Properties

Label 1700.1.p.a
Level $1700$
Weight $1$
Character orbit 1700.p
Analytic conductor $0.848$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1700 = 2^{2} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1700.p (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.848410521476\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 68)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.19652.1
Artin image: $C_2\times C_4\wr C_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} -i q^{8} -i q^{9} +O(q^{10})\) \( q + i q^{2} - q^{4} -i q^{8} -i q^{9} + q^{16} + q^{17} + q^{18} + ( 1 - i ) q^{29} + i q^{32} + i q^{34} + i q^{36} + ( 1 - i ) q^{37} + ( 1 + i ) q^{41} + i q^{49} + ( 1 + i ) q^{58} + ( -1 - i ) q^{61} - q^{64} - q^{68} - q^{72} + ( -1 + i ) q^{73} + ( 1 + i ) q^{74} - q^{81} + ( -1 + i ) q^{82} + ( 1 - i ) q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + O(q^{10}) \) \( 2 q - 2 q^{4} + 2 q^{16} + 2 q^{17} + 2 q^{18} + 2 q^{29} + 2 q^{37} + 2 q^{41} + 2 q^{58} - 2 q^{61} - 2 q^{64} - 2 q^{68} - 2 q^{72} - 2 q^{73} + 2 q^{74} - 2 q^{81} - 2 q^{82} + 2 q^{97} - 2 q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1700\mathbb{Z}\right)^\times\).

\(n\) \(477\) \(851\) \(1601\)
\(\chi(n)\) \(1\) \(-1\) \(-i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
1.00000i
1.00000i
1.00000i 0 −1.00000 0 0 0 1.00000i 1.00000i 0
1551.1 1.00000i 0 −1.00000 0 0 0 1.00000i 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
17.c even 4 1 inner
68.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1700.1.p.a 2
4.b odd 2 1 CM 1700.1.p.a 2
5.b even 2 1 68.1.f.a 2
5.c odd 4 1 1700.1.n.a 2
5.c odd 4 1 1700.1.n.b 2
15.d odd 2 1 612.1.l.a 2
17.c even 4 1 inner 1700.1.p.a 2
20.d odd 2 1 68.1.f.a 2
20.e even 4 1 1700.1.n.a 2
20.e even 4 1 1700.1.n.b 2
35.c odd 2 1 3332.1.m.b 2
35.i odd 6 2 3332.1.bc.b 4
35.j even 6 2 3332.1.bc.c 4
40.e odd 2 1 1088.1.p.a 2
40.f even 2 1 1088.1.p.a 2
60.h even 2 1 612.1.l.a 2
68.f odd 4 1 inner 1700.1.p.a 2
85.c even 2 1 1156.1.f.b 2
85.f odd 4 1 1700.1.n.a 2
85.i odd 4 1 1700.1.n.b 2
85.j even 4 1 68.1.f.a 2
85.j even 4 1 1156.1.f.b 2
85.m even 8 2 1156.1.c.b 2
85.m even 8 2 1156.1.d.a 2
85.p odd 16 8 1156.1.g.b 8
140.c even 2 1 3332.1.m.b 2
140.p odd 6 2 3332.1.bc.c 4
140.s even 6 2 3332.1.bc.b 4
255.i odd 4 1 612.1.l.a 2
340.d odd 2 1 1156.1.f.b 2
340.i even 4 1 1700.1.n.b 2
340.n odd 4 1 68.1.f.a 2
340.n odd 4 1 1156.1.f.b 2
340.s even 4 1 1700.1.n.a 2
340.ba odd 8 2 1156.1.c.b 2
340.ba odd 8 2 1156.1.d.a 2
340.bg even 16 8 1156.1.g.b 8
595.u odd 4 1 3332.1.m.b 2
595.bk even 12 2 3332.1.bc.c 4
595.bl odd 12 2 3332.1.bc.b 4
680.bc odd 4 1 1088.1.p.a 2
680.be even 4 1 1088.1.p.a 2
1020.ba even 4 1 612.1.l.a 2
2380.bd even 4 1 3332.1.m.b 2
2380.dc odd 12 2 3332.1.bc.c 4
2380.dj even 12 2 3332.1.bc.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.f.a 2 5.b even 2 1
68.1.f.a 2 20.d odd 2 1
68.1.f.a 2 85.j even 4 1
68.1.f.a 2 340.n odd 4 1
612.1.l.a 2 15.d odd 2 1
612.1.l.a 2 60.h even 2 1
612.1.l.a 2 255.i odd 4 1
612.1.l.a 2 1020.ba even 4 1
1088.1.p.a 2 40.e odd 2 1
1088.1.p.a 2 40.f even 2 1
1088.1.p.a 2 680.bc odd 4 1
1088.1.p.a 2 680.be even 4 1
1156.1.c.b 2 85.m even 8 2
1156.1.c.b 2 340.ba odd 8 2
1156.1.d.a 2 85.m even 8 2
1156.1.d.a 2 340.ba odd 8 2
1156.1.f.b 2 85.c even 2 1
1156.1.f.b 2 85.j even 4 1
1156.1.f.b 2 340.d odd 2 1
1156.1.f.b 2 340.n odd 4 1
1156.1.g.b 8 85.p odd 16 8
1156.1.g.b 8 340.bg even 16 8
1700.1.n.a 2 5.c odd 4 1
1700.1.n.a 2 20.e even 4 1
1700.1.n.a 2 85.f odd 4 1
1700.1.n.a 2 340.s even 4 1
1700.1.n.b 2 5.c odd 4 1
1700.1.n.b 2 20.e even 4 1
1700.1.n.b 2 85.i odd 4 1
1700.1.n.b 2 340.i even 4 1
1700.1.p.a 2 1.a even 1 1 trivial
1700.1.p.a 2 4.b odd 2 1 CM
1700.1.p.a 2 17.c even 4 1 inner
1700.1.p.a 2 68.f odd 4 1 inner
3332.1.m.b 2 35.c odd 2 1
3332.1.m.b 2 140.c even 2 1
3332.1.m.b 2 595.u odd 4 1
3332.1.m.b 2 2380.bd even 4 1
3332.1.bc.b 4 35.i odd 6 2
3332.1.bc.b 4 140.s even 6 2
3332.1.bc.b 4 595.bl odd 12 2
3332.1.bc.b 4 2380.dj even 12 2
3332.1.bc.c 4 35.j even 6 2
3332.1.bc.c 4 140.p odd 6 2
3332.1.bc.c 4 595.bk even 12 2
3332.1.bc.c 4 2380.dc odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1700, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -1 + T )^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 2 - 2 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 2 - 2 T + T^{2} \)
$41$ \( 2 - 2 T + T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 2 + 2 T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 2 + 2 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 2 - 2 T + T^{2} \)
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