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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

Character values

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

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

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 \(S_{2}^{\mathrm{new}}(1300, [\chi])\):

\( T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display
\( T_{19}^{2} - 7T_{19} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

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