Properties

Label 1350.3.d.g
Level $1350$
Weight $3$
Character orbit 1350.d
Analytic conductor $36.785$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1350.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7848356886\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 2 q^{4} + 5 q^{7} - 2 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - 2 q^{4} + 5 q^{7} - 2 \beta q^{8} + \beta q^{11} - 9 q^{13} + 5 \beta q^{14} + 4 q^{16} - 8 \beta q^{17} - 21 q^{19} - 2 q^{22} + \beta q^{23} - 9 \beta q^{26} - 10 q^{28} - 27 \beta q^{29} + 40 q^{31} + 4 \beta q^{32} + 16 q^{34} - 25 q^{37} - 21 \beta q^{38} + 37 \beta q^{41} - 64 q^{43} - 2 \beta q^{44} - 2 q^{46} - 16 \beta q^{47} - 24 q^{49} + 18 q^{52} + 51 \beta q^{53} - 10 \beta q^{56} + 54 q^{58} - 64 \beta q^{59} - 97 q^{61} + 40 \beta q^{62} - 8 q^{64} - 131 q^{67} + 16 \beta q^{68} - 63 \beta q^{71} + 17 q^{73} - 25 \beta q^{74} + 42 q^{76} + 5 \beta q^{77} + 117 q^{79} - 74 q^{82} - 41 \beta q^{83} - 64 \beta q^{86} + 4 q^{88} - 104 \beta q^{89} - 45 q^{91} - 2 \beta q^{92} + 32 q^{94} - 41 q^{97} - 24 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} + 10 q^{7} - 18 q^{13} + 8 q^{16} - 42 q^{19} - 4 q^{22} - 20 q^{28} + 80 q^{31} + 32 q^{34} - 50 q^{37} - 128 q^{43} - 4 q^{46} - 48 q^{49} + 36 q^{52} + 108 q^{58} - 194 q^{61} - 16 q^{64} - 262 q^{67} + 34 q^{73} + 84 q^{76} + 234 q^{79} - 148 q^{82} + 8 q^{88} - 90 q^{91} + 64 q^{94} - 82 q^{97}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
1.41421i
1.41421i
1.41421i 0 −2.00000 0 0 5.00000 2.82843i 0 0
701.2 1.41421i 0 −2.00000 0 0 5.00000 2.82843i 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.3.d.g 2
3.b odd 2 1 inner 1350.3.d.g 2
5.b even 2 1 1350.3.d.f 2
5.c odd 4 2 270.3.b.c 4
15.d odd 2 1 1350.3.d.f 2
15.e even 4 2 270.3.b.c 4
20.e even 4 2 2160.3.c.i 4
45.k odd 12 4 810.3.j.e 8
45.l even 12 4 810.3.j.e 8
60.l odd 4 2 2160.3.c.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.b.c 4 5.c odd 4 2
270.3.b.c 4 15.e even 4 2
810.3.j.e 8 45.k odd 12 4
810.3.j.e 8 45.l even 12 4
1350.3.d.f 2 5.b even 2 1
1350.3.d.f 2 15.d odd 2 1
1350.3.d.g 2 1.a even 1 1 trivial
1350.3.d.g 2 3.b odd 2 1 inner
2160.3.c.i 4 20.e even 4 2
2160.3.c.i 4 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1350, [\chi])\):

\( T_{7} - 5 \) Copy content Toggle raw display
\( T_{11}^{2} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 5)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 2 \) Copy content Toggle raw display
$13$ \( (T + 9)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 128 \) Copy content Toggle raw display
$19$ \( (T + 21)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2 \) Copy content Toggle raw display
$29$ \( T^{2} + 1458 \) Copy content Toggle raw display
$31$ \( (T - 40)^{2} \) Copy content Toggle raw display
$37$ \( (T + 25)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2738 \) Copy content Toggle raw display
$43$ \( (T + 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 512 \) Copy content Toggle raw display
$53$ \( T^{2} + 5202 \) Copy content Toggle raw display
$59$ \( T^{2} + 8192 \) Copy content Toggle raw display
$61$ \( (T + 97)^{2} \) Copy content Toggle raw display
$67$ \( (T + 131)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 7938 \) Copy content Toggle raw display
$73$ \( (T - 17)^{2} \) Copy content Toggle raw display
$79$ \( (T - 117)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 3362 \) Copy content Toggle raw display
$89$ \( T^{2} + 21632 \) Copy content Toggle raw display
$97$ \( (T + 41)^{2} \) Copy content Toggle raw display
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