Properties

Label 1300.2.bn.b
Level $1300$
Weight $2$
Character orbit 1300.bn
Analytic conductor $10.381$
Analytic rank $0$
Dimension $4$
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.bn (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + 1) q^{3} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12} - 2) q^{7} + ( - \zeta_{12}^{2} + \zeta_{12} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + 1) q^{3} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12} - 2) q^{7} + ( - \zeta_{12}^{2} + \zeta_{12} - 1) q^{9} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 3 \zeta_{12} + 1) q^{11} + ( - 3 \zeta_{12}^{3} - 2) q^{13} + ( - 3 \zeta_{12}^{3} + \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{17} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12} - 1) q^{19} + (\zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12} + 1) q^{21} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + 4 \zeta_{12} + 3) q^{23} + ( - 2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 2) q^{27} + ( - 4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 4 \zeta_{12} + 6) q^{29} + ( - 5 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 5) q^{31} + (2 \zeta_{12}^{2} + \zeta_{12} + 2) q^{33} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12}) q^{37} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 3 \zeta_{12} - 5) q^{39} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + \zeta_{12} - 3) q^{41} + (9 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{43} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 4) q^{47} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{49} + ( - 5 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{51} + (7 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 7) q^{53} + (2 \zeta_{12}^{2} - 1) q^{57} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 9 \zeta_{12} - 9) q^{59} + ( - \zeta_{12}^{2} + 1) q^{61} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 2) q^{63} + ( - 9 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 9 \zeta_{12} - 4) q^{67} + ( - 8 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 4 \zeta_{12} + 7) q^{69} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12} + 1) q^{71} + (6 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4) q^{73} + ( - 2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12} - 2) q^{77} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{79} + ( - 5 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 5 \zeta_{12}) q^{81} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{83} + ( - 7 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{87} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 9 \zeta_{12} - 5) q^{89} + ( - 4 \zeta_{12}^{3} - \zeta_{12}^{2} + 8 \zeta_{12} + 7) q^{91} + ( - 5 \zeta_{12}^{3} - 7 \zeta_{12}^{2} - 5 \zeta_{12}) q^{93} + ( - 2 \zeta_{12}^{2} - \zeta_{12} - 2) q^{97} + (\zeta_{12}^{3} - 6 \zeta_{12}^{2} + 6 \zeta_{12} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 4 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 4 q^{7} - 6 q^{9} + 8 q^{11} - 8 q^{13} + 10 q^{17} + 10 q^{21} + 14 q^{23} + 2 q^{27} + 18 q^{29} + 16 q^{31} + 12 q^{33} + 4 q^{37} - 16 q^{39} - 4 q^{41} + 26 q^{43} + 16 q^{47} + 20 q^{53} - 24 q^{59} + 2 q^{61} + 6 q^{63} - 12 q^{67} + 14 q^{69} + 8 q^{71} - 14 q^{77} - 4 q^{81} - 8 q^{87} - 28 q^{89} + 26 q^{91} - 14 q^{93} - 12 q^{97} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(\zeta_{12}\) \(1\) \(-\zeta_{12}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0.500000 1.86603i 0 0 0 −1.86603 + 3.23205i 0 −0.633975 0.366025i 0
557.1 0 0.500000 + 0.133975i 0 0 0 −0.133975 + 0.232051i 0 −2.36603 1.36603i 0
657.1 0 0.500000 + 1.86603i 0 0 0 −1.86603 3.23205i 0 −0.633975 + 0.366025i 0
1293.1 0 0.500000 0.133975i 0 0 0 −0.133975 0.232051i 0 −2.36603 + 1.36603i 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
65.t even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.bn.b 4
5.b even 2 1 260.2.bf.a 4
5.c odd 4 1 260.2.bk.b yes 4
5.c odd 4 1 1300.2.bs.a 4
13.f odd 12 1 1300.2.bs.a 4
65.o even 12 1 260.2.bf.a 4
65.s odd 12 1 260.2.bk.b yes 4
65.t even 12 1 inner 1300.2.bn.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.bf.a 4 5.b even 2 1
260.2.bf.a 4 65.o even 12 1
260.2.bk.b yes 4 5.c odd 4 1
260.2.bk.b yes 4 65.s odd 12 1
1300.2.bn.b 4 1.a even 1 1 trivial
1300.2.bn.b 4 65.t even 12 1 inner
1300.2.bs.a 4 5.c odd 4 1
1300.2.bs.a 4 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 2T_{3}^{3} + 5T_{3}^{2} - 4T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + 5 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} + 15 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} - 8 T^{3} + 17 T^{2} - 22 T + 121 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 10 T^{3} + 41 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$19$ \( T^{4} + 9 T^{2} + 18 T + 9 \) Copy content Toggle raw display
$23$ \( T^{4} - 14 T^{3} + 53 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$29$ \( T^{4} - 18 T^{3} + 119 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$31$ \( T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 676 \) Copy content Toggle raw display
$37$ \( T^{4} - 4 T^{3} + 15 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$41$ \( T^{4} + 4 T^{3} + 53 T^{2} + 14 T + 1 \) Copy content Toggle raw display
$43$ \( T^{4} - 26 T^{3} + 365 T^{2} + \cdots + 14641 \) Copy content Toggle raw display
$47$ \( (T^{2} - 8 T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 20 T^{3} + 200 T^{2} + \cdots + 676 \) Copy content Toggle raw display
$59$ \( T^{4} + 24 T^{3} + 153 T^{2} + \cdots + 9801 \) Copy content Toggle raw display
$61$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 12 T^{3} - 21 T^{2} + \cdots + 4761 \) Copy content Toggle raw display
$71$ \( T^{4} - 8 T^{3} + 41 T^{2} - 130 T + 169 \) Copy content Toggle raw display
$73$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$79$ \( T^{4} + 152T^{2} + 2704 \) Copy content Toggle raw display
$83$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 28 T^{3} + 197 T^{2} + \cdots + 2209 \) Copy content Toggle raw display
$97$ \( T^{4} + 12 T^{3} + 59 T^{2} + \cdots + 121 \) Copy content Toggle raw display
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