Properties

Label 2300.2.c.f
Level $2300$
Weight $2$
Character orbit 2300.c
Analytic conductor $18.366$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 92)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{3} -2 i q^{7} + 2 q^{9} +O(q^{10})\) \( q + i q^{3} -2 i q^{7} + 2 q^{9} -i q^{13} + 6 i q^{17} -2 q^{19} + 2 q^{21} -i q^{23} + 5 i q^{27} + 3 q^{29} + 5 q^{31} -8 i q^{37} + q^{39} + 3 q^{41} + 8 i q^{43} -9 i q^{47} + 3 q^{49} -6 q^{51} + 6 i q^{53} -2 i q^{57} + 12 q^{59} + 14 q^{61} -4 i q^{63} -8 i q^{67} + q^{69} -15 q^{71} -7 i q^{73} + 10 q^{79} + q^{81} + 6 i q^{83} + 3 i q^{87} -2 q^{91} + 5 i q^{93} + 10 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{9} + O(q^{10}) \) \( 2q + 4q^{9} - 4q^{19} + 4q^{21} + 6q^{29} + 10q^{31} + 2q^{39} + 6q^{41} + 6q^{49} - 12q^{51} + 24q^{59} + 28q^{61} + 2q^{69} - 30q^{71} + 20q^{79} + 2q^{81} - 4q^{91} + O(q^{100}) \)

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
1.00000i
1.00000i
0 1.00000i 0 0 0 2.00000i 0 2.00000 0
1749.2 0 1.00000i 0 0 0 2.00000i 0 2.00000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2300.2.c.f 2
5.b even 2 1 inner 2300.2.c.f 2
5.c odd 4 1 92.2.a.b 1
5.c odd 4 1 2300.2.a.c 1
15.e even 4 1 828.2.a.b 1
20.e even 4 1 368.2.a.b 1
20.e even 4 1 9200.2.a.ba 1
35.f even 4 1 4508.2.a.a 1
40.i odd 4 1 1472.2.a.c 1
40.k even 4 1 1472.2.a.j 1
60.l odd 4 1 3312.2.a.g 1
115.e even 4 1 2116.2.a.d 1
460.k odd 4 1 8464.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.2.a.b 1 5.c odd 4 1
368.2.a.b 1 20.e even 4 1
828.2.a.b 1 15.e even 4 1
1472.2.a.c 1 40.i odd 4 1
1472.2.a.j 1 40.k even 4 1
2116.2.a.d 1 115.e even 4 1
2300.2.a.c 1 5.c odd 4 1
2300.2.c.f 2 1.a even 1 1 trivial
2300.2.c.f 2 5.b even 2 1 inner
3312.2.a.g 1 60.l odd 4 1
4508.2.a.a 1 35.f even 4 1
8464.2.a.f 1 460.k odd 4 1
9200.2.a.ba 1 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 \(S_{2}^{\mathrm{new}}(2300, [\chi])\):

\( T_{3}^{2} + 1 \)
\( T_{7}^{2} + 4 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 1 + T^{2} \)
$29$ \( ( -3 + T )^{2} \)
$31$ \( ( -5 + T )^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( ( -3 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( 81 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( -12 + T )^{2} \)
$61$ \( ( -14 + T )^{2} \)
$67$ \( 64 + T^{2} \)
$71$ \( ( 15 + T )^{2} \)
$73$ \( 49 + T^{2} \)
$79$ \( ( -10 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 100 + T^{2} \)
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