Properties

Label 225.6.a.v
Level $225$
Weight $6$
Character orbit 225.a
Self dual yes
Analytic conductor $36.086$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.0863594579\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{5}, \sqrt{14})\)
Defining polynomial: \( x^{4} - 2x^{3} - 29x^{2} + 30x + 155 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 12 q^{4} - \beta_{2} q^{7} + 44 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 12 q^{4} - \beta_{2} q^{7} + 44 \beta_1 q^{8} - \beta_{3} q^{11} - \beta_{2} q^{13} + 2 \beta_{3} q^{14} - 496 q^{16} - 373 \beta_1 q^{17} + 484 q^{19} + 10 \beta_{2} q^{22} - 506 \beta_1 q^{23} + 2 \beta_{3} q^{26} + 12 \beta_{2} q^{28} - 11 \beta_{3} q^{29} + 3608 q^{31} - 912 \beta_1 q^{32} + 7460 q^{34} - 33 \beta_{2} q^{37} - 484 \beta_1 q^{38} + 22 \beta_{3} q^{41} - 56 \beta_{2} q^{43} + 12 \beta_{3} q^{44} + 10120 q^{46} - 2134 \beta_1 q^{47} + 33593 q^{49} + 12 \beta_{2} q^{52} - 1067 \beta_1 q^{53} - 88 \beta_{3} q^{56} + 110 \beta_{2} q^{58} + 11 \beta_{3} q^{59} + 21362 q^{61} - 3608 \beta_1 q^{62} + 34112 q^{64} + 154 \beta_{2} q^{67} + 4476 \beta_1 q^{68} + 66 \beta_{3} q^{71} + 12 \beta_{2} q^{73} + 66 \beta_{3} q^{74} - 5808 q^{76} + 25200 \beta_1 q^{77} + 99616 q^{79} - 220 \beta_{2} q^{82} + 12772 \beta_1 q^{83} + 112 \beta_{3} q^{86} - 440 \beta_{2} q^{88} - 240 \beta_{3} q^{89} + 50400 q^{91} + 6072 \beta_1 q^{92} + 42680 q^{94} - 286 \beta_{2} q^{97} - 33593 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 48 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 48 q^{4} - 1984 q^{16} + 1936 q^{19} + 14432 q^{31} + 29840 q^{34} + 40480 q^{46} + 134372 q^{49} + 85448 q^{61} + 136448 q^{64} - 23232 q^{76} + 398464 q^{79} + 201600 q^{91} + 170720 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 29x^{2} + 30x + 155 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 8\nu^{3} - 12\nu^{2} - 136\nu + 70 ) / 51 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -40\nu^{3} + 60\nu^{2} + 1700\nu - 860 ) / 17 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 60\nu^{2} - 60\nu - 900 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 15\beta _1 + 30 ) / 60 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 15\beta _1 + 930 ) / 60 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{3} + 37\beta_{2} + 1320\beta _1 + 2760 ) / 120 \) 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
5.35969
−2.12362
3.12362
−4.35969
−4.47214 0 −12.0000 0 0 −224.499 196.774 0 0
1.2 −4.47214 0 −12.0000 0 0 224.499 196.774 0 0
1.3 4.47214 0 −12.0000 0 0 −224.499 −196.774 0 0
1.4 4.47214 0 −12.0000 0 0 224.499 −196.774 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.6.a.v 4
3.b odd 2 1 inner 225.6.a.v 4
5.b even 2 1 inner 225.6.a.v 4
5.c odd 4 2 45.6.b.d 4
15.d odd 2 1 inner 225.6.a.v 4
15.e even 4 2 45.6.b.d 4
20.e even 4 2 720.6.f.k 4
60.l odd 4 2 720.6.f.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.6.b.d 4 5.c odd 4 2
45.6.b.d 4 15.e even 4 2
225.6.a.v 4 1.a even 1 1 trivial
225.6.a.v 4 3.b odd 2 1 inner
225.6.a.v 4 5.b even 2 1 inner
225.6.a.v 4 15.d odd 2 1 inner
720.6.f.k 4 20.e even 4 2
720.6.f.k 4 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2}^{2} - 20 \) Copy content Toggle raw display
\( T_{7}^{2} - 50400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 50400)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 252000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 50400)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2782580)^{2} \) Copy content Toggle raw display
$19$ \( (T - 484)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 5120720)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 30492000)^{2} \) Copy content Toggle raw display
$31$ \( (T - 3608)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 54885600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 121968000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 158054400)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 91079120)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 22769780)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 30492000)^{2} \) Copy content Toggle raw display
$61$ \( (T - 21362)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 1195286400)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 1097712000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 7257600)^{2} \) Copy content Toggle raw display
$79$ \( (T - 99616)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 3262479680)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 14515200000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 4122518400)^{2} \) Copy content Toggle raw display
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