Properties

Label 450.7.b.a
Level $450$
Weight $7$
Character orbit 450.b
Analytic conductor $103.524$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 450.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(103.524337629\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{2} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 \beta_{3} q^{2} + 32 q^{4} + 242 \beta_1 q^{7} - 128 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 \beta_{3} q^{2} + 32 q^{4} + 242 \beta_1 q^{7} - 128 \beta_{3} q^{8} - 948 \beta_{2} q^{11} + 1684 \beta_1 q^{13} - 1936 \beta_{2} q^{14} + 1024 q^{16} - 9 \beta_{3} q^{17} - 5744 q^{19} + 3792 \beta_1 q^{22} - 2388 \beta_{3} q^{23} - 13472 \beta_{2} q^{26} + 7744 \beta_1 q^{28} + 20757 \beta_{2} q^{29} - 39796 q^{31} - 4096 \beta_{3} q^{32} + 72 q^{34} - 26263 \beta_1 q^{37} + 22976 \beta_{3} q^{38} + 26193 \beta_{2} q^{41} + 1900 \beta_1 q^{43} - 30336 \beta_{2} q^{44} + 19104 q^{46} - 54300 \beta_{3} q^{47} - 116607 q^{49} + 53888 \beta_1 q^{52} + 168813 \beta_{3} q^{53} - 61952 \beta_{2} q^{56} - 83028 \beta_1 q^{58} - 176664 \beta_{2} q^{59} + 13250 q^{61} + 159184 \beta_{3} q^{62} + 32768 q^{64} - 84484 \beta_1 q^{67} - 288 \beta_{3} q^{68} + 375804 \beta_{2} q^{71} + 118072 \beta_1 q^{73} + 210104 \beta_{2} q^{74} - 183808 q^{76} + 458832 \beta_{3} q^{77} + 35116 q^{79} - 104772 \beta_1 q^{82} - 7764 \beta_{3} q^{83} - 15200 \beta_{2} q^{86} + 121344 \beta_1 q^{88} - 91449 \beta_{2} q^{89} - 1630112 q^{91} - 76416 \beta_{3} q^{92} + 434400 q^{94} + 160712 \beta_1 q^{97} + 466428 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 128 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 128 q^{4} + 4096 q^{16} - 22976 q^{19} - 159184 q^{31} + 288 q^{34} + 76416 q^{46} - 466428 q^{49} + 53000 q^{61} + 131072 q^{64} - 735232 q^{76} + 140464 q^{79} - 6520448 q^{91} + 1737600 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
−5.65685 0 32.0000 0 0 484.000i −181.019 0 0
449.2 −5.65685 0 32.0000 0 0 484.000i −181.019 0 0
449.3 5.65685 0 32.0000 0 0 484.000i 181.019 0 0
449.4 5.65685 0 32.0000 0 0 484.000i 181.019 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
5.b even 2 1 inner
15.d 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.b.a 4
3.b odd 2 1 inner 450.7.b.a 4
5.b even 2 1 inner 450.7.b.a 4
5.c odd 4 1 18.7.b.a 2
5.c odd 4 1 450.7.d.a 2
15.d odd 2 1 inner 450.7.b.a 4
15.e even 4 1 18.7.b.a 2
15.e even 4 1 450.7.d.a 2
20.e even 4 1 144.7.e.d 2
40.i odd 4 1 576.7.e.b 2
40.k even 4 1 576.7.e.k 2
45.k odd 12 2 162.7.d.d 4
45.l even 12 2 162.7.d.d 4
60.l odd 4 1 144.7.e.d 2
120.q odd 4 1 576.7.e.k 2
120.w even 4 1 576.7.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.7.b.a 2 5.c odd 4 1
18.7.b.a 2 15.e even 4 1
144.7.e.d 2 20.e even 4 1
144.7.e.d 2 60.l odd 4 1
162.7.d.d 4 45.k odd 12 2
162.7.d.d 4 45.l even 12 2
450.7.b.a 4 1.a even 1 1 trivial
450.7.b.a 4 3.b odd 2 1 inner
450.7.b.a 4 5.b even 2 1 inner
450.7.b.a 4 15.d odd 2 1 inner
450.7.d.a 2 5.c odd 4 1
450.7.d.a 2 15.e even 4 1
576.7.e.b 2 40.i odd 4 1
576.7.e.b 2 120.w even 4 1
576.7.e.k 2 40.k even 4 1
576.7.e.k 2 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 234256 \) acting on \(S_{7}^{\mathrm{new}}(450, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 234256)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1797408)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 11343424)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 162)^{2} \) Copy content Toggle raw display
$19$ \( (T + 5744)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 11405088)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 861706098)^{2} \) Copy content Toggle raw display
$31$ \( (T + 39796)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2758980676)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 1372146498)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 14440000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 5896980000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 56995657938)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 62420337792)^{2} \) Copy content Toggle raw display
$61$ \( (T - 13250)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 28550185024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 282457292832)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 55763988736)^{2} \) Copy content Toggle raw display
$79$ \( (T - 35116)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 120559392)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 16725839202)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 103313387776)^{2} \) Copy content Toggle raw display
show more
show less