Properties

Label 1300.2.bb.e
Level $1300$
Weight $2$
Character orbit 1300.bb
Analytic conductor $10.381$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.bb (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \zeta_{12} q^{3} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + 6 \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + 3 \zeta_{12} q^{3} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + 6 \zeta_{12}^{2} q^{9} + ( -3 + 3 \zeta_{12}^{2} ) q^{11} + ( -4 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{13} + ( 7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{17} + \zeta_{12}^{2} q^{19} -9 q^{21} + 7 \zeta_{12} q^{23} + 9 \zeta_{12}^{3} q^{27} + ( -5 + 5 \zeta_{12}^{2} ) q^{29} -4 q^{31} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{33} -3 \zeta_{12} q^{37} + ( -9 - 3 \zeta_{12}^{2} ) q^{39} + ( -7 + 7 \zeta_{12}^{2} ) q^{41} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{43} -8 \zeta_{12}^{3} q^{47} + ( 2 - 2 \zeta_{12}^{2} ) q^{49} + 21 q^{51} -6 \zeta_{12}^{3} q^{53} + 3 \zeta_{12}^{3} q^{57} + 5 \zeta_{12}^{2} q^{59} + 5 \zeta_{12}^{2} q^{61} -18 \zeta_{12} q^{63} + 13 \zeta_{12} q^{67} + 21 \zeta_{12}^{2} q^{69} + 3 \zeta_{12}^{2} q^{71} -14 \zeta_{12}^{3} q^{73} -9 \zeta_{12}^{3} q^{77} + 8 q^{79} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} + 12 \zeta_{12}^{3} q^{83} + ( -15 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{87} + ( 7 - 7 \zeta_{12}^{2} ) q^{89} + ( 12 - 9 \zeta_{12}^{2} ) q^{91} -12 \zeta_{12} q^{93} + ( 11 \zeta_{12} - 11 \zeta_{12}^{3} ) q^{97} -18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} - 6q^{11} + 2q^{19} - 36q^{21} - 10q^{29} - 16q^{31} - 42q^{39} - 14q^{41} + 4q^{49} + 84q^{51} + 10q^{59} + 10q^{61} + 42q^{69} + 6q^{71} + 32q^{79} - 18q^{81} + 14q^{89} + 30q^{91} - 72q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
549.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 −2.59808 + 1.50000i 0 0 0 2.59808 + 1.50000i 0 3.00000 5.19615i 0
549.2 0 2.59808 1.50000i 0 0 0 −2.59808 1.50000i 0 3.00000 5.19615i 0
1049.1 0 −2.59808 1.50000i 0 0 0 2.59808 1.50000i 0 3.00000 + 5.19615i 0
1049.2 0 2.59808 + 1.50000i 0 0 0 −2.59808 + 1.50000i 0 3.00000 + 5.19615i 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
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.bb.e 4
5.b even 2 1 inner 1300.2.bb.e 4
5.c odd 4 1 260.2.i.d 2
5.c odd 4 1 1300.2.i.a 2
13.c even 3 1 inner 1300.2.bb.e 4
15.e even 4 1 2340.2.q.a 2
20.e even 4 1 1040.2.q.b 2
65.n even 6 1 inner 1300.2.bb.e 4
65.o even 12 1 3380.2.f.a 2
65.q odd 12 1 260.2.i.d 2
65.q odd 12 1 1300.2.i.a 2
65.q odd 12 1 3380.2.a.b 1
65.r odd 12 1 3380.2.a.a 1
65.t even 12 1 3380.2.f.a 2
195.bl even 12 1 2340.2.q.a 2
260.bj even 12 1 1040.2.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.d 2 5.c odd 4 1
260.2.i.d 2 65.q odd 12 1
1040.2.q.b 2 20.e even 4 1
1040.2.q.b 2 260.bj even 12 1
1300.2.i.a 2 5.c odd 4 1
1300.2.i.a 2 65.q odd 12 1
1300.2.bb.e 4 1.a even 1 1 trivial
1300.2.bb.e 4 5.b even 2 1 inner
1300.2.bb.e 4 13.c even 3 1 inner
1300.2.bb.e 4 65.n even 6 1 inner
2340.2.q.a 2 15.e even 4 1
2340.2.q.a 2 195.bl even 12 1
3380.2.a.a 1 65.r odd 12 1
3380.2.a.b 1 65.q odd 12 1
3380.2.f.a 2 65.o even 12 1
3380.2.f.a 2 65.t even 12 1

Hecke kernels

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

\( T_{3}^{4} - 9 T_{3}^{2} + 81 \)
\( T_{7}^{4} - 9 T_{7}^{2} + 81 \)
\( T_{19}^{2} - T_{19} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 81 - 9 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 81 - 9 T^{2} + T^{4} \)
$11$ \( ( 9 + 3 T + T^{2} )^{2} \)
$13$ \( 169 - 22 T^{2} + T^{4} \)
$17$ \( 2401 - 49 T^{2} + T^{4} \)
$19$ \( ( 1 - T + T^{2} )^{2} \)
$23$ \( 2401 - 49 T^{2} + T^{4} \)
$29$ \( ( 25 + 5 T + T^{2} )^{2} \)
$31$ \( ( 4 + T )^{4} \)
$37$ \( 81 - 9 T^{2} + T^{4} \)
$41$ \( ( 49 + 7 T + T^{2} )^{2} \)
$43$ \( 6561 - 81 T^{2} + T^{4} \)
$47$ \( ( 64 + T^{2} )^{2} \)
$53$ \( ( 36 + T^{2} )^{2} \)
$59$ \( ( 25 - 5 T + T^{2} )^{2} \)
$61$ \( ( 25 - 5 T + T^{2} )^{2} \)
$67$ \( 28561 - 169 T^{2} + T^{4} \)
$71$ \( ( 9 - 3 T + T^{2} )^{2} \)
$73$ \( ( 196 + T^{2} )^{2} \)
$79$ \( ( -8 + T )^{4} \)
$83$ \( ( 144 + T^{2} )^{2} \)
$89$ \( ( 49 - 7 T + T^{2} )^{2} \)
$97$ \( 14641 - 121 T^{2} + T^{4} \)
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