# Properties

 Label 32.7.d.a Level $32$ Weight $7$ Character orbit 32.d Self dual yes Analytic conductor $7.362$ Analytic rank $0$ Dimension $1$ CM discriminant -8 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$32 = 2^{5}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 32.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.36173067584$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 8) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 46 q^{3} + 1387 q^{9}+O(q^{10})$$ q - 46 * q^3 + 1387 * q^9 $$q - 46 q^{3} + 1387 q^{9} + 2338 q^{11} - 1726 q^{17} + 2482 q^{19} + 15625 q^{25} - 30268 q^{27} - 107548 q^{33} + 134642 q^{41} + 74914 q^{43} + 117649 q^{49} + 79396 q^{51} - 114172 q^{57} - 304958 q^{59} + 596626 q^{67} - 593134 q^{73} - 718750 q^{75} + 381205 q^{81} - 678926 q^{83} - 357262 q^{89} + 1822754 q^{97} + 3242806 q^{99}+O(q^{100})$$ q - 46 * q^3 + 1387 * q^9 + 2338 * q^11 - 1726 * q^17 + 2482 * q^19 + 15625 * q^25 - 30268 * q^27 - 107548 * q^33 + 134642 * q^41 + 74914 * q^43 + 117649 * q^49 + 79396 * q^51 - 114172 * q^57 - 304958 * q^59 + 596626 * q^67 - 593134 * q^73 - 718750 * q^75 + 381205 * q^81 - 678926 * q^83 - 357262 * q^89 + 1822754 * q^97 + 3242806 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$31$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1
 0
0 −46.0000 0 0 0 0 0 1387.00 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.7.d.a 1
3.b odd 2 1 288.7.b.a 1
4.b odd 2 1 8.7.d.a 1
8.b even 2 1 8.7.d.a 1
8.d odd 2 1 CM 32.7.d.a 1
12.b even 2 1 72.7.b.a 1
16.e even 4 2 256.7.c.d 2
16.f odd 4 2 256.7.c.d 2
24.f even 2 1 288.7.b.a 1
24.h odd 2 1 72.7.b.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.7.d.a 1 4.b odd 2 1
8.7.d.a 1 8.b even 2 1
32.7.d.a 1 1.a even 1 1 trivial
32.7.d.a 1 8.d odd 2 1 CM
72.7.b.a 1 12.b even 2 1
72.7.b.a 1 24.h odd 2 1
256.7.c.d 2 16.e even 4 2
256.7.c.d 2 16.f odd 4 2
288.7.b.a 1 3.b odd 2 1
288.7.b.a 1 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 46$$ acting on $$S_{7}^{\mathrm{new}}(32, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 46$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T - 2338$$
$13$ $$T$$
$17$ $$T + 1726$$
$19$ $$T - 2482$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T - 134642$$
$43$ $$T - 74914$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T + 304958$$
$61$ $$T$$
$67$ $$T - 596626$$
$71$ $$T$$
$73$ $$T + 593134$$
$79$ $$T$$
$83$ $$T + 678926$$
$89$ $$T + 357262$$
$97$ $$T - 1822754$$