Properties

Label 900.6.a.n
Level $900$
Weight $6$
Character orbit 900.a
Self dual yes
Analytic conductor $144.345$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 900.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(144.345437832\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 300)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 60\sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 11) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 11) q^{7} + ( - 3 \beta - 186) q^{11} + (4 \beta - 5) q^{13} + (3 \beta + 606) q^{17} + ( - 11 \beta - 775) q^{19} + (3 \beta + 1326) q^{23} + ( - 45 \beta + 906) q^{29} + (\beta + 5459) q^{31} + (18 \beta - 1658) q^{37} + ( - 3 \beta - 9540) q^{41} + ( - 29 \beta - 6131) q^{43} + (72 \beta - 7626) q^{47} + ( - 22 \beta + 8514) q^{49} + ( - 165 \beta + 10392) q^{53} + (48 \beta - 9354) q^{59} + ( - 42 \beta + 17867) q^{61} + ( - 3 \beta + 49081) q^{67} + ( - 75 \beta - 57060) q^{71} + ( - 162 \beta - 54938) q^{73} + ( - 153 \beta - 73554) q^{77} + (216 \beta + 47768) q^{79} + (528 \beta + 1866) q^{83} + (132 \beta - 46800) q^{89} + ( - 49 \beta + 100855) q^{91} + ( - 352 \beta + 45943) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 22 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 22 q^{7} - 372 q^{11} - 10 q^{13} + 1212 q^{17} - 1550 q^{19} + 2652 q^{23} + 1812 q^{29} + 10918 q^{31} - 3316 q^{37} - 19080 q^{41} - 12262 q^{43} - 15252 q^{47} + 17028 q^{49} + 20784 q^{53} - 18708 q^{59} + 35734 q^{61} + 98162 q^{67} - 114120 q^{71} - 109876 q^{73} - 147108 q^{77} + 95536 q^{79} + 3732 q^{83} - 93600 q^{89} + 201710 q^{91} + 91886 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.64575
2.64575
0 0 0 0 0 −169.745 0 0 0
1.2 0 0 0 0 0 147.745 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.6.a.n 2
3.b odd 2 1 300.6.a.g 2
5.b even 2 1 900.6.a.r 2
5.c odd 4 2 900.6.d.j 4
15.d odd 2 1 300.6.a.h yes 2
15.e even 4 2 300.6.d.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.6.a.g 2 3.b odd 2 1
300.6.a.h yes 2 15.d odd 2 1
300.6.d.f 4 15.e even 4 2
900.6.a.n 2 1.a even 1 1 trivial
900.6.a.r 2 5.b even 2 1
900.6.d.j 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(900))\):

\( T_{7}^{2} + 22T_{7} - 25079 \) Copy content Toggle raw display
\( T_{11}^{2} + 372T_{11} - 192204 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 22T - 25079 \) Copy content Toggle raw display
$11$ \( T^{2} + 372T - 192204 \) Copy content Toggle raw display
$13$ \( T^{2} + 10T - 403175 \) Copy content Toggle raw display
$17$ \( T^{2} - 1212 T + 140436 \) Copy content Toggle raw display
$19$ \( T^{2} + 1550 T - 2448575 \) Copy content Toggle raw display
$23$ \( T^{2} - 2652 T + 1531476 \) Copy content Toggle raw display
$29$ \( T^{2} - 1812 T - 50209164 \) Copy content Toggle raw display
$31$ \( T^{2} - 10918 T + 29775481 \) Copy content Toggle raw display
$37$ \( T^{2} + 3316 T - 5415836 \) Copy content Toggle raw display
$41$ \( T^{2} + 19080 T + 90784800 \) Copy content Toggle raw display
$43$ \( T^{2} + 12262 T + 16395961 \) Copy content Toggle raw display
$47$ \( T^{2} + 15252 T - 72480924 \) Copy content Toggle raw display
$53$ \( T^{2} - 20784 T - 578076336 \) Copy content Toggle raw display
$59$ \( T^{2} + 18708 T + 29436516 \) Copy content Toggle raw display
$61$ \( T^{2} - 35734 T + 274776889 \) Copy content Toggle raw display
$67$ \( T^{2} - 98162 T + 2408717761 \) Copy content Toggle raw display
$71$ \( T^{2} + 114120 T + 3114093600 \) Copy content Toggle raw display
$73$ \( T^{2} + 109876 T + 2356835044 \) Copy content Toggle raw display
$79$ \( T^{2} - 95536 T + 1106050624 \) Copy content Toggle raw display
$83$ \( T^{2} - 3732 T - 7021874844 \) Copy content Toggle raw display
$89$ \( T^{2} + 93600 T + 1751155200 \) Copy content Toggle raw display
$97$ \( T^{2} - 91886 T - 1011621551 \) Copy content Toggle raw display
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