# Properties

 Label 4176.2.a.bq Level $4176$ Weight $2$ Character orbit 4176.a Self dual yes Analytic conductor $33.346$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4176 = 2^{4} \cdot 3^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4176.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$33.3455278841$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 29) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{5} - 2 \beta q^{7} +O(q^{10})$$ q + q^5 - 2*b * q^7 $$q + q^{5} - 2 \beta q^{7} + (\beta + 1) q^{11} + (2 \beta - 1) q^{13} + (2 \beta + 2) q^{17} - 6 q^{19} + ( - 4 \beta - 2) q^{23} - 4 q^{25} - q^{29} + (5 \beta - 3) q^{31} - 2 \beta q^{35} - 4 q^{37} + ( - 6 \beta - 4) q^{41} + ( - \beta - 5) q^{43} + (3 \beta + 1) q^{47} + q^{49} + (6 \beta - 1) q^{53} + (\beta + 1) q^{55} + (4 \beta + 2) q^{59} + (2 \beta - 2) q^{61} + (2 \beta - 1) q^{65} + 4 \beta q^{67} + (2 \beta - 6) q^{71} + 4 q^{73} + ( - 2 \beta - 4) q^{77} + ( - \beta + 1) q^{79} + ( - 4 \beta + 2) q^{83} + (2 \beta + 2) q^{85} + ( - 6 \beta + 4) q^{89} + (2 \beta - 8) q^{91} - 6 q^{95} + ( - 6 \beta - 4) q^{97} +O(q^{100})$$ q + q^5 - 2*b * q^7 + (b + 1) * q^11 + (2*b - 1) * q^13 + (2*b + 2) * q^17 - 6 * q^19 + (-4*b - 2) * q^23 - 4 * q^25 - q^29 + (5*b - 3) * q^31 - 2*b * q^35 - 4 * q^37 + (-6*b - 4) * q^41 + (-b - 5) * q^43 + (3*b + 1) * q^47 + q^49 + (6*b - 1) * q^53 + (b + 1) * q^55 + (4*b + 2) * q^59 + (2*b - 2) * q^61 + (2*b - 1) * q^65 + 4*b * q^67 + (2*b - 6) * q^71 + 4 * q^73 + (-2*b - 4) * q^77 + (-b + 1) * q^79 + (-4*b + 2) * q^83 + (2*b + 2) * q^85 + (-6*b + 4) * q^89 + (2*b - 8) * q^91 - 6 * q^95 + (-6*b - 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5}+O(q^{10})$$ 2 * q + 2 * q^5 $$2 q + 2 q^{5} + 2 q^{11} - 2 q^{13} + 4 q^{17} - 12 q^{19} - 4 q^{23} - 8 q^{25} - 2 q^{29} - 6 q^{31} - 8 q^{37} - 8 q^{41} - 10 q^{43} + 2 q^{47} + 2 q^{49} - 2 q^{53} + 2 q^{55} + 4 q^{59} - 4 q^{61} - 2 q^{65} - 12 q^{71} + 8 q^{73} - 8 q^{77} + 2 q^{79} + 4 q^{83} + 4 q^{85} + 8 q^{89} - 16 q^{91} - 12 q^{95} - 8 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 + 2 * q^11 - 2 * q^13 + 4 * q^17 - 12 * q^19 - 4 * q^23 - 8 * q^25 - 2 * q^29 - 6 * q^31 - 8 * q^37 - 8 * q^41 - 10 * q^43 + 2 * q^47 + 2 * q^49 - 2 * q^53 + 2 * q^55 + 4 * q^59 - 4 * q^61 - 2 * q^65 - 12 * q^71 + 8 * q^73 - 8 * q^77 + 2 * q^79 + 4 * q^83 + 4 * q^85 + 8 * q^89 - 16 * q^91 - 12 * q^95 - 8 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.41421 −1.41421
0 0 0 1.00000 0 −2.82843 0 0 0
1.2 0 0 0 1.00000 0 2.82843 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$29$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4176.2.a.bq 2
3.b odd 2 1 464.2.a.h 2
4.b odd 2 1 261.2.a.d 2
12.b even 2 1 29.2.a.a 2
20.d odd 2 1 6525.2.a.o 2
24.f even 2 1 1856.2.a.r 2
24.h odd 2 1 1856.2.a.w 2
60.h even 2 1 725.2.a.b 2
60.l odd 4 2 725.2.b.b 4
84.h odd 2 1 1421.2.a.j 2
116.d odd 2 1 7569.2.a.c 2
132.d odd 2 1 3509.2.a.j 2
156.h even 2 1 4901.2.a.g 2
204.h even 2 1 8381.2.a.e 2
348.b even 2 1 841.2.a.d 2
348.k odd 4 2 841.2.b.a 4
348.s even 14 6 841.2.d.j 12
348.t even 14 6 841.2.d.f 12
348.v odd 28 12 841.2.e.k 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.a.a 2 12.b even 2 1
261.2.a.d 2 4.b odd 2 1
464.2.a.h 2 3.b odd 2 1
725.2.a.b 2 60.h even 2 1
725.2.b.b 4 60.l odd 4 2
841.2.a.d 2 348.b even 2 1
841.2.b.a 4 348.k odd 4 2
841.2.d.f 12 348.t even 14 6
841.2.d.j 12 348.s even 14 6
841.2.e.k 24 348.v odd 28 12
1421.2.a.j 2 84.h odd 2 1
1856.2.a.r 2 24.f even 2 1
1856.2.a.w 2 24.h odd 2 1
3509.2.a.j 2 132.d odd 2 1
4176.2.a.bq 2 1.a even 1 1 trivial
4901.2.a.g 2 156.h even 2 1
6525.2.a.o 2 20.d odd 2 1
7569.2.a.c 2 116.d odd 2 1
8381.2.a.e 2 204.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4176))$$:

 $$T_{5} - 1$$ T5 - 1 $$T_{7}^{2} - 8$$ T7^2 - 8 $$T_{11}^{2} - 2T_{11} - 1$$ T11^2 - 2*T11 - 1 $$T_{17}^{2} - 4T_{17} - 4$$ T17^2 - 4*T17 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 8$$
$11$ $$T^{2} - 2T - 1$$
$13$ $$T^{2} + 2T - 7$$
$17$ $$T^{2} - 4T - 4$$
$19$ $$(T + 6)^{2}$$
$23$ $$T^{2} + 4T - 28$$
$29$ $$(T + 1)^{2}$$
$31$ $$T^{2} + 6T - 41$$
$37$ $$(T + 4)^{2}$$
$41$ $$T^{2} + 8T - 56$$
$43$ $$T^{2} + 10T + 23$$
$47$ $$T^{2} - 2T - 17$$
$53$ $$T^{2} + 2T - 71$$
$59$ $$T^{2} - 4T - 28$$
$61$ $$T^{2} + 4T - 4$$
$67$ $$T^{2} - 32$$
$71$ $$T^{2} + 12T + 28$$
$73$ $$(T - 4)^{2}$$
$79$ $$T^{2} - 2T - 1$$
$83$ $$T^{2} - 4T - 28$$
$89$ $$T^{2} - 8T - 56$$
$97$ $$T^{2} + 8T - 56$$