Properties

Label 1620.4.i.v
Level $1620$
Weight $4$
Character orbit 1620.i
Analytic conductor $95.583$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1620.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(95.5830942093\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - x^{5} + 91x^{4} + 570x^{3} + 7860x^{2} + 21600x + 57600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (5 \beta_1 + 5) q^{5} + ( - \beta_{4} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (5 \beta_1 + 5) q^{5} + ( - \beta_{4} - \beta_1) q^{7} + (\beta_{5} + \beta_{4} + 8 \beta_1) q^{11} + (2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{13} + (2 \beta_{3} + 2 \beta_{2} + 10) q^{17} + ( - \beta_{3} - 4 \beta_{2} - 9) q^{19} + ( - \beta_{4} - \beta_{2} + 33 \beta_1 + 33) q^{23} + 25 \beta_1 q^{25} + (\beta_{4} + 18 \beta_1) q^{29} + (\beta_{5} - 5 \beta_{4} + \beta_{3} - 5 \beta_{2} - 24 \beta_1 - 24) q^{31} + (5 \beta_{2} + 5) q^{35} + (2 \beta_{3} - 8 \beta_{2} - 6) q^{37} + (4 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 137 \beta_1 + 137) q^{41} + (4 \beta_{5} + 10 \beta_{4} + 148 \beta_1) q^{43} + ( - 10 \beta_{5} + 3 \beta_{4} + 25 \beta_1) q^{47} + ( - 10 \beta_{5} - 7 \beta_{4} - 10 \beta_{3} - 7 \beta_{2} - 68 \beta_1 - 68) q^{49} + ( - \beta_{2} - 57) q^{53} + ( - 5 \beta_{3} - 5 \beta_{2} - 40) q^{55} + ( - 9 \beta_{5} - 28 \beta_{4} - 9 \beta_{3} - 28 \beta_{2} + 99 \beta_1 + 99) q^{59} + ( - 16 \beta_{5} - 2 \beta_{4} + 228 \beta_1) q^{61} + (10 \beta_{5} - 5 \beta_{4} - 5 \beta_1) q^{65} + ( - 10 \beta_{5} + 10 \beta_{4} - 10 \beta_{3} + 10 \beta_{2} - 4 \beta_1 - 4) q^{67} + ( - 29 \beta_{3} - \beta_{2} - 214) q^{71} + (18 \beta_{3} + 30 \beta_{2} - 22) q^{73} + (6 \beta_{5} + 32 \beta_{4} + 6 \beta_{3} + 32 \beta_{2} + 348 \beta_1 + 348) q^{77} + (12 \beta_{5} - 6 \beta_{4} + 374 \beta_1) q^{79} + (28 \beta_{4} - 30 \beta_1) q^{83} + (10 \beta_{5} + 10 \beta_{4} + 10 \beta_{3} + 10 \beta_{2} + 50 \beta_1 + 50) q^{85} + (18 \beta_{3} - 33 \beta_{2} - 252) q^{89} + ( - 18 \beta_{3} + 29 \beta_{2} - 551) q^{91} + ( - 5 \beta_{5} - 20 \beta_{4} - 5 \beta_{3} - 20 \beta_{2} - 45 \beta_1 - 45) q^{95} + (6 \beta_{5} - 26 \beta_{4} + 512 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 15 q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 15 q^{5} + 3 q^{7} - 24 q^{11} - 3 q^{13} + 60 q^{17} - 54 q^{19} + 99 q^{23} - 75 q^{25} - 54 q^{29} - 72 q^{31} + 30 q^{35} - 36 q^{37} + 411 q^{41} - 444 q^{43} - 75 q^{47} - 204 q^{49} - 342 q^{53} - 240 q^{55} + 297 q^{59} - 684 q^{61} + 15 q^{65} - 12 q^{67} - 1284 q^{71} - 132 q^{73} + 1044 q^{77} - 1122 q^{79} + 90 q^{83} + 150 q^{85} - 1512 q^{89} - 3306 q^{91} - 135 q^{95} - 1536 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 91x^{4} + 570x^{3} + 7860x^{2} + 21600x + 57600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -273\nu^{5} + 281\nu^{4} - 25571\nu^{3} - 89362\nu^{2} - 2208660\nu - 6069600 ) / 5894880 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 91\nu^{5} - 8281\nu^{4} + 16711\nu^{3} - 715260\nu^{2} - 1965600\nu - 42188400 ) / 1473720 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -97\nu^{5} + 8827\nu^{4} - 66397\nu^{3} + 762420\nu^{2} + 2095200\nu + 28921080 ) / 736860 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4093\nu^{5} - 4033\nu^{4} + 367003\nu^{3} + 2829870\nu^{2} + 31699380\nu + 87112800 ) / 2947440 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7889\nu^{5} + 5601\nu^{4} - 509691\nu^{3} - 5759018\nu^{2} - 44023860\nu - 120981600 ) / 2947440 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + 2\beta_{4} + \beta_{3} + 2\beta_{2} + 2\beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 14\beta_{4} + 362\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -91\beta_{3} - 194\beta_{2} - 1982 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -421\beta_{5} - 1934\beta_{4} - 421\beta_{3} - 1934\beta_{2} - 35042\beta _1 - 35042 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -8851\beta_{5} - 22754\beta_{4} - 300302\beta_1 ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
541.1
−1.55365 2.69100i
−3.48579 6.03757i
5.53944 + 9.59460i
−1.55365 + 2.69100i
−3.48579 + 6.03757i
5.53944 9.59460i
0 0 0 2.50000 + 4.33013i 0 −11.3093 + 19.5883i 0 0 0
541.2 0 0 0 2.50000 + 4.33013i 0 −0.606369 + 1.05026i 0 0 0
541.3 0 0 0 2.50000 + 4.33013i 0 13.4157 23.2367i 0 0 0
1081.1 0 0 0 2.50000 4.33013i 0 −11.3093 19.5883i 0 0 0
1081.2 0 0 0 2.50000 4.33013i 0 −0.606369 1.05026i 0 0 0
1081.3 0 0 0 2.50000 4.33013i 0 13.4157 + 23.2367i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1081.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.4.i.v 6
3.b odd 2 1 1620.4.i.t 6
9.c even 3 1 1620.4.a.c 3
9.c even 3 1 inner 1620.4.i.v 6
9.d odd 6 1 1620.4.a.e yes 3
9.d odd 6 1 1620.4.i.t 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.4.a.c 3 9.c even 3 1
1620.4.a.e yes 3 9.d odd 6 1
1620.4.i.t 6 3.b odd 2 1
1620.4.i.t 6 9.d odd 6 1
1620.4.i.v 6 1.a even 1 1 trivial
1620.4.i.v 6 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{6} - 3T_{7}^{5} + 621T_{7}^{4} + 3308T_{7}^{3} + 372336T_{7}^{2} + 450432T_{7} + 541696 \) Copy content Toggle raw display
\( T_{11}^{6} + 24T_{11}^{5} + 2001T_{11}^{4} + 216T_{11}^{3} + 2443617T_{11}^{2} + 24521400T_{11} + 296115264 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} + 621 T^{4} + \cdots + 541696 \) Copy content Toggle raw display
$11$ \( T^{6} + 24 T^{5} + \cdots + 296115264 \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} + 5889 T^{4} + \cdots + 50694400 \) Copy content Toggle raw display
$17$ \( (T^{3} - 30 T^{2} - 6168 T + 101952)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 27 T^{2} - 9969 T + 289237)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} - 99 T^{5} + \cdots + 204261264 \) Copy content Toggle raw display
$29$ \( T^{6} + 54 T^{5} + 2559 T^{4} + \cdots + 15116544 \) Copy content Toggle raw display
$31$ \( T^{6} + 72 T^{5} + \cdots + 785953171600 \) Copy content Toggle raw display
$37$ \( (T^{3} + 18 T^{2} - 47460 T - 2963720)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 411 T^{5} + \cdots + 1388407386249 \) Copy content Toggle raw display
$43$ \( T^{6} + 444 T^{5} + \cdots + 69780505600 \) Copy content Toggle raw display
$47$ \( T^{6} + 75 T^{5} + \cdots + 10844507610000 \) Copy content Toggle raw display
$53$ \( (T^{3} + 171 T^{2} + 9132 T + 151488)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 297 T^{5} + \cdots + 39\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( T^{6} + 684 T^{5} + \cdots + 13780131865600 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 552309409597696 \) Copy content Toggle raw display
$71$ \( (T^{3} + 642 T^{2} - 876429 T - 547252470)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 66 T^{2} - 831336 T - 197601920)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 709033342182400 \) Copy content Toggle raw display
$83$ \( T^{6} - 90 T^{5} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{3} + 756 T^{2} - 996651 T - 780931530)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 1536 T^{5} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
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