Properties

Label 4600.2.e.a
Level $4600$
Weight $2$
Character orbit 4600.e
Analytic conductor $36.731$
Analytic rank $1$
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: \(1\)
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 184)
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.

\(f(q)\) \(=\) \( q + 3 i q^{3} + 2 i q^{7} -6 q^{9} +O(q^{10})\) \( q + 3 i q^{3} + 2 i q^{7} -6 q^{9} -5 i q^{13} + 6 i q^{17} -6 q^{19} -6 q^{21} + i q^{23} -9 i q^{27} -9 q^{29} + 3 q^{31} + 8 i q^{37} + 15 q^{39} + 3 q^{41} -8 i q^{43} -7 i q^{47} + 3 q^{49} -18 q^{51} -2 i q^{53} -18 i q^{57} -4 q^{59} -10 q^{61} -12 i q^{63} -8 i q^{67} -3 q^{69} + 7 q^{71} + 9 i q^{73} + 6 q^{79} + 9 q^{81} -14 i q^{83} -27 i q^{87} -16 q^{89} + 10 q^{91} + 9 i q^{93} -6 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{9} + O(q^{10}) \) \( 2 q - 12 q^{9} - 12 q^{19} - 12 q^{21} - 18 q^{29} + 6 q^{31} + 30 q^{39} + 6 q^{41} + 6 q^{49} - 36 q^{51} - 8 q^{59} - 20 q^{61} - 6 q^{69} + 14 q^{71} + 12 q^{79} + 18 q^{81} - 32 q^{89} + 20 q^{91} + O(q^{100}) \)

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 3.00000i 0 0 0 2.00000i 0 −6.00000 0
4049.2 0 3.00000i 0 0 0 2.00000i 0 −6.00000 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.a 2
5.b even 2 1 inner 4600.2.e.a 2
5.c odd 4 1 184.2.a.d 1
5.c odd 4 1 4600.2.a.a 1
15.e even 4 1 1656.2.a.c 1
20.e even 4 1 368.2.a.a 1
20.e even 4 1 9200.2.a.bj 1
35.f even 4 1 9016.2.a.b 1
40.i odd 4 1 1472.2.a.a 1
40.k even 4 1 1472.2.a.m 1
60.l odd 4 1 3312.2.a.i 1
115.e even 4 1 4232.2.a.j 1
460.k odd 4 1 8464.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.2.a.d 1 5.c odd 4 1
368.2.a.a 1 20.e even 4 1
1472.2.a.a 1 40.i odd 4 1
1472.2.a.m 1 40.k even 4 1
1656.2.a.c 1 15.e even 4 1
3312.2.a.i 1 60.l odd 4 1
4232.2.a.j 1 115.e even 4 1
4600.2.a.a 1 5.c odd 4 1
4600.2.e.a 2 1.a even 1 1 trivial
4600.2.e.a 2 5.b even 2 1 inner
8464.2.a.b 1 460.k odd 4 1
9016.2.a.b 1 35.f even 4 1
9200.2.a.bj 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 \(S_{2}^{\mathrm{new}}(4600, [\chi])\):

\( T_{3}^{2} + 9 \)
\( T_{7}^{2} + 4 \)
\( T_{11} \)
\( T_{13}^{2} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 25 + T^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( ( 6 + T )^{2} \)
$23$ \( 1 + T^{2} \)
$29$ \( ( 9 + T )^{2} \)
$31$ \( ( -3 + T )^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( ( -3 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( 49 + T^{2} \)
$53$ \( 4 + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( ( 10 + T )^{2} \)
$67$ \( 64 + T^{2} \)
$71$ \( ( -7 + T )^{2} \)
$73$ \( 81 + T^{2} \)
$79$ \( ( -6 + T )^{2} \)
$83$ \( 196 + T^{2} \)
$89$ \( ( 16 + T )^{2} \)
$97$ \( 36 + T^{2} \)
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