Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2300.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3655924649\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 14 x^{6} + 53 x^{4} + 29 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -\beta_{1} - \beta_{5} ) q^{7} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -\beta_{1} - \beta_{5} ) q^{7} + ( -1 + \beta_{2} ) q^{9} + ( -\beta_{3} + \beta_{6} ) q^{11} + ( \beta_{1} + \beta_{5} + \beta_{7} ) q^{13} + ( -2 \beta_{1} - 2 \beta_{4} ) q^{17} + ( -2 - \beta_{3} + \beta_{6} ) q^{19} + 2 q^{21} -\beta_{4} q^{23} + ( -\beta_{4} + \beta_{5} ) q^{27} + ( 2 - \beta_{2} - \beta_{6} ) q^{29} + ( 4 - \beta_{2} - \beta_{3} ) q^{31} + ( -6 \beta_{4} - 2 \beta_{7} ) q^{33} + ( -2 \beta_{1} + 2 \beta_{7} ) q^{37} + ( -3 - 2 \beta_{3} + \beta_{6} ) q^{39} + ( 2 \beta_{2} - 2 \beta_{3} - \beta_{6} ) q^{41} + ( -\beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{43} + ( -4 \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{47} + ( 1 - 2 \beta_{2} - \beta_{3} - \beta_{6} ) q^{49} + ( 8 - 2 \beta_{2} + 2 \beta_{3} ) q^{51} + ( -2 \beta_{4} - 2 \beta_{5} ) q^{53} + ( -2 \beta_{1} - 6 \beta_{4} - 2 \beta_{7} ) q^{57} + ( -3 - \beta_{3} + \beta_{6} ) q^{59} + ( 6 - 2 \beta_{6} ) q^{61} + ( -\beta_{1} - 3 \beta_{5} ) q^{63} + ( -4 \beta_{4} - 2 \beta_{7} ) q^{67} + \beta_{3} q^{69} + ( 4 + \beta_{2} - 2 \beta_{3} ) q^{71} + ( 2 \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{73} + ( -\beta_{1} - 2 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{77} + ( -6 - 3 \beta_{3} - \beta_{6} ) q^{79} + ( -1 + 2 \beta_{2} + \beta_{3} ) q^{81} + ( -3 \beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{83} + ( 4 \beta_{1} + 3 \beta_{4} - \beta_{5} + \beta_{7} ) q^{87} + ( 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{6} ) q^{89} + ( 6 + 2 \beta_{2} - \beta_{3} - 3 \beta_{6} ) q^{91} + ( 6 \beta_{1} - 3 \beta_{4} - \beta_{5} - \beta_{7} ) q^{93} + ( -2 \beta_{4} - 4 \beta_{5} - 2 \beta_{7} ) q^{97} + ( 2 + 7 \beta_{3} + \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{9} + O(q^{10}) \) \( 8q - 4q^{9} + 2q^{11} - 14q^{19} + 16q^{21} + 10q^{29} + 28q^{31} - 22q^{39} + 6q^{41} - 2q^{49} + 56q^{51} - 22q^{59} + 44q^{61} + 36q^{71} - 50q^{79} + 50q^{91} + 18q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 14 x^{6} + 53 x^{4} + 29 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 4 \)
\(\beta_{3}\)\(=\)\( \nu^{4} + 7 \nu^{2} + 2 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 14 \nu^{5} + 51 \nu^{3} + 15 \nu \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 14 \nu^{5} + 53 \nu^{3} + 27 \nu \)\()/2\)
\(\beta_{6}\)\(=\)\( \nu^{6} + 13 \nu^{4} + 44 \nu^{2} + 13 \)
\(\beta_{7}\)\(=\)\( -2 \nu^{7} - 27 \nu^{5} - 95 \nu^{3} - 28 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 4\)
\(\nu^{3}\)\(=\)\(\beta_{5} - \beta_{4} - 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{3} - 7 \beta_{2} + 26\)
\(\nu^{5}\)\(=\)\(\beta_{7} - 7 \beta_{5} + 11 \beta_{4} + 40 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{6} - 13 \beta_{3} + 47 \beta_{2} - 175\)
\(\nu^{7}\)\(=\)\(-14 \beta_{7} + 47 \beta_{5} - 101 \beta_{4} - 269 \beta_{1}\)

Character values

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

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1749.1
2.66337i
2.50653i
0.631352i
0.474520i
0.474520i
0.631352i
2.50653i
2.66337i
0 2.66337i 0 0 0 0.750930i 0 −4.09352 0
1749.2 0 2.50653i 0 0 0 0.797915i 0 −3.28271 0
1749.3 0 0.631352i 0 0 0 3.16780i 0 2.60139 0
1749.4 0 0.474520i 0 0 0 4.21479i 0 2.77483 0
1749.5 0 0.474520i 0 0 0 4.21479i 0 2.77483 0
1749.6 0 0.631352i 0 0 0 3.16780i 0 2.60139 0
1749.7 0 2.50653i 0 0 0 0.797915i 0 −3.28271 0
1749.8 0 2.66337i 0 0 0 0.750930i 0 −4.09352 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1749.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2300.2.c.j 8
5.b even 2 1 inner 2300.2.c.j 8
5.c odd 4 1 2300.2.a.l 4
5.c odd 4 1 2300.2.a.m yes 4
20.e even 4 1 9200.2.a.cm 4
20.e even 4 1 9200.2.a.co 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2300.2.a.l 4 5.c odd 4 1
2300.2.a.m yes 4 5.c odd 4 1
2300.2.c.j 8 1.a even 1 1 trivial
2300.2.c.j 8 5.b even 2 1 inner
9200.2.a.cm 4 20.e even 4 1
9200.2.a.co 4 20.e even 4 1

Hecke kernels

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

\( T_{3}^{8} + 14 T_{3}^{6} + 53 T_{3}^{4} + 29 T_{3}^{2} + 4 \)
\( T_{7}^{8} + 29 T_{7}^{6} + 212 T_{7}^{4} + 224 T_{7}^{2} + 64 \)
\( T_{11}^{4} - T_{11}^{3} - 26 T_{11}^{2} + 24 T_{11} + 120 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 4 + 29 T^{2} + 53 T^{4} + 14 T^{6} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( 64 + 224 T^{2} + 212 T^{4} + 29 T^{6} + T^{8} \)
$11$ \( ( 120 + 24 T - 26 T^{2} - T^{3} + T^{4} )^{2} \)
$13$ \( 289 + 1650 T^{2} + 823 T^{4} + 63 T^{6} + T^{8} \)
$17$ \( 2304 + 4800 T^{2} + 1072 T^{4} + 72 T^{6} + T^{8} \)
$19$ \( ( 72 - 60 T - 8 T^{2} + 7 T^{3} + T^{4} )^{2} \)
$23$ \( ( 1 + T^{2} )^{4} \)
$29$ \( ( -93 + 132 T - 31 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$31$ \( ( -10 - 27 T + 49 T^{2} - 14 T^{3} + T^{4} )^{2} \)
$37$ \( 173056 + 243008 T^{2} + 12560 T^{4} + 200 T^{6} + T^{8} \)
$41$ \( ( 381 + 174 T - 107 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$43$ \( 5184 + 4752 T^{2} + 1048 T^{4} + 65 T^{6} + T^{8} \)
$47$ \( 1542564 + 281637 T^{2} + 13077 T^{4} + 210 T^{6} + T^{8} \)
$53$ \( 82944 + 37440 T^{2} + 4192 T^{4} + 132 T^{6} + T^{8} \)
$59$ \( ( 12 - 51 T + 19 T^{2} + 11 T^{3} + T^{4} )^{2} \)
$61$ \( ( -416 - 40 T + 128 T^{2} - 22 T^{3} + T^{4} )^{2} \)
$67$ \( 984064 + 196928 T^{2} + 11792 T^{4} + 200 T^{6} + T^{8} \)
$71$ \( ( -1800 + 327 T + 67 T^{2} - 18 T^{3} + T^{4} )^{2} \)
$73$ \( 19321 + 19682 T^{2} + 2927 T^{4} + 119 T^{6} + T^{8} \)
$79$ \( ( 40 + 244 T + 176 T^{2} + 25 T^{3} + T^{4} )^{2} \)
$83$ \( 5184 + 52560 T^{2} + 8776 T^{4} + 201 T^{6} + T^{8} \)
$89$ \( ( -3840 - 1848 T - 236 T^{2} + T^{4} )^{2} \)
$97$ \( 1081600 + 3769536 T^{2} + 79216 T^{4} + 504 T^{6} + T^{8} \)
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