# Properties

 Label 3179.2.a.a Level $3179$ Weight $2$ Character orbit 3179.a Self dual yes Analytic conductor $25.384$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3179 = 11 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3179.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.3844428026$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + q^{3} + 2 q^{4} - q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{9}+O(q^{10})$$ q - 2 * q^2 + q^3 + 2 * q^4 - q^5 - 2 * q^6 + 2 * q^7 - 2 * q^9 $$q - 2 q^{2} + q^{3} + 2 q^{4} - q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{9} + 2 q^{10} - q^{11} + 2 q^{12} + 4 q^{13} - 4 q^{14} - q^{15} - 4 q^{16} + 4 q^{18} - 2 q^{20} + 2 q^{21} + 2 q^{22} + q^{23} - 4 q^{25} - 8 q^{26} - 5 q^{27} + 4 q^{28} + 2 q^{30} - 7 q^{31} + 8 q^{32} - q^{33} - 2 q^{35} - 4 q^{36} - 3 q^{37} + 4 q^{39} + 8 q^{41} - 4 q^{42} - 6 q^{43} - 2 q^{44} + 2 q^{45} - 2 q^{46} + 8 q^{47} - 4 q^{48} - 3 q^{49} + 8 q^{50} + 8 q^{52} - 6 q^{53} + 10 q^{54} + q^{55} + 5 q^{59} - 2 q^{60} - 12 q^{61} + 14 q^{62} - 4 q^{63} - 8 q^{64} - 4 q^{65} + 2 q^{66} - 7 q^{67} + q^{69} + 4 q^{70} + 3 q^{71} - 4 q^{73} + 6 q^{74} - 4 q^{75} - 2 q^{77} - 8 q^{78} + 10 q^{79} + 4 q^{80} + q^{81} - 16 q^{82} - 6 q^{83} + 4 q^{84} + 12 q^{86} + 15 q^{89} - 4 q^{90} + 8 q^{91} + 2 q^{92} - 7 q^{93} - 16 q^{94} + 8 q^{96} + 7 q^{97} + 6 q^{98} + 2 q^{99}+O(q^{100})$$ q - 2 * q^2 + q^3 + 2 * q^4 - q^5 - 2 * q^6 + 2 * q^7 - 2 * q^9 + 2 * q^10 - q^11 + 2 * q^12 + 4 * q^13 - 4 * q^14 - q^15 - 4 * q^16 + 4 * q^18 - 2 * q^20 + 2 * q^21 + 2 * q^22 + q^23 - 4 * q^25 - 8 * q^26 - 5 * q^27 + 4 * q^28 + 2 * q^30 - 7 * q^31 + 8 * q^32 - q^33 - 2 * q^35 - 4 * q^36 - 3 * q^37 + 4 * q^39 + 8 * q^41 - 4 * q^42 - 6 * q^43 - 2 * q^44 + 2 * q^45 - 2 * q^46 + 8 * q^47 - 4 * q^48 - 3 * q^49 + 8 * q^50 + 8 * q^52 - 6 * q^53 + 10 * q^54 + q^55 + 5 * q^59 - 2 * q^60 - 12 * q^61 + 14 * q^62 - 4 * q^63 - 8 * q^64 - 4 * q^65 + 2 * q^66 - 7 * q^67 + q^69 + 4 * q^70 + 3 * q^71 - 4 * q^73 + 6 * q^74 - 4 * q^75 - 2 * q^77 - 8 * q^78 + 10 * q^79 + 4 * q^80 + q^81 - 16 * q^82 - 6 * q^83 + 4 * q^84 + 12 * q^86 + 15 * q^89 - 4 * q^90 + 8 * q^91 + 2 * q^92 - 7 * q^93 - 16 * q^94 + 8 * q^96 + 7 * q^97 + 6 * q^98 + 2 * q^99

## 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
 0
−2.00000 1.00000 2.00000 −1.00000 −2.00000 2.00000 0 −2.00000 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$1$$
$$17$$ $$1$$

## 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 3179.2.a.a 1
17.b even 2 1 11.2.a.a 1
51.c odd 2 1 99.2.a.d 1
68.d odd 2 1 176.2.a.b 1
85.c even 2 1 275.2.a.b 1
85.g odd 4 2 275.2.b.a 2
119.d odd 2 1 539.2.a.a 1
119.h odd 6 2 539.2.e.g 2
119.j even 6 2 539.2.e.h 2
136.e odd 2 1 704.2.a.c 1
136.h even 2 1 704.2.a.h 1
153.h even 6 2 891.2.e.k 2
153.i odd 6 2 891.2.e.b 2
187.b odd 2 1 121.2.a.d 1
187.j even 10 4 121.2.c.e 4
187.l odd 10 4 121.2.c.a 4
204.h even 2 1 1584.2.a.g 1
221.b even 2 1 1859.2.a.b 1
255.h odd 2 1 2475.2.a.a 1
255.o even 4 2 2475.2.c.a 2
272.k odd 4 2 2816.2.c.f 2
272.r even 4 2 2816.2.c.j 2
323.c odd 2 1 3971.2.a.b 1
340.d odd 2 1 4400.2.a.i 1
340.r even 4 2 4400.2.b.h 2
357.c even 2 1 4851.2.a.t 1
391.c odd 2 1 5819.2.a.a 1
408.b odd 2 1 6336.2.a.br 1
408.h even 2 1 6336.2.a.bu 1
476.e even 2 1 8624.2.a.j 1
493.c even 2 1 9251.2.a.d 1
561.h even 2 1 1089.2.a.b 1
748.f even 2 1 1936.2.a.i 1
935.h odd 2 1 3025.2.a.a 1
1309.e even 2 1 5929.2.a.h 1
1496.g even 2 1 7744.2.a.k 1
1496.p odd 2 1 7744.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 17.b even 2 1
99.2.a.d 1 51.c odd 2 1
121.2.a.d 1 187.b odd 2 1
121.2.c.a 4 187.l odd 10 4
121.2.c.e 4 187.j even 10 4
176.2.a.b 1 68.d odd 2 1
275.2.a.b 1 85.c even 2 1
275.2.b.a 2 85.g odd 4 2
539.2.a.a 1 119.d odd 2 1
539.2.e.g 2 119.h odd 6 2
539.2.e.h 2 119.j even 6 2
704.2.a.c 1 136.e odd 2 1
704.2.a.h 1 136.h even 2 1
891.2.e.b 2 153.i odd 6 2
891.2.e.k 2 153.h even 6 2
1089.2.a.b 1 561.h even 2 1
1584.2.a.g 1 204.h even 2 1
1859.2.a.b 1 221.b even 2 1
1936.2.a.i 1 748.f even 2 1
2475.2.a.a 1 255.h odd 2 1
2475.2.c.a 2 255.o even 4 2
2816.2.c.f 2 272.k odd 4 2
2816.2.c.j 2 272.r even 4 2
3025.2.a.a 1 935.h odd 2 1
3179.2.a.a 1 1.a even 1 1 trivial
3971.2.a.b 1 323.c odd 2 1
4400.2.a.i 1 340.d odd 2 1
4400.2.b.h 2 340.r even 4 2
4851.2.a.t 1 357.c even 2 1
5819.2.a.a 1 391.c odd 2 1
5929.2.a.h 1 1309.e even 2 1
6336.2.a.br 1 408.b odd 2 1
6336.2.a.bu 1 408.h even 2 1
7744.2.a.k 1 1496.g even 2 1
7744.2.a.x 1 1496.p odd 2 1
8624.2.a.j 1 476.e even 2 1
9251.2.a.d 1 493.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3179))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{3} - 1$$ T3 - 1 $$T_{5} + 1$$ T5 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T - 2$$
$11$ $$T + 1$$
$13$ $$T - 4$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T - 1$$
$29$ $$T$$
$31$ $$T + 7$$
$37$ $$T + 3$$
$41$ $$T - 8$$
$43$ $$T + 6$$
$47$ $$T - 8$$
$53$ $$T + 6$$
$59$ $$T - 5$$
$61$ $$T + 12$$
$67$ $$T + 7$$
$71$ $$T - 3$$
$73$ $$T + 4$$
$79$ $$T - 10$$
$83$ $$T + 6$$
$89$ $$T - 15$$
$97$ $$T - 7$$