Properties

Label 1350.4.c.s
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: no (minimal twist has level 54)
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.

\(f(q)\) \(=\) \( q - 2 i q^{2} - 4 q^{4} + 29 i q^{7} + 8 i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 i q^{2} - 4 q^{4} + 29 i q^{7} + 8 i q^{8} + 57 q^{11} - 20 i q^{13} + 58 q^{14} + 16 q^{16} + 72 i q^{17} + 106 q^{19} - 114 i q^{22} + 174 i q^{23} - 40 q^{26} - 116 i q^{28} - 210 q^{29} + 47 q^{31} - 32 i q^{32} + 144 q^{34} + 2 i q^{37} - 212 i q^{38} + 6 q^{41} - 218 i q^{43} - 228 q^{44} + 348 q^{46} - 474 i q^{47} - 498 q^{49} + 80 i q^{52} + 81 i q^{53} - 232 q^{56} + 420 i q^{58} + 84 q^{59} + 56 q^{61} - 94 i q^{62} - 64 q^{64} - 142 i q^{67} - 288 i q^{68} - 360 q^{71} + 1159 i q^{73} + 4 q^{74} - 424 q^{76} + 1653 i q^{77} + 160 q^{79} - 12 i q^{82} + 735 i q^{83} - 436 q^{86} + 456 i q^{88} - 954 q^{89} + 580 q^{91} - 696 i q^{92} - 948 q^{94} + 191 i q^{97} + 996 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 114 q^{11} + 116 q^{14} + 32 q^{16} + 212 q^{19} - 80 q^{26} - 420 q^{29} + 94 q^{31} + 288 q^{34} + 12 q^{41} - 456 q^{44} + 696 q^{46} - 996 q^{49} - 464 q^{56} + 168 q^{59} + 112 q^{61} - 128 q^{64} - 720 q^{71} + 8 q^{74} - 848 q^{76} + 320 q^{79} - 872 q^{86} - 1908 q^{89} + 1160 q^{91} - 1896 q^{94}+O(q^{100}) \) 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 29.0000i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 29.0000i 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.s 2
3.b odd 2 1 1350.4.c.b 2
5.b even 2 1 inner 1350.4.c.s 2
5.c odd 4 1 54.4.a.b 1
5.c odd 4 1 1350.4.a.o 1
15.d odd 2 1 1350.4.c.b 2
15.e even 4 1 54.4.a.c yes 1
15.e even 4 1 1350.4.a.a 1
20.e even 4 1 432.4.a.e 1
40.i odd 4 1 1728.4.a.v 1
40.k even 4 1 1728.4.a.u 1
45.k odd 12 2 162.4.c.g 2
45.l even 12 2 162.4.c.b 2
60.l odd 4 1 432.4.a.j 1
120.q odd 4 1 1728.4.a.k 1
120.w even 4 1 1728.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.4.a.b 1 5.c odd 4 1
54.4.a.c yes 1 15.e even 4 1
162.4.c.b 2 45.l even 12 2
162.4.c.g 2 45.k odd 12 2
432.4.a.e 1 20.e even 4 1
432.4.a.j 1 60.l odd 4 1
1350.4.a.a 1 15.e even 4 1
1350.4.a.o 1 5.c odd 4 1
1350.4.c.b 2 3.b odd 2 1
1350.4.c.b 2 15.d odd 2 1
1350.4.c.s 2 1.a even 1 1 trivial
1350.4.c.s 2 5.b even 2 1 inner
1728.4.a.k 1 120.q odd 4 1
1728.4.a.l 1 120.w even 4 1
1728.4.a.u 1 40.k even 4 1
1728.4.a.v 1 40.i odd 4 1

Hecke kernels

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

\( T_{7}^{2} + 841 \) Copy content Toggle raw display
\( T_{11} - 57 \) 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
$7$ \( T^{2} + 841 \) Copy content Toggle raw display
$11$ \( (T - 57)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 400 \) Copy content Toggle raw display
$17$ \( T^{2} + 5184 \) Copy content Toggle raw display
$19$ \( (T - 106)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 30276 \) Copy content Toggle raw display
$29$ \( (T + 210)^{2} \) Copy content Toggle raw display
$31$ \( (T - 47)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 47524 \) Copy content Toggle raw display
$47$ \( T^{2} + 224676 \) Copy content Toggle raw display
$53$ \( T^{2} + 6561 \) Copy content Toggle raw display
$59$ \( (T - 84)^{2} \) Copy content Toggle raw display
$61$ \( (T - 56)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 20164 \) Copy content Toggle raw display
$71$ \( (T + 360)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1343281 \) Copy content Toggle raw display
$79$ \( (T - 160)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 540225 \) Copy content Toggle raw display
$89$ \( (T + 954)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 36481 \) Copy content Toggle raw display
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