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Properties

Label 17.3.e.b
Level $17$
Weight $3$
Character orbit 17.e
Analytic conductor $0.463$
Analytic rank $0$
Dimension $8$
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] = [17,3,Mod(3,17)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(17, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.3");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 17.e (of order \(16\), degree \(8\), 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: \(0.463216449413\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-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 primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{16}^{7} + \cdots - \zeta_{16}^{2}) q^{2}+ \cdots + (2 \zeta_{16}^{7} - 7 \zeta_{16}^{6} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{16}^{7} + \cdots - \zeta_{16}^{2}) q^{2}+ \cdots + (18 \zeta_{16}^{7} - 15 \zeta_{16}^{6} + \cdots + 17) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} - 24 q^{5} - 16 q^{7} + 16 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 24 q^{5} - 16 q^{7} + 16 q^{8} + 8 q^{9} + 40 q^{11} + 40 q^{12} + 16 q^{14} + 32 q^{15} - 16 q^{17} - 136 q^{18} - 32 q^{19} - 40 q^{20} - 64 q^{21} - 8 q^{23} + 24 q^{24} + 16 q^{25} + 96 q^{27} + 80 q^{28} + 24 q^{29} + 168 q^{30} + 32 q^{31} - 24 q^{32} + 64 q^{34} + 80 q^{35} - 104 q^{36} - 168 q^{37} + 8 q^{38} - 72 q^{39} - 200 q^{40} - 72 q^{42} + 96 q^{43} - 96 q^{44} - 88 q^{45} - 80 q^{47} + 88 q^{48} + 8 q^{49} - 176 q^{51} + 240 q^{52} + 96 q^{53} + 208 q^{54} - 8 q^{55} + 72 q^{56} + 248 q^{57} + 8 q^{59} + 16 q^{60} + 264 q^{61} - 136 q^{62} + 8 q^{63} - 120 q^{64} - 32 q^{65} + 8 q^{66} - 176 q^{68} - 208 q^{69} - 80 q^{70} + 32 q^{71} + 24 q^{72} + 24 q^{73} + 176 q^{74} - 192 q^{75} - 80 q^{76} - 216 q^{77} - 368 q^{78} - 96 q^{79} + 24 q^{80} - 224 q^{81} - 408 q^{82} - 88 q^{83} + 512 q^{85} + 288 q^{86} + 312 q^{87} + 176 q^{88} + 288 q^{89} + 256 q^{90} - 24 q^{91} + 336 q^{92} + 280 q^{93} - 8 q^{94} - 152 q^{95} + 328 q^{96} - 344 q^{97} + 16 q^{98} + 136 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(\zeta_{16}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.923880 + 0.382683i
−0.382683 + 0.923880i
0.923880 − 0.382683i
−0.382683 − 0.923880i
0.382683 + 0.923880i
−0.923880 + 0.382683i
0.382683 − 0.923880i
−0.923880 − 0.382683i
−1.63099 + 0.675577i 3.96908 + 0.789499i −0.624715 + 0.624715i −4.29916 − 6.43416i −7.00688 + 1.39376i −2.27356 + 3.40262i 3.29916 − 7.96489i 6.81537 + 2.82302i 11.3586 + 7.58960i
5.1 1.08979 + 2.63099i −2.88669 − 4.32023i −2.90602 + 2.90602i −0.711297 + 3.57593i 8.22059 − 12.3030i −0.644047 − 3.23784i −0.288703 − 0.119585i −6.88730 + 16.6274i −10.1834 + 2.02560i
6.1 −1.63099 − 0.675577i 3.96908 − 0.789499i −0.624715 − 0.624715i −4.29916 + 6.43416i −7.00688 − 1.39376i −2.27356 − 3.40262i 3.29916 + 7.96489i 6.81537 − 2.82302i 11.3586 − 7.58960i
7.1 1.08979 − 2.63099i −2.88669 + 4.32023i −2.90602 − 2.90602i −0.711297 − 3.57593i 8.22059 + 12.3030i −0.644047 + 3.23784i −0.288703 + 0.119585i −6.88730 − 16.6274i −10.1834 − 2.02560i
10.1 0.324423 − 0.783227i −1.35595 − 0.906019i 2.32023 + 2.32023i −6.70292 + 1.33329i −1.14952 + 0.768086i 0.886687 + 0.176373i 5.70292 − 2.36223i −2.42641 − 5.85788i −1.13031 + 5.68246i
11.1 0.216773 + 0.0897902i 0.273561 + 1.37529i −2.78950 − 2.78950i −0.286621 − 0.191514i −0.0641865 + 0.322688i −5.96908 + 3.98841i −0.713379 − 1.72225i 6.49834 − 2.69170i −0.0449356 − 0.0672509i
12.1 0.324423 + 0.783227i −1.35595 + 0.906019i 2.32023 − 2.32023i −6.70292 − 1.33329i −1.14952 − 0.768086i 0.886687 − 0.176373i 5.70292 + 2.36223i −2.42641 + 5.85788i −1.13031 − 5.68246i
14.1 0.216773 − 0.0897902i 0.273561 − 1.37529i −2.78950 + 2.78950i −0.286621 + 0.191514i −0.0641865 − 0.322688i −5.96908 − 3.98841i −0.713379 + 1.72225i 6.49834 + 2.69170i −0.0449356 + 0.0672509i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.e odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.3.e.b ✓ 8
3.b odd 2 1 153.3.p.a 8
4.b odd 2 1 272.3.bh.b 8
5.b even 2 1 425.3.u.a 8
5.c odd 4 1 425.3.t.b 8
5.c odd 4 1 425.3.t.d 8
17.b even 2 1 289.3.e.g 8
17.c even 4 1 289.3.e.f 8
17.c even 4 1 289.3.e.h 8
17.d even 8 1 289.3.e.a 8
17.d even 8 1 289.3.e.e 8
17.d even 8 1 289.3.e.j 8
17.d even 8 1 289.3.e.n 8
17.e odd 16 1 inner 17.3.e.b ✓ 8
17.e odd 16 1 289.3.e.a 8
17.e odd 16 1 289.3.e.e 8
17.e odd 16 1 289.3.e.f 8
17.e odd 16 1 289.3.e.g 8
17.e odd 16 1 289.3.e.h 8
17.e odd 16 1 289.3.e.j 8
17.e odd 16 1 289.3.e.n 8
51.i even 16 1 153.3.p.a 8
68.i even 16 1 272.3.bh.b 8
85.o even 16 1 425.3.t.d 8
85.p odd 16 1 425.3.u.a 8
85.r even 16 1 425.3.t.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.3.e.b ✓ 8 1.a even 1 1 trivial
17.3.e.b ✓ 8 17.e odd 16 1 inner
153.3.p.a 8 3.b odd 2 1
153.3.p.a 8 51.i even 16 1
272.3.bh.b 8 4.b odd 2 1
272.3.bh.b 8 68.i even 16 1
289.3.e.a 8 17.d even 8 1
289.3.e.a 8 17.e odd 16 1
289.3.e.e 8 17.d even 8 1
289.3.e.e 8 17.e odd 16 1
289.3.e.f 8 17.c even 4 1
289.3.e.f 8 17.e odd 16 1
289.3.e.g 8 17.b even 2 1
289.3.e.g 8 17.e odd 16 1
289.3.e.h 8 17.c even 4 1
289.3.e.h 8 17.e odd 16 1
289.3.e.j 8 17.d even 8 1
289.3.e.j 8 17.e odd 16 1
289.3.e.n 8 17.d even 8 1
289.3.e.n 8 17.e odd 16 1
425.3.t.b 8 5.c odd 4 1
425.3.t.b 8 85.r even 16 1
425.3.t.d 8 5.c odd 4 1
425.3.t.d 8 85.o even 16 1
425.3.u.a 8 5.b even 2 1
425.3.u.a 8 85.p odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 4T_{2}^{6} + 16T_{2}^{5} + 8T_{2}^{4} - 8T_{2}^{3} + 20T_{2}^{2} - 8T_{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(17, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{6} + \cdots + 2312 \) Copy content Toggle raw display
$5$ \( T^{8} + 24 T^{7} + \cdots + 4418 \) Copy content Toggle raw display
$7$ \( T^{8} + 16 T^{7} + \cdots + 7688 \) Copy content Toggle raw display
$11$ \( T^{8} - 40 T^{7} + \cdots + 2221832 \) Copy content Toggle raw display
$13$ \( T^{8} + 784 T^{5} + \cdots + 16273156 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 34090452496 \) Copy content Toggle raw display
$23$ \( T^{8} + 8 T^{7} + \cdots + 564614408 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 55846825218 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 182700453128 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 368317129538 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 126445152962 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 6626611216 \) Copy content Toggle raw display
$47$ \( T^{8} + 80 T^{7} + \cdots + 298321984 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 41458660996 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 2347596304 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 22880163635522 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 883915868224 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 53847249369608 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 18825456624578 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 376288804594952 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 128947789048324 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 18\!\cdots\!42 \) Copy content Toggle raw display
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