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Properties

Label 17.6.c.a
Level $17$
Weight $6$
Character orbit 17.c
Analytic conductor $2.727$
Analytic rank $0$
Dimension $12$
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] = [17,6,Mod(4,17)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(17, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.4");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 17.c (of order \(4\), degree \(2\), 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: \(2.72652493682\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 248x^{10} + 21830x^{8} + 802540x^{6} + 10668257x^{4} + 7196228x^{2} + 1183744 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{8} - 2 \beta_{3} - 2) q^{3} + ( - 2 \beta_{4} + \beta_{2} - 10) q^{4} + (\beta_{10} + \beta_{4} - 4 \beta_{3} + \cdots - 4) q^{5}+ \cdots + (2 \beta_{11} - 2 \beta_{10} + \cdots + 9 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{8} - 2 \beta_{3} - 2) q^{3} + ( - 2 \beta_{4} + \beta_{2} - 10) q^{4} + (\beta_{10} + \beta_{4} - 4 \beta_{3} + \cdots - 4) q^{5}+ \cdots + (160 \beta_{10} - 133 \beta_{8} + \cdots - 54644) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 24 q^{3} - 132 q^{4} - 40 q^{5} + 250 q^{6} - 120 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 24 q^{3} - 132 q^{4} - 40 q^{5} + 250 q^{6} - 120 q^{7} - 334 q^{10} - 504 q^{11} + 1722 q^{12} + 1600 q^{13} - 2364 q^{14} + 2564 q^{16} + 2432 q^{17} - 6052 q^{18} - 4218 q^{20} + 7144 q^{21} - 2590 q^{22} - 2728 q^{23} - 7246 q^{24} - 3024 q^{27} + 26164 q^{28} - 992 q^{29} + 18072 q^{30} - 14680 q^{31} - 96 q^{33} - 20962 q^{34} - 43744 q^{35} + 33552 q^{37} + 9024 q^{38} + 2720 q^{39} + 30922 q^{40} + 20716 q^{41} - 13682 q^{44} + 44848 q^{45} + 4416 q^{46} + 32032 q^{47} - 144770 q^{48} - 42636 q^{50} - 83368 q^{51} - 132556 q^{52} + 80728 q^{54} + 81040 q^{55} + 122460 q^{56} + 36840 q^{57} + 165766 q^{58} - 143200 q^{61} + 222580 q^{62} + 7592 q^{63} + 18428 q^{64} - 175016 q^{65} - 83648 q^{67} - 231150 q^{68} - 341160 q^{69} + 116088 q^{71} + 347100 q^{72} + 83892 q^{73} - 79282 q^{74} + 245864 q^{75} - 128572 q^{78} + 291672 q^{79} + 195426 q^{80} + 387028 q^{81} - 452240 q^{82} - 798600 q^{84} - 200704 q^{85} - 736076 q^{86} + 431258 q^{88} + 456424 q^{89} - 65406 q^{90} + 36224 q^{91} + 476504 q^{92} - 413488 q^{95} + 884318 q^{96} + 356284 q^{97} + 444704 q^{98} - 640440 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 248x^{10} + 21830x^{8} + 802540x^{6} + 10668257x^{4} + 7196228x^{2} + 1183744 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 41 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1001 \nu^{11} + 252208 \nu^{9} + 22665614 \nu^{7} + 857504052 \nu^{5} + 11895531185 \nu^{3} + 11904513348 \nu ) / 4553425792 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -495\nu^{10} - 101723\nu^{8} - 6770189\nu^{6} - 152075741\nu^{4} - 587636140\nu^{2} + 148115968 ) / 569178224 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3527 \nu^{10} - 796668 \nu^{8} - 58875465 \nu^{6} - 1526056634 \nu^{4} - 142294556 \nu^{3} + \cdots + 2775546752 ) / 284589112 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3527 \nu^{10} - 796668 \nu^{8} - 58875465 \nu^{6} - 1526056634 \nu^{4} + 142294556 \nu^{3} + \cdots + 2775546752 ) / 284589112 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 45001 \nu^{11} - 11154312 \nu^{9} - 983451686 \nu^{7} - 36374272060 \nu^{5} + \cdots - 486900119524 \nu ) / 4553425792 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 116603 \nu^{11} - 80567 \nu^{10} - 28941921 \nu^{9} - 19862433 \nu^{8} - 2549446865 \nu^{7} + \cdots - 215503789568 ) / 9106851584 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 116603 \nu^{11} + 80567 \nu^{10} - 28941921 \nu^{9} + 19862433 \nu^{8} - 2549446865 \nu^{7} + \cdots + 215503789568 ) / 9106851584 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1203637 \nu^{11} + 715203 \nu^{10} + 298043697 \nu^{9} + 176727437 \nu^{8} + \cdots + 2461586965760 ) / 9106851584 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1203637 \nu^{11} + 715203 \nu^{10} - 298043697 \nu^{9} + 176727437 \nu^{8} + \cdots + 2461586965760 ) / 9106851584 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 41 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} - 71\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{11} + 4\beta_{10} - 36\beta_{9} + 36\beta_{8} - 10\beta_{4} - 83\beta_{2} + 2947 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{11} + 4 \beta_{10} + 52 \beta_{9} + 52 \beta_{8} - 10 \beta_{7} - 123 \beta_{6} + \cdots + 5557 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 496 \beta_{11} - 496 \beta_{10} + 4496 \beta_{9} - 4496 \beta_{8} + 46 \beta_{6} + 46 \beta_{5} + \cdots - 234049 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 680 \beta_{11} - 680 \beta_{10} - 8136 \beta_{9} - 8136 \beta_{8} + 28 \beta_{7} + 12025 \beta_{6} + \cdots - 459523 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 48780 \beta_{11} + 48780 \beta_{10} - 443756 \beta_{9} + 443756 \beta_{8} - 8788 \beta_{6} + \cdots + 19551279 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 83932 \beta_{11} + 83932 \beta_{10} + 955244 \beta_{9} + 955244 \beta_{8} + 137618 \beta_{7} + \cdots + 39050961 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4469368 \beta_{11} - 4469368 \beta_{10} + 40759896 \beta_{9} - 40759896 \beta_{8} + 1176794 \beta_{6} + \cdots - 1673100605 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 9176544 \beta_{11} - 9176544 \beta_{10} - 101001952 \beta_{9} - 101001952 \beta_{8} + \cdots - 3362702943 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
− 9.48424i
− 6.30410i
− 0.656304i
− 0.521304i
5.80736i
9.15858i
− 9.15858i
− 5.80736i
0.521304i
0.656304i
6.30410i
9.48424i
− 10.4842i −15.8317 + 15.8317i −77.9193 28.0082 − 28.0082i 165.983 + 165.983i −131.243 − 131.243i 481.429i − 258.283i −293.645 − 293.645i
4.2 − 7.30410i 11.9937 − 11.9937i −21.3498 −21.7675 + 21.7675i −87.6029 − 87.6029i 6.10724 + 6.10724i − 77.7899i − 44.6964i 158.992 + 158.992i
4.3 − 1.65630i −15.0964 + 15.0964i 29.2567 −67.8362 + 67.8362i 25.0042 + 25.0042i 81.3402 + 81.3402i − 101.460i − 212.800i 112.357 + 112.357i
4.4 − 1.52130i −1.82066 + 1.82066i 29.6856 52.9767 − 52.9767i 2.76978 + 2.76978i −25.6330 − 25.6330i − 93.8426i 236.370i −80.5937 − 80.5937i
4.5 4.80736i 15.6906 − 15.6906i 8.88927 −8.57737 + 8.57737i 75.4302 + 75.4302i 38.9646 + 38.9646i 196.570i − 249.387i −41.2345 − 41.2345i
4.6 8.15858i −6.93554 + 6.93554i −34.5625 −2.80390 + 2.80390i −56.5842 − 56.5842i −29.5355 − 29.5355i − 20.9060i 146.796i −22.8759 − 22.8759i
13.1 − 8.15858i −6.93554 − 6.93554i −34.5625 −2.80390 − 2.80390i −56.5842 + 56.5842i −29.5355 + 29.5355i 20.9060i − 146.796i −22.8759 + 22.8759i
13.2 − 4.80736i 15.6906 + 15.6906i 8.88927 −8.57737 − 8.57737i 75.4302 − 75.4302i 38.9646 − 38.9646i − 196.570i 249.387i −41.2345 + 41.2345i
13.3 1.52130i −1.82066 − 1.82066i 29.6856 52.9767 + 52.9767i 2.76978 − 2.76978i −25.6330 + 25.6330i 93.8426i − 236.370i −80.5937 + 80.5937i
13.4 1.65630i −15.0964 − 15.0964i 29.2567 −67.8362 − 67.8362i 25.0042 − 25.0042i 81.3402 − 81.3402i 101.460i 212.800i 112.357 − 112.357i
13.5 7.30410i 11.9937 + 11.9937i −21.3498 −21.7675 − 21.7675i −87.6029 + 87.6029i 6.10724 − 6.10724i 77.7899i 44.6964i 158.992 − 158.992i
13.6 10.4842i −15.8317 − 15.8317i −77.9193 28.0082 + 28.0082i 165.983 − 165.983i −131.243 + 131.243i − 481.429i 258.283i −293.645 + 293.645i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.6.c.a ✓ 12
3.b odd 2 1 153.6.f.a 12
4.b odd 2 1 272.6.o.c 12
17.c even 4 1 inner 17.6.c.a ✓ 12
17.d even 8 2 289.6.a.g 12
51.f odd 4 1 153.6.f.a 12
68.f odd 4 1 272.6.o.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.6.c.a ✓ 12 1.a even 1 1 trivial
17.6.c.a ✓ 12 17.c even 4 1 inner
153.6.f.a 12 3.b odd 2 1
153.6.f.a 12 51.f odd 4 1
272.6.o.c 12 4.b odd 2 1
272.6.o.c 12 68.f odd 4 1
289.6.a.g 12 17.d even 8 2

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(17, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 258 T^{10} + \cdots + 57274624 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 20643283206144 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 948700375418624)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 81\!\cdots\!49 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 50\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 74\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 42\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 17\!\cdots\!16)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 25\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 24\!\cdots\!28)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 15\!\cdots\!12)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 63\!\cdots\!36 \) Copy content Toggle raw display
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