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Properties

Label 712.4.a.b
Level $712$
Weight $4$
Character orbit 712.a
Self dual yes
Analytic conductor $42.009$
Analytic rank $0$
Dimension $16$
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: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} - 286 x^{14} + 154 x^{13} + 31801 x^{12} - 7913 x^{11} - 1747636 x^{10} + \cdots - 136223856 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{19} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} - \beta_{4} q^{5} + \beta_{6} q^{7} + (\beta_{2} - \beta_1 + 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} - \beta_{4} q^{5} + \beta_{6} q^{7} + (\beta_{2} - \beta_1 + 10) q^{9} + ( - \beta_{5} - \beta_{4} + 6) q^{11} + (\beta_{13} + \beta_{6} - \beta_1 + 1) q^{13} + ( - \beta_{15} + \beta_{13} - \beta_{11} + \cdots + 2) q^{15}+ \cdots + ( - 8 \beta_{14} + 11 \beta_{13} + \cdots + 538) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 15 q^{3} + 7 q^{5} + 4 q^{7} + 155 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 15 q^{3} + 7 q^{5} + 4 q^{7} + 155 q^{9} + 96 q^{11} + 22 q^{13} + 37 q^{15} + 55 q^{17} + 327 q^{19} - 184 q^{21} + 323 q^{23} + 375 q^{25} + 525 q^{27} + 248 q^{29} + 127 q^{31} + 104 q^{33} + 732 q^{35} - 112 q^{37} + 312 q^{39} + 104 q^{41} + 1145 q^{43} + 516 q^{45} + 822 q^{47} + 1664 q^{49} + 1717 q^{51} + 1517 q^{53} + 1476 q^{55} + 2143 q^{57} + 2164 q^{59} + 606 q^{61} + 1644 q^{63} + 1864 q^{65} + 2942 q^{67} + 1599 q^{69} + 1250 q^{71} + 359 q^{73} + 3528 q^{75} + 2560 q^{77} + 2186 q^{79} + 2896 q^{81} + 1444 q^{83} + 1827 q^{85} + 2778 q^{87} + 1424 q^{89} + 4392 q^{91} - 1069 q^{93} + 3149 q^{95} + 69 q^{97} + 8436 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} - 286 x^{14} + 154 x^{13} + 31801 x^{12} - 7913 x^{11} - 1747636 x^{10} + \cdots - 136223856 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 16\!\cdots\!95 \nu^{15} + \cdots - 19\!\cdots\!36 ) / 31\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 88\!\cdots\!19 \nu^{15} + \cdots + 19\!\cdots\!08 ) / 15\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 93\!\cdots\!43 \nu^{15} + \cdots + 15\!\cdots\!04 ) / 15\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 34\!\cdots\!34 \nu^{15} + \cdots + 11\!\cdots\!56 ) / 52\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12\!\cdots\!15 \nu^{15} + \cdots - 40\!\cdots\!28 ) / 79\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 56\!\cdots\!11 \nu^{15} + \cdots - 53\!\cdots\!76 ) / 35\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 71\!\cdots\!91 \nu^{15} + \cdots - 19\!\cdots\!52 ) / 31\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 43\!\cdots\!28 \nu^{15} + \cdots + 18\!\cdots\!12 ) / 15\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 91\!\cdots\!65 \nu^{15} + \cdots - 82\!\cdots\!00 ) / 31\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 56\!\cdots\!58 \nu^{15} + \cdots + 47\!\cdots\!04 ) / 15\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13\!\cdots\!89 \nu^{15} + \cdots + 92\!\cdots\!72 ) / 31\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 26\!\cdots\!85 \nu^{15} + \cdots + 29\!\cdots\!20 ) / 52\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 42\!\cdots\!69 \nu^{15} + \cdots + 14\!\cdots\!00 ) / 79\!\cdots\!68 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{14} + 2 \beta_{13} - \beta_{11} + \beta_{10} - \beta_{9} + 3 \beta_{8} + \beta_{7} + \cdots + 21 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 5 \beta_{15} - 8 \beta_{14} + 21 \beta_{13} + 15 \beta_{12} - 14 \beta_{11} + \beta_{10} + \cdots + 2327 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 116 \beta_{15} - 161 \beta_{14} + 325 \beta_{13} + 87 \beta_{12} - 128 \beta_{11} + 125 \beta_{10} + \cdots + 4176 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 491 \beta_{15} - 1275 \beta_{14} + 3649 \beta_{13} + 2367 \beta_{12} - 2079 \beta_{11} + \cdots + 179491 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11062 \beta_{15} - 20391 \beta_{14} + 42148 \beta_{13} + 16164 \beta_{12} - 15540 \beta_{11} + \cdots + 563191 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 32351 \beta_{15} - 164280 \beta_{14} + 477184 \beta_{13} + 292506 \beta_{12} - 246144 \beta_{11} + \cdots + 15310606 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 996078 \beta_{15} - 2357963 \beta_{14} + 5036873 \beta_{13} + 2234898 \beta_{12} - 1862792 \beta_{11} + \cdots + 68029564 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1379887 \beta_{15} - 19679274 \beta_{14} + 56415734 \beta_{13} + 33348582 \beta_{12} + \cdots + 1396574318 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 89328288 \beta_{15} - 261513121 \beta_{14} + 577330963 \beta_{13} + 275909325 \beta_{12} + \cdots + 7865853011 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 26189117 \beta_{15} - 2269703131 \beta_{14} + 6372623193 \beta_{13} + 3675242730 \beta_{12} + \cdots + 133585496518 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8161229947 \beta_{15} - 28430745376 \beta_{14} + 64612490672 \beta_{13} + 32211956769 \beta_{12} + \cdots + 889868927889 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 16340348226 \beta_{15} - 255843972196 \beta_{14} + 703602008344 \beta_{13} + 398793540930 \beta_{12} + \cdots + 13222362670148 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 765980982318 \beta_{15} - 3062278223531 \beta_{14} + 7128373925561 \beta_{13} + 3646697650389 \beta_{12} + \cdots + 99360747577243 \) 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
10.3767
8.14715
7.96706
5.10621
4.74681
2.68209
1.64796
0.864408
0.0255279
−1.42838
−3.58491
−5.69957
−5.76508
−6.91478
−8.16281
−9.00841
0 −9.37674 0 −5.60860 0 10.4511 0 60.9233 0
1.2 0 −7.14715 0 9.07481 0 34.3570 0 24.0818 0
1.3 0 −6.96706 0 −4.67234 0 −4.80746 0 21.5399 0
1.4 0 −4.10621 0 0.305391 0 2.18013 0 −10.1391 0
1.5 0 −3.74681 0 16.2199 0 −14.3250 0 −12.9614 0
1.6 0 −1.68209 0 −2.54863 0 −35.1918 0 −24.1706 0
1.7 0 −0.647957 0 −20.3225 0 −12.2567 0 −26.5802 0
1.8 0 0.135592 0 18.7953 0 8.88526 0 −26.9816 0
1.9 0 0.974472 0 −10.1860 0 21.6359 0 −26.0504 0
1.10 0 2.42838 0 −4.92221 0 −26.2744 0 −21.1030 0
1.11 0 4.58491 0 14.3100 0 16.9529 0 −5.97859 0
1.12 0 6.69957 0 −20.5692 0 −15.7852 0 17.8843 0
1.13 0 6.76508 0 9.85117 0 6.06482 0 18.7663 0
1.14 0 7.91478 0 −11.7947 0 31.1281 0 35.6437 0
1.15 0 9.16281 0 13.3823 0 15.7921 0 56.9572 0
1.16 0 10.0084 0 5.68541 0 −34.8067 0 73.1684 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
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.b ✓ 16
4.b odd 2 1 1424.4.a.l 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
712.4.a.b ✓ 16 1.a even 1 1 trivial
1424.4.a.l 16 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 15 T_{3}^{15} - 181 T_{3}^{14} + 3395 T_{3}^{13} + 9142 T_{3}^{12} - 284610 T_{3}^{11} + \cdots - 378910400 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(712))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots - 378910400 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 149605912308640 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots - 32\!\cdots\!40 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots - 37\!\cdots\!88 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots - 10\!\cdots\!80 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots - 26\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots - 53\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 13\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 59\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 18\!\cdots\!88 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots - 50\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 18\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 55\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 33\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots - 23\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots - 15\!\cdots\!40 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots - 54\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 96\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 21\!\cdots\!80 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots - 41\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots - 50\!\cdots\!80 \) Copy content Toggle raw display
$89$ \( (T - 89)^{16} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots - 13\!\cdots\!32 \) Copy content Toggle raw display
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