Properties

Label 1350.4.c.ba
Level $1350$
Weight $4$
Character orbit 1350.c
Analytic conductor $79.653$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1350.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(79.6525785077\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_1 q^{2} - 4 q^{4} + (\beta_{2} - 5 \beta_1) q^{7} + 8 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_1 q^{2} - 4 q^{4} + (\beta_{2} - 5 \beta_1) q^{7} + 8 \beta_1 q^{8} + (4 \beta_{3} + 15) q^{11} + ( - \beta_{2} + 20 \beta_1) q^{13} + (2 \beta_{3} - 10) q^{14} + 16 q^{16} + ( - 5 \beta_{2} - 15 \beta_1) q^{17} + (5 \beta_{3} - 23) q^{19} + ( - 8 \beta_{2} - 30 \beta_1) q^{22} + (5 \beta_{2} - 12 \beta_1) q^{23} + ( - 2 \beta_{3} + 40) q^{26} + ( - 4 \beta_{2} + 20 \beta_1) q^{28} + (7 \beta_{3} - 45) q^{29} + ( - 20 \beta_{3} - 10) q^{31} - 32 \beta_1 q^{32} + ( - 10 \beta_{3} - 30) q^{34} + (3 \beta_{2} - 20 \beta_1) q^{37} + ( - 10 \beta_{2} + 46 \beta_1) q^{38} + ( - 12 \beta_{3} + 150) q^{41} + (5 \beta_{2} + 305 \beta_1) q^{43} + ( - 16 \beta_{3} - 60) q^{44} + (10 \beta_{3} - 24) q^{46} + (5 \beta_{2} + 312 \beta_1) q^{47} + (10 \beta_{3} + 129) q^{49} + (4 \beta_{2} - 80 \beta_1) q^{52} + ( - 5 \beta_{2} + 123 \beta_1) q^{53} + ( - 8 \beta_{3} + 40) q^{56} + ( - 14 \beta_{2} + 90 \beta_1) q^{58} + ( - 26 \beta_{3} - 45) q^{59} + (45 \beta_{3} + 68) q^{61} + (40 \beta_{2} + 20 \beta_1) q^{62} - 64 q^{64} + ( - 9 \beta_{2} - 695 \beta_1) q^{67} + (20 \beta_{2} + 60 \beta_1) q^{68} + (15 \beta_{3} + 390) q^{71} + (12 \beta_{2} + 290 \beta_1) q^{73} + (6 \beta_{3} - 40) q^{74} + ( - 20 \beta_{3} + 92) q^{76} + ( - 5 \beta_{2} + 681 \beta_1) q^{77} + ( - 15 \beta_{3} + 223) q^{79} + (24 \beta_{2} - 300 \beta_1) q^{82} + ( - 10 \beta_{2} + 954 \beta_1) q^{83} + (10 \beta_{3} + 610) q^{86} + (32 \beta_{2} + 120 \beta_1) q^{88} + (69 \beta_{3} - 285) q^{89} + ( - 25 \beta_{3} + 289) q^{91} + ( - 20 \beta_{2} + 48 \beta_1) q^{92} + (10 \beta_{3} + 624) q^{94} + (26 \beta_{2} - 1175 \beta_1) q^{97} + ( - 20 \beta_{2} - 258 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{4} + 60 q^{11} - 40 q^{14} + 64 q^{16} - 92 q^{19} + 160 q^{26} - 180 q^{29} - 40 q^{31} - 120 q^{34} + 600 q^{41} - 240 q^{44} - 96 q^{46} + 516 q^{49} + 160 q^{56} - 180 q^{59} + 272 q^{61} - 256 q^{64} + 1560 q^{71} - 160 q^{74} + 368 q^{76} + 892 q^{79} + 2440 q^{86} - 1140 q^{89} + 1156 q^{91} + 2496 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 11x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 6\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{3} + 48\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 6\nu^{2} + 33 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 33 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 8\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
2.79129i
1.79129i
1.79129i
2.79129i
2.00000i 0 −4.00000 0 0 18.7477i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 8.74773i 8.00000i 0 0
649.3 2.00000i 0 −4.00000 0 0 8.74773i 8.00000i 0 0
649.4 2.00000i 0 −4.00000 0 0 18.7477i 8.00000i 0 0
\(n\): e.g. 2-40 or 990-1000
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 1350.4.c.ba 4
3.b odd 2 1 1350.4.c.v 4
5.b even 2 1 inner 1350.4.c.ba 4
5.c odd 4 1 1350.4.a.be 2
5.c odd 4 1 1350.4.a.bn yes 2
15.d odd 2 1 1350.4.c.v 4
15.e even 4 1 1350.4.a.bg yes 2
15.e even 4 1 1350.4.a.bl yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1350.4.a.be 2 5.c odd 4 1
1350.4.a.bg yes 2 15.e even 4 1
1350.4.a.bl yes 2 15.e even 4 1
1350.4.a.bn yes 2 5.c odd 4 1
1350.4.c.v 4 3.b odd 2 1
1350.4.c.v 4 15.d odd 2 1
1350.4.c.ba 4 1.a even 1 1 trivial
1350.4.c.ba 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1350, [\chi])\):

\( T_{7}^{4} + 428T_{7}^{2} + 26896 \) Copy content Toggle raw display
\( T_{11}^{2} - 30T_{11} - 2799 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 428 T^{2} + 26896 \) Copy content Toggle raw display
$11$ \( (T^{2} - 30 T - 2799)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 1178 T^{2} + 44521 \) Copy content Toggle raw display
$17$ \( T^{4} + 9900 T^{2} + \cdots + 20250000 \) Copy content Toggle raw display
$19$ \( (T^{2} + 46 T - 4196)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 9738 T^{2} + \cdots + 20985561 \) Copy content Toggle raw display
$29$ \( (T^{2} + 90 T - 7236)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 20 T - 75500)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 4202 T^{2} + \cdots + 1692601 \) Copy content Toggle raw display
$41$ \( (T^{2} - 300 T - 4716)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 195500 T^{2} + \cdots + 7796890000 \) Copy content Toggle raw display
$47$ \( T^{4} + 204138 T^{2} + \cdots + 8578279161 \) Copy content Toggle raw display
$53$ \( T^{4} + 39708 T^{2} + \cdots + 108243216 \) Copy content Toggle raw display
$59$ \( (T^{2} + 90 T - 125739)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 136 T - 378101)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 996668 T^{2} + \cdots + 218758256656 \) Copy content Toggle raw display
$71$ \( (T^{2} - 780 T + 109575)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 222632 T^{2} + \cdots + 3235789456 \) Copy content Toggle raw display
$79$ \( (T^{2} - 446 T + 7204)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1858032 T^{2} + \cdots + 794265958656 \) Copy content Toggle raw display
$89$ \( (T^{2} + 570 T - 818604)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 3016778 T^{2} + \cdots + 1569660685321 \) Copy content Toggle raw display
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