Properties

Label 2548.2.i.m
Level 25482548
Weight 22
Character orbit 2548.i
Analytic conductor 20.34620.346
Analytic rank 00
Dimension 1212
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2548,2,Mod(165,2548)]
 
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, 4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2548.165");
 
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.i (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: 1212
Relative dimension: 66 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q[x]/(x12+)\mathbb{Q}[x]/(x^{12} + \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x12+9x10+66x8+127x6+189x4+60x2+16 x^{12} + 9x^{10} + 66x^{8} + 127x^{6} + 189x^{4} + 60x^{2} + 16 Copy content Toggle raw display
Coefficient ring: Z[a1,,a9]\Z[a_1, \ldots, a_{9}]
Coefficient ring index: 32 3^{2}
Twist minimal: yes
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,,β111,\beta_1,\ldots,\beta_{11} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q3+β7q5+β9q9+(β10+β41)q11+(β11β6+β5)q13+(β9+β4)q15+(2β11β5β3)q17++(2β8+β23)q99+O(q100) q + \beta_1 q^{3} + \beta_{7} q^{5} + \beta_{9} q^{9} + ( - \beta_{10} + \beta_{4} - 1) q^{11} + (\beta_{11} - \beta_{6} + \beta_{5}) q^{13} + ( - \beta_{9} + \beta_{4}) q^{15} + ( - 2 \beta_{11} - \beta_{5} - \beta_{3}) q^{17}+ \cdots + ( - 2 \beta_{8} + \beta_{2} - 3) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 12q6q11+6q15+6q25+12q2912q376q396q436q51+6q53+36q5730q65+12q6712q7112q79+6q81+36q85+12q9312q95+36q99+O(q100) 12 q - 6 q^{11} + 6 q^{15} + 6 q^{25} + 12 q^{29} - 12 q^{37} - 6 q^{39} - 6 q^{43} - 6 q^{51} + 6 q^{53} + 36 q^{57} - 30 q^{65} + 12 q^{67} - 12 q^{71} - 12 q^{79} + 6 q^{81} + 36 q^{85} + 12 q^{93} - 12 q^{95}+ \cdots - 36 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x12+9x10+66x8+127x6+189x4+60x2+16 x^{12} + 9x^{10} + 66x^{8} + 127x^{6} + 189x^{4} + 60x^{2} + 16 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (9ν10+66ν8+484ν6+189ν4+60ν23538)/1326 ( 9\nu^{10} + 66\nu^{8} + 484\nu^{6} + 189\nu^{4} + 60\nu^{2} - 3538 ) / 1326 Copy content Toggle raw display
β3\beta_{3}== (3ν11+39ν9+286ν7+981ν5+819ν3+260ν)/408 ( 3\nu^{11} + 39\nu^{9} + 286\nu^{7} + 981\nu^{5} + 819\nu^{3} + 260\nu ) / 408 Copy content Toggle raw display
β4\beta_{4}== (55ν10477ν83498ν66017ν410017ν2528)/2652 ( -55\nu^{10} - 477\nu^{8} - 3498\nu^{6} - 6017\nu^{4} - 10017\nu^{2} - 528 ) / 2652 Copy content Toggle raw display
β5\beta_{5}== (24ν11+176ν9+1217ν7+504ν5+160ν3521ν)/1326 ( 24\nu^{11} + 176\nu^{9} + 1217\nu^{7} + 504\nu^{5} + 160\nu^{3} - 521\nu ) / 1326 Copy content Toggle raw display
β6\beta_{6}== (55ν11477ν93498ν76017ν510017ν3528ν)/2652 ( -55\nu^{11} - 477\nu^{9} - 3498\nu^{7} - 6017\nu^{5} - 10017\nu^{3} - 528\nu ) / 2652 Copy content Toggle raw display
β7\beta_{7}== (33ν11242ν91701ν7693ν5220ν3+6711ν)/1326 ( -33\nu^{11} - 242\nu^{9} - 1701\nu^{7} - 693\nu^{5} - 220\nu^{3} + 6711\nu ) / 1326 Copy content Toggle raw display
β8\beta_{8}== (57ν10418ν82918ν61197ν4380ν2+5906)/1326 ( -57\nu^{10} - 418\nu^{8} - 2918\nu^{6} - 1197\nu^{4} - 380\nu^{2} + 5906 ) / 1326 Copy content Toggle raw display
β9\beta_{9}== (55ν10477ν83498ν66017ν49133ν2528)/884 ( -55\nu^{10} - 477\nu^{8} - 3498\nu^{6} - 6017\nu^{4} - 9133\nu^{2} - 528 ) / 884 Copy content Toggle raw display
β10\beta_{10}== (241ν10+2283ν8+16742ν6+36443ν4+47943ν2+15220)/2652 ( 241\nu^{10} + 2283\nu^{8} + 16742\nu^{6} + 36443\nu^{4} + 47943\nu^{2} + 15220 ) / 2652 Copy content Toggle raw display
β11\beta_{11}== (685ν11+5981ν9+43566ν7+74939ν5+106153ν3+6576ν)/5304 ( 685\nu^{11} + 5981\nu^{9} + 43566\nu^{7} + 74939\nu^{5} + 106153\nu^{3} + 6576\nu ) / 5304 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β93β4 \beta_{9} - 3\beta_{4} Copy content Toggle raw display
ν3\nu^{3}== β115β6+β5+β3 -\beta_{11} - 5\beta_{6} + \beta_{5} + \beta_{3} Copy content Toggle raw display
ν4\nu^{4}== β108β9+17β48β217 -\beta_{10} - 8\beta_{9} + 17\beta_{4} - 8\beta_{2} - 17 Copy content Toggle raw display
ν5\nu^{5}== 7β11+7β7+32β69β332β1 7\beta_{11} + 7\beta_{7} + 32\beta_{6} - 9\beta_{3} - 32\beta_1 Copy content Toggle raw display
ν6\nu^{6}== 9β8+57β2+112 9\beta_{8} + 57\beta_{2} + 112 Copy content Toggle raw display
ν7\nu^{7}== 48β766β5+217β1 -48\beta_{7} - 66\beta_{5} + 217\beta_1 Copy content Toggle raw display
ν8\nu^{8}== 66β10+397β966β8765β4 66\beta_{10} + 397\beta_{9} - 66\beta_{8} - 765\beta_{4} Copy content Toggle raw display
ν9\nu^{9}== 331β111493β6+463β5+463β3 -331\beta_{11} - 1493\beta_{6} + 463\beta_{5} + 463\beta_{3} Copy content Toggle raw display
ν10\nu^{10}== 463β102750β9+5273β42750β25273 -463\beta_{10} - 2750\beta_{9} + 5273\beta_{4} - 2750\beta_{2} - 5273 Copy content Toggle raw display
ν11\nu^{11}== 2287β11+2287β7+10310β63213β310310β1 2287\beta_{11} + 2287\beta_{7} + 10310\beta_{6} - 3213\beta_{3} - 10310\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) 1+β4-1 + \beta_{4} 1+β4-1 + \beta_{4} 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}
165.1
−1.31475 2.27721i
−0.662644 1.14773i
−0.286958 0.497025i
0.286958 + 0.497025i
0.662644 + 1.14773i
1.31475 + 2.27721i
−1.31475 + 2.27721i
−0.662644 + 1.14773i
−0.286958 + 0.497025i
0.286958 0.497025i
0.662644 1.14773i
1.31475 2.27721i
0 −1.31475 2.27721i 0 0.934445 + 1.61851i 0 0 0 −1.95712 + 3.38982i 0
165.2 0 −0.662644 1.14773i 0 −0.0919085 0.159190i 0 0 0 0.621805 1.07700i 0
165.3 0 −0.286958 0.497025i 0 −1.45546 2.52093i 0 0 0 1.33531 2.31283i 0
165.4 0 0.286958 + 0.497025i 0 1.45546 + 2.52093i 0 0 0 1.33531 2.31283i 0
165.5 0 0.662644 + 1.14773i 0 0.0919085 + 0.159190i 0 0 0 0.621805 1.07700i 0
165.6 0 1.31475 + 2.27721i 0 −0.934445 1.61851i 0 0 0 −1.95712 + 3.38982i 0
1745.1 0 −1.31475 + 2.27721i 0 0.934445 1.61851i 0 0 0 −1.95712 3.38982i 0
1745.2 0 −0.662644 + 1.14773i 0 −0.0919085 + 0.159190i 0 0 0 0.621805 + 1.07700i 0
1745.3 0 −0.286958 + 0.497025i 0 −1.45546 + 2.52093i 0 0 0 1.33531 + 2.31283i 0
1745.4 0 0.286958 0.497025i 0 1.45546 2.52093i 0 0 0 1.33531 + 2.31283i 0
1745.5 0 0.662644 1.14773i 0 0.0919085 0.159190i 0 0 0 0.621805 + 1.07700i 0
1745.6 0 1.31475 2.27721i 0 −0.934445 + 1.61851i 0 0 0 −1.95712 3.38982i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 165.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
91.h even 3 1 inner
91.v odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2548.2.i.m 12
7.b odd 2 1 inner 2548.2.i.m 12
7.c even 3 1 2548.2.k.g 12
7.c even 3 1 2548.2.l.m 12
7.d odd 6 1 2548.2.k.g 12
7.d odd 6 1 2548.2.l.m 12
13.c even 3 1 2548.2.l.m 12
91.g even 3 1 2548.2.k.g 12
91.h even 3 1 inner 2548.2.i.m 12
91.m odd 6 1 2548.2.k.g 12
91.n odd 6 1 2548.2.l.m 12
91.v odd 6 1 inner 2548.2.i.m 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2548.2.i.m 12 1.a even 1 1 trivial
2548.2.i.m 12 7.b odd 2 1 inner
2548.2.i.m 12 91.h even 3 1 inner
2548.2.i.m 12 91.v odd 6 1 inner
2548.2.k.g 12 7.c even 3 1
2548.2.k.g 12 7.d odd 6 1
2548.2.k.g 12 91.g even 3 1
2548.2.k.g 12 91.m odd 6 1
2548.2.l.m 12 7.c even 3 1
2548.2.l.m 12 7.d odd 6 1
2548.2.l.m 12 13.c even 3 1
2548.2.l.m 12 91.n odd 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]):

T312+9T310+66T38+127T36+189T34+60T32+16 T_{3}^{12} + 9T_{3}^{10} + 66T_{3}^{8} + 127T_{3}^{6} + 189T_{3}^{4} + 60T_{3}^{2} + 16 Copy content Toggle raw display
T512+12T510+114T58+358T56+888T54+30T52+1 T_{5}^{12} + 12T_{5}^{10} + 114T_{5}^{8} + 358T_{5}^{6} + 888T_{5}^{4} + 30T_{5}^{2} + 1 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T12 T^{12} Copy content Toggle raw display
33 T12+9T10++16 T^{12} + 9 T^{10} + \cdots + 16 Copy content Toggle raw display
55 T12+12T10++1 T^{12} + 12 T^{10} + \cdots + 1 Copy content Toggle raw display
77 T12 T^{12} Copy content Toggle raw display
1111 (T6+3T5+24T4++64)2 (T^{6} + 3 T^{5} + 24 T^{4} + \cdots + 64)^{2} Copy content Toggle raw display
1313 T12+3T10++4826809 T^{12} + 3 T^{10} + \cdots + 4826809 Copy content Toggle raw display
1717 (T651T4+169)2 (T^{6} - 51 T^{4} + \cdots - 169)^{2} Copy content Toggle raw display
1919 T12+9T10++16 T^{12} + 9 T^{10} + \cdots + 16 Copy content Toggle raw display
2323 (T348T+104)4 (T^{3} - 48 T + 104)^{4} Copy content Toggle raw display
2929 (T66T5++84681)2 (T^{6} - 6 T^{5} + \cdots + 84681)^{2} Copy content Toggle raw display
3131 T12++12444741136 T^{12} + \cdots + 12444741136 Copy content Toggle raw display
3737 (T3+3T218T52)4 (T^{3} + 3 T^{2} - 18 T - 52)^{4} Copy content Toggle raw display
4141 T12++136048896 T^{12} + \cdots + 136048896 Copy content Toggle raw display
4343 (T6+3T5++53824)2 (T^{6} + 3 T^{5} + \cdots + 53824)^{2} Copy content Toggle raw display
4747 T12+36T10++104976 T^{12} + 36 T^{10} + \cdots + 104976 Copy content Toggle raw display
5353 (T63T5++22801)2 (T^{6} - 3 T^{5} + \cdots + 22801)^{2} Copy content Toggle raw display
5959 (T6240T4+215296)2 (T^{6} - 240 T^{4} + \cdots - 215296)^{2} Copy content Toggle raw display
6161 T12++479785216 T^{12} + \cdots + 479785216 Copy content Toggle raw display
6767 (T66T5++68644)2 (T^{6} - 6 T^{5} + \cdots + 68644)^{2} Copy content Toggle raw display
7171 (T6+6T5++984064)2 (T^{6} + 6 T^{5} + \cdots + 984064)^{2} Copy content Toggle raw display
7373 T12++723394816 T^{12} + \cdots + 723394816 Copy content Toggle raw display
7979 (T6+6T5++256)2 (T^{6} + 6 T^{5} + \cdots + 256)^{2} Copy content Toggle raw display
8383 (T6480T4+2598544)2 (T^{6} - 480 T^{4} + \cdots - 2598544)^{2} Copy content Toggle raw display
8989 (T6369T4+24336)2 (T^{6} - 369 T^{4} + \cdots - 24336)^{2} Copy content Toggle raw display
9797 T12++723394816 T^{12} + \cdots + 723394816 Copy content Toggle raw display
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