Properties

Label 1800.4.f.s
Level $1800$
Weight $4$
Character orbit 1800.f
Analytic conductor $106.203$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(106.203438010\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 360)
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.

\(f(q)\) \(=\) \( q + 9 \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 \beta q^{7} + 34 q^{11} + 6 \beta q^{13} + 51 \beta q^{17} - 164 q^{19} + 24 \beta q^{23} - 146 q^{29} + 100 q^{31} - 164 \beta q^{37} - 288 q^{41} + 60 \beta q^{43} - 8 \beta q^{47} + 19 q^{49} - 63 \beta q^{53} - 642 q^{59} + 602 q^{61} - 218 \beta q^{67} + 652 q^{71} + 531 \beta q^{73} + 306 \beta q^{77} - 388 q^{79} - 222 \beta q^{83} + 820 q^{89} - 216 q^{91} + 383 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 68 q^{11} - 328 q^{19} - 292 q^{29} + 200 q^{31} - 576 q^{41} + 38 q^{49} - 1284 q^{59} + 1204 q^{61} + 1304 q^{71} - 776 q^{79} + 1640 q^{89} - 432 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
0 0 0 0 0 18.0000i 0 0 0
649.2 0 0 0 0 0 18.0000i 0 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 1800.4.f.s 2
3.b odd 2 1 1800.4.f.e 2
5.b even 2 1 inner 1800.4.f.s 2
5.c odd 4 1 360.4.a.j yes 1
5.c odd 4 1 1800.4.a.be 1
15.d odd 2 1 1800.4.f.e 2
15.e even 4 1 360.4.a.a 1
15.e even 4 1 1800.4.a.bc 1
20.e even 4 1 720.4.a.z 1
60.l odd 4 1 720.4.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.4.a.a 1 15.e even 4 1
360.4.a.j yes 1 5.c odd 4 1
720.4.a.m 1 60.l odd 4 1
720.4.a.z 1 20.e even 4 1
1800.4.a.bc 1 15.e even 4 1
1800.4.a.be 1 5.c odd 4 1
1800.4.f.e 2 3.b odd 2 1
1800.4.f.e 2 15.d odd 2 1
1800.4.f.s 2 1.a even 1 1 trivial
1800.4.f.s 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1800, [\chi])\):

\( T_{7}^{2} + 324 \) Copy content Toggle raw display
\( T_{11} - 34 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 324 \) Copy content Toggle raw display
$11$ \( (T - 34)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 144 \) Copy content Toggle raw display
$17$ \( T^{2} + 10404 \) Copy content Toggle raw display
$19$ \( (T + 164)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2304 \) Copy content Toggle raw display
$29$ \( (T + 146)^{2} \) Copy content Toggle raw display
$31$ \( (T - 100)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 107584 \) Copy content Toggle raw display
$41$ \( (T + 288)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 14400 \) Copy content Toggle raw display
$47$ \( T^{2} + 256 \) Copy content Toggle raw display
$53$ \( T^{2} + 15876 \) Copy content Toggle raw display
$59$ \( (T + 642)^{2} \) Copy content Toggle raw display
$61$ \( (T - 602)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 190096 \) Copy content Toggle raw display
$71$ \( (T - 652)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1127844 \) Copy content Toggle raw display
$79$ \( (T + 388)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 197136 \) Copy content Toggle raw display
$89$ \( (T - 820)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 586756 \) Copy content Toggle raw display
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