Properties

Label 2600.1.o.d
Level 2600
Weight 1
Character orbit 2600.o
Self dual yes
Analytic conductor 1.298
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM discriminant -104
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2600.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.29756903285\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 104)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.104.1
Artin image $D_6$
Artin field Galois closure of 6.0.10816000.2

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{6} - q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{3} + q^{4} + q^{6} - q^{7} + q^{8} + q^{12} + q^{13} - q^{14} + q^{16} + q^{17} - q^{21} + q^{24} + q^{26} - q^{27} - q^{28} - 2q^{31} + q^{32} + q^{34} - q^{37} + q^{39} - q^{42} + q^{43} - q^{47} + q^{48} + q^{51} + q^{52} - q^{54} - q^{56} - 2q^{62} + q^{64} + q^{68} + q^{71} - q^{74} + q^{78} - q^{81} - q^{84} + q^{86} - q^{91} - 2q^{93} - q^{94} + q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times\).

\(n\) \(1301\) \(1601\) \(1951\) \(1977\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
0
1.00000 1.00000 1.00000 0 1.00000 −1.00000 1.00000 0 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
104.h odd 2 1 CM by \(\Q(\sqrt{-26}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2600.1.o.d 1
5.b even 2 1 104.1.h.a 1
5.c odd 4 2 2600.1.b.b 2
8.d odd 2 1 2600.1.o.b 1
13.b even 2 1 2600.1.o.b 1
15.d odd 2 1 936.1.o.b 1
20.d odd 2 1 416.1.h.b 1
40.e odd 2 1 104.1.h.b yes 1
40.f even 2 1 416.1.h.a 1
40.k even 4 2 2600.1.b.a 2
60.h even 2 1 3744.1.o.a 1
65.d even 2 1 104.1.h.b yes 1
65.g odd 4 2 1352.1.g.a 2
65.h odd 4 2 2600.1.b.a 2
65.l even 6 2 1352.1.p.a 2
65.n even 6 2 1352.1.p.b 2
65.s odd 12 4 1352.1.n.a 4
80.k odd 4 2 3328.1.c.e 2
80.q even 4 2 3328.1.c.a 2
104.h odd 2 1 CM 2600.1.o.d 1
120.i odd 2 1 3744.1.o.b 1
120.m even 2 1 936.1.o.a 1
195.e odd 2 1 936.1.o.a 1
260.g odd 2 1 416.1.h.a 1
520.b odd 2 1 104.1.h.a 1
520.p even 2 1 416.1.h.b 1
520.t even 4 2 1352.1.g.a 2
520.bc even 4 2 2600.1.b.b 2
520.bx odd 6 2 1352.1.p.a 2
520.cd odd 6 2 1352.1.p.b 2
520.cz even 12 4 1352.1.n.a 4
780.d even 2 1 3744.1.o.b 1
1040.be even 4 2 3328.1.c.e 2
1040.cb odd 4 2 3328.1.c.a 2
1560.n even 2 1 936.1.o.b 1
1560.y odd 2 1 3744.1.o.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.1.h.a 1 5.b even 2 1
104.1.h.a 1 520.b odd 2 1
104.1.h.b yes 1 40.e odd 2 1
104.1.h.b yes 1 65.d even 2 1
416.1.h.a 1 40.f even 2 1
416.1.h.a 1 260.g odd 2 1
416.1.h.b 1 20.d odd 2 1
416.1.h.b 1 520.p even 2 1
936.1.o.a 1 120.m even 2 1
936.1.o.a 1 195.e odd 2 1
936.1.o.b 1 15.d odd 2 1
936.1.o.b 1 1560.n even 2 1
1352.1.g.a 2 65.g odd 4 2
1352.1.g.a 2 520.t even 4 2
1352.1.n.a 4 65.s odd 12 4
1352.1.n.a 4 520.cz even 12 4
1352.1.p.a 2 65.l even 6 2
1352.1.p.a 2 520.bx odd 6 2
1352.1.p.b 2 65.n even 6 2
1352.1.p.b 2 520.cd odd 6 2
2600.1.b.a 2 40.k even 4 2
2600.1.b.a 2 65.h odd 4 2
2600.1.b.b 2 5.c odd 4 2
2600.1.b.b 2 520.bc even 4 2
2600.1.o.b 1 8.d odd 2 1
2600.1.o.b 1 13.b even 2 1
2600.1.o.d 1 1.a even 1 1 trivial
2600.1.o.d 1 104.h odd 2 1 CM
3328.1.c.a 2 80.q even 4 2
3328.1.c.a 2 1040.cb odd 4 2
3328.1.c.e 2 80.k odd 4 2
3328.1.c.e 2 1040.be even 4 2
3744.1.o.a 1 60.h even 2 1
3744.1.o.a 1 1560.y odd 2 1
3744.1.o.b 1 120.i odd 2 1
3744.1.o.b 1 780.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2600, [\chi])\):

\( T_{3} - 1 \)
\( T_{7} + 1 \)

Hecke characteristic polynomials

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