Properties

Label 1300.2.f.e
Level $1300$
Weight $2$
Character orbit 1300.f
Analytic conductor $10.381$
Analytic rank $0$
Dimension $6$
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.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.9144576.1
Defining polynomial: \(x^{6} + 12 x^{4} + 36 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{3} q^{3} + \beta_{5} q^{7} + ( 1 - \beta_{4} ) q^{9} +O(q^{10})\) \( q -\beta_{3} q^{3} + \beta_{5} q^{7} + ( 1 - \beta_{4} ) q^{9} + ( \beta_{2} - \beta_{5} ) q^{11} + ( \beta_{1} - \beta_{3} - \beta_{4} ) q^{13} + ( -2 \beta_{1} + \beta_{2} - \beta_{5} ) q^{19} + 2 \beta_{1} q^{21} + ( 2 - \beta_{3} - 2 \beta_{4} ) q^{23} -2 q^{27} + ( -2 - 2 \beta_{3} - \beta_{4} ) q^{29} + ( -\beta_{2} - \beta_{5} ) q^{31} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{5} ) q^{33} + ( -2 \beta_{2} - \beta_{5} ) q^{37} + ( 2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{39} + ( 2 \beta_{1} + 2 \beta_{5} ) q^{41} + ( -2 - 3 \beta_{3} ) q^{43} + ( -4 \beta_{1} - \beta_{5} ) q^{47} + ( -1 - 2 \beta_{3} - 2 \beta_{4} ) q^{49} + ( 2 + 2 \beta_{3} - 2 \beta_{4} ) q^{53} + ( 2 \beta_{1} + 4 \beta_{2} - 3 \beta_{5} ) q^{57} + ( -2 \beta_{1} - 3 \beta_{2} + \beta_{5} ) q^{59} + ( -2 + 3 \beta_{4} ) q^{61} + ( -4 \beta_{1} - 2 \beta_{2} - \beta_{5} ) q^{63} + ( -2 \beta_{2} + 3 \beta_{5} ) q^{67} + ( -6 \beta_{3} - \beta_{4} ) q^{69} + ( -2 \beta_{1} + \beta_{2} + 3 \beta_{5} ) q^{71} + ( -4 \beta_{1} - 2 \beta_{2} + \beta_{5} ) q^{73} + 6 q^{77} + ( -4 + 4 \beta_{3} - 2 \beta_{4} ) q^{79} + ( -3 + 2 \beta_{3} + 3 \beta_{4} ) q^{81} + ( 2 \beta_{2} + \beta_{5} ) q^{83} + ( 6 - 2 \beta_{4} ) q^{87} + ( 4 \beta_{2} + 2 \beta_{5} ) q^{89} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{91} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{5} ) q^{93} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} ) q^{97} + ( 2 \beta_{1} + 3 \beta_{2} - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{9} + O(q^{10}) \) \( 6 q + 6 q^{9} + 12 q^{23} - 12 q^{27} - 12 q^{29} + 12 q^{39} - 12 q^{43} - 6 q^{49} + 12 q^{53} - 12 q^{61} + 36 q^{77} - 24 q^{79} - 18 q^{81} + 36 q^{87} + 12 q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 12 x^{4} + 36 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\( \nu^{3} + 7 \nu \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} + 6 \nu^{2} \)\()/2\)
\(\beta_{4}\)\(=\)\( \nu^{2} + 4 \)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} + 10 \nu^{3} + 24 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 2 \beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\(\beta_{4} - 4\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{2} + 14 \beta_{1}\)\()/3\)
\(\nu^{4}\)\(=\)\(-6 \beta_{4} + 2 \beta_{3} + 24\)
\(\nu^{5}\)\(=\)\((\)\(6 \beta_{5} + 16 \beta_{2} - 92 \beta_{1}\)\()/3\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
2.60168i
2.60168i
0.339877i
0.339877i
2.26180i
2.26180i
0 −2.60168 0 0 0 2.76873i 0 3.76873 0
701.2 0 −2.60168 0 0 0 2.76873i 0 3.76873 0
701.3 0 0.339877 0 0 0 3.88448i 0 −2.88448 0
701.4 0 0.339877 0 0 0 3.88448i 0 −2.88448 0
701.5 0 2.26180 0 0 0 1.11575i 0 2.11575 0
701.6 0 2.26180 0 0 0 1.11575i 0 2.11575 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 701.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.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.f.e 6
5.b even 2 1 260.2.f.a 6
5.c odd 4 1 1300.2.d.c 6
5.c odd 4 1 1300.2.d.d 6
13.b even 2 1 inner 1300.2.f.e 6
15.d odd 2 1 2340.2.c.d 6
20.d odd 2 1 1040.2.k.c 6
65.d even 2 1 260.2.f.a 6
65.g odd 4 1 3380.2.a.m 3
65.g odd 4 1 3380.2.a.n 3
65.h odd 4 1 1300.2.d.c 6
65.h odd 4 1 1300.2.d.d 6
195.e odd 2 1 2340.2.c.d 6
260.g odd 2 1 1040.2.k.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.f.a 6 5.b even 2 1
260.2.f.a 6 65.d even 2 1
1040.2.k.c 6 20.d odd 2 1
1040.2.k.c 6 260.g odd 2 1
1300.2.d.c 6 5.c odd 4 1
1300.2.d.c 6 65.h odd 4 1
1300.2.d.d 6 5.c odd 4 1
1300.2.d.d 6 65.h odd 4 1
1300.2.f.e 6 1.a even 1 1 trivial
1300.2.f.e 6 13.b even 2 1 inner
2340.2.c.d 6 15.d odd 2 1
2340.2.c.d 6 195.e odd 2 1
3380.2.a.m 3 65.g odd 4 1
3380.2.a.n 3 65.g odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 6 T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(1300, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( ( 2 - 6 T + T^{3} )^{2} \)
$5$ \( T^{6} \)
$7$ \( 144 + 144 T^{2} + 24 T^{4} + T^{6} \)
$11$ \( 324 + 216 T^{2} + 36 T^{4} + T^{6} \)
$13$ \( 2197 - 117 T^{2} + 16 T^{3} - 9 T^{4} + T^{6} \)
$17$ \( T^{6} \)
$19$ \( 12996 + 2304 T^{2} + 96 T^{4} + T^{6} \)
$23$ \( ( 174 - 30 T - 6 T^{2} + T^{3} )^{2} \)
$29$ \( ( -84 - 12 T + 6 T^{2} + T^{3} )^{2} \)
$31$ \( 4356 + 936 T^{2} + 60 T^{4} + T^{6} \)
$37$ \( 63504 + 6048 T^{2} + 144 T^{4} + T^{6} \)
$41$ \( 5184 + 2160 T^{2} + 108 T^{4} + T^{6} \)
$43$ \( ( -46 - 42 T + 6 T^{2} + T^{3} )^{2} \)
$47$ \( 219024 + 12528 T^{2} + 216 T^{4} + T^{6} \)
$53$ \( ( -24 - 84 T - 6 T^{2} + T^{3} )^{2} \)
$59$ \( 171396 + 12528 T^{2} + 216 T^{4} + T^{6} \)
$61$ \( ( -532 - 96 T + 6 T^{2} + T^{3} )^{2} \)
$67$ \( 345744 + 16416 T^{2} + 240 T^{4} + T^{6} \)
$71$ \( 492804 + 46656 T^{2} + 432 T^{4} + T^{6} \)
$73$ \( 63504 + 8640 T^{2} + 288 T^{4} + T^{6} \)
$79$ \( ( -1696 - 144 T + 12 T^{2} + T^{3} )^{2} \)
$83$ \( 63504 + 6048 T^{2} + 144 T^{4} + T^{6} \)
$89$ \( 4064256 + 96768 T^{2} + 576 T^{4} + T^{6} \)
$97$ \( 576 + 2736 T^{2} + 204 T^{4} + T^{6} \)
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