Properties

Label 17.2.d.a
Level 17
Weight 2
Character orbit 17.d
Analytic conductor 0.136
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 17 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 17.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.135745683436\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{2} + ( -1 - \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( 2 \zeta_{8} - \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{4} + ( -\zeta_{8}^{2} - \zeta_{8}^{3} ) q^{5} + ( 1 - \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{6} + ( -1 + \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{7} + ( 1 - 3 \zeta_{8} + \zeta_{8}^{2} ) q^{8} + ( 2 + \zeta_{8} + 2 \zeta_{8}^{2} ) q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{2} + ( -1 - \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( 2 \zeta_{8} - \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{4} + ( -\zeta_{8}^{2} - \zeta_{8}^{3} ) q^{5} + ( 1 - \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{6} + ( -1 + \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{7} + ( 1 - 3 \zeta_{8} + \zeta_{8}^{2} ) q^{8} + ( 2 + \zeta_{8} + 2 \zeta_{8}^{2} ) q^{9} + ( 1 + \zeta_{8}^{3} ) q^{10} + ( -1 + \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{11} + ( -1 + \zeta_{8} - 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{12} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{13} + ( 1 + \zeta_{8} - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{14} + ( -2 + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{15} -3 q^{16} + ( -2 \zeta_{8} + 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{17} + ( -3 + \zeta_{8} - \zeta_{8}^{3} ) q^{18} + ( 2 - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{19} + ( 1 + \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{20} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{21} + ( 3 - 3 \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{22} + ( 1 + 3 \zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{23} + ( 3 + \zeta_{8} + \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{24} + ( -1 + 3 \zeta_{8} - \zeta_{8}^{2} ) q^{25} + ( -1 + 2 \zeta_{8} - \zeta_{8}^{2} ) q^{26} + ( 2 - 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{27} + ( 1 - 5 \zeta_{8} + 5 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{28} + ( -1 + \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{29} -2 \zeta_{8}^{2} q^{30} + ( -3 - 3 \zeta_{8} + 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{31} + ( 1 - \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{32} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{33} + ( -5 + 3 \zeta_{8} - \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{34} + 2 q^{35} + 7 \zeta_{8}^{3} q^{36} + ( -5 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{37} + 2 \zeta_{8}^{2} q^{38} + ( 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{39} + ( -2 + \zeta_{8} - \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{40} + ( -1 - 4 \zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{41} + ( -2 + 4 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{42} + ( -2 - 2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{43} + ( -5 + \zeta_{8} + \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{44} + ( 3 + 2 \zeta_{8} - 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{45} + ( 5 - 5 \zeta_{8} + 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{46} + ( 2 \zeta_{8} + 8 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{47} + ( 3 + 3 \zeta_{8} + 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{48} + ( 2 - 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{49} + ( 5 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{50} + ( 7 + \zeta_{8} - 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{51} + ( 4 - \zeta_{8} + \zeta_{8}^{3} ) q^{52} + ( -1 + \zeta_{8}^{2} ) q^{53} + ( -2 - 2 \zeta_{8} + 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{54} + 2 \zeta_{8}^{2} q^{55} + ( -5 + 5 \zeta_{8} - 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{56} + ( -6 - 2 \zeta_{8} + 2 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{57} + ( -\zeta_{8} - \zeta_{8}^{2} ) q^{58} -6 \zeta_{8} q^{59} + ( -2 - 6 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{60} + ( 5 \zeta_{8} + 5 \zeta_{8}^{2} ) q^{61} + ( -3 + 9 \zeta_{8} - 9 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{62} + ( -3 + 3 \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{63} + ( 2 \zeta_{8} - 7 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{64} + ( -1 - \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{65} + ( 2 - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{66} + ( 4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{67} + ( 3 - 4 \zeta_{8} - 8 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{68} + ( -8 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{69} + ( -2 + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{70} + ( 5 + 5 \zeta_{8} + 5 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{71} + ( -5 \zeta_{8} + \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{72} + ( -7 + 7 \zeta_{8} ) q^{73} + ( 5 - 10 \zeta_{8} + 10 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{74} + ( -3 - \zeta_{8} - \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{75} + ( 2 + 6 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{76} + ( 2 - 4 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{77} + ( -2 - 2 \zeta_{8}^{3} ) q^{78} + ( -1 - 3 \zeta_{8} + 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{79} + ( 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{80} + ( -2 \zeta_{8} - 3 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{81} + ( 1 + \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{82} + ( 4 - 4 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{83} + ( 8 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{84} + ( 1 + 5 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{85} + 2 q^{86} + ( 4 - 4 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{87} + ( 3 + 3 \zeta_{8} - 5 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{88} + ( -\zeta_{8} - 8 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{89} + ( 1 - \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{90} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{91} + ( -7 + 7 \zeta_{8} + 7 \zeta_{8}^{2} - 7 \zeta_{8}^{3} ) q^{92} + ( 6 + 6 \zeta_{8}^{2} ) q^{93} + ( -6 + 4 \zeta_{8} - 6 \zeta_{8}^{2} ) q^{94} + ( -2 - 4 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{95} + ( -5 - 3 \zeta_{8} + 3 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{96} + ( 6 - 6 \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{97} + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{98} + ( -1 - \zeta_{8} - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 4q^{3} + 4q^{6} - 4q^{7} + 4q^{8} + 8q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - 4q^{3} + 4q^{6} - 4q^{7} + 4q^{8} + 8q^{9} + 4q^{10} - 4q^{11} - 4q^{12} + 4q^{14} - 8q^{15} - 12q^{16} - 12q^{18} + 8q^{19} + 4q^{20} + 12q^{22} + 4q^{23} + 12q^{24} - 4q^{25} - 4q^{26} + 8q^{27} + 4q^{28} - 4q^{29} - 12q^{31} + 4q^{32} - 20q^{34} + 8q^{35} - 8q^{40} - 4q^{41} - 8q^{42} - 8q^{43} - 20q^{44} + 12q^{45} + 20q^{46} + 12q^{48} + 8q^{49} + 20q^{50} + 28q^{51} + 16q^{52} - 4q^{53} - 8q^{54} - 20q^{56} - 24q^{57} - 8q^{60} - 12q^{62} - 12q^{63} - 4q^{65} + 8q^{66} + 16q^{67} + 12q^{68} - 32q^{69} - 8q^{70} + 20q^{71} - 28q^{73} + 20q^{74} - 12q^{75} + 8q^{76} + 8q^{77} - 8q^{78} - 4q^{79} + 4q^{82} + 16q^{83} + 32q^{84} + 4q^{85} + 8q^{86} + 16q^{87} + 12q^{88} + 4q^{90} - 28q^{92} + 24q^{93} - 24q^{94} - 8q^{95} - 20q^{96} + 24q^{97} - 4q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(\zeta_{8}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
−0.292893 0.292893i −2.41421 + 1.00000i 1.82843i 0.707107 + 1.70711i 1.00000 + 0.414214i 0.414214 1.00000i −1.12132 + 1.12132i 2.70711 2.70711i 0.292893 0.707107i
8.1 −1.70711 + 1.70711i 0.414214 1.00000i 3.82843i −0.707107 0.292893i 1.00000 + 2.41421i −2.41421 + 1.00000i 3.12132 + 3.12132i 1.29289 + 1.29289i 1.70711 0.707107i
9.1 −0.292893 + 0.292893i −2.41421 1.00000i 1.82843i 0.707107 1.70711i 1.00000 0.414214i 0.414214 + 1.00000i −1.12132 1.12132i 2.70711 + 2.70711i 0.292893 + 0.707107i
15.1 −1.70711 1.70711i 0.414214 + 1.00000i 3.82843i −0.707107 + 0.292893i 1.00000 2.41421i −2.41421 1.00000i 3.12132 3.12132i 1.29289 1.29289i 1.70711 + 0.707107i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.2.d.a 4
3.b odd 2 1 153.2.l.c 4
4.b odd 2 1 272.2.v.d 4
5.b even 2 1 425.2.m.a 4
5.c odd 4 1 425.2.n.a 4
5.c odd 4 1 425.2.n.b 4
7.b odd 2 1 833.2.l.a 4
7.c even 3 2 833.2.v.b 8
7.d odd 6 2 833.2.v.a 8
17.b even 2 1 289.2.d.a 4
17.c even 4 1 289.2.d.b 4
17.c even 4 1 289.2.d.c 4
17.d even 8 1 inner 17.2.d.a 4
17.d even 8 1 289.2.d.a 4
17.d even 8 1 289.2.d.b 4
17.d even 8 1 289.2.d.c 4
17.e odd 16 2 289.2.a.f 4
17.e odd 16 2 289.2.b.b 4
17.e odd 16 4 289.2.c.c 8
51.g odd 8 1 153.2.l.c 4
51.i even 16 2 2601.2.a.bb 4
68.g odd 8 1 272.2.v.d 4
68.i even 16 2 4624.2.a.bp 4
85.k odd 8 1 425.2.n.b 4
85.m even 8 1 425.2.m.a 4
85.n odd 8 1 425.2.n.a 4
85.p odd 16 2 7225.2.a.u 4
119.l odd 8 1 833.2.l.a 4
119.q even 24 2 833.2.v.b 8
119.r odd 24 2 833.2.v.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.d.a 4 1.a even 1 1 trivial
17.2.d.a 4 17.d even 8 1 inner
153.2.l.c 4 3.b odd 2 1
153.2.l.c 4 51.g odd 8 1
272.2.v.d 4 4.b odd 2 1
272.2.v.d 4 68.g odd 8 1
289.2.a.f 4 17.e odd 16 2
289.2.b.b 4 17.e odd 16 2
289.2.c.c 8 17.e odd 16 4
289.2.d.a 4 17.b even 2 1
289.2.d.a 4 17.d even 8 1
289.2.d.b 4 17.c even 4 1
289.2.d.b 4 17.d even 8 1
289.2.d.c 4 17.c even 4 1
289.2.d.c 4 17.d even 8 1
425.2.m.a 4 5.b even 2 1
425.2.m.a 4 85.m even 8 1
425.2.n.a 4 5.c odd 4 1
425.2.n.a 4 85.n odd 8 1
425.2.n.b 4 5.c odd 4 1
425.2.n.b 4 85.k odd 8 1
833.2.l.a 4 7.b odd 2 1
833.2.l.a 4 119.l odd 8 1
833.2.v.a 8 7.d odd 6 2
833.2.v.a 8 119.r odd 24 2
833.2.v.b 8 7.c even 3 2
833.2.v.b 8 119.q even 24 2
2601.2.a.bb 4 51.i even 16 2
4624.2.a.bp 4 68.i even 16 2
7225.2.a.u 4 85.p odd 16 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(17, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 8 T^{2} + 12 T^{3} + 17 T^{4} + 24 T^{5} + 32 T^{6} + 32 T^{7} + 16 T^{8} \)
$3$ \( 1 + 4 T + 4 T^{2} - 12 T^{3} - 40 T^{4} - 36 T^{5} + 36 T^{6} + 108 T^{7} + 81 T^{8} \)
$5$ \( 1 + 2 T^{2} - 16 T^{3} + 2 T^{4} - 80 T^{5} + 50 T^{6} + 625 T^{8} \)
$7$ \( 1 + 4 T + 4 T^{2} - 28 T^{3} - 104 T^{4} - 196 T^{5} + 196 T^{6} + 1372 T^{7} + 2401 T^{8} \)
$11$ \( 1 + 4 T + 12 T^{2} + 60 T^{3} + 184 T^{4} + 660 T^{5} + 1452 T^{6} + 5324 T^{7} + 14641 T^{8} \)
$13$ \( ( 1 - 24 T^{2} + 169 T^{4} )^{2} \)
$17$ \( 1 + 2 T^{2} + 289 T^{4} \)
$19$ \( 1 - 8 T + 32 T^{2} - 184 T^{3} + 1042 T^{4} - 3496 T^{5} + 11552 T^{6} - 54872 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 4 T + 12 T^{2} + 164 T^{3} - 712 T^{4} + 3772 T^{5} + 6348 T^{6} - 48668 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 4 T + 22 T^{2} + 244 T^{3} + 930 T^{4} + 7076 T^{5} + 18502 T^{6} + 97556 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 12 T + 108 T^{2} + 804 T^{3} + 5112 T^{4} + 24924 T^{5} + 103788 T^{6} + 357492 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 50 T^{2} - 240 T^{3} + 1250 T^{4} - 8880 T^{5} + 68450 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2}( 1 + 16 T + 128 T^{2} + 656 T^{3} + 1681 T^{4} ) \)
$43$ \( 1 + 8 T + 32 T^{2} + 376 T^{3} + 4402 T^{4} + 16168 T^{5} + 59168 T^{6} + 636056 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 44 T^{2} + 2854 T^{4} - 97196 T^{6} + 4879681 T^{8} \)
$53$ \( ( 1 + 2 T + 2 T^{2} + 106 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 - 20 T + 200 T^{2} - 1180 T^{3} + 3481 T^{4} )( 1 + 20 T + 200 T^{2} + 1180 T^{3} + 3481 T^{4} ) \)
$61$ \( 1 + 50 T^{2} - 720 T^{3} + 1250 T^{4} - 43920 T^{5} + 186050 T^{6} + 13845841 T^{8} \)
$67$ \( ( 1 - 8 T + 142 T^{2} - 536 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( 1 - 20 T + 100 T^{2} + 1420 T^{3} - 23400 T^{4} + 100820 T^{5} + 504100 T^{6} - 7158220 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 28 T + 294 T^{2} + 1372 T^{3} + 4802 T^{4} + 100156 T^{5} + 1566726 T^{6} + 10892476 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 4 T + 12 T^{2} - 836 T^{3} - 3400 T^{4} - 66044 T^{5} + 74892 T^{6} + 1972156 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 16 T + 128 T^{2} - 1264 T^{3} + 12466 T^{4} - 104912 T^{5} + 881792 T^{6} - 9148592 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 224 T^{2} + 27874 T^{4} - 1774304 T^{6} + 62742241 T^{8} \)
$97$ \( 1 - 24 T + 242 T^{2} - 1704 T^{3} + 13730 T^{4} - 165288 T^{5} + 2276978 T^{6} - 21904152 T^{7} + 88529281 T^{8} \)
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