Properties

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

Related objects

Downloads

Learn more

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: \(\Q(i, \sqrt{209})\)
Defining polynomial: \( x^{4} + 105x^{2} + 2704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q + 2 \beta_1 q^{2} - 4 q^{4} + ( - \beta_{2} - 7 \beta_1) q^{7} - 8 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_1 q^{2} - 4 q^{4} + ( - \beta_{2} - 7 \beta_1) q^{7} - 8 \beta_1 q^{8} + ( - 2 \beta_{3} - 8) q^{11} + (\beta_{2} + 4 \beta_1) q^{13} + (2 \beta_{3} + 12) q^{14} + 16 q^{16} + ( - 2 \beta_{2} + 2 \beta_1) q^{17} + (\beta_{3} - 22) q^{19} + ( - 4 \beta_{2} - 20 \beta_1) q^{22} + ( - 4 \beta_{2} - 68 \beta_1) q^{23} + ( - 2 \beta_{3} - 6) q^{26} + (4 \beta_{2} + 28 \beta_1) q^{28} + ( - 2 \beta_{3} + 16) q^{29} + (2 \beta_{3} + 73) q^{31} + 32 \beta_1 q^{32} + (4 \beta_{3} - 8) q^{34} + (12 \beta_{2} + 121 \beta_1) q^{37} + (2 \beta_{2} - 42 \beta_1) q^{38} + ( - 6 \beta_{3} - 12) q^{41} + ( - 14 \beta_{2} + 7 \beta_1) q^{43} + (8 \beta_{3} + 32) q^{44} + (8 \beta_{3} + 128) q^{46} + ( - 10 \beta_{2} + 154 \beta_1) q^{47} + ( - 13 \beta_{3} - 163) q^{49} + ( - 4 \beta_{2} - 16 \beta_1) q^{52} + ( - 14 \beta_{2} - 430 \beta_1) q^{53} + ( - 8 \beta_{3} - 48) q^{56} + ( - 4 \beta_{2} + 28 \beta_1) q^{58} + ( - 8 \beta_{3} - 8) q^{59} + ( - 27 \beta_{3} + 320) q^{61} + (4 \beta_{2} + 150 \beta_1) q^{62} - 64 q^{64} + (15 \beta_{2} + 622 \beta_1) q^{67} + (8 \beta_{2} - 8 \beta_1) q^{68} + (36 \beta_{3} + 60) q^{71} + (12 \beta_{2} + 701 \beta_1) q^{73} + ( - 24 \beta_{3} - 218) q^{74} + ( - 4 \beta_{3} + 88) q^{76} + (22 \beta_{2} + 1010 \beta_1) q^{77} + 259 q^{79} + ( - 12 \beta_{2} - 36 \beta_1) q^{82} + (62 \beta_{2} - 2 \beta_1) q^{83} + (28 \beta_{3} - 42) q^{86} + (16 \beta_{2} + 80 \beta_1) q^{88} + (24 \beta_{3} - 780) q^{89} + (10 \beta_{3} + 488) q^{91} + (16 \beta_{2} + 272 \beta_1) q^{92} + (20 \beta_{3} - 328) q^{94} + ( - 23 \beta_{2} + 525 \beta_1) q^{97} + ( - 26 \beta_{2} - 352 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{4} - 36 q^{11} + 52 q^{14} + 64 q^{16} - 86 q^{19} - 28 q^{26} + 60 q^{29} + 296 q^{31} - 24 q^{34} - 60 q^{41} + 144 q^{44} + 528 q^{46} - 678 q^{49} - 208 q^{56} - 48 q^{59} + 1226 q^{61} - 256 q^{64} + 312 q^{71} - 920 q^{74} + 344 q^{76} + 1036 q^{79} - 112 q^{86} - 3072 q^{89} + 1972 q^{91} - 1272 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 105x^{2} + 2704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 53\nu ) / 52 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 209\nu ) / 52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{2} + 158 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 158 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -53\beta_{2} + 209\beta_1 ) / 3 \) 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
7.72842i
6.72842i
6.72842i
7.72842i
2.00000i 0 −4.00000 0 0 15.1852i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 28.1852i 8.00000i 0 0
649.3 2.00000i 0 −4.00000 0 0 28.1852i 8.00000i 0 0
649.4 2.00000i 0 −4.00000 0 0 15.1852i 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.x 4
3.b odd 2 1 1350.4.c.y 4
5.b even 2 1 inner 1350.4.c.x 4
5.c odd 4 1 1350.4.a.bh yes 2
5.c odd 4 1 1350.4.a.bk yes 2
15.d odd 2 1 1350.4.c.y 4
15.e even 4 1 1350.4.a.bd 2
15.e even 4 1 1350.4.a.bo yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1350.4.a.bd 2 15.e even 4 1
1350.4.a.bh yes 2 5.c odd 4 1
1350.4.a.bk yes 2 5.c odd 4 1
1350.4.a.bo yes 2 15.e even 4 1
1350.4.c.x 4 1.a even 1 1 trivial
1350.4.c.x 4 5.b even 2 1 inner
1350.4.c.y 4 3.b odd 2 1
1350.4.c.y 4 15.d odd 2 1

Hecke kernels

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

\( T_{7}^{4} + 1025T_{7}^{2} + 183184 \) Copy content Toggle raw display
\( T_{11}^{2} + 18T_{11} - 1800 \) 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
$7$ \( T^{4} + 1025 T^{2} + 183184 \) Copy content Toggle raw display
$11$ \( (T^{2} + 18 T - 1800)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 965 T^{2} + 209764 \) Copy content Toggle raw display
$17$ \( T^{4} + 3780 T^{2} + \cdots + 3504384 \) Copy content Toggle raw display
$19$ \( (T^{2} + 43 T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 23760 T^{2} + \cdots + 10036224 \) Copy content Toggle raw display
$29$ \( (T^{2} - 30 T - 1656)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 148 T + 3595)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 161882 T^{2} + \cdots + 2969269081 \) Copy content Toggle raw display
$41$ \( (T^{2} + 30 T - 16704)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 184730 T^{2} + \cdots + 8459032729 \) Copy content Toggle raw display
$47$ \( T^{4} + 144612 T^{2} + \cdots + 472801536 \) Copy content Toggle raw display
$53$ \( T^{4} + 542196 T^{2} + \cdots + 7527297600 \) Copy content Toggle raw display
$59$ \( (T^{2} + 24 T - 29952)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 613 T - 248870)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 966833 T^{2} + \cdots + 73877414416 \) Copy content Toggle raw display
$71$ \( (T^{2} - 156 T - 603360)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 1101482 T^{2} + \cdots + 172481565481 \) Copy content Toggle raw display
$79$ \( (T - 259)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 3617460 T^{2} + \cdots + 3263630128704 \) Copy content Toggle raw display
$89$ \( (T^{2} + 1536 T + 318960)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 1073189 T^{2} + \cdots + 1526464900 \) Copy content Toggle raw display
show more
show less