Properties

Label 1300.2.c.c
Level $1300$
Weight $2$
Character orbit 1300.c
Analytic conductor $10.381$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{7} + 3 q^{9} - 2 q^{11} - i q^{13} - 6 i q^{17} + 6 q^{19} + 8 i q^{23} - 2 q^{29} + 10 q^{31} + 6 i q^{37} - 6 q^{41} + 4 i q^{43} + 2 i q^{47} + 3 q^{49} + 6 i q^{53} + 10 q^{59} - 2 q^{61} + 6 i q^{63} - 10 i q^{67} + 10 q^{71} + 2 i q^{73} - 4 i q^{77} + 4 q^{79} + 9 q^{81} - 6 i q^{83} + 6 q^{89} + 2 q^{91} - 2 i q^{97} - 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{9} - 4 q^{11} + 12 q^{19} - 4 q^{29} + 20 q^{31} - 12 q^{41} + 6 q^{49} + 20 q^{59} - 4 q^{61} + 20 q^{71} + 8 q^{79} + 18 q^{81} + 12 q^{89} + 4 q^{91} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
1.00000i
1.00000i
0 0 0 0 0 2.00000i 0 3.00000 0
1249.2 0 0 0 0 0 2.00000i 0 3.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.c.c 2
5.b even 2 1 inner 1300.2.c.c 2
5.c odd 4 1 52.2.a.a 1
5.c odd 4 1 1300.2.a.d 1
15.e even 4 1 468.2.a.b 1
20.e even 4 1 208.2.a.c 1
20.e even 4 1 5200.2.a.q 1
35.f even 4 1 2548.2.a.e 1
35.k even 12 2 2548.2.j.f 2
35.l odd 12 2 2548.2.j.e 2
40.i odd 4 1 832.2.a.e 1
40.k even 4 1 832.2.a.f 1
45.k odd 12 2 4212.2.i.d 2
45.l even 12 2 4212.2.i.i 2
55.e even 4 1 6292.2.a.g 1
60.l odd 4 1 1872.2.a.f 1
65.f even 4 1 676.2.d.c 2
65.h odd 4 1 676.2.a.c 1
65.k even 4 1 676.2.d.c 2
65.o even 12 2 676.2.h.c 4
65.q odd 12 2 676.2.e.c 2
65.r odd 12 2 676.2.e.b 2
65.t even 12 2 676.2.h.c 4
80.i odd 4 1 3328.2.b.q 2
80.j even 4 1 3328.2.b.e 2
80.s even 4 1 3328.2.b.e 2
80.t odd 4 1 3328.2.b.q 2
120.q odd 4 1 7488.2.a.bw 1
120.w even 4 1 7488.2.a.bn 1
195.j odd 4 1 6084.2.b.m 2
195.s even 4 1 6084.2.a.m 1
195.u odd 4 1 6084.2.b.m 2
260.l odd 4 1 2704.2.f.f 2
260.p even 4 1 2704.2.a.g 1
260.s odd 4 1 2704.2.f.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.a.a 1 5.c odd 4 1
208.2.a.c 1 20.e even 4 1
468.2.a.b 1 15.e even 4 1
676.2.a.c 1 65.h odd 4 1
676.2.d.c 2 65.f even 4 1
676.2.d.c 2 65.k even 4 1
676.2.e.b 2 65.r odd 12 2
676.2.e.c 2 65.q odd 12 2
676.2.h.c 4 65.o even 12 2
676.2.h.c 4 65.t even 12 2
832.2.a.e 1 40.i odd 4 1
832.2.a.f 1 40.k even 4 1
1300.2.a.d 1 5.c odd 4 1
1300.2.c.c 2 1.a even 1 1 trivial
1300.2.c.c 2 5.b even 2 1 inner
1872.2.a.f 1 60.l odd 4 1
2548.2.a.e 1 35.f even 4 1
2548.2.j.e 2 35.l odd 12 2
2548.2.j.f 2 35.k even 12 2
2704.2.a.g 1 260.p even 4 1
2704.2.f.f 2 260.l odd 4 1
2704.2.f.f 2 260.s odd 4 1
3328.2.b.e 2 80.j even 4 1
3328.2.b.e 2 80.s even 4 1
3328.2.b.q 2 80.i odd 4 1
3328.2.b.q 2 80.t odd 4 1
4212.2.i.d 2 45.k odd 12 2
4212.2.i.i 2 45.l even 12 2
5200.2.a.q 1 20.e even 4 1
6084.2.a.m 1 195.s even 4 1
6084.2.b.m 2 195.j odd 4 1
6084.2.b.m 2 195.u odd 4 1
6292.2.a.g 1 55.e even 4 1
7488.2.a.bn 1 120.w even 4 1
7488.2.a.bw 1 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1300, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T - 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 64 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( (T - 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 10)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 100 \) Copy content Toggle raw display
$71$ \( (T - 10)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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