Properties

Label 450.4.c.g
Level $450$
Weight $4$
Character orbit 450.c
Analytic conductor $26.551$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.5508595026\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 90)
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.

\(f(q)\) \(=\) \( q - \beta q^{2} - 4 q^{4} + 7 \beta q^{7} + 4 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} - 4 q^{4} + 7 \beta q^{7} + 4 \beta q^{8} + 6 q^{11} - 34 \beta q^{13} + 28 q^{14} + 16 q^{16} + 39 \beta q^{17} - 44 q^{19} - 6 \beta q^{22} - 60 \beta q^{23} - 136 q^{26} - 28 \beta q^{28} - 126 q^{29} - 244 q^{31} - 16 \beta q^{32} + 156 q^{34} - 152 \beta q^{37} + 44 \beta q^{38} - 480 q^{41} - 52 \beta q^{43} - 24 q^{44} - 240 q^{46} + 300 \beta q^{47} + 147 q^{49} + 136 \beta q^{52} + 129 \beta q^{53} - 112 q^{56} + 126 \beta q^{58} - 534 q^{59} + 362 q^{61} + 244 \beta q^{62} - 64 q^{64} - 134 \beta q^{67} - 156 \beta q^{68} - 972 q^{71} - 235 \beta q^{73} - 608 q^{74} + 176 q^{76} + 42 \beta q^{77} - 1244 q^{79} + 480 \beta q^{82} - 198 \beta q^{83} - 208 q^{86} + 24 \beta q^{88} + 972 q^{89} + 952 q^{91} + 240 \beta q^{92} + 1200 q^{94} - 23 \beta q^{97} - 147 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 12 q^{11} + 56 q^{14} + 32 q^{16} - 88 q^{19} - 272 q^{26} - 252 q^{29} - 488 q^{31} + 312 q^{34} - 960 q^{41} - 48 q^{44} - 480 q^{46} + 294 q^{49} - 224 q^{56} - 1068 q^{59} + 724 q^{61} - 128 q^{64} - 1944 q^{71} - 1216 q^{74} + 352 q^{76} - 2488 q^{79} - 416 q^{86} + 1944 q^{89} + 1904 q^{91} + 2400 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
2.00000i 0 −4.00000 0 0 14.0000i 8.00000i 0 0
199.2 2.00000i 0 −4.00000 0 0 14.0000i 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 450.4.c.g 2
3.b odd 2 1 450.4.c.f 2
5.b even 2 1 inner 450.4.c.g 2
5.c odd 4 1 90.4.a.b 1
5.c odd 4 1 450.4.a.m 1
15.d odd 2 1 450.4.c.f 2
15.e even 4 1 90.4.a.e yes 1
15.e even 4 1 450.4.a.c 1
20.e even 4 1 720.4.a.e 1
45.k odd 12 2 810.4.e.u 2
45.l even 12 2 810.4.e.a 2
60.l odd 4 1 720.4.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.4.a.b 1 5.c odd 4 1
90.4.a.e yes 1 15.e even 4 1
450.4.a.c 1 15.e even 4 1
450.4.a.m 1 5.c odd 4 1
450.4.c.f 2 3.b odd 2 1
450.4.c.f 2 15.d odd 2 1
450.4.c.g 2 1.a even 1 1 trivial
450.4.c.g 2 5.b even 2 1 inner
720.4.a.e 1 20.e even 4 1
720.4.a.t 1 60.l odd 4 1
810.4.e.a 2 45.l even 12 2
810.4.e.u 2 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{2} + 196 \) Copy content Toggle raw display
\( T_{11} - 6 \) 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
$7$ \( T^{2} + 196 \) Copy content Toggle raw display
$11$ \( (T - 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4624 \) Copy content Toggle raw display
$17$ \( T^{2} + 6084 \) Copy content Toggle raw display
$19$ \( (T + 44)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 14400 \) Copy content Toggle raw display
$29$ \( (T + 126)^{2} \) Copy content Toggle raw display
$31$ \( (T + 244)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 92416 \) Copy content Toggle raw display
$41$ \( (T + 480)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 10816 \) Copy content Toggle raw display
$47$ \( T^{2} + 360000 \) Copy content Toggle raw display
$53$ \( T^{2} + 66564 \) Copy content Toggle raw display
$59$ \( (T + 534)^{2} \) Copy content Toggle raw display
$61$ \( (T - 362)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 71824 \) Copy content Toggle raw display
$71$ \( (T + 972)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 220900 \) Copy content Toggle raw display
$79$ \( (T + 1244)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 156816 \) Copy content Toggle raw display
$89$ \( (T - 972)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2116 \) Copy content Toggle raw display
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