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Properties

Label 712.4.a.a
Level $712$
Weight $4$
Character orbit 712.a
Self dual yes
Analytic conductor $42.009$
Analytic rank $1$
Dimension $14$
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] = [712,4,Mod(1,712)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(712, 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("712.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 712 = 2^{3} \cdot 89 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 712.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: \(42.0093599241\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 3 x^{13} - 215 x^{12} + 523 x^{11} + 17984 x^{10} - 29814 x^{9} - 756606 x^{8} + \cdots + 604902904 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{4} - 1) q^{5} + ( - \beta_{6} - 2) q^{7} + (\beta_{4} + \beta_{3} + \beta_1 + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{4} - 1) q^{5} + ( - \beta_{6} - 2) q^{7} + (\beta_{4} + \beta_{3} + \beta_1 + 4) q^{9} + (\beta_{13} + \beta_{6} + \beta_{4} + \cdots + 2) q^{11}+ \cdots + (4 \beta_{13} + 3 \beta_{11} + \cdots - 61) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 3 q^{3} - 17 q^{5} - 28 q^{7} + 61 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 3 q^{3} - 17 q^{5} - 28 q^{7} + 61 q^{9} + 36 q^{11} - 108 q^{13} - 39 q^{15} - 125 q^{17} + 153 q^{19} - 56 q^{21} - 229 q^{23} + 201 q^{25} - 231 q^{27} - 526 q^{29} + 23 q^{31} - 488 q^{33} - 60 q^{35} - 542 q^{37} - 224 q^{39} - 728 q^{41} + 235 q^{43} - 1084 q^{45} - 874 q^{47} - 1042 q^{49} + 257 q^{51} - 1519 q^{53} - 1564 q^{55} - 2093 q^{57} - 534 q^{59} - 2068 q^{61} - 1380 q^{63} - 2116 q^{65} - 1438 q^{67} - 2689 q^{69} - 2558 q^{71} - 2177 q^{73} - 1526 q^{75} - 2368 q^{77} - 1230 q^{79} - 3294 q^{81} - 1242 q^{83} - 4253 q^{85} - 1782 q^{87} + 1246 q^{89} - 600 q^{91} - 5517 q^{93} - 671 q^{95} - 3451 q^{97} - 656 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} - 215 x^{12} + 523 x^{11} + 17984 x^{10} - 29814 x^{9} - 756606 x^{8} + \cdots + 604902904 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\!\cdots\!35 \nu^{13} + \cdots + 16\!\cdots\!16 ) / 23\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 18\!\cdots\!31 \nu^{13} + \cdots + 27\!\cdots\!80 ) / 10\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18\!\cdots\!31 \nu^{13} + \cdots - 27\!\cdots\!68 ) / 10\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 62\!\cdots\!95 \nu^{13} + \cdots - 89\!\cdots\!04 ) / 10\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13\!\cdots\!05 \nu^{13} + \cdots - 19\!\cdots\!60 ) / 20\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18\!\cdots\!50 \nu^{13} + \cdots + 27\!\cdots\!46 ) / 26\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 64\!\cdots\!53 \nu^{13} + \cdots + 93\!\cdots\!28 ) / 69\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 30\!\cdots\!07 \nu^{13} + \cdots + 44\!\cdots\!76 ) / 20\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 47\!\cdots\!07 \nu^{13} + \cdots + 69\!\cdots\!90 ) / 26\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 44\!\cdots\!71 \nu^{13} + \cdots - 64\!\cdots\!20 ) / 20\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 19\!\cdots\!76 \nu^{13} + \cdots + 28\!\cdots\!00 ) / 86\!\cdots\!79 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 58\!\cdots\!19 \nu^{13} + \cdots - 85\!\cdots\!48 ) / 23\!\cdots\!44 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + \beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 3 \beta_{12} - 4 \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - 2 \beta_{3} + \cdots + 17 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{13} - \beta_{12} + 8 \beta_{11} + 3 \beta_{10} + 7 \beta_{9} + 9 \beta_{8} + 6 \beta_{7} + \cdots + 1553 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{13} - 239 \beta_{12} - 407 \beta_{11} - 154 \beta_{10} + 82 \beta_{9} - 107 \beta_{8} + \cdots + 495 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 419 \beta_{13} - 49 \beta_{12} + 932 \beta_{11} + 368 \beta_{10} + 729 \beta_{9} + 971 \beta_{8} + \cdots + 92851 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1555 \beta_{13} - 16648 \beta_{12} - 33933 \beta_{11} - 15332 \beta_{10} + 5796 \beta_{9} + \cdots - 24280 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 47497 \beta_{13} + 1253 \beta_{12} + 91402 \beta_{11} + 37810 \beta_{10} + 61390 \beta_{9} + \cdots + 6024203 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 174158 \beta_{13} - 1144203 \beta_{12} - 2682758 \beta_{11} - 1326620 \beta_{10} + 390920 \beta_{9} + \cdots - 5676790 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4433674 \beta_{13} + 559922 \beta_{12} + 8362268 \beta_{11} + 3611484 \beta_{10} + 4796254 \beta_{9} + \cdots + 408752767 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 16690350 \beta_{13} - 79353753 \beta_{12} - 207874274 \beta_{11} - 108015978 \beta_{10} + \cdots - 667285527 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 379272858 \beta_{13} + 79388996 \beta_{12} + 733180217 \beta_{11} + 328418491 \beta_{10} + \cdots + 28563941230 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1488615209 \beta_{13} - 5578597178 \beta_{12} - 15960702740 \beta_{11} - 8530728431 \beta_{10} + \cdots - 65660729580 \) 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
8.34630
7.53948
7.14136
5.71307
4.65267
3.49480
−0.848775
−1.06746
−3.47256
−4.17367
−4.17437
−4.20645
−7.12761
−8.81678
0 −8.34630 0 −4.33865 0 −23.8898 0 42.6608 0
1.2 0 −7.53948 0 10.7801 0 21.2352 0 29.8438 0
1.3 0 −7.14136 0 −20.9731 0 −7.92224 0 23.9991 0
1.4 0 −5.71307 0 2.90146 0 5.35991 0 5.63914 0
1.5 0 −4.65267 0 −11.7642 0 −3.78992 0 −5.35266 0
1.6 0 −3.49480 0 17.1557 0 7.37477 0 −14.7863 0
1.7 0 0.848775 0 2.21394 0 −21.9499 0 −26.2796 0
1.8 0 1.06746 0 12.8792 0 −18.8997 0 −25.8605 0
1.9 0 3.47256 0 6.23252 0 −2.68189 0 −14.9413 0
1.10 0 4.17367 0 −18.9159 0 9.76438 0 −9.58048 0
1.11 0 4.17437 0 −13.9544 0 25.9958 0 −9.57461 0
1.12 0 4.20645 0 3.52757 0 17.5429 0 −9.30578 0
1.13 0 7.12761 0 7.47653 0 −18.1075 0 23.8028 0
1.14 0 8.81678 0 −10.2207 0 −18.0321 0 50.7357 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(89\) \(-1\)

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 712.4.a.a ✓ 14
4.b odd 2 1 1424.4.a.k 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
712.4.a.a ✓ 14 1.a even 1 1 trivial
1424.4.a.k 14 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 3 T_{3}^{13} - 215 T_{3}^{12} - 523 T_{3}^{11} + 17984 T_{3}^{10} + 29814 T_{3}^{9} + \cdots + 604902904 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(712))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 604902904 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 7263414334520 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 973941092732928 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 71\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 24\!\cdots\!20 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 15\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 11\!\cdots\!48 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 11\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 51\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 57\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 26\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 62\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 83\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 25\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 57\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 39\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 31\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T - 89)^{14} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 43\!\cdots\!48 \) Copy content Toggle raw display
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