Properties

Label 1350.4.c.d
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 270)
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.

\(f(q)\) \(=\) \( q + 2 i q^{2} - 4 q^{4} + 4 i q^{7} - 8 i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{2} - 4 q^{4} + 4 i q^{7} - 8 i q^{8} - 42 q^{11} + 20 i q^{13} - 8 q^{14} + 16 q^{16} + 93 i q^{17} - 59 q^{19} - 84 i q^{22} - 9 i q^{23} - 40 q^{26} - 16 i q^{28} + 120 q^{29} + 47 q^{31} + 32 i q^{32} - 186 q^{34} + 262 i q^{37} - 118 i q^{38} - 126 q^{41} - 178 i q^{43} + 168 q^{44} + 18 q^{46} + 144 i q^{47} + 327 q^{49} - 80 i q^{52} - 741 i q^{53} + 32 q^{56} + 240 i q^{58} - 444 q^{59} + 221 q^{61} + 94 i q^{62} - 64 q^{64} + 538 i q^{67} - 372 i q^{68} - 690 q^{71} - 1126 i q^{73} - 524 q^{74} + 236 q^{76} - 168 i q^{77} - 665 q^{79} - 252 i q^{82} - 75 i q^{83} + 356 q^{86} + 336 i q^{88} - 1086 q^{89} - 80 q^{91} + 36 i q^{92} - 288 q^{94} - 1544 i q^{97} + 654 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} - 84 q^{11} - 16 q^{14} + 32 q^{16} - 118 q^{19} - 80 q^{26} + 240 q^{29} + 94 q^{31} - 372 q^{34} - 252 q^{41} + 336 q^{44} + 36 q^{46} + 654 q^{49} + 64 q^{56} - 888 q^{59} + 442 q^{61} - 128 q^{64} - 1380 q^{71} - 1048 q^{74} + 472 q^{76} - 1330 q^{79} + 712 q^{86} - 2172 q^{89} - 160 q^{91} - 576 q^{94}+O(q^{100}) \) 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 4.00000i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 4.00000i 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.d 2
3.b odd 2 1 1350.4.c.q 2
5.b even 2 1 inner 1350.4.c.d 2
5.c odd 4 1 270.4.a.e 1
5.c odd 4 1 1350.4.a.u 1
15.d odd 2 1 1350.4.c.q 2
15.e even 4 1 270.4.a.i yes 1
15.e even 4 1 1350.4.a.g 1
20.e even 4 1 2160.4.a.o 1
45.k odd 12 2 810.4.e.q 2
45.l even 12 2 810.4.e.h 2
60.l odd 4 1 2160.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.e 1 5.c odd 4 1
270.4.a.i yes 1 15.e even 4 1
810.4.e.h 2 45.l even 12 2
810.4.e.q 2 45.k odd 12 2
1350.4.a.g 1 15.e even 4 1
1350.4.a.u 1 5.c odd 4 1
1350.4.c.d 2 1.a even 1 1 trivial
1350.4.c.d 2 5.b even 2 1 inner
1350.4.c.q 2 3.b odd 2 1
1350.4.c.q 2 15.d odd 2 1
2160.4.a.e 1 60.l odd 4 1
2160.4.a.o 1 20.e even 4 1

Hecke kernels

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

\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} + 42 \) 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
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T + 42)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 400 \) Copy content Toggle raw display
$17$ \( T^{2} + 8649 \) Copy content Toggle raw display
$19$ \( (T + 59)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 81 \) Copy content Toggle raw display
$29$ \( (T - 120)^{2} \) Copy content Toggle raw display
$31$ \( (T - 47)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 68644 \) Copy content Toggle raw display
$41$ \( (T + 126)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 31684 \) Copy content Toggle raw display
$47$ \( T^{2} + 20736 \) Copy content Toggle raw display
$53$ \( T^{2} + 549081 \) Copy content Toggle raw display
$59$ \( (T + 444)^{2} \) Copy content Toggle raw display
$61$ \( (T - 221)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 289444 \) Copy content Toggle raw display
$71$ \( (T + 690)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1267876 \) Copy content Toggle raw display
$79$ \( (T + 665)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 5625 \) Copy content Toggle raw display
$89$ \( (T + 1086)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2383936 \) Copy content Toggle raw display
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