Properties

Label 4600.2.e.r
Level $4600$
Weight $2$
Character orbit 4600.e
Analytic conductor $36.731$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.24681024.1
Defining polynomial: \( x^{6} + 12x^{4} + 36x^{2} + 9 \) Copy content Toggle raw display
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} - \beta_1) q^{7} + (\beta_{4} - \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{2} - \beta_1) q^{7} + (\beta_{4} - \beta_{3} - 1) q^{9} + ( - \beta_{4} - \beta_{3} + 1) q^{11} + ( - \beta_{5} + \beta_1) q^{13} + (\beta_{5} - \beta_{2} - \beta_1) q^{17} + ( - 2 \beta_{4} + \beta_{3} - 1) q^{19} + ( - \beta_{4} + 2 \beta_{3} + 4) q^{21} - \beta_{2} q^{23} + 3 \beta_{2} q^{27} + (\beta_{4} - 1) q^{29} + (\beta_{4} - 3 \beta_{3} + 2) q^{31} + ( - 2 \beta_{5} + 5 \beta_{2} + \beta_1) q^{33} + 4 \beta_{2} q^{37} + (2 \beta_{4} - 5) q^{39} + (2 \beta_{4} - 3 \beta_{3} - 2) q^{41} + (2 \beta_{5} - 6 \beta_{2}) q^{43} + ( - \beta_{5} + 5 \beta_{2}) q^{47} + (\beta_{4} - 3 \beta_{3} + 2) q^{49} + ( - 2 \beta_{4} - \beta_{3} + 5) q^{51} + (4 \beta_{5} - 2 \beta_{2}) q^{53} + ( - \beta_{5} - 2 \beta_{2} + 2 \beta_1) q^{57} + ( - 2 \beta_{4} + 2 \beta_{3} - 2) q^{59} + ( - \beta_{4} - \beta_{3} + 9) q^{61} + (\beta_{5} - 4 \beta_{2} + 4 \beta_1) q^{63} + (4 \beta_{5} + 4 \beta_{2} - 4 \beta_1) q^{67} - \beta_{3} q^{69} + (2 \beta_{4} - 3 \beta_{3} + 2) q^{71} + (\beta_{5} + 5 \beta_{2} - 4 \beta_1) q^{73} + (\beta_{5} - 4 \beta_{2}) q^{77} + (3 \beta_{4} - 3) q^{81} + ( - 2 \beta_{5} + 4 \beta_{2} - 4 \beta_1) q^{83} + (\beta_{5} - \beta_{2} - 2 \beta_1) q^{87} + ( - 2 \beta_{4} + 10) q^{89} + ( - \beta_{4} + \beta_{3} + 5) q^{91} + ( - 2 \beta_{5} + 11 \beta_{2} - 2 \beta_1) q^{93} + ( - 3 \beta_{5} + 3 \beta_{2} + \beta_1) q^{97} + (3 \beta_{3} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{9} + 6 q^{11} - 6 q^{19} + 24 q^{21} - 6 q^{29} + 12 q^{31} - 30 q^{39} - 12 q^{41} + 12 q^{49} + 30 q^{51} - 12 q^{59} + 54 q^{61} + 12 q^{71} - 18 q^{81} + 60 q^{89} + 30 q^{91} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 12x^{4} + 36x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 6\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 6\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 9\nu^{2} + 12 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + 10\nu^{3} + 21\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{4} + 9\beta_{3} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{5} - 30\beta_{2} + 39\beta_1 \) Copy content Toggle raw display

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
2.66908i
2.14510i
0.523976i
0.523976i
2.14510i
2.66908i
0 2.66908i 0 0 0 3.66908i 0 −4.12398 0
4049.2 0 2.14510i 0 0 0 1.14510i 0 −1.60147 0
4049.3 0 0.523976i 0 0 0 0.476024i 0 2.72545 0
4049.4 0 0.523976i 0 0 0 0.476024i 0 2.72545 0
4049.5 0 2.14510i 0 0 0 1.14510i 0 −1.60147 0
4049.6 0 2.66908i 0 0 0 3.66908i 0 −4.12398 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4049.6
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.r 6
5.b even 2 1 inner 4600.2.e.r 6
5.c odd 4 1 920.2.a.g 3
5.c odd 4 1 4600.2.a.y 3
15.e even 4 1 8280.2.a.bo 3
20.e even 4 1 1840.2.a.t 3
20.e even 4 1 9200.2.a.cd 3
40.i odd 4 1 7360.2.a.cb 3
40.k even 4 1 7360.2.a.ca 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.g 3 5.c odd 4 1
1840.2.a.t 3 20.e even 4 1
4600.2.a.y 3 5.c odd 4 1
4600.2.e.r 6 1.a even 1 1 trivial
4600.2.e.r 6 5.b even 2 1 inner
7360.2.a.ca 3 40.k even 4 1
7360.2.a.cb 3 40.i odd 4 1
8280.2.a.bo 3 15.e even 4 1
9200.2.a.cd 3 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 \(S_{2}^{\mathrm{new}}(4600, [\chi])\):

\( T_{3}^{6} + 12T_{3}^{4} + 36T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{6} + 15T_{7}^{4} + 21T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{3} - 3T_{11}^{2} - 15T_{11} - 12 \) Copy content Toggle raw display
\( T_{13}^{6} + 36T_{13}^{4} + 324T_{13}^{2} + 841 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 12 T^{4} + 36 T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 15 T^{4} + 21 T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T^{3} - 3 T^{2} - 15 T - 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 36 T^{4} + 324 T^{2} + \cdots + 841 \) Copy content Toggle raw display
$17$ \( T^{6} + 39 T^{4} + 153 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$19$ \( (T^{3} + 3 T^{2} - 33 T + 48)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$29$ \( (T^{3} + 3 T^{2} - 6 T - 12)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 6 T^{2} - 42 T + 249)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 16)^{3} \) Copy content Toggle raw display
$41$ \( (T^{3} + 6 T^{2} - 60 T - 69)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 180 T^{4} + 6336 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$47$ \( T^{6} + 93 T^{4} + 2076 T^{2} + \cdots + 5776 \) Copy content Toggle raw display
$53$ \( T^{6} + 300 T^{4} + 23856 T^{2} + \cdots + 287296 \) Copy content Toggle raw display
$59$ \( (T^{3} + 6 T^{2} - 36 T - 32)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 27 T^{2} + 225 T - 596)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 624 T^{4} + 128256 T^{2} + \cdots + 8667136 \) Copy content Toggle raw display
$71$ \( (T^{3} - 6 T^{2} - 60 T + 203)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 309 T^{4} + 19404 T^{2} + \cdots + 345744 \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} + 264 T^{4} + 5136 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$89$ \( (T^{3} - 30 T^{2} + 264 T - 608)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 219 T^{4} + 7245 T^{2} + \cdots + 19044 \) Copy content Toggle raw display
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