Properties

Label 735.4.a.bc
Level 735735
Weight 44
Character orbit 735.a
Self dual yes
Analytic conductor 43.36643.366
Analytic rank 00
Dimension 88
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [735,4,Mod(1,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 735=3572 735 = 3 \cdot 5 \cdot 7^{2}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 735.a (trivial)

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: yes
Analytic conductor: 43.366403854243.3664038542
Analytic rank: 00
Dimension: 88
Coefficient field: Q[x]/(x8)\mathbb{Q}[x]/(x^{8} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x82x755x6+80x5+969x4866x35783x2+2328x+9992 x^{8} - 2x^{7} - 55x^{6} + 80x^{5} + 969x^{4} - 866x^{3} - 5783x^{2} + 2328x + 9992 Copy content Toggle raw display
Coefficient ring: Z[a1,,a19]\Z[a_1, \ldots, a_{19}]
Coefficient ring index: 272 2\cdot 7^{2}
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

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,,β71,\beta_1,\ldots,\beta_{7} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2+3q3+(β2+β1+6)q4+5q5+3β1q6+(β5+β4+β2++7)q8+9q9+5β1q10+(β4+β3β2++7)q11++(9β4+9β39β2++63)q99+O(q100) q + \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + \beta_1 + 6) q^{4} + 5 q^{5} + 3 \beta_1 q^{6} + (\beta_{5} + \beta_{4} + \beta_{2} + \cdots + 7) q^{8} + 9 q^{9} + 5 \beta_1 q^{10} + (\beta_{4} + \beta_{3} - \beta_{2} + \cdots + 7) q^{11}+ \cdots + (9 \beta_{4} + 9 \beta_{3} - 9 \beta_{2} + \cdots + 63) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+2q2+24q3+50q4+40q5+6q6+66q8+72q9+10q10+64q11+150q12+120q15+206q16+48q17+18q18+80q19+250q20+452q22++576q99+O(q100) 8 q + 2 q^{2} + 24 q^{3} + 50 q^{4} + 40 q^{5} + 6 q^{6} + 66 q^{8} + 72 q^{9} + 10 q^{10} + 64 q^{11} + 150 q^{12} + 120 q^{15} + 206 q^{16} + 48 q^{17} + 18 q^{18} + 80 q^{19} + 250 q^{20} + 452 q^{22}+ \cdots + 576 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x82x755x6+80x5+969x4866x35783x2+2328x+9992 x^{8} - 2x^{7} - 55x^{6} + 80x^{5} + 969x^{4} - 866x^{3} - 5783x^{2} + 2328x + 9992 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2ν14 \nu^{2} - \nu - 14 Copy content Toggle raw display
β3\beta_{3}== (9ν7+109ν6844ν54156ν4+17325ν3+45225ν270072ν136312)/3088 ( 9\nu^{7} + 109\nu^{6} - 844\nu^{5} - 4156\nu^{4} + 17325\nu^{3} + 45225\nu^{2} - 70072\nu - 136312 ) / 3088 Copy content Toggle raw display
β4\beta_{4}== (61ν7+119ν6+2804ν54084ν430961ν3+28523ν26624ν10056)/6176 ( -61\nu^{7} + 119\nu^{6} + 2804\nu^{5} - 4084\nu^{4} - 30961\nu^{3} + 28523\nu^{2} - 6624\nu - 10056 ) / 6176 Copy content Toggle raw display
β5\beta_{5}== (61ν7119ν62804ν5+4084ν4+37137ν334699ν2116896ν+53288)/6176 ( 61\nu^{7} - 119\nu^{6} - 2804\nu^{5} + 4084\nu^{4} + 37137\nu^{3} - 34699\nu^{2} - 116896\nu + 53288 ) / 6176 Copy content Toggle raw display
β6\beta_{6}== (19ν7113ν6924ν5+4436ν4+13415ν345029ν242080ν+102176)/1544 ( 19\nu^{7} - 113\nu^{6} - 924\nu^{5} + 4436\nu^{4} + 13415\nu^{3} - 45029\nu^{2} - 42080\nu + 102176 ) / 1544 Copy content Toggle raw display
β7\beta_{7}== (19ν7113ν6924ν5+5980ν4+11871ν388261ν223552ν+264296)/1544 ( 19\nu^{7} - 113\nu^{6} - 924\nu^{5} + 5980\nu^{4} + 11871\nu^{3} - 88261\nu^{2} - 23552\nu + 264296 ) / 1544 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2+β1+14 \beta_{2} + \beta _1 + 14 Copy content Toggle raw display
ν3\nu^{3}== β5+β4+β2+21β1+7 \beta_{5} + \beta_{4} + \beta_{2} + 21\beta _1 + 7 Copy content Toggle raw display
ν4\nu^{4}== β7β6+β5+β4+29β2+37β1+294 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 29\beta_{2} + 37\beta _1 + 294 Copy content Toggle raw display
ν5\nu^{5}== β76β6+38β5+30β46β3+45β2+512β1+312 \beta_{7} - 6\beta_{6} + 38\beta_{5} + 30\beta_{4} - 6\beta_{3} + 45\beta_{2} + 512\beta _1 + 312 Copy content Toggle raw display
ν6\nu^{6}== 41β758β6+66β5+46β4+4β3+752β2+1186β1+7030 41\beta_{7} - 58\beta_{6} + 66\beta_{5} + 46\beta_{4} + 4\beta_{3} + 752\beta_{2} + 1186\beta _1 + 7030 Copy content Toggle raw display
ν7\nu^{7}== 59β7322β6+1301β5+793β4268β3+1554β2+13072β1+11201 59\beta_{7} - 322\beta_{6} + 1301\beta_{5} + 793\beta_{4} - 268\beta_{3} + 1554\beta_{2} + 13072\beta _1 + 11201 Copy content Toggle raw display

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}
1.1
−4.86877
−4.22998
−2.32865
−1.52943
1.83170
2.78988
4.95150
5.38374
−4.86877 3.00000 15.7049 5.00000 −14.6063 0 −37.5134 9.00000 −24.3438
1.2 −4.22998 3.00000 9.89271 5.00000 −12.6899 0 −8.00613 9.00000 −21.1499
1.3 −2.32865 3.00000 −2.57741 5.00000 −6.98594 0 24.6310 9.00000 −11.6432
1.4 −1.52943 3.00000 −5.66083 5.00000 −4.58830 0 20.8933 9.00000 −7.64717
1.5 1.83170 3.00000 −4.64489 5.00000 5.49509 0 −23.1616 9.00000 9.15848
1.6 2.78988 3.00000 −0.216543 5.00000 8.36965 0 −22.9232 9.00000 13.9494
1.7 4.95150 3.00000 16.5174 5.00000 14.8545 0 42.1739 9.00000 24.7575
1.8 5.38374 3.00000 20.9847 5.00000 16.1512 0 69.9061 9.00000 26.9187
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 1 -1
55 1 -1
77 +1 +1

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.4.a.bc yes 8
3.b odd 2 1 2205.4.a.cd 8
7.b odd 2 1 735.4.a.bb 8
21.c even 2 1 2205.4.a.ce 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.4.a.bb 8 7.b odd 2 1
735.4.a.bc yes 8 1.a even 1 1 trivial
2205.4.a.cd 8 3.b odd 2 1
2205.4.a.ce 8 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(Γ0(735))S_{4}^{\mathrm{new}}(\Gamma_0(735)):

T282T2755T26+80T25+969T24866T235783T22+2328T2+9992 T_{2}^{8} - 2T_{2}^{7} - 55T_{2}^{6} + 80T_{2}^{5} + 969T_{2}^{4} - 866T_{2}^{3} - 5783T_{2}^{2} + 2328T_{2} + 9992 Copy content Toggle raw display
T11864T1173258T116+304584T1154199384T114+9346926848 T_{11}^{8} - 64 T_{11}^{7} - 3258 T_{11}^{6} + 304584 T_{11}^{5} - 4199384 T_{11}^{4} + \cdots - 9346926848 Copy content Toggle raw display
T13810464T136+36160T135+22566928T134+176712704T133++211872661504 T_{13}^{8} - 10464 T_{13}^{6} + 36160 T_{13}^{5} + 22566928 T_{13}^{4} + 176712704 T_{13}^{3} + \cdots + 211872661504 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T82T7++9992 T^{8} - 2 T^{7} + \cdots + 9992 Copy content Toggle raw display
33 (T3)8 (T - 3)^{8} Copy content Toggle raw display
55 (T5)8 (T - 5)^{8} Copy content Toggle raw display
77 T8 T^{8} Copy content Toggle raw display
1111 T8+9346926848 T^{8} + \cdots - 9346926848 Copy content Toggle raw display
1313 T8++211872661504 T^{8} + \cdots + 211872661504 Copy content Toggle raw display
1717 T8+6565556795392 T^{8} + \cdots - 6565556795392 Copy content Toggle raw display
1919 T8+129732409433664 T^{8} + \cdots - 129732409433664 Copy content Toggle raw display
2323 T8++84 ⁣ ⁣84 T^{8} + \cdots + 84\!\cdots\!84 Copy content Toggle raw display
2929 T8++17 ⁣ ⁣96 T^{8} + \cdots + 17\!\cdots\!96 Copy content Toggle raw display
3131 T8++17 ⁣ ⁣04 T^{8} + \cdots + 17\!\cdots\!04 Copy content Toggle raw display
3737 T8++22 ⁣ ⁣88 T^{8} + \cdots + 22\!\cdots\!88 Copy content Toggle raw display
4141 T8+60 ⁣ ⁣12 T^{8} + \cdots - 60\!\cdots\!12 Copy content Toggle raw display
4343 T8+12 ⁣ ⁣00 T^{8} + \cdots - 12\!\cdots\!00 Copy content Toggle raw display
4747 T8++28 ⁣ ⁣48 T^{8} + \cdots + 28\!\cdots\!48 Copy content Toggle raw display
5353 T8+18 ⁣ ⁣68 T^{8} + \cdots - 18\!\cdots\!68 Copy content Toggle raw display
5959 T8++13 ⁣ ⁣16 T^{8} + \cdots + 13\!\cdots\!16 Copy content Toggle raw display
6161 T8++76 ⁣ ⁣04 T^{8} + \cdots + 76\!\cdots\!04 Copy content Toggle raw display
6767 T8+46 ⁣ ⁣56 T^{8} + \cdots - 46\!\cdots\!56 Copy content Toggle raw display
7171 T8+79 ⁣ ⁣44 T^{8} + \cdots - 79\!\cdots\!44 Copy content Toggle raw display
7373 T8++36 ⁣ ⁣72 T^{8} + \cdots + 36\!\cdots\!72 Copy content Toggle raw display
7979 T8++50 ⁣ ⁣64 T^{8} + \cdots + 50\!\cdots\!64 Copy content Toggle raw display
8383 T8++24 ⁣ ⁣32 T^{8} + \cdots + 24\!\cdots\!32 Copy content Toggle raw display
8989 T8+36 ⁣ ⁣32 T^{8} + \cdots - 36\!\cdots\!32 Copy content Toggle raw display
9797 T8+23 ⁣ ⁣48 T^{8} + \cdots - 23\!\cdots\!48 Copy content Toggle raw display
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