Properties

Label 272.4.o.f
Level 272272
Weight 44
Character orbit 272.o
Analytic conductor 16.04916.049
Analytic rank 00
Dimension 1414
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [272,4,Mod(81,272)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(272, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("272.81");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 272=2417 272 = 2^{4} \cdot 17
Weight: k k == 4 4
Character orbit: [χ][\chi] == 272.o (of order 44, 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: 16.048519521616.0485195216
Analytic rank: 00
Dimension: 1414
Relative dimension: 77 over Q(i)\Q(i)
Coefficient field: Q[x]/(x14+)\mathbb{Q}[x]/(x^{14} + \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x14+119x12+5319x10+112122x8+1120191x6+4382607x4+1699337x2+2704 x^{14} + 119x^{12} + 5319x^{10} + 112122x^{8} + 1120191x^{6} + 4382607x^{4} + 1699337x^{2} + 2704 Copy content Toggle raw display
Coefficient ring: Z[a1,,a9]\Z[a_1, \ldots, a_{9}]
Coefficient ring index: 21534 2^{15}\cdot 3^{4}
Twist minimal: no (minimal twist has level 136)
Sato-Tate group: SU(2)[C4]\mathrm{SU}(2)[C_{4}]

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,,β131,\beta_1,\ldots,\beta_{13} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β2q3+β6q5+(β9β51)q7+(β116β5+β2)q9+(β13+β7+6β5++6)q11+(β3+β2β18)q13++(12β1219β11+211)q99+O(q100) q + \beta_{2} q^{3} + \beta_{6} q^{5} + ( - \beta_{9} - \beta_{5} - 1) q^{7} + (\beta_{11} - 6 \beta_{5} + \cdots - \beta_{2}) q^{9} + (\beta_{13} + \beta_{7} + 6 \beta_{5} + \cdots + 6) q^{11} + (\beta_{3} + \beta_{2} - \beta_1 - 8) q^{13}+ \cdots + ( - 12 \beta_{12} - 19 \beta_{11} + \cdots - 211) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 14q6q3+6q510q7+66q11124q13+130q17148q21+162q23+204q27+158q29+350q31+116q33236q35+582q37+320q39+878q41+3870q99+O(q100) 14 q - 6 q^{3} + 6 q^{5} - 10 q^{7} + 66 q^{11} - 124 q^{13} + 130 q^{17} - 148 q^{21} + 162 q^{23} + 204 q^{27} + 158 q^{29} + 350 q^{31} + 116 q^{33} - 236 q^{35} + 582 q^{37} + 320 q^{39} + 878 q^{41}+ \cdots - 3870 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x14+119x12+5319x10+112122x8+1120191x6+4382607x4+1699337x2+2704 x^{14} + 119x^{12} + 5319x^{10} + 112122x^{8} + 1120191x^{6} + 4382607x^{4} + 1699337x^{2} + 2704 : Copy content Toggle raw display

β1\beta_{1}== (16553ν122958786ν10553258413ν835042935761ν6+1462819625392)/53247141288 ( - 16553 \nu^{12} - 2958786 \nu^{10} - 553258413 \nu^{8} - 35042935761 \nu^{6} + \cdots - 1462819625392 ) / 53247141288 Copy content Toggle raw display
β2\beta_{2}== (9031ν12+574494ν102543421ν8708095937ν612509813490ν4+1244553232)/29043895248 ( 9031 \nu^{12} + 574494 \nu^{10} - 2543421 \nu^{8} - 708095937 \nu^{6} - 12509813490 \nu^{4} + \cdots - 1244553232 ) / 29043895248 Copy content Toggle raw display
β3\beta_{3}== (9031ν12+574494ν102543421ν8708095937ν612509813490ν4+1244553232)/29043895248 ( 9031 \nu^{12} + 574494 \nu^{10} - 2543421 \nu^{8} - 708095937 \nu^{6} - 12509813490 \nu^{4} + \cdots - 1244553232 ) / 29043895248 Copy content Toggle raw display
β4\beta_{4}== (9031ν12574494ν10+2543421ν8+708095937ν6+12509813490ν4++480468824824)/14521947624 ( - 9031 \nu^{12} - 574494 \nu^{10} + 2543421 \nu^{8} + 708095937 \nu^{6} + 12509813490 \nu^{4} + \cdots + 480468824824 ) / 14521947624 Copy content Toggle raw display
β5\beta_{5}== (5983429ν13712145454ν1131833327273ν9670840961865ν7+9487413190220ν)/377570638224 ( - 5983429 \nu^{13} - 712145454 \nu^{11} - 31833327273 \nu^{9} - 670840961865 \nu^{7} + \cdots - 9487413190220 \nu ) / 377570638224 Copy content Toggle raw display
β6\beta_{6}== (16718259ν1319735339ν121997695269ν112196992499ν10+2125260748364)/1038319255116 ( - 16718259 \nu^{13} - 19735339 \nu^{12} - 1997695269 \nu^{11} - 2196992499 \nu^{10} + \cdots - 2125260748364 ) / 1038319255116 Copy content Toggle raw display
β7\beta_{7}== (16718259ν13+19735339ν121997695269ν11+2196992499ν10++2125260748364)/1038319255116 ( - 16718259 \nu^{13} + 19735339 \nu^{12} - 1997695269 \nu^{11} + 2196992499 \nu^{10} + \cdots + 2125260748364 ) / 1038319255116 Copy content Toggle raw display
β8\beta_{8}== (394782247ν13+30952792ν12+46939392582ν11+2586850656ν10++21859444421360)/4153277020464 ( 394782247 \nu^{13} + 30952792 \nu^{12} + 46939392582 \nu^{11} + 2586850656 \nu^{10} + \cdots + 21859444421360 ) / 4153277020464 Copy content Toggle raw display
β9\beta_{9}== (394782247ν1330952792ν12+46939392582ν112586850656ν10+21859444421360)/4153277020464 ( 394782247 \nu^{13} - 30952792 \nu^{12} + 46939392582 \nu^{11} - 2586850656 \nu^{10} + \cdots - 21859444421360 ) / 4153277020464 Copy content Toggle raw display
β10\beta_{10}== (778104491ν1393193189950ν114199517040215ν9+14 ⁣ ⁣88ν)/2076638510232 ( - 778104491 \nu^{13} - 93193189950 \nu^{11} - 4199517040215 \nu^{9} + \cdots - 14\!\cdots\!88 \nu ) / 2076638510232 Copy content Toggle raw display
β11\beta_{11}== (197687963ν1323515736826ν111050433671063ν9+313807418169676ν)/377570638224 ( - 197687963 \nu^{13} - 23515736826 \nu^{11} - 1050433671063 \nu^{9} + \cdots - 313807418169676 \nu ) / 377570638224 Copy content Toggle raw display
β12\beta_{12}== (143212757ν1311120005ν1217048541404ν111318184036ν10+10332452117624)/230737612248 ( - 143212757 \nu^{13} - 11120005 \nu^{12} - 17048541404 \nu^{11} - 1318184036 \nu^{10} + \cdots - 10332452117624 ) / 230737612248 Copy content Toggle raw display
β13\beta_{13}== (143212757ν1311120005ν12+17048541404ν111318184036ν10+10332452117624)/230737612248 ( 143212757 \nu^{13} - 11120005 \nu^{12} + 17048541404 \nu^{11} - 1318184036 \nu^{10} + \cdots - 10332452117624 ) / 230737612248 Copy content Toggle raw display
ν\nu== (β3β2)/2 ( \beta_{3} - \beta_{2} ) / 2 Copy content Toggle raw display
ν2\nu^{2}== (β4+β3+β233)/2 ( \beta_{4} + \beta_{3} + \beta_{2} - 33 ) / 2 Copy content Toggle raw display
ν3\nu^{3}== (β13+β124β114β94β85β7++60β2)/4 ( - \beta_{13} + \beta_{12} - 4 \beta_{11} - 4 \beta_{9} - 4 \beta_{8} - 5 \beta_{7} + \cdots + 60 \beta_{2} ) / 4 Copy content Toggle raw display
ν4\nu^{4}== (7β137β1216β9+16β823β7+23β675β4++1851)/4 ( - 7 \beta_{13} - 7 \beta_{12} - 16 \beta_{9} + 16 \beta_{8} - 23 \beta_{7} + 23 \beta_{6} - 75 \beta_{4} + \cdots + 1851 ) / 4 Copy content Toggle raw display
ν5\nu^{5}== (59β1359β12+267β11+9β10+212β9+212β8+2104β2)/4 ( 59 \beta_{13} - 59 \beta_{12} + 267 \beta_{11} + 9 \beta_{10} + 212 \beta_{9} + 212 \beta_{8} + \cdots - 2104 \beta_{2} ) / 4 Copy content Toggle raw display
ν6\nu^{6}== (400β13+400β12+1000β91000β8+1328β71328β6+61061)/4 ( 400 \beta_{13} + 400 \beta_{12} + 1000 \beta_{9} - 1000 \beta_{8} + 1328 \beta_{7} - 1328 \beta_{6} + \cdots - 61061 ) / 4 Copy content Toggle raw display
ν7\nu^{7}== (1389β13+1389β126950β11336β104776β9++40585β2)/2 ( - 1389 \beta_{13} + 1389 \beta_{12} - 6950 \beta_{11} - 336 \beta_{10} - 4776 \beta_{9} + \cdots + 40585 \beta_{2} ) / 2 Copy content Toggle raw display
ν8\nu^{8}== (9472β139472β1225150β9+25150β831688β7++1124836)/2 ( - 9472 \beta_{13} - 9472 \beta_{12} - 25150 \beta_{9} + 25150 \beta_{8} - 31688 \beta_{7} + \cdots + 1124836 ) / 2 Copy content Toggle raw display
ν9\nu^{9}== (125251β13125251β12+665760β11+37224β10+415996β9+3320388β2)/4 ( 125251 \beta_{13} - 125251 \beta_{12} + 665760 \beta_{11} + 37224 \beta_{10} + 415996 \beta_{9} + \cdots - 3320388 \beta_{2} ) / 4 Copy content Toggle raw display
ν10\nu^{10}== (855515β13+855515β12+2370092β92370092β8+2880691β7+89151457)/4 ( 855515 \beta_{13} + 855515 \beta_{12} + 2370092 \beta_{9} - 2370092 \beta_{8} + 2880691 \beta_{7} + \cdots - 89151457 ) / 4 Copy content Toggle raw display
ν11\nu^{11}== (5594021β13+5594021β1230731683β111859493β10++140628400β2)/4 ( - 5594021 \beta_{13} + 5594021 \beta_{12} - 30731683 \beta_{11} - 1859493 \beta_{10} + \cdots + 140628400 \beta_{2} ) / 4 Copy content Toggle raw display
ν12\nu^{12}== (38091144β1338091144β12108692148β9+108692148β8++3699241435)/4 ( - 38091144 \beta_{13} - 38091144 \beta_{12} - 108692148 \beta_{9} + 108692148 \beta_{8} + \cdots + 3699241435 ) / 4 Copy content Toggle raw display
ν13\nu^{13}== (124623174β13124623174β12+696412656β11+44274816β10+3038627173β2)/2 ( 124623174 \beta_{13} - 124623174 \beta_{12} + 696412656 \beta_{11} + 44274816 \beta_{10} + \cdots - 3038627173 \beta_{2} ) / 2 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/272Z)×\left(\mathbb{Z}/272\mathbb{Z}\right)^\times.

nn 6969 239239 241241
χ(n)\chi(n) 11 11 β5-\beta_{5}

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}
81.1
6.66182i
3.80701i
3.13314i
0.0399724i
0.657265i
4.87012i
5.11455i
6.66182i
3.80701i
3.13314i
0.0399724i
0.657265i
4.87012i
5.11455i
0 −6.66182 6.66182i 0 3.32200 + 3.32200i 0 12.8505 12.8505i 0 61.7597i 0
81.2 0 −3.80701 3.80701i 0 2.78003 + 2.78003i 0 −25.7370 + 25.7370i 0 1.98658i 0
81.3 0 −3.13314 3.13314i 0 −13.6954 13.6954i 0 0.366316 0.366316i 0 7.36691i 0
81.4 0 −0.0399724 0.0399724i 0 −1.97903 1.97903i 0 4.30175 4.30175i 0 26.9968i 0
81.5 0 0.657265 + 0.657265i 0 13.8302 + 13.8302i 0 14.0096 14.0096i 0 26.1360i 0
81.6 0 4.87012 + 4.87012i 0 −8.46115 8.46115i 0 8.46698 8.46698i 0 20.4362i 0
81.7 0 5.11455 + 5.11455i 0 7.20340 + 7.20340i 0 −19.2581 + 19.2581i 0 25.3172i 0
225.1 0 −6.66182 + 6.66182i 0 3.32200 3.32200i 0 12.8505 + 12.8505i 0 61.7597i 0
225.2 0 −3.80701 + 3.80701i 0 2.78003 2.78003i 0 −25.7370 25.7370i 0 1.98658i 0
225.3 0 −3.13314 + 3.13314i 0 −13.6954 + 13.6954i 0 0.366316 + 0.366316i 0 7.36691i 0
225.4 0 −0.0399724 + 0.0399724i 0 −1.97903 + 1.97903i 0 4.30175 + 4.30175i 0 26.9968i 0
225.5 0 0.657265 0.657265i 0 13.8302 13.8302i 0 14.0096 + 14.0096i 0 26.1360i 0
225.6 0 4.87012 4.87012i 0 −8.46115 + 8.46115i 0 8.46698 + 8.46698i 0 20.4362i 0
225.7 0 5.11455 5.11455i 0 7.20340 7.20340i 0 −19.2581 19.2581i 0 25.3172i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 272.4.o.f 14
4.b odd 2 1 136.4.k.b 14
17.c even 4 1 inner 272.4.o.f 14
68.f odd 4 1 136.4.k.b 14
68.g odd 8 2 2312.4.a.l 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.k.b 14 4.b odd 2 1
136.4.k.b 14 68.f odd 4 1
272.4.o.f 14 1.a even 1 1 trivial
272.4.o.f 14 17.c even 4 1 inner
2312.4.a.l 14 68.g odd 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T314+6T313+18T312140T311+6044T310+26400T39++346112 T_{3}^{14} + 6 T_{3}^{13} + 18 T_{3}^{12} - 140 T_{3}^{11} + 6044 T_{3}^{10} + 26400 T_{3}^{9} + \cdots + 346112 acting on S4new(272,[χ])S_{4}^{\mathrm{new}}(272, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T14 T^{14} Copy content Toggle raw display
33 T14+6T13++346112 T^{14} + 6 T^{13} + \cdots + 346112 Copy content Toggle raw display
55 T14++5698471938048 T^{14} + \cdots + 5698471938048 Copy content Toggle raw display
77 T14++181426562039808 T^{14} + \cdots + 181426562039808 Copy content Toggle raw display
1111 T14++27 ⁣ ⁣28 T^{14} + \cdots + 27\!\cdots\!28 Copy content Toggle raw display
1313 (T7+62T6+237436285696)2 (T^{7} + 62 T^{6} + \cdots - 237436285696)^{2} Copy content Toggle raw display
1717 T14++69 ⁣ ⁣17 T^{14} + \cdots + 69\!\cdots\!17 Copy content Toggle raw display
1919 T14++19 ⁣ ⁣04 T^{14} + \cdots + 19\!\cdots\!04 Copy content Toggle raw display
2323 T14++88 ⁣ ⁣12 T^{14} + \cdots + 88\!\cdots\!12 Copy content Toggle raw display
2929 T14++58 ⁣ ⁣32 T^{14} + \cdots + 58\!\cdots\!32 Copy content Toggle raw display
3131 T14++20 ⁣ ⁣72 T^{14} + \cdots + 20\!\cdots\!72 Copy content Toggle raw display
3737 T14++19 ⁣ ⁣12 T^{14} + \cdots + 19\!\cdots\!12 Copy content Toggle raw display
4141 T14++20 ⁣ ⁣28 T^{14} + \cdots + 20\!\cdots\!28 Copy content Toggle raw display
4343 T14++18 ⁣ ⁣76 T^{14} + \cdots + 18\!\cdots\!76 Copy content Toggle raw display
4747 (T7+16 ⁣ ⁣60)2 (T^{7} + \cdots - 16\!\cdots\!60)^{2} Copy content Toggle raw display
5353 T14++84 ⁣ ⁣84 T^{14} + \cdots + 84\!\cdots\!84 Copy content Toggle raw display
5959 T14++48 ⁣ ⁣00 T^{14} + \cdots + 48\!\cdots\!00 Copy content Toggle raw display
6161 T14++77 ⁣ ⁣92 T^{14} + \cdots + 77\!\cdots\!92 Copy content Toggle raw display
6767 (T7++11 ⁣ ⁣64)2 (T^{7} + \cdots + 11\!\cdots\!64)^{2} Copy content Toggle raw display
7171 T14++12 ⁣ ⁣28 T^{14} + \cdots + 12\!\cdots\!28 Copy content Toggle raw display
7373 T14++80 ⁣ ⁣92 T^{14} + \cdots + 80\!\cdots\!92 Copy content Toggle raw display
7979 T14++10 ⁣ ⁣92 T^{14} + \cdots + 10\!\cdots\!92 Copy content Toggle raw display
8383 T14++72 ⁣ ⁣04 T^{14} + \cdots + 72\!\cdots\!04 Copy content Toggle raw display
8989 (T7+34 ⁣ ⁣40)2 (T^{7} + \cdots - 34\!\cdots\!40)^{2} Copy content Toggle raw display
9797 T14++53 ⁣ ⁣08 T^{14} + \cdots + 53\!\cdots\!08 Copy content Toggle raw display
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