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: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

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

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

$p$ $F_p(T)$
$2$ \( T^{4} + 4 T^{3} + 8 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} + 4 T^{2} + 8 \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{2} + 4 T + 2 \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} + 4 T^{2} + 8 \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} + 12 T^{2} + 16 T + 8 \) Copy content Toggle raw display
$13$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 2T^{2} + 289 \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} + 32 T^{2} - 32 T + 16 \) Copy content Toggle raw display
$23$ \( T^{4} - 4 T^{3} + 12 T^{2} - 112 T + 392 \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} + 22 T^{2} + 12 T + 2 \) Copy content Toggle raw display
$31$ \( T^{4} + 12 T^{3} + 108 T^{2} + \cdots + 648 \) Copy content Toggle raw display
$37$ \( T^{4} + 50 T^{2} + 500 T + 1250 \) Copy content Toggle raw display
$41$ \( T^{4} + 4 T^{3} + 54 T^{2} - 140 T + 98 \) Copy content Toggle raw display
$43$ \( T^{4} + 8 T^{3} + 32 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$47$ \( T^{4} + 144T^{2} + 3136 \) Copy content Toggle raw display
$53$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 1296 \) Copy content Toggle raw display
$61$ \( T^{4} + 50 T^{2} + 500 T + 1250 \) Copy content Toggle raw display
$67$ \( (T^{2} - 8 T + 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 20 T^{3} + 100 T^{2} + \cdots + 5000 \) Copy content Toggle raw display
$73$ \( T^{4} + 28 T^{3} + 294 T^{2} + \cdots + 4802 \) Copy content Toggle raw display
$79$ \( T^{4} + 4 T^{3} + 12 T^{2} + 112 T + 392 \) Copy content Toggle raw display
$83$ \( T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$89$ \( T^{4} + 132T^{2} + 3844 \) Copy content Toggle raw display
$97$ \( T^{4} - 24 T^{3} + 242 T^{2} + \cdots + 4418 \) Copy content Toggle raw display
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