Properties

Label 3400.1.bc.a
Level $3400$
Weight $1$
Character orbit 3400.bc
Analytic conductor $1.697$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3400 = 2^{3} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3400.bc (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.69682104295\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 136)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.314432.1
Artin image: $C_4^3.C_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{32} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{2} + ( 1 - i ) q^{3} + q^{4} + ( -1 + i ) q^{6} - q^{8} -i q^{9} +O(q^{10})\) \( q - q^{2} + ( 1 - i ) q^{3} + q^{4} + ( -1 + i ) q^{6} - q^{8} -i q^{9} + ( 1 - i ) q^{11} + ( 1 - i ) q^{12} + q^{16} - q^{17} + i q^{18} -2 i q^{19} + ( -1 + i ) q^{22} + ( -1 + i ) q^{24} - q^{32} -2 i q^{33} + q^{34} -i q^{36} + 2 i q^{38} + ( -1 + i ) q^{41} + ( 1 - i ) q^{44} + ( 1 - i ) q^{48} + i q^{49} + ( -1 + i ) q^{51} + ( -2 - 2 i ) q^{57} + q^{64} + 2 i q^{66} - q^{68} + i q^{72} + ( 1 - i ) q^{73} -2 i q^{76} + q^{81} + ( 1 - i ) q^{82} + ( -1 + i ) q^{88} + ( -1 + i ) q^{96} + ( 1 - i ) q^{97} -i q^{98} + ( -1 - i ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 2q^{8} + 2q^{11} + 2q^{12} + 2q^{16} - 2q^{17} - 2q^{22} - 2q^{24} - 2q^{32} + 2q^{34} - 2q^{41} + 2q^{44} + 2q^{48} - 2q^{51} - 4q^{57} + 2q^{64} - 2q^{68} + 2q^{73} + 2q^{81} + 2q^{82} - 2q^{88} - 2q^{96} + 2q^{97} - 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3400\mathbb{Z}\right)^\times\).

\(n\) \(1601\) \(1701\) \(2177\) \(2551\)
\(\chi(n)\) \(i\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2299.1
1.00000i
1.00000i
−1.00000 1.00000 + 1.00000i 1.00000 0 −1.00000 1.00000i 0 −1.00000 1.00000i 0
2699.1 −1.00000 1.00000 1.00000i 1.00000 0 −1.00000 + 1.00000i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
85.j even 4 1 inner
680.bc odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3400.1.bc.a 2
5.b even 2 1 3400.1.bc.b 2
5.c odd 4 1 136.1.j.a 2
5.c odd 4 1 3400.1.y.a 2
8.d odd 2 1 CM 3400.1.bc.a 2
15.e even 4 1 1224.1.s.a 2
17.c even 4 1 3400.1.bc.b 2
20.e even 4 1 544.1.n.a 2
40.e odd 2 1 3400.1.bc.b 2
40.i odd 4 1 544.1.n.a 2
40.k even 4 1 136.1.j.a 2
40.k even 4 1 3400.1.y.a 2
85.f odd 4 1 136.1.j.a 2
85.g odd 4 1 2312.1.j.b 2
85.i odd 4 1 2312.1.j.b 2
85.i odd 4 1 3400.1.y.a 2
85.j even 4 1 inner 3400.1.bc.a 2
85.k odd 8 2 2312.1.f.b 2
85.n odd 8 2 2312.1.e.a 2
85.o even 16 4 2312.1.p.e 8
85.r even 16 4 2312.1.p.e 8
120.q odd 4 1 1224.1.s.a 2
136.j odd 4 1 3400.1.bc.b 2
255.k even 4 1 1224.1.s.a 2
340.s even 4 1 544.1.n.a 2
680.t even 4 1 136.1.j.a 2
680.u even 4 1 2312.1.j.b 2
680.bc odd 4 1 inner 3400.1.bc.a 2
680.bk odd 4 1 544.1.n.a 2
680.bl even 4 1 2312.1.j.b 2
680.bl even 4 1 3400.1.y.a 2
680.bw even 8 2 2312.1.f.b 2
680.bz even 8 2 2312.1.e.a 2
680.ch odd 16 4 2312.1.p.e 8
680.cr odd 16 4 2312.1.p.e 8
2040.da odd 4 1 1224.1.s.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.1.j.a 2 5.c odd 4 1
136.1.j.a 2 40.k even 4 1
136.1.j.a 2 85.f odd 4 1
136.1.j.a 2 680.t even 4 1
544.1.n.a 2 20.e even 4 1
544.1.n.a 2 40.i odd 4 1
544.1.n.a 2 340.s even 4 1
544.1.n.a 2 680.bk odd 4 1
1224.1.s.a 2 15.e even 4 1
1224.1.s.a 2 120.q odd 4 1
1224.1.s.a 2 255.k even 4 1
1224.1.s.a 2 2040.da odd 4 1
2312.1.e.a 2 85.n odd 8 2
2312.1.e.a 2 680.bz even 8 2
2312.1.f.b 2 85.k odd 8 2
2312.1.f.b 2 680.bw even 8 2
2312.1.j.b 2 85.g odd 4 1
2312.1.j.b 2 85.i odd 4 1
2312.1.j.b 2 680.u even 4 1
2312.1.j.b 2 680.bl even 4 1
2312.1.p.e 8 85.o even 16 4
2312.1.p.e 8 85.r even 16 4
2312.1.p.e 8 680.ch odd 16 4
2312.1.p.e 8 680.cr odd 16 4
3400.1.y.a 2 5.c odd 4 1
3400.1.y.a 2 40.k even 4 1
3400.1.y.a 2 85.i odd 4 1
3400.1.y.a 2 680.bl even 4 1
3400.1.bc.a 2 1.a even 1 1 trivial
3400.1.bc.a 2 8.d odd 2 1 CM
3400.1.bc.a 2 85.j even 4 1 inner
3400.1.bc.a 2 680.bc odd 4 1 inner
3400.1.bc.b 2 5.b even 2 1
3400.1.bc.b 2 17.c even 4 1
3400.1.bc.b 2 40.e odd 2 1
3400.1.bc.b 2 136.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2 T_{3} + 2 \) acting on \(S_{1}^{\mathrm{new}}(3400, [\chi])\).

Hecke characteristic polynomials

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