Properties

Label 712.4.a.c
Level $712$
Weight $4$
Character orbit 712.a
Self dual yes
Analytic conductor $42.009$
Analytic rank $1$
Dimension $17$
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: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 8 x^{16} - 270 x^{15} + 2004 x^{14} + 29685 x^{13} - 201006 x^{12} - 1715610 x^{11} + 10238540 x^{10} + 56474974 x^{9} - 276273174 x^{8} + \cdots + 31547006496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18} \)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} + (\beta_{9} - 1) q^{5} + ( - \beta_{4} - 1) q^{7} + (\beta_{2} - \beta_1 + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} + (\beta_{9} - 1) q^{5} + ( - \beta_{4} - 1) q^{7} + (\beta_{2} - \beta_1 + 9) q^{9} + ( - \beta_{9} + \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 - 7) q^{11} + (\beta_{11} - \beta_{9} - \beta_{5} + \beta_{4} + 1) q^{13} + ( - \beta_{11} - 3 \beta_{9} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{2} - 2 \beta_1 - 5) q^{15} + ( - \beta_{11} - 2 \beta_{9} - \beta_{8} + \beta_{7} + \beta_{4} - 5 \beta_1 - 2) q^{17} + ( - \beta_{12} - 2 \beta_{9} + \beta_{8} + \beta_{7} - \beta_{2} - 4 \beta_1 - 15) q^{19} + (\beta_{15} + 2 \beta_{12} - 3 \beta_{9} + 2 \beta_{8} + \beta_{7} + \beta_{6} + 4 \beta_{4} - 2 \beta_{2} + \cdots - 3) q^{21}+ \cdots + (5 \beta_{16} + 10 \beta_{15} - 11 \beta_{14} - 15 \beta_{13} + 29 \beta_{12} + \cdots - 538) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q - 9 q^{3} - 13 q^{5} - 24 q^{7} + 146 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q - 9 q^{3} - 13 q^{5} - 24 q^{7} + 146 q^{9} - 124 q^{11} + 22 q^{13} - 113 q^{15} - 81 q^{17} - 281 q^{19} - 100 q^{21} - 229 q^{23} + 400 q^{25} - 339 q^{27} + 16 q^{29} - 59 q^{31} - 226 q^{33} - 668 q^{35} - 186 q^{37} - 156 q^{39} - 60 q^{41} - 833 q^{43} - 642 q^{45} - 1018 q^{47} - 315 q^{49} - 2305 q^{51} - 1391 q^{53} - 1644 q^{55} - 1715 q^{57} - 2456 q^{59} - 66 q^{61} - 2146 q^{63} - 1826 q^{65} - 3298 q^{67} - 3255 q^{69} - 2306 q^{71} - 1869 q^{73} - 3836 q^{75} - 1520 q^{77} - 1534 q^{79} - 775 q^{81} - 3396 q^{83} - 1835 q^{85} - 4786 q^{87} - 1513 q^{89} - 4896 q^{91} - 543 q^{93} - 5181 q^{95} + 585 q^{97} - 8968 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 8 x^{16} - 270 x^{15} + 2004 x^{14} + 29685 x^{13} - 201006 x^{12} - 1715610 x^{11} + 10238540 x^{10} + 56474974 x^{9} - 276273174 x^{8} + \cdots + 31547006496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 35 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 26\!\cdots\!47 \nu^{16} + \cdots + 12\!\cdots\!92 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11\!\cdots\!77 \nu^{16} + \cdots - 10\!\cdots\!92 ) / 67\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 23\!\cdots\!06 \nu^{16} + \cdots - 27\!\cdots\!36 ) / 11\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24\!\cdots\!54 \nu^{16} + \cdots - 60\!\cdots\!54 ) / 84\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11\!\cdots\!61 \nu^{16} + \cdots - 35\!\cdots\!36 ) / 33\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!37 \nu^{16} + \cdots - 18\!\cdots\!52 ) / 33\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 75\!\cdots\!35 \nu^{16} + \cdots + 14\!\cdots\!08 ) / 20\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 76\!\cdots\!59 \nu^{16} + \cdots + 97\!\cdots\!68 ) / 20\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 58\!\cdots\!67 \nu^{16} + \cdots + 95\!\cdots\!92 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 30\!\cdots\!28 \nu^{16} + \cdots + 54\!\cdots\!88 ) / 50\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 12\!\cdots\!12 \nu^{16} + \cdots + 18\!\cdots\!76 ) / 20\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11\!\cdots\!57 \nu^{16} + \cdots - 15\!\cdots\!52 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 16\!\cdots\!46 \nu^{16} + \cdots + 27\!\cdots\!16 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 20\!\cdots\!31 \nu^{16} + \cdots - 34\!\cdots\!76 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 35 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} + 2 \beta_{15} + \beta_{14} + 2 \beta_{11} - 3 \beta_{9} - \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 59 \beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4 \beta_{16} + 2 \beta_{15} + 6 \beta_{14} + 13 \beta_{13} - 7 \beta_{12} + 3 \beta_{11} + 6 \beta_{10} + 8 \beta_{8} - 12 \beta_{7} - 2 \beta_{6} + 6 \beta_{5} - 22 \beta_{4} - 25 \beta_{3} + 92 \beta_{2} + 153 \beta _1 + 2060 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 109 \beta_{16} + 210 \beta_{15} + 132 \beta_{14} + 106 \beta_{13} - 23 \beta_{12} + 158 \beta_{11} + 21 \beta_{10} - 324 \beta_{9} + 105 \beta_{8} - 175 \beta_{7} + 91 \beta_{6} + 74 \beta_{5} - 105 \beta_{4} - 353 \beta_{3} + \cdots + 4678 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 685 \beta_{16} + 584 \beta_{15} + 1181 \beta_{14} + 2164 \beta_{13} - 1035 \beta_{12} + 324 \beta_{11} + 882 \beta_{10} + 56 \beta_{9} + 1354 \beta_{8} - 1901 \beta_{7} + 30 \beta_{6} + 733 \beta_{5} - 3400 \beta_{4} + \cdots + 148043 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11432 \beta_{16} + 20636 \beta_{15} + 16233 \beta_{14} + 19257 \beta_{13} - 5577 \beta_{12} + 11465 \beta_{11} + 5061 \beta_{10} - 27555 \beta_{9} + 17622 \beta_{8} - 22859 \beta_{7} + 8829 \beta_{6} + \cdots + 620723 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 88982 \beta_{16} + 95273 \beta_{15} + 164687 \beta_{14} + 278686 \beta_{13} - 126808 \beta_{12} + 29812 \beta_{11} + 109294 \beta_{10} + 2725 \beta_{9} + 179760 \beta_{8} - 241784 \beta_{7} + \cdots + 12323181 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1209991 \beta_{16} + 2061547 \beta_{15} + 1932465 \beta_{14} + 2617900 \beta_{13} - 892474 \beta_{12} + 839799 \beta_{11} + 798907 \beta_{10} - 2185008 \beta_{9} + 2236999 \beta_{8} + \cdots + 76279299 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 10624786 \beta_{16} + 12943849 \beta_{15} + 20355173 \beta_{14} + 32908451 \beta_{13} - 14593321 \beta_{12} + 2758802 \beta_{11} + 12772793 \beta_{10} - 679405 \beta_{9} + \cdots + 1147236474 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 129831442 \beta_{16} + 211883773 \beta_{15} + 224821964 \beta_{14} + 320983303 \beta_{13} - 120179830 \beta_{12} + 63927801 \beta_{11} + 107023631 \beta_{10} + \cdots + 8970823073 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1225751455 \beta_{16} + 1616620317 \beta_{15} + 2390328038 \beta_{14} + 3752811065 \beta_{13} - 1639772688 \beta_{12} + 266883114 \beta_{11} + 1452920802 \beta_{10} + \cdots + 115228114048 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 14104753880 \beta_{16} + 22346301287 \beta_{15} + 25753222449 \beta_{14} + 37590849184 \beta_{13} - 14867147389 \beta_{12} + 5148285784 \beta_{11} + \cdots + 1030127032384 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 139199636797 \beta_{16} + 192949436693 \beta_{15} + 274005238563 \beta_{14} + 421898390080 \beta_{13} - 182735655155 \beta_{12} + 27096967259 \beta_{11} + \cdots + 12124016943267 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1546982050436 \beta_{16} + 2403471170977 \beta_{15} + 2920838688738 \beta_{14} + 4304906082101 \beta_{13} - 1758424527424 \beta_{12} + \cdots + 116789912825559 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 15686348712801 \beta_{16} + 22431268705131 \beta_{15} + 31030919441578 \beta_{14} + 47169838331755 \beta_{13} - 20324180151634 \beta_{12} + \cdots + 13\!\cdots\!42 \) Copy content Toggle raw display

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
−8.14902
−7.35383
−7.31322
−6.98412
−3.96761
−2.93690
−1.90936
−1.88737
−0.331336
1.61252
3.57394
5.70735
5.80074
6.40738
7.20736
7.95543
10.5680
0 −9.14902 0 17.1383 0 8.96361 0 56.7045 0
1.2 0 −8.35383 0 12.5667 0 −25.5549 0 42.7865 0
1.3 0 −8.31322 0 −15.9287 0 19.0592 0 42.1097 0
1.4 0 −7.98412 0 −15.8056 0 −31.0443 0 36.7462 0
1.5 0 −4.96761 0 7.37046 0 −11.0966 0 −2.32289 0
1.6 0 −3.93690 0 −4.29869 0 8.63544 0 −11.5008 0
1.7 0 −2.90936 0 12.7069 0 28.2050 0 −18.5356 0
1.8 0 −2.88737 0 −17.5713 0 19.8110 0 −18.6631 0
1.9 0 −1.33134 0 −9.94223 0 −6.25939 0 −25.2275 0
1.10 0 0.612524 0 9.90092 0 −12.2964 0 −26.6248 0
1.11 0 2.57394 0 0.163399 0 27.8111 0 −20.3748 0
1.12 0 4.70735 0 20.8629 0 −27.9281 0 −4.84085 0
1.13 0 4.80074 0 −6.96804 0 4.65735 0 −3.95286 0
1.14 0 5.40738 0 0.569841 0 −5.34520 0 2.23974 0
1.15 0 6.20736 0 1.65217 0 −14.5313 0 11.5314 0
1.16 0 6.95543 0 −15.8718 0 −7.98935 0 21.3780 0
1.17 0 9.56803 0 −9.54536 0 0.902912 0 64.5473 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
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.c 17
4.b odd 2 1 1424.4.a.m 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
712.4.a.c 17 1.a even 1 1 trivial
1424.4.a.m 17 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{17} + 9 T_{3}^{16} - 262 T_{3}^{15} - 2326 T_{3}^{14} + 27291 T_{3}^{13} + 236041 T_{3}^{12} - 1473914 T_{3}^{11} - 12259078 T_{3}^{10} + 44098727 T_{3}^{9} + 352568779 T_{3}^{8} + \cdots + 88306808312 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(712))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} \) Copy content Toggle raw display
$3$ \( T^{17} + 9 T^{16} + \cdots + 88306808312 \) Copy content Toggle raw display
$5$ \( T^{17} + \cdots - 127930736530512 \) Copy content Toggle raw display
$7$ \( T^{17} + 24 T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{17} + 124 T^{16} + \cdots + 44\!\cdots\!52 \) Copy content Toggle raw display
$13$ \( T^{17} - 22 T^{16} + \cdots + 14\!\cdots\!48 \) Copy content Toggle raw display
$17$ \( T^{17} + 81 T^{16} + \cdots + 14\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( T^{17} + 281 T^{16} + \cdots - 23\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{17} + 229 T^{16} + \cdots + 10\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{17} - 16 T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{17} + 59 T^{16} + \cdots - 36\!\cdots\!88 \) Copy content Toggle raw display
$37$ \( T^{17} + 186 T^{16} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{17} + 60 T^{16} + \cdots + 82\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{17} + 833 T^{16} + \cdots - 73\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{17} + 1018 T^{16} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{17} + 1391 T^{16} + \cdots + 85\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{17} + 2456 T^{16} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{17} + 66 T^{16} + \cdots - 35\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{17} + 3298 T^{16} + \cdots + 65\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{17} + 2306 T^{16} + \cdots - 32\!\cdots\!08 \) Copy content Toggle raw display
$73$ \( T^{17} + 1869 T^{16} + \cdots - 63\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{17} + 1534 T^{16} + \cdots + 34\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{17} + 3396 T^{16} + \cdots + 54\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( (T + 89)^{17} \) Copy content Toggle raw display
$97$ \( T^{17} - 585 T^{16} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
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