Properties

Label 4275.2.a.br
Level $4275$
Weight $2$
Character orbit 4275.a
Self dual yes
Analytic conductor $34.136$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4275.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.1360468641\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.66064384.1
Defining polynomial: \( x^{6} - 9x^{4} + 13x^{2} - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_{3} + 1) q^{4} + (\beta_{5} + \beta_{4}) q^{7} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + ( - \beta_{3} + 1) q^{4} + (\beta_{5} + \beta_{4}) q^{7} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{8} + (\beta_{3} + \beta_{2}) q^{11} + ( - \beta_{5} + \beta_1) q^{13} + (2 \beta_{3} - 2) q^{14} + ( - \beta_{3} - \beta_{2} - 1) q^{16} + (\beta_{5} + \beta_{4} - 2 \beta_1) q^{17} - q^{19} + (2 \beta_{4} - 4 \beta_1) q^{22} + 2 \beta_1 q^{23} + ( - 2 \beta_{3} - \beta_{2} - 2) q^{26} + (4 \beta_{4} - 2 \beta_1) q^{28} - 6 q^{29} - 2 \beta_{2} q^{31} + (2 \beta_{5} + \beta_{4} + 2 \beta_1) q^{32} + (4 \beta_{3} + 2 \beta_{2}) q^{34} + ( - \beta_{5} - 2 \beta_{4} - \beta_1) q^{37} + \beta_{4} q^{38} + ( - 2 \beta_{2} - 2) q^{41} + ( - \beta_{5} - \beta_{4}) q^{43} + (4 \beta_{3} + 2 \beta_{2} - 2) q^{44} + ( - 2 \beta_{3} - 2 \beta_{2} - 2) q^{46} + ( - \beta_{5} + 3 \beta_{4} - 4 \beta_1) q^{47} + ( - \beta_{3} - \beta_{2} - 1) q^{49} + (\beta_{5} - 2 \beta_{4} + 3 \beta_1) q^{52} + (\beta_{5} + 4 \beta_{4} - \beta_1) q^{53} + (2 \beta_{3} + 2 \beta_{2} - 6) q^{56} + 6 \beta_{4} q^{58} + ( - 2 \beta_{3} - 4) q^{59} + (\beta_{3} + 3 \beta_{2} - 2) q^{61} + (2 \beta_{5} + 6 \beta_1) q^{62} + (3 \beta_{3} - 1) q^{64} + ( - 5 \beta_{5} - 2 \beta_{4} - 3 \beta_1) q^{67} + (6 \beta_{4} - 6 \beta_1) q^{68} + (2 \beta_{3} - 8) q^{71} + ( - 3 \beta_{5} + \beta_{4} - 2 \beta_1) q^{73} + ( - 2 \beta_{3} + \beta_{2} + 6) q^{74} + (\beta_{3} - 1) q^{76} + ( - \beta_{5} - 3 \beta_{4} + 4 \beta_1) q^{77} + ( - 2 \beta_{2} - 4) q^{79} + (2 \beta_{5} + 2 \beta_{4} + 6 \beta_1) q^{82} + ( - 4 \beta_{4} - 2 \beta_1) q^{83} + ( - 2 \beta_{3} + 2) q^{86} + (2 \beta_{5} + 6 \beta_{4} - 2 \beta_1) q^{88} + (6 \beta_{3} - 2 \beta_{2} - 2) q^{89} + (2 \beta_{2} - 4) q^{91} + ( - 2 \beta_{4} + 4 \beta_1) q^{92} + (6 \beta_{3} + 4 \beta_{2} - 6) q^{94} + ( - \beta_{5} + \beta_1) q^{97} + ( - \beta_{4} + 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 8 q^{4} - 2 q^{11} - 16 q^{14} - 4 q^{16} - 6 q^{19} - 8 q^{26} - 36 q^{29} - 8 q^{34} - 12 q^{41} - 20 q^{44} - 8 q^{46} - 4 q^{49} - 40 q^{56} - 20 q^{59} - 14 q^{61} - 12 q^{64} - 52 q^{71} + 40 q^{74} - 8 q^{76} - 24 q^{79} + 16 q^{86} - 24 q^{89} - 24 q^{91} - 48 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 9x^{4} + 13x^{2} - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 8\nu^{2} + 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 8\nu^{3} + 7\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{5} + 9\nu^{3} - 13\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 2\beta_{4} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{3} + 8\beta_{2} + 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{5} + 18\beta_{4} + 41\beta_1 \) 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
−1.30397
2.68667
0.285442
−0.285442
−2.68667
1.30397
−2.41987 0 3.85577 0 0 3.18676 −4.49073 0 0
1.2 −1.82254 0 1.32164 0 0 1.45033 1.23634 0 0
1.3 −0.906968 0 −1.17741 0 0 −2.59637 2.88181 0 0
1.4 0.906968 0 −1.17741 0 0 2.59637 −2.88181 0 0
1.5 1.82254 0 1.32164 0 0 −1.45033 −1.23634 0 0
1.6 2.41987 0 3.85577 0 0 −3.18676 4.49073 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(19\) \(1\)

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 4275.2.a.br 6
3.b odd 2 1 475.2.a.j 6
5.b even 2 1 inner 4275.2.a.br 6
5.c odd 4 2 855.2.c.d 6
12.b even 2 1 7600.2.a.ck 6
15.d odd 2 1 475.2.a.j 6
15.e even 4 2 95.2.b.b 6
57.d even 2 1 9025.2.a.bx 6
60.h even 2 1 7600.2.a.ck 6
60.l odd 4 2 1520.2.d.h 6
285.b even 2 1 9025.2.a.bx 6
285.j odd 4 2 1805.2.b.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.b 6 15.e even 4 2
475.2.a.j 6 3.b odd 2 1
475.2.a.j 6 15.d odd 2 1
855.2.c.d 6 5.c odd 4 2
1520.2.d.h 6 60.l odd 4 2
1805.2.b.e 6 285.j odd 4 2
4275.2.a.br 6 1.a even 1 1 trivial
4275.2.a.br 6 5.b even 2 1 inner
7600.2.a.ck 6 12.b even 2 1
7600.2.a.ck 6 60.h even 2 1
9025.2.a.bx 6 57.d even 2 1
9025.2.a.bx 6 285.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4275))\):

\( T_{2}^{6} - 10T_{2}^{4} + 27T_{2}^{2} - 16 \) Copy content Toggle raw display
\( T_{7}^{6} - 19T_{7}^{4} + 104T_{7}^{2} - 144 \) Copy content Toggle raw display
\( T_{11}^{3} + T_{11}^{2} - 16T_{11} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 10 T^{4} + 27 T^{2} - 16 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 19 T^{4} + 104 T^{2} + \cdots - 144 \) Copy content Toggle raw display
$11$ \( (T^{3} + T^{2} - 16 T - 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 28 T^{4} + 236 T^{2} + \cdots - 576 \) Copy content Toggle raw display
$17$ \( T^{6} - 59 T^{4} + 1008 T^{2} + \cdots - 5184 \) Copy content Toggle raw display
$19$ \( (T + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 36 T^{4} + 208 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$29$ \( (T + 6)^{6} \) Copy content Toggle raw display
$31$ \( (T^{3} - 56 T + 128)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 56 T^{4} + 764 T^{2} + \cdots - 1296 \) Copy content Toggle raw display
$41$ \( (T^{3} + 6 T^{2} - 44 T + 24)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} - 19 T^{4} + 104 T^{2} + \cdots - 144 \) Copy content Toggle raw display
$47$ \( T^{6} - 187 T^{4} + 7464 T^{2} + \cdots - 85264 \) Copy content Toggle raw display
$53$ \( T^{6} - 156 T^{4} + 2476 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$59$ \( (T^{3} + 10 T^{2} + 8 T - 48)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 7 T^{2} - 104 T - 776)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} - 340 T^{4} + 28556 T^{2} + \cdots - 484416 \) Copy content Toggle raw display
$71$ \( (T^{3} + 26 T^{2} + 200 T + 432)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 131 T^{4} + 1616 T^{2} + \cdots - 5184 \) Copy content Toggle raw display
$79$ \( (T^{3} + 12 T^{2} - 8 T - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} - 228 T^{4} + 11728 T^{2} + \cdots - 141376 \) Copy content Toggle raw display
$89$ \( (T^{3} + 12 T^{2} - 284 T - 3456)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} - 28 T^{4} + 236 T^{2} + \cdots - 576 \) Copy content Toggle raw display
show more
show less