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Properties

Label 786.4.a.e
Level $786$
Weight $4$
Character orbit 786.a
Self dual yes
Analytic conductor $46.376$
Analytic rank $1$
Dimension $7$
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] = [786,4,Mod(1,786)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(786, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("786.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 786 = 2 \cdot 3 \cdot 131 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 786.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: \(46.3755012645\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 240x^{5} + 928x^{4} + 13257x^{3} - 62028x^{2} - 103218x + 537912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + 3 q^{3} + 4 q^{4} + ( - \beta_{4} - \beta_{2} - 1) q^{5} - 6 q^{6} + (\beta_{6} + \beta_{5} - \beta_{3} - 7) q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 3 q^{3} + 4 q^{4} + ( - \beta_{4} - \beta_{2} - 1) q^{5} - 6 q^{6} + (\beta_{6} + \beta_{5} - \beta_{3} - 7) q^{7} - 8 q^{8} + 9 q^{9} + (2 \beta_{4} + 2 \beta_{2} + 2) q^{10} + (2 \beta_{6} + \beta_{5} + 3 \beta_{4} + \cdots - 2) q^{11}+ \cdots + (18 \beta_{6} + 9 \beta_{5} + \cdots - 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 14 q^{2} + 21 q^{3} + 28 q^{4} - q^{5} - 42 q^{6} - 47 q^{7} - 56 q^{8} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 14 q^{2} + 21 q^{3} + 28 q^{4} - q^{5} - 42 q^{6} - 47 q^{7} - 56 q^{8} + 63 q^{9} + 2 q^{10} - 27 q^{11} + 84 q^{12} - 41 q^{13} + 94 q^{14} - 3 q^{15} + 112 q^{16} + 17 q^{17} - 126 q^{18} - 151 q^{19} - 4 q^{20} - 141 q^{21} + 54 q^{22} + 87 q^{23} - 168 q^{24} + 102 q^{25} + 82 q^{26} + 189 q^{27} - 188 q^{28} + 8 q^{29} + 6 q^{30} - 314 q^{31} - 224 q^{32} - 81 q^{33} - 34 q^{34} - 88 q^{35} + 252 q^{36} - 697 q^{37} + 302 q^{38} - 123 q^{39} + 8 q^{40} - 697 q^{41} + 282 q^{42} - 1201 q^{43} - 108 q^{44} - 9 q^{45} - 174 q^{46} - 636 q^{47} + 336 q^{48} - 814 q^{49} - 204 q^{50} + 51 q^{51} - 164 q^{52} + 229 q^{53} - 378 q^{54} - 1856 q^{55} + 376 q^{56} - 453 q^{57} - 16 q^{58} - 954 q^{59} - 12 q^{60} - 1590 q^{61} + 628 q^{62} - 423 q^{63} + 448 q^{64} - 565 q^{65} + 162 q^{66} - 1215 q^{67} + 68 q^{68} + 261 q^{69} + 176 q^{70} + 912 q^{71} - 504 q^{72} - 620 q^{73} + 1394 q^{74} + 306 q^{75} - 604 q^{76} + 754 q^{77} + 246 q^{78} - 1778 q^{79} - 16 q^{80} + 567 q^{81} + 1394 q^{82} + 805 q^{83} - 564 q^{84} - 286 q^{85} + 2402 q^{86} + 24 q^{87} + 216 q^{88} + 1728 q^{89} + 18 q^{90} - 1283 q^{91} + 348 q^{92} - 942 q^{93} + 1272 q^{94} + 612 q^{95} - 672 q^{96} - 1426 q^{97} + 1628 q^{98} - 243 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 240x^{5} + 928x^{4} + 13257x^{3} - 62028x^{2} - 103218x + 537912 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 612725 \nu^{6} - 21169455 \nu^{5} - 28094940 \nu^{4} + 4289892286 \nu^{3} + \cdots - 512983239570 ) / 36079891086 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1566553 \nu^{6} + 7605303 \nu^{5} - 406310340 \nu^{4} - 836737736 \nu^{3} + 23719624023 \nu^{2} + \cdots - 65859911412 ) / 36079891086 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1686887 \nu^{6} - 7886254 \nu^{5} + 440637784 \nu^{4} + 1436299194 \nu^{3} + \cdots + 479783417559 ) / 18039945543 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5458372 \nu^{6} + 22138700 \nu^{5} - 1232181959 \nu^{4} - 3077092503 \nu^{3} + \cdots - 687498977115 ) / 18039945543 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11062732 \nu^{6} - 22430736 \nu^{5} + 2505709797 \nu^{4} + 2431172015 \nu^{3} + \cdots + 1151184139755 ) / 18039945543 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 3\beta_{5} - 6\beta_{3} + 2\beta_{2} + \beta _1 + 68 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 8\beta_{6} + 24\beta_{5} + 30\beta_{4} + 17\beta_{3} + 17\beta_{2} + 131\beta _1 - 121 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 147\beta_{6} + 520\beta_{5} + 146\beta_{4} - 1131\beta_{3} + 261\beta_{2} + 190\beta _1 + 8200 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 1918\beta_{6} + 4942\beta_{5} + 5033\beta_{4} + 4432\beta_{3} + 2410\beta_{2} + 18866\beta _1 - 25555 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17947\beta_{6} + 78273\beta_{5} + 29457\beta_{4} - 191900\beta_{3} + 34792\beta_{2} + 24577\beta _1 + 1198669 \) 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
−3.13478
−13.0993
12.1429
−8.54006
6.88143
3.49009
5.25964
−2.00000 3.00000 4.00000 −21.8074 −6.00000 −22.2603 −8.00000 9.00000 43.6149
1.2 −2.00000 3.00000 4.00000 −4.61712 −6.00000 4.67795 −8.00000 9.00000 9.23423
1.3 −2.00000 3.00000 4.00000 −4.40855 −6.00000 11.0333 −8.00000 9.00000 8.81710
1.4 −2.00000 3.00000 4.00000 −0.423836 −6.00000 7.11006 −8.00000 9.00000 0.847673
1.5 −2.00000 3.00000 4.00000 −0.0744044 −6.00000 −15.0634 −8.00000 9.00000 0.148809
1.6 −2.00000 3.00000 4.00000 14.6642 −6.00000 −7.81204 −8.00000 9.00000 −29.3285
1.7 −2.00000 3.00000 4.00000 15.6671 −6.00000 −24.6855 −8.00000 9.00000 −31.3342
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(131\) \(1\)

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 786.4.a.e ✓ 7
3.b odd 2 1 2358.4.a.g 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
786.4.a.e ✓ 7 1.a even 1 1 trivial
2358.4.a.g 7 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{7} + T_{5}^{6} - 488T_{5}^{5} + 697T_{5}^{4} + 36886T_{5}^{3} + 120163T_{5}^{2} + 51960T_{5} + 3216 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(786))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{7} \) Copy content Toggle raw display
$3$ \( (T - 3)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} + T^{6} + \cdots + 3216 \) Copy content Toggle raw display
$7$ \( T^{7} + 47 T^{6} + \cdots - 23729832 \) Copy content Toggle raw display
$11$ \( T^{7} + 27 T^{6} + \cdots + 630365625 \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots - 2036024406 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots - 99367068231 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots + 96615083988 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots + 84933443422968 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots - 550247549936958 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots - 269977041511034 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots - 38\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots + 80\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots + 102152462670612 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots + 39\!\cdots\!33 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots - 17\!\cdots\!82 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots - 81\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots - 30\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots - 58\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots - 15\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots - 17\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots - 81\!\cdots\!56 \) Copy content Toggle raw display
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