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Properties

Label 735.4.a.bb
Level $735$
Weight $4$
Character orbit 735.a
Self dual yes
Analytic conductor $43.366$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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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: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 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: \(43.3664038542\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 55x^{6} + 80x^{5} + 969x^{4} - 866x^{3} - 5783x^{2} + 2328x + 9992 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + \beta_1 + 6) q^{4} - 5 q^{5} - 3 \beta_1 q^{6} + (\beta_{5} + \beta_{4} + \beta_{2} + \cdots + 7) q^{8}+ \cdots + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + \beta_1 + 6) q^{4} - 5 q^{5} - 3 \beta_1 q^{6} + (\beta_{5} + \beta_{4} + \beta_{2} + \cdots + 7) q^{8}+ \cdots + (9 \beta_{4} + 9 \beta_{3} - 9 \beta_{2} + \cdots + 63) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 24 q^{3} + 50 q^{4} - 40 q^{5} - 6 q^{6} + 66 q^{8} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 24 q^{3} + 50 q^{4} - 40 q^{5} - 6 q^{6} + 66 q^{8} + 72 q^{9} - 10 q^{10} + 64 q^{11} - 150 q^{12} + 120 q^{15} + 206 q^{16} - 48 q^{17} + 18 q^{18} - 80 q^{19} - 250 q^{20} + 452 q^{22} + 120 q^{23} - 198 q^{24} + 200 q^{25} - 272 q^{26} - 216 q^{27} + 76 q^{29} + 30 q^{30} - 20 q^{31} + 770 q^{32} - 192 q^{33} - 320 q^{34} + 450 q^{36} + 348 q^{37} + 236 q^{38} - 330 q^{40} - 944 q^{41} + 1116 q^{43} + 172 q^{44} - 360 q^{45} + 496 q^{46} + 208 q^{47} - 618 q^{48} + 50 q^{50} + 144 q^{51} + 2272 q^{52} + 1144 q^{53} - 54 q^{54} - 320 q^{55} + 240 q^{57} + 560 q^{58} - 596 q^{59} + 750 q^{60} - 740 q^{61} + 1184 q^{62} + 1298 q^{64} - 1356 q^{66} + 1964 q^{67} - 96 q^{68} - 360 q^{69} - 4 q^{71} + 594 q^{72} + 1500 q^{73} + 3368 q^{74} - 600 q^{75} + 2912 q^{76} + 816 q^{78} - 460 q^{79} - 1030 q^{80} + 648 q^{81} + 5644 q^{82} - 700 q^{83} + 240 q^{85} - 1396 q^{86} - 228 q^{87} + 6892 q^{88} - 90 q^{90} + 644 q^{92} + 60 q^{93} + 6692 q^{94} + 400 q^{95} - 2310 q^{96} + 2052 q^{97} + 576 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{7} + 109\nu^{6} - 844\nu^{5} - 4156\nu^{4} + 17325\nu^{3} + 45225\nu^{2} - 70072\nu - 136312 ) / 3088 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -61\nu^{7} + 119\nu^{6} + 2804\nu^{5} - 4084\nu^{4} - 30961\nu^{3} + 28523\nu^{2} - 6624\nu - 10056 ) / 6176 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 61\nu^{7} - 119\nu^{6} - 2804\nu^{5} + 4084\nu^{4} + 37137\nu^{3} - 34699\nu^{2} - 116896\nu + 53288 ) / 6176 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\nu^{7} - 113\nu^{6} - 924\nu^{5} + 4436\nu^{4} + 13415\nu^{3} - 45029\nu^{2} - 42080\nu + 102176 ) / 1544 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\nu^{7} - 113\nu^{6} - 924\nu^{5} + 5980\nu^{4} + 11871\nu^{3} - 88261\nu^{2} - 23552\nu + 264296 ) / 1544 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{2} + 21\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 29\beta_{2} + 37\beta _1 + 294 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} - 6\beta_{6} + 38\beta_{5} + 30\beta_{4} - 6\beta_{3} + 45\beta_{2} + 512\beta _1 + 312 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 41\beta_{7} - 58\beta_{6} + 66\beta_{5} + 46\beta_{4} + 4\beta_{3} + 752\beta_{2} + 1186\beta _1 + 7030 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 59\beta_{7} - 322\beta_{6} + 1301\beta_{5} + 793\beta_{4} - 268\beta_{3} + 1554\beta_{2} + 13072\beta _1 + 11201 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(1\)

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

Hecke kernels

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

\( T_{2}^{8} - 2T_{2}^{7} - 55T_{2}^{6} + 80T_{2}^{5} + 969T_{2}^{4} - 866T_{2}^{3} - 5783T_{2}^{2} + 2328T_{2} + 9992 \) Copy content Toggle raw display
\( T_{11}^{8} - 64 T_{11}^{7} - 3258 T_{11}^{6} + 304584 T_{11}^{5} - 4199384 T_{11}^{4} + \cdots - 9346926848 \) Copy content Toggle raw display
\( T_{13}^{8} - 10464 T_{13}^{6} - 36160 T_{13}^{5} + 22566928 T_{13}^{4} - 176712704 T_{13}^{3} + \cdots + 211872661504 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + \cdots + 9992 \) Copy content Toggle raw display
$3$ \( (T + 3)^{8} \) Copy content Toggle raw display
$5$ \( (T + 5)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots - 9346926848 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 211872661504 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots - 6565556795392 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots - 129732409433664 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 84\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 22\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots - 60\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 28\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots - 18\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 76\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 46\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots - 79\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 36\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 50\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 24\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots - 36\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots - 23\!\cdots\!48 \) Copy content Toggle raw display
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