Properties

Label 1350.4.c.l
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.

\(f(q)\) \(=\) \( q + 2 i q^{2} - 4 q^{4} + 22 i q^{7} - 8 i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{2} - 4 q^{4} + 22 i q^{7} - 8 i q^{8} + 12 q^{11} + 38 i q^{13} - 44 q^{14} + 16 q^{16} - 105 i q^{17} + 157 q^{19} + 24 i q^{22} + 117 i q^{23} - 76 q^{26} - 88 i q^{28} + 66 q^{29} - 25 q^{31} + 32 i q^{32} + 210 q^{34} - 314 i q^{37} + 314 i q^{38} + 504 q^{41} + 380 i q^{43} - 48 q^{44} - 234 q^{46} - 252 i q^{47} - 141 q^{49} - 152 i q^{52} - 3 i q^{53} + 176 q^{56} + 132 i q^{58} - 318 q^{59} + 293 q^{61} - 50 i q^{62} - 64 q^{64} + 322 i q^{67} + 420 i q^{68} + 120 q^{71} + 44 i q^{73} + 628 q^{74} - 628 q^{76} + 264 i q^{77} - 917 q^{79} + 1008 i q^{82} - 309 i q^{83} - 760 q^{86} - 96 i q^{88} + 1272 q^{89} - 836 q^{91} - 468 i q^{92} + 504 q^{94} - 1328 i q^{97} - 282 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 24 q^{11} - 88 q^{14} + 32 q^{16} + 314 q^{19} - 152 q^{26} + 132 q^{29} - 50 q^{31} + 420 q^{34} + 1008 q^{41} - 96 q^{44} - 468 q^{46} - 282 q^{49} + 352 q^{56} - 636 q^{59} + 586 q^{61} - 128 q^{64} + 240 q^{71} + 1256 q^{74} - 1256 q^{76} - 1834 q^{79} - 1520 q^{86} + 2544 q^{89} - 1672 q^{91} + 1008 q^{94}+O(q^{100}) \) 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 22.0000i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 22.0000i 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.l 2
3.b odd 2 1 1350.4.c.i 2
5.b even 2 1 inner 1350.4.c.l 2
5.c odd 4 1 270.4.a.c 1
5.c odd 4 1 1350.4.a.z 1
15.d odd 2 1 1350.4.c.i 2
15.e even 4 1 270.4.a.g yes 1
15.e even 4 1 1350.4.a.l 1
20.e even 4 1 2160.4.a.r 1
45.k odd 12 2 810.4.e.s 2
45.l even 12 2 810.4.e.k 2
60.l odd 4 1 2160.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.c 1 5.c odd 4 1
270.4.a.g yes 1 15.e even 4 1
810.4.e.k 2 45.l even 12 2
810.4.e.s 2 45.k odd 12 2
1350.4.a.l 1 15.e even 4 1
1350.4.a.z 1 5.c odd 4 1
1350.4.c.i 2 3.b odd 2 1
1350.4.c.i 2 15.d odd 2 1
1350.4.c.l 2 1.a even 1 1 trivial
1350.4.c.l 2 5.b even 2 1 inner
2160.4.a.i 1 60.l odd 4 1
2160.4.a.r 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 \(S_{4}^{\mathrm{new}}(1350, [\chi])\):

\( T_{7}^{2} + 484 \) Copy content Toggle raw display
\( T_{11} - 12 \) 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
$7$ \( T^{2} + 484 \) Copy content Toggle raw display
$11$ \( (T - 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1444 \) Copy content Toggle raw display
$17$ \( T^{2} + 11025 \) Copy content Toggle raw display
$19$ \( (T - 157)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 13689 \) Copy content Toggle raw display
$29$ \( (T - 66)^{2} \) Copy content Toggle raw display
$31$ \( (T + 25)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 98596 \) Copy content Toggle raw display
$41$ \( (T - 504)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 144400 \) Copy content Toggle raw display
$47$ \( T^{2} + 63504 \) Copy content Toggle raw display
$53$ \( T^{2} + 9 \) Copy content Toggle raw display
$59$ \( (T + 318)^{2} \) Copy content Toggle raw display
$61$ \( (T - 293)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 103684 \) Copy content Toggle raw display
$71$ \( (T - 120)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1936 \) Copy content Toggle raw display
$79$ \( (T + 917)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 95481 \) Copy content Toggle raw display
$89$ \( (T - 1272)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1763584 \) Copy content Toggle raw display
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