raw display
ν14\nu^{14}== 178561β17+215088β16101756β15489242β14342616β13+437948β1 - 178561 \beta_{17} + 215088 \beta_{16} - 101756 \beta_{15} - 489242 \beta_{14} - 342616 \beta_{13} + \cdots - 437948 \beta_1 Copy content Toggle raw display
ν15\nu^{15}== 1428192β171428192β15590998β12+590998β111199103β10++1448905 1428192 \beta_{17} - 1428192 \beta_{15} - 590998 \beta_{12} + 590998 \beta_{11} - 1199103 \beta_{10} + \cdots + 1448905 Copy content Toggle raw display
ν16\nu^{16}== 823504β172382291β16+1848118β15+5368078β14+3581528β13+16360906 823504 \beta_{17} - 2382291 \beta_{16} + 1848118 \beta_{15} + 5368078 \beta_{14} + 3581528 \beta_{13} + \cdots - 16360906 Copy content Toggle raw display
ν17\nu^{17}== 6191582β17+12043875β16+7913018β1527060034β14+24751717β1 - 6191582 \beta_{17} + 12043875 \beta_{16} + 7913018 \beta_{15} - 27060034 \beta_{14} + \cdots - 24751717 \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) β5-\beta_{5} 1+β5-1 + \beta_{5} 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}
373.1
1.49468 2.58887i
1.26295 2.18749i
0.789914 1.36817i
0.734808 1.27273i
−0.210627 + 0.364817i
−0.326032 + 0.564704i
−0.520270 + 0.901135i
−1.11606 + 1.93307i
−1.60936 + 2.78750i
1.49468 + 2.58887i
1.26295 + 2.18749i
0.789914 + 1.36817i
0.734808 + 1.27273i
−0.210627 0.364817i
−0.326032 0.564704i
−0.520270 0.901135i
−1.11606 1.93307i
−1.60936 2.78750i
0 −2.98937 0 2.11635 3.66562i 0 0 0 5.93631 0
373.2 0 −2.52589 0 −1.33439 + 2.31123i 0 0 0 3.38014 0
373.3 0 −1.57983 0 −0.985205 + 1.70642i 0 0 0 −0.504145 0
373.4 0 −1.46962 0 0.879363 1.52310i 0 0 0 −0.840227 0
373.5 0 0.421254 0 0.466726 0.808393i 0 0 0 −2.82254 0
373.6 0 0.652064 0 1.14588 1.98472i 0 0 0 −2.57481 0
373.7 0 1.04054 0 −1.59004 + 2.75404i 0 0 0 −1.91727 0
373.8 0 2.23212 0 −0.660291 + 1.14366i 0 0 0 1.98235 0
373.9 0 3.21873 0 −0.0383870 + 0.0664882i 0 0 0 7.36021 0
1537.1 0 −2.98937 0 2.11635 + 3.66562i 0 0 0 5.93631 0
1537.2 0 −2.52589 0 −1.33439 2.31123i 0 0 0 3.38014 0
1537.3 0 −1.57983 0 −0.985205 1.70642i 0 0 0 −0.504145 0
1537.4 0 −1.46962 0 0.879363 + 1.52310i 0 0 0 −0.840227 0
1537.5 0 0.421254 0 0.466726 + 0.808393i 0 0 0 −2.82254 0
1537.6 0 0.652064 0 1.14588 + 1.98472i 0 0 0 −2.57481 0
1537.7 0 1.04054 0 −1.59004 2.75404i 0 0 0 −1.91727 0
1537.8 0 2.23212 0 −0.660291 1.14366i 0 0 0 1.98235 0
1537.9 0 3.21873 0 −0.0383870 0.0664882i 0 0 0 7.36021 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 373.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2548.2.l.n 18
7.b odd 2 1 364.2.l.a yes 18
7.c even 3 1 2548.2.i.n 18
7.c even 3 1 2548.2.k.i 18
7.d odd 6 1 364.2.i.a 18
7.d odd 6 1 2548.2.k.h 18
13.c even 3 1 2548.2.i.n 18
21.c even 2 1 3276.2.x.k 18
21.g even 6 1 3276.2.u.k 18
91.g even 3 1 inner 2548.2.l.n 18
91.h even 3 1 2548.2.k.i 18
91.m odd 6 1 364.2.l.a yes 18
91.n odd 6 1 364.2.i.a 18
91.v odd 6 1 2548.2.k.h 18
273.bf even 6 1 3276.2.x.k 18
273.bn even 6 1 3276.2.u.k 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.i.a 18 7.d odd 6 1
364.2.i.a 18 91.n odd 6 1
364.2.l.a yes 18 7.b odd 2 1
364.2.l.a yes 18 91.m odd 6 1
2548.2.i.n 18 7.c even 3 1
2548.2.i.n 18 13.c even 3 1
2548.2.k.h 18 7.d odd 6 1
2548.2.k.h 18 91.v odd 6 1
2548.2.k.i 18 7.c even 3 1
2548.2.k.i 18 91.h even 3 1
2548.2.l.n 18 1.a even 1 1 trivial
2548.2.l.n 18 91.g even 3 1 inner
3276.2.u.k 18 21.g even 6 1
3276.2.u.k 18 273.bn even 6 1
3276.2.x.k 18 21.c even 2 1
3276.2.x.k 18 273.bf even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(2548,[χ])S_{2}^{\mathrm{new}}(2548, [\chi]):

T39+T3818T3717T36+97T35+69T34183T3345T32+129T336 T_{3}^{9} + T_{3}^{8} - 18T_{3}^{7} - 17T_{3}^{6} + 97T_{3}^{5} + 69T_{3}^{4} - 183T_{3}^{3} - 45T_{3}^{2} + 129T_{3} - 36 Copy content Toggle raw display
T518+25T516+22T515+445T514+361T513+3697T512++729 T_{5}^{18} + 25 T_{5}^{16} + 22 T_{5}^{15} + 445 T_{5}^{14} + 361 T_{5}^{13} + 3697 T_{5}^{12} + \cdots + 729 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T18 T^{18} Copy content Toggle raw display
33 (T9+T818T7+36)2 (T^{9} + T^{8} - 18 T^{7} + \cdots - 36)^{2} Copy content Toggle raw display
55 T18+25T16++729 T^{18} + 25 T^{16} + \cdots + 729 Copy content Toggle raw display
77 T18 T^{18} Copy content Toggle raw display
1111 (T9+2T8+468)2 (T^{9} + 2 T^{8} + \cdots - 468)^{2} Copy content Toggle raw display
1313 T18++10604499373 T^{18} + \cdots + 10604499373 Copy content Toggle raw display
1717 T18++38805848064 T^{18} + \cdots + 38805848064 Copy content Toggle raw display
1919 (T9+T877T7++1896)2 (T^{9} + T^{8} - 77 T^{7} + \cdots + 1896)^{2} Copy content Toggle raw display
2323 T18++141580617984 T^{18} + \cdots + 141580617984 Copy content Toggle raw display
2929 T182T17++927369 T^{18} - 2 T^{17} + \cdots + 927369 Copy content Toggle raw display
3131 T18+7T17++2064969 T^{18} + 7 T^{17} + \cdots + 2064969 Copy content Toggle raw display
3737 T18++287364612096 T^{18} + \cdots + 287364612096 Copy content Toggle raw display
4141 T18++291104490681 T^{18} + \cdots + 291104490681 Copy content Toggle raw display
4343 T18++256754010681 T^{18} + \cdots + 256754010681 Copy content Toggle raw display
4747 T18++215391594609 T^{18} + \cdots + 215391594609 Copy content Toggle raw display
5353 T18++1455498801 T^{18} + \cdots + 1455498801 Copy content Toggle raw display
5959 T18++6347497291776 T^{18} + \cdots + 6347497291776 Copy content Toggle raw display
6161 (T9+T8++10140682)2 (T^{9} + T^{8} + \cdots + 10140682)^{2} Copy content Toggle raw display
6767 (T9+29T8++7838036)2 (T^{9} + 29 T^{8} + \cdots + 7838036)^{2} Copy content Toggle raw display
7171 T18++3994485901641 T^{18} + \cdots + 3994485901641 Copy content Toggle raw display
7373 T18++123708070942569 T^{18} + \cdots + 123708070942569 Copy content Toggle raw display
7979 T18++55833279675889 T^{18} + \cdots + 55833279675889 Copy content Toggle raw display
8383 (T915T8+874800)2 (T^{9} - 15 T^{8} + \cdots - 874800)^{2} Copy content Toggle raw display
8989 T18++1557623810304 T^{18} + \cdots + 1557623810304 Copy content Toggle raw display
9797 T18++54 ⁣ ⁣09 T^{18} + \cdots + 54\!\cdots\!09 Copy content Toggle raw display
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