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Properties

Label 17.3.e.a
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{16}^{7} + \zeta_{16}^{6} + \cdots - 1) q^{2}+ \cdots + ( - 4 \zeta_{16}^{7} + \zeta_{16}^{6} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{16}^{7} + \zeta_{16}^{6} + \cdots - 1) q^{2}+ \cdots + ( - 16 \zeta_{16}^{7} - 27 \zeta_{16}^{6} + \cdots - 28) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 8 q^{3} + 16 q^{5} - 8 q^{6} + 8 q^{7} - 24 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} - 8 q^{3} + 16 q^{5} - 8 q^{6} + 8 q^{7} - 24 q^{8} - 16 q^{9} + 16 q^{10} - 8 q^{11} + 48 q^{12} + 16 q^{13} + 8 q^{14} - 16 q^{15} + 56 q^{18} - 80 q^{20} - 64 q^{21} - 104 q^{22} - 56 q^{23} - 80 q^{24} + 64 q^{25} + 176 q^{26} + 40 q^{27} + 152 q^{28} + 48 q^{29} + 16 q^{30} + 24 q^{31} + 88 q^{32} - 136 q^{34} - 160 q^{35} - 128 q^{36} + 32 q^{37} - 120 q^{38} + 48 q^{39} + 64 q^{40} + 48 q^{41} + 16 q^{42} - 232 q^{43} + 120 q^{44} - 88 q^{46} + 192 q^{47} + 136 q^{48} + 16 q^{49} + 136 q^{51} - 384 q^{52} - 32 q^{53} + 8 q^{54} + 224 q^{55} - 120 q^{56} + 24 q^{57} + 240 q^{58} - 48 q^{59} + 64 q^{60} - 160 q^{61} - 168 q^{62} + 56 q^{63} - 64 q^{64} - 96 q^{65} - 8 q^{66} + 272 q^{68} + 240 q^{69} + 224 q^{70} + 40 q^{71} + 40 q^{72} + 48 q^{73} - 160 q^{74} - 296 q^{75} + 80 q^{76} - 48 q^{77} - 400 q^{78} - 136 q^{79} - 240 q^{80} - 424 q^{81} - 64 q^{82} - 264 q^{83} - 272 q^{85} + 832 q^{86} + 208 q^{87} + 264 q^{88} + 160 q^{89} + 448 q^{90} + 320 q^{91} + 24 q^{92} - 64 q^{93} + 32 q^{94} + 272 q^{95} - 56 q^{96} + 48 q^{97} - 120 q^{98} - 224 q^{99}+O(q^{100}) \) 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
1.23044 − 0.509666i −2.63099 − 0.523336i −1.57420 + 1.57420i 0.902812 + 1.35115i −3.50400 + 0.696990i 4.92562 − 7.37170i −3.17331 + 7.66104i −1.66671 − 0.690373i 1.79949 + 1.20238i
5.1 −0.841487 − 2.03153i 0.0897902 + 0.134381i −0.590587 + 0.590587i −1.04667 + 5.26197i 0.197441 − 0.295491i 1.21824 + 6.12453i −6.42935 − 2.66313i 3.43416 − 8.29078i 11.5706 − 2.30154i
6.1 1.23044 + 0.509666i −2.63099 + 0.523336i −1.57420 − 1.57420i 0.902812 − 1.35115i −3.50400 − 0.696990i 4.92562 + 7.37170i −3.17331 − 7.66104i −1.66671 + 0.690373i 1.79949 − 1.20238i
7.1 −0.841487 + 2.03153i 0.0897902 − 0.134381i −0.590587 − 0.590587i −1.04667 − 5.26197i 0.197441 + 0.295491i 1.21824 − 6.12453i −6.42935 + 2.66313i 3.43416 + 8.29078i 11.5706 + 2.30154i
10.1 −1.15851 + 2.79690i −0.675577 − 0.451406i −3.65205 − 3.65205i 7.87510 − 1.56645i 2.04520 − 1.36656i −7.70353 − 1.53233i 3.25778 − 1.34942i −3.19151 − 7.70500i −4.74219 + 23.8406i
11.1 −3.23044 − 1.33809i −0.783227 − 3.93755i 5.81684 + 5.81684i 0.268761 + 0.179580i −2.73864 + 13.7681i 5.55967 − 3.71485i −5.65512 − 13.6527i −6.57593 + 2.72384i −0.627922 − 0.939752i
12.1 −1.15851 − 2.79690i −0.675577 + 0.451406i −3.65205 + 3.65205i 7.87510 + 1.56645i 2.04520 + 1.36656i −7.70353 + 1.53233i 3.25778 + 1.34942i −3.19151 + 7.70500i −4.74219 − 23.8406i
14.1 −3.23044 + 1.33809i −0.783227 + 3.93755i 5.81684 − 5.81684i 0.268761 − 0.179580i −2.73864 − 13.7681i 5.55967 + 3.71485i −5.65512 + 13.6527i −6.57593 − 2.72384i −0.627922 + 0.939752i
\(n\): 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.a ✓ 8
3.b odd 2 1 153.3.p.b 8
4.b odd 2 1 272.3.bh.c 8
5.b even 2 1 425.3.u.b 8
5.c odd 4 1 425.3.t.a 8
5.c odd 4 1 425.3.t.c 8
17.b even 2 1 289.3.e.c 8
17.c even 4 1 289.3.e.i 8
17.c even 4 1 289.3.e.m 8
17.d even 8 1 289.3.e.b 8
17.d even 8 1 289.3.e.d 8
17.d even 8 1 289.3.e.k 8
17.d even 8 1 289.3.e.l 8
17.e odd 16 1 inner 17.3.e.a ✓ 8
17.e odd 16 1 289.3.e.b 8
17.e odd 16 1 289.3.e.c 8
17.e odd 16 1 289.3.e.d 8
17.e odd 16 1 289.3.e.i 8
17.e odd 16 1 289.3.e.k 8
17.e odd 16 1 289.3.e.l 8
17.e odd 16 1 289.3.e.m 8
51.i even 16 1 153.3.p.b 8
68.i even 16 1 272.3.bh.c 8
85.o even 16 1 425.3.t.a 8
85.p odd 16 1 425.3.u.b 8
85.r even 16 1 425.3.t.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.3.e.a ✓ 8 1.a even 1 1 trivial
17.3.e.a ✓ 8 17.e odd 16 1 inner
153.3.p.b 8 3.b odd 2 1
153.3.p.b 8 51.i even 16 1
272.3.bh.c 8 4.b odd 2 1
272.3.bh.c 8 68.i even 16 1
289.3.e.b 8 17.d even 8 1
289.3.e.b 8 17.e odd 16 1
289.3.e.c 8 17.b even 2 1
289.3.e.c 8 17.e odd 16 1
289.3.e.d 8 17.d even 8 1
289.3.e.d 8 17.e odd 16 1
289.3.e.i 8 17.c even 4 1
289.3.e.i 8 17.e odd 16 1
289.3.e.k 8 17.d even 8 1
289.3.e.k 8 17.e odd 16 1
289.3.e.l 8 17.d even 8 1
289.3.e.l 8 17.e odd 16 1
289.3.e.m 8 17.c even 4 1
289.3.e.m 8 17.e odd 16 1
425.3.t.a 8 5.c odd 4 1
425.3.t.a 8 85.o even 16 1
425.3.t.c 8 5.c odd 4 1
425.3.t.c 8 85.r even 16 1
425.3.u.b 8 5.b even 2 1
425.3.u.b 8 85.p odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 8T_{2}^{7} + 32T_{2}^{6} + 72T_{2}^{5} + 64T_{2}^{4} - 120T_{2}^{3} - 192T_{2}^{2} - 248T_{2} + 961 \) acting on \(S_{3}^{\mathrm{new}}(17, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 8 T^{7} + \cdots + 961 \) Copy content Toggle raw display
$3$ \( T^{8} + 8 T^{7} + \cdots + 2 \) Copy content Toggle raw display
$5$ \( T^{8} - 16 T^{7} + \cdots + 512 \) Copy content Toggle raw display
$7$ \( T^{8} - 8 T^{7} + \cdots + 8454272 \) Copy content Toggle raw display
$11$ \( T^{8} + 8 T^{7} + \cdots + 27572738 \) Copy content Toggle raw display
$13$ \( T^{8} - 16 T^{7} + \cdots + 9048064 \) Copy content Toggle raw display
$17$ \( T^{8} + 6975757441 \) Copy content Toggle raw display
$19$ \( T^{8} + 368 T^{6} + \cdots + 929296 \) Copy content Toggle raw display
$23$ \( T^{8} + 56 T^{7} + \cdots + 859299968 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 4240836608 \) Copy content Toggle raw display
$31$ \( T^{8} - 24 T^{7} + \cdots + 14536832 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 120057840128 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 59058658562 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 6201305218564 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 2259754549504 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 26754490624 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 2675455605124 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 106680004247552 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 83643718741636 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 71246079996032 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 51301810562 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 3948319764608 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 117588822570244 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 21682310986562 \) Copy content Toggle raw display
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