Properties

Label 1300.2.i.e
Level $1300$
Weight $2$
Character orbit 1300.i
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{7} + 2 \zeta_{6} q^{9} + ( -3 + 3 \zeta_{6} ) q^{11} + ( -3 + 4 \zeta_{6} ) q^{13} -3 \zeta_{6} q^{17} + 7 \zeta_{6} q^{19} - q^{21} + ( -3 + 3 \zeta_{6} ) q^{23} + 5 q^{27} + ( -3 + 3 \zeta_{6} ) q^{29} -4 q^{31} + 3 \zeta_{6} q^{33} + ( -7 + 7 \zeta_{6} ) q^{37} + ( 1 + 3 \zeta_{6} ) q^{39} + ( 9 - 9 \zeta_{6} ) q^{41} + 11 \zeta_{6} q^{43} + ( 6 - 6 \zeta_{6} ) q^{49} -3 q^{51} + 6 q^{53} + 7 q^{57} + 3 \zeta_{6} q^{59} -11 \zeta_{6} q^{61} + ( 2 - 2 \zeta_{6} ) q^{63} + ( -7 + 7 \zeta_{6} ) q^{67} + 3 \zeta_{6} q^{69} + 3 \zeta_{6} q^{71} -2 q^{73} + 3 q^{77} + 8 q^{79} + ( -1 + \zeta_{6} ) q^{81} + 12 q^{83} + 3 \zeta_{6} q^{87} + ( -15 + 15 \zeta_{6} ) q^{89} + ( 4 - \zeta_{6} ) q^{91} + ( -4 + 4 \zeta_{6} ) q^{93} -7 \zeta_{6} q^{97} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + q^{3} - q^{7} + 2q^{9} - 3q^{11} - 2q^{13} - 3q^{17} + 7q^{19} - 2q^{21} - 3q^{23} + 10q^{27} - 3q^{29} - 8q^{31} + 3q^{33} - 7q^{37} + 5q^{39} + 9q^{41} + 11q^{43} + 6q^{49} - 6q^{51} + 12q^{53} + 14q^{57} + 3q^{59} - 11q^{61} + 2q^{63} - 7q^{67} + 3q^{69} + 3q^{71} - 4q^{73} + 6q^{77} + 16q^{79} - q^{81} + 24q^{83} + 3q^{87} - 15q^{89} + 7q^{91} - 4q^{93} - 7q^{97} - 12q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
601.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i 0 0 0 −0.500000 + 0.866025i 0 1.00000 1.73205i 0
1101.1 0 0.500000 0.866025i 0 0 0 −0.500000 0.866025i 0 1.00000 + 1.73205i 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.i.e 2
5.b even 2 1 260.2.i.b 2
5.c odd 4 2 1300.2.bb.a 4
13.c even 3 1 inner 1300.2.i.e 2
15.d odd 2 1 2340.2.q.b 2
20.d odd 2 1 1040.2.q.j 2
65.l even 6 1 3380.2.a.g 1
65.n even 6 1 260.2.i.b 2
65.n even 6 1 3380.2.a.h 1
65.q odd 12 2 1300.2.bb.a 4
65.s odd 12 2 3380.2.f.e 2
195.x odd 6 1 2340.2.q.b 2
260.v odd 6 1 1040.2.q.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.b 2 5.b even 2 1
260.2.i.b 2 65.n even 6 1
1040.2.q.j 2 20.d odd 2 1
1040.2.q.j 2 260.v odd 6 1
1300.2.i.e 2 1.a even 1 1 trivial
1300.2.i.e 2 13.c even 3 1 inner
1300.2.bb.a 4 5.c odd 4 2
1300.2.bb.a 4 65.q odd 12 2
2340.2.q.b 2 15.d odd 2 1
2340.2.q.b 2 195.x odd 6 1
3380.2.a.g 1 65.l even 6 1
3380.2.a.h 1 65.n even 6 1
3380.2.f.e 2 65.s odd 12 2

Hecke kernels

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

\( T_{3}^{2} - T_{3} + 1 \)
\( T_{19}^{2} - 7 T_{19} + 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T + T^{2} \)
$11$ \( 9 + 3 T + T^{2} \)
$13$ \( 13 + 2 T + T^{2} \)
$17$ \( 9 + 3 T + T^{2} \)
$19$ \( 49 - 7 T + T^{2} \)
$23$ \( 9 + 3 T + T^{2} \)
$29$ \( 9 + 3 T + T^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 49 + 7 T + T^{2} \)
$41$ \( 81 - 9 T + T^{2} \)
$43$ \( 121 - 11 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( 9 - 3 T + T^{2} \)
$61$ \( 121 + 11 T + T^{2} \)
$67$ \( 49 + 7 T + T^{2} \)
$71$ \( 9 - 3 T + T^{2} \)
$73$ \( ( 2 + T )^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( ( -12 + T )^{2} \)
$89$ \( 225 + 15 T + T^{2} \)
$97$ \( 49 + 7 T + T^{2} \)
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