Properties

Label 450.7.d.a
Level $450$
Weight $7$
Character orbit 450.d
Analytic conductor $103.524$
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: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 450.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(103.524337629\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
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.

\(f(q)\) \(=\) \( q - 4 \beta q^{2} - 32 q^{4} + 484 q^{7} + 128 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - 4 \beta q^{2} - 32 q^{4} + 484 q^{7} + 128 \beta q^{8} + 948 \beta q^{11} - 3368 q^{13} - 1936 \beta q^{14} + 1024 q^{16} - 9 \beta q^{17} + 5744 q^{19} + 7584 q^{22} + 2388 \beta q^{23} + 13472 \beta q^{26} - 15488 q^{28} + 20757 \beta q^{29} - 39796 q^{31} - 4096 \beta q^{32} - 72 q^{34} - 52526 q^{37} - 22976 \beta q^{38} - 26193 \beta q^{41} - 3800 q^{43} - 30336 \beta q^{44} + 19104 q^{46} - 54300 \beta q^{47} + 116607 q^{49} + 107776 q^{52} - 168813 \beta q^{53} + 61952 \beta q^{56} + 166056 q^{58} - 176664 \beta q^{59} + 13250 q^{61} + 159184 \beta q^{62} - 32768 q^{64} - 168968 q^{67} + 288 \beta q^{68} - 375804 \beta q^{71} - 236144 q^{73} + 210104 \beta q^{74} - 183808 q^{76} + 458832 \beta q^{77} - 35116 q^{79} - 209544 q^{82} + 7764 \beta q^{83} + 15200 \beta q^{86} - 242688 q^{88} - 91449 \beta q^{89} - 1630112 q^{91} - 76416 \beta q^{92} - 434400 q^{94} + 321424 q^{97} - 466428 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 64 q^{4} + 968 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 64 q^{4} + 968 q^{7} - 6736 q^{13} + 2048 q^{16} + 11488 q^{19} + 15168 q^{22} - 30976 q^{28} - 79592 q^{31} - 144 q^{34} - 105052 q^{37} - 7600 q^{43} + 38208 q^{46} + 233214 q^{49} + 215552 q^{52} + 332112 q^{58} + 26500 q^{61} - 65536 q^{64} - 337936 q^{67} - 472288 q^{73} - 367616 q^{76} - 70232 q^{79} - 419088 q^{82} - 485376 q^{88} - 3260224 q^{91} - 868800 q^{94} + 642848 q^{97}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
1.41421i
1.41421i
5.65685i 0 −32.0000 0 0 484.000 181.019i 0 0
251.2 5.65685i 0 −32.0000 0 0 484.000 181.019i 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 450.7.d.a 2
3.b odd 2 1 inner 450.7.d.a 2
5.b even 2 1 18.7.b.a 2
5.c odd 4 2 450.7.b.a 4
15.d odd 2 1 18.7.b.a 2
15.e even 4 2 450.7.b.a 4
20.d odd 2 1 144.7.e.d 2
40.e odd 2 1 576.7.e.k 2
40.f even 2 1 576.7.e.b 2
45.h odd 6 2 162.7.d.d 4
45.j even 6 2 162.7.d.d 4
60.h even 2 1 144.7.e.d 2
120.i odd 2 1 576.7.e.b 2
120.m even 2 1 576.7.e.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.7.b.a 2 5.b even 2 1
18.7.b.a 2 15.d odd 2 1
144.7.e.d 2 20.d odd 2 1
144.7.e.d 2 60.h even 2 1
162.7.d.d 4 45.h odd 6 2
162.7.d.d 4 45.j even 6 2
450.7.b.a 4 5.c odd 4 2
450.7.b.a 4 15.e even 4 2
450.7.d.a 2 1.a even 1 1 trivial
450.7.d.a 2 3.b odd 2 1 inner
576.7.e.b 2 40.f even 2 1
576.7.e.b 2 120.i odd 2 1
576.7.e.k 2 40.e odd 2 1
576.7.e.k 2 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 484 \) acting on \(S_{7}^{\mathrm{new}}(450, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 32 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 484)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1797408 \) Copy content Toggle raw display
$13$ \( (T + 3368)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 162 \) Copy content Toggle raw display
$19$ \( (T - 5744)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 11405088 \) Copy content Toggle raw display
$29$ \( T^{2} + 861706098 \) Copy content Toggle raw display
$31$ \( (T + 39796)^{2} \) Copy content Toggle raw display
$37$ \( (T + 52526)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 1372146498 \) Copy content Toggle raw display
$43$ \( (T + 3800)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 5896980000 \) Copy content Toggle raw display
$53$ \( T^{2} + 56995657938 \) Copy content Toggle raw display
$59$ \( T^{2} + 62420337792 \) Copy content Toggle raw display
$61$ \( (T - 13250)^{2} \) Copy content Toggle raw display
$67$ \( (T + 168968)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 282457292832 \) Copy content Toggle raw display
$73$ \( (T + 236144)^{2} \) Copy content Toggle raw display
$79$ \( (T + 35116)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 120559392 \) Copy content Toggle raw display
$89$ \( T^{2} + 16725839202 \) Copy content Toggle raw display
$97$ \( (T - 321424)^{2} \) Copy content Toggle raw display
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