Properties

Label 2600.1.b.a
Level $2600$
Weight $1$
Character orbit 2600.b
Analytic conductor $1.298$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -104
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2600.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.29756903285\)
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 104)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.104.1
Artin image: $C_2\times C_4\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1299.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 1.00000i 0 0
1299.2 1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 1.00000i 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 \(\Q(\sqrt{-26}) \)
5.b even 2 1 inner
520.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2600.1.b.a 2
5.b even 2 1 inner 2600.1.b.a 2
5.c odd 4 1 104.1.h.b yes 1
5.c odd 4 1 2600.1.o.b 1
8.d odd 2 1 2600.1.b.b 2
13.b even 2 1 2600.1.b.b 2
15.e even 4 1 936.1.o.a 1
20.e even 4 1 416.1.h.a 1
40.e odd 2 1 2600.1.b.b 2
40.i odd 4 1 416.1.h.b 1
40.k even 4 1 104.1.h.a 1
40.k even 4 1 2600.1.o.d 1
60.l odd 4 1 3744.1.o.b 1
65.d even 2 1 2600.1.b.b 2
65.f even 4 1 1352.1.g.a 2
65.h odd 4 1 104.1.h.a 1
65.h odd 4 1 2600.1.o.d 1
65.k even 4 1 1352.1.g.a 2
65.o even 12 2 1352.1.n.a 4
65.q odd 12 2 1352.1.p.a 2
65.r odd 12 2 1352.1.p.b 2
65.t even 12 2 1352.1.n.a 4
80.i odd 4 1 3328.1.c.e 2
80.j even 4 1 3328.1.c.a 2
80.s even 4 1 3328.1.c.a 2
80.t odd 4 1 3328.1.c.e 2
104.h odd 2 1 CM 2600.1.b.a 2
120.q odd 4 1 936.1.o.b 1
120.w even 4 1 3744.1.o.a 1
195.s even 4 1 936.1.o.b 1
260.p even 4 1 416.1.h.b 1
520.b odd 2 1 inner 2600.1.b.a 2
520.x odd 4 1 1352.1.g.a 2
520.bc even 4 1 104.1.h.b yes 1
520.bc even 4 1 2600.1.o.b 1
520.bg odd 4 1 416.1.h.a 1
520.bk odd 4 1 1352.1.g.a 2
520.ci odd 12 2 1352.1.n.a 4
520.cm even 12 2 1352.1.p.b 2
520.cs even 12 2 1352.1.p.a 2
520.cv odd 12 2 1352.1.n.a 4
780.w odd 4 1 3744.1.o.a 1
1040.w odd 4 1 3328.1.c.a 2
1040.y even 4 1 3328.1.c.e 2
1040.co odd 4 1 3328.1.c.a 2
1040.cq even 4 1 3328.1.c.e 2
1560.bq even 4 1 3744.1.o.b 1
1560.cs odd 4 1 936.1.o.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.1.h.a 1 40.k even 4 1
104.1.h.a 1 65.h odd 4 1
104.1.h.b yes 1 5.c odd 4 1
104.1.h.b yes 1 520.bc even 4 1
416.1.h.a 1 20.e even 4 1
416.1.h.a 1 520.bg odd 4 1
416.1.h.b 1 40.i odd 4 1
416.1.h.b 1 260.p even 4 1
936.1.o.a 1 15.e even 4 1
936.1.o.a 1 1560.cs odd 4 1
936.1.o.b 1 120.q odd 4 1
936.1.o.b 1 195.s even 4 1
1352.1.g.a 2 65.f even 4 1
1352.1.g.a 2 65.k even 4 1
1352.1.g.a 2 520.x odd 4 1
1352.1.g.a 2 520.bk odd 4 1
1352.1.n.a 4 65.o even 12 2
1352.1.n.a 4 65.t even 12 2
1352.1.n.a 4 520.ci odd 12 2
1352.1.n.a 4 520.cv odd 12 2
1352.1.p.a 2 65.q odd 12 2
1352.1.p.a 2 520.cs even 12 2
1352.1.p.b 2 65.r odd 12 2
1352.1.p.b 2 520.cm even 12 2
2600.1.b.a 2 1.a even 1 1 trivial
2600.1.b.a 2 5.b even 2 1 inner
2600.1.b.a 2 104.h odd 2 1 CM
2600.1.b.a 2 520.b odd 2 1 inner
2600.1.b.b 2 8.d odd 2 1
2600.1.b.b 2 13.b even 2 1
2600.1.b.b 2 40.e odd 2 1
2600.1.b.b 2 65.d even 2 1
2600.1.o.b 1 5.c odd 4 1
2600.1.o.b 1 520.bc even 4 1
2600.1.o.d 1 40.k even 4 1
2600.1.o.d 1 65.h odd 4 1
3328.1.c.a 2 80.j even 4 1
3328.1.c.a 2 80.s even 4 1
3328.1.c.a 2 1040.w odd 4 1
3328.1.c.a 2 1040.co odd 4 1
3328.1.c.e 2 80.i odd 4 1
3328.1.c.e 2 80.t odd 4 1
3328.1.c.e 2 1040.y even 4 1
3328.1.c.e 2 1040.cq even 4 1
3744.1.o.a 1 120.w even 4 1
3744.1.o.a 1 780.w odd 4 1
3744.1.o.b 1 60.l odd 4 1
3744.1.o.b 1 1560.bq even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{31} - 2 \) acting on \(S_{1}^{\mathrm{new}}(2600, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) 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
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 1 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 1 \) Copy content Toggle raw display
$53$ \( T^{2} \) 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
$71$ \( (T + 1)^{2} \) 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
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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