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Properties

Label 735.4.a.bc
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.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(735))\):

\( T_{2}^{8} - 2T_{2}^{7} - 55T_{2}^{6} + 80T_{2}^{5} + 969T_{2}^{4} - 866T_{2}^{3} - 5783T_{2}^{2} + 2328T_{2} + 9992 \) 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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