# Properties

 Label 7623.2.a.ci 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

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## 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: 4.4.725.1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} + \beta_{2} ) q^{2} + ( \beta_{1} + \beta_{3} ) q^{4} + 2 \beta_{2} q^{5} - q^{7} + ( 2 + \beta_{1} ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta_{1} + \beta_{2} ) q^{2} + ( \beta_{1} + \beta_{3} ) q^{4} + 2 \beta_{2} q^{5} - q^{7} + ( 2 + \beta_{1} ) q^{8} + ( 2 + 2 \beta_{1} ) q^{10} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{13} + ( 1 - \beta_{1} - \beta_{2} ) q^{14} + ( -2 + \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{16} + ( -2 + 4 \beta_{1} - 2 \beta_{3} ) q^{17} + ( -2 + 2 \beta_{1} - 2 \beta_{3} ) q^{19} + ( -2 + 4 \beta_{1} + 2 \beta_{3} ) q^{20} + ( 3 + 4 \beta_{1} - 3 \beta_{3} ) q^{23} + ( 3 + 4 \beta_{2} - 4 \beta_{3} ) q^{25} + ( 2 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{26} + ( -\beta_{1} - \beta_{3} ) q^{28} + ( -2 - 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{29} + ( -4 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{31} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{32} + ( 2 - 2 \beta_{1} + 6 \beta_{3} ) q^{34} -2 \beta_{2} q^{35} + ( -3 + 4 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{37} + ( 2 - 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{38} + ( -2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{40} + ( 2 + 2 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} ) q^{41} + ( 6 \beta_{1} - 5 \beta_{3} ) q^{43} + ( -3 + \beta_{1} + 4 \beta_{2} + 7 \beta_{3} ) q^{46} + ( -2 - 4 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} ) q^{47} + q^{49} + ( 1 - \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{50} + ( 8 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{52} + ( 2 + 2 \beta_{1} - 5 \beta_{3} ) q^{53} + ( -2 - \beta_{1} ) q^{56} + ( 6 + 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{58} + ( 6 - 2 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 2 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{61} + ( 6 - 8 \beta_{1} - 8 \beta_{2} ) q^{62} + ( 1 - \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{64} + ( 4 + 4 \beta_{1} + 4 \beta_{2} ) q^{65} + ( -8 - 2 \beta_{1} + 3 \beta_{3} ) q^{67} + ( 2 + 4 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{68} + ( -2 - 2 \beta_{1} ) q^{70} + ( 6 - 4 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{71} + ( -2 + 2 \beta_{2} ) q^{73} + ( 7 + 3 \beta_{1} + 5 \beta_{3} ) q^{74} + ( 2 \beta_{2} - 4 \beta_{3} ) q^{76} + ( -4 - 2 \beta_{1} - \beta_{3} ) q^{79} + ( 10 + 2 \beta_{2} - 4 \beta_{3} ) q^{80} + ( -6 - 12 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} ) q^{82} + ( -2 - 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{83} + ( -8 + 4 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{85} + ( -4 \beta_{1} + \beta_{2} + 11 \beta_{3} ) q^{86} + ( 2 + 4 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{89} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{91} + ( 1 + 8 \beta_{1} + 5 \beta_{2} ) q^{92} + ( 4 \beta_{1} - 10 \beta_{3} ) q^{94} + ( -4 - 4 \beta_{2} + 4 \beta_{3} ) q^{95} + ( 8 + 4 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{97} + ( -1 + \beta_{1} + \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - q^{2} + 3q^{4} + 4q^{5} - 4q^{7} + 9q^{8} + O(q^{10})$$ $$4q - q^{2} + 3q^{4} + 4q^{5} - 4q^{7} + 9q^{8} + 10q^{10} + 6q^{13} + q^{14} - 3q^{16} - 8q^{17} - 10q^{19} + 10q^{23} + 12q^{25} + 20q^{26} - 3q^{28} - 18q^{31} + 2q^{32} + 18q^{34} - 4q^{35} - 2q^{37} + 8q^{38} + 6q^{40} - 10q^{41} - 4q^{43} + 11q^{46} - 4q^{47} + 4q^{49} + 9q^{50} + 20q^{52} - 9q^{56} + 14q^{58} + 16q^{59} + 14q^{61} - 11q^{64} + 28q^{65} - 28q^{67} + 16q^{68} - 10q^{70} + 18q^{71} - 4q^{73} + 41q^{74} - 4q^{76} - 20q^{79} + 36q^{80} - 24q^{82} - 6q^{83} - 20q^{85} + 20q^{86} + 16q^{89} - 6q^{91} + 22q^{92} - 16q^{94} - 16q^{95} + 32q^{97} - q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 3 x^{2} + x + 1$$:

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

## 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
 −0.477260 0.737640 −1.35567 2.09529
−1.77222 0 1.14077 −0.589926 0 −1.00000 1.52274 0 1.04548
1.2 −1.45589 0 0.119606 −2.38705 0 −1.00000 2.73764 0 3.47528
1.3 −0.162147 0 −1.97371 4.38705 0 −1.00000 0.644326 0 −0.711349
1.4 2.39026 0 3.71333 2.58993 0 −1.00000 4.09529 0 6.19059
 $$n$$: 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.ci 4
3.b odd 2 1 2541.2.a.bn 4
11.b odd 2 1 7623.2.a.cl 4
11.c even 5 2 693.2.m.f 8
33.d even 2 1 2541.2.a.bm 4
33.h odd 10 2 231.2.j.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.f 8 33.h odd 10 2
693.2.m.f 8 11.c even 5 2
2541.2.a.bm 4 33.d even 2 1
2541.2.a.bn 4 3.b odd 2 1
7623.2.a.ci 4 1.a even 1 1 trivial
7623.2.a.cl 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2}^{4} + T_{2}^{3} - 5 T_{2}^{2} - 7 T_{2} - 1$$ $$T_{5}^{4} - 4 T_{5}^{3} - 8 T_{5}^{2} + 24 T_{5} + 16$$ $$T_{13}^{4} - 6 T_{13}^{3} - 8 T_{13}^{2} + 16 T_{13} + 16$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 3 T^{2} - T^{3} + 3 T^{4} - 2 T^{5} + 12 T^{6} + 8 T^{7} + 16 T^{8}$$
$3$ 
$5$ $$1 - 4 T + 12 T^{2} - 36 T^{3} + 86 T^{4} - 180 T^{5} + 300 T^{6} - 500 T^{7} + 625 T^{8}$$
$7$ $$( 1 + T )^{4}$$
$11$ 
$13$ $$1 - 6 T + 44 T^{2} - 218 T^{3} + 822 T^{4} - 2834 T^{5} + 7436 T^{6} - 13182 T^{7} + 28561 T^{8}$$
$17$ $$1 + 8 T + 48 T^{2} + 264 T^{3} + 1358 T^{4} + 4488 T^{5} + 13872 T^{6} + 39304 T^{7} + 83521 T^{8}$$
$19$ $$1 + 10 T + 100 T^{2} + 570 T^{3} + 3062 T^{4} + 10830 T^{5} + 36100 T^{6} + 68590 T^{7} + 130321 T^{8}$$
$23$ $$1 - 10 T + 83 T^{2} - 500 T^{3} + 2869 T^{4} - 11500 T^{5} + 43907 T^{6} - 121670 T^{7} + 279841 T^{8}$$
$29$ $$1 + 37 T^{2} + 493 T^{4} + 31117 T^{6} + 707281 T^{8}$$
$31$ $$1 + 18 T + 192 T^{2} + 1362 T^{3} + 8238 T^{4} + 42222 T^{5} + 184512 T^{6} + 536238 T^{7} + 923521 T^{8}$$
$37$ $$1 + 2 T + 71 T^{2} + 4 T^{3} + 2797 T^{4} + 148 T^{5} + 97199 T^{6} + 101306 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 10 T + 60 T^{2} - 290 T^{3} - 2938 T^{4} - 11890 T^{5} + 100860 T^{6} + 689210 T^{7} + 2825761 T^{8}$$
$43$ $$1 + 4 T + 69 T^{2} + 392 T^{3} + 4097 T^{4} + 16856 T^{5} + 127581 T^{6} + 318028 T^{7} + 3418801 T^{8}$$
$47$ $$1 + 4 T + 108 T^{2} - 4 T^{3} + 4758 T^{4} - 188 T^{5} + 238572 T^{6} + 415292 T^{7} + 4879681 T^{8}$$
$53$ $$1 + 161 T^{2} + 20 T^{3} + 11657 T^{4} + 1060 T^{5} + 452249 T^{6} + 7890481 T^{8}$$
$59$ $$1 - 16 T + 308 T^{2} - 2936 T^{3} + 29398 T^{4} - 173224 T^{5} + 1072148 T^{6} - 3286064 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 14 T + 260 T^{2} - 2306 T^{3} + 24294 T^{4} - 140666 T^{5} + 967460 T^{6} - 3177734 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 28 T + 541 T^{2} + 6696 T^{3} + 64817 T^{4} + 448632 T^{5} + 2428549 T^{6} + 8421364 T^{7} + 20151121 T^{8}$$
$71$ $$1 - 18 T + 327 T^{2} - 3632 T^{3} + 36173 T^{4} - 257872 T^{5} + 1648407 T^{6} - 6442398 T^{7} + 25411681 T^{8}$$
$73$ $$1 + 4 T + 284 T^{2} + 852 T^{3} + 30822 T^{4} + 62196 T^{5} + 1513436 T^{6} + 1556068 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 20 T + 445 T^{2} + 5040 T^{3} + 57977 T^{4} + 398160 T^{5} + 2777245 T^{6} + 9860780 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 6 T + 212 T^{2} + 982 T^{3} + 24278 T^{4} + 81506 T^{5} + 1460468 T^{6} + 3430722 T^{7} + 47458321 T^{8}$$
$89$ $$1 - 16 T + 308 T^{2} - 3096 T^{3} + 38678 T^{4} - 275544 T^{5} + 2439668 T^{6} - 11279504 T^{7} + 62742241 T^{8}$$
$97$ $$1 - 32 T + 656 T^{2} - 8944 T^{3} + 99982 T^{4} - 867568 T^{5} + 6172304 T^{6} - 29205536 T^{7} + 88529281 T^{8}$$
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