Properties

Label 1350.4.c.z
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: \(\Q(i, \sqrt{6})\)
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q - 2 \beta_1 q^{2} - 4 q^{4} + ( - \beta_{2} - 10 \beta_1) q^{7} + 8 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_1 q^{2} - 4 q^{4} + ( - \beta_{2} - 10 \beta_1) q^{7} + 8 \beta_1 q^{8} + (2 \beta_{3} + 15) q^{11} + ( - 5 \beta_{2} - 5 \beta_1) q^{13} + ( - 2 \beta_{3} - 20) q^{14} + 16 q^{16} + ( - 5 \beta_{2} - 18 \beta_1) q^{17} + (5 \beta_{3} + 16) q^{19} + ( - 4 \beta_{2} - 30 \beta_1) q^{22} + (5 \beta_{2} + 9 \beta_1) q^{23} + ( - 10 \beta_{3} - 10) q^{26} + (4 \beta_{2} + 40 \beta_1) q^{28} + (5 \beta_{3} - 60) q^{29} + (10 \beta_{3} + 122) q^{31} - 32 \beta_1 q^{32} + ( - 10 \beta_{3} - 36) q^{34} + ( - 3 \beta_{2} - 205 \beta_1) q^{37} + ( - 10 \beta_{2} - 32 \beta_1) q^{38} + (6 \beta_{3} + 210) q^{41} + (7 \beta_{2} + 160 \beta_1) q^{43} + ( - 8 \beta_{3} - 60) q^{44} + (10 \beta_{3} + 18) q^{46} + (35 \beta_{2} + 81 \beta_1) q^{47} + ( - 20 \beta_{3} + 27) q^{49} + (20 \beta_{2} + 20 \beta_1) q^{52} + ( - 5 \beta_{2} - 384 \beta_1) q^{53} + (8 \beta_{3} + 80) q^{56} + ( - 10 \beta_{2} + 120 \beta_1) q^{58} + (14 \beta_{3} + 165) q^{59} + ( - 15 \beta_{3} - 229) q^{61} + ( - 20 \beta_{2} - 244 \beta_1) q^{62} - 64 q^{64} + (3 \beta_{2} - 430 \beta_1) q^{67} + (20 \beta_{2} + 72 \beta_1) q^{68} + (15 \beta_{3} + 555) q^{71} + ( - 6 \beta_{2} + 430 \beta_1) q^{73} + ( - 6 \beta_{3} - 410) q^{74} + ( - 20 \beta_{3} - 64) q^{76} + ( - 35 \beta_{2} - 582 \beta_1) q^{77} + (45 \beta_{3} - 290) q^{79} + ( - 12 \beta_{2} - 420 \beta_1) q^{82} + ( - 40 \beta_{2} - 600 \beta_1) q^{83} + (14 \beta_{3} + 320) q^{86} + (16 \beta_{2} + 120 \beta_1) q^{88} + ( - 9 \beta_{3} + 330) q^{89} + ( - 55 \beta_{3} - 1130) q^{91} + ( - 20 \beta_{2} - 36 \beta_1) q^{92} + (70 \beta_{3} + 162) q^{94} + ( - 14 \beta_{2} - 655 \beta_1) q^{97} + (40 \beta_{2} - 54 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{4} + 60 q^{11} - 80 q^{14} + 64 q^{16} + 64 q^{19} - 40 q^{26} - 240 q^{29} + 488 q^{31} - 144 q^{34} + 840 q^{41} - 240 q^{44} + 72 q^{46} + 108 q^{49} + 320 q^{56} + 660 q^{59} - 916 q^{61} - 256 q^{64} + 2220 q^{71} - 1640 q^{74} - 256 q^{76} - 1160 q^{79} + 1280 q^{86} + 1320 q^{89} - 4520 q^{91} + 648 q^{94}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\nu^{3} + 6\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 4 \) 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
1.22474 + 1.22474i
−1.22474 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
2.00000i 0 −4.00000 0 0 24.6969i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 4.69694i 8.00000i 0 0
649.3 2.00000i 0 −4.00000 0 0 4.69694i 8.00000i 0 0
649.4 2.00000i 0 −4.00000 0 0 24.6969i 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.z 4
3.b odd 2 1 1350.4.c.w 4
5.b even 2 1 inner 1350.4.c.z 4
5.c odd 4 1 1350.4.a.bc 2
5.c odd 4 1 1350.4.a.bp yes 2
15.d odd 2 1 1350.4.c.w 4
15.e even 4 1 1350.4.a.bi yes 2
15.e even 4 1 1350.4.a.bj yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1350.4.a.bc 2 5.c odd 4 1
1350.4.a.bi yes 2 15.e even 4 1
1350.4.a.bj yes 2 15.e even 4 1
1350.4.a.bp yes 2 5.c odd 4 1
1350.4.c.w 4 3.b odd 2 1
1350.4.c.w 4 15.d odd 2 1
1350.4.c.z 4 1.a even 1 1 trivial
1350.4.c.z 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 \(S_{4}^{\mathrm{new}}(1350, [\chi])\):

\( T_{7}^{4} + 632T_{7}^{2} + 13456 \) Copy content Toggle raw display
\( T_{11}^{2} - 30T_{11} - 639 \) 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
$7$ \( T^{4} + 632 T^{2} + 13456 \) Copy content Toggle raw display
$11$ \( (T^{2} - 30 T - 639)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 10850 T^{2} + \cdots + 28890625 \) Copy content Toggle raw display
$17$ \( T^{4} + 11448 T^{2} + \cdots + 25765776 \) Copy content Toggle raw display
$19$ \( (T^{2} - 32 T - 5144)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 10962 T^{2} + \cdots + 28291761 \) Copy content Toggle raw display
$29$ \( (T^{2} + 120 T - 1800)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 244 T - 6716)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 87938 T^{2} + \cdots + 1606486561 \) Copy content Toggle raw display
$41$ \( (T^{2} - 420 T + 36324)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 72368 T^{2} + \cdots + 225480256 \) Copy content Toggle raw display
$47$ \( T^{4} + 542322 T^{2} + \cdots + 66584125521 \) Copy content Toggle raw display
$53$ \( T^{4} + 305712 T^{2} + \cdots + 20179907136 \) Copy content Toggle raw display
$59$ \( (T^{2} - 330 T - 15111)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 458 T + 3841)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 373688 T^{2} + \cdots + 33472897936 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1110 T + 259425)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 385352 T^{2} + \cdots + 31372911376 \) Copy content Toggle raw display
$79$ \( (T^{2} + 580 T - 353300)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1411200 T^{2} + \cdots + 207360000 \) Copy content Toggle raw display
$89$ \( (T^{2} - 660 T + 91404)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 942722 T^{2} + \cdots + 149528382721 \) Copy content Toggle raw display
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