Properties

Label 1700.1.n.b
Level $1700$
Weight $1$
Character orbit 1700.n
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.n (of order \(4\), degree \(2\), not 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: \(\Z[a_1, \ldots, a_{9}]\)
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: $C_4\times C_4{\rm wrC}_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{32} - \cdots)\)

$q$-expansion

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
599.1
1.00000i
1.00000i
1.00000 0 1.00000 0 0 0 1.00000 1.00000i 0
999.1 1.00000 0 1.00000 0 0 0 1.00000 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 \(\Q(\sqrt{-1}) \)
85.j even 4 1 inner
340.n 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.n.b 2
4.b odd 2 1 CM 1700.1.n.b 2
5.b even 2 1 1700.1.n.a 2
5.c odd 4 1 68.1.f.a 2
5.c odd 4 1 1700.1.p.a 2
15.e even 4 1 612.1.l.a 2
17.c even 4 1 1700.1.n.a 2
20.d odd 2 1 1700.1.n.a 2
20.e even 4 1 68.1.f.a 2
20.e even 4 1 1700.1.p.a 2
35.f even 4 1 3332.1.m.b 2
35.k even 12 2 3332.1.bc.b 4
35.l odd 12 2 3332.1.bc.c 4
40.i odd 4 1 1088.1.p.a 2
40.k even 4 1 1088.1.p.a 2
60.l odd 4 1 612.1.l.a 2
68.f odd 4 1 1700.1.n.a 2
85.f odd 4 1 68.1.f.a 2
85.g odd 4 1 1156.1.f.b 2
85.i odd 4 1 1156.1.f.b 2
85.i odd 4 1 1700.1.p.a 2
85.j even 4 1 inner 1700.1.n.b 2
85.k odd 8 2 1156.1.c.b 2
85.n odd 8 2 1156.1.d.a 2
85.o even 16 4 1156.1.g.b 8
85.r even 16 4 1156.1.g.b 8
140.j odd 4 1 3332.1.m.b 2
140.w even 12 2 3332.1.bc.c 4
140.x odd 12 2 3332.1.bc.b 4
255.k even 4 1 612.1.l.a 2
340.i even 4 1 1156.1.f.b 2
340.i even 4 1 1700.1.p.a 2
340.n odd 4 1 inner 1700.1.n.b 2
340.r even 4 1 1156.1.f.b 2
340.s even 4 1 68.1.f.a 2
340.w even 8 2 1156.1.d.a 2
340.z even 8 2 1156.1.c.b 2
340.bc odd 16 4 1156.1.g.b 8
340.bj odd 16 4 1156.1.g.b 8
595.r even 4 1 3332.1.m.b 2
595.bn even 12 2 3332.1.bc.b 4
595.bs odd 12 2 3332.1.bc.c 4
680.t even 4 1 1088.1.p.a 2
680.bk odd 4 1 1088.1.p.a 2
1020.bl odd 4 1 612.1.l.a 2
2380.t odd 4 1 3332.1.m.b 2
2380.cx even 12 2 3332.1.bc.c 4
2380.dp odd 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.c odd 4 1
68.1.f.a 2 20.e even 4 1
68.1.f.a 2 85.f odd 4 1
68.1.f.a 2 340.s even 4 1
612.1.l.a 2 15.e even 4 1
612.1.l.a 2 60.l odd 4 1
612.1.l.a 2 255.k even 4 1
612.1.l.a 2 1020.bl odd 4 1
1088.1.p.a 2 40.i odd 4 1
1088.1.p.a 2 40.k even 4 1
1088.1.p.a 2 680.t even 4 1
1088.1.p.a 2 680.bk odd 4 1
1156.1.c.b 2 85.k odd 8 2
1156.1.c.b 2 340.z even 8 2
1156.1.d.a 2 85.n odd 8 2
1156.1.d.a 2 340.w even 8 2
1156.1.f.b 2 85.g odd 4 1
1156.1.f.b 2 85.i odd 4 1
1156.1.f.b 2 340.i even 4 1
1156.1.f.b 2 340.r even 4 1
1156.1.g.b 8 85.o even 16 4
1156.1.g.b 8 85.r even 16 4
1156.1.g.b 8 340.bc odd 16 4
1156.1.g.b 8 340.bj odd 16 4
1700.1.n.a 2 5.b even 2 1
1700.1.n.a 2 17.c even 4 1
1700.1.n.a 2 20.d odd 2 1
1700.1.n.a 2 68.f odd 4 1
1700.1.n.b 2 1.a even 1 1 trivial
1700.1.n.b 2 4.b odd 2 1 CM
1700.1.n.b 2 85.j even 4 1 inner
1700.1.n.b 2 340.n odd 4 1 inner
1700.1.p.a 2 5.c odd 4 1
1700.1.p.a 2 20.e even 4 1
1700.1.p.a 2 85.i odd 4 1
1700.1.p.a 2 340.i even 4 1
3332.1.m.b 2 35.f even 4 1
3332.1.m.b 2 140.j odd 4 1
3332.1.m.b 2 595.r even 4 1
3332.1.m.b 2 2380.t odd 4 1
3332.1.bc.b 4 35.k even 12 2
3332.1.bc.b 4 140.x odd 12 2
3332.1.bc.b 4 595.bn even 12 2
3332.1.bc.b 4 2380.dp odd 12 2
3332.1.bc.c 4 35.l odd 12 2
3332.1.bc.c 4 140.w even 12 2
3332.1.bc.c 4 595.bs odd 12 2
3332.1.bc.c 4 2380.cx even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{37}^{2} + 2 T_{37} + 2 \) acting on \(S_{1}^{\mathrm{new}}(1700, [\chi])\).

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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