Properties

Label 2300.2.i.b
Level $2300$
Weight $2$
Character orbit 2300.i
Analytic conductor $18.366$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2300.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3655924649\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.157351936.1
Defining polynomial: \(x^{8} + x^{4} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} -\beta_{5} q^{7} + 4 \beta_{3} q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} -\beta_{5} q^{7} + 4 \beta_{3} q^{9} -\beta_{1} q^{11} + \beta_{2} q^{13} + \beta_{6} q^{19} -\beta_{1} q^{21} + ( -\beta_{2} + 2 \beta_{4} ) q^{23} + \beta_{7} q^{27} -\beta_{3} q^{29} + 3 q^{31} + 7 \beta_{4} q^{33} + 5 \beta_{5} q^{37} + 7 \beta_{3} q^{39} + 9 q^{41} -3 \beta_{4} q^{43} + \beta_{7} q^{47} -3 \beta_{3} q^{49} -2 \beta_{4} q^{53} + 7 \beta_{5} q^{57} -\beta_{1} q^{61} + 4 \beta_{4} q^{63} + \beta_{5} q^{67} + ( -7 \beta_{3} + 2 \beta_{6} ) q^{69} -9 q^{71} + 5 \beta_{2} q^{73} + 4 \beta_{7} q^{77} + \beta_{6} q^{79} + 5 q^{81} + 6 \beta_{4} q^{83} -\beta_{7} q^{87} -3 \beta_{6} q^{89} -\beta_{1} q^{91} + 3 \beta_{2} q^{93} + \beta_{5} q^{97} + 4 \beta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 24q^{31} + 72q^{41} - 72q^{71} + 40q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + x^{4} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 4 \nu^{4} + 2 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 11 \nu \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 5 \nu^{2} \)\()/12\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} + \nu \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + 7 \nu^{3} \)\()/12\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} + 3 \nu^{2} \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( 5 \nu^{7} + 13 \nu^{3} \)\()/24\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + 2 \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + 6 \beta_{3}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{7} + 5 \beta_{5}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{1} - 2\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-11 \beta_{4} + 2 \beta_{2}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{6} + 18 \beta_{3}\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(14 \beta_{7} - 13 \beta_{5}\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-\beta_{3}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1057.1
−0.581861 + 1.28897i
−1.28897 + 0.581861i
1.28897 0.581861i
0.581861 1.28897i
−0.581861 1.28897i
−1.28897 0.581861i
1.28897 + 0.581861i
0.581861 + 1.28897i
0 −1.87083 + 1.87083i 0 0 0 −1.41421 + 1.41421i 0 4.00000i 0
1057.2 0 −1.87083 + 1.87083i 0 0 0 1.41421 1.41421i 0 4.00000i 0
1057.3 0 1.87083 1.87083i 0 0 0 −1.41421 + 1.41421i 0 4.00000i 0
1057.4 0 1.87083 1.87083i 0 0 0 1.41421 1.41421i 0 4.00000i 0
1793.1 0 −1.87083 1.87083i 0 0 0 −1.41421 1.41421i 0 4.00000i 0
1793.2 0 −1.87083 1.87083i 0 0 0 1.41421 + 1.41421i 0 4.00000i 0
1793.3 0 1.87083 + 1.87083i 0 0 0 −1.41421 1.41421i 0 4.00000i 0
1793.4 0 1.87083 + 1.87083i 0 0 0 1.41421 + 1.41421i 0 4.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1793.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
23.b odd 2 1 inner
115.c odd 2 1 inner
115.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2300.2.i.b 8
5.b even 2 1 inner 2300.2.i.b 8
5.c odd 4 2 inner 2300.2.i.b 8
23.b odd 2 1 inner 2300.2.i.b 8
115.c odd 2 1 inner 2300.2.i.b 8
115.e even 4 2 inner 2300.2.i.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2300.2.i.b 8 1.a even 1 1 trivial
2300.2.i.b 8 5.b even 2 1 inner
2300.2.i.b 8 5.c odd 4 2 inner
2300.2.i.b 8 23.b odd 2 1 inner
2300.2.i.b 8 115.c odd 2 1 inner
2300.2.i.b 8 115.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2300, [\chi])\):

\( T_{3}^{4} + 49 \)
\( T_{19}^{2} - 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 49 + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( ( 16 + T^{4} )^{2} \)
$11$ \( ( 28 + T^{2} )^{4} \)
$13$ \( ( 49 + T^{4} )^{2} \)
$17$ \( T^{8} \)
$19$ \( ( -28 + T^{2} )^{4} \)
$23$ \( 279841 - 734 T^{4} + T^{8} \)
$29$ \( ( 1 + T^{2} )^{4} \)
$31$ \( ( -3 + T )^{8} \)
$37$ \( ( 10000 + T^{4} )^{2} \)
$41$ \( ( -9 + T )^{8} \)
$43$ \( ( 1296 + T^{4} )^{2} \)
$47$ \( ( 49 + T^{4} )^{2} \)
$53$ \( ( 256 + T^{4} )^{2} \)
$59$ \( T^{8} \)
$61$ \( ( 28 + T^{2} )^{4} \)
$67$ \( ( 16 + T^{4} )^{2} \)
$71$ \( ( 9 + T )^{8} \)
$73$ \( ( 30625 + T^{4} )^{2} \)
$79$ \( ( -28 + T^{2} )^{4} \)
$83$ \( ( 20736 + T^{4} )^{2} \)
$89$ \( ( -252 + T^{2} )^{4} \)
$97$ \( ( 16 + T^{4} )^{2} \)
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