Properties

Label 2300.2.i.a
Level $2300$
Weight $2$
Character orbit 2300.i
Analytic conductor $18.366$
Analytic rank $0$
Dimension $8$
CM discriminant -115
Inner twists $8$

Related objects

Downloads

Learn more

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: 8.0.928445276160000.122
Defining polynomial: \(x^{8} + 353 x^{4} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{7} -3 \beta_{2} q^{9} +O(q^{10})\) \( q -\beta_{1} q^{7} -3 \beta_{2} q^{9} + ( \beta_{1} - 2 \beta_{5} ) q^{17} + ( \beta_{3} + \beta_{6} ) q^{23} -\beta_{7} q^{29} + ( 2 - \beta_{4} ) q^{31} + ( -3 \beta_{1} + 2 \beta_{5} ) q^{37} + ( -4 + \beta_{4} ) q^{41} + ( 2 \beta_{3} + 2 \beta_{6} ) q^{43} + ( 3 \beta_{2} + \beta_{7} ) q^{49} + ( \beta_{3} + 2 \beta_{6} ) q^{53} + ( 6 \beta_{2} + \beta_{7} ) q^{59} + 3 \beta_{3} q^{63} + ( 3 \beta_{1} - 4 \beta_{5} ) q^{67} + ( 6 + \beta_{4} ) q^{71} -9 q^{81} + ( -\beta_{3} + 4 \beta_{6} ) q^{83} + ( -4 \beta_{1} - 4 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 12q^{31} - 28q^{41} + 52q^{71} - 72q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 353 x^{4} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 357 \nu^{2} \)\()/76\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 357 \nu^{3} \)\()/76\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} + 186 \)\()/19\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} + 357 \nu \)\()/38\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} + 353 \nu^{3} \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{6} - 1747 \nu^{2} \)\()/38\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} + 10 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-2 \beta_{6} + 19 \beta_{3}\)
\(\nu^{4}\)\(=\)\(19 \beta_{4} - 186\)
\(\nu^{5}\)\(=\)\(38 \beta_{5} - 357 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-357 \beta_{7} - 3494 \beta_{2}\)
\(\nu^{7}\)\(=\)\(714 \beta_{6} - 6707 \beta_{3}\)

Character values

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

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1057.1
3.06489 3.06489i
0.326276 0.326276i
−0.326276 + 0.326276i
−3.06489 + 3.06489i
3.06489 + 3.06489i
0.326276 + 0.326276i
−0.326276 0.326276i
−3.06489 3.06489i
0 0 0 0 0 −3.06489 + 3.06489i 0 3.00000i 0
1057.2 0 0 0 0 0 −0.326276 + 0.326276i 0 3.00000i 0
1057.3 0 0 0 0 0 0.326276 0.326276i 0 3.00000i 0
1057.4 0 0 0 0 0 3.06489 3.06489i 0 3.00000i 0
1793.1 0 0 0 0 0 −3.06489 3.06489i 0 3.00000i 0
1793.2 0 0 0 0 0 −0.326276 0.326276i 0 3.00000i 0
1793.3 0 0 0 0 0 0.326276 + 0.326276i 0 3.00000i 0
1793.4 0 0 0 0 0 3.06489 + 3.06489i 0 3.00000i 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
115.c odd 2 1 CM by \(\Q(\sqrt{-115}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
23.b 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.a 8
5.b even 2 1 inner 2300.2.i.a 8
5.c odd 4 2 inner 2300.2.i.a 8
23.b odd 2 1 inner 2300.2.i.a 8
115.c odd 2 1 CM 2300.2.i.a 8
115.e even 4 2 inner 2300.2.i.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2300.2.i.a 8 1.a even 1 1 trivial
2300.2.i.a 8 5.b even 2 1 inner
2300.2.i.a 8 5.c odd 4 2 inner
2300.2.i.a 8 23.b odd 2 1 inner
2300.2.i.a 8 115.c odd 2 1 CM
2300.2.i.a 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 \(S_{2}^{\mathrm{new}}(2300, [\chi])\):

\( T_{3} \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( 16 + 353 T^{4} + T^{8} \)
$11$ \( T^{8} \)
$13$ \( T^{8} \)
$17$ \( 614656 + 4673 T^{4} + T^{8} \)
$19$ \( T^{8} \)
$23$ \( ( 529 + T^{4} )^{2} \)
$29$ \( ( 7396 + 173 T^{2} + T^{4} )^{2} \)
$31$ \( ( -84 - 3 T + T^{2} )^{4} \)
$37$ \( 59969536 + 24113 T^{4} + T^{8} \)
$41$ \( ( -74 + 7 T + T^{2} )^{4} \)
$43$ \( ( 8464 + T^{4} )^{2} \)
$47$ \( T^{8} \)
$53$ \( 5308416 + 7713 T^{4} + T^{8} \)
$59$ \( ( 3136 + 233 T^{2} + T^{4} )^{2} \)
$61$ \( T^{8} \)
$67$ \( 1003875856 + 80273 T^{4} + T^{8} \)
$71$ \( ( -44 - 13 T + T^{2} )^{4} \)
$73$ \( T^{8} \)
$79$ \( T^{8} \)
$83$ \( 3111696 + 81153 T^{4} + T^{8} \)
$89$ \( T^{8} \)
$97$ \( ( 135424 + T^{4} )^{2} \)
show more
show less