Properties

Label 1350.4.c.n
Level $1350$
Weight $4$
Character orbit 1350.c
Analytic conductor $79.653$
Analytic rank $1$
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: \(1\)
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.

\(f(q)\) \(=\) \( q + 2 i q^{2} - 4 q^{4} + 8 i q^{7} - 8 i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{2} - 4 q^{4} + 8 i q^{7} - 8 i q^{8} + 18 q^{11} - 8 i q^{13} - 16 q^{14} + 16 q^{16} + 15 i q^{17} - 23 q^{19} + 36 i q^{22} - 63 i q^{23} + 16 q^{26} - 32 i q^{28} - 156 q^{29} - 85 q^{31} + 32 i q^{32} - 30 q^{34} + 74 i q^{37} - 46 i q^{38} + 246 q^{41} + 190 i q^{43} - 72 q^{44} + 126 q^{46} + 288 i q^{47} + 279 q^{49} + 32 i q^{52} + 177 i q^{53} + 64 q^{56} - 312 i q^{58} - 792 q^{59} - 907 q^{61} - 170 i q^{62} - 64 q^{64} - 322 i q^{67} - 60 i q^{68} - 270 q^{71} - 254 i q^{73} - 148 q^{74} + 92 q^{76} + 144 i q^{77} + 1123 q^{79} + 492 i q^{82} + 771 i q^{83} - 380 q^{86} - 144 i q^{88} + 198 q^{89} + 64 q^{91} + 252 i q^{92} - 576 q^{94} - 1192 i q^{97} + 558 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 36 q^{11} - 32 q^{14} + 32 q^{16} - 46 q^{19} + 32 q^{26} - 312 q^{29} - 170 q^{31} - 60 q^{34} + 492 q^{41} - 144 q^{44} + 252 q^{46} + 558 q^{49} + 128 q^{56} - 1584 q^{59} - 1814 q^{61} - 128 q^{64} - 540 q^{71} - 296 q^{74} + 184 q^{76} + 2246 q^{79} - 760 q^{86} + 396 q^{89} + 128 q^{91} - 1152 q^{94}+O(q^{100}) \) 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 8.00000i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 8.00000i 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.n 2
3.b odd 2 1 1350.4.c.g 2
5.b even 2 1 inner 1350.4.c.n 2
5.c odd 4 1 270.4.a.l yes 1
5.c odd 4 1 1350.4.a.f 1
15.d odd 2 1 1350.4.c.g 2
15.e even 4 1 270.4.a.b 1
15.e even 4 1 1350.4.a.t 1
20.e even 4 1 2160.4.a.m 1
45.k odd 12 2 810.4.e.b 2
45.l even 12 2 810.4.e.v 2
60.l odd 4 1 2160.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.b 1 15.e even 4 1
270.4.a.l yes 1 5.c odd 4 1
810.4.e.b 2 45.k odd 12 2
810.4.e.v 2 45.l even 12 2
1350.4.a.f 1 5.c odd 4 1
1350.4.a.t 1 15.e even 4 1
1350.4.c.g 2 3.b odd 2 1
1350.4.c.g 2 15.d odd 2 1
1350.4.c.n 2 1.a even 1 1 trivial
1350.4.c.n 2 5.b even 2 1 inner
2160.4.a.c 1 60.l odd 4 1
2160.4.a.m 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 \(S_{4}^{\mathrm{new}}(1350, [\chi])\):

\( T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{11} - 18 \) 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
$7$ \( T^{2} + 64 \) Copy content Toggle raw display
$11$ \( (T - 18)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 64 \) Copy content Toggle raw display
$17$ \( T^{2} + 225 \) Copy content Toggle raw display
$19$ \( (T + 23)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3969 \) Copy content Toggle raw display
$29$ \( (T + 156)^{2} \) Copy content Toggle raw display
$31$ \( (T + 85)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 5476 \) Copy content Toggle raw display
$41$ \( (T - 246)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36100 \) Copy content Toggle raw display
$47$ \( T^{2} + 82944 \) Copy content Toggle raw display
$53$ \( T^{2} + 31329 \) Copy content Toggle raw display
$59$ \( (T + 792)^{2} \) Copy content Toggle raw display
$61$ \( (T + 907)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 103684 \) Copy content Toggle raw display
$71$ \( (T + 270)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 64516 \) Copy content Toggle raw display
$79$ \( (T - 1123)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 594441 \) Copy content Toggle raw display
$89$ \( (T - 198)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1420864 \) Copy content Toggle raw display
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