Properties

Label 4600.2.e.p
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: 6.0.431642176.2
Defining polynomial: \( x^{6} + 19x^{4} + 97x^{2} + 64 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{2} + \beta_1) q^{7} + (\beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{2} + \beta_1) q^{7} + (\beta_{3} - 3) q^{9} + (\beta_{5} + 2) q^{11} - \beta_{4} q^{13} + ( - \beta_{4} - 3 \beta_{2}) q^{17} + ( - \beta_{3} + 4) q^{19} + (\beta_{5} + \beta_{3} - 6) q^{21} + \beta_{2} q^{23} + ( - \beta_{4} + 2 \beta_{2} - 3 \beta_1) q^{27} + (\beta_{5} - \beta_{3} - 5) q^{29} + (\beta_{5} - 3) q^{31} + (\beta_{4} + 6 \beta_{2} + 2 \beta_1) q^{33} + (\beta_{4} - 3 \beta_{2} + 3 \beta_1) q^{37} + (3 \beta_{5} - \beta_{3} - 2) q^{39} + (\beta_{3} + 3) q^{41} + 8 \beta_{2} q^{43} - 2 \beta_{4} q^{47} + (2 \beta_{5} + \beta_{3}) q^{49} + (6 \beta_{5} - \beta_{3} - 2) q^{51} + (\beta_{4} - \beta_{2} + 3 \beta_1) q^{53} + (\beta_{4} - 2 \beta_{2} + 7 \beta_1) q^{57} + (\beta_{5} - \beta_{3} + 5) q^{59} + ( - \beta_{5} - 2 \beta_{3} + 4) q^{61} + (5 \beta_{2} - 6 \beta_1) q^{63} + (\beta_{4} - 3 \beta_{2} + 3 \beta_1) q^{67} - \beta_{5} q^{69} + ( - 2 \beta_{5} - \beta_{3} - 7) q^{71} + ( - 2 \beta_{4} + 6 \beta_{2} - 4 \beta_1) q^{73} + (\beta_{4} + 4 \beta_{2} + \beta_1) q^{77} + 4 \beta_{5} q^{79} + (\beta_{5} - \beta_{3} + 7) q^{81} + (\beta_{4} - 5 \beta_{2} - \beta_1) q^{83} + (2 \beta_{4} + 4 \beta_{2} - 2 \beta_1) q^{87} + ( - 4 \beta_{5} + 4) q^{89} + (3 \beta_{5} - 2) q^{91} + (\beta_{4} + 6 \beta_{2} - 3 \beta_1) q^{93} + (2 \beta_{2} - 3 \beta_1) q^{97} + ( - 6 \beta_{5} + 3 \beta_{3} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 20 q^{9} + 14 q^{11} + 26 q^{19} - 36 q^{21} - 26 q^{29} - 16 q^{31} - 4 q^{39} + 16 q^{41} + 2 q^{49} + 2 q^{51} + 34 q^{59} + 26 q^{61} - 2 q^{69} - 44 q^{71} + 8 q^{79} + 46 q^{81} + 16 q^{89} - 6 q^{91} - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 19x^{4} + 97x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 11\nu^{3} + 17\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 7\nu^{3} - 19\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{4} + 10\nu^{2} + 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + 2\beta_{2} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - 10\beta_{3} + 52 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{4} - 14\beta_{2} + 82\beta_1 \) 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
3.07912i
2.95759i
0.878468i
0.878468i
2.95759i
3.07912i
0 3.07912i 0 0 0 2.07912i 0 −6.48097 0
4049.2 0 2.95759i 0 0 0 3.95759i 0 −5.74732 0
4049.3 0 0.878468i 0 0 0 0.121532i 0 2.22829 0
4049.4 0 0.878468i 0 0 0 0.121532i 0 2.22829 0
4049.5 0 2.95759i 0 0 0 3.95759i 0 −5.74732 0
4049.6 0 3.07912i 0 0 0 2.07912i 0 −6.48097 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.p 6
5.b even 2 1 inner 4600.2.e.p 6
5.c odd 4 1 920.2.a.h 3
5.c odd 4 1 4600.2.a.x 3
15.e even 4 1 8280.2.a.bj 3
20.e even 4 1 1840.2.a.s 3
20.e even 4 1 9200.2.a.ce 3
40.i odd 4 1 7360.2.a.by 3
40.k even 4 1 7360.2.a.cc 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.h 3 5.c odd 4 1
1840.2.a.s 3 20.e even 4 1
4600.2.a.x 3 5.c odd 4 1
4600.2.e.p 6 1.a even 1 1 trivial
4600.2.e.p 6 5.b even 2 1 inner
7360.2.a.by 3 40.i odd 4 1
7360.2.a.cc 3 40.k even 4 1
8280.2.a.bj 3 15.e even 4 1
9200.2.a.ce 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 \(S_{2}^{\mathrm{new}}(4600, [\chi])\):

\( T_{3}^{6} + 19T_{3}^{4} + 97T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{6} + 20T_{7}^{4} + 68T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{3} - 7T_{11}^{2} + 7T_{11} + 14 \) Copy content Toggle raw display
\( T_{13}^{6} + 47T_{13}^{4} + 629T_{13}^{2} + 2500 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 19 T^{4} + 97 T^{2} + 64 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 20 T^{4} + 68 T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T^{3} - 7 T^{2} + 7 T + 14)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 47 T^{4} + 629 T^{2} + \cdots + 2500 \) Copy content Toggle raw display
$17$ \( T^{6} + 80 T^{4} + 1760 T^{2} + \cdots + 6889 \) Copy content Toggle raw display
$19$ \( (T^{3} - 13 T^{2} + 33 T + 62)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$29$ \( (T^{3} + 13 T^{2} + 26 T - 76)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 8 T^{2} + 12 T - 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 209 T^{4} + 13424 T^{2} + \cdots + 246016 \) Copy content Toggle raw display
$41$ \( (T^{3} - 8 T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 64)^{3} \) Copy content Toggle raw display
$47$ \( T^{6} + 188 T^{4} + 10064 T^{2} + \cdots + 160000 \) Copy content Toggle raw display
$53$ \( T^{6} + 201 T^{4} + 9400 T^{2} + \cdots + 90000 \) Copy content Toggle raw display
$59$ \( (T^{3} - 17 T^{2} + 66 T - 36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 13 T^{2} - 51 T + 720)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 209 T^{4} + 13424 T^{2} + \cdots + 246016 \) Copy content Toggle raw display
$71$ \( (T^{3} + 22 T^{2} + 96 T - 225)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 496 T^{4} + 81856 T^{2} + \cdots + 4494400 \) Copy content Toggle raw display
$79$ \( (T^{3} - 4 T^{2} - 144 T + 512)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 145 T^{4} + 2680 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$89$ \( (T^{3} - 8 T^{2} - 128 T + 64)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 171 T^{4} + 6261 T^{2} + \cdots + 2500 \) Copy content Toggle raw display
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