Properties

Label 296.2.i.c
Level 296296
Weight 22
Character orbit 296.i
Analytic conductor 2.3642.364
Analytic rank 00
Dimension 1010
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [296,2,Mod(121,296)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(296, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("296.121"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 296=2337 296 = 2^{3} \cdot 37
Weight: k k == 2 2
Character orbit: [χ][\chi] == 296.i (of order 33, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 2.363571899832.36357189983
Analytic rank: 00
Dimension: 1010
Relative dimension: 55 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q[x]/(x10)\mathbb{Q}[x]/(x^{10} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x10x9+9x8+4x7+54x6+6x5+98x48x3+148x224x+4 x^{10} - x^{9} + 9x^{8} + 4x^{7} + 54x^{6} + 6x^{5} + 98x^{4} - 8x^{3} + 148x^{2} - 24x + 4 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 24 2^{4}
Twist minimal: yes
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,,β91,\beta_1,\ldots,\beta_{9} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == qβ9q3+(β9+β6+β2)q5+(β8β7+1)q7+(β9β8+β4)q9+(β3+2)q11+(β7β6β2++1)q13++(2β92β8++β1)q99+O(q100) q - \beta_{9} q^{3} + (\beta_{9} + \beta_{6} + \beta_{2}) q^{5} + ( - \beta_{8} - \beta_{7} + 1) q^{7} + (\beta_{9} - \beta_{8} + \cdots - \beta_{4}) q^{9} + ( - \beta_{3} + 2) q^{11} + ( - \beta_{7} - \beta_{6} - \beta_{2} + \cdots + 1) q^{13}+ \cdots + (2 \beta_{9} - 2 \beta_{8} + \cdots + \beta_1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 10qq3+3q5+3q76q9+16q11+q13+13q15+5q173q19+3q2112q2312q25+26q2716q31+4q33+5q3512q37+3q39+36q97+O(q100) 10 q - q^{3} + 3 q^{5} + 3 q^{7} - 6 q^{9} + 16 q^{11} + q^{13} + 13 q^{15} + 5 q^{17} - 3 q^{19} + 3 q^{21} - 12 q^{23} - 12 q^{25} + 26 q^{27} - 16 q^{31} + 4 q^{33} + 5 q^{35} - 12 q^{37} + 3 q^{39}+ \cdots - 36 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x10x9+9x8+4x7+54x6+6x5+98x48x3+148x224x+4 x^{10} - x^{9} + 9x^{8} + 4x^{7} + 54x^{6} + 6x^{5} + 98x^{4} - 8x^{3} + 148x^{2} - 24x + 4 : Copy content Toggle raw display

β1\beta_{1}== 2ν 2\nu Copy content Toggle raw display
β2\beta_{2}== (13ν9605ν8+1210ν77028ν6+1815ν527830ν4++5505)/15127 ( - 13 \nu^{9} - 605 \nu^{8} + 1210 \nu^{7} - 7028 \nu^{6} + 1815 \nu^{5} - 27830 \nu^{4} + \cdots + 5505 ) / 15127 Copy content Toggle raw display
β3\beta_{3}== (40ν9+33ν866ν7+1012ν699ν5+1518ν415676ν3++5004)/15127 ( 40 \nu^{9} + 33 \nu^{8} - 66 \nu^{7} + 1012 \nu^{6} - 99 \nu^{5} + 1518 \nu^{4} - 15676 \nu^{3} + \cdots + 5004 ) / 15127 Copy content Toggle raw display
β4\beta_{4}== (146ν9+1309ν82618ν7+17092ν63927ν5+60214ν4++49383)/15127 ( 146 \nu^{9} + 1309 \nu^{8} - 2618 \nu^{7} + 17092 \nu^{6} - 3927 \nu^{5} + 60214 \nu^{4} + \cdots + 49383 ) / 15127 Copy content Toggle raw display
β5\beta_{5}== (153ν9+360ν8720ν7+235ν61080ν5+16560ν4++50660)/15127 ( - 153 \nu^{9} + 360 \nu^{8} - 720 \nu^{7} + 235 \nu^{6} - 1080 \nu^{5} + 16560 \nu^{4} + \cdots + 50660 ) / 15127 Copy content Toggle raw display
β6\beta_{6}== (863ν91395ν8+7112ν7+4762ν6+27956ν5+7143ν4+11542)/30254 ( 863 \nu^{9} - 1395 \nu^{8} + 7112 \nu^{7} + 4762 \nu^{6} + 27956 \nu^{5} + 7143 \nu^{4} + \cdots - 11542 ) / 30254 Copy content Toggle raw display
β7\beta_{7}== (1251ν9+1291ν811226ν75070ν666542ν57605ν4++29562)/30254 ( - 1251 \nu^{9} + 1291 \nu^{8} - 11226 \nu^{7} - 5070 \nu^{6} - 66542 \nu^{5} - 7605 \nu^{4} + \cdots + 29562 ) / 30254 Copy content Toggle raw display
β8\beta_{8}== (2432ν91142ν817165ν736462ν6156488ν5123845ν4+3432)/30254 ( - 2432 \nu^{9} - 1142 \nu^{8} - 17165 \nu^{7} - 36462 \nu^{6} - 156488 \nu^{5} - 123845 \nu^{4} + \cdots - 3432 ) / 30254 Copy content Toggle raw display
β9\beta_{9}== (4505ν94117ν8+40649ν7+22134ν6+243578ν5+65616ν4++2796)/30254 ( 4505 \nu^{9} - 4117 \nu^{8} + 40649 \nu^{7} + 22134 \nu^{6} + 243578 \nu^{5} + 65616 \nu^{4} + \cdots + 2796 ) / 30254 Copy content Toggle raw display
ν\nu== (β1)/2 ( \beta_1 ) / 2 Copy content Toggle raw display
ν2\nu^{2}== (β9+β87β72β6+β5+β4β3+β1)/2 ( -\beta_{9} + \beta_{8} - 7\beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 ) / 2 Copy content Toggle raw display
ν3\nu^{3}== β43β3+2β23 \beta_{4} - 3\beta_{3} + 2\beta_{2} - 3 Copy content Toggle raw display
ν4\nu^{4}== 2β94β8+21β7+9β6+9β26β121 2\beta_{9} - 4\beta_{8} + 21\beta_{7} + 9\beta_{6} + 9\beta_{2} - 6\beta _1 - 21 Copy content Toggle raw display
ν5\nu^{5}== β911β8+40β7+23β6β511β4+23β323β1 \beta_{9} - 11\beta_{8} + 40\beta_{7} + 23\beta_{6} - \beta_{5} - 11\beta_{4} + 23\beta_{3} - 23\beta_1 Copy content Toggle raw display
ν6\nu^{6}== 11β535β4+60β379β2+160 -11\beta_{5} - 35\beta_{4} + 60\beta_{3} - 79\beta_{2} + 160 Copy content Toggle raw display
ν7\nu^{7}== 16β9+104β8409β7223β6223β2+197β1+409 -16\beta_{9} + 104\beta_{8} - 409\beta_{7} - 223\beta_{6} - 223\beta_{2} + 197\beta _1 + 409 Copy content Toggle raw display
ν8\nu^{8}== 78β9+316β81363β7705β6+78β5+316β4565β3+565β1 -78\beta_{9} + 316\beta_{8} - 1363\beta_{7} - 705\beta_{6} + 78\beta_{5} + 316\beta_{4} - 565\beta_{3} + 565\beta_1 Copy content Toggle raw display
ν9\nu^{9}== 176β5+954β41757β3+2073β23877 176\beta_{5} + 954\beta_{4} - 1757\beta_{3} + 2073\beta_{2} - 3877 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/296Z)×\left(\mathbb{Z}/296\mathbb{Z}\right)^\times.

nn 113113 149149 223223
χ(n)\chi(n) 1+β7-1 + \beta_{7} 11 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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}
121.1
0.0825878 + 0.143046i
1.50928 + 2.61415i
−0.831426 1.44007i
−0.917643 1.58940i
0.657199 + 1.13830i
0.0825878 0.143046i
1.50928 2.61415i
−0.831426 + 1.44007i
−0.917643 + 1.58940i
0.657199 1.13830i
0 −1.64462 2.84856i 0 1.87297 + 3.24408i 0 2.03658 + 3.52745i 0 −3.90954 + 6.77153i 0
121.2 0 −0.603312 1.04497i 0 −0.717824 1.24331i 0 −0.0542045 0.0938850i 0 0.772029 1.33719i 0
121.3 0 −0.261675 0.453235i 0 1.19680 + 2.07292i 0 −2.55985 4.43380i 0 1.36305 2.36088i 0
121.4 0 0.788575 + 1.36585i 0 −1.73584 3.00655i 0 1.47954 + 2.56263i 0 0.256299 0.443924i 0
121.5 0 1.22103 + 2.11489i 0 0.883889 + 1.53094i 0 0.597948 + 1.03568i 0 −1.48184 + 2.56662i 0
137.1 0 −1.64462 + 2.84856i 0 1.87297 3.24408i 0 2.03658 3.52745i 0 −3.90954 6.77153i 0
137.2 0 −0.603312 + 1.04497i 0 −0.717824 + 1.24331i 0 −0.0542045 + 0.0938850i 0 0.772029 + 1.33719i 0
137.3 0 −0.261675 + 0.453235i 0 1.19680 2.07292i 0 −2.55985 + 4.43380i 0 1.36305 + 2.36088i 0
137.4 0 0.788575 1.36585i 0 −1.73584 + 3.00655i 0 1.47954 2.56263i 0 0.256299 + 0.443924i 0
137.5 0 1.22103 2.11489i 0 0.883889 1.53094i 0 0.597948 1.03568i 0 −1.48184 2.56662i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 296.2.i.c 10
3.b odd 2 1 2664.2.r.n 10
4.b odd 2 1 592.2.i.h 10
37.c even 3 1 inner 296.2.i.c 10
111.i odd 6 1 2664.2.r.n 10
148.i odd 6 1 592.2.i.h 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
296.2.i.c 10 1.a even 1 1 trivial
296.2.i.c 10 37.c even 3 1 inner
592.2.i.h 10 4.b odd 2 1
592.2.i.h 10 148.i odd 6 1
2664.2.r.n 10 3.b odd 2 1
2664.2.r.n 10 111.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T310+T39+11T382T37+88T36+16T35+168T34+96T33+288T32+128T3+64 T_{3}^{10} + T_{3}^{9} + 11T_{3}^{8} - 2T_{3}^{7} + 88T_{3}^{6} + 16T_{3}^{5} + 168T_{3}^{4} + 96T_{3}^{3} + 288T_{3}^{2} + 128T_{3} + 64 acting on S2new(296,[χ])S_{2}^{\mathrm{new}}(296, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T10 T^{10} Copy content Toggle raw display
33 T10+T9++64 T^{10} + T^{9} + \cdots + 64 Copy content Toggle raw display
55 T103T9++6241 T^{10} - 3 T^{9} + \cdots + 6241 Copy content Toggle raw display
77 T103T9++64 T^{10} - 3 T^{9} + \cdots + 64 Copy content Toggle raw display
1111 (T58T4+128)2 (T^{5} - 8 T^{4} + \cdots - 128)^{2} Copy content Toggle raw display
1313 T10T9++2304 T^{10} - T^{9} + \cdots + 2304 Copy content Toggle raw display
1717 (T2T+1)5 (T^{2} - T + 1)^{5} Copy content Toggle raw display
1919 T10+3T9++256 T^{10} + 3 T^{9} + \cdots + 256 Copy content Toggle raw display
2323 (T5+6T4++1152)2 (T^{5} + 6 T^{4} + \cdots + 1152)^{2} Copy content Toggle raw display
2929 (T572T3++358)2 (T^{5} - 72 T^{3} + \cdots + 358)^{2} Copy content Toggle raw display
3131 (T5+8T4++128)2 (T^{5} + 8 T^{4} + \cdots + 128)^{2} Copy content Toggle raw display
3737 T10+12T9++69343957 T^{10} + 12 T^{9} + \cdots + 69343957 Copy content Toggle raw display
4141 T109T9++8543929 T^{10} - 9 T^{9} + \cdots + 8543929 Copy content Toggle raw display
4343 (T52T416T3++32)2 (T^{5} - 2 T^{4} - 16 T^{3} + \cdots + 32)^{2} Copy content Toggle raw display
4747 (T516T4++23424)2 (T^{5} - 16 T^{4} + \cdots + 23424)^{2} Copy content Toggle raw display
5353 T105T9++2304 T^{10} - 5 T^{9} + \cdots + 2304 Copy content Toggle raw display
5959 T10+5T9++147456 T^{10} + 5 T^{9} + \cdots + 147456 Copy content Toggle raw display
6161 T1015T9++40921609 T^{10} - 15 T^{9} + \cdots + 40921609 Copy content Toggle raw display
6767 T10T9++20736 T^{10} - T^{9} + \cdots + 20736 Copy content Toggle raw display
7171 T10+17T9++40000 T^{10} + 17 T^{9} + \cdots + 40000 Copy content Toggle raw display
7373 (T5+18T4++1184)2 (T^{5} + 18 T^{4} + \cdots + 1184)^{2} Copy content Toggle raw display
7979 T10++12908595456 T^{10} + \cdots + 12908595456 Copy content Toggle raw display
8383 T10+15T9++45481536 T^{10} + 15 T^{9} + \cdots + 45481536 Copy content Toggle raw display
8989 T10++1372480209 T^{10} + \cdots + 1372480209 Copy content Toggle raw display
9797 (T5+18T4++88282)2 (T^{5} + 18 T^{4} + \cdots + 88282)^{2} Copy content Toggle raw display
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