Properties

Label 1444.2.e.h
Level 14441444
Weight 22
Character orbit 1444.e
Analytic conductor 11.53011.530
Analytic rank 00
Dimension 1212
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1444,2,Mod(429,1444)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1444, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1444.429");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1444=22192 1444 = 2^{2} \cdot 19^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1444.e (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: 11.530398051911.5303980519
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: x126x113x10+70x915x8426x7+64x6+1659x5+267x4++4161 x^{12} - 6 x^{11} - 3 x^{10} + 70 x^{9} - 15 x^{8} - 426 x^{7} + 64 x^{6} + 1659 x^{5} + 267 x^{4} + \cdots + 4161 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 32 3^{2}
Twist minimal: no (minimal twist has level 76)
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+(β11β5)q3+β9q5+(β11β8β5+1)q7+(2β11β10+2)q9+(β11β8β7+1)q11++(8β11+β10+7β8++8)q99+O(q100) q + ( - \beta_{11} - \beta_{5}) q^{3} + \beta_{9} q^{5} + ( - \beta_{11} - \beta_{8} - \beta_{5} + \cdots - 1) q^{7} + ( - 2 \beta_{11} - \beta_{10} + \cdots - 2) q^{9} + ( - \beta_{11} - \beta_{8} - \beta_{7} + \cdots - 1) q^{11}+ \cdots + (8 \beta_{11} + \beta_{10} + 7 \beta_{8} + \cdots + 8) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 12q+3q3+3q56q79q96q11+12q13+6q17+21q219q2518q27+21q29+12q31+9q333q35+12q37+60q39+36q41+18q43++33q99+O(q100) 12 q + 3 q^{3} + 3 q^{5} - 6 q^{7} - 9 q^{9} - 6 q^{11} + 12 q^{13} + 6 q^{17} + 21 q^{21} - 9 q^{25} - 18 q^{27} + 21 q^{29} + 12 q^{31} + 9 q^{33} - 3 q^{35} + 12 q^{37} + 60 q^{39} + 36 q^{41} + 18 q^{43}+ \cdots + 33 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x126x113x10+70x915x8426x7+64x6+1659x5+267x4++4161 x^{12} - 6 x^{11} - 3 x^{10} + 70 x^{9} - 15 x^{8} - 426 x^{7} + 64 x^{6} + 1659 x^{5} + 267 x^{4} + \cdots + 4161 : Copy content Toggle raw display

β1\beta_{1}== (ν10+5ν9+9ν866ν7+49ν6+105ν5553ν4+5578)/1853 ( - \nu^{10} + 5 \nu^{9} + 9 \nu^{8} - 66 \nu^{7} + 49 \nu^{6} + 105 \nu^{5} - 553 \nu^{4} + \cdots - 5578 ) / 1853 Copy content Toggle raw display
β2\beta_{2}== (22ν10110ν989ν8+1016ν7+230ν64708ν51786ν4+13970)/1853 ( 22 \nu^{10} - 110 \nu^{9} - 89 \nu^{8} + 1016 \nu^{7} + 230 \nu^{6} - 4708 \nu^{5} - 1786 \nu^{4} + \cdots - 13970 ) / 1853 Copy content Toggle raw display
β3\beta_{3}== (995ν11+7831ν10+46995ν9326814ν8+54349ν7+1703408ν6++10936061)/8740601 ( - 995 \nu^{11} + 7831 \nu^{10} + 46995 \nu^{9} - 326814 \nu^{8} + 54349 \nu^{7} + 1703408 \nu^{6} + \cdots + 10936061 ) / 8740601 Copy content Toggle raw display
β4\beta_{4}== (3014ν11+25876ν10250822ν984243ν8+2839567ν7+91337490)/8740601 ( 3014 \nu^{11} + 25876 \nu^{10} - 250822 \nu^{9} - 84243 \nu^{8} + 2839567 \nu^{7} + \cdots - 91337490 ) / 8740601 Copy content Toggle raw display
β5\beta_{5}== (3014ν11+59030ν10173708ν9679911ν8+2764229ν7+108083119)/8740601 ( - 3014 \nu^{11} + 59030 \nu^{10} - 173708 \nu^{9} - 679911 \nu^{8} + 2764229 \nu^{7} + \cdots - 108083119 ) / 8740601 Copy content Toggle raw display
β6\beta_{6}== (5549ν11+28161ν10+42520ν988867ν81402242ν7++13272986)/8740601 ( - 5549 \nu^{11} + 28161 \nu^{10} + 42520 \nu^{9} - 88867 \nu^{8} - 1402242 \nu^{7} + \cdots + 13272986 ) / 8740601 Copy content Toggle raw display
β7\beta_{7}== (5549ν11+32878ν10+18935ν9645473ν8+965692ν7+41137609)/8740601 ( - 5549 \nu^{11} + 32878 \nu^{10} + 18935 \nu^{9} - 645473 \nu^{8} + 965692 \nu^{7} + \cdots - 41137609 ) / 8740601 Copy content Toggle raw display
β8\beta_{8}== (24220ν1181323ν10427002ν9+1543220ν8+2521526ν7+9775054)/8740601 ( 24220 \nu^{11} - 81323 \nu^{10} - 427002 \nu^{9} + 1543220 \nu^{8} + 2521526 \nu^{7} + \cdots - 9775054 ) / 8740601 Copy content Toggle raw display
β9\beta_{9}== (24441ν11+84897ν10+198319ν9340479ν82357880ν7++137176039)/8740601 ( - 24441 \nu^{11} + 84897 \nu^{10} + 198319 \nu^{9} - 340479 \nu^{8} - 2357880 \nu^{7} + \cdots + 137176039 ) / 8740601 Copy content Toggle raw display
β10\beta_{10}== (24441ν11+183954ν10296966ν91231992ν8+4179882ν7+114419307)/8740601 ( - 24441 \nu^{11} + 183954 \nu^{10} - 296966 \nu^{9} - 1231992 \nu^{8} + 4179882 \nu^{7} + \cdots - 114419307 ) / 8740601 Copy content Toggle raw display
β11\beta_{11}== (1506ν11+8283ν10+6301ν990477ν81422ν7+485184ν6+1259576)/514153 ( - 1506 \nu^{11} + 8283 \nu^{10} + 6301 \nu^{9} - 90477 \nu^{8} - 1422 \nu^{7} + 485184 \nu^{6} + \cdots - 1259576 ) / 514153 Copy content Toggle raw display
ν\nu== (3β11+4β8+3β53β4+2β32β2+β1+3)/3 ( 3\beta_{11} + 4\beta_{8} + 3\beta_{5} - 3\beta_{4} + 2\beta_{3} - 2\beta_{2} + \beta _1 + 3 ) / 3 Copy content Toggle raw display
ν2\nu^{2}== (3β11+β10β9+4β8β7+β6+3β53β4++12)/3 ( 3 \beta_{11} + \beta_{10} - \beta_{9} + 4 \beta_{8} - \beta_{7} + \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + \cdots + 12 ) / 3 Copy content Toggle raw display
ν3\nu^{3}== 4β11+β10+7β8+β7+2β6+3β53β4+8β3++7 4 \beta_{11} + \beta_{10} + 7 \beta_{8} + \beta_{7} + 2 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 8 \beta_{3} + \cdots + 7 Copy content Toggle raw display
ν4\nu^{4}== (21β11+8β102β9+38β8+β7+17β6+18β5++39)/3 ( 21 \beta_{11} + 8 \beta_{10} - 2 \beta_{9} + 38 \beta_{8} + \beta_{7} + 17 \beta_{6} + 18 \beta_{5} + \cdots + 39 ) / 3 Copy content Toggle raw display
ν5\nu^{5}== (21β11+25β10+5β9+85β8+29β7+64β6+27β5++57)/3 ( 21 \beta_{11} + 25 \beta_{10} + 5 \beta_{9} + 85 \beta_{8} + 29 \beta_{7} + 64 \beta_{6} + 27 \beta_{5} + \cdots + 57 ) / 3 Copy content Toggle raw display
ν6\nu^{6}== 4β11+15β10+10β9+54β8+21β7+57β6+24β5++9 4 \beta_{11} + 15 \beta_{10} + 10 \beta_{9} + 54 \beta_{8} + 21 \beta_{7} + 57 \beta_{6} + 24 \beta_{5} + \cdots + 9 Copy content Toggle raw display
ν7\nu^{7}== (270β11+119β10+133β9+155β8+211β7+470β6+207)/3 ( - 270 \beta_{11} + 119 \beta_{10} + 133 \beta_{9} + 155 \beta_{8} + 211 \beta_{7} + 470 \beta_{6} + \cdots - 207 ) / 3 Copy content Toggle raw display
ν8\nu^{8}== (1089β11+166β10+506β950β8+470β7+1204β6+1071)/3 ( - 1089 \beta_{11} + 166 \beta_{10} + 506 \beta_{9} - 50 \beta_{8} + 470 \beta_{7} + 1204 \beta_{6} + \cdots - 1071 ) / 3 Copy content Toggle raw display
ν9\nu^{9}== 1477β11+86β10+542β9627β8+306β7+936β6+1329 - 1477 \beta_{11} + 86 \beta_{10} + 542 \beta_{9} - 627 \beta_{8} + 306 \beta_{7} + 936 \beta_{6} + \cdots - 1329 Copy content Toggle raw display
ν10\nu^{10}== (14004β11130β10+5005β98080β8+1003β7+6620β6+12339)/3 ( - 14004 \beta_{11} - 130 \beta_{10} + 5005 \beta_{9} - 8080 \beta_{8} + 1003 \beta_{7} + 6620 \beta_{6} + \cdots - 12339 ) / 3 Copy content Toggle raw display
ν11\nu^{11}== (43776β112189β10+13871β930497β82023β7+36225)/3 ( - 43776 \beta_{11} - 2189 \beta_{10} + 13871 \beta_{9} - 30497 \beta_{8} - 2023 \beta_{7} + \cdots - 36225 ) / 3 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1444Z)×\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times.

nn 723723 10851085
χ(n)\chi(n) 11 1β11-1 - \beta_{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}
429.1
2.75227 0.342020i
2.26253 + 0.984808i
2.25236 0.642788i
−1.75227 0.342020i
−1.26253 + 0.984808i
−1.25236 0.642788i
2.75227 + 0.342020i
2.26253 0.984808i
2.25236 + 0.642788i
−1.75227 + 0.342020i
−1.26253 0.984808i
−1.25236 + 0.642788i
0 −1.34598 + 2.33131i 0 0.641102 1.11042i 0 −3.34467 0 −2.12333 3.67771i 0
429.2 0 −0.544440 + 0.942998i 0 1.60017 2.77158i 0 −0.556791 0 0.907170 + 1.57126i 0
429.3 0 −0.243160 + 0.421165i 0 −1.39668 + 2.41913i 0 −3.36570 0 1.38175 + 2.39326i 0
429.4 0 0.906288 1.56974i 0 −0.141102 + 0.244396i 0 1.15987 0 −0.142716 0.247192i 0
429.5 0 1.21809 2.10979i 0 −1.10017 + 1.90556i 0 2.96827 0 −1.46748 2.54175i 0
429.6 0 1.50920 2.61402i 0 1.89668 3.28515i 0 0.139023 0 −3.05539 5.29210i 0
653.1 0 −1.34598 2.33131i 0 0.641102 + 1.11042i 0 −3.34467 0 −2.12333 + 3.67771i 0
653.2 0 −0.544440 0.942998i 0 1.60017 + 2.77158i 0 −0.556791 0 0.907170 1.57126i 0
653.3 0 −0.243160 0.421165i 0 −1.39668 2.41913i 0 −3.36570 0 1.38175 2.39326i 0
653.4 0 0.906288 + 1.56974i 0 −0.141102 0.244396i 0 1.15987 0 −0.142716 + 0.247192i 0
653.5 0 1.21809 + 2.10979i 0 −1.10017 1.90556i 0 2.96827 0 −1.46748 + 2.54175i 0
653.6 0 1.50920 + 2.61402i 0 1.89668 + 3.28515i 0 0.139023 0 −3.05539 + 5.29210i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 429.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1444.2.e.h 12
19.b odd 2 1 1444.2.e.g 12
19.c even 3 1 1444.2.a.g 6
19.c even 3 1 inner 1444.2.e.h 12
19.d odd 6 1 1444.2.a.h 6
19.d odd 6 1 1444.2.e.g 12
19.f odd 18 2 76.2.i.a 12
57.j even 18 2 684.2.bo.c 12
76.f even 6 1 5776.2.a.bw 6
76.g odd 6 1 5776.2.a.by 6
76.k even 18 2 304.2.u.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.i.a 12 19.f odd 18 2
304.2.u.e 12 76.k even 18 2
684.2.bo.c 12 57.j even 18 2
1444.2.a.g 6 19.c even 3 1
1444.2.a.h 6 19.d odd 6 1
1444.2.e.g 12 19.b odd 2 1
1444.2.e.g 12 19.d odd 6 1
1444.2.e.h 12 1.a even 1 1 trivial
1444.2.e.h 12 19.c even 3 1 inner
5776.2.a.bw 6 76.f even 6 1
5776.2.a.by 6 76.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3123T311+18T31027T39+147T38192T37+709T36++361 T_{3}^{12} - 3 T_{3}^{11} + 18 T_{3}^{10} - 27 T_{3}^{9} + 147 T_{3}^{8} - 192 T_{3}^{7} + 709 T_{3}^{6} + \cdots + 361 acting on S2new(1444,[χ])S_{2}^{\mathrm{new}}(1444, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T12 T^{12} Copy content Toggle raw display
33 T123T11++361 T^{12} - 3 T^{11} + \cdots + 361 Copy content Toggle raw display
55 T123T11++729 T^{12} - 3 T^{11} + \cdots + 729 Copy content Toggle raw display
77 (T6+3T512T4+3)2 (T^{6} + 3 T^{5} - 12 T^{4} + \cdots - 3)^{2} Copy content Toggle raw display
1111 (T6+3T524T4+27)2 (T^{6} + 3 T^{5} - 24 T^{4} + \cdots - 27)^{2} Copy content Toggle raw display
1313 T1212T11++130321 T^{12} - 12 T^{11} + \cdots + 130321 Copy content Toggle raw display
1717 T126T11++8982009 T^{12} - 6 T^{11} + \cdots + 8982009 Copy content Toggle raw display
1919 T12 T^{12} Copy content Toggle raw display
2323 T12++151585344 T^{12} + \cdots + 151585344 Copy content Toggle raw display
2929 T1221T11++263169 T^{12} - 21 T^{11} + \cdots + 263169 Copy content Toggle raw display
3131 (T66T5+361)2 (T^{6} - 6 T^{5} + \cdots - 361)^{2} Copy content Toggle raw display
3737 (T66T5++11096)2 (T^{6} - 6 T^{5} + \cdots + 11096)^{2} Copy content Toggle raw display
4141 T12++360278361 T^{12} + \cdots + 360278361 Copy content Toggle raw display
4343 T1218T11++1203409 T^{12} - 18 T^{11} + \cdots + 1203409 Copy content Toggle raw display
4747 T12+30T11++729 T^{12} + 30 T^{11} + \cdots + 729 Copy content Toggle raw display
5353 T1218T11++95004009 T^{12} - 18 T^{11} + \cdots + 95004009 Copy content Toggle raw display
5959 T12++12621848409 T^{12} + \cdots + 12621848409 Copy content Toggle raw display
6161 T12++1418049649 T^{12} + \cdots + 1418049649 Copy content Toggle raw display
6767 T12++6080256576 T^{12} + \cdots + 6080256576 Copy content Toggle raw display
7171 T12+12T11++16842816 T^{12} + 12 T^{11} + \cdots + 16842816 Copy content Toggle raw display
7373 T1224T11++36864 T^{12} - 24 T^{11} + \cdots + 36864 Copy content Toggle raw display
7979 T12++1548343801 T^{12} + \cdots + 1548343801 Copy content Toggle raw display
8383 (T6+3T5+109269)2 (T^{6} + 3 T^{5} + \cdots - 109269)^{2} Copy content Toggle raw display
8989 T12++7695324729 T^{12} + \cdots + 7695324729 Copy content Toggle raw display
9797 T12++16983563041 T^{12} + \cdots + 16983563041 Copy content Toggle raw display
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