# Properties

 Label 7623.2.a.co Level $7623$ Weight $2$ Character orbit 7623.a Self dual yes Analytic conductor $60.870$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7623 = 3^{2} \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7623.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$60.8699614608$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.2525.1 Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 5x + 5$$ x^4 - 2*x^3 - 4*x^2 + 5*x + 5 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_1 + 1) q^{5} + q^{7} + (\beta_{3} + \beta_{2} + 3) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + b1 + 1) * q^4 + (b1 + 1) * q^5 + q^7 + (b3 + b2 + 3) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_1 + 1) q^{5} + q^{7} + (\beta_{3} + \beta_{2} + 3) q^{8} + (\beta_{2} + 2 \beta_1 + 3) q^{10} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{13} + \beta_1 q^{14} + (2 \beta_{3} + \beta_1 - 1) q^{16} + ( - \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{17} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{19} + (\beta_{3} + 2 \beta_{2} + 3 \beta_1 + 4) q^{20} + (2 \beta_{3} + \beta_{2} + 3) q^{23} + (\beta_{2} + 3 \beta_1 - 1) q^{25} + ( - 3 \beta_{2} + \beta_1 + 1) q^{26} + (\beta_{2} + \beta_1 + 1) q^{28} - 3 \beta_{3} q^{29} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{31} + (3 \beta_{2} - 1) q^{32} + (2 \beta_{3} - 3 \beta_{2} - 4) q^{34} + (\beta_1 + 1) q^{35} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{37} + (2 \beta_{3} + 4 \beta_{2} + \beta_1 + 7) q^{38} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 4) q^{40} + ( - 2 \beta_{3} - 3 \beta_{2} - 1) q^{41} + ( - 6 \beta_{2} - 5) q^{43} + (3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 2) q^{46} + ( - 4 \beta_{3} + 2 \beta_{2} + \beta_1 + 3) q^{47} + q^{49} + (\beta_{3} + 3 \beta_{2} + 2 \beta_1 + 9) q^{50} + (\beta_{3} - 3 \beta_{2} + 3) q^{52} + ( - 3 \beta_{3} + \beta_{2} - 4 \beta_1 + 4) q^{53} + (\beta_{3} + \beta_{2} + 3) q^{56} + ( - 3 \beta_{3} - 6 \beta_{2} - 3) q^{58} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 6) q^{59} + ( - \beta_{3} - 5 \beta_1 + 7) q^{61} + (2 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{62} + ( - \beta_{3} - 3 \beta_1 + 2) q^{64} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{65} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{67} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{68} + (\beta_{2} + 2 \beta_1 + 3) q^{70} + (3 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 1) q^{71} + ( - 5 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 4) q^{73} + ( - \beta_{2} - \beta_1 + 2) q^{74} + (4 \beta_{3} + 3 \beta_{2} + 4 \beta_1 + 7) q^{76} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{79} + (4 \beta_{3} + 5 \beta_{2} + \beta_1 + 4) q^{80} + ( - 5 \beta_{3} - 4 \beta_{2} - \beta_1 - 2) q^{82} + (5 \beta_{3} - 7 \beta_{2} + 3 \beta_1 - 4) q^{83} + (\beta_{3} - \beta_1 - 3) q^{85} + ( - 6 \beta_{3} - 5 \beta_1) q^{86} + ( - 5 \beta_{3} + 3 \beta_{2} + \beta_1 + 4) q^{89} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{91} + (3 \beta_{3} + 7 \beta_{2} + 5 \beta_1 + 6) q^{92} + ( - 2 \beta_{3} - 7 \beta_{2} + 4 \beta_1 - 1) q^{94} + (3 \beta_{3} + 5 \beta_{2} + 3 \beta_1 + 6) q^{95} + ( - 5 \beta_{3} - 2 \beta_{2} - 6) q^{97} + \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + b1 + 1) * q^4 + (b1 + 1) * q^5 + q^7 + (b3 + b2 + 3) * q^8 + (b2 + 2*b1 + 3) * q^10 + (-2*b3 + 2*b2 + b1) * q^13 + b1 * q^14 + (2*b3 + b1 - 1) * q^16 + (-b3 + 3*b2 - b1 + 1) * q^17 + (b3 + b2 + 2*b1 - 1) * q^19 + (b3 + 2*b2 + 3*b1 + 4) * q^20 + (2*b3 + b2 + 3) * q^23 + (b2 + 3*b1 - 1) * q^25 + (-3*b2 + b1 + 1) * q^26 + (b2 + b1 + 1) * q^28 - 3*b3 * q^29 + (-b3 + 3*b2 + b1) * q^31 + (3*b2 - 1) * q^32 + (2*b3 - 3*b2 - 4) * q^34 + (b1 + 1) * q^35 + (-b3 + b2 + b1 - 2) * q^37 + (2*b3 + 4*b2 + b1 + 7) * q^38 + (3*b3 + 3*b2 + 3*b1 + 4) * q^40 + (-2*b3 - 3*b2 - 1) * q^41 + (-6*b2 - 5) * q^43 + (3*b3 + 4*b2 + 3*b1 + 2) * q^46 + (-4*b3 + 2*b2 + b1 + 3) * q^47 + q^49 + (b3 + 3*b2 + 2*b1 + 9) * q^50 + (b3 - 3*b2 + 3) * q^52 + (-3*b3 + b2 - 4*b1 + 4) * q^53 + (b3 + b2 + 3) * q^56 + (-3*b3 - 6*b2 - 3) * q^58 + (-b3 - 2*b2 - 2*b1 + 6) * q^59 + (-b3 - 5*b1 + 7) * q^61 + (2*b3 - b2 + b1 + 2) * q^62 + (-b3 - 3*b1 + 2) * q^64 + (-2*b3 - b2 + 2*b1 + 1) * q^65 + (b3 - 2*b2 - b1 - 4) * q^67 + (b3 - 2*b2 - 2*b1) * q^68 + (b2 + 2*b1 + 3) * q^70 + (3*b3 - 5*b2 + 2*b1 - 1) * q^71 + (-5*b3 + 3*b2 - 2*b1 + 4) * q^73 + (-b2 - b1 + 2) * q^74 + (4*b3 + 3*b2 + 4*b1 + 7) * q^76 + (2*b3 - 2*b2 - b1 - 2) * q^79 + (4*b3 + 5*b2 + b1 + 4) * q^80 + (-5*b3 - 4*b2 - b1 - 2) * q^82 + (5*b3 - 7*b2 + 3*b1 - 4) * q^83 + (b3 - b1 - 3) * q^85 + (-6*b3 - 5*b1) * q^86 + (-5*b3 + 3*b2 + b1 + 4) * q^89 + (-2*b3 + 2*b2 + b1) * q^91 + (3*b3 + 7*b2 + 5*b1 + 6) * q^92 + (-2*b3 - 7*b2 + 4*b1 - 1) * q^94 + (3*b3 + 5*b2 + 3*b1 + 6) * q^95 + (-5*b3 - 2*b2 - 6) * q^97 + b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{7} + 9 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 4 * q^7 + 9 * q^8 $$4 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{7} + 9 q^{8} + 14 q^{10} + 2 q^{14} - 4 q^{16} - 3 q^{17} - 3 q^{19} + 17 q^{20} + 8 q^{23} + 12 q^{26} + 4 q^{28} + 3 q^{29} - 3 q^{31} - 10 q^{32} - 12 q^{34} + 6 q^{35} - 7 q^{37} + 20 q^{38} + 13 q^{40} + 4 q^{41} - 8 q^{43} + 3 q^{46} + 14 q^{47} + 4 q^{49} + 33 q^{50} + 17 q^{52} + 9 q^{53} + 9 q^{56} + 3 q^{58} + 25 q^{59} + 19 q^{61} + 10 q^{62} + 3 q^{64} + 12 q^{65} - 15 q^{67} - q^{68} + 14 q^{70} + 7 q^{71} + 11 q^{73} + 8 q^{74} + 26 q^{76} - 8 q^{79} + 4 q^{80} + 3 q^{82} - q^{83} - 15 q^{85} - 4 q^{86} + 17 q^{89} + 17 q^{92} + 20 q^{94} + 17 q^{95} - 15 q^{97} + 2 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 4 * q^7 + 9 * q^8 + 14 * q^10 + 2 * q^14 - 4 * q^16 - 3 * q^17 - 3 * q^19 + 17 * q^20 + 8 * q^23 + 12 * q^26 + 4 * q^28 + 3 * q^29 - 3 * q^31 - 10 * q^32 - 12 * q^34 + 6 * q^35 - 7 * q^37 + 20 * q^38 + 13 * q^40 + 4 * q^41 - 8 * q^43 + 3 * q^46 + 14 * q^47 + 4 * q^49 + 33 * q^50 + 17 * q^52 + 9 * q^53 + 9 * q^56 + 3 * q^58 + 25 * q^59 + 19 * q^61 + 10 * q^62 + 3 * q^64 + 12 * q^65 - 15 * q^67 - q^68 + 14 * q^70 + 7 * q^71 + 11 * q^73 + 8 * q^74 + 26 * q^76 - 8 * q^79 + 4 * q^80 + 3 * q^82 - q^83 - 15 * q^85 - 4 * q^86 + 17 * q^89 + 17 * q^92 + 20 * q^94 + 17 * q^95 - 15 * q^97 + 2 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 4x^{2} + 5x + 5$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3\nu$$ v^3 - v^2 - 3*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4\beta _1 + 3$$ b3 + b2 + 4*b1 + 3

## 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
 −1.46673 −0.777484 1.77748 2.46673
−1.46673 0 0.151302 −0.466732 0 1.00000 2.71154 0 0.684570
1.2 −0.777484 0 −1.39552 0.222516 0 1.00000 2.63996 0 −0.173002
1.3 1.77748 0 1.15945 2.77748 0 1.00000 −1.49406 0 4.93693
1.4 2.46673 0 4.08477 3.46673 0 1.00000 5.14256 0 8.55150
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 7623.2.a.co 4
3.b odd 2 1 847.2.a.k 4
11.b odd 2 1 7623.2.a.ch 4
11.d odd 10 2 693.2.m.g 8
21.c even 2 1 5929.2.a.bb 4
33.d even 2 1 847.2.a.l 4
33.f even 10 2 77.2.f.a 8
33.f even 10 2 847.2.f.p 8
33.h odd 10 2 847.2.f.q 8
33.h odd 10 2 847.2.f.s 8
231.h odd 2 1 5929.2.a.bi 4
231.r odd 10 2 539.2.f.d 8
231.be even 30 4 539.2.q.c 16
231.bf odd 30 4 539.2.q.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.a 8 33.f even 10 2
539.2.f.d 8 231.r odd 10 2
539.2.q.b 16 231.bf odd 30 4
539.2.q.c 16 231.be even 30 4
693.2.m.g 8 11.d odd 10 2
847.2.a.k 4 3.b odd 2 1
847.2.a.l 4 33.d even 2 1
847.2.f.p 8 33.f even 10 2
847.2.f.q 8 33.h odd 10 2
847.2.f.s 8 33.h odd 10 2
5929.2.a.bb 4 21.c even 2 1
5929.2.a.bi 4 231.h odd 2 1
7623.2.a.ch 4 11.b odd 2 1
7623.2.a.co 4 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2}^{4} - 2T_{2}^{3} - 4T_{2}^{2} + 5T_{2} + 5$$ T2^4 - 2*T2^3 - 4*T2^2 + 5*T2 + 5 $$T_{5}^{4} - 6T_{5}^{3} + 8T_{5}^{2} + 3T_{5} - 1$$ T5^4 - 6*T5^3 + 8*T5^2 + 3*T5 - 1 $$T_{13}^{4} - 32T_{13}^{2} + 65T_{13} - 29$$ T13^4 - 32*T13^2 + 65*T13 - 29

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} - 4 T^{2} + 5 T + 5$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 6 T^{3} + 8 T^{2} + 3 T - 1$$
$7$ $$(T - 1)^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 32 T^{2} + 65 T - 29$$
$17$ $$T^{4} + 3 T^{3} - 23 T^{2} - 91 T - 71$$
$19$ $$T^{4} + 3 T^{3} - 29 T^{2} - 135 T - 145$$
$23$ $$T^{4} - 8 T^{3} - 9 T^{2} + 150 T - 205$$
$29$ $$T^{4} - 3 T^{3} - 54 T^{2} + 405$$
$31$ $$T^{4} + 3 T^{3} - 24 T^{2} + 20 T + 5$$
$37$ $$T^{4} + 7 T^{3} + 6 T^{2} - 10 T - 5$$
$41$ $$T^{4} - 4 T^{3} - 57 T^{2} + 22 T + 79$$
$43$ $$(T^{2} + 4 T - 41)^{2}$$
$47$ $$T^{4} - 14 T^{3} - 26 T^{2} + \cdots - 305$$
$53$ $$T^{4} - 9 T^{3} - 104 T^{2} + \cdots - 869$$
$59$ $$T^{4} - 25 T^{3} + 192 T^{2} + \cdots - 1189$$
$61$ $$T^{4} - 19 T^{3} - 6 T^{2} + 1030 T - 995$$
$67$ $$T^{4} + 15 T^{3} + 67 T^{2} + \cdots - 199$$
$71$ $$T^{4} - 7 T^{3} - 83 T^{2} + 679 T - 991$$
$73$ $$T^{4} - 11 T^{3} - 116 T^{2} + \cdots + 4975$$
$79$ $$T^{4} + 8 T^{3} - 8 T^{2} - 161 T - 271$$
$83$ $$T^{4} + T^{3} - 236 T^{2} + 900 T + 2245$$
$89$ $$T^{4} - 17 T^{3} - 44 T^{2} + \cdots - 755$$
$97$ $$T^{4} + 15 T^{3} - 110 T^{2} + \cdots - 4225$$