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Properties

Label 12.17.d.a
Level $12$
Weight $17$
Character orbit 12.d
Analytic conductor $19.479$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [12,17,Mod(7,12)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(12, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("12.7");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 12.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(19.4789452628\)
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: \( x^{16} - x^{15} + 37115 x^{14} + 433616 x^{13} + 965822723 x^{12} + 11579264195 x^{11} + \cdots + 43\!\cdots\!49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{106}\cdot 3^{51} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 12) q^{2} + ( - \beta_{3} + \beta_1) q^{3} + ( - \beta_{4} + 2 \beta_{3} + \cdots + 8542) q^{4}+ \cdots - 14348907 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 12) q^{2} + ( - \beta_{3} + \beta_1) q^{3} + ( - \beta_{4} + 2 \beta_{3} + \cdots + 8542) q^{4}+ \cdots + ( - 43046721 \beta_{15} + \cdots - 1149964453701) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 186 q^{2} + 136588 q^{4} + 354144 q^{5} + 1561518 q^{6} + 14683680 q^{8} - 229582512 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 186 q^{2} + 136588 q^{4} + 354144 q^{5} + 1561518 q^{6} + 14683680 q^{8} - 229582512 q^{9} - 49800172 q^{10} + 425284020 q^{12} - 906419296 q^{13} - 806064072 q^{14} - 2108540816 q^{16} + 12240765600 q^{17} + 2668896702 q^{18} + 1002788712 q^{20} + 39806479296 q^{21} + 216706355928 q^{22} - 111931394832 q^{24} + 206381182512 q^{25} + 1054507182588 q^{26} - 1526063922288 q^{28} + 327679573728 q^{29} + 1192344308100 q^{30} - 5158730488416 q^{32} + 679591529280 q^{33} + 9473293385948 q^{34} - 1959888509316 q^{36} - 8149494749152 q^{37} + 23318999782920 q^{38} - 28671795971776 q^{40} - 25536724613472 q^{41} + 5103781482168 q^{42} - 11442227373552 q^{44} - 5081579320608 q^{45} + 9929654732736 q^{46} + 29246734238832 q^{48} - 93287012964080 q^{49} - 133601044957998 q^{50} + 302261844234872 q^{52} - 86928436629792 q^{53} - 22406076560826 q^{54} + 530930989929024 q^{56} + 48687411524544 q^{57} - 189801665049916 q^{58} + 268455359263896 q^{60} + 476028596468000 q^{61} - 419080420491096 q^{62} + 305944925720704 q^{64} - 12\!\cdots\!92 q^{65}+ \cdots - 33\!\cdots\!90 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 37115 x^{14} + 433616 x^{13} + 965822723 x^{12} + 11579264195 x^{11} + \cdots + 43\!\cdots\!49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 17\!\cdots\!75 \nu^{15} + \cdots + 89\!\cdots\!75 ) / 36\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 74\!\cdots\!63 \nu^{15} + \cdots - 33\!\cdots\!79 ) / 84\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 58\!\cdots\!65 \nu^{15} + \cdots - 35\!\cdots\!65 ) / 36\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25\!\cdots\!13 \nu^{15} + \cdots - 21\!\cdots\!37 ) / 36\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 51\!\cdots\!03 \nu^{15} + \cdots - 33\!\cdots\!87 ) / 36\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 20\!\cdots\!53 \nu^{15} + \cdots + 15\!\cdots\!14 ) / 60\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 33\!\cdots\!39 \nu^{15} + \cdots - 16\!\cdots\!21 ) / 84\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 21\!\cdots\!81 \nu^{15} + \cdots + 16\!\cdots\!79 ) / 28\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 23\!\cdots\!93 \nu^{15} + \cdots - 22\!\cdots\!69 ) / 84\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 37\!\cdots\!31 \nu^{15} + \cdots + 17\!\cdots\!77 ) / 84\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 57\!\cdots\!03 \nu^{15} + \cdots - 70\!\cdots\!11 ) / 84\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 68\!\cdots\!65 \nu^{15} + \cdots - 12\!\cdots\!29 ) / 84\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 71\!\cdots\!47 \nu^{15} + \cdots - 15\!\cdots\!43 ) / 84\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 50\!\cdots\!37 \nu^{15} + \cdots + 13\!\cdots\!29 ) / 28\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 52\!\cdots\!93 \nu^{15} + \cdots + 52\!\cdots\!85 ) / 28\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 8 \beta_{15} - \beta_{13} + 5 \beta_{12} - 2 \beta_{10} + 4 \beta_{9} - 6 \beta_{8} + \cdots + 1339491 ) / 8957952 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 1092 \beta_{15} + 108 \beta_{14} - 795 \beta_{13} - 489 \beta_{12} - 396 \beta_{11} + \cdots - 41554325728 ) / 8957952 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 124658 \beta_{15} - 20682 \beta_{14} + 13465 \beta_{13} - 44291 \beta_{12} + 34362 \beta_{11} + \cdots - 395773131918 ) / 4478976 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 8165460 \beta_{15} - 1121580 \beta_{14} + 35184171 \beta_{13} - 6733311 \beta_{12} + \cdots - 621854512408688 ) / 8957952 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2335437428 \beta_{15} - 206698932 \beta_{14} + 2491387439 \beta_{13} + 147278681 \beta_{12} + \cdots + 13\!\cdots\!57 ) / 8957952 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4189257874 \beta_{15} - 865853118 \beta_{14} - 7721017532 \beta_{13} + 6873420994 \beta_{12} + \cdots + 58\!\cdots\!21 ) / 248832 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 32865728557664 \beta_{15} + 29503330762536 \beta_{14} - 70554016885345 \beta_{13} + \cdots + 38\!\cdots\!07 ) / 8957952 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 47\!\cdots\!08 \beta_{15} + \cdots - 19\!\cdots\!68 ) / 8957952 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 69\!\cdots\!50 \beta_{15} + \cdots - 95\!\cdots\!24 ) / 4478976 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 19\!\cdots\!56 \beta_{15} + \cdots - 36\!\cdots\!40 ) / 8957952 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 15\!\cdots\!92 \beta_{15} + \cdots + 22\!\cdots\!37 ) / 8957952 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 23\!\cdots\!86 \beta_{15} + \cdots + 19\!\cdots\!21 ) / 124416 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 20\!\cdots\!28 \beta_{15} + \cdots + 53\!\cdots\!19 ) / 8957952 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 39\!\cdots\!40 \beta_{15} + \cdots - 14\!\cdots\!64 ) / 8957952 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 52\!\cdots\!22 \beta_{15} + \cdots - 12\!\cdots\!78 ) / 4478976 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.857908 − 1.48594i
0.857908 + 1.48594i
36.4487 + 63.1310i
36.4487 − 63.1310i
−54.1541 + 93.7976i
−54.1541 − 93.7976i
72.8587 + 126.195i
72.8587 − 126.195i
−66.3279 + 114.883i
−66.3279 − 114.883i
54.8669 + 95.0323i
54.8669 − 95.0323i
−39.4461 + 68.3227i
−39.4461 − 68.3227i
−4.60413 + 7.97459i
−4.60413 − 7.97459i
−254.964 − 23.0089i 3788.00i 64477.2 + 11732.9i −363465. 87157.5 − 965802.i 2.07768e6i −1.61694e7 − 4.47501e6i −1.43489e7 9.26704e7 + 8.36292e6i
7.2 −254.964 + 23.0089i − 3788.00i 64477.2 − 11732.9i −363465. 87157.5 + 965802.i − 2.07768e6i −1.61694e7 + 4.47501e6i −1.43489e7 9.26704e7 − 8.36292e6i
7.3 −235.541 − 100.281i − 3788.00i 45423.4 + 47240.7i 699204. −379865. + 892229.i 4.77594e6i −5.96172e6 − 1.56822e7i −1.43489e7 −1.64691e8 − 7.01170e7i
7.4 −235.541 + 100.281i 3788.00i 45423.4 − 47240.7i 699204. −379865. − 892229.i − 4.77594e6i −5.96172e6 + 1.56822e7i −1.43489e7 −1.64691e8 + 7.01170e7i
7.5 −141.143 − 213.576i 3788.00i −25693.4 + 60289.4i 110106. 809025. − 534648.i 7.35151e6i 1.65028e7 − 3.02193e6i −1.43489e7 −1.55406e7 − 2.35159e7i
7.6 −141.143 + 213.576i − 3788.00i −25693.4 − 60289.4i 110106. 809025. + 534648.i − 7.35151e6i 1.65028e7 + 3.02193e6i −1.43489e7 −1.55406e7 + 2.35159e7i
7.7 −119.478 − 226.409i − 3788.00i −36986.2 + 54101.7i −481576. −857637. + 452581.i 515341.i 1.66681e7 + 1.91007e6i −1.43489e7 5.75375e7 + 1.09033e8i
7.8 −119.478 + 226.409i 3788.00i −36986.2 − 54101.7i −481576. −857637. − 452581.i − 515341.i 1.66681e7 − 1.91007e6i −1.43489e7 5.75375e7 − 1.09033e8i
7.9 11.3734 − 255.747i 3788.00i −65277.3 − 5817.42i 86591.3 968769. + 43082.3i − 1.07917e7i −2.23021e6 + 1.66283e7i −1.43489e7 984836. − 2.21455e7i
7.10 11.3734 + 255.747i − 3788.00i −65277.3 + 5817.42i 86591.3 968769. − 43082.3i 1.07917e7i −2.23021e6 − 1.66283e7i −1.43489e7 984836. + 2.21455e7i
7.11 196.501 − 164.084i − 3788.00i 11689.0 − 64485.1i −45026.4 −621548. − 744344.i − 3.24149e6i −8.28406e6 − 1.45894e7i −1.43489e7 −8.84773e6 + 7.38811e6i
7.12 196.501 + 164.084i 3788.00i 11689.0 + 64485.1i −45026.4 −621548. + 744344.i 3.24149e6i −8.28406e6 + 1.45894e7i −1.43489e7 −8.84773e6 − 7.38811e6i
7.13 197.709 − 162.626i 3788.00i 12641.5 − 64305.2i 551815. 616027. + 748920.i 6.23463e6i −7.95836e6 − 1.47695e7i −1.43489e7 1.09099e8 − 8.97395e7i
7.14 197.709 + 162.626i − 3788.00i 12641.5 + 64305.2i 551815. 616027. − 748920.i − 6.23463e6i −7.95836e6 + 1.47695e7i −1.43489e7 1.09099e8 + 8.97395e7i
7.15 252.543 − 41.9299i 3788.00i 62019.8 − 21178.2i −380576. 158830. + 956631.i − 8.07663e6i 1.47746e7 − 7.94889e6i −1.43489e7 −9.61118e7 + 1.59575e7i
7.16 252.543 + 41.9299i − 3788.00i 62019.8 + 21178.2i −380576. 158830. − 956631.i 8.07663e6i 1.47746e7 + 7.94889e6i −1.43489e7 −9.61118e7 − 1.59575e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 12.17.d.a ✓ 16
3.b odd 2 1 36.17.d.d 16
4.b odd 2 1 inner 12.17.d.a ✓ 16
12.b even 2 1 36.17.d.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.17.d.a ✓ 16 1.a even 1 1 trivial
12.17.d.a ✓ 16 4.b odd 2 1 inner
36.17.d.d 16 3.b odd 2 1
36.17.d.d 16 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{17}^{\mathrm{new}}(12, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 34\!\cdots\!56 \) Copy content Toggle raw display
$3$ \( (T^{2} + 14348907)^{8} \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 43\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 36\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 19\!\cdots\!76)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 67\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 62\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 38\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 50\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 55\!\cdots\!96)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 43\!\cdots\!68)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 48\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 64\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 10\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 62\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 14\!\cdots\!88)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 41\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 12\!\cdots\!28)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 19\!\cdots\!68)^{2} \) Copy content Toggle raw display
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