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: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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.

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

Character values

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

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

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

\( T_{3}^{4} - 9T_{3}^{2} + 81 \) Copy content Toggle raw display
\( T_{7}^{4} - 9T_{7}^{2} + 81 \) Copy content Toggle raw display
\( T_{19}^{2} - T_{19} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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