Properties

Label 2600.1.b.b
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: \(x^{2} + 1\)
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.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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.b 2
5.b even 2 1 inner 2600.1.b.b 2
5.c odd 4 1 104.1.h.a 1
5.c odd 4 1 2600.1.o.d 1
8.d odd 2 1 2600.1.b.a 2
13.b even 2 1 2600.1.b.a 2
15.e even 4 1 936.1.o.b 1
20.e even 4 1 416.1.h.b 1
40.e odd 2 1 2600.1.b.a 2
40.i odd 4 1 416.1.h.a 1
40.k even 4 1 104.1.h.b yes 1
40.k even 4 1 2600.1.o.b 1
60.l odd 4 1 3744.1.o.a 1
65.d even 2 1 2600.1.b.a 2
65.f even 4 1 1352.1.g.a 2
65.h odd 4 1 104.1.h.b yes 1
65.h odd 4 1 2600.1.o.b 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.b 2
65.r odd 12 2 1352.1.p.a 2
65.t even 12 2 1352.1.n.a 4
80.i odd 4 1 3328.1.c.a 2
80.j even 4 1 3328.1.c.e 2
80.s even 4 1 3328.1.c.e 2
80.t odd 4 1 3328.1.c.a 2
104.h odd 2 1 CM 2600.1.b.b 2
120.q odd 4 1 936.1.o.a 1
120.w even 4 1 3744.1.o.b 1
195.s even 4 1 936.1.o.a 1
260.p even 4 1 416.1.h.a 1
520.b odd 2 1 inner 2600.1.b.b 2
520.x odd 4 1 1352.1.g.a 2
520.bc even 4 1 104.1.h.a 1
520.bc even 4 1 2600.1.o.d 1
520.bg odd 4 1 416.1.h.b 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.a 2
520.cs even 12 2 1352.1.p.b 2
520.cv odd 12 2 1352.1.n.a 4
780.w odd 4 1 3744.1.o.b 1
1040.w odd 4 1 3328.1.c.e 2
1040.y even 4 1 3328.1.c.a 2
1040.co odd 4 1 3328.1.c.e 2
1040.cq even 4 1 3328.1.c.a 2
1560.bq even 4 1 3744.1.o.a 1
1560.cs odd 4 1 936.1.o.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.1.h.a 1 5.c odd 4 1
104.1.h.a 1 520.bc even 4 1
104.1.h.b yes 1 40.k even 4 1
104.1.h.b yes 1 65.h odd 4 1
416.1.h.a 1 40.i odd 4 1
416.1.h.a 1 260.p even 4 1
416.1.h.b 1 20.e even 4 1
416.1.h.b 1 520.bg odd 4 1
936.1.o.a 1 120.q odd 4 1
936.1.o.a 1 195.s even 4 1
936.1.o.b 1 15.e even 4 1
936.1.o.b 1 1560.cs odd 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.r odd 12 2
1352.1.p.a 2 520.cm even 12 2
1352.1.p.b 2 65.q odd 12 2
1352.1.p.b 2 520.cs even 12 2
2600.1.b.a 2 8.d odd 2 1
2600.1.b.a 2 13.b even 2 1
2600.1.b.a 2 40.e odd 2 1
2600.1.b.a 2 65.d even 2 1
2600.1.b.b 2 1.a even 1 1 trivial
2600.1.b.b 2 5.b even 2 1 inner
2600.1.b.b 2 104.h odd 2 1 CM
2600.1.b.b 2 520.b odd 2 1 inner
2600.1.o.b 1 40.k even 4 1
2600.1.o.b 1 65.h odd 4 1
2600.1.o.d 1 5.c odd 4 1
2600.1.o.d 1 520.bc even 4 1
3328.1.c.a 2 80.i odd 4 1
3328.1.c.a 2 80.t odd 4 1
3328.1.c.a 2 1040.y even 4 1
3328.1.c.a 2 1040.cq even 4 1
3328.1.c.e 2 80.j even 4 1
3328.1.c.e 2 80.s even 4 1
3328.1.c.e 2 1040.w odd 4 1
3328.1.c.e 2 1040.co odd 4 1
3744.1.o.a 1 60.l odd 4 1
3744.1.o.a 1 1560.bq even 4 1
3744.1.o.b 1 120.w even 4 1
3744.1.o.b 1 780.w odd 4 1

Hecke kernels

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

Hecke characteristic polynomials

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