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Properties

Label 1629.4.a.b
Level $1629$
Weight $4$
Character orbit 1629.a
Self dual yes
Analytic conductor $96.114$
Analytic rank $1$
Dimension $17$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1629,4,Mod(1,1629)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1629.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1629, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1629 = 3^{2} \cdot 181 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1629.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [17] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(96.1141113994\)
Analytic rank: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 5 x^{16} - 71 x^{15} + 350 x^{14} + 1993 x^{13} - 9702 x^{12} - 28280 x^{11} + 136606 x^{10} + \cdots - 19616 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 543)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 2) q^{4} + (\beta_{11} - \beta_1 + 2) q^{5} + ( - \beta_{16} + \beta_{11} - \beta_{2} - 4) q^{7} + (\beta_{13} - 2 \beta_{11} - \beta_{10} + \cdots + 3) q^{8}+ \cdots + ( - 7 \beta_{16} + 33 \beta_{15} + \cdots - 122) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 5 q^{2} + 31 q^{4} + 21 q^{5} - 62 q^{7} + 60 q^{8} - 127 q^{10} + 65 q^{11} - 246 q^{13} + 121 q^{14} - 85 q^{16} + 433 q^{17} - 459 q^{19} + 208 q^{20} - 270 q^{22} + 165 q^{23} - 146 q^{25} + 89 q^{26}+ \cdots - 1803 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 5 x^{16} - 71 x^{15} + 350 x^{14} + 1993 x^{13} - 9702 x^{12} - 28280 x^{11} + 136606 x^{10} + \cdots - 19616 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 34\!\cdots\!49 \nu^{16} + \cdots - 82\!\cdots\!80 ) / 47\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23\!\cdots\!81 \nu^{16} + \cdots - 63\!\cdots\!08 ) / 23\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 34\!\cdots\!63 \nu^{16} + \cdots + 10\!\cdots\!84 ) / 23\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{16} - 6 \nu^{15} - 65 \nu^{14} + 415 \nu^{13} + 1578 \nu^{12} - 11280 \nu^{11} - 17000 \nu^{10} + \cdots + 1094496 ) / 65536 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 87\!\cdots\!11 \nu^{16} + \cdots - 19\!\cdots\!44 ) / 47\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 25\!\cdots\!85 \nu^{16} + \cdots + 11\!\cdots\!20 ) / 47\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 26\!\cdots\!25 \nu^{16} + \cdots + 25\!\cdots\!04 ) / 47\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 27\!\cdots\!79 \nu^{16} + \cdots + 65\!\cdots\!40 ) / 47\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 27\!\cdots\!07 \nu^{16} + \cdots + 32\!\cdots\!00 ) / 47\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 31\!\cdots\!19 \nu^{16} + \cdots - 75\!\cdots\!76 ) / 47\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 43\!\cdots\!95 \nu^{16} + \cdots - 65\!\cdots\!92 ) / 47\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 31\!\cdots\!99 \nu^{16} + \cdots - 80\!\cdots\!60 ) / 23\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 17\!\cdots\!89 \nu^{16} + \cdots + 21\!\cdots\!00 ) / 11\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 13\!\cdots\!89 \nu^{16} + \cdots + 29\!\cdots\!92 ) / 47\!\cdots\!76 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - 2\beta_{11} - \beta_{10} + \beta_{9} - \beta_{7} - \beta_{4} + \beta_{2} + 18\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - \beta_{14} + 2 \beta_{13} + \beta_{12} - 4 \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \cdots + 168 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{16} + 2 \beta_{15} - 4 \beta_{14} + 29 \beta_{13} + \beta_{12} - 63 \beta_{11} - 40 \beta_{10} + \cdots + 149 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12 \beta_{16} + 54 \beta_{15} - 44 \beta_{14} + 83 \beta_{13} + 36 \beta_{12} - 176 \beta_{11} + \cdots + 3423 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 170 \beta_{16} + 157 \beta_{15} - 207 \beta_{14} + 759 \beta_{13} + 45 \beta_{12} - 1802 \beta_{11} + \cdots + 5377 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 741 \beta_{16} + 2177 \beta_{15} - 1499 \beta_{14} + 2810 \beta_{13} + 1000 \beta_{12} - 6217 \beta_{11} + \cdots + 78258 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6727 \beta_{16} + 7665 \beta_{15} - 7779 \beta_{14} + 20452 \beta_{13} + 1672 \beta_{12} - 51561 \beta_{11} + \cdots + 174922 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 31131 \beta_{16} + 76731 \beta_{15} - 47393 \beta_{14} + 89311 \beta_{13} + 25988 \beta_{12} + \cdots + 1926597 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 231583 \beta_{16} + 303546 \beta_{15} - 259494 \beta_{14} + 573006 \beta_{13} + 56745 \beta_{12} + \cdots + 5459320 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1117706 \beta_{16} + 2513725 \beta_{15} - 1459211 \beta_{14} + 2758369 \beta_{13} + 669981 \beta_{12} + \cdots + 49881825 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 7454407 \beta_{16} + 10768041 \beta_{15} - 8186521 \beta_{14} + 16492233 \beta_{13} + 1820174 \beta_{12} + \cdots + 167172973 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 37073207 \beta_{16} + 78971714 \beta_{15} - 44393518 \beta_{14} + 83855661 \beta_{13} + \cdots + 1338309611 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 231495158 \beta_{16} + 358387120 \beta_{15} - 250899374 \beta_{14} + 481917536 \beta_{13} + \cdots + 5067727640 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1176668636 \beta_{16} + 2420432688 \beta_{15} - 1340747114 \beta_{14} + 2524747606 \beta_{13} + \cdots + 36838732096 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−4.72315
−3.94169
−3.40834
−3.20897
−2.38067
−2.16787
−0.471795
−0.316194
0.0290924
0.896539
2.22090
2.26849
3.00134
3.15430
4.10282
4.51242
5.43277
−4.72315 0 14.3081 16.8438 0 −9.67650 −29.7943 0 −79.5556
1.2 −3.94169 0 7.53692 4.29766 0 −28.8908 1.82531 0 −16.9401
1.3 −3.40834 0 3.61680 14.7565 0 3.97638 14.9394 0 −50.2951
1.4 −3.20897 0 2.29750 −10.0377 0 −24.0685 18.2992 0 32.2108
1.5 −2.38067 0 −2.33241 4.22127 0 18.3952 24.5981 0 −10.0495
1.6 −2.16787 0 −3.30035 −5.97361 0 2.39713 24.4977 0 12.9500
1.7 −0.471795 0 −7.77741 7.35814 0 10.1800 7.44370 0 −3.47153
1.8 −0.316194 0 −7.90002 −17.3963 0 −26.6761 5.02749 0 5.50060
1.9 0.0290924 0 −7.99915 8.65432 0 5.99540 −0.465454 0 0.251775
1.10 0.896539 0 −7.19622 −2.78179 0 −13.3798 −13.6240 0 −2.49399
1.11 2.22090 0 −3.06759 −12.5951 0 24.1343 −24.5800 0 −27.9724
1.12 2.26849 0 −2.85395 19.5441 0 −6.50289 −24.6221 0 44.3357
1.13 3.00134 0 1.00807 −9.81976 0 −20.6701 −20.9852 0 −29.4725
1.14 3.15430 0 1.94961 11.7085 0 5.77387 −19.0847 0 36.9322
1.15 4.10282 0 8.83313 3.39405 0 18.0677 3.41817 0 13.9252
1.16 4.51242 0 12.3619 −8.52997 0 −0.164119 19.6828 0 −38.4908
1.17 5.43277 0 21.5150 −2.64409 0 −20.8910 73.4240 0 −14.3648
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(181\) \( +1 \)

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 1629.4.a.b 17
3.b odd 2 1 543.4.a.a ✓ 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
543.4.a.a ✓ 17 3.b odd 2 1
1629.4.a.b 17 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} - 5 T_{2}^{16} - 71 T_{2}^{15} + 350 T_{2}^{14} + 1993 T_{2}^{13} - 9702 T_{2}^{12} + \cdots - 19616 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1629))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} - 5 T^{16} + \cdots - 19616 \) Copy content Toggle raw display
$3$ \( T^{17} \) Copy content Toggle raw display
$5$ \( T^{17} + \cdots - 18\!\cdots\!04 \) Copy content Toggle raw display
$7$ \( T^{17} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{17} + \cdots - 29\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{17} + \cdots - 22\!\cdots\!20 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots - 47\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots + 51\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots + 19\!\cdots\!60 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots - 96\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots - 19\!\cdots\!80 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots + 32\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots + 99\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots + 51\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots - 14\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots - 27\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots - 85\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots + 63\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots - 68\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots + 41\!\cdots\!71 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots - 23\!\cdots\!16 \) Copy content Toggle raw display
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