# Properties

 Label 7623.2.a.ch 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 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4 \beta_{1} + 3$$

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

## 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.ch 4
3.b odd 2 1 847.2.a.l 4
11.b odd 2 1 7623.2.a.co 4
11.c even 5 2 693.2.m.g 8
21.c even 2 1 5929.2.a.bi 4
33.d even 2 1 847.2.a.k 4
33.f even 10 2 847.2.f.q 8
33.f even 10 2 847.2.f.s 8
33.h odd 10 2 77.2.f.a 8
33.h odd 10 2 847.2.f.p 8
231.h odd 2 1 5929.2.a.bb 4
231.u even 10 2 539.2.f.d 8
231.z odd 30 4 539.2.q.c 16
231.bc even 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.h odd 10 2
539.2.f.d 8 231.u even 10 2
539.2.q.b 16 231.bc even 30 4
539.2.q.c 16 231.z odd 30 4
693.2.m.g 8 11.c even 5 2
847.2.a.k 4 33.d even 2 1
847.2.a.l 4 3.b odd 2 1
847.2.f.p 8 33.h odd 10 2
847.2.f.q 8 33.f even 10 2
847.2.f.s 8 33.f even 10 2
5929.2.a.bb 4 231.h odd 2 1
5929.2.a.bi 4 21.c even 2 1
7623.2.a.ch 4 1.a even 1 1 trivial
7623.2.a.co 4 11.b odd 2 1

## Atkin-Lehner signs

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

## Hecke kernels

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

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 4 T^{2} + 7 T^{3} + 13 T^{4} + 14 T^{5} + 16 T^{6} + 16 T^{7} + 16 T^{8}$$
$3$ 
$5$ $$1 - 6 T + 28 T^{2} - 87 T^{3} + 229 T^{4} - 435 T^{5} + 700 T^{6} - 750 T^{7} + 625 T^{8}$$
$7$ $$( 1 + T )^{4}$$
$11$ 
$13$ $$1 + 20 T^{2} - 65 T^{3} + 153 T^{4} - 845 T^{5} + 3380 T^{6} + 28561 T^{8}$$
$17$ $$1 - 3 T + 45 T^{2} - 62 T^{3} + 881 T^{4} - 1054 T^{5} + 13005 T^{6} - 14739 T^{7} + 83521 T^{8}$$
$19$ $$1 - 3 T + 47 T^{2} - 36 T^{3} + 919 T^{4} - 684 T^{5} + 16967 T^{6} - 20577 T^{7} + 130321 T^{8}$$
$23$ $$1 - 8 T + 83 T^{2} - 402 T^{3} + 2555 T^{4} - 9246 T^{5} + 43907 T^{6} - 97336 T^{7} + 279841 T^{8}$$
$29$ $$1 + 3 T + 62 T^{2} + 261 T^{3} + 2319 T^{4} + 7569 T^{5} + 52142 T^{6} + 73167 T^{7} + 707281 T^{8}$$
$31$ $$1 + 3 T + 100 T^{2} + 299 T^{3} + 4283 T^{4} + 9269 T^{5} + 96100 T^{6} + 89373 T^{7} + 923521 T^{8}$$
$37$ $$1 + 7 T + 154 T^{2} + 767 T^{3} + 8653 T^{4} + 28379 T^{5} + 210826 T^{6} + 354571 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 4 T + 107 T^{2} + 470 T^{3} + 5491 T^{4} + 19270 T^{5} + 179867 T^{6} + 275684 T^{7} + 2825761 T^{8}$$
$43$ $$( 1 - 4 T + 45 T^{2} - 172 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$1 - 14 T + 162 T^{2} - 1449 T^{3} + 10505 T^{4} - 68103 T^{5} + 357858 T^{6} - 1453522 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 9 T + 108 T^{2} - 725 T^{3} + 4961 T^{4} - 38425 T^{5} + 303372 T^{6} - 1339893 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 25 T + 428 T^{2} - 4725 T^{3} + 42353 T^{4} - 278775 T^{5} + 1489868 T^{6} - 5134475 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 19 T + 238 T^{2} + 2447 T^{3} + 20599 T^{4} + 149267 T^{5} + 885598 T^{6} + 4312639 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 15 T + 335 T^{2} + 3060 T^{3} + 35713 T^{4} + 205020 T^{5} + 1503815 T^{6} + 4511445 T^{7} + 20151121 T^{8}$$
$71$ $$1 - 7 T + 201 T^{2} - 812 T^{3} + 17469 T^{4} - 57652 T^{5} + 1013241 T^{6} - 2505377 T^{7} + 25411681 T^{8}$$
$73$ $$1 + 11 T + 176 T^{2} + 1649 T^{3} + 20013 T^{4} + 120377 T^{5} + 937904 T^{6} + 4279187 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 8 T + 308 T^{2} - 1735 T^{3} + 35911 T^{4} - 137065 T^{5} + 1922228 T^{6} - 3944312 T^{7} + 38950081 T^{8}$$
$83$ $$1 - T + 96 T^{2} - 1149 T^{3} + 4403 T^{4} - 95367 T^{5} + 661344 T^{6} - 571787 T^{7} + 47458321 T^{8}$$
$89$ $$1 - 17 T + 312 T^{2} - 3419 T^{3} + 38939 T^{4} - 304291 T^{5} + 2471352 T^{6} - 11984473 T^{7} + 62742241 T^{8}$$
$97$ $$1 + 15 T + 278 T^{2} + 2415 T^{3} + 30889 T^{4} + 234255 T^{5} + 2615702 T^{6} + 13690095 T^{7} + 88529281 T^{8}$$