Properties

Label 4600.2.e.j
Level $4600$
Weight $2$
Character orbit 4600.e
Analytic conductor $36.731$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
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.

\(f(q)\) \(=\) \( q + i q^{7} + 3 q^{9} +O(q^{10})\) \( q + i q^{7} + 3 q^{9} -6 q^{11} + 2 i q^{13} -3 i q^{17} + 6 q^{19} -i q^{23} -3 q^{29} -3 q^{31} + i q^{37} + 9 q^{41} + 8 i q^{43} + 4 i q^{47} + 6 q^{49} -i q^{53} - q^{59} + 8 q^{61} + 3 i q^{63} -7 i q^{67} -5 q^{71} + 6 i q^{73} -6 i q^{77} + 9 q^{81} + 11 i q^{83} -4 q^{89} -2 q^{91} + 6 i q^{97} -18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{9} + O(q^{10}) \) \( 2q + 6q^{9} - 12q^{11} + 12q^{19} - 6q^{29} - 6q^{31} + 18q^{41} + 12q^{49} - 2q^{59} + 16q^{61} - 10q^{71} + 18q^{81} - 8q^{89} - 4q^{91} - 36q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4049.1
1.00000i
1.00000i
0 0 0 0 0 1.00000i 0 3.00000 0
4049.2 0 0 0 0 0 1.00000i 0 3.00000 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 4600.2.e.j 2
5.b even 2 1 inner 4600.2.e.j 2
5.c odd 4 1 920.2.a.c 1
5.c odd 4 1 4600.2.a.h 1
15.e even 4 1 8280.2.a.j 1
20.e even 4 1 1840.2.a.f 1
20.e even 4 1 9200.2.a.w 1
40.i odd 4 1 7360.2.a.l 1
40.k even 4 1 7360.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.c 1 5.c odd 4 1
1840.2.a.f 1 20.e even 4 1
4600.2.a.h 1 5.c odd 4 1
4600.2.e.j 2 1.a even 1 1 trivial
4600.2.e.j 2 5.b even 2 1 inner
7360.2.a.k 1 40.k even 4 1
7360.2.a.l 1 40.i odd 4 1
8280.2.a.j 1 15.e even 4 1
9200.2.a.w 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 \(S_{2}^{\mathrm{new}}(4600, [\chi])\):

\( T_{3} \)
\( T_{7}^{2} + 1 \)
\( T_{11} + 6 \)
\( T_{13}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 6 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 9 + T^{2} \)
$19$ \( ( -6 + T )^{2} \)
$23$ \( 1 + T^{2} \)
$29$ \( ( 3 + T )^{2} \)
$31$ \( ( 3 + T )^{2} \)
$37$ \( 1 + T^{2} \)
$41$ \( ( -9 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( 1 + T^{2} \)
$59$ \( ( 1 + T )^{2} \)
$61$ \( ( -8 + T )^{2} \)
$67$ \( 49 + T^{2} \)
$71$ \( ( 5 + T )^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 121 + T^{2} \)
$89$ \( ( 4 + T )^{2} \)
$97$ \( 36 + T^{2} \)
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