Properties

Label 1350.4.c.p
Level $1350$
Weight $4$
Character orbit 1350.c
Analytic conductor $79.653$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 i q^{2} - 4 q^{4} + 23 i q^{7} + 8 i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 i q^{2} - 4 q^{4} + 23 i q^{7} + 8 i q^{8} + 30 q^{11} + 34 i q^{13} + 46 q^{14} + 16 q^{16} - 42 i q^{17} + 139 q^{19} - 60 i q^{22} - 192 i q^{23} + 68 q^{26} - 92 i q^{28} - 234 q^{29} - 55 q^{31} - 32 i q^{32} - 84 q^{34} + 191 i q^{37} - 278 i q^{38} + 138 q^{41} - 53 i q^{43} - 120 q^{44} - 384 q^{46} + 366 i q^{47} - 186 q^{49} - 136 i q^{52} + 330 i q^{53} - 184 q^{56} + 468 i q^{58} + 396 q^{59} + 23 q^{61} + 110 i q^{62} - 64 q^{64} + 452 i q^{67} + 168 i q^{68} + 204 q^{71} + 691 i q^{73} + 382 q^{74} - 556 q^{76} + 690 i q^{77} + 709 q^{79} - 276 i q^{82} - 1098 i q^{83} - 106 q^{86} + 240 i q^{88} + 816 q^{89} - 782 q^{91} + 768 i q^{92} + 732 q^{94} + 905 i q^{97} + 372 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 60 q^{11} + 92 q^{14} + 32 q^{16} + 278 q^{19} + 136 q^{26} - 468 q^{29} - 110 q^{31} - 168 q^{34} + 276 q^{41} - 240 q^{44} - 768 q^{46} - 372 q^{49} - 368 q^{56} + 792 q^{59} + 46 q^{61} - 128 q^{64} + 408 q^{71} + 764 q^{74} - 1112 q^{76} + 1418 q^{79} - 212 q^{86} + 1632 q^{89} - 1564 q^{91} + 1464 q^{94}+O(q^{100}) \) 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
1.00000i
1.00000i
2.00000i 0 −4.00000 0 0 23.0000i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 23.0000i 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.p 2
3.b odd 2 1 1350.4.c.e 2
5.b even 2 1 inner 1350.4.c.p 2
5.c odd 4 1 1350.4.a.m yes 1
5.c odd 4 1 1350.4.a.p yes 1
15.d odd 2 1 1350.4.c.e 2
15.e even 4 1 1350.4.a.b 1
15.e even 4 1 1350.4.a.ba yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1350.4.a.b 1 15.e even 4 1
1350.4.a.m yes 1 5.c odd 4 1
1350.4.a.p yes 1 5.c odd 4 1
1350.4.a.ba yes 1 15.e even 4 1
1350.4.c.e 2 3.b odd 2 1
1350.4.c.e 2 15.d odd 2 1
1350.4.c.p 2 1.a even 1 1 trivial
1350.4.c.p 2 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 \(S_{4}^{\mathrm{new}}(1350, [\chi])\):

\( T_{7}^{2} + 529 \) Copy content Toggle raw display
\( T_{11} - 30 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 529 \) Copy content Toggle raw display
$11$ \( (T - 30)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1156 \) Copy content Toggle raw display
$17$ \( T^{2} + 1764 \) Copy content Toggle raw display
$19$ \( (T - 139)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36864 \) Copy content Toggle raw display
$29$ \( (T + 234)^{2} \) Copy content Toggle raw display
$31$ \( (T + 55)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36481 \) Copy content Toggle raw display
$41$ \( (T - 138)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 2809 \) Copy content Toggle raw display
$47$ \( T^{2} + 133956 \) Copy content Toggle raw display
$53$ \( T^{2} + 108900 \) Copy content Toggle raw display
$59$ \( (T - 396)^{2} \) Copy content Toggle raw display
$61$ \( (T - 23)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 204304 \) Copy content Toggle raw display
$71$ \( (T - 204)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 477481 \) Copy content Toggle raw display
$79$ \( (T - 709)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1205604 \) Copy content Toggle raw display
$89$ \( (T - 816)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 819025 \) Copy content Toggle raw display
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