Properties

Label 4600.2.e.m
Level $4600$
Weight $2$
Character orbit 4600.e
Analytic conductor $36.731$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -2 \beta_{1} q^{7} + ( -2 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -2 \beta_{1} q^{7} + ( -2 + \beta_{3} ) q^{9} -4 q^{11} + ( -\beta_{1} - 2 \beta_{2} ) q^{13} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{17} -4 q^{19} + ( 10 - 2 \beta_{3} ) q^{21} -\beta_{2} q^{23} + ( \beta_{1} + 4 \beta_{2} ) q^{27} + ( 7 - \beta_{3} ) q^{29} + ( 3 + \beta_{3} ) q^{31} -4 \beta_{1} q^{33} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{37} + ( 3 + \beta_{3} ) q^{39} + ( -1 - \beta_{3} ) q^{41} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{43} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{47} + ( -13 + 4 \beta_{3} ) q^{49} -8 q^{51} + ( -4 \beta_{1} - 6 \beta_{2} ) q^{53} -4 \beta_{1} q^{57} -4 \beta_{3} q^{59} + ( 8 - 2 \beta_{3} ) q^{61} + ( 4 \beta_{1} - 8 \beta_{2} ) q^{63} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{67} + ( -1 + \beta_{3} ) q^{69} + ( -1 - 3 \beta_{3} ) q^{71} + ( \beta_{1} - 14 \beta_{2} ) q^{73} + 8 \beta_{1} q^{77} + ( 4 - 4 \beta_{3} ) q^{79} -7 q^{81} -12 \beta_{2} q^{83} + ( 7 \beta_{1} - 4 \beta_{2} ) q^{87} -10 q^{89} + ( -6 - 2 \beta_{3} ) q^{91} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{93} + ( 4 \beta_{1} - 6 \beta_{2} ) q^{97} + ( 8 - 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9} + O(q^{10}) \) \( 4 q - 6 q^{9} - 16 q^{11} - 16 q^{19} + 36 q^{21} + 26 q^{29} + 14 q^{31} + 14 q^{39} - 6 q^{41} - 44 q^{49} - 32 q^{51} - 8 q^{59} + 28 q^{61} - 2 q^{69} - 10 q^{71} + 8 q^{79} - 28 q^{81} - 40 q^{89} - 28 q^{91} + 24 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 5 \nu \)\()/4\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 5\)
\(\nu^{3}\)\(=\)\(4 \beta_{2} - 5 \beta_{1}\)

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
2.56155i
1.56155i
1.56155i
2.56155i
0 2.56155i 0 0 0 5.12311i 0 −3.56155 0
4049.2 0 1.56155i 0 0 0 3.12311i 0 0.561553 0
4049.3 0 1.56155i 0 0 0 3.12311i 0 0.561553 0
4049.4 0 2.56155i 0 0 0 5.12311i 0 −3.56155 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.m 4
5.b even 2 1 inner 4600.2.e.m 4
5.c odd 4 1 920.2.a.f 2
5.c odd 4 1 4600.2.a.r 2
15.e even 4 1 8280.2.a.bb 2
20.e even 4 1 1840.2.a.k 2
20.e even 4 1 9200.2.a.bx 2
40.i odd 4 1 7360.2.a.bj 2
40.k even 4 1 7360.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.f 2 5.c odd 4 1
1840.2.a.k 2 20.e even 4 1
4600.2.a.r 2 5.c odd 4 1
4600.2.e.m 4 1.a even 1 1 trivial
4600.2.e.m 4 5.b even 2 1 inner
7360.2.a.bj 2 40.i odd 4 1
7360.2.a.bm 2 40.k even 4 1
8280.2.a.bb 2 15.e even 4 1
9200.2.a.bx 2 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 \(S_{2}^{\mathrm{new}}(4600, [\chi])\):

\( T_{3}^{4} + 9 T_{3}^{2} + 16 \)
\( T_{7}^{4} + 36 T_{7}^{2} + 256 \)
\( T_{11} + 4 \)
\( T_{13}^{4} + 13 T_{13}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 16 + 9 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 256 + 36 T^{2} + T^{4} \)
$11$ \( ( 4 + T )^{4} \)
$13$ \( 4 + 13 T^{2} + T^{4} \)
$17$ \( 256 + 36 T^{2} + T^{4} \)
$19$ \( ( 4 + T )^{4} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( 38 - 13 T + T^{2} )^{2} \)
$31$ \( ( 8 - 7 T + T^{2} )^{2} \)
$37$ \( 64 + 52 T^{2} + T^{4} \)
$41$ \( ( -2 + 3 T + T^{2} )^{2} \)
$43$ \( 64 + 84 T^{2} + T^{4} \)
$47$ \( 1024 + 89 T^{2} + T^{4} \)
$53$ \( 2704 + 168 T^{2} + T^{4} \)
$59$ \( ( -64 + 4 T + T^{2} )^{2} \)
$61$ \( ( 32 - 14 T + T^{2} )^{2} \)
$67$ \( 4096 + 144 T^{2} + T^{4} \)
$71$ \( ( -32 + 5 T + T^{2} )^{2} \)
$73$ \( 42436 + 429 T^{2} + T^{4} \)
$79$ \( ( -64 - 4 T + T^{2} )^{2} \)
$83$ \( ( 144 + T^{2} )^{2} \)
$89$ \( ( 10 + T )^{4} \)
$97$ \( 16 + 264 T^{2} + T^{4} \)
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