Properties

Label 297.2.f.b
Level 297297
Weight 22
Character orbit 297.f
Analytic conductor 2.3722.372
Analytic rank 00
Dimension 1616
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [297,2,Mod(82,297)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, base_ring=CyclotomicField(10))
 
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("297.82");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 297=3311 297 = 3^{3} \cdot 11
Weight: k k == 2 2
Character orbit: [χ][\chi] == 297.f (of order 55, degree 44, 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: 2.371556940032.37155694003
Analytic rank: 00
Dimension: 1616
Relative dimension: 44 over Q(ζ5)\Q(\zeta_{5})
Coefficient field: Q[x]/(x16)\mathbb{Q}[x]/(x^{16} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x169x14+51x12249x10+1476x82875x6+2335x4+125x2+25 x^{16} - 9x^{14} + 51x^{12} - 249x^{10} + 1476x^{8} - 2875x^{6} + 2335x^{4} + 125x^{2} + 25 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C5]\mathrm{SU}(2)[C_{5}]

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,,β151,\beta_1,\ldots,\beta_{15} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β12q2+(β9β7+β2)q4+(β15+β13++β1)q5+(β6β5+β3++1)q7+(β152β14+2β1)q8++(β15+10β13++7β1)q98+O(q100) q + \beta_{12} q^{2} + (\beta_{9} - \beta_{7} + \cdots - \beta_{2}) q^{4} + ( - \beta_{15} + \beta_{13} + \cdots + \beta_1) q^{5} + (\beta_{6} - \beta_{5} + \beta_{3} + \cdots + 1) q^{7} + ( - \beta_{15} - 2 \beta_{14} + \cdots - 2 \beta_1) q^{8}+ \cdots + ( - \beta_{15} + 10 \beta_{13} + \cdots + 7 \beta_1) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 16q10q4+12q1010q162q1936q22+32q25+42q2826q3148q3424q3720q40+24q4316q46+24q4940q5216q55+106q58++72q97+O(q100) 16 q - 10 q^{4} + 12 q^{10} - 10 q^{16} - 2 q^{19} - 36 q^{22} + 32 q^{25} + 42 q^{28} - 26 q^{31} - 48 q^{34} - 24 q^{37} - 20 q^{40} + 24 q^{43} - 16 q^{46} + 24 q^{49} - 40 q^{52} - 16 q^{55} + 106 q^{58}+ \cdots + 72 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x169x14+51x12249x10+1476x82875x6+2335x4+125x2+25 x^{16} - 9x^{14} + 51x^{12} - 249x^{10} + 1476x^{8} - 2875x^{6} + 2335x^{4} + 125x^{2} + 25 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (16594091ν14+132889182ν12669375495ν10+3344289682ν8++109896768237)/48030072872 ( - 16594091 \nu^{14} + 132889182 \nu^{12} - 669375495 \nu^{10} + 3344289682 \nu^{8} + \cdots + 109896768237 ) / 48030072872 Copy content Toggle raw display
β3\beta_{3}== (399459369ν143872461014ν12+22604705541ν10111418189614ν8+163815397855)/240150364360 ( 399459369 \nu^{14} - 3872461014 \nu^{12} + 22604705541 \nu^{10} - 111418189614 \nu^{8} + \cdots - 163815397855 ) / 240150364360 Copy content Toggle raw display
β4\beta_{4}== (641538645ν145845087188ν12+33077583727ν10161239139198ν8++220551349875)/240150364360 ( 641538645 \nu^{14} - 5845087188 \nu^{12} + 33077583727 \nu^{10} - 161239139198 \nu^{8} + \cdots + 220551349875 ) / 240150364360 Copy content Toggle raw display
β5\beta_{5}== (420560697ν144175189607ν12+24562062708ν10121476379662ν8+186816081040)/120075182180 ( 420560697 \nu^{14} - 4175189607 \nu^{12} + 24562062708 \nu^{10} - 121476379662 \nu^{8} + \cdots - 186816081040 ) / 120075182180 Copy content Toggle raw display
β6\beta_{6}== (1070034741ν149471203848ν12+53263591527ν10259312649798ν8++118775784515)/240150364360 ( 1070034741 \nu^{14} - 9471203848 \nu^{12} + 53263591527 \nu^{10} - 259312649798 \nu^{8} + \cdots + 118775784515 ) / 240150364360 Copy content Toggle raw display
β7\beta_{7}== (724570675ν14+6672080139ν1237857976976ν10+184639030474ν8+251290182000)/120075182180 ( - 724570675 \nu^{14} + 6672080139 \nu^{12} - 37857976976 \nu^{10} + 184639030474 \nu^{8} + \cdots - 251290182000 ) / 120075182180 Copy content Toggle raw display
β8\beta_{8}== (399459369ν153872461014ν13+22604705541ν11111418189614ν9+163815397855ν)/240150364360 ( 399459369 \nu^{15} - 3872461014 \nu^{13} + 22604705541 \nu^{11} - 111418189614 \nu^{9} + \cdots - 163815397855 \nu ) / 240150364360 Copy content Toggle raw display
β9\beta_{9}== (921163875ν148325322917ν12+47647278148ν10233617980722ν8+28361391040)/120075182180 ( 921163875 \nu^{14} - 8325322917 \nu^{12} + 47647278148 \nu^{10} - 233617980722 \nu^{8} + \cdots - 28361391040 ) / 120075182180 Copy content Toggle raw display
β10\beta_{10}== (827955465ν157498577674ν13+42790713341ν11209491700214ν9+25440598855ν)/240150364360 ( 827955465 \nu^{15} - 7498577674 \nu^{13} + 42790713341 \nu^{11} - 209491700214 \nu^{9} + \cdots - 25440598855 \nu ) / 240150364360 Copy content Toggle raw display
β11\beta_{11}== (1070034741ν159471203848ν13+53263591527ν11259312649798ν9++118775784515ν)/240150364360 ( 1070034741 \nu^{15} - 9471203848 \nu^{13} + 53263591527 \nu^{11} - 259312649798 \nu^{9} + \cdots + 118775784515 \nu ) / 240150364360 Copy content Toggle raw display
β12\beta_{12}== (914980841ν15+8770447689ν1351196668716ν11+253062739159ν9++962789857750ν)/150093977725 ( - 914980841 \nu^{15} + 8770447689 \nu^{13} - 51196668716 \nu^{11} + 253062739159 \nu^{9} + \cdots + 962789857750 \nu ) / 150093977725 Copy content Toggle raw display
β13\beta_{13}== (12695929441ν15+111086725194ν13620796945061ν11+5798667404425ν)/1200751821800 ( - 12695929441 \nu^{15} + 111086725194 \nu^{13} - 620796945061 \nu^{11} + \cdots - 5798667404425 \nu ) / 1200751821800 Copy content Toggle raw display
β14\beta_{14}== (13576283373ν15+121399649892ν13682891277823ν11+8793919655675ν)/1200751821800 ( - 13576283373 \nu^{15} + 121399649892 \nu^{13} - 682891277823 \nu^{11} + \cdots - 8793919655675 \nu ) / 1200751821800 Copy content Toggle raw display
β15\beta_{15}== (24923473777ν15228729353598ν13+1310185729437ν11+3637970480775ν)/1200751821800 ( 24923473777 \nu^{15} - 228729353598 \nu^{13} + 1310185729437 \nu^{11} + \cdots - 3637970480775 \nu ) / 1200751821800 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β9+3β64β4+3β3+4 -\beta_{9} + 3\beta_{6} - 4\beta_{4} + 3\beta_{3} + 4 Copy content Toggle raw display
ν3\nu^{3}== 2β15β142β12β11+7β10β8 -2\beta_{15} - \beta_{14} - 2\beta_{12} - \beta_{11} + 7\beta_{10} - \beta_{8} Copy content Toggle raw display
ν4\nu^{4}== β96β7+6β6+β522β4+6 -\beta_{9} - 6\beta_{7} + 6\beta_{6} + \beta_{5} - 22\beta_{4} + 6 Copy content Toggle raw display
ν5\nu^{5}== 8β1514β14+β1315β1234β11+43β1042β833β1 -8\beta_{15} - 14\beta_{14} + \beta_{13} - 15\beta_{12} - 34\beta_{11} + 43\beta_{10} - 42\beta_{8} - 33\beta_1 Copy content Toggle raw display
ν6\nu^{6}== 9β9+39β6+43β539β489β3+9β2 -9\beta_{9} + 39\beta_{6} + 43\beta_{5} - 39\beta_{4} - 89\beta_{3} + 9\beta_{2} Copy content Toggle raw display
ν7\nu^{7}== 9β1552β14+52β1343β12+66β10205β814β1 -9\beta_{15} - 52\beta_{14} + 52\beta_{13} - 43\beta_{12} + 66\beta_{10} - 205\beta_{8} - 14\beta_1 Copy content Toggle raw display
ν8\nu^{8}== 191β9+191β7+501β6+257β5256β3+257β2256 -191\beta_{9} + 191\beta_{7} + 501\beta_{6} + 257\beta_{5} - 256\beta_{3} + 257\beta_{2} - 256 Copy content Toggle raw display
ν9\nu^{9}== 66β1566β14+514β13+257β12+1074β11+60β8+60β1 66\beta_{15} - 66\beta_{14} + 514\beta_{13} + 257\beta_{12} + 1074\beta_{11} + 60\beta_{8} + 60\beta_1 Copy content Toggle raw display
ν10\nu^{10}== 454β7+454β5+1664β41664β3+1528β24503 454\beta_{7} + 454\beta_{5} + 1664\beta_{4} - 1664\beta_{3} + 1528\beta_{2} - 4503 Copy content Toggle raw display
ν11\nu^{11}== 1982β15+454β14+1982β13+3510β12+3026β11+4079β1 1982 \beta_{15} + 454 \beta_{14} + 1982 \beta_{13} + 3510 \beta_{12} + 3026 \beta_{11} + \cdots - 4079 \beta_1 Copy content Toggle raw display
ν12\nu^{12}== 6061β9+3026β716171β6+26862β416171β3+3026β226862 6061\beta_{9} + 3026\beta_{7} - 16171\beta_{6} + 26862\beta_{4} - 16171\beta_{3} + 3026\beta_{2} - 26862 Copy content Toggle raw display
ν13\nu^{13}== 18174β15+12113β14+3026β13+24226β12+19769β11++3026β1 18174 \beta_{15} + 12113 \beta_{14} + 3026 \beta_{13} + 24226 \beta_{12} + 19769 \beta_{11} + \cdots + 3026 \beta_1 Copy content Toggle raw display
ν14\nu^{14}== 19769β9+34354β768004β619769β5+160548β468004 19769\beta_{9} + 34354\beta_{7} - 68004\beta_{6} - 19769\beta_{5} + 160548\beta_{4} - 68004 Copy content Toggle raw display
ν15\nu^{15}== 73892β15+108246β1419769β13+128015β12+195606β11++175837β1 73892 \beta_{15} + 108246 \beta_{14} - 19769 \beta_{13} + 128015 \beta_{12} + 195606 \beta_{11} + \cdots + 175837 \beta_1 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/297Z)×\left(\mathbb{Z}/297\mathbb{Z}\right)^\times.

nn 5656 244244
χ(n)\chi(n) 11 β6\beta_{6}

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}
82.1
−1.18970 0.386556i
−2.33991 0.760284i
2.33991 + 0.760284i
1.18970 + 0.386556i
−0.185814 0.255752i
−1.35089 1.85934i
1.35089 + 1.85934i
0.185814 + 0.255752i
−1.18970 + 0.386556i
−2.33991 + 0.760284i
2.33991 0.760284i
1.18970 0.386556i
−0.185814 + 0.255752i
−1.35089 + 1.85934i
1.35089 1.85934i
0.185814 0.255752i
−0.760284 2.33991i 0 −3.27913 + 2.38243i −0.305860 + 0.941339i 0 −3.49672 + 2.54052i 4.08684 + 2.96926i 0 2.43519
82.2 −0.386556 1.18970i 0 0.352078 0.255800i 0.507211 1.56103i 0 2.37869 1.72822i −2.46446 1.79053i 0 −2.05323
82.3 0.386556 + 1.18970i 0 0.352078 0.255800i −0.507211 + 1.56103i 0 2.37869 1.72822i 2.46446 + 1.79053i 0 −2.05323
82.4 0.760284 + 2.33991i 0 −3.27913 + 2.38243i 0.305860 0.941339i 0 −3.49672 + 2.54052i −4.08684 2.96926i 0 2.43519
136.1 −1.85934 1.35089i 0 1.01420 + 3.12140i −1.37287 + 0.997447i 0 0.0641710 + 0.197498i 0.910502 2.80224i 0 3.90006
136.2 −0.255752 0.185814i 0 −0.587152 1.80707i 3.28092 2.38373i 0 1.05386 + 3.24346i −0.380991 + 1.17257i 0 −1.28203
136.3 0.255752 + 0.185814i 0 −0.587152 1.80707i −3.28092 + 2.38373i 0 1.05386 + 3.24346i 0.380991 1.17257i 0 −1.28203
136.4 1.85934 + 1.35089i 0 1.01420 + 3.12140i 1.37287 0.997447i 0 0.0641710 + 0.197498i −0.910502 + 2.80224i 0 3.90006
163.1 −0.760284 + 2.33991i 0 −3.27913 2.38243i −0.305860 0.941339i 0 −3.49672 2.54052i 4.08684 2.96926i 0 2.43519
163.2 −0.386556 + 1.18970i 0 0.352078 + 0.255800i 0.507211 + 1.56103i 0 2.37869 + 1.72822i −2.46446 + 1.79053i 0 −2.05323
163.3 0.386556 1.18970i 0 0.352078 + 0.255800i −0.507211 1.56103i 0 2.37869 + 1.72822i 2.46446 1.79053i 0 −2.05323
163.4 0.760284 2.33991i 0 −3.27913 2.38243i 0.305860 + 0.941339i 0 −3.49672 2.54052i −4.08684 + 2.96926i 0 2.43519
190.1 −1.85934 + 1.35089i 0 1.01420 3.12140i −1.37287 0.997447i 0 0.0641710 0.197498i 0.910502 + 2.80224i 0 3.90006
190.2 −0.255752 + 0.185814i 0 −0.587152 + 1.80707i 3.28092 + 2.38373i 0 1.05386 3.24346i −0.380991 1.17257i 0 −1.28203
190.3 0.255752 0.185814i 0 −0.587152 + 1.80707i −3.28092 2.38373i 0 1.05386 3.24346i 0.380991 + 1.17257i 0 −1.28203
190.4 1.85934 1.35089i 0 1.01420 3.12140i 1.37287 + 0.997447i 0 0.0641710 0.197498i −0.910502 2.80224i 0 3.90006
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.2.f.b 16
3.b odd 2 1 inner 297.2.f.b 16
9.c even 3 2 891.2.n.h 32
9.d odd 6 2 891.2.n.h 32
11.c even 5 1 inner 297.2.f.b 16
11.c even 5 1 3267.2.a.bj 8
11.d odd 10 1 3267.2.a.bi 8
33.f even 10 1 3267.2.a.bi 8
33.h odd 10 1 inner 297.2.f.b 16
33.h odd 10 1 3267.2.a.bj 8
99.m even 15 2 891.2.n.h 32
99.n odd 30 2 891.2.n.h 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.f.b 16 1.a even 1 1 trivial
297.2.f.b 16 3.b odd 2 1 inner
297.2.f.b 16 11.c even 5 1 inner
297.2.f.b 16 33.h odd 10 1 inner
891.2.n.h 32 9.c even 3 2
891.2.n.h 32 9.d odd 6 2
891.2.n.h 32 99.m even 15 2
891.2.n.h 32 99.n odd 30 2
3267.2.a.bi 8 11.d odd 10 1
3267.2.a.bi 8 33.f even 10 1
3267.2.a.bj 8 11.c even 5 1
3267.2.a.bj 8 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T216+9T214+51T212+249T210+1476T28+2875T26+2335T24125T22+25 T_{2}^{16} + 9T_{2}^{14} + 51T_{2}^{12} + 249T_{2}^{10} + 1476T_{2}^{8} + 2875T_{2}^{6} + 2335T_{2}^{4} - 125T_{2}^{2} + 25 acting on S2new(297,[χ])S_{2}^{\mathrm{new}}(297, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T16+9T14++25 T^{16} + 9 T^{14} + \cdots + 25 Copy content Toggle raw display
33 T16 T^{16} Copy content Toggle raw display
55 T166T14++15625 T^{16} - 6 T^{14} + \cdots + 15625 Copy content Toggle raw display
77 (T8+T6+10T5++81)2 (T^{8} + T^{6} + 10 T^{5} + \cdots + 81)^{2} Copy content Toggle raw display
1111 T16++214358881 T^{16} + \cdots + 214358881 Copy content Toggle raw display
1313 (T4+10T2++25)4 (T^{4} + 10 T^{2} + \cdots + 25)^{4} Copy content Toggle raw display
1717 T16++25665642025 T^{16} + \cdots + 25665642025 Copy content Toggle raw display
1919 (T8+T7++450241)2 (T^{8} + T^{7} + \cdots + 450241)^{2} Copy content Toggle raw display
2323 (T842T6++405)2 (T^{8} - 42 T^{6} + \cdots + 405)^{2} Copy content Toggle raw display
2929 T1684T14++25 T^{16} - 84 T^{14} + \cdots + 25 Copy content Toggle raw display
3131 (T8+13T7++44521)2 (T^{8} + 13 T^{7} + \cdots + 44521)^{2} Copy content Toggle raw display
3737 (T4+6T3+16T2++16)4 (T^{4} + 6 T^{3} + 16 T^{2} + \cdots + 16)^{4} Copy content Toggle raw display
4141 T16++43 ⁣ ⁣25 T^{16} + \cdots + 43\!\cdots\!25 Copy content Toggle raw display
4343 (T46T316T2++5)4 (T^{4} - 6 T^{3} - 16 T^{2} + \cdots + 5)^{4} Copy content Toggle raw display
4747 T16++980347515625 T^{16} + \cdots + 980347515625 Copy content Toggle raw display
5353 T16++5358972025 T^{16} + \cdots + 5358972025 Copy content Toggle raw display
5959 T16++415257804025 T^{16} + \cdots + 415257804025 Copy content Toggle raw display
6161 (T8+3T7++194481)2 (T^{8} + 3 T^{7} + \cdots + 194481)^{2} Copy content Toggle raw display
6767 (T2+5T5)8 (T^{2} + 5 T - 5)^{8} Copy content Toggle raw display
7171 T16++64072265625 T^{16} + \cdots + 64072265625 Copy content Toggle raw display
7373 (T8+164T6++14622976)2 (T^{8} + 164 T^{6} + \cdots + 14622976)^{2} Copy content Toggle raw display
7979 (T832T7++421201)2 (T^{8} - 32 T^{7} + \cdots + 421201)^{2} Copy content Toggle raw display
8383 T16++463495448025 T^{16} + \cdots + 463495448025 Copy content Toggle raw display
8989 (T8357T6++25515405)2 (T^{8} - 357 T^{6} + \cdots + 25515405)^{2} Copy content Toggle raw display
9797 (T836T7++160801)2 (T^{8} - 36 T^{7} + \cdots + 160801)^{2} Copy content Toggle raw display
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