# Properties

 Label 7623.2.a.cu Level $7623$ Weight $2$ Character orbit 7623.a Self dual yes Analytic conductor $60.870$ Analytic rank $1$ Dimension $8$ CM no Inner twists $2$

# 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: $$1$$ Dimension: $$8$$ Coefficient field: 8.8.6988960000.1 Defining polynomial: $$x^{8} - 9x^{6} + 22x^{4} - 11x^{2} + 1$$ x^8 - 9*x^6 + 22*x^4 - 11*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 693) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{2} q^{4} + (\beta_{7} + \beta_1) q^{5} - q^{7} + (\beta_{6} + \beta_{5} + \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + b2 * q^4 + (b7 + b1) * q^5 - q^7 + (b6 + b5 + b1) * q^8 $$q + \beta_1 q^{2} + \beta_{2} q^{4} + (\beta_{7} + \beta_1) q^{5} - q^{7} + (\beta_{6} + \beta_{5} + \beta_1) q^{8} + ( - \beta_{3} + \beta_{2} + 1) q^{10} + (\beta_{4} + \beta_{3}) q^{13} - \beta_1 q^{14} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{16} + (\beta_{7} - 2 \beta_{5} - 2 \beta_1) q^{17} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2}) q^{19} + ( - \beta_{7} + \beta_{6} + 2 \beta_1) q^{20} + ( - \beta_{7} - \beta_{6} - 3 \beta_1) q^{23} + ( - \beta_{3} - 2) q^{25} + ( - 2 \beta_{7} + 3 \beta_{5} + \beta_1) q^{26} - \beta_{2} q^{28} + ( - 3 \beta_{6} - 3 \beta_1) q^{29} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{31} + ( - 2 \beta_{7} - 3 \beta_{6} - 4 \beta_1) q^{32} + ( - 2 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - 3) q^{34} + ( - \beta_{7} - \beta_1) q^{35} + (\beta_{4} - \beta_{2} - 5) q^{37} + (3 \beta_{7} - \beta_{6} - 6 \beta_{5} - 5 \beta_1) q^{38} + (3 \beta_{3} + 2) q^{40} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 4 \beta_1) q^{41} + (3 \beta_{4} + 3 \beta_{3} + 2) q^{43} + (\beta_{3} - 3 \beta_{2} - 4) q^{46} + (\beta_{6} - 5 \beta_{5}) q^{47} + q^{49} + (\beta_{7} - \beta_{5} - 2 \beta_1) q^{50} + (\beta_{4} + 3 \beta_{3} + \beta_{2} + 1) q^{52} + ( - 4 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_1) q^{53} + ( - \beta_{6} - \beta_{5} - \beta_1) q^{56} + ( - 3 \beta_{2} - 3) q^{58} + ( - 4 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} - \beta_1) q^{59} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 4) q^{61} + (\beta_{7} - \beta_{6} - 4 \beta_{5} - 4 \beta_1) q^{62} + ( - 2 \beta_{4} - 2 \beta_{2} - 3) q^{64} + ( - 2 \beta_{7} - \beta_{6} + 3 \beta_{5}) q^{65} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 6) q^{67} + (3 \beta_{7} - 2 \beta_{6} - 5 \beta_{5} - 7 \beta_1) q^{68} + (\beta_{3} - \beta_{2} - 1) q^{70} + ( - \beta_{6} + 4 \beta_{5} + 3 \beta_1) q^{71} + ( - 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 3) q^{73} + ( - \beta_{7} - \beta_{6} + \beta_{5} - 7 \beta_1) q^{74} + ( - 2 \beta_{4} - 7 \beta_{3} - 3 \beta_{2} - 6) q^{76} + ( - \beta_{4} - 7 \beta_{3} + 3 \beta_{2} - 1) q^{79} + ( - \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - 2 \beta_1) q^{80} + (2 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} + 4) q^{82} + ( - 2 \beta_{7} - 4 \beta_{6} + 2 \beta_{5} - 2 \beta_1) q^{83} + ( - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2) q^{85} + ( - 6 \beta_{7} + 9 \beta_{5} + 5 \beta_1) q^{86} + (3 \beta_{7} + 6 \beta_{6} + \beta_{5} - \beta_1) q^{89} + ( - \beta_{4} - \beta_{3}) q^{91} + (\beta_{7} - \beta_{6} - 2 \beta_{5} - 7 \beta_1) q^{92} + ( - 5 \beta_{4} - 5 \beta_{3} + 4) q^{94} + (4 \beta_{7} + 2 \beta_{6} - 5 \beta_{5} - 2 \beta_1) q^{95} + ( - \beta_{4} + 5 \beta_{3} - 4 \beta_{2} - 2) q^{97} + \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 + b2 * q^4 + (b7 + b1) * q^5 - q^7 + (b6 + b5 + b1) * q^8 + (-b3 + b2 + 1) * q^10 + (b4 + b3) * q^13 - b1 * q^14 + (b4 + b3 - b2) * q^16 + (b7 - 2*b5 - 2*b1) * q^17 + (-2*b4 - b3 - b2) * q^19 + (-b7 + b6 + 2*b1) * q^20 + (-b7 - b6 - 3*b1) * q^23 + (-b3 - 2) * q^25 + (-2*b7 + 3*b5 + b1) * q^26 - b2 * q^28 + (-3*b6 - 3*b1) * q^29 + (-2*b4 + b3 - b2 + 1) * q^31 + (-2*b7 - 3*b6 - 4*b1) * q^32 + (-2*b4 - 3*b3 - 2*b2 - 3) * q^34 + (-b7 - b1) * q^35 + (b4 - b2 - 5) * q^37 + (3*b7 - b6 - 6*b5 - 5*b1) * q^38 + (3*b3 + 2) * q^40 + (-b7 + 3*b6 + 2*b5 + 4*b1) * q^41 + (3*b4 + 3*b3 + 2) * q^43 + (b3 - 3*b2 - 4) * q^46 + (b6 - 5*b5) * q^47 + q^49 + (b7 - b5 - 2*b1) * q^50 + (b4 + 3*b3 + b2 + 1) * q^52 + (-4*b7 - b6 + b5 - 2*b1) * q^53 + (-b6 - b5 - b1) * q^56 + (-3*b2 - 3) * q^58 + (-4*b7 + 2*b6 + 3*b5 - b1) * q^59 + (-b4 - b3 - 2*b2 + 4) * q^61 + (b7 - b6 - 4*b5 - 4*b1) * q^62 + (-2*b4 - 2*b2 - 3) * q^64 + (-2*b7 - b6 + 3*b5) * q^65 + (2*b4 - 2*b3 - 2*b2 - 6) * q^67 + (3*b7 - 2*b6 - 5*b5 - 7*b1) * q^68 + (b3 - b2 - 1) * q^70 + (-b6 + 4*b5 + 3*b1) * q^71 + (-2*b4 + 2*b3 + 3*b2 + 3) * q^73 + (-b7 - b6 + b5 - 7*b1) * q^74 + (-2*b4 - 7*b3 - 3*b2 - 6) * q^76 + (-b4 - 7*b3 + 3*b2 - 1) * q^79 + (-b7 - 2*b6 + 3*b5 - 2*b1) * q^80 + (2*b4 + 3*b3 + 4*b2 + 4) * q^82 + (-2*b7 - 4*b6 + 2*b5 - 2*b1) * q^83 + (-2*b4 - 2*b3 - b2 - 2) * q^85 + (-6*b7 + 9*b5 + 5*b1) * q^86 + (3*b7 + 6*b6 + b5 - b1) * q^89 + (-b4 - b3) * q^91 + (b7 - b6 - 2*b5 - 7*b1) * q^92 + (-5*b4 - 5*b3 + 4) * q^94 + (4*b7 + 2*b6 - 5*b5 - 2*b1) * q^95 + (-b4 + 5*b3 - 4*b2 - 2) * q^97 + b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{4} - 8 q^{7}+O(q^{10})$$ 8 * q + 2 * q^4 - 8 * q^7 $$8 q + 2 q^{4} - 8 q^{7} + 14 q^{10} - 2 q^{16} - 6 q^{19} - 12 q^{25} - 2 q^{28} - 6 q^{31} - 24 q^{34} - 38 q^{37} + 4 q^{40} + 16 q^{43} - 42 q^{46} + 8 q^{49} + 2 q^{52} - 30 q^{58} + 28 q^{61} - 36 q^{64} - 36 q^{67} - 14 q^{70} + 14 q^{73} - 34 q^{76} + 22 q^{79} + 36 q^{82} - 18 q^{85} + 32 q^{94} - 48 q^{97}+O(q^{100})$$ 8 * q + 2 * q^4 - 8 * q^7 + 14 * q^10 - 2 * q^16 - 6 * q^19 - 12 * q^25 - 2 * q^28 - 6 * q^31 - 24 * q^34 - 38 * q^37 + 4 * q^40 + 16 * q^43 - 42 * q^46 + 8 * q^49 + 2 * q^52 - 30 * q^58 + 28 * q^61 - 36 * q^64 - 36 * q^67 - 14 * q^70 + 14 * q^73 - 34 * q^76 + 22 * q^79 + 36 * q^82 - 18 * q^85 + 32 * q^94 - 48 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 9x^{6} + 22x^{4} - 11x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$( \nu^{6} - 8\nu^{4} + 16\nu^{2} - 5 ) / 2$$ (v^6 - 8*v^4 + 16*v^2 - 5) / 2 $$\beta_{4}$$ $$=$$ $$( -\nu^{6} + 10\nu^{4} - 26\nu^{2} + 9 ) / 2$$ (-v^6 + 10*v^4 - 26*v^2 + 9) / 2 $$\beta_{5}$$ $$=$$ $$-\nu^{7} + 9\nu^{5} - 21\nu^{3} + 6\nu$$ -v^7 + 9*v^5 - 21*v^3 + 6*v $$\beta_{6}$$ $$=$$ $$\nu^{7} - 9\nu^{5} + 22\nu^{3} - 11\nu$$ v^7 - 9*v^5 + 22*v^3 - 11*v $$\beta_{7}$$ $$=$$ $$( -3\nu^{7} + 26\nu^{5} - 58\nu^{3} + 17\nu ) / 2$$ (-3*v^7 + 26*v^5 - 58*v^3 + 17*v) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} + 5\beta_1$$ b6 + b5 + 5*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 5\beta_{2} + 8$$ b4 + b3 + 5*b2 + 8 $$\nu^{5}$$ $$=$$ $$-2\beta_{7} + 5\beta_{6} + 8\beta_{5} + 24\beta_1$$ -2*b7 + 5*b6 + 8*b5 + 24*b1 $$\nu^{6}$$ $$=$$ $$8\beta_{4} + 10\beta_{3} + 24\beta_{2} + 37$$ 8*b4 + 10*b3 + 24*b2 + 37 $$\nu^{7}$$ $$=$$ $$-18\beta_{7} + 24\beta_{6} + 50\beta_{5} + 117\beta_1$$ -18*b7 + 24*b6 + 50*b5 + 117*b1

## 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
 −2.25947 −1.80692 −0.716111 −0.342036 0.342036 0.716111 1.80692 2.25947
−2.25947 0 3.10522 −1.54336 0 −1.00000 −2.49721 0 3.48718
1.2 −1.80692 0 1.26498 −2.14896 0 −1.00000 1.32813 0 3.88301
1.3 −0.716111 0 −1.48718 1.54336 0 −1.00000 2.49721 0 −1.10522
1.4 −0.342036 0 −1.88301 −2.14896 0 −1.00000 1.32813 0 0.735023
1.5 0.342036 0 −1.88301 2.14896 0 −1.00000 −1.32813 0 0.735023
1.6 0.716111 0 −1.48718 −1.54336 0 −1.00000 −2.49721 0 −1.10522
1.7 1.80692 0 1.26498 2.14896 0 −1.00000 −1.32813 0 3.88301
1.8 2.25947 0 3.10522 1.54336 0 −1.00000 2.49721 0 3.48718
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7623.2.a.cu 8
3.b odd 2 1 inner 7623.2.a.cu 8
11.b odd 2 1 7623.2.a.cv 8
11.c even 5 2 693.2.m.h 16
33.d even 2 1 7623.2.a.cv 8
33.h odd 10 2 693.2.m.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.m.h 16 11.c even 5 2
693.2.m.h 16 33.h odd 10 2
7623.2.a.cu 8 1.a even 1 1 trivial
7623.2.a.cu 8 3.b odd 2 1 inner
7623.2.a.cv 8 11.b odd 2 1
7623.2.a.cv 8 33.d even 2 1

## Hecke kernels

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

 $$T_{2}^{8} - 9T_{2}^{6} + 22T_{2}^{4} - 11T_{2}^{2} + 1$$ T2^8 - 9*T2^6 + 22*T2^4 - 11*T2^2 + 1 $$T_{5}^{4} - 7T_{5}^{2} + 11$$ T5^4 - 7*T5^2 + 11 $$T_{13}^{4} - 11T_{13}^{2} + 10T_{13} + 4$$ T13^4 - 11*T13^2 + 10*T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 9 T^{6} + 22 T^{4} - 11 T^{2} + \cdots + 1$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 7 T^{2} + 11)^{2}$$
$7$ $$(T + 1)^{8}$$
$11$ $$T^{8}$$
$13$ $$(T^{4} - 11 T^{2} + 10 T + 4)^{2}$$
$17$ $$T^{8} - 49 T^{6} + 246 T^{4} - 299 T^{2} + \cdots + 1$$
$19$ $$(T^{4} + 3 T^{3} - 41 T^{2} - 18 T + 236)^{2}$$
$23$ $$T^{8} - 61 T^{6} + 991 T^{4} + \cdots + 16$$
$29$ $$T^{8} - 108 T^{6} + 3483 T^{4} + \cdots + 104976$$
$31$ $$(T^{4} + 3 T^{3} - 36 T^{2} - 73 T + 281)^{2}$$
$37$ $$(T^{4} + 19 T^{3} + 117 T^{2} + 236 T + 16)^{2}$$
$41$ $$T^{8} - 148 T^{6} + 5329 T^{4} + \cdots + 121$$
$43$ $$(T^{4} - 8 T^{3} - 75 T^{2} + 634 T - 596)^{2}$$
$47$ $$T^{8} - 311 T^{6} + \cdots + 11262736$$
$53$ $$T^{8} - 115 T^{6} + 3975 T^{4} + \cdots + 10000$$
$59$ $$T^{8} - 241 T^{6} + 14227 T^{4} + \cdots + 256$$
$61$ $$(T^{4} - 14 T^{3} + 27 T^{2} + 314 T - 1124)^{2}$$
$67$ $$(T^{4} + 18 T^{3} + 28 T^{2} - 912 T - 3904)^{2}$$
$71$ $$T^{8} - 212 T^{6} + 14083 T^{4} + \cdots + 2085136$$
$73$ $$(T^{4} - 7 T^{3} - 125 T^{2} + 1106 T - 1436)^{2}$$
$79$ $$(T^{4} - 11 T^{3} - 113 T^{2} + 506 T - 484)^{2}$$
$83$ $$T^{8} - 176 T^{6} + 8976 T^{4} + \cdots + 256$$
$89$ $$T^{8} - 524 T^{6} + \cdots + 71554681$$
$97$ $$(T^{4} + 24 T^{3} + 67 T^{2} - 1584 T - 7744)^{2}$$