Properties

Label 1700.1.d.b
Level $1700$
Weight $1$
Character orbit 1700.d
Analytic conductor $0.848$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -4, -68, 17
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1700 = 2^{2} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1700.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.848410521476\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 68)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{17})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.1156000000.2

$q$-expansion

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

\(f(q)\) \(=\) \( q - i q^{2} - q^{4} + i q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} - q^{4} + i q^{8} + q^{9} + i q^{13} + q^{16} + i q^{17} - i q^{18} + 2 q^{26} - i q^{32} + q^{34} - q^{36} + q^{49} - 2 i q^{52} - i q^{53} - q^{64} - i q^{68} + i q^{72} + q^{81} + q^{89} - i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{9} + 2 q^{16} + 4 q^{26} + 2 q^{34} - 2 q^{36} + 2 q^{49} - 2 q^{64} + 2 q^{81} + 4 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1699.1
1.00000i
1.00000i
1.00000i 0 −1.00000 0 0 0 1.00000i 1.00000 0
1699.2 1.00000i 0 −1.00000 0 0 0 1.00000i 1.00000 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.b even 2 1 RM by \(\Q(\sqrt{17}) \)
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)
5.b even 2 1 inner
20.d odd 2 1 inner
85.c even 2 1 inner
340.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1700.1.d.b 2
4.b odd 2 1 CM 1700.1.d.b 2
5.b even 2 1 inner 1700.1.d.b 2
5.c odd 4 1 68.1.d.a 1
5.c odd 4 1 1700.1.h.d 1
15.e even 4 1 612.1.e.a 1
17.b even 2 1 RM 1700.1.d.b 2
20.d odd 2 1 inner 1700.1.d.b 2
20.e even 4 1 68.1.d.a 1
20.e even 4 1 1700.1.h.d 1
35.f even 4 1 3332.1.g.a 1
35.k even 12 2 3332.1.o.d 2
35.l odd 12 2 3332.1.o.c 2
40.i odd 4 1 1088.1.g.a 1
40.k even 4 1 1088.1.g.a 1
60.l odd 4 1 612.1.e.a 1
68.d odd 2 1 CM 1700.1.d.b 2
85.c even 2 1 inner 1700.1.d.b 2
85.f odd 4 1 1156.1.c.a 1
85.g odd 4 1 68.1.d.a 1
85.g odd 4 1 1700.1.h.d 1
85.i odd 4 1 1156.1.c.a 1
85.k odd 8 2 1156.1.f.a 2
85.n odd 8 2 1156.1.f.a 2
85.o even 16 4 1156.1.g.a 4
85.r even 16 4 1156.1.g.a 4
140.j odd 4 1 3332.1.g.a 1
140.w even 12 2 3332.1.o.c 2
140.x odd 12 2 3332.1.o.d 2
255.o even 4 1 612.1.e.a 1
340.d odd 2 1 inner 1700.1.d.b 2
340.i even 4 1 1156.1.c.a 1
340.r even 4 1 68.1.d.a 1
340.r even 4 1 1700.1.h.d 1
340.s even 4 1 1156.1.c.a 1
340.w even 8 2 1156.1.f.a 2
340.z even 8 2 1156.1.f.a 2
340.bc odd 16 4 1156.1.g.a 4
340.bj odd 16 4 1156.1.g.a 4
595.p even 4 1 3332.1.g.a 1
595.bp odd 12 2 3332.1.o.c 2
595.br even 12 2 3332.1.o.d 2
680.u even 4 1 1088.1.g.a 1
680.bi odd 4 1 1088.1.g.a 1
1020.x odd 4 1 612.1.e.a 1
2380.bi odd 4 1 3332.1.g.a 1
2380.cz even 12 2 3332.1.o.c 2
2380.db odd 12 2 3332.1.o.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.d.a 1 5.c odd 4 1
68.1.d.a 1 20.e even 4 1
68.1.d.a 1 85.g odd 4 1
68.1.d.a 1 340.r even 4 1
612.1.e.a 1 15.e even 4 1
612.1.e.a 1 60.l odd 4 1
612.1.e.a 1 255.o even 4 1
612.1.e.a 1 1020.x odd 4 1
1088.1.g.a 1 40.i odd 4 1
1088.1.g.a 1 40.k even 4 1
1088.1.g.a 1 680.u even 4 1
1088.1.g.a 1 680.bi odd 4 1
1156.1.c.a 1 85.f odd 4 1
1156.1.c.a 1 85.i odd 4 1
1156.1.c.a 1 340.i even 4 1
1156.1.c.a 1 340.s even 4 1
1156.1.f.a 2 85.k odd 8 2
1156.1.f.a 2 85.n odd 8 2
1156.1.f.a 2 340.w even 8 2
1156.1.f.a 2 340.z even 8 2
1156.1.g.a 4 85.o even 16 4
1156.1.g.a 4 85.r even 16 4
1156.1.g.a 4 340.bc odd 16 4
1156.1.g.a 4 340.bj odd 16 4
1700.1.d.b 2 1.a even 1 1 trivial
1700.1.d.b 2 4.b odd 2 1 CM
1700.1.d.b 2 5.b even 2 1 inner
1700.1.d.b 2 17.b even 2 1 RM
1700.1.d.b 2 20.d odd 2 1 inner
1700.1.d.b 2 68.d odd 2 1 CM
1700.1.d.b 2 85.c even 2 1 inner
1700.1.d.b 2 340.d odd 2 1 inner
1700.1.h.d 1 5.c odd 4 1
1700.1.h.d 1 20.e even 4 1
1700.1.h.d 1 85.g odd 4 1
1700.1.h.d 1 340.r even 4 1
3332.1.g.a 1 35.f even 4 1
3332.1.g.a 1 140.j odd 4 1
3332.1.g.a 1 595.p even 4 1
3332.1.g.a 1 2380.bi odd 4 1
3332.1.o.c 2 35.l odd 12 2
3332.1.o.c 2 140.w even 12 2
3332.1.o.c 2 595.bp odd 12 2
3332.1.o.c 2 2380.cz even 12 2
3332.1.o.d 2 35.k even 12 2
3332.1.o.d 2 140.x odd 12 2
3332.1.o.d 2 595.br even 12 2
3332.1.o.d 2 2380.db odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1700, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 1 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 4 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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